1$ and $\\\\mathcal {R}^{(1)}=1+r(y)+ r^{\\\\prime }(y)y$ .\",\n \"Herein $r(y)$ is the regulator shape function with $y= \\\\frac{p^2}{ k^2}$ .\",\n \"This structure arises, since all diagrams where the external derivative with respect to the momentum hits the propagator, can be organised in a way that only derivatives of the scalar propagator, and not the graviton propagator, need to be taken.\"\n ],\n [\n \"Scalar anomalous dimension\",\n \"Here we give our result for the anomalous dimension $\\\\eta _{\\\\phi }$ for a spectrally adjusted cutoff with exponential shape function: $\\\\eta _{\\\\phi }&=&\\\\frac{1}{96 \\\\pi g \\\\lambda \\\\left(e^{4 \\\\lambda /3}\\\\text{Ei}\\\\left(\\\\frac{4 \\\\lambda }{3}\\\\right)-\\\\left(e^{4 \\\\lambda /3}+1\\\\right)\\\\text{Ei}\\\\left(\\\\frac{8 \\\\lambda }{3}\\\\right)+\\\\text{Ei}(4 \\\\lambda )\\\\right)+12 \\\\pi ge^{4 \\\\lambda }-9 e^{8 \\\\lambda /3} (4 \\\\pi (g+24 \\\\pi )+9 (4 \\\\rho _b-2\\\\rho _c+3 \\\\rho _a))} 4 e^{4 \\\\lambda /3} \\\\cdot \\\\nonumber \\\\\\\\&{}&\\\\Bigl (8 \\\\pi g \\\\left(\\\\text{Ei}\\\\left(\\\\frac{8 \\\\lambda }{3}\\\\right) (\\\\partial _t \\\\lambda (3-12 \\\\lambda )+\\\\lambda (3 \\\\eta _N-32\\\\lambda +6))+3 \\\\text{Ei}(4 \\\\lambda ) (\\\\partial _t \\\\lambda (4 \\\\lambda -1)+\\\\lambda (-\\\\eta _N+8 \\\\lambda -2))+8 \\\\lambda ^2 \\\\text{Ei}\\\\left(\\\\frac{4 \\\\lambda }{3}\\\\right)\\\\right)\\\\nonumber \\\\\\\\&{}&+4 e^{4 \\\\lambda /3} \\\\left(2 \\\\pi g \\\\left(\\\\text{Ei}\\\\left(\\\\frac{4\\\\lambda }{3}\\\\right)-\\\\text{Ei}\\\\left(\\\\frac{8 \\\\lambda }{3}\\\\right)\\\\right)(\\\\partial _t \\\\lambda (8 \\\\lambda -3)+\\\\lambda (-3 \\\\eta _N+16 \\\\lambda -6))-3(\\\\pi g (4 \\\\lambda +3)+9 (4 \\\\rho _b-2 \\\\rho _c+3 \\\\rho _a))\\\\right)\\\\nonumber \\\\\\\\&{}&-3 \\\\pi g e^{8 \\\\lambda /3} (4 \\\\partial _t \\\\lambda -3 \\\\eta _N+6)-3 \\\\pi g \\\\eta _N e^{4 \\\\lambda }\\\\Bigr )$\"\n ]\n]"},"id":{"kind":"string","value":"1204.0965"}}},{"rowIdx":1867485,"cells":{"text":{"kind":"list like","value":[["Andreev current induced by ferromagnetic resonance"],["Abstract We study charge transport through a metallic dot coupled to a superconducting and a ferromagnetic lead with a precessing magnetization due to ferromagnetic resonance.","Using the quasiclassical theory, we find that the magnetization precession induces a dc current in the subgap regime even in the absence of a bias voltage.","This effect is due to the rectification of the ac spin currents at the interface with the ferromagnet; it exists in the absence of spin current in the superconductor.","When the dot is strongly coupled to the superconductor, we find a strong enhancement in a wide range of parameters as compared to the induced current in the normal state."],["Andreev current induced by ferromagnetic resonance Caroline Richard Manuel Houzet Julia S. Meyer SPSMS, UMR-E 9001 CEA/UJF-Grenoble 1, INAC, Grenoble, F-38054, France We study charge transport through a metallic dot coupled to a superconducting and a ferromagnetic lead with a precessing magnetization due to ferromagnetic resonance.","Using the quasiclassical theory, we find that the magnetization precession induces a dc current in the subgap regime even in the absence of a bias voltage.","This effect is due to the rectification of the ac spin currents at the interface with the ferromagnet; it exists in the absence of spin current in the superconductor.","When the dot is strongly coupled to the superconductor, we find a strong enhancement in a wide range of parameters as compared to the induced current in the normal state.","74.45.+c, 75.76.+j, 72.25.-b Spin-transfer torque allows one to manipulate the magnetization of a ferromagnetic (F) layer by means of a spin-polarized current [1], [2].","Random-access memories using this effect in order to induce magnetization reversal of the active elements are on their way to commercialization.","The reverse effect, namely the generation of a spin current in a normal metal (N) by means of a dynamically precessing ferromagnetic metal, has also been predicted [3].","In the absence of direct spin probes, this effect may be measured by using a second ferromagnet as an analyzer that converts the spin current into a charge current.","However, it was pointed out theoretically [4] and measured experimentally [5], [6] that a single F/N junction is enough to both generate and detect the spin current through the generation of a dc voltage at ferromagnetic resonance (FMR) in an open-circuit geometry.","At the origin of this phenomenon is the spin accumulation on the normal side of the junction – due to the precession-induced spin current – which is typically different from that on the ferromagnetic side.","If transmissions for the majority and minority electron species through the junction are different, the difference in spin accumulation generates a net charge current which must be compensated by a difference in electrochemical potentials such that no charge accumulation occurs.","Spin relaxation inhibits spin accumulation and, thus, suppresses the effect.","The aim of our work is to explore how this effect is modified in a ferromagnet/superconductor junction.","The combination of ferromagnetic and superconducting (S) materials has been shown to lead to a variety of interesting spin phenomena [7], [8].","However, the study of the interplay between magnetization dynamics and superconductivity is a relatively new topic.","Experimentally, a narrowing of the FMR width at the superconducting transition was observed in an F/S bilayer [9].","Theoretically, it was proposed that a dynamically precessing ferromagnet may generate a long-range proximity effect [10].","This effect would manifest itself in the enhancement of the critical current in a phase-biased ferromagnetic Josephson junction under FMR conditions.","Related signatures in the tunneling density of states of the F layer have also been investigated [11].","However, these works disregard interface effects and, therefore, do not take into account the possible generation of an FMR-induced dc voltage.","Finally, let us note that the first experiments on voltage generation by FMR of Refs.","[5], [6] were performed with Al as the normal metal, which becomes superconducting at low temperatures.","As the FMR-generated charge current in an F/N junction is typically associated with a spin current, one may wonder what happens in an F/S junction in the subgap regime, where transport is mediated by Andreev processes [12].","We show that the generation of charge current in the absence of a spin current in a conventional singlet superconductor is possible.","In fact, the absence of spin currents in the superconductor may even lead to a strong enhancement of the induced charge current as compared to the normal state.","Figure: Setup of the junction.","A metallic dot is coupled to a ferromagnetic lead with precessing magnetization m(t){m}(t) on the left and to a normal or superconducting lead on the right.","The left barrier is characterized by the conductances G l G_l and G m G_m, defined above Eq.","(), whereas the right barrier is characterized by the conductance G r G_r.As in the normal state, the two main ingredients necessary to generate the effect are spin-dependent transmissions through the junction and a spin accumulation region The effect is absent in the theory of Ref.","[18], which did not include such a region..","The simplest setup meeting these requirements is a metallic dot coupled through tunnel barriers to a ferromagnet and to a superconductor (see Fig.","REF ).","Coulomb blockade effects are neglected, assuming that the conductances of the barriers largely exceed the conductance quantum.","The magnetization precession in the ferromagnetic lead is described by a time-dependent exchange field, ${J}(t)=J{m}(t)$ with ${m}(t)=(\\sin \\theta \\cos \\Omega t,\\sin \\theta \\sin \\Omega t,\\cos \\theta ),$ acting on the spin of the conduction electrons.","Here the precession frequency, $\\Omega $ , and the tilt angle, $\\theta $ , are both tunable with external dc and rf fields under standard FMR conditions [13].","We will consider them as externally fixed parameters.","The precession of the magnetization drives the system out of equilibrium and, thus, may generate a current.","To describe the system, we use the quasiclassical Keldysh theory [14].","In particular, the current through the junction can be expressed in terms of the quasiclassical Green function $\\check{g}$ of the dot.","Here $\\check{g}$ is a matrix in Keldysh, Nambu, and spin space.","In Keldysh space, it has a triangular structure with retarded ($\\hat{g}^R$ ), advanced ($\\hat{g}^A$ ), and Keldysh ($\\hat{g}^K$ ) components.","Furthermore, it satisfies the normalization condition $\\check{g}^2=1$ .","The equations determining the Green functions are most conveniently written in the rotational frame for the magnetization precession, where the problem is stationary [10].","Assuming that the conductance of the dot largely exceeds the conductances of the junctions with the leads, the equation determining $\\check{g}$ may be cast in the form $-i\\frac{2\\pi G_Q}{\\delta }\\left[\\left(E+\\frac{\\Omega }{2}\\sigma _z \\right)\\tau _z, \\check{g}\\right]+\\check{I}_l+ \\check{I}_r=0.$ Here $\\sigma _i$ and $\\tau _i$ are Pauli matrices in Nambu and spin space, respectively ($i=x,y,z$ ).","Furthermore, $G_Q=e^2/\\pi $ is the conductance quantum (in units where $\\hbar =1$ ), and $\\delta $ is the mean level spacing in the dot.","The spin-dependent energy shift $\\pm \\Omega /2$ is a spin-resolved chemical potential induced by the transformation from the laboratory to the rotational frame.","The boundary conditions with the ferromagnetic ($l$ = left) and superconducting ($r$ = right) leads are represented by the matrix currents $\\check{I}_{l/r}$ and depend on the Green functions $\\check{g}_{l/r}$ describing the non-equilibrium state in the leads due to the magnetization precession.","Tunneling through an F/N interface is generally spin-dependent.","The relevant processes can be characterized by the total conductance of the junction, $G_l$ , and the difference between the conductances for the majority and minority electrons, $G_m$ In addition one may take into account the imaginary part of the spin-mixing conductance, $G_\\phi $ , which accounts for a spin-dependent phase shift upon reflection.","In good F/N contacts with large Fermi velocity mismatch, $G_\\phi $ is typically small and therefore neglected.","It may, however, be large in tunnel junctions.","As $G_\\phi $ leads to spin relaxation, it would suppress the effect studied in this work..","The matrix current at the tunnel interface between the dot and the ferromagnet then takes the form [15] $\\check{I}_l=\\frac{G_l}{2}[ \\check{g}_l,\\check{g}]+\\frac{G_m}{4}[ \\lbrace {m}\\cdot {{\\sigma }}\\tau _z,\\check{g}_l\\rbrace ,\\check{g}],$ where ${m}\\equiv {m}(0)$ .","Within the quasiclassical approximation, we assume $|G_m|\\ll G_l$ .","Thus, $G_m$ can be treated perturbatively.","The Green function in the F lead, $\\check{g}_l$ , is determined by $\\left[\\left(E+\\frac{\\Omega }{2}\\sigma _z+J{{m}}\\cdot {\\sigma }\\right)\\tau _z-\\check{\\Sigma }, \\check{g}_l \\right]=0,$ where the self-energy $\\check{\\Sigma }=-i\\Gamma \\check{g}_N(E+(\\Omega /2)\\sigma _z)$ accounts for inelastic scattering in the relaxation time approximation.","Here, $1/\\Gamma $ is the inelastic scattering time and $\\check{g}_N$ is the equilibrium Green function in a normal metal.","Namely, $\\hat{g}_N^{R(A)}(E)=\\pm \\tau _z$ and $\\hat{g}_N^{K}(E)=2\\tau _z f(E)$ , where $f(E)=\\tanh (E/2T)$ is related to the Fermi distribution at temperature $T$ .","For a large exchange field, $J\\gg \\Omega ,\\Gamma $ , the solution of Eq.","(REF ) takes the form $\\hat{g}_l^{R(A)}=\\pm \\tau _z$ and $\\hat{g}_l^{K}=2\\tau _z(f_++f_-\\cos \\theta {m}\\!\\cdot \\!","{{\\sigma }})$ , where $f_\\pm (E)=[f(E+\\Omega /2)\\pm f(E-\\Omega /2)]/2$ .","The matrix current at the dot-superconductor tunnel interface is given as $\\check{I}_r=\\frac{G_r}{2}[ \\check{g}_r,\\check{g}],$ where $G_r$ is the conductance of the junction.","The Green function in the S lead reads $\\check{g}_r=\\check{g}_S(E+(\\Omega /2)\\sigma _z)$ , where $\\check{g}_S$ is the equilibrium Green function in a superconductor.","Namely, $\\hat{g}^{R(A)}_S(E)=(-iE \\tau _z+\\Delta \\tau _x)/\\sqrt{\\Delta ^2-(E\\pm i0^+)^2}$ and $\\hat{g}^{K}_S(E)=[\\hat{g}^{R}_S(E)-\\hat{g}^{A}_S(E)]f(E)$ , where $\\Delta $ is the superconducting order parameter (taken to be real).","Now we have all the ingredients necessary to determine the Green function in the dot and subsequently the spin and charge currents at both interfaces.","The charge currents are given by $I_{l/r}=\\frac{1}{16e}\\int dE\\; \\mathrm {Tr}[\\tau _z \\hat{I}_{l/r}^K].$ Current conservation ensures that $I\\equiv I_l=-I_r$ .","The spin currents in the rotational frame are given by ${I}_{l/r}=-\\frac{1}{32e^2}\\int dE \\;\\mathrm {Tr}[ {\\sigma } \\hat{I}_{l/r}^K].$ In the laboratory frame, they decompose into a dc contribution along the precession axis, $I_{\\alpha ,z}$ , and ac components in the perpendicular plane, $I_{\\alpha ,x/y}(t)=I_{\\alpha ,x/y}\\cos \\Omega t\\mp I_{\\alpha ,y/x}\\sin \\Omega t$ .","Contrarily to the charge current, the spin currents do not need to be conserved.","Eq.","(REF ) yields ${I}_{l}+{I}_{r}-\\frac{\\Omega }{16\\delta }\\int dE\\;\\mathrm {Tr}[(\\hat{\\bf z}\\times {\\sigma })\\tau _z\\hat{g}^K]=0.$ Thus, only the dc spin current along $\\hat{\\bf z}$ is conserved.","While our main interest are the FMR-induced currents in the subgap regime of an F-dot-S junction, we first study the simpler case of an F-dot-N junction for comparison.","For better readibility, in the following, we will normalize conductances by $G_\\Sigma =G_l+G_r$ and energies by the Thouless energy $E_g=G_\\Sigma \\delta /(4\\pi G_Q)$ .","In particular, we introduce the dimensionless conductances $\\gamma _\\alpha =G_\\alpha /G_\\Sigma $ ($\\alpha =l,r,m$ ) as well as the dimensionless energies $\\epsilon =E/E_g$ and $\\omega =\\Omega /(2E_g)$ .","A normal lead is described by setting $\\Delta =0$ in the above equations for $\\check{g}_r$ .","In the absence of superconductivity, the retarded and advanced Green functions in the dot are trivial, $\\hat{g}^{R(A)}=\\pm \\tau _z$ .","The Keldysh component is obtained with the help of Eqs.","(REF ), (REF ), and (REF ).","There is a remarkable relation between the spin current at the left contact with the F lead and the charge current to lowest order in $\\gamma _m$ , $I=\\frac{2e\\gamma _m\\gamma _r}{\\gamma _l}{m}(t)\\cdot {I}_l(t)=\\frac{2e\\gamma _m\\gamma _r}{\\gamma _l}(I_{l,x}\\sin \\theta +I_{l,z}\\cos \\theta ),$ namely the charge current is proportional to the projection of the spin current onto the instantaneous magnetization axis of the barrier due to the spin-dependent conductance $G_m$ .","That is, the charge current originates from two effects: (i) the rectification of the ac in-plane spin current pumped from the ferromagnet, and (ii) the conversion of the dc spin current along the $\\hat{\\bf z}$ -axis into a charge current.","It turns out that the two effects have opposite sign, and that the former dominates over the latter.","Namely, we find $I_{l,x}&=&\\frac{G_lE_g}{2e^2}\\frac{\\omega (\\gamma _r+\\omega ^2)}{1+\\omega ^2}\\sin \\theta \\cos \\theta ,\\\\I_{l,z}&=&-\\frac{G_lE_g}{2e^2}\\omega \\gamma _r\\sin ^2\\theta .$ Note that, in the limit $\\gamma _r\\ll \\gamma _l$ , the spin current along the $\\hat{\\bf z}$ -axis is negligible.","By contrast, in the limit $\\gamma _l\\ll \\gamma _r$ , the two components are of comparable magnitude and almost complete cancellation between the competing effects takes place.","The charge current reads $I=\\frac{G_m E_g}{e}\\frac{\\gamma _r\\gamma _l\\omega ^3\\sin ^2\\theta \\cos \\theta }{1+\\omega ^2}.$ At large precession frequency, $\\omega \\gg 1$ , the current scales linearly with frequency, $I\\simeq (G_lG_rG_m/2eG_\\Sigma ^2)\\Omega \\sin ^2\\theta \\cos \\theta $ .","In particular, in an open-circuit geometry, this would correspond to an FMR-induced dc voltage $eV=(G_m/2G_\\Sigma )\\Omega \\sin ^2\\theta \\cos \\theta $ in accordance with Refs.","[4], [5], [6].","At $\\omega \\ll 1$ , spin-relaxation mechanisms induced by the tunnel coupling of the dot to the leads tend to suppress the effect.","We now turn to the F-dot-S junction.","In the subgap regime, the spin current at the interface with the superconductor vanishes, ${I}_r=0$ .","Thus, $I_{z,l}=0$ .","However, an ac spin current is present at the interface with the ferromagnet.","Then, the Andreev charge current originates entirely from the rectification of this ac spin current.","Restricting ourselves to energy scales much smaller than $\\Delta $ , the Green function in the superconducting lead takes the simple form $\\hat{g}_r^{R(A)}=\\tau _x$ and $\\hat{g}_r^K=0$ .","Taking $\\gamma _m$ as a small parameter, we search for a perturbative solution of equation (REF ) in the form $\\check{g} =\\check{g}_0+\\gamma _m\\check{g}_1+\\dots $ Due to the proximity effect, now the retarded and advanced Green functions of the dot are modified as well.","To zeroth order in $\\gamma _m$ , an explicit solution is given by $\\hat{g}_0^{R(A)}=\\frac{\\gamma _r\\tau _x+[-i(\\epsilon +\\omega \\sigma _z)\\pm \\gamma _l]\\tau _z}{\\sqrt{\\gamma _r^2-(\\epsilon +\\omega \\sigma _z\\pm i\\gamma _l)^2}}.$ Here, $\\gamma _r$ is the effective minigap due to the coupling with the S lead [16], $\\omega $ an effective exchange field, and $\\gamma _l$ yields a broadening of the energy levels due to the coupling with the F lead.","The Keldysh Green function can be cast in the form $\\hat{g}_0^K=\\hat{g}^R\\hat{\\varphi }-\\hat{\\varphi }\\hat{g}^A$ with $\\hat{\\varphi }=f_++f_-\\cos \\theta \\left[\\frac{\\gamma _l\\sin \\theta }{\\omega ^2+\\gamma _l^2}(\\gamma _l\\sigma _x\\!-\\!\\omega \\sigma _y)+\\cos \\theta \\sigma _z\\right]\\!\\!.$ The function $\\hat{\\varphi }$ can be interpreted as a matrix-distribution function.","Note that it does not depend on $\\gamma _r$ as subgap electrons only thermalize with the F lead.","To first order in $\\gamma _m$ , a solution which satisfies the normalization condition, $\\check{g}^2=1$ , is obtained in the form $\\check{g}_1=\\check{g}_0 \\check{X} -\\check{X} \\check{g}_0$ .","For the advanced and retarded components, one finds $\\hat{X}^{R(A)}=\\mp (\\sin \\theta /2 \\omega )[i\\gamma _r/(\\epsilon \\pm i\\gamma _l) \\tau _x+\\tau _z]\\sigma _y$ .","The Keldysh component can be decomposed as $\\hat{X}^K=X^K_x\\tau _x+X^K_z\\tau _z$ , where $X^K_x$ and $X^K_z$ solve the coupled equations $&&2 \\gamma _l X^K_z -i\\omega [\\sigma _z,X^K_z]\\\\&=&2 \\sin \\theta [ \\cos \\theta \\sin \\theta f_-+ f_+(\\sigma _x-\\frac{ \\gamma _l}{ \\omega }\\sigma _y)],\\nonumber \\\\&&2\\epsilon X^K_x-2i \\gamma _r X^K_z+\\omega \\lbrace \\sigma _z,X^K_x\\rbrace \\\\&=&2i\\frac{\\gamma _l\\gamma _r\\sin \\theta }{ \\omega (\\gamma _l^2+ \\epsilon ^2)}[\\gamma _l f_+\\sigma _y-\\epsilon f_- \\cos \\theta (\\cos \\theta \\sigma _x-\\sin \\theta \\sigma _z) ].\\nonumber $ Evaluating the current at the right interface, Eq.","(REF ) yields $I=-i\\gamma _m G_r/(16e)\\int dE\\; \\mathrm {Tr}[\\tau _y \\hat{g}_1^K]$ .","Inserting the solution for $\\hat{g}_1^K$ and using the property $\\hat{g}_0^R(-\\epsilon )=-\\sigma _x\\tau _z\\hat{g}_0^A(\\epsilon )\\sigma _x\\tau _z$ , we obtain the current $I&=& \\frac{1}{2}I_0\\frac{\\gamma _r^2\\omega }{\\gamma _l^2+\\omega ^2}\\int d\\epsilon \\;\\frac{\\epsilon f_-}{(\\gamma _l^2+\\epsilon ^2)(\\epsilon +\\omega )}\\\\&&\\qquad \\qquad \\qquad \\times \\sum _\\pm \\frac{-\\gamma _l(\\epsilon +\\omega )\\pm i(\\gamma _l^2-\\epsilon \\omega )}{\\sqrt{\\gamma _r^2-(\\epsilon +\\omega \\pm i \\gamma _l)^2}},\\nonumber $ where $I_0= (G_mE_g/e)\\sin ^2\\theta \\cos \\theta $ .","The current as a function of frequency for different values of $\\gamma _l=1-\\gamma _r$ is shown in Fig.","REF .","Simple analytic expressions can be found in different asymptotic regimes.","Figure: Andreev current induced by ferromagnetic resonance as a function of precession frequency for different values of γ l =G l /(G r +G l )\\gamma _l=G_l/(G_r+G_l).Here I 0 =(G m E g /e)sin 2 θcosθI_0=(G_mE_g/e)\\sin ^2\\theta \\cos \\theta .In particular, at temperature $T=0$ and low frequency $|\\omega |\\ll \\gamma $ , where $\\gamma =(\\gamma _l^2+\\gamma _r^2)^{1/2}$ , the FMR-induced current is given as $I\\simeq \\frac{10}{3}I_0\\frac{\\gamma _r^2\\gamma _l}{\\gamma ^7}\\omega ^5.$ The large power $\\omega ^5$ indicates the strong suppression of the effect.","At large frequencies $|\\omega |\\gg \\gamma $ , the current saturates.","The frequency-independent value is given by $I\\simeq \\frac{\\pi }{2}I_0\\,\\mathrm {sign}(\\omega )\\times \\left\\lbrace \\begin{array}{lll}\\gamma _r^2, & \\,& \\gamma _r\\ll \\gamma _l, \\\\1, & \\,& \\gamma _l\\ll \\gamma _r.\\end{array}\\right.$ The saturation can be understood as Andreev processes become inefficient at energies larger than the minigap [17].","Depending whether the dot is more strongly coupled to the ferromagnet or to the superconductor, the crossover between these asymptotic regimes is different.","If the dot is weakly coupled to the superconductor, $\\gamma _r\\ll \\gamma _l$ , a smooth crossover happens at $\\omega \\sim 1\\gg \\gamma _r$ with a typical current $I/I_0\\sim \\gamma _r^2$ .","By contrast, if the dot is weakly coupled to the ferromagnet, $\\gamma _l\\ll \\gamma _r$ , the crossover in the region $\\omega \\sim 1/2$ is described by $I\\simeq I_0\\times \\!\\left\\lbrace \\begin{array}{lll}\\gamma _l/(2\\sqrt{-\\delta \\omega }),&\\,&\\!-1\\ll \\delta \\omega \\ll -\\gamma _l,\\\\2\\sqrt{\\delta \\omega },&\\,&\\!\\gamma _l\\ll \\delta \\omega \\ll 1,\\end{array}\\right.$ where $\\delta \\omega \\equiv \\omega -1/2$ , with a typical current $I/I_0\\sim \\sqrt{\\gamma _l}$ at $\\omega =1/2$ .","While in the asymptotic regimes of $\\omega $ very small or very large, the current is suppressed as compared to the normal state, there is in fact a wide intermediate regime where it may be strongly enhanced.","Comparing Eqs.","(REF ) and (REF ), one notices that, if the dot is strongly coupled to the superconductor, in the regime $1/2<\\omega <\\gamma _l^{-1}$ the induced current in the superconducting state exceeds the induced current in the normal state.","This effect may be understood due to the absence of a dc spin current along along the $\\hat{\\bf z}$ -axis which leads to a strong suppression of the effect in the normal state.","Fig.","REF shows the current in the superconducting and normal state as well the contribution due to rectification only in the normal state.","The ratio between the current in the superconducting state and the latter contribution in the normal state reflects the ratio between Andreev and normal state conductances in an N-dot-S junction.","Figure: Induced current in the superconducting (dotted line) and normal state for γ l =0.4\\gamma _l=0.4.The thin line shows the contribution to the normal state current due to rectification only.So far we assumed that the magnetization in the ferromagnet is uniform.","However, boundary effects may lead to a suppression of the magnetization in the vicinity of the F/N interface.","This would result in a different resonance frequency at the barrier than in the ferromagnetic reservoir and, consequently, in a tilt angle $\\theta _B\\ne \\theta $ at FMR.","The effect can be accounted for by replacing ${m}$ with ${m}_B=(\\sin \\theta _B,0,\\cos \\theta _B)$ in Eq.","(REF ).","In particular, at $\\theta _B=0$ , the spin dependent conductance $G_m$ refers to the constant axis $\\hat{\\bf z}$ .","In the normal case, the relation between spin and charge currents, Eq.","(REF ), now reads $I=(2e\\gamma _m\\gamma _r/\\gamma _l){m}_B(t)\\cdot {I}_l(t)$ .","While at $\\theta _B=\\theta $ the rectification of the in-plane ac spin currents always dominates over the conversion of the dc spin current along $\\hat{\\bf z}$ into a charge current, this effect is completely suppressed at $\\theta _B=0$ .","As a consequence, at $\\theta _B=0$ , the charge current, $I=-({G_m E_g}/{e})\\omega \\gamma _r^2\\sin ^2\\theta $ , has the opposite sign compared to Eq.","(REF ).","In general, both effects are important.","The sign reversal occurs at $\\tan \\theta _B=[1-\\gamma _l\\omega ^2/(\\gamma _r+\\omega ^2)]\\tan \\theta $ .","In the superconducting case, the dc spin current along $\\hat{\\bf z}$ is always zero, and the charge current is due entirely to the rectification of the in-plane ac spin currents.","As a consequence, we find that the charge current vanishes at $\\theta _B=0$ .","The general result is obtained from Eq.","(REF ) by replacing $I_0$ with $I_0^B=(G_mE_g/e)\\sin \\theta _B\\sin \\theta \\cos \\theta $ .","In summary, we demonstrate that a subgap charge current in an F/S junction may be induced by ferromagnetic resonance.","The effect is due to the rectification of ac spin currents generated by the precessing magnetization in the ferromagnet.","In the normal case, a competing effect of conversion of a dc spin current into a charge current exists.","This effect is absent in an F/S junction as the superconductor cannot carry a subgap spin current.","As a consequence, the induced current in the superconducting state may be strongly enhanced as compared to the normal state.","Interesting non-equilibrium phenomena should be expected in ferromagnetic Josephson junctions under ferromagnetic resonance conditions.","We acknowledge funding through an ANR grant (ANR-11-JS04-003-01) and an EU-FP7 Marie Curie IRG."]],"string":"[\n [\n \"Andreev current induced by ferromagnetic resonance\"\n ],\n [\n \"Abstract We study charge transport through a metallic dot coupled to a superconducting and a ferromagnetic lead with a precessing magnetization due to ferromagnetic resonance.\",\n \"Using the quasiclassical theory, we find that the magnetization precession induces a dc current in the subgap regime even in the absence of a bias voltage.\",\n \"This effect is due to the rectification of the ac spin currents at the interface with the ferromagnet; it exists in the absence of spin current in the superconductor.\",\n \"When the dot is strongly coupled to the superconductor, we find a strong enhancement in a wide range of parameters as compared to the induced current in the normal state.\"\n ],\n [\n \"Andreev current induced by ferromagnetic resonance Caroline Richard Manuel Houzet Julia S. Meyer SPSMS, UMR-E 9001 CEA/UJF-Grenoble 1, INAC, Grenoble, F-38054, France We study charge transport through a metallic dot coupled to a superconducting and a ferromagnetic lead with a precessing magnetization due to ferromagnetic resonance.\",\n \"Using the quasiclassical theory, we find that the magnetization precession induces a dc current in the subgap regime even in the absence of a bias voltage.\",\n \"This effect is due to the rectification of the ac spin currents at the interface with the ferromagnet; it exists in the absence of spin current in the superconductor.\",\n \"When the dot is strongly coupled to the superconductor, we find a strong enhancement in a wide range of parameters as compared to the induced current in the normal state.\",\n \"74.45.+c, 75.76.+j, 72.25.-b Spin-transfer torque allows one to manipulate the magnetization of a ferromagnetic (F) layer by means of a spin-polarized current [1], [2].\",\n \"Random-access memories using this effect in order to induce magnetization reversal of the active elements are on their way to commercialization.\",\n \"The reverse effect, namely the generation of a spin current in a normal metal (N) by means of a dynamically precessing ferromagnetic metal, has also been predicted [3].\",\n \"In the absence of direct spin probes, this effect may be measured by using a second ferromagnet as an analyzer that converts the spin current into a charge current.\",\n \"However, it was pointed out theoretically [4] and measured experimentally [5], [6] that a single F/N junction is enough to both generate and detect the spin current through the generation of a dc voltage at ferromagnetic resonance (FMR) in an open-circuit geometry.\",\n \"At the origin of this phenomenon is the spin accumulation on the normal side of the junction – due to the precession-induced spin current – which is typically different from that on the ferromagnetic side.\",\n \"If transmissions for the majority and minority electron species through the junction are different, the difference in spin accumulation generates a net charge current which must be compensated by a difference in electrochemical potentials such that no charge accumulation occurs.\",\n \"Spin relaxation inhibits spin accumulation and, thus, suppresses the effect.\",\n \"The aim of our work is to explore how this effect is modified in a ferromagnet/superconductor junction.\",\n \"The combination of ferromagnetic and superconducting (S) materials has been shown to lead to a variety of interesting spin phenomena [7], [8].\",\n \"However, the study of the interplay between magnetization dynamics and superconductivity is a relatively new topic.\",\n \"Experimentally, a narrowing of the FMR width at the superconducting transition was observed in an F/S bilayer [9].\",\n \"Theoretically, it was proposed that a dynamically precessing ferromagnet may generate a long-range proximity effect [10].\",\n \"This effect would manifest itself in the enhancement of the critical current in a phase-biased ferromagnetic Josephson junction under FMR conditions.\",\n \"Related signatures in the tunneling density of states of the F layer have also been investigated [11].\",\n \"However, these works disregard interface effects and, therefore, do not take into account the possible generation of an FMR-induced dc voltage.\",\n \"Finally, let us note that the first experiments on voltage generation by FMR of Refs.\",\n \"[5], [6] were performed with Al as the normal metal, which becomes superconducting at low temperatures.\",\n \"As the FMR-generated charge current in an F/N junction is typically associated with a spin current, one may wonder what happens in an F/S junction in the subgap regime, where transport is mediated by Andreev processes [12].\",\n \"We show that the generation of charge current in the absence of a spin current in a conventional singlet superconductor is possible.\",\n \"In fact, the absence of spin currents in the superconductor may even lead to a strong enhancement of the induced charge current as compared to the normal state.\",\n \"Figure: Setup of the junction.\",\n \"A metallic dot is coupled to a ferromagnetic lead with precessing magnetization m(t){m}(t) on the left and to a normal or superconducting lead on the right.\",\n \"The left barrier is characterized by the conductances G l G_l and G m G_m, defined above Eq.\",\n \"(), whereas the right barrier is characterized by the conductance G r G_r.As in the normal state, the two main ingredients necessary to generate the effect are spin-dependent transmissions through the junction and a spin accumulation region The effect is absent in the theory of Ref.\",\n \"[18], which did not include such a region..\",\n \"The simplest setup meeting these requirements is a metallic dot coupled through tunnel barriers to a ferromagnet and to a superconductor (see Fig.\",\n \"REF ).\",\n \"Coulomb blockade effects are neglected, assuming that the conductances of the barriers largely exceed the conductance quantum.\",\n \"The magnetization precession in the ferromagnetic lead is described by a time-dependent exchange field, ${J}(t)=J{m}(t)$ with ${m}(t)=(\\\\sin \\\\theta \\\\cos \\\\Omega t,\\\\sin \\\\theta \\\\sin \\\\Omega t,\\\\cos \\\\theta ),$ acting on the spin of the conduction electrons.\",\n \"Here the precession frequency, $\\\\Omega $ , and the tilt angle, $\\\\theta $ , are both tunable with external dc and rf fields under standard FMR conditions [13].\",\n \"We will consider them as externally fixed parameters.\",\n \"The precession of the magnetization drives the system out of equilibrium and, thus, may generate a current.\",\n \"To describe the system, we use the quasiclassical Keldysh theory [14].\",\n \"In particular, the current through the junction can be expressed in terms of the quasiclassical Green function $\\\\check{g}$ of the dot.\",\n \"Here $\\\\check{g}$ is a matrix in Keldysh, Nambu, and spin space.\",\n \"In Keldysh space, it has a triangular structure with retarded ($\\\\hat{g}^R$ ), advanced ($\\\\hat{g}^A$ ), and Keldysh ($\\\\hat{g}^K$ ) components.\",\n \"Furthermore, it satisfies the normalization condition $\\\\check{g}^2=1$ .\",\n \"The equations determining the Green functions are most conveniently written in the rotational frame for the magnetization precession, where the problem is stationary [10].\",\n \"Assuming that the conductance of the dot largely exceeds the conductances of the junctions with the leads, the equation determining $\\\\check{g}$ may be cast in the form $-i\\\\frac{2\\\\pi G_Q}{\\\\delta }\\\\left[\\\\left(E+\\\\frac{\\\\Omega }{2}\\\\sigma _z \\\\right)\\\\tau _z, \\\\check{g}\\\\right]+\\\\check{I}_l+ \\\\check{I}_r=0.$ Here $\\\\sigma _i$ and $\\\\tau _i$ are Pauli matrices in Nambu and spin space, respectively ($i=x,y,z$ ).\",\n \"Furthermore, $G_Q=e^2/\\\\pi $ is the conductance quantum (in units where $\\\\hbar =1$ ), and $\\\\delta $ is the mean level spacing in the dot.\",\n \"The spin-dependent energy shift $\\\\pm \\\\Omega /2$ is a spin-resolved chemical potential induced by the transformation from the laboratory to the rotational frame.\",\n \"The boundary conditions with the ferromagnetic ($l$ = left) and superconducting ($r$ = right) leads are represented by the matrix currents $\\\\check{I}_{l/r}$ and depend on the Green functions $\\\\check{g}_{l/r}$ describing the non-equilibrium state in the leads due to the magnetization precession.\",\n \"Tunneling through an F/N interface is generally spin-dependent.\",\n \"The relevant processes can be characterized by the total conductance of the junction, $G_l$ , and the difference between the conductances for the majority and minority electrons, $G_m$ In addition one may take into account the imaginary part of the spin-mixing conductance, $G_\\\\phi $ , which accounts for a spin-dependent phase shift upon reflection.\",\n \"In good F/N contacts with large Fermi velocity mismatch, $G_\\\\phi $ is typically small and therefore neglected.\",\n \"It may, however, be large in tunnel junctions.\",\n \"As $G_\\\\phi $ leads to spin relaxation, it would suppress the effect studied in this work..\",\n \"The matrix current at the tunnel interface between the dot and the ferromagnet then takes the form [15] $\\\\check{I}_l=\\\\frac{G_l}{2}[ \\\\check{g}_l,\\\\check{g}]+\\\\frac{G_m}{4}[ \\\\lbrace {m}\\\\cdot {{\\\\sigma }}\\\\tau _z,\\\\check{g}_l\\\\rbrace ,\\\\check{g}],$ where ${m}\\\\equiv {m}(0)$ .\",\n \"Within the quasiclassical approximation, we assume $|G_m|\\\\ll G_l$ .\",\n \"Thus, $G_m$ can be treated perturbatively.\",\n \"The Green function in the F lead, $\\\\check{g}_l$ , is determined by $\\\\left[\\\\left(E+\\\\frac{\\\\Omega }{2}\\\\sigma _z+J{{m}}\\\\cdot {\\\\sigma }\\\\right)\\\\tau _z-\\\\check{\\\\Sigma }, \\\\check{g}_l \\\\right]=0,$ where the self-energy $\\\\check{\\\\Sigma }=-i\\\\Gamma \\\\check{g}_N(E+(\\\\Omega /2)\\\\sigma _z)$ accounts for inelastic scattering in the relaxation time approximation.\",\n \"Here, $1/\\\\Gamma $ is the inelastic scattering time and $\\\\check{g}_N$ is the equilibrium Green function in a normal metal.\",\n \"Namely, $\\\\hat{g}_N^{R(A)}(E)=\\\\pm \\\\tau _z$ and $\\\\hat{g}_N^{K}(E)=2\\\\tau _z f(E)$ , where $f(E)=\\\\tanh (E/2T)$ is related to the Fermi distribution at temperature $T$ .\",\n \"For a large exchange field, $J\\\\gg \\\\Omega ,\\\\Gamma $ , the solution of Eq.\",\n \"(REF ) takes the form $\\\\hat{g}_l^{R(A)}=\\\\pm \\\\tau _z$ and $\\\\hat{g}_l^{K}=2\\\\tau _z(f_++f_-\\\\cos \\\\theta {m}\\\\!\\\\cdot \\\\!\",\n \"{{\\\\sigma }})$ , where $f_\\\\pm (E)=[f(E+\\\\Omega /2)\\\\pm f(E-\\\\Omega /2)]/2$ .\",\n \"The matrix current at the dot-superconductor tunnel interface is given as $\\\\check{I}_r=\\\\frac{G_r}{2}[ \\\\check{g}_r,\\\\check{g}],$ where $G_r$ is the conductance of the junction.\",\n \"The Green function in the S lead reads $\\\\check{g}_r=\\\\check{g}_S(E+(\\\\Omega /2)\\\\sigma _z)$ , where $\\\\check{g}_S$ is the equilibrium Green function in a superconductor.\",\n \"Namely, $\\\\hat{g}^{R(A)}_S(E)=(-iE \\\\tau _z+\\\\Delta \\\\tau _x)/\\\\sqrt{\\\\Delta ^2-(E\\\\pm i0^+)^2}$ and $\\\\hat{g}^{K}_S(E)=[\\\\hat{g}^{R}_S(E)-\\\\hat{g}^{A}_S(E)]f(E)$ , where $\\\\Delta $ is the superconducting order parameter (taken to be real).\",\n \"Now we have all the ingredients necessary to determine the Green function in the dot and subsequently the spin and charge currents at both interfaces.\",\n \"The charge currents are given by $I_{l/r}=\\\\frac{1}{16e}\\\\int dE\\\\; \\\\mathrm {Tr}[\\\\tau _z \\\\hat{I}_{l/r}^K].$ Current conservation ensures that $I\\\\equiv I_l=-I_r$ .\",\n \"The spin currents in the rotational frame are given by ${I}_{l/r}=-\\\\frac{1}{32e^2}\\\\int dE \\\\;\\\\mathrm {Tr}[ {\\\\sigma } \\\\hat{I}_{l/r}^K].$ In the laboratory frame, they decompose into a dc contribution along the precession axis, $I_{\\\\alpha ,z}$ , and ac components in the perpendicular plane, $I_{\\\\alpha ,x/y}(t)=I_{\\\\alpha ,x/y}\\\\cos \\\\Omega t\\\\mp I_{\\\\alpha ,y/x}\\\\sin \\\\Omega t$ .\",\n \"Contrarily to the charge current, the spin currents do not need to be conserved.\",\n \"Eq.\",\n \"(REF ) yields ${I}_{l}+{I}_{r}-\\\\frac{\\\\Omega }{16\\\\delta }\\\\int dE\\\\;\\\\mathrm {Tr}[(\\\\hat{\\\\bf z}\\\\times {\\\\sigma })\\\\tau _z\\\\hat{g}^K]=0.$ Thus, only the dc spin current along $\\\\hat{\\\\bf z}$ is conserved.\",\n \"While our main interest are the FMR-induced currents in the subgap regime of an F-dot-S junction, we first study the simpler case of an F-dot-N junction for comparison.\",\n \"For better readibility, in the following, we will normalize conductances by $G_\\\\Sigma =G_l+G_r$ and energies by the Thouless energy $E_g=G_\\\\Sigma \\\\delta /(4\\\\pi G_Q)$ .\",\n \"In particular, we introduce the dimensionless conductances $\\\\gamma _\\\\alpha =G_\\\\alpha /G_\\\\Sigma $ ($\\\\alpha =l,r,m$ ) as well as the dimensionless energies $\\\\epsilon =E/E_g$ and $\\\\omega =\\\\Omega /(2E_g)$ .\",\n \"A normal lead is described by setting $\\\\Delta =0$ in the above equations for $\\\\check{g}_r$ .\",\n \"In the absence of superconductivity, the retarded and advanced Green functions in the dot are trivial, $\\\\hat{g}^{R(A)}=\\\\pm \\\\tau _z$ .\",\n \"The Keldysh component is obtained with the help of Eqs.\",\n \"(REF ), (REF ), and (REF ).\",\n \"There is a remarkable relation between the spin current at the left contact with the F lead and the charge current to lowest order in $\\\\gamma _m$ , $I=\\\\frac{2e\\\\gamma _m\\\\gamma _r}{\\\\gamma _l}{m}(t)\\\\cdot {I}_l(t)=\\\\frac{2e\\\\gamma _m\\\\gamma _r}{\\\\gamma _l}(I_{l,x}\\\\sin \\\\theta +I_{l,z}\\\\cos \\\\theta ),$ namely the charge current is proportional to the projection of the spin current onto the instantaneous magnetization axis of the barrier due to the spin-dependent conductance $G_m$ .\",\n \"That is, the charge current originates from two effects: (i) the rectification of the ac in-plane spin current pumped from the ferromagnet, and (ii) the conversion of the dc spin current along the $\\\\hat{\\\\bf z}$ -axis into a charge current.\",\n \"It turns out that the two effects have opposite sign, and that the former dominates over the latter.\",\n \"Namely, we find $I_{l,x}&=&\\\\frac{G_lE_g}{2e^2}\\\\frac{\\\\omega (\\\\gamma _r+\\\\omega ^2)}{1+\\\\omega ^2}\\\\sin \\\\theta \\\\cos \\\\theta ,\\\\\\\\I_{l,z}&=&-\\\\frac{G_lE_g}{2e^2}\\\\omega \\\\gamma _r\\\\sin ^2\\\\theta .$ Note that, in the limit $\\\\gamma _r\\\\ll \\\\gamma _l$ , the spin current along the $\\\\hat{\\\\bf z}$ -axis is negligible.\",\n \"By contrast, in the limit $\\\\gamma _l\\\\ll \\\\gamma _r$ , the two components are of comparable magnitude and almost complete cancellation between the competing effects takes place.\",\n \"The charge current reads $I=\\\\frac{G_m E_g}{e}\\\\frac{\\\\gamma _r\\\\gamma _l\\\\omega ^3\\\\sin ^2\\\\theta \\\\cos \\\\theta }{1+\\\\omega ^2}.$ At large precession frequency, $\\\\omega \\\\gg 1$ , the current scales linearly with frequency, $I\\\\simeq (G_lG_rG_m/2eG_\\\\Sigma ^2)\\\\Omega \\\\sin ^2\\\\theta \\\\cos \\\\theta $ .\",\n \"In particular, in an open-circuit geometry, this would correspond to an FMR-induced dc voltage $eV=(G_m/2G_\\\\Sigma )\\\\Omega \\\\sin ^2\\\\theta \\\\cos \\\\theta $ in accordance with Refs.\",\n \"[4], [5], [6].\",\n \"At $\\\\omega \\\\ll 1$ , spin-relaxation mechanisms induced by the tunnel coupling of the dot to the leads tend to suppress the effect.\",\n \"We now turn to the F-dot-S junction.\",\n \"In the subgap regime, the spin current at the interface with the superconductor vanishes, ${I}_r=0$ .\",\n \"Thus, $I_{z,l}=0$ .\",\n \"However, an ac spin current is present at the interface with the ferromagnet.\",\n \"Then, the Andreev charge current originates entirely from the rectification of this ac spin current.\",\n \"Restricting ourselves to energy scales much smaller than $\\\\Delta $ , the Green function in the superconducting lead takes the simple form $\\\\hat{g}_r^{R(A)}=\\\\tau _x$ and $\\\\hat{g}_r^K=0$ .\",\n \"Taking $\\\\gamma _m$ as a small parameter, we search for a perturbative solution of equation (REF ) in the form $\\\\check{g} =\\\\check{g}_0+\\\\gamma _m\\\\check{g}_1+\\\\dots $ Due to the proximity effect, now the retarded and advanced Green functions of the dot are modified as well.\",\n \"To zeroth order in $\\\\gamma _m$ , an explicit solution is given by $\\\\hat{g}_0^{R(A)}=\\\\frac{\\\\gamma _r\\\\tau _x+[-i(\\\\epsilon +\\\\omega \\\\sigma _z)\\\\pm \\\\gamma _l]\\\\tau _z}{\\\\sqrt{\\\\gamma _r^2-(\\\\epsilon +\\\\omega \\\\sigma _z\\\\pm i\\\\gamma _l)^2}}.$ Here, $\\\\gamma _r$ is the effective minigap due to the coupling with the S lead [16], $\\\\omega $ an effective exchange field, and $\\\\gamma _l$ yields a broadening of the energy levels due to the coupling with the F lead.\",\n \"The Keldysh Green function can be cast in the form $\\\\hat{g}_0^K=\\\\hat{g}^R\\\\hat{\\\\varphi }-\\\\hat{\\\\varphi }\\\\hat{g}^A$ with $\\\\hat{\\\\varphi }=f_++f_-\\\\cos \\\\theta \\\\left[\\\\frac{\\\\gamma _l\\\\sin \\\\theta }{\\\\omega ^2+\\\\gamma _l^2}(\\\\gamma _l\\\\sigma _x\\\\!-\\\\!\\\\omega \\\\sigma _y)+\\\\cos \\\\theta \\\\sigma _z\\\\right]\\\\!\\\\!.$ The function $\\\\hat{\\\\varphi }$ can be interpreted as a matrix-distribution function.\",\n \"Note that it does not depend on $\\\\gamma _r$ as subgap electrons only thermalize with the F lead.\",\n \"To first order in $\\\\gamma _m$ , a solution which satisfies the normalization condition, $\\\\check{g}^2=1$ , is obtained in the form $\\\\check{g}_1=\\\\check{g}_0 \\\\check{X} -\\\\check{X} \\\\check{g}_0$ .\",\n \"For the advanced and retarded components, one finds $\\\\hat{X}^{R(A)}=\\\\mp (\\\\sin \\\\theta /2 \\\\omega )[i\\\\gamma _r/(\\\\epsilon \\\\pm i\\\\gamma _l) \\\\tau _x+\\\\tau _z]\\\\sigma _y$ .\",\n \"The Keldysh component can be decomposed as $\\\\hat{X}^K=X^K_x\\\\tau _x+X^K_z\\\\tau _z$ , where $X^K_x$ and $X^K_z$ solve the coupled equations $&&2 \\\\gamma _l X^K_z -i\\\\omega [\\\\sigma _z,X^K_z]\\\\\\\\&=&2 \\\\sin \\\\theta [ \\\\cos \\\\theta \\\\sin \\\\theta f_-+ f_+(\\\\sigma _x-\\\\frac{ \\\\gamma _l}{ \\\\omega }\\\\sigma _y)],\\\\nonumber \\\\\\\\&&2\\\\epsilon X^K_x-2i \\\\gamma _r X^K_z+\\\\omega \\\\lbrace \\\\sigma _z,X^K_x\\\\rbrace \\\\\\\\&=&2i\\\\frac{\\\\gamma _l\\\\gamma _r\\\\sin \\\\theta }{ \\\\omega (\\\\gamma _l^2+ \\\\epsilon ^2)}[\\\\gamma _l f_+\\\\sigma _y-\\\\epsilon f_- \\\\cos \\\\theta (\\\\cos \\\\theta \\\\sigma _x-\\\\sin \\\\theta \\\\sigma _z) ].\\\\nonumber $ Evaluating the current at the right interface, Eq.\",\n \"(REF ) yields $I=-i\\\\gamma _m G_r/(16e)\\\\int dE\\\\; \\\\mathrm {Tr}[\\\\tau _y \\\\hat{g}_1^K]$ .\",\n \"Inserting the solution for $\\\\hat{g}_1^K$ and using the property $\\\\hat{g}_0^R(-\\\\epsilon )=-\\\\sigma _x\\\\tau _z\\\\hat{g}_0^A(\\\\epsilon )\\\\sigma _x\\\\tau _z$ , we obtain the current $I&=& \\\\frac{1}{2}I_0\\\\frac{\\\\gamma _r^2\\\\omega }{\\\\gamma _l^2+\\\\omega ^2}\\\\int d\\\\epsilon \\\\;\\\\frac{\\\\epsilon f_-}{(\\\\gamma _l^2+\\\\epsilon ^2)(\\\\epsilon +\\\\omega )}\\\\\\\\&&\\\\qquad \\\\qquad \\\\qquad \\\\times \\\\sum _\\\\pm \\\\frac{-\\\\gamma _l(\\\\epsilon +\\\\omega )\\\\pm i(\\\\gamma _l^2-\\\\epsilon \\\\omega )}{\\\\sqrt{\\\\gamma _r^2-(\\\\epsilon +\\\\omega \\\\pm i \\\\gamma _l)^2}},\\\\nonumber $ where $I_0= (G_mE_g/e)\\\\sin ^2\\\\theta \\\\cos \\\\theta $ .\",\n \"The current as a function of frequency for different values of $\\\\gamma _l=1-\\\\gamma _r$ is shown in Fig.\",\n \"REF .\",\n \"Simple analytic expressions can be found in different asymptotic regimes.\",\n \"Figure: Andreev current induced by ferromagnetic resonance as a function of precession frequency for different values of γ l =G l /(G r +G l )\\\\gamma _l=G_l/(G_r+G_l).Here I 0 =(G m E g /e)sin 2 θcosθI_0=(G_mE_g/e)\\\\sin ^2\\\\theta \\\\cos \\\\theta .In particular, at temperature $T=0$ and low frequency $|\\\\omega |\\\\ll \\\\gamma $ , where $\\\\gamma =(\\\\gamma _l^2+\\\\gamma _r^2)^{1/2}$ , the FMR-induced current is given as $I\\\\simeq \\\\frac{10}{3}I_0\\\\frac{\\\\gamma _r^2\\\\gamma _l}{\\\\gamma ^7}\\\\omega ^5.$ The large power $\\\\omega ^5$ indicates the strong suppression of the effect.\",\n \"At large frequencies $|\\\\omega |\\\\gg \\\\gamma $ , the current saturates.\",\n \"The frequency-independent value is given by $I\\\\simeq \\\\frac{\\\\pi }{2}I_0\\\\,\\\\mathrm {sign}(\\\\omega )\\\\times \\\\left\\\\lbrace \\\\begin{array}{lll}\\\\gamma _r^2, & \\\\,& \\\\gamma _r\\\\ll \\\\gamma _l, \\\\\\\\1, & \\\\,& \\\\gamma _l\\\\ll \\\\gamma _r.\\\\end{array}\\\\right.$ The saturation can be understood as Andreev processes become inefficient at energies larger than the minigap [17].\",\n \"Depending whether the dot is more strongly coupled to the ferromagnet or to the superconductor, the crossover between these asymptotic regimes is different.\",\n \"If the dot is weakly coupled to the superconductor, $\\\\gamma _r\\\\ll \\\\gamma _l$ , a smooth crossover happens at $\\\\omega \\\\sim 1\\\\gg \\\\gamma _r$ with a typical current $I/I_0\\\\sim \\\\gamma _r^2$ .\",\n \"By contrast, if the dot is weakly coupled to the ferromagnet, $\\\\gamma _l\\\\ll \\\\gamma _r$ , the crossover in the region $\\\\omega \\\\sim 1/2$ is described by $I\\\\simeq I_0\\\\times \\\\!\\\\left\\\\lbrace \\\\begin{array}{lll}\\\\gamma _l/(2\\\\sqrt{-\\\\delta \\\\omega }),&\\\\,&\\\\!-1\\\\ll \\\\delta \\\\omega \\\\ll -\\\\gamma _l,\\\\\\\\2\\\\sqrt{\\\\delta \\\\omega },&\\\\,&\\\\!\\\\gamma _l\\\\ll \\\\delta \\\\omega \\\\ll 1,\\\\end{array}\\\\right.$ where $\\\\delta \\\\omega \\\\equiv \\\\omega -1/2$ , with a typical current $I/I_0\\\\sim \\\\sqrt{\\\\gamma _l}$ at $\\\\omega =1/2$ .\",\n \"While in the asymptotic regimes of $\\\\omega $ very small or very large, the current is suppressed as compared to the normal state, there is in fact a wide intermediate regime where it may be strongly enhanced.\",\n \"Comparing Eqs.\",\n \"(REF ) and (REF ), one notices that, if the dot is strongly coupled to the superconductor, in the regime $1/2<\\\\omega <\\\\gamma _l^{-1}$ the induced current in the superconducting state exceeds the induced current in the normal state.\",\n \"This effect may be understood due to the absence of a dc spin current along along the $\\\\hat{\\\\bf z}$ -axis which leads to a strong suppression of the effect in the normal state.\",\n \"Fig.\",\n \"REF shows the current in the superconducting and normal state as well the contribution due to rectification only in the normal state.\",\n \"The ratio between the current in the superconducting state and the latter contribution in the normal state reflects the ratio between Andreev and normal state conductances in an N-dot-S junction.\",\n \"Figure: Induced current in the superconducting (dotted line) and normal state for γ l =0.4\\\\gamma _l=0.4.The thin line shows the contribution to the normal state current due to rectification only.So far we assumed that the magnetization in the ferromagnet is uniform.\",\n \"However, boundary effects may lead to a suppression of the magnetization in the vicinity of the F/N interface.\",\n \"This would result in a different resonance frequency at the barrier than in the ferromagnetic reservoir and, consequently, in a tilt angle $\\\\theta _B\\\\ne \\\\theta $ at FMR.\",\n \"The effect can be accounted for by replacing ${m}$ with ${m}_B=(\\\\sin \\\\theta _B,0,\\\\cos \\\\theta _B)$ in Eq.\",\n \"(REF ).\",\n \"In particular, at $\\\\theta _B=0$ , the spin dependent conductance $G_m$ refers to the constant axis $\\\\hat{\\\\bf z}$ .\",\n \"In the normal case, the relation between spin and charge currents, Eq.\",\n \"(REF ), now reads $I=(2e\\\\gamma _m\\\\gamma _r/\\\\gamma _l){m}_B(t)\\\\cdot {I}_l(t)$ .\",\n \"While at $\\\\theta _B=\\\\theta $ the rectification of the in-plane ac spin currents always dominates over the conversion of the dc spin current along $\\\\hat{\\\\bf z}$ into a charge current, this effect is completely suppressed at $\\\\theta _B=0$ .\",\n \"As a consequence, at $\\\\theta _B=0$ , the charge current, $I=-({G_m E_g}/{e})\\\\omega \\\\gamma _r^2\\\\sin ^2\\\\theta $ , has the opposite sign compared to Eq.\",\n \"(REF ).\",\n \"In general, both effects are important.\",\n \"The sign reversal occurs at $\\\\tan \\\\theta _B=[1-\\\\gamma _l\\\\omega ^2/(\\\\gamma _r+\\\\omega ^2)]\\\\tan \\\\theta $ .\",\n \"In the superconducting case, the dc spin current along $\\\\hat{\\\\bf z}$ is always zero, and the charge current is due entirely to the rectification of the in-plane ac spin currents.\",\n \"As a consequence, we find that the charge current vanishes at $\\\\theta _B=0$ .\",\n \"The general result is obtained from Eq.\",\n \"(REF ) by replacing $I_0$ with $I_0^B=(G_mE_g/e)\\\\sin \\\\theta _B\\\\sin \\\\theta \\\\cos \\\\theta $ .\",\n \"In summary, we demonstrate that a subgap charge current in an F/S junction may be induced by ferromagnetic resonance.\",\n \"The effect is due to the rectification of ac spin currents generated by the precessing magnetization in the ferromagnet.\",\n \"In the normal case, a competing effect of conversion of a dc spin current into a charge current exists.\",\n \"This effect is absent in an F/S junction as the superconductor cannot carry a subgap spin current.\",\n \"As a consequence, the induced current in the superconducting state may be strongly enhanced as compared to the normal state.\",\n \"Interesting non-equilibrium phenomena should be expected in ferromagnetic Josephson junctions under ferromagnetic resonance conditions.\",\n \"We acknowledge funding through an ANR grant (ANR-11-JS04-003-01) and an EU-FP7 Marie Curie IRG.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1491"}}},{"rowIdx":1867486,"cells":{"text":{"kind":"list like","value":[["The Galactic Center Weather Forecast"],["Abstract In accretion-based models for Sgr A* the X-ray, infrared, and millimeter emission arise in a hot, geometrically thick accretion flow close to the black hole.","The spectrum and size of the source depend on the black hole mass accretion rate $\\dot{M}$.","Since Gillessen et al.","have recently discovered a cloud moving toward Sgr A* that will arrive in summer 2013, $\\dot{M}$ may increase from its present value $\\dot{M}_0$.","We therefore reconsider the \"best-bet\" accretion model of Moscibrodzka et al., which is based on a general relativistic MHD flow model and fully relativistic radiative transfer, for a range of $\\dot{M}$.","We find that for modest increases in $\\dot{M}$ the characteristic ring of emission due to the photon orbit becomes brighter, more extended, and easier to detect by the planned Event Horizon Telescope submm VLBI experiment.","If $\\dot{M} \\gtrsim 8 \\dot{M}_0$ this \"silhouette of the black hole will be hidden beneath the synchrotron photosphere at 230 GHz, and for $\\dot{M} \\gtrsim 16 \\dot{M}_0$ the silhouette is hidden at 345 GHz.","We also find that for $\\dot{M} > 2 \\dot{M}_0$ the near-horizon accretion flow becomes a persistent X-ray and mid-infrared source, and in the near-infrared Sgr A* will acquire a persistent component that is brighter than currently observed flares."],["Introduction","The recent discovery of a cloud moving towards the Galactic Center [18] creates a potential opportunity for testing models of Sgr A*.","The $> 3 M_{\\oplus }$ cloud will interact strongly with gas near nominal pericenter at $r_p \\simeq 300 {\\rm AU} \\simeq 8000GM/c^2$ ($M \\equiv $ black hole mass), and may change the black hole accretion rate $\\dot{M}$ .","Since the structure of the cloud and the surrounding medium are uncertain, possible outcomes range from very small changes in the accretion rate over timescales of decades to rapid, large changes in the accretion rate.","The dynamical timescale at $r_p$ is $t_d = (r_p^3/(G M))^{1/2} = 0.5$ yr, and the viscous timescale $t_{vis} = (\\alpha \\Omega )^{-1} (R/H)^2\\approx 100$ yr $\\gg t_{PhD}$ , assuming $\\alpha = 0.05$ and $H/R = 0.3$ , i.e.","a hot, radiatively inefficient accretion flow.","After an initial transient phase while the flow circularizes—accompanied by transient emission—it is natural to think the flow will settle into a steady state.","The settling timescale could be as little as a few $t_d$ , and so the steady state may arrive as soon as mid-2014.","If the resulting flow can be modeled as a steady disk, the excess mass will drain away on the viscous timescale, i.e.","the source will remain bright well into the 22nd century.","It is therefore interesting to ask how changes in $\\dot{M}$ will manifest themselves observationally.","Current observations of Sgr A* show $F_\\nu = 0.5-1$ Jy at 1-50 GHz with a nearly flat spectral slope [10]; $F_\\nu \\sim \\nu ^{p=0.17-0.3}$ .","The spectral slope becomes flatter and variable at 230-690 GHz, with $p= -0.46 - 0.08$ , [22]), which is commonly interpreted as signaling a transition from optically thick to optically thin synchrotron emission.","The discovery of polarized emission (polarization fraction at level of a few per cent) at $\\lambda =1.3$ mm and subsequent measurement of Faraday rotation imply a model dependent limit $2 \\times 10^{-7} > \\dot{M} > 2 \\times 10^{-9}{\\rm M_\\odot yr^{-1}}$ in the inner accretion flow ([4], [23]).","Sgr A* is resolved by VLBI at $1.3$ mm where interstellar scattering is comparable to the intrinsic source FWHM $37^{+16}_{-10} \\mu {\\rm as}$ [7].","This is smaller than the apparent diameter of the event horizon $\\sim 55 \\mu as$ .","Sgr A* fluctuates rapidly in the near-infrared (NIR), with average $F_\\nu \\sim 1 {\\rm mJy}$ ([6]).","It is not yet detected at mid-infrared (MIR) wavelengths ($\\lambda \\le 8.6 \\mu {\\rm m}$ , e.g.","[33]).","In the X-ray Sgr A* exhibits flares with characteristic duration of order an hour and a duty cycle of order $5\\%$ .","The upper limit for quiescent state X-ray emission is $\\nu L_\\nu < 2.4 \\times 10^{33} {\\rm erg s^{-1}}$ [3].","We are not aware of any secular trends in these observed properties of Sgr A*.","One model of Sgr A* that fits most observational constraints is our relativistic accretion model [27], where submillimeter, IR, and X-ray emission arise in an optically thin, geometrically thick disk close to the event horizon; radio emission is assumed to arise nonthermally in a synchrotron photosphere at larger radius [30], but is not predicted by our model, which focuses on the inner parts of the accretion flow.","The underlying flow model is a general relativistic magnetohydrodynamic (GRMHD) simulation ([14], [28], [36]).","The emerging radiation is calculated using Monte Carlo [8] and ray-tracing schemes.","The $1.3$ mm flux originates as Doppler boosted synchrotron emission from the approaching side of the disk between $\\sim 10 GM/c^2$ and the innermost stable circular orbit (ISCO).","Our relativistic disk model assumes a thermal electron distribution function, neglects thermal conduction, and assumes a constant ratio of ion to electron temperature $T_i/T_e$ .","The model does not produce the observed IR flaring at the correct amplitude, but can with the addition of a small nonthermal component in the distribution function [19].","If we fix the black hole mass ($M_{BH}=4.5 \\times 10^6 M_{\\odot }$ , [16]) and distance ($D=8.4\\, {\\rm kpc}$ , [17]), the remaining model parameters are the source inclination $i$ , black hole spin $a_*$ ($0 \\le a_* \\le 1$ ), $T_i/T_e$ , and $\\dot{M}$ .","We fix $\\dot{M}$ so that the $1.3$ mm flux matches the observed $\\simeq 3$ Jy.","The relativistic accretion models are not tightly constrained by the data, but they reveal the following: (1) face-on models that reproduce the millimeter flux would look like rings and therefore, in VLBI data, have dips in visibilities on fixed intermediate baselines, while existing observations suggest that the ring radius would need to vary to fit the data ([13]; see also the discussion of [5]).","More nearly edge-on models are therefore favored; (2) models with $a_* \\gtrsim 0.98$ that reproduce the millimeter flux have a hot, dense inner disk that would overproduce X-rays via inverse Compton scattering.","Lower spin, $a_* \\sim 0.9$ , models are therefore favored; (3) the observed source size and flux fix the temperature of the emitting electrons $T_e = F_\\nu c^2/(4\\pi k \\nu ^2\\sigma ^2)$ ($\\sigma \\equiv $ the RMS size of the source on the sky) and this turns out to favor $T_i/T_e \\simeq 3$ models.","The “best-bet” model from [27] has $a_* \\simeq 0.94$ , $i = 85$ deg, $\\dot{M} \\equiv \\dot{M}_0 \\simeq 2 \\times 10^{-9} {\\rm M_{\\odot }yr^{-1}}$ , and $T_i/T_e = 3$ .","The relativistic disk model uses self-consistent dynamics and radiative transfer but is not unique.","The electron distribution function is particularly poorly constrained.","It is likely anisotropic, may contain multiple temperature components ([32]) and power-law components, and vary in basic functional form with time and position.","Alternative accretion models (e.g.","[5], [35]) make different assumptions about the flow and/or distribution function and favor slightly different $\\dot{M}, a_*,$ and $i$ .","These models may respond differently to an increase in $\\dot{M}$ .","Indeed, radically different models may also fit the data.","The model dynamics depends on the initial magnetic field distribution, particularly the distribution of vertical flux through the accretion disk; models with large vertical magnetic flux (e.g.","[24]) are likely to respond quite differently to variations in the mass flux.","Also, jet models for Sgr A* ([11], [21], [12]) posit a luminous jet and a comparatively dim accretion disk.","Again, these may respond differently to an influx of mass.","How, then, does our relativistic disk model respond to an increase in $\\dot{M}$ ?","In this Letter we calculate the $1.3$ and $0.87$ mm flux and source size as well as the spectrum that would result for the best-bet model over a range of $\\dot{M}$ .","One key question we seek to answer is whether a small increase in $\\dot{M}$ would hide the event horizon (and the signature ring-like appearance of the photon orbit, also known as the shadow or silhouette of the event horizon) underneath a synchrotron photosphere.","This might prevent detection of the photon orbit by the planned Event Horizon Telescope [7].","Another key question is whether the increased $\\dot{M}$ would make Sgr A* detectable in its quiescent state in the IR and X-ray.","Below we describe variation of the flux and source morphology at 230GHz ($1.3$ mm) and 345GHz ($0.87$ mm) (§ ), describe variation of the spectrum (§ ), and finally discuss which features of the results are likely to be most robust (§ )."],["Change of Sgr A* sub-mm luminosity and size for enhanced $\\dot{M}$","How do we naively expect the mm disk image size and flux to respond to changes in $\\dot{M}$ ?","In our model, $1.3$ mm emission in Sgr A* is thermal synchrotron emission from plasma with optical depth $\\tau _\\nu \\sim 1$ , near the ISCO.","The true electron distribution undoubtedly contains nonthermal components (see, Riquelme et al.","2012).","Models with thermal + power-law distribution functions (e.g.","[5]) contain an O(1/3) nonthermal contribution to the flux at $1.3$ mm, which hints at how uncertainty in the distribution function translates into uncertainties in the spectrum.","The thermal synchrotron absorptivity is $\\alpha _{\\nu ,a}=j_\\nu /B_\\nu $ , where $j_{\\nu } = \\frac{\\sqrt{2}\\pi e^2 n_e \\nu _s}{3cK_2(\\Theta _e^{-1})}(X^{1/2} + 2^{11/12} X^{1/6} )^2 \\exp (-X^{1/3})$ , $X = \\nu /\\nu _s$ , $\\nu _s=2/9 (eB/2\\pi m_ec) \\Theta _e^2 \\sin \\theta $ , $\\theta $ is an angle between the magnetic field vector and emitted photon, $K_2$ is a modified Bessel function of the second kind [20] and $B_\\nu \\simeq 2 \\nu ^2 \\Theta _e m_e$ .","Near $1.3$ mm, $X \\sim 1$ , and the emissivity is nearly independent of frequency, so $j_\\nu \\propto \\nu ^0n_e B$ and $\\alpha _{\\nu ,a} \\propto \\nu ^{-2} n_e B \\Theta _e^{-1}$ .","We will assume that $n_e \\propto \\dot{M} r^{-3/2}$ , $\\Theta _e \\propto 1/r$ , and $\\beta \\sim $ const., so that $B \\propto \\dot{M}^{1/2}r^{-5/4}$ for $r > G M/c^2$ , and ignore relativistic corrections.","Then for $\\tau _\\nu \\ll 1$ (or $\\dot{M} \\ll \\dot{M}_0$ ) the source has size $\\sim G M/c^2$ and the flux $F_\\nu \\sim (4/3)\\pi (G M/c^2)^3 j_\\nu \\propto \\dot{M}^{3/2}$ .","For $\\tau _\\nu \\gg 1$ the source size is set by the photosphere radius $r_{ph}$ where $\\int _{r_{ph}}^\\infty \\alpha _a(r)dr = 1$ (i.e.","for $\\dot{M} \\gg \\dot{M}_0$ , but not so large that $\\Theta _e(r_{ph}) < 0.5$ so that our emissivity approximation fails).","Then $r_{ph} \\propto \\dot{M}^{3/2}/\\nu ^2$ , and the flux $F_\\nu \\propto r_{ph}^2 B_\\nu (r_{ph}) \\sim \\dot{M}^{9/4}/\\nu $ .","These simple scaling laws, unfortunately, are not a good description of the variation of source size and flux with $\\dot{M}$ .","There are at least three reasons for this.","First, relativistic effects are very important; for $\\dot{M} \\sim \\dot{M}_0$ emission comes from close to the photon orbit and the source size is determined by Doppler beaming and gravitational lensing.","Second, for the best-bet model at $1.3$ mm $\\tau _\\nu \\sim 1$ , so in a turbulent flow there is a complicated variation of the size of the effective photosphere with $\\dot{M}$ .","Third, the emissivity is not precisely frequency independent near peak.","We therefore need to turn to numerical models.","The best-best model is taken from a survey of 2D models.","Here we adopt the best-bet model parameters ($a_*=0.94$ , $i=85 \\deg $ and $T_i/T_e=3$ ) and use them to set parameters for a fully 3D model [36] A 3D GRMHD model parameter survey is still too computationally expensive.","It also has a poor return on investment given the electron distribution function uncertainties.. We use the same data set as [9], and choose three representative snapshots taken at times when the flow is quiescent ($t=5000, 9000$ and $13000 GM/c^3$ , where $GM/c^3 = 20$ s).","We then recalculate disk images and spectra for $\\dot{M} = (0.5, 1, 2, 4, 8, 16,32,64) \\dot{M}_0$ .","The 3D model with $\\dot{M} = \\dot{M}_0$ is broadly consistent with observational data but is slightly more luminous at higher energies than 2D models ($\\beta $ is lower in the 3D models, and this changes the X-ray to millimeter color).","The model is self-consistent only for $\\dot{M} <64 \\dot{M}_0$ .","At higher $\\dot{M}$ the efficiency of the flow is $>0.1$ and therefore our neglect of cooling in the underlying GRMHD model is not justified.","At higher $\\dot{M}$ the $1.3$ mm photosphere also lies outside the limited range in radius where $\\langle d\\dot{M}/dr \\rangle =0$ , so the flow is not in a steady state.","The images and total fluxes emitted by the disk at 230 and 345GHz are calculated using a ray tracing scheme [29].","To estimate the size of the emitting region we calculate the eigenvalues of the matrix formed by taking the second angular moments of the image on the sky (the principal axis lengths).","The major and minor axis eigenvalues, $\\sigma _1$ and $\\sigma _2$ respectively, are related by $\\sigma ={\\rm FWHM}/2.3$ to the FWHM of the axisymmetric Gaussian model used to interpret the VLBI observations.","We use $\\langle \\sigma \\rangle = (\\sigma _1+\\sigma _2)/2$ to measure the average radius of the emitting spot.","Figures REF and REF show the variation of 230 and 345 GHz Sgr A* model images with $\\dot{M}$ (based on a single snapshot from the 3D GRMHD simulation).","Evidently modest increases $\\dot{M}$ will make the ring-like feature that is the observational signature of the photon orbit easier to detect.","For $\\dot{M} \\gtrsim 8 \\dot{M}_0$ , however, the ring (or black hole silhouette) is hidden beneath the synchrotron photosphere at 230 GHz.","The silhouette survives for higher $\\dot{M}$ at 345 GHz, disappearing only at $\\dot{M} \\gtrsim 16\\dot{M}_0$ .","For low $\\dot{M}$ the silhouette is also difficult to detect because the emitting region is too small.","Figure: Horizon silhouette detectability at 230 GHz for variousM ˙\\dot{M}.","Panels from a) to h) show the imagesof Sgr A* calculated for M ˙=(0.5,1,2,4,8,16,32,64)M ˙ 0 \\dot{M} =(0.5, 1, 2, 4, 8, 16, 32, 64) \\dot{M}_0 ,respectively.","The center of the circle is positioned at the image centroidand its radius, r=(σ 1 +σ 2 )/2r=(\\sigma _1+\\sigma _2)/2 is the RMS radius ofthe emitting region.Figure: Same as in Figure but for ν=345\\nu =345 GHz.Figure: Sizes (in terms of σ\\sigma ) of the emitting region and the total flux as afunction of M ˙\\dot{M} at 230 (upper panels) and 345GHz (lowerpanels).","Different point types correspond to three different snapshots fromthe 3D GRMHD simulation.","The dashed/dotted lines show the best fit to the databy Equations , , , and.Figure REF , shows the accretion flow image size ($\\langle \\sigma \\rangle $ , circle radii in Figure REF and REF ) and flux at 230 and 345GHz as a function of $\\dot{M}$ .","Different point types correspond to three different snapshots from the 3D GRMHD simulation.","The size of the emission region and flux increase with $\\dot{M}$ .","The 345/230 GHz flux ratio increases with the increasing $\\dot{M}$ ; this is caused by the shift of the synchrotron peak towards higher energies at higher $\\dot{M}$ .","The following simple fitting formulas describe how $\\sigma $ and $F_\\nu $ depend on $\\dot{M}$ : $\\langle \\sigma \\rangle _{230{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}15.2 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{0.38},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\21.1 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 13,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.","}\\, \\, {\\rm [\\mu as]}$ $\\langle \\sigma \\rangle _{345{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}12 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{0.31},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\19.7 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 8.3,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.","}\\, \\, {\\rm [\\mu as]}$ $F_{230{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}3.17 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{1.03},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\10.62 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 3.8,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.","}\\, \\, {\\rm [Jy]} $ $F_{345{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}3.2 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{1.3},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\25.13 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 0.54,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.","}\\, \\, {\\rm [Jy]} $ The above constants are nontrivial to interpret because they encapsulate the complexities of the accretion flow structure and relativistic effects in the radiation transport.","The advantage of the above formulas is their simplicity.","The fitting functions are shown in Figure REF as dashed and dotted lines.","Figure: Relation between two observables: the flux and the size of theimage at ν=\\nu =230 and 345 GHz.","The dashed and dotted lines are thebest fits to the data points.Figure REF shows the relation between two observables, $\\sigma $ and $F_\\nu $ .","The size is a linear function of the flux and increases more steeply at 230 GHz than at 345 GHz.","We also provide two phenomenological scaling laws fitted to the data: $\\langle \\sigma \\rangle _{230GHz} = 1.72 (F_{230{\\rm GHz}}/{\\rm Jy}) + 9.3 \\,\\,\\,{\\rm [\\mu as]} $ $\\langle \\sigma \\rangle _{345GHz} = 0.73 (F_{345{\\rm GHz}}/{\\rm Jy}) + 9.2 \\,\\,\\,{\\rm [\\mu as]} $ Notice that these fits apply to the best-bet model only.","For other $a_*$ , $T_i/T_e$ , or $i$ the scalings will be slightly different, although if the source is optically thick the linear scaling follows directly from $T_e \\propto 1/r$ .","Notice that Equation REF predicts the change of Sgr A* size by 9 per cent when 230 GHz flux changes from 2 to 2.7 Jy.","Taking into account observational and theoretical uncertainties this is consistent with the observed variations of the size of the source which increases by a few per cent as the flux increases from 2 to 2.7 Jy [13]."],["Spectra","Our model spectra are generated by thermal synchrotron emission in the submillimeter/far-IR bump, and by Compton scattering in the X-rays.","The spectral slope of flaring NIR emission, and its high degree of linear polarization [6], imply that it is synchrotron from a small, nonthermal tail of high energy electrons that is not (but can be; see [19]) included in our best-bet model.","What are the expected scaling laws?","Again, $j_\\nu \\sim n_e B \\sim \\dot{M}^{3/2}$ .","The luminosity around the synchrotron peak is $L_{peak}\\sim 4 \\pi \\nu _{peak} j_{\\nu _{peak}} (GM/c^2)^3 \\sim \\dot{M}^{9/4}$ , where $\\nu _{peak} \\sim \\dot{M}^{3/4}$ is such that $\\alpha _\\nu GM/c^2 =1$ .","The emission rightward of the MIR/NIR is produced by Compton up-scattered synchrotron radiation.","The Thomson depth $\\tau _{sc} =\\sigma _{TH} n_e GM/c^2 \\sim \\dot{M}$ , the X-ray luminosity is expected to scale as: $\\nu L_\\nu (\\nu \\approx 10^{18} {\\rm Hz} ) \\sim L_{peak}\\tau _{sc} \\sim \\dot{M}^{13/4}$ , assuming that X-rays are produced primarily by singly scattered synchrotron photons.","What do the numerical models show?","Figure REF shows spectra emitted from the 3D disk model as observed at $i\\approx 85 \\deg $ .","The SEDs are calculated using a general relativistic Monte Carlo scheme [8].","The NIR luminosity ${\\nu }L_{\\nu }(\\nu =10^{14} {\\rm Hz}) \\sim \\dot{M}^{2.5}$ , which is only slightly steeper than the expected dependence for the synchrotron peak.","${\\nu }L_{\\nu }(\\nu \\approx 10^{18} {\\rm Hz}) \\sim \\dot{M}^{3.25}$ agrees well with the expected scaling.","We conclude that Sgr A* would become a persistent MIR and X-ray source (above the present upper limits of $84 {\\rm mJy}$ in MIR and $2.4 \\times 10^{33} {\\rm erg s^{-1}}$ in X-rays) if $\\dot{M} > 2 \\dot{M}_0$ .","This is conservative in the sense that our models are strictly thermal.","The addition of a high energy nonthermal tail would only increase the MIR/NIR and X-ray flux.","We do not consider higher accretion rate models because for $\\dot{M} =64 \\dot{M}_0$ (${\\nu }L_{\\nu }(\\nu =10^{18} {\\rm Hz})=10^{39} {\\rm ergs^{-1}}$ ) the model becomes radiatively efficient, $\\epsilon =L_{Bol}/\\dot{M} c^2 > 0.1$ , and our neglect of radiative cooling in the underlying 3D GRMHD model is unjustified.","Finally, notice that the MeV flux increases sharply with $\\dot{M}$ .","This suggests that electron-positron pair production by photon-photon collisions in the funnel over the poles of the hole would increase sharply (the pair production $\\dot{n}_{e^\\pm } \\sim L_{\\gamma }^2$ , [26]).","If this pair production is connected to jet production, it is reasonable to think that a high $\\dot{M}$ Sgr A* might also produce a jet.","Figure: Spectrum emitted by 3D disk model for various M ˙\\dot{M}.","TheM ˙/M ˙ 0 \\dot{M}/\\dot{M}_0 is shown on the righthand side.","Observational pointsand upper limits are taken from: , ,, , ,.","The black symbols in the NIR showing the flaringstate are from .","An example of X-ray flare is taken from ."],["Discussion","In summary, we have used general relativistic disk models and relativistic radiative transfer to recalculate millimeter images and spectra for Sgr A* at a range of $\\dot{M} > \\dot{M}_0$ .","Our models predict the following: (1) if the 230 GHz flux increases by more than a factor of 2, corresponding to an increase of $\\dot{M}$ by more than a factor of 2, the central accretion flow will become a persistent, detectable, MIR and X-ray source; (2) the photon orbit, which produces the narrow ring of emission visible in Figures 1 and 2, becomes easier to detect for modest increases in the 230 GHz flux; (3) the photon orbit is cloaked beneath the synchrotron photosphere at 230 GHz for $\\dot{M}\\gtrsim 8 \\dot{M}_0$ , or 230 GHz flux $\\gtrsim 13$ Jy; (4) the photon orbit is cloaked at 345 GHz only at higher $\\dot{M} \\gtrsim 16\\dot{M}_0$ , or 230 GHz flux $\\gtrsim 17$ Jy; (5) the size of the source increases in proportion to the flux at both 230 and 345 GHz.","We suspect that almost any accretion model for Sgr A* with a spatially uniform model for the plasma distribution function will reach qualitatively similar conclusions, but that jet models may differ significantly.","There are order-unity uncertainties in our model predictions due to uncertainties in the plasma model.","What range of changes in $\\dot{M}$ are reasonable?","In our best-bet model the mass at radii within a factor of two of the pericenter radius is $\\approx 10^{-2.5}$ $M_{\\oplus }$ , assuming steady mass inflow from $r_p$ to the event horizon.","The addition of even a fraction of the inferred cloud mass to the accretion flow in a ring near $r_p$ would (eventually) increase $\\dot{M}$ by a factor of $\\sim 300$ .","On the other hand, stellar winds supply mass in the neighborhood of the central black hole at $\\sim 10^{-3} M_{\\odot } yr^{-1}$ .","Models by [31] and [34] suggest that most of this mass is ejected in the form of a wind, and that $\\sim 10^{-4.5}M_{\\odot } yr^{-1}$ to $10^{-7.3} M_{\\odot } yr^{-1}$ flows inward.","A reasonable extrapolation of these models suggest the accretion flow at $r < r_p$ has a mass of $\\sim 2 M_{\\oplus }$ ; this is comparable to estimates of the mass of the inflowing cloud, so in this case we might expect a factor of 2 increase in $\\dot{M}$ .","This work was supported by the National Science Foundation under grant AST 07-09246 and by NASA under grant NNX10AD03G, through TeraGrid resources provided by NCSA and TACC."]],"string":"[\n [\n \"The Galactic Center Weather Forecast\"\n ],\n [\n \"Abstract In accretion-based models for Sgr A* the X-ray, infrared, and millimeter emission arise in a hot, geometrically thick accretion flow close to the black hole.\",\n \"The spectrum and size of the source depend on the black hole mass accretion rate $\\\\dot{M}$.\",\n \"Since Gillessen et al.\",\n \"have recently discovered a cloud moving toward Sgr A* that will arrive in summer 2013, $\\\\dot{M}$ may increase from its present value $\\\\dot{M}_0$.\",\n \"We therefore reconsider the \\\"best-bet\\\" accretion model of Moscibrodzka et al., which is based on a general relativistic MHD flow model and fully relativistic radiative transfer, for a range of $\\\\dot{M}$.\",\n \"We find that for modest increases in $\\\\dot{M}$ the characteristic ring of emission due to the photon orbit becomes brighter, more extended, and easier to detect by the planned Event Horizon Telescope submm VLBI experiment.\",\n \"If $\\\\dot{M} \\\\gtrsim 8 \\\\dot{M}_0$ this \\\"silhouette of the black hole will be hidden beneath the synchrotron photosphere at 230 GHz, and for $\\\\dot{M} \\\\gtrsim 16 \\\\dot{M}_0$ the silhouette is hidden at 345 GHz.\",\n \"We also find that for $\\\\dot{M} > 2 \\\\dot{M}_0$ the near-horizon accretion flow becomes a persistent X-ray and mid-infrared source, and in the near-infrared Sgr A* will acquire a persistent component that is brighter than currently observed flares.\"\n ],\n [\n \"Introduction\",\n \"The recent discovery of a cloud moving towards the Galactic Center [18] creates a potential opportunity for testing models of Sgr A*.\",\n \"The $> 3 M_{\\\\oplus }$ cloud will interact strongly with gas near nominal pericenter at $r_p \\\\simeq 300 {\\\\rm AU} \\\\simeq 8000GM/c^2$ ($M \\\\equiv $ black hole mass), and may change the black hole accretion rate $\\\\dot{M}$ .\",\n \"Since the structure of the cloud and the surrounding medium are uncertain, possible outcomes range from very small changes in the accretion rate over timescales of decades to rapid, large changes in the accretion rate.\",\n \"The dynamical timescale at $r_p$ is $t_d = (r_p^3/(G M))^{1/2} = 0.5$ yr, and the viscous timescale $t_{vis} = (\\\\alpha \\\\Omega )^{-1} (R/H)^2\\\\approx 100$ yr $\\\\gg t_{PhD}$ , assuming $\\\\alpha = 0.05$ and $H/R = 0.3$ , i.e.\",\n \"a hot, radiatively inefficient accretion flow.\",\n \"After an initial transient phase while the flow circularizes—accompanied by transient emission—it is natural to think the flow will settle into a steady state.\",\n \"The settling timescale could be as little as a few $t_d$ , and so the steady state may arrive as soon as mid-2014.\",\n \"If the resulting flow can be modeled as a steady disk, the excess mass will drain away on the viscous timescale, i.e.\",\n \"the source will remain bright well into the 22nd century.\",\n \"It is therefore interesting to ask how changes in $\\\\dot{M}$ will manifest themselves observationally.\",\n \"Current observations of Sgr A* show $F_\\\\nu = 0.5-1$ Jy at 1-50 GHz with a nearly flat spectral slope [10]; $F_\\\\nu \\\\sim \\\\nu ^{p=0.17-0.3}$ .\",\n \"The spectral slope becomes flatter and variable at 230-690 GHz, with $p= -0.46 - 0.08$ , [22]), which is commonly interpreted as signaling a transition from optically thick to optically thin synchrotron emission.\",\n \"The discovery of polarized emission (polarization fraction at level of a few per cent) at $\\\\lambda =1.3$ mm and subsequent measurement of Faraday rotation imply a model dependent limit $2 \\\\times 10^{-7} > \\\\dot{M} > 2 \\\\times 10^{-9}{\\\\rm M_\\\\odot yr^{-1}}$ in the inner accretion flow ([4], [23]).\",\n \"Sgr A* is resolved by VLBI at $1.3$ mm where interstellar scattering is comparable to the intrinsic source FWHM $37^{+16}_{-10} \\\\mu {\\\\rm as}$ [7].\",\n \"This is smaller than the apparent diameter of the event horizon $\\\\sim 55 \\\\mu as$ .\",\n \"Sgr A* fluctuates rapidly in the near-infrared (NIR), with average $F_\\\\nu \\\\sim 1 {\\\\rm mJy}$ ([6]).\",\n \"It is not yet detected at mid-infrared (MIR) wavelengths ($\\\\lambda \\\\le 8.6 \\\\mu {\\\\rm m}$ , e.g.\",\n \"[33]).\",\n \"In the X-ray Sgr A* exhibits flares with characteristic duration of order an hour and a duty cycle of order $5\\\\%$ .\",\n \"The upper limit for quiescent state X-ray emission is $\\\\nu L_\\\\nu < 2.4 \\\\times 10^{33} {\\\\rm erg s^{-1}}$ [3].\",\n \"We are not aware of any secular trends in these observed properties of Sgr A*.\",\n \"One model of Sgr A* that fits most observational constraints is our relativistic accretion model [27], where submillimeter, IR, and X-ray emission arise in an optically thin, geometrically thick disk close to the event horizon; radio emission is assumed to arise nonthermally in a synchrotron photosphere at larger radius [30], but is not predicted by our model, which focuses on the inner parts of the accretion flow.\",\n \"The underlying flow model is a general relativistic magnetohydrodynamic (GRMHD) simulation ([14], [28], [36]).\",\n \"The emerging radiation is calculated using Monte Carlo [8] and ray-tracing schemes.\",\n \"The $1.3$ mm flux originates as Doppler boosted synchrotron emission from the approaching side of the disk between $\\\\sim 10 GM/c^2$ and the innermost stable circular orbit (ISCO).\",\n \"Our relativistic disk model assumes a thermal electron distribution function, neglects thermal conduction, and assumes a constant ratio of ion to electron temperature $T_i/T_e$ .\",\n \"The model does not produce the observed IR flaring at the correct amplitude, but can with the addition of a small nonthermal component in the distribution function [19].\",\n \"If we fix the black hole mass ($M_{BH}=4.5 \\\\times 10^6 M_{\\\\odot }$ , [16]) and distance ($D=8.4\\\\, {\\\\rm kpc}$ , [17]), the remaining model parameters are the source inclination $i$ , black hole spin $a_*$ ($0 \\\\le a_* \\\\le 1$ ), $T_i/T_e$ , and $\\\\dot{M}$ .\",\n \"We fix $\\\\dot{M}$ so that the $1.3$ mm flux matches the observed $\\\\simeq 3$ Jy.\",\n \"The relativistic accretion models are not tightly constrained by the data, but they reveal the following: (1) face-on models that reproduce the millimeter flux would look like rings and therefore, in VLBI data, have dips in visibilities on fixed intermediate baselines, while existing observations suggest that the ring radius would need to vary to fit the data ([13]; see also the discussion of [5]).\",\n \"More nearly edge-on models are therefore favored; (2) models with $a_* \\\\gtrsim 0.98$ that reproduce the millimeter flux have a hot, dense inner disk that would overproduce X-rays via inverse Compton scattering.\",\n \"Lower spin, $a_* \\\\sim 0.9$ , models are therefore favored; (3) the observed source size and flux fix the temperature of the emitting electrons $T_e = F_\\\\nu c^2/(4\\\\pi k \\\\nu ^2\\\\sigma ^2)$ ($\\\\sigma \\\\equiv $ the RMS size of the source on the sky) and this turns out to favor $T_i/T_e \\\\simeq 3$ models.\",\n \"The “best-bet” model from [27] has $a_* \\\\simeq 0.94$ , $i = 85$ deg, $\\\\dot{M} \\\\equiv \\\\dot{M}_0 \\\\simeq 2 \\\\times 10^{-9} {\\\\rm M_{\\\\odot }yr^{-1}}$ , and $T_i/T_e = 3$ .\",\n \"The relativistic disk model uses self-consistent dynamics and radiative transfer but is not unique.\",\n \"The electron distribution function is particularly poorly constrained.\",\n \"It is likely anisotropic, may contain multiple temperature components ([32]) and power-law components, and vary in basic functional form with time and position.\",\n \"Alternative accretion models (e.g.\",\n \"[5], [35]) make different assumptions about the flow and/or distribution function and favor slightly different $\\\\dot{M}, a_*,$ and $i$ .\",\n \"These models may respond differently to an increase in $\\\\dot{M}$ .\",\n \"Indeed, radically different models may also fit the data.\",\n \"The model dynamics depends on the initial magnetic field distribution, particularly the distribution of vertical flux through the accretion disk; models with large vertical magnetic flux (e.g.\",\n \"[24]) are likely to respond quite differently to variations in the mass flux.\",\n \"Also, jet models for Sgr A* ([11], [21], [12]) posit a luminous jet and a comparatively dim accretion disk.\",\n \"Again, these may respond differently to an influx of mass.\",\n \"How, then, does our relativistic disk model respond to an increase in $\\\\dot{M}$ ?\",\n \"In this Letter we calculate the $1.3$ and $0.87$ mm flux and source size as well as the spectrum that would result for the best-bet model over a range of $\\\\dot{M}$ .\",\n \"One key question we seek to answer is whether a small increase in $\\\\dot{M}$ would hide the event horizon (and the signature ring-like appearance of the photon orbit, also known as the shadow or silhouette of the event horizon) underneath a synchrotron photosphere.\",\n \"This might prevent detection of the photon orbit by the planned Event Horizon Telescope [7].\",\n \"Another key question is whether the increased $\\\\dot{M}$ would make Sgr A* detectable in its quiescent state in the IR and X-ray.\",\n \"Below we describe variation of the flux and source morphology at 230GHz ($1.3$ mm) and 345GHz ($0.87$ mm) (§ ), describe variation of the spectrum (§ ), and finally discuss which features of the results are likely to be most robust (§ ).\"\n ],\n [\n \"Change of Sgr A* sub-mm luminosity and size for enhanced $\\\\dot{M}$\",\n \"How do we naively expect the mm disk image size and flux to respond to changes in $\\\\dot{M}$ ?\",\n \"In our model, $1.3$ mm emission in Sgr A* is thermal synchrotron emission from plasma with optical depth $\\\\tau _\\\\nu \\\\sim 1$ , near the ISCO.\",\n \"The true electron distribution undoubtedly contains nonthermal components (see, Riquelme et al.\",\n \"2012).\",\n \"Models with thermal + power-law distribution functions (e.g.\",\n \"[5]) contain an O(1/3) nonthermal contribution to the flux at $1.3$ mm, which hints at how uncertainty in the distribution function translates into uncertainties in the spectrum.\",\n \"The thermal synchrotron absorptivity is $\\\\alpha _{\\\\nu ,a}=j_\\\\nu /B_\\\\nu $ , where $j_{\\\\nu } = \\\\frac{\\\\sqrt{2}\\\\pi e^2 n_e \\\\nu _s}{3cK_2(\\\\Theta _e^{-1})}(X^{1/2} + 2^{11/12} X^{1/6} )^2 \\\\exp (-X^{1/3})$ , $X = \\\\nu /\\\\nu _s$ , $\\\\nu _s=2/9 (eB/2\\\\pi m_ec) \\\\Theta _e^2 \\\\sin \\\\theta $ , $\\\\theta $ is an angle between the magnetic field vector and emitted photon, $K_2$ is a modified Bessel function of the second kind [20] and $B_\\\\nu \\\\simeq 2 \\\\nu ^2 \\\\Theta _e m_e$ .\",\n \"Near $1.3$ mm, $X \\\\sim 1$ , and the emissivity is nearly independent of frequency, so $j_\\\\nu \\\\propto \\\\nu ^0n_e B$ and $\\\\alpha _{\\\\nu ,a} \\\\propto \\\\nu ^{-2} n_e B \\\\Theta _e^{-1}$ .\",\n \"We will assume that $n_e \\\\propto \\\\dot{M} r^{-3/2}$ , $\\\\Theta _e \\\\propto 1/r$ , and $\\\\beta \\\\sim $ const., so that $B \\\\propto \\\\dot{M}^{1/2}r^{-5/4}$ for $r > G M/c^2$ , and ignore relativistic corrections.\",\n \"Then for $\\\\tau _\\\\nu \\\\ll 1$ (or $\\\\dot{M} \\\\ll \\\\dot{M}_0$ ) the source has size $\\\\sim G M/c^2$ and the flux $F_\\\\nu \\\\sim (4/3)\\\\pi (G M/c^2)^3 j_\\\\nu \\\\propto \\\\dot{M}^{3/2}$ .\",\n \"For $\\\\tau _\\\\nu \\\\gg 1$ the source size is set by the photosphere radius $r_{ph}$ where $\\\\int _{r_{ph}}^\\\\infty \\\\alpha _a(r)dr = 1$ (i.e.\",\n \"for $\\\\dot{M} \\\\gg \\\\dot{M}_0$ , but not so large that $\\\\Theta _e(r_{ph}) < 0.5$ so that our emissivity approximation fails).\",\n \"Then $r_{ph} \\\\propto \\\\dot{M}^{3/2}/\\\\nu ^2$ , and the flux $F_\\\\nu \\\\propto r_{ph}^2 B_\\\\nu (r_{ph}) \\\\sim \\\\dot{M}^{9/4}/\\\\nu $ .\",\n \"These simple scaling laws, unfortunately, are not a good description of the variation of source size and flux with $\\\\dot{M}$ .\",\n \"There are at least three reasons for this.\",\n \"First, relativistic effects are very important; for $\\\\dot{M} \\\\sim \\\\dot{M}_0$ emission comes from close to the photon orbit and the source size is determined by Doppler beaming and gravitational lensing.\",\n \"Second, for the best-bet model at $1.3$ mm $\\\\tau _\\\\nu \\\\sim 1$ , so in a turbulent flow there is a complicated variation of the size of the effective photosphere with $\\\\dot{M}$ .\",\n \"Third, the emissivity is not precisely frequency independent near peak.\",\n \"We therefore need to turn to numerical models.\",\n \"The best-best model is taken from a survey of 2D models.\",\n \"Here we adopt the best-bet model parameters ($a_*=0.94$ , $i=85 \\\\deg $ and $T_i/T_e=3$ ) and use them to set parameters for a fully 3D model [36] A 3D GRMHD model parameter survey is still too computationally expensive.\",\n \"It also has a poor return on investment given the electron distribution function uncertainties.. We use the same data set as [9], and choose three representative snapshots taken at times when the flow is quiescent ($t=5000, 9000$ and $13000 GM/c^3$ , where $GM/c^3 = 20$ s).\",\n \"We then recalculate disk images and spectra for $\\\\dot{M} = (0.5, 1, 2, 4, 8, 16,32,64) \\\\dot{M}_0$ .\",\n \"The 3D model with $\\\\dot{M} = \\\\dot{M}_0$ is broadly consistent with observational data but is slightly more luminous at higher energies than 2D models ($\\\\beta $ is lower in the 3D models, and this changes the X-ray to millimeter color).\",\n \"The model is self-consistent only for $\\\\dot{M} <64 \\\\dot{M}_0$ .\",\n \"At higher $\\\\dot{M}$ the efficiency of the flow is $>0.1$ and therefore our neglect of cooling in the underlying GRMHD model is not justified.\",\n \"At higher $\\\\dot{M}$ the $1.3$ mm photosphere also lies outside the limited range in radius where $\\\\langle d\\\\dot{M}/dr \\\\rangle =0$ , so the flow is not in a steady state.\",\n \"The images and total fluxes emitted by the disk at 230 and 345GHz are calculated using a ray tracing scheme [29].\",\n \"To estimate the size of the emitting region we calculate the eigenvalues of the matrix formed by taking the second angular moments of the image on the sky (the principal axis lengths).\",\n \"The major and minor axis eigenvalues, $\\\\sigma _1$ and $\\\\sigma _2$ respectively, are related by $\\\\sigma ={\\\\rm FWHM}/2.3$ to the FWHM of the axisymmetric Gaussian model used to interpret the VLBI observations.\",\n \"We use $\\\\langle \\\\sigma \\\\rangle = (\\\\sigma _1+\\\\sigma _2)/2$ to measure the average radius of the emitting spot.\",\n \"Figures REF and REF show the variation of 230 and 345 GHz Sgr A* model images with $\\\\dot{M}$ (based on a single snapshot from the 3D GRMHD simulation).\",\n \"Evidently modest increases $\\\\dot{M}$ will make the ring-like feature that is the observational signature of the photon orbit easier to detect.\",\n \"For $\\\\dot{M} \\\\gtrsim 8 \\\\dot{M}_0$ , however, the ring (or black hole silhouette) is hidden beneath the synchrotron photosphere at 230 GHz.\",\n \"The silhouette survives for higher $\\\\dot{M}$ at 345 GHz, disappearing only at $\\\\dot{M} \\\\gtrsim 16\\\\dot{M}_0$ .\",\n \"For low $\\\\dot{M}$ the silhouette is also difficult to detect because the emitting region is too small.\",\n \"Figure: Horizon silhouette detectability at 230 GHz for variousM ˙\\\\dot{M}.\",\n \"Panels from a) to h) show the imagesof Sgr A* calculated for M ˙=(0.5,1,2,4,8,16,32,64)M ˙ 0 \\\\dot{M} =(0.5, 1, 2, 4, 8, 16, 32, 64) \\\\dot{M}_0 ,respectively.\",\n \"The center of the circle is positioned at the image centroidand its radius, r=(σ 1 +σ 2 )/2r=(\\\\sigma _1+\\\\sigma _2)/2 is the RMS radius ofthe emitting region.Figure: Same as in Figure but for ν=345\\\\nu =345 GHz.Figure: Sizes (in terms of σ\\\\sigma ) of the emitting region and the total flux as afunction of M ˙\\\\dot{M} at 230 (upper panels) and 345GHz (lowerpanels).\",\n \"Different point types correspond to three different snapshots fromthe 3D GRMHD simulation.\",\n \"The dashed/dotted lines show the best fit to the databy Equations , , , and.Figure REF , shows the accretion flow image size ($\\\\langle \\\\sigma \\\\rangle $ , circle radii in Figure REF and REF ) and flux at 230 and 345GHz as a function of $\\\\dot{M}$ .\",\n \"Different point types correspond to three different snapshots from the 3D GRMHD simulation.\",\n \"The size of the emission region and flux increase with $\\\\dot{M}$ .\",\n \"The 345/230 GHz flux ratio increases with the increasing $\\\\dot{M}$ ; this is caused by the shift of the synchrotron peak towards higher energies at higher $\\\\dot{M}$ .\",\n \"The following simple fitting formulas describe how $\\\\sigma $ and $F_\\\\nu $ depend on $\\\\dot{M}$ : $\\\\langle \\\\sigma \\\\rangle _{230{\\\\rm GHz}}={\\\\left\\\\lbrace \\\\begin{array}{ll}15.2 \\\\times (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})^{0.38},& \\\\mbox{for}\\\\frac{\\\\dot{M}}{\\\\dot{M}_0} < 2\\\\\\\\21.1 \\\\times \\\\log _{10} (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})+ 13,& \\\\mbox{for} \\\\frac{\\\\dot{M}}{\\\\dot{M}_0} \\\\ge 2\\\\end{array}\\\\right.\",\n \"}\\\\, \\\\, {\\\\rm [\\\\mu as]}$ $\\\\langle \\\\sigma \\\\rangle _{345{\\\\rm GHz}}={\\\\left\\\\lbrace \\\\begin{array}{ll}12 \\\\times (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})^{0.31},& \\\\mbox{for}\\\\frac{\\\\dot{M}}{\\\\dot{M}_0} < 2\\\\\\\\19.7 \\\\times \\\\log _{10} (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})+ 8.3,& \\\\mbox{for} \\\\frac{\\\\dot{M}}{\\\\dot{M}_0} \\\\ge 2\\\\end{array}\\\\right.\",\n \"}\\\\, \\\\, {\\\\rm [\\\\mu as]}$ $F_{230{\\\\rm GHz}}={\\\\left\\\\lbrace \\\\begin{array}{ll}3.17 \\\\times (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})^{1.03},& \\\\mbox{for}\\\\frac{\\\\dot{M}}{\\\\dot{M}_0} < 2\\\\\\\\10.62 \\\\times \\\\log _{10} (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})+ 3.8,& \\\\mbox{for} \\\\frac{\\\\dot{M}}{\\\\dot{M}_0} \\\\ge 2\\\\end{array}\\\\right.\",\n \"}\\\\, \\\\, {\\\\rm [Jy]} $ $F_{345{\\\\rm GHz}}={\\\\left\\\\lbrace \\\\begin{array}{ll}3.2 \\\\times (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})^{1.3},& \\\\mbox{for}\\\\frac{\\\\dot{M}}{\\\\dot{M}_0} < 2\\\\\\\\25.13 \\\\times \\\\log _{10} (\\\\frac{\\\\dot{M}}{\\\\dot{M}_0})+ 0.54,& \\\\mbox{for} \\\\frac{\\\\dot{M}}{\\\\dot{M}_0} \\\\ge 2\\\\end{array}\\\\right.\",\n \"}\\\\, \\\\, {\\\\rm [Jy]} $ The above constants are nontrivial to interpret because they encapsulate the complexities of the accretion flow structure and relativistic effects in the radiation transport.\",\n \"The advantage of the above formulas is their simplicity.\",\n \"The fitting functions are shown in Figure REF as dashed and dotted lines.\",\n \"Figure: Relation between two observables: the flux and the size of theimage at ν=\\\\nu =230 and 345 GHz.\",\n \"The dashed and dotted lines are thebest fits to the data points.Figure REF shows the relation between two observables, $\\\\sigma $ and $F_\\\\nu $ .\",\n \"The size is a linear function of the flux and increases more steeply at 230 GHz than at 345 GHz.\",\n \"We also provide two phenomenological scaling laws fitted to the data: $\\\\langle \\\\sigma \\\\rangle _{230GHz} = 1.72 (F_{230{\\\\rm GHz}}/{\\\\rm Jy}) + 9.3 \\\\,\\\\,\\\\,{\\\\rm [\\\\mu as]} $ $\\\\langle \\\\sigma \\\\rangle _{345GHz} = 0.73 (F_{345{\\\\rm GHz}}/{\\\\rm Jy}) + 9.2 \\\\,\\\\,\\\\,{\\\\rm [\\\\mu as]} $ Notice that these fits apply to the best-bet model only.\",\n \"For other $a_*$ , $T_i/T_e$ , or $i$ the scalings will be slightly different, although if the source is optically thick the linear scaling follows directly from $T_e \\\\propto 1/r$ .\",\n \"Notice that Equation REF predicts the change of Sgr A* size by 9 per cent when 230 GHz flux changes from 2 to 2.7 Jy.\",\n \"Taking into account observational and theoretical uncertainties this is consistent with the observed variations of the size of the source which increases by a few per cent as the flux increases from 2 to 2.7 Jy [13].\"\n ],\n [\n \"Spectra\",\n \"Our model spectra are generated by thermal synchrotron emission in the submillimeter/far-IR bump, and by Compton scattering in the X-rays.\",\n \"The spectral slope of flaring NIR emission, and its high degree of linear polarization [6], imply that it is synchrotron from a small, nonthermal tail of high energy electrons that is not (but can be; see [19]) included in our best-bet model.\",\n \"What are the expected scaling laws?\",\n \"Again, $j_\\\\nu \\\\sim n_e B \\\\sim \\\\dot{M}^{3/2}$ .\",\n \"The luminosity around the synchrotron peak is $L_{peak}\\\\sim 4 \\\\pi \\\\nu _{peak} j_{\\\\nu _{peak}} (GM/c^2)^3 \\\\sim \\\\dot{M}^{9/4}$ , where $\\\\nu _{peak} \\\\sim \\\\dot{M}^{3/4}$ is such that $\\\\alpha _\\\\nu GM/c^2 =1$ .\",\n \"The emission rightward of the MIR/NIR is produced by Compton up-scattered synchrotron radiation.\",\n \"The Thomson depth $\\\\tau _{sc} =\\\\sigma _{TH} n_e GM/c^2 \\\\sim \\\\dot{M}$ , the X-ray luminosity is expected to scale as: $\\\\nu L_\\\\nu (\\\\nu \\\\approx 10^{18} {\\\\rm Hz} ) \\\\sim L_{peak}\\\\tau _{sc} \\\\sim \\\\dot{M}^{13/4}$ , assuming that X-rays are produced primarily by singly scattered synchrotron photons.\",\n \"What do the numerical models show?\",\n \"Figure REF shows spectra emitted from the 3D disk model as observed at $i\\\\approx 85 \\\\deg $ .\",\n \"The SEDs are calculated using a general relativistic Monte Carlo scheme [8].\",\n \"The NIR luminosity ${\\\\nu }L_{\\\\nu }(\\\\nu =10^{14} {\\\\rm Hz}) \\\\sim \\\\dot{M}^{2.5}$ , which is only slightly steeper than the expected dependence for the synchrotron peak.\",\n \"${\\\\nu }L_{\\\\nu }(\\\\nu \\\\approx 10^{18} {\\\\rm Hz}) \\\\sim \\\\dot{M}^{3.25}$ agrees well with the expected scaling.\",\n \"We conclude that Sgr A* would become a persistent MIR and X-ray source (above the present upper limits of $84 {\\\\rm mJy}$ in MIR and $2.4 \\\\times 10^{33} {\\\\rm erg s^{-1}}$ in X-rays) if $\\\\dot{M} > 2 \\\\dot{M}_0$ .\",\n \"This is conservative in the sense that our models are strictly thermal.\",\n \"The addition of a high energy nonthermal tail would only increase the MIR/NIR and X-ray flux.\",\n \"We do not consider higher accretion rate models because for $\\\\dot{M} =64 \\\\dot{M}_0$ (${\\\\nu }L_{\\\\nu }(\\\\nu =10^{18} {\\\\rm Hz})=10^{39} {\\\\rm ergs^{-1}}$ ) the model becomes radiatively efficient, $\\\\epsilon =L_{Bol}/\\\\dot{M} c^2 > 0.1$ , and our neglect of radiative cooling in the underlying 3D GRMHD model is unjustified.\",\n \"Finally, notice that the MeV flux increases sharply with $\\\\dot{M}$ .\",\n \"This suggests that electron-positron pair production by photon-photon collisions in the funnel over the poles of the hole would increase sharply (the pair production $\\\\dot{n}_{e^\\\\pm } \\\\sim L_{\\\\gamma }^2$ , [26]).\",\n \"If this pair production is connected to jet production, it is reasonable to think that a high $\\\\dot{M}$ Sgr A* might also produce a jet.\",\n \"Figure: Spectrum emitted by 3D disk model for various M ˙\\\\dot{M}.\",\n \"TheM ˙/M ˙ 0 \\\\dot{M}/\\\\dot{M}_0 is shown on the righthand side.\",\n \"Observational pointsand upper limits are taken from: , ,, , ,.\",\n \"The black symbols in the NIR showing the flaringstate are from .\",\n \"An example of X-ray flare is taken from .\"\n ],\n [\n \"Discussion\",\n \"In summary, we have used general relativistic disk models and relativistic radiative transfer to recalculate millimeter images and spectra for Sgr A* at a range of $\\\\dot{M} > \\\\dot{M}_0$ .\",\n \"Our models predict the following: (1) if the 230 GHz flux increases by more than a factor of 2, corresponding to an increase of $\\\\dot{M}$ by more than a factor of 2, the central accretion flow will become a persistent, detectable, MIR and X-ray source; (2) the photon orbit, which produces the narrow ring of emission visible in Figures 1 and 2, becomes easier to detect for modest increases in the 230 GHz flux; (3) the photon orbit is cloaked beneath the synchrotron photosphere at 230 GHz for $\\\\dot{M}\\\\gtrsim 8 \\\\dot{M}_0$ , or 230 GHz flux $\\\\gtrsim 13$ Jy; (4) the photon orbit is cloaked at 345 GHz only at higher $\\\\dot{M} \\\\gtrsim 16\\\\dot{M}_0$ , or 230 GHz flux $\\\\gtrsim 17$ Jy; (5) the size of the source increases in proportion to the flux at both 230 and 345 GHz.\",\n \"We suspect that almost any accretion model for Sgr A* with a spatially uniform model for the plasma distribution function will reach qualitatively similar conclusions, but that jet models may differ significantly.\",\n \"There are order-unity uncertainties in our model predictions due to uncertainties in the plasma model.\",\n \"What range of changes in $\\\\dot{M}$ are reasonable?\",\n \"In our best-bet model the mass at radii within a factor of two of the pericenter radius is $\\\\approx 10^{-2.5}$ $M_{\\\\oplus }$ , assuming steady mass inflow from $r_p$ to the event horizon.\",\n \"The addition of even a fraction of the inferred cloud mass to the accretion flow in a ring near $r_p$ would (eventually) increase $\\\\dot{M}$ by a factor of $\\\\sim 300$ .\",\n \"On the other hand, stellar winds supply mass in the neighborhood of the central black hole at $\\\\sim 10^{-3} M_{\\\\odot } yr^{-1}$ .\",\n \"Models by [31] and [34] suggest that most of this mass is ejected in the form of a wind, and that $\\\\sim 10^{-4.5}M_{\\\\odot } yr^{-1}$ to $10^{-7.3} M_{\\\\odot } yr^{-1}$ flows inward.\",\n \"A reasonable extrapolation of these models suggest the accretion flow at $r < r_p$ has a mass of $\\\\sim 2 M_{\\\\oplus }$ ; this is comparable to estimates of the mass of the inflowing cloud, so in this case we might expect a factor of 2 increase in $\\\\dot{M}$ .\",\n \"This work was supported by the National Science Foundation under grant AST 07-09246 and by NASA under grant NNX10AD03G, through TeraGrid resources provided by NCSA and TACC.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1371"}}},{"rowIdx":1867487,"cells":{"text":{"kind":"list like","value":[["On the longest length of arithmetic progressions"],["Abstract Suppose that $\\xi^{(n)}_1,\\xi^{(n)}_2,...,\\xi^{(n)}_n$ are i.i.d with $P(\\xi^{(n)}_i=1)=p_n=1-P(\\xi^{(n)}_i=0)$.","Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to $\\xi^{(n)}_1,\\xi^{(n)}_2,..., \\xi^{(n)}_n$ respectively.","Firstly, the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are given.","Simultaneously, the errors are estimated by using Chen-Stein method.","Next, the almost surely limits are discussed when all $p_n$ are equal and when considered on a common probability space.","Finally, we consider the case that $\\lim_{n\\to\\infty}p_n=0$ and $\\lim_{n\\to\\infty}{np_n}=\\infty$.","We prove that as $n$ tends to $\\infty$, the probability that $U^{(n)}$ takes two numbers and $W^{(n)}$ takes three numbers tends to 1."],["Introduction and main results","Limit distributions for the longest length of runs with respect to Bernoulli sequence have been investigated for a long time, see, e.g., [5], [6],[7], [8] and [9].","But what about the longest length of arithmetic progressions?","Problems connected to arithmetic progressions are very important in number theory, see [10].","For example, Roth's theory says that every set of integers of positive density contained infinitely many progressions of length three.","Suppose that $\\xi _1,\\xi _2,\\cdots $ is a Bernoulli sequence with $P(\\xi _i=1)=p=1-q$ , where $00,\\sum _n b_n=\\infty , \\lim _{n\\rightarrow \\infty }\\frac{\\sum _{k=1}^n b_k}{n\\max _{1\\le k\\le n} b_k}=1 $ and $a\\le \\liminf _{n\\rightarrow \\infty } \\frac{a_n}{b_n}\\le \\limsup _{n\\rightarrow \\infty } \\frac{a_n}{b_n}<\\infty .$ Then $\\lim _{n\\rightarrow \\infty } \\frac{\\sum _{k=1}^n a_k}{\\sum _{k=1}^n b_k}=a$ if and only if for all $c>a$ , $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n I_{\\lbrace {a_k}/{b_k}a$ , let $A_{n}=\\lbrace 1\\le k\\le n: {a_k}/{b_k}\\ge c\\rbrace $ .","Then (REF ) implies that $\\lim _{n\\rightarrow \\infty } {|A_{n}|}/{n}=0$ .","Hence $\\frac{\\sum _{k=1}^n a_k}{\\sum _{k=1}^n b_k}\\le c+ \\frac{(d-c)\\sum _{k\\in A_{n}}b_k}{\\sum _{k=1}^n b_k}\\le c +\\frac{(d-c)|A_{n}|\\max _{1\\le k\\le n} b_k}{\\sum _{k=1}^n b_k}\\rightarrow c.$ The arbitrary of $c>a$ yields that $\\limsup _{n\\rightarrow \\infty } {\\sum _{k=1}^n a_k}/{\\sum _{k=1}^n b_k}\\le a$ .","This, together with (REF ) and (REF ), gives (REF ).","Next, we shall show the necessity.","By (REF ), for any $\\varepsilon >0$ , there is $K$ such that $ {a_k}/{b_k}>a-\\varepsilon $ for all $k\\ge K$ .","For any $c>a$ , let $B_n=\\lbrace KK$ , $\\frac{\\sum _{k=1}^n a_k}{\\sum _{k=1}^n b_k}&\\ge & c+\\frac{\\sum _{k=1}^{K} (a_k-cb_k)+(a-\\varepsilon -c)\\sum _{k\\in B_n} b_k}{\\sum _{k=1}^n b_k}\\\\&\\ge & c+\\frac{\\sum _{k=1}^{K} (a_k-cb_k)+(a-\\varepsilon -c)|B_{n}|\\max _{1\\le k\\le n} b_k}{\\sum _{k=1}^n b_k}.$ Letting $n$ tends to $\\infty $ , we get that $\\liminf _{n\\rightarrow \\infty }\\frac{|B_{n}|\\max _{1\\le k\\le n} b_k}{\\sum _{k=1}^n b_k}\\ge \\frac{c-a}{c+\\varepsilon -a}.$ The arbitrary of $\\varepsilon >0$ , together with (REF ), implies (REF ) and completes our proof.","$\\Box $ As an application of Lemma REF .","Suppose that (REF ) holds.","In addition, suppose that $b_nc_n>0$ , $\\sum _{n} b_nc_n=\\infty $ and $ \\lim _{n\\rightarrow \\infty }\\frac{\\sum _{k=1}^n b_kc_k}{n\\max _{1\\le k\\le n} b_kc_k}=1$ .","Then $\\lim _{n\\rightarrow \\infty } {\\sum _{k=1}^n a_kc_k}/{\\sum _{k=1}^n b_kc_k}=a$ if and only if (REF ) holds for all $c>a$ .","Particularly, by letting $c_n=1/b_n$ , we see that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n \\frac{a_k}{b_k}=a$ if and only if (REF ) holds for all $c>a$ .","Therefore if (REF ) and (REF ) hold, then $\\lim _{n\\rightarrow \\infty } {\\sum _{k=1}^n a_k}/{\\sum _{k=1}^n b_k}=a$ if and only if $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n \\frac{a_k}{b_k}=a$ , if and only if (REF ) holds for all $c>a$ .","Note that $n\\ln n-n\\le \\sum _{k=1}^n \\ln k \\le n\\ln n$ .","By Lemma REF , we have the following propositions.","Proposition 2.1 If (REF ) holds, then () holds if and only if $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} {\\sum _{k=2}^n \\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}}=D \\,\\,a.s, $ and also if and only if for any $1>\\varepsilon >0$ , $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\sum _{k=1}^n I_{\\lbrace U^{(2^k)}0$ , $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\sum _{k=1}^n I_{\\lbrace W^{(2^k)}s\\rbrace =\\lbrace (a,s):1\\le s\\le [ {n-1\\over r}], ss\\rbrace |\\le \\frac{n^2}{2r}.","$ Clearly, $I=p^r|B_n|-p^{r+1}|B_n\\cap \\lbrace (a,s):a>s\\rbrace |$ .","This, together with (REF ) and (REF ), gives (REF ).","$\\Box $ Lemma 3.2 It holds that $e^{(n,r)}\\le 9(n^3 p^{2r-1}+n^2 r^3 p^{\\frac{5}{3} r-1}+n^2 p^{\\frac{3}{2} r-1})$ .","Let $c_1&=&|\\lbrace (a,s,b,t)\\in B_n\\times B_n: (a,s)\\sim (b,t)\\rbrace |,\\\\c_2&=&|\\lbrace (a,s,b,t)\\in B_n\\times B_n: |B_{a,s}\\cap B_{b,t}|\\ge 2\\rbrace |$ and $c_3=|\\lbrace (a,s,b,t)\\in B_n\\times B_n: (a,s)\\sim (b,t),t=2s\\text{ or } s=2t\\rbrace |.$ Then $c_1\\le &&|\\lbrace (a,s,b,t): (a,s)\\in B_n, 1\\le t\\le [\\frac{n-1}{r-1}], b=a+is-jt,\\\\&&-1\\le i,j\\le r-1\\rbrace |\\le 9n^3/2$ and $c_3&\\le & 2|\\lbrace (a,s,b,t): (a,s)\\in B_n, t=2s , b=a+is-jt,-1\\le i,j\\le r-1\\rbrace |\\\\&=&2|\\lbrace (a,s,b):(a,s)\\in B_n, b=a+ks,-2r+1\\le k\\le r+1\\rbrace |\\le 7n^2.$ Suppose that $|B_{a,s}\\cap B_{b,t}|\\ge 2$ and $x_0$ is the minimal number of the set $B_{a,s}\\cap B_{b,t}$ .","Then $x_0=a+is=b+jt$ for some $-1\\le i,j\\le r-1$ .","If $x\\in B_{a,s}\\cap B_{b,t}$ and $x>x_0$ , then $x=a+i^{\\prime }s=b+j^{\\prime }t$ for some $-1\\le i^{\\prime },j^{\\prime }\\le r-1$ .","It follows that $x-x_0=(i^{\\prime }-i)s=(j^{\\prime }-j)t$ .","Thus $t=\\frac{k_1}{k_2}s$ for some $1\\le k_1,k_2 \\le r$ .","In addition, there is positive integer $k$ such that $i^{\\prime }-i=kt_0, j^{\\prime }-j=ks_0$ and $x-x_0=kst_0$ , where $s_0={s}/{\\gcd (s,t)}$ and $t_0={t}/{\\gcd (s,t)}$ .","Since $i^{\\prime }-i\\le r$ , $k\\le {r}/{t_0}$ .","Similarly, $k\\le {r}/{s_0}$ .","Therefore $|B_{a,s}\\cap B_{b,t}|\\le {r}/\\max (s_0, t_0)+1.$ Consequently, $|B_{a,s}\\cap B_{b,t}|\\le {r}/{3}+1$ whenever $\\max (s_0, t_0)\\ge 3$ .","When $\\max (s_0, t_0)=2$ , $|B_{a,s}\\cap B_{b,t}|\\le {r}/{2}+1$ .","Actually, in this case, $s=2t$ or $t=2s$ .","When $\\max (s_0, t_0)=1$ , $s=t$ .","We shall show that $A_{a,s}\\cap A_{b,s}=\\emptyset $ whenever $a\\ne b$ and $B_{a,s}\\cap B_{b,s}\\ne \\emptyset $ .","Assume that $b>a$ without loss of generality.","Since $B_{a,s}\\cap B_{b,s}\\ne \\emptyset $ , $a+is=b+js$ for some $-1\\le i,j\\le r-1$ and hence $b=a+ks$ for some $1\\le k \\le r$ .","Thus $A_{a,s}\\subseteq \\lbrace \\xi _{a+(k-1)s}=1\\rbrace $ and $A_{b,s}\\subseteq \\lbrace \\xi _{b-s}=0\\rbrace =\\lbrace \\xi _{a+(k-1)s}=0\\rbrace $ .","It implies that $A_{a,s}\\cap A_{b,s}=\\emptyset $ as desired.","In view of the discussion above, we have $c_2\\le && |\\lbrace (a,s,b,t): (a,s)\\in B_n, t= {sk_1}/{k_2}, b=a+is-jt,\\\\&&-1\\le i,j\\le r-1,1\\le k_1,k_2\\le r\\rbrace |\\le 9n^2r^3/4$ and $e^{(n,r)}\\le 2c_1p^{2r-1}+c_2p^{\\frac{5r}{3}-1}+c_3p^{\\frac{3r}{2}-1}\\le &9(n^3 p^{2r-1}+n^2 r^3 p^{\\frac{5}{3} r-1}+n^2 p^{\\frac{3}{2} r-1})$ as desired.","$\\Box $ Similar as the proof of Lemma REF , we may show that for any $m$ , $n$ and $2\\le r_m\\le r_n$ , with $H=\\lbrace (a,s,b,t):(a,s)\\in B^{(r_m)}_m, (b,t)\\in B^{(r_n)}_n\\rbrace $ , $|\\lbrace (a,s,b,t)\\in H:|B^{(r_m)}_{a,s}\\cap B^{(r_n)}_{b,t}|\\ge 1\\rbrace |\\le 9m^2n/2 $ and $|\\lbrace (a,s,b,t)\\in H:|B^{(r_m)}_{a,s}\\cap B^{(r_n)}_{b,t}|\\ge 2\\rbrace |\\le 9m^2r_mr^2_n/4.$ Also we can show that if $|B^{(r_m)}_{a,s}\\cap B^{(r_n)}_{b,t}|>\\frac{r_n}{2}+1$ and $A^{(r_m)}_{a,s}\\cap A^{(r_n)}_{b,t}\\ne \\emptyset $ , then $(b,t)=(a,s)$ .","Therefore $|\\lbrace (a,s,b,t)\\in H:|B^{(r_k)}_{a,s}\\cap B^{(r_m)}_{b,t}|>\\frac{r_n}{2}+1,A^{(r_m)}_{a,s}\\cap A^{(r_n)}_{b,t}\\ne \\emptyset \\rbrace |\\le k^2/r_k.$ Proof of Theorem 1.1 (1) Lemma REF and (REF ) imply that $|e^{-I_{n,r}}-e^{-\\lambda _{n,r}}|\\le |I_{n,r}- \\lambda _{n,r}|\\le 2np^r.$ Let $r_n=[\\frac{-2\\ln n}{\\ln p}+2 \\frac{\\ln \\ln n}{\\ln p}+\\frac{\\ln \\ln \\ln n}{2\\ln p}]$ and $R_n=[\\frac{-3\\ln n}{\\ln p}]$ .","By (REF ), (REF ) and Lemma REF , $&&\\max _{r_n\\le r\\le R_n}|P(U^{(n)}R_n$ , $&&|P(U^{(n)} -D\\ln \\ln n+1}\\epsilon _{n,x}\\le p^{-D\\ln \\ln n}O(1)=O(\\frac{1}{\\ln n})$ and $\\max _{ aD\\ln \\ln n\\le x \\le -D\\ln \\ln n+1}\\epsilon _{n,x}\\le p^{ aD\\ln \\ln n}O(\\frac{\\ln \\ln n}{\\ln n})=O(\\frac{ \\ln \\ln n}{\\ln ^{1-a} n}).$ Now (REF ) follows by (REF )–(REF ).","(3) For any $\\varepsilon >0$ , (REF ) and (REF ) imply that $P(U^{(n)}\\ge C\\ln n +(1-\\varepsilon )D\\ln \\ln n)\\le O(\\ln ^{-\\varepsilon } n).$ On the other hand, by (REF ) and (REF ), $P(U^{(n)}< C\\ln n +(1+\\varepsilon )D\\ln \\ln n)\\le e^{-O(\\ln ^{\\varepsilon } n)}+O(\\frac{\\ln ^{4} n\\ln \\ln n}{n}).$ Hence (REF ) holds.","(4) By (REF ), $\\sum _{k=1}^\\infty P(U^{(2^k)}< C\\ln 2^k +(1+\\varepsilon )D\\ln \\ln 2^k)<\\infty $ for any $\\varepsilon >0$ .","One then deduces from the Borel-Cantelli Lemma that $P(U^{(2^k)}2^{K(\\omega )}$ , then $2^k\\le n<2^{k+1}$ for some $k\\ge K(\\omega )$ .","Hence $U^{(2^k)}(\\omega )\\le U^{(n)}(\\omega )\\le U^{(2^{k+1})}(\\omega )$ .","This, together with (REF ), gives that $U^{(n)}(\\omega )\\ge C\\ln n-C\\ln 2+(1+\\varepsilon )D\\ln \\ln n.$ Now the arbitrary of $\\varepsilon >0$ yields that $\\liminf _{n\\rightarrow \\infty } \\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}\\ge D$ .","Therefore $\\liminf _{n\\rightarrow \\infty } \\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}= D$ by considering (REF ).","Let $T_k$ be the longest length of arithmetic progressions relative to $\\xi _{2^{k-1}+1},\\\\ \\xi _{2^{k-1}+2},\\cdots ,\\xi _{2^{k}}$ .","Then $T_1,T_2,\\cdots $ are independent.","In addition, $T_k$ has the same distribution with $U^{(2^{k-1})}$ .","By (REF ), $P(T_k\\ge C\\ln 2^k)=P(U^{(2^{k-1})}\\ge C\\ln 2^k)\\le O(1/k).$ Hence $\\sum _{k=1}^\\infty P(T_k\\ge C\\ln 2^k)=\\infty $ .","By Borel-Cantelli Lemma, $P(T_k\\ge C\\ln 2^k\\,\\,\\, i.o.",")=1$ .","Consequently, $\\limsup _{k\\rightarrow \\infty } \\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}\\ge 0$ by noting that $U^{(2^k)}\\ge T_k$ .","On the other hand, (REF ) yields that $\\sum _{k=1}^\\infty P(U^{(2^k)}\\ge C\\ln 2^k +(1-\\varepsilon )D \\ln \\ln 2^k)<\\infty $ whenever $\\varepsilon >1$ .","Hence $\\limsup _{k\\rightarrow \\infty } \\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}\\le 0$ .","Therefore $\\limsup _{k\\rightarrow \\infty } \\\\\\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}= 0$ .","Furthermore, we can deduce that $\\limsup _{n\\rightarrow \\infty } \\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}= 0$ by the fact that $U^{(n)}$ is increasing.","This completes the proof of (REF ).","Now we come to prove ().","By Proposition REF , we need only to show (REF ).","Let $r_k= [C\\ln 2^k+D(1-\\varepsilon )\\ln \\ln 2^k]$ , $V_n=\\lbrace (a,s,k):1\\le k\\le n, (a,s)\\in B_{2^k}^{(r_k)}\\rbrace $ and $\\Lambda (n)=\\sum _{(a,s,k)\\in V_n}I_{ A_{a,s}^{(r_k)}}$ .","Then it suffices to show that $\\lim _{n\\rightarrow \\infty }\\Lambda (n)/n=0$ a.s. Clearly, $E\\Lambda (n)=\\sum _{k=1}^n I_{2^k,r_k}=O(n^{1-\\varepsilon })$ and $D\\Lambda (n)$ is less than the sum of $p^{r_k+r_m-|B^{(r_k)}_{a,s}\\cap B^{(r_m)}_{b,t}|}$ with $(a,s,k),(b,t,m)\\in V_n$ , $B^{(r_k)}_{a,s}\\cap B^{(r_m)}_{b,t}\\ne \\emptyset $ and $A^{(r_k)}_{a,s}\\cap A^{(r_m)}_{b,t}\\ne \\emptyset $ .","By (REF )–(REF ), $D\\Lambda (n)\\le &&\\sum _{1\\le i\\le j\\le n} O(2^{-j}ij+2^{-j}i^{2}j^3+2^{2i-2j}j^{1-\\varepsilon }i^{-1})\\\\\\le && O(\\sum _{j=1}^\\infty 2^{-j} j^{6})+O(\\sum _{j=1}^n j^{-\\varepsilon }\\sum _{k=0}^{j-1} 2^{-2k}j(j-k)^{-1})\\\\\\le &&O(1)+\\sum _{k=0}^{\\infty } 2^{-2k}(k+1)O(\\sum _{j=1}^n j^{-\\varepsilon })=O(n^{1-\\varepsilon }).$ Then by Tchebychev's inequality, for any $\\delta >0$ , $\\sum _{n=1}^\\infty P(|\\Lambda _n/n-E(\\Lambda _n/n)|>\\delta )\\le \\sum _{n=1}^\\infty O(\\frac{1}{n^{1+\\varepsilon }\\delta ^2})<\\infty .$ The Borel-Cantelli Lemma yields that $\\Lambda _n/n\\rightarrow 0\\,\\,a.s.$ .","Hence (REF ) holds as desired.","Finally, we shall prove (REF ).","Let $c_n=[\\ln n/\\ln 2]$ .","Then $2^{c_n}\\le n<2^{c_n+1}$ .","For any integers $1\\le a\\le b$ , $\\ln \\frac{b+1}{a}=\\int _{a}^{b+1} \\frac{1}{x}\\,dx\\le \\sum _{i=a}^b\\frac{1}{i}\\le \\int _{a-1}^b \\frac{1}{x}\\,dx=\\ln \\frac{b}{a-1}.$ Thus $\\sum _{k=1}^n \\frac{U^{(k)}}{k}\\ge \\sum _{i=0}^{c_n-1} U^{(2^i)}\\sum _{j=2^i}^{2^{i+1}-1}\\frac{1}{j} \\ge \\ln 2\\sum _{i=0}^{c_n-1} U^{(2^i)}$ and $\\sum _{k=2}^n \\frac{U^{(k)}}{k}\\le \\sum _{i=0}^{c_n} U^{(2^{i+1})}\\sum _{j=2^i+1}^{2^{i+1}}\\frac{1}{j} \\le \\ln 2\\sum _{i=1}^{c_n+1} U^{(2^i)}.$ Hence by (), $\\lim _{n\\rightarrow \\infty }\\frac{1}{c_n\\ln c_n}(\\frac{1}{\\ln 2}\\sum _{k=1}^n \\frac{U^{(k)}}{k}+\\frac{\\ln 2}{\\ln p} c^2_n)=D\\,\\,a.s.$ It follows (REF ) immediately and completes the proof of Theorem 1.1.","$\\Box $"],["The asymptotic distribution of $W^{(n)}$","Suppose that $2\\le r\\le n$ .","For any $1\\le a,s \\le n$ , let $\\tilde{A}_{a,s}=\\tilde{A}^{(n,r)}_{a,s}=\\lbrace \\xi _a=0,\\xi _{a+s \\mod {n}}=1,\\cdots , \\xi _{a+rs \\mod {n}}=1\\rbrace .$ Let $\\tilde{B}_n=\\tilde{B}^{(r)}_n=\\lbrace (a,s): 1\\le a\\le n,1\\le s\\le [ n/2],\\gcd (n,s) r$ .","Therefore $\\lbrace W^{(n)}\\ge r\\rbrace = \\big (\\bigcup _{(i,s)\\in \\tilde{B}_n}\\tilde{A}_{i,s}\\big )\\bigcup \\big (\\bigcup _{1\\le a\\le \\gcd (s,n),1\\le s \\le [ \\frac{n}{2}], \\frac{n}{\\gcd (s,n)}\\ge r}\\lbrace \\xi _i=1,\\forall i\\in C_{a,s}\\rbrace \\big ).$ This, together with the fact that $C_{a,s}=C_{a,\\gcd (s,n)}$ , yields (REF ).","$\\Box $ Let $\\tilde{I}=\\tilde{I}_{n,r}=\\sum _{(a,s)\\in \\tilde{B}_n} P(\\tilde{A}_{a,s})$ .","By Lemma (REF ), we have $P(A_1)\\le P(W^{(n)}\\ge r)\\le P(A_1)+P(A_2)\\le \\tilde{I}+P(A_2).$ Set $\\tilde{B}_{a,s}=\\tilde{B}^{(n,r)}_{a,s}=\\lbrace a,a+s \\mod {n},\\cdots , a+rs \\mod {n}\\rbrace .$ Define $\\tilde{G}$ to be the graph with vertex set $\\tilde{B}_n$ and edges defined by $(a,s) \\sim (b,t)$ if and only if $\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}\\ne \\emptyset $ .","Then $\\tilde{G}$ is a dependency graph of $\\lbrace I_{\\tilde{A}_{a,s}}: (a,s)\\in \\tilde{B}_n\\rbrace $ .","Put $\\tilde{e}^{(n,r)}=\\sum _{(a,s)\\in \\tilde{B}_n}\\sum _{(b,t)\\sim (a,s)} P(\\tilde{A}_{a,s})P(\\tilde{A}_{b,t})+\\sum _{(a,s)\\in \\tilde{B}_n}\\sum _{(b,t)\\sim (a,s),(b,t)\\ne (a,s)} P(\\tilde{A}_{a,s} \\tilde{A}_{b,t}).$ Then $|P(A^c_1)-e^{-\\tilde{I}}|\\le \\tilde{e}^{(n,r)}.$ The estimations of $P(A_2)$ , $\\tilde{I}$ and $\\tilde{e}^{(n,r)}$ are given in the following two lemmas.","Lemma 4.2 We have $P(A_2)\\le \\frac{np^r}{qr}$ and $(1 -\\frac{(r+1)^2}{2n})\\frac{qn^2p^r}{2}\\le \\tilde{I} \\le \\frac{qn^2 p^r}{2}.$ Obviously, $P(A_2)&\\le & \\sum _{s|n,s\\le n/r}sp^{\\frac{n}{s}}\\le \\sum _{i=r}^n \\frac{n}{i}p^i\\le \\frac{n}{r} \\sum _{i=r}^n p^i\\le \\frac{np^r}{qr}.$ Clearly, $\\lbrace 1\\le s\\le [\\frac{n}{2}]: \\gcd (n,s)\\ge \\frac{n}{r}\\rbrace \\subseteq \\lbrace s=\\frac{n}{i} j:2\\le i\\le r,1\\le j\\le [\\frac{i}{2}]\\rbrace $ .","It implies that $\\frac{n^2}{2}\\ge |B_n|\\ge n([\\frac{n}{2}]-\\sum _{i=2}^r [\\frac{i}{2}])\\ge \\frac{n^2}{2}(1 -\\frac{(r+1)^2}{2n}).$ Now the fact that $\\tilde{I}=|\\tilde{B}_n|qp^r$ yields (REF ) immediately.","$\\Box $ Lemma 4.3 It holds that $\\tilde{e}^{(n,r)}\\le 4(n^3r^2p^{2r-1}+n^2r^5p^{\\frac{3}{2} r-1}+nr^6 p^r)$ .","Let $H=\\lbrace (a,s,b,t)\\in \\tilde{B}_n\\times \\tilde{B}_n: (a,s)\\sim (b,t)\\rbrace $ , $\\tilde{c}_1=|H|$ , $\\tilde{c}_2&=&|\\lbrace (a,s,b,t)\\in H: |\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|\\ge 2\\rbrace |$ and $\\tilde{c}_3=|\\lbrace (a,s,b,t)\\in H: (a,s)\\ne (b,t),|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|>\\frac{r}{2}+1,\\tilde{A}_{a,s}\\cap \\tilde{A}_{b,t}\\ne \\emptyset \\rbrace |.$ Then $\\tilde{c}_1\\le && |\\lbrace (a,s,b,t): (a,s)\\in \\tilde{B}_n, 1\\le t\\le [\\frac{n}{2}], b=a+is-jt \\mod {n},\\nonumber \\\\&&0\\le i,j\\le r\\rbrace |\\le n^3r^2.$ Suppose that $|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|\\ge 2$ .","Then there is $0\\le j_1 r/2+1$ and $|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|=\\lbrace a+i_0s,a+i_1s,\\cdots ,a+i_ks\\rbrace $ with $0\\le i_03n/r_n$ , then $|\\tilde{B}^{(m,r_m)}_{a,s}\\cap \\tilde{B}^{(n,r_n)}_{b,t}|<\\frac{3}{4} r_n$ .","If $1\\le x\\le m$ , $A=\\lbrace x,x+t,\\cdots ,x+kt\\rbrace \\subseteq \\lbrace 1,2,\\cdots ,n\\rbrace $ and $x+(k+1)t>n$ , then $k\\ge 1$ and $x+\\frac{k+1}{2}t>n/2\\ge m$ .","Thus $|A\\cap \\lbrace 1,2,\\cdots ,m\\rbrace |\\le (k+1)/2$ when $k$ is odd, or $|A\\cap \\lbrace 1,2,\\cdots ,m\\rbrace |\\le k/2+1$ when $k$ is even.","Hence $|A\\cap \\lbrace 1,2,\\cdots ,m\\rbrace |/|A|\\le 2/3$ .","Since $t>3n/r_n$ , there is $h\\ge 3$ , $0\\le i_1n\\ge b+i_1 t, b+(i_2+1)t>2n\\ge b+i_2 t, \\cdots , b+(i_h+1)t>hn\\ge b+i_h t$ and $b+r_nt\\le (h+1)n$ .","Let $i_0=-1$ , $i_{h+1}=r_n$ and $A_j=\\lbrace b+(i_{j}+1)t \\mod {n}, b+(i_{j}+2)t\\mod {n},\\cdots , b+i_{j+1} t\\mod {n}\\rbrace .$ Then $|A_j\\cap \\lbrace 1,2,\\cdots m\\rbrace |\\le \\frac{2}{3}|A_j|$ when $0\\le j\\le h-1$ , and $\\jmath =|A_{h+1}\\cap \\lbrace 1,2\\cdots ,m\\rbrace |\\le m/t+1$ .","Thus $&&|\\lbrace b,b+t\\mod {n},\\cdots , b+r_n t\\mod {n}\\rbrace \\cap \\lbrace 1,2,\\cdots , m\\rbrace |/r_n\\\\&&\\le \\frac{\\frac{2}{3} \\sum _{j=0}^{h-1}|A_j|+\\jmath }{r_n}\\le \\frac{2(i_h+1+\\jmath )}{3r_n}+\\frac{\\jmath }{3r_n}\\le \\frac{2}{3}+\\frac{m}{3r_nt}+\\frac{1}{r_n}<\\frac{3}{4}.\\Box $ Proof of Theorem 1.2 (1)Firstly, by (REF ) and (REF ), $|e^{-\\tilde{I}_{n,r}}-e^{-\\mu _{n,r}}|\\le |\\tilde{I}_{n,r}- \\mu _{n,r}|\\le qnp^r r^2.$ Next, let $r_n=[\\frac{-2\\ln n}{\\ln p}+ \\frac{\\ln \\ln n}{\\ln p}+\\frac{\\ln 2-\\ln q}{\\ln p}-1]$ and $R_n=[\\frac{-3\\ln n}{\\ln p}]$ .","Lemma REF , together with (REF ), (REF ), (REF ) and (REF ), yields that $&&\\max _{r_n\\le r\\le R_n}|P(W^{(n)}0$.","}\\end{array}\\right.$ It follows (REF ) immediately.","(4) In view of (REF ), to prove (REF ), it remains only to show that $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}\\le \\frac{3}{2} \\,\\, a.s. $ For any $\\varepsilon >0$ , by (REF ),(REF ) and (REF ), $P(W^{(n)}>(1+\\varepsilon )C\\ln n) \\le O(n^{-2\\varepsilon })+O(\\frac{n^{-1-2\\varepsilon }}{\\ln n}).$ Hence $\\sum _{n=1}^\\infty P(W^{(n)}>(1+\\varepsilon )C\\ln n) <\\infty $ whenever $\\varepsilon >\\frac{1}{2}$ .","Therefore, $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}\\le \\frac{3}{2} \\,\\,\\,a.s$ .","as desired.","Since $W^{(n)}\\ge U^{(n)}$ , by (REF ), to prove (REF ), we need only to show that $\\liminf _{n\\rightarrow \\infty } \\frac{W^{(n)}-C\\ln n}{\\ln \\ln n}\\le D\\,\\,\\, a.s. $ Fix any $0<\\varepsilon <1$ .","Let $r_n=[C\\ln n+\\varepsilon D\\ln \\ln n]$ , $H_n=\\lbrace W^{(n)}0)\\ge \\frac{(EX_k)^2}{EX^2_k}=\\frac{(\\sum _{n=k+1}^{2k} P(H_n))^2}{\\sum _{m,n=k+1}^{2n} P(H_mH_n)}.","$ Clearly, $\\mu _{n,r_n}=O(\\ln ^\\varepsilon n)=o(\\ln n/4)$ .","Together with (REF ), it implies that $\\sum _{n=k+1}^{2k} P(H_n)\\ge \\sum _{n=k+1}^{2k}e^{-\\mu _{n,r_n}}-O(\\ln ^7k)\\ge O(k^{3/4}).","$ Set $E_n=\\lbrace (a,s):1\\le a\\le n, [\\frac{3n}{C\\ln n}]+1\\le s\\le [\\frac{n}{2}],\\gcd (n,s)<\\frac{n}{r_n}\\rbrace $ .","By (REF ) and (REF ), $P((H_mH_n)^c)\\ge P(\\cup _{(a,s)\\in E_m}\\tilde{A}^{(m,r_m)}_{a,s}\\cup _{(b,t)\\in E_n}\\tilde{A}^{(n,r_n)}_{b,t}).$ Let $V$ be the graph with vertex set $V_{k}=\\lbrace (a,s,n):k+1\\le n\\le 2k, (a,s)\\in E_n\\rbrace $ and edges defined by $(a,s,m)\\sim (b,t,n)$ if and only if $\\tilde{B}^{(m,r_m)}_{a,s}\\cap \\tilde{B}^{(n,r_n)}_{b,t}\\ne \\emptyset $ .","Then $V$ is a dependency graph of $\\lbrace I_{\\tilde{A}^{(n,r_n)}_{a,s}}:(a,s,n)\\in V_k\\rbrace $ .","Let $J_{m}=\\sum _{(a,s)\\in E_m} P(\\tilde{A}^{(m,r_m)}_{a,s})$ .","Then $P(H_mH_n)\\le e^{-J_m-J_n}+\\tilde{e}^{(m,r_m)}+\\tilde{e}^{(n,r_n)}+2 \\tilde{e}^{(m,r_m,n,r_n)}, $ where $\\tilde{e}^{(m,r_m,n,r_n)}=\\sum _{(a,s,m)\\sim (b,t,n)}\\big (P(\\tilde{A}^{(m,r_m)}_{a,s})P(\\tilde{A}^{(n,r_n)}_{b,t})+P(\\tilde{A}^{(m,r_m)}_{a,s}\\tilde{A}^{(n,r_n)}_{b,t})\\big )$ .","It follows that $\\sum _{m,n=k+1}^{2k} P(H_mH_n)\\le (\\sum _{n=k+1}^{2k} e^{-J_n})^2+{\\cal L}_k, $ where ${\\cal L}_k=2k\\sum _{n=k+1}^{2k}\\tilde{e}^{(n,r_n)}+2\\sum _{m,n=k+1}^{2k}\\tilde{e}^{(m,r_m,n,r_n)} $ .","One deduces from (REF ) that $J_n\\ge \\mu _{n,r_n}(1-\\frac{(r_n+1)^2}{2n}-\\frac{6}{C\\ln n})$ .","Hence $\\sum _{n=k+1}^{2k} e^{-J_n}\\le e^{O(\\ln ^{\\varepsilon -1}k)} \\sum _{n=k+1}^{2k} e^{- \\mu _{n,r_n}} $ by noting that $\\mu _{n,r_n}=O(\\ln ^\\varepsilon n)$ .","If we have showed that ${\\cal L}_k\\le O(k\\ln ^8 k)$ , then $\\lim _{k\\rightarrow \\infty }P(\\bigcup _{n=k+1}^{2k}H_n)=1$ and hence (REF ) holds, in view of (REF )–(REF ).","Now we shall show that ${\\cal L}_k\\le O(k \\ln ^8 k)$ .","By Lemma REF , $ k\\sum _{n=k+1}^{2k}\\tilde{e}^{(n,r_n)}\\le O(k\\ln ^7k).$ Fix any $(a,s,m)\\in V_k$ , define $\\gamma (a,s,m)$ to be the set of all triples $(b,t,n)\\in V_k$ such that $|\\tilde{B}^{(n,r_n)}_{b,t}\\cap \\tilde{B}^{(m,r_m)}_{a,s}|> {C\\ln {(2k)}}/2$ .","Suppose that $k$ is sufficiently large and $(b,t,n)\\in \\gamma (a,s,m)$ .","Let $i_l=\\min \\lbrace j\\ge 0:b+jt>ln\\rbrace $ , $\\jmath =\\max \\lbrace l:i_l\\le r_m\\rbrace $ and $Z_l=\\lbrace b+it-ln:i_l\\le i\\le min(i_{l+1}-1,r_n)\\rbrace $ .","Then $\\tilde{B}^{(n,r_n)}_{b,t}$ is the disjoint union of $Z_i$ with $0\\le i\\le \\jmath $ .","Since $[\\frac{3n}{C\\ln n}]+1\\le t\\le [\\frac{n}{2}]$ , $|Z_i|\\le C\\ln n/3+1$ for all $i\\le \\jmath $ , and $|Z_i|\\ge 2$ for $0 {C\\ln {(2k)}}/2$ and $|Z_i|\\le C\\ln n/3+1$ , there are $l$ and $\\imath $ such that $\\imath \\ne l$ , $|Z_l\\cap \\tilde{B}^{(m,r_m)}_{a,s}|\\ge 2$ and $|Z_\\imath \\cap \\tilde{B}^{(m,r_m)}_{a,s}|\\ge 1$ .","That is to say, there is $0\\le i,j,\\ell \\le r_n$ and $x,y,z\\in \\tilde{B}_{a,s,m}$ such that $b+it-ln=x$ , $b+jt-ln=y$ and $b+\\ell t-\\imath n=z$ .","It leads to $(j-i)t=y-x$ and $(\\imath -l)n=(\\ell -i)t+x-z$ .","Therefore $|\\gamma (a,s,m)|\\le &&|\\lbrace (b,t,n): t=(y-x)/i,n=(\\ell t+x-z)/j, b+\\imath t \\mod {n}=x,\\\\&&|i|,|j|,|\\ell |,|\\imath |\\le C\\ln 2k, i,j\\ne 0,x,y,z\\in \\tilde{B}_{a,s,m}\\rbrace |=O(\\ln ^7 k).$ Combining with (REF ) and (REF ), one then deduces that $\\sum _{m,n=k+1}^{2k}\\tilde{e}^{(m,r_m,n,r_n)} &\\le & \\sum _{m,n=k+1}^{2k}\\big (2p^{r_m+r_n-1}m^2nr_mr_n+3p^{r_m+r_n-C\\ln (2k)/2}m^2r^2_m r^3_n\\big )\\\\&&+\\sum _{(a,s,m)\\in V_k}p^{r_m}|\\gamma (a,s,m)|\\le O(k\\ln ^8 k)$ as desired.","It completes the proof of (REF ).","(5) Now we come to the proof of (REF ).","By (REF ), for any $0<\\varepsilon <1$ , $P(W^{(2^n)}C\\ln 2^n-(1+\\varepsilon ) D\\ln \\ln 2^n)=O(n^{-(1+\\varepsilon )}).$ Hence $\\liminf _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}\\ge 0$ and $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}\\le -D$ by the Borel-Cantelli Lemma.","In view of (REF ), it remains only to show that $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}\\ge -D. $ For any $0<\\varepsilon <1$ , let $r_n=[C\\ln n-\\varepsilon D\\ln \\ln n]$ and $F_{n}=\\lbrace (a,s):1\\le a\\le n, 3n/r_n0)\\ge \\frac{(EX_k)^2}{DX_k+(EX_k)^2}\\rightarrow 1.$ As to (REF ), by Proposition REF , we need only to show (REF ).","It suffices to show that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n \\sum _{(a,s)\\in F_{2^k}}I_{\\tilde{A}^{(2^k,r_{2^k})}_{a,s}}=0 \\,\\, a.s. $ and $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n I_{H_k}=0 \\,\\, a.s.,$ where $H_k=\\lbrace W^{(2^k)}\\ge r_{2^k}\\rbrace \\setminus (\\bigcup _{(a,s)\\in F_{2^k}}\\tilde{A}^{(2^k,r_{2^k})}_{a,s})$ .","Similar as the proof of (REF ), we may show (REF ).","Note that $P(H_k)=O(k^{-1-\\varepsilon })$ .","By Borel-Cantelli Lemma, $\\lim _{k\\rightarrow \\infty }I_{H_k}=0 \\,\\,a.s$ .","It follows (REF ) and completes our proof.","$\\Box $"],["The asymptotic distributions when $p_n=o(1)$","Proof of Theorem 1.3 (i) Set $q_n=1-p_n$ and $r_n=[\\frac{2\\ln n}{-\\ln p_n}+\\frac{\\ln \\ln n}{\\ln p_n}]$ .","Clearly, $p^{r_n+1}_n\\le n^{-2}\\ln n \\le p^{r_n}_n$ .","Similar as the proof of (REF ), we may show that there is a constant $c>0$ such that $\\max _{0\\le k\\le 2}|P(U^{(n,p_n)}N$ .","It follows that for all $n>N$ , $p^{\\frac{-b+\\varepsilon }{2}}_nN$ , $P(W^{(n,p_n)}\\ge [b]+1)\\le \\frac{p_n^{[b]+1-b/2-\\varepsilon /2}}{q_n([b]+1])}+\\frac{q_n p_n^{[b]+1-b-\\varepsilon }}{2}\\rightarrow 0.$ Suppose that $r0,\\\\sum _n b_n=\\\\infty , \\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{\\\\sum _{k=1}^n b_k}{n\\\\max _{1\\\\le k\\\\le n} b_k}=1 $ and $a\\\\le \\\\liminf _{n\\\\rightarrow \\\\infty } \\\\frac{a_n}{b_n}\\\\le \\\\limsup _{n\\\\rightarrow \\\\infty } \\\\frac{a_n}{b_n}<\\\\infty .$ Then $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{\\\\sum _{k=1}^n a_k}{\\\\sum _{k=1}^n b_k}=a$ if and only if for all $c>a$ , $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{n} \\\\sum _{k=1}^n I_{\\\\lbrace {a_k}/{b_k}a$ , let $A_{n}=\\\\lbrace 1\\\\le k\\\\le n: {a_k}/{b_k}\\\\ge c\\\\rbrace $ .\",\n \"Then (REF ) implies that $\\\\lim _{n\\\\rightarrow \\\\infty } {|A_{n}|}/{n}=0$ .\",\n \"Hence $\\\\frac{\\\\sum _{k=1}^n a_k}{\\\\sum _{k=1}^n b_k}\\\\le c+ \\\\frac{(d-c)\\\\sum _{k\\\\in A_{n}}b_k}{\\\\sum _{k=1}^n b_k}\\\\le c +\\\\frac{(d-c)|A_{n}|\\\\max _{1\\\\le k\\\\le n} b_k}{\\\\sum _{k=1}^n b_k}\\\\rightarrow c.$ The arbitrary of $c>a$ yields that $\\\\limsup _{n\\\\rightarrow \\\\infty } {\\\\sum _{k=1}^n a_k}/{\\\\sum _{k=1}^n b_k}\\\\le a$ .\",\n \"This, together with (REF ) and (REF ), gives (REF ).\",\n \"Next, we shall show the necessity.\",\n \"By (REF ), for any $\\\\varepsilon >0$ , there is $K$ such that $ {a_k}/{b_k}>a-\\\\varepsilon $ for all $k\\\\ge K$ .\",\n \"For any $c>a$ , let $B_n=\\\\lbrace KK$ , $\\\\frac{\\\\sum _{k=1}^n a_k}{\\\\sum _{k=1}^n b_k}&\\\\ge & c+\\\\frac{\\\\sum _{k=1}^{K} (a_k-cb_k)+(a-\\\\varepsilon -c)\\\\sum _{k\\\\in B_n} b_k}{\\\\sum _{k=1}^n b_k}\\\\\\\\&\\\\ge & c+\\\\frac{\\\\sum _{k=1}^{K} (a_k-cb_k)+(a-\\\\varepsilon -c)|B_{n}|\\\\max _{1\\\\le k\\\\le n} b_k}{\\\\sum _{k=1}^n b_k}.$ Letting $n$ tends to $\\\\infty $ , we get that $\\\\liminf _{n\\\\rightarrow \\\\infty }\\\\frac{|B_{n}|\\\\max _{1\\\\le k\\\\le n} b_k}{\\\\sum _{k=1}^n b_k}\\\\ge \\\\frac{c-a}{c+\\\\varepsilon -a}.$ The arbitrary of $\\\\varepsilon >0$ , together with (REF ), implies (REF ) and completes our proof.\",\n \"$\\\\Box $ As an application of Lemma REF .\",\n \"Suppose that (REF ) holds.\",\n \"In addition, suppose that $b_nc_n>0$ , $\\\\sum _{n} b_nc_n=\\\\infty $ and $ \\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{\\\\sum _{k=1}^n b_kc_k}{n\\\\max _{1\\\\le k\\\\le n} b_kc_k}=1$ .\",\n \"Then $\\\\lim _{n\\\\rightarrow \\\\infty } {\\\\sum _{k=1}^n a_kc_k}/{\\\\sum _{k=1}^n b_kc_k}=a$ if and only if (REF ) holds for all $c>a$ .\",\n \"Particularly, by letting $c_n=1/b_n$ , we see that $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{n} \\\\sum _{k=1}^n \\\\frac{a_k}{b_k}=a$ if and only if (REF ) holds for all $c>a$ .\",\n \"Therefore if (REF ) and (REF ) hold, then $\\\\lim _{n\\\\rightarrow \\\\infty } {\\\\sum _{k=1}^n a_k}/{\\\\sum _{k=1}^n b_k}=a$ if and only if $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{n} \\\\sum _{k=1}^n \\\\frac{a_k}{b_k}=a$ , if and only if (REF ) holds for all $c>a$ .\",\n \"Note that $n\\\\ln n-n\\\\le \\\\sum _{k=1}^n \\\\ln k \\\\le n\\\\ln n$ .\",\n \"By Lemma REF , we have the following propositions.\",\n \"Proposition 2.1 If (REF ) holds, then () holds if and only if $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n} {\\\\sum _{k=2}^n \\\\frac{U^{(2^k)}-C\\\\ln 2^k}{\\\\ln \\\\ln 2^k}}=D \\\\,\\\\,a.s, $ and also if and only if for any $1>\\\\varepsilon >0$ , $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n} \\\\sum _{k=1}^n I_{\\\\lbrace U^{(2^k)}0$ , $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{n} \\\\sum _{k=1}^n I_{\\\\lbrace W^{(2^k)}s\\\\rbrace =\\\\lbrace (a,s):1\\\\le s\\\\le [ {n-1\\\\over r}], ss\\\\rbrace |\\\\le \\\\frac{n^2}{2r}.\",\n \"$ Clearly, $I=p^r|B_n|-p^{r+1}|B_n\\\\cap \\\\lbrace (a,s):a>s\\\\rbrace |$ .\",\n \"This, together with (REF ) and (REF ), gives (REF ).\",\n \"$\\\\Box $ Lemma 3.2 It holds that $e^{(n,r)}\\\\le 9(n^3 p^{2r-1}+n^2 r^3 p^{\\\\frac{5}{3} r-1}+n^2 p^{\\\\frac{3}{2} r-1})$ .\",\n \"Let $c_1&=&|\\\\lbrace (a,s,b,t)\\\\in B_n\\\\times B_n: (a,s)\\\\sim (b,t)\\\\rbrace |,\\\\\\\\c_2&=&|\\\\lbrace (a,s,b,t)\\\\in B_n\\\\times B_n: |B_{a,s}\\\\cap B_{b,t}|\\\\ge 2\\\\rbrace |$ and $c_3=|\\\\lbrace (a,s,b,t)\\\\in B_n\\\\times B_n: (a,s)\\\\sim (b,t),t=2s\\\\text{ or } s=2t\\\\rbrace |.$ Then $c_1\\\\le &&|\\\\lbrace (a,s,b,t): (a,s)\\\\in B_n, 1\\\\le t\\\\le [\\\\frac{n-1}{r-1}], b=a+is-jt,\\\\\\\\&&-1\\\\le i,j\\\\le r-1\\\\rbrace |\\\\le 9n^3/2$ and $c_3&\\\\le & 2|\\\\lbrace (a,s,b,t): (a,s)\\\\in B_n, t=2s , b=a+is-jt,-1\\\\le i,j\\\\le r-1\\\\rbrace |\\\\\\\\&=&2|\\\\lbrace (a,s,b):(a,s)\\\\in B_n, b=a+ks,-2r+1\\\\le k\\\\le r+1\\\\rbrace |\\\\le 7n^2.$ Suppose that $|B_{a,s}\\\\cap B_{b,t}|\\\\ge 2$ and $x_0$ is the minimal number of the set $B_{a,s}\\\\cap B_{b,t}$ .\",\n \"Then $x_0=a+is=b+jt$ for some $-1\\\\le i,j\\\\le r-1$ .\",\n \"If $x\\\\in B_{a,s}\\\\cap B_{b,t}$ and $x>x_0$ , then $x=a+i^{\\\\prime }s=b+j^{\\\\prime }t$ for some $-1\\\\le i^{\\\\prime },j^{\\\\prime }\\\\le r-1$ .\",\n \"It follows that $x-x_0=(i^{\\\\prime }-i)s=(j^{\\\\prime }-j)t$ .\",\n \"Thus $t=\\\\frac{k_1}{k_2}s$ for some $1\\\\le k_1,k_2 \\\\le r$ .\",\n \"In addition, there is positive integer $k$ such that $i^{\\\\prime }-i=kt_0, j^{\\\\prime }-j=ks_0$ and $x-x_0=kst_0$ , where $s_0={s}/{\\\\gcd (s,t)}$ and $t_0={t}/{\\\\gcd (s,t)}$ .\",\n \"Since $i^{\\\\prime }-i\\\\le r$ , $k\\\\le {r}/{t_0}$ .\",\n \"Similarly, $k\\\\le {r}/{s_0}$ .\",\n \"Therefore $|B_{a,s}\\\\cap B_{b,t}|\\\\le {r}/\\\\max (s_0, t_0)+1.$ Consequently, $|B_{a,s}\\\\cap B_{b,t}|\\\\le {r}/{3}+1$ whenever $\\\\max (s_0, t_0)\\\\ge 3$ .\",\n \"When $\\\\max (s_0, t_0)=2$ , $|B_{a,s}\\\\cap B_{b,t}|\\\\le {r}/{2}+1$ .\",\n \"Actually, in this case, $s=2t$ or $t=2s$ .\",\n \"When $\\\\max (s_0, t_0)=1$ , $s=t$ .\",\n \"We shall show that $A_{a,s}\\\\cap A_{b,s}=\\\\emptyset $ whenever $a\\\\ne b$ and $B_{a,s}\\\\cap B_{b,s}\\\\ne \\\\emptyset $ .\",\n \"Assume that $b>a$ without loss of generality.\",\n \"Since $B_{a,s}\\\\cap B_{b,s}\\\\ne \\\\emptyset $ , $a+is=b+js$ for some $-1\\\\le i,j\\\\le r-1$ and hence $b=a+ks$ for some $1\\\\le k \\\\le r$ .\",\n \"Thus $A_{a,s}\\\\subseteq \\\\lbrace \\\\xi _{a+(k-1)s}=1\\\\rbrace $ and $A_{b,s}\\\\subseteq \\\\lbrace \\\\xi _{b-s}=0\\\\rbrace =\\\\lbrace \\\\xi _{a+(k-1)s}=0\\\\rbrace $ .\",\n \"It implies that $A_{a,s}\\\\cap A_{b,s}=\\\\emptyset $ as desired.\",\n \"In view of the discussion above, we have $c_2\\\\le && |\\\\lbrace (a,s,b,t): (a,s)\\\\in B_n, t= {sk_1}/{k_2}, b=a+is-jt,\\\\\\\\&&-1\\\\le i,j\\\\le r-1,1\\\\le k_1,k_2\\\\le r\\\\rbrace |\\\\le 9n^2r^3/4$ and $e^{(n,r)}\\\\le 2c_1p^{2r-1}+c_2p^{\\\\frac{5r}{3}-1}+c_3p^{\\\\frac{3r}{2}-1}\\\\le &9(n^3 p^{2r-1}+n^2 r^3 p^{\\\\frac{5}{3} r-1}+n^2 p^{\\\\frac{3}{2} r-1})$ as desired.\",\n \"$\\\\Box $ Similar as the proof of Lemma REF , we may show that for any $m$ , $n$ and $2\\\\le r_m\\\\le r_n$ , with $H=\\\\lbrace (a,s,b,t):(a,s)\\\\in B^{(r_m)}_m, (b,t)\\\\in B^{(r_n)}_n\\\\rbrace $ , $|\\\\lbrace (a,s,b,t)\\\\in H:|B^{(r_m)}_{a,s}\\\\cap B^{(r_n)}_{b,t}|\\\\ge 1\\\\rbrace |\\\\le 9m^2n/2 $ and $|\\\\lbrace (a,s,b,t)\\\\in H:|B^{(r_m)}_{a,s}\\\\cap B^{(r_n)}_{b,t}|\\\\ge 2\\\\rbrace |\\\\le 9m^2r_mr^2_n/4.$ Also we can show that if $|B^{(r_m)}_{a,s}\\\\cap B^{(r_n)}_{b,t}|>\\\\frac{r_n}{2}+1$ and $A^{(r_m)}_{a,s}\\\\cap A^{(r_n)}_{b,t}\\\\ne \\\\emptyset $ , then $(b,t)=(a,s)$ .\",\n \"Therefore $|\\\\lbrace (a,s,b,t)\\\\in H:|B^{(r_k)}_{a,s}\\\\cap B^{(r_m)}_{b,t}|>\\\\frac{r_n}{2}+1,A^{(r_m)}_{a,s}\\\\cap A^{(r_n)}_{b,t}\\\\ne \\\\emptyset \\\\rbrace |\\\\le k^2/r_k.$ Proof of Theorem 1.1 (1) Lemma REF and (REF ) imply that $|e^{-I_{n,r}}-e^{-\\\\lambda _{n,r}}|\\\\le |I_{n,r}- \\\\lambda _{n,r}|\\\\le 2np^r.$ Let $r_n=[\\\\frac{-2\\\\ln n}{\\\\ln p}+2 \\\\frac{\\\\ln \\\\ln n}{\\\\ln p}+\\\\frac{\\\\ln \\\\ln \\\\ln n}{2\\\\ln p}]$ and $R_n=[\\\\frac{-3\\\\ln n}{\\\\ln p}]$ .\",\n \"By (REF ), (REF ) and Lemma REF , $&&\\\\max _{r_n\\\\le r\\\\le R_n}|P(U^{(n)}R_n$ , $&&|P(U^{(n)} -D\\\\ln \\\\ln n+1}\\\\epsilon _{n,x}\\\\le p^{-D\\\\ln \\\\ln n}O(1)=O(\\\\frac{1}{\\\\ln n})$ and $\\\\max _{ aD\\\\ln \\\\ln n\\\\le x \\\\le -D\\\\ln \\\\ln n+1}\\\\epsilon _{n,x}\\\\le p^{ aD\\\\ln \\\\ln n}O(\\\\frac{\\\\ln \\\\ln n}{\\\\ln n})=O(\\\\frac{ \\\\ln \\\\ln n}{\\\\ln ^{1-a} n}).$ Now (REF ) follows by (REF )–(REF ).\",\n \"(3) For any $\\\\varepsilon >0$ , (REF ) and (REF ) imply that $P(U^{(n)}\\\\ge C\\\\ln n +(1-\\\\varepsilon )D\\\\ln \\\\ln n)\\\\le O(\\\\ln ^{-\\\\varepsilon } n).$ On the other hand, by (REF ) and (REF ), $P(U^{(n)}< C\\\\ln n +(1+\\\\varepsilon )D\\\\ln \\\\ln n)\\\\le e^{-O(\\\\ln ^{\\\\varepsilon } n)}+O(\\\\frac{\\\\ln ^{4} n\\\\ln \\\\ln n}{n}).$ Hence (REF ) holds.\",\n \"(4) By (REF ), $\\\\sum _{k=1}^\\\\infty P(U^{(2^k)}< C\\\\ln 2^k +(1+\\\\varepsilon )D\\\\ln \\\\ln 2^k)<\\\\infty $ for any $\\\\varepsilon >0$ .\",\n \"One then deduces from the Borel-Cantelli Lemma that $P(U^{(2^k)}2^{K(\\\\omega )}$ , then $2^k\\\\le n<2^{k+1}$ for some $k\\\\ge K(\\\\omega )$ .\",\n \"Hence $U^{(2^k)}(\\\\omega )\\\\le U^{(n)}(\\\\omega )\\\\le U^{(2^{k+1})}(\\\\omega )$ .\",\n \"This, together with (REF ), gives that $U^{(n)}(\\\\omega )\\\\ge C\\\\ln n-C\\\\ln 2+(1+\\\\varepsilon )D\\\\ln \\\\ln n.$ Now the arbitrary of $\\\\varepsilon >0$ yields that $\\\\liminf _{n\\\\rightarrow \\\\infty } \\\\frac{U^{(n)}-C\\\\ln n}{\\\\ln \\\\ln n}\\\\ge D$ .\",\n \"Therefore $\\\\liminf _{n\\\\rightarrow \\\\infty } \\\\frac{U^{(n)}-C\\\\ln n}{\\\\ln \\\\ln n}= D$ by considering (REF ).\",\n \"Let $T_k$ be the longest length of arithmetic progressions relative to $\\\\xi _{2^{k-1}+1},\\\\\\\\ \\\\xi _{2^{k-1}+2},\\\\cdots ,\\\\xi _{2^{k}}$ .\",\n \"Then $T_1,T_2,\\\\cdots $ are independent.\",\n \"In addition, $T_k$ has the same distribution with $U^{(2^{k-1})}$ .\",\n \"By (REF ), $P(T_k\\\\ge C\\\\ln 2^k)=P(U^{(2^{k-1})}\\\\ge C\\\\ln 2^k)\\\\le O(1/k).$ Hence $\\\\sum _{k=1}^\\\\infty P(T_k\\\\ge C\\\\ln 2^k)=\\\\infty $ .\",\n \"By Borel-Cantelli Lemma, $P(T_k\\\\ge C\\\\ln 2^k\\\\,\\\\,\\\\, i.o.\",\n \")=1$ .\",\n \"Consequently, $\\\\limsup _{k\\\\rightarrow \\\\infty } \\\\frac{U^{(2^k)}-C\\\\ln 2^k}{\\\\ln \\\\ln 2^k}\\\\ge 0$ by noting that $U^{(2^k)}\\\\ge T_k$ .\",\n \"On the other hand, (REF ) yields that $\\\\sum _{k=1}^\\\\infty P(U^{(2^k)}\\\\ge C\\\\ln 2^k +(1-\\\\varepsilon )D \\\\ln \\\\ln 2^k)<\\\\infty $ whenever $\\\\varepsilon >1$ .\",\n \"Hence $\\\\limsup _{k\\\\rightarrow \\\\infty } \\\\frac{U^{(2^k)}-C\\\\ln 2^k}{\\\\ln \\\\ln 2^k}\\\\le 0$ .\",\n \"Therefore $\\\\limsup _{k\\\\rightarrow \\\\infty } \\\\\\\\\\\\frac{U^{(2^k)}-C\\\\ln 2^k}{\\\\ln \\\\ln 2^k}= 0$ .\",\n \"Furthermore, we can deduce that $\\\\limsup _{n\\\\rightarrow \\\\infty } \\\\frac{U^{(n)}-C\\\\ln n}{\\\\ln \\\\ln n}= 0$ by the fact that $U^{(n)}$ is increasing.\",\n \"This completes the proof of (REF ).\",\n \"Now we come to prove ().\",\n \"By Proposition REF , we need only to show (REF ).\",\n \"Let $r_k= [C\\\\ln 2^k+D(1-\\\\varepsilon )\\\\ln \\\\ln 2^k]$ , $V_n=\\\\lbrace (a,s,k):1\\\\le k\\\\le n, (a,s)\\\\in B_{2^k}^{(r_k)}\\\\rbrace $ and $\\\\Lambda (n)=\\\\sum _{(a,s,k)\\\\in V_n}I_{ A_{a,s}^{(r_k)}}$ .\",\n \"Then it suffices to show that $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\Lambda (n)/n=0$ a.s. Clearly, $E\\\\Lambda (n)=\\\\sum _{k=1}^n I_{2^k,r_k}=O(n^{1-\\\\varepsilon })$ and $D\\\\Lambda (n)$ is less than the sum of $p^{r_k+r_m-|B^{(r_k)}_{a,s}\\\\cap B^{(r_m)}_{b,t}|}$ with $(a,s,k),(b,t,m)\\\\in V_n$ , $B^{(r_k)}_{a,s}\\\\cap B^{(r_m)}_{b,t}\\\\ne \\\\emptyset $ and $A^{(r_k)}_{a,s}\\\\cap A^{(r_m)}_{b,t}\\\\ne \\\\emptyset $ .\",\n \"By (REF )–(REF ), $D\\\\Lambda (n)\\\\le &&\\\\sum _{1\\\\le i\\\\le j\\\\le n} O(2^{-j}ij+2^{-j}i^{2}j^3+2^{2i-2j}j^{1-\\\\varepsilon }i^{-1})\\\\\\\\\\\\le && O(\\\\sum _{j=1}^\\\\infty 2^{-j} j^{6})+O(\\\\sum _{j=1}^n j^{-\\\\varepsilon }\\\\sum _{k=0}^{j-1} 2^{-2k}j(j-k)^{-1})\\\\\\\\\\\\le &&O(1)+\\\\sum _{k=0}^{\\\\infty } 2^{-2k}(k+1)O(\\\\sum _{j=1}^n j^{-\\\\varepsilon })=O(n^{1-\\\\varepsilon }).$ Then by Tchebychev's inequality, for any $\\\\delta >0$ , $\\\\sum _{n=1}^\\\\infty P(|\\\\Lambda _n/n-E(\\\\Lambda _n/n)|>\\\\delta )\\\\le \\\\sum _{n=1}^\\\\infty O(\\\\frac{1}{n^{1+\\\\varepsilon }\\\\delta ^2})<\\\\infty .$ The Borel-Cantelli Lemma yields that $\\\\Lambda _n/n\\\\rightarrow 0\\\\,\\\\,a.s.$ .\",\n \"Hence (REF ) holds as desired.\",\n \"Finally, we shall prove (REF ).\",\n \"Let $c_n=[\\\\ln n/\\\\ln 2]$ .\",\n \"Then $2^{c_n}\\\\le n<2^{c_n+1}$ .\",\n \"For any integers $1\\\\le a\\\\le b$ , $\\\\ln \\\\frac{b+1}{a}=\\\\int _{a}^{b+1} \\\\frac{1}{x}\\\\,dx\\\\le \\\\sum _{i=a}^b\\\\frac{1}{i}\\\\le \\\\int _{a-1}^b \\\\frac{1}{x}\\\\,dx=\\\\ln \\\\frac{b}{a-1}.$ Thus $\\\\sum _{k=1}^n \\\\frac{U^{(k)}}{k}\\\\ge \\\\sum _{i=0}^{c_n-1} U^{(2^i)}\\\\sum _{j=2^i}^{2^{i+1}-1}\\\\frac{1}{j} \\\\ge \\\\ln 2\\\\sum _{i=0}^{c_n-1} U^{(2^i)}$ and $\\\\sum _{k=2}^n \\\\frac{U^{(k)}}{k}\\\\le \\\\sum _{i=0}^{c_n} U^{(2^{i+1})}\\\\sum _{j=2^i+1}^{2^{i+1}}\\\\frac{1}{j} \\\\le \\\\ln 2\\\\sum _{i=1}^{c_n+1} U^{(2^i)}.$ Hence by (), $\\\\lim _{n\\\\rightarrow \\\\infty }\\\\frac{1}{c_n\\\\ln c_n}(\\\\frac{1}{\\\\ln 2}\\\\sum _{k=1}^n \\\\frac{U^{(k)}}{k}+\\\\frac{\\\\ln 2}{\\\\ln p} c^2_n)=D\\\\,\\\\,a.s.$ It follows (REF ) immediately and completes the proof of Theorem 1.1.\",\n \"$\\\\Box $\"\n ],\n [\n \"The asymptotic distribution of $W^{(n)}$\",\n \"Suppose that $2\\\\le r\\\\le n$ .\",\n \"For any $1\\\\le a,s \\\\le n$ , let $\\\\tilde{A}_{a,s}=\\\\tilde{A}^{(n,r)}_{a,s}=\\\\lbrace \\\\xi _a=0,\\\\xi _{a+s \\\\mod {n}}=1,\\\\cdots , \\\\xi _{a+rs \\\\mod {n}}=1\\\\rbrace .$ Let $\\\\tilde{B}_n=\\\\tilde{B}^{(r)}_n=\\\\lbrace (a,s): 1\\\\le a\\\\le n,1\\\\le s\\\\le [ n/2],\\\\gcd (n,s) r$ .\",\n \"Therefore $\\\\lbrace W^{(n)}\\\\ge r\\\\rbrace = \\\\big (\\\\bigcup _{(i,s)\\\\in \\\\tilde{B}_n}\\\\tilde{A}_{i,s}\\\\big )\\\\bigcup \\\\big (\\\\bigcup _{1\\\\le a\\\\le \\\\gcd (s,n),1\\\\le s \\\\le [ \\\\frac{n}{2}], \\\\frac{n}{\\\\gcd (s,n)}\\\\ge r}\\\\lbrace \\\\xi _i=1,\\\\forall i\\\\in C_{a,s}\\\\rbrace \\\\big ).$ This, together with the fact that $C_{a,s}=C_{a,\\\\gcd (s,n)}$ , yields (REF ).\",\n \"$\\\\Box $ Let $\\\\tilde{I}=\\\\tilde{I}_{n,r}=\\\\sum _{(a,s)\\\\in \\\\tilde{B}_n} P(\\\\tilde{A}_{a,s})$ .\",\n \"By Lemma (REF ), we have $P(A_1)\\\\le P(W^{(n)}\\\\ge r)\\\\le P(A_1)+P(A_2)\\\\le \\\\tilde{I}+P(A_2).$ Set $\\\\tilde{B}_{a,s}=\\\\tilde{B}^{(n,r)}_{a,s}=\\\\lbrace a,a+s \\\\mod {n},\\\\cdots , a+rs \\\\mod {n}\\\\rbrace .$ Define $\\\\tilde{G}$ to be the graph with vertex set $\\\\tilde{B}_n$ and edges defined by $(a,s) \\\\sim (b,t)$ if and only if $\\\\tilde{B}_{a,s}\\\\cap \\\\tilde{B}_{b,t}\\\\ne \\\\emptyset $ .\",\n \"Then $\\\\tilde{G}$ is a dependency graph of $\\\\lbrace I_{\\\\tilde{A}_{a,s}}: (a,s)\\\\in \\\\tilde{B}_n\\\\rbrace $ .\",\n \"Put $\\\\tilde{e}^{(n,r)}=\\\\sum _{(a,s)\\\\in \\\\tilde{B}_n}\\\\sum _{(b,t)\\\\sim (a,s)} P(\\\\tilde{A}_{a,s})P(\\\\tilde{A}_{b,t})+\\\\sum _{(a,s)\\\\in \\\\tilde{B}_n}\\\\sum _{(b,t)\\\\sim (a,s),(b,t)\\\\ne (a,s)} P(\\\\tilde{A}_{a,s} \\\\tilde{A}_{b,t}).$ Then $|P(A^c_1)-e^{-\\\\tilde{I}}|\\\\le \\\\tilde{e}^{(n,r)}.$ The estimations of $P(A_2)$ , $\\\\tilde{I}$ and $\\\\tilde{e}^{(n,r)}$ are given in the following two lemmas.\",\n \"Lemma 4.2 We have $P(A_2)\\\\le \\\\frac{np^r}{qr}$ and $(1 -\\\\frac{(r+1)^2}{2n})\\\\frac{qn^2p^r}{2}\\\\le \\\\tilde{I} \\\\le \\\\frac{qn^2 p^r}{2}.$ Obviously, $P(A_2)&\\\\le & \\\\sum _{s|n,s\\\\le n/r}sp^{\\\\frac{n}{s}}\\\\le \\\\sum _{i=r}^n \\\\frac{n}{i}p^i\\\\le \\\\frac{n}{r} \\\\sum _{i=r}^n p^i\\\\le \\\\frac{np^r}{qr}.$ Clearly, $\\\\lbrace 1\\\\le s\\\\le [\\\\frac{n}{2}]: \\\\gcd (n,s)\\\\ge \\\\frac{n}{r}\\\\rbrace \\\\subseteq \\\\lbrace s=\\\\frac{n}{i} j:2\\\\le i\\\\le r,1\\\\le j\\\\le [\\\\frac{i}{2}]\\\\rbrace $ .\",\n \"It implies that $\\\\frac{n^2}{2}\\\\ge |B_n|\\\\ge n([\\\\frac{n}{2}]-\\\\sum _{i=2}^r [\\\\frac{i}{2}])\\\\ge \\\\frac{n^2}{2}(1 -\\\\frac{(r+1)^2}{2n}).$ Now the fact that $\\\\tilde{I}=|\\\\tilde{B}_n|qp^r$ yields (REF ) immediately.\",\n \"$\\\\Box $ Lemma 4.3 It holds that $\\\\tilde{e}^{(n,r)}\\\\le 4(n^3r^2p^{2r-1}+n^2r^5p^{\\\\frac{3}{2} r-1}+nr^6 p^r)$ .\",\n \"Let $H=\\\\lbrace (a,s,b,t)\\\\in \\\\tilde{B}_n\\\\times \\\\tilde{B}_n: (a,s)\\\\sim (b,t)\\\\rbrace $ , $\\\\tilde{c}_1=|H|$ , $\\\\tilde{c}_2&=&|\\\\lbrace (a,s,b,t)\\\\in H: |\\\\tilde{B}_{a,s}\\\\cap \\\\tilde{B}_{b,t}|\\\\ge 2\\\\rbrace |$ and $\\\\tilde{c}_3=|\\\\lbrace (a,s,b,t)\\\\in H: (a,s)\\\\ne (b,t),|\\\\tilde{B}_{a,s}\\\\cap \\\\tilde{B}_{b,t}|>\\\\frac{r}{2}+1,\\\\tilde{A}_{a,s}\\\\cap \\\\tilde{A}_{b,t}\\\\ne \\\\emptyset \\\\rbrace |.$ Then $\\\\tilde{c}_1\\\\le && |\\\\lbrace (a,s,b,t): (a,s)\\\\in \\\\tilde{B}_n, 1\\\\le t\\\\le [\\\\frac{n}{2}], b=a+is-jt \\\\mod {n},\\\\nonumber \\\\\\\\&&0\\\\le i,j\\\\le r\\\\rbrace |\\\\le n^3r^2.$ Suppose that $|\\\\tilde{B}_{a,s}\\\\cap \\\\tilde{B}_{b,t}|\\\\ge 2$ .\",\n \"Then there is $0\\\\le j_1 r/2+1$ and $|\\\\tilde{B}_{a,s}\\\\cap \\\\tilde{B}_{b,t}|=\\\\lbrace a+i_0s,a+i_1s,\\\\cdots ,a+i_ks\\\\rbrace $ with $0\\\\le i_03n/r_n$ , then $|\\\\tilde{B}^{(m,r_m)}_{a,s}\\\\cap \\\\tilde{B}^{(n,r_n)}_{b,t}|<\\\\frac{3}{4} r_n$ .\",\n \"If $1\\\\le x\\\\le m$ , $A=\\\\lbrace x,x+t,\\\\cdots ,x+kt\\\\rbrace \\\\subseteq \\\\lbrace 1,2,\\\\cdots ,n\\\\rbrace $ and $x+(k+1)t>n$ , then $k\\\\ge 1$ and $x+\\\\frac{k+1}{2}t>n/2\\\\ge m$ .\",\n \"Thus $|A\\\\cap \\\\lbrace 1,2,\\\\cdots ,m\\\\rbrace |\\\\le (k+1)/2$ when $k$ is odd, or $|A\\\\cap \\\\lbrace 1,2,\\\\cdots ,m\\\\rbrace |\\\\le k/2+1$ when $k$ is even.\",\n \"Hence $|A\\\\cap \\\\lbrace 1,2,\\\\cdots ,m\\\\rbrace |/|A|\\\\le 2/3$ .\",\n \"Since $t>3n/r_n$ , there is $h\\\\ge 3$ , $0\\\\le i_1n\\\\ge b+i_1 t, b+(i_2+1)t>2n\\\\ge b+i_2 t, \\\\cdots , b+(i_h+1)t>hn\\\\ge b+i_h t$ and $b+r_nt\\\\le (h+1)n$ .\",\n \"Let $i_0=-1$ , $i_{h+1}=r_n$ and $A_j=\\\\lbrace b+(i_{j}+1)t \\\\mod {n}, b+(i_{j}+2)t\\\\mod {n},\\\\cdots , b+i_{j+1} t\\\\mod {n}\\\\rbrace .$ Then $|A_j\\\\cap \\\\lbrace 1,2,\\\\cdots m\\\\rbrace |\\\\le \\\\frac{2}{3}|A_j|$ when $0\\\\le j\\\\le h-1$ , and $\\\\jmath =|A_{h+1}\\\\cap \\\\lbrace 1,2\\\\cdots ,m\\\\rbrace |\\\\le m/t+1$ .\",\n \"Thus $&&|\\\\lbrace b,b+t\\\\mod {n},\\\\cdots , b+r_n t\\\\mod {n}\\\\rbrace \\\\cap \\\\lbrace 1,2,\\\\cdots , m\\\\rbrace |/r_n\\\\\\\\&&\\\\le \\\\frac{\\\\frac{2}{3} \\\\sum _{j=0}^{h-1}|A_j|+\\\\jmath }{r_n}\\\\le \\\\frac{2(i_h+1+\\\\jmath )}{3r_n}+\\\\frac{\\\\jmath }{3r_n}\\\\le \\\\frac{2}{3}+\\\\frac{m}{3r_nt}+\\\\frac{1}{r_n}<\\\\frac{3}{4}.\\\\Box $ Proof of Theorem 1.2 (1)Firstly, by (REF ) and (REF ), $|e^{-\\\\tilde{I}_{n,r}}-e^{-\\\\mu _{n,r}}|\\\\le |\\\\tilde{I}_{n,r}- \\\\mu _{n,r}|\\\\le qnp^r r^2.$ Next, let $r_n=[\\\\frac{-2\\\\ln n}{\\\\ln p}+ \\\\frac{\\\\ln \\\\ln n}{\\\\ln p}+\\\\frac{\\\\ln 2-\\\\ln q}{\\\\ln p}-1]$ and $R_n=[\\\\frac{-3\\\\ln n}{\\\\ln p}]$ .\",\n \"Lemma REF , together with (REF ), (REF ), (REF ) and (REF ), yields that $&&\\\\max _{r_n\\\\le r\\\\le R_n}|P(W^{(n)}0$.\",\n \"}\\\\end{array}\\\\right.$ It follows (REF ) immediately.\",\n \"(4) In view of (REF ), to prove (REF ), it remains only to show that $\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\frac{W^{(n)}}{C\\\\ln n}\\\\le \\\\frac{3}{2} \\\\,\\\\, a.s. $ For any $\\\\varepsilon >0$ , by (REF ),(REF ) and (REF ), $P(W^{(n)}>(1+\\\\varepsilon )C\\\\ln n) \\\\le O(n^{-2\\\\varepsilon })+O(\\\\frac{n^{-1-2\\\\varepsilon }}{\\\\ln n}).$ Hence $\\\\sum _{n=1}^\\\\infty P(W^{(n)}>(1+\\\\varepsilon )C\\\\ln n) <\\\\infty $ whenever $\\\\varepsilon >\\\\frac{1}{2}$ .\",\n \"Therefore, $\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\frac{W^{(n)}}{C\\\\ln n}\\\\le \\\\frac{3}{2} \\\\,\\\\,\\\\,a.s$ .\",\n \"as desired.\",\n \"Since $W^{(n)}\\\\ge U^{(n)}$ , by (REF ), to prove (REF ), we need only to show that $\\\\liminf _{n\\\\rightarrow \\\\infty } \\\\frac{W^{(n)}-C\\\\ln n}{\\\\ln \\\\ln n}\\\\le D\\\\,\\\\,\\\\, a.s. $ Fix any $0<\\\\varepsilon <1$ .\",\n \"Let $r_n=[C\\\\ln n+\\\\varepsilon D\\\\ln \\\\ln n]$ , $H_n=\\\\lbrace W^{(n)}0)\\\\ge \\\\frac{(EX_k)^2}{EX^2_k}=\\\\frac{(\\\\sum _{n=k+1}^{2k} P(H_n))^2}{\\\\sum _{m,n=k+1}^{2n} P(H_mH_n)}.\",\n \"$ Clearly, $\\\\mu _{n,r_n}=O(\\\\ln ^\\\\varepsilon n)=o(\\\\ln n/4)$ .\",\n \"Together with (REF ), it implies that $\\\\sum _{n=k+1}^{2k} P(H_n)\\\\ge \\\\sum _{n=k+1}^{2k}e^{-\\\\mu _{n,r_n}}-O(\\\\ln ^7k)\\\\ge O(k^{3/4}).\",\n \"$ Set $E_n=\\\\lbrace (a,s):1\\\\le a\\\\le n, [\\\\frac{3n}{C\\\\ln n}]+1\\\\le s\\\\le [\\\\frac{n}{2}],\\\\gcd (n,s)<\\\\frac{n}{r_n}\\\\rbrace $ .\",\n \"By (REF ) and (REF ), $P((H_mH_n)^c)\\\\ge P(\\\\cup _{(a,s)\\\\in E_m}\\\\tilde{A}^{(m,r_m)}_{a,s}\\\\cup _{(b,t)\\\\in E_n}\\\\tilde{A}^{(n,r_n)}_{b,t}).$ Let $V$ be the graph with vertex set $V_{k}=\\\\lbrace (a,s,n):k+1\\\\le n\\\\le 2k, (a,s)\\\\in E_n\\\\rbrace $ and edges defined by $(a,s,m)\\\\sim (b,t,n)$ if and only if $\\\\tilde{B}^{(m,r_m)}_{a,s}\\\\cap \\\\tilde{B}^{(n,r_n)}_{b,t}\\\\ne \\\\emptyset $ .\",\n \"Then $V$ is a dependency graph of $\\\\lbrace I_{\\\\tilde{A}^{(n,r_n)}_{a,s}}:(a,s,n)\\\\in V_k\\\\rbrace $ .\",\n \"Let $J_{m}=\\\\sum _{(a,s)\\\\in E_m} P(\\\\tilde{A}^{(m,r_m)}_{a,s})$ .\",\n \"Then $P(H_mH_n)\\\\le e^{-J_m-J_n}+\\\\tilde{e}^{(m,r_m)}+\\\\tilde{e}^{(n,r_n)}+2 \\\\tilde{e}^{(m,r_m,n,r_n)}, $ where $\\\\tilde{e}^{(m,r_m,n,r_n)}=\\\\sum _{(a,s,m)\\\\sim (b,t,n)}\\\\big (P(\\\\tilde{A}^{(m,r_m)}_{a,s})P(\\\\tilde{A}^{(n,r_n)}_{b,t})+P(\\\\tilde{A}^{(m,r_m)}_{a,s}\\\\tilde{A}^{(n,r_n)}_{b,t})\\\\big )$ .\",\n \"It follows that $\\\\sum _{m,n=k+1}^{2k} P(H_mH_n)\\\\le (\\\\sum _{n=k+1}^{2k} e^{-J_n})^2+{\\\\cal L}_k, $ where ${\\\\cal L}_k=2k\\\\sum _{n=k+1}^{2k}\\\\tilde{e}^{(n,r_n)}+2\\\\sum _{m,n=k+1}^{2k}\\\\tilde{e}^{(m,r_m,n,r_n)} $ .\",\n \"One deduces from (REF ) that $J_n\\\\ge \\\\mu _{n,r_n}(1-\\\\frac{(r_n+1)^2}{2n}-\\\\frac{6}{C\\\\ln n})$ .\",\n \"Hence $\\\\sum _{n=k+1}^{2k} e^{-J_n}\\\\le e^{O(\\\\ln ^{\\\\varepsilon -1}k)} \\\\sum _{n=k+1}^{2k} e^{- \\\\mu _{n,r_n}} $ by noting that $\\\\mu _{n,r_n}=O(\\\\ln ^\\\\varepsilon n)$ .\",\n \"If we have showed that ${\\\\cal L}_k\\\\le O(k\\\\ln ^8 k)$ , then $\\\\lim _{k\\\\rightarrow \\\\infty }P(\\\\bigcup _{n=k+1}^{2k}H_n)=1$ and hence (REF ) holds, in view of (REF )–(REF ).\",\n \"Now we shall show that ${\\\\cal L}_k\\\\le O(k \\\\ln ^8 k)$ .\",\n \"By Lemma REF , $ k\\\\sum _{n=k+1}^{2k}\\\\tilde{e}^{(n,r_n)}\\\\le O(k\\\\ln ^7k).$ Fix any $(a,s,m)\\\\in V_k$ , define $\\\\gamma (a,s,m)$ to be the set of all triples $(b,t,n)\\\\in V_k$ such that $|\\\\tilde{B}^{(n,r_n)}_{b,t}\\\\cap \\\\tilde{B}^{(m,r_m)}_{a,s}|> {C\\\\ln {(2k)}}/2$ .\",\n \"Suppose that $k$ is sufficiently large and $(b,t,n)\\\\in \\\\gamma (a,s,m)$ .\",\n \"Let $i_l=\\\\min \\\\lbrace j\\\\ge 0:b+jt>ln\\\\rbrace $ , $\\\\jmath =\\\\max \\\\lbrace l:i_l\\\\le r_m\\\\rbrace $ and $Z_l=\\\\lbrace b+it-ln:i_l\\\\le i\\\\le min(i_{l+1}-1,r_n)\\\\rbrace $ .\",\n \"Then $\\\\tilde{B}^{(n,r_n)}_{b,t}$ is the disjoint union of $Z_i$ with $0\\\\le i\\\\le \\\\jmath $ .\",\n \"Since $[\\\\frac{3n}{C\\\\ln n}]+1\\\\le t\\\\le [\\\\frac{n}{2}]$ , $|Z_i|\\\\le C\\\\ln n/3+1$ for all $i\\\\le \\\\jmath $ , and $|Z_i|\\\\ge 2$ for $0 {C\\\\ln {(2k)}}/2$ and $|Z_i|\\\\le C\\\\ln n/3+1$ , there are $l$ and $\\\\imath $ such that $\\\\imath \\\\ne l$ , $|Z_l\\\\cap \\\\tilde{B}^{(m,r_m)}_{a,s}|\\\\ge 2$ and $|Z_\\\\imath \\\\cap \\\\tilde{B}^{(m,r_m)}_{a,s}|\\\\ge 1$ .\",\n \"That is to say, there is $0\\\\le i,j,\\\\ell \\\\le r_n$ and $x,y,z\\\\in \\\\tilde{B}_{a,s,m}$ such that $b+it-ln=x$ , $b+jt-ln=y$ and $b+\\\\ell t-\\\\imath n=z$ .\",\n \"It leads to $(j-i)t=y-x$ and $(\\\\imath -l)n=(\\\\ell -i)t+x-z$ .\",\n \"Therefore $|\\\\gamma (a,s,m)|\\\\le &&|\\\\lbrace (b,t,n): t=(y-x)/i,n=(\\\\ell t+x-z)/j, b+\\\\imath t \\\\mod {n}=x,\\\\\\\\&&|i|,|j|,|\\\\ell |,|\\\\imath |\\\\le C\\\\ln 2k, i,j\\\\ne 0,x,y,z\\\\in \\\\tilde{B}_{a,s,m}\\\\rbrace |=O(\\\\ln ^7 k).$ Combining with (REF ) and (REF ), one then deduces that $\\\\sum _{m,n=k+1}^{2k}\\\\tilde{e}^{(m,r_m,n,r_n)} &\\\\le & \\\\sum _{m,n=k+1}^{2k}\\\\big (2p^{r_m+r_n-1}m^2nr_mr_n+3p^{r_m+r_n-C\\\\ln (2k)/2}m^2r^2_m r^3_n\\\\big )\\\\\\\\&&+\\\\sum _{(a,s,m)\\\\in V_k}p^{r_m}|\\\\gamma (a,s,m)|\\\\le O(k\\\\ln ^8 k)$ as desired.\",\n \"It completes the proof of (REF ).\",\n \"(5) Now we come to the proof of (REF ).\",\n \"By (REF ), for any $0<\\\\varepsilon <1$ , $P(W^{(2^n)}C\\\\ln 2^n-(1+\\\\varepsilon ) D\\\\ln \\\\ln 2^n)=O(n^{-(1+\\\\varepsilon )}).$ Hence $\\\\liminf _{n\\\\rightarrow \\\\infty }\\\\frac{W^{(2^n)}-C\\\\ln 2^n}{\\\\ln \\\\ln 2^n}\\\\ge 0$ and $\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\frac{W^{(2^n)}-C\\\\ln 2^n}{\\\\ln \\\\ln 2^n}\\\\le -D$ by the Borel-Cantelli Lemma.\",\n \"In view of (REF ), it remains only to show that $\\\\limsup _{n\\\\rightarrow \\\\infty }\\\\frac{W^{(2^n)}-C\\\\ln 2^n}{\\\\ln \\\\ln 2^n}\\\\ge -D. $ For any $0<\\\\varepsilon <1$ , let $r_n=[C\\\\ln n-\\\\varepsilon D\\\\ln \\\\ln n]$ and $F_{n}=\\\\lbrace (a,s):1\\\\le a\\\\le n, 3n/r_n0)\\\\ge \\\\frac{(EX_k)^2}{DX_k+(EX_k)^2}\\\\rightarrow 1.$ As to (REF ), by Proposition REF , we need only to show (REF ).\",\n \"It suffices to show that $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{n} \\\\sum _{k=1}^n \\\\sum _{(a,s)\\\\in F_{2^k}}I_{\\\\tilde{A}^{(2^k,r_{2^k})}_{a,s}}=0 \\\\,\\\\, a.s. $ and $\\\\lim _{n\\\\rightarrow \\\\infty } \\\\frac{1}{n} \\\\sum _{k=1}^n I_{H_k}=0 \\\\,\\\\, a.s.,$ where $H_k=\\\\lbrace W^{(2^k)}\\\\ge r_{2^k}\\\\rbrace \\\\setminus (\\\\bigcup _{(a,s)\\\\in F_{2^k}}\\\\tilde{A}^{(2^k,r_{2^k})}_{a,s})$ .\",\n \"Similar as the proof of (REF ), we may show (REF ).\",\n \"Note that $P(H_k)=O(k^{-1-\\\\varepsilon })$ .\",\n \"By Borel-Cantelli Lemma, $\\\\lim _{k\\\\rightarrow \\\\infty }I_{H_k}=0 \\\\,\\\\,a.s$ .\",\n \"It follows (REF ) and completes our proof.\",\n \"$\\\\Box $\"\n ],\n [\n \"The asymptotic distributions when $p_n=o(1)$\",\n \"Proof of Theorem 1.3 (i) Set $q_n=1-p_n$ and $r_n=[\\\\frac{2\\\\ln n}{-\\\\ln p_n}+\\\\frac{\\\\ln \\\\ln n}{\\\\ln p_n}]$ .\",\n \"Clearly, $p^{r_n+1}_n\\\\le n^{-2}\\\\ln n \\\\le p^{r_n}_n$ .\",\n \"Similar as the proof of (REF ), we may show that there is a constant $c>0$ such that $\\\\max _{0\\\\le k\\\\le 2}|P(U^{(n,p_n)}N$ .\",\n \"It follows that for all $n>N$ , $p^{\\\\frac{-b+\\\\varepsilon }{2}}_nN$ , $P(W^{(n,p_n)}\\\\ge [b]+1)\\\\le \\\\frac{p_n^{[b]+1-b/2-\\\\varepsilon /2}}{q_n([b]+1])}+\\\\frac{q_n p_n^{[b]+1-b-\\\\varepsilon }}{2}\\\\rightarrow 0.$ Suppose that $r> h_m$ .","In first order approximation we can write the CTE of ULE around its zero crossing temperature $T_0(ULE)$ to be $\\alpha _{ULE}(T)=\\beta (T-T_0(ULE)),$ where $\\beta $ is a linear temperature coefficient and is taken to be $2\\times 10^{-9}$ K$^{-2}$ [31].","Thus the effective CTE can be expressed as: $\\alpha _{eff}&\\approx &\\beta [T-T_0(ULE)-\\frac{2\\alpha _{FS}h_m+2\\delta R(\\alpha _{FS}-\\overline{\\alpha _{ULE}})}{\\beta L}]\\nonumber \\\\&\\equiv &\\beta [T-T_0(eff)].$ Here we have used the fact that $2\\delta R << L$ to replace the $\\alpha _{ULE}$ in the numerator with temperature independent value of $\\overline{\\alpha _{ULE}}$ .","The effective zero crossing temperature of CTE of the cavity can then be expressed as $T_0(eff)=T_0(ULE)+\\frac{2\\alpha _{FS}h_m+2\\delta R(\\alpha _{FS}-\\overline{\\alpha _{ULE}})}{\\beta L}.$ According to the above equations, to make sure the temperature difference of $\\Delta T=T_0(eff)-T_0(ULE)$ to be smaller than 23 $^\\circ $ C, the spacer length $L$ must be at least 14 cm even without considering the effect of deformation coefficient $\\delta $ .","If $\\delta $ is positive, then the mirrors bulge in the same direction of its thermal expansion direction, as shown in Fig.","REF (a), the required cavity length will be even longer, typically longer than 30 cm.","Currently spacer length longer than 30 cm is very difficult to obtain, and has higher vibration sensitivity.","To obtain a more flexible tuning range of $\\Delta T$ with a moderate cavity length, the coefficient $\\delta $ in Eq.","(REF ) needs to be negative to compensate the huge difference between the CTE of FS and ULE.","Therefore, a modified design of re-entrant cavity is applied, in which the FS mirrors are optically contacted to FS rings instead of ULE rings.","In this configuration, the deformation of CTE mismatch is between the FS rings and ULE spacer, causing the FS rings bulge in the opposite direction of the mirror thermal expansion direction, as shown in Fig.","REF (b).","In another word, this cavity structure has a negative $\\delta $ .","Simulations of cavities with this new structure are shown in Fig.","REF .","Figure: FEA simulations of the deformations caused by CTE mismatch of re-entrant cavities; (a) deformation between FS mirror and ULE ring causing the mirror bulges in the same direction as its thermal expansion direction; (b) deformation between FS ring and ULE spacer causing the ring bulges in the opposite direction of the mirror thermal expansion direction.","Color scale shows the axial displacement.Figure: Tuning range of CTE zero crossing temperature of re-entrant cavities with FS mirror optically contacted to outer FS rings instead of ULE rings.","The definitions of dimensional parameters are the same as those in Fig.",".According to the simulations, the modified re-entrant cavity yields a larger tuning range of $\\Delta T$ from -7 $^\\circ $ C to 25 $^\\circ $ C simply by varying the thickness $h$ of FS rings, and this range can be adjusted further by altering the outer diameter $d_2$ according to the characteristics of ULE batch used and the set-point temperature of the thermal control system.","With this design, the flexibility requirement of $-10\\ ^\\circ \\mathrm {C}\\le \\Delta T\\le 23\\ ^\\circ \\mathrm {C}$ is easily achieved.","Through comparison, cavities with FS rings of relatively smaller outer diameter $d_2$ are more suitable because of the less area of optical contact as well as the compatibility with vibration immunity design which will be discussed in Sec.","4.","As a result, in the following simulations and analysis, an outer diameter of 80 mm of FS ring is adopted.","Figure REF shows the zero crossing temperature tuning capabilities with different cavity lengths between the original and the modified re-entrant cavity.","The zero crossing temperature tuning ranges are much narrower for the original design (only about 5 $^\\circ $ C) than those of the new design (more than 30 $^\\circ $ C).","Furthermore, to obtain a moderate temperature tuning (less than 25 $^\\circ $ C), the cavity length must be longer than 30 cm with the original design.","In comparison, for the new design, variation of ULE spacer lengths from 10 cm to any longer length can achieve a moderate temperature tuning range.","It is interesting to note that all curves except $L=10$ cm in Fig.","REF (b) intersect and cross zero at the same position where the ring thickness $h$ is 11.2 mm.","Referring to Eq.","(REF ), this intersection with zero reveals the fact that when the ring thickness is at this particular value ($h$ =11.2 mm) the thermal expansions of FS mirrors are exactly compensated by the deformation strain of FS rings no matter how long the cavity is.","In this case, the effective CTE of the cavity is the same as that of the ULE spacer material.","For $L=10$ cm case, the requirement $2\\delta R << L$ is not valid, which causes the zero crossing point to be different with that of longer cavities.","Figure: Comparison of the zero crossing temperature tuning capabilities for different cavity lengths between the original re-entrant cavity (a) and the modified cavity with FS rings (b).","The inner diameters of rings d 1 d_1 are 15 mm for both cases.In this work, the linear temperature coefficient of the effective CTE and mirror deformations of the modified cavity are also studied.","The results are shown in Fig.","REF and listed in Table REF .","According to Eq.","(REF ), the linear temperature coefficient $\\beta $ of the effective CTE also depends on cavity structure.","Figure REF (a) shows the variation of $\\beta $ with respect to ring thickness $h$ of different cavity designs, including the sandwich cavity and the two re-entrant cavities.","The coefficients are all near $2\\times 10^{-9}$ K$^{-2}$ , which is the original linear temperature coefficient of ULE material, and the exact values depend on the detailed structure parameters.","In addition, the deformations of mirror caused by CTE mismatch are also analyzed and are compared between different cavity designs.","The results are shown in Fig.","REF (b).","Mirrors of the “sandwich” design and re-entrant designs all deform in a convex shape at the mirror centers, and the added radius of curvature (negative in these cases) has a magnitude of approximately $10^4$ m to $10^5$ m, which are 4 to 5 orders of magnitude larger than those of the cavity mirrors (0.5 m or 1 m), and thus has negligible contribution.","As can be seen from the figure, “sandwich” structure causes more abrupt distortions within the mirror.","Details of the properties of cavities with different structure designs are listed in Table REF , including their thermal noise limits.","It should be noted here that in the “sandwich” design and re-entrant designs, the introduction of ULE or FS rings increases the thermal noise limit by a very small amount.","This effect is taken into account in the thermal noise limits calculations in Table REF .","Figure: Comparison of properties between different designs of composite cavities: (a) linear temperature coefficient β\\beta of the effective CTE; (b) mirror deformations due to CTE mismatch of sandwich design (upper) and re-entrant designs (below), respectively.Table: Properties of reference cavities with different structure designs.","Relative deformation is defined as the ratio between the original mirror radius of curvature and the extra radius of curvature caused by mirror deformations (both horizontal and vertical directions have similar results)."],["Design of Vibration Insensitive Cavity","The cavities designed with re-entrant FS mirrors and FS rings have an excellent CTE zero crossing temperature tuning property with a very low thermal noise limit.","In order to reach this thermal noise limit, the cavity should be further designed to be vibration insensitive.","A quantitative analysis of the elastic deformation of reference cavities provides valuable guidance for cavity design.","Since the time-dependent vibration perturbation consists of a broad spectrum, in principle a full dynamic analysis of the cavity deformation is required to correctly predict the cavity vibration sensitivity.","This dynamic analysis to our knowledge is not available in the literature.","However, we can greatly simplify the process by performing static analysis.","A static analysis can be used to approximate dynamic analysis because the following two conditions are satisfied.","First, only the vibration modes with frequency lower than 100 Hz are of interest, because the modes with higher frequencies will be significantly damped by either passive or active vibration isolations used in all reference cavities systems.","Second, the considered cavity dimension is much smaller than the cavity vibration characteristic wavelength, meaning that all particles move in-phase inside each eigenmode.","Thus the dynamic response of the cavity can be mimicked by applying a static force with the random acceleration frozen at that moment.","For example, consider ULE spacer material (with $\\rho $ of 2.21 g/cm$^3$ and $E$ of 67.6 GPa), its dispersion relation is [17] $f=\\sqrt{\\frac{E}{\\rho }}\\frac{1}{\\lambda }.$ For oscillation frequency of 100 Hz, the characteristic wavelength $\\lambda $ of 55 m is much greater than the simulated 0.3 m cavity length.","Consequently, at these low frequencies considered here the static analysis is a reliable substitute for the dynamic analysis.","In static analysis the elastic deformation of cavities caused by a gravity-like acceleration are studied.","To understand the principle, for simplicity we assume the cavity is held in a plane instead of several mounting points.","If the cavity is horizontally held on a plane from bottom, compression caused by acceleration normal to the supporting plane will cause expansion in length because of the non-zero Poisson's ratio; on the other hand, if the cavity is hung on the plane from top, the deformation is reversed, the cavity elongates in height and shortens in length.","When the cavity is held on its symmetry plane, the lengthening of the upper part and the shortening of the bottom part compensate each other and the axial length of the cavity remains unchanged under vertical accelerations.","In practice, cavities are held on discrete points instead of the complete plane, which will cause additional bending deformation as well.","The bending effect along with the simple compression or expansion, contribute to the total cavity length variation.","In this case, the optimal supporting plane is no-longer the symmetry plane and the cavity length variation under acceleration is dependent on both the discrete supporting positions and the mounting plane of those discrete points.","The bending of cavity will cause mirror tilting as well.","If the cavity is supported on points locating at the two opposite sides along the horizontal axial length, it will bend downwards under vertical acceleration and the mirrors at both sides tilt correspondingly; if the cavity is supported on the middle of its length, as an extreme, it will bend upwards, and the mirrors tilt in an opposite way.","There must be a critical position along the length of the cavity (Airy points), on which the cavity is supported, so that the mirror tilting is eliminated by the compensation of both bending processes.","To eliminate both cavity length variation and mirrors tilting simultaneously in the case of our cavity design, we apply a notched structure [17], [18], [19], [22] to the cavities as shown in Fig.","REF .","In simulations, we implement FEA analysis by applying a gravity-like force on the cavity vertically with $g=9.8$ m$/\\mathrm {s}^2$ , and the support with $1\\times 1$ mm$^2$ square on the notch surface is constrained to simulate the mounting condition.","Optimal dimensions of the notched structure are searched so that the two kinds of critical mounting positions with zero length variation and zero mirrors tilting of the cavity overlap, while maintaining a less rigorous requirement of mounting accuracy and machining tolerance.","The definitions of parameters about the dimensional and mounting configurations are also illustrated in Fig.","REF .","Figure: (a) Dimensions (C x C_x and C y C_y) of the notched structure, and positions of mounting supports (X p X_p and Z p Z_p) on the notch surface.","(b) FEA simulations of the deformation of a quarter section of the cavity with optimal notch sizes and supporting positions under vertical vibration.During simulations, cavities with different notch sizes ($C_x$ and $C_y$ ) are examined to search for an optimal dimension with which an overlap of the above two critical mounting positions is available.","In most cases, the mounting positions satisfying those two conditions do not overlap, which means only one of the effects can be eliminated.","Analysis results are shown in Fig.","REF in which the mounting positions with zero length variations and zero mirrors tilting of different notch sizes are compared.","Calculations reveal that the mounting positions with zero mirror tilting angles are not sensitive to the changes of the notch sizes, while the zero crossing mounting positions (where the length variations of cavities are immune to vertical vibrations) are much more sensitive to the notch sizes, as shown in Fig.","REF (a) and its inset.","It is also found that at a particular value of $C_y$ (or $C_x$ ) there is a corresponding $C_x$ (or $C_y$ ) that can result in the overlap of these two critical positions.","Another important parameter is the slope of the cavity length variation curve, which reflects the sensitivity of vibration immunity to mounting positions accuracy and machining tolerance.","Different slopes of cavities with different notch sizes are shown in Fig.","REF (b).","It indicates that the sensitivity of the frequency variation as mounting positions changes in the transverse ($X$ ) direction is not sensitive to the notch sizes, neither $C_x$ nor $C_y$ , and has a value of approximately 42 kHz/g/mm.","In contrast, the sensitivity with mounting positions changes in the axial ($Z$ ) direction steadily decreases with decreasing notch sizes of both $C_x$ and $C_y$ .","As a result, smaller notches are much more preferred when optimizing the cavity dimensions.","We choose dimensions of $C_x$ =28 mm and $C_y$ =49 mm and the corresponding optimized vibration sensitivity of the cavity are shown in Fig.","REF .","Figure: (a) Comparison of critical positions, Z p Z_p, for cavity mounting with different notch sizes.","(b) the slopes near the zero crossing positions of curves related to the dependence of resonance frequency deviation on mounting positions under vertical vibrations with different notch sizes.Figure: (a) FEA simulations of the axial length variations at the mirror center and mirror tilting angles at different mounting positions of Z p Z_p.","(b) FEA simulations of the axial length variations at the mirror center and mirror tilting angles at different mounting positions of X p X_p.","The cavity has notch sizes of C x C_x=28 mm and C y C_y=49 mm, under a gravitational acceleration of g=9.8g=9.8 m/s 2 /\\mathrm {s}^2.With the optimized notch dimensions, both the length variation and the mirror tilting have a value of zero at the same mounting position.","As can be seen in Fig.","REF , at the zero crossing point, the frequency variation has a very small slope against mounting position deviation in $Z$ direction, which is 13.1 kHz/g/mm, corresponding to a relative frequency instability of $4.3\\times 10^{-11}$ /g/mm at a laser wavelength of 1064 nm.","The slope of mirror tilting in the axial direction is about $2.3$ nrad/g/mm, corresponding to about $4.5$ kHz/g/mm, thus the relative frequency instability of $1.5\\times 10^{-11}$ /g/mm if the optical axis is displaced from the mechanical axis by $\\pm 1$ mm.","Comparably, the slope of frequency variation in transverse ($X$ ) direction is four times larger than that in $Z$ direction, as shown in Fig.","REF (b), indicating that the cavity length variation is more sensitive and critical to the accuracy of mounting positions in $X$ direction.","The slope of $42.7$ kHz/g/mm corresponds to a relative frequency instability of $1.4\\times 10^{-10}$ /g/mm at 1064 nm.","The sensitivity of mirror tilting to mounting accuracy in $X$ direction is very small ($6.1\\times 10^{-12}$ /g/mm at $\\pm 1$ mm optical axis displacement) and can be neglected."],["Conclusion","In conclusion, we have proposed a new ultra-stable optical reference cavity design with ULE spacer, re-entrant FS mirrors and FS rings based on extensive FEA simulations for different cavity structures.","The designed cavity has an ultra-low thermal noise limit at the levels of $1\\times 10^{-16}$ .","The zero crossing temperature of the effective CTE of the designed cavity can be easily tuned to above room temperature with any ULE spacer materials, as long as the CTE zero crossing temperature of the selected ULE spacer is measured beforehand.","Currently the CTE zero crossing temperature of the ULE spacer can be measured with an accuracy of $\\pm 1$ $^\\circ $ C at reasonable cost (Stable Laser Systems).","This accuracy level is more than enough for our design purpose.","No matter what the zero crossing temperature of the selected ULE spacer is measured to be, one can always design a cavity with compatible FS rings thickness such that the zero crossing temperature of the effective CTE is above room temperature.","The design is applicable for cavities with different lengths.","In addition, we have designed the reference cavity to be vibration insensitive through FEA structure optimization.","Ultra-low thermal noise limit reference cavities play a key role in realizing ultra-narrow linewidth lasers.","Reference cavities based on ULE and FS materials are still very popular choices for lab-based or portable ultra-narrow linewidth lasers.","To further reduce the thermal noise limit to below $1\\times 10^{-16}$ , ULE based reference cavities are no longer adequate.","To this end, there are currently strong interests in the developments of cryogenic cavities based on single crystal silicon or sapphire [36], [37].","We hope the technique discussed in this paper can also stimulate ideas in the design of these cryogenic cavities."],["Acknowledgments","We thank Yan-Yi Jiang and Long-sheng Ma for helpful discussions, and M. Notcutt for technical suggestions.","The project is partially supported by the National Basic Research Program of China (Grant No.2012CB821300), the National Natural Science Foundation of China (Grant Number 61108025 and 11174095), and Program for New Century Excellent Talents by the Ministry of Education."]],"string":"[\n [\n \"Design of an optical reference cavity with low thermal noise and\\n flexible thermal expansion properties\"\n ],\n [\n \"Abstract An ultrastable optical reference cavity with re-entrant fused silica mirrors and a ULE spacer structure is designed through finite element analysis.\",\n \"The designed cavity has a low thermal noise limit of $1\\\\times10^{-16}$ and a flexible zero crossing temperature of the effective coefficient of thermal expansion (CTE).\",\n \"The CTE zero crossing temperature difference between a composite cavity and a pure ULE cavity can be tuned from $-10\\\\ ^{\\\\circ}$C to $23\\\\ ^{\\\\circ}$C, which enables operation of the designed reference cavity near room temperature without worrying about the CTE zero crossing temperature of the ULE spacer.\",\n \"The design can be applied to cavities with different lengths.\",\n \"Vibration immunity of the cavity is also achieved through structure optimization.\"\n ],\n [\n \"Introduction\",\n \"Ultra-stable lasers [1] are essential in high-resolution laser spectroscopy [2], optical frequency standards [3], [4], [5], [6], [7], [8], [9], [10], [11], gravitational wave detection [12], and fundamental tests of physics [13], [14].\",\n \"To achieve such a frequency stabilized local oscillator, the laser is typically servo locked to a high-finesse reference cavity by using Pound-Drever-Hall (PDH) technique [15].\",\n \"Within the servo locking bandwidth, the stability of the locked laser is determined by the stability of the optical path-length of the reference cavity.\",\n \"At low frequencies (below 100 Hz) the most severe fluctuations of the cavity length are caused by environmental disturbances such as seismic and acoustic noise.\",\n \"At higher frequencies these disturbances are usually damped by active vibration isolation.\",\n \"It can be shown that the sensitivity of the reference cavity to the low frequency environmental disturbances can be minimized by optimizing cavity geometry and mounting method [16], [17], [18], [19], [20], [21], [22], [23].\",\n \"While the sensitivity of a reference cavity to mechanical vibrations can be suppressed such that the influence of these perturbations can be ignored, there still exists a fundamental thermal noise limit to the level of stability achievable with the reference cavity [24].\",\n \"This limit is attributed to Brownian motion of the reference cavity spacer, mirror substrates, and optical coatings [24].\",\n \"As have been widely reported before [25], [26], [27], [28], [29], using short length cavities ($\\\\sim 10$ cm) constructed entirely from ULE glass, the relative frequency stability characterized by Allan deviation was limited by thermal noise to the level of $10^{-15}$ .\",\n \"Several methods can be used to lower the thermal noise limit, such as increasing the reference cavity spacer length, increasing the laser spot size, and using high mechanical $Q$ materials.\",\n \"Of these methods, using high $Q$ materials offer the biggest improvements.\",\n \"To lower the thermal noise limit of reference cavities, a commonly used high $Q$ material is fused silica (FS).\",\n \"Because FS has a much larger coefficient of thermal expansion (CTE) (500 ppb/K) than that of ULE ($\\\\pm 50$ ppb/K) [30], it is not appropriate to use FS as cavity spacer material.\",\n \"Since the dominant thermal noise contribution comes from the mirror substrates, it would be more suitable to just use FS as mirror substrates, and keep ULE as cavity spacer material.\",\n \"Replacing the ULE mirrors of a short ULE cavity by FS mirrors can lower the thermal noise limit approximately by a factor of two [28].\",\n \"Combining with a longer length cavity spacer, it is then possible to reach a thermal noise limit of $10^{-16}$ .\",\n \"However, the CTE mismatch between FS and ULE can cause a problem.\",\n \"Temperature changes inside the reference cavity can create stress which bends the mirror substrates, and introduces a larger effective CTE for the composite cavity.\",\n \"This problem can be alleviated by setting the operating temperature of the reference cavity at the zero crossing temperature of the effective CTE.\",\n \"If a ULE spacer has its zero crossing temperature of CTE at room temperature, then the zero crossing temperature of the effective CTE will be much lower.\",\n \"However, this is not very convenient, because cooling a reference cavity is technically more challenging than simply heating the reference cavity.\",\n \"If the temperature is set too low, it can introduce extra problems, like water condensation on cavity housing, etc.\",\n \"For practical purpose, it is preferable to stabilize the reference cavity temperature a little bit above room temperature.\",\n \"Presently there are several approaches to achieve this objective.\",\n \"One is to find a ULE piece that has negative CTE at room temperature, and in combination with positive CTE of FS, it is possible to make the zero crossing temperature of the effective CTE to be above room temperature [30].\",\n \"Another method is to contact ULE rings to the back side of FS mirrors to prevent the mirror bending due to CTE mismatch [31].\",\n \"As a consequence, the zero crossing temperature of the effective CTE can be above room temperature.\",\n \"Both methods are used by Jiang et al., and a fractional frequency instability of $2\\\\times 10^{-16}$ was achieved [32].\",\n \"The above two approaches have their limitations.\",\n \"The first approach relies on the availability of ULE material with negative CTE at room temperature.\",\n \"This requires the good will of the ULE manufacturer, and might not be always available.\",\n \"The second approach in principle can be used to tune the zero crossing temperature over a range of about 30 $^{\\\\circ }$ C, but it only works for a smaller diameter ULE spacer [31].\",\n \"To lower the thermal noise of the reference cavity, it is common to choose a longer cavity length with bigger diameter.\",\n \"The tuning range then is limited to only about $5\\\\sim 6$ $^{\\\\circ }$ C for this geometry according to our calculations.\",\n \"Normal ULE has the zero crossing temperature of CTE varying between 5 $^\\\\circ $ C and 35 $^\\\\circ $ C [30], [33], so with this limited tuning range it is not always possible to raise the zero crossing temperature of the effective CTE to above room temperature.\",\n \"Another approach is to use a re-entrant mirror design for the reference cavity [34].\",\n \"By using a re-entrant mirror design, the direction of FS mirror thermal expansion is opposite to that of ULE spacer, so in principle they can be compensated better, and introduce less stress to the mirrors.\",\n \"In reality the tuning range of the zero crossing temperature difference is about 20 $^\\\\circ $ C $\\\\sim $ 30 $^\\\\circ $ C. Consequently, this cavity design is only suitable for cases when ULE spacers have a zero crossing temperature lower than 5 $^\\\\circ $ C. In this paper, we analyze a modified re-entrant mirror design for the reference cavity, where the ULE rings for the re-entrant FS mirrors are replaced with FS rings.\",\n \"With this design modification, the zero crossing temperature of the effective CTE can be tuned to much greater range, and it is very easy to design a composite cavity so that the zero crossing temperature of the effective CTE is above room temperature.\",\n \"Since a low thermal noise reference cavity design is central to this paper, a detailed discussion of the thermal noise limit of a ULE cavity is presented in Sec.\",\n \"2.\",\n \"Based on our calculation results, a particular geometry of the reference cavity is chosen, and re-entrant FS mirrors are used to replace the traditional ULE mirrors.\",\n \"With this geometry, in Sec.\",\n \"3 we analyze the zero crossing temperature of the composite cavity with finite elements analysis (FEA), and compare the results with that of the other approaches.\",\n \"For completeness, in Sec.\",\n \"4 we also design the cavity to be vibration-insensitive.\",\n \"Finally we concluded in Sec.\",\n \"5.\"\n ],\n [\n \"Calculation of Thermal Noise Limits\",\n \"The stability of a reference cavity is limited by an inevitable mechanical thermal fluctuation due to the Brownian motions of materials.\",\n \"From the fluctuation-dissipation theorem, the most significant parameters in the calculation of the thermal noise spectrum for a given structure are the mechanical quality $Q$ factors of materials.\",\n \"Materials with higher $Q$ factors generate less mechanical dissipation, thus contribute smaller thermal noise.\",\n \"To get an order of magnitude estimation of their contributions, using the formulas given in [24], the power spectral density of mirror displacement due to thermal noise, $G_m(f)$ , which typically dominates the spacer contribution, is given by $G_m(f)=\\\\frac{4k_BT}{2\\\\pi f}\\\\frac{1-\\\\sigma _{sub}^2}{\\\\sqrt{\\\\pi }E_{sub} w_0}\\\\frac{1}{Q_{sub}}(1+\\\\frac{2}{\\\\sqrt{\\\\pi }}\\\\frac{1-2\\\\sigma _{sub}}{1-\\\\sigma _{sub}}\\\\frac{Q_{sub}}{Q_{coating}}\\\\frac{d}{w_0}),$ where $k_B$ is the Boltzmann's constant, $T$ is the temperature, $\\\\sigma _{sub}$ is the Poisson's ratio of mirror substrate, $E_{sub}$ is the Young's modulus of mirror substrate, $w_0$ is the beam radius, $Q_{sub}(Q_{coating})$ is the mechanical quality factor of the mirror substrate (coating), and $d$ is the coating thickness ($\\\\sim 5$ $\\\\mu $ m).\",\n \"In addition, the spacer contribution can be written as $G_{sp}(f)=\\\\frac{4k_BT}{2\\\\pi f}\\\\frac{L}{3\\\\pi R^2E_{sp}}\\\\frac{1}{Q_{sp}},$ where $L$ and $R$ are the spacer length and radius, respectively, $E_{sp}$ and $Q_{sp}$ is the Young's modulus and mechanical quality factor of the spacer, respectively.\",\n \"The total reference cavity displacement power spectral density due to thermal noise is $G_t(f)=2G_m(f)+G_{sp}(f).$ From which we can calculate the Allan deviation of the relative stability of the cavity as $\\\\sigma _y=\\\\frac{\\\\sqrt{2\\\\ln 2f}\\\\cdot \\\\sqrt{G_t(f)}}{L}.$ From Eq.\",\n \"(REF ) and Eq.\",\n \"(REF ), we can see that by choosing materials of higher $Q$ factors, the thermal noise limit can be efficiently lowered.\",\n \"Among the low CTE materials that are commonly used in ultra-stable cavities, ULE has a $Q$ factor of $6.1\\\\times 10^4$ that is more than one order of magnitude larger than that of Zerodur ($3.1\\\\times 10^3$ ).\",\n \"Furthermore, FS has an even larger $Q$ factor of about $10^6$ [35].\",\n \"In cases where thermal noise limits are the ultimate limiting factor of achievable cavity stability, it is very effective to use a higher $Q$ factor material (like FS) as mirror substrate while keeping ULE as spacer material.\",\n \"Figure: Variation of vibration sensitivity of a 30 cm long cavity with respect to spacer diameter.\",\n \"The vibration sensitivity becomes much smoother at diameter larger than 15 cm.\",\n \"Consequently, a spacer diameter of 15 cm is chosen in all following simulationsAccording to Eq.\",\n \"(REF ), using longer cavity can also have the benefit of suppressing the thermal noise.\",\n \"In addition, the total thermal noise limit also depends on the laser beam sizes on the reflecting mirror surfaces.\",\n \"Larger beam radius contributes a lower thermal noise.\",\n \"Taking into account of all these effects, we calculated the thermal noise limits of cavities with different dimensions and configurations, expressed in both Allan deviation and frequency spectral density, and the data are listed in Table REF .\",\n \"By replacing ULE mirrors with FS mirrors and increasing the cavity length from 10 cm to 30 cm, the thermal noise limit can be suppressed by nearly one order of magnitude.\",\n \"Consequently in the following we will consider the design of a long, FS mirror substrate cavity.\",\n \"Notice here that for longer cavity, the spacer radius is also increased.\",\n \"This is due to the fact that longer cavity is more susceptible to vibration influence, and larger cavity can help to suppress this effect.\",\n \"The result is shown in Fig.\",\n \"REF .\",\n \"The detail is explained in Sec.\",\n \"4.\",\n \"Table: Thermal noise limits of optical reference cavities with different configurations and dimensions.\",\n \"LL, RR, and w 01 /w 02 w_{01}/w_{02} are the spacer length, spacer radius and approximate mode spot sizes (intensity 1/e 2 1/e^2 radius) on the plano-confocal cavity mirrors, respectively.\",\n \"σ\\\\sigma is the Allan deviation of cavity stability and S ν (1Hz)\\\\sqrt{S_{\\\\nu }(1Hz)} is the root cavity frequency noise spectral density at 1 Hz for the 1064 nm light.\"\n ],\n [\n \"Compensation of Thermal Distortion\",\n \"By replacing the ULE mirrors with FS mirrors, CTE mismatch between the two materials will cause structure deformation when environmental temperature changes, and thus change the effective CTE of the composite cavity, $\\\\alpha _{eff}$ , and zero crossing temperature of the effective CTE, $T_0(eff)$ .\",\n \"To tune this zero crossing temperature to near room temperature, strategies include attaching ULE rings to FS mirrors, and using re-entrant FS mirrors inside the spacer [31], [34].\",\n \"In this section, both strategies are analyzed with FEA for a 30 cm long reference cavity with a diameter of 15 cm.\",\n \"Based on the analysis, we make modifications to the design and also give the simulation results.\",\n \"In our calculations, we assume the ULE spacer has length of $L$ , and optical contacted FS mirrors have radius of $R$ .\",\n \"Due to the CTE difference, a temperature variation $dT$ results in a difference of the radial expansion between $dR=(\\\\alpha _{FS}-\\\\alpha _{ULE})RdT$ .\",\n \"This results in an axial expansion $dA$ of the mirrors if we assume the optical contact to be perfectly rigid.\",\n \"Assuming a linear stress-strain relation, the radial expansion $dR$ and the axial expansion $dA$ are related by a constant $\\\\delta $ to be $dA=\\\\delta dR$ .\",\n \"The effective cavity CTE is then $\\\\alpha _{eff}&=&\\\\frac{\\\\alpha _{ULE}LdT+2dA}{LdT}\\\\nonumber \\\\\\\\&=&\\\\alpha _{ULE}+2\\\\delta \\\\frac{R}{L}(\\\\alpha _{FS}-\\\\alpha _{ULE}).$ We are primarily interested in the difference of zero crossing temperatures, $\\\\Delta T=T_0(eff)-T_0(ULE),$ between the effective CTE of cavity and the original CTE of ULE material.\",\n \"We aim to find a structure design which can tune the effective zero crossing temperature of the cavity CTE, $T_0(eff)$ , to be slightly higher than room temperature (25 $^\\\\circ $ C $\\\\sim $ 28 $^\\\\circ $ C) for better thermal control of the system.\",\n \"Since normal ULE material has zero crossing temperature of CTE, $T_0(ULE)$ , in the range between 5 $^\\\\circ $ C and 35 $^\\\\circ $ C [30], [33], this means that the tuning ability of an optimal design should at least achieve $-10\\\\ ^\\\\circ \\\\mathrm {C}\\\\le \\\\Delta T\\\\le 23\\\\ ^\\\\circ \\\\mathrm {C}$ .\",\n \"Following Ref.\",\n \"[31], we add ULE rings to the back side of FS mirrors to form a “sandwich” structure and analyze the thermal properties of the combined cavity using both Comsol multiphysics and ANSYS FEA packages.\",\n \"Both analyses give consistent results.\",\n \"Numerical analysis indicates that the temperature difference $\\\\Delta T$ to be only slightly higher than zero for a 30 cm long cavity, even if many efforts have been paid to optimize the cavity structures, including the introduction of a tapered structure to reduce the areas of the ends of spacer and the adoption of larger mirrors (1.5 inch diameter).\",\n \"Cavity geometries and FEA analysis results are shown in Fig.\",\n \"REF and Fig.\",\n \"REF , respectively.\",\n \"As shown in Fig.\",\n \"REF , for cavities with simple cylindrical structures, it is impossible to tune the temperature difference $\\\\Delta T$ to be greater than zero.\",\n \"For tapered cavities, an upper limit of 3 $^\\\\circ $ C of temperature difference in the case of 1.5 inch mirrors structure can be achieved.\",\n \"Although this might be good enough for some cases when the ULE spacer has a $T_0(ULE)$ near or higher than room temperature, it is not flexible enough because you are not normally able to choose the very batch of material with the characteristics you want.\",\n \"It should be noticed that although smaller inner diameter of the ULE ring $d$ can further increase the range of $\\\\Delta T$ , it has a practical limit and should not be too small, otherwise the entrance of the laser beam will be affected, and the scattered light caused by laser diffraction at small hole will introduce unwanted noise.\",\n \"Figure: Geometries of sandwich cavities and FEA simulations of the deformations caused by thermal expansion mismatch when temperature is increased by 1 ∘ ^\\\\circ C. (a) cavity with a cylindrical spacer.\",\n \"(b) cavity with a tapered spacer.\",\n \"Color scale shows the axial displacement.\",\n \"dd and hh are the inner diameter and thickness of the ULE rings, respectively.\",\n \"The notch structure is designed for vibration suppression purpose, and will be explained in Sec.\",\n \"4.Figure: FEA simulations of the CTE zero crossing temperature tuning capabilities of sandwich cavities with different dimensions and configurations, including cylindrical and tapered structures (upper panel) as well as designs of 1.5 inch mirror structures (lower panel).\",\n \"dd and hh are the inner diameter and thickness of the ULE rings, respectively, other parameters of D ' D^{\\\\prime } and L ' L^{\\\\prime } are defined in Fig.1.\",\n \"Right vertical axis shows the coupling coefficient δ\\\\delta as defined in Ref.\",\n \"that directly connects the axial mirror displacement with the radial expansion between mirror and spacer.Cavities with FS mirrors re-entrant within the bores of ULE spacers have the advantage of compensation of opposite thermal expansion between spacers and mirrors inside.\",\n \"Geometry of the cavity is illustrated in Fig.\",\n \"REF .\",\n \"In this structure, FS mirrors are optically contacted to ULE rings which are in turn optically contacted to the ends of ULE spacer.\",\n \"Simulations show that temperature difference between $T_0(eff)$ and $T_0(ULE)$ can be as large as 30 $^\\\\circ $ C, and the results are shown in Fig.\",\n \"REF .\",\n \"Again, the inner diameter of the ring $d_1$ cannot be too small to affect the entrance of laser beam.\",\n \"In contrast with that of the “sandwich” structure, it is difficult to tune the $T_0(eff)$ lower.\",\n \"The tuning range of the zero crossing temperature difference, $\\\\Delta T$ , is about 20 $^\\\\circ $ C $\\\\sim $ 30 $^\\\\circ $ C. Consequently, this cavity design is only suitable for cases when ULE materials have a $T_0(ULE)$ lower than 5 $^\\\\circ $ C. Figure: Geometry and dimensions of cavity with re-entrant FS mirrors optically contacted to ULE rings.\",\n \"The ULE ring has an inner diameter of d 1 d_1, an outer diameter of d 2 d_2, and hh is the thickness of rings.Figure: FEA simulations of the CTE zero crossing temperature tuning range of re-entrant cavities with FS mirrors optically contacted to outer ULE rings with different structure parameters.\",\n \"d 1 d_1 and d 2 d_2 are the inner and outer diameters of ULE rings, respectively, and hh is the thickness of rings.\",\n \"Definitions of these dimensional parameters are shown in Fig.\",\n \".One of the reasons for such a high temperature difference is due to huge difference of CTE between FS and ULE.\",\n \"FS has a CTE of about $5.2\\\\times 10^{-7}$ K$^{-1}$ at room temperature while ULE has CTE of less than $10^{-8}$ K$^{-1}$ (depending on temperature $T$ ) [30], [33].\",\n \"However, the more important reason is the deformation of mirrors caused by CTE mismatch.\",\n \"This deformation makes the FS mirrors bulge in the same direction of its thermal expansion direction, as shown in Fig.\",\n \"REF (a), leading to a result that only long cavities are suitable for this design.\",\n \"This process can be understood theoretically using Eq.\",\n \"REF .\",\n \"For the re-entrant cavity structure, the effective CTE of the cavity can be expressed as: $\\\\alpha _{eff}&=&\\\\frac{\\\\alpha _{ULE}L-2\\\\alpha _{FS}h_m}{L-2h_m}-2\\\\delta \\\\frac{R}{L}(\\\\alpha _{FS}-\\\\alpha _{ULE})\\\\nonumber \\\\\\\\&\\\\approx &\\\\alpha _{ULE}-2\\\\alpha _{FS}\\\\frac{h_m}{L}-2\\\\delta \\\\frac{R}{L}(\\\\alpha _{FS}-\\\\alpha _{ULE}),$ where $h_m$ is the thickness of the FS mirror substrates, and is assumed to be $6.2$ mm in our calculations.\",\n \"Here we have used the assumption that $L >> h_m$ .\",\n \"In first order approximation we can write the CTE of ULE around its zero crossing temperature $T_0(ULE)$ to be $\\\\alpha _{ULE}(T)=\\\\beta (T-T_0(ULE)),$ where $\\\\beta $ is a linear temperature coefficient and is taken to be $2\\\\times 10^{-9}$ K$^{-2}$ [31].\",\n \"Thus the effective CTE can be expressed as: $\\\\alpha _{eff}&\\\\approx &\\\\beta [T-T_0(ULE)-\\\\frac{2\\\\alpha _{FS}h_m+2\\\\delta R(\\\\alpha _{FS}-\\\\overline{\\\\alpha _{ULE}})}{\\\\beta L}]\\\\nonumber \\\\\\\\&\\\\equiv &\\\\beta [T-T_0(eff)].$ Here we have used the fact that $2\\\\delta R << L$ to replace the $\\\\alpha _{ULE}$ in the numerator with temperature independent value of $\\\\overline{\\\\alpha _{ULE}}$ .\",\n \"The effective zero crossing temperature of CTE of the cavity can then be expressed as $T_0(eff)=T_0(ULE)+\\\\frac{2\\\\alpha _{FS}h_m+2\\\\delta R(\\\\alpha _{FS}-\\\\overline{\\\\alpha _{ULE}})}{\\\\beta L}.$ According to the above equations, to make sure the temperature difference of $\\\\Delta T=T_0(eff)-T_0(ULE)$ to be smaller than 23 $^\\\\circ $ C, the spacer length $L$ must be at least 14 cm even without considering the effect of deformation coefficient $\\\\delta $ .\",\n \"If $\\\\delta $ is positive, then the mirrors bulge in the same direction of its thermal expansion direction, as shown in Fig.\",\n \"REF (a), the required cavity length will be even longer, typically longer than 30 cm.\",\n \"Currently spacer length longer than 30 cm is very difficult to obtain, and has higher vibration sensitivity.\",\n \"To obtain a more flexible tuning range of $\\\\Delta T$ with a moderate cavity length, the coefficient $\\\\delta $ in Eq.\",\n \"(REF ) needs to be negative to compensate the huge difference between the CTE of FS and ULE.\",\n \"Therefore, a modified design of re-entrant cavity is applied, in which the FS mirrors are optically contacted to FS rings instead of ULE rings.\",\n \"In this configuration, the deformation of CTE mismatch is between the FS rings and ULE spacer, causing the FS rings bulge in the opposite direction of the mirror thermal expansion direction, as shown in Fig.\",\n \"REF (b).\",\n \"In another word, this cavity structure has a negative $\\\\delta $ .\",\n \"Simulations of cavities with this new structure are shown in Fig.\",\n \"REF .\",\n \"Figure: FEA simulations of the deformations caused by CTE mismatch of re-entrant cavities; (a) deformation between FS mirror and ULE ring causing the mirror bulges in the same direction as its thermal expansion direction; (b) deformation between FS ring and ULE spacer causing the ring bulges in the opposite direction of the mirror thermal expansion direction.\",\n \"Color scale shows the axial displacement.Figure: Tuning range of CTE zero crossing temperature of re-entrant cavities with FS mirror optically contacted to outer FS rings instead of ULE rings.\",\n \"The definitions of dimensional parameters are the same as those in Fig.\",\n \".According to the simulations, the modified re-entrant cavity yields a larger tuning range of $\\\\Delta T$ from -7 $^\\\\circ $ C to 25 $^\\\\circ $ C simply by varying the thickness $h$ of FS rings, and this range can be adjusted further by altering the outer diameter $d_2$ according to the characteristics of ULE batch used and the set-point temperature of the thermal control system.\",\n \"With this design, the flexibility requirement of $-10\\\\ ^\\\\circ \\\\mathrm {C}\\\\le \\\\Delta T\\\\le 23\\\\ ^\\\\circ \\\\mathrm {C}$ is easily achieved.\",\n \"Through comparison, cavities with FS rings of relatively smaller outer diameter $d_2$ are more suitable because of the less area of optical contact as well as the compatibility with vibration immunity design which will be discussed in Sec.\",\n \"4.\",\n \"As a result, in the following simulations and analysis, an outer diameter of 80 mm of FS ring is adopted.\",\n \"Figure REF shows the zero crossing temperature tuning capabilities with different cavity lengths between the original and the modified re-entrant cavity.\",\n \"The zero crossing temperature tuning ranges are much narrower for the original design (only about 5 $^\\\\circ $ C) than those of the new design (more than 30 $^\\\\circ $ C).\",\n \"Furthermore, to obtain a moderate temperature tuning (less than 25 $^\\\\circ $ C), the cavity length must be longer than 30 cm with the original design.\",\n \"In comparison, for the new design, variation of ULE spacer lengths from 10 cm to any longer length can achieve a moderate temperature tuning range.\",\n \"It is interesting to note that all curves except $L=10$ cm in Fig.\",\n \"REF (b) intersect and cross zero at the same position where the ring thickness $h$ is 11.2 mm.\",\n \"Referring to Eq.\",\n \"(REF ), this intersection with zero reveals the fact that when the ring thickness is at this particular value ($h$ =11.2 mm) the thermal expansions of FS mirrors are exactly compensated by the deformation strain of FS rings no matter how long the cavity is.\",\n \"In this case, the effective CTE of the cavity is the same as that of the ULE spacer material.\",\n \"For $L=10$ cm case, the requirement $2\\\\delta R << L$ is not valid, which causes the zero crossing point to be different with that of longer cavities.\",\n \"Figure: Comparison of the zero crossing temperature tuning capabilities for different cavity lengths between the original re-entrant cavity (a) and the modified cavity with FS rings (b).\",\n \"The inner diameters of rings d 1 d_1 are 15 mm for both cases.In this work, the linear temperature coefficient of the effective CTE and mirror deformations of the modified cavity are also studied.\",\n \"The results are shown in Fig.\",\n \"REF and listed in Table REF .\",\n \"According to Eq.\",\n \"(REF ), the linear temperature coefficient $\\\\beta $ of the effective CTE also depends on cavity structure.\",\n \"Figure REF (a) shows the variation of $\\\\beta $ with respect to ring thickness $h$ of different cavity designs, including the sandwich cavity and the two re-entrant cavities.\",\n \"The coefficients are all near $2\\\\times 10^{-9}$ K$^{-2}$ , which is the original linear temperature coefficient of ULE material, and the exact values depend on the detailed structure parameters.\",\n \"In addition, the deformations of mirror caused by CTE mismatch are also analyzed and are compared between different cavity designs.\",\n \"The results are shown in Fig.\",\n \"REF (b).\",\n \"Mirrors of the “sandwich” design and re-entrant designs all deform in a convex shape at the mirror centers, and the added radius of curvature (negative in these cases) has a magnitude of approximately $10^4$ m to $10^5$ m, which are 4 to 5 orders of magnitude larger than those of the cavity mirrors (0.5 m or 1 m), and thus has negligible contribution.\",\n \"As can be seen from the figure, “sandwich” structure causes more abrupt distortions within the mirror.\",\n \"Details of the properties of cavities with different structure designs are listed in Table REF , including their thermal noise limits.\",\n \"It should be noted here that in the “sandwich” design and re-entrant designs, the introduction of ULE or FS rings increases the thermal noise limit by a very small amount.\",\n \"This effect is taken into account in the thermal noise limits calculations in Table REF .\",\n \"Figure: Comparison of properties between different designs of composite cavities: (a) linear temperature coefficient β\\\\beta of the effective CTE; (b) mirror deformations due to CTE mismatch of sandwich design (upper) and re-entrant designs (below), respectively.Table: Properties of reference cavities with different structure designs.\",\n \"Relative deformation is defined as the ratio between the original mirror radius of curvature and the extra radius of curvature caused by mirror deformations (both horizontal and vertical directions have similar results).\"\n ],\n [\n \"Design of Vibration Insensitive Cavity\",\n \"The cavities designed with re-entrant FS mirrors and FS rings have an excellent CTE zero crossing temperature tuning property with a very low thermal noise limit.\",\n \"In order to reach this thermal noise limit, the cavity should be further designed to be vibration insensitive.\",\n \"A quantitative analysis of the elastic deformation of reference cavities provides valuable guidance for cavity design.\",\n \"Since the time-dependent vibration perturbation consists of a broad spectrum, in principle a full dynamic analysis of the cavity deformation is required to correctly predict the cavity vibration sensitivity.\",\n \"This dynamic analysis to our knowledge is not available in the literature.\",\n \"However, we can greatly simplify the process by performing static analysis.\",\n \"A static analysis can be used to approximate dynamic analysis because the following two conditions are satisfied.\",\n \"First, only the vibration modes with frequency lower than 100 Hz are of interest, because the modes with higher frequencies will be significantly damped by either passive or active vibration isolations used in all reference cavities systems.\",\n \"Second, the considered cavity dimension is much smaller than the cavity vibration characteristic wavelength, meaning that all particles move in-phase inside each eigenmode.\",\n \"Thus the dynamic response of the cavity can be mimicked by applying a static force with the random acceleration frozen at that moment.\",\n \"For example, consider ULE spacer material (with $\\\\rho $ of 2.21 g/cm$^3$ and $E$ of 67.6 GPa), its dispersion relation is [17] $f=\\\\sqrt{\\\\frac{E}{\\\\rho }}\\\\frac{1}{\\\\lambda }.$ For oscillation frequency of 100 Hz, the characteristic wavelength $\\\\lambda $ of 55 m is much greater than the simulated 0.3 m cavity length.\",\n \"Consequently, at these low frequencies considered here the static analysis is a reliable substitute for the dynamic analysis.\",\n \"In static analysis the elastic deformation of cavities caused by a gravity-like acceleration are studied.\",\n \"To understand the principle, for simplicity we assume the cavity is held in a plane instead of several mounting points.\",\n \"If the cavity is horizontally held on a plane from bottom, compression caused by acceleration normal to the supporting plane will cause expansion in length because of the non-zero Poisson's ratio; on the other hand, if the cavity is hung on the plane from top, the deformation is reversed, the cavity elongates in height and shortens in length.\",\n \"When the cavity is held on its symmetry plane, the lengthening of the upper part and the shortening of the bottom part compensate each other and the axial length of the cavity remains unchanged under vertical accelerations.\",\n \"In practice, cavities are held on discrete points instead of the complete plane, which will cause additional bending deformation as well.\",\n \"The bending effect along with the simple compression or expansion, contribute to the total cavity length variation.\",\n \"In this case, the optimal supporting plane is no-longer the symmetry plane and the cavity length variation under acceleration is dependent on both the discrete supporting positions and the mounting plane of those discrete points.\",\n \"The bending of cavity will cause mirror tilting as well.\",\n \"If the cavity is supported on points locating at the two opposite sides along the horizontal axial length, it will bend downwards under vertical acceleration and the mirrors at both sides tilt correspondingly; if the cavity is supported on the middle of its length, as an extreme, it will bend upwards, and the mirrors tilt in an opposite way.\",\n \"There must be a critical position along the length of the cavity (Airy points), on which the cavity is supported, so that the mirror tilting is eliminated by the compensation of both bending processes.\",\n \"To eliminate both cavity length variation and mirrors tilting simultaneously in the case of our cavity design, we apply a notched structure [17], [18], [19], [22] to the cavities as shown in Fig.\",\n \"REF .\",\n \"In simulations, we implement FEA analysis by applying a gravity-like force on the cavity vertically with $g=9.8$ m$/\\\\mathrm {s}^2$ , and the support with $1\\\\times 1$ mm$^2$ square on the notch surface is constrained to simulate the mounting condition.\",\n \"Optimal dimensions of the notched structure are searched so that the two kinds of critical mounting positions with zero length variation and zero mirrors tilting of the cavity overlap, while maintaining a less rigorous requirement of mounting accuracy and machining tolerance.\",\n \"The definitions of parameters about the dimensional and mounting configurations are also illustrated in Fig.\",\n \"REF .\",\n \"Figure: (a) Dimensions (C x C_x and C y C_y) of the notched structure, and positions of mounting supports (X p X_p and Z p Z_p) on the notch surface.\",\n \"(b) FEA simulations of the deformation of a quarter section of the cavity with optimal notch sizes and supporting positions under vertical vibration.During simulations, cavities with different notch sizes ($C_x$ and $C_y$ ) are examined to search for an optimal dimension with which an overlap of the above two critical mounting positions is available.\",\n \"In most cases, the mounting positions satisfying those two conditions do not overlap, which means only one of the effects can be eliminated.\",\n \"Analysis results are shown in Fig.\",\n \"REF in which the mounting positions with zero length variations and zero mirrors tilting of different notch sizes are compared.\",\n \"Calculations reveal that the mounting positions with zero mirror tilting angles are not sensitive to the changes of the notch sizes, while the zero crossing mounting positions (where the length variations of cavities are immune to vertical vibrations) are much more sensitive to the notch sizes, as shown in Fig.\",\n \"REF (a) and its inset.\",\n \"It is also found that at a particular value of $C_y$ (or $C_x$ ) there is a corresponding $C_x$ (or $C_y$ ) that can result in the overlap of these two critical positions.\",\n \"Another important parameter is the slope of the cavity length variation curve, which reflects the sensitivity of vibration immunity to mounting positions accuracy and machining tolerance.\",\n \"Different slopes of cavities with different notch sizes are shown in Fig.\",\n \"REF (b).\",\n \"It indicates that the sensitivity of the frequency variation as mounting positions changes in the transverse ($X$ ) direction is not sensitive to the notch sizes, neither $C_x$ nor $C_y$ , and has a value of approximately 42 kHz/g/mm.\",\n \"In contrast, the sensitivity with mounting positions changes in the axial ($Z$ ) direction steadily decreases with decreasing notch sizes of both $C_x$ and $C_y$ .\",\n \"As a result, smaller notches are much more preferred when optimizing the cavity dimensions.\",\n \"We choose dimensions of $C_x$ =28 mm and $C_y$ =49 mm and the corresponding optimized vibration sensitivity of the cavity are shown in Fig.\",\n \"REF .\",\n \"Figure: (a) Comparison of critical positions, Z p Z_p, for cavity mounting with different notch sizes.\",\n \"(b) the slopes near the zero crossing positions of curves related to the dependence of resonance frequency deviation on mounting positions under vertical vibrations with different notch sizes.Figure: (a) FEA simulations of the axial length variations at the mirror center and mirror tilting angles at different mounting positions of Z p Z_p.\",\n \"(b) FEA simulations of the axial length variations at the mirror center and mirror tilting angles at different mounting positions of X p X_p.\",\n \"The cavity has notch sizes of C x C_x=28 mm and C y C_y=49 mm, under a gravitational acceleration of g=9.8g=9.8 m/s 2 /\\\\mathrm {s}^2.With the optimized notch dimensions, both the length variation and the mirror tilting have a value of zero at the same mounting position.\",\n \"As can be seen in Fig.\",\n \"REF , at the zero crossing point, the frequency variation has a very small slope against mounting position deviation in $Z$ direction, which is 13.1 kHz/g/mm, corresponding to a relative frequency instability of $4.3\\\\times 10^{-11}$ /g/mm at a laser wavelength of 1064 nm.\",\n \"The slope of mirror tilting in the axial direction is about $2.3$ nrad/g/mm, corresponding to about $4.5$ kHz/g/mm, thus the relative frequency instability of $1.5\\\\times 10^{-11}$ /g/mm if the optical axis is displaced from the mechanical axis by $\\\\pm 1$ mm.\",\n \"Comparably, the slope of frequency variation in transverse ($X$ ) direction is four times larger than that in $Z$ direction, as shown in Fig.\",\n \"REF (b), indicating that the cavity length variation is more sensitive and critical to the accuracy of mounting positions in $X$ direction.\",\n \"The slope of $42.7$ kHz/g/mm corresponds to a relative frequency instability of $1.4\\\\times 10^{-10}$ /g/mm at 1064 nm.\",\n \"The sensitivity of mirror tilting to mounting accuracy in $X$ direction is very small ($6.1\\\\times 10^{-12}$ /g/mm at $\\\\pm 1$ mm optical axis displacement) and can be neglected.\"\n ],\n [\n \"Conclusion\",\n \"In conclusion, we have proposed a new ultra-stable optical reference cavity design with ULE spacer, re-entrant FS mirrors and FS rings based on extensive FEA simulations for different cavity structures.\",\n \"The designed cavity has an ultra-low thermal noise limit at the levels of $1\\\\times 10^{-16}$ .\",\n \"The zero crossing temperature of the effective CTE of the designed cavity can be easily tuned to above room temperature with any ULE spacer materials, as long as the CTE zero crossing temperature of the selected ULE spacer is measured beforehand.\",\n \"Currently the CTE zero crossing temperature of the ULE spacer can be measured with an accuracy of $\\\\pm 1$ $^\\\\circ $ C at reasonable cost (Stable Laser Systems).\",\n \"This accuracy level is more than enough for our design purpose.\",\n \"No matter what the zero crossing temperature of the selected ULE spacer is measured to be, one can always design a cavity with compatible FS rings thickness such that the zero crossing temperature of the effective CTE is above room temperature.\",\n \"The design is applicable for cavities with different lengths.\",\n \"In addition, we have designed the reference cavity to be vibration insensitive through FEA structure optimization.\",\n \"Ultra-low thermal noise limit reference cavities play a key role in realizing ultra-narrow linewidth lasers.\",\n \"Reference cavities based on ULE and FS materials are still very popular choices for lab-based or portable ultra-narrow linewidth lasers.\",\n \"To further reduce the thermal noise limit to below $1\\\\times 10^{-16}$ , ULE based reference cavities are no longer adequate.\",\n \"To this end, there are currently strong interests in the developments of cryogenic cavities based on single crystal silicon or sapphire [36], [37].\",\n \"We hope the technique discussed in this paper can also stimulate ideas in the design of these cryogenic cavities.\"\n ],\n [\n \"Acknowledgments\",\n \"We thank Yan-Yi Jiang and Long-sheng Ma for helpful discussions, and M. Notcutt for technical suggestions.\",\n \"The project is partially supported by the National Basic Research Program of China (Grant No.2012CB821300), the National Natural Science Foundation of China (Grant Number 61108025 and 11174095), and Program for New Century Excellent Talents by the Ministry of Education.\"\n ]\n]"},"id":{"kind":"string","value":"1204.1209"}}},{"rowIdx":1867491,"cells":{"text":{"kind":"list like","value":[["Transverse-Momentum Dependence of the J/psi Nuclear Modification in d+Au\n Collisions at sqrt(s_NN)=200 GeV"],["Abstract We present measured J/psi production rates in d+Au collisions at sqrt(s_NN) = 200 GeV over a broad range of transverse momentum (p_T=0-14 GeV/c) and rapidity (-2.21) for p_T>2 GeV/c.","The observed enhancement at negative rapidity has implications for the observed modification in heavy-ion collisions at high p_T."],["Introduction ","Modifications of quarkonia yields when production takes place in a nuclear target, often termed cold-nuclear-matter (CNM) effects, give insight into the production and evolution of $q\\bar{q}$ pairs.","A number of effects are predicted to occur in the presence of nuclear matter (for a recent review, see [1]).","These include nuclear breakup, modification of the parton-distribution functions, initial-state parton-energy loss and, more recently, coherent gluon saturation.","Measuring the production rate of quarkonia in a nuclear environment over a broad range of collision energies, and as a function of all kinematic variables, is the best way to disentangle these different mechanisms.","The measurement of $J/\\psi $ production rates over a broad range of rapidity ($y$ ) and transverse momentum ($p_T$) samples a wide range of parton momentum fraction ($x$ ) and energy transfer ($Q^{2}$ ), providing a simultaneous constraint on the modification of parton-distribution functions inside nuclei (nPDF's).","The production of $J/\\psi $ mesons, which at RHIC occurs mainly through gluon fusion, can provide critical input on the modification of the gluon distribution, which is probed only indirectly by the deep-inelastic scattering (DIS) data that forms the bulk of the current constraints on the nPDF parametrizations.","Measuring the $p_T$ distribution of $J/\\psi $ production allows access to $p_T$-broadening effects, which are not constrained by measurements of the rapidity dependence alone.","The $p_T$-broadening effects on quarkonia production at high energies are not well constrained by current data.","New data for $J/\\psi $ production over a broad range in $p_T$ is necessary to provide guidance for theoretical calculations.","The CNM effects on $J/\\psi $ production have been studied in fixed-target $p+A$ experiments at SPS, FNAL, and HERA [2], [3], [4], [5], [6], [7], [8] spanning the center of mass energy range $\\mbox{$\\sqrt{s}$} \\approx 17-42$ GeV.","The fixed-target results at midrapidity show greater suppression of $J/\\psi $ production at lower collision energy [6].","This has been interpreted [9] as an increase of the nuclear breakup of the $J/\\psi $ through collisions with nuclei.","At lower collision energy the crossing time of the nuclei is long enough for the $J/\\psi $ to fully form.","The fully formed $J/\\psi $ has an increased probability of interacting with other nucleons in the collision, which can cause the breakup of the $J/\\psi $ into heavy-meson pairs.","At higher collision energies it is likely that the time required for the $J/\\psi $ to fully evolve is as long, or longer than the crossing time of the collision.","This may result in a decrease in the probability of collisions with other nucleons, leading to less suppression of the $J/\\psi $ production.","The E866 [7] and HERA-B [8] experiments have measured $J/\\psi $ production as a function of $p_T$ in fixed target $p+A$ experiments.","Results are presented in terms of the nuclear-suppression factor, $\\alpha $ , which is obtained assuming that the cross section for $p+A$ collisions scales as $\\sigma _{pA}=\\sigma _{pN}\\times A^{\\alpha }$ , where $\\sigma _{pN}$ is the proton-nucleon cross section and $A$ is the mass number.","They find a $p_T$ dependence of $\\alpha $ , which is similar across a range of Feynman-$x$ ($x_F$ ) and $p_T$.","At $\\mbox{$p_T$} <2$ GeV/$c~$they find a suppression in the $J/\\psi $ production that transitions to an excess in the $J/\\psi $ production at higher $p_T$, which is characteristic of multiple scattering of the incident parton [7].","It is crucial to test these conclusions at the higher energies provided by $d+$ Au collisions at RHIC in order to better understand the $J/\\psi $ production mechanisms.","Measuring, and understanding, the CNM effects on quarkonia production is critical to interpreting the results for $J/\\psi $ production in nucleus-nucleus ($A+A$ ) collisions.","In 1986 Matsui and Satz predicted that the suppression of $J/\\psi $ production in heavy-ion collisions would be a clear signature of the formation of a quark-gluon plasma [10].","The Debye color screening of the dense medium produced is expected to cause the dissociation of bound states, thereby causing a decrease in the observed production.","Since then suppression of quarkonia production has been observed for a number of states, including the $J/\\psi $ and $\\Upsilon $ , over a wide range in collision energy [11], [12], [13], [14].","However, the interpretation of these results is still unclear.","Before the modification due to the produced medium can be determined, the CNM effects must first be corrected.","This has been done at lower energies [11], but accurate data on CNM effects are still absent at the higher energies of RHIC and the LHC.","Here we report new high-precision measurements of the $J/\\psi $ production as a function of $p_T$ and collision centrality in $d+$ Au collisions at $\\sqrt{s_{_{NN}}}$ = 200 GeV.","We also present measurements of the $J/\\psi $ $R_{d{\\rm Au}}$ as a function of $p_T$, rapidity, and collision centrality using data for $J/\\psi $ production in $p$ +$p$ collisions published in [15].","PHENIX has previously measured the $J/\\psi $ yield in $d+$ Au collisions [16], [17] with data recorded in 2003.","The data presented here, recorded in 2008, feature an increase in statistics of 30–50 times over those used in the previously published results, as well as a significant reduction of the systematic uncertainties.","The rapidity dependence of $J/\\psi $ production in $d+$ Au collisions from this data set has been previously published in [18].","This paper presents results for the $p_T$ dependence of the $J/\\psi $ yield from the same data set."],["Experimental Apparatus & Data Sets ","The PHENIX detector [19] comprises three separate spectrometers in three pseudorapidity ($\\eta $ ) ranges.","Two central arms at midrapidity cover $\\left|\\eta \\right|<0.35$ and have an azimuthal coverage ($\\phi $ ) of $\\pi /2$ rad each, while muon arms at backward/forward rapidity cover $-2.2<\\eta <-1.2$ (Au going direction) and $1.2<\\eta <2.4$ ($d$ going direction), with full azimuthal coverage.","In the central arms the $J/\\psi $ yield is measured via dielectron decays.","Charged particle tracks are reconstructed using the drift chamber and pad chambers.","Electron candidates are selected by matching charged tracks to hits in the ring imaging Čerenkov (RICH) counters and clusters in the Electromagnetic Calorimeters (EMCal).","In $d+$ Au collisions, a charged track is identified as an electron by requiring at least two matching RICH phototube hits within a radius of $3.4$) is determined using a Glauber calculation [21] combined with a simulation of the BBC response (as described in [18]).","The resulting $\\left< N_{\\rm coll}\\right>$ values for the centrality categories used in this analysis are shown in Table REF .","The $N_{\\rm coll}$ distributions within each centrality bin are shown in Fig.","REF .","There is a significant overlap between the $N_{\\rm coll}$ distributions for different centralities.","Also shown in Table REF is the correction factor $c$ , which accounts for the correlation between the detection of a $J/\\psi $ in the final state and an increase in the total charge collected in the BBC [16].","This correlation affects both the MB-trigger efficiency and the determination of the centrality of a given collision.","The correction factors for each centrality bin are obtained within the same Glauber framework as the $\\left< N_{\\rm coll}\\right>$ values by assuming that one of the N binary collisions produces a charge in the BBC that is characteristic of a hard-scattering process (the remaining N-1 binary collisions maintain a BBC charge distribution characteristic of soft scattering processes).","The increase in the BBC charge from a hard process is tuned using real data.","Since both $c$ and $\\left< N_{\\rm coll}\\right>$ are calculated in the same Glauber framework there are correlations between their uncertainties.","These correlations are removed in the ratio of $c/\\mbox{$\\left< N_{\\rm coll}\\right>$} $ , which occurs in the calculation of $R_{d{\\rm Au}}$.","The resulting values and uncertainties are given in the third column of Table REF .","The correction factor for 0–100% centrality contains an additional factor to extrapolate the measured yield, which covers only 88% of all $d+$ Au collisions, to 100% of the $d+$ Au inelastic cross section, essentially correcting for the efficiency of the BBC trigger.","This correction is again determined within the Glauber framework using the parametrization of the BBC trigger efficiency."],["The procedure for analyzing the $\\mbox{$J/\\psi $} \\rightarrow e^+e^-$ signal and the results in the central arms are discussed in this section.","The extraction of the correlated $e^+e^-$ yield is discussed in Sec.","REF .","The estimation of the correlated background and losses due to the radiative tail in the $J/\\psi $ mass distribution is discussed in Sec.","REF .","The estimation of the detector efficiencies is described in Sec.","REF .","The calculation of the $J/\\psi $ invariant yield is detailed in Sec.","REF .","The $p$ +$p$ baseline used in calculating $R_{d{\\rm Au}}$ is described in Sec.","REF ."],["Correlated $e^+e^-$ Signal extraction ","The $\\mbox{$J/\\psi $} \\rightarrow e^+e^-$ yield is measured using the invariant mass spectrum for all dielectron pairs where at least one of the electrons fired the ERT trigger.","This selection is necessary to match the conditions under which the $J/\\psi $ trigger efficiency is calculated (see Sec.","REF ).","An example of the dielectron mass spectrum is shown in Fig.","REF for 0–20% central collisions.","In a given bin of $p_T$, rapidity, and collision centrality, the correlated $e^+e^-$ yield ($N_{e^+e^-}$ ) is determined by counting over a fixed mass window of $2.8 < M_{ee}\\,[\\mbox{GeV/$c^{2}~$}] < 3.3$ the number of unlike-sign dielectrons, after the subtraction of the like-sign dielectrons, which arise by random association and so are representative of the combinatorial background within the unlike sign dielectron distribution.","This method assumes that the acceptance is the same for $e^-$ and $e^+$ , which, while untrue at lower masses, is a good assumption in the $J/\\psi $ mass range.","At higher $p_T$, where statistical precision is limited, the yield, along with the statistical uncertainties, are derived from Poisson statistics.","Assuming both the unlike-sign (foreground) and like-sign (background) are independent, and assuming no negative signal, the combined distribution $ P(s) =\\sum _{k=0}^{fg}\\frac{(bg+fg-k)!}{bg!(fg-k)!}\\frac{1}{2}\\left(\\frac{1}{2}\\right)^{bg+fg-k}\\frac{s^ke^{-s}}{k!","}, $ represents the probability of a signal ($s$ ) given a number of unlike-sign dielectrons ($fg$ ) and a number of like-sign dielectrons ($bg$ ) (see [15] for derivation).","The mean and standard deviation of Eq.","REF are then used as the yield and uncertainties.","A correlated $e^{+}e^{-}$ yield in the mass window $2.8 < M_{ee}\\,[\\mbox{GeV/$c^{2}~$}] < 3.3$ of approximately 8600 is obtained across all $p_T$ and collision centralities.","Figure: (Color online)(top) Invariant mass distribution of unlike-sign (filled circles) andlike-sign (filled boxes) dielectron pairs in central d+d+Au collisions,integrated over p T p_T and rapidity.","Dashed vertical lines represent themass range used to determine the correlated e + e - e^+e^- yield.","(bottom)Correlated dielectron invariant mass distribution for MB d+d+Au collisions.","The line shapes are those used to extract the continuum andradiative tail contributions to the correlated e + e - e^+e^- yield in the massrange 2.8$}}\\frac{d^2N_{J/\\psi }^{d+\\rm Au}(i)/dyd\\mbox{$p_T$}}{d^2N_{J/\\psi }^{p+p}/dyd\\mbox{$p_T$}},$ where $d^2N_{J/\\psi }^{d+\\rm Au}(i)/dyd\\mbox{$p_T$} $ is the $d+$ Au invariant yield for the $i^{\\rm th}$ centrality bin, $d^2N_{J/\\psi }^{p+p}/dyd\\mbox{$p_T$} $ is the $p$ +$p$ invariant yield for the same $p_T$ and $y$ bin, and $\\left< N_{\\rm coll}(i)\\right>$ is the average number of binary collisions for the given centrality bin, as listed in Table REF .","The 0–100% centrality integrated $J/\\psi $ $R_{d{\\rm Au}}$ as a function of $p_T$ is shown in Fig.","REF for each of the three rapidity regions.","The numerical values can be found in Table REF , REF , and REF for backward, mid and forward rapidity, respectively.","Figure REF shows a different behavior for $R_{d{\\rm Au}}$ at backward ($-2.24$ GeV/$c~$by fitting each distribution with a constant.","The results are shown in Table REF along with the fit uncertainties, which take into account only the type A uncertainties on the data.","Since the type B uncertainties are roughly consistent in the fit range, we have chosen here to add the average type B uncertainty for $\\mbox{$p_T$} >4$ GeV/$c~$in quadrature with the type C uncertainty.","We find that at mid and forward rapidity the average $R_{d{\\rm Au}}$ for $\\mbox{$p_T$} >4$ GeV/$c~$is consistent with 1.0, while at backward rapidity the average $R_{d{\\rm Au}}$ is greater than 1.0.","The production of a $J/\\psi $ at forward rapidity in $A+A$ collisions involves a low-$x$ gluon colliding with a high-$x$ gluon.","The symmetry due to identical colliding nuclei results, essentially, in the folding of the forward and backward rapidity $R_{d{\\rm Au}}$.","The production of a $J/\\psi $ at midrapidity results, essentially, in the folding of the midrapidity $R_{d{\\rm Au}}$ with itself.","This picture is simplistic and leaves out many details, but it gives some expectation for the result of the modification of $J/\\psi $ production in $A+A$ collisions due to CNM effects.","If we extrapolate the observed behavior of $R_{d{\\rm Au}}$ to the modification of $J/\\psi $'s produced at forward rapidity in $A+A$ collisions, we would expect a $R_{\\rm AA}$ contribution from CNM effects to be similar to, or greater than, 1.0 at high $p_T$ and a modification similar to 1.0 at midrapidity.","The observation at midrapidity of a $J/\\psi $ $R_{\\rm AA}$ in Cu+Cu collisions that exceeds, but is consistent with, 1.0 at high $p_T$ [26] may therefore be largely accounted for by the contribution from CNM effects.","Further work is needed to understand the detailed propagation of measured results in $d+$ Au collisions to an expected CNM contribution in $A+A$ collisions before this can be fully understood.","Table: The average 0–100% R d Au R_{d{\\rm Au}} for p T >4\\mbox{$p_T$} >4 GeV/cc~where the first quoteduncertainty corresponds to the fit uncertainty and the second correspondsto the combined type B and C systematic uncertainties.Figures REF , REF , and REF show $R_{d{\\rm Au}}$ vs $p_T$ in four centrality bins for backward rapidity, midrapidity, and forward rapidity, respectively.","Numerical values can be found in Tables REF , REF , REF and REF for 0–20%, 20–40%, 40–60% and 60–88% central collisions respectively.","For peripheral collisions the $R_{d{\\rm Au}}$ remains consistent with 1.0 within statistical and systematic uncertainties across all $p_T$ in all rapidity regions.","Figure: (Color Online)J/ψ→μ + μ - J/\\psi \\rightarrow \\mu ^+\\mu ^- R d Au R_{d{\\rm Au}}, as afunction of p T p_T for a) central, b) midcentral, c)midperipheral, and d) peripheral events at -2.20.25$ ), while for lower values, power suppressed coherent final-state scattering leads to a modification of parton $x$ [41].","Initial state energy loss, which accounts for the radiative energy loss of the incoming particles through multiple interactions with the target nucleus is included.","This effect reduces the energy of the incoming parton, so, to achieve the same final-state kinematics the parton must have a greater momentum, and therefore a larger value of $x$ .","This effectively shifts the portion of the gluon distribution sampled to higher $x$ .","Also included is a calculation of the Cronin effect.","The solid curve in Fig.","REF shows the full calculation including the Cronin effect.","The dashed curve in Figure REF is the same calculation without the Cronin effect.","This comparison gives a direct indication of the contribution from the Cronin effect, which is evidently over predicted when compared to the data.","The results presented here will hopefully provide a much needed constraint on the Cronin effect at RHIC energies.","The calculation including the Cronin effect indicate an $R_{d{\\rm Au}}$ that decreases at higher $p_T$.","This is consistent with data, however the current statistical and systematic uncertainties make determining the precise trend of $R_{d{\\rm Au}}$ difficult at high $p_T$.","Better data with a larger $p_T$ coverage is needed to determine the $J/\\psi $ modification at higher $p_T$."],["Summary & Conclusions ","We have measured the $J/\\psi $ invariant yield and $R_{d{\\rm Au}}$ , as a function of $p_T$ over three rapidity ranges in $d+$ Au collisions at $\\mbox{$\\sqrt{s_{_{NN}}}$} =200$ GeV using the PHENIX detector.","These measurements provide a large improvement in statistical precision and $p_T$ reach over the previously published PHENIX $d+$ Au results [16], [17], and are the first measurements of the centrality dependence of the $J/\\psi $ $p_T$ distribution in $d+$ Au collisions by PHENIX.","The $\\Delta \\mbox{$\\langle p_T^2 \\rangle $} $ values determined from the data show a marked increase with $N_{\\rm coll}$ that is similar at all rapidities.","The $R_{d{\\rm Au}}$ vs $p_T$ displays similar behavior at mid and forward rapidity, showing suppression at low $p_T$ with a gradual increase to a value consistent with 1.0.","The $R_{d{\\rm Au}}$ at backward rapidity has a different distribution with $p_T$, showing a more rapid increase from suppression to a value of 1.0, and transitioning to $\\mbox{$R_{d{\\rm Au}}$} >1.0$ above 2 GeV/$c~$.","These trends are greater for central collisions, while the peripheral collisions show $R_{d{\\rm Au}}$ consistent with 1.0 across all rapidities.","We find an average $R_{d{\\rm Au}}$ for $\\mbox{$p_T$} >4$ GeV/$c~$of $1.27\\pm 0.06\\pm 0.11$ at backward rapidity, and an $R_{d{\\rm Au}}$ consistent with 1.0 at mid and forward rapidity.","This implies a CNM contribution in $A+A$ collisions that is likely consistent with 1.0 at high $p_T$ across all rapidity.","This could potentially explain the reported increase in $R_{\\rm AA}$ with increasing $p_T$ [26].","However more data and further work to understand the propagation of $R_{d{\\rm Au}}$ to $R_{\\rm AA}$ is needed to confirm this.","A comparison of the measured $R_{d{\\rm Au}}$ with three types of theoretical calculations was shown.","The parameter independent dipole model of $J/\\psi $ production in $p+A$ collisions agrees well with the shape of the data at mid and forward rapidities, while the shape of the predicted $p_T$ dependence is different from the data at backward rapidity.","However the suppression is over-predicted at all rapidities.","The second model uses $2\\rightarrow 2$ $J/\\psi $ production kinematics coupled with shadowing taken from both EKS98 and nDSg nPDF sets as well as an effective absorption cross section of 4.2 mb.","The calculations with both EKS98 and nDSg show good agreement with the data at midrapidity in each centrality bin, as well as the centrality integrated case.","At forward rapidity the shape of the distribution is in reasonable agreement with the data, while the overall level of suppression seems to be greater in the model calculations than the data.","At backward rapidity, the model calculations using both EKS98 and nDSG nPDF sets are in strong disagreement with the data.","At backward rapidity calculations using the nDSg nPDF set show a suppression that is constant with $p_T$, while those using the EKS98 nPDF set predict an increase of suppression with increasing $p_T$.","The data show the opposite trend.","The third model, an NRQCD calculation of high $p_T$ $J/\\psi $ production show a Cronin effect, which although generally consistent with the data, is significantly larger than observed in the data, and a suppression at high $p_T$ that cannot be confirmed due to the large uncertainties at high $p_T$ and the limited $p_T$ reach of the current data.","In summary, the data presented here cover a large range in $x$ and $Q^2$ , providing a further constraint on the modification of the gluon distribution in nuclei, as well as providing constraints on the size of the Cronin effect on $J/\\psi $ production at RHIC."],["ACKNOWLEDGMENTS","We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.","We thank Jean-Philippe Lansberg, Nicolas Matagne, Boris Kopeliovich, Ivan Vitev, and Rishi Sharma for useful discussions and theoretical calculations.","We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Ministry of Industry, Science and Tekhnologies, Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), Hungarian National Science Fund, OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), National Research Foundation and WCU program of the Ministry Education Science and Technology (Korea), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and the Wallenberg Foundation (Sweden), the U.S.","Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the US-Hungarian Fulbright Foundation for Educational Exchange, and the US-Israel Binational Science Foundation."],["Fitting The ","The $J/\\psi $ invariant yields as a function of $p_T$ were fitted with a modified Kaplan function of the form $f(\\mbox{$p_T$})=p_0\\left(1-\\left(\\frac{\\mbox{$p_T$}}{p_1}\\right)^2\\right)^{p_2}.$ The data points were compared to the integral of the function over the $p_T$ bin when calculating the $\\chi ^2$ .","The fit results, along with the ratio of the data to the fit are shown in Fig.","REF for $p$ +$p$ collisions and Figure REF for 0–100% centrality integrated $d+$ Au collisions.","The fit results for each centrality bin are shown in Figs.","REF , REF , and REF for backward, mid, and forward rapidities, respectively."],["Calculating the Correction Factor $k$","To account for the fact that the experimental upper $p_T$ limits on the $J/\\psi $ invariant yield distributions vary with rapidity and centrality, a correction factor was calculated using the fits described in Sec.","REF .","The ratio $k=\\frac{\\mbox{$\\langle p_T^2 \\rangle $} [0,\\infty ]}{\\mbox{$\\langle p_T^2 \\rangle $} [0,\\mbox{$p_T$} ^{\\rm max}]}$ was calculated from the fit and applied to the numerically calculated $\\langle p_T^2 \\rangle |_{p_T\\le p_T^{\\rm max}}$.","The correction factors are shown in Table REF , and are in all cases small ($k<1.03$ ).","The uncertainty on $k$ is derived from the fit uncertainty by varying the data points within their statistical uncertainties, refitting, and thereby finding the variation in $k$ .","Figure: (Color Online)Results of modified Kaplan fits to backward rapidity 0–100% d+d+Au (Left), midrapidity 0-100% d+d+Au (Center), and forward rapidity 0-100%d+d+Au (Right).Figure: (Color Online)Results of modified Kaplan fits to midrapidity d+d+Au collisions for eachcentrality.Figure: (Color Online)Results of modified Kaplan fits to forward rapidity d+d+Au collisions foreach centrality.Table: 〈p T 2 〉\\langle p_T^2 \\rangle correction factors, kk, for pp+pp andd+d+Au collisions."],["Propagating the Type B uncertainties to ","When propagating the Type B systematic uncertainties on the $J/\\psi $ invariant yields to the calculated $\\langle p_T^2 \\rangle $ values, the type B uncertainties are assumed to be normally distributed.","With this assumption we independently sample the uncertainty distribution of the first and the last data point of the $p_T$ distribution.","We then assume the Type B uncertainties are linearly correlated between these two values.","The resulting distribution of the $\\langle p_T^2 \\rangle $ values that arises from this procedure gives an estimate of the effect of the Type B uncertainties on the value of $\\langle p_T^2 \\rangle $.","The Type C systematic uncertainties on the $J/\\psi $ invariant yields do not affect the calculation of $\\langle p_T^2 \\rangle $.","The Type C uncertainties are a global uncertainty, which cancels in the calculation."]],"string":"[\n [\n \"Transverse-Momentum Dependence of the J/psi Nuclear Modification in d+Au\\n Collisions at sqrt(s_NN)=200 GeV\"\n ],\n [\n \"Abstract We present measured J/psi production rates in d+Au collisions at sqrt(s_NN) = 200 GeV over a broad range of transverse momentum (p_T=0-14 GeV/c) and rapidity (-2.21) for p_T>2 GeV/c.\",\n \"The observed enhancement at negative rapidity has implications for the observed modification in heavy-ion collisions at high p_T.\"\n ],\n [\n \"Introduction \",\n \"Modifications of quarkonia yields when production takes place in a nuclear target, often termed cold-nuclear-matter (CNM) effects, give insight into the production and evolution of $q\\\\bar{q}$ pairs.\",\n \"A number of effects are predicted to occur in the presence of nuclear matter (for a recent review, see [1]).\",\n \"These include nuclear breakup, modification of the parton-distribution functions, initial-state parton-energy loss and, more recently, coherent gluon saturation.\",\n \"Measuring the production rate of quarkonia in a nuclear environment over a broad range of collision energies, and as a function of all kinematic variables, is the best way to disentangle these different mechanisms.\",\n \"The measurement of $J/\\\\psi $ production rates over a broad range of rapidity ($y$ ) and transverse momentum ($p_T$) samples a wide range of parton momentum fraction ($x$ ) and energy transfer ($Q^{2}$ ), providing a simultaneous constraint on the modification of parton-distribution functions inside nuclei (nPDF's).\",\n \"The production of $J/\\\\psi $ mesons, which at RHIC occurs mainly through gluon fusion, can provide critical input on the modification of the gluon distribution, which is probed only indirectly by the deep-inelastic scattering (DIS) data that forms the bulk of the current constraints on the nPDF parametrizations.\",\n \"Measuring the $p_T$ distribution of $J/\\\\psi $ production allows access to $p_T$-broadening effects, which are not constrained by measurements of the rapidity dependence alone.\",\n \"The $p_T$-broadening effects on quarkonia production at high energies are not well constrained by current data.\",\n \"New data for $J/\\\\psi $ production over a broad range in $p_T$ is necessary to provide guidance for theoretical calculations.\",\n \"The CNM effects on $J/\\\\psi $ production have been studied in fixed-target $p+A$ experiments at SPS, FNAL, and HERA [2], [3], [4], [5], [6], [7], [8] spanning the center of mass energy range $\\\\mbox{$\\\\sqrt{s}$} \\\\approx 17-42$ GeV.\",\n \"The fixed-target results at midrapidity show greater suppression of $J/\\\\psi $ production at lower collision energy [6].\",\n \"This has been interpreted [9] as an increase of the nuclear breakup of the $J/\\\\psi $ through collisions with nuclei.\",\n \"At lower collision energy the crossing time of the nuclei is long enough for the $J/\\\\psi $ to fully form.\",\n \"The fully formed $J/\\\\psi $ has an increased probability of interacting with other nucleons in the collision, which can cause the breakup of the $J/\\\\psi $ into heavy-meson pairs.\",\n \"At higher collision energies it is likely that the time required for the $J/\\\\psi $ to fully evolve is as long, or longer than the crossing time of the collision.\",\n \"This may result in a decrease in the probability of collisions with other nucleons, leading to less suppression of the $J/\\\\psi $ production.\",\n \"The E866 [7] and HERA-B [8] experiments have measured $J/\\\\psi $ production as a function of $p_T$ in fixed target $p+A$ experiments.\",\n \"Results are presented in terms of the nuclear-suppression factor, $\\\\alpha $ , which is obtained assuming that the cross section for $p+A$ collisions scales as $\\\\sigma _{pA}=\\\\sigma _{pN}\\\\times A^{\\\\alpha }$ , where $\\\\sigma _{pN}$ is the proton-nucleon cross section and $A$ is the mass number.\",\n \"They find a $p_T$ dependence of $\\\\alpha $ , which is similar across a range of Feynman-$x$ ($x_F$ ) and $p_T$.\",\n \"At $\\\\mbox{$p_T$} <2$ GeV/$c~$they find a suppression in the $J/\\\\psi $ production that transitions to an excess in the $J/\\\\psi $ production at higher $p_T$, which is characteristic of multiple scattering of the incident parton [7].\",\n \"It is crucial to test these conclusions at the higher energies provided by $d+$ Au collisions at RHIC in order to better understand the $J/\\\\psi $ production mechanisms.\",\n \"Measuring, and understanding, the CNM effects on quarkonia production is critical to interpreting the results for $J/\\\\psi $ production in nucleus-nucleus ($A+A$ ) collisions.\",\n \"In 1986 Matsui and Satz predicted that the suppression of $J/\\\\psi $ production in heavy-ion collisions would be a clear signature of the formation of a quark-gluon plasma [10].\",\n \"The Debye color screening of the dense medium produced is expected to cause the dissociation of bound states, thereby causing a decrease in the observed production.\",\n \"Since then suppression of quarkonia production has been observed for a number of states, including the $J/\\\\psi $ and $\\\\Upsilon $ , over a wide range in collision energy [11], [12], [13], [14].\",\n \"However, the interpretation of these results is still unclear.\",\n \"Before the modification due to the produced medium can be determined, the CNM effects must first be corrected.\",\n \"This has been done at lower energies [11], but accurate data on CNM effects are still absent at the higher energies of RHIC and the LHC.\",\n \"Here we report new high-precision measurements of the $J/\\\\psi $ production as a function of $p_T$ and collision centrality in $d+$ Au collisions at $\\\\sqrt{s_{_{NN}}}$ = 200 GeV.\",\n \"We also present measurements of the $J/\\\\psi $ $R_{d{\\\\rm Au}}$ as a function of $p_T$, rapidity, and collision centrality using data for $J/\\\\psi $ production in $p$ +$p$ collisions published in [15].\",\n \"PHENIX has previously measured the $J/\\\\psi $ yield in $d+$ Au collisions [16], [17] with data recorded in 2003.\",\n \"The data presented here, recorded in 2008, feature an increase in statistics of 30–50 times over those used in the previously published results, as well as a significant reduction of the systematic uncertainties.\",\n \"The rapidity dependence of $J/\\\\psi $ production in $d+$ Au collisions from this data set has been previously published in [18].\",\n \"This paper presents results for the $p_T$ dependence of the $J/\\\\psi $ yield from the same data set.\"\n ],\n [\n \"Experimental Apparatus & Data Sets \",\n \"The PHENIX detector [19] comprises three separate spectrometers in three pseudorapidity ($\\\\eta $ ) ranges.\",\n \"Two central arms at midrapidity cover $\\\\left|\\\\eta \\\\right|<0.35$ and have an azimuthal coverage ($\\\\phi $ ) of $\\\\pi /2$ rad each, while muon arms at backward/forward rapidity cover $-2.2<\\\\eta <-1.2$ (Au going direction) and $1.2<\\\\eta <2.4$ ($d$ going direction), with full azimuthal coverage.\",\n \"In the central arms the $J/\\\\psi $ yield is measured via dielectron decays.\",\n \"Charged particle tracks are reconstructed using the drift chamber and pad chambers.\",\n \"Electron candidates are selected by matching charged tracks to hits in the ring imaging Čerenkov (RICH) counters and clusters in the Electromagnetic Calorimeters (EMCal).\",\n \"In $d+$ Au collisions, a charged track is identified as an electron by requiring at least two matching RICH phototube hits within a radius of $3.4$) is determined using a Glauber calculation [21] combined with a simulation of the BBC response (as described in [18]).\",\n \"The resulting $\\\\left< N_{\\\\rm coll}\\\\right>$ values for the centrality categories used in this analysis are shown in Table REF .\",\n \"The $N_{\\\\rm coll}$ distributions within each centrality bin are shown in Fig.\",\n \"REF .\",\n \"There is a significant overlap between the $N_{\\\\rm coll}$ distributions for different centralities.\",\n \"Also shown in Table REF is the correction factor $c$ , which accounts for the correlation between the detection of a $J/\\\\psi $ in the final state and an increase in the total charge collected in the BBC [16].\",\n \"This correlation affects both the MB-trigger efficiency and the determination of the centrality of a given collision.\",\n \"The correction factors for each centrality bin are obtained within the same Glauber framework as the $\\\\left< N_{\\\\rm coll}\\\\right>$ values by assuming that one of the N binary collisions produces a charge in the BBC that is characteristic of a hard-scattering process (the remaining N-1 binary collisions maintain a BBC charge distribution characteristic of soft scattering processes).\",\n \"The increase in the BBC charge from a hard process is tuned using real data.\",\n \"Since both $c$ and $\\\\left< N_{\\\\rm coll}\\\\right>$ are calculated in the same Glauber framework there are correlations between their uncertainties.\",\n \"These correlations are removed in the ratio of $c/\\\\mbox{$\\\\left< N_{\\\\rm coll}\\\\right>$} $ , which occurs in the calculation of $R_{d{\\\\rm Au}}$.\",\n \"The resulting values and uncertainties are given in the third column of Table REF .\",\n \"The correction factor for 0–100% centrality contains an additional factor to extrapolate the measured yield, which covers only 88% of all $d+$ Au collisions, to 100% of the $d+$ Au inelastic cross section, essentially correcting for the efficiency of the BBC trigger.\",\n \"This correction is again determined within the Glauber framework using the parametrization of the BBC trigger efficiency.\"\n ],\n [\n \"The procedure for analyzing the $\\\\mbox{$J/\\\\psi $} \\\\rightarrow e^+e^-$ signal and the results in the central arms are discussed in this section.\",\n \"The extraction of the correlated $e^+e^-$ yield is discussed in Sec.\",\n \"REF .\",\n \"The estimation of the correlated background and losses due to the radiative tail in the $J/\\\\psi $ mass distribution is discussed in Sec.\",\n \"REF .\",\n \"The estimation of the detector efficiencies is described in Sec.\",\n \"REF .\",\n \"The calculation of the $J/\\\\psi $ invariant yield is detailed in Sec.\",\n \"REF .\",\n \"The $p$ +$p$ baseline used in calculating $R_{d{\\\\rm Au}}$ is described in Sec.\",\n \"REF .\"\n ],\n [\n \"Correlated $e^+e^-$ Signal extraction \",\n \"The $\\\\mbox{$J/\\\\psi $} \\\\rightarrow e^+e^-$ yield is measured using the invariant mass spectrum for all dielectron pairs where at least one of the electrons fired the ERT trigger.\",\n \"This selection is necessary to match the conditions under which the $J/\\\\psi $ trigger efficiency is calculated (see Sec.\",\n \"REF ).\",\n \"An example of the dielectron mass spectrum is shown in Fig.\",\n \"REF for 0–20% central collisions.\",\n \"In a given bin of $p_T$, rapidity, and collision centrality, the correlated $e^+e^-$ yield ($N_{e^+e^-}$ ) is determined by counting over a fixed mass window of $2.8 < M_{ee}\\\\,[\\\\mbox{GeV/$c^{2}~$}] < 3.3$ the number of unlike-sign dielectrons, after the subtraction of the like-sign dielectrons, which arise by random association and so are representative of the combinatorial background within the unlike sign dielectron distribution.\",\n \"This method assumes that the acceptance is the same for $e^-$ and $e^+$ , which, while untrue at lower masses, is a good assumption in the $J/\\\\psi $ mass range.\",\n \"At higher $p_T$, where statistical precision is limited, the yield, along with the statistical uncertainties, are derived from Poisson statistics.\",\n \"Assuming both the unlike-sign (foreground) and like-sign (background) are independent, and assuming no negative signal, the combined distribution $ P(s) =\\\\sum _{k=0}^{fg}\\\\frac{(bg+fg-k)!}{bg!(fg-k)!}\\\\frac{1}{2}\\\\left(\\\\frac{1}{2}\\\\right)^{bg+fg-k}\\\\frac{s^ke^{-s}}{k!\",\n \"}, $ represents the probability of a signal ($s$ ) given a number of unlike-sign dielectrons ($fg$ ) and a number of like-sign dielectrons ($bg$ ) (see [15] for derivation).\",\n \"The mean and standard deviation of Eq.\",\n \"REF are then used as the yield and uncertainties.\",\n \"A correlated $e^{+}e^{-}$ yield in the mass window $2.8 < M_{ee}\\\\,[\\\\mbox{GeV/$c^{2}~$}] < 3.3$ of approximately 8600 is obtained across all $p_T$ and collision centralities.\",\n \"Figure: (Color online)(top) Invariant mass distribution of unlike-sign (filled circles) andlike-sign (filled boxes) dielectron pairs in central d+d+Au collisions,integrated over p T p_T and rapidity.\",\n \"Dashed vertical lines represent themass range used to determine the correlated e + e - e^+e^- yield.\",\n \"(bottom)Correlated dielectron invariant mass distribution for MB d+d+Au collisions.\",\n \"The line shapes are those used to extract the continuum andradiative tail contributions to the correlated e + e - e^+e^- yield in the massrange 2.8$}}\\\\frac{d^2N_{J/\\\\psi }^{d+\\\\rm Au}(i)/dyd\\\\mbox{$p_T$}}{d^2N_{J/\\\\psi }^{p+p}/dyd\\\\mbox{$p_T$}},$ where $d^2N_{J/\\\\psi }^{d+\\\\rm Au}(i)/dyd\\\\mbox{$p_T$} $ is the $d+$ Au invariant yield for the $i^{\\\\rm th}$ centrality bin, $d^2N_{J/\\\\psi }^{p+p}/dyd\\\\mbox{$p_T$} $ is the $p$ +$p$ invariant yield for the same $p_T$ and $y$ bin, and $\\\\left< N_{\\\\rm coll}(i)\\\\right>$ is the average number of binary collisions for the given centrality bin, as listed in Table REF .\",\n \"The 0–100% centrality integrated $J/\\\\psi $ $R_{d{\\\\rm Au}}$ as a function of $p_T$ is shown in Fig.\",\n \"REF for each of the three rapidity regions.\",\n \"The numerical values can be found in Table REF , REF , and REF for backward, mid and forward rapidity, respectively.\",\n \"Figure REF shows a different behavior for $R_{d{\\\\rm Au}}$ at backward ($-2.24$ GeV/$c~$by fitting each distribution with a constant.\",\n \"The results are shown in Table REF along with the fit uncertainties, which take into account only the type A uncertainties on the data.\",\n \"Since the type B uncertainties are roughly consistent in the fit range, we have chosen here to add the average type B uncertainty for $\\\\mbox{$p_T$} >4$ GeV/$c~$in quadrature with the type C uncertainty.\",\n \"We find that at mid and forward rapidity the average $R_{d{\\\\rm Au}}$ for $\\\\mbox{$p_T$} >4$ GeV/$c~$is consistent with 1.0, while at backward rapidity the average $R_{d{\\\\rm Au}}$ is greater than 1.0.\",\n \"The production of a $J/\\\\psi $ at forward rapidity in $A+A$ collisions involves a low-$x$ gluon colliding with a high-$x$ gluon.\",\n \"The symmetry due to identical colliding nuclei results, essentially, in the folding of the forward and backward rapidity $R_{d{\\\\rm Au}}$.\",\n \"The production of a $J/\\\\psi $ at midrapidity results, essentially, in the folding of the midrapidity $R_{d{\\\\rm Au}}$ with itself.\",\n \"This picture is simplistic and leaves out many details, but it gives some expectation for the result of the modification of $J/\\\\psi $ production in $A+A$ collisions due to CNM effects.\",\n \"If we extrapolate the observed behavior of $R_{d{\\\\rm Au}}$ to the modification of $J/\\\\psi $'s produced at forward rapidity in $A+A$ collisions, we would expect a $R_{\\\\rm AA}$ contribution from CNM effects to be similar to, or greater than, 1.0 at high $p_T$ and a modification similar to 1.0 at midrapidity.\",\n \"The observation at midrapidity of a $J/\\\\psi $ $R_{\\\\rm AA}$ in Cu+Cu collisions that exceeds, but is consistent with, 1.0 at high $p_T$ [26] may therefore be largely accounted for by the contribution from CNM effects.\",\n \"Further work is needed to understand the detailed propagation of measured results in $d+$ Au collisions to an expected CNM contribution in $A+A$ collisions before this can be fully understood.\",\n \"Table: The average 0–100% R d Au R_{d{\\\\rm Au}} for p T >4\\\\mbox{$p_T$} >4 GeV/cc~where the first quoteduncertainty corresponds to the fit uncertainty and the second correspondsto the combined type B and C systematic uncertainties.Figures REF , REF , and REF show $R_{d{\\\\rm Au}}$ vs $p_T$ in four centrality bins for backward rapidity, midrapidity, and forward rapidity, respectively.\",\n \"Numerical values can be found in Tables REF , REF , REF and REF for 0–20%, 20–40%, 40–60% and 60–88% central collisions respectively.\",\n \"For peripheral collisions the $R_{d{\\\\rm Au}}$ remains consistent with 1.0 within statistical and systematic uncertainties across all $p_T$ in all rapidity regions.\",\n \"Figure: (Color Online)J/ψ→μ + μ - J/\\\\psi \\\\rightarrow \\\\mu ^+\\\\mu ^- R d Au R_{d{\\\\rm Au}}, as afunction of p T p_T for a) central, b) midcentral, c)midperipheral, and d) peripheral events at -2.20.25$ ), while for lower values, power suppressed coherent final-state scattering leads to a modification of parton $x$ [41].\",\n \"Initial state energy loss, which accounts for the radiative energy loss of the incoming particles through multiple interactions with the target nucleus is included.\",\n \"This effect reduces the energy of the incoming parton, so, to achieve the same final-state kinematics the parton must have a greater momentum, and therefore a larger value of $x$ .\",\n \"This effectively shifts the portion of the gluon distribution sampled to higher $x$ .\",\n \"Also included is a calculation of the Cronin effect.\",\n \"The solid curve in Fig.\",\n \"REF shows the full calculation including the Cronin effect.\",\n \"The dashed curve in Figure REF is the same calculation without the Cronin effect.\",\n \"This comparison gives a direct indication of the contribution from the Cronin effect, which is evidently over predicted when compared to the data.\",\n \"The results presented here will hopefully provide a much needed constraint on the Cronin effect at RHIC energies.\",\n \"The calculation including the Cronin effect indicate an $R_{d{\\\\rm Au}}$ that decreases at higher $p_T$.\",\n \"This is consistent with data, however the current statistical and systematic uncertainties make determining the precise trend of $R_{d{\\\\rm Au}}$ difficult at high $p_T$.\",\n \"Better data with a larger $p_T$ coverage is needed to determine the $J/\\\\psi $ modification at higher $p_T$.\"\n ],\n [\n \"Summary & Conclusions \",\n \"We have measured the $J/\\\\psi $ invariant yield and $R_{d{\\\\rm Au}}$ , as a function of $p_T$ over three rapidity ranges in $d+$ Au collisions at $\\\\mbox{$\\\\sqrt{s_{_{NN}}}$} =200$ GeV using the PHENIX detector.\",\n \"These measurements provide a large improvement in statistical precision and $p_T$ reach over the previously published PHENIX $d+$ Au results [16], [17], and are the first measurements of the centrality dependence of the $J/\\\\psi $ $p_T$ distribution in $d+$ Au collisions by PHENIX.\",\n \"The $\\\\Delta \\\\mbox{$\\\\langle p_T^2 \\\\rangle $} $ values determined from the data show a marked increase with $N_{\\\\rm coll}$ that is similar at all rapidities.\",\n \"The $R_{d{\\\\rm Au}}$ vs $p_T$ displays similar behavior at mid and forward rapidity, showing suppression at low $p_T$ with a gradual increase to a value consistent with 1.0.\",\n \"The $R_{d{\\\\rm Au}}$ at backward rapidity has a different distribution with $p_T$, showing a more rapid increase from suppression to a value of 1.0, and transitioning to $\\\\mbox{$R_{d{\\\\rm Au}}$} >1.0$ above 2 GeV/$c~$.\",\n \"These trends are greater for central collisions, while the peripheral collisions show $R_{d{\\\\rm Au}}$ consistent with 1.0 across all rapidities.\",\n \"We find an average $R_{d{\\\\rm Au}}$ for $\\\\mbox{$p_T$} >4$ GeV/$c~$of $1.27\\\\pm 0.06\\\\pm 0.11$ at backward rapidity, and an $R_{d{\\\\rm Au}}$ consistent with 1.0 at mid and forward rapidity.\",\n \"This implies a CNM contribution in $A+A$ collisions that is likely consistent with 1.0 at high $p_T$ across all rapidity.\",\n \"This could potentially explain the reported increase in $R_{\\\\rm AA}$ with increasing $p_T$ [26].\",\n \"However more data and further work to understand the propagation of $R_{d{\\\\rm Au}}$ to $R_{\\\\rm AA}$ is needed to confirm this.\",\n \"A comparison of the measured $R_{d{\\\\rm Au}}$ with three types of theoretical calculations was shown.\",\n \"The parameter independent dipole model of $J/\\\\psi $ production in $p+A$ collisions agrees well with the shape of the data at mid and forward rapidities, while the shape of the predicted $p_T$ dependence is different from the data at backward rapidity.\",\n \"However the suppression is over-predicted at all rapidities.\",\n \"The second model uses $2\\\\rightarrow 2$ $J/\\\\psi $ production kinematics coupled with shadowing taken from both EKS98 and nDSg nPDF sets as well as an effective absorption cross section of 4.2 mb.\",\n \"The calculations with both EKS98 and nDSg show good agreement with the data at midrapidity in each centrality bin, as well as the centrality integrated case.\",\n \"At forward rapidity the shape of the distribution is in reasonable agreement with the data, while the overall level of suppression seems to be greater in the model calculations than the data.\",\n \"At backward rapidity, the model calculations using both EKS98 and nDSG nPDF sets are in strong disagreement with the data.\",\n \"At backward rapidity calculations using the nDSg nPDF set show a suppression that is constant with $p_T$, while those using the EKS98 nPDF set predict an increase of suppression with increasing $p_T$.\",\n \"The data show the opposite trend.\",\n \"The third model, an NRQCD calculation of high $p_T$ $J/\\\\psi $ production show a Cronin effect, which although generally consistent with the data, is significantly larger than observed in the data, and a suppression at high $p_T$ that cannot be confirmed due to the large uncertainties at high $p_T$ and the limited $p_T$ reach of the current data.\",\n \"In summary, the data presented here cover a large range in $x$ and $Q^2$ , providing a further constraint on the modification of the gluon distribution in nuclei, as well as providing constraints on the size of the Cronin effect on $J/\\\\psi $ production at RHIC.\"\n ],\n [\n \"ACKNOWLEDGMENTS\",\n \"We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.\",\n \"We thank Jean-Philippe Lansberg, Nicolas Matagne, Boris Kopeliovich, Ivan Vitev, and Rishi Sharma for useful discussions and theoretical calculations.\",\n \"We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Ministry of Industry, Science and Tekhnologies, Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), Hungarian National Science Fund, OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), National Research Foundation and WCU program of the Ministry Education Science and Technology (Korea), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and the Wallenberg Foundation (Sweden), the U.S.\",\n \"Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the US-Hungarian Fulbright Foundation for Educational Exchange, and the US-Israel Binational Science Foundation.\"\n ],\n [\n \"Fitting The \",\n \"The $J/\\\\psi $ invariant yields as a function of $p_T$ were fitted with a modified Kaplan function of the form $f(\\\\mbox{$p_T$})=p_0\\\\left(1-\\\\left(\\\\frac{\\\\mbox{$p_T$}}{p_1}\\\\right)^2\\\\right)^{p_2}.$ The data points were compared to the integral of the function over the $p_T$ bin when calculating the $\\\\chi ^2$ .\",\n \"The fit results, along with the ratio of the data to the fit are shown in Fig.\",\n \"REF for $p$ +$p$ collisions and Figure REF for 0–100% centrality integrated $d+$ Au collisions.\",\n \"The fit results for each centrality bin are shown in Figs.\",\n \"REF , REF , and REF for backward, mid, and forward rapidities, respectively.\"\n ],\n [\n \"Calculating the Correction Factor $k$\",\n \"To account for the fact that the experimental upper $p_T$ limits on the $J/\\\\psi $ invariant yield distributions vary with rapidity and centrality, a correction factor was calculated using the fits described in Sec.\",\n \"REF .\",\n \"The ratio $k=\\\\frac{\\\\mbox{$\\\\langle p_T^2 \\\\rangle $} [0,\\\\infty ]}{\\\\mbox{$\\\\langle p_T^2 \\\\rangle $} [0,\\\\mbox{$p_T$} ^{\\\\rm max}]}$ was calculated from the fit and applied to the numerically calculated $\\\\langle p_T^2 \\\\rangle |_{p_T\\\\le p_T^{\\\\rm max}}$.\",\n \"The correction factors are shown in Table REF , and are in all cases small ($k<1.03$ ).\",\n \"The uncertainty on $k$ is derived from the fit uncertainty by varying the data points within their statistical uncertainties, refitting, and thereby finding the variation in $k$ .\",\n \"Figure: (Color Online)Results of modified Kaplan fits to backward rapidity 0–100% d+d+Au (Left), midrapidity 0-100% d+d+Au (Center), and forward rapidity 0-100%d+d+Au (Right).Figure: (Color Online)Results of modified Kaplan fits to midrapidity d+d+Au collisions for eachcentrality.Figure: (Color Online)Results of modified Kaplan fits to forward rapidity d+d+Au collisions foreach centrality.Table: 〈p T 2 〉\\\\langle p_T^2 \\\\rangle correction factors, kk, for pp+pp andd+d+Au collisions.\"\n ],\n [\n \"Propagating the Type B uncertainties to \",\n \"When propagating the Type B systematic uncertainties on the $J/\\\\psi $ invariant yields to the calculated $\\\\langle p_T^2 \\\\rangle $ values, the type B uncertainties are assumed to be normally distributed.\",\n \"With this assumption we independently sample the uncertainty distribution of the first and the last data point of the $p_T$ distribution.\",\n \"We then assume the Type B uncertainties are linearly correlated between these two values.\",\n \"The resulting distribution of the $\\\\langle p_T^2 \\\\rangle $ values that arises from this procedure gives an estimate of the effect of the Type B uncertainties on the value of $\\\\langle p_T^2 \\\\rangle $.\",\n \"The Type C systematic uncertainties on the $J/\\\\psi $ invariant yields do not affect the calculation of $\\\\langle p_T^2 \\\\rangle $.\",\n \"The Type C uncertainties are a global uncertainty, which cancels in the calculation.\"\n ]\n]"},"id":{"kind":"string","value":"1204.0777"}}},{"rowIdx":1867492,"cells":{"text":{"kind":"list like","value":[["The roles of charge exchange and dissociation in spreading Saturn's\n neutral clouds"],["Abstract Neutrals sourced directly from Enceladus's plumes are initially confined to a dense neutral torus in Enceladus's orbit around Saturn.","This neutral torus is redistributed by charge exchange, impact/photodissociation, and neutral-neutral collisions to produce Saturn's neutral clouds.","Here we consider the former processes in greater detail than in previous studies.","In the case of dissociation, models have assumed that OH is produced with a single speed of 1 km/s, whereas laboratory measurements suggest a range of speeds between 1 and 1.6 km/s.","We show that the high-speed case increases dissociation's range of influence from 9 to 15 Rs.","For charge exchange, we present a new modeling approach, where the ions are followed within a neutral background, whereas neutral cloud models are conventionally constructed from the neutrals' point of view.","This approach allows us to comment on the significance of the ions' gyrophase at the moment charge exchange occurs.","Accounting for gyrophase: (1) has no consequence on the H2O cloud; (2) doubles the local density of OH at the orbit of Enceladus; and (3) decreases the oxygen densities at Enceladus's orbit by less than 10%.","Finally, we consider velocity-dependent, as well as species-dependent cross sections and find that the oxygen cloud produced from charge exchange is spread out more than H2O, whereas the OH cloud is the most confined."],["Introduction","The Enceladus plumes directly produce a dense H$_2$ O torus centered on Enceladus's orbit, within which charge exchange and dissociation subsequently produce neutrals that either feed Saturn's extended neutral clouds, collide (absorb) with Saturn and its rings, or leave the system altogether on escape orbits.","This paper is a report on the results of a sensitivity study of low-velocity charge exchange and dissociation within the neutral torus.","Several decades before Cassini arrived at Saturn and the Enceladus water plumes were discovered [12], neutral hydrogen was observed in Saturn's magnetosphere, both from Earth [43] and from Voyagers 1 and 2 [33].","Hydroxyl was later discovered by [34] with HST, and more recently, [8] detected atomic oxygen.","These observations collectively hinted at the presence of a source of water, and models predicted its location to be near the orbit of Enceladus (c.f., [21]).","After identifying the Enceladus plumes as the dominant source of the water-group neutrals (O, OH, H$_2$ O)—and indeed the co-rotating plasma itself via electron impact and photoionization [46], [35], [36]—researchers have been attempting to understand how neutrals are transported from Enceladus to 20 Saturn radii ($\\textrm {R}_\\textrm {S}$ = 6 $\\times 10^9$ cm) and beyond, as observed by [34], [8], and most recently by [25].","Early on, [21] mentioned the role of charge exchange in this inflation process.","[18] later showed that if magnetospheric plasma is slowed sufficiently with respect to neutrals in the Enceladus torus, then charge exchange produces a sufficient number of particles with velocities capable of spreading the dense H$_2$ O Enceladus torus into the cloud observed by [34].","[9] pointed out the importance of dipole–dipole interactions in collisions involving H$_2$ O molecules.","She showed that collisions inside the dense Enceladus torus (parameterized by macroscopic viscosity) are alone sufficient for the creation of the extended component of Saturn's neutral cloud.","[5] later argued that Farmer's fluid treatment is inappropriate for neutral–neutral collisions in the Enceladus torus, where the mean free path is on the order of the torus size itself.","Instead, [5] approached the problem with a direct simulation Monte Carlo (DSMC) model.","Their model self-consistently included losses due to charge exchange, dissociation, and ionization, whereas [9] accounted for losses to charge exchange and ionization by evolving the neutral cloud for the time scales (months to a few years) found in [36].","Both studies agree that neutral–neutral collisions are necessary for the inflation of Saturn's neutral cloud.","Collisions between neutrals occur at a rate proportional to the square of the neutral density.","Thus, where neutral densities peak near the orbit of Enceladus, neutral–neutral collisions occur more often than either charge exchange or dissociation, whereas near 6 $\\textrm {R}_\\textrm {S}$ , neutral densities drop and all three processes become comparable (see Fig.","3, [5]).","Models involving neutral collisions have recently been validated with Herschel observations by [14], who attribute a warm and broadened Enceladus torus to heating via neutral–neutral collisions; the effect of these interactions should therefore be included in any attempt to fully model Saturn's neutral clouds.","Nevertheless, several first-order conclusions can be drawn by revisiting charge exchange and dissociation.","Previous neutral cloud models approach charge exchange from the neutrals' point of view, whereas we follow the ion along its trajectory, thus allowing us to identify the gyrophase at which an ion undergoes charge exchange.","We find that including the phase dependence doubles OH densities at the orbit of Enceladus, decreases oxygen density by $\\lesssim $ 10%, and has no effect on H$_2$ O (section REF ).","Also, the velocity-dependence of charge exchange varies by species.","Previous studies ($e.g.$ , [18], [5]) have considered velocity-dependence, but have used a single cross section to represent all charge exchanges.","We show in section REF that symmetric reactions such as $\\mathrm {H_2O+H_2O^+\\rightarrow H_2O^++H_2O^*}$ (the asterisk identifies a neutral released with the speed of the reacting ion) tend to distribute neutrals closer to Saturn, while asymmetric exchanges such as $\\mathrm {H_2O+O^+\\rightarrow H_2O^++O^*}$ populate a more extended cloud, with less absorption on Saturn.","With regard to dissociation, OH produced by impact/photodissociation of H$_2$ O has previously been modeled with an initial speed of 1 $\\textrm {km\\ s}^{-1}$ [19], [5], whereas recent laboratory measurements span 1 to 1.6 $\\textrm {km\\ s}^{-1}$ , depending on the molecule's internal energy [45], [24].","We model this parameter space and find that, relative to charge exchange, most OH found inside 9 $\\textrm {R}_\\textrm {S}$ is produced by dissociation when OH is dissociated from H$_2$ O at 1 $\\textrm {km\\ s}^{-1}$ , with that location extended to 15 $\\textrm {R}_\\textrm {S}$ when OH is dissociated from H$_2$ O at 1.6 $\\textrm {km\\ s}^{-1}$ instead.","This paper is organized as follows.","The model for the production of neutrals via dissociation and three illustrative charge exchanges is explained in section .","Our results are found in section , followed by a discussion in section .","The important points are summarized in section ."],["Model","We begin with a few words on nomenclature.","The neutral torus in this paper pertains to the primary neutral torus (not plasma torus) supplied directly by Enceladus's plumes.","The neutral clouds refer to the secondary neutrals produced from charge exchange and dissociation in the neutral torus.","The production of Saturn's neutral cloud is modeled in two steps.","We first construct a dense H$_2$ O torus from a plume positioned at Enceladus's south pole with specifications based on several Cassini Enceladus flybys ([39]; see also [38], [37], and [40]).","Secondary neutrals are then produced from the primary neutral torus by charge exchange and dissociation, some of which remain gravitationally bound to Saturn and form the neutral clouds.","On the basis that they spend most of the time outside the neutral torus and plasma sheet, we assume their lifetimes to be determined solely by photo-processes, though in section REF , we consider the effects of charge exchange and electron impact."],["Neutral torus model","The neutral torus and Enceladus plume models are described in the following subsections.","Our aim is to study the effects of several important reactions occurring in Enceladus's orbit.","The primary neutral torus, fed directly by Enceladus, is produced in the model by evolving water molecules released from Enceladus into a dense neutral torus centered on Enceladus's orbit (3.95 $\\textrm {R}_\\textrm {S}$ ).","The assumption is that all H$_2$ O is initially produced by a single plume at Enceladus's south pole.","In reality, more than one plume has been observed [26], and researchers such as [31] and [39] have studied their signature on flyby observations.","For our purposes, the detailed influence of multiple plumes can be neglected.","The plume particles' radial speed distribution is prescribed as a one-dimensional Maxwellian with temperature $T=180$ K [41], [12]: $f(v)=\\left(\\frac{m_\\mathrm {H_2O}}{2\\pi kT}\\right)^{1/2}\\exp \\left[-\\frac{m_\\mathrm {H_2O}}{2kT}(v-v_\\mathrm {bulk})^2\\right],$ where $v_\\mathrm {bulk}$ is the bulk speed, equal to 720 m s$^{-1}$ , 1.8$\\times $ the thermal speed estimated by [39] ($v_\\mathrm {therm}=\\sqrt{2kT/m_\\mathrm {H_2O}}=400$ $\\textrm {m\\ s}^{-1}$ ).","Additionally, a raised cosine distribution is used to determine where the molecules are released: $g(\\theta )=\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\theta _0}\\left[1+\\cos \\left(\\frac{\\theta }{\\theta _0}\\pi \\right)\\right]&\\mbox{ if $\\theta < \\theta _0=30^\\circ $} \\\\0 &\\mbox{ otherwise.","}\\end{array} \\right.$ Co-latitude theta is measured from Enceladus's south pole, and $\\theta _0=30^\\circ $ is the plume half-width, based on INMS in situ observations [39].","We assume no azimuthal dependence.","Enceladus's gravity is ignored since the escape velocity, $v_\\mathrm {esc}\\,=\\,240 \\mathrm {\\ m\\ s^{-1}}$ , is greatly exceeded for most molecules (99%), where $v_\\mathrm {bulk}>v_\\mathrm {therm}>v_\\mathrm {esc}$ ; our results differ by less than one percent whether Enceladus's gravity is considered or not.","Particles released from Enceladus are thus assumed to move on Keplerian orbits with respect to Saturn.","Each water molecule is allowed to orbit inside the torus for a period determined by the collective lifetimes against photodissociation, electron-impact dissociation, and charge exchange.","To be clear, the molecules forming the neutral torus are subject to all of the losses stated, while the neutral cloud is subject to photo-processes only (section REF ).","Further reaction details are important for modeling plasma characteristics but only the timescales given below are required to model the neutral torus.","The photodissociation lifetime for H$_2$ O, $\\tau _\\mathrm {phot}=9.1\\times 10^6$ s, comes directly from [15], scaled to Saturn's distance from the Sun.","At peak solar activity, neutral abundances attributed to dissociation double [16]."],["Impact dissociation","In the case of impact dissociation, suprathermal (hot) electrons dominate [11].","Assuming the conditions near Enceladus's orbit apply throughout the neutral torus, we estimate the hot electron density and temperature to be 160 eV and 0.3 cm$^{-3}$ from [11], which fit within a range of recent observations (cf., [46], [36]).","The rate coefficient for impact dissocation of water is found by convolving $\\sigma (v)v$ with a 160 eV Maxwellian distribution to find $\\kappa _\\mathrm {imp}=1.5\\times 10^{-6}\\mbox{ cm$^3$ s$^{-1}$}$ ([11], table S9).","Above 100 eV, $\\kappa _\\mathrm {imp} $ is insensitive to temperature, making this estimate valid for a range of observations.","Assuming the hot electron density, $n_\\mathrm {eh}$ , is constant over the neutral torus, the lifetime against impact dissociation is $\\tau _\\mathrm {imp}\\approx [\\kappa _\\mathrm {imp}n_\\mathrm {eh}]^{-1}=2.2 \\times 10^6$ s. Dissociation $via$ thermal electrons—whose temperature and density are 2 eV and 60 cm$^{-3}$ —is also expected, but such collisions occur 5$\\times $ less often ([11], table S9) and are thus ignored here."],["Charge exchange","The following three reactions are included in this study: $\\mathrm {H_2O+H_2O^+} &\\rightarrow \\mathrm {H_2O^+ +H_2O^*} \\\\\\mathrm {H_2O+H_2O^+} &\\rightarrow \\mathrm {H_3O^+ +OH^*} \\\\\\mathrm {H_2O+O^+} &\\rightarrow \\mathrm {H_2O^++O^*} .$ Other charge exchanges are important in the neutral torus, some of which involve ions reacting with secondary neutrals such as H, O, and OH.","To model their effect properly, one would calculate these neutral densities as in a conventional time-dependent neutral cloud model.","We estimate that including all such reactions would increase our estimates on neutral cloud densities by approximately a factor of two.","The primary purpose of choosing this combination of reactions is to study three classes of charge exchanges, for which the collision frequency decreases with, increases with, or is independent of the relative speed of the reacting pair (reactions , , and REF , respectively).","We return to this point in section REF .","An estimate of the charge exchange lifetime can be made by adding the rate coefficients for reactions REF –.","Multiplying by the observed H$_2$ O$^+$ and O$^+$ densities near the orbit of Enceladus (6 and 12 cm$^{-3}$ , [36]), we find $\\tau _\\mathrm {chex}=[\\kappa _\\mathrm {exch}^\\mathrm {H_2O^+}n_\\mathrm {H_2O^+}+\\kappa _\\mathrm {exch}^\\mathrm {O^+}n_\\mathrm {O^+}]^{-1}=[6.0+2.8]^{-1}\\times 10^8 \\mathrm {\\ s\\ }=1.1\\times 10^7$ s. The reaction rates are from [23] and [1] for $\\kappa ^\\mathrm {H_2O^+}_\\mathrm {exch}$ and $\\kappa ^\\mathrm {O^+}_\\mathrm {exch}$ , respectively.","The lifetime of H$_2$ O against the sum of these processes is then $\\tau _\\mathrm {torus}=\\left[\\frac{1}{\\tau _\\mathrm {phot}} +\\frac{1}{\\tau _\\mathrm {imp}} +\\frac{1}{\\tau _\\mathrm {chex}}\\right]^{-1} &=&(1.1+4.5+0.91)^{-1}\\times 10^7 \\mathrm {\\ s} \\nonumber \\\\&=& 1.6\\times 10^6 \\mathrm {\\ s}\\approx 20 \\mbox{ days.}","$ These estimates can be compared to Fig.","3 of [5]: for example, our lifetime against dissociation is 1.7 $\\times 10^6$ s, while they use $7\\times 10^6$ s near 4 $\\textrm {R}_\\textrm {S}$ .","The discrepancy comes mostly from the impact dissociation timescale.","Cassidy's dissociation rate was calculated using [32], wherein CAPS ELS data were fitted and extapolated down from 5.5 $\\textrm {R}_\\textrm {S}$ , while our own estimate hinges on a hot electron density derived from our chemistry model [11].","For charge exchange, we have a lifetime of 1.1 $\\times 10^7$ s, and [5] have a comparable $8\\times 10^6$ s. Particles are created and tracked in each of our model runs, and the results are scaled to the number of water molecules in the real neutral torus.","The total number is estimated from an assumed neutral source rate from Enceladus of $\\dot{M}=200\\ \\textrm {kg\\ s}^{-1}$ [19], [12], [13] and lifetime, $\\tau _\\mathrm {torus}$ (Eq.","REF ): $N_\\mathrm {torus}=\\dot{M}\\tau _\\mathrm {torus}/m_\\mathrm {H_{2}O}\\approx 1.1\\times 10^{34} \\mathrm {\\ H_2O\\ molecules}.$ We bin and azimuthally average the results to find a 2-D density function, $n_\\mathrm {torus}(r,\\theta )$ (radius and latitude), through which to introduce ions for charge exchange.","This function also determines from where dissociated neutrals are produced."],["Plume model","Before describing dissociation and charge exchange within the torus, we address a calculation with the purpose of comparing the neutral production near Enceladus with that from the entire torus.","In doing so, we prescribe a plume whose density is consistent with Eqs.","REF and REF .","In this case, the ambient neutral density can be ignored compared to the neutrals leaving the surface of Enceladus directly.","The densities are determined everywhere by imposing integrated flux ($\\int n(r,\\theta )v(r,\\theta ) dA$ ) and energy ($mv^2/2-mM_\\mathrm {E}G/r$ ) conservation at a given distance $r$ .","The picture can be simplified, however, since most neutrals have at least twice Enceladus's escape velocity.","The speeds are thus independent of $r$ in the immediate vicinity of Enceladus.","By equating the integrated flux at the surface of Enceladus to the same integral at another distance $r>\\mathrm {R_E}$ , we find a familiar $1/r^2$ dependence: $n_\\mathrm {plume}(r,\\theta )=n(\\theta )\\left(\\frac{\\mathrm {R_E}}{r}\\right)^2\\exp \\left[-\\left(\\frac{r-\\mathrm {R_E}}{H_r}\\right)\\right].$ The trailing exponential factor is imposed $ad$ $hoc$ to keep the total plume content finite, and reflects Saturn's influence as the molecules leave Enceladus.","Consistent with [31], $H_r$ is set at 4 times the Hill radius of 948 km.","The angular dependence is consistent with our velocity distribution in the previous section (Eq.","REF ), $n(\\theta )=\\left\\lbrace \\begin{array}{ll}\\frac{n_0}{2}\\left[1+\\cos \\left(\\frac{\\theta }{\\theta _0}\\pi \\right)\\right]&\\mbox{ if $\\theta < \\theta _0=30^\\circ $} \\\\0 &\\mbox{ otherwise,}\\end{array} \\right.$ which is normalized such that $n(0)=n_0$ , where the plume strength $n_0$ is found to be $5.9\\times 10^8\\ \\mathrm {cm}^{-3}$ by integrating $n(\\theta )v_\\mathrm {bulk}$ over the area spanning the south pole of Enceladus from $\\theta =0^\\circ $ to $\\theta =\\theta _0=30^\\circ $ , and setting that result equal to the plume production rate of $\\dot{M}/m_\\mathrm {H_2O}=(200\\ \\textrm {kg\\ s}^{-1})/m_\\mathrm {H_2O}=6.7\\times 10^{27}$ molecules per second.","In section , we compare the results of charge exchange with the plume ($n_\\mathrm {plume}$ ) to that with the entire torus ($n_\\mathrm {torus}$ ).","Production of Saturn's neutral clouds entails following the neutrals produced by dissociation and charge exchange occurring within the neutral torus.","The treatment of each of these processes are described below."],["Dissociation","The hydroxyl radical, OH, produced largely by dissociated H$_2$ O, has previously been modeled with a single speed of 1 $\\textrm {km\\ s}^{-1}$ (i.e., [19], [5]).","Dissociated OH has been measured however with speeds between 1 and 1.6 $\\textrm {km\\ s}^{-1}$ [45], [24].","Here we bound this range by modeling the OH neutral clouds produced from an azimuthally-symmetric source (with respect to Saturn) with velocities drawn from Maxwellian distributions with temperatures $T=\\frac{1}{2}m_\\mathrm {OH}v_\\mathrm {mp}^2$ , where the most probable speed, $v_\\mathrm {mp}$ , is set to 1 and 1.6 $\\textrm {km\\ s}^{-1}$ , representing the low- and high-speed limits.","The initial locations of the ejected OH are determined by the spatial distribution of neutrals in the Enceladus torus ($n_\\mathrm {torus}(r,\\theta )$ , section REF ), and the directions of their release are chosen randomly and isotropically.","The molecules orbit Saturn until they are photodissociated and removed from the system.","By assuming a volume over which dissociations occur, the number of modeled OH molecules can be scaled to a realistic value.","We take the volume to be a torus centered on Enceladus (3.95 $\\textrm {R}_\\textrm {S}$ ), with a minor radius of 1 $\\textrm {R}_\\textrm {S}$ : $ V\\approx 2\\pi (4\\,\\textrm {R}_\\textrm {S})(2\\,\\textrm {R}_\\textrm {S})^2=2\\times 10^{31}\\,\\mathrm {cm}^3.$ For impact dissociation, we then expect a contribution of $ N^\\mathrm {imp}_\\mathrm {cloud}=k_\\mathrm {imp}\\tau _\\mathrm {phot}^\\mathrm {OH}V=2.8\\times 10^{34} \\mbox{\\ OH molecules},$ where $k_\\mathrm {imp}=7.9\\times 10^{-5}$ cm$^{-3}$ s$^{-1}$ is the rate (per volume) of impact dissociations occurring between suprathermal electrons and H$_2$ O molecules in the torus [11] and $\\tau _\\mathrm {phot}^\\mathrm {OH}=1.8\\times 10^7$ s is the photodissociation lifetime of OH at Saturn [15].","The number of OH molecules produced by photodissociation in the torus is similarly given by $ N^\\mathrm {phot}_\\mathrm {cloud}=k_\\mathrm {phot}\\tau _\\mathrm {phot}^\\mathrm {OH}V=7.6\\times 10^{33} \\mbox{\\ OH molecules},$ where $k_\\mathrm {phot}=2.1\\times 10^{-5}$ cm$^{-3}$ s$^{-1}$ is the rate (per volume) of H$_2$ O photodissociations occurring in the Enceladus torus ([11], Table S9).","The total abundance attributed to dissociation is then given by the sum of Eqs.","REF and REF .","[5] constrained their study with HST observations [25] and found a similar OH content (see comparison in Fig.","REF c, this paper).","That neutral production by photo- and impact dissociation are comparable in magnitude is itself noteworthy.","This condition is not shared by systems with hotter and denser plasma.","For example, electron impact dissociation and ionization dominate over photon-driven processes in Jupiter's Io torus, where the plasma is warmer where the pick-up energies are four times higher than at Enceladus [6], [11].","We also note that unlike with Io, long neutral lifetimes in the Enceladus neutral torus inhibit the response of Saturn's neutral clouds to short-term plume variability, though variability on the order of months has been studied by [39]."],["Charge exchange","We now describe the model for producing and following neutrals from charge exchange.","[5] and [19] also considered velocity-dependent charge exchange, but unlike these previous studies, we capture the gyrophase at which the reactions occur by following ions along their trajectories (section REF ).","We also prescribe cross sections specific to each reaction, being particularly interested in the effects of low-velocity charge exchange.","At very high speeds, the cross sections go to zero for all charge exchanges [17].","At low relative velocities, however (few $\\textrm {km\\ s}^{-1}$ ), the details of the collision are determined by the nature of the reacting species.","If the reactants and products are identical, apart from an electron ($i.e.$ , $\\mathrm {H_2O + H_2O^+}\\rightarrow \\mathrm {H_2O^+ + H_2O^*}$ ), the reaction is termed resonant, or symmetric, and the cross sections grow as the inverse of the relative speed.","If the reactants differ, as with $\\mathrm {H_2O + O^+}\\rightarrow \\mathrm {H_2O^+ + O^*}$ , the cross sections are likely to vanish at low speeds [27]—the difference being that the energy of the electron configurations is unchanged for symmetric-type charge exchanges [17].","Neutrals produced from resonant charge exchange therefore tend to have lower velocities than do neutrals produced from non-resonant (asymmetric) charge exchange.","This is a key point central to much of our discussion in section .","Individual ions are followed as they traverse the neutral torus (section REF ).","This approach allows their gyrophase to be determined the instant that charge exchanges occur (see Fig.","REF ).","The implicit assumption is that the collision is elastic, and that the neutral product has an initial velocity given by the ion velocity just before the exchange takes place.","The ions are introduced into the model from two Maxwellian speed distributions, $f_\\perp (v_\\perp )&=\\frac{m_\\mathrm {ion}}{kT_\\perp }v_\\perp \\exp \\left[-\\frac{m_\\mathrm {ion}v_\\perp ^2}{2kT_\\perp } \\right]\\mbox{ (speeds perpendicular to $B$)}\\\\f_\\parallel (v_\\parallel )&=\\sqrt{\\frac{m_\\mathrm {ion}}{2\\pi kT_\\parallel }}\\exp \\left[-\\frac{m_\\mathrm {ion}v_\\parallel ^2}{2kT_\\parallel } \\right]\\mbox{ (speeds parallel to $B$)}$ with a temperature anisotropy of $\\frac{kT_\\perp }{kT_\\parallel }=\\frac{27\\,\\mathrm {eV}}{5.4\\,\\mathrm {eV}}=5$ for both O$^+$ and H$_2$ O$^+$ [36].","The perpendicular temperature is derived from the pick-up ion velocity at the orbit of Enceladus, determined from CAPS data by [44] [$kT_\\perp =\\frac{1}{2}m_\\mathrm {W^+}(v_\\phi -v_\\mathrm {Kep})^2$ ].","The ions also rotate around a guiding center (field line) moving at $v_\\phi =18$ $\\textrm {km\\ s}^{-1}$ in a frame rotating with the neutrals.","For the component of our study aimed at estimating local neutral production (section REF ), ions passing near Enceladus are diverted (treating Enceladus as a rigid cylinder) and are slowed to 10% of the ambient flow speed to account for the effects of mass-loading (see [10]).","Time steps are taken at less than 1% of an ion's gyroperiod: $\\Delta t=R\\,\\times \\,T_\\mathrm {gyro}=R\\,\\times \\,\\frac{2\\pi m}{qB},$ where $R$ is a random number between 0 and 0.01, $T_\\mathrm {gyro}$ is the ion's gyroperiod (3.6 s for H$_2$ O$^+$ ), $B$ = 325 nT, and $q$ and $m$ are the charge and mass, respectively, of the reacting ion.","Such resolution is necessary in order to capture the significance of the energy dependence at low relative speeds.","After each time step, the collision frequency $\\nu $ is calculated from $\\nu (r,\\theta ,v_\\mathrm {rel})=n(r,\\theta ) \\sigma (v_\\mathrm {rel})v_\\mathrm {rel},$ where $n(r,\\theta )$ is the local H$_2$ O density (section REF ), $v_\\mathrm {rel}$ is the relative velocity between the reacting ion and neutral, and $\\sigma (v_\\mathrm {rel})$ is the velocity-dependent cross section.","Poisson statistics are used to test the likelihood of one or more reactions having occurred within $\\Delta t$ .","If $\\exp (-\\nu \\Delta t)$ is less than a second random number between 0 and 1, then a reaction occurs.","The possibility of multiple reactions occurring over $\\Delta t$ is taken into account, but it is neglectable (appendix A).","As with OH produced by dissociation (section REF ), neutrals produced by charge exchange are followed under the influence of Saturn's gravity until they are photodissociated or photoionized.","Their initial location and velocity are taken to be that of the reacting ion, pre-transfer.","The model runs are centered on the orbit of Enceladus spanning 10 R$_\\mathrm {E}$ in the direction of corotation (R$_\\mathrm {E}$ = 250 km = radius of Enceladus) and $\\pm $ 120 R$_\\mathrm {E}$ (0.5 $\\textrm {R}_\\textrm {S}$ ) in both the radial and $z$ directions to adequately sample the H$_2$ O torus (section REF ).","Ions are introduced into the model on the upstream boundary, and their guiding centers flow downstream at a speed $v_\\mathrm {plasma}=18$ $\\textrm {km\\ s}^{-1}$ relative to the neutrals.","Their starting location in ($r,z$ ) is chosen randomly."],["Scaling","The neutral clouds formed $via$ charge exchange are done so in our model by following a relatively small number of ions, and must thus be scaled to facilitate comparison with observations and other models.","The number of neutrals in our modeled clouds have been scaled by accounting for the following.","First, the number of representative ions used to produce the neutral clouds $via$ charge exchange falls short of, and must be scaled to, the number of ions present in the actual plasma torus, $n_\\mathrm {ion}V$ .","The volume of the plasma torus, $V$ , is given in Eq.","REF , and $n=12$ and 6 cm$^{-3}$ for O$^+$ and H$_2$ O$^+$ , respectively [36].","Second, we have argued that photo-processes are more likely to occur than either charge exchange or electron-impact processes throughout the neutral clouds with the exception of very near the neutral torus.","In keeping with this assumption, the plasma torus thus feeds the extended neutral clouds $via$ charge exchange for a photodissociation (photoionization in the case of oxygen) time scale before equilibrium of the neutral cloud is achieved: $\\tau _\\mathrm {phot}= $ 14, 0.6, 0.3 years for O, OH, and H$_2$ O, respectively [11].","Our model runs followed 10$^5$ ions for 100 seconds, and the resulting neutral clouds were scaled as described."],["Results","In the following sections, we present and discuss the neutral clouds resulting from dissociation and charge exchange in our model."],["Charge exchange","In the neutrals' reference frame, ions oscillate between $\\approx $ 0 $\\textrm {km\\ s}^{-1}$ and twice the local pick-up speed ($v_\\phi \\approx 18\\ \\textrm {km\\ s}^{-1}$ ) due to gyro-motion.","A cartoon of this can be seen in Fig.","REF , where $v_\\mathrm {rel}\\approx 0$ at the cusp of the ion trajectory and reaches a maximum of $v_\\mathrm {rel}\\approx 2 v_\\phi $ along the flow direction.","Shown are several trajectories for which $v_\\perp $ is either less than, greater than, or approximately equal to the bulk flow velocity.","The neutrals formed $via$ charge exchange follow the trajectories indicated in red.","The velocity dependence of reactions REF – are determined by the details of the reacting species [17].","Essentially, symmetric charge exchanges have cross sections that increase monotonically with decreasing velocity, whereas cross sections for asymmetric exchange peak and then vanish at low relative speeds.","The implication is that symmetric exchanges produce lower velocity neutrals and a more compact neutral cloud than do asymmetric reactions.","With symmetric charge exchange, the cross sections go as ${v_\\mathrm {rel}}^{-1}$ , so that the collision frequency ($n\\sigma v$ ) is independent of $v$ , as with reaction REF , whereas asymmetric exchanges are defined by cross sections (and collision frequencies) which tend rapidly toward zero at low relative velocities ($\\sim {v_\\mathrm {rel}}^4$ , [27]).","The cross sections (10$^{-16}$ cm$^2$ ) used in this paper to study reactions REF – plotted in Fig.","REF a are given by $\\mbox{Reaction \\ref {h2o_chex}} :&\\ \\mathrm {H_2O + H_2O^+} \\rightarrow \\mathrm {H_2O^+ + H_2O^*} \\nonumber \\\\\\sigma _\\mathrm {H_2O} =&\\ 38E_\\mathrm {rel}^{-0.5} \\\\\\mbox{Reaction \\ref {oh_chex}} :&\\ \\mathrm {H_2O + H_2O^+} \\rightarrow \\mathrm {H_3O^+ + OH^*} \\nonumber \\\\\\sigma _\\mathrm {OH} =&\\ 38E_\\mathrm {rel}^{-0.88}-0.39\\exp \\left[{-\\frac{1}{2}\\left(\\frac{E_\\mathrm {rel}-57}{12}\\right)^2}\\right] \\\\\\mbox{Reaction \\ref {o_chex}} :&\\ \\mathrm {H_2O + O^+} \\rightarrow \\mathrm {H_2O^+ + O^*} \\nonumber \\\\\\sigma _\\mathrm {O} =&\\ 69E_\\mathrm {rel}^{-0.29}+30\\exp \\left[{-\\frac{1}{2}\\left(\\frac{E_\\mathrm {rel}-65}{18}\\right)^2}\\right].$ The Gaussian terms in Eqs.","and account for downward and upward trends in the associated data sets near 30 $\\textrm {km\\ s}^{-1}$ , but have little consequence on the neutral cloud, given that most bound particles are produced at lower velocities.","Symmetric exchanges occur between like species by definition, although unlike species also exhibit symmetric behavior on occasion.","Therefore, we explore several hypothetical behaviors for the OH$^*$ -producing reaction at low energies.","This test is separate from, but related to, the comparison between reactions REF – themselves, and it motivates the point that both high and low energy behaviors have an important effect on the neutral cloud.","With $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ , we have extropolated the best-fit curve (Eq. )","to the lowest energies.","Symmetric and asymmetric behaviors are explored with $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ and $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ [27], [17].","$\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ is the same as $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ except that below 1.5 eV, $\\sigma ^\\mathrm {symmetric}_\\mathrm {OH}=30E_\\mathrm {rel}^{-0.5}\\times 10^{-16}$ cm$^2$ .","Notice that a similar energy dependence also applies to Eq.","REF , consistent with symmetric charge exchange.","$\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ is the same as $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ except that below 1.5 eV, $\\sigma ^\\mathrm {asymmetric}_\\mathrm {OH}=11E_\\mathrm {rel}^2\\times 10^{-16}$ cm$^2$ .","Although it could be argued that $\\sigma _\\mathrm {OH}^\\mathrm {symmetric}$ better fits the data if the two measurements at 2 eV are ignored, our results for reaction were obtained with $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ unless noted otherwise.","We will discuss the implications of choosing $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ over $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ and $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ shortly.","The collision frequencies ($n\\sigma v$ ) are plotted in Fig.","REF b for a given neutral density—in this case for $n_\\mathrm {H_2O}=10^3$ cm$^{-3}$ .","The collision frequency for oxygen increases with relative speed, while it is constant for water, and peaks at low velocities for OH.","The significance is that the oxygen cloud tends to be more extended than either the OH or H$_2$ O clouds.","The average collision frequency is also much higher for oxygen ($\\times $ 10) than for either OH or H$_2$ O, resulting in greater oxygen abundance.","The equatorial neutral cloud densities resulting from reactions REF – are plotted in Fig.","REF .","Only neutrals produced from charge exchange are shown; neither the Enceladus neutral torus, nor the neutrals produced via dissociation have been included.","Oxygen is two orders of magnitude more abundant than either OH or H$_2$ O because of the higher rate of production, but also because oxygen has a longer lifetime against photoionization than either OH or H$_2$ O have against photodissociation.","Unlike [5], dissociated neutrals from the latter processes are not tracked in our model.","Beyond the scope of the present study, this additional heating source would serve to further inflate the oxygen and OH clouds.","Fig.","REF b is the same as REF a, except that the profiles are normalized to the peak density at the orbit of Enceladus.","The oxygen cloud is seen to be the most extended, followed by water, and finally by OH, with an order of magnitude separating the three species at 20 $\\mathrm {R_S}$ .","The effects of low-velocity charge exchange are shown in Fig.","REF .","In Fig.","REF a, we see that the peak density (as well as the total neutral cloud content) is the highest with $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ because more low-velocity neutrals are produced than with either $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ or $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ .","Conversely, fewer low-velocity neutrals are available to populate the region near Enceladus's orbit with $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ when compared to either $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ or $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ .","Stated another way, $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ yields a neutral cloud with the steepest slope, and $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ , the shallowest.","Fig.","REF b is identical to Fig.","REF a, apart from normalization.","In this case, the slope of the density profile should not be confused with the effect of inflating (spreading) the OH cloud.","It should be viewed, rather, as the enhancement or depletion of low velocity neutrals to fill the region inside of $\\approx 10$ $\\textrm {R}_\\textrm {S}$ .","In other words, neutrals beyond 10 $\\textrm {R}_\\textrm {S}$ are mostly formed in charge exchanges at high velocities, for which all $\\sigma _\\mathrm {OH}$ converge to the same curve (Fig.","REF ).","We have assumed to this point that the plasma is sub-corotating in Enceladus's orbit (18 $\\textrm {km\\ s}^{-1}$ , [44]).","One might expect, however, that the neutral clould would be affected in a measurable way if instead, the plasma corotates at 26 $\\textrm {km\\ s}^{-1}$ .","The H$_2$ O cloud would be least affected, given that the collision frequency of reaction REF is independent of speed (Fig.","REF b), but what about reactions such as and , whose collision frequencies are velocity-dependent?","Increasing the plasma speed amounts to shifting the spread of ion velocities in Fig.","REF to the right, which would on average increase the speed of the neutral products.","This is indeed the case, and in such a test where we increased the plasma speed from 18 to 26 $\\textrm {km\\ s}^{-1}$ , the oxygen cloud increased in abundance and became even more extended.","The OH cloud also expanded somewhat, but decreased in total abundance.","Unfortunately, the differences were less than 10% in both the slope of the distribution and in total oxygen abundance, suggesting that neutral cloud observations are in this way unlikely to predict plasma speeds in the torus.","We described in section REF the production in our model of the neutral H$_2$ O torus from the Enceladus plumes.","The plumes themselves have also been prescribed as a separate background density ($n_\\mathrm {plume}$ , section REF ) so that we can compare charge exchange occurring throughout the neutral torus to that occurring only within the Enceladus plumes.","The results are shown in Fig.","REF , where we have plotted the oxygen clouds produced from charge exchange within both the Enceladus plumes (local) and the entire neutral torus (global).","The results are for reaction , but the same test with reactions REF and produces similar results.","Immediately noticable is that the local production is $\\approx 0.1$ % of the overall neutral production.","The torus's dominance of neutral production can be explained as follows.","First, the volume of the torus where reactions are occurring can be estimated as $2\\pi (4\\mathrm {R_S})(0.2\\mathrm {R_S})^2$ , where $0.1\\mathrm {R_S}$ is roughly the torus's scale height.","The volume of the plume can be estimated from Eq.","REF , where the dimensions are on the order of a cylinder with width 2$\\mathrm {R_E}$ and height $H_r\\approx 16\\mathrm {R_E}$ .","Dividing these volumes gives roughly $250(\\mathrm {R_E}/\\mathrm {R_S})^3\\approx 10^{-5}$ .","Further, the collision frequencies are proportional to the neutral density, which in the plume are on the order of 10$^7$ cm$^{-3}$ , whereas typical torus densities are 10$^5$ cm$^{-3}$ , making collisions in the plume 100$\\times $ more frequent per volume than in the torus.","All told, the ratio of the volumes ($10^{-5}$ ) combined with the ratio of densities ($10^2$ ) explain the local-to-global neutral production ratio of $10^{-3}$ shown in Fig.","REF a.","A similar pattern has been shown to exist at Jupiter by [2] and [7], where the majority of plasma is produced throughout the neutral torus, rather than near the interaction at Io itself.","The slopes of the neutral clouds from the plume and torus are most easily compared in Fig.","REF b, in which the density profiles have been normalized.","The local source produces a more confined neutral cloud because the ions from which they originate have been slowed near the plume to account for the effect of mass-loading ([10]).","Nevertheless, such a signature would be difficult to untangle in the data since global exceeds local production so overwhelmingly."],["Dissociation","A major component of the OH cloud is produced by dissociation within the neutral torus, whereby the initial velocities of the OH products range from 1 to 1.6 $\\textrm {km\\ s}^{-1}$ [45], [24].","In Fig.","REF , the clouds resulting from the high- and low-speed cases are plotted along with the result from velocity-dependent charge exchange in section REF .","First note that dissociation contributes 100$\\times $ more OH than does charge exchange at the orbit of Enceladus (4 $\\textrm {R}_\\textrm {S}$ ); the total cloud mass is almost 100$\\times $ greater as well.","Second, dissociation dominates over charge exchange from the Enceladus torus out to 9 and 15 $\\textrm {R}_\\textrm {S}$ in the low- and high-speed cases, respectively.","The OH cloud content will only be marginally affected by variable solar activity [16], given that impact dissociation contributes 4$\\times $ more neutrals than does photodissociation, by virtue of the respective reaction rates (section REF ).","In both cases, few neutrals are absorbed by the rings, and even less by Saturn itself.","The same is not true of charge exchange, where $\\approx 50$ % of the neutrals are absorbed by Saturn (section REF ).","Fig.","REF c is a two-dimensional version of Fig.","REF , where the dissociation results have been averaged and added to the results from charge exchange.","Saturn is at the left, and the Enceladus's orbit is located on the equator at 4 $\\textrm {R}_\\textrm {S}$ .","In addition to being confined radially, the dissociated neutrals are also bound tightly to the equator, while neutrals from charge exchange tenuously fill the magnetosphere.","Fig.","REF a shows the hydrogen cloud that accompanies the dissociated OH clouds ($\\mathrm {H_2O + e,\\gamma }\\rightarrow \\mathrm {OH^*+H^*}$ ).","To conserve momentum, the hydrogen atoms have 17$\\times $ the speed of the dissociated OH molecules, and thus range between 17 and 27 $\\textrm {km\\ s}^{-1}$ , with a relatively large, diffuse neutral cloud.","Shown is the result for the low-speed case, which produces more bound particles and thus a more substantial neutral cloud.","Charge exchange from reactions such as $\\mathrm {H_2O+H^+}\\rightarrow \\mathrm {H_2O^++H^*}$ are also responsible for H-cloud production, and deserve attention in future studies."],["Fates of neutral atoms and molecules","In our model, neutrals created by dissociation and charge exchange are eventually either absorbed by Saturn, escape the system, or orbit until they are destroyed (ionized) by photons.","In Fig.","REF a the fates for each species are given by percentage.","In the case of hydrogen, the results are from the dissociation model, described in section REF .","The enormous amount of escape (84%) is due to the high velocities ($\\approx 17$ $\\textrm {km\\ s}^{-1}$ ) with which hydrogen is created following H$_2$ O dissociation, and the 8% absorption is largely comprised of hydrogen which would escape the system otherwise.","Oxygen is produced purely from charge exchange in our model (reaction ).","About one-half escapes, one-third is absorbed, and the remaining 13% contributes to the neutral cloud before being photoionized.","Water is also produced purely by charge exchange (reaction REF ) with 18% contributing to the neutral cloud.","Percentage-wise, more water is absorbed than oxygen because oxygen is produced with higher speeds and generally larger orbits (section REF ).","The fate of OH is dominated by dissociation: 96% feed the neutral cloud (ultimately ionized), 4% are absorbed, and virtually none escape.","The reason for the large percentage of bound and unabsorbed neutrals is that dissociated OH has a velocity spread of 1 to 1.6 $\\textrm {km\\ s}^{-1}$ in the neutral frame, compared to the escape speed of $\\approx 5$ $\\textrm {km\\ s}^{-1}$ in the same frame.","Looking only at OH produced by charge exchange (minor compared to dissociation), 58% are absorbed, 23% supply the neutral cloud, and 20% escape.","Compared to H$_2$ O, an even greater percentage of charge-exchanged OH is absorbed because the cross sections favor production of low-velocity OH molecules (Fig.","REF b).","The production of oxygen via dissociation of H$_2$ O has been ignored in this paper on the grounds that, unlike OH, oxygen is largely produced by charge exchange.","The cross section for oxygen-producing charge exchange is an order of magnitude higher than that for the OH-producing reaction near the plasma flow speed of $v_\\mathrm {plasma}=18$ $\\textrm {km\\ s}^{-1}$ (Fig.","REF a), while the photodissociation rates are an order of magnitude smaller [15].","We estimate that including oxygen produced from dissociation would increase the total oxygen cloud content by less than 20%.","Charge exchange and dissociation play a large role in creating Saturn's neutral clouds from the plume-fed neutral torus.","The reactions we have included have been chosen to demonstrate the effects of low velocity charge exchange and dissociation, but they are also among the most important.","The neutral cloud densities presented in this paper are expected to undershoot the results from models which include the additional reactions found in Fig.","3, of [11] by no more than a factor of two.","With this caveat in mind, we now compare the present results with several other recent models."],["Comparison with other models","Fig.","REF b: J06 is the work of [18], where they also investigated the neutral clouds created from low-velocity charge exchange in the stagnated flows in Enceladus's orbit.","Fig.","REF b: J07 is from [20], where the authors were primarily interested in the interaction between the neutral cloud and Saturn's rings.","The most recent model comes from [5] (C10), where they investigated the spreading of the neutral cloud from neutral–neutral collisions.","To compare with these studies, we first had to weight our H, O, OH, and H$_2$ O clouds.","We did this for two limiting cases.","In the first case ($\\tau _\\mathrm {phot}$ , Fig.","REF b), we assume, as we have thus far, that the neutral clouds evolve until destroyed by either photoionization or photodissociation: H, O, OH, H$_2$ O = 40, 14, 0.6, 0.3 years, respectively.","These lifetimes yield an upper limit since charge exchange and electron impact are not included as losses.","In the second case ($\\tau _\\mathrm {all}$ ) , we derived a lower limit to the lifetimes from Table 2 of [11] by summing the additional losses due to charge exchange and electron impact, finding: H, O, OH, H$_2$ O = 0.4, 0.4, 0.2, 0.03 years, respectively.","Notice in particular the drastically different times scales for H and O, where including the additional sinks reduce the size of the H cloud by a factor of 40/0.4 = 100, and the oxygen cloud by 14/0.4 = 35.","This case represents an extreme limit, given that the neutrals spend almost all of their time orbiting outside of the Enceladus torus, where compared to photo-processes, the chances of charge exchange and electron impact are relatively unlikely.","We mention, however, that [30], [29] has shown that circulation patterns inside of 12 $\\textrm {R}_\\textrm {S}$ at Saturn gives rise to `butterfly' hot electron pitch angle distributions, related to low temperature anisotropy ($T_\\perp /T_\\parallel $ ), on which proton field-aligned distributions depend [36].","The individual clouds (excluding hydrogen) were weighted by the stated time scales and totaled in Fig.","REF b.","When only losses to photodissociation/ionization are considered ($\\tau _\\mathrm {phot}$ ), the neutral cloud is dominated by oxygen, whose fate thus determines that of the neutral cloud.","When charge exchange and electron impact are also included ($\\tau _\\mathrm {all}$ ), dissociated OH contributes significantly, driving the neutral cloud (ionized) percentage up, and the escape percentage down.","We note that the neutral fates presented in [3] (escape = 44%, ionized = 17%, absorbed = 39%) were based on an earlier version of our model which only included H$_2$ O.","The particles that are neither absorbed nor lost by escape make up the neutral clouds.","In the case where the cloud evolves for $\\tau _\\mathrm {phot}$ , oxygen and hydrogen dominate since they are far less likely to be photoionized than are OH and H$_2$ O to be photodissociated.","With charge exchange and electron impact included ($\\tau _\\mathrm {all}$ ), however, more oxygen and hydrogen are removed from the system, which then tends to favor a molecular OH–H$_2$ O cloud.","In terms of total mass the same applies, although hydrogen accounts for only a few percent at most.","We find that the total cloud mass is bounded between $\\approx 1$ and 10 Mtons, for $\\tau _\\mathrm {all}$ and $\\tau _\\mathrm {phot}$ , respectively.","It is worth pausing to re-emphasize that the system is in reality better represented by the $\\tau _\\mathrm {phot}$ case, from which all neutral clouds in this paper have been derived.","The $\\tau _\\mathrm {all}$ case is strictly valid only for neutrals within the Enceladus torus, though reactions with electrons and protons may also prove important, as discussed above.","What is illustrated, however, is that Saturn's magnetosphere is less oxygen-dominated than suggested by looking at losses from photo-processes alone.","These results suggest that our oxygen abundances are somewhat overestimated, likely by less than a factor of two."],["Neutral absorption","The particles absorbed by Saturn and its rings are plotted by species and latitude in Fig.","REF .","In Fig.","REF b, we see that most absorption comes from oxygen (74%), followed by H$_2$ O (11%), OH (9%), and finally by hydrogen (6%).","Absorption is equally divided between Saturn and its rings except in the case of OH, where twice as much falls on Saturn's rings.","This is because OH is largely produced by impact dissociation, which creates slower neutrals than does charge exchange, whereby in our model, H$_2$ O and oxygen arise exclusively.","In Fig.","REF b, absorption is plotted against Saturn's latitude.","Because the model is symmetric about the equator, the results apply to either hemisphere.","Oxygen, water, and OH follow the same trends because they all originate from charge exchange (dissociated OH is slow and does not reach Saturn), and have been created from ions with similar velocity distributions.","Any second-order differences due to the velocity-dependence of the respective cross sections are not immediately apparent.","Hydrogen, on the other hand, is produced entirely by dissociation in the model and exhibits a more uniform flux across Saturn.","The explanation is that the velocity distribution from which hydrogen is produced is isotropic, whereas that which produces charge-exchanged neutrals is bi-Maxwellian (section REF ).","The fluxes shown in Fig.","REF b are consistent with [14], who modeled recent Herschel observations of Saturn's water torus and found an average flux of $6\\times 10^5$ cm$^{-2}$ s$^{-1}$ for H$_2$ O + OH impinging on Saturn."],["Discussion","Some useful conclusions can be drawn by further contrasting our results with [5] (C10).","It is important that we first mention a profound difference between our models.","The model of C10 effectively carries out resonant charge exchange only, which does not chemically alter the neutral population; neutrals in their model are produced either directly from Enceladus or from subsequent dissociations.","Neutrals in our model, on the other hand, originate from Enceladus (H$_2$ O).","OH is then created $via$ dissociation (as with C10), but secondary O, OH, and H$_2$ O populations are $created$ from H$_2$ O $via$ charge exchange with the dense plume-fed Enceladus torus.","The C10 model redistributes neutrals around Saturn, while we redistribute and chemically re-assign neutral abundances by allowing for asymmetric charge exchanges.","Thus, it may well be a coincidence that our models are similar in total abundance.","While it may be difficult to compare our total abundances, the slope of our radial density profiles can be contrasted directly because our redistribution mechanisms (charge exchange and dissociation) are similar.","Differences are due largely to C10's inclusion of neutral collisions and our prescribing unique velocity-dependent charge exchange for each of the O-, OH-, and H$_2$ O-producing reactions (reactions REF –).","Our neutral clouds are compared with C10 in Fig.","REF .","All of our clouds include contributions from charge exchange, but the H$_2$ O cloud is mostly comprised of water sourced directly from Enceladus (3.95 $\\textrm {R}_\\textrm {S}$ ), and OH includes the additional source from dissociation.","In the C10 model, the water molecules were spread due to neutral–neutral collisions, which explains our higher H$_2$ O densities near Enceladus's orbit (Fig.","REF a).","The slope of the oxygen profile agrees best with C10 because their charge exchange cross section most resembles our own (Eq.",").","Our H$_2$ O profiles agree less, and our OH slopes, the least, due mainly to the strong effect that neutral collisions have on those more polar molecules.","In particular, C10 used a much larger cross section for neutral collisions involving H$_2$ O and OH [42] than for atomic oxygen [4].","This helps to further explain our agreement with their oxygen profile since we exclude neutral–neutral collisions from our model altogether.","We conclude that neutral–neutral collisions appear to play a less significant role with atomic species, such as oxygen and hydrogen.","The column densities (Fig.","REF b) are similar to C10, who constrained their O and OH clouds with the most recent Cassini UVIS results of [25].","Our oxygen density—as well as our total oxygen content (Fig.","REF c)—is higher for two reasons.","First, we use a larger cross section than does C10 for reaction , and second, the clouds presented here have been limited only by photoionization.","Charge exchange and electron impact are second order losses beyond 6 $\\mathrm {R_S}$ , but including them would favorably reduce the oxygen content more than OH and H$_2$ O (section REF ), bringing our models into better agreement.","Our total H$_2$ O content is 4$\\times $ less than C10 found (Fig.","REF c).","This is partly because we have subjected H$_2$ O molecules in the primary (plume-fed) neutral torus to the shortest lifetimes possible (section REF ), whereas C10 tracks molecules that get kicked out of the densest plasma $via$ neutral collisions, and thus survive longer, being less susceptible to both charge exchange and electron impact.","That their total H$_2$ O content is higher than ours (Fig.","REF c), does not contradict the fact that their H$_2$ O column density is lower; neutral–neutral collisions would spread out the torus, lowering the column density, while allowing neutrals to survive longer, increasing the total abundance.","Our model would benefit by including the redistribution attributed to neutral collisions by allowing particles to interact in a direct simulation Monte Carlo (DSMC) model such as in C10.","Likewise, DSMC models would benefit by including charge exchange cross sections specific to each reaction.","Such models should also take into account asymmetric charge exchanges, which affects neutral cloud composition.","The reactions modeled in this study were chosen in order to measure the effect of symmetric and asymmetric charge exchange at low velocities.","Building upon our findings, future studies should include additional neutral-producing charge exchanges, such as $\\mathrm {OH^+ + H_2O} \\rightarrow \\mathrm {OH^* + H_2O^+}$ , $\\mathrm {H^+ + H_2O} \\rightarrow \\mathrm {H^* + H_2O^+}$ , and $\\mathrm {OH^+ + H_2O} \\rightarrow \\mathrm {O^* + H_3O^+}$ , as well as dissociative recombination of H$_2$ O$^+$ ."],["Conclusions","We have modeled low-velocity charge exchange from the point of view of the ions, allowing us to study the effects of velocity as well as gyrophase.","With reactions REF –, we have been able to offer an estimate on the size and shape of the neutral clouds at Saturn, while simultaneously exploring the sensitivity of the neutral clouds to a variety of velocity-dependent reactions.","We have also re-visited the production of OH following H$_2$ O dissociation in the primary neutral torus.","Previous models have used 1 $\\textrm {km\\ s}^{-1}$ as the initial velocity for OH, while measurements suggest a range of speeds from 1 to 1.6 $\\textrm {km\\ s}^{-1}$ .","In our model, the higher speed increases the range within which dissociation dominates neutral production from 9 to 15 $\\textrm {R}_\\textrm {S}$ .","Additional findings are: (1.)","Charge exchange cross sections that increase steeply at low speeds tend to produce neutral clouds more confined to the orbit of Enceladus, implying the most spreading for oxygen, moderate spreading for H$_2$ O, and the least for OH (Fig.","REF ).","Accounting for gyrophase doubles the local OH density within Encelacus's orbit, has $\\approx $ no effect on H$_2$ O, and decreases oxygen density by less than 10%.","(2.)","Enceladus is solely responsible for the creation of the neutral H$_2$ O torus $via$ thermal ejection from its plumes.","However, Saturn's neutral clouds are overwhelmingly produced by charge exchange and dissociation occurring throughout the torus (99%), and not near Enceladus itself (Fig.","REF ).","(3.)","We estimate that roughly half of all neutrals escape the system, with the remaining equally divided between absorption by the rings/planet and the neutral clouds (Fig.","REF ).","Less than 50 $\\textrm {kg\\ s}^{-1}$ is thus ionized and transported out of the system as plasma.","This number is expected to represent an upper limit, given we have assumed that all particles forming the neutral clouds are ultimately ionized; a more accurate result would require modeling the detailed effects of charge exchange and neutral–neutral collisions within the neutral clouds.","This estimate can be compared to [36], whose Figs.","14 and 17 give roughly $([NL^2]_\\mathrm {W^+}/L^2)\\times m_\\mathrm {W^+}/\\tau _\\mathrm {transport}\\approx 3\\times 10^{31}\\times m_\\mathrm {W^+}/10^5$ s $\\approx 10$ $\\textrm {kg\\ s}^{-1}$ at $L=10$ .","(4.)","Saturn's neutral cloud has a total mass of at least 1 Mton, but likely much closer to 10 Mtons.","The primary plume-fed neutral torus (0.3 Mtons) is comprised entirely of water in our model, while the secondary neutral clouds are broken down into H ($\\lesssim 5$ %), O ($\\lesssim 82$ %), OH ($\\gtrsim 13$ %), and H$_2$ O ($\\approx 1$ %).","Atomic oxygen dominates the composition both because of a high production rate from charge exchange as well as a long lifetime against photoionization.","Charge exchange and reactions with electrons favorably remove hydrogen and oxygen, but are secondary loss mechanisms throughout the majority of the magnetosphere.","(5.)","Our model predicts fluxes on Saturn from charge exchange of $\\approx 6\\times 10^5$ cm$^{-2}$ s$^{-1}$ for both OH and H$_2$ O (consistent with Herschel observations by [14]), and oxygen is about 5$\\times $ higher.","Absorption is divided equally between Saturn and its rings (Fig.","REF a).","(6.)","Our total neutral abundances are similar to [5] (C10) for both OH and H$_2$ O, and 4$\\times $ higher for oxygen (Fig.","REF ).","Differences in the slopes of our equatorial density profiles are in part due to our not including neutral–neutral collisions, while this fact appears to have no effect on the oxygen profile.","On the other hand, C10 did not include the effects on neutral chemistry following asymmetric charge exchanges, nor did they use velocity-dependent cross sections particular to each reaction.","Herschel observations by [14] confirm the importance of neutral–neutral collisions for H$_2$ O, but if oxygen is the dominant neutral species in Saturn's magnetosphere, as our model predicts, neutral–neutral collisions may play a smaller role in Saturn's neutral cloud than previously expected.","Given the effect on both the size and shape of the neutral clouds, we suggest that future neutral cloud models include charge exchange cross sections unique to each reaction.","Asymmetric charge exchange also has an important effect on neutral chemistry that should be implemented.","Regarding the ions' gyrophase, Monte Carlo models can account for its effect by using phase-dependent probability distributions.","Finally, the range of OH velocities studied here should be considered when modeling dissociation.","Moving forward, we plan to implement these suggestions into the neutral cloud model of C10 and to couple that model with the chemistry model of [11].","Constrained by Cassini plasma observations, the chemistry model uses C10's neutrals as input, and provides ion temperatures and densities throughout the magnetosphere ($<$ 20 $\\textrm {R}_\\textrm {S}$ ), which C10 in turn uses to update neutral densities.","An improved understanding of two issues is planned: (1) Where does plasma transport become important?","(2) What is the role of hot electrons with regard to ion–neutral chemistry inside 20 $\\textrm {R}_\\textrm {S}$ ?"],["Collision probability","The average number of collisions occurring during a time interval $\\Delta t$ is given by $\\lambda \\equiv \\nu \\Delta t$ , where $\\nu =n_\\mathrm {neutrals}\\sigma (v_\\mathrm {rel})v_\\mathrm {rel}$ is the local collision frequency, assumed to be constant during $\\Delta t$ .","Statistics are applied to determine if and how many reactions occur during $\\Delta t$ .","The Poisson distribution [47], [28] gives the probability of suffering exactly $n$ collisions for a given $\\lambda $ : $ f(n;\\lambda )=\\frac{e^{-\\lambda }\\lambda ^n}{n!","}.$ Notice that Eq.","REF peaks at $n=\\lambda $ if one treats $n$ as a continuous variable.","Summing Eq.","REF discretely from $n=k$ to $n=\\infty $ gives the probability of suffering at least $k$ collisions during $\\Delta t$ , $ P_k(\\lambda )=e^{-\\lambda }\\sum _{n=k}^\\infty \\frac{\\lambda ^n}{n!","}.$ Because Eq.","REF is normalized ($e^{-\\lambda }\\sum _{n=0}^\\infty \\lambda ^n/n!=e^{-\\lambda }e^\\lambda =1$ ), Eq.","REF can be conveniently rewritten as $ P_k(\\lambda )=1-e^{-\\lambda }\\sum _{n=0}^{k-1} \\frac{\\lambda ^n}{n!","}.$ A random number ($0 N$ determines how many fast neutrals (collisions), $k$ , are produced during $\\Delta t$ .","In practice, it is only necessary to compare to the first few $P_k$ when $\\lambda \\ll 1$ , made evident by the leading terms in Eq.","REF for $k+1$ and $k$ : $ \\frac{P_{k+1}}{P_k} \\approx \\frac{f(k+1;\\lambda )}{f(k;\\lambda )}= \\frac{\\lambda ^{k+1}/(k+1)!","}{\\lambda ^k/k!","}=\\frac{\\lambda }{k+1}\\xrightarrow[\\lambda \\rightarrow 0]{}0.$ Multiple collisions are thus increasingly unlikely when $\\lambda \\ll 1$ .","In such cases, comparison with $P_1=1-e^{-\\lambda }\\approx \\lambda = \\nu \\Delta t$ is sufficient.","This work was supported under the NESSF program, fellowship number 11-Planet11R-0005.","BF thanks two reviewers for their feedback and many useful suggestions.","Figure: Sketch of gyrating ions in the neutral framewith guiding centers moving along a prescribedflow field, shown here near Enceladus for scale.Warm ions (v ⊥ >v flow v_\\perp > v_\\mathrm {flow}) move on trajectories that coilaround themselves and do not reach zero relative velocity with respect to theneutrals at any point.Cool ions (v ⊥ v_\\\\mathrm {therm}>v_\\\\mathrm {esc}$ ; our results differ by less than one percent whether Enceladus's gravity is considered or not.\",\n \"Particles released from Enceladus are thus assumed to move on Keplerian orbits with respect to Saturn.\",\n \"Each water molecule is allowed to orbit inside the torus for a period determined by the collective lifetimes against photodissociation, electron-impact dissociation, and charge exchange.\",\n \"To be clear, the molecules forming the neutral torus are subject to all of the losses stated, while the neutral cloud is subject to photo-processes only (section REF ).\",\n \"Further reaction details are important for modeling plasma characteristics but only the timescales given below are required to model the neutral torus.\",\n \"The photodissociation lifetime for H$_2$ O, $\\\\tau _\\\\mathrm {phot}=9.1\\\\times 10^6$ s, comes directly from [15], scaled to Saturn's distance from the Sun.\",\n \"At peak solar activity, neutral abundances attributed to dissociation double [16].\"\n ],\n [\n \"Impact dissociation\",\n \"In the case of impact dissociation, suprathermal (hot) electrons dominate [11].\",\n \"Assuming the conditions near Enceladus's orbit apply throughout the neutral torus, we estimate the hot electron density and temperature to be 160 eV and 0.3 cm$^{-3}$ from [11], which fit within a range of recent observations (cf., [46], [36]).\",\n \"The rate coefficient for impact dissocation of water is found by convolving $\\\\sigma (v)v$ with a 160 eV Maxwellian distribution to find $\\\\kappa _\\\\mathrm {imp}=1.5\\\\times 10^{-6}\\\\mbox{ cm$^3$ s$^{-1}$}$ ([11], table S9).\",\n \"Above 100 eV, $\\\\kappa _\\\\mathrm {imp} $ is insensitive to temperature, making this estimate valid for a range of observations.\",\n \"Assuming the hot electron density, $n_\\\\mathrm {eh}$ , is constant over the neutral torus, the lifetime against impact dissociation is $\\\\tau _\\\\mathrm {imp}\\\\approx [\\\\kappa _\\\\mathrm {imp}n_\\\\mathrm {eh}]^{-1}=2.2 \\\\times 10^6$ s. Dissociation $via$ thermal electrons—whose temperature and density are 2 eV and 60 cm$^{-3}$ —is also expected, but such collisions occur 5$\\\\times $ less often ([11], table S9) and are thus ignored here.\"\n ],\n [\n \"Charge exchange\",\n \"The following three reactions are included in this study: $\\\\mathrm {H_2O+H_2O^+} &\\\\rightarrow \\\\mathrm {H_2O^+ +H_2O^*} \\\\\\\\\\\\mathrm {H_2O+H_2O^+} &\\\\rightarrow \\\\mathrm {H_3O^+ +OH^*} \\\\\\\\\\\\mathrm {H_2O+O^+} &\\\\rightarrow \\\\mathrm {H_2O^++O^*} .$ Other charge exchanges are important in the neutral torus, some of which involve ions reacting with secondary neutrals such as H, O, and OH.\",\n \"To model their effect properly, one would calculate these neutral densities as in a conventional time-dependent neutral cloud model.\",\n \"We estimate that including all such reactions would increase our estimates on neutral cloud densities by approximately a factor of two.\",\n \"The primary purpose of choosing this combination of reactions is to study three classes of charge exchanges, for which the collision frequency decreases with, increases with, or is independent of the relative speed of the reacting pair (reactions , , and REF , respectively).\",\n \"We return to this point in section REF .\",\n \"An estimate of the charge exchange lifetime can be made by adding the rate coefficients for reactions REF –.\",\n \"Multiplying by the observed H$_2$ O$^+$ and O$^+$ densities near the orbit of Enceladus (6 and 12 cm$^{-3}$ , [36]), we find $\\\\tau _\\\\mathrm {chex}=[\\\\kappa _\\\\mathrm {exch}^\\\\mathrm {H_2O^+}n_\\\\mathrm {H_2O^+}+\\\\kappa _\\\\mathrm {exch}^\\\\mathrm {O^+}n_\\\\mathrm {O^+}]^{-1}=[6.0+2.8]^{-1}\\\\times 10^8 \\\\mathrm {\\\\ s\\\\ }=1.1\\\\times 10^7$ s. The reaction rates are from [23] and [1] for $\\\\kappa ^\\\\mathrm {H_2O^+}_\\\\mathrm {exch}$ and $\\\\kappa ^\\\\mathrm {O^+}_\\\\mathrm {exch}$ , respectively.\",\n \"The lifetime of H$_2$ O against the sum of these processes is then $\\\\tau _\\\\mathrm {torus}=\\\\left[\\\\frac{1}{\\\\tau _\\\\mathrm {phot}} +\\\\frac{1}{\\\\tau _\\\\mathrm {imp}} +\\\\frac{1}{\\\\tau _\\\\mathrm {chex}}\\\\right]^{-1} &=&(1.1+4.5+0.91)^{-1}\\\\times 10^7 \\\\mathrm {\\\\ s} \\\\nonumber \\\\\\\\&=& 1.6\\\\times 10^6 \\\\mathrm {\\\\ s}\\\\approx 20 \\\\mbox{ days.}\",\n \"$ These estimates can be compared to Fig.\",\n \"3 of [5]: for example, our lifetime against dissociation is 1.7 $\\\\times 10^6$ s, while they use $7\\\\times 10^6$ s near 4 $\\\\textrm {R}_\\\\textrm {S}$ .\",\n \"The discrepancy comes mostly from the impact dissociation timescale.\",\n \"Cassidy's dissociation rate was calculated using [32], wherein CAPS ELS data were fitted and extapolated down from 5.5 $\\\\textrm {R}_\\\\textrm {S}$ , while our own estimate hinges on a hot electron density derived from our chemistry model [11].\",\n \"For charge exchange, we have a lifetime of 1.1 $\\\\times 10^7$ s, and [5] have a comparable $8\\\\times 10^6$ s. Particles are created and tracked in each of our model runs, and the results are scaled to the number of water molecules in the real neutral torus.\",\n \"The total number is estimated from an assumed neutral source rate from Enceladus of $\\\\dot{M}=200\\\\ \\\\textrm {kg\\\\ s}^{-1}$ [19], [12], [13] and lifetime, $\\\\tau _\\\\mathrm {torus}$ (Eq.\",\n \"REF ): $N_\\\\mathrm {torus}=\\\\dot{M}\\\\tau _\\\\mathrm {torus}/m_\\\\mathrm {H_{2}O}\\\\approx 1.1\\\\times 10^{34} \\\\mathrm {\\\\ H_2O\\\\ molecules}.$ We bin and azimuthally average the results to find a 2-D density function, $n_\\\\mathrm {torus}(r,\\\\theta )$ (radius and latitude), through which to introduce ions for charge exchange.\",\n \"This function also determines from where dissociated neutrals are produced.\"\n ],\n [\n \"Plume model\",\n \"Before describing dissociation and charge exchange within the torus, we address a calculation with the purpose of comparing the neutral production near Enceladus with that from the entire torus.\",\n \"In doing so, we prescribe a plume whose density is consistent with Eqs.\",\n \"REF and REF .\",\n \"In this case, the ambient neutral density can be ignored compared to the neutrals leaving the surface of Enceladus directly.\",\n \"The densities are determined everywhere by imposing integrated flux ($\\\\int n(r,\\\\theta )v(r,\\\\theta ) dA$ ) and energy ($mv^2/2-mM_\\\\mathrm {E}G/r$ ) conservation at a given distance $r$ .\",\n \"The picture can be simplified, however, since most neutrals have at least twice Enceladus's escape velocity.\",\n \"The speeds are thus independent of $r$ in the immediate vicinity of Enceladus.\",\n \"By equating the integrated flux at the surface of Enceladus to the same integral at another distance $r>\\\\mathrm {R_E}$ , we find a familiar $1/r^2$ dependence: $n_\\\\mathrm {plume}(r,\\\\theta )=n(\\\\theta )\\\\left(\\\\frac{\\\\mathrm {R_E}}{r}\\\\right)^2\\\\exp \\\\left[-\\\\left(\\\\frac{r-\\\\mathrm {R_E}}{H_r}\\\\right)\\\\right].$ The trailing exponential factor is imposed $ad$ $hoc$ to keep the total plume content finite, and reflects Saturn's influence as the molecules leave Enceladus.\",\n \"Consistent with [31], $H_r$ is set at 4 times the Hill radius of 948 km.\",\n \"The angular dependence is consistent with our velocity distribution in the previous section (Eq.\",\n \"REF ), $n(\\\\theta )=\\\\left\\\\lbrace \\\\begin{array}{ll}\\\\frac{n_0}{2}\\\\left[1+\\\\cos \\\\left(\\\\frac{\\\\theta }{\\\\theta _0}\\\\pi \\\\right)\\\\right]&\\\\mbox{ if $\\\\theta < \\\\theta _0=30^\\\\circ $} \\\\\\\\0 &\\\\mbox{ otherwise,}\\\\end{array} \\\\right.$ which is normalized such that $n(0)=n_0$ , where the plume strength $n_0$ is found to be $5.9\\\\times 10^8\\\\ \\\\mathrm {cm}^{-3}$ by integrating $n(\\\\theta )v_\\\\mathrm {bulk}$ over the area spanning the south pole of Enceladus from $\\\\theta =0^\\\\circ $ to $\\\\theta =\\\\theta _0=30^\\\\circ $ , and setting that result equal to the plume production rate of $\\\\dot{M}/m_\\\\mathrm {H_2O}=(200\\\\ \\\\textrm {kg\\\\ s}^{-1})/m_\\\\mathrm {H_2O}=6.7\\\\times 10^{27}$ molecules per second.\",\n \"In section , we compare the results of charge exchange with the plume ($n_\\\\mathrm {plume}$ ) to that with the entire torus ($n_\\\\mathrm {torus}$ ).\",\n \"Production of Saturn's neutral clouds entails following the neutrals produced by dissociation and charge exchange occurring within the neutral torus.\",\n \"The treatment of each of these processes are described below.\"\n ],\n [\n \"Dissociation\",\n \"The hydroxyl radical, OH, produced largely by dissociated H$_2$ O, has previously been modeled with a single speed of 1 $\\\\textrm {km\\\\ s}^{-1}$ (i.e., [19], [5]).\",\n \"Dissociated OH has been measured however with speeds between 1 and 1.6 $\\\\textrm {km\\\\ s}^{-1}$ [45], [24].\",\n \"Here we bound this range by modeling the OH neutral clouds produced from an azimuthally-symmetric source (with respect to Saturn) with velocities drawn from Maxwellian distributions with temperatures $T=\\\\frac{1}{2}m_\\\\mathrm {OH}v_\\\\mathrm {mp}^2$ , where the most probable speed, $v_\\\\mathrm {mp}$ , is set to 1 and 1.6 $\\\\textrm {km\\\\ s}^{-1}$ , representing the low- and high-speed limits.\",\n \"The initial locations of the ejected OH are determined by the spatial distribution of neutrals in the Enceladus torus ($n_\\\\mathrm {torus}(r,\\\\theta )$ , section REF ), and the directions of their release are chosen randomly and isotropically.\",\n \"The molecules orbit Saturn until they are photodissociated and removed from the system.\",\n \"By assuming a volume over which dissociations occur, the number of modeled OH molecules can be scaled to a realistic value.\",\n \"We take the volume to be a torus centered on Enceladus (3.95 $\\\\textrm {R}_\\\\textrm {S}$ ), with a minor radius of 1 $\\\\textrm {R}_\\\\textrm {S}$ : $ V\\\\approx 2\\\\pi (4\\\\,\\\\textrm {R}_\\\\textrm {S})(2\\\\,\\\\textrm {R}_\\\\textrm {S})^2=2\\\\times 10^{31}\\\\,\\\\mathrm {cm}^3.$ For impact dissociation, we then expect a contribution of $ N^\\\\mathrm {imp}_\\\\mathrm {cloud}=k_\\\\mathrm {imp}\\\\tau _\\\\mathrm {phot}^\\\\mathrm {OH}V=2.8\\\\times 10^{34} \\\\mbox{\\\\ OH molecules},$ where $k_\\\\mathrm {imp}=7.9\\\\times 10^{-5}$ cm$^{-3}$ s$^{-1}$ is the rate (per volume) of impact dissociations occurring between suprathermal electrons and H$_2$ O molecules in the torus [11] and $\\\\tau _\\\\mathrm {phot}^\\\\mathrm {OH}=1.8\\\\times 10^7$ s is the photodissociation lifetime of OH at Saturn [15].\",\n \"The number of OH molecules produced by photodissociation in the torus is similarly given by $ N^\\\\mathrm {phot}_\\\\mathrm {cloud}=k_\\\\mathrm {phot}\\\\tau _\\\\mathrm {phot}^\\\\mathrm {OH}V=7.6\\\\times 10^{33} \\\\mbox{\\\\ OH molecules},$ where $k_\\\\mathrm {phot}=2.1\\\\times 10^{-5}$ cm$^{-3}$ s$^{-1}$ is the rate (per volume) of H$_2$ O photodissociations occurring in the Enceladus torus ([11], Table S9).\",\n \"The total abundance attributed to dissociation is then given by the sum of Eqs.\",\n \"REF and REF .\",\n \"[5] constrained their study with HST observations [25] and found a similar OH content (see comparison in Fig.\",\n \"REF c, this paper).\",\n \"That neutral production by photo- and impact dissociation are comparable in magnitude is itself noteworthy.\",\n \"This condition is not shared by systems with hotter and denser plasma.\",\n \"For example, electron impact dissociation and ionization dominate over photon-driven processes in Jupiter's Io torus, where the plasma is warmer where the pick-up energies are four times higher than at Enceladus [6], [11].\",\n \"We also note that unlike with Io, long neutral lifetimes in the Enceladus neutral torus inhibit the response of Saturn's neutral clouds to short-term plume variability, though variability on the order of months has been studied by [39].\"\n ],\n [\n \"Charge exchange\",\n \"We now describe the model for producing and following neutrals from charge exchange.\",\n \"[5] and [19] also considered velocity-dependent charge exchange, but unlike these previous studies, we capture the gyrophase at which the reactions occur by following ions along their trajectories (section REF ).\",\n \"We also prescribe cross sections specific to each reaction, being particularly interested in the effects of low-velocity charge exchange.\",\n \"At very high speeds, the cross sections go to zero for all charge exchanges [17].\",\n \"At low relative velocities, however (few $\\\\textrm {km\\\\ s}^{-1}$ ), the details of the collision are determined by the nature of the reacting species.\",\n \"If the reactants and products are identical, apart from an electron ($i.e.$ , $\\\\mathrm {H_2O + H_2O^+}\\\\rightarrow \\\\mathrm {H_2O^+ + H_2O^*}$ ), the reaction is termed resonant, or symmetric, and the cross sections grow as the inverse of the relative speed.\",\n \"If the reactants differ, as with $\\\\mathrm {H_2O + O^+}\\\\rightarrow \\\\mathrm {H_2O^+ + O^*}$ , the cross sections are likely to vanish at low speeds [27]—the difference being that the energy of the electron configurations is unchanged for symmetric-type charge exchanges [17].\",\n \"Neutrals produced from resonant charge exchange therefore tend to have lower velocities than do neutrals produced from non-resonant (asymmetric) charge exchange.\",\n \"This is a key point central to much of our discussion in section .\",\n \"Individual ions are followed as they traverse the neutral torus (section REF ).\",\n \"This approach allows their gyrophase to be determined the instant that charge exchanges occur (see Fig.\",\n \"REF ).\",\n \"The implicit assumption is that the collision is elastic, and that the neutral product has an initial velocity given by the ion velocity just before the exchange takes place.\",\n \"The ions are introduced into the model from two Maxwellian speed distributions, $f_\\\\perp (v_\\\\perp )&=\\\\frac{m_\\\\mathrm {ion}}{kT_\\\\perp }v_\\\\perp \\\\exp \\\\left[-\\\\frac{m_\\\\mathrm {ion}v_\\\\perp ^2}{2kT_\\\\perp } \\\\right]\\\\mbox{ (speeds perpendicular to $B$)}\\\\\\\\f_\\\\parallel (v_\\\\parallel )&=\\\\sqrt{\\\\frac{m_\\\\mathrm {ion}}{2\\\\pi kT_\\\\parallel }}\\\\exp \\\\left[-\\\\frac{m_\\\\mathrm {ion}v_\\\\parallel ^2}{2kT_\\\\parallel } \\\\right]\\\\mbox{ (speeds parallel to $B$)}$ with a temperature anisotropy of $\\\\frac{kT_\\\\perp }{kT_\\\\parallel }=\\\\frac{27\\\\,\\\\mathrm {eV}}{5.4\\\\,\\\\mathrm {eV}}=5$ for both O$^+$ and H$_2$ O$^+$ [36].\",\n \"The perpendicular temperature is derived from the pick-up ion velocity at the orbit of Enceladus, determined from CAPS data by [44] [$kT_\\\\perp =\\\\frac{1}{2}m_\\\\mathrm {W^+}(v_\\\\phi -v_\\\\mathrm {Kep})^2$ ].\",\n \"The ions also rotate around a guiding center (field line) moving at $v_\\\\phi =18$ $\\\\textrm {km\\\\ s}^{-1}$ in a frame rotating with the neutrals.\",\n \"For the component of our study aimed at estimating local neutral production (section REF ), ions passing near Enceladus are diverted (treating Enceladus as a rigid cylinder) and are slowed to 10% of the ambient flow speed to account for the effects of mass-loading (see [10]).\",\n \"Time steps are taken at less than 1% of an ion's gyroperiod: $\\\\Delta t=R\\\\,\\\\times \\\\,T_\\\\mathrm {gyro}=R\\\\,\\\\times \\\\,\\\\frac{2\\\\pi m}{qB},$ where $R$ is a random number between 0 and 0.01, $T_\\\\mathrm {gyro}$ is the ion's gyroperiod (3.6 s for H$_2$ O$^+$ ), $B$ = 325 nT, and $q$ and $m$ are the charge and mass, respectively, of the reacting ion.\",\n \"Such resolution is necessary in order to capture the significance of the energy dependence at low relative speeds.\",\n \"After each time step, the collision frequency $\\\\nu $ is calculated from $\\\\nu (r,\\\\theta ,v_\\\\mathrm {rel})=n(r,\\\\theta ) \\\\sigma (v_\\\\mathrm {rel})v_\\\\mathrm {rel},$ where $n(r,\\\\theta )$ is the local H$_2$ O density (section REF ), $v_\\\\mathrm {rel}$ is the relative velocity between the reacting ion and neutral, and $\\\\sigma (v_\\\\mathrm {rel})$ is the velocity-dependent cross section.\",\n \"Poisson statistics are used to test the likelihood of one or more reactions having occurred within $\\\\Delta t$ .\",\n \"If $\\\\exp (-\\\\nu \\\\Delta t)$ is less than a second random number between 0 and 1, then a reaction occurs.\",\n \"The possibility of multiple reactions occurring over $\\\\Delta t$ is taken into account, but it is neglectable (appendix A).\",\n \"As with OH produced by dissociation (section REF ), neutrals produced by charge exchange are followed under the influence of Saturn's gravity until they are photodissociated or photoionized.\",\n \"Their initial location and velocity are taken to be that of the reacting ion, pre-transfer.\",\n \"The model runs are centered on the orbit of Enceladus spanning 10 R$_\\\\mathrm {E}$ in the direction of corotation (R$_\\\\mathrm {E}$ = 250 km = radius of Enceladus) and $\\\\pm $ 120 R$_\\\\mathrm {E}$ (0.5 $\\\\textrm {R}_\\\\textrm {S}$ ) in both the radial and $z$ directions to adequately sample the H$_2$ O torus (section REF ).\",\n \"Ions are introduced into the model on the upstream boundary, and their guiding centers flow downstream at a speed $v_\\\\mathrm {plasma}=18$ $\\\\textrm {km\\\\ s}^{-1}$ relative to the neutrals.\",\n \"Their starting location in ($r,z$ ) is chosen randomly.\"\n ],\n [\n \"Scaling\",\n \"The neutral clouds formed $via$ charge exchange are done so in our model by following a relatively small number of ions, and must thus be scaled to facilitate comparison with observations and other models.\",\n \"The number of neutrals in our modeled clouds have been scaled by accounting for the following.\",\n \"First, the number of representative ions used to produce the neutral clouds $via$ charge exchange falls short of, and must be scaled to, the number of ions present in the actual plasma torus, $n_\\\\mathrm {ion}V$ .\",\n \"The volume of the plasma torus, $V$ , is given in Eq.\",\n \"REF , and $n=12$ and 6 cm$^{-3}$ for O$^+$ and H$_2$ O$^+$ , respectively [36].\",\n \"Second, we have argued that photo-processes are more likely to occur than either charge exchange or electron-impact processes throughout the neutral clouds with the exception of very near the neutral torus.\",\n \"In keeping with this assumption, the plasma torus thus feeds the extended neutral clouds $via$ charge exchange for a photodissociation (photoionization in the case of oxygen) time scale before equilibrium of the neutral cloud is achieved: $\\\\tau _\\\\mathrm {phot}= $ 14, 0.6, 0.3 years for O, OH, and H$_2$ O, respectively [11].\",\n \"Our model runs followed 10$^5$ ions for 100 seconds, and the resulting neutral clouds were scaled as described.\"\n ],\n [\n \"Results\",\n \"In the following sections, we present and discuss the neutral clouds resulting from dissociation and charge exchange in our model.\"\n ],\n [\n \"Charge exchange\",\n \"In the neutrals' reference frame, ions oscillate between $\\\\approx $ 0 $\\\\textrm {km\\\\ s}^{-1}$ and twice the local pick-up speed ($v_\\\\phi \\\\approx 18\\\\ \\\\textrm {km\\\\ s}^{-1}$ ) due to gyro-motion.\",\n \"A cartoon of this can be seen in Fig.\",\n \"REF , where $v_\\\\mathrm {rel}\\\\approx 0$ at the cusp of the ion trajectory and reaches a maximum of $v_\\\\mathrm {rel}\\\\approx 2 v_\\\\phi $ along the flow direction.\",\n \"Shown are several trajectories for which $v_\\\\perp $ is either less than, greater than, or approximately equal to the bulk flow velocity.\",\n \"The neutrals formed $via$ charge exchange follow the trajectories indicated in red.\",\n \"The velocity dependence of reactions REF – are determined by the details of the reacting species [17].\",\n \"Essentially, symmetric charge exchanges have cross sections that increase monotonically with decreasing velocity, whereas cross sections for asymmetric exchange peak and then vanish at low relative speeds.\",\n \"The implication is that symmetric exchanges produce lower velocity neutrals and a more compact neutral cloud than do asymmetric reactions.\",\n \"With symmetric charge exchange, the cross sections go as ${v_\\\\mathrm {rel}}^{-1}$ , so that the collision frequency ($n\\\\sigma v$ ) is independent of $v$ , as with reaction REF , whereas asymmetric exchanges are defined by cross sections (and collision frequencies) which tend rapidly toward zero at low relative velocities ($\\\\sim {v_\\\\mathrm {rel}}^4$ , [27]).\",\n \"The cross sections (10$^{-16}$ cm$^2$ ) used in this paper to study reactions REF – plotted in Fig.\",\n \"REF a are given by $\\\\mbox{Reaction \\\\ref {h2o_chex}} :&\\\\ \\\\mathrm {H_2O + H_2O^+} \\\\rightarrow \\\\mathrm {H_2O^+ + H_2O^*} \\\\nonumber \\\\\\\\\\\\sigma _\\\\mathrm {H_2O} =&\\\\ 38E_\\\\mathrm {rel}^{-0.5} \\\\\\\\\\\\mbox{Reaction \\\\ref {oh_chex}} :&\\\\ \\\\mathrm {H_2O + H_2O^+} \\\\rightarrow \\\\mathrm {H_3O^+ + OH^*} \\\\nonumber \\\\\\\\\\\\sigma _\\\\mathrm {OH} =&\\\\ 38E_\\\\mathrm {rel}^{-0.88}-0.39\\\\exp \\\\left[{-\\\\frac{1}{2}\\\\left(\\\\frac{E_\\\\mathrm {rel}-57}{12}\\\\right)^2}\\\\right] \\\\\\\\\\\\mbox{Reaction \\\\ref {o_chex}} :&\\\\ \\\\mathrm {H_2O + O^+} \\\\rightarrow \\\\mathrm {H_2O^+ + O^*} \\\\nonumber \\\\\\\\\\\\sigma _\\\\mathrm {O} =&\\\\ 69E_\\\\mathrm {rel}^{-0.29}+30\\\\exp \\\\left[{-\\\\frac{1}{2}\\\\left(\\\\frac{E_\\\\mathrm {rel}-65}{18}\\\\right)^2}\\\\right].$ The Gaussian terms in Eqs.\",\n \"and account for downward and upward trends in the associated data sets near 30 $\\\\textrm {km\\\\ s}^{-1}$ , but have little consequence on the neutral cloud, given that most bound particles are produced at lower velocities.\",\n \"Symmetric exchanges occur between like species by definition, although unlike species also exhibit symmetric behavior on occasion.\",\n \"Therefore, we explore several hypothetical behaviors for the OH$^*$ -producing reaction at low energies.\",\n \"This test is separate from, but related to, the comparison between reactions REF – themselves, and it motivates the point that both high and low energy behaviors have an important effect on the neutral cloud.\",\n \"With $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ , we have extropolated the best-fit curve (Eq. )\",\n \"to the lowest energies.\",\n \"Symmetric and asymmetric behaviors are explored with $\\\\sigma ^\\\\textrm {symmetric}_\\\\textrm {OH}$ and $\\\\sigma ^\\\\textrm {asymmetric}_\\\\textrm {OH}$ [27], [17].\",\n \"$\\\\sigma ^\\\\textrm {symmetric}_\\\\textrm {OH}$ is the same as $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ except that below 1.5 eV, $\\\\sigma ^\\\\mathrm {symmetric}_\\\\mathrm {OH}=30E_\\\\mathrm {rel}^{-0.5}\\\\times 10^{-16}$ cm$^2$ .\",\n \"Notice that a similar energy dependence also applies to Eq.\",\n \"REF , consistent with symmetric charge exchange.\",\n \"$\\\\sigma ^\\\\textrm {asymmetric}_\\\\textrm {OH}$ is the same as $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ except that below 1.5 eV, $\\\\sigma ^\\\\mathrm {asymmetric}_\\\\mathrm {OH}=11E_\\\\mathrm {rel}^2\\\\times 10^{-16}$ cm$^2$ .\",\n \"Although it could be argued that $\\\\sigma _\\\\mathrm {OH}^\\\\mathrm {symmetric}$ better fits the data if the two measurements at 2 eV are ignored, our results for reaction were obtained with $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ unless noted otherwise.\",\n \"We will discuss the implications of choosing $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ over $\\\\sigma ^\\\\textrm {symmetric}_\\\\textrm {OH}$ and $\\\\sigma ^\\\\textrm {asymmetric}_\\\\textrm {OH}$ shortly.\",\n \"The collision frequencies ($n\\\\sigma v$ ) are plotted in Fig.\",\n \"REF b for a given neutral density—in this case for $n_\\\\mathrm {H_2O}=10^3$ cm$^{-3}$ .\",\n \"The collision frequency for oxygen increases with relative speed, while it is constant for water, and peaks at low velocities for OH.\",\n \"The significance is that the oxygen cloud tends to be more extended than either the OH or H$_2$ O clouds.\",\n \"The average collision frequency is also much higher for oxygen ($\\\\times $ 10) than for either OH or H$_2$ O, resulting in greater oxygen abundance.\",\n \"The equatorial neutral cloud densities resulting from reactions REF – are plotted in Fig.\",\n \"REF .\",\n \"Only neutrals produced from charge exchange are shown; neither the Enceladus neutral torus, nor the neutrals produced via dissociation have been included.\",\n \"Oxygen is two orders of magnitude more abundant than either OH or H$_2$ O because of the higher rate of production, but also because oxygen has a longer lifetime against photoionization than either OH or H$_2$ O have against photodissociation.\",\n \"Unlike [5], dissociated neutrals from the latter processes are not tracked in our model.\",\n \"Beyond the scope of the present study, this additional heating source would serve to further inflate the oxygen and OH clouds.\",\n \"Fig.\",\n \"REF b is the same as REF a, except that the profiles are normalized to the peak density at the orbit of Enceladus.\",\n \"The oxygen cloud is seen to be the most extended, followed by water, and finally by OH, with an order of magnitude separating the three species at 20 $\\\\mathrm {R_S}$ .\",\n \"The effects of low-velocity charge exchange are shown in Fig.\",\n \"REF .\",\n \"In Fig.\",\n \"REF a, we see that the peak density (as well as the total neutral cloud content) is the highest with $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ because more low-velocity neutrals are produced than with either $\\\\sigma ^\\\\textrm {symmetric}_\\\\textrm {OH}$ or $\\\\sigma ^\\\\textrm {asymmetric}_\\\\textrm {OH}$ .\",\n \"Conversely, fewer low-velocity neutrals are available to populate the region near Enceladus's orbit with $\\\\sigma ^\\\\textrm {asymmetric}_\\\\textrm {OH}$ when compared to either $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ or $\\\\sigma ^\\\\textrm {symmetric}_\\\\textrm {OH}$ .\",\n \"Stated another way, $\\\\sigma ^\\\\textrm {extrapolated}_\\\\textrm {OH}$ yields a neutral cloud with the steepest slope, and $\\\\sigma ^\\\\textrm {asymmetric}_\\\\textrm {OH}$ , the shallowest.\",\n \"Fig.\",\n \"REF b is identical to Fig.\",\n \"REF a, apart from normalization.\",\n \"In this case, the slope of the density profile should not be confused with the effect of inflating (spreading) the OH cloud.\",\n \"It should be viewed, rather, as the enhancement or depletion of low velocity neutrals to fill the region inside of $\\\\approx 10$ $\\\\textrm {R}_\\\\textrm {S}$ .\",\n \"In other words, neutrals beyond 10 $\\\\textrm {R}_\\\\textrm {S}$ are mostly formed in charge exchanges at high velocities, for which all $\\\\sigma _\\\\mathrm {OH}$ converge to the same curve (Fig.\",\n \"REF ).\",\n \"We have assumed to this point that the plasma is sub-corotating in Enceladus's orbit (18 $\\\\textrm {km\\\\ s}^{-1}$ , [44]).\",\n \"One might expect, however, that the neutral clould would be affected in a measurable way if instead, the plasma corotates at 26 $\\\\textrm {km\\\\ s}^{-1}$ .\",\n \"The H$_2$ O cloud would be least affected, given that the collision frequency of reaction REF is independent of speed (Fig.\",\n \"REF b), but what about reactions such as and , whose collision frequencies are velocity-dependent?\",\n \"Increasing the plasma speed amounts to shifting the spread of ion velocities in Fig.\",\n \"REF to the right, which would on average increase the speed of the neutral products.\",\n \"This is indeed the case, and in such a test where we increased the plasma speed from 18 to 26 $\\\\textrm {km\\\\ s}^{-1}$ , the oxygen cloud increased in abundance and became even more extended.\",\n \"The OH cloud also expanded somewhat, but decreased in total abundance.\",\n \"Unfortunately, the differences were less than 10% in both the slope of the distribution and in total oxygen abundance, suggesting that neutral cloud observations are in this way unlikely to predict plasma speeds in the torus.\",\n \"We described in section REF the production in our model of the neutral H$_2$ O torus from the Enceladus plumes.\",\n \"The plumes themselves have also been prescribed as a separate background density ($n_\\\\mathrm {plume}$ , section REF ) so that we can compare charge exchange occurring throughout the neutral torus to that occurring only within the Enceladus plumes.\",\n \"The results are shown in Fig.\",\n \"REF , where we have plotted the oxygen clouds produced from charge exchange within both the Enceladus plumes (local) and the entire neutral torus (global).\",\n \"The results are for reaction , but the same test with reactions REF and produces similar results.\",\n \"Immediately noticable is that the local production is $\\\\approx 0.1$ % of the overall neutral production.\",\n \"The torus's dominance of neutral production can be explained as follows.\",\n \"First, the volume of the torus where reactions are occurring can be estimated as $2\\\\pi (4\\\\mathrm {R_S})(0.2\\\\mathrm {R_S})^2$ , where $0.1\\\\mathrm {R_S}$ is roughly the torus's scale height.\",\n \"The volume of the plume can be estimated from Eq.\",\n \"REF , where the dimensions are on the order of a cylinder with width 2$\\\\mathrm {R_E}$ and height $H_r\\\\approx 16\\\\mathrm {R_E}$ .\",\n \"Dividing these volumes gives roughly $250(\\\\mathrm {R_E}/\\\\mathrm {R_S})^3\\\\approx 10^{-5}$ .\",\n \"Further, the collision frequencies are proportional to the neutral density, which in the plume are on the order of 10$^7$ cm$^{-3}$ , whereas typical torus densities are 10$^5$ cm$^{-3}$ , making collisions in the plume 100$\\\\times $ more frequent per volume than in the torus.\",\n \"All told, the ratio of the volumes ($10^{-5}$ ) combined with the ratio of densities ($10^2$ ) explain the local-to-global neutral production ratio of $10^{-3}$ shown in Fig.\",\n \"REF a.\",\n \"A similar pattern has been shown to exist at Jupiter by [2] and [7], where the majority of plasma is produced throughout the neutral torus, rather than near the interaction at Io itself.\",\n \"The slopes of the neutral clouds from the plume and torus are most easily compared in Fig.\",\n \"REF b, in which the density profiles have been normalized.\",\n \"The local source produces a more confined neutral cloud because the ions from which they originate have been slowed near the plume to account for the effect of mass-loading ([10]).\",\n \"Nevertheless, such a signature would be difficult to untangle in the data since global exceeds local production so overwhelmingly.\"\n ],\n [\n \"Dissociation\",\n \"A major component of the OH cloud is produced by dissociation within the neutral torus, whereby the initial velocities of the OH products range from 1 to 1.6 $\\\\textrm {km\\\\ s}^{-1}$ [45], [24].\",\n \"In Fig.\",\n \"REF , the clouds resulting from the high- and low-speed cases are plotted along with the result from velocity-dependent charge exchange in section REF .\",\n \"First note that dissociation contributes 100$\\\\times $ more OH than does charge exchange at the orbit of Enceladus (4 $\\\\textrm {R}_\\\\textrm {S}$ ); the total cloud mass is almost 100$\\\\times $ greater as well.\",\n \"Second, dissociation dominates over charge exchange from the Enceladus torus out to 9 and 15 $\\\\textrm {R}_\\\\textrm {S}$ in the low- and high-speed cases, respectively.\",\n \"The OH cloud content will only be marginally affected by variable solar activity [16], given that impact dissociation contributes 4$\\\\times $ more neutrals than does photodissociation, by virtue of the respective reaction rates (section REF ).\",\n \"In both cases, few neutrals are absorbed by the rings, and even less by Saturn itself.\",\n \"The same is not true of charge exchange, where $\\\\approx 50$ % of the neutrals are absorbed by Saturn (section REF ).\",\n \"Fig.\",\n \"REF c is a two-dimensional version of Fig.\",\n \"REF , where the dissociation results have been averaged and added to the results from charge exchange.\",\n \"Saturn is at the left, and the Enceladus's orbit is located on the equator at 4 $\\\\textrm {R}_\\\\textrm {S}$ .\",\n \"In addition to being confined radially, the dissociated neutrals are also bound tightly to the equator, while neutrals from charge exchange tenuously fill the magnetosphere.\",\n \"Fig.\",\n \"REF a shows the hydrogen cloud that accompanies the dissociated OH clouds ($\\\\mathrm {H_2O + e,\\\\gamma }\\\\rightarrow \\\\mathrm {OH^*+H^*}$ ).\",\n \"To conserve momentum, the hydrogen atoms have 17$\\\\times $ the speed of the dissociated OH molecules, and thus range between 17 and 27 $\\\\textrm {km\\\\ s}^{-1}$ , with a relatively large, diffuse neutral cloud.\",\n \"Shown is the result for the low-speed case, which produces more bound particles and thus a more substantial neutral cloud.\",\n \"Charge exchange from reactions such as $\\\\mathrm {H_2O+H^+}\\\\rightarrow \\\\mathrm {H_2O^++H^*}$ are also responsible for H-cloud production, and deserve attention in future studies.\"\n ],\n [\n \"Fates of neutral atoms and molecules\",\n \"In our model, neutrals created by dissociation and charge exchange are eventually either absorbed by Saturn, escape the system, or orbit until they are destroyed (ionized) by photons.\",\n \"In Fig.\",\n \"REF a the fates for each species are given by percentage.\",\n \"In the case of hydrogen, the results are from the dissociation model, described in section REF .\",\n \"The enormous amount of escape (84%) is due to the high velocities ($\\\\approx 17$ $\\\\textrm {km\\\\ s}^{-1}$ ) with which hydrogen is created following H$_2$ O dissociation, and the 8% absorption is largely comprised of hydrogen which would escape the system otherwise.\",\n \"Oxygen is produced purely from charge exchange in our model (reaction ).\",\n \"About one-half escapes, one-third is absorbed, and the remaining 13% contributes to the neutral cloud before being photoionized.\",\n \"Water is also produced purely by charge exchange (reaction REF ) with 18% contributing to the neutral cloud.\",\n \"Percentage-wise, more water is absorbed than oxygen because oxygen is produced with higher speeds and generally larger orbits (section REF ).\",\n \"The fate of OH is dominated by dissociation: 96% feed the neutral cloud (ultimately ionized), 4% are absorbed, and virtually none escape.\",\n \"The reason for the large percentage of bound and unabsorbed neutrals is that dissociated OH has a velocity spread of 1 to 1.6 $\\\\textrm {km\\\\ s}^{-1}$ in the neutral frame, compared to the escape speed of $\\\\approx 5$ $\\\\textrm {km\\\\ s}^{-1}$ in the same frame.\",\n \"Looking only at OH produced by charge exchange (minor compared to dissociation), 58% are absorbed, 23% supply the neutral cloud, and 20% escape.\",\n \"Compared to H$_2$ O, an even greater percentage of charge-exchanged OH is absorbed because the cross sections favor production of low-velocity OH molecules (Fig.\",\n \"REF b).\",\n \"The production of oxygen via dissociation of H$_2$ O has been ignored in this paper on the grounds that, unlike OH, oxygen is largely produced by charge exchange.\",\n \"The cross section for oxygen-producing charge exchange is an order of magnitude higher than that for the OH-producing reaction near the plasma flow speed of $v_\\\\mathrm {plasma}=18$ $\\\\textrm {km\\\\ s}^{-1}$ (Fig.\",\n \"REF a), while the photodissociation rates are an order of magnitude smaller [15].\",\n \"We estimate that including oxygen produced from dissociation would increase the total oxygen cloud content by less than 20%.\",\n \"Charge exchange and dissociation play a large role in creating Saturn's neutral clouds from the plume-fed neutral torus.\",\n \"The reactions we have included have been chosen to demonstrate the effects of low velocity charge exchange and dissociation, but they are also among the most important.\",\n \"The neutral cloud densities presented in this paper are expected to undershoot the results from models which include the additional reactions found in Fig.\",\n \"3, of [11] by no more than a factor of two.\",\n \"With this caveat in mind, we now compare the present results with several other recent models.\"\n ],\n [\n \"Comparison with other models\",\n \"Fig.\",\n \"REF b: J06 is the work of [18], where they also investigated the neutral clouds created from low-velocity charge exchange in the stagnated flows in Enceladus's orbit.\",\n \"Fig.\",\n \"REF b: J07 is from [20], where the authors were primarily interested in the interaction between the neutral cloud and Saturn's rings.\",\n \"The most recent model comes from [5] (C10), where they investigated the spreading of the neutral cloud from neutral–neutral collisions.\",\n \"To compare with these studies, we first had to weight our H, O, OH, and H$_2$ O clouds.\",\n \"We did this for two limiting cases.\",\n \"In the first case ($\\\\tau _\\\\mathrm {phot}$ , Fig.\",\n \"REF b), we assume, as we have thus far, that the neutral clouds evolve until destroyed by either photoionization or photodissociation: H, O, OH, H$_2$ O = 40, 14, 0.6, 0.3 years, respectively.\",\n \"These lifetimes yield an upper limit since charge exchange and electron impact are not included as losses.\",\n \"In the second case ($\\\\tau _\\\\mathrm {all}$ ) , we derived a lower limit to the lifetimes from Table 2 of [11] by summing the additional losses due to charge exchange and electron impact, finding: H, O, OH, H$_2$ O = 0.4, 0.4, 0.2, 0.03 years, respectively.\",\n \"Notice in particular the drastically different times scales for H and O, where including the additional sinks reduce the size of the H cloud by a factor of 40/0.4 = 100, and the oxygen cloud by 14/0.4 = 35.\",\n \"This case represents an extreme limit, given that the neutrals spend almost all of their time orbiting outside of the Enceladus torus, where compared to photo-processes, the chances of charge exchange and electron impact are relatively unlikely.\",\n \"We mention, however, that [30], [29] has shown that circulation patterns inside of 12 $\\\\textrm {R}_\\\\textrm {S}$ at Saturn gives rise to `butterfly' hot electron pitch angle distributions, related to low temperature anisotropy ($T_\\\\perp /T_\\\\parallel $ ), on which proton field-aligned distributions depend [36].\",\n \"The individual clouds (excluding hydrogen) were weighted by the stated time scales and totaled in Fig.\",\n \"REF b.\",\n \"When only losses to photodissociation/ionization are considered ($\\\\tau _\\\\mathrm {phot}$ ), the neutral cloud is dominated by oxygen, whose fate thus determines that of the neutral cloud.\",\n \"When charge exchange and electron impact are also included ($\\\\tau _\\\\mathrm {all}$ ), dissociated OH contributes significantly, driving the neutral cloud (ionized) percentage up, and the escape percentage down.\",\n \"We note that the neutral fates presented in [3] (escape = 44%, ionized = 17%, absorbed = 39%) were based on an earlier version of our model which only included H$_2$ O.\",\n \"The particles that are neither absorbed nor lost by escape make up the neutral clouds.\",\n \"In the case where the cloud evolves for $\\\\tau _\\\\mathrm {phot}$ , oxygen and hydrogen dominate since they are far less likely to be photoionized than are OH and H$_2$ O to be photodissociated.\",\n \"With charge exchange and electron impact included ($\\\\tau _\\\\mathrm {all}$ ), however, more oxygen and hydrogen are removed from the system, which then tends to favor a molecular OH–H$_2$ O cloud.\",\n \"In terms of total mass the same applies, although hydrogen accounts for only a few percent at most.\",\n \"We find that the total cloud mass is bounded between $\\\\approx 1$ and 10 Mtons, for $\\\\tau _\\\\mathrm {all}$ and $\\\\tau _\\\\mathrm {phot}$ , respectively.\",\n \"It is worth pausing to re-emphasize that the system is in reality better represented by the $\\\\tau _\\\\mathrm {phot}$ case, from which all neutral clouds in this paper have been derived.\",\n \"The $\\\\tau _\\\\mathrm {all}$ case is strictly valid only for neutrals within the Enceladus torus, though reactions with electrons and protons may also prove important, as discussed above.\",\n \"What is illustrated, however, is that Saturn's magnetosphere is less oxygen-dominated than suggested by looking at losses from photo-processes alone.\",\n \"These results suggest that our oxygen abundances are somewhat overestimated, likely by less than a factor of two.\"\n ],\n [\n \"Neutral absorption\",\n \"The particles absorbed by Saturn and its rings are plotted by species and latitude in Fig.\",\n \"REF .\",\n \"In Fig.\",\n \"REF b, we see that most absorption comes from oxygen (74%), followed by H$_2$ O (11%), OH (9%), and finally by hydrogen (6%).\",\n \"Absorption is equally divided between Saturn and its rings except in the case of OH, where twice as much falls on Saturn's rings.\",\n \"This is because OH is largely produced by impact dissociation, which creates slower neutrals than does charge exchange, whereby in our model, H$_2$ O and oxygen arise exclusively.\",\n \"In Fig.\",\n \"REF b, absorption is plotted against Saturn's latitude.\",\n \"Because the model is symmetric about the equator, the results apply to either hemisphere.\",\n \"Oxygen, water, and OH follow the same trends because they all originate from charge exchange (dissociated OH is slow and does not reach Saturn), and have been created from ions with similar velocity distributions.\",\n \"Any second-order differences due to the velocity-dependence of the respective cross sections are not immediately apparent.\",\n \"Hydrogen, on the other hand, is produced entirely by dissociation in the model and exhibits a more uniform flux across Saturn.\",\n \"The explanation is that the velocity distribution from which hydrogen is produced is isotropic, whereas that which produces charge-exchanged neutrals is bi-Maxwellian (section REF ).\",\n \"The fluxes shown in Fig.\",\n \"REF b are consistent with [14], who modeled recent Herschel observations of Saturn's water torus and found an average flux of $6\\\\times 10^5$ cm$^{-2}$ s$^{-1}$ for H$_2$ O + OH impinging on Saturn.\"\n ],\n [\n \"Discussion\",\n \"Some useful conclusions can be drawn by further contrasting our results with [5] (C10).\",\n \"It is important that we first mention a profound difference between our models.\",\n \"The model of C10 effectively carries out resonant charge exchange only, which does not chemically alter the neutral population; neutrals in their model are produced either directly from Enceladus or from subsequent dissociations.\",\n \"Neutrals in our model, on the other hand, originate from Enceladus (H$_2$ O).\",\n \"OH is then created $via$ dissociation (as with C10), but secondary O, OH, and H$_2$ O populations are $created$ from H$_2$ O $via$ charge exchange with the dense plume-fed Enceladus torus.\",\n \"The C10 model redistributes neutrals around Saturn, while we redistribute and chemically re-assign neutral abundances by allowing for asymmetric charge exchanges.\",\n \"Thus, it may well be a coincidence that our models are similar in total abundance.\",\n \"While it may be difficult to compare our total abundances, the slope of our radial density profiles can be contrasted directly because our redistribution mechanisms (charge exchange and dissociation) are similar.\",\n \"Differences are due largely to C10's inclusion of neutral collisions and our prescribing unique velocity-dependent charge exchange for each of the O-, OH-, and H$_2$ O-producing reactions (reactions REF –).\",\n \"Our neutral clouds are compared with C10 in Fig.\",\n \"REF .\",\n \"All of our clouds include contributions from charge exchange, but the H$_2$ O cloud is mostly comprised of water sourced directly from Enceladus (3.95 $\\\\textrm {R}_\\\\textrm {S}$ ), and OH includes the additional source from dissociation.\",\n \"In the C10 model, the water molecules were spread due to neutral–neutral collisions, which explains our higher H$_2$ O densities near Enceladus's orbit (Fig.\",\n \"REF a).\",\n \"The slope of the oxygen profile agrees best with C10 because their charge exchange cross section most resembles our own (Eq.\",\n \").\",\n \"Our H$_2$ O profiles agree less, and our OH slopes, the least, due mainly to the strong effect that neutral collisions have on those more polar molecules.\",\n \"In particular, C10 used a much larger cross section for neutral collisions involving H$_2$ O and OH [42] than for atomic oxygen [4].\",\n \"This helps to further explain our agreement with their oxygen profile since we exclude neutral–neutral collisions from our model altogether.\",\n \"We conclude that neutral–neutral collisions appear to play a less significant role with atomic species, such as oxygen and hydrogen.\",\n \"The column densities (Fig.\",\n \"REF b) are similar to C10, who constrained their O and OH clouds with the most recent Cassini UVIS results of [25].\",\n \"Our oxygen density—as well as our total oxygen content (Fig.\",\n \"REF c)—is higher for two reasons.\",\n \"First, we use a larger cross section than does C10 for reaction , and second, the clouds presented here have been limited only by photoionization.\",\n \"Charge exchange and electron impact are second order losses beyond 6 $\\\\mathrm {R_S}$ , but including them would favorably reduce the oxygen content more than OH and H$_2$ O (section REF ), bringing our models into better agreement.\",\n \"Our total H$_2$ O content is 4$\\\\times $ less than C10 found (Fig.\",\n \"REF c).\",\n \"This is partly because we have subjected H$_2$ O molecules in the primary (plume-fed) neutral torus to the shortest lifetimes possible (section REF ), whereas C10 tracks molecules that get kicked out of the densest plasma $via$ neutral collisions, and thus survive longer, being less susceptible to both charge exchange and electron impact.\",\n \"That their total H$_2$ O content is higher than ours (Fig.\",\n \"REF c), does not contradict the fact that their H$_2$ O column density is lower; neutral–neutral collisions would spread out the torus, lowering the column density, while allowing neutrals to survive longer, increasing the total abundance.\",\n \"Our model would benefit by including the redistribution attributed to neutral collisions by allowing particles to interact in a direct simulation Monte Carlo (DSMC) model such as in C10.\",\n \"Likewise, DSMC models would benefit by including charge exchange cross sections specific to each reaction.\",\n \"Such models should also take into account asymmetric charge exchanges, which affects neutral cloud composition.\",\n \"The reactions modeled in this study were chosen in order to measure the effect of symmetric and asymmetric charge exchange at low velocities.\",\n \"Building upon our findings, future studies should include additional neutral-producing charge exchanges, such as $\\\\mathrm {OH^+ + H_2O} \\\\rightarrow \\\\mathrm {OH^* + H_2O^+}$ , $\\\\mathrm {H^+ + H_2O} \\\\rightarrow \\\\mathrm {H^* + H_2O^+}$ , and $\\\\mathrm {OH^+ + H_2O} \\\\rightarrow \\\\mathrm {O^* + H_3O^+}$ , as well as dissociative recombination of H$_2$ O$^+$ .\"\n ],\n [\n \"Conclusions\",\n \"We have modeled low-velocity charge exchange from the point of view of the ions, allowing us to study the effects of velocity as well as gyrophase.\",\n \"With reactions REF –, we have been able to offer an estimate on the size and shape of the neutral clouds at Saturn, while simultaneously exploring the sensitivity of the neutral clouds to a variety of velocity-dependent reactions.\",\n \"We have also re-visited the production of OH following H$_2$ O dissociation in the primary neutral torus.\",\n \"Previous models have used 1 $\\\\textrm {km\\\\ s}^{-1}$ as the initial velocity for OH, while measurements suggest a range of speeds from 1 to 1.6 $\\\\textrm {km\\\\ s}^{-1}$ .\",\n \"In our model, the higher speed increases the range within which dissociation dominates neutral production from 9 to 15 $\\\\textrm {R}_\\\\textrm {S}$ .\",\n \"Additional findings are: (1.)\",\n \"Charge exchange cross sections that increase steeply at low speeds tend to produce neutral clouds more confined to the orbit of Enceladus, implying the most spreading for oxygen, moderate spreading for H$_2$ O, and the least for OH (Fig.\",\n \"REF ).\",\n \"Accounting for gyrophase doubles the local OH density within Encelacus's orbit, has $\\\\approx $ no effect on H$_2$ O, and decreases oxygen density by less than 10%.\",\n \"(2.)\",\n \"Enceladus is solely responsible for the creation of the neutral H$_2$ O torus $via$ thermal ejection from its plumes.\",\n \"However, Saturn's neutral clouds are overwhelmingly produced by charge exchange and dissociation occurring throughout the torus (99%), and not near Enceladus itself (Fig.\",\n \"REF ).\",\n \"(3.)\",\n \"We estimate that roughly half of all neutrals escape the system, with the remaining equally divided between absorption by the rings/planet and the neutral clouds (Fig.\",\n \"REF ).\",\n \"Less than 50 $\\\\textrm {kg\\\\ s}^{-1}$ is thus ionized and transported out of the system as plasma.\",\n \"This number is expected to represent an upper limit, given we have assumed that all particles forming the neutral clouds are ultimately ionized; a more accurate result would require modeling the detailed effects of charge exchange and neutral–neutral collisions within the neutral clouds.\",\n \"This estimate can be compared to [36], whose Figs.\",\n \"14 and 17 give roughly $([NL^2]_\\\\mathrm {W^+}/L^2)\\\\times m_\\\\mathrm {W^+}/\\\\tau _\\\\mathrm {transport}\\\\approx 3\\\\times 10^{31}\\\\times m_\\\\mathrm {W^+}/10^5$ s $\\\\approx 10$ $\\\\textrm {kg\\\\ s}^{-1}$ at $L=10$ .\",\n \"(4.)\",\n \"Saturn's neutral cloud has a total mass of at least 1 Mton, but likely much closer to 10 Mtons.\",\n \"The primary plume-fed neutral torus (0.3 Mtons) is comprised entirely of water in our model, while the secondary neutral clouds are broken down into H ($\\\\lesssim 5$ %), O ($\\\\lesssim 82$ %), OH ($\\\\gtrsim 13$ %), and H$_2$ O ($\\\\approx 1$ %).\",\n \"Atomic oxygen dominates the composition both because of a high production rate from charge exchange as well as a long lifetime against photoionization.\",\n \"Charge exchange and reactions with electrons favorably remove hydrogen and oxygen, but are secondary loss mechanisms throughout the majority of the magnetosphere.\",\n \"(5.)\",\n \"Our model predicts fluxes on Saturn from charge exchange of $\\\\approx 6\\\\times 10^5$ cm$^{-2}$ s$^{-1}$ for both OH and H$_2$ O (consistent with Herschel observations by [14]), and oxygen is about 5$\\\\times $ higher.\",\n \"Absorption is divided equally between Saturn and its rings (Fig.\",\n \"REF a).\",\n \"(6.)\",\n \"Our total neutral abundances are similar to [5] (C10) for both OH and H$_2$ O, and 4$\\\\times $ higher for oxygen (Fig.\",\n \"REF ).\",\n \"Differences in the slopes of our equatorial density profiles are in part due to our not including neutral–neutral collisions, while this fact appears to have no effect on the oxygen profile.\",\n \"On the other hand, C10 did not include the effects on neutral chemistry following asymmetric charge exchanges, nor did they use velocity-dependent cross sections particular to each reaction.\",\n \"Herschel observations by [14] confirm the importance of neutral–neutral collisions for H$_2$ O, but if oxygen is the dominant neutral species in Saturn's magnetosphere, as our model predicts, neutral–neutral collisions may play a smaller role in Saturn's neutral cloud than previously expected.\",\n \"Given the effect on both the size and shape of the neutral clouds, we suggest that future neutral cloud models include charge exchange cross sections unique to each reaction.\",\n \"Asymmetric charge exchange also has an important effect on neutral chemistry that should be implemented.\",\n \"Regarding the ions' gyrophase, Monte Carlo models can account for its effect by using phase-dependent probability distributions.\",\n \"Finally, the range of OH velocities studied here should be considered when modeling dissociation.\",\n \"Moving forward, we plan to implement these suggestions into the neutral cloud model of C10 and to couple that model with the chemistry model of [11].\",\n \"Constrained by Cassini plasma observations, the chemistry model uses C10's neutrals as input, and provides ion temperatures and densities throughout the magnetosphere ($<$ 20 $\\\\textrm {R}_\\\\textrm {S}$ ), which C10 in turn uses to update neutral densities.\",\n \"An improved understanding of two issues is planned: (1) Where does plasma transport become important?\",\n \"(2) What is the role of hot electrons with regard to ion–neutral chemistry inside 20 $\\\\textrm {R}_\\\\textrm {S}$ ?\"\n ],\n [\n \"Collision probability\",\n \"The average number of collisions occurring during a time interval $\\\\Delta t$ is given by $\\\\lambda \\\\equiv \\\\nu \\\\Delta t$ , where $\\\\nu =n_\\\\mathrm {neutrals}\\\\sigma (v_\\\\mathrm {rel})v_\\\\mathrm {rel}$ is the local collision frequency, assumed to be constant during $\\\\Delta t$ .\",\n \"Statistics are applied to determine if and how many reactions occur during $\\\\Delta t$ .\",\n \"The Poisson distribution [47], [28] gives the probability of suffering exactly $n$ collisions for a given $\\\\lambda $ : $ f(n;\\\\lambda )=\\\\frac{e^{-\\\\lambda }\\\\lambda ^n}{n!\",\n \"}.$ Notice that Eq.\",\n \"REF peaks at $n=\\\\lambda $ if one treats $n$ as a continuous variable.\",\n \"Summing Eq.\",\n \"REF discretely from $n=k$ to $n=\\\\infty $ gives the probability of suffering at least $k$ collisions during $\\\\Delta t$ , $ P_k(\\\\lambda )=e^{-\\\\lambda }\\\\sum _{n=k}^\\\\infty \\\\frac{\\\\lambda ^n}{n!\",\n \"}.$ Because Eq.\",\n \"REF is normalized ($e^{-\\\\lambda }\\\\sum _{n=0}^\\\\infty \\\\lambda ^n/n!=e^{-\\\\lambda }e^\\\\lambda =1$ ), Eq.\",\n \"REF can be conveniently rewritten as $ P_k(\\\\lambda )=1-e^{-\\\\lambda }\\\\sum _{n=0}^{k-1} \\\\frac{\\\\lambda ^n}{n!\",\n \"}.$ A random number ($0 N$ determines how many fast neutrals (collisions), $k$ , are produced during $\\\\Delta t$ .\",\n \"In practice, it is only necessary to compare to the first few $P_k$ when $\\\\lambda \\\\ll 1$ , made evident by the leading terms in Eq.\",\n \"REF for $k+1$ and $k$ : $ \\\\frac{P_{k+1}}{P_k} \\\\approx \\\\frac{f(k+1;\\\\lambda )}{f(k;\\\\lambda )}= \\\\frac{\\\\lambda ^{k+1}/(k+1)!\",\n \"}{\\\\lambda ^k/k!\",\n \"}=\\\\frac{\\\\lambda }{k+1}\\\\xrightarrow[\\\\lambda \\\\rightarrow 0]{}0.$ Multiple collisions are thus increasingly unlikely when $\\\\lambda \\\\ll 1$ .\",\n \"In such cases, comparison with $P_1=1-e^{-\\\\lambda }\\\\approx \\\\lambda = \\\\nu \\\\Delta t$ is sufficient.\",\n \"This work was supported under the NESSF program, fellowship number 11-Planet11R-0005.\",\n \"BF thanks two reviewers for their feedback and many useful suggestions.\",\n \"Figure: Sketch of gyrating ions in the neutral framewith guiding centers moving along a prescribedflow field, shown here near Enceladus for scale.Warm ions (v ⊥ >v flow v_\\\\perp > v_\\\\mathrm {flow}) move on trajectories that coilaround themselves and do not reach zero relative velocity with respect to theneutrals at any point.Cool ions (v ⊥ 3\\mbox{ GeV}\\approx 2m_c$ and that interference contributions are small.","Also one can try to avoid double counting by subtracting the part due to the direct gluon splitting from the total $c$ -quark structure function: $\\tilde{f}_c(x,Q^2)=f_c(x,Q^2)-\\frac{\\alpha _s(Q^2)}{4\\pi }\\int _x^1 \\frac{dz}{z}\\left[\\Bigl (\\frac{x}{z}\\Bigr )^2+\\Bigl (1-\\frac{x}{z}\\Bigr )^2 + \\frac{2m_c^2 x (z-x)}{z^2Q^2}\\right] f_g(z,Q^2),$ where splitting function is taken from [59].","The cross section value of the subprocesses $cg \\rightarrow J/\\psi c$ was found to be about $10\\div 40~\\mathrm {nb}$ .","Therefore the contribution of such corrections to the $J/\\psi +D$ associated production is of the same order as the contribution of the main subprocess $gg\\rightarrow J/\\psi c \\bar{c}$ .","Thus the calculations within pQCD lead to the cross section value of about $30\\div 100$ nb for the $J/\\psi +D$ production in the LHCb fiducial region.","It should be noticed that in contrast to the charmonia pairs production in the associated charm production there are no $C$ -parity selection rules.","So one should expect not only feed-down from the $J/\\psi +\\psi (2S)$ production but also from the $J/\\psi +\\chi _C$ one.","This contributions can increase observed $J/\\psi + D$ cross section by up to 50%.","In the Fig.","REF Figure: Distribution over the transverse momentum of the J/ψJ/\\psi -meson in the J/ψ+DJ/\\psi + D productioncompared with the LHCb measurement (points for J/ψJ/\\psi produced together with D 0 D^0 or D + D^+-meson are shown).Solid curves were obtained with the hard scale value of 1·m T J/ψ 1\\cdot m_T^{J/\\psi }, dashed — 2·m T J/ψ 2 \\cdot m_T^{J/\\psi } and dotted —0.5·m T J/ψ 0.5 \\cdot m_T^{J/\\psi }.","Dot-dashed curve corresponds to the collinear gluon approximation.cross section distribution over the $J/\\psi $ meson transverse momentum in the $gg \\rightarrow J/\\psi +c\\bar{c}$ subprocess is shown in comparison with the LHCb experimental data.","The $d \\ln \\sigma /d p_T$ distributions are plotted, i.e.","spectra are normalized to unity.","Both $J/\\psi $ and associated charmed hadron produced in the events plotted are limited to the LHCb fiducial region.","It can be seen that at least in the high $p_T^{J/\\psi }$ region the predicted slope is in a good agreement with the experimental data.","It should be noticed that $p_T^{J/\\psi }$ distribution in the inclusive $J/\\psi $ production measured by LHCb exhibits significantly more rapid decrease with the $p_T^{J/\\psi }$ growth.","Cross section distribution over the $D$ -meson transverse momentum for the same $gg \\rightarrow J/\\psi +c\\bar{c}$ subprocess is given in the Fig.","REF Figure: Distribution over the transverse momentum of the DD-meson in the J/ψ+DJ/\\psi + D productioncompared with the LHCb measurement (points for D 0 D^0 and D + D^+-mesons are shown).Curve designations coincide with Fig.",".and demonstrates good agreement with the LHCb measurement.","As in the previous figure, both spectra are normalized to unity.","In contrast to the $J/\\psi $ signal, both predicted and measured spectra are similar to those in the inclusive $D$ -meson production at LHCb [48].","As in double $J/\\psi $ production, essential angle and rapidity correlations in the $gg \\rightarrow J/\\psi c \\bar{c}$ process are predicted by pQCD.","Within collinear approach $J/\\psi $ and $D$ mesons move in the opposite directions in most cases.","However no concrete prediction can be made when taking into account transverse gluon motion in the framework of the Pythia generator as the distribution is highly sensitive on the scale selection (see Fig.","REF ).","Figure: Distribution over the difference of the azimuthal angles of the J/ψJ/\\psi and DD-meson in the J/ψ+DJ/\\psi + D production.Curve designations coincide with Fig.",".What concerns distribution over the rapidity difference between $J/\\psi $ and $D$ -meson produced, from Fig.","REF Figure: Distribution over the difference of rapidities of the J/ψJ/\\psi and DD-meson in the J/ψ+DJ/\\psi + D production.Solid curve corresponds to the LHCb kinematic limits imposed, dashed — to the absence of kinematic limits.one can see that contrary to the two $J/\\psi $ -meson production, LHCb rapidity window appears to be too narrow to observe rapidity correlations predicted in the $gg \\rightarrow J/\\psi c \\bar{c}$ subprocess.","We omit discussion of correlations between $D$ and $\\bar{D}$ mesons in the $J/\\psi + D$ associated production as LHCb analysis focuses on events in which one co-produced $D$ meson is observed.","The cross section value obtained by LHCb collaboration, $\\sigma ^\\mathrm {exp}(pp \\rightarrow J/\\psi + D^0(D^+,D^+_s,\\Lambda ^+_c)+X)\\approx 300 \\mbox{ nb},$ is several times larger than the SPS prediction of $30\\div 100~\\mathrm {nb}$ .","Let us now address to the simultaneous production of $J/\\psi $ and open charm in two gluon-gluon interactions.","Within the DPS approach cross section of the associated $J/\\psi $ and $D$ -meson production can be expressed as follows: $\\sigma ^{\\rm DPS}_{J/\\psi D} = \\frac{\\sigma _{J/\\psi } \\sigma _D}{\\sigma _{\\rm eff.","}},$ where $\\sigma _{J/\\psi }$ and $\\sigma _D$ are cross sections of the inclusive $J/\\psi $ and $D$ -meson production in the LHCb acceptance correspondingly.","Recalculated for the fiducial region discussed ($23~\\mathrm {GeV}$ ), these values are $9~\\mathrm {\\mu b}$ and $380~\\mathrm {\\mu b}$ respectively [60], [48].","As always summation with the charge conjugate state is assumed.","Unpublished cross section of the $\\Lambda _C$ inclusive production is not included in consideration.","Thus the associated $J/\\psi $ and $D$ meson production cross section for the LHCb kinematical region within the DPS model can be estimated as $\\sigma ^{\\rm DPS}_{J/\\psi D} = 240~\\mathrm {nb}.$ As earlier, numerical value of $\\sigma _{\\rm eff.","}=14.5~\\mathrm {mb}$ [45], [46] was used.","One can see that DPS prediction is several times larger than the SPS one and within uncertainty limits agrees with the experimental value (REF )."],["Four $D$ -meson production in the LHCb detector","In the same LHCb studies [15] production of four $c$ -quarks is investigated.","Events in which two open charm hadrons both containing $c$ -quark (or both containing $\\bar{c}$ -quark) are produced in the fiducial region $2.020$ GeV can be explained by cut on the maximum transverse momenta: $m_{cc}^\\mathrm {cut}\\approx 2p_T^\\mathrm {max}.$ Let us now turn to the DPS contribution to the different $D$ -meson pairs production.","Expression (REF ) has to be modified as experimentally observed quantities are inclusive production cross sections of particular types of $D$ -mesons summed together with anti-mesons of the same type.","These cross sections can be written down as follows: $\\sigma ^{incl.","}_i = \\sigma _1 p^{c\\vee \\bar{c}}_i + \\sigma _2 (2 p^{c\\vee \\bar{c}}_i - (p^{c\\vee \\bar{c}}_i)^2),$ where $\\sigma _1$ and $\\sigma _2$ are cross sections of one and two $c\\bar{c}$ pair production in a single proton-proton collision respectively and $p^{c\\vee \\bar{c}}_i$ is probability that $c$ or $\\bar{c}$ quark transits into detected hadron of type $i$ .","In the following we will be interested in events in which both $c$ and $\\bar{c}$ quarks form two $D$ -mesons of particular type in the detector acceptance, or it is done by pairs of identical quarks — $cc$ or $\\bar{c}\\bar{c}$ .","In the first case cross section of the $i$ type meson pair production can be written down as $\\sigma ^{diff.","}_{i,i} = \\sigma _1 p^{c\\wedge \\bar{c}}_{i,i} + \\sigma _2 (2 p^{c\\wedge \\bar{c}}_{i,i} - (p^{c\\wedge \\bar{c}}_{i,i})^2+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,i})^2/2 ),$ and in the second — as $\\sigma ^{same}_{i,i} = \\sigma _2 ((p^{c\\wedge \\bar{c}}_{i,i})^2+ 2 (p^{c\\wedge \\bar{c}}_{i,i})(p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,i})+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,i})^2/2 ).$ Here $p^{c\\wedge \\bar{c}}_{i,j}$ stands for the probability for $c$ and $\\bar{c}$ quarks from one pair to transit into mesons of type $i$ and $j$ observed in the detector and probabilities for quarks from the different pairs are assumed independent.","For the different $i$ and $j$ types of mesons analogous quantities are written down as $\\sigma ^{diff.","}_{i,j} &=& \\sigma _1 p^{c\\wedge \\bar{c}}_{i,j} + \\sigma _2 (2 p^{c\\wedge \\bar{c}}_{i,j}- (p^{c\\wedge \\bar{c}}_{i,j})^2+ 2 p^{c\\wedge \\bar{c}}_{i,i}p^{c\\wedge \\bar{c}}_{j,j}+ 2 p^{c\\wedge \\bar{c}}_{i,i} (p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j}) +\\nonumber \\\\&+& 2 p^{c\\wedge \\bar{c}}_{j,j} (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})(p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j})),\\\\\\sigma ^{same}_{i,j} &=& \\sigma _2 (0.5 (p^{c\\wedge \\bar{c}}_{i,j})^2+ 2 p^{c\\wedge \\bar{c}}_{i,i}p^{c\\wedge \\bar{c}}_{j,j}+ 2 p^{c\\wedge \\bar{c}}_{i,i} (p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j}) +\\nonumber \\\\&+& 2 p^{c\\wedge \\bar{c}}_{j,j} (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})(p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j})).$ To solve the equations adduced we will use known inclusive production cross sections of particular $D$ -meson types [48].","As LHCb collaboration presents these cross sections in bins of rapidity and transverse momenta, they can be recalculated into the fiducial region discussed ($23\\\\mbox{ GeV}\\\\approx 2m_c$ and that interference contributions are small.\",\n \"Also one can try to avoid double counting by subtracting the part due to the direct gluon splitting from the total $c$ -quark structure function: $\\\\tilde{f}_c(x,Q^2)=f_c(x,Q^2)-\\\\frac{\\\\alpha _s(Q^2)}{4\\\\pi }\\\\int _x^1 \\\\frac{dz}{z}\\\\left[\\\\Bigl (\\\\frac{x}{z}\\\\Bigr )^2+\\\\Bigl (1-\\\\frac{x}{z}\\\\Bigr )^2 + \\\\frac{2m_c^2 x (z-x)}{z^2Q^2}\\\\right] f_g(z,Q^2),$ where splitting function is taken from [59].\",\n \"The cross section value of the subprocesses $cg \\\\rightarrow J/\\\\psi c$ was found to be about $10\\\\div 40~\\\\mathrm {nb}$ .\",\n \"Therefore the contribution of such corrections to the $J/\\\\psi +D$ associated production is of the same order as the contribution of the main subprocess $gg\\\\rightarrow J/\\\\psi c \\\\bar{c}$ .\",\n \"Thus the calculations within pQCD lead to the cross section value of about $30\\\\div 100$ nb for the $J/\\\\psi +D$ production in the LHCb fiducial region.\",\n \"It should be noticed that in contrast to the charmonia pairs production in the associated charm production there are no $C$ -parity selection rules.\",\n \"So one should expect not only feed-down from the $J/\\\\psi +\\\\psi (2S)$ production but also from the $J/\\\\psi +\\\\chi _C$ one.\",\n \"This contributions can increase observed $J/\\\\psi + D$ cross section by up to 50%.\",\n \"In the Fig.\",\n \"REF Figure: Distribution over the transverse momentum of the J/ψJ/\\\\psi -meson in the J/ψ+DJ/\\\\psi + D productioncompared with the LHCb measurement (points for J/ψJ/\\\\psi produced together with D 0 D^0 or D + D^+-meson are shown).Solid curves were obtained with the hard scale value of 1·m T J/ψ 1\\\\cdot m_T^{J/\\\\psi }, dashed — 2·m T J/ψ 2 \\\\cdot m_T^{J/\\\\psi } and dotted —0.5·m T J/ψ 0.5 \\\\cdot m_T^{J/\\\\psi }.\",\n \"Dot-dashed curve corresponds to the collinear gluon approximation.cross section distribution over the $J/\\\\psi $ meson transverse momentum in the $gg \\\\rightarrow J/\\\\psi +c\\\\bar{c}$ subprocess is shown in comparison with the LHCb experimental data.\",\n \"The $d \\\\ln \\\\sigma /d p_T$ distributions are plotted, i.e.\",\n \"spectra are normalized to unity.\",\n \"Both $J/\\\\psi $ and associated charmed hadron produced in the events plotted are limited to the LHCb fiducial region.\",\n \"It can be seen that at least in the high $p_T^{J/\\\\psi }$ region the predicted slope is in a good agreement with the experimental data.\",\n \"It should be noticed that $p_T^{J/\\\\psi }$ distribution in the inclusive $J/\\\\psi $ production measured by LHCb exhibits significantly more rapid decrease with the $p_T^{J/\\\\psi }$ growth.\",\n \"Cross section distribution over the $D$ -meson transverse momentum for the same $gg \\\\rightarrow J/\\\\psi +c\\\\bar{c}$ subprocess is given in the Fig.\",\n \"REF Figure: Distribution over the transverse momentum of the DD-meson in the J/ψ+DJ/\\\\psi + D productioncompared with the LHCb measurement (points for D 0 D^0 and D + D^+-mesons are shown).Curve designations coincide with Fig.\",\n \".and demonstrates good agreement with the LHCb measurement.\",\n \"As in the previous figure, both spectra are normalized to unity.\",\n \"In contrast to the $J/\\\\psi $ signal, both predicted and measured spectra are similar to those in the inclusive $D$ -meson production at LHCb [48].\",\n \"As in double $J/\\\\psi $ production, essential angle and rapidity correlations in the $gg \\\\rightarrow J/\\\\psi c \\\\bar{c}$ process are predicted by pQCD.\",\n \"Within collinear approach $J/\\\\psi $ and $D$ mesons move in the opposite directions in most cases.\",\n \"However no concrete prediction can be made when taking into account transverse gluon motion in the framework of the Pythia generator as the distribution is highly sensitive on the scale selection (see Fig.\",\n \"REF ).\",\n \"Figure: Distribution over the difference of the azimuthal angles of the J/ψJ/\\\\psi and DD-meson in the J/ψ+DJ/\\\\psi + D production.Curve designations coincide with Fig.\",\n \".What concerns distribution over the rapidity difference between $J/\\\\psi $ and $D$ -meson produced, from Fig.\",\n \"REF Figure: Distribution over the difference of rapidities of the J/ψJ/\\\\psi and DD-meson in the J/ψ+DJ/\\\\psi + D production.Solid curve corresponds to the LHCb kinematic limits imposed, dashed — to the absence of kinematic limits.one can see that contrary to the two $J/\\\\psi $ -meson production, LHCb rapidity window appears to be too narrow to observe rapidity correlations predicted in the $gg \\\\rightarrow J/\\\\psi c \\\\bar{c}$ subprocess.\",\n \"We omit discussion of correlations between $D$ and $\\\\bar{D}$ mesons in the $J/\\\\psi + D$ associated production as LHCb analysis focuses on events in which one co-produced $D$ meson is observed.\",\n \"The cross section value obtained by LHCb collaboration, $\\\\sigma ^\\\\mathrm {exp}(pp \\\\rightarrow J/\\\\psi + D^0(D^+,D^+_s,\\\\Lambda ^+_c)+X)\\\\approx 300 \\\\mbox{ nb},$ is several times larger than the SPS prediction of $30\\\\div 100~\\\\mathrm {nb}$ .\",\n \"Let us now address to the simultaneous production of $J/\\\\psi $ and open charm in two gluon-gluon interactions.\",\n \"Within the DPS approach cross section of the associated $J/\\\\psi $ and $D$ -meson production can be expressed as follows: $\\\\sigma ^{\\\\rm DPS}_{J/\\\\psi D} = \\\\frac{\\\\sigma _{J/\\\\psi } \\\\sigma _D}{\\\\sigma _{\\\\rm eff.\",\n \"}},$ where $\\\\sigma _{J/\\\\psi }$ and $\\\\sigma _D$ are cross sections of the inclusive $J/\\\\psi $ and $D$ -meson production in the LHCb acceptance correspondingly.\",\n \"Recalculated for the fiducial region discussed ($23~\\\\mathrm {GeV}$ ), these values are $9~\\\\mathrm {\\\\mu b}$ and $380~\\\\mathrm {\\\\mu b}$ respectively [60], [48].\",\n \"As always summation with the charge conjugate state is assumed.\",\n \"Unpublished cross section of the $\\\\Lambda _C$ inclusive production is not included in consideration.\",\n \"Thus the associated $J/\\\\psi $ and $D$ meson production cross section for the LHCb kinematical region within the DPS model can be estimated as $\\\\sigma ^{\\\\rm DPS}_{J/\\\\psi D} = 240~\\\\mathrm {nb}.$ As earlier, numerical value of $\\\\sigma _{\\\\rm eff.\",\n \"}=14.5~\\\\mathrm {mb}$ [45], [46] was used.\",\n \"One can see that DPS prediction is several times larger than the SPS one and within uncertainty limits agrees with the experimental value (REF ).\"\n ],\n [\n \"Four $D$ -meson production in the LHCb detector\",\n \"In the same LHCb studies [15] production of four $c$ -quarks is investigated.\",\n \"Events in which two open charm hadrons both containing $c$ -quark (or both containing $\\\\bar{c}$ -quark) are produced in the fiducial region $2.020$ GeV can be explained by cut on the maximum transverse momenta: $m_{cc}^\\\\mathrm {cut}\\\\approx 2p_T^\\\\mathrm {max}.$ Let us now turn to the DPS contribution to the different $D$ -meson pairs production.\",\n \"Expression (REF ) has to be modified as experimentally observed quantities are inclusive production cross sections of particular types of $D$ -mesons summed together with anti-mesons of the same type.\",\n \"These cross sections can be written down as follows: $\\\\sigma ^{incl.\",\n \"}_i = \\\\sigma _1 p^{c\\\\vee \\\\bar{c}}_i + \\\\sigma _2 (2 p^{c\\\\vee \\\\bar{c}}_i - (p^{c\\\\vee \\\\bar{c}}_i)^2),$ where $\\\\sigma _1$ and $\\\\sigma _2$ are cross sections of one and two $c\\\\bar{c}$ pair production in a single proton-proton collision respectively and $p^{c\\\\vee \\\\bar{c}}_i$ is probability that $c$ or $\\\\bar{c}$ quark transits into detected hadron of type $i$ .\",\n \"In the following we will be interested in events in which both $c$ and $\\\\bar{c}$ quarks form two $D$ -mesons of particular type in the detector acceptance, or it is done by pairs of identical quarks — $cc$ or $\\\\bar{c}\\\\bar{c}$ .\",\n \"In the first case cross section of the $i$ type meson pair production can be written down as $\\\\sigma ^{diff.\",\n \"}_{i,i} = \\\\sigma _1 p^{c\\\\wedge \\\\bar{c}}_{i,i} + \\\\sigma _2 (2 p^{c\\\\wedge \\\\bar{c}}_{i,i} - (p^{c\\\\wedge \\\\bar{c}}_{i,i})^2+ (p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,i})^2/2 ),$ and in the second — as $\\\\sigma ^{same}_{i,i} = \\\\sigma _2 ((p^{c\\\\wedge \\\\bar{c}}_{i,i})^2+ 2 (p^{c\\\\wedge \\\\bar{c}}_{i,i})(p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,i})+ (p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,i})^2/2 ).$ Here $p^{c\\\\wedge \\\\bar{c}}_{i,j}$ stands for the probability for $c$ and $\\\\bar{c}$ quarks from one pair to transit into mesons of type $i$ and $j$ observed in the detector and probabilities for quarks from the different pairs are assumed independent.\",\n \"For the different $i$ and $j$ types of mesons analogous quantities are written down as $\\\\sigma ^{diff.\",\n \"}_{i,j} &=& \\\\sigma _1 p^{c\\\\wedge \\\\bar{c}}_{i,j} + \\\\sigma _2 (2 p^{c\\\\wedge \\\\bar{c}}_{i,j}- (p^{c\\\\wedge \\\\bar{c}}_{i,j})^2+ 2 p^{c\\\\wedge \\\\bar{c}}_{i,i}p^{c\\\\wedge \\\\bar{c}}_{j,j}+ 2 p^{c\\\\wedge \\\\bar{c}}_{i,i} (p^{c\\\\vee \\\\bar{c}}_j - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{j,j}) +\\\\nonumber \\\\\\\\&+& 2 p^{c\\\\wedge \\\\bar{c}}_{j,j} (p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{i,i})+ (p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{i,i})(p^{c\\\\vee \\\\bar{c}}_j - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{j,j})),\\\\\\\\\\\\sigma ^{same}_{i,j} &=& \\\\sigma _2 (0.5 (p^{c\\\\wedge \\\\bar{c}}_{i,j})^2+ 2 p^{c\\\\wedge \\\\bar{c}}_{i,i}p^{c\\\\wedge \\\\bar{c}}_{j,j}+ 2 p^{c\\\\wedge \\\\bar{c}}_{i,i} (p^{c\\\\vee \\\\bar{c}}_j - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{j,j}) +\\\\nonumber \\\\\\\\&+& 2 p^{c\\\\wedge \\\\bar{c}}_{j,j} (p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{i,i})+ (p^{c\\\\vee \\\\bar{c}}_i - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{i,i})(p^{c\\\\vee \\\\bar{c}}_j - p^{c\\\\wedge \\\\bar{c}}_{i,j} - p^{c\\\\wedge \\\\bar{c}}_{j,j})).$ To solve the equations adduced we will use known inclusive production cross sections of particular $D$ -meson types [48].\",\n \"As LHCb collaboration presents these cross sections in bins of rapidity and transverse momenta, they can be recalculated into the fiducial region discussed ($20 \\\\-sign(\\Delta _{hyst}(k)), & \\quad s_{14}(k-1) \\le 0\\end{array}\\right.","}\\end{array}}$ where $k_{hyst}$ is set around $\\frac{1 (s)}{dt}$ and it is used to provide some hysteresis: the yaw damping moment is increased about 1 second before the turning rate command starts slowing down.","Additionally, the following parameters are chosen: ${\\begin{array}{c}P_{err}=0.01^2 \\\\Qu=Pu_{k=0} \\cdot 0.1\\end{array}}$"],["Sigma Points","At each simulation step we choose $2m+1$ sigma points $\\vec{U^i}$ and their associated weights $W^i$ in the following way, where the value of $W^0$ was chosen by tuning: ${\\begin{array}{c}\\vec{U^0}=\\vec{U}_{k/k} \\\\\\vec{U^i}=\\vec{U}_{k/k} \\pm \\sqrt{\\frac{n}{1-W^0}}\\vec{S}_j, \\qquad i=1,..2m, \\quad j=1,..m \\\\W^i=\\frac{1-W^0}{2m}, \\qquad W^0=0.25\\end{array}}$ where $\\vec{S}_j$ is column $j$ of matrix $S$ that satisfies $S \\cdot S = Pu_{k/k}$ ."],["Actuation Constraints","After the sigma points are selected they are bounded by the minimum and maximum values of the actuation constraints: $ {\\begin{array}{c}\\vec{U^i}_{min}=[0.01, 0, 0, -1.5]^T \\\\\\vec{U^i}_{max}=[6, 5, 5, 1.5]^T\\end{array}}$ Additionally, the turning pattern angle command $\\alpha $ has a slew rate limit, according to (REF ).","This is taken into account when the plant model is propagated: the turning pattern angle at each step is computed from $\\vec{U^i}$ and the slew rate constraint.","The sigma points are prevented from being identical, what can happen after imposing the constraints, by modifying the relevant elements of sigma points.","If an element $j$ of $ \\vec{U}_{k/k}$ equals to its min or max boundary and $\\vec{U^i}(j)<\\vec{U^i}_{min}(j)$ or $\\vec{U^i}(j)>\\vec{U^i}_{max}(j)$ , respectively, then this element is moved to the other side of the boundary: $ \\vec{U^i}(j)=1.5 \\cdot \\vec{U}_{k/k}(j)-0.5 \\cdot \\vec{U^i}(j)$"],["Observation Model: Propagation of the plant dynamics","Each sigma point drives a skydiver model, since the control variables contained in $\\vec{U^i}$ are the plant actuators.","The plant equations are propagated from the plant's state at step $k$ ($\\vec{X_k}$ ) until the prediction horizon ($t_{pred}$ ).","The plant's state contains the linear and angular velocities of the skydiver.","The final value of yaw rate $Y^i_{k+\\frac{t_{pred}}{dt}}$ , computed for each sigma point $i$ and saved as $Y^i$ , is then weighted, (REF ), and used for the update step.","${\\begin{array}{c}Y^i \\text{ is } Y^i_{k+\\frac{t_{pred}}{dt}}\\\\Ypred_{k}= \\displaystyle \\sum _{i=0}^{2m}W^iY^i \\\\ W^i=\\frac{1-W^0}{2m}\\end{array}}$ The prediction horizon $t_{pred}$ is a tuning parameter and will be discussed later in this section."],["Prediction Step","The computation flow is schematically shown in Figure REF .","Sigma points are selected according to (REF ), constrained according to (REF ), (REF ), propagated until the prediction horizon $\\vec{U^i}_{k+\\frac{t_{pred}}{dt}}$ , saved as $\\vec{U^i}$ , and summarized as: ${\\begin{array}{c}\\vec{U^i} \\text{ is } \\vec{U^i}_{k+\\frac{t_{pred}}{dt}} \\\\\\vec{U}_{k+1/k}=\\displaystyle \\sum _{i=0}^{2m}W^i\\vec{U^i}\\end{array}}$ $Pu_{k+1/k}=Qu+\\displaystyle \\sum _{i=0}^{2m}W^i(\\vec{U^i}-\\vec{U}_{k+1/k})\\cdot (\\vec{U^i}-\\vec{U}_{k+1/k})^T$"],["Update Step","The updated controller $\\vec{U}_{k+1/k+1}$ is bounded according to (REF ), and the slew rate constraint (REF )."],["Simulation Results","The fast turning maneuver defined in Sect.","is now tracked utilizing the UTC, while different solutions are obtained by the means of configuring the controller parameters.","First of all, the input moment coefficients $k_{right}$ , $k_{left}$ are considered.","Their desired dynamics is not known and, therefore, the behavior is driven by the process noise $Qu$ .","Thus, in order to get qualitatively different solutions the off-diagonal values of $Qu$ are modeled in different ways, e.g.","(REF ), (REF ).","${\\begin{array}{c}Qu_{1,2} = Qu_{2,1} = q_{1,2} \\cdot sign(yr_{ref}) \\\\Qu_{1,3} = Qu_{3,1} = -q_{1,3} \\cdot sign(yr_{ref})\\end{array}}$ where $1,2,3$ are the indices of $Cm_{damp}^{yaw}$ , $k_{right}$ , $k_{left}$ , respectively, and $q_{1,2}, q_{1,3}$ are tuning parameters that can be assumed equal.","${\\begin{array}{c}Qu_{1,2} = Qu_{2,1} = q_{1,2} \\cdot sign(\\frac{d}{dt} yr_{ref}) \\\\Qu_{1,3} = Qu_{3,1} = q_{1,3} \\cdot sign(\\frac{d}{dt}yr_{ref})\\end{array}}$ The coupling in (REF ) means that when $Cm_{damp}^{yaw}$ is reducing, making the turn faster, the appropriate input moment is increased to further accelerate the turn.","Thus, the coupling expressed by $Qu$ should be negative for $ k_{left}$ during left turns ($sign(yr_{ref})>0$ ) and for $ k_{right}$ during right turns, and positive otherwise.","The desired dynamics of the turning pattern angle command ($\\alpha $ ) can be assumed unknown, and left to be resolved by the UTC.","Optionally, the dynamics given in (REF ) can be utilized.","The gains of the proportional and the feed-forward parts can be tuning parameters.","The computation of the controller command during the prediction window can be implemented as shown in (REF ).","This enforces the slew rate constraint on $\\alpha (t)$ , see (REF ).","$\\begin{split}\\alpha (t)= \\alpha (t-dt)+ & K_p \\cdot (yr_{ref}(t)-yr_{ref}(t-dt))- \\\\ -&K_p \\cdot (yr_{real}(t)-yr_{real}(t-dt))+ \\\\ +&K_{FF} \\cdot (yr_{ref}(t+\\tau )-yr_{ref}(t+\\tau -dt))\\end{split}$ Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming unknown controller dynamics and correlation between input moment and yaw damping moment coefficients according to ().Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming controller dynamics according to () and correlation between input moment and yaw damping moment coefficients according to ()Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming controller dynamics according to () and correlation between input moment and yaw damping moment coefficients according to ()When a desired dynamics for the turning pattern angle is incorporated in the prediction step, a better accuracy is achieved: compare Figures REF , REF .","Also a smaller prediction horizon can be used.","For $dt=0.0042 \\, (s)$ 20 prediction steps were used (i.e.","$t_{pred}\\approx 0.08 \\, (s)$ ) when assuming unknown dynamics for $\\alpha (t)$ , versus 5 steps ($t_{pred}\\approx 0.02 \\, (s)$ ) when incorporating (REF ).","Notice the qualitatively different behavior of the input moment coefficients, shown in Figs.","REF , REF .","If $\\alpha (t)$ dynamics is unknown, the best accuracy is achieved when the correlation between the yaw damping moment and the input moment coefficients is defined according to (REF ), see Fig.","REF .","Notice, that (REF ) is a more intuitive model than (REF ).","If $\\alpha (t)$ dynamics is reasonably modeled, tuning the correlation between these variables is less significant.","Accurate tracking is achieved even if no correlation is assumed.","This, and other interesting solutions, are shown in [4].","The minimal control effort is achieved when combining (REF ) and (REF ), see Fig.","REF ."],["CONCLUSIONS AND FUTURE WORK","From simulations we observe the following advantages of the UTC: 1.","Fast convergence (around 2 (s)); 2.","Not much sensitivity to the tuning parameters: if the covariances are set to reasonable values, i.e.","the same order of magnitude as the allowed ranges of the actuators involved, UTC always converges.","3.","Convenient tuning: achieved by the means of setting the elements of $Pu_0$ and $Qu$ , which have a clear physical meaning.","Therefore, it is possible to obtain different solutions that implement the desired/ expected behavior of the control variables.","Further investigating the UTC properties remains an issue for future work: sensitivity to model uncertainty and measurement noise, and operation with only partial state feedback.","Additionally, it is important to formulate and prove conditions for the global/ local stability of systems controlled by the UTC.","Initial efforts in this regard are presented in [4]: for linear systems a stability condition takes the form of an Algebraic Riccati Equation, while for the number of prediction steps $n>1$ it has one additional term."]],"string":"[\n [\n \"The Unscented Transform Controller: a new model predictive control law\\n for highly nonlinear systems\"\n ],\n [\n \"Abstract The Unscented Transform which is the basis of the Unscented Kalman Filter, UKF, is used here to develop a novel predictive controller for non-linear plants, called the Unscented Transform Controller, UTC.\",\n \"The UTC can be seen as the dual of the UKF, the same way as the LQG regulator and the Kalman Filter are related.\",\n \"The UTC is demonstrated on the control of complex maneuvers in free fall of a virtual skydiver the model of which was verified in wind tunnel and free fall experiments.\"\n ],\n [\n \"INTRODUCTION\",\n \"Even after more than one century of research, non-linear control is still more of an art than an orderly engineering discipline.\",\n \"A plethora of methods and approaches exist, [5], from various linearization approaches to more recent neural network controllers, and non-linear model predictive control, [3].\",\n \"As a general rule, the control designer must profoundly understand the controlled plant and its particular non-linear structure in order to succeed in choosing an appropriate control method.\",\n \"In a similar way, non-linear filtering and estimation, [7], is an art in contrast to the systematic way linear estimation is done with the Kalman filter, [9].\",\n \"For non-linear processes, developments of the Kalman filter include the Extended Kalman Filter (EKF), particle filters, and in particular the numerically efficient Unscented Kalman Filter, UKF, [6], [8].\",\n \"A well known feature of linear control and estimation is the duality of the LQG controller and the Kalman filter, [10].\",\n \"In this paper, a novel controller dual of the UKF, called the Unscented Transform Controller, UTC, is proposed for the control of nonlinear plants.\",\n \"The UTC is model based, and can be seen as an original model predictive controller.\",\n \"The inspiration behind the UTC is the successful use of the UKF for the estimation of inputs to a highly non-linear skydiver model in complex free-fall maneuvers, and the attempt to control that system.\",\n \"While simple maneuvers for such a model can be efficiently followed using robust linear control, [11], such an approach fails for complex maneuvers.\",\n \"Another less than successful attempt was the use of a Recurrent Neural Network Controller, as reported in [4].\",\n \"The UTC, however, accurately tracks the desired maneuvers, as demonstrated in this paper, and in [4] for additional maneuvers such as front and back layouts: transitions between belly-to-earth and back-to-earth equilibria through the head-to-earth orientation.\",\n \"The paper is organized as follows: Section describes the skydiver model, the control input, and the control task.\",\n \"In Section the Unscented Transform Controller is presented in detail together with some illuminating simulations.\",\n \"Section contains the conclusion with suggestions for future work, and the remark that [4] contains a rudimentary stability proof for the case the UTC is used to control a linear plant.\"\n ],\n [\n \"SYSTEM UNDER INVESTIGATION\",\n \"The system under investigation is a skydiver performing aerial maneuvers in free-fall.\",\n \"A model of skydiver's body, aerodynamic forces and moments acting on it at terminal velocity (around 60 (m/s)), and equations of motion were developed in [1].\",\n \"The model was validated in free-fall and wind tunnel experiments with different skydivers performing a variety of maneuvers.\",\n \"The model is driven by the following inputs: 1. body posture; 2. roll, pitch, and yaw damping moment coefficients; and 3. input moment coefficients, associated with body segments.\",\n \"The damping moment coefficients reflect an approximate body resistance (e.g.\",\n \"muscle stiffness) to the developing rotation rates.\",\n \"The input moment coefficients reflect the consciously applied physical resistance of each limb to the aerodynamic force acting on it.\",\n \"The body posture is defined by the relative orientation of body limbs, while each joint has 3 rotational degrees-of-freedom (DOFs).\",\n \"During skilled actions the human central nervous system organizes these DOFs into movement patterns: combinations of DOFs that are activated synchronously and proportionally, as a single unit.\",\n \"These movement patterns can be extracted from free-fall experiments by recording a sequence of measured postures during the maneuver time and conducting Principal Component Analysis (PCA).\",\n \"It was discovered that experienced skydivers can perform most maneuvers by utilizing just one movement pattern [2].\",\n \"These maneuvers can be reconstructed in simulation by actuating the skydiver model with the chosen movement pattern, the amplitude of which is the control variable.\",\n \"For example, in [4] the aerial rotations maneuver (360 degrees right and left turns in a belly-to-earth position) is reconstructed in simulation by applying a proportional controller with a feed-forward part, as shown in (REF ).\",\n \"${\\\\begin{array}{c}u(t) =1.5 \\\\cdot (yr_{ref}(t)-yr_{real}(t))+0.1\\\\cdot yr_{ref}(t) \\\\\\\\-1.5 \\\\le u(t) \\\\le 1.5 (rad), \\\\quad -3.5 \\\\le \\\\frac{du(t)}{dt} \\\\le 3.5 (rad/s) \\\\\\\\pose(t) = NeutralPose + u(t) \\\\cdot TurningPattern\\\\end{array}}$ where $yr_{ref}(t)$ $(rad/s)$ is the reference yaw rate profile; $yr_{real}(t)$ $(rad/s)$ is the yaw rate of the virtual skydiver in simulation; $u(t)$ $(rad)$ is the turning pattern angle command; $NeutralPose$ $(rad)$ is the trimmed pose of a skydiver falling straight down, defined by a vector containing values of body DOFs for all modeled joints; and $TurningPattern$ is an eigenvector defining the movement pattern utilized for turning.\",\n \"If the reference signal has a slow dynamics, the damping moment coefficients may remain constant during the maneuver, and zero input moments can be assumed.\",\n \"However, as the amplitude and/or the frequency of the desired yaw rate increases it becomes impossible to track the reference signal just by means of posture adjustments.\",\n \"The reason is that the magnitude of the movement pattern is limited due to the natural constraints of the human body.\",\n \"Additionally, the range of orientations of each limb relative to the airflow, within which the posture preserves aerodynamic efficiency, is also limited.\",\n \"Therefore, the adjustment of input and damping moments becomes necessary.\",\n \"Consider, for example, the yaw rate reference signal in (REF ).\",\n \"$yr_{ref}(t)=400\\\\cdot \\\\frac{\\\\pi }{180}sin(2\\\\cdot \\\\pi \\\\cdot 0.2 \\\\cdot t)$ Figure: Tracking the yaw rate reference profile with constant yaw damping moment coefficient (0.5) and pose controller in ().This signal can not be accurately tracked by utilizing only a turning movement pattern, while the yaw damping moment coefficient remains constant, see Fig.\",\n \"REF .\",\n \"Thus, for the purpose of an illustrating example of the control method described in the following section, three additional control variables are utilized: the yaw damping moment coefficient, and two input moment coefficients.\",\n \"For the purpose of simulating aerial rotations, the following simplified model for the input moments is used: ${\\\\begin{array}{c}Mz_{hands} = K_{N}\\\\cdot lx_{hand} \\\\cdot (k_{right}-k_{left}) \\\\\\\\Fy_{hands} = -K_{N}\\\\cdot (k_{right}+k_{left}) \\\\end{array}}$ where $Mz$ (Nm) is the roll moment acting about the longitudinal body axis, $Fy$ (N) is the force acting along the sagittal axis, $lx_{hand}$ =0.35 (m) is the characteristic (for neutral pose) distance of hands from the longitudinal axis projected to the frontal plane, $K_{N}$ =100 (N) is a scaling coefficient, and $k_{right}$ , $k_{left}$ are the user input dimensionless coefficients related to the right and left hands, accordingly.\",\n \"They represent pressure applied on the airflow with the hands.\",\n \"This creates a roll moment, which will cause the skydiver to develop a roll angle and therefore partially expose his torso to the airflow during the turn.\",\n \"This will create a force along a frontal axis, which will create a yaw moment, desired for the turning maneuver.\"\n ],\n [\n \"UNSCENTED TRANSFORM CONTROLLER\",\n \"This control law was inspired by a modification to the Unscented Kalman Filter developed in [1] for estimation of user inputs from experiments of advanced skydiving maneuvers in free-fall.\",\n \"At the prediction step the sigma points and the skydiver state were propagated during the prediction horizon $t_{pred}=0.25$ (s).\",\n \"Allowing each sigma point to drive the skydiver dynamics during the prediction window was essential in order to determine what influence a certain combination of user inputs has on the skydiver state.\",\n \"The prediction time reflected the skydiver model time constant.\",\n \"Notice, that estimating the user inputs that explain the measured plant dynamics is a very similar problem to predicting the user inputs that will provide the desired maneuvers.\",\n \"The latter is basically the definition of the control problem under investigation: designing a control algorithm, which will compute commands for body posture, input moment coefficients, and damping moment coefficients in order to track the reference angular/linear velocity signals.\",\n \"Therefore, we suggest the novel control scheme outlined below, which was termed the Unscented Transform Controller (UTC).\",\n \"It has much in common with the concept of standard Model Predictive Control, MPC, [12].\",\n \"However it has some important differences: The control output is not a result of optimization of a cost function, as in MPC, but a weighted average of sigma points.\",\n \"This average also represents an optimal solution in the sense of Minimum Mean Square Error (MMSE), given the sigma points.\",\n \"Propagation of the plant for 4 control variables (i.e.\",\n \"9 sigma points) is less computationally intensive than solving a nonlinear optimization problem at each step for the same prediction horizon.\",\n \"Additionally, there are no convergence problems that optimization engines have to deal with.\",\n \"The control outputs do not need to be parameterized as in non-linear MPC [3], what also contributes to the simplicity of the problem formulation, tuning, and computations.\",\n \"It is possible to incorporate the desired/expected dynamics of the control signals into the prediction step, when the sigma points are propagated.\",\n \"For example, it can be enforced that the turning pattern angle command has the dynamics of Proportional-Integral (PI) control.\",\n \"These features are demonstrated subsequently.\"\n ],\n [\n \"Formulation\",\n \"The structure of this controller is shown in block diagram in Fig.\",\n \"REF : it includes the definition of initial conditions, prediction step, propagation of the plant model up to the prediction horizon, and update step.\",\n \"Figure: Block diagram of the Unscented Transform Controller, comprising the computation of sigma points, prediction step, observation model, and update step.The control variables include the input moment coefficients, damping moment coefficients, and angles of the involved movement patterns.\",\n \"Specifically, in the case of rotation maneuvers: $\\\\vec{U}_k=[Cm_{damp}^{yaw}, k_{right}, k_{left}, \\\\alpha ]^T \\\\in \\\\textbf {R}^m,\\\\; m=4$ where $k$ is the simulation step such that $t=k \\\\cdot dt$ ; $k_{right}, k_{left}$ are the input moment coefficients, see (REF ); $Cm_{damp}^{yaw}$ is the yaw damping moment coefficient; and $\\\\alpha $ (rad) is the angle of the turning movement pattern, so that the body posture is then computed as: $pose_k=NeutralPose+\\\\alpha _k \\\\cdot TurningPattern$ If no a-priori knowledge is assumed about the dynamics of these control variables, the controller dynamics becomes: $\\\\vec{U}_{k+1}=\\\\vec{U}_k+\\\\vec{w_u}_k, \\\\qquad \\\\vec{w_u}_k \\\\sim N(0,Qu)$ where $\\\\vec{w_u}_k$ is the process noise, which will be discussed later, as it is the primary tuning parameter used for obtaining different solutions to the tracking problem.\"\n ],\n [\n \"Reference Signal\",\n \"The desired angular/linear velocities play in UTC the same role as the measurement vector in a conventional UKF.\",\n \"In this section it is the yaw rate reference signal (dimension $l=1$ ), which has to be tracked, defined in (REF ).\",\n \"Thus, $ Y_{ref}(k \\\\cdot dt + t_{pred})=yr_{ref}(k \\\\cdot dt + t_{pred})$ where $dt$ is the time between step $k$ and $k+1$ , and $t_{pred}$ is the prediction horizon.\",\n \"The measurement noise covariance matrix present in UKF, is denoted in UTC by $P_{err}$ and defines how accurately the reference signal will be tracked, i.e.\",\n \"the trade-off between accuracy and control effort.\"\n ],\n [\n \"Initial Conditions\",\n \"The initial conditions (step $k=0$ ) are chosen as follows: ${\\\\begin{array}{c}\\\\vec{U}_{k/k}=[0.5, 0, 0, 0]^T \\\\\\\\Pu_{k/k}=\\\\begin{bmatrix}6.25 & s_{12}\\\\cdot 0.74 & s_{13}\\\\cdot 0.74 & s_{14}\\\\cdot 0.56 \\\\\\\\ s_{12}\\\\cdot 0.74 & 0.01 & 0 & 0 \\\\\\\\ s_{13}\\\\cdot 0.74 & 0 & 0.01 & 0 \\\\\\\\ s_{14}\\\\cdot 0.56 & 0 & 0 & 0.01\\\\end{bmatrix}\\\\end{array}}$ where $s_{12}, s_{13}, s_{14}$ are the signs of the relevant covariance elements.\",\n \"There are several options for defining $s_{12}, s_{13}$ as a function of $k$ , which will be discussed later.\",\n \"The initial $s_{14}=-1$ , meaning that in order to start turning the yaw damping moment must be reduced.\",\n \"Equation (REF ) below defines $s_{14}$ as a function of $k$ : ${\\\\begin{array}{c}\\\\Delta (k)=|yr_{ref}(k+1)|-|yr_{ref}(k)| \\\\\\\\\\\\Delta _{hyst}(k)=|yr_{ref}(k+k_{hyst}+1)|-|yr_{ref}(k+k_{hyst})|\\\\\\\\s_{14}(k) = {\\\\left\\\\lbrace \\\\begin{array}{ll}-sign(\\\\Delta (k)), & \\\\quad s_{14}(k-1)>0 \\\\\\\\-sign(\\\\Delta _{hyst}(k)), & \\\\quad s_{14}(k-1) \\\\le 0\\\\end{array}\\\\right.\",\n \"}\\\\end{array}}$ where $k_{hyst}$ is set around $\\\\frac{1 (s)}{dt}$ and it is used to provide some hysteresis: the yaw damping moment is increased about 1 second before the turning rate command starts slowing down.\",\n \"Additionally, the following parameters are chosen: ${\\\\begin{array}{c}P_{err}=0.01^2 \\\\\\\\Qu=Pu_{k=0} \\\\cdot 0.1\\\\end{array}}$\"\n ],\n [\n \"Sigma Points\",\n \"At each simulation step we choose $2m+1$ sigma points $\\\\vec{U^i}$ and their associated weights $W^i$ in the following way, where the value of $W^0$ was chosen by tuning: ${\\\\begin{array}{c}\\\\vec{U^0}=\\\\vec{U}_{k/k} \\\\\\\\\\\\vec{U^i}=\\\\vec{U}_{k/k} \\\\pm \\\\sqrt{\\\\frac{n}{1-W^0}}\\\\vec{S}_j, \\\\qquad i=1,..2m, \\\\quad j=1,..m \\\\\\\\W^i=\\\\frac{1-W^0}{2m}, \\\\qquad W^0=0.25\\\\end{array}}$ where $\\\\vec{S}_j$ is column $j$ of matrix $S$ that satisfies $S \\\\cdot S = Pu_{k/k}$ .\"\n ],\n [\n \"Actuation Constraints\",\n \"After the sigma points are selected they are bounded by the minimum and maximum values of the actuation constraints: $ {\\\\begin{array}{c}\\\\vec{U^i}_{min}=[0.01, 0, 0, -1.5]^T \\\\\\\\\\\\vec{U^i}_{max}=[6, 5, 5, 1.5]^T\\\\end{array}}$ Additionally, the turning pattern angle command $\\\\alpha $ has a slew rate limit, according to (REF ).\",\n \"This is taken into account when the plant model is propagated: the turning pattern angle at each step is computed from $\\\\vec{U^i}$ and the slew rate constraint.\",\n \"The sigma points are prevented from being identical, what can happen after imposing the constraints, by modifying the relevant elements of sigma points.\",\n \"If an element $j$ of $ \\\\vec{U}_{k/k}$ equals to its min or max boundary and $\\\\vec{U^i}(j)<\\\\vec{U^i}_{min}(j)$ or $\\\\vec{U^i}(j)>\\\\vec{U^i}_{max}(j)$ , respectively, then this element is moved to the other side of the boundary: $ \\\\vec{U^i}(j)=1.5 \\\\cdot \\\\vec{U}_{k/k}(j)-0.5 \\\\cdot \\\\vec{U^i}(j)$\"\n ],\n [\n \"Observation Model: Propagation of the plant dynamics\",\n \"Each sigma point drives a skydiver model, since the control variables contained in $\\\\vec{U^i}$ are the plant actuators.\",\n \"The plant equations are propagated from the plant's state at step $k$ ($\\\\vec{X_k}$ ) until the prediction horizon ($t_{pred}$ ).\",\n \"The plant's state contains the linear and angular velocities of the skydiver.\",\n \"The final value of yaw rate $Y^i_{k+\\\\frac{t_{pred}}{dt}}$ , computed for each sigma point $i$ and saved as $Y^i$ , is then weighted, (REF ), and used for the update step.\",\n \"${\\\\begin{array}{c}Y^i \\\\text{ is } Y^i_{k+\\\\frac{t_{pred}}{dt}}\\\\\\\\Ypred_{k}= \\\\displaystyle \\\\sum _{i=0}^{2m}W^iY^i \\\\\\\\ W^i=\\\\frac{1-W^0}{2m}\\\\end{array}}$ The prediction horizon $t_{pred}$ is a tuning parameter and will be discussed later in this section.\"\n ],\n [\n \"Prediction Step\",\n \"The computation flow is schematically shown in Figure REF .\",\n \"Sigma points are selected according to (REF ), constrained according to (REF ), (REF ), propagated until the prediction horizon $\\\\vec{U^i}_{k+\\\\frac{t_{pred}}{dt}}$ , saved as $\\\\vec{U^i}$ , and summarized as: ${\\\\begin{array}{c}\\\\vec{U^i} \\\\text{ is } \\\\vec{U^i}_{k+\\\\frac{t_{pred}}{dt}} \\\\\\\\\\\\vec{U}_{k+1/k}=\\\\displaystyle \\\\sum _{i=0}^{2m}W^i\\\\vec{U^i}\\\\end{array}}$ $Pu_{k+1/k}=Qu+\\\\displaystyle \\\\sum _{i=0}^{2m}W^i(\\\\vec{U^i}-\\\\vec{U}_{k+1/k})\\\\cdot (\\\\vec{U^i}-\\\\vec{U}_{k+1/k})^T$\"\n ],\n [\n \"Update Step\",\n \"The updated controller $\\\\vec{U}_{k+1/k+1}$ is bounded according to (REF ), and the slew rate constraint (REF ).\"\n ],\n [\n \"Simulation Results\",\n \"The fast turning maneuver defined in Sect.\",\n \"is now tracked utilizing the UTC, while different solutions are obtained by the means of configuring the controller parameters.\",\n \"First of all, the input moment coefficients $k_{right}$ , $k_{left}$ are considered.\",\n \"Their desired dynamics is not known and, therefore, the behavior is driven by the process noise $Qu$ .\",\n \"Thus, in order to get qualitatively different solutions the off-diagonal values of $Qu$ are modeled in different ways, e.g.\",\n \"(REF ), (REF ).\",\n \"${\\\\begin{array}{c}Qu_{1,2} = Qu_{2,1} = q_{1,2} \\\\cdot sign(yr_{ref}) \\\\\\\\Qu_{1,3} = Qu_{3,1} = -q_{1,3} \\\\cdot sign(yr_{ref})\\\\end{array}}$ where $1,2,3$ are the indices of $Cm_{damp}^{yaw}$ , $k_{right}$ , $k_{left}$ , respectively, and $q_{1,2}, q_{1,3}$ are tuning parameters that can be assumed equal.\",\n \"${\\\\begin{array}{c}Qu_{1,2} = Qu_{2,1} = q_{1,2} \\\\cdot sign(\\\\frac{d}{dt} yr_{ref}) \\\\\\\\Qu_{1,3} = Qu_{3,1} = q_{1,3} \\\\cdot sign(\\\\frac{d}{dt}yr_{ref})\\\\end{array}}$ The coupling in (REF ) means that when $Cm_{damp}^{yaw}$ is reducing, making the turn faster, the appropriate input moment is increased to further accelerate the turn.\",\n \"Thus, the coupling expressed by $Qu$ should be negative for $ k_{left}$ during left turns ($sign(yr_{ref})>0$ ) and for $ k_{right}$ during right turns, and positive otherwise.\",\n \"The desired dynamics of the turning pattern angle command ($\\\\alpha $ ) can be assumed unknown, and left to be resolved by the UTC.\",\n \"Optionally, the dynamics given in (REF ) can be utilized.\",\n \"The gains of the proportional and the feed-forward parts can be tuning parameters.\",\n \"The computation of the controller command during the prediction window can be implemented as shown in (REF ).\",\n \"This enforces the slew rate constraint on $\\\\alpha (t)$ , see (REF ).\",\n \"$\\\\begin{split}\\\\alpha (t)= \\\\alpha (t-dt)+ & K_p \\\\cdot (yr_{ref}(t)-yr_{ref}(t-dt))- \\\\\\\\ -&K_p \\\\cdot (yr_{real}(t)-yr_{real}(t-dt))+ \\\\\\\\ +&K_{FF} \\\\cdot (yr_{ref}(t+\\\\tau )-yr_{ref}(t+\\\\tau -dt))\\\\end{split}$ Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming unknown controller dynamics and correlation between input moment and yaw damping moment coefficients according to ().Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming controller dynamics according to () and correlation between input moment and yaw damping moment coefficients according to ()Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming controller dynamics according to () and correlation between input moment and yaw damping moment coefficients according to ()When a desired dynamics for the turning pattern angle is incorporated in the prediction step, a better accuracy is achieved: compare Figures REF , REF .\",\n \"Also a smaller prediction horizon can be used.\",\n \"For $dt=0.0042 \\\\, (s)$ 20 prediction steps were used (i.e.\",\n \"$t_{pred}\\\\approx 0.08 \\\\, (s)$ ) when assuming unknown dynamics for $\\\\alpha (t)$ , versus 5 steps ($t_{pred}\\\\approx 0.02 \\\\, (s)$ ) when incorporating (REF ).\",\n \"Notice the qualitatively different behavior of the input moment coefficients, shown in Figs.\",\n \"REF , REF .\",\n \"If $\\\\alpha (t)$ dynamics is unknown, the best accuracy is achieved when the correlation between the yaw damping moment and the input moment coefficients is defined according to (REF ), see Fig.\",\n \"REF .\",\n \"Notice, that (REF ) is a more intuitive model than (REF ).\",\n \"If $\\\\alpha (t)$ dynamics is reasonably modeled, tuning the correlation between these variables is less significant.\",\n \"Accurate tracking is achieved even if no correlation is assumed.\",\n \"This, and other interesting solutions, are shown in [4].\",\n \"The minimal control effort is achieved when combining (REF ) and (REF ), see Fig.\",\n \"REF .\"\n ],\n [\n \"CONCLUSIONS AND FUTURE WORK\",\n \"From simulations we observe the following advantages of the UTC: 1.\",\n \"Fast convergence (around 2 (s)); 2.\",\n \"Not much sensitivity to the tuning parameters: if the covariances are set to reasonable values, i.e.\",\n \"the same order of magnitude as the allowed ranges of the actuators involved, UTC always converges.\",\n \"3.\",\n \"Convenient tuning: achieved by the means of setting the elements of $Pu_0$ and $Qu$ , which have a clear physical meaning.\",\n \"Therefore, it is possible to obtain different solutions that implement the desired/ expected behavior of the control variables.\",\n \"Further investigating the UTC properties remains an issue for future work: sensitivity to model uncertainty and measurement noise, and operation with only partial state feedback.\",\n \"Additionally, it is important to formulate and prove conditions for the global/ local stability of systems controlled by the UTC.\",\n \"Initial efforts in this regard are presented in [4]: for linear systems a stability condition takes the form of an Algebraic Riccati Equation, while for the number of prediction steps $n>1$ it has one additional term.\"\n ]\n]"},"id":{"kind":"string","value":"2207.10496"}}},{"rowIdx":1867499,"cells":{"text":{"kind":"list like","value":[["Towards Efficient Adversarial Training on Vision Transformers"],["Abstract Vision Transformer (ViT), as a powerful alternative to Convolutional Neural Network (CNN), has received much attention.","Recent work showed that ViTs are also vulnerable to adversarial examples like CNNs.","To build robust ViTs, an intuitive way is to apply adversarial training since it has been shown as one of the most effective ways to accomplish robust CNNs.","However, one major limitation of adversarial training is its heavy computational cost.","The self-attention mechanism adopted by ViTs is a computationally intense operation whose expense increases quadratically with the number of input patches, making adversarial training on ViTs even more time-consuming.","In this work, we first comprehensively study fast adversarial training on a variety of vision transformers and illustrate the relationship between the efficiency and robustness.","Then, to expediate adversarial training on ViTs, we propose an efficient Attention Guided Adversarial Training mechanism.","Specifically, relying on the specialty of self-attention, we actively remove certain patch embeddings of each layer with an attention-guided dropping strategy during adversarial training.","The slimmed self-attention modules accelerate the adversarial training on ViTs significantly.","With only 65\\% of the fast adversarial training time, we match the state-of-the-art results on the challenging ImageNet benchmark."],["Introduction","Vision Transformers with the self-attention mechanism have been broadly studied and become de facto state-of-the-art models for many benchmarks.","Recent works broadly investigated the traits of this new genre of architectures on computer vision tasks.","Meanwhile, the adversarial robustness of Vision Transformers has also been intensively studied [11], [51], [9], [43], [7], [1], [59], [83], [29], [45], [44], [47], [63], [32], [22].","To build robust Vision Transformers, an intuitive way is to apply adversarial training [62], [25] since it has been shown to be one of the most effective ways to achieve robust CNNs [23], [42], [5].","However, one major limitation of adversarial training is its expensive computational cost.","Adversarial training is known for requiring no extra cost during testing but greatly increasing the training cost.","Huge efforts have been devoted to overcome this deficit [61], [80], [75], [3], [65], [50], [30], [31].","However, the philosophy of designing stronger ViTs has largely lifted up the computation intensity and weakened the performance of previously-proposed techniques.","The specialty of the newly-proposed self-attention design [19], [13] in ViTs also introduces new challenges for accelerating adversarial training.","In this paper, we study the problem of how to efficiently carry out adversarial training on Vision Transformers.","We first apply the state-of-the-art Fast Adversarial Training (Fast AT) algorithm [61], [80] on a variety of vision transformers and analyze how factors like attention mechanism, computational complexity, and parameter size influence the training quality.","To the best of our knowledge, we are the first to accomplish a broad investigation on this topic.","Our survey shows that, although ViTs outperform CNNs by a great margin on robustness, they have hugely increased the computational complexity.","The self-attention mechanism adopted by ViTs is a computationally intense operation whose cost increases quadratically with the number of input patches.","This newly-emerged module hampers the utilization of several techniques for accelerating adversarial training.","Meanwhile, we find that large ViTs models also suffer from obvious catastrophic over-fitting problems [55].","This eventually leads to the degradation of robustness on ViTs with increasingly large capacity.","Figure: (a) AGAT chooses to drop a certain proportion of image embeddings based on the attention information at each self-attention layer.","(b) We plot the robust accuracy against the training time (hours) for various ViTs.","AGAT substantially accelerates adversarial training, while maintaining or improving the robustness at the same time.To make adversarial training efficient on the heavy-weight vision transformers, we investigate in accelerating adversarial training for vision transformers.","Particularly, we leverage the specialty that the self-attention mechanism of transformers is capable of processing variable-length inputs.","This specialty of ViTs has been utilized in a wide range of applications, including processing variable-length word sequences for translations [19], mining graphs with unlimited edges [74], etc.","Recently, on vision tasks, several works [69], [54], [16] explored the possibility of dropping input image patches during training or testing for acceleration purposes.","However, randomly dropping a certain number of input patches will inevitably hurt the training quality.","Thus, many works have proposed adaptive designs [49], [82] in the scenarios of a variety of targeted tasks.","Enlightened by the above works, we propose an Attention-Guided Adversarial Training (AGAT) mechanism, where we drop the patches based on the attention information.","As illustrated by Fig.","REF , our method intends to drop image embeddings after each layer of self-attention.","Note that the self-attention layer is non-parametric and thus is not limited to a static number of inputs.","We drop the embeddings with lower attention and keep the higher ones.","Such a design will better preserve the feed-forward process and therefore guard the backward gradient computation of generating the adversarial examples.","As shown in Fig.","REF , AGAT gets to keep the training quality mostly unchanged or be improved by taking only $65\\%$ training time.","Our work matches the state-of-the-art results of adversarial robustness on the challenging benchmark of ImageNet."],["Relate Works","Adversarial Training.","Adversarial attacks [66], [23], [42], [79], [37], [38] intend to endanger the performance of deep networks via repeatedly optimizing the input images with repect to the output of the model.","To counter this unwanted deficit, various defensive approaches were proposed [46], [20], [39], [81], [27], [64], [60], [40].","Among these defenses, the methodology of adversarial training withstands most kinds of examinations and has become one of a few defenses that can consistently improve the robustness of deep networks when facing most attacks [5].","However, adversarial training is known to suffer from complexity issues [61], [80], [53].","Particularly, Fast AT [80] enhances the single-step adversarial training with random initialization.","Fast AT shows promising results on benchmark datasets.","Later works [4], [33], [75], [3], [65], [50] also proposed improved variants of Fast AT.","Vision Transformer.","The Transformer architecture and its self-attention mechanism were first proposed in the field of natural language processing (NLP) [73], [19], [13] and then adopted in the scenario of computer vision [84], [17], [78].","After the huge efforts of a surge of explorations [21], [70], [76], the Vision Transformer (ViT) has shown the potential to surpass the traditional convolutional neural networks.","Then, researchers keep pushing this new philosophy of model design into a wide range of fields like high-resolution vision tasks [41], [77].","Meanwhile, to reduce the huge computational expense that is brought by the densely modeled self-attention mechanism, various techniques have been proposed [76], [17].","Adversarial Robustness of ViT.","The adversarial robustness of ViT has also achieved great attention due to its impressive performance [12], [62], [52], [48], [26], [8], [10], [63].","Some works [12], [62], [10] first reported positive results where they showed that standard ViTs perform more robust than standard CNNs under adversarial attacks.","The later works [8], [26] revealed that ViTs are not more robust than CNNs if both are trained in the same training framework.","By adopting Transformers’ training recipes, CNNs can become as robust as Transformers on defending against adversarial attacks.","In both sides, we can observe that the clean accuracy of standard models can be easily reduced to near zero under standard attack protocols.","In addtion, Fu et al.","[22] studied attacking ViTs in a patch-wise approach, which reveals the unique vulnerability of ViTs.","To boost the adversarial robustness of ViTs, recent works [67], [6] explored multiple-step adversarial training to ViTs.","Shao et al.","[62], tested the vanilla adversarial training on CIFAR10.","However, multi-step adversarial training is computationally expensive.","And in this work, we take the step of exploring fast single-step adversarial training on ViT models."],["Fast Adversarial Training on Vision Transformers","We first comprehensively study the Fast AT [80] algorithm on vision transformers.","Fast AT is designed to be efficient so that it can be applied to large models (e.g., ResNet-101 [28]) on large-scale datasets (e.g., ImageNet [57]).","Specifically, Fast AT refines the standard single-step FGSM [23] algorithm by adopting a large random perturbation as the starting point for searching adversarial examples.","By doing so, Fast AT is supposed to effectively resist the catastrophic overfitting [80] of training with the plain FGSM.","Thus, Fast AT preserves the effectiveness while significantly improves the efficiency over the multi-step PGD [42] algorithm.","To better understand the robustness of the newly-developed ViT models, in this section, we apply Fast AT to a wide range of ViTs.","We select nineteen models of different sizes from five vision transformer families, including ViT [21], CaiT [72], LeViT [24], SwinTransformer [41], and CrossFormer [77].","The selected models cover a wide range of model designs, including hybrid models [24], slim models [24], [77], constrained attention [41], [77], and multi-scale attention [24], [77].","Following the settings of Fast AT [80], we set the perturbation radius to $2/255$ for ImageNet and test the adversarially trained models with the 100-step PGD attack.","Our training schedule aligns with SwinTransformer [41] and DeiT [71].","We keep all hyper-parameters, e.g., image size, training epoch, and data augmentation, identical for all models.","We also present the results of Fast AT on the CNN models of ResNet-50 and ResNet-101 for comparison.","As shown by Fig.","REF , ViTs are consistently more robust than CNNs.","This aligns with concurrent researches on model robustness [68].","We conclude draw novel observations as follows: Figure: (a) Robust accuracy for various ViTs.","(b) Large ViTs like CaiT-S36 may suffer from obvious unstable training and over-fitting, while designs like window attention can alleviate the issue.","(c) The perfromance of ViTs on Fast AT may not align with their natural accuracy of natural training.","(d) In contrast, the natural and robust accuracies of ViTs align with each other when both are under Fast AT.","(e) Large ViTs greatly increase the computational complexity.","Within the same transformer family, larger transformers do not always result in better robustness.","In Fig.","REF , the transformer families of LeViT, CaiT, and ViT exhibit a pattern of over-fitting.","Namely, as the models get larger, the network robustness learned by Fast AT degrades conversely.","For instance, Cait-S36 performs worse than CaiT-XXS24.","Fig.","REF provides more details of the above over-fitting issue.","The optimization of the CaiT-S36 model gradually degrades after a certain point.","In contrast, the CaiT-XXS24 model possesses a monotonically increasing training curve.","This aligns with previous findings that ViTs may suffer from more severe over-fitting on natural tasks and need the assistance of aggressive data augmentation schedules [71].","Moreover, transformers with a constrained attention mechanism can alleviate the over-fitting problem.","In Fig.","REF , unlike CaiT-S36, the large model of Swin-Base shows a steady training curve.","Among different transformer families, the attention mechanism designed for better natural performance not necessarily results in better robustness.","In Fig.","REF , for each transformer architecture, we plot its natural performance under natural training against its robust performance under Fast AT.","Among different transformer architectures, the two metrics of natural accuracy and robust accuracy approximately form a line, indicating the close relation between natural and robust performances.","Compared with the line formed by CNNs, vision transformers consistently achieve better robustness on models with similar natural performance.","However, we can observe a few outliers of models from the LeViT and CaiT families.","Specifically, the hybrid design of LeViT can achieve high robustness with very small models.","Large CaiTs, despite being effective on the natural task, result in obvious inferiority on robustness.","Notice that, for each point in Fig.","REF , the two metrics of the vertical axis and the horizontal axis are evaluated on two models, either adversarially-trained or naturally-trained, of the same architecture.","When we plot the natural accuracy and robust accuracy, both of which are under Fast AT, as shown by Fig.","REF , the two metrics consistently align with each other without any outlier, revealing the difference between Fast AT and standard training.","SOTA ViTs suffer from a severe efficiency issue and require much more training time than SOTA CNNs.","The above over-fitting problem mainly shows on large ViTs like Cait-S36 or LeViT-384, but not on the small ones.","This is reasonable since models like Cait-S36 are consistently larger than commonly-used CNNs.","However, the over-fitting is not the only problem of adopting larger and larger ViTs.","These increasingly large models have hugely lifted up the computation intensity.","In Fig.","REF , we plot robustness (Robust Accuracy) against computation intensity (GFLOPs).","State-of-the-art ViTs can be a few times larger than CNNs.","This makes utilizing Fast AT even harder since the self-attention module is more intricate to accelerate."],["Efficient Adversarial Training on Vision Transformers","As discussed in the last section, one major problem that hinders the deployment of adversarial training on vision transformers is the efficiency issue.","Adversarial training is known to be computationally intensive.","This greatly hampers its usage on large-scale models or datasets.","Various techniques have been proposed to mitigate this problem.","However, with the revolution brought by vision transformers, many existing techniques such as variable-resolution training [80] have been unusable.","More importantly, there is a rising trend of adopting increasingly large ViTs for better performance.","The complexity of these enormous ViTs is too large to afford, even for efficient algorithms like Fast AT.","In this section, we first analyze the computational complexity of popular ViTs.","Our analysis shows that ViT requires a much longer time to finish adversarial training, which is caused by the large computational cost of ViT brought by a large number of input patches.","Then, we explore the input patches to reduce the brought computational cost.","Given the fact that the flexibility of self-attention allows ViTs to process an arbitrary length of image patches, we explore a random patch dropping strategy to reduce the computation.","The dropping operation with the reduced number of patches can accelerate adversarial training, as expected.","However, the naive dropping strategy will also hurt robustness.","To address the above issues, we propose our Attention-Guided Adversarial Training algorithm, which selectively drops patches based on attention magnitude."],["Computation Intensity of ViTs","We first formally formulate our task.","For each matrix, we present its shape in the lower right corner and its index in the upper right corner.","Denote the input feature as $_{p \\times d}$ , which consists of a sequence of $p$ embeddings with the dimension being $d$ : $=[X^1_{d}, X^2_{d}, ..., X^p_d]$ .","Each embedding relates to a specific non-overlapped patch of the input image.","Vision transformer consists of a list of blocks, each of which consists of two kinds of computation, i.e., the Multi-head Self-Attention layer (MSA) and the Multi-Layer Perceptron layer (MLP).","In the MSA module, $$ is first normalized via Layer Normalization and then transformed to the query, key, and value matrices ($,\\ ,\\ $ ).","$[_{p\\times d},\\ _{p\\times d},\\ _{p\\times d} ] = \\text{LayerNorm}(_{p\\times d})^{1}_{d\\times 3d} .$ For the multi-head design, we partition the $,\\ ,\\ $ matrices of shape $p\\times d$ into $h$ heads, with each part having a shape of $p\\times \\frac{d}{h}$ .","Then, taking the first head as an example, $^1$ will be re-weighted by $^1$ with the following form: $\\text{Attn}(^1_{p\\times \\frac{d}{h}}, ^1_{p\\times \\frac{d}{h}}, ^1_{p\\times \\frac{d}{h}})=\\text{SoftMax}(^1{^1}^{\\top } / \\sqrt{d} + )^1=^1_{p\\times p}^1 .$ $$ is a learnable bias.","Note that all the column vectors of $^1$ are normalized by Softmax and thus have a summation of 1 in each.","Then, $$ value of each head will be concatenated and transformed to the output of MSA.","$^{\\prime }_{p\\times d} = \\text{Concat}(^1^1, ^2^2,..., ^h^h)_{p\\times d} ^2_{d\\times d} .$ A following MLP module will take in the output of MSA, $^{\\prime }$ , and transform each embedding with Layer Normalization and GELU activation.","$^{\\prime \\prime }_{p \\times c} =\\text{GELU}\\Big [\\text{LayerNorm}(^{\\prime }_{p \\times d})^{3}_{d\\times 4d}\\Big ]^{4}_{4d\\times d} .$ The computational complexity of the above process is: $\\mathrm {\\Omega }(\\text{MSA}) = 4pd^2+2p^2d+pd ;\\quad \\mathrm {\\Omega }(\\text{MLP}) = 8pd^2+pd .$ A typical vision transformer will consecutively conduct the above process to generate the final image representation for prediction.","Take the ViT-Base model as an example.","Each layer has $d=768$ .","Therefore, we have $\\mathrm {\\Omega }(\\text{MSA})+\\mathrm {\\Omega }(\\text{MLP})=7\\times 10^6p+1.5\\times 10^3p^2$ .","Since $7\\times 10^6\\gg 1.5\\times 10^3$ and $p$ is mostly around $2\\times 10^2$ , the computational complexity of the entire ViT is approximately linear to $p$ ."],["Dropping Patch: The Flexibility of Self-Attention","Different from the convolutional operation where the hyperparameters (e.g., kernel size, padding size) are supposed to be fixed, the self-attention operation does not require the inputs with fixed length.","For instance, this flexibility of self-attention is leveraged to process an arbitrary length of words in NLP tasks.","Similarly, the flexibility makes its adaption to graph data feasible, in which different nodes can have a different number of connected edges [74].","When transformers with self-attention mechanisms have been introduced into computer vision tasks, researchers also investigate dynamically dropping the patches or the embeddings in the forward pass of a ViT model [69], [54], [16].","It is found that, when a constrained quantity of patches are dropped, the forward inference can be significantly accelerated.","Meanwhile, the performance of the model will be only slightly degraded [49], [82].","Several works utilize this feature to design new mechanisms for their own unique purposes.","In this work, we also explore a patch dropping strategy to accelerate adversarial training given the excellent trade-off it achieves.","We first test the scheme of randomly dropping a certain number of input patches to see how it influences the training quality of Fast AT.","We report the results in Fig REF , where we plot the robustness against the training epoch for different ratios of dropping.","Note that no patches will be dropped during inference in the testing stage.","When the number of input patches is reduced, the forward inference of ViT can be accelerated.","Surprisingly, from the figure, we also observe that the dropping operation also stabilizes the adversarial training and alleviates the phenomenon of catastrophic over-fitting [80], [58], [33].","As shown by Fig REF , we repeat Fast AT without any dropping for three times.","The training procedure can be very unstable and occasionally drop to zero accuracy.","We conjecture that it is the regularization effect brought by the patch dropping operation that stabilizes Fast AT.","To further verify this conjecture, we test ViTs equipped with the dropout operation as in DeiT [71].","The dropout module is applied right after the self-attention module.","As shown in Fig REF , like dropping patches, the attention dropout module also stabilizes Fast AT.","Unlike dropping patches, the dropout module cannot save computation.","However, the final robust accuracy can be reduced in both cases when dropping is applied.","The random patch dropping strategy poses a dilemma.","Namely, it brings both acceleration and performance degradation.","In the following section, we will present our attention-guided patch dropping strategy, where we achieve a better trade-off between efficiency and effectiveness.","Figure: (a) We repeat the baseline without any dropping for three times.","We can observe the training is very unstable.","(b) Training curve when we drop various rates of input patches.","(c)Training curve when we adopt various rates of attention dropout.Figure: The illustration of the plain self-attention layer and the attention layer of AGAT.","The embedding mask is directly generated from and is non-parametric.","Thus, for the same model, one can adopt our AGAT during training and the plain self-attention during testing, respectively."],["Attention-Guided Adversarial Training","It is known that adversarial training utilizes adversarial attacks to generate examples so that the network can learn to fit the generated adversarial examples.","This is a typical hard example mining framework.","The more powerful the adversarial examples are, the more robust the learned network will be.","The quality of the adversarial examples relies on the adversarial attacking algorithm.","And the attack algorithm depends on accurately estimating the gradient of the input pixels with respect to the loss function.","Thus, a major insight of achieving our goal is to drop the patches that will barely hamper the gradient estimation.","A similar philosophy has been utilized for sparse attacks or black-box attacks.","Recent works studied the learned attention and found that the magnitude of the attention can reveal how salient an embedding is [15].","This enlightens us that we can utilize this ready-made information to filter salient embeddings.","Thus, we sum $\\frac{1}{h}\\sum _{i=1}^{h} ^i$ by row and generate an index vector $$ .","Note that the column of $\\frac{1}{h}\\sum _{i=1}^{h} ^i$ is the weighted-average parameter and thus always equal to 1.","It indicates how much a generated embedding receives the information of each input embedding.","In contrast, the row of $\\frac{1}{h}\\sum _{i=1}^{h} ^i$ reveals how much each input embedding influences the output embeddings.","This value differs from embedding to embedding.","Thus, we choose to select the top-k embeddings based on their magnitude in $$ .","Then, the formulation of (REF ) becomes: $^{\\prime }_{k\\times d} =\\text{MaskBy}\\Big [\\text{Concat}(^1^1, ..., ^h^h),\\text{Topk}( )\\Big ]_{k\\times d}^2_{d\\times d} .$ We drop the embeddings after the weighted average calculation of $$ so that the magnitude of embeddings will be kept stable.","The number of embeddings will reduce from $p$ to $k$ .","To fully utilize the attention information in each layer, we propose a layer-wise exponential dropping scheme.","Namely, in each layer, we drop a constant proportion of patches.","Thus, this scheme will drop more embeddings on deeper layers, where the embeddings are consistently more redundant [54].","We set the dropping rate to $0.9$ .","On a 12-layer ViT-Base model, the final layer will process only $31\\%$ number of embeddings and save more than $40\\%$ FLOPs of the entire model.","A detailed implementation of our Attention-Guided Adversarial Training is shown in Algorithm REF .","Our AGAT only modifies the training process.","During testing, we use the original model for prediction.","The class token will not be dropped when it involves the feed-forward procedure.","[t] AGAT attention code (PyTorch-like) whitefullflexiblebcodeblue def init(num_heads, dim, drop_rate): head_dim = dim // num_heads # num_heads: the number of heads, dim: embedding dimension scale = head_dim ** -0.5 qkv = nn.Linear(dim, dim * 3) proj = nn.Linear(dim, dim) def forward(self, x): # x: input tensor with the shape of (b, p, d); b, p, d = x.shape # b: batch size; p: patch number q, k, v = qkv(x).reshape(b, p, 3, num_heads, head_dim).permute(2, 0, 3, 1, 4).unbind(0) attn = (q @ k.transpose(-2, -1)) * scale attn = attn.softmax(dim=-1) # b num_heads p p own_attn = torch.sum(torch.sum(attn, dim=1), dim=-2) # b p kept_num = int(p * drop_rate) - 1 # compute the number of kept embeddings _, rank_indices = torch.topk(own_attn[:,1:], k=kept_num, dim=-1) # b k rank_indices = rank_indices.unsqueeze(-1).repeat(1,1,dim) # for the API of torch.gather x = (attn @ v).transpose(1, 2).reshape(B, N, C) x = torch.gather(x, dim=1, index=rank_indices) # drop embeddings return proj(x)"],["Experimental Setup","Dataset We evaluate our method on the challenging ImageNet [56] dataset.","Due to the huge computational expense of adversarial training, most adversarial training approaches are only verified on relatively small datasets such as CIFAR10 [35] or MNIST [36].","Our efforts on improving the efficiency of adversarial training allow us to apply adversarial training to large-scale datasets.","The input image size is 224 on all models for a fair comparison.","Training Schedule Following previous works [71], [77], we use the AdamW [34] optimizer for training for 300 epochs with a cosine decay learning rate schedule.","The initial learning rate is set to $0.001$ .","The first 20 epochs adopt the linear warm-up strategy.","The batch size is 1024 split on 8 NVIDIA A100 GPUs.","Evaluation Metrics We report our major results on two metrics, robust accuracy and GFLOPs.","We mainly focus on improving the speed of training and keeping the learned robustness unchanged at the mean time.","For a direct impression of training speed, we also record the training time for each method.","Note that the training time is not only determined by the efficiency of the training algorithm, but also the IO speed and many other nonnegligible factors.","Adversarial Attack We choose the powerful multi-step PGD attack [42] with the perturbation radius being $2/255$ or $4/255$ and the optimization step being 20 or 100 [81].","We also test different kinds of attacks, including black-box attacks, to rule out the possibility of obfuscated gradient [5].","Table: Adversarial Training on The ImageNet Dataset.","In most cases, our Attention-Guided Adversarial Training on ViTs achieves comparable clean performance and robust accuracy to Fast-AT with much less time.","The conclusion still holds when different perturbation ranges are applied.Vision Transformers Our AGAT can be directly used on the self-attention model and most of its variants.","We apply our AGAT to three commonly-used models ViT [21], CaiT [72], and LeViT [24].","All the three models are built on the original self-attention module and can fully reveal the effectiveness of our AGAT.","In future work, we will explore combining our AGAT with more sophisticated attention mechanism like window attention [41] or multi-scale attention [77].","Training Algorithm For vision transformers, we compare our AGAT with FastAT and the random dropping strategy (Random).","we also provide results of Free AT [61] and Grad Align [4] algorithms on the ResNet [28] models."],["Improved Efficiency of Adversarial Training on ImageNet","We present the performance of AGAT in Table REF .","Because AGAT drops a static rate of embeddings for each self-attention layer, the depth of the ViTs will decide the total number of dropped features.","Thus, we adjust the dropping rate for ViTs with different numbers of blocks so that the complexity of the feed-forward process will be approximately reduced by $40\\%$ of the baseline.","For instance, we set the dropping rate to $0.1$ for models with 12 blocks but $0.03$ for models with 36 blocks.","For the baseline of randomly dropping input patches, we can always set the dropping rate to $0.4$ since the total amount of saved computation will not be affected by the number of blocks.","As shown by Table REF , for each vision transformer, the Fast AT baseline achieves high robustness but is time-consuming, while the Random dropping strategy saves training time but achieves inferior robustness.","In contrast, AGAT achieves comparable robustness with Fast AT using much less training time.","Particularly, on the ViT-Base model, AGAT achieves similar robustness to Fast AT but only takes $65\\%$ of training time.","Meanwhile, since slim models such as ViT-Tiny and LeViT-128 do not possess the same level of model redundancy as their large-sized counterparts, the robustness of these slim models degrades more dramatically than larger ones when we randomly drop patches.","When trained with AGAT, the robustness of slim models matches the plain Fast AT .","Table: Evaluation of Adversarially Trained Models under Various Attacks.","Robust accuracy of adversarially trained models is reported in this table.","The robust accuracy achieved by our AGAT is comparable to that by Fast-AT under various attack evaluations and different perturbation ranges."],["Results under Various Attacks.","Adversarial robustness is known to be hard to examine.","Several defensive algorithms were found to be vulnerable to tailored attacks.","One of the most important and typical representatives of such phenomena is the obfuscated gradient problem.","Our AGAT does not fall into this category, considering the algorithm neither utilizes any stochastic process nor hampers the gradient computation.","In fact, our AGAT only takes effect on the training stage and does not modify any procedure during evaluation.","To further show the robustness of the learned ViTs, we present the robust accuracy of our learned models under various different attacks in Table REF .","We select the attacking criteria of C&W [14], Square [2], APGD-CE [18], and APGD-DLR [18].","The AGAT models achieve the same level of robustness as Fast AT."],["Robustness and Dropping Rate.","We cross-validate the rate of dropping of our AGAT in Table REF .","To provide a clear comparison with the random dropping strategy, we compare their learned robust accuracy when both dropping strategies reduce approximately the same amount of computation.","Due to the difference between the two algorithms (layer-wise vs input-wise), the actual learning rates are different across the two methods.","It can be told that AGAT can maintain the learned robustness in a wide range of dropping rates.","In contrast, the random dropping strategy significantly degrades the performance, especially for the slim model of ViT-Tiny.","Dropping more than $40\\%$ computation will bring obvious degradation on robustness, even for AGAT.","Thus, we consider this dropping rate as a good trade-off between effectiveness and efficiency.","Table: Ablation Study on Dropping Strategy.","We compare our attention-guided dropping strategy with random dropping.","Ours outperforms random dropping constantly in different dropping rates."],["Visualization.","To better get an insight of how AGAT takes effect, we visualize the internal results of the ViT-Base model.","For each of the 12 blocks in ViT-Base, we show the position of the dropped embeddings by masking out the corresponding image patches.","In Fig.","REF , the position of the dropped embedding mainly concentrates on the relatively unimportant positions like background, while the patches of the main object are mostly kept.","This indicates that our AGAT successfully guards the feed-forward procedure and thus secures the generation of adversarial examples.","We also visualize the corresponding value of attention for each patch.","The darkness of each patch position indicates how much the corresponding embedding of this patch influences the other embeddings.","For each block, we normalize all the values of attention by dividing the maximum value of attention.","This visualization also demonstrates that the dropping rate of AGAT gets to cover the embeddings on the position of the main object.","Therefore, dropping rates larger than the chosen value may lose crucial information for inference and thus hamper the generation of adversarial examples for training.","Figure: Visualizations of dropped image patches and the distribution of attention.","AGAT mainly cuts off the computation of non-object embeddings and thus maintains the performance and gradient computation."],["Conclusions","Adversarial training is one of the most effective defense methods to boost the adversarial robustness of models.","However, it is computationally expensive, even after many efforts have been made to address it.","The emergence of ViTs, whose computational cost increases quadratically with the number of input patches, makes adversarial training more challenging.","In this work, we first thoroughly examined the most popular fast adversarial training on various ViTs.","Our investigation shows that ViT achieves higher robust accuracy than ResNet, while it does suffer from a large computation burden, as expected.","Our further exploration showed that random input patch dropping can accelerate and stabilize the adversarial training, which, however, sacrifices the final robust accuracy.","To overcome the dilemma, we proposed an Attention-Guided Adversarial Training (AGAT) mechanism based on the specialty of the self-attention mechanism.","Our AGAT leverages the attention to guide the patch dropping process, which accelerates the adversarial training significantly and maintains the high robust accuracy of ViTs.","We hope that this work can serve the community as a baseline for research on efficient adversarial training on vision transformers.","Acknowledgements.","This work was supported in part by The National Key Research and Development Program of China (Grant Nos: 2018AAA0101400), in part by The National Nature Science Foundation of China (Grant Nos: 62036009, U1909203, 61936006, 62133013), in part by Innovation Capability Support Program of Shaanxi (Program No.","2021TD-05)."]],"string":"[\n [\n \"Towards Efficient Adversarial Training on Vision Transformers\"\n ],\n [\n \"Abstract Vision Transformer (ViT), as a powerful alternative to Convolutional Neural Network (CNN), has received much attention.\",\n \"Recent work showed that ViTs are also vulnerable to adversarial examples like CNNs.\",\n \"To build robust ViTs, an intuitive way is to apply adversarial training since it has been shown as one of the most effective ways to accomplish robust CNNs.\",\n \"However, one major limitation of adversarial training is its heavy computational cost.\",\n \"The self-attention mechanism adopted by ViTs is a computationally intense operation whose expense increases quadratically with the number of input patches, making adversarial training on ViTs even more time-consuming.\",\n \"In this work, we first comprehensively study fast adversarial training on a variety of vision transformers and illustrate the relationship between the efficiency and robustness.\",\n \"Then, to expediate adversarial training on ViTs, we propose an efficient Attention Guided Adversarial Training mechanism.\",\n \"Specifically, relying on the specialty of self-attention, we actively remove certain patch embeddings of each layer with an attention-guided dropping strategy during adversarial training.\",\n \"The slimmed self-attention modules accelerate the adversarial training on ViTs significantly.\",\n \"With only 65\\\\% of the fast adversarial training time, we match the state-of-the-art results on the challenging ImageNet benchmark.\"\n ],\n [\n \"Introduction\",\n \"Vision Transformers with the self-attention mechanism have been broadly studied and become de facto state-of-the-art models for many benchmarks.\",\n \"Recent works broadly investigated the traits of this new genre of architectures on computer vision tasks.\",\n \"Meanwhile, the adversarial robustness of Vision Transformers has also been intensively studied [11], [51], [9], [43], [7], [1], [59], [83], [29], [45], [44], [47], [63], [32], [22].\",\n \"To build robust Vision Transformers, an intuitive way is to apply adversarial training [62], [25] since it has been shown to be one of the most effective ways to achieve robust CNNs [23], [42], [5].\",\n \"However, one major limitation of adversarial training is its expensive computational cost.\",\n \"Adversarial training is known for requiring no extra cost during testing but greatly increasing the training cost.\",\n \"Huge efforts have been devoted to overcome this deficit [61], [80], [75], [3], [65], [50], [30], [31].\",\n \"However, the philosophy of designing stronger ViTs has largely lifted up the computation intensity and weakened the performance of previously-proposed techniques.\",\n \"The specialty of the newly-proposed self-attention design [19], [13] in ViTs also introduces new challenges for accelerating adversarial training.\",\n \"In this paper, we study the problem of how to efficiently carry out adversarial training on Vision Transformers.\",\n \"We first apply the state-of-the-art Fast Adversarial Training (Fast AT) algorithm [61], [80] on a variety of vision transformers and analyze how factors like attention mechanism, computational complexity, and parameter size influence the training quality.\",\n \"To the best of our knowledge, we are the first to accomplish a broad investigation on this topic.\",\n \"Our survey shows that, although ViTs outperform CNNs by a great margin on robustness, they have hugely increased the computational complexity.\",\n \"The self-attention mechanism adopted by ViTs is a computationally intense operation whose cost increases quadratically with the number of input patches.\",\n \"This newly-emerged module hampers the utilization of several techniques for accelerating adversarial training.\",\n \"Meanwhile, we find that large ViTs models also suffer from obvious catastrophic over-fitting problems [55].\",\n \"This eventually leads to the degradation of robustness on ViTs with increasingly large capacity.\",\n \"Figure: (a) AGAT chooses to drop a certain proportion of image embeddings based on the attention information at each self-attention layer.\",\n \"(b) We plot the robust accuracy against the training time (hours) for various ViTs.\",\n \"AGAT substantially accelerates adversarial training, while maintaining or improving the robustness at the same time.To make adversarial training efficient on the heavy-weight vision transformers, we investigate in accelerating adversarial training for vision transformers.\",\n \"Particularly, we leverage the specialty that the self-attention mechanism of transformers is capable of processing variable-length inputs.\",\n \"This specialty of ViTs has been utilized in a wide range of applications, including processing variable-length word sequences for translations [19], mining graphs with unlimited edges [74], etc.\",\n \"Recently, on vision tasks, several works [69], [54], [16] explored the possibility of dropping input image patches during training or testing for acceleration purposes.\",\n \"However, randomly dropping a certain number of input patches will inevitably hurt the training quality.\",\n \"Thus, many works have proposed adaptive designs [49], [82] in the scenarios of a variety of targeted tasks.\",\n \"Enlightened by the above works, we propose an Attention-Guided Adversarial Training (AGAT) mechanism, where we drop the patches based on the attention information.\",\n \"As illustrated by Fig.\",\n \"REF , our method intends to drop image embeddings after each layer of self-attention.\",\n \"Note that the self-attention layer is non-parametric and thus is not limited to a static number of inputs.\",\n \"We drop the embeddings with lower attention and keep the higher ones.\",\n \"Such a design will better preserve the feed-forward process and therefore guard the backward gradient computation of generating the adversarial examples.\",\n \"As shown in Fig.\",\n \"REF , AGAT gets to keep the training quality mostly unchanged or be improved by taking only $65\\\\%$ training time.\",\n \"Our work matches the state-of-the-art results of adversarial robustness on the challenging benchmark of ImageNet.\"\n ],\n [\n \"Relate Works\",\n \"Adversarial Training.\",\n \"Adversarial attacks [66], [23], [42], [79], [37], [38] intend to endanger the performance of deep networks via repeatedly optimizing the input images with repect to the output of the model.\",\n \"To counter this unwanted deficit, various defensive approaches were proposed [46], [20], [39], [81], [27], [64], [60], [40].\",\n \"Among these defenses, the methodology of adversarial training withstands most kinds of examinations and has become one of a few defenses that can consistently improve the robustness of deep networks when facing most attacks [5].\",\n \"However, adversarial training is known to suffer from complexity issues [61], [80], [53].\",\n \"Particularly, Fast AT [80] enhances the single-step adversarial training with random initialization.\",\n \"Fast AT shows promising results on benchmark datasets.\",\n \"Later works [4], [33], [75], [3], [65], [50] also proposed improved variants of Fast AT.\",\n \"Vision Transformer.\",\n \"The Transformer architecture and its self-attention mechanism were first proposed in the field of natural language processing (NLP) [73], [19], [13] and then adopted in the scenario of computer vision [84], [17], [78].\",\n \"After the huge efforts of a surge of explorations [21], [70], [76], the Vision Transformer (ViT) has shown the potential to surpass the traditional convolutional neural networks.\",\n \"Then, researchers keep pushing this new philosophy of model design into a wide range of fields like high-resolution vision tasks [41], [77].\",\n \"Meanwhile, to reduce the huge computational expense that is brought by the densely modeled self-attention mechanism, various techniques have been proposed [76], [17].\",\n \"Adversarial Robustness of ViT.\",\n \"The adversarial robustness of ViT has also achieved great attention due to its impressive performance [12], [62], [52], [48], [26], [8], [10], [63].\",\n \"Some works [12], [62], [10] first reported positive results where they showed that standard ViTs perform more robust than standard CNNs under adversarial attacks.\",\n \"The later works [8], [26] revealed that ViTs are not more robust than CNNs if both are trained in the same training framework.\",\n \"By adopting Transformers’ training recipes, CNNs can become as robust as Transformers on defending against adversarial attacks.\",\n \"In both sides, we can observe that the clean accuracy of standard models can be easily reduced to near zero under standard attack protocols.\",\n \"In addtion, Fu et al.\",\n \"[22] studied attacking ViTs in a patch-wise approach, which reveals the unique vulnerability of ViTs.\",\n \"To boost the adversarial robustness of ViTs, recent works [67], [6] explored multiple-step adversarial training to ViTs.\",\n \"Shao et al.\",\n \"[62], tested the vanilla adversarial training on CIFAR10.\",\n \"However, multi-step adversarial training is computationally expensive.\",\n \"And in this work, we take the step of exploring fast single-step adversarial training on ViT models.\"\n ],\n [\n \"Fast Adversarial Training on Vision Transformers\",\n \"We first comprehensively study the Fast AT [80] algorithm on vision transformers.\",\n \"Fast AT is designed to be efficient so that it can be applied to large models (e.g., ResNet-101 [28]) on large-scale datasets (e.g., ImageNet [57]).\",\n \"Specifically, Fast AT refines the standard single-step FGSM [23] algorithm by adopting a large random perturbation as the starting point for searching adversarial examples.\",\n \"By doing so, Fast AT is supposed to effectively resist the catastrophic overfitting [80] of training with the plain FGSM.\",\n \"Thus, Fast AT preserves the effectiveness while significantly improves the efficiency over the multi-step PGD [42] algorithm.\",\n \"To better understand the robustness of the newly-developed ViT models, in this section, we apply Fast AT to a wide range of ViTs.\",\n \"We select nineteen models of different sizes from five vision transformer families, including ViT [21], CaiT [72], LeViT [24], SwinTransformer [41], and CrossFormer [77].\",\n \"The selected models cover a wide range of model designs, including hybrid models [24], slim models [24], [77], constrained attention [41], [77], and multi-scale attention [24], [77].\",\n \"Following the settings of Fast AT [80], we set the perturbation radius to $2/255$ for ImageNet and test the adversarially trained models with the 100-step PGD attack.\",\n \"Our training schedule aligns with SwinTransformer [41] and DeiT [71].\",\n \"We keep all hyper-parameters, e.g., image size, training epoch, and data augmentation, identical for all models.\",\n \"We also present the results of Fast AT on the CNN models of ResNet-50 and ResNet-101 for comparison.\",\n \"As shown by Fig.\",\n \"REF , ViTs are consistently more robust than CNNs.\",\n \"This aligns with concurrent researches on model robustness [68].\",\n \"We conclude draw novel observations as follows: Figure: (a) Robust accuracy for various ViTs.\",\n \"(b) Large ViTs like CaiT-S36 may suffer from obvious unstable training and over-fitting, while designs like window attention can alleviate the issue.\",\n \"(c) The perfromance of ViTs on Fast AT may not align with their natural accuracy of natural training.\",\n \"(d) In contrast, the natural and robust accuracies of ViTs align with each other when both are under Fast AT.\",\n \"(e) Large ViTs greatly increase the computational complexity.\",\n \"Within the same transformer family, larger transformers do not always result in better robustness.\",\n \"In Fig.\",\n \"REF , the transformer families of LeViT, CaiT, and ViT exhibit a pattern of over-fitting.\",\n \"Namely, as the models get larger, the network robustness learned by Fast AT degrades conversely.\",\n \"For instance, Cait-S36 performs worse than CaiT-XXS24.\",\n \"Fig.\",\n \"REF provides more details of the above over-fitting issue.\",\n \"The optimization of the CaiT-S36 model gradually degrades after a certain point.\",\n \"In contrast, the CaiT-XXS24 model possesses a monotonically increasing training curve.\",\n \"This aligns with previous findings that ViTs may suffer from more severe over-fitting on natural tasks and need the assistance of aggressive data augmentation schedules [71].\",\n \"Moreover, transformers with a constrained attention mechanism can alleviate the over-fitting problem.\",\n \"In Fig.\",\n \"REF , unlike CaiT-S36, the large model of Swin-Base shows a steady training curve.\",\n \"Among different transformer families, the attention mechanism designed for better natural performance not necessarily results in better robustness.\",\n \"In Fig.\",\n \"REF , for each transformer architecture, we plot its natural performance under natural training against its robust performance under Fast AT.\",\n \"Among different transformer architectures, the two metrics of natural accuracy and robust accuracy approximately form a line, indicating the close relation between natural and robust performances.\",\n \"Compared with the line formed by CNNs, vision transformers consistently achieve better robustness on models with similar natural performance.\",\n \"However, we can observe a few outliers of models from the LeViT and CaiT families.\",\n \"Specifically, the hybrid design of LeViT can achieve high robustness with very small models.\",\n \"Large CaiTs, despite being effective on the natural task, result in obvious inferiority on robustness.\",\n \"Notice that, for each point in Fig.\",\n \"REF , the two metrics of the vertical axis and the horizontal axis are evaluated on two models, either adversarially-trained or naturally-trained, of the same architecture.\",\n \"When we plot the natural accuracy and robust accuracy, both of which are under Fast AT, as shown by Fig.\",\n \"REF , the two metrics consistently align with each other without any outlier, revealing the difference between Fast AT and standard training.\",\n \"SOTA ViTs suffer from a severe efficiency issue and require much more training time than SOTA CNNs.\",\n \"The above over-fitting problem mainly shows on large ViTs like Cait-S36 or LeViT-384, but not on the small ones.\",\n \"This is reasonable since models like Cait-S36 are consistently larger than commonly-used CNNs.\",\n \"However, the over-fitting is not the only problem of adopting larger and larger ViTs.\",\n \"These increasingly large models have hugely lifted up the computation intensity.\",\n \"In Fig.\",\n \"REF , we plot robustness (Robust Accuracy) against computation intensity (GFLOPs).\",\n \"State-of-the-art ViTs can be a few times larger than CNNs.\",\n \"This makes utilizing Fast AT even harder since the self-attention module is more intricate to accelerate.\"\n ],\n [\n \"Efficient Adversarial Training on Vision Transformers\",\n \"As discussed in the last section, one major problem that hinders the deployment of adversarial training on vision transformers is the efficiency issue.\",\n \"Adversarial training is known to be computationally intensive.\",\n \"This greatly hampers its usage on large-scale models or datasets.\",\n \"Various techniques have been proposed to mitigate this problem.\",\n \"However, with the revolution brought by vision transformers, many existing techniques such as variable-resolution training [80] have been unusable.\",\n \"More importantly, there is a rising trend of adopting increasingly large ViTs for better performance.\",\n \"The complexity of these enormous ViTs is too large to afford, even for efficient algorithms like Fast AT.\",\n \"In this section, we first analyze the computational complexity of popular ViTs.\",\n \"Our analysis shows that ViT requires a much longer time to finish adversarial training, which is caused by the large computational cost of ViT brought by a large number of input patches.\",\n \"Then, we explore the input patches to reduce the brought computational cost.\",\n \"Given the fact that the flexibility of self-attention allows ViTs to process an arbitrary length of image patches, we explore a random patch dropping strategy to reduce the computation.\",\n \"The dropping operation with the reduced number of patches can accelerate adversarial training, as expected.\",\n \"However, the naive dropping strategy will also hurt robustness.\",\n \"To address the above issues, we propose our Attention-Guided Adversarial Training algorithm, which selectively drops patches based on attention magnitude.\"\n ],\n [\n \"Computation Intensity of ViTs\",\n \"We first formally formulate our task.\",\n \"For each matrix, we present its shape in the lower right corner and its index in the upper right corner.\",\n \"Denote the input feature as $_{p \\\\times d}$ , which consists of a sequence of $p$ embeddings with the dimension being $d$ : $=[X^1_{d}, X^2_{d}, ..., X^p_d]$ .\",\n \"Each embedding relates to a specific non-overlapped patch of the input image.\",\n \"Vision transformer consists of a list of blocks, each of which consists of two kinds of computation, i.e., the Multi-head Self-Attention layer (MSA) and the Multi-Layer Perceptron layer (MLP).\",\n \"In the MSA module, $$ is first normalized via Layer Normalization and then transformed to the query, key, and value matrices ($,\\\\ ,\\\\ $ ).\",\n \"$[_{p\\\\times d},\\\\ _{p\\\\times d},\\\\ _{p\\\\times d} ] = \\\\text{LayerNorm}(_{p\\\\times d})^{1}_{d\\\\times 3d} .$ For the multi-head design, we partition the $,\\\\ ,\\\\ $ matrices of shape $p\\\\times d$ into $h$ heads, with each part having a shape of $p\\\\times \\\\frac{d}{h}$ .\",\n \"Then, taking the first head as an example, $^1$ will be re-weighted by $^1$ with the following form: $\\\\text{Attn}(^1_{p\\\\times \\\\frac{d}{h}}, ^1_{p\\\\times \\\\frac{d}{h}}, ^1_{p\\\\times \\\\frac{d}{h}})=\\\\text{SoftMax}(^1{^1}^{\\\\top } / \\\\sqrt{d} + )^1=^1_{p\\\\times p}^1 .$ $$ is a learnable bias.\",\n \"Note that all the column vectors of $^1$ are normalized by Softmax and thus have a summation of 1 in each.\",\n \"Then, $$ value of each head will be concatenated and transformed to the output of MSA.\",\n \"$^{\\\\prime }_{p\\\\times d} = \\\\text{Concat}(^1^1, ^2^2,..., ^h^h)_{p\\\\times d} ^2_{d\\\\times d} .$ A following MLP module will take in the output of MSA, $^{\\\\prime }$ , and transform each embedding with Layer Normalization and GELU activation.\",\n \"$^{\\\\prime \\\\prime }_{p \\\\times c} =\\\\text{GELU}\\\\Big [\\\\text{LayerNorm}(^{\\\\prime }_{p \\\\times d})^{3}_{d\\\\times 4d}\\\\Big ]^{4}_{4d\\\\times d} .$ The computational complexity of the above process is: $\\\\mathrm {\\\\Omega }(\\\\text{MSA}) = 4pd^2+2p^2d+pd ;\\\\quad \\\\mathrm {\\\\Omega }(\\\\text{MLP}) = 8pd^2+pd .$ A typical vision transformer will consecutively conduct the above process to generate the final image representation for prediction.\",\n \"Take the ViT-Base model as an example.\",\n \"Each layer has $d=768$ .\",\n \"Therefore, we have $\\\\mathrm {\\\\Omega }(\\\\text{MSA})+\\\\mathrm {\\\\Omega }(\\\\text{MLP})=7\\\\times 10^6p+1.5\\\\times 10^3p^2$ .\",\n \"Since $7\\\\times 10^6\\\\gg 1.5\\\\times 10^3$ and $p$ is mostly around $2\\\\times 10^2$ , the computational complexity of the entire ViT is approximately linear to $p$ .\"\n ],\n [\n \"Dropping Patch: The Flexibility of Self-Attention\",\n \"Different from the convolutional operation where the hyperparameters (e.g., kernel size, padding size) are supposed to be fixed, the self-attention operation does not require the inputs with fixed length.\",\n \"For instance, this flexibility of self-attention is leveraged to process an arbitrary length of words in NLP tasks.\",\n \"Similarly, the flexibility makes its adaption to graph data feasible, in which different nodes can have a different number of connected edges [74].\",\n \"When transformers with self-attention mechanisms have been introduced into computer vision tasks, researchers also investigate dynamically dropping the patches or the embeddings in the forward pass of a ViT model [69], [54], [16].\",\n \"It is found that, when a constrained quantity of patches are dropped, the forward inference can be significantly accelerated.\",\n \"Meanwhile, the performance of the model will be only slightly degraded [49], [82].\",\n \"Several works utilize this feature to design new mechanisms for their own unique purposes.\",\n \"In this work, we also explore a patch dropping strategy to accelerate adversarial training given the excellent trade-off it achieves.\",\n \"We first test the scheme of randomly dropping a certain number of input patches to see how it influences the training quality of Fast AT.\",\n \"We report the results in Fig REF , where we plot the robustness against the training epoch for different ratios of dropping.\",\n \"Note that no patches will be dropped during inference in the testing stage.\",\n \"When the number of input patches is reduced, the forward inference of ViT can be accelerated.\",\n \"Surprisingly, from the figure, we also observe that the dropping operation also stabilizes the adversarial training and alleviates the phenomenon of catastrophic over-fitting [80], [58], [33].\",\n \"As shown by Fig REF , we repeat Fast AT without any dropping for three times.\",\n \"The training procedure can be very unstable and occasionally drop to zero accuracy.\",\n \"We conjecture that it is the regularization effect brought by the patch dropping operation that stabilizes Fast AT.\",\n \"To further verify this conjecture, we test ViTs equipped with the dropout operation as in DeiT [71].\",\n \"The dropout module is applied right after the self-attention module.\",\n \"As shown in Fig REF , like dropping patches, the attention dropout module also stabilizes Fast AT.\",\n \"Unlike dropping patches, the dropout module cannot save computation.\",\n \"However, the final robust accuracy can be reduced in both cases when dropping is applied.\",\n \"The random patch dropping strategy poses a dilemma.\",\n \"Namely, it brings both acceleration and performance degradation.\",\n \"In the following section, we will present our attention-guided patch dropping strategy, where we achieve a better trade-off between efficiency and effectiveness.\",\n \"Figure: (a) We repeat the baseline without any dropping for three times.\",\n \"We can observe the training is very unstable.\",\n \"(b) Training curve when we drop various rates of input patches.\",\n \"(c)Training curve when we adopt various rates of attention dropout.Figure: The illustration of the plain self-attention layer and the attention layer of AGAT.\",\n \"The embedding mask is directly generated from and is non-parametric.\",\n \"Thus, for the same model, one can adopt our AGAT during training and the plain self-attention during testing, respectively.\"\n ],\n [\n \"Attention-Guided Adversarial Training\",\n \"It is known that adversarial training utilizes adversarial attacks to generate examples so that the network can learn to fit the generated adversarial examples.\",\n \"This is a typical hard example mining framework.\",\n \"The more powerful the adversarial examples are, the more robust the learned network will be.\",\n \"The quality of the adversarial examples relies on the adversarial attacking algorithm.\",\n \"And the attack algorithm depends on accurately estimating the gradient of the input pixels with respect to the loss function.\",\n \"Thus, a major insight of achieving our goal is to drop the patches that will barely hamper the gradient estimation.\",\n \"A similar philosophy has been utilized for sparse attacks or black-box attacks.\",\n \"Recent works studied the learned attention and found that the magnitude of the attention can reveal how salient an embedding is [15].\",\n \"This enlightens us that we can utilize this ready-made information to filter salient embeddings.\",\n \"Thus, we sum $\\\\frac{1}{h}\\\\sum _{i=1}^{h} ^i$ by row and generate an index vector $$ .\",\n \"Note that the column of $\\\\frac{1}{h}\\\\sum _{i=1}^{h} ^i$ is the weighted-average parameter and thus always equal to 1.\",\n \"It indicates how much a generated embedding receives the information of each input embedding.\",\n \"In contrast, the row of $\\\\frac{1}{h}\\\\sum _{i=1}^{h} ^i$ reveals how much each input embedding influences the output embeddings.\",\n \"This value differs from embedding to embedding.\",\n \"Thus, we choose to select the top-k embeddings based on their magnitude in $$ .\",\n \"Then, the formulation of (REF ) becomes: $^{\\\\prime }_{k\\\\times d} =\\\\text{MaskBy}\\\\Big [\\\\text{Concat}(^1^1, ..., ^h^h),\\\\text{Topk}( )\\\\Big ]_{k\\\\times d}^2_{d\\\\times d} .$ We drop the embeddings after the weighted average calculation of $$ so that the magnitude of embeddings will be kept stable.\",\n \"The number of embeddings will reduce from $p$ to $k$ .\",\n \"To fully utilize the attention information in each layer, we propose a layer-wise exponential dropping scheme.\",\n \"Namely, in each layer, we drop a constant proportion of patches.\",\n \"Thus, this scheme will drop more embeddings on deeper layers, where the embeddings are consistently more redundant [54].\",\n \"We set the dropping rate to $0.9$ .\",\n \"On a 12-layer ViT-Base model, the final layer will process only $31\\\\%$ number of embeddings and save more than $40\\\\%$ FLOPs of the entire model.\",\n \"A detailed implementation of our Attention-Guided Adversarial Training is shown in Algorithm REF .\",\n \"Our AGAT only modifies the training process.\",\n \"During testing, we use the original model for prediction.\",\n \"The class token will not be dropped when it involves the feed-forward procedure.\",\n \"[t] AGAT attention code (PyTorch-like) whitefullflexiblebcodeblue def init(num_heads, dim, drop_rate): head_dim = dim // num_heads # num_heads: the number of heads, dim: embedding dimension scale = head_dim ** -0.5 qkv = nn.Linear(dim, dim * 3) proj = nn.Linear(dim, dim) def forward(self, x): # x: input tensor with the shape of (b, p, d); b, p, d = x.shape # b: batch size; p: patch number q, k, v = qkv(x).reshape(b, p, 3, num_heads, head_dim).permute(2, 0, 3, 1, 4).unbind(0) attn = (q @ k.transpose(-2, -1)) * scale attn = attn.softmax(dim=-1) # b num_heads p p own_attn = torch.sum(torch.sum(attn, dim=1), dim=-2) # b p kept_num = int(p * drop_rate) - 1 # compute the number of kept embeddings _, rank_indices = torch.topk(own_attn[:,1:], k=kept_num, dim=-1) # b k rank_indices = rank_indices.unsqueeze(-1).repeat(1,1,dim) # for the API of torch.gather x = (attn @ v).transpose(1, 2).reshape(B, N, C) x = torch.gather(x, dim=1, index=rank_indices) # drop embeddings return proj(x)\"\n ],\n [\n \"Experimental Setup\",\n \"Dataset We evaluate our method on the challenging ImageNet [56] dataset.\",\n \"Due to the huge computational expense of adversarial training, most adversarial training approaches are only verified on relatively small datasets such as CIFAR10 [35] or MNIST [36].\",\n \"Our efforts on improving the efficiency of adversarial training allow us to apply adversarial training to large-scale datasets.\",\n \"The input image size is 224 on all models for a fair comparison.\",\n \"Training Schedule Following previous works [71], [77], we use the AdamW [34] optimizer for training for 300 epochs with a cosine decay learning rate schedule.\",\n \"The initial learning rate is set to $0.001$ .\",\n \"The first 20 epochs adopt the linear warm-up strategy.\",\n \"The batch size is 1024 split on 8 NVIDIA A100 GPUs.\",\n \"Evaluation Metrics We report our major results on two metrics, robust accuracy and GFLOPs.\",\n \"We mainly focus on improving the speed of training and keeping the learned robustness unchanged at the mean time.\",\n \"For a direct impression of training speed, we also record the training time for each method.\",\n \"Note that the training time is not only determined by the efficiency of the training algorithm, but also the IO speed and many other nonnegligible factors.\",\n \"Adversarial Attack We choose the powerful multi-step PGD attack [42] with the perturbation radius being $2/255$ or $4/255$ and the optimization step being 20 or 100 [81].\",\n \"We also test different kinds of attacks, including black-box attacks, to rule out the possibility of obfuscated gradient [5].\",\n \"Table: Adversarial Training on The ImageNet Dataset.\",\n \"In most cases, our Attention-Guided Adversarial Training on ViTs achieves comparable clean performance and robust accuracy to Fast-AT with much less time.\",\n \"The conclusion still holds when different perturbation ranges are applied.Vision Transformers Our AGAT can be directly used on the self-attention model and most of its variants.\",\n \"We apply our AGAT to three commonly-used models ViT [21], CaiT [72], and LeViT [24].\",\n \"All the three models are built on the original self-attention module and can fully reveal the effectiveness of our AGAT.\",\n \"In future work, we will explore combining our AGAT with more sophisticated attention mechanism like window attention [41] or multi-scale attention [77].\",\n \"Training Algorithm For vision transformers, we compare our AGAT with FastAT and the random dropping strategy (Random).\",\n \"we also provide results of Free AT [61] and Grad Align [4] algorithms on the ResNet [28] models.\"\n ],\n [\n \"Improved Efficiency of Adversarial Training on ImageNet\",\n \"We present the performance of AGAT in Table REF .\",\n \"Because AGAT drops a static rate of embeddings for each self-attention layer, the depth of the ViTs will decide the total number of dropped features.\",\n \"Thus, we adjust the dropping rate for ViTs with different numbers of blocks so that the complexity of the feed-forward process will be approximately reduced by $40\\\\%$ of the baseline.\",\n \"For instance, we set the dropping rate to $0.1$ for models with 12 blocks but $0.03$ for models with 36 blocks.\",\n \"For the baseline of randomly dropping input patches, we can always set the dropping rate to $0.4$ since the total amount of saved computation will not be affected by the number of blocks.\",\n \"As shown by Table REF , for each vision transformer, the Fast AT baseline achieves high robustness but is time-consuming, while the Random dropping strategy saves training time but achieves inferior robustness.\",\n \"In contrast, AGAT achieves comparable robustness with Fast AT using much less training time.\",\n \"Particularly, on the ViT-Base model, AGAT achieves similar robustness to Fast AT but only takes $65\\\\%$ of training time.\",\n \"Meanwhile, since slim models such as ViT-Tiny and LeViT-128 do not possess the same level of model redundancy as their large-sized counterparts, the robustness of these slim models degrades more dramatically than larger ones when we randomly drop patches.\",\n \"When trained with AGAT, the robustness of slim models matches the plain Fast AT .\",\n \"Table: Evaluation of Adversarially Trained Models under Various Attacks.\",\n \"Robust accuracy of adversarially trained models is reported in this table.\",\n \"The robust accuracy achieved by our AGAT is comparable to that by Fast-AT under various attack evaluations and different perturbation ranges.\"\n ],\n [\n \"Results under Various Attacks.\",\n \"Adversarial robustness is known to be hard to examine.\",\n \"Several defensive algorithms were found to be vulnerable to tailored attacks.\",\n \"One of the most important and typical representatives of such phenomena is the obfuscated gradient problem.\",\n \"Our AGAT does not fall into this category, considering the algorithm neither utilizes any stochastic process nor hampers the gradient computation.\",\n \"In fact, our AGAT only takes effect on the training stage and does not modify any procedure during evaluation.\",\n \"To further show the robustness of the learned ViTs, we present the robust accuracy of our learned models under various different attacks in Table REF .\",\n \"We select the attacking criteria of C&W [14], Square [2], APGD-CE [18], and APGD-DLR [18].\",\n \"The AGAT models achieve the same level of robustness as Fast AT.\"\n ],\n [\n \"Robustness and Dropping Rate.\",\n \"We cross-validate the rate of dropping of our AGAT in Table REF .\",\n \"To provide a clear comparison with the random dropping strategy, we compare their learned robust accuracy when both dropping strategies reduce approximately the same amount of computation.\",\n \"Due to the difference between the two algorithms (layer-wise vs input-wise), the actual learning rates are different across the two methods.\",\n \"It can be told that AGAT can maintain the learned robustness in a wide range of dropping rates.\",\n \"In contrast, the random dropping strategy significantly degrades the performance, especially for the slim model of ViT-Tiny.\",\n \"Dropping more than $40\\\\%$ computation will bring obvious degradation on robustness, even for AGAT.\",\n \"Thus, we consider this dropping rate as a good trade-off between effectiveness and efficiency.\",\n \"Table: Ablation Study on Dropping Strategy.\",\n \"We compare our attention-guided dropping strategy with random dropping.\",\n \"Ours outperforms random dropping constantly in different dropping rates.\"\n ],\n [\n \"Visualization.\",\n \"To better get an insight of how AGAT takes effect, we visualize the internal results of the ViT-Base model.\",\n \"For each of the 12 blocks in ViT-Base, we show the position of the dropped embeddings by masking out the corresponding image patches.\",\n \"In Fig.\",\n \"REF , the position of the dropped embedding mainly concentrates on the relatively unimportant positions like background, while the patches of the main object are mostly kept.\",\n \"This indicates that our AGAT successfully guards the feed-forward procedure and thus secures the generation of adversarial examples.\",\n \"We also visualize the corresponding value of attention for each patch.\",\n \"The darkness of each patch position indicates how much the corresponding embedding of this patch influences the other embeddings.\",\n \"For each block, we normalize all the values of attention by dividing the maximum value of attention.\",\n \"This visualization also demonstrates that the dropping rate of AGAT gets to cover the embeddings on the position of the main object.\",\n \"Therefore, dropping rates larger than the chosen value may lose crucial information for inference and thus hamper the generation of adversarial examples for training.\",\n \"Figure: Visualizations of dropped image patches and the distribution of attention.\",\n \"AGAT mainly cuts off the computation of non-object embeddings and thus maintains the performance and gradient computation.\"\n ],\n [\n \"Conclusions\",\n \"Adversarial training is one of the most effective defense methods to boost the adversarial robustness of models.\",\n \"However, it is computationally expensive, even after many efforts have been made to address it.\",\n \"The emergence of ViTs, whose computational cost increases quadratically with the number of input patches, makes adversarial training more challenging.\",\n \"In this work, we first thoroughly examined the most popular fast adversarial training on various ViTs.\",\n \"Our investigation shows that ViT achieves higher robust accuracy than ResNet, while it does suffer from a large computation burden, as expected.\",\n \"Our further exploration showed that random input patch dropping can accelerate and stabilize the adversarial training, which, however, sacrifices the final robust accuracy.\",\n \"To overcome the dilemma, we proposed an Attention-Guided Adversarial Training (AGAT) mechanism based on the specialty of the self-attention mechanism.\",\n \"Our AGAT leverages the attention to guide the patch dropping process, which accelerates the adversarial training significantly and maintains the high robust accuracy of ViTs.\",\n \"We hope that this work can serve the community as a baseline for research on efficient adversarial training on vision transformers.\",\n \"Acknowledgements.\",\n \"This work was supported in part by The National Key Research and Development Program of China (Grant Nos: 2018AAA0101400), in part by The National Nature Science Foundation of China (Grant Nos: 62036009, U1909203, 61936006, 62133013), in part by Innovation Capability Support Program of Shaanxi (Program No.\",\n \"2021TD-05).\"\n ]\n]"},"id":{"kind":"string","value":"2207.10498"}}}],"truncated":false,"partial":false},"paginationData":{"pageIndex":18674,"numItemsPerPage":100,"numTotalItems":1867602,"offset":1867400,"length":100}},"jwt":"eyJhbGciOiJFZERTQSJ9.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc0MTgwMzc1MSwic3ViIjoiL2RhdGFzZXRzL2hvd2V5L3VuYXJYaXZlIiwiZXhwIjoxNzQxODA3MzUxLCJpc3MiOiJodHRwczovL2h1Z2dpbmdmYWNlLmNvIn0.LqMuJnciPPaHd8BaFRj0tuwpEQxXgmYXdiiZsM0Ku76JNXkqZiB8QDEtsbjf1sav0mNItaceBddjUnAAdiuICg","displayUrls":true},"discussionsStats":{"closed":0,"open":1,"total":1},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false}">
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[
[
"Extension of the Measurement Capabilities of the Quadrupole Resonator"
],
[
"Abstract The Quadrupole Resonator, designed to measure the surface resistance of superconducting samples at 400 MHz has been refurbished.",
"The accuracy of its RF-DC compensation measurement technique is tested by an independent method.",
"It is shown that the device enables also measurements at 800 and 1200 MHz and is capable to probe the critical RF magnetic field.",
"The electric and magnetic field configuration of the Quadrupole Resonator are dependent on the excited mode.",
"It is shown how this can be used to distinguish between electric and magnetic losses."
],
[
"Introduction",
"The power dissipated in superconducting cavities is directly proportional to their surface resistance $R_{\\text{S}}$ , which shows a complex behavior on the external parameters: frequency $f$ , temperature $T$ , magnetic field $\\vec{B}$ and electric field $\\vec{E}$ .",
"In particular there is no widely accepted model which can describe the increase of the surface resistance with applied field.",
"There is strong evidence that there are several different loss mechanisms, some only relevant if certain surface preparations are applied [1].",
"If not limited by a quench at a local defect, the maximum accelerating gradient of a superconducting cavity is set by the critical RF magnetic field.",
"Its exact value and correlation to the surface properties of the material are not fully understood yet.",
"Surface resistance and critical RF field can be directly measured in a superconducting cavity.",
"However for the former its value obtained is the average $R_{\\text{S}}$ over the whole surface.",
"A convenient way to investigate the surface resistance and critical field of superconducting materials is to examine small samples, which can be manufactured at low cost, duplicated easily and used for further surface analyses.",
"The Quadrupole Resonator [2] was opted for measuring the surface resistance of superconducting niobium film samples at [400]MHz, the technology and RF frequency chosen for the Large Hadron Collider (LHC) at CERN.",
"The device is a four-wire transmission line half-wave resonator using a TE$_{21}$ -like mode.",
"The samples are thermally decoupled from the host cavity and their surface resistance is derived by a calorimetric RF-DC compensation technique.",
"In this paper the extension of the Quadrupole Resonator to additionally cover the frequencies of 800 and [1200]MHz and to probe the critical RF magnetic field of the samples is presented.",
"It is shown how the frequency dependent field configuration on the sample surface can be used to distinguish between losses caused by the RF electric and magnetic field.",
"Figure: Technical drawing of the Quadrupole Resonator with attached sample and thermometry chamber housing a DC heater and temperature sensors ."
],
[
"Excitation at multiple frequencies",
"The Quadrupole Resonator is a four-wire transmission line half-wave resonator.",
"It was designed for excitation in a TE$_{210}$ -like mode at [400]MHz.",
"The geometry also allows for excitation at multiple integers of [400]MHz (TE$_{211}$ , TE$_{212}$ -like...).",
"In the following it will be discussed whether these modes are also suited for surface resistance measurements on the attached samples.",
"In the Quadrupole Resonator the cover plate of a cylinder attached to the cavity in a coaxial structure serves as the sample, see Fig.REF .",
"For the Quadrupole mode at [400]MHz this design yields exponentially decaying RF fields between the outer wall of the sample cylinder and the host cavity.",
"Therefore, the power dissipated inside this [1]mm gap and especially at the end flange and joint of the sample cylinder is negligible.",
"Additionally to the [400]MHz design mode the fields are also exponentially decaying for all other excitable quadrupole (TE$_{21}$ -like) modes up to [2.0]GHz, as can be shown by analytical calculations [3].",
"In principal five quadrupole modes could be excited and used for RF measurements.",
"At CERN equipment for 400, 800 and [1200]MHz is available for the test stand.",
"The Quadrupole Resonator consists of two [2]mm thick niobium cans for convenient handling and cleaning of the device, see Fig.",
"REF .",
"These cans are flanged to each other in the middle of the resonator, where the screening current on the cavity surface vanishes for the modes at 400 and [1200]MHz.",
"For the [800]MHz mode the screening current has a maximum at this position.",
"However, since the field is strongly concentrated around the rods in the middle of the resonator, excitation and measurements at [800]MHz are not perturbed by losses at this flange.",
"In fact the magnetic field at the flange between the upper and the lower can is only [0.5]% of the maximum field on the sample as has been calculated by Microwave Studio ® (MWS) [4].",
"The Quadrupole Resonator is equipped with two identical strongly overcoupled antennas.",
"One serves as the input, the other as the output.",
"Due to this configuration almost the whole power transmitted to the cavity is coupled out and only about [1]% is dissipated in the cavity walls and on the sample surface.",
"The system acts like a narrow band filter with minor losses.",
"At [800]MHz it remains in this strongly overcoupled state up to the highest field level reached, which is about [40]mT for this frequency.",
"No deviation of the coupling strength at higher field levels was ever observed if sample and cavity remained in a superconducting state.",
"Therefore the design comprising the two cans does not perturb the measurements at [800]MHz.",
"At [1200]MHz measurements are more cumbersome due to [69]Hz oscillations of the resonator rods.",
"It could be shown that these vibrations are excited by helium bubbles forming in the resonator rods, because they are suppressed, but not completely avoided, when measurements are performed inside a superfluid helium bath."
],
[
"Low field surface resistance",
"For magnetic fields below [15]mT the surface resistance $R_{\\text{S}}$ is assumed to be independent of the field strength and thus can be written as a sum of BCS and residual surface resistance, $R_{\\text{S}}=R_{\\text{BCS}}(f,T)+R_{\\text{Res}}(f).$ Figure REF displays $R_{\\text{S}}$ for an applied RF magnetic field of approx.",
"[15]mT in the temperature range between 2 and [10.6]K for a reactor grade bulk niobium sample, which was prepared by buffered chemical polishing (BCP).",
"Figure: Surface resistance of a reactor grade bulk niobium sample for three frequencies as a function of temperature, measured at a magnetic field of about [15]mT.",
"The solid lines show least squares fits to BCS theory.",
"In the normal conducting regime above T c T_{\\text{c}} the surface resistance was also derived from [3]dB bandwidth measurements.In the normal conducting regime above $T_{\\text{c}}$ , the surface resistance depends only slightly on temperature as can be seen from the flat curves in this area.",
"For these temperatures the normal conducting surface resistance of the sample $R_{\\text{N}}$ can also be derived by a non-calorimetric [3]dB bandwidth method [5].",
"The measurement is only possible when the sample is normal conducting and the coupling changes from strongly overcoupled to strongly undercoupled.",
"This means that in the latter case the majority of the power coupled in is dissipated on the sample surface instead of being coupled out.",
"The loaded quality factor $Q_{\\text{L}}$ consists of the quality factors of the host cavity $Q_{\\text{C}}$ , the sample $Q_{\\text{Sample}}$ and the two couplers combined $Q_{\\text{ext}}$ $\\frac{1}{Q_{\\text{L}}}=\\frac{1}{Q_{\\text{ext}}}+\\frac{1}{Q_{\\text{C}}}+\\frac{1}{Q_{\\text{Sample}}}.$ In the superconducting state $Q_{\\text{L}}$ was measured to be about $10^6$ , dependent on frequency.",
"The quality factor of the sample $Q_{\\text{Sample}}$ can be derived from the calorimetric measurement: $R{_{\\text{S}}}=\\frac{G_{\\text{Sample}}}{Q_{\\text{Sample}}},$ where $G_{\\text{Sample}}$ is the geometry factor of the sample.",
"It relates the losses on the sample surface to the stored energy in the cavity $U$ : $G_{\\text{Sample}}= \\mu _0^2\\frac{2\\omega U}{\\int \\limits _{\\text{\\hbox{t}o 0pt{\\hss $\\mathrm {Sample}$\\hss }}}{{B^2}\\mathrm {d}S}}.$ At [2]K $R_{\\text{S}}$ is several tens of nanoohms, corresponding to $Q_{\\text{Sample}}$ values of several times $10^9$ .",
"The host cavity is made of the same material as the sample.",
"Therefore, its surface resistance is considered to be almost identical.",
"From this assumption and the field configuration of the Quadrupole Resonator (about ten percent of the power is dissipated on the sample surface) one can estimate $Q_{\\text{C}}$ to be about ten times lower than $Q_{\\text{Sample}}$ .",
"This is still two orders of magnitude higher than $Q_{\\text{L}}$ , allowing to simplify Eq.",
"REF to $\\frac{1}{Q_{\\text{L}}}=\\frac{1}{Q_{\\text{ext}}}.$ If the sample is normal conducting the system becomes undercoupled.",
"The host cavity remains superconducting, since it is thermally decoupled from the sample.",
"Thus, $1/Q_{\\text{C}}$ remains negligible and Eq.",
"(REF ) reads $\\frac{1}{Q_{\\text{L}}}=\\frac{1}{Q_{\\text{ext}}}+\\frac{1}{Q_{\\text{Sample}}}.$ $Q_{\\text{Sample}}$ can now be calculated with the value of $Q_{\\text{ext}}$ obtained from the measurement in the superconducting state.",
"From $Q_{\\text{Sample}}$ $R_{\\text{S}}$ is derived using Eq.",
"(REF ).",
"The results from these non-calorimetric [3]dB bandwidth measurements agree within [4]% with the results obtained from the calorimetric measurements, see Fig.",
"REF .",
"The surface resistance in this normal conducting regime is found to be proportional to $\\sqrt{f}$ as expected in case of normal skin effect.",
"These results confirm the validity of the calorimetric approach and therefore also give confidence in the measurements at lower temperatures.",
"The solid lines in Fig.",
"REF show the predictions from least squares fits to BCS theory performed with Win Super Fit [6], [7].",
"The program uses the Levenberg-Marquardt algorithm [8], [9] for $\\chi ^2$ minimization and is based on the widely used Halbritter code for the calculation of the surface resistance [10].",
"For the data presented here the superconducting energy gap $\\Delta $ and the residual resistance $R_{\\text{Res}}$ were varied to minimize $\\chi ^2$ .",
"The derived values for both samples are given in Tab.",
"REF .",
"Other input parameters, which were set constant in the program are the critical temperature $T_{\\text{c}}$ =[9.25]K and the mean free path $l$ =[110]nm derived from penetration depth measurements, while the BCS coherence length $\\xi _0$ =[39]nm and the London penetration depth $\\lambda _L$ =[33]nm were taken from the literature [11].",
"The energy gap $\\Delta $ is found consistent for the three frequencies.",
"The values are also consistent with theory and other measurements [11].",
"The obtained residual resistance $R_{\\text{Res}}$ is as expected for reactor grade niobium material [12].",
"It is obviously dependent on frequency, as has also been pointed out in other publications, where $R_{\\text{Res}}$ was derived with a multimode cavity [13] and for a large batch of elliptical cavities of same surface treatment but different resonant frequency [14].",
"The former approach is limited by the different field configurations for each mode and the latter can give only a statistical result, since every cavity surface is different.",
"In contrast the results from the Quadrupole Resonator are obtained for the same sample under almost identical magnetic field configuration on the surface.",
"Table: Material parameters derived from low field surface resistance measurements below [4.5]K"
],
[
"Electric and Magnetic Field Configuration",
"Cylindrical cavities operated in a TE mode are often used for material characterization.",
"These cavities expose the samples attached only to an RF magnetic field.",
"The Quadrupole Resonator with its different field configuration exposes the samples to electric and magnetic fields simultaneously.",
"The electric field $\\vec{E}$ on the Quadrupole Resonator sample surface scales linearly with frequency for a given magnetic field $\\vec{B}$ , as required by the law of induction when applied to the geometry in between the crooked endings of the rods and the sample.",
"For a peak magnetic field $B_{\\text{p}}$ =[10]mT, the peak electric field is $E_{\\text{p}}=$ [0.52, 1.04, 1.56]MV/m for [400, 800 and 1200]MHz respectively as has been calculated using Microwave Studio.",
"These values are small compared to $E_{\\text{p}}$ -field levels on elliptical cavities, but the area of high electric field is larger.",
"In elliptical cavities the surface electric field is mainly concentrated around the iris of the cavity.",
"In the Quadrupole Resonator it is approximately spread over the same area on the sample surface as the magnetic field.",
"The fact that the ratio of the mean values $E_{\\text{mean}}$ /$B_{\\text{mean}}$ for elliptical cavities and the Quadrupole Resonator are comparable is a valuable feature if real accelerator cavity surfaces are to be studied.",
"In the following the implications of the field dependent ratio $E/B$ on the interpretation on measurement results is discussed.",
"The Quadrupole Resonator measures the power dissipated on the surface of the attached sample by a calorimetric RF-DC compensation measurement consisting of two steps: The temperature of interest is set by applying a current to the resistor on the back side of the sample.",
"The power dissipated $P_{\\text{DC,1}}$ is derived from measuring the voltage across the resistor.",
"The RF is switched on and the current applied to the resistor is lowered to keep the sample temperature and the total power dissipated constant.",
"The power dissipated by RF, $P_{\\mathrm {RF}}$ is the difference between the DC power applied without RF, $P_{\\mathrm {DC1}}$ and the DC power applied with RF, $P_{\\mathrm {DC2}}$ .",
"In general these losses are caused by the RF magnetic and electric fields with the two contributions being additive, $P_{\\text{RF}}=P_{\\text{DC1}}-P_{\\text{DC2}}=\\frac{1}{2\\mu _0^2} \\int \\limits _{\\text{\\hbox{t}o 0pt{\\hss $\\mathrm {Sample}$\\hss }}}R_{\\mathrm {S}}\\vec{B}^2\\mathrm {d}S+\\frac{\\varepsilon _0}{2\\mu _0} \\int \\limits _{\\text{\\hbox{t}o 0pt{\\hss $\\mathrm {Sample}$\\hss }}} R_{\\mathrm {S}}^{\\text{E}}\\vec{E}^2\\mathrm {d}S,$ where $R_{\\mathrm {S}}^{\\text{E}}$ is the electrical surface resistance.",
"Usually, when the surface resistance of superconducting cavities is investigated, the losses are assumed to be caused by the RF magnetic field $B$ , since the contribution from the electric field $E$ is negligible, even for normal conducting metals [15].",
"However, for oxidized surfaces additional loss mechanisms need to be taken into account.",
"To relate electric and magnetic losses to each other the constant $c$ is introduced $c=\\frac{G_{\\text{Sample}}^{\\text{E}}}{G_{\\text{Sample}}}$ This ratio between the magnetic and the electric geometry factor scales quadratically with frequency.",
"This follows directly from the law of induction for the Quadrupole Resonator geometry and has been verified with an agreement better than 1% using MWS.",
"This allows to normalize $c$ to [400]MHz $c(f)=\\frac{G^{\\text{E}}_{\\text{Sample}}(f)}{G_{\\text{Sample}}(f)}=c_{\\text{400}}\\frac{(400\\,\\textrm {MHz})^2}{f^2}.$ Microwave Studio was used to calculate $c_{400}$ =53.5.",
"This implies that a power dissipated by the RF field on the sample surface $P_{\\text{RF}}$ , corresponding to a magnetic surface resistance of [1]n$\\Omega $ , is equivalent to an electric surface resistance of [53.5]n$\\Omega $ at [400]MHz, while a measured RF heating corresponding to $R_{\\text{S}}$ =[1]n$\\Omega $ at [800]MHz is only equivalent to $R_{\\text{S}}^{\\text{E}}$ =[13.4]n$\\Omega $ .",
"The Quadrupole Resonator does not allow to measure the magnetic and the electric losses independently.",
"One has to measure the complete losses and then the interpret the data.",
"For example the surface resistance as shown in Tab.",
"REF has been calculated assuming the residual resistance is completely caused by the RF magnetic field.",
"Assuming that it is caused by the electric field, the values are $R_{\\text{S}}^{\\text{E}}$ =[(1060$\\pm $ 40)]n$\\Omega $ at [400]MHz, $R_{\\text{S}}^{\\text{E}}$ =[(840$\\pm $ 8)]n$\\Omega $ at [800]MHz and $R_{\\text{S}}^{\\text{E}}$ =[(593$\\pm $ 4)]n$\\Omega $ at [1200]MHz.",
"This would imply an unphysical higher electric surface resistance at lower frequency, which allows to conclude that $R_{\\text{Res}}$ is at least mainly caused by the magnetic field.",
"The frequency dependent ratio between $E_{\\text{p}}$ and $B_{\\text{p}}$ enabled to reveal electric losses due to interface tunnel exchange to be the cause for a field dependent surface resistance at RF electric fields of a few MV/m on oxidized granular surfaces, see..."
],
[
"Maximum RF Field",
"In the following it will be shown that the Quadrupole Resonator, designed for surface resistance measurements, is also suited to probe the intrinsic maximum RF magnetic field $B_{\\text{max,RF}}$ of the samples.",
"A quench is detected from a sudden drop of the transmitted power by several orders of magnitude.",
"One can easily determine, without further diagnostics, if it happened on the host cavity or on the sample by measuring the sample temperature at the moment the quench occurs.",
"If the temperature rises above the critical temperature $T_{\\text{c}}$ it was on the sample, otherwise it must have been on the host cavity.",
"The critical field under RF exposure has been investigated using pulses just long enough that the stored energy in the cavity reaches steady state (pulse length approx.",
"[2]ms) and also in continuous wave (CW) operation.",
"Different field levels and dependencies on frequency have been found for each case.",
"In the analysis of the CW measurements it is assumed that $B_{\\text{max,RF}}$ has the same dependence on temperature as the critical thermodynamic field $B_{\\text{c}}$ and can therefore be written as $B_{\\text{max,RF}}(T)=B_{\\text{max,RF}}(0)\\left(1-\\left(\\frac{T}{T_{\\text{c}}}\\right)^2\\right).$ In order to measure the critical field in continuous wave (CW), first the magnetic field on the sample surface $B_{\\text{p}}$ is set to a fixed level.",
"Then the sample temperature is slowly raised until the quench occurs.",
"Usually a sudden temperature rise above $T_{\\text{c}}$ is observed at the moment the quench occurs.",
"When measured in CW the quench field is dependent on frequency and surface properties.",
"In early tests a bulk niobium sample quenched at relatively low field levels due to a local defect.",
"A second etching (BCP 100 $$ m) yielded higher field levels for the clean sample, see Fig.",
"REF .",
"The fact that $B_{\\text{p}}$ vs. $T^2$ gives a straight line is an indication that an intrinsic superconducting field limitation is found for all curves.",
"This can be explained by a local defect heating its surrounding area.",
"When the temperature in the vicinity of the defect exceeds the field dependent critical temperature the quench occurs.",
"Figure: Quench field B max,RF B_{\\text{max,RF}} of the bulk niobium sample in clean and dirty condition, measured in CW.Fitting a straight line to the data displayed in Fig.REF allows to derive $T_{\\text{c}}$ and $B_{\\text{max,RF}}(0)$ .",
"The intersection of each line with the x-axis gives $T_{\\text{c}}^2$ and the slope $B_{\\text{max,RF}}(0)/T_{\\text{c}}^2$ .",
"The parameter values found for all curves are listed in Tab.",
"REF .",
"Table: Critical RF field of the bulk niobium sampleThe quench field in CW is about twice as high at the lower frequency.",
"This cannot be explained by pure thermal breakdown caused by a local normal conducting defect [16].",
"In this case the quench field would scale with $f^{-0.25}$ assuming normal skin effect.",
"This would imply that the quench field at [400]MHz is only 1.3 times higher than at [1200]MHz.",
"This prediction is clearly in contradiction to the measurement results presented here.",
"It can also be excluded that the quench is a complete magnetic effect.",
"In this case the same maximum field should be reached independent of frequency and duty cycle.",
"The RF breakdown can be explained by taking both magnetic and thermal effects into account.",
"That both effects are relevant for RF breakdown was also recently found by Eremeev et al.",
"[17].",
"Measured in pulsed operation a lower critical temperature compared to the value derived from the low field surface resistance measurements was obtained.",
"This is due to the fact that the position of highest field value is located about [1.5]mm closer to the heater than the position of the temperature sensors, which consequently indicates a lower temperature than prevails at the region of maximum field.",
"For the following comparison of the maximum RF field with theory the value of $T_{\\text{c}}$ =[9.11]K derived from the pulsed measurement will be used.",
"Figure: Critical field under RF (short pulses) of the clean bulk niobium sample.Figure REF shows the maximum RF field normalized to the thermodynamic critical field $B_{\\text{c}}$ =[199]mT [18], [19] as a function of temperature.",
"Here it can be seen that the maximum RF field systematically exceeds $B_{\\text{c}}$ .",
"Critical RF fields above $B_{\\text{c}}$ have also been measured in several other publications [20], [21], [22].",
"Their results can be explained by a superheating field $B_{\\text{sh}}$ either derived from considering a metastable state preventing flux entry in the superconductor by a surface barrier [23], [24] or by considering a thermodynamic energy balance at the interface between the superconductor and the adjacent vacuum [25], [26], [27].",
"Figure REF shows the predictions from the vortex line nucleation model (VLNM) [27] and the approximate formulas from [24] based on Ginsburg-Landau theory, therefore in the following named the Ginsburg-Landau model (GLM).",
"The VLNM is based on a thermodynamic energy balance, while the GLM considers metastability.",
"Both models relate the superheating field to the Ginsburg-Landau parameter $\\kappa $ and the critical thermodynamic field $B_{\\text{c}}$ .",
"The latter parameter was taken from literature, while $\\kappa $ has been calculated from penetration depth measurements.",
"The error bands are directly correlated to the uncertainties of $\\kappa $ .",
"For low values of $\\kappa $ (long mean free path) the Ginsburg-Landau model predicts $B_{\\text{sh}}$ smaller than the VLNM.",
"This is the case for the bulk niobium sample in clean condition, for which $B_{\\text{max,RF}}$ was measured with values consistent with the Ginsburg-Landau model, see Fig.",
"REF .",
"Figure: Critical field under RF (short pulses) of the niobium film sample.For high values of $\\kappa $ (low mean free path) the Ginsburg-Landau model predicts $B_{\\text{sh}}$ higher than the VLNM.",
"This condition was found for a niobium film sample, which was prepared by DC magnetron sputtering onto a copper substrate.",
"The critical temperature of this sample was measured to be $T_{\\text{c}}$ =[9.28]$\\pm $ 0.03 by the same approach as described above for the bulk niobium sample, while the Ginsburg-Landau parameter $\\kappa $ was again derived from penetration depth measurements.",
"The maximum RF field for this sample does not exceed $B_{\\text{c}}$ .",
"The values are consistent with the VLNM, see Fig.",
"REF .",
"For all three frequencies the same value of $B_{\\text{max,RF}}$ is found if sufficiently short pulses are used.",
"This shows that the intrinsic critical RF field can be measured with the Quadrupole Resonator.",
"From the analysis of the critical field of one sample of low and one of high purity it seems that the GLM and the VLNM can both limit the maximum field under RF in superconducting cavities.",
"For both samples the lower barrier could not be exceeded.",
"It can however not be excluded that the limitation set by the VLNM can be overcome for cleaner surfaces.",
"It has been stated that for very high values $\\kappa $ this model predicts unrealistic low values for the superheating field [28]."
],
[
"Summary",
"The Quadrupole Resonator has been refurbished and its measurement capabilities have been extended.",
"The calorimetric results were thereby verified by an independent technique.",
"It was shown that the device can be used for surface resistance measurements at multiple integers of its design frequency.",
"The almost identical magnetic field configurations in combination with the mode dependent electrical field configuration enable unique possibilities to test theoretical surface resistance models.",
"Additionally it was shown that the intrinsic critical RF magnetic field can be measured with the device."
],
[
"Acknowledgment",
"The authors would like to thank everybody who contributed to the refurbishment and operation of the Quadrupole Resonator.",
"The work of S. Calatroni and S. Forel (preparation of samples) and the operators from the CERN cryogenics group is highly appreciated.",
"We thank Ernst Haebel, now retired, for explaining to us the original idea and conception of the Quadrupole Resonator.",
"The help of N. Schwerg performing the [3]dB bandwidth measurements is appreciated."
]
] | 1204.1018 |
[
[
"The Nori fundamental gerbe of a fibered category"
],
[
"Abstract We give a condition that ensures that a fibered category over a field admits a universal morphism to a profinite gerbe.",
"This fundamental gerbe generalizes both Nori's fundamental group scheme and Deligne's relative fundamental groupoid.",
"Using a simplified notion of essentially finite bundle, we also give a tannakian construction.",
"As an application, we show how the fundamental gerbe enables to formulate a version of Grothendieck's section conjecture in arbitrary characteristic.",
"We then study various natural quotients of the fundamental gerbe."
],
[
"Introduction",
"Let $X$ be a reduced proper connected scheme over a perfect field $\\kappa $ , with a rational point $x_{0} \\in X(\\kappa )$ .",
"The celebrated result of Nori [17] says the following.",
"There is a profinite group scheme $\\pi (X, x_{0})$ , the Nori fundamental group scheme, with a $\\pi (X,x_{0})$ -torsor $P \\rightarrow X$ with a trivialization $P\\mid _{x_{0}} \\simeq \\pi (X, x_{0})$ such that for every finite group scheme $G \\rightarrow \\operatorname{Spec}\\kappa $ and every $G$ torsor $Q \\rightarrow X$ with a trivialization $\\alpha \\colon Q \\mid _{x_{0}} \\simeq G$ , there is a unique homomorphism of group schemes $\\pi (X, x_{0}) \\rightarrow G$ inducing $Q$ and $\\alpha $ .",
"There is an equivalence of Tannaka categories between representations of the group scheme $\\pi (X, x_{0})$ and essentially finite locally free sheaves on $X$ .",
"Let us recall that a locally free sheaf $E$ on $X$ is finite if there exist two polynomials $f$ and $g$ in one variable with natural numbers as coefficients, with $f \\ne g$ , such that $f(E) \\simeq g(E)$ (here we evaluate $f$ on $E$ by using direct sums and tensor powers).",
"The notion of essentially finite is more delicate, and we refer to [17] for the definition, which uses a notion of semistable locally free sheaf.",
"In characteristic 0, it turns out, as a consequence of the theorem, that every essentially finite sheaf is in fact finite.",
"In this paper we generalize Nori's construction by removing the assumption that $X$ has a rational point $x_{0} \\in X(\\kappa )$ , and we give a simpler and more direct approach to Nori's correspondence between representations and essentially finite locally free sheaves.",
"We also show how our formalism allows to give a natural interpretation of Grothendieck's Section Conjecture, and a natural formulation of the conjecture in arbitrary characteristic.",
"As to the first point, we replace the group with a gerbe (thus we use the language of gerbes, not that of groupoids).",
"In characteristic 0 this gerbe is the one associated with the fundamental groupoid of Deligne [6]; an alternative construction in the smooth case is due to Esnault and Hai [8].",
"The gerbes that appear are not of finite type, so we need to consider gerbes in the uncomfortably large fpqc topology.",
"Fortunately, we don't need to use dubious notions such as the fpqc stackification.",
"In our approach we don't need $X$ to be a scheme, we work with fibered categories over a fixed field $\\kappa $ (not necessarily stacks, the sheaf conditions on $X$ don't play any role).",
"After some preliminary work on fpqc gerbes and projective limits, and on finite stacks, in Section we introduce a class of fibered categories, called inflexible (Definition REF ).",
"This notion has some geometric content: the class of inflexible algebraic stacks is properly contained in the class of geometrically connected algebraic stacks, and properly contains the class of geometrically connected and geometrically reduced algebraic stacks.",
"Then we define a fundamental gerbe of a fibered category $X$ as a profinite gerbe $\\Pi _{X/\\kappa }$ with a morphism $X \\rightarrow \\Pi _{X/\\kappa }$ , such that every morphism from $X$ to a finite stack $\\Gamma $ factors uniquely through $\\Pi _{X/\\kappa }$ .",
"The fundamental gerbe is unique.",
"Our main result in this section is that a fibered category has a fundamental gerbe if and only if it is inflexible.",
"In Section we show that the fundamental base gerbe has a very useful base-change property with respect to algebraic separable extensions.",
"Next we discuss the tannakian interpretation of the fundamental gerbe, generalizing Nori's approach.",
"Let us recall that Grothendieck, Saavedra Rivano and Deligne ([25], [7]) have shown that there is an equivalence between non-neutral Tannaka categories on one side and affine fpqc gerbes on the other; thus it is natural to ask whether the gerbe $\\Pi _{X/\\kappa }$ has a tannakian interpretation.",
"For this we need to work with finite locally free sheaves on $X$ , and this only works if we impose a finiteness condition on $X$ .",
"In Section we introduce a class of fibered categories, that we call pseudo-proper: a fibered category over $\\kappa $ is pseudo-proper when it has a fpqc cover by a quasi-compact and quasi-separated scheme, and furthermore for any locally free sheaf $E$ on $X$ the dimension of the $\\kappa $ -vector space $\\operatorname{H}^0(X,E)$ is finite.",
"This last condition is a very natural one to impose, as it ensures that the Krull–Schmidt theorem holds for locally free sheaves on $X$ .",
"Then we discuss finite locally free sheaves on a pseudo-proper fibered category, copying Nori's definition.",
"We also define essentially finite locally free sheaves, with a definition that is more general and simpler than Nori's, and does not use semistable locally free sheaves at all.",
"(In the case of a complete scheme over $\\kappa $ our notion and Nori's turn out to coincide.)",
"We denote by $\\operatorname{EFin}X$ the category of essentially finite locally free sheaves.",
"The following is our main result.",
"Main theorem (Theorems REF and REF ) Let $X$ be a pseudo-proper fibered category.",
"The fibered category $X$ is inflexible if and only if $\\operatorname{EFin}X$ is tannakian.",
"If $X$ is inflexible, then $\\operatorname{EFin}X$ is equivalent as a Tannaka category to the category of representations of the fundamental gerbe $\\Pi _{X/\\kappa }$ .",
"If $\\operatorname{char}\\kappa = 0$ and $X$ is inflexible, then every essentially finite locally free on $X$ is in fact finite.",
"We find it interesting that the condition of being inflexible is in fact equivalent to the completely different condition that $\\operatorname{EFin}X$ be tannakian; this seems to suggest that indeed this inflexibility condition is a very natural one.",
"Next we study two natural quotients of $\\Pi _{X/\\kappa }$ .",
"The first is the largest étale quotient $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ of $\\Pi _{X/\\kappa }$ ; in Section we show that this coincides with the stack associated with Deligne's relative fundamental groupoid, introduced in [6].",
"In Section , we show how Grothendieck's famous Section Conjecture can be interpreted as a statement about the étale fundamental gerbe (Conjecture REF ).",
"Furthermore, we formulate a version of the Section Conjecture in arbitrary characteristic (Conjecture REF ).",
"The second quotient is the largest quotient of $\\Pi _{X/\\kappa }$ that is tame, in the sense of [1].",
"Of course in characteristic 0 we have $\\Pi _{X/\\kappa } = \\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}} = \\Pi _{X/\\kappa }^{\\mathrm {tame}}$ .",
"It is natural to ask what are the tannakian subcategories of $\\operatorname{EFin}X$ that correspond to $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ and $\\Pi _{X/\\kappa }^{\\mathrm {tame}}$ .",
"The first one does not seem to have a natural answer (although there is a tannakian interpretation of $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ in terms of local systems, see Section ).",
"In Section we show that the category of representations of $\\Pi _{X/\\kappa }^{\\mathrm {tame}}$ is equivalent to the category of finite tame locally free sheaves on $X$ : these are the finite locally free sheaves $E$ on $X$ such that all tensor powers $E^{\\otimes n}$ are semisimple.",
"Finally, Section contains some examples and applications to illustrate the theory."
],
[
"Acknowledgments",
"The authors would like to thank Bertrand Töen for an extremely stimulating conversation.",
"They are also grateful to Michel Emsalem, Damian Rössler, and Jakob Stix, for several discussions, especially about Section , and to the referee for his useful comments.",
"The content of Section started out as joint work with Sylvain Brochard, whom we thank heartily.",
"Finally, we also thank the ENS Lyon for its hospitality while much of this research was done."
],
[
"Conventions",
"We will work over a fixed field $\\kappa $ ; all schemes and all morphisms of schemes will be over $\\kappa $ , unless explicit mention to the contrary is made.",
"All fibered categories will be fibered in groupoids over the category $(\\mathrm {Aff}/\\kappa )$ of affine schemes over $\\kappa $ , with the same proviso.",
"As usual, we will identify a functor $(\\mathrm {Aff}/\\kappa )^{\\mathrm {op}}\\rightarrow \\mathrm {(Set)}$ with the corresponding fibered category, and a scheme with the corresponding functor.",
"A fibered product (of fibered categories, or of schemes) $X \\times _{\\operatorname{Spec}\\kappa }Y$ will be denoted simply by $X \\times Â~Y$ .",
"If $\\kappa ^{\\prime }$ is an extension of $\\kappa $ and $X$ is a fibered category over $\\kappa $ , we set $X_{\\kappa ^{\\prime }} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}(\\mathrm {Aff}/\\kappa ^{\\prime }) \\times _{(\\mathrm {Aff}/\\kappa )}X$ .",
"If $p\\colon X \\rightarrow (\\mathrm {Aff}/\\kappa )$ is a fibered category, we will consider it as a site by putting on it the fpqc topology, generated by the collection of morphisms $\\lbrace \\xi _{i} \\rightarrow \\xi \\rbrace $ such that $\\lbrace p(\\xi _{i}) \\rightarrow p(\\xi )\\rbrace $ is an fpqc cover.",
"There is a sheaf $\\mathcal {O}_{X}$ on $X$ , sending each object $\\xi $ into $\\mathcal {O}\\bigl (p(\\xi )\\bigr )$ .",
"Let $f\\colon X \\rightarrow Y$ be a morphism of fibered categories, where $Y$ is an algebraic stack.",
"We call the scheme-theoretic image $Y^{\\prime }$ of $X$ in $Y$ the intersection of all the closed substacks $Z$ of $Y$ such that $f$ factors, necessarily uniquely, though $Z$ .",
"Alternatively, $Y^{\\prime } \\subseteq Y$ is the closed substack associated with the largest quasi-coherent sheaf of ideals of $\\mathcal {O}_{Y}$ contained in the kernel of the natural homomorphism $\\mathcal {O}_{Y} \\rightarrow f_{*}\\mathcal {O}_{X}$ .",
"It is easy to see that $f$ factors uniquely through $Y^{\\prime }$ (this is clear when $X$ is a scheme, and the general case reduces to this).",
"If $X$ is an algebraic stack over $\\kappa $ , we will call $\\operatorname{Vect}X$ the category of locally free sheaves of $\\mathcal {O}_{X}$ -modules of finite constant rank on $X$ , which we will also call vector bundles.",
"If $X$ is a locally noetherian algebraic stack, we denote by $\\operatorname{Coh}X$ the category of coherent sheaves on $X$ .",
"We also denote by $\\operatorname{Vect}_{\\kappa }$ the category $\\operatorname{Vect}(\\operatorname{Spec}\\kappa )$ of vector spaces on $\\kappa $ .",
"All 2-categories appearing in this paper will be strict $(2,1)$ -categories, unless we mention otherwise.",
"Likewise, all functors will be strict.",
"We will use the symbol $*$ for the Godement product."
],
[
"Projective limits of fpqc gerbes",
"We will work with the 2-category $(\\mathrm {Aff\\,Ger}/\\kappa )$ of affine fpqc gerbes over $\\kappa $ , called tannakian gerbes in [25].",
"These are fpqc gerbes with a flat presentation $R \\mathbin {\\rightrightarrows }U$ , where $R$ and $U$ are affine $\\kappa $ -schemes.",
"Equivalently they can be defined as fpqc gerbes over $(\\mathrm {Sch}/\\kappa )$ with affine diagonal, and an affine chart.",
"Suppose that $R \\mathbin {\\rightrightarrows }U$ is a flat groupoid, where $R$ and $U$ are affine; then the associated fpqc stack is a gerbe if and only if the diagonal $R \\rightarrow U \\times U$ is faithfully flat, and $U$ is nonempty.",
"If an affine fpqc gerbe $\\Phi $ has an object $\\xi $ defined over $\\kappa $ , then it is equivalent to the classifying $\\mathcal {B}_{k}G$ , where $G$ is the automorphism group scheme of $\\xi $ over $\\kappa $ .",
"Proposition 3.1 Let $\\Phi $ be an affine fpqc gerbe.",
"$\\Phi $ has an fpqc presentation of the type $R \\mathbin {\\rightrightarrows }U$ , where $R$ is affine and $U$ is the spectrum of a field.",
"Any morphism from a non-empty algebraic stack $X$ to $\\Phi $ is faithfully flat, and an fpqc cover.",
"If $X$ is a scheme, it is moreover representable.",
"The diagonal $\\Phi \\rightarrow \\Phi \\times \\Phi $ is representable, faithfully flat, and affine.",
"For part (REF ), one can make a field extension; so we can assume that $\\Phi (\\kappa ) \\ne \\emptyset $ , so that $\\Phi \\simeq \\mathcal {B}_{\\kappa }G$ , where $G$ is a affine group scheme.",
"Then the morphism $\\operatorname{Spec}\\kappa \\rightarrow \\mathcal {B}_{\\kappa }G$ corresponding to the trivial torsor $G \\rightarrow \\operatorname{Spec}\\kappa $ is the universal torsor over $\\mathcal {B}_{\\kappa }G$ ; in particular, it is affine and faithfully flat, hence an fpqc cover.",
"Hence, if $S$ is a non-empty scheme and $S \\rightarrow \\mathcal {B}_{\\kappa }G$ is a morphism, the fibered product $P \\overset{\\mathrm {\\scriptscriptstyle def}}{=}S \\times _{\\mathcal {B}_{\\kappa }G} \\operatorname{Spec}\\kappa $ is a $G$ -torsor over $S$ , so it is non-empty, and $P \\rightarrow \\operatorname{Spec}\\kappa $ is an fpqc cover.",
"Hence $S \\rightarrow \\mathcal {B}_{\\kappa }G$ is an fpqc cover, as claimed.",
"For part (REF ) we can also make an extension of base field, since affine morphisms satisfy fpqc descent, so that $\\Phi (\\kappa ) \\ne \\emptyset $ .",
"In this case $\\Phi \\simeq \\mathcal {B}_{\\kappa }G$ for an affine group $G$ , and the statement is clear.",
"For (REF ), take a field extension $K$ of $\\kappa $ such that $\\Phi (K) \\ne \\emptyset $ , and set $U \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\operatorname{Spec}K$ and $R \\overset{\\mathrm {\\scriptscriptstyle def}}{=}U \\times _{\\Phi } U$ .",
"Since the diagonal $\\Phi \\rightarrow \\Phi \\times \\Phi $ is affine, we have that $R$ is an affine scheme.",
"Because of (REF ), the groupoid $R \\mathbin {\\rightrightarrows }U$ gives an fpqc presentation of $\\Phi $ .",
"Definition 3.2 A boolean cofiltered 2-category $I$ is a small 2-category such that, for any two objects $i$ and $j$ of $I$ , there exists another object $k$ with arrows $k \\rightarrow i$ and $k \\rightarrow j$ , and given any two 1-arrows $a$ , $b\\colon j \\rightarrow i$ , there exists a unique 2-arrow $a \\Rightarrow b$ .",
"One sees immediately that a 2-category is cofiltered if and only if it is equivalent to a cofiltered partially ordered set, considered as a 2-category.",
"From now we will call a boolean cofiltered 2-category simply a cofiltered 2-category.",
"The reason for the adjective “boolean” is that, as was pointed out by the referee, a general cofiltered 1-category is not equivalent to a partially ordered set, and so it can not be a boolean cofiltered 2-category.",
"Definition 3.3 A projective system in $(\\mathrm {Aff\\,Ger}/\\kappa )$ consists of a cofiltered 2-category $I$ and a strict 2-functor $\\Gamma \\colon I \\rightarrow (\\mathrm {Aff\\,Ger}/\\kappa )$ .",
"Remark 3.4 There are two possible ways of defining a projective system.",
"One as in the definition above; the other is to take $I$ to be a cofiltered partially ordered set, and $\\Gamma $ to be a pseudo-functor.",
"The two methods are essentially equivalent, but the one above works better for our purposes.",
"Given a projective system $\\Gamma \\colon I \\rightarrow (\\mathrm {Aff\\,Ger}/\\kappa )$ , we denote by $\\Gamma _{i}$ the image of an object $i$ of $I$ , and by $\\Gamma _{a}\\colon \\Gamma _{j} \\rightarrow \\Gamma _{i}$ the cartesian functor corresponding to a 1-arrow $a\\colon j \\rightarrow i$ .",
"Finally, if $a$ , $b\\colon j \\rightarrow i$ are 1-arrows, we denote by $\\Gamma _{a,b}\\colon \\Gamma _{a} \\rightarrow \\Gamma _{b}$ the natural isomorphism corresponding to the unique 2-arrow $a \\rightarrow b$ .",
"Definition 3.5 Let $\\Gamma \\colon I \\rightarrow (\\mathrm {Aff\\,Ger}/\\kappa )$ be a projective system of affine gerbes.",
"The projective limit $\\varprojlim \\Gamma $ is the category fibered in groupoids over $(\\mathrm {Sch}/\\kappa )$ defined as follows.",
"An object $\\xi $ of $\\varprojlim \\Gamma $ consists of the following data.",
"A scheme $T$ over $\\kappa $ , and an object $\\xi _{i}$ of $\\Gamma _{i}(T)$ for each object $i$ of $\\Gamma $ .",
"For each 1-arrow $a\\colon j \\rightarrow i$ in $I$ , an arrow $\\xi _{a}\\colon \\Gamma _{a}(\\xi _{j}) \\rightarrow \\xi _{i}$ in $\\Gamma _{i}(T)$ .",
"These are required to satisfy the following conditions.",
"If $a\\colon j \\rightarrow i$ and $b\\colon k \\rightarrow j$ are 1-arrows in $I$ , the diagram $@C+15pt{\\Gamma _{ab}(\\xi _{k})[r]^{\\Gamma _{a}(\\xi _{b})}[rd]_{\\xi _{ab}}&\\Gamma _{a}(\\xi _{j}) [d]^{\\xi _{a}}\\\\&\\xi _{i}}$ commutes.",
"If $a$ and $b$ are 1-arrows from $j$ to $i$ , the diagram ${\\Gamma _{a}(\\xi _{j}) [rr]^{\\Gamma _{a,b}}[rd]_{\\xi _{a}}&& \\Gamma _{b}(\\xi _{j})[ld]^{\\xi _{b}}\\\\& \\xi _{i}}$ commutes.",
"An arrow $f\\colon \\xi \\rightarrow \\eta $ consists of a morphism of $\\kappa $ -schemes $\\phi \\colon T \\rightarrow T^{\\prime }$ and an arrow $f_{i}\\colon \\xi _{i} \\rightarrow \\eta _{i}$ for each $i$ , such that $f_{i}$ maps to $\\phi $ in $(\\mathrm {Sch}/k)$ for all $i$ , and for each 2-arrow $a\\colon j \\rightarrow i$ , the diagram $@C+15pt{\\Gamma _{a}(\\xi _{j}) [r]^{\\Gamma _{a}(f_{j})} [d]^{\\xi _{a}}&\\Gamma _{a}(\\eta _{j})[d]^{\\eta _{a}}\\\\\\xi _{i} [r]^{f_{i}} & \\eta _{i}}$ commutes.",
"For the projective limit $\\varprojlim \\Gamma $ we will also use the notation $\\varprojlim _{i \\in I} \\Gamma _{i}$ , or $\\varprojlim _{i} \\Gamma _{i}$ .",
"It is a straightforward exercise in descent theory to show that $\\varprojlim \\Gamma $ is an fpqc stack.",
"Remark 3.6 Here is different description of $\\varprojlim _{i \\in I} \\Gamma _{i}$ from the point of view of groupoids.",
"Suppose that $\\Phi \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim \\Gamma $ is non-empty (it is not clear to us whether it can happen that it is empty); let $X$ be a non-empty scheme with a morphism $X \\rightarrow \\varprojlim \\Gamma $ , corresponding to an object $\\xi = \\lbrace \\xi _{i}\\rbrace $ of $\\Phi (X)$ .",
"We claim that $X \\rightarrow \\Phi $ is representable, faithfully flat, and an fpqc cover.",
"For this, let $T$ be an affine scheme with a morphism $T \\rightarrow \\Phi $ , corresponding to an object $\\tau = \\lbrace \\tau _{i}\\rbrace $ of $\\Phi (T)$ .",
"The fibered product $X \\times _{\\Gamma _{i}}T$ is represented by the scheme $P_{i} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\mathop {\\underline{\\mathrm {Isom}}}\\nolimits _{X \\times T}(\\operatorname{pr}_{2}^{*}\\tau _{i}, \\operatorname{pr}_{1}^{*}\\xi _{i})$ ; by Proposition REF (REF ), $P_{i}$ is affine and faithfully flat over $X \\times T$ .",
"If $a\\colon j \\rightarrow i$ is an arrow in $I$ , the corresponding isomorphisms $\\xi _{a}\\colon \\Gamma _{a}(\\xi _{j}) \\rightarrow \\xi _{i}$ and $\\tau _{a}\\colon \\Gamma _{a}(\\tau _{j}) \\rightarrow \\tau _{i}$ induce a morphism of $X \\times T$ -schemes $P_{j} \\rightarrow P_{i}$ ; this defines a functor from $I$ to the category $(\\mathrm {Aff}/X \\times T)$ of affine schemes over $X \\times T$ .",
"Since the target is a 1-category, this functor factors through the preordered set $\\overline{I}$ corresponding to $I$ (that is, the objects are the objects of $I$ , and we set $j \\le i$ when there exists an arrow $j \\rightarrow i$ ).",
"Set $P \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim _{\\overline{I}}P_{i}$ ; it follows from the definition of projective limit that $P$ represents the functor $\\mathop {\\underline{\\mathrm {Isom}}}\\nolimits _{X \\times T}(\\operatorname{pr}_{2}^{*}\\tau , \\operatorname{pr}_{1}^{*}\\xi ) = X \\times _{\\Phi } T$ .",
"Since the limit of affine faithfully flat schemes over $X \\times T$ is again affine and faithfully flat, we see that $P$ is affine and faithfully flat over $X \\times T$ ; hence $P$ is an fpqc cover of $T$ , and the result follows.",
"If we set $R_{i} = X \\times _{\\Gamma _{i}} X$ and $R = \\varprojlim R_{i}$ , it results from the above that $\\varprojlim \\Gamma $ is the fpqc quotient of the groupoid $R \\mathbin {\\rightrightarrows }X$ ; this will be used in Remark REF to compare our construction with Deligne's construction of the étale fundamental groupoid.",
"Also, since $R \\rightarrow X \\times X$ is faithfully flat and affine, and $X$ is nonempty, we have that $\\Phi $ is a fpqc gerbe.",
"We record this in a Proposition.",
"Proposition 3.7 Let $\\Gamma \\colon I \\rightarrow (\\mathrm {Aff\\,Ger}/\\kappa )$ a projective system of affine fpqc gerbes.",
"If the limit $\\varprojlim \\Gamma $ is not empty, it is an fpqc gerbe.",
"Proposition 3.8 Let $\\Phi = \\varprojlim _{i}\\Gamma _{i}$ be a projective limit of affine fpqc gerbes, $\\Delta $ an affine fpqc stack with a flat presentation by affine $\\kappa $ -schemes of finite type.",
"Then the natural functor $\\varinjlim _{i}\\operatorname{Hom}(\\Gamma _{i}, \\Delta ) \\rightarrow \\operatorname{Hom}(\\Phi , \\Delta )$ is an equivalence.",
"The stack $\\Delta $ is finitely presented over $\\kappa $ .",
"This implies that if $\\lbrace T_{i}\\rbrace $ is a projective system of affine $\\kappa $ -schemes, the natural functor $\\varinjlim _{i}\\Delta (T_{i}) \\rightarrow \\Delta (\\varprojlim T_{i})$ is an equivalence.",
"Let $S \\rightarrow \\Phi $ be a morphism, where $S = \\operatorname{Spec}K$ is the spectrum of a field.",
"Set $R \\overset{\\mathrm {\\scriptscriptstyle def}}{=}S \\times _{\\Phi }S$ and $R_{i}\\overset{\\mathrm {\\scriptscriptstyle def}}{=}S\\times _{\\Gamma _{i}} S$ for each $i$ ; as in the proof of Proposition REF , we have $R = \\varprojlim R_{i}$ .",
"There is an equivalence of categories between $\\operatorname{Hom}(\\Phi , \\Delta )$ and the category of objects of $\\Delta (S)$ with descent data on the flat groupoid $R \\mathbin {\\rightrightarrows }S$ , and, analogously, an equivalence of $\\operatorname{Hom}(\\Gamma _{i}, \\Delta )$ with the category of objects of $\\Delta (S)$ with descent data on $R_{i} \\mathbin {\\rightrightarrows }S$ .",
"But the equivalence of $\\varinjlim _{i}\\Delta (R_{i})$ with $\\Delta (R)$ is easily seen to yield an equivalence between the category of objects with descent data on $R \\mathbin {\\rightrightarrows }S$ and the colimit of the categories of objects with descent data on $R_{i} \\mathbin {\\rightrightarrows }S$ .",
"If $\\Phi $ is an fpqc gerbe over $\\kappa $ , we denote by $\\operatorname{Rep}\\Phi $ the category of coherent sheaves on $\\Phi $ ; these are all locally free, because $\\Phi $ admits a faithfully flat morphism $\\operatorname{Spec}K \\rightarrow \\Phi $ , where $K$ is a field.",
"So $\\operatorname{Rep}\\Phi = \\operatorname{Vect}\\Phi $ .",
"The category $\\operatorname{Rep}\\Phi $ is tannakian.",
"Proposition 3.9 Suppose that $\\Phi = \\varprojlim _{i}\\Gamma _{i}$ is a non-empty limit of fpqc gerbes.",
"The natural functor $\\varinjlim _{i} \\operatorname{Rep}\\Gamma _{i} \\rightarrow \\operatorname{Rep}\\Phi $ is an equivalence.",
"Let $\\Delta $ be the fibered category of locally free sheaves on $(\\mathrm {Sch}/\\kappa )$ , that is, the fibered category whose objects are pairs $(T, E)$ , where $T$ is a $\\kappa $ -scheme and $E$ is a locally free sheaf on $T$ , and in which the arrows are given by arbitrary homomorphism of locally free sheaves (thus, $\\Delta $ is not fibered in groupoids).",
"Then $\\Delta $ is still finitely presented, in the sense that the natural functor $\\varinjlim _{i}\\Delta (T_{i}) \\rightarrow \\Delta (\\varprojlim T_{i})$ is an equivalence for any projective system of affine $\\kappa $ -schemes $\\lbrace T_{i}\\rbrace $ .",
"Hence, the proof of Proposition REF goes through in this case."
],
[
"Finite stacks",
"Definition 4.1 A finite stack over $\\kappa $ is an fppf stack that is represented by a flat groupoid $R \\mathbin {\\rightrightarrows }X$ , where $R$ and $X$ are finite $\\kappa $ -schemes.",
"A finite gerbe is a finite stack over $\\kappa $ that is a gerbe in the fppf topology.",
"By a well known theorem of M. Artin [13], a finite stack is algebraic; it can be defined as an algebraic stack $\\Gamma $ over $\\kappa $ with finite diagonal, which admits a flat surjective map $U \\rightarrow \\Gamma $ , where $U$ is finite over $\\kappa $ .",
"Notice that a finite stack is always an fpqc stack, as it can be interpreted as the stack of $R \\mathbin {\\rightrightarrows }X$ -torsors (if $R\\mathbin {\\rightrightarrows }X$ is an fppf groupoid, then an fpqc torsor is also an fppf torsor, as it is trivialized by one the projections $R \\rightarrow X$ ).",
"We have the following useful characterization of finite stacks.",
"Proposition 4.2 Let $\\Gamma $ be an algebraic stack over $\\kappa $ .",
"Then $\\Gamma $ is finite stack if and only if the following conditions are satisfied.",
"$\\Gamma $ is of finite type over $\\kappa $ .",
"The diagonal of $\\Gamma $ is quasi-finite.",
"The category $\\Gamma (\\overline{\\kappa })$ has finitely many isomorphism classes.",
"It is immediate to check that if $\\Gamma $ is finite, it has the properties above.",
"Conversely, assume that the conditions are satisfied.",
"If $\\Gamma $ is an algebraic space of finite type over $k$ , and $\\Gamma (\\overline{k})$ is finite, then it is immediate to see that $\\Gamma $ is in fact the spectrum of a finite $k$ -algebra.",
"Let us reduce the general case to this one.",
"By [24] (see also the second paragraph of the proof of [19]) there exists a scheme $X$ and a quasi-finite flat surjective morphism $X \\rightarrow \\Gamma $ of finite type; then $\\Gamma $ has an fppf presentation $X\\times _{\\Gamma }X \\mathbin {\\rightrightarrows }X$ .",
"Since $X(\\overline{\\kappa })$ is immediately seen to be finite, it follows that $X$ is finite; and since $X\\times _{\\Gamma }X \\rightarrow X \\times X$ is quasi-finite, $(X\\times _{\\Gamma }X)(\\overline{k})$ is also finite, so $X\\times _{\\Gamma }X$ is finite.",
"The result follows.",
"Proposition 4.3 A finite stack over $\\kappa $ is a finite gerbe if and only if it is geometrically connected and geometrically reduced.",
"It is obvious that a finite gerbe is connected; it is also reduced, since it admits a faithfully flat map from the spectrum of a field (Proposition REF (REF )).",
"Since by extending the base field a finite gerbe stays a finite gerbe, it is also geometrically connected and geometrically reduced.",
"Conversely, suppose that $\\Gamma $ is a finite stack that is geometrically connected and geometrically reduced.",
"Being an fppf gerbe is a local property in the fppf topology, so we may base change to a finite extension of $\\kappa $ , and assume that $\\Gamma $ admits a section $\\operatorname{Spec}\\kappa \\rightarrow \\Gamma $ , corresponding to an object $\\xi $ of $\\Gamma (\\kappa )$ .",
"The fibered product $G \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\operatorname{Spec}\\kappa \\times _{\\Gamma }\\operatorname{Spec}\\kappa $ is the group scheme of automorphisms of $\\xi $ ; if we show that $\\operatorname{Spec}\\kappa \\rightarrow \\Gamma $ is flat and surjective, then $\\Gamma $ will have a presentation $G \\mathbin {\\rightrightarrows }\\operatorname{Spec}\\kappa $ , that is, $\\Gamma \\simeq \\mathcal {B}_{\\kappa }G$ , so it is indeed an fppf gerbe.",
"The morphism $\\operatorname{Spec}\\kappa \\rightarrow \\Gamma $ is of finite type.",
"By the theorem on generic flatness, it is generically flat, since $\\Gamma $ is connected; but since $\\operatorname{Spec}\\kappa $ consists of a point, it is in fact flat.",
"Hence its image in $\\Gamma $ is open.",
"Since $\\Gamma $ is finite, it is also closed, so it is surjective, because $\\Gamma $ is connected.",
"If $G$ and $H$ are finite group schemes over $\\kappa $ , every homomorphism of group schemes $G \\rightarrow H$ induces a morphism of gerbes $\\mathcal {B}_{\\kappa }G \\rightarrow \\mathcal {B}_{\\kappa }H$ .",
"However, the category $\\operatorname{Hom}_{\\kappa }(\\mathcal {B}_{\\kappa }G, \\mathcal {B}_{\\kappa }H)$ is not equivalent to the set $\\operatorname{Hom}_{\\kappa }(G,H)$ .",
"Denote by $\\mathop {\\underline{\\mathrm {Hom}}}\\nolimits _{\\kappa }(G, H)$ the fibered category over $\\kappa $ of homomorphisms $G \\rightarrow H$ , that is, the sheaf sending each $\\kappa $ -scheme $T$ into the set $\\operatorname{Hom}_{T}(G_{T}, H_{T})$ .",
"There is a natural action by conjugation of $H$ on $\\mathop {\\underline{\\mathrm {Hom}}}\\nolimits _{\\kappa }(G, H)$ ; then the fibered category $\\mathop {\\underline{\\mathrm {Hom}}}\\nolimits _{\\kappa }(\\mathcal {B}_{\\kappa }G, \\mathcal {B}_{\\kappa }H)$ is equivalent to the category of $\\kappa $ -valued objects of the quotient stack $[\\mathop {\\underline{\\mathrm {Hom}}}\\nolimits _{\\kappa }(G, H)/H]$ (see [9]).",
"Definition 4.4 The degree $\\deg \\Gamma $ of a finite gerbe $\\Gamma $ is the degree of the automorphism group scheme $\\mathop {\\underline{\\mathrm {Aut}}}\\nolimits _{\\kappa ^{\\prime }}\\xi $ , where $\\kappa ^{\\prime }$ is an extension of $\\kappa $ and $\\xi $ is an object of $\\Gamma (\\operatorname{Spec}\\kappa ^{\\prime })$ .",
"In the definition above, it is straightforward to show that $\\deg \\Gamma $ does not depend on $\\kappa ^{\\prime }$ nor on $\\xi $ .",
"Furthermore, one should notice that the degree in this sense is not the degree of the proper quasi-finite morphism $\\Gamma \\rightarrow \\operatorname{Spec}\\kappa $ , which equals $1/\\deg \\Gamma $ .",
"Proposition 4.5 Let $f\\colon \\Gamma \\rightarrow \\Delta $ be a representable morphism of finite gerbes.",
"Then $\\deg \\Gamma $ divides $\\deg \\Delta $ , and $f$ is an isomorphism if and only if $\\deg \\Gamma = \\deg \\Delta $ .",
"The property of being representable is stable under extension of base field, as is the degree; hence we may assume that $\\Gamma (\\operatorname{Spec}\\kappa )$ is not empty.",
"Then $f\\colon \\Gamma \\rightarrow \\Delta $ is equivalent to a morphism $\\phi \\colon \\mathcal {B}_{\\kappa }G \\rightarrow \\mathcal {B}_{\\kappa }H$ induced by a homomorphism $G \\rightarrow H$ of finite $\\kappa $ -group schemes.",
"If $f$ is representable, then $\\phi $ is injective, and the result is standard in this case.",
"Definition 4.6 A profinite gerbe over $\\kappa $ is an fpqc gerbe that is equivalent to a projective limit of finite gerbes over $\\kappa $ ."
],
[
"The Nori fundamental gerbe",
"Let $X$ be a fibered category over $\\kappa $ .",
"Definition 5.1 A fundamental gerbe for $X$ is a profinite gerbe $\\Pi _{X/\\kappa }$ with a morphism $X \\rightarrow \\Pi _{X/\\kappa }$ of algebraic stacks over $\\operatorname{Spec}\\kappa $ , such that, if $\\Gamma $ is a finite stack over $\\operatorname{Spec}\\kappa $ , the induced functor $\\operatorname{Hom}_{\\kappa }(\\Pi _{X/\\kappa }, \\Gamma ) \\longrightarrow \\operatorname{Hom}_{\\kappa }(X, \\Gamma )$ is an equivalence of categories.",
"Proposition 5.2 Let $X \\rightarrow \\Pi _{X/\\kappa }$ be a fundamental gerbe.",
"Then for any profinite gerbe $\\Phi $ the induced functor $\\operatorname{Hom}_{\\kappa }(\\Pi _{X/\\kappa }, \\Phi ) \\longrightarrow \\operatorname{Hom}_{\\kappa }(X, \\Phi )$ is equivalence of categories.",
"In particular, a fundamental gerbe is unique up to a canonical equivalence.",
"This follows easily from the definitions of a profinite gerbe and of a projective limit.",
"Definition 5.3 We say that a fibered category $X$ is inflexible if it is non-empty, and for any morphism $X \\rightarrow \\Gamma $ to a finite stack, there exists a closed substack $\\Gamma ^{\\prime } \\subseteq \\Gamma $ that is a gerbe, and a factorization $X \\rightarrow \\Gamma ^{\\prime } \\rightarrow \\Gamma $ .",
"Notice that in the statement above, if $\\Gamma ^{\\prime }$ exists then it must be the scheme-theoretic image of $X$ in $\\Gamma $ ; hence it is unique.",
"Here are some properties of this notion.",
"Proposition 5.4 Let $X$ be a fibered category over $\\kappa $ .",
"If $X$ is inflexible, the only $\\kappa $ -subalgebra of $\\operatorname{H}^{0}(X, \\mathcal {O})$ that is finite over $\\kappa $ is $\\kappa $ itself.",
"If $X \\ne Â~\\emptyset $ and every morphism $X \\rightarrow \\Gamma $ to a finite stack $\\Gamma $ factors through an affine fpqc gerbe, then $X$ is inflexible.",
"In particular, an affine fpqc gerbe is inflexible.",
"(REF ): Let $A$ be a finite $\\kappa $ -subalgebra of $\\operatorname{H}^{0}(X, \\mathcal {O})$ .",
"Then $\\operatorname{Spec}A$ is a finite stack, so corresponding morphism $X \\rightarrow \\operatorname{Spec}A$ factors trough a closed subgerbe of $\\operatorname{Spec}A$ .",
"But the only scheme over $\\kappa $ that is a gerbe is $\\operatorname{Spec}\\kappa $ ; hence the embedding $A \\subseteq \\operatorname{H}^{0}(X, \\mathcal {O})$ factors through $\\kappa $ , and $A = \\kappa $ .",
"(REF ): Let $f\\colon X \\rightarrow \\Gamma $ be a morphism to a finite stack, and suppose that there is a factorization $X \\xrightarrow{} \\Phi \\xrightarrow{} \\Gamma $ , where $\\Phi $ is an affine fpqc gerbe.",
"The adjunction homomorphism $\\mathcal {O}_{\\Phi } \\rightarrow g_{*}\\mathcal {O}_{X}$ is injective, because $g$ is faithfully flat, by REF (REF ).",
"Hence the kernel of $\\mathcal {O}_{\\Gamma } \\rightarrow f_{*}\\mathcal {O}_{X}$ coincides with the kernel of $\\mathcal {O}_{\\Gamma } \\rightarrow h_{*}\\mathcal {O}_{\\Phi }$ .",
"So $X$ and $\\Phi $ have the same scheme-theoretic image in $\\Gamma $ , and we may assume $X = \\Phi $ .",
"By Proposition REF (REF ), there exists an extension $K$ of $\\kappa $ and an affine faithfully flat morphism $\\operatorname{Spec}K \\rightarrow \\Phi $ ; the scheme-theoretic images of $\\Phi $ and of $\\operatorname{Spec}K$ are evidently the same.",
"Since $\\operatorname{Spec}K$ is reduced and connected, we deduce that the scheme-theoretic image of $\\Phi $ in $\\Gamma $ is reduced and connected.",
"On the other hand, the formation of the scheme-theoretic image commutes with extensions of the base field $\\kappa $ , by Remark REF below, and by extending the base field an affine fpqc gerbe remains an affine fpqc gerbe; hence the scheme theoretic image is geometrically connected and geometrically reduced, and we conclude by Proposition REF .",
"This strange condition of being inflexible has some geometric content.",
"Proposition 5.5 Suppose that $X$ is an algebraic stack of finite type over $\\kappa $ .",
"If $X$ is inflexible, it is geometrically connected.",
"If $X$ is geometrically connected and geometrically reduced, it is inflexible.",
"(REF ): Suppose that $X$ is inflexible, but not geometrically connected; then there exists a finite separable extension $\\kappa ^{\\prime }$ of $\\kappa $ and a surjective morphism of $\\kappa ^{\\prime }$ -schemes $X_{\\kappa ^{\\prime }} \\rightarrow \\operatorname{Spec}\\kappa ^{\\prime } \\sqcup \\operatorname{Spec}\\kappa ^{\\prime }$ .",
"Let $\\Gamma $ be the $\\kappa $ -scheme obtained by Weil restriction from $\\operatorname{Spec}\\kappa ^{\\prime } \\sqcup \\operatorname{Spec}\\kappa ^{\\prime }$ ; it is well known, and easy to see, that the Weil restriction of a finite $\\kappa ^{\\prime }$ -scheme along a finite separable field extension is finite.",
"The morphism $X \\rightarrow \\Gamma $ corresponding to the given morphism $X_{\\kappa ^{\\prime }} \\rightarrow \\operatorname{Spec}\\kappa ^{\\prime } \\sqcup \\operatorname{Spec}\\kappa ^{\\prime }$ factors through $\\operatorname{Spec}\\kappa $ ; hence $X_{\\kappa ^{\\prime }} \\rightarrow \\operatorname{Spec}\\kappa ^{\\prime } \\sqcup \\operatorname{Spec}\\kappa ^{\\prime }$ factors through $\\operatorname{Spec}\\kappa ^{\\prime }$ , by Proposition REF (REF ), and this is a contradiction.",
"(REF ): Let $\\Gamma ^{\\prime }$ be the scheme-theoretic image of $f\\colon X\\rightarrow \\Gamma $ , where $\\Gamma $ is a finite stack.",
"The stack $\\Gamma ^{\\prime }$ is geometrically connected, since $X$ is.",
"We claim that $\\Gamma ^{\\prime }$ is geometrically reduced.",
"If $\\operatorname{char}\\kappa = 0$ , then $\\Gamma ^{\\prime }$ is étale.",
"If $\\operatorname{char}\\kappa > 0$ , the canonical morphism $\\mathcal {O}_{\\Gamma ^{\\prime }} \\rightarrow f_{*}\\mathcal {O}_{X}$ is injective, by definition; and this injectivity is preserved under finite field extension.",
"Hence in any case $\\Gamma ^{\\prime }$ is geometrically connected and geometrically reduced, so by Proposition REF it is a gerbe.",
"The following examples seem to show that the geometric content in the condition of being inflexible is somewhat subtle.",
"A geometric characterization of inflexible algebraic stacks eludes us.",
"Examples 5.6 The condition of Proposition REF (REF ) is not sufficient for an algebraic stack, or even a scheme, of finite type to be inflexible.",
"For example, let $X_{0}$ be a geometrically connected and geometrically reduced projective scheme over $\\kappa $ with a non-trivial invertible sheaf $L$ and an isomorphism $L^{\\otimes 2} \\simeq \\mathcal {O}_{X_{0}}$ .",
"Consider the relative spectrum $X$ of the finite sheaf of algebras $\\mathcal {O}_{X_{0}} \\oplus L$ over $X_{0}$ , where the product of two sections of $L$ is always 0.",
"Then $\\operatorname{H}^{0}(X, \\mathcal {O}) = \\kappa $ , but we claim that $X$ is not inflexible.",
"The invertible sheaf $L$ correspond to a $\\mu _{2}$ -torsor $Y_{0} \\rightarrow X_{0}$ ; call $Y$ the relative scheme of the sheaf $\\mathcal {O}_{Y_{0}} \\oplus \\mathcal {O}_{Y_{0}}\\epsilon $ , with $\\epsilon ^{2} = 0$ .",
"There is a free action of $\\mu _{2}$ on $Y$ , extending the given action on $Y_{0} \\subseteq Y$ , changing the sign of $\\epsilon $ .",
"There is a tautological $\\mu _{2}$ -equivariant morphism $Y \\rightarrow \\operatorname{Spec}\\kappa [\\epsilon ]$ , where $\\kappa [\\epsilon ]$ is the ring of dual numbers, and $\\mu _{2}$ acts on $\\kappa [\\epsilon ]$ by changing the sign of $\\epsilon $ .",
"This induces a morphism $X \\rightarrow \\bigl [\\operatorname{Spec}k[\\epsilon ] / \\mu _{2}\\bigr ]$ , which does not factor through a gerbe.",
"On the other hand, there are examples of projective schemes over $\\kappa $ that are inflexible without being reduced.",
"Suppose that $X_{0}$ is a geometrically connected and geometrically reduced positive-dimensional projective scheme over $\\kappa $ , and let $L$ be the dual of an ample invertible sheaf on $X_{0}$ .",
"Consider the relative spectrum $X$ of the finite sheaf of algebras $\\mathcal {O}_{X_{0}} \\oplus L$ over $X_{0}$ , where the product of two sections of $L$ is always 0.",
"Assume also the characteristic of $\\kappa $ is 0 (this is not necessary, but makes the proof somewhat easier).",
"Then we claim that $X$ is inflexible.",
"The scheme $X_{0}$ is a closed subscheme of $X$ with sheaf of ideals $L$ .",
"Take a morphism $f\\colon X \\rightarrow \\Gamma $ to a finite stack.",
"Assume that the homomorphism $\\mathcal {O}_{\\Gamma } \\rightarrow f_{*}\\mathcal {O}_{X}$ is injective (that is, the scheme-theoretic image of $X$ in $\\Gamma $ is $\\Gamma $ itself); we need to show that $\\Gamma $ is a gerbe.",
"Call $\\Gamma _{0}$ the reduced substack of $\\Gamma $ , and $N$ the sheaf of ideals of $\\Gamma _{0}$ in $\\Gamma $ .",
"We have $N^{2} = 0$ , because $N^{2}$ pulls back to $L^{2} = 0$ .",
"The morphism $f$ restricts to a morphism $f_{0}\\colon X_{0} \\rightarrow \\Gamma _{0}$ with scheme-theoretic image $\\Gamma _{0}$ ; hence $\\Gamma _{0}$ is a gerbe, since $X_{0}$ is inflexible; so we need to show that $N = 0$ .",
"The pullback $f^{*}N = f_{0}^{*}N$ maps to $L$ , and it is enough to show that this map is 0.",
"Choose a morphism $\\phi \\colon \\operatorname{Spec}K \\rightarrow \\Gamma _{0}$ , where $K$ is a finite extension of $\\kappa $ ; this map is étale, since we are in characteristic 0.",
"Consider the cartesian diagram ${Y [r]^-{g}[d]^{\\psi } & \\operatorname{Spec}K [d]^{\\phi }\\\\X_{0} [r]^{f_{0}} & \\Gamma _{0}{}{\\,.",
"}\\,.\\hspace{0.0pt}}$ It is enough to show that $\\operatorname{H}^{0}(X_{0}, f^{*}N^{\\vee } \\otimes L) = 0$ ; this follows if we prove that $\\operatorname{H}^{0}\\bigl (Y, \\psi ^{*}(f^{*}N^{\\vee } \\otimes L)\\bigr ) =\\operatorname{H}^{0}(Y, g^{*}\\phi ^{*}N^{\\vee } \\otimes \\psi ^{*}L))$ is 0.",
"But this is clear, because $\\phi ^{*}N^{\\vee }$ is free, the sheaf $\\psi ^{*}L$ is the dual of an ample invertible sheaf on $Y$ , and $Y$ is étale over $X_{0}$ , hence projective and reduced.",
"The following is the main result of this section.",
"Theorem 5.7 A fibered category over $\\kappa $ has a fundamental gerbe if and only if it is inflexible.",
"From this and Proposition REF (REF ) we obtain the following.",
"Corollary 5.8 A geometrically connected and geometrically reduced algebraic stack of finite type over $k$ has a fundamental gerbe.",
"Remark 5.9 Of course one could relax the definition of fundamental gerbe and only require that it be universal for maps from $X$ to finite (or, equivalently, profinite) gerbes.",
"However, still $\\Pi _{X/\\kappa }$ would not exist in general.",
"For example, it is easy to see that $\\operatorname{Spec}\\kappa \\sqcup \\operatorname{Spec}\\kappa $ can not have a universal gerbe in this sense.",
"When $\\operatorname{char}\\kappa = 0$ , one can show that an algebraic stack of finite type has a fundamental gerbe in this weaker sense if and only if it is geometrically connected.",
"The main point is that in characteristic 0 every finite gerbe is étale, so for any finite gerbe $\\Gamma $ the restriction functor $\\operatorname{Hom}(X, \\Gamma ) \\rightarrow \\operatorname{Hom}(X_{\\mathrm {red}}, \\Gamma )$ is an equivalence; hence the fundamental gerbe for $X_{\\mathrm {red}}$ is also a fundamental gerbe for $X$ .",
"However, in positive characteristic the exact conditions for the existence of a fundamental gerbe in this weaker sense are not clear.",
"Let us prove Theorem REF .",
"First, assume that $X$ has a fundamental gerbe $\\Pi _{X/\\kappa }$ , and take a morphism $X \\rightarrow \\Gamma $ to a finite stack.",
"This factors through $\\Pi _{X/\\kappa }$ , and the result follows from REF (REF ).",
"Now assume that $X$ is inflexible.",
"Definition 5.10 Let $\\Gamma $ be a finite gerbe.",
"A morphism of fibered categories $X \\rightarrow \\Gamma $ is Nori-reduced if for any factorization $X \\rightarrow \\Gamma ^{\\prime } \\rightarrow \\Gamma $ , where $\\Gamma ^{\\prime }$ is a finite gerbe and $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ is faithful, then $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ is an isomorphism.",
"Remark 5.11 The notion of Nori-reduced morphism is perhaps clarified by the following fact.",
"Suppose that $G$ is a finite étale group scheme over $\\kappa $ and $X \\rightarrow \\mathcal {B}_{\\kappa }G$ is a morphism, where $X$ is a geometrically connected and geometrically reduced stack of finite type over $\\kappa $ .",
"This morphism corresponds to a $G$ -torsor $Y \\rightarrow X$ .",
"We claim that $X \\rightarrow \\mathcal {B}_{\\kappa }G$ is Nori-reduced if and only if $Y$ is geometrically connected.",
"In fact, a representable map $\\Gamma \\rightarrow \\mathcal {B}_{\\kappa }G$ , where $\\Gamma $ is a finite gerbe, is given by the projection $[U/G] \\rightarrow \\mathcal {B}_{\\kappa }G$ , where $U$ is a finite étale scheme on which $G$ acts transitively.",
"So, $X \\rightarrow \\mathcal {B}_{\\kappa }G$ is not Nori reduced if and only if there exists a finite étale scheme $U$ , different from $\\operatorname{Spec}\\kappa $ , on which $G$ acts transitively, and a $G$ -equivariant morphism $Y \\rightarrow U$ .",
"If this exists, $Y$ cannot by geometrically connected, since $Y \\rightarrow U$ must be surjective.",
"Conversely, if $Y$ is not geometrically connected, you can take as $U$ the spectrum of the algebraic closure of $\\kappa $ in $\\operatorname{H}^0(Y, \\mathcal {O})$ , with the natural action of $G$ .",
"Lemma 5.12 Let $\\Gamma $ be a finite gerbe, $X$ an inflexible fibered category, $X \\rightarrow \\Gamma $ a morphism.",
"Then there exists a factorization $X \\rightarrow \\Delta \\rightarrow \\Gamma $ , where $\\Delta $ is a finite gerbe, $\\Delta \\rightarrow \\Gamma $ is representable, and $X \\rightarrow \\Delta $ is Nori-reduced.",
"Furthermore, $X \\rightarrow \\Delta $ is unique up to equivalence.",
"Take a factorization $X \\rightarrow \\Delta \\rightarrow \\Gamma $ with $\\Delta \\rightarrow \\Gamma $ representable, and suppose that $X \\rightarrow \\Delta $ is not Nori-reduced.",
"By definition there will be a factorization $X \\rightarrow \\Delta ^{\\prime } \\rightarrow \\Delta $ , where $\\Delta ^{\\prime } \\rightarrow \\Delta $ is representable, but not an isomorphism.",
"By Proposition REF , the degree of $\\Delta ^{\\prime }$ is less than the degree of $\\Delta $ .",
"The proof is concluded by induction on the degree of $\\Delta $ .",
"For the second part, suppose that $X \\rightarrow \\Delta \\rightarrow \\Gamma $ and $X \\rightarrow \\Delta ^{\\prime } \\rightarrow \\Gamma $ are two factorizations as in the statement.",
"Then $\\Delta \\times _{\\Gamma }\\Delta ^{\\prime }$ is a finite stack, and its two projections onto $\\Delta $ and $\\Delta ^{\\prime }$ are representable.",
"Consider the morphism $X \\rightarrow \\Delta \\times _{\\Gamma } \\Delta ^{\\prime }$ induced by the two morphism above; since $X$ is inflexible, the scheme-theoretic image $\\Delta ^{\\prime \\prime }$ of $X$ in $\\Delta \\times _{\\Gamma }\\Delta ^{\\prime }$ is a finite gerbe, and the two morphisms $\\Delta ^{\\prime \\prime } \\rightarrow \\Delta $ and $\\Delta ^{\\prime \\prime } \\rightarrow \\Delta ^{\\prime }$ are representable.",
"Since $X \\rightarrow \\Delta $ and $X \\rightarrow \\Delta ^{\\prime }$ are Nori-reduced, $\\Delta ^{\\prime \\prime } \\rightarrow \\Delta $ and $\\Delta ^{\\prime \\prime } \\rightarrow \\Delta ^{\\prime }$ are equivalences, and the result follows.",
"Here is the key lemma.",
"Lemma 5.13 Let $f\\colon X \\rightarrow \\Gamma $ and $g\\colon X \\rightarrow \\Delta $ be morphisms of fibered categories, where $\\Gamma $ and $\\Delta $ are finite gerbes and $f$ is Nori-reduced.",
"Suppose that $u$ , $v\\colon \\Gamma \\rightarrow \\Delta $ are morphism of fibered categories, and $\\alpha \\colon u \\circ f \\simeq g$ and $\\beta \\colon v \\circ f \\simeq g$ are isomorphisms.",
"Then there exists a unique isomorphism $\\gamma \\colon u \\simeq v$ such that $ \\beta \\circ (\\gamma * \\mathrm {id}_{f})=\\alpha $ .",
"This can be expressed by saying that, given two 2-commutative diagrams ${X [r]^{f}[rd]_{g} &\\Gamma [d]^{u}\\\\&\\Delta }\\quad \\text{and}\\quad {X [r]^{f}[rd]_{g} &\\Gamma [d]^{v}\\\\&\\Delta }$ in which $f$ is Nori-reduced, there exists a unique isomorphism $u \\simeq v$ making the diagram $@C+15pt{X [r]^{f}[rd]_{g} &\\Gamma @/_/[d]_{u}@/^/[d]^{v}\\\\&\\Delta }$ 2-commutative.",
"Consider the category $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ fibered in sets over $\\Gamma $ , whose objects over a $\\kappa $ -scheme $T$ are pairs $(\\xi , \\rho )$ , where $\\xi $ is an object of $\\Gamma (T)$ and $\\rho $ is an isomorphism of $u(\\xi )$ with $v(\\xi )$ in $\\Delta (T)$ .",
"This can be written as a fibered product ${@{}[rd]|{\\square }\\Gamma ^{\\prime } [r][d] & \\Gamma [d]^{\\langle u,v \\rangle }\\\\\\Delta [r] & \\Delta \\times \\Delta \\,,}$ where the morphism $\\Delta \\rightarrow \\Delta \\times \\Delta $ is the diagonal.",
"So $\\Gamma ^{\\prime }$ is a fibered product of finite stacks, hence it is a finite stack.",
"An isomorphism $u \\simeq v$ corresponds to a section of the projection $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ , or, again, to a substack $\\Gamma ^{\\prime \\prime } \\subseteq \\Gamma ^{\\prime }$ such that the restriction $\\Gamma ^{\\prime \\prime } \\rightarrow \\Gamma $ of the projection is an isomorphism.",
"The composite isomorphism $u \\circ f \\xrightarrow{} g \\xrightarrow{} v \\circ f$ yields a lifting $X \\rightarrow \\Gamma ^{\\prime }$ of $f\\colon X \\rightarrow \\Gamma $ ; the thesis can be translated into the condition that there exists a unique substack $\\Gamma ^{\\prime \\prime } \\subseteq \\Gamma ^{\\prime }$ as above, such that $X \\rightarrow \\Gamma ^{\\prime }$ factors through $\\Gamma ^{\\prime \\prime }$ .",
"Since $X$ is inflexible, there is a unique substack $\\Gamma ^{\\prime \\prime }$ of $\\Gamma ^{\\prime }$ that is a gerbe, such that $X \\rightarrow \\Gamma ^{\\prime }$ factors through $\\Gamma ^{\\prime \\prime }$ .",
"However, $\\Gamma ^{\\prime \\prime } \\rightarrow \\Gamma $ is representable, because $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ is, so $\\Gamma ^{\\prime \\prime } \\rightarrow \\Gamma $ is an isomorphism, since $f$ is Nori-reduced.",
"Consider the 2-category whose objects are Nori-reduced morphisms $X \\rightarrow \\Gamma $ , and whose 1-arrows from $f\\colon X \\rightarrow \\Gamma $ to $g\\colon X \\rightarrow \\Delta $ are pairs $(u, \\alpha )$ , where $u\\colon \\Gamma \\rightarrow \\Delta $ is a morphism of finite stacks and $\\alpha \\colon u \\circ f \\simeq g$ is an isomorphism A 2-arrow from $(u, \\alpha )$ to $(v, \\beta )$ is an isomorphism $\\gamma \\colon u \\simeq v$ such that $\\beta \\circ (\\gamma * \\mathrm {id}_{f}) =\\alpha $ .",
"The composites and the Godement products are defined in the obvious way.",
"Let $I$ be a skeleton of this category; it is a small 2-category.",
"We claim that $I$ is a cofiltered 2-category.",
"The fact that given any 1-arrows between two fixed objects there is a unique arrow between them is the content of Lemma REF .",
"Let us check that given two objects $X \\rightarrow \\Gamma _{i}$ and $X \\rightarrow \\Gamma _{j}$ , there is an object $X \\rightarrow \\Gamma _{k}$ with an arrow to both.",
"Since $X$ is inflexible, the morphism $X \\rightarrow \\Gamma _{i} \\times \\Gamma _{j}$ induced by two objects can be factored through a finite gerbe $\\Gamma ^{\\prime } \\subseteq \\Gamma _{i} \\times \\Gamma _{j}$ ; from Lemma REF we see that $X \\rightarrow \\Gamma ^{\\prime }$ can be lifted to a morphism $X \\rightarrow \\Gamma _{k}$ , which is an object in $I$ .",
"We set $\\Pi _{X/\\kappa } \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim \\Gamma $ .",
"The morphisms $X \\rightarrow \\Gamma _{i}$ induce a morphism of fibered categories $X \\rightarrow \\Pi _{X/\\kappa }$ .",
"If $\\Delta $ is a finite stack over $\\kappa $ , we need to show that the functor $\\operatorname{Hom}(\\Pi _{X/\\kappa }, \\Delta ) \\rightarrow \\operatorname{Hom}(X, \\Delta )$ is an equivalence.",
"It follows from Lemma REF that it is essentially surjective, so we need to show that it is fully faithful.",
"By Proposition REF we see that it is enough to show that given a Nori-reduced morphism $X \\rightarrow \\Gamma _{i}$ , the induced functor $\\operatorname{Hom}(\\Gamma _{i}, \\Delta ) \\rightarrow \\operatorname{Hom}(X, \\Delta )$ is fully faithful.",
"This follows immediately from Lemma REF .",
"Remark 5.14 It was pointed out to us by Bertrand Töen that an alternate proof of Theorem REF can be given along the following lines.",
"Consider the embedding of the 2-category of finite stacks into the category of all algebraic stacks.",
"This embedding preserves 2-limits, hence, it extends to a functor from the 2-category of pro-objects in the category of all algebraic stacks.",
"By a 2-categorical analogue of the adjoint functor theorem, this has a right adjoint, which associates with each algebraic stack a universal pro-object in the 2-category of finite stacks.",
"From the definition of inflexible stack, the universal pro-object of an inflexible stack is a pro-object in the category of finite gerbes.",
"By the results of Section , this can be thought of a profinite gerbe.",
"This clarifies considerably the meaning of Theorem REF .",
"Unfortunately we don't know a reference for the 2-categorical result used above.",
"Furthermore, we find the direct construction, via Nori-reduced morphisms, both useful and enlightening.",
"Remark 5.15 Given an affine group scheme $G$ over $\\kappa $ , we obtain an fpqc gerbe $\\mathcal {B}_{\\kappa }G$ , together with a preferred object $G \\rightarrow \\operatorname{Spec}\\kappa $ in $\\mathcal {B}_{\\kappa }G(\\kappa )$ , the trivial torsor.",
"Conversely, given an affine fpqc gerbe $\\Phi $ and an object $\\xi $ of $\\Phi (\\kappa )$ , we obtain an affine group scheme $\\mathop {\\underline{\\mathrm {Aut}}}\\nolimits _{\\kappa }\\xi $ , with a canonical equivalence $\\Phi \\simeq \\mathcal {B}_{\\kappa }\\mathop {\\underline{\\mathrm {Aut}}}\\nolimits _{\\kappa }\\xi $ .",
"Consider the 2-category whose objects are pairs $(\\Phi , \\xi )$ , where $\\Phi $ is an fpqc gerbe over $\\kappa $ and $\\xi $ is an object of $\\Phi (\\kappa )$ .",
"The 1-arrows $(\\Phi , \\xi ) \\rightarrow (\\Psi , \\eta )$ are pairs $(F, a)$ , where $F\\colon \\Phi \\rightarrow \\Psi $ is a cartesian functor, and $a\\colon F(\\xi ) \\simeq \\eta $ is an isomorphism in $\\Psi (\\kappa )$ .",
"The 2-arrows $(F, a) \\rightarrow (F^{\\prime }, a^{\\prime })$ are the base-preserving natural transformations $F \\rightarrow F^{\\prime }$ , which are compatible with $a$ and $a^{\\prime }$ , in the obvious sense.",
"It is easy to see that this 2-category is equivalent to the 1-category of affine group schemes; an affine group scheme $G$ corresponds to the pair $(\\mathcal {B}_{\\kappa }G, G \\rightarrow \\operatorname{Spec}\\kappa )$ , while an object $(\\Phi , \\xi )$ is carried to $\\mathop {\\underline{\\mathrm {Aut}}}\\nolimits _{\\kappa }\\xi $ .",
"In this correspondence profinite gerbes correspond to profinite group schemes.",
"Now, assume that $X$ is inflexible, and that we are given an object $x_{0}$ of $X(\\kappa )$ , corresponding to a section $x_{0}\\colon \\operatorname{Spec}\\kappa \\rightarrow X$ .",
"The image $\\xi _{0}$ of $x_{0}$ in $\\Pi _{X/\\kappa }(\\kappa )$ gives a profinite group $\\pi (X, x_{0}) \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\mathop {\\underline{\\mathrm {Aut}}}\\nolimits _{\\kappa }\\xi _{0}$ ; we claim that this is the fundamental group scheme in the sense of Nori.",
"This means the following.",
"Denote by $P_{0} \\rightarrow X$ the $\\pi (X, x_{0})$ -torsor corresponding to the morphism $X \\rightarrow \\Pi _{X/\\kappa } \\simeq \\mathcal {B}_{\\kappa }\\pi (X, x_{0})$ ; by construction we have a trivialization $x_{0}^{*}P_{0} \\simeq \\pi (X, x_{0})$ .",
"Suppose that we are given a finite group scheme $G$ with a homomorphism $\\pi (X, x_{0}) \\rightarrow G$ ; by transport of structure we obtain a $G$ -torsor $P \\rightarrow X$ with a trivialization $x_{0}^{*}P \\simeq G$ .",
"Conversely, suppose we are given a finite group scheme $G$ , a $G$ -torsor $P \\rightarrow X$ , and a trivialization $x_{0}^{*}P \\simeq G$ .",
"This gives a factorization of the morphism $\\operatorname{Spec}\\kappa \\rightarrow \\mathcal {B}_{\\kappa }G$ corresponding to the trivial torsor as $\\operatorname{Spec}\\kappa \\rightarrow X \\rightarrow \\mathcal {B}_{\\kappa }G$ ; by definition of $\\Pi _{X/\\kappa }$ , we obtain a morphism $\\Pi _{X/\\kappa } \\rightarrow \\mathcal {B}_{\\kappa }G$ , together with an isomorphism of the image of the object in $\\Pi _{X/\\kappa }(\\kappa )$ corresponding to the composite $\\operatorname{Spec}\\kappa \\xrightarrow{} X \\rightarrow \\Pi _{X/\\kappa }$ with the trivial torsor in $\\mathcal {B}_{\\kappa }G(\\kappa )$ .",
"By the discussion above, this yields a homomorphism of group schemes $\\pi (X, x_{0}) \\rightarrow G$ .",
"This gives a bijective correspondence between isomorphism classes of $G$ -torsors $P \\rightarrow X$ with a trivialization $x_{0}^{*}P \\simeq G$ , and homomorphism of group schemes $\\pi (X, x_{0}) \\rightarrow G$ .",
"Thus, in the case of stacks with a given rational point, the Nori fundamental gerbe corresponds to the Nori fundamental group."
],
[
"Base change for the Nori fundamental gerbe",
"Nori showed in [17] that the formation of the fundamental group scheme satisfies base change for algebraic separable extensions.",
"Here we prove the analogous result for fundamental gerbes under finite separable extensions, which is very useful for applications and calculations (see Section ).",
"Nori used the action of the Galois group on the fundamental group scheme; this can be made to work in our case too, using the theory of group actions on stacks (see [23]); however, this is technically rather involved, so we prefer a different method, based on the Weil restriction of algebraic stacks.",
"Proposition 6.1 Let $\\kappa ^{\\prime }/\\kappa $ be a separable extension, $X$ an inflexible fibered category over $(\\mathrm {Aff}/\\kappa )$ .",
"Suppose that either $\\kappa ^{\\prime }$ is finite over $\\kappa $ , or there exists a quasi-compact scheme $U$ and a morphism $U \\rightarrow X$ which is representable, faithfully flat, quasi-compact and quasi-separated.",
"Then $X_{\\kappa ^{\\prime }}$ is inflexible over $\\kappa ^{\\prime }$ , and $\\Pi _{X_{\\kappa ^{\\prime }}/\\kappa ^{\\prime }} = \\operatorname{Spec}\\kappa ^{\\prime } \\times \\Pi _{X/\\kappa }$ .",
"Let us describe the essential features of the Weil restriction of stacks.",
"If $A$ is a finite $\\kappa $ -algebra and $Y \\rightarrow (\\mathrm {Aff}/A)$ is a fibered category, we define the Weil restriction $\\mathrm {R}_{A/\\kappa }Y$ , as usual, as the fibered product $(\\mathrm {Aff}/\\kappa )\\times _{(\\mathrm {Aff}/A)} Y$ , where the functor $(\\mathrm {Aff}/\\kappa )\\rightarrow (\\mathrm {Aff}/A)$ is defined by $S \\mapsto S_{A} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\operatorname{Spec}A\\times _{\\operatorname{Spec}\\kappa }S$ .",
"As a pseudo-functor, $\\mathrm {R}_{A/\\kappa }Y$ is defined by $S \\rightarrow Y(S_{A})$ .",
"When $Y$ is represented by a scheme, then $\\mathrm {R}_{A/\\kappa }Y$ is represented by its Weil restriction, which is a scheme when $A$ is finite over $\\kappa $ .",
"Furthermore, it is immediate to check that if $Y$ is a stack, say in the étale topology, so is $\\mathrm {R}_{A/\\kappa }Y$ .",
"This construction is clearly functorial, that is, every morphism $f\\colon Y \\rightarrow Y^{\\prime }$ of fibered categories over $(\\mathrm {Aff}/A)$ induces a morphism $\\mathrm {R}_{A/\\kappa }f\\colon \\mathrm {R}_{A/\\kappa }Y \\rightarrow \\mathrm {R}_{A/\\kappa }Y^{\\prime }$ .",
"When $X \\rightarrow (\\mathrm {Aff}/\\kappa )$ is fibered category, we have a natural equivalence of categories of base-preserving functors $\\operatorname{Hom}_{\\kappa }(X, \\mathrm {R}_{A/\\kappa }Y) \\simeq \\operatorname{Hom}_{A}(X_{A}, Y)\\,,$ where $X_{A}$ is the fibered product $(\\mathrm {Aff}/A) \\times _{(\\mathrm {Aff}/\\kappa )}X$ (i.e., the Weil restriction is left adjoint to the pullback functor of fibered categories, in the 2-categorical sense).",
"Because of this, one sees immediately that Weil restriction preserves fibered products, that is, if $Y^{\\prime } \\rightarrow Y$ and $Y^{\\prime \\prime } \\rightarrow Y$ are morphisms of fibered categories over $(\\mathrm {Aff}/A)$ , the natural morphism $\\mathrm {R}_{A/\\kappa }(Y^{\\prime } \\times _{Y}Y^{\\prime \\prime }) \\rightarrow \\mathrm {R}_{A/\\kappa }Y^{\\prime } \\times _{\\mathrm {R}_{A/\\kappa }Y} \\mathrm {R}_{A/\\kappa }Y^{\\prime \\prime }$ is an equivalence.",
"If $\\ell $ is an extension of $\\kappa $ , and $A_{\\ell } \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\ell \\otimes _{\\kappa }A$ , it is easy to see that $(\\mathrm {R}_{A/\\kappa }Y)_{\\ell } \\simeq \\mathrm {R}_{A_{\\ell }/\\ell }(Y_{A_{\\ell }})$ .",
"Lemma 6.2 Suppose that $\\kappa ^{\\prime }$ is a finite separable extension of $\\kappa $ , and $\\Gamma $ is a finite stack over $\\kappa ^{\\prime }$ .",
"Then $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma $ is finite stack over $\\kappa $ .",
"Call $n$ the degree of $\\kappa ^{\\prime }$ over $\\kappa $ .",
"Let $\\kappa ^{\\mathrm {sep}}$ be the separable closure of $\\kappa $ , and call $\\nu _{1}$ , ..., $\\nu _{n}$ the embeddings of $\\kappa ^{\\prime }$ into $\\kappa ^{\\mathrm {sep}}$ .",
"Set $A \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\kappa ^{\\mathrm {sep}}\\otimes _{\\kappa }\\kappa ^{\\prime }$ ; we have an isomorphism of $\\kappa $ -algebras $A \\simeq (\\kappa ^{\\mathrm {sep}})^{n}$ , such that for each $i \\in I$ the composite of the embedding $\\kappa ^{\\prime } \\subseteq A$ with the $i^\\text{th}$ projection is $\\nu _{i}$ .",
"For each stack $Y$ over $\\kappa ^{\\prime }$ , denote by $Y_{i}$ the fibered product $\\operatorname{Spec}\\kappa ^{\\mathrm {sep}}\\times _{\\operatorname{Spec}\\kappa ^{\\prime }} Y$ , where the morphism $\\operatorname{Spec}\\kappa ^{\\mathrm {sep}}\\rightarrow \\operatorname{Spec}\\kappa ^{\\prime }$ is induced by $\\nu _{i}\\colon \\kappa ^{\\prime } \\rightarrow \\kappa ^{\\mathrm {sep}}$ .",
"If $Y$ is a stack over $\\kappa ^{\\prime }$ , then we have $Y_{A} = \\coprod _{i=1}^{n}Y_{i}$ ; hence it is easy to see that $(\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }Y)_{\\kappa ^{\\mathrm {sep}}} = \\mathrm {R}_{A/\\kappa ^{\\mathrm {sep}}}\\biggl (\\,\\coprod _{i=1}^{n}Y_{i}\\biggr )= \\prod _{i=1}^{n}Y_{i}\\,;$ hence if $Y$ is a finite scheme over $\\kappa $ , then $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }Y$ is a finite scheme over $\\kappa $ .",
"In the general case, take an fppf presentation $R \\mathbin {\\rightrightarrows }U$ of $\\Gamma $ , where $U$ and $R$ are finite schemes over $\\kappa ^{\\prime }$ .",
"Then $\\prod _{i}R_{i} \\mathbin {\\rightrightarrows }\\prod _{i}U_{i}$ is an fppf presentation of $\\prod _{i}\\Gamma _{i}$ , hence $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }R \\mathbin {\\rightrightarrows }\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }U$ is an fppf presentation of $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma $ .",
"Since $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }R$ and $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }U$ are finite schemes, the proof is complete.",
"Let us show that $(\\Pi _{X/\\kappa })_{\\kappa ^{\\prime }}$ is a universal gerbe for $X_{\\kappa ^{\\prime }}$ .",
"For this, first assume that $\\kappa ^{\\prime }$ is finite over $\\kappa $ .",
"Let $\\Gamma $ be a finite stack over $\\kappa ^{\\prime }$ , and consider the functor $\\operatorname{Hom}_{\\kappa ^{\\prime }}\\bigl ((\\Pi _{X/\\kappa })_{\\kappa ^{\\prime }}, \\Gamma \\bigr ) \\longrightarrow \\operatorname{Hom}_{\\kappa ^{\\prime }}\\bigl (X_{\\kappa ^{\\prime }}, \\Gamma \\bigr )$ induced by $X_{\\kappa ^{\\prime }} \\rightarrow (\\Pi _{X/\\kappa })_{\\kappa ^{\\prime }}$ .",
"Under the adjunction between Weil restrictions and pullbacks, we have equivalences $\\operatorname{Hom}_{\\kappa ^{\\prime }}\\bigl ((\\Pi _{X/\\kappa })_{\\kappa ^{\\prime }}, \\Gamma \\bigr ) \\simeq \\operatorname{Hom}_{\\kappa }\\bigl (\\Pi _{X/\\kappa }, \\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma \\bigr )$ and $\\operatorname{Hom}_{\\kappa ^{\\prime }}\\bigl (X_{\\kappa ^{\\prime }}, \\Gamma \\bigr ) \\simeq \\operatorname{Hom}_{\\kappa }\\bigl (X, \\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma \\bigr )\\,.$ However, by the Lemma above $\\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma $ is a finite stack, so the natural functor $\\operatorname{Hom}_{\\kappa }\\bigl (\\Pi _{X/\\kappa }, \\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma \\bigr ) \\longrightarrow \\operatorname{Hom}_{\\kappa }\\bigl (X, \\mathrm {R}_{\\kappa ^{\\prime }/\\kappa }\\Gamma \\bigr )$ is an equivalence.",
"This completes the proof in the case that $\\kappa ^{\\prime }$ is finite over $\\kappa $ .",
"Assume that we are under the hypothesis (b): choose a morphism $U \\rightarrow X$ as in condition (b), and set $R \\overset{\\mathrm {\\scriptscriptstyle def}}{=}U \\times _{X} U$ .",
"Suppose that we are given a finite stack $\\Gamma $ over $\\kappa ^{\\prime }$ ; we need to show that the functor $\\operatorname{Hom}_{\\kappa ^{\\prime }}\\bigl ((\\Pi _{X/\\kappa })_{\\kappa ^{\\prime }}, \\Gamma \\bigr )\\longrightarrow \\operatorname{Hom}_{\\kappa ^{\\prime }}\\bigl (X_{\\kappa ^{\\prime }}, \\Gamma \\bigr )$ is an equivalence.",
"Let us show that the functor is essentially surjective; for this, choose a morphism $X_{\\kappa ^{\\prime }} \\rightarrow \\Gamma $ .",
"By descent theory, this corresponds to an object $\\xi $ of $\\Gamma (U_{\\kappa ^{\\prime }})$ , with descent data, that is, with an isomorphism of the two pullback of $\\xi $ to $\\Gamma (R_{\\kappa ^{\\prime }})$ , satisfying the cocycle condition.",
"Since the stack $\\Gamma $ is finitely presented, there exists an intermediate extension $\\kappa \\subseteq \\ell \\subseteq \\kappa ^{\\prime }$ finite over $\\kappa $ , and a finite stack $\\Delta $ on $\\ell $ , such that $\\Gamma \\simeq \\Delta _{\\kappa ^{\\prime }}$ .",
"Furthermore, by enlarging $\\ell $ we may assume that there exists an object $\\eta $ of $\\Delta (U_{\\ell })$ with descent data in $\\Delta (R_{\\ell })$ , whose pullback to $\\Gamma (U_{\\kappa ^{\\prime }})$ is isomorphic to $\\xi $ , as an object with descent data.",
"This gives a morphism $X_{\\ell } \\rightarrow \\Delta $ which pulls back to the given morphism $X_{\\kappa ^{\\prime }} \\rightarrow \\Gamma $ .",
"Since $\\ell $ is finite over $\\kappa $ we have that $X_{\\ell } \\rightarrow \\Delta $ factors through $(\\Pi _{X/\\kappa })_{\\ell }$ ; hence $X_{\\kappa ^{\\prime }} \\rightarrow \\Gamma $ factors through $\\bigl ((\\Pi _{X/\\kappa })_{\\ell }\\bigr )_{\\kappa ^{\\prime }} = (\\Pi _{X/\\kappa })_{\\kappa ^{\\prime }}$ , as claimed.",
"The proof that the functor is fully faithful is similar, and left to the reader.",
"Remark 6.3 The base-change result fails for inseparable extensions; see [17], [16] and [21]."
],
[
"The tannakian interpretation of the fundamental gerbe",
"In this section all schemes, and all algebraic spaces, will be quasi-separated.",
"A morphism of fibered categories is called representable when it is represented by algebraic spaces.",
"Definition 7.1 A fibered category $X$ over $\\kappa $ is pseudo-proper if it satisfies the following two conditions.",
"There exists a quasi-compact scheme $U$ and a morphism $U \\rightarrow X$ which is representable, faithfully flat, quasi-compact and quasi-separated.",
"For any locally free sheaf of $\\mathcal {O}_{X}$ -modules $E$ on $X$ , the $\\kappa $ -vector space $\\operatorname{H}^{0}(X, E)$ is finite-dimensional.",
"Notice that in the definition above we don't assume that $X$ is a stack, in any topology.",
"Given a morphism $U \\rightarrow X$ as in part (REF ), we obtain an fpqc groupoid $R \\mathbin {\\rightrightarrows }U$ , where $R \\overset{\\mathrm {\\scriptscriptstyle def}}{=}U \\times _{X} U$ .",
"If $X$ is a stack in the fpqc topology, then it is equivalent to the quotient stack of $(R \\mathbin {\\rightrightarrows }U)$ -torsors.",
"Examples 7.2 A finite algebraic stack is pseudo-proper.",
"An affine fpqc gerbe is pseudo-proper.",
"This is clear for a finite stack.",
"If $X$ is an affine fpqc gerbe, condition (REF ) of Definition REF is obviously verified.",
"To prove condition (REF ), let $E$ be a locally free sheaf on $X$ .",
"Because of Remark REF below we can make a base extension, and assume that $X(\\kappa ) \\ne \\emptyset $ .",
"Then $X = \\mathcal {B}_{\\kappa }G$ , where $G$ is an affine group scheme; but then a locally free sheaf on $X$ is a finite-dimensional representation of $G$ , $\\operatorname{H}^{0}(X, E) = E^{G}$ , and the result is obvious.",
"Remark 7.3 We will use the following fact.",
"Suppose that $X$ is a fibered category, and let $U \\rightarrow X$ be a morphism as in Definition REF (REF ); set $R \\overset{\\mathrm {\\scriptscriptstyle def}}{=}U \\times _{X} U$ .",
"If $E$ is a locally free sheaf on $X$ we denote by $E_{U}$ and $E_{R}$ the restrictions of $E$ to $U$ and $R$ respectively.",
"Then $\\operatorname{H}^{0}(X, E)$ is the equalizer of the pullbacks $\\operatorname{H}^{0}(U, E_{U}) \\mathbin {\\rightrightarrows }\\operatorname{H}^{0}(R, E_{R})$ (this is a straightforward application of descent theory).",
"Furthermore, in the situation above, let $\\kappa ^{\\prime }$ be a field extension.",
"Then the induced morphism $U_{\\kappa ^{\\prime }} \\rightarrow X_{\\kappa ^{\\prime }}$ is also representable, faithfully flat, quasi-compact and quasi-separated, and $R_{\\kappa ^{\\prime }} = U_{\\kappa ^{\\prime }}\\times _{X_{\\kappa ^{\\prime }}} U_{\\kappa ^{\\prime }}$ .",
"Since $U$ is quasi-compact and quasi-separated we have $\\operatorname{H}^{0}(U, E_{U}) \\otimes _{\\kappa }\\kappa ^{\\prime } = \\operatorname{H}^{0}(U_{\\kappa ^{\\prime }}, E_{U_{\\kappa ^{\\prime }}})$ , and analogously for $R$ .",
"Hence we also have $\\operatorname{H}^{0}(X, E) \\otimes _{\\kappa } \\kappa ^{\\prime } = \\operatorname{H}^{0}(X_{\\kappa ^{\\prime }}, E_{\\kappa ^{\\prime }})$ .",
"Lemma 7.4 If $X$ is inflexible and pseudo-proper, then $\\operatorname{H}^{0}(X, \\mathcal {O}) = \\kappa $ .",
"Since $X$ is pseudo-proper, the $\\kappa $ -vector space $\\operatorname{H}^{0}(X, \\mathcal {O})$ is finite-dimensional.",
"The result follows from Proposition REF (REF ).",
"Let $\\mathcal {A}$ be a $\\kappa $ -linear rigid tensor category, with finite-dimensional Hom vector spaces, in which the idempotents split.",
"(For example, if $X$ is a pseudo-proper fibered category we can take $\\mathcal {A}= \\operatorname{Vect}X$ , see [3], Lemma 6).",
"We can define the indecomposable K-theory ring $\\widetilde{\\mathrm {K}}_{0}\\mathcal {A}$ as the Grothendieck group associated with the monoid of isomorphism classes of objects of $\\mathcal {A}$ , with the product given by tensor product.",
"The Krull–Schmidt theorem holds in the category $\\mathcal {A}$ (see [22], §2.2); hence $\\widetilde{\\mathrm {K}}_{0}\\mathcal {A}$ is a free abelian group on the isomorphism classes of indecomposable objects of $\\mathcal {A}$ , and two objects of $\\mathcal {A}$ are isomorphic if and only if their classes in $\\widetilde{\\mathrm {K}}_{0}\\mathcal {A}$ coincide.",
"Notice that if $f \\in \\mathbb {N}[x]$ is a polynomial with natural numbers as coefficient and $E$ is an object of $\\mathcal {A}$ , we can define $f(E)$ , by interpreting the sum as a direct sum, and a power as a tensor power.",
"Definition 7.5 We say that an object $E$ of $\\mathcal {A}$ is finite when one of the following equivalent conditions is satisfied.",
"The class of $E$ in $\\widetilde{\\mathrm {K}}_{0}\\mathcal {A}$ is integral over $\\mathbb {Z}$ .",
"The class of $E$ in $\\widetilde{\\mathrm {K}}_{0}\\mathcal {A}\\otimes _{\\mathbb {Z}} \\mathbb {Q}$ is algebraic over $\\mathbb {Q}$ .",
"There exist $f$ and $g$ in $\\mathbb {N}[x]$ with $f \\ne g$ and $f(E) \\simeq g(E)$ .",
"The set of isomorphism classes of indecomposable components of all the powers of $E$ is finite.",
"The equivalence of these conditions is proved as in [17].",
"Proposition 7.6 ([17]) Finite sums, tensor products and duals of finite objects are finite.",
"If the direct sum of two objects is finite, both objects are finite.",
"Our definition of essentially finite sheaf is more elementary and direct than Nori's, and works more generally.",
"Definition 7.7 An object $E$ of $\\mathcal {A}$ is essentially finite if it is the kernel of a homomorphism between two finite objects.",
"If $X$ is a pseudo-proper fibered category and $E$ is a locally free sheaf on $X$ , then we say that $E$ is finite, or essentially finite, when it has the corresponding property when viewed as an object of $\\operatorname{Vect}X$ .",
"Proposition 7.8 Let $\\Phi $ be a profinite gerbe over $\\kappa $ .",
"Then all representations of $\\Phi $ are essentially finite.",
"Furthermore, if the characteristic of $\\kappa $ is 0 all representations are finite, and the category $\\operatorname{Rep}\\Phi $ is semisimple.",
"Since $\\operatorname{Rep}\\Phi $ is the colimit of categories $\\operatorname{Rep}\\Gamma $ , where $\\Gamma $ is finite, it is enough to show both parts when $\\Phi = \\Gamma $ is finite.",
"Then the first part follows from Lemma REF (REF ).",
"For the second part it is enough to check that the functor $\\operatorname{H}^{0}\\colon \\operatorname{Rep}\\Gamma \\rightarrow \\operatorname{Vect}_{\\kappa }$ is exact.",
"For this we can make a finite extension of the base field, and assume that there exists a section $\\operatorname{Spec}\\kappa \\rightarrow \\Gamma $ .",
"Then $\\Gamma $ is of the type $\\mathcal {B}_{\\kappa }G$ , where $G$ is a finite group scheme on $\\kappa $ , and the result is standard.",
"Theorem 7.9 Suppose that $X$ is inflexible and pseudo-proper over $\\kappa $ .",
"Then the pullback functor $\\operatorname{Rep}\\Pi _{X/\\kappa } \\rightarrow \\operatorname{Vect}X$ gives an equivalence of tensor categories of $\\operatorname{Rep}\\Pi _{X/\\kappa }$ with $\\operatorname{EFin}X$ .",
"Furthermore, if the characteristic of $\\kappa $ is 0 we have $\\operatorname{Fin}X = \\operatorname{EFin}X$ .",
"Here is a purely tannakian consequence of the Theorem.",
"Corollary 7.10 Let $\\mathcal {A}$ be a Tannaka category.",
"The full subcategory of $\\mathcal {A}$ consisting of essentially finite objects is tannakian.",
"Furthermore, if $\\operatorname{char}\\kappa = 0$ , then every essentially finite object of $\\mathcal {A}$ is finite.",
"We don't know a purely tannakian proof of this, that does not use the formalism of affine gerbes.",
"Let $\\Phi $ be an affine fpqc gerbe, such that $\\mathcal {A}$ is equivalent to $\\operatorname{Rep}\\Phi $ .",
"Then $\\Phi $ is inflexible (Proposition REF (REF )) and pseudo-proper (Examples REF ).",
"By Theorem REF , the category of essentially finite objects in $\\mathcal {A}$ is equivalent to the category of representations of the universal gerbe $\\Pi _{\\Phi /\\kappa }$ , and the result follows.",
"For the proof of the Theorem REF , call $F\\colon X \\rightarrow \\Pi _{X/\\kappa }$ the morphism, and consider $F^{*}\\colon \\operatorname{Rep}\\Pi _{X/\\kappa } \\rightarrow \\operatorname{Vect}X$ .",
"The functor $F^{*}$ is an exact functor of tensor categories; as such, it carries finite representations into finite locally free sheaves, and essentially finite representations into essentially finite locally free sheaves.",
"By Proposition REF the image of $F^{*}$ is contained in $\\operatorname{EFin}X$ .",
"Let us check that $F^{*}$ is fully faithful.",
"The category $\\operatorname{Rep}\\Pi _{X/\\kappa }$ is the colimit of the categories $\\operatorname{Rep}\\Gamma $ over the category of Nori-reduced morphism $X \\rightarrow \\Gamma $ ; hence it is enough to show that if $f\\colon X \\rightarrow \\Gamma $ is a Nori-reduced morphism, the pullback functor $f^{*}\\colon \\operatorname{Rep}\\Gamma \\rightarrow \\operatorname{Vect}X$ is fully faithful; this follows from Lemma REF and from the following.",
"Lemma 7.11 Let $f\\colon X \\rightarrow \\Gamma $ be a Nori-reduced morphism.",
"Then $f_{*}\\mathcal {O}_{X} = \\mathcal {O}_{\\Gamma }$ .",
"From Lemma REF below, we have that $f_{*}\\mathcal {O}_{X}$ is a coherent sheaf of $\\mathcal {O}_{\\Gamma }$ -algebras; let $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ be its relative spectrum.",
"Then $\\Gamma ^{\\prime }$ is a finite stack, the morphism $\\Gamma ^{\\prime } \\rightarrow \\Gamma $ is finite and representable, and $f$ factors as $X \\xrightarrow{}\\Gamma ^{\\prime } \\rightarrow \\Gamma $ .",
"Since $X$ is inflexible, there exists a closed subgerbe $\\Gamma ^{\\prime \\prime } \\subseteq \\Gamma ^{\\prime }$ such that $f^{\\prime }$ factors through $\\Gamma ^{\\prime \\prime }$ .",
"If $I \\subseteq \\mathcal {O}_{\\Gamma ^{\\prime }}$ is the sheaf of ideals of $\\Gamma ^{\\prime \\prime }$ in $\\Gamma ^{\\prime }$ , we have that all the elements of $I$ pull back to zero on $X$ .",
"By the definition of $\\Gamma ^{\\prime }$ it follows that $I = 0$ , so $\\Gamma ^{\\prime } = \\Gamma ^{\\prime \\prime }$ is a gerbe.",
"But $f$ is Nori-reduced, so $\\Gamma ^{\\prime } = \\Gamma $ , and $f_{*}\\mathcal {O}_{X} = \\mathcal {O}_{\\Gamma }$ .",
"Next we need to show that every essentially finite locally free sheaf is isomorphic to a pullback from $\\Pi _{X/\\kappa }$ .",
"Since the functor $F^{*}$ is exact and the category $\\operatorname{Rep}\\Pi _{X/\\kappa }$ is abelian, we may assume that $E$ is finite.",
"Let $f$ and $g$ be distinct polynomials in $\\mathbb {N}[x]$ , such that there exists an isomorphism $\\sigma $ of $f(E)$ with $g(E)$ .",
"Let $r$ be the rank of $E$ .",
"Set $V \\overset{\\mathrm {\\scriptscriptstyle def}}{=}k^{r}$ , and denote by $I$ the scheme representing the isomorphisms of $f(V)$ with $g(V)$ .",
"It is isomorphic to $\\mathrm {GL}_{N}$ , where $N \\overset{\\mathrm {\\scriptscriptstyle def}}{=}f(r) = g(r)$ ; in particular, it is affine.",
"There is a natural left action of $\\mathrm {GL}_{r}$ on $I$ ; the isomorphism $\\sigma $ gives a lifting $X \\rightarrow [I/\\mathrm {GL}_{r}]$ of the morphism $X \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ corresponding to the locally free sheaf $E$ .",
"We need to show that the morphism $X \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ factors through a finite gerbe; for this it is sufficient to prove that the scheme-theoretic image of $X$ in $[I/\\mathrm {GL}_{r}]$ is a finite stack.",
"Lemma 7.12 The action of $\\mathrm {GL}_{r}$ on $I$ has finite stabilizers.",
"Let us take this for granted for the time being, and let us complete the proof.",
"By the Lemma, all geometric orbits of $\\mathrm {GL}_{r}$ on $I$ are closed.",
"Since $I$ is affine and $\\mathrm {GL}_{r}$ is geometrically reductive, there exists an affine geometric quotient $I \\rightarrow I/\\mathrm {GL}_{r}$ , whose geometric fibers are, set-theoretically, the geometric orbits of the action of $\\mathrm {GL}_{r}$ on $I$ .",
"The composite $X \\rightarrow [I/\\mathrm {GL}_{r}] \\rightarrow I/\\mathrm {GL}_{r}$ must factor through a rational point $\\operatorname{Spec}\\kappa \\rightarrow I/\\mathrm {GL}_{r}$ , since $\\operatorname{H}^{0}(X, \\mathcal {O}_{X}) = \\kappa $ and $I/\\mathrm {GL}_{r}$ is affine.",
"Call $\\Omega $ the fiber of $I$ over this rational point; the morphism $X \\rightarrow [I/\\mathrm {GL}_{r}]$ factors through $[\\Omega /\\mathrm {GL}_{r}]$ .",
"To conclude it is enough to show that $[\\Omega /\\mathrm {GL}_{r}]$ is a finite stack.",
"But $[\\Omega /\\mathrm {GL}_{r}](\\overline{\\kappa })$ is a connected groupoid, and has quasi-finite diagonal, since the stabilizers are finite, so the result follows from Proposition REF .",
"To prove the Lemma we may extend the base field $\\kappa $ , and assume that it is algebraically closed.",
"Let $\\phi \\colon f(V) \\simeq g(V)$ be an isomorphism, and call $G$ its stabilizer; we need to show that $G$ is finite.",
"Since the Krull–Schmidt property holds, we may assume that $\\deg f \\ne \\deg g$ .",
"Let $H$ be a subgroup of $G$ ; then $V$ has the property that $f(V)$ is isomorphic to $g(V)$ as a representation of $H$ .",
"If $G$ is positive-dimensional, then it must contain either a copy of $\\mathbb {G}_{\\mathrm {a}}$ or of $\\mathbb {G}_{\\mathrm {m}}$ ; hence it is enough to show that if $H = \\mathbb {G}_{\\mathrm {a}}$ or $H = \\mathbb {G}_{\\mathrm {m}}$ and $V$ is a faithful representation of $H$ , then $f(V) \\lnot \\simeq g(V)$ .",
"For $H = \\mathbb {G}_{\\mathrm {m}}$ this is easy; since every representation of $\\mathbb {G}_{\\mathrm {m}}$ is semisimple, two representations of $\\mathbb {G}_{\\mathrm {m}}$ that have the same class in the ring of representations of $\\mathbb {G}_{\\mathrm {m}}$ are isomorphic.",
"The ring of representations of $\\mathbb {G}_{\\mathrm {m}}$ is well known to be isomorphic to $\\mathbb {Z}[t^{\\pm 1}]$ ; since $\\mathbb {Z}$ is algebraically closed in $\\mathbb {Z}[t^{\\pm 1}]$ and the class of $V$ is not in $\\mathbb {Z}$ , because $V$ is not trivial, $f(V) \\lnot \\simeq g(V)$ .",
"Suppose that $H = \\mathbb {G}_{\\mathrm {a}}$ .",
"For any representation $V$ of $\\mathbb {G}_{\\mathrm {a}}$ , define the $\\delta $ -invariant $\\delta (V)$ as follows.",
"Fix a basis of $V$ , giving an isomorphism $\\mathrm {GL}(V) \\simeq \\mathrm {GL}_{n}$ .",
"Write the action as an invertible matrix whose entries are polynomials in $\\kappa [t]$ ; the largest degree of one of these polynomials is $\\delta (V)$ .",
"It is immediate to see that $\\delta (V)$ does not depend on the basis.",
"Furthermore, we have that $\\delta (V \\oplus W) = \\max \\bigl (\\delta (V), \\delta (W)\\bigr )$ , $\\delta (V \\otimes W) = \\delta (V) + \\delta (W)$ , and $\\delta (V) = 0$ if and only if $V$ is trivial.",
"Hence we have $\\delta \\bigl (f(V)\\bigr ) = (\\deg f)\\delta (V)$ and $\\delta \\bigl (g(V)\\bigr ) = (\\deg g)\\delta (V)$ .",
"Since $\\delta (V) > 0$ and $\\deg f \\ne \\deg g$ we have $\\delta \\bigl (f(V)\\bigr ) \\ne \\delta \\bigl (g(V)\\bigr )$ , and $f(V) \\lnot \\simeq g(V)$ .",
"This completes the proof of Theorem REF .",
"It is interesting to observe this theorem has a converse, showing that indeed the concept of inflexible stack is a very natural one.",
"Theorem 7.13 Let $X$ be a pseudo-proper fibered category over $\\operatorname{Spec}\\kappa $ .",
"Then $X$ is inflexible if and only if the tensor category $\\operatorname{EFin}X$ is tannakian.",
"We have already seen that if $X$ is inflexible, then $\\operatorname{EFin}X$ is tannakian.",
"So, suppose that $\\operatorname{EFin}X$ is tannakian.",
"Let $\\Gamma $ be a finite stack with a morphism $X \\rightarrow \\Gamma $ ; we need to show that the stack-theoretic image $\\Gamma ^{\\prime }$ of $X$ in $\\Gamma $ is a gerbe.",
"By Lemma REF , the sheaf $f_{*}\\mathcal {O}_{X}$ is coherent.",
"If $\\Gamma ^{\\prime }$ is its relative spectrum over $\\Gamma $ , the morphism $f\\colon X \\rightarrow \\Gamma $ factors through $\\Gamma ^{\\prime }$ ; by Proposition REF (REF ), it is enough to prove that $\\Gamma ^{\\prime }$ is a gerbe.",
"By replacing $\\Gamma $ with $\\Gamma ^{\\prime }$ , we may assume that $\\mathcal {O}_{\\Gamma } = f_{*}\\mathcal {O}_{X}$ .",
"Lemma 7.14 If the category $\\operatorname{EFin}X$ is tannakian, it is an exact abelian subcategory of the category of sheaves of $\\mathcal {O}_{X}$ -modules.",
"Let $\\Phi $ be an fpqc gerbe with an equivalence of Tannaka categories $\\operatorname{Rep}\\Phi \\simeq \\operatorname{EFin}X$ .",
"This equivalence is realized by a morphism $f\\colon X \\rightarrow \\Phi $ .",
"The pullback from $\\operatorname{Rep}\\Phi $ to sheaves of $\\mathcal {O}_{X}$ -modules is exact, by Proposition REF (REF ), and this proves the Lemma.",
"We have $\\operatorname{H}^{0}(X, \\mathcal {O}) = \\kappa $ , because $\\mathcal {O}_{X}$ is the unit in the tannakian category $\\operatorname{EFin}X$ .",
"Since $\\operatorname{H}^{0}(\\Gamma , \\mathcal {O}) = \\operatorname{H}^{0}(X, \\mathcal {O}) = \\kappa $ , we see that $\\Gamma $ is geometrically connected.",
"Now let us show that $\\Gamma $ is reduced.",
"Let $N \\subseteq \\mathcal {O}_{\\Gamma }$ the sheaf of nilpotent sections.",
"Let $\\rho \\colon U \\rightarrow \\Gamma $ a faithfully flat morphism from a finite connected scheme $U$ , and call $U_{0}$ the inverse image of $\\Gamma _{\\mathrm {red}}$ in $U$ ; this is the subscheme whose sheaf of ideals is $\\rho ^{*}N$ .",
"Choose a surjective homomorphism $\\mathcal {O}_{U}^{n} \\rightarrow \\rho ^{*}N$ ; applying $\\rho _{*}$ , which is exact, because $\\rho $ is representable and finite, and then pulling back along $f$ we obtain an exact sequence $f^{*}\\rho _{*}\\mathcal {O}_{U}^{n} \\longrightarrow f^{*}\\rho _{*}\\mathcal {O}_{U} \\longrightarrow f^{*}\\rho _{*}\\mathcal {O}_{U_{0}} \\longrightarrow 0\\,.$ From Lemmas REF and REF (REF ), we have that $f^{*}\\rho _{*}\\mathcal {O}_{U_{0}}$ is locally free.",
"Restricting to a point of $X$ we see that it can not be 0; hence its annihilator in $\\mathcal {O}_{X}$ must be 0.",
"Since $\\mathcal {O}_{\\Gamma } = f_{*}\\mathcal {O}_{X}$ , the annihilator of $\\rho _{*}\\mathcal {O}_{U_{0}}$ in $\\mathcal {O}_{\\Gamma }$ , which is $N$ , must also be 0.",
"So $\\Gamma $ is reduced, as claimed.",
"If $\\kappa $ is perfect, this is enough to conclude, by Lemma REF .",
"When $\\kappa $ is not perfect we need some additional work to show that $\\Gamma $ is geometrically reduced.",
"From Lemma REF , we have the equality $\\operatorname{EFin}\\Gamma = \\operatorname{Coh}\\Gamma $ .",
"From the equality $\\operatorname{H}^{0}(\\Gamma , \\mathcal {O}) = \\kappa $ and from the fact that $\\operatorname{EFin}\\Gamma $ is an abelian category, we see that $\\operatorname{EFin}\\Gamma $ is tannakian.",
"Let $\\Phi $ be the corresponding fpqc gerbe; the equivalence $\\operatorname{Coh}\\Phi = \\operatorname{Rep}\\Phi \\simeq \\operatorname{EFin}\\Gamma $ is realized as pullback along a morphism $\\phi \\colon \\Gamma \\rightarrow \\Phi $ .",
"Let $\\kappa ^{\\prime }$ be a finite extension of $\\kappa $ .",
"The category of coherent sheaves on $\\Gamma _{\\kappa ^{\\prime }}$ is equivalent to the category of coherent sheaves $F$ on $\\Gamma $ , with a homomorphism of $\\kappa $ -algebras $\\kappa ^{\\prime } \\rightarrow \\operatorname{End}_{\\mathcal {O}_{\\Gamma }}(F)$ , and analogously for $\\Phi $ .",
"Hence pullback along the natural morphism $\\Gamma _{\\kappa ^{\\prime }} \\rightarrow \\Phi _{\\kappa ^{\\prime }}$ yields an equivalence of categories between $\\operatorname{Coh}\\Gamma _{\\kappa ^{\\prime }}$ and $\\operatorname{Coh}\\Phi _{\\kappa ^{\\prime }}$ .",
"Since $\\Phi _\\kappa ^{\\prime }$ is a gerbe we have $\\operatorname{Coh}\\Phi _{\\kappa ^{\\prime }} = \\operatorname{Vect}\\Phi _{\\kappa ^{\\prime }}$ ; it follows that every coherent sheaf on $\\Gamma _{\\kappa ^{\\prime }}$ is locally free.",
"This implies that $\\Gamma _{\\kappa ^{\\prime }}$ is reduced (for otherwise the structure sheaf of $(\\Gamma _{\\kappa ^{\\prime }})_{\\mathrm {red}}$ would not be locally free).",
"This shows that $\\Gamma $ is geometrically reduced, and completes the proof of the Theorem."
],
[
"Some technical lemmas",
"Here we collect some lemmas that were used in the proof of the results above, in order to unclutter the exposition.",
"Lemma 7.15 Let $\\Gamma $ be a finite stack over $\\kappa $ .",
"Let $\\rho \\colon T \\rightarrow \\Gamma $ be a faithfully flat morphism, where $T$ is a connected finite scheme over $\\kappa $ .",
"Then every locally free sheaf on $\\Gamma $ is a subsheaf of $(\\rho _{*}\\mathcal {O}_{T})^{\\oplus r}$ for some $r \\ge 0$ , and $\\rho _{*}\\mathcal {O}_{T}$ is a finite locally free sheaf.",
"Assume moreover that $\\Gamma $ is reduced.",
"Then every coherent sheaf on $\\Gamma $ is locally free, and essentially finite.",
"Assume again that $\\Gamma $ is reduced.",
"Every morphism from an algebraic stack to $\\Gamma $ is flat.",
"For all this, we may assume that $\\Gamma $ is connected.",
"For (REF ), call $d$ the degree of $\\rho $ .",
"Since $F$ is locally free, the sheaf $\\rho ^{*}F$ is free over $T$ (since $T$ is the spectrum of a local artinian ring, every locally free sheaf on $T$ is free); fix an isomorphism $\\rho ^{*}F \\simeq \\mathcal {O}_{T}^{\\oplus r}$ .",
"The adjunction homomorphism $F \\rightarrow \\rho _{*}\\rho ^{*}F \\simeq (\\rho _{*}\\mathcal {O}_{T})^{\\oplus r}$ is injective.",
"This proves the first part of the statement.",
"For the second part, it follows from the projection formula that $\\rho _{*}\\mathcal {O}_{T} \\otimes \\rho _{*}\\mathcal {O}_{T} &\\simeq \\rho _{*}(\\mathcal {O}_{T} \\otimes \\rho ^{*}\\rho _{*}\\mathcal {O}_{T})\\\\& \\simeq \\rho _{*}(\\mathcal {O}_{T} \\otimes \\mathcal {O}_{T}^{\\oplus d})\\\\& \\simeq \\rho _{*}(\\mathcal {O}_{T})^{\\oplus d}\\,.\\\\$ For (REF ), take a smooth surjective morphism $\\pi \\colon U \\rightarrow \\Gamma $ , where $U$ is a scheme.",
"Let $F$ be a coherent sheaf on $\\Gamma $ .",
"Then $\\pi ^{*}F$ is a coherent sheaf on the reduced noetherian scheme $U$ ; such a sheaf is locally free on an open dense subscheme $V \\subseteq U$ .",
"But $\\Gamma $ is finite, so $V$ also surjects onto $\\Gamma $ , hence $F$ is locally free.",
"Now take a faithfully flat morphism $\\rho \\colon T \\rightarrow \\Gamma $ , where $T$ is a connected finite scheme over $\\kappa $ .",
"By embedding the cokernel of $F \\rightarrow \\rho _{*}\\rho ^{*} F$ into a finite representation, we see that $F$ is essentially finite.",
"As to (REF ), notice that it is enough to prove that every morphism from an affine scheme to $\\Gamma $ is flat.",
"The diagonal of $\\Gamma $ is affine, so such a morphism is affine, and it is enough to show that all quasi-coherent sheaves on $\\Gamma $ are flat.",
"By [13] every quasi-coherent sheaf on $\\Gamma $ is a colimit of coherent sheaves, so the statement follows from (REF ).",
"Lemma 7.16 Suppose that $f\\colon X \\rightarrow \\Gamma $ is a morphism of $\\kappa $ -algebraic stacks, where $X$ is pseudo-proper and $\\Gamma $ is finite.",
"Then $f_{*}\\mathcal {O}_{X}$ is a coherent sheaf of $\\mathcal {O}_{\\Gamma }$ -modules.",
"First of all, let us show that $f_{*}\\mathcal {O}_{X}$ is quasi-coherent.",
"Choose a faithfully flat quasi-compact morphism $U \\rightarrow X$ , and set $R \\overset{\\mathrm {\\scriptscriptstyle def}}{=}U \\times _{X} U$ .",
"Call $f_{U}\\colon U \\rightarrow \\Gamma $ and $f_{R}\\colon R \\rightarrow \\Gamma $ the composites of $f$ with the morphisms $U \\rightarrow X$ and $R \\rightarrow X$ .",
"Then it is easily checked that $f_{*}\\mathcal {O}_{X}$ is the equalizer of the two morphisms ${f_{U}}_{*}\\mathcal {O}_{U} \\mathbin {\\rightrightarrows }{f_{R}}_{*}\\mathcal {O}_{R}$ resulting from the projections $R \\mathbin {\\rightrightarrows }U$ .",
"Since $U$ and $R$ are quasi-compact and quasi-separated, ${f_{U}}_{*}\\mathcal {O}_{U}$ and ${f_{R}}_{*}\\mathcal {O}_{R}$ are quasi-coherent, so the result follows.",
"To prove that $f_{*}\\mathcal {O}_{X}$ is coherent, choose a faithfully flat morphism $\\pi \\colon T \\rightarrow \\Gamma $ , where $T$ is a finite $\\kappa $ -scheme, and consider the cartesian diagram ${X^{\\prime } [r] ^-{f^{\\prime }} [d]^{\\rho } &T [d]^\\pi \\\\X [r] ^-{f} &\\Gamma \\,.",
"}$ Clearly, $X^{\\prime }$ is pseudo-proper.",
"Since $\\mathcal {O}_{X}$ is contained in $\\rho _{*}\\mathcal {O}_{X^{\\prime }}$ , because $\\rho $ is faithfully flat, it is enough to show that $f_{*}\\rho _{*}\\mathcal {O}_{X^{\\prime }} = \\pi _{*}f^{\\prime }_{*}\\mathcal {O}_{X^{\\prime }}$ is coherent.",
"But $f^{\\prime }_{*}\\mathcal {O}_{X^{\\prime }}$ is a coherent sheaf of $\\mathcal {O}_{T}$ -algebras, because $\\operatorname{H}^{0}(X^{\\prime }, \\mathcal {O})$ is a finite dimensional vector space over $\\kappa $ .",
"Since $\\pi $ is finite, we have that $\\pi _{*}f^{\\prime }_{*}\\mathcal {O}_{X^{\\prime }}$ is coherent.",
"Lemma 7.17 Suppose that $f\\colon X \\rightarrow Y$ is a morphism of fibered categories.",
"Suppose that the natural homomorphism $\\mathcal {O}_{Y} \\rightarrow f_{*}\\mathcal {O}_{X}$ is an isomorphism.",
"Then the pullback functor $f^{*}\\colon \\operatorname{Vect}Y \\rightarrow \\operatorname{Vect}X$ is fully faithful.",
"This is a standard application of the projection formula."
],
[
"The étale fundamental gerbe",
"Let $X$ be an inflexible algebraic stack, $\\Pi _{X/\\kappa } = \\varprojlim _{i} \\Gamma _{i}$ its fundamental gerbe, expressed as a projective limit along the category $I$ whose objects are Nori-reduced morphisms $X \\rightarrow \\Gamma _{i}$ .",
"Consider the full subcategory $I^{{\\rm \\acute{e}t}}$ of $I$ whose objects consist of Nori-reduced morphisms $X \\rightarrow \\Gamma _{j}$ in which $\\Gamma _{j}$ is étale.",
"In characteristic 0 we have $I^{{\\rm \\acute{e}t}} = I$ .",
"Since the fibered product of two étale gerbes is an étale stack, we have that $I^{{\\rm \\acute{e}t}}$ is a 2-cofiltered category.",
"Definition 8.1 The étale fundamental gerbe of $X$ is the profinite gerbe $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim _{i \\in I^{{\\rm \\acute{e}t}}} \\Gamma _{i}$ .",
"Since $I^{{\\rm \\acute{e}t}}$ is a subcategory of $I$ , there is a natural projection $\\Pi _{X/\\kappa } \\rightarrow \\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ .",
"The natural morphism $X \\rightarrow \\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ is easily seen to be universal among all morphisms to a pro-étale gerbe.",
"The profinite étale gerbe $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ is isomorphic to the gerbe associated with the relative fundamental groupoid introduced by Deligne in [6].",
"First of all, let us recall Deligne's construction.",
"Suppose that $X$ is a geometrically connected algebraic stack over $\\kappa $ .",
"Let $Y \\rightarrow X$ be a connected Galois coverBy cover we mean a representable morphism that is finite and étale..",
"This corresponds to a morphism $X \\rightarrow \\mathcal {B}_{\\kappa }(\\operatorname{Aut}Y)$ , where $\\operatorname{Aut}Y \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\operatorname{Aut}_{X}Y$ is the Galois group, which is an fpqc cover; hence $\\mathcal {B}_{\\kappa }(\\operatorname{Aut}Y)$ is equivalent to the stack of fpqc torsors under the groupoid $X \\times _{\\mathcal {B}_{\\kappa }(\\operatorname{Aut}Y)} X \\mathbin {\\rightrightarrows }X$ .",
"The fibered product $X \\times _{\\mathcal {B}_{\\kappa }(\\operatorname{Aut}Y)} X$ is equivalent to the $X$ -scheme $P_{Y} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\mathop {\\underline{\\mathrm {Isom}}}\\nolimits _{X\\times _S X}^{\\operatorname{Aut}Y}(\\operatorname{pr}_{2}^{*}Y, \\operatorname{pr}_{1}^{*}Y)$ of $\\operatorname{Aut}Y$ -equivariant isomorphisms of the two pullbacks of $Y$ .",
"Let $P_{Y}^{0}$ be the connected component of $P_{Y}$ containing the image of the diagonal $X \\rightarrow X \\times _{\\mathcal {B}_{\\kappa }(\\operatorname{Aut}Y)} X$ ; then $P^{0}_{Y} \\mathbin {\\rightrightarrows }X$ is an fpqc groupoid.",
"Now, let $Z \\rightarrow X$ be another connected Galois étale cover of $X$ , and let $f\\colon Z \\rightarrow Y$ be a morphism of coverings.",
"Then $f$ induces a morphism $P_{Z} \\rightarrow P_{Y}$ , which is easily seen to be independent of $f$ ; this sends $P^{0}_{Z}$ into $P^{0}_{Y}$ , giving a morphism of groupoids from $P^{0}_{Z}\\mathbin {\\rightrightarrows }X$ to $P^{0}_{Y}\\mathbin {\\rightrightarrows }X$ .",
"Let $\\lbrace Y_{j}\\rbrace _{j \\in J}$ be a set of representatives for isomorphism classes of Galois connected covers of $X$ ; we introduce a partial ordering on $J$ , saying that $j \\le k$ if there exists a morphism of coverings $Y_{j} \\rightarrow Y_{k}$ .",
"So $J$ becomes a cofiltered set.",
"Deligne's absolute groupoid is $\\widehat{P}_{X} \\mathbin {\\rightrightarrows }X$ , where $\\widehat{P}_{X} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim _{j \\in J}P_{Y_{j}}$ , and Deligne's relative groupoid is $\\widehat{P}^0_{X} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim _{j \\in J}P^{0}_{Y_{j}}$ .",
"Definition 8.2 Let $X$ be a geometrically connected algebraic stack over $\\kappa $ .",
"We define Deligne's absolute fundamental gerbe $\\Pi _X^\\mathrm {D}$ (respectively Deligne's relative fundamental gerbe $\\Pi _{X/\\kappa }^\\mathrm {D}$ ) as the stack associated with the groupoid $\\widehat{P}_{X} \\mathbin {\\rightrightarrows }X$ (respectively $\\widehat{P}^0_{X} \\mathbin {\\rightrightarrows }X$ ).",
"Notice that, by construction, we are given natural morphisms $X \\rightarrow \\Pi _{X/\\kappa }^\\mathrm {D}\\rightarrow \\Pi _{X}^\\mathrm {D}$ .",
"Since $\\Pi _{X/\\kappa }^\\mathrm {D}$ is an étale profinite gerbe, the composite $X \\rightarrow \\Pi _{X/\\kappa }^\\mathrm {D}$ induces a morphism $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}} \\rightarrow \\Pi _{X/\\kappa }^\\mathrm {D}$ .",
"Theorem 8.3 Assume that $X$ is inflexible.",
"The natural morphism $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}} \\simeq \\Pi _{X/\\kappa }^\\mathrm {D}$ is an isomorphism.",
"We start with a useful lemma.",
"Lemma 8.4 Let $Y \\rightarrow X$ be a connected étale Galois cover and $G \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\operatorname{Aut}_{X} Y$ .",
"The morphism $X \\rightarrow [X/P^{0}_{Y}]$ is Nori-reduced.",
"Suppose that $X \\rightarrow [X/P^{0}_{Y}]=\\mathcal {G}$ factors trough a representable morphism $\\mathcal {G}^{\\prime } \\rightarrow \\mathcal {G}$ , where $\\mathcal {G}^{\\prime }$ is a finite gerbe.",
"Since the diagonal of $\\mathcal {G}$ is étale, so is the diagonal of $\\mathcal {G}^{\\prime }$ , and we deduce that both morphisms $X\\times _{\\mathcal {G}} X \\rightarrow X\\times _S X$ and $X\\times _{\\mathcal {G}^{\\prime }} X \\rightarrow X\\times _S X$ are also étale.",
"Hence the natural morphism $X\\times _{\\mathcal {G}^{\\prime }} X \\rightarrow X\\times _{\\mathcal {G}} X$ is étale.",
"Since $\\mathcal {G}^{\\prime } \\rightarrow \\mathcal {G}$ is representable, this morphism is an immersion, since $X\\times _{\\mathcal {G}} X=P_Y^0$ is connected, this must be an isomorphism.",
"Hence with the notations of the lemma, the factorization $X \\rightarrow [X/P^{0}_{Y}]$ of $X \\rightarrow \\mathcal {B}_{\\kappa }G$ is the canonical factorization of Lemma REF .",
"This defines a lax 2-functor $J \\rightarrow I^{{\\rm \\acute{e}t}}$ , which induces a homomorphism of preordered sets $\\lambda \\colon J \\rightarrow \\overline{I^{{\\rm \\acute{e}t}}}$ , where $\\overline{I^{{\\rm \\acute{e}t}}}$ is the preordered set associated with $I^{{\\rm \\acute{e}t}}$ , as in Remark REF , such that for each $j \\in J$ we have a canonical isomorphism $P^{0}_{Y_{j}} \\simeq X \\times _{\\Gamma _{\\lambda (j)}} X$ .",
"We need to show that the induced morphism $\\varprojlim P^{0}_{Y_{j}} = \\varprojlim (X \\times _{\\Gamma _{\\lambda (j)}} X)$ into $X \\times _{\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}}X$ , which, by Remark REF coincides with $\\varprojlim _{\\overline{I^{{\\rm \\acute{e}t}}}}(X \\times _{\\Gamma _{i}} X)$ , is an isomorphism.",
"In fact, let us show that the image of $J$ is cofinal in $\\overline{I^{{\\rm \\acute{e}t}}}$ ; in other words, we need to show that given a Nori-reduced morphism $X \\rightarrow \\Gamma $ , where $\\Gamma $ is a finite étale gerbe, there exists a finite group $G$ and a representable morphism $\\Gamma \\rightarrow \\mathcal {B}_{\\kappa }G$ , such that the composite $X \\rightarrow \\mathcal {B}_{\\kappa }G$ corresponds to a connected $G$ -cover of $X$ .",
"Since $\\Gamma $ is finite and étale, there is a finite separable extension $\\kappa ^{\\prime }$ of $\\kappa $ and a morphism $\\operatorname{Spec}\\kappa ^{\\prime } \\rightarrow \\Gamma $ , which is an étale covering; call $n$ its degree.",
"The covering above corresponds to a representable morphism $\\Gamma \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {S}_{n}$ (where $\\mathrm {S}_{n}$ is the symmetric group on $n$ letters).",
"Consider the $\\mathrm {S}_{n}$ -torsor $P \\rightarrow X$ corresponding to the composite $X \\rightarrow \\Gamma \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {S}_{n}$ , and take a connected component $Y \\subseteq P$ ; this is a connected $G$ -torsor for a subgroup $G \\subseteq \\mathrm {S}_{n}$ .",
"There is a commutative diagram ${&&\\mathcal {B}_{\\kappa }G[d]\\\\X[r][urr]& \\Gamma [r] @{-->}[ur] & \\mathcal {B}_{\\kappa }\\mathrm {S}_{n}\\,;}$ we need to show that we can insert a dashed arrow into it.",
"Consider the morphism $X \\rightarrow \\Gamma \\times _{\\mathcal {B}_{\\kappa }\\mathrm {S}_{n}}\\mathcal {B}_{\\kappa }G$ induced by the diagram above: since $X$ is inflexible, the scheme-theoretic image $\\Delta $ of $X$ is a finite gerbe.",
"The projection $\\Delta \\rightarrow \\Gamma $ is representable, since $\\mathcal {B}_{\\kappa }G$ is representable over $\\mathcal {B}_{\\kappa }\\mathrm {S}_{n}$ ; since $X \\rightarrow \\Gamma $ is Nori-reduced, the morphism $\\Delta \\rightarrow \\Gamma $ must be an isomorphism.",
"The inverse $\\Gamma \\rightarrow \\Delta $ , followed by the embedding $\\Delta \\subseteq \\Gamma \\times _{\\mathcal {B}_{\\kappa }\\mathrm {S}_{n}}\\mathcal {B}_{\\kappa }G$ and the projection $\\Gamma \\times _{\\mathcal {B}_{\\kappa }\\mathrm {S}_{n}}\\mathcal {B}_{\\kappa }G \\rightarrow \\mathcal {B}_{\\kappa }G$ gives the required morphism."
],
[
"The fundamental gerbe and the section conjecture",
"As an application of the formalism in Section , let us show that the rational points of the gerbe $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ have a natural interpretation in terms of sections of Grothendieck's fundamental exact sequence.",
"Let us first recall what this means.",
"Theorem 9.1 ([10] IX, Théorème 6.1) Let $X/\\kappa $ be a quasi-compact, quasi-separated and geometrically connected algebraic stack, and fix a geometric point $\\overline{x}:\\operatorname{Spec}\\Omega \\rightarrow X$ .",
"Then the following sequence is exact: $1 \\rightarrow \\pi _1(X_{\\overline{\\kappa }},\\overline{x})\\rightarrow \\pi _1(X,\\overline{x}) \\rightarrow G_\\kappa \\rightarrow 1$ where $G_\\kappa $ denotes the absolute Galois group of $\\kappa $ relative to the separable closure of $k$ in $\\Omega $ .",
"By functoriality, a rational point of $X$ induces a section of this exact sequence, well defined up to conjugacy by an element of $\\pi _1(X_{\\overline{\\kappa }},\\overline{x})$ .",
"Denoting by $\\operatorname{IsCl}$ the set of isomorphism classes of a small category, and by $\\operatorname{{Hom-ext}}_{G_\\kappa }(G_\\kappa ,\\pi _1(X,\\overline{x}))$ this set of equivalence classes of sections, we thus obtain an application: $s_X: \\operatorname{IsCl}(X(\\kappa )) \\rightarrow \\operatorname{{Hom-ext}}_{G_\\kappa }(G_\\kappa ,\\pi _1(X,\\overline{x})) $ Grothendieck's famous section conjecture is stated as follows: Conjecture 9.2 ([11]) If $X$ is a proper, smooth and geometrically connected curve of genus at least 2 over a finitely generated extension $\\kappa $ of $\\mathbb {Q}$ , the application $s_X$ is one to one.",
"As we will explain now, this conjecture can be stated in our terms by saying that the natural morphism $X\\rightarrow \\Pi _{X/\\kappa }^\\mathrm {D}$ induces a bijection of isomorphism classes of $\\kappa $ -rational points.",
"Rather than working with isomorphism classes, it is natural to consider the category $X(\\kappa )$ and the category $\\operatorname{\\mathbf {Hom-ext}}_{G_\\kappa }(G_\\kappa ,\\pi _1(X,\\overline{x}))$ whose objects are sections, and morphisms are given by conjugacy by elements of $\\pi _1(X_{\\overline{\\kappa }},\\overline{x})$ , in the natural way.",
"Proposition 9.3 Let $X/\\kappa $ be a quasi-compact, quasi-separated and inflexible algebraic stack, and fix a geometric point $\\overline{x}:\\operatorname{Spec}\\Omega \\rightarrow X$ .",
"There is a (non canonical) equivalence of categories $\\Pi _{X/\\kappa }^\\mathrm {D}(\\kappa ) \\rightarrow \\operatorname{\\mathbf {Hom-ext}}_{G_\\kappa }(G_\\kappa ,\\pi _1(X,\\overline{x}))$ that composed with the canonical functor $X(\\kappa )\\rightarrow \\Pi _{X/\\kappa }^\\mathrm {D}(\\kappa )$ is a lifting of $s_X$ at the level of categories.",
"This will follow from the next lemma.",
"Lemma 9.4 The natural diagram: ${\\Pi _{X/\\kappa }^\\mathrm {D}[d] [r] & \\Pi _X^\\mathrm {D}[d] \\\\\\operatorname{Spec}\\kappa [r] & \\Pi _{\\operatorname{Spec}\\kappa }^\\mathrm {D}}$ is cartesian.",
"Indeed, let us admit this for a while.",
"If we fix a geometric point $\\overline{x}:\\operatorname{Spec}\\Omega \\rightarrow X$ , then by definition of Deligne absolute fundamental gerbe, there is a canonical isomorphism $\\operatorname{Aut}_{\\Pi _X^\\mathrm {D}}(\\overline{x})\\simeq \\pi _1(X,\\overline{x})$ , and it follows that there is a compatible isomorphism $\\operatorname{Aut}_{\\Pi _{X/\\kappa }^\\mathrm {D}}(\\overline{x})\\simeq \\pi _1(X_{\\overline{\\kappa }},\\overline{x})$ .",
"In other words, the fundamental exact sequence can be interpreted as the exact sequence of $\\operatorname{Aut}$ -groups associated with the fibration at $\\overline{x}$ .",
"We note that we can safely replace $X$ by its étale fundamental gerbe: indeed, if $\\overline{x}:\\operatorname{Spec}\\Omega \\rightarrow X$ is a geometric point, it induces a morphism of the corresponding fundamental exact sequences, that is in fact an isomorphism of exact sequences.",
"Since $\\Pi _{X/\\kappa }^\\mathrm {D}$ is a profinite étale gerbe, the morphism $\\Pi _{X/\\kappa }^\\mathrm {D}\\rightarrow \\Pi _{\\Pi _{X/\\kappa }^\\mathrm {D}/\\kappa }^\\mathrm {D}$ is an isomorphism, and the lemma implies then that the morphism $\\Pi _{X}^\\mathrm {D}\\rightarrow \\Pi _{\\Pi _{X}^\\mathrm {D}/\\kappa }^\\mathrm {D}$ is an isomorphism.",
"So let us assume $X=\\mathcal {G}$ is a profinite étale gerbe, and fix a point $x\\in \\mathcal {G}(S)$ .",
"It induces a canonical equivalence $ \\mathcal {G}(S)\\simeq \\operatorname{Tors}_\\kappa \\operatorname{Aut}_\\kappa x$ , the category of torsors over $\\kappa $ under the $\\kappa $ -group $\\operatorname{Aut}_\\kappa x$ .",
"It induces also a canonical section of the exact sequence of $\\operatorname{Aut}$ -groups associated with the fibration at the corresponding geometric point $\\overline{x}$ .",
"This section induces, in turn, as is well known, a canonical equivalence $\\operatorname{\\mathbf {Hom-ext}}_{\\operatorname{Aut}_{\\Pi _S}(\\overline{x})}(\\operatorname{Aut}_{\\Pi _S}(\\overline{x}),\\operatorname{Aut}_{\\Pi _{\\mathcal {G}}}(\\overline{x}))\\simeq \\operatorname{Tors}_{\\operatorname{Aut}_{\\Pi _S}(\\overline{x})-\\mathrm {sets}}(\\operatorname{Aut}_{\\mathcal {G}}(\\overline{x}))$ , the category of torsors under the $\\operatorname{Aut}_{\\Pi _S}(\\overline{x})$ -group $\\operatorname{Aut}_{\\mathcal {G}}(\\overline{x})$ in the category of $\\operatorname{Aut}_{\\Pi _S}(\\overline{x})$ -sets.",
"Hence we get the (non canonical) equivalence we needed.",
"Let $Y\\rightarrow X$ be a connected Galois cover, with Galois group $G$ .",
"Let $\\kappa ^{\\prime }/\\kappa $ be the largest separable extension such that $X_{\\kappa ^{\\prime }}\\rightarrow X$ is a quotient of $Y\\rightarrow X$ , and let $H$ be its Galois group.",
"It is enough to show that the following diagram is cartesian: ${[X/P^{0}_{Y}] [d] [r] & \\mathcal {B}_{\\kappa }G [d] \\\\\\operatorname{Spec}\\kappa [r] & \\mathcal {B}_{\\kappa }H}$ Denote by $\\mathcal {G}\\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\operatorname{Spec}\\kappa \\times _{\\mathcal {B}_{\\kappa }H}\\mathcal {B}_{\\kappa }G$ , this is clearly a gerbe, since $G\\rightarrow H$ is an epimorphism.",
"Because of uniqueness of Nori-reduced factorization (Lemma REF ) and the fact that $X\\rightarrow [X/P^{0}_{Y}]$ is Nori-reduced (Lemma REF ), it is enough to show that the natural morphism $X\\rightarrow \\mathcal {G}$ is Nori-reduced.",
"So assume that $X \\rightarrow \\mathcal {G}$ factors trough a representable morphism $\\mathcal {G}^{\\prime } \\rightarrow \\mathcal {G}$ , where $\\mathcal {G}^{\\prime }$ is a finite gerbe.",
"Let $U\\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\mathcal {G}^{\\prime }\\times _{\\mathcal {B}_{\\kappa }G}\\operatorname{Spec}\\kappa $ , this is a finite $\\kappa $ -scheme, endowed with a transitive action of $G$ , and a $G$ -equivariant map $Y\\rightarrow U$ .",
"Since $Y$ is non-empty, $Y\\rightarrow U$ is an epimorphism.",
"The scheme $U$ is also endowed with a natural morphism $U\\rightarrow \\operatorname{Spec}\\kappa ^{\\prime }=\\mathcal {G}\\times _{\\mathcal {B}_{\\kappa }G}\\operatorname{Spec}\\kappa $ .",
"Since $Y$ is geometrically connected over $\\kappa ^{\\prime }$ by construction, so is $U$ .",
"Moreover the morphism $U\\rightarrow \\mathcal {G}^{\\prime }$ is étale, and so by Proposition REF the scheme $U$ is reduced, hence is étale.",
"We conclude that $U\\rightarrow \\operatorname{Spec}\\kappa ^{\\prime }$ is an isomorphism, and so is $\\mathcal {G}^{\\prime } \\rightarrow \\mathcal {G}$ .",
"So according to Proposition REF , we can reformulate Conjecture REF into the following one, that can be stated without choosing a geometric point: Conjecture 9.5 If $X$ is a proper, smooth and geometrically connected curve of genus at least 2 over a finitely generated extension $\\kappa $ of $\\mathbb {Q}$ , the natural morphism $X\\rightarrow \\Pi _{X/\\kappa }^\\mathrm {D}$ induces a bijection of isomorphism classes of $\\kappa $ -rational points.",
"The injectivity is known as a consequence of the Mordell-Weil theorem (see for instance [26], Appendix B).",
"This goes trough over a function field if one uses the Nori fundamental gerbe: Proposition 9.6 Let $\\kappa $ be a field that is finitely generated over its prime subfield, and $X/\\kappa $ a proper, smooth, and geometrically connected curve of genus at least 1.",
"Then the natural functor $ X(\\kappa ) \\rightarrow \\Pi _{X/\\kappa }(\\kappa )$ is injective on isomorphism classes.",
"Let us denote by $s_X(x)$ the image of a rational point $x\\in X(\\kappa )$ in $\\Pi _{X/\\kappa }(\\kappa )$ .",
"We have to show that if two rational points $x,y \\in X(\\kappa )$ give rise to isomorphic sections $s_X(x)\\simeq s_X(y)$ , then $x=y$ .",
"Since we can embed $X$ into an abelian variety $A/\\kappa $ , and $\\Pi _{X/\\kappa }$ is covariant in $X$ , it is enough to show the corresponding statement for $A$ .",
"So let $a,b \\in A(\\kappa )$ such that $s_A(a)\\simeq s_A(b)$ .",
"Nori's generalization of Lang-Serre theorem [18] and Remark REF imply that $ \\Pi _{A/S}\\simeq \\varprojlim _n \\mathcal {B}_\\kappa A[n] \\, .$ Since isomorphism classes in $\\mathcal {B}_\\kappa A[n](\\kappa )$ are in one to one correspondence with $1(\\kappa , A[n])$ , it follows, for any non negative integer $n$ , that $a-b$ lies in $\\ker (A(\\kappa )\\rightarrow 1(\\kappa , A[n]))=nA(\\kappa )$ , in other words $a-b$ is divisible in $A(\\kappa )$ .",
"But the Mordell-Weil theorem, that asserts that $A(\\kappa )$ is of finite type, holds for $A/\\kappa $ ([14], Theorem 1, see also [5], Corollary 7.2).",
"Remark 9.7 It is unclear whether Proposition REF remains true when one replaces $\\Pi _{X/\\kappa }$ by $\\Pi _{X/\\kappa }^{{\\rm \\acute{e}t}}$ .",
"The critical point is to identify $\\ker (A(\\kappa )\\rightarrow 1(\\kappa , A[n]^{\\rm \\acute{e}t}))$ , which does not seem to be an easy task.",
"Also, when $k$ is a finite field, $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}(\\kappa )$ is never empty, because the Galois group of $k$ is free.",
"However, there examples of hyperbolic curves over a finite field with no rational points.",
"Hence for finitely generated fields of positive characteristic $X(\\kappa ) \\rightarrow \\Pi _{X/\\kappa }(\\kappa )$ is in general not surjective on isomorphism classes, as and the section conjecture, as originally stated, fails.",
"This leads naturally to the following extension of the Grothendieck section conjecture to arbitrary characteristic.",
"Conjecture 9.8 If $X$ is a proper, smooth and geometrically connected curve of genus at least 2 over a finitely generated field $\\kappa $ , the natural morphism $X\\rightarrow \\Pi _{X/\\kappa }$ induces a bijection of isomorphism classes of $\\kappa $ -rational points."
],
[
"The tame fundamental gerbe",
"Unfortunately, the étale fundamental gerbe does not seem to have a natural tannakian interpretation purely in terms of vector bundles.",
"There is another quotient of $\\Pi _{X/\\kappa }$ , however, that does.",
"Definition 10.1 A finite stack $\\Gamma $ over $\\kappa $ is tame if the functor $\\operatorname{Coh}\\Gamma \\rightarrow \\operatorname{Vect}_{\\kappa }$ given by $F \\mapsto \\operatorname{H}^{0}(\\Gamma , F)$ is exact.",
"A finite stack $\\Gamma $ has a moduli space $\\pi \\colon \\Gamma \\rightarrow M$ , which is finite over $\\kappa $ , hence $\\Gamma $ is tame if and only if $\\pi _{*}\\colon \\operatorname{Coh}\\Gamma \\rightarrow \\operatorname{Coh}M$ is exact; so this is a particular case of the notion of tame stack in [1].",
"From [1] we obtain the following.",
"Proposition 10.2 A finite stack $\\Gamma $ is tame if and only if for any object $\\xi $ in $\\Gamma (\\Omega )$ , where $\\Omega $ is an algebraically closed extension of $\\kappa $ , the group scheme $\\mathop {\\underline{\\mathrm {Aut}}}\\nolimits _{\\Omega }\\xi \\rightarrow \\operatorname{Spec}\\Omega $ is linearly reductive.",
"Recall, again from [1], that a finite group scheme $G$ over a field is linearly reductive if every finite-dimensional representation of $G$ is a sum of irreducible representations.",
"Over an algebraically closed field $\\Omega $ , the finite group scheme $G$ is linearly reductive if and only if it is a semidirect product $H \\ltimes D$ , where $H$ is a finite constant group of order prime to the characteristic of $\\Omega $ , and $D$ is a finite diagonalizable group ([1]).",
"Of course in characteristic 0 every finite group scheme is linearly reductive, so every finite stack is tame.",
"Proposition 10.3 If $\\Delta \\rightarrow \\Gamma $ is a representable morphism of tame stacks, and $\\Gamma $ is tame, then so is $\\Delta $ .",
"If $\\Delta _{1} \\rightarrow \\Gamma $ and $\\Delta _{2}\\rightarrow \\Gamma $ are morphisms of tame finite stacks, the fibered product $\\Delta _{1} \\times _{\\Gamma } \\Delta _{2}$ is also tame.",
"If $\\kappa ^{\\prime }$ is an extension of $\\kappa $ , then $\\Gamma _{\\kappa ^{\\prime }}$ is tame over $\\kappa ^{\\prime }$ if and only if $\\Gamma $ is tame over $\\kappa $ .",
"If $G$ is a finite group scheme over $\\kappa $ , then $\\mathcal {B}_{\\kappa }G$ is tame if and only if $G$ is linearly reductive.",
"A profinite gerbe $\\Phi $ is tame if and only if the category $\\operatorname{Rep}\\Phi $ is semisimple.",
"Part (REF ) follows immediately from the definitions.",
"Parts (REF ) and (REF ) are easy consequences of the following facts.",
"A subgroup of a finite linearly reductive group scheme is linearly reductive [1].",
"A product of finite linearly reductive group schemes is linearly reductive [1].",
"(REF ) is a particular case of [1].",
"(REF ) is easy and left to the reader.",
"Let $X$ be an inflexible algebraic stack, $\\Pi _{X/\\kappa } = \\varprojlim _{i} \\Gamma _{i}$ as above.",
"Consider the full subcategory $I^{\\rm tame}$ of $I$ whose objects consist of Nori-reduced morphisms $X \\rightarrow \\Gamma _{j}$ in which $\\Gamma _{j}$ is tame.",
"Since the fibered product of two tame gerbes is a tame stack, we have that $I^{\\rm tame}$ is a 2-cofiltered category.",
"Definition 10.4 Let $X$ be an algebraic stack over $\\kappa $ .",
"The tame fundamental gerbe of $X$ is the profinite gerbe $\\Pi _{X/\\kappa }^{\\mathrm {tame}} \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\varprojlim _{j \\in I^{\\rm tame}} \\Gamma _{j}$ .",
"Again, there is a natural morphism $\\Pi _{X/\\kappa } \\rightarrow \\Pi _{X/\\kappa }^{\\mathrm {tame}}$ .",
"It is easy to see that for any tame finite stack $\\Gamma $ over $\\kappa $ , the natural functor $\\operatorname{Hom}(\\Pi _{X/\\kappa }^{\\mathrm {tame}}, \\Gamma ) \\longrightarrow \\operatorname{Hom}(X, \\Gamma )$ is an equivalence.",
"Also, the morphism $X \\rightarrow \\Pi _{X/\\kappa }^{\\mathrm {tame}}$ is universal among all maps from $X$ to a tame profinite gerbe."
],
[
"The tannakian interpretation of the étale fundamental gerbe",
"Let $X$ be a connected algebraic stack.",
"We endow it with the finite étale topology, where coverings are given by surjective families of finite étale morphisms.",
"We denote by $X_{\\rm f\\acute{e}t}$ the corresponding site.",
"Let $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ be the category of local systems of finite dimensional $\\kappa $ -vector spaces.",
"This is a Tannaka category.",
"The aim of this section is to relate it to $\\operatorname{EFin}X$ , when $X$ is defined over $\\kappa $ , inflexible and pseudo-proper.",
"We can describe the objects of $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ in terms of gerbes in the following way.",
"Since $X$ is connected, we can stick to covers consisting of a single finite étale morphism $Y\\rightarrow X$ , that we can suppose Galois, with Galois group $G$ .",
"A local system $V$ in $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ trivialized by such a $G$ -torsor $Y\\rightarrow X$ corresponds to a morphism $X\\rightarrow \\mathcal {B}_\\mathbb {Z}G$ , and a local system $V_0$ on $\\mathcal {B}_\\mathbb {Z}G$ , which is nothing else than a finite dimensional $\\kappa $ -linear representation $\\rho _0\\colon G\\rightarrow \\mathrm {GL}(V_0)$ of the constant group $G$ .",
"In view of this, it is clear that the gerbe associated with the Tannaka category $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ is canonically isomorphic to Deligne's absolute fundamental gerbe $\\Pi _X^\\mathrm {D}$ (Definition REF ).",
"Definition 11.1 Let $X$ be a connected algebraic stack over $\\kappa $ .",
"The Riemann-Hilbert functor $\\operatorname{RH}_{X/\\kappa }:\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )\\rightarrow \\operatorname{Vect}X$ is defined by the formula $\\operatorname{RH}_{X/\\kappa }(V)=\\mathcal {O}_X\\otimes _\\kappa V$ .",
"In this description, it is implicit that we use descent theory to get a vector bundle on $X$ .",
"In other terms, if $V$ comes from a morphism $X\\rightarrow B_\\mathbb {Z}G$ and a representation $\\rho _0:G\\rightarrow \\mathrm {GL}(V_0)$ as above, we can use the diagram: ${&\\operatorname{Spec}\\kappa [r][d] &\\operatorname{Spec}\\mathbb {Z}[d] \\\\X [r] &B_\\kappa G [r] &B_\\mathbb {Z}G}$ to descend the trivial bundle $\\mathcal {O}_{\\operatorname{Spec}\\kappa }\\otimes _\\kappa V_0$ to $B_\\kappa G$ , and then pull it back to $X$ , to get $\\operatorname{RH}_{X/\\kappa }(V)$ .",
"This construction is independent (up to canonical isomorphism) of the choice of the pair $(X\\rightarrow B_\\mathbb {Z}G,\\rho _0:G\\rightarrow \\mathrm {GL}(V_0))$ .",
"Moreover, this description makes it clear that, when $X$ is inflexible and pseudo-proper, the functor $\\operatorname{RH}_{X/\\kappa }$ factors trough $\\operatorname{EFin}X$ (see Lemma REF (REF )).",
"It is also clear that the functor $\\operatorname{RH}_{X/\\kappa }$ is functorial in $X/\\kappa $ .",
"To state our result we need two additional definitions.",
"Definition 11.2 Let $X/\\kappa $ be an inflexible and pseudo-proper algebraic stack.",
"We denote by $\\operatorname{EFin}^{\\rm \\acute{e}t}X$ the full subcategory of $\\operatorname{EFin}X$ whose objects are essentially finite locally free sheaves with étale holonomy gerbe.",
"In other words, $\\operatorname{EFin}^{\\rm \\acute{e}t}X$ is the category of representations of $\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ .",
"It is clear that the functor $\\operatorname{RH}_{X/\\kappa }$ factors trough $\\operatorname{EFin}^{\\rm \\acute{e}t}X$ .",
"Definition 11.3 ([15]) Let $\\mathcal {C}$ be a Tannaka category over $\\kappa $ , and $\\mathcal {B}$ be a Tannaka subcategory.",
"We say that a tensor functor $F:\\mathcal {C} \\rightarrow \\mathcal {D}$ to a Tannaka category $\\mathcal {D}$ identifies $\\mathcal {D}$ with the quotient of $\\mathcal {C}$ by $\\mathcal {B}$ if: the objects of $\\mathcal {C}$ whose image by $F$ is trivial are exactly those of $\\mathcal {B}$ , any object of $\\mathcal {D}$ is a subquotient of an object in the image of $F$ .",
"Theorem 11.4 Let $X/\\kappa $ be a inflexible and pseudo-proper stack, and denote by $s:X\\rightarrow \\operatorname{Spec}\\kappa $ the structural morphism.",
"Then the functor $s^*:\\operatorname{LC}( ( \\operatorname{Spec}\\kappa )_{\\rm f\\acute{e}t},\\kappa )\\rightarrow \\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ is fully faithful and the functor $\\operatorname{RH}_{X/\\kappa }$ identifies $\\operatorname{EFin}^{\\rm \\acute{e}t}X$ with the quotient of $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ by $\\operatorname{LC}( (\\operatorname{Spec}\\kappa )_{\\rm f\\acute{e}t},\\kappa )$ .",
"According to Proposition REF (REF ), the stack $X/\\kappa $ is geometrically connected, and one deduces that, if $\\kappa _X$ denotes the 1-dimensional constant sheaf on $X$ , we have $s_* \\kappa _X=\\kappa _{\\operatorname{Spec}\\kappa }$ .",
"It now follows from the projection formula that the functor $s^*$ is fully faithful.",
"To check the first condition of Definition REF , we have to prove that, given an object $V$ of $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ , its image $\\operatorname{RH}_{X/\\kappa }(V)$ is trivial (that is, a free vector bundle) if and only if there exists an object $W$ of $\\operatorname{LC}( ( \\operatorname{Spec}\\kappa )_{\\rm f\\acute{e}t},\\kappa )$ such that $V\\simeq s^*W$ .",
"The “if” direction is clear, because of the functoriality of $\\operatorname{RH}_{X/\\kappa }$ in $X/\\kappa $ .",
"To prove the “only if” direction, let us start from a local system $V$ in $\\operatorname{LC}(X_{\\rm f\\acute{e}t},\\kappa )$ such that $\\operatorname{RH}_{X/\\kappa }(V)$ is a trivial vector bundle.",
"We can assume that $V$ comes from a morphism $X \\rightarrow B_\\kappa G$ , and a local system $V_0$ on $B_\\kappa G$ , as above.",
"Since $X$ is inflexible, we can use Lemma REF to factor $X \\rightarrow B_\\kappa G$ trough a Nori-reduced morphism to a finite gerbe $f\\colon X \\rightarrow \\Gamma $ .",
"Let $V_1$ be the pull-back of $V_0$ along the morphism $\\Gamma \\rightarrow B_\\kappa G$ .",
"Using the functoriality of $\\operatorname{RH}_{X/\\kappa }$ in $X/\\kappa $ again, and the fact that the functor $f^*: \\operatorname{Vect}\\Gamma \\rightarrow \\operatorname{Vect}X$ is fully faithful (Lemmas REF , REF ), we see that $\\operatorname{RH}_{\\Gamma /\\kappa }(V_1)$ is trivial.",
"So we can finally assume that $X=\\Gamma $ is a finite gerbe, and $V=V_1$ .",
"Lemma 11.5 Let $\\Gamma /\\kappa $ be a finite gerbe, $\\kappa ^{\\rm sep}$ a separable closure of $\\kappa $ , and $V$ an object of $\\operatorname{LC}(\\Gamma _{\\rm f\\acute{e}t},\\kappa )$ .",
"Then $V$ descends to an object of $\\operatorname{LC}(( \\operatorname{Spec}\\kappa )_{\\rm f\\acute{e}t},\\kappa )$ if and only if $V_{|\\Gamma _{\\kappa ^{\\rm sep}}}$ is constant.",
"This is a straightforward consequence of descent theory, since we can see $(\\Gamma \\rightarrow \\operatorname{Spec}\\kappa , \\operatorname{Spec}\\kappa ^{\\rm sep}\\rightarrow \\operatorname{Spec}\\kappa )$ as a cover of $\\operatorname{Spec}\\kappa $ .",
"So the last thing to do to prove the first condition of Definition REF is to show that if $\\operatorname{RH}_{\\Gamma /\\kappa }(V)$ is trivial, then $V_{|\\Gamma _{\\kappa ^{\\rm sep}}}$ is constant.",
"This is a consequence of the functoriality of $\\operatorname{RH}_{X/\\kappa }$ in $X/k$ and of the following obvious lemma.",
"Lemma 11.6 The functor $\\operatorname{RH}_{\\Gamma _{\\kappa ^{\\rm sep}}/\\kappa ^{\\rm sep}}:\\operatorname{LC}({\\Gamma _{\\kappa ^{\\rm sep}}}_{\\rm f\\acute{e}t},\\kappa ^{\\rm sep})\\rightarrow \\operatorname{Vect}\\Gamma _{\\kappa ^{\\rm sep}}$ is an equivalence.",
"This is clear, since $\\Gamma _{\\kappa ^{\\rm sep}}\\simeq B_{k^{\\rm sep}}G$ for a finite constant group $G$ , and both categories then identify with the category of finite dimensional representations of $G$ with values in $\\kappa ^{\\rm sep}$ .",
"Note that since $V\\otimes _\\kappa \\kappa ^{\\rm sep}$ is constant, the same holds for $V_{|\\Gamma _{\\kappa ^{\\rm sep}}}$ , and this finishes the proof of the first condition of Definition REF .",
"We now prove the second condition.",
"Let $E$ be an object of $ \\operatorname{EFin}^{\\rm \\acute{e}t}X=\\operatorname{Rep}\\Pi _{X/\\kappa }^{\\mathrm {\\acute{e}t}}$ .",
"According to Remark REF , we can choose a connected Galois cover $Y\\rightarrow X$ of group $G$ so that $E$ comes from a representation of the groupoid $[X/P^{0}_{Y}]$ along the Nori-reduced morphism $X \\rightarrow [X/P^{0}_{Y}]$ .",
"This morphism is the canonical factorization of the given morphism $X\\rightarrow \\mathcal {B}_\\kappa {G}$ , in particular, the morphism $f\\colon [X/P^{0}_{Y}]\\rightarrow \\mathcal {B}_\\kappa {G}$ is representable.",
"Lemma 11.7 Let $f\\colon \\Gamma \\rightarrow \\Delta $ be a representable morphism of finite gerbes.",
"Then every representation of $\\Gamma $ is a quotient of a representation of $\\Delta $ .",
"It is enough to show that $f$ is affine, since then for any vector bundle $E$ on $\\Gamma $ , the canonical morphism $f^*f_* E\\rightarrow E$ is an epimorphism.",
"We may base change so that $\\Gamma (\\operatorname{Spec}\\kappa )$ is not empty.",
"Then $f\\colon \\Gamma \\rightarrow \\Delta $ can be identified with a morphism $\\phi \\colon \\mathcal {B}_{\\kappa }H \\rightarrow \\mathcal {B}_{\\kappa }G$ induced by an injective homomorphism $H \\rightarrow G$ of finite $\\kappa $ -group schemes.",
"But then the base change of $\\phi $ in the chart $\\operatorname{Spec}\\kappa \\rightarrow \\mathcal {B}_{\\kappa }G$ is $G/H\\rightarrow \\operatorname{Spec}\\kappa $ , which is affine, since $H$ and $G$ are finite.",
"So the vector bundle $E$ , seen as a representation of $[X/P^{0}_{Y}]$ , is a quotient of $f^*f_*E$ , where $f$ is the natural morphism: $f\\colon [X/P^{0}_{Y}]\\rightarrow \\mathcal {B}_\\kappa {G}$ .",
"But $f_*E$ is trivialized by the $G$ -torsor $\\operatorname{Spec}\\kappa \\rightarrow \\mathcal {B}_{\\kappa }G$ , hence corresponds to a representation of the finite constant group $G$ with values in finite dimensional $\\kappa $ -vector spaces.",
"This gives rise to a local system on $X$ which is sent to $f_*E$ by $\\operatorname{RH}_{X/\\kappa }$ .",
"This proves the second condition of Definition REF , hence the Theorem."
],
[
"The tannakian interpretation of the tame fundamental gerbe",
"Let us fix an inflexible pseudo-proper algebraic stack $X$ over $\\kappa $ .",
"Then we have an equivalence of Tannaka categories $\\operatorname{Rep}\\Pi _{X/\\kappa } \\rightarrow \\operatorname{EFin}X$ .",
"Definition 12.1 A locally free sheaf $E$ on $X$ is tamely finite when it is finite, and all the indecomposable components of all the tensor powers $E^{\\otimes n}$ are irreducible in $\\operatorname{EFin}X$ .",
"We denote by $\\operatorname{TFin}X$ the full subcategory of $\\operatorname{EFin}X$ consisting of tamely finite objects.",
"It is easy to see that $\\operatorname{TFin}X$ is an exact abelian subcategory of $\\operatorname{EFin}X$ ; however, it does not seem obvious to us that it is a tannakian subcategory, that is, that the tensor product of two tamely finite sheaves is tamely finite.",
"Clearly, if $\\Gamma $ is a tame finite gerbe every object of $\\operatorname{Rep}\\Gamma $ is tamely finite; hence the pullback $\\operatorname{Rep}\\Pi _{X/\\kappa }^{\\mathrm {tame}} \\rightarrow \\operatorname{Rep}\\Pi _{X/\\kappa }$ , which according to Lemma REF is fully faithful, gives an equivalence of $\\operatorname{Rep}\\Pi _{X/\\kappa }^{\\mathrm {tame}}$ with a tannakian semisimple subcategory of $\\operatorname{Rep}\\Pi _{X/\\kappa }$ , which is contained in $\\operatorname{TFin}X$ .",
"The following theorem says that this is an equivalence.",
"Theorem 12.2 The pullback $\\operatorname{Rep}\\Pi _{X/\\kappa }^{\\mathrm {tame}} \\rightarrow \\operatorname{EFin}X$ induces an equivalence of the Tannaka category $\\operatorname{Rep}\\Pi _{X/\\kappa }^{\\mathrm {tame}}$ with $\\operatorname{TFin}X$ .",
"In particular, $\\operatorname{TFin}X$ is a tannakian subcategory of $\\operatorname{EFin}X$ .",
"It follows from the Theorem that $\\operatorname{TFin}X$ is the largest tannakian semisimple subcategory of $\\operatorname{EFin}X$ .",
"The proof of the Theorem is based on the following fact.",
"Let $V$ be a representation of degree $r$ of a finite gerbe $\\Gamma $ , corresponding to a morphism $\\Gamma \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ .",
"We say that $V$ is faithful if the morphism is representable.",
"Suppose that $\\kappa ^{\\prime }$ is extension of $\\kappa $ , such that $\\Gamma (\\kappa ^{\\prime }) \\ne \\emptyset $ ; then $\\Gamma _{\\kappa ^{\\prime }} \\simeq \\mathcal {B}_{\\kappa ^{\\prime }}G$ for a finite group scheme $G$ over $\\kappa ^{\\prime }$ .",
"The pullback $V_{\\kappa ^{\\prime }}$ of $V$ to $\\Gamma _{\\kappa ^{\\prime }}$ comes from a representation $G \\rightarrow \\mathrm {GL}_{r}$ ; then $V$ is faithful if and only if the representation $G \\rightarrow \\mathrm {GL}_{r}$ is faithful, in the sense that it has trivial kernel.",
"Lemma 12.3 Suppose that a finite gerbe $\\Gamma $ has a faithful tamely finite representation.",
"Then $\\Gamma $ is tame.",
"Let $V$ be a faithful representation of degree $r$ of $\\Gamma $ ; this corresponds to a morphism $f\\colon \\Gamma \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ .",
"Notice that $f$ is affine; for this we can extend the base field and assume that $\\Gamma = \\mathcal {B}_{\\kappa }G$ for a certain finite group scheme $G$ , so that $V$ corresponds to a faithful representation $G \\rightarrow \\mathrm {GL}_{r}$ .",
"In this case the fiber of $f$ over $\\operatorname{Spec}\\kappa $ is the quotient $\\mathrm {GL}_{r}/G$ ; and since the quotient of an affine variety by a finite group scheme is affine, we have the result.",
"Let $\\kappa ^{\\prime }$ be a finite extension of $\\kappa $ such that $\\Gamma (\\kappa ^{\\prime }) \\ne \\emptyset $ , and choose a morphism $\\rho \\colon \\operatorname{Spec}\\kappa ^{\\prime } \\rightarrow \\Gamma $ .",
"By Lemma REF (REF ) every representation of $\\Gamma $ is contained in a sum of copies of $E \\overset{\\mathrm {\\scriptscriptstyle def}}{=}\\rho _{*}\\mathcal {O}_{\\operatorname{Spec}\\kappa ^{\\prime }}$ ; hence it is enough to prove that $E$ is semisimple.",
"Set $\\sigma \\overset{\\mathrm {\\scriptscriptstyle def}}{=}f\\rho \\colon \\operatorname{Spec}\\kappa ^{\\prime } \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ .",
"Consider the adjunction homomorphism $f^{*}f_{*}E \\rightarrow E$ ; this is surjective, because $f$ is affine, so it is enough to show that the infinite dimensional representation $f^{*}f_{*}E = f^{*}\\sigma _{*}\\mathcal {O}_{\\operatorname{Spec}\\kappa ^{\\prime }}$ is semisimple.",
"Notice that any two morphisms $\\operatorname{Spec}\\kappa ^{\\prime } \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ are isomorphic; hence $\\sigma \\simeq \\pi \\alpha $ , where $\\alpha \\colon \\operatorname{Spec}\\kappa ^{\\prime } \\rightarrow \\operatorname{Spec}\\kappa $ is the canonical morphism and $\\pi \\colon \\operatorname{Spec}\\kappa \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ corresponds the trivial $\\mathrm {GL}_{r}$ -torsor on $\\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ .",
"If $d$ denotes the degree of the extension $\\kappa ^{\\prime }/\\kappa $ , then $\\alpha _{*}\\mathcal {O}_{\\operatorname{Spec}\\kappa ^{\\prime }} \\simeq \\mathcal {O}_{\\operatorname{Spec}\\kappa }^{\\oplus d}$ , so $\\sigma _{*}\\mathcal {O}_{\\operatorname{Spec}\\kappa ^{\\prime }} \\simeq (\\pi _{*}\\mathcal {O}_{\\operatorname{Spec}k})^{\\oplus d}$ ; so it is enough to prove that $V \\overset{\\mathrm {\\scriptscriptstyle def}}{=}f^{*}\\pi _{*}\\mathcal {O}_{\\operatorname{Spec}k}$ is semisimple.",
"The representation $\\pi _{*}\\mathcal {O}_{\\operatorname{Spec}k}$ corresponds to the standard regular representation of $\\mathrm {GL}_{r}$ , that is, to the action of $\\mathrm {GL}_{r}$ on the vector space $k[\\mathrm {GL}_{r}]$ induced by right translation.",
"Let $W$ be the tautological representation of $\\mathrm {GL}_{r}$ of degree $r$ ; the pullback of $W$ to $\\Gamma $ is exactly $V$ .",
"On the other hand we have that $k[\\mathrm {GL}_{r}]$ is a quotient of the representation $\\bigoplus _{m, n \\ge 0} \\det ( W^{\\vee })^{\\otimes m}\\otimes \\operatorname{Sym}^{n}(W \\otimes W^{\\vee })\\,,$ from which we obtain that $f^{*}\\pi _{*}\\mathcal {O}_{\\operatorname{Spec}k}$ is a quotient of $\\bigoplus _{m, n \\ge 0} \\det ( V^{\\vee })^{\\otimes m}\\otimes \\operatorname{Sym}^{n}(V \\otimes V^{\\vee })\\,.$ Since each of the summands $\\det (V^{\\vee })^{\\otimes m} \\otimes \\operatorname{Sym}^{n}(V \\otimes V^{\\vee })$ is a quotient of a tensor product of tensor powers of $V$ and $V^{\\vee }$ , and all these tensor powers are semi-simple, because $V$ , and hence $V^{\\vee }$ , is tamely finite.",
"This concludes the proof.",
"Let us proceed with the proof of the theorem.",
"According to the discussion above, it is enough to show that every tamely finite sheaf $E$ on $X$ is in the essential image of $\\operatorname{Rep}\\Pi _{X/\\kappa }^{\\mathrm {tame}}$ .",
"Choose a morphism $f\\colon X \\rightarrow \\Gamma $ to a finite gerbe and a representation $V$ of $\\Gamma $ with $f^{*}V \\simeq E$ .",
"The representation $V$ corresponds to a morphism $\\Gamma \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ , where $r$ is the rank of $E$ .",
"We claim that there is a factorization $\\Gamma \\rightarrow \\Delta \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ , where $\\Delta $ is a finite gerbe and the morphism $\\Delta \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ is representable.",
"Let $I_{\\Gamma }$ be the inertia stack of $\\Gamma $ , which is a group scheme over $\\Gamma $ , and let $G \\subseteq I_{\\Gamma }$ be the kernel of the induced homomorphism of relative group schemes $I_{\\Gamma } \\rightarrow \\Gamma \\times _{\\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}} I_{\\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}} \\subseteq \\Gamma \\times \\mathrm {GL}_{r}$ ; the morphism $\\Gamma \\rightarrow \\mathcal {B}_{\\kappa }\\mathrm {GL}_{r}$ is representable if and only if $G$ is trivial.",
"Now we can take $\\Delta $ to be the rigidification $\\Gamma \\mathbin {\\!\\!",
"}G$ , as in [1].",
"Then the tamely finite sheaf $E$ is a pullback from a representation of $\\Delta $ , which is a tame finite gerbe, according to Lemma REF ."
],
[
"Examples and applications",
"We conclude with some examples to illustrate the theory.",
"We will use without comments some standard facts in the theory of Brauer–Severi varieties; as a general reference, see [2] or [12].",
"Proposition 13.1 Let $P$ be a Brauer–Severi variety over $\\kappa $ .",
"Then $\\Pi _{P/\\kappa } = \\operatorname{Spec}\\kappa $ .",
"There exists some finite separable extension $\\kappa ^{\\prime }$ of $\\kappa $ such that $P(\\kappa ^{\\prime }) \\ne Â~\\emptyset $ , so that $P_\\kappa ^{\\prime } = \\mathbb {P}^{n}_{\\kappa ^{\\prime }}$ .",
"From Proposition REF we see that we may assume $\\kappa = \\kappa ^{\\prime }$ , that is, $P= \\mathbb {P}^{n}_{\\kappa }$ .",
"Also, we can assume that $\\kappa $ is infinite (if not, pass to a separable closure).",
"It is enough to show that every finite bundle $E$ on $\\mathbb {P}^{n}_{\\kappa }$ is trivial, as this implies immediately that $\\operatorname{EFin}\\mathbb {P}^{n}_{\\kappa } = \\operatorname{Vect}_{\\kappa }$ , which implies the thesis, by Tannaka duality.",
"By [20], it is enough to show that the restriction of $E$ to each line is trivial (in the reference given the the result is only stated over $\\mathbb {C}$ , but the proof works over any infinite field).",
"Since the restriction of $E$ to a line is again finite, we are reduced to the case $n =1$ , which follows from Proposition REF , and Grothendieck's theorem on the structure of vector bundles on $\\mathbb {P}^{1}_{\\kappa }$ .",
"Let us give examples of schemes $X$ over $\\kappa $ in which $\\Pi _{X/\\kappa }(\\kappa ) = \\emptyset $ .",
"Here is a general method for producing examples.",
"Proposition 13.2 Let $P$ be a Brauer–Severi variety $\\kappa $ .",
"Call $r$ its exponent; then $\\operatorname{Pic}P$ is generated by a sheaf of degree $r$ , which we call $\\mathcal {O}_{P}(r)$ .",
"Let $f\\colon X \\rightarrow P$ be a morphism, where $X$ is an inflexible algebraic stack.",
"Assume that there exists a prime $p$ dividing $r$ and an invertible sheaf $\\Lambda $ on $X$ , such that $\\Lambda ^{\\otimes p} \\simeq f^{*}\\mathcal {O}_{P}(r)$ .",
"Then $\\Pi _{X/\\kappa }(\\kappa ) = \\emptyset $ .",
"Let $P^{\\vee }$ be the dual Brauer–Severi variety, that is, the Hilbert scheme of hyperplanes in $P$ .",
"Then $P^{\\vee }$ has also exponent $r$ .",
"Let $\\Gamma \\rightarrow \\operatorname{Spec}\\kappa $ be the stack, whose sections over a $\\kappa $ -scheme $S$ consist of invertible sheaves $E$ on $S \\times P^{\\vee }$ , with an isomorphism $E^{\\otimes p} \\simeq \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(r)$ .",
"It is easy to check that $\\Gamma $ is a gerbe banded by $\\mu _{p}$ .",
"Clearly $\\Gamma (\\kappa ) = \\emptyset $ , because $\\mathcal {O}_{P^{\\vee }}(r)$ has no $p^\\text{th}$ root on $P^{\\vee }$ .",
"Denote by $H \\subseteq P \\times P^{\\vee }$ the tautological divisor; the invertible sheaf $\\mathcal {O}(H)$ has bidegree $(1,1)$ .",
"Notice that there is an isomorphism $\\mathcal {O}(rH) \\simeq \\operatorname{pr}_{1}^{*}\\mathcal {O}_{P}(r) \\otimes \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(r)\\,.$ Consider the morphism $f\\times \\mathrm {id}\\colon X \\times P^{\\vee } \\rightarrow P \\times P^{\\vee }$ , and the invertible sheaf $E \\overset{\\mathrm {\\scriptscriptstyle def}}{=}(f \\times \\mathrm {id})^{*}\\mathcal {O}(H)^{\\otimes r/p} \\otimes \\operatorname{pr}_{1}^{*}\\Lambda ^{\\vee }$ on $X \\times P^{\\vee }$ .",
"We claim that $E^{\\otimes p}$ is isomorphic to $\\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(r)$ ; this gives a morphism $X \\rightarrow \\Gamma $ , and shows that $\\Pi _{X/\\kappa }(\\kappa ) = \\emptyset $ .",
"If $F$ is a geometric fiber of the projection $\\operatorname{pr}_{1}\\colon X \\times P^{\\vee } \\rightarrow X$ , the invertible sheaf $E^{\\otimes p} \\otimes \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(-r)$ is trivial along $F$ ; since $\\operatorname{H}^{1}(F, \\mathcal {O}) = 0$ , we have, by the standard base change theorems, that $\\operatorname{pr}_{1*}\\bigl (E^{\\otimes p} \\otimes \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(-r)\\bigr )$ is an invertible sheaf on $X$ , and that $E^{\\otimes p} \\otimes \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(-r) &=\\operatorname{pr}_{1}^{*}\\operatorname{pr}_{1*}\\bigl (E^{\\otimes p} \\otimes \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(-r)\\bigr )\\\\&= \\operatorname{pr}_{1}^{*}\\operatorname{pr}_{1*}\\bigl ((f \\times \\mathrm {id})^{*}\\mathcal {O}(rH)\\otimes \\operatorname{pr}_{1}^{*}\\Lambda ^{\\otimes - p} \\otimes \\operatorname{pr}_{2}^{*}\\mathcal {O}_{P^{\\vee }}(-r)\\bigr )\\\\&= \\operatorname{pr}_{1}^{*}\\operatorname{pr}_{1*}\\bigl (\\operatorname{pr}_{1}^{*}(f^*\\mathcal {O}_{P}(r)\\otimes \\Lambda ^{\\otimes - p}) \\otimes \\operatorname{pr}_{2}^{*}(\\mathcal {O}_{P^{\\vee }}(r)\\otimes \\mathcal {O}_{P^{\\vee }}(-r))\\bigr )\\\\&= \\operatorname{pr}_{1}^{*}\\operatorname{pr}_{1*} \\mathcal {O}_{X \\times P^{\\vee }}\\\\&= \\mathcal {O}_{X}\\,.$ This concludes the proof.",
"Remark 13.3 In applying Proposition REF , it is useful to notice that if $f^{*}\\mathcal {O}_{P}(mr)$ is a $p^\\text{th}$ power in $\\operatorname{Pic}X$ , where $m$ is not divisible by $p$ , then $f^{*}\\mathcal {O}_{P}(r)$ is also a $p^\\text{th}$ power.",
"Using the relative Brauer group, Jakob Stix has shown the following similar result ([27], §10.3).",
"Assume that $P$ is a non trivial Severi-Brauer variety of dimension $n-1$ .",
"Let $\\mathcal {L}$ be an invertible sheaf on $P$ whose order in $\\operatorname{Pic}P/n$ is a multiple of the period $r$ .",
"If $\\mathcal {L}$ admits a $n$ -th root on $X\\rightarrow P$ , then $\\Pi _{X/\\kappa }(\\kappa ) = \\emptyset $ .",
"In our result however, the dimension of $P$ does not play any role.",
"Sylvain Brochard has given an independent proof of Proposition REF using the torsion of the Picard scheme and an interesting duality theory for commutative group stacks (see [4]).",
"From this it is easy to give examples of smooth projective geometrically connected curves of genus at least 2 over a finitely generated field $\\kappa $ such that $\\Pi _{X/\\kappa }(\\kappa ) = \\emptyset $ , over any field with non-trivial Brauer group.",
"Proposition 13.4 Let $P$ be a Brauer-Severi variety over $\\kappa $ with $P(\\kappa ) = \\emptyset $ , let $r$ be its exponent, and $p$ a prime factor of $r$ .",
"Then there exists a smooth geometrically connected projective curve $X$ with a morphism $f\\colon X \\rightarrow P$ , and an invertible sheaf $\\Lambda $ on $X$ such that $\\Lambda ^{\\otimes p} \\simeq f^{*}\\mathcal {O}_{P}(r)$ .",
"The result follows from the next lemma, applied to a smooth geometrically connected projective curve $Y \\subseteq P$ (for example, a complete intersection in $P$ ), and to the restriction of $\\mathcal {O}_{P}(r)$ to $Y$ .",
"Lemma 13.5 Let $Y$ be a smooth geometrically connected projective curve over $\\kappa $ , let $L$ be an invertible sheaf on $Y$ , and $n$ a positive integer.",
"Then there exists a morphism of smooth geometrically connected projective curves $f\\colon X \\rightarrow Y$ and an invertible sheaf $\\Lambda $ on $X$ with $\\Lambda ^{\\otimes n} \\simeq f^{*}L$ .",
"Suppose that $f\\colon X \\rightarrow Y$ is a morphism of smooth geometrically connected projective curves over $\\kappa $ .",
"If $y\\in Y$ is a closed point, and we set $f^{*}y = \\sum _{x \\in f^{-1}(y)} e_{x} x$ (where the equality is an equality of divisors), with $e_{x} \\in \\mathbb {N}$ .",
"We call the branch index of $f$ at $y$ the greatest common divisor $\\mathrm {b}_{y}(f)$ of the $e_{x}$ .",
"Now, by multiplying $L$ by the $n^\\text{th}$ power of a sufficiently ample invertible sheaf on $Y$ , we may assume that $L$ is very ample.",
"Let $D$ be a smooth divisor in its linear system (which exists, since the field $\\kappa $ is infinite, because it has non-trivial Brauer group).",
"It is sufficient to show that there exists a morphism of smooth geometrically connected projective curves $f\\colon X \\rightarrow Y$ with $\\mathrm {b}_{y}(f) = n$ for each $y \\in D$ .",
"Let $D^{\\prime }$ be a smooth divisor in the linear system of $L^{\\otimes (n-1)}$ which is disjoint from $D$ ; then $D + D^{\\prime }$ is a smooth divisor in the linear system of $L^{\\otimes n}$ .",
"We obtain $X$ as the ramified cover of $n^\\text{th}$ roots of $D + D^{\\prime }$ in the usual fashion."
]
] | 1204.1260 |
[
[
"Combinatorial specification of permutation classes"
],
[
"Abstract This article presents a methodology that automatically derives a combinatorial specification for the permutation class C = Av(B), given its basis B of excluded patterns and the set of simple permutations in C, when these sets are both finite.",
"This is achieved considering both pattern avoidance and pattern containment constraints in permutations.The obtained specification yields a system of equations satisfied by the generating function of C, this system being always positiveand algebraic.",
"It also yields a uniform random sampler of permutations in C. The method presentedis fully algorithmic."
],
[
"Introduction",
"Initiated by [15] almost forty years ago, the study of permutation classes has since received a lot of attention, mostly with respect to enumerative questions (see [7], [12], [14] and their references among many others).",
"Most articles are focused on a given class $\\mathcal {C} = Av(B)$ where the basis $B$ of excluded patterns characterizing $\\mathcal {C}$ is finite, explicit, and in most cases contains only patterns of size 3 or 4.",
"Recently, some results of a rather different nature have been obtained, and have in common that they describe general properties of permutation classes – see [1], [3], [4], [5], [9], [10], [17] for example.",
"Our work falls into this new line of research.",
"Our goal in this article is to provide a general algorithmic method to obtain a combinatorial specification for any permutation class $\\mathcal {C}$ from its basis $B$ and the set $\\mathcal {S}_{\\mathcal {C}}$ of simple permutations in $\\mathcal {C}$ , and assuming these two sets are finite.",
"Notice that by previous works to be detailed in Section , it is enough to know the finite basis $B$ of the class to decide whether the set $\\mathcal {S}_{\\mathcal {C}}$ is finite and (in the affirmative) to compute $\\mathcal {S}_{\\mathcal {C}}$ .",
"By combinatorial specification of a class (see [13]), we mean an unambiguous system of combinatorial equations that describe recursively the permutations of $\\mathcal {C}$ using only combinatorial constructors (disjoint union, cartesian product, sequence, ...) and permutations of size 1.",
"Notice the major difference with the results of [1]: our specifications are unambiguous, whereas [1] obtain combinatorial systems of equations characterizing permutations classes that are ambiguous in general.",
"We believe that our purpose of obtaining algorithmically combinatorial specifications of permutation classes is of interest per se but also because it then allows to obtain by routine algorithms a system of equations satisfied by the generating function of $\\mathcal {C}$ and a Boltzmann uniform random sampler of permutations in $\\mathcal {C}$ , using the methods of [13] and [11] respectively.",
"The paper is organized as follows.",
"Section proceeds with some background on permutation classes, simple permutations and substitution decomposition, and Section sets the algorithmic context of our study.",
"Section then explains how to obtain a system of combinatorial equations describing $\\mathcal {C}$ from the set of simple permutations in $\\mathcal {C}$ , that we assume to be finite.",
"The system so obtained may be ambiguous and Section describes a disambiguation algorithm to obtain a combinatorial specification for $\\mathcal {C}$ .",
"The most important idea of this disambiguation procedure is to transform ambiguous unions into disjoint unions of terms that involve both pattern avoidance and pattern containment constraints.",
"This somehow allows to interpret on the combinatorial objects themselves the result of applying the inclusion-exclusion on their generating functions.",
"Finally, Section concludes the whole algorithmic process by explaining how this specification can be plugged into the general methodologies of [13] and [11] to obtain a system of equations satisfied by the generating function of $\\mathcal {C}$ and a Boltzmann uniform random sampler of permutations in $\\mathcal {C}$ .",
"We also give a number of perspectives opened by our algorithm.",
"A permutation $\\sigma = \\sigma _1 \\sigma _2 \\ldots \\sigma _n$ of size $|\\sigma | = n$ is a bijective map from $\\lbrace 1,\\ldots ,n\\rbrace $ to itself, each $\\sigma _i$ denoting the image of $i$ under $\\sigma $ .",
"A permutation $\\pi = \\pi _1 \\pi _2 \\ldots \\pi _k$ is a pattern of a permutation ${\\sigma = \\sigma _1 \\sigma _2 \\ldots \\sigma _n}$ (denoted $\\pi \\preceq \\sigma $ ) if and only if $k\\le n$ and there exist integers $1\\le i_1 < i_2 < \\ldots < i_k \\le n$ such that $\\sigma _{i_1}\\ldots \\sigma _{i_k}$ is order-isomorphic to $\\pi $ , i.e.",
"such that $\\sigma _{i_{\\ell }} < \\sigma _{i_m}$ whenever $\\pi _{\\ell } < \\pi _{m}$ .",
"A permutation $\\sigma $ that does not contain $\\pi $ as a pattern is said to avoid $\\pi $ .",
"For example the permutation $\\sigma =316452$ contains $\\pi = 2431$ as a pattern, whose occurrences are 3642 and 3652.",
"But $\\sigma $ avoids the pattern 2413 as none of its subsequences of length 4 is order-isomorphic to 2413.",
"The pattern containment relation $\\preceq $ is a partial order on permutations, and a permutation class $\\mathcal {C}$ is a downset under this order: for any $\\sigma \\in \\mathcal {C}$ , if $\\pi \\preceq \\sigma $ , then we also have $\\pi \\in \\mathcal {C}$ .",
"For every set $B$ , the set $Av(B)$ of permutations avoiding any pattern of $B$ is a class.",
"Furthermore every class $\\mathcal {C}$ can be rewritten as $\\mathcal {C} = Av(B)$ for a unique antichain $B$ (i.e., a unique set of pairwise incomparable elements) called the basis of $\\mathcal {C}$ .",
"The basis of a class $\\mathcal {C}$ may be finite or infinite; it is described as the set of permutations that do not belong to $\\mathcal {C}$ and that are minimal in the sense of $\\preceq $ for this criterion.",
"In the following, we only consider classes whose basis $B$ is given explicitly, and is finite.",
"This does not cover the whole range of permutation classes, but it is a reasonable assumption when dealing with algorithms on permutation classes, that take a finite description of a permutation class as input.",
"Moreover, as proved by [1], it is necessary that $B$ is finite as soon as the set $\\mathcal {S}_{\\mathcal {C}}$ of simple permutations in $\\mathcal {C}= Av(B)$ is finite.",
"Consequently the assumption of the finiteness of $B$ is not a restriction when working on permutation classes such that $\\mathcal {S}_{\\mathcal {C}}$ is finite, which is the context of our study."
],
[
"Simple permutations and substitution decomposition of permutations",
"An interval (or block) of a permutation $\\sigma $ of size $n$ is a subset $\\lbrace i,\\ldots ,(i+\\ell -1)\\rbrace $ of consecutive integers of $\\lbrace 1,\\ldots ,n\\rbrace $ whose images by $\\sigma $ also form an interval of $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"The integer $\\ell $ is called the size of the interval.",
"A permutation $\\sigma $ is simple when it is of size at least 4 and it contains no interval, except the trivial ones: those of size 1 (the singletons) or of size $n$ ($\\sigma $ itself).",
"The permutations 1, 12 and 21 also have only trivial intervals, nevertheless they are not considered to be simple here.",
"Moreover no permutation of size 3 has only trivial intervals.",
"For a detailed study of simple permutations, in particular from an enumerative point of view, we refer the reader to [1], [2], [8].",
"Let $\\sigma $ be a permutation of size $n$ and $\\pi ^{1},\\ldots ,\\pi ^{n}$ be $n$ permutations of size $p_1, \\ldots , p_n$ respectively.",
"Define the substitution $\\sigma [\\pi ^{1}, \\pi ^{2},\\ldots , \\pi ^{n}]$ of $\\pi ^{1},\\pi ^{2} , \\ldots , \\pi ^{n}$ in $\\sigma $ to be the permutation of size $p_1 + \\ldots + p_n$ obtained by concatenation of $n$ sequences of integers $S^1, \\ldots , S^n$ from left to right, such that for every $i,j$ , the integers of $S^i$ form an interval, are ordered in a sequence order-isomorphic to $\\pi ^{i}$ , and $S^i$ consists of integers smaller than $S^j$ if and only if $\\sigma _i < \\sigma _j$ .",
"For instance, the substitution $ 1\\, 3\\, 2[2\\, 1, 1\\, 3\\, 2, 1]$ gives the permutation $ 2\\, 1\\, \\, 4\\, 6\\, 5\\,\\, 3$ .",
"We say that a permutation $\\pi $ is 12-indecomposable (resp.",
"21-indecomposable) if it cannot be written as $12[\\pi ^{1},\\pi ^{2}]$ (resp.",
"$21[\\pi ^{1},\\pi ^{2}]$ ), for any permutations $\\pi ^{1}$ and $\\pi ^{2}$ .",
"Simple permutations allow to describe all permutations through their substitution decomposition.",
"Theorem 2.1 ([1]) Every permutation $\\pi $ of size $n$ with $n \\ge 2$ can be uniquely decomposed as follows, 12 (resp.",
"21, $\\sigma $ ) being called the root of $\\pi $ : $12[\\pi ^{1},\\pi ^{2}]$ , with $\\pi ^{1}$ 12-indecomposable, $21[\\pi ^{1},\\pi ^{2}]$ , with $\\pi ^{1}$ 21-indecomposable, $\\sigma [\\pi ^{1},\\pi ^{2},\\ldots ,\\pi ^{k}]$ , with $\\sigma $ a simple permutation of size $k$ .",
"To account for the first two items of Theorem REF in later discussions, we furthermore introduce the following notations: For any set $ ofpermutations, $ +$ (resp.",
"$ -$) denotes the set of permutations of$ that are 12-indecomposable (resp.",
"21-indecomposable).",
"Notice that even when $ is a permutation class, this is not the case for$ +$ and $ -$ in general.$ Theorem REF provides the first step in the decomposition of a permutation $\\pi $ .",
"To obtain its full decomposition, we can recursively decompose the permutations $\\pi ^{i}$ in the same fashion, until we reach permutations of size 1.",
"This recursive decomposition can naturally be represented by a tree, that is called the substitution decomposition tree (or decomposition tree for short) of $\\pi $ .",
"Each internal node of the tree is labeled by $12,21$ or by a simple permutation and the leaves represent permutation 1.",
"Notice that in decomposition trees, the left child of a node labeled 12 (resp.",
"21) is never labeled by 12 (resp.",
"21), since $\\pi ^{1}$ is 12-indecomposable (resp.",
"21-indecomposable) in the first (resp.",
"second) item of Theorem REF .",
"Example 2.2 The permutation $\\pi = 8\\ 9\\ 5\\ 11\\ 7\\ 6\\ 10\\ 17\\ 2\\ 1\\ 3\\ 4\\ 14\\ 16\\ 13\\ 15\\ 12$ is recursively decomposed as $\\pi =2413[4517326,1,2134,35241] =2413[31524[12[1,1],1,1,21[1,1],1]],1,12[21[1,1],12[1,1]],\\\\21[2413[1,1,1,1],1]] $ and its decomposition tree is given in Figure REF .",
"Figure: Decomposition tree of π\\pi (from Ex.",
").The substitution closure ${\\hat{\\mathcal {C}}}$ of a permutation classthat contains permutations 12 and 21.",
"We will assume so in the rest of this article to avoid trivial cases.",
"$\\mathcal {C}$ is defined as the set of permutations whose decomposition trees have internal nodes labeled by either $12, 21$ or a simple permutation of $\\mathcal {C}$ .",
"Notice that $\\mathcal {C}$ and ${\\hat{\\mathcal {C}}}$ therefore contain the same simple permutations.",
"Obviously, for any class $\\mathcal {C}$ , we have $\\mathcal {C} \\subseteq {\\hat{\\mathcal {C}}}$ .",
"When the equality holds, the class $\\mathcal {C}$ is said to be substitution-closed (or sometimes wreath-closed).",
"But this is not always the case, and the simplest example is given by $\\mathcal {C}= Av(213)$ .",
"This class contains no simple permutation hence its substitution closure is the class of separable permutations of [6], i.e.",
"of permutations whose decomposition trees have internal nodes labeled by 12 and 21.",
"It is immediate to notice that $213 \\in {\\hat{\\mathcal {C}}}$ whereas of course $213 \\notin \\mathcal {C}$ .",
"A characterization of substitution-closed classes useful for our purpose is given in [1]: A class is substitution-closed if and only if its basis contains only simple permutations."
],
[
"Algorithmic context of our work",
"Putting together the work reported in this article and recent algorithms from the litterature provides a full algorithmic chain starting with the finite basis $B$ of a permutation class $\\mathcal {C}$ , and computing a specification for $\\mathcal {C}$ .",
"The hope for such a very general algorithm is of course very tenuous, and the algorithm we describe below will compute its output only when some hypothesis are satisfied, which are also tested algorithmically.",
"Figure REF summarizes the main steps of the algorithm.",
"Figure: Automatic process from the basis of apermutation class to generating function andBoltzmann sampler.The algorithms performing the first two steps of the algorithmic process of Figure REF are as follows.",
"First step : Finite number of simple permutations First, we check whether $\\mathcal {C} = Av(B)$ contains only a finite number of simple permutations.",
"This is achieved using algorithms of [4] when the class is substitution-closed and of [5] otherwise.",
"The complexity of these algorithms are respectively $\\mathcal {O}(n \\log n)$ and $\\mathcal {O}(n^{4k})$ , where $n = \\sum _{\\beta \\in B} |\\beta |$ and $k = |B|$ .",
"Second step : Computing simple permutations The second step of the algorithm is the computation of the set of simple permutations $\\mathcal {S}_{\\mathcal {C}}$ contained in $\\mathcal {C} = Av(B)$ , when we know it is finite.",
"Again, when $\\mathcal {C}$ is substitution-closed, $\\mathcal {S}_{\\mathcal {C}}$ can be computed by an algorithm that is more efficient than in the general case.",
"The two algorithms are described in [16], and their complexity depends on the output: $\\mathcal {O}(N \\cdot \\ell ^{p+2}\\cdot |B|)$ in general and $\\mathcal {O}(N \\cdot \\ell ^{4})$ for substitution-closed classes, with $N = |\\mathcal {S}_{\\mathcal {C}}|$ , $p = \\max \\lbrace |\\beta | : \\beta \\in B\\rbrace $ and $\\ell = \\max \\lbrace |\\pi | : \\pi \\in \\mathcal {S}_{\\mathcal {C}}\\rbrace $ .",
"Sections and will then explain how to derive a specification for $\\mathcal {C}$ from $\\mathcal {S}_{\\mathcal {C}}$ ."
],
[
"Ambiguous combinatorial system describing $\\mathcal {C}$",
"We describe here an algorithm that takes as input the set $\\mathcal {S}_{\\mathcal {C}}$ of simple permutations in a class $\\mathcal {C}$ and the basis $B$ of $\\mathcal {C}$ , and that produces in output a (possibly ambiguous) system of combinatorial equations describing the permutations of $\\mathcal {C}$ through their decomposition trees.",
"The main ideas are those of Theorem 10 of [1], but unlike this work, we make the whole process fully algorithmic."
],
[
"The simple case of substitution-closed classes",
"Recall that $\\mathcal {C}$ is a substitution-closed permutation class when $\\mathcal {C}={\\hat{\\mathcal {C}}}$ , or equivalently when the permutations in $\\mathcal {C}$ are exactly the ones whose decomposition trees have internal nodes labeled by $12, 21$ or any simple permutation of $$ .",
"Then Theorem REF directly yields the following system $\\mathcal {E} _{{\\hat{\\mathcal {C}}}}$ : ${\\hat{\\mathcal {C}}}&=& 1\\ \\uplus \\ 12[{\\hat{\\mathcal {C}}}^+, {\\hat{\\mathcal {C}}}]\\ \\uplus \\ 21[{\\hat{\\mathcal {C}}}^-, {\\hat{\\mathcal {C}}}] \\ \\textstyle \\uplus \\biguplus _{\\pi \\in {\\mathcal {S}}_{\\hat{\\mathcal {C}}}} \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}] \\\\{\\hat{\\mathcal {C}}}^+ &=& 1\\ \\uplus \\ 21[{\\hat{\\mathcal {C}}}^-, {\\hat{\\mathcal {C}}}]\\ \\uplus \\ \\textstyle \\biguplus _{\\pi \\in {\\mathcal {S}}_{\\hat{\\mathcal {C}}}} \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}] \\\\{\\hat{\\mathcal {C}}}^- &=& 1\\ \\uplus \\ 12[{\\hat{\\mathcal {C}}}^+, {\\hat{\\mathcal {C}}}]\\ \\uplus \\ \\textstyle \\biguplus _{\\pi \\in {\\mathcal {S}}_{\\hat{\\mathcal {C}}}} \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}].",
"$ By uniqueness of substitution decomposition, unions are disjoint and so Equations (REF ) to () describe unambiguously the substitution closure ${\\hat{\\mathcal {C}}}$ of a permutation class $\\mathcal {C}$ .",
"For a substitution-closed class (and the substitution closure of any class), this description gives a combinatorial specification.",
"Hence, it provides an efficient way to compute the generating function of the class, and to generate uniformly at random a permutation of a given size in the class."
],
[
"Adding constraints for classes that are not substitution-closed",
"When ${\\mathcal {C}}$ is not substitution-closed, we compute a new system by adding constraints to the system obtained for ${\\hat{\\mathcal {C}}}$ , as in [1].",
"Denoting by $X\\langle Y\\rangle $ the set of permutations of $X$ that avoid the patterns in $Y$ , we have $ {\\hat{\\mathcal {C}}}\\langle {B^\\star }\\rangle $ where ${B^\\star }$ is the subset of non-simple permutations of $B$ .",
"Noticing that ${\\mathcal {S}}_{\\hat{\\mathcal {C}}}= {\\mathcal {S}}_ (by definition of $${\\hat{\\mathcal {C}}}$$), and since$ = ${\\hat{\\mathcal {C}}}$ ${B^\\star }$$for $ { , +, -}$, Equations~(\\ref {eqn:Wc1}) to~(\\ref {eqn:Wc3}) give\\begin{eqnarray}{\\hat{\\mathcal {C}}}\\langle {B^\\star }\\rangle &=& 1\\ \\uplus \\ 12[{\\hat{\\mathcal {C}}}^+, {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle \\ \\uplus \\ 21[{\\hat{\\mathcal {C}}}^-, {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle \\ \\uplus \\textstyle \\biguplus _{\\pi \\in {\\mathcal {S}}_ \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle \\\\{\\hat{\\mathcal {C}}}^+ \\langle {B^\\star }\\rangle &=& 1\\ \\uplus \\ 21[{\\hat{\\mathcal {C}}}^-, {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle \\ \\uplus \\ \\textstyle \\biguplus _{\\pi \\in {\\mathcal {S}}_ \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle \\\\{\\hat{\\mathcal {C}}}^- \\langle {B^\\star }\\rangle &=& 1\\ \\uplus \\ 12[{\\hat{\\mathcal {C}}}^+, {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle \\ \\uplus \\ \\textstyle \\biguplus _{\\pi \\in {\\mathcal {S}}_ \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle , }all these unions being disjoint.This specification is not complete, since sets of the form \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle are not immediately described from {\\hat{\\mathcal {C}}}\\langle {B^\\star }\\rangle .Theorem 10 of \\cite {AA05} explains how sets such as \\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle can be expressed as union of smaller sets:\\pi [{\\hat{\\mathcal {C}}}, \\dots , {\\hat{\\mathcal {C}}}]\\langle {B^\\star }\\rangle = \\textstyle \\bigcup _{i=1}^{k} \\pi [{\\hat{\\mathcal {C}}}\\langle E_{i,1} \\rangle ,{\\hat{\\mathcal {C}}}\\langle E_{i,2} \\rangle ,\\ldots ,{\\hat{\\mathcal {C}}}\\langle E_{i,k} \\rangle ]where E_{i,j} are sets of permutations which are patterns of somepermutations of {B^\\star }.This introduces sets of the form {\\hat{\\mathcal {C}}}\\langle E_{i,j} \\rangle on the right-hand side of an equation of the system that do not appear on the left-hand side of any equation.",
"We will call such sets \\emph {right-only} sets.Taking E_{i,j} instead of {B^\\star } in Equations~(\\ref {eqn:const1}) to~(\\ref {eqn:const3}), we can recursively compute these right-only sets by introducing new equations in the system.This process terminates since there exists only a finite number of sets of patterns of elements of {B^\\star } (as B is finite).",
"Let us introduce some definitions to describe these sets E_{i,j}.",
"}}A \\emph {generalized substitution} \\sigma \\lbrace \\pi ^{1}, \\pi ^{2} ,\\ldots ,\\pi ^{n}\\rbrace is defined as a substitution (seep.\\pageref {thm:decomp_perm_AA05}) with the particularity that any\\pi ^i may be the empty permutation (denoted by~0).",
"Specifically\\sigma [\\pi ^{1}, \\pi ^{2} ,\\ldots , \\pi ^{n}] necessarily contains\\sigma whereas \\sigma \\lbrace \\pi ^{1}, \\pi ^{2} ,\\ldots , \\pi ^{n}\\rbrace mayavoid \\sigma .",
"For instance, 1\\, 3\\, 2 \\lbrace 2\\, 1, 0, 1\\rbrace = 2\\, 1\\,3\\in Av(132).\\end{eqnarray}An {\\em embedding of \\gamma in~{\\pi =\\pi _1\\dots \\pi _n}} is amap $$ from $ {1, ..., n}$ to the set of (possibly empty)blocks\\footnote {Recall that here blocks of a permutation are sets of \\emph {indices}.",
"}of $$ such that:\\vspace{-6.5pt}\\begin{itemize}\\item if blocks \\alpha (i) and \\alpha (j) are not empty, and i<j,then \\alpha (i) consists of smaller indices than \\alpha (j);\\end{itemize}\\item as a word, $ (1) ...(n)$ is a factorization of the word$ 1...||$ (which may include empty factors).$ denoting $\\gamma _I$ the pattern corresponding to $\\gamma _{i_1} \\ldots \\gamma _{i_{\\ell }}$ for any block $I$ of indices from $i_1$ to $i_{\\ell }$ in increasing order, we have $\\pi \\lbrace \\gamma _{\\alpha (1)},\\dots ,\\gamma _{\\alpha (n)}\\rbrace =\\gamma $ .",
"There are 11 embeddings of $\\gamma = 5\\,4\\,6\\,3\\,1\\,2$ into $\\pi = 3\\,1\\,4\\,2$ , which correspond for instance to the generalized substitutions $\\pi \\lbrace 3241,12,0,0\\rbrace $ , $\\pi \\lbrace 3241,0,0,12\\rbrace $ and $\\pi \\lbrace 0,0,3241,12\\rbrace $ for the same expression of $\\gamma $ as the substitution ${21[3241,12]}$ , or $\\pi \\lbrace 3241,1,0,1\\rbrace $ which is the only one corresponding to $312[3241,1,1]$ .",
"Notice that this definition of embeddings conveys the same notion than in [1], but it is formally different and it will turn to be more adapted to the definition of the sets $E_{i,j}$ .",
"Equations () to () can be viewed as Equations (REF ) to () “decorated” with pattern avoidance constraints.",
"These constraints apply to every set $\\pi [{\\hat{\\mathcal {C}}}_1, \\dots , {\\hat{\\mathcal {C}}}_n]$ that appears in a disjoint union on the right-hand side of an equation.",
"For each such set, the pattern avoidance constraints can be expressed by pushing constraints into the subtrees, using embeddings of excluded patterns in the root $\\pi $ .",
"For instance, assume that $\\gamma = 5\\,4\\,6\\,3\\,1\\,2 \\in B^\\star $ and $\\mathcal {S}_{\\mathcal {C}}=\\lbrace 3142\\rbrace $ , and consider $3142[{\\hat{\\mathcal {C}}}, {\\hat{\\mathcal {C}}},{\\hat{\\mathcal {C}}},{\\hat{\\mathcal {C}}}]\\langle \\gamma \\rangle $ .",
"The embeddings of $\\gamma $ in 3142 indicates how pattern $\\gamma $ can be found in the subtrees in $3142[{\\hat{\\mathcal {C}}}, {\\hat{\\mathcal {C}}},{\\hat{\\mathcal {C}}},{\\hat{\\mathcal {C}}}]$ .",
"As example the last embedding of the previous example tells that $\\gamma $ can spread over all the subtrees of 3142 except the third.",
"In order to avoid this particular embedding of $\\gamma $ , it is enough to avoid one of the induced pattern $\\gamma _I$ on one of the subtrees.",
"However, in order to ensure that $\\gamma $ is avoided, the constraints resulting from all the embeddings must be considered and merged.",
"More precisely, consider a set $\\pi [1, \\dots ,n]\\langle \\gamma \\rangle $ , $\\pi $ being a simple permutation.",
"Let $\\lbrace \\alpha _1, \\dots , \\alpha _\\ell \\rbrace $ be the set of embeddings of $\\gamma $ in $\\pi $ , each $\\alpha _i$ being associated to a generalized substitution $\\gamma = \\pi \\lbrace \\gamma _{\\alpha _i(1)},\\dots ,\\gamma _{\\alpha _i(n)}\\rbrace $ where $\\gamma _{\\alpha _i(k)}$ is embedded in $\\pi _k$ .",
"Then the constraints are propagated according to the following equation: $ \\pi [1, \\dots ,n] \\langle \\gamma \\rangle =\\textstyle \\bigcup _{(k_1, \\dots , k_\\ell ) \\in K^\\pi _\\gamma } \\pi [1 \\langle E_{1,k_1 \\dots k_\\ell } \\rangle , \\dots ,n \\langle E_{n,k_1 \\dots k_\\ell } \\rangle ]$ where $K^\\pi _\\gamma =\\lbrace (k_1, \\dots , k_\\ell ) \\in [1..n]^\\ell \\ |\\ \\forall i,\\ \\gamma _{\\alpha _i(k_i)} \\ne 0\\rbrace $ and $E_{m,k_1 \\dots k_\\ell }= \\lbrace \\gamma _{\\alpha _i(k_i)}\\ |\\ i \\in [1..\\ell ] \\text{ and } k_i=m \\rbrace $ is a set containing at least $\\gamma $ for $(k_1, \\dots , k_\\ell ) \\in K^\\pi _\\gamma $ .",
"In a tuple $(k_1, \\ldots , k_{\\ell })$ of $K^\\pi _\\gamma $ , $k_i$ indicates a subtree of $\\pi $ where the pattern avoidance constraint ($\\gamma _{\\alpha _i(k_i)}$ excluded) forbids any occurrence of $\\gamma $ that could result from the embedding $\\alpha _i$ .",
"The set $E_{m,k_1 \\dots k_\\ell }$ represents the pattern avoidance constraints that have been pushed into the $m$ -th subtree of $\\pi $ by embeddings $\\alpha _i$ of $\\gamma $ in $\\pi $ where the block $\\alpha _i(k_i)$ of $\\gamma $ is embedded into $\\pi _m$ .",
"Starting from a finite basis of patterns $B$ , Algorithm REF describes the whole process to compute an ambiguous system defining the class $ Av(B)$ knowing its set of simple permutations ${\\mathcal {S}}_.The propagation of the constraints expressed by Equation~(\\ref {eq:propagate}) is performed by theprocedure~\\textsc {AddConstraints}.",
"It is applied to every set ofthe form $ [1, ..., n] B' $ that appears in the equation defining some$${\\hat{\\mathcal {C}}}$ B' $ by the procedure~\\textsc {ComputeEqn}.",
"Finally,Algorithm~\\ref {alg:sys-ambigu} computes an ambiguous system for apermutation class $ Av(B)$ containing a finite number of simplepermutations: it starts from Equations~(\\ref {eqn:const1}) to~(\\ref {eqn:const3}),and adds new equations to this system callingprocedure~\\textsc {ComputeEqn}, until every $ [1, ..., n] B' $ is replaced by some $ [1, ..., n]$ and until every $ i = ${\\hat{\\mathcal {C}}}$ B'i $ is defined by an equation of the system.",
"All thesets $ B'$ are sets ofpatterns of some permutations in $ B$.",
"Since there is only afinite number of patterns of elements of $ B$, there is afinite number of possible $ B'$, and Algorithm~\\ref {alg:sys-ambigu} terminates.$ PGfuncAddConstraintsend EfuncComputeEqnend [h!]",
"$B$ is a finite basis of patterns defining ${\\mathcal {C}}=Av(B)$ such that $\\mathcal {S}_ is known and finite.$ A system of equations of the form $ \\bigcup \\pi [1, \\dots ,n]$ defining $.$ $\\mathcal {E}\\leftarrow $ ComputeEqn($({\\hat{\\mathcal {C}}},{B^\\star })$ ) $\\cup $ ComputeEqn($({\\hat{\\mathcal {C}}}^+,{B^\\star })$ ) $\\cup $ ComputeEqn($({\\hat{\\mathcal {C}}}^-,{B^\\star })$ ) there is a right-only ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle B^{\\prime }\\rangle $ in some equation of $\\mathcal {E} $ $\\mathcal {E}\\leftarrow \\mathcal {E}~\\cup $ ComputeEqn(${\\hat{\\mathcal {C}}}^{\\varepsilon }$ , $B^{\\prime }$ ) /* Returns an equation defining ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle B^{\\prime }\\rangle $ as a union of $\\pi [1, \\dots , n]$ */ /* $B^{\\prime }$ is a set of permutations, ${\\hat{\\mathcal {C}}}^{\\varepsilon }$ is given by ${\\mathcal {S}}_{\\hat{\\mathcal {C}}}$ and $\\varepsilon \\in \\lbrace ~~ , +, -\\rbrace $ */ ($($${\\hat{\\mathcal {C}}}^\\varepsilon ,B^{\\prime }$$($) $\\mathcal {E} \\leftarrow $ Equation () or () or () (depending on $\\varepsilon $ ) written with $B^{\\prime }$ instead of ${B^\\star }$ $t=\\pi [1, \\dots , n] \\langle B^{\\prime } \\rangle $ that appears in $\\mathcal {E}$ $t\\leftarrow $ AddConstraints$(\\pi [1, \\dots , n], B^{\\prime } )$ $\\mathcal {E}$ /* Returns a rewriting of $\\pi [1 \\dots n] \\langle E \\rangle $ as a union $\\bigcup \\pi [1, \\dots n]$ */ ($($$(\\pi [1 \\ldots n] , E)$$($) $E = \\emptyset $ return $\\pi [1 \\dots n]$ choose $\\gamma \\in E$ and compute all the embeddings of $\\gamma $ in $\\pi $ compute $K^\\pi _\\gamma $ and sets $E_{m,k_1 \\dots k_\\ell }$ defined in Equation (REF ) return $\\bigcup _{(k_1, \\dots , k_\\ell ) \\in K^\\pi _\\gamma } \\textsc {AddConstraints}(\\pi [1 \\langle E_{1,k_1 \\dots k_\\ell } \\rangle , \\dots ,n \\langle E_{n,k_1 \\dots k_\\ell } \\rangle ], E \\setminus \\gamma )$ .",
"AmbiguousSystem($B$ ) Consider for instance the class $Av(B)$ for $B=\\lbrace 1243,2413,531642,41352\\rbrace $ : $ contains only one simple permutation (namely $ 3142$), and $${B^\\star }$ = {1243}$.",
"Applying Algorithm~\\ref {alg:sys-ambigu} to this class $ gives the following system of equations: ${\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle &=& 1 \\ \\cup \\ 12[{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\cup \\ 12[{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ] \\ \\cup \\ 21[{\\hat{\\mathcal {C}}}^{-}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle ] \\nonumber \\\\&\\ \\cup \\ & 3 1 4 2[{\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\cup \\ 3 1 4 2[{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\\\{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle &=& 1 \\ \\cup \\ 21[{\\hat{\\mathcal {C}}}^{-}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle ]\\\\{\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle &=& 1 \\ \\cup \\ 12[{\\hat{\\mathcal {C}}}^{+}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ] \\ \\cup \\ 21[{\\hat{\\mathcal {C}}}^{-}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ]\\\\{\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle &=& 1 \\ \\cup \\ 1 2 [{\\hat{\\mathcal {C}}}^{+}\\langle 2 1 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ].$ Disambiguation of the system In the above, Equation (REF ) gives an ambiguous description of the class ${\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle $ .",
"As noticed in [1], we can derive an unambiguous equation using the inclusion-exclusion principle: ${\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle = 1 \\ \\cup \\ 1 2 [{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\cup \\ 1 2[{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 4 3 \\rangle ,{\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ] \\ \\setminus \\ 1 2 [{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ] \\ \\cup \\ 2 1 [{\\hat{\\mathcal {C}}}^{-}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle ] \\ \\cup \\ $ $3 1 4 2 [{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle ,{\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\cup \\ 3 1 42 [{\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 21 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\setminus \\ 3 1 42[{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1\\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ]$.",
"The system so obtained contains negative terms in general.",
"This still gives a system of equations allowing to compute the generating function of the class.",
"However, this cannot be easily used for random generation, as the subtraction of combinatorial objects is not handled by random samplers.",
"In this section we disambiguate this system to obtain a new positive one: the key idea is to replace the negative terms by complement sets, hereby transforming pattern avoidance constraints into pattern containment constraints.",
"General framework The starting point of the disambiguation is to rewrite ambiguous terms like $A \\cup B \\cup C$ as a disjoint union $(A \\cap B\\cap C) \\uplus (\\bar{A} \\cap B \\cap C) \\uplus (\\bar{A} \\cap \\bar{B}\\cap C) \\uplus (\\bar{A} \\cap B \\cap \\bar{C}) \\uplus (A \\cap \\bar{B}\\cap C) \\uplus (A \\cap \\bar{B} \\cap \\bar{C}) \\uplus (A \\cap B \\cap \\bar{C})\\textrm {.",
"}$ By disambiguating the union $A \\cup B \\cup C$ using complement sets instead of negative terms, we obtain an unambiguous description of the union with only positive terms.",
"But when taking the complement of a set defined by pattern avoidance constraints, these are transformed into pattern containment constraints.",
"Therefore, for any set $\\mathcal {P}$ of permutations, we define the restriction $\\mathcal {P}\\langle E \\rangle (A)$ of $\\mathcal {P}$ as the set of permutations that belong to $\\mathcal {P}$ and that avoid every pattern of $E$ and contain every pattern of $A$ .",
"This notation will be used when $\\mathcal {P} = {\\hat{\\mathcal {C}}}^{\\varepsilon }$ , for $\\varepsilon \\in \\lbrace ~~ , +,-\\rbrace $ and $ a permutation class.",
"With this notation, notice alsothat for $ A=$, $ E = E ()$ is a standard permutation class.",
"Restrictionshave the nice feature of being stable by intersection as$ PE (A) PE' (A') = PE E' (A A')$.",
"We also define a {\\em restriction term} to be a set of permutationsdescribed as $ [S1,S2,...,Sn]$ where $$ is a simplepermutation or $ 12$ or $ 21$ and the $ Si$ are restrictions.",
"By uniqueness of the substitution decomposition of a permutation, restriction terms are stable by intersection as well and the intersection is performed componentwise for terms sharing the same root: $ [S1,S2,...,Sn] [T1,T2,...,Tn] = [S1T1,S2T2,...,SnTn]$.$ Disambiguate The disambiguation of the system obtained by Algorithm REF is performed by Algorithm REF .",
"It consists in two main operations.",
"One is the disambiguation of an equation according to the root of the terms that induce ambiguity, which may introduce right-only restrictions.",
"This leads to the second procedure which computes new equations (that are added to the system) to describe these new restrictions (Algorithm REF ).",
"As stated in Section , every equation $F$ of our system can be written as $t =1 \\cup t_{1} \\cup t_{2} \\cup t_{3} \\ldots \\cup t_{k}$ where the $t_{i}$ are restriction terms and $t$ is a restriction.",
"By uniqueness of the substitution decomposition of a permutation, terms of this union which have different roots $\\pi $ are disjoint.",
"Thus for an equation we only need to disambiguate unions of terms with same root.",
"[t] A ambiguous system $\\mathcal {E} $ of combinatorial equations /* obtained by Algo.",
"REF */ An unambiguous system of combinatorial equations equivalent to $\\mathcal {E} $ there is an ambiguous equation $F$ in $\\mathcal {E} $ Take $\\pi $ a root that appears several times in $F$ in an ambiguous way Replace the restriction terms of F whose root is $\\pi $ by a disjoint union using Eq.",
"(REF ) – (REF ) there exists a right-only restriction ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ in some equation of $\\mathcal {E} $ $\\mathcal {E} \\longleftarrow \\mathcal {E} \\bigcup $ ComputeEqnForRestriction(${\\hat{\\mathcal {C}}}^{\\varepsilon }$ ,$E$ ,$A$ ).",
"/* See Algo.",
"REF */ $\\mathcal {E} $ DisambiguateSystem($\\mathcal {E} $ ) For example in Equation (REF ), there are two pairs of ambiguous terms which are terms with root 3142 and terms with root 12.",
"Every ambiguous union can be written in the following unambiguous way: $\\textstyle \\bigcup _{i=1}^{k} t_{i}=\\textstyle \\biguplus _{X\\subseteq [1\\ldots k], X \\ne \\emptyset } \\bigcap _{i \\in X} t_{i}\\cap \\bigcap _{i \\in \\overline{X}} \\overline{t_{i}}, $ where the complement $\\overline{t_{i}}$ of a restriction term $t_{i}$ is defined as the set of permutations of ${\\hat{\\mathcal {C}}}$ whose decomposition tree has the same root than $t_{i}$ but that do not belong to $t_{i}$ .",
"Equation REF below shows that $\\overline{t_{i}}$ is not a term in general but can be expressed as a disjoint union of terms.",
"By distributivity of $\\cap $ over $\\uplus $ , the above expression can therefore be rewritten as a disjoint union of intersection of terms.",
"Because terms are stable by intersection, the right-hand side of Equation REF is hereby written as a disjoint union of terms.",
"For instance, consider terms with root 3142 in Equation (REF ): $t_{1} = 3 1 4 2 [{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle ,{\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ]$ and $t_{2} = 3 14 2 [{\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 21 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2\\rangle ]$ .",
"Equation (REF ) applied to $t_1$ and $t_2$ gives an expression of the form ${\\hat{\\mathcal {C}}}\\langle 1243\\rangle = 1 \\cup 12[\\ldots ] \\cup 12[\\ldots ] \\cup 21[\\ldots ] \\cup (t_{1} \\cap t_{2}) \\uplus (t_{1} \\cap \\overline{t_{2}}) \\uplus (\\overline{t_{1}} \\cap t_{2})\\textrm {.",
"}$ To compute the complement of a term $t$ , it is enough to write that $\\overline{t}=\\biguplus _{X \\subseteq \\lbrace 1,\\ldots ,n\\rbrace , X \\ne \\emptyset } \\pi [{\\mathcal {S}}^{\\prime }_{1},\\ldots ,{\\mathcal {S}}^{\\prime }_{n}] \\mbox{ where }{\\mathcal {S}}^{\\prime }_{i} = \\overline{{\\mathcal {S}}_{i}}\\mbox{ if }i \\in X\\mbox{ and }{\\mathcal {S}}^{\\prime }_{i} = {\\mathcal {S}}_{i}\\mbox{ otherwise},$ with the convention that $\\overline{{\\mathcal {S}}_{i}} = {\\hat{\\mathcal {C}}}^{\\varepsilon } \\setminus {\\mathcal {S}}_{i}$ for ${\\mathcal {S}}_{i} = {\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ .",
"Indeed, by uniqueness of substitution decomposition, the set of permutations of ${\\hat{\\mathcal {C}}}$ that do not belong to $t$ but whose decomposition tree has root $\\pi $ can be written as the union of terms $u = \\pi [{\\mathcal {S}}^{\\prime }_{1},{\\mathcal {S}}^{\\prime }_{2},\\ldots ,{\\mathcal {S}}^{\\prime }_{n}]$ where ${\\mathcal {S}}^{\\prime }_{i} = {\\mathcal {S}}_{i}$ or ${\\mathcal {S}}^{\\prime }_{i}=\\overline{{\\mathcal {S}}_{i}}$ and at least one restriction ${\\mathcal {S}}_{i}$ must be complemented.",
"For example $\\overline{21[{\\mathcal {S}}_{1},{\\mathcal {S}}_{2}]} =21[{\\mathcal {S}}_{1},\\overline{{\\mathcal {S}}_{2}}] \\uplus 21[\\overline{{\\mathcal {S}}_{1}},{\\mathcal {S}}_{2}]\\uplus 21[\\overline{{\\mathcal {S}}_{1}},\\overline{{\\mathcal {S}}_{2}}]$ .",
"The complement operation being pushed from restriction terms down to restrictions, we now compute $\\overline{{\\mathcal {S}}}$ , for a given restriction ${\\mathcal {S}} = {\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ , $\\overline{{\\mathcal {S}}}$ denoting the set of permutations of ${\\hat{\\mathcal {C}}}^{\\varepsilon }$ that are not in ${\\mathcal {S}}$ .",
"Notice that given a permutation $\\sigma $ of $A$ , then any permutation $\\tau $ of ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\sigma \\rangle $ is in $\\overline{{\\mathcal {S}}}$ because $\\tau $ avoids $\\sigma $ whereas permutations of ${\\mathcal {S}}$ must contain $\\sigma $ .",
"Symmetrically, if a permutation $\\sigma $ is in $E$ then permutations of ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\rangle (\\sigma )$ are in $\\overline{{\\mathcal {S}}}$ .",
"It is straightforward to check that $\\textstyle \\overline{{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)} = \\big [\\bigcup _{\\sigma \\in E} {\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\rangle (\\sigma )\\big ]\\bigcup \\big [ \\bigcup _{\\sigma \\in A}{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\sigma \\rangle ()\\big ]$ .",
"Unfortunately this expression is ambiguous.",
"Like before we can rewrite it as an unambiguous union $ \\overline{{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)}= \\biguplus _{\\underset{X \\times Y \\ne \\emptyset \\times \\emptyset }{{X\\subseteq A, Y \\subseteq E}}}{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle X\\cup \\overline{Y} \\rangle (Y \\cup \\overline{X}) \\textrm {, where } \\overline{X} = A \\setminus X \\textrm { and }\\overline{Y} = E \\setminus Y \\textrm {.",
"}$ In our example (Equations (REF ) to ()), only trivial complements appear as every restriction is of the form ${\\hat{\\mathcal {C}}}\\langle \\sigma \\rangle ()$ or ${\\hat{\\mathcal {C}}}\\langle \\rangle (\\sigma )$ for which complements are respectively ${\\hat{\\mathcal {C}}}\\langle \\rangle (\\sigma )$ and ${\\hat{\\mathcal {C}}}\\langle \\sigma \\rangle ()$ .",
"All together, for any equation of our system, we are able to rewrite it unambiguously as a disjoint union of restriction terms.",
"As noticed before, some new right-only restrictions may appear during this process, for example as the result of the intersection of several restrictions or when complementing restrictions.",
"To obtain a complete system we must compute iteratively equations defining these new restrictions using Algorithm REF described below.",
"Finally, the terminaison of Algorithm REF is easily proved.",
"Indeed, for all the restrictions ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ that are considered in the inner loop of Algorithm REF , every permutation in the sets $E$ and $A$ is a pattern of some element of the basis $B$ of $\\mathcal {C}$ .",
"And since $B$ is finite, there is a finite number of such restrictions.",
"Compute an equation for a restriction Let ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)$ be a restriction.",
"Our goal here is to find a combinatorial specification of this restriction in terms of smaller restriction terms (smaller w.r.t.",
"inclusion).",
"If $A = \\emptyset $ , this is exactly the problem addressed in Section REF and solved by pushing down the pattern avoidance constraints in the procedure AddConstraints of Algorithm REF .",
"Algorithm REF below shows how to propagate also the pattern containment constraints induced by $A \\ne \\emptyset $ .",
"AMfuncAddMandatoryend [H] ${\\hat{\\mathcal {C}}}^{\\varepsilon }, E,A$ with $E,A$ sets of permutations, ${\\hat{\\mathcal {C}}}^{\\varepsilon }$ given by ${\\mathcal {S}}_{\\hat{\\mathcal {C}}}$ and $\\varepsilon \\in \\lbrace ~~ , +, -\\rbrace $ .",
"An equation defining ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)$ as a union of restriction terms.",
"$F \\leftarrow $ Equation (REF ) or () or () (depending on $\\varepsilon $ ) $\\sigma \\in E$ /* This step modifies $F$ !",
"*/ Replace any restriction term $t$ in $F$ by AddConstraints$(t, \\lbrace \\sigma \\rbrace )$/* See Algo.",
"REF */ $\\sigma \\in A$ /* This step modifies $F$ !",
"*/ Replace any restriction term $t$ in $F$ by AddMandatory$(t, \\sigma )$ $F$ ($($$\\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n],\\gamma $$($) a rewriting of $\\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n] (\\gamma )$ as a union of restriction terms using Equation (REF ).",
"ComputeEqnForRestriction$({\\hat{\\mathcal {C}}}^{\\varepsilon },E,A)$ The pattern containment constraints are propagated by AddMandatory, in a very similar fashion to the pattern avoidance constraints propagated by AddConstraints.",
"To compute $t(\\gamma )$ for $\\gamma $ a permutation and $t = \\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n]$ a restriction term, we first compute all embeddings of $\\gamma $ into $\\pi $ .",
"In this case, a permutation belongs to $t(\\gamma )$ if and only if at least one embedding is satisfied.",
"Hence, any restriction term $t = \\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n](\\gamma )$ rewrites as a (possibly ambiguous) union as follows: $\\textstyle \\bigcup _{i=1}^{\\ell } \\pi [{\\mathcal {S}}_{1}(\\gamma _{\\alpha _{i}(1)}),{\\mathcal {S}}_{2}(\\gamma _{\\alpha _{i}(2)}),\\ldots ,{\\mathcal {S}}_{n}(\\gamma _{\\alpha _{i}(n)})], $ where the $(\\alpha _{i})_{i \\in \\lbrace 1, \\ldots , \\ell \\rbrace }$ are all the embeddings of $\\gamma $ in $\\pi $ and if $\\gamma _{\\alpha _{i}(j)}=0$ , then ${\\mathcal {S}}_{j}(\\gamma _{\\alpha _{i}(j)}) = {\\mathcal {S}}_j$ .",
"For instance, for $t = 2413[{\\mathcal {S}}_{1},{\\mathcal {S}}_{2},{\\mathcal {S}}_{3},{\\mathcal {S}}_{4}]$ and $\\gamma = 3214$ , there are 9 embeddings of $\\gamma $ into 2413, and the embedding $2413\\lbrace 321,1,0,0\\rbrace $ contributes to the above union with the term $2413[{\\mathcal {S}}_{1}(321),{\\mathcal {S}}_{2}(1),{\\mathcal {S}}_{3},{\\mathcal {S}}_{4}]$ .",
"Notice that although the unions of Equation REF may be ambiguous, they will be transformed into disjoint unions by the outer loop of Algorithm REF .",
"Finally, the algorithm produces an unambiguous system which is the result of a finite number of iterations of computing equations followed by their disambiguation.",
"Conclusion We provide an algorithm to compute a combinatorial specification for a permutation class $Av(B)$ , when its basis $B$ and the set of its simple permutations are finite and given as input.",
"The complexity of this algorithm is however still to analyse.",
"In particular, we observe a combinatorial explosion of the number of equations in the system obtained, that needs to be quantified.",
"Combined with existing algorithms, our procedure provides a full algorithmic chain from the basis (when finite) of a permutation class t̏o a specification for This procedure may fail to compute its result, when c̏ontains an infinite number of simple permutations, this condition being tested algorithmically.",
"This procedure has two natural algorithmic continuations.",
"First, with the dictionnary of [13], the constructors in the specification of $\\mathcal {C}$ can be directly translated into operators on the generating function $C(z)$ of $\\mathcal {C}$ , turning the specification into a system of (possibly implicit) equations defining $C(z)$ .",
"Notice that, using the inclusion-exclusion principle as in [1], a system defining $C(z)$ could also be obtained from an ambiguous system describing $\\mathcal {C}$ .",
"Second, the specification can be translated directly into a Boltzmann uniform random sampler of permutations in $\\mathcal {C}$ , in the same fashion as the above dictionnary (see [11]).",
"This second translation is possible only from an unambiguous system: indeed, whereas adapted when considering enumeration sequences, the inclusion-exclusion principle does not apply when working on the combinatorial objects themselves.",
"When generating permutations with a Boltzmann sampler, complexity is measured w.r.t.",
"the size of the permutation produced (and is linear if we allow a small variation on the size of the output permutation; quadratic otherwise) and not at all w.r.t.",
"the number of equations in the specification.",
"In our context, this dependency is of course relevant, and opens a new direction in the study of Boltzmann random samplers.",
"With a complete implementation of the algorithmic chain from $B$ to the specification and the Boltzmann sampler, one should be able to test conjectures on and study permutation classes.",
"One direction would be to somehow measure the randomness of permutations in a given class, by comparing random permutations with random permutations in a class, or random permutations in two different classes, w.r.t.",
"well-known statistics on permutations.",
"Another perspective would be to use the specifications obtained to compute or estimate the growth rates of permutation classes, to provide improvements on the known bounds on these growth rates.",
"We could also explore the possible use the computed specifications to provide more efficient algorithms to test membership of a permutation to a class.",
"However, a weekness of our procedure that we must acknowledge is that it fails to be completely general.",
"Although the method is generic and algorithmic, the classes that are fully handled by the algorithmic process are those containing a finite number of simple permutations.",
"By [1], such classes have finite basis (which is a restriction we imposed already), but they also have an algebraic generating function.",
"Of course, this is not the case for every permutation class.",
"We may wonder how restrictive this framework is, depending on which problems are studied.",
"First, does it often happen that a permutation class contains finitely many simple permutations?",
"To properly express what often means, a probability distribution on permutation classes should be defined, which is a direction of research yet to be explored.",
"Second, we may want to describe some problems (maybe like the distribution of some statistics) for which algebraic permutation classes are representative of all permutation classes.",
"To enlarge the framework of application of our algorithm, we could explore the possibility of extending it to permutation classes that contain an infinite number of simple permutations, but that are finitely described (like the family of oscillations of [10] for instance).",
"With such an improvement, more classes would enter our framework, but it would be hard to leave the algebraic case.",
"This is however a promising direction for the construction of Boltzmann random samplers for such permutation classes."
],
[
"Disambiguation of the system",
"In the above, Equation (REF ) gives an ambiguous description of the class ${\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle $ .",
"As noticed in [1], we can derive an unambiguous equation using the inclusion-exclusion principle: ${\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle = 1 \\ \\cup \\ 1 2 [{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\cup \\ 1 2[{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 4 3 \\rangle ,{\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ] \\ \\setminus \\ 1 2 [{\\hat{\\mathcal {C}}}^{+}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1 \\rangle ] \\ \\cup \\ 2 1 [{\\hat{\\mathcal {C}}}^{-}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle ] \\ \\cup \\ $ $3 1 4 2 [{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle ,{\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\cup \\ 3 1 42 [{\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 21 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ] \\ \\setminus \\ 3 1 42[{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 2 1\\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ]$.",
"The system so obtained contains negative terms in general.",
"This still gives a system of equations allowing to compute the generating function of the class.",
"However, this cannot be easily used for random generation, as the subtraction of combinatorial objects is not handled by random samplers.",
"In this section we disambiguate this system to obtain a new positive one: the key idea is to replace the negative terms by complement sets, hereby transforming pattern avoidance constraints into pattern containment constraints."
],
[
"General framework",
"The starting point of the disambiguation is to rewrite ambiguous terms like $A \\cup B \\cup C$ as a disjoint union $(A \\cap B\\cap C) \\uplus (\\bar{A} \\cap B \\cap C) \\uplus (\\bar{A} \\cap \\bar{B}\\cap C) \\uplus (\\bar{A} \\cap B \\cap \\bar{C}) \\uplus (A \\cap \\bar{B}\\cap C) \\uplus (A \\cap \\bar{B} \\cap \\bar{C}) \\uplus (A \\cap B \\cap \\bar{C})\\textrm {.",
"}$ By disambiguating the union $A \\cup B \\cup C$ using complement sets instead of negative terms, we obtain an unambiguous description of the union with only positive terms.",
"But when taking the complement of a set defined by pattern avoidance constraints, these are transformed into pattern containment constraints.",
"Therefore, for any set $\\mathcal {P}$ of permutations, we define the restriction $\\mathcal {P}\\langle E \\rangle (A)$ of $\\mathcal {P}$ as the set of permutations that belong to $\\mathcal {P}$ and that avoid every pattern of $E$ and contain every pattern of $A$ .",
"This notation will be used when $\\mathcal {P} = {\\hat{\\mathcal {C}}}^{\\varepsilon }$ , for $\\varepsilon \\in \\lbrace ~~ , +,-\\rbrace $ and $ a permutation class.",
"With this notation, notice alsothat for $ A=$, $ E = E ()$ is a standard permutation class.",
"Restrictionshave the nice feature of being stable by intersection as$ PE (A) PE' (A') = PE E' (A A')$.",
"We also define a {\\em restriction term} to be a set of permutationsdescribed as $ [S1,S2,...,Sn]$ where $$ is a simplepermutation or $ 12$ or $ 21$ and the $ Si$ are restrictions.",
"By uniqueness of the substitution decomposition of a permutation, restriction terms are stable by intersection as well and the intersection is performed componentwise for terms sharing the same root: $ [S1,S2,...,Sn] [T1,T2,...,Tn] = [S1T1,S2T2,...,SnTn]$.$"
],
[
"Disambiguate",
"The disambiguation of the system obtained by Algorithm REF is performed by Algorithm REF .",
"It consists in two main operations.",
"One is the disambiguation of an equation according to the root of the terms that induce ambiguity, which may introduce right-only restrictions.",
"This leads to the second procedure which computes new equations (that are added to the system) to describe these new restrictions (Algorithm REF ).",
"As stated in Section , every equation $F$ of our system can be written as $t =1 \\cup t_{1} \\cup t_{2} \\cup t_{3} \\ldots \\cup t_{k}$ where the $t_{i}$ are restriction terms and $t$ is a restriction.",
"By uniqueness of the substitution decomposition of a permutation, terms of this union which have different roots $\\pi $ are disjoint.",
"Thus for an equation we only need to disambiguate unions of terms with same root.",
"[t] A ambiguous system $\\mathcal {E} $ of combinatorial equations /* obtained by Algo.",
"REF */ An unambiguous system of combinatorial equations equivalent to $\\mathcal {E} $ there is an ambiguous equation $F$ in $\\mathcal {E} $ Take $\\pi $ a root that appears several times in $F$ in an ambiguous way Replace the restriction terms of F whose root is $\\pi $ by a disjoint union using Eq.",
"(REF ) – (REF ) there exists a right-only restriction ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ in some equation of $\\mathcal {E} $ $\\mathcal {E} \\longleftarrow \\mathcal {E} \\bigcup $ ComputeEqnForRestriction(${\\hat{\\mathcal {C}}}^{\\varepsilon }$ ,$E$ ,$A$ ).",
"/* See Algo.",
"REF */ $\\mathcal {E} $ DisambiguateSystem($\\mathcal {E} $ ) For example in Equation (REF ), there are two pairs of ambiguous terms which are terms with root 3142 and terms with root 12.",
"Every ambiguous union can be written in the following unambiguous way: $\\textstyle \\bigcup _{i=1}^{k} t_{i}=\\textstyle \\biguplus _{X\\subseteq [1\\ldots k], X \\ne \\emptyset } \\bigcap _{i \\in X} t_{i}\\cap \\bigcap _{i \\in \\overline{X}} \\overline{t_{i}}, $ where the complement $\\overline{t_{i}}$ of a restriction term $t_{i}$ is defined as the set of permutations of ${\\hat{\\mathcal {C}}}$ whose decomposition tree has the same root than $t_{i}$ but that do not belong to $t_{i}$ .",
"Equation REF below shows that $\\overline{t_{i}}$ is not a term in general but can be expressed as a disjoint union of terms.",
"By distributivity of $\\cap $ over $\\uplus $ , the above expression can therefore be rewritten as a disjoint union of intersection of terms.",
"Because terms are stable by intersection, the right-hand side of Equation REF is hereby written as a disjoint union of terms.",
"For instance, consider terms with root 3142 in Equation (REF ): $t_{1} = 3 1 4 2 [{\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle ,{\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2 \\rangle ]$ and $t_{2} = 3 14 2 [{\\hat{\\mathcal {C}}}\\langle 1 2 4 3 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 2 \\rangle , {\\hat{\\mathcal {C}}}\\langle 21 \\rangle , {\\hat{\\mathcal {C}}}\\langle 1 3 2\\rangle ]$ .",
"Equation (REF ) applied to $t_1$ and $t_2$ gives an expression of the form ${\\hat{\\mathcal {C}}}\\langle 1243\\rangle = 1 \\cup 12[\\ldots ] \\cup 12[\\ldots ] \\cup 21[\\ldots ] \\cup (t_{1} \\cap t_{2}) \\uplus (t_{1} \\cap \\overline{t_{2}}) \\uplus (\\overline{t_{1}} \\cap t_{2})\\textrm {.",
"}$ To compute the complement of a term $t$ , it is enough to write that $\\overline{t}=\\biguplus _{X \\subseteq \\lbrace 1,\\ldots ,n\\rbrace , X \\ne \\emptyset } \\pi [{\\mathcal {S}}^{\\prime }_{1},\\ldots ,{\\mathcal {S}}^{\\prime }_{n}] \\mbox{ where }{\\mathcal {S}}^{\\prime }_{i} = \\overline{{\\mathcal {S}}_{i}}\\mbox{ if }i \\in X\\mbox{ and }{\\mathcal {S}}^{\\prime }_{i} = {\\mathcal {S}}_{i}\\mbox{ otherwise},$ with the convention that $\\overline{{\\mathcal {S}}_{i}} = {\\hat{\\mathcal {C}}}^{\\varepsilon } \\setminus {\\mathcal {S}}_{i}$ for ${\\mathcal {S}}_{i} = {\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ .",
"Indeed, by uniqueness of substitution decomposition, the set of permutations of ${\\hat{\\mathcal {C}}}$ that do not belong to $t$ but whose decomposition tree has root $\\pi $ can be written as the union of terms $u = \\pi [{\\mathcal {S}}^{\\prime }_{1},{\\mathcal {S}}^{\\prime }_{2},\\ldots ,{\\mathcal {S}}^{\\prime }_{n}]$ where ${\\mathcal {S}}^{\\prime }_{i} = {\\mathcal {S}}_{i}$ or ${\\mathcal {S}}^{\\prime }_{i}=\\overline{{\\mathcal {S}}_{i}}$ and at least one restriction ${\\mathcal {S}}_{i}$ must be complemented.",
"For example $\\overline{21[{\\mathcal {S}}_{1},{\\mathcal {S}}_{2}]} =21[{\\mathcal {S}}_{1},\\overline{{\\mathcal {S}}_{2}}] \\uplus 21[\\overline{{\\mathcal {S}}_{1}},{\\mathcal {S}}_{2}]\\uplus 21[\\overline{{\\mathcal {S}}_{1}},\\overline{{\\mathcal {S}}_{2}}]$ .",
"The complement operation being pushed from restriction terms down to restrictions, we now compute $\\overline{{\\mathcal {S}}}$ , for a given restriction ${\\mathcal {S}} = {\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ , $\\overline{{\\mathcal {S}}}$ denoting the set of permutations of ${\\hat{\\mathcal {C}}}^{\\varepsilon }$ that are not in ${\\mathcal {S}}$ .",
"Notice that given a permutation $\\sigma $ of $A$ , then any permutation $\\tau $ of ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\sigma \\rangle $ is in $\\overline{{\\mathcal {S}}}$ because $\\tau $ avoids $\\sigma $ whereas permutations of ${\\mathcal {S}}$ must contain $\\sigma $ .",
"Symmetrically, if a permutation $\\sigma $ is in $E$ then permutations of ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\rangle (\\sigma )$ are in $\\overline{{\\mathcal {S}}}$ .",
"It is straightforward to check that $\\textstyle \\overline{{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)} = \\big [\\bigcup _{\\sigma \\in E} {\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\rangle (\\sigma )\\big ]\\bigcup \\big [ \\bigcup _{\\sigma \\in A}{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle \\sigma \\rangle ()\\big ]$ .",
"Unfortunately this expression is ambiguous.",
"Like before we can rewrite it as an unambiguous union $ \\overline{{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)}= \\biguplus _{\\underset{X \\times Y \\ne \\emptyset \\times \\emptyset }{{X\\subseteq A, Y \\subseteq E}}}{\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle X\\cup \\overline{Y} \\rangle (Y \\cup \\overline{X}) \\textrm {, where } \\overline{X} = A \\setminus X \\textrm { and }\\overline{Y} = E \\setminus Y \\textrm {.",
"}$ In our example (Equations (REF ) to ()), only trivial complements appear as every restriction is of the form ${\\hat{\\mathcal {C}}}\\langle \\sigma \\rangle ()$ or ${\\hat{\\mathcal {C}}}\\langle \\rangle (\\sigma )$ for which complements are respectively ${\\hat{\\mathcal {C}}}\\langle \\rangle (\\sigma )$ and ${\\hat{\\mathcal {C}}}\\langle \\sigma \\rangle ()$ .",
"All together, for any equation of our system, we are able to rewrite it unambiguously as a disjoint union of restriction terms.",
"As noticed before, some new right-only restrictions may appear during this process, for example as the result of the intersection of several restrictions or when complementing restrictions.",
"To obtain a complete system we must compute iteratively equations defining these new restrictions using Algorithm REF described below.",
"Finally, the terminaison of Algorithm REF is easily proved.",
"Indeed, for all the restrictions ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E\\rangle (A)$ that are considered in the inner loop of Algorithm REF , every permutation in the sets $E$ and $A$ is a pattern of some element of the basis $B$ of $\\mathcal {C}$ .",
"And since $B$ is finite, there is a finite number of such restrictions."
],
[
"Compute an equation for a restriction",
"Let ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)$ be a restriction.",
"Our goal here is to find a combinatorial specification of this restriction in terms of smaller restriction terms (smaller w.r.t.",
"inclusion).",
"If $A = \\emptyset $ , this is exactly the problem addressed in Section REF and solved by pushing down the pattern avoidance constraints in the procedure AddConstraints of Algorithm REF .",
"Algorithm REF below shows how to propagate also the pattern containment constraints induced by $A \\ne \\emptyset $ .",
"AMfuncAddMandatoryend [H] ${\\hat{\\mathcal {C}}}^{\\varepsilon }, E,A$ with $E,A$ sets of permutations, ${\\hat{\\mathcal {C}}}^{\\varepsilon }$ given by ${\\mathcal {S}}_{\\hat{\\mathcal {C}}}$ and $\\varepsilon \\in \\lbrace ~~ , +, -\\rbrace $ .",
"An equation defining ${\\hat{\\mathcal {C}}}^{\\varepsilon }\\langle E \\rangle (A)$ as a union of restriction terms.",
"$F \\leftarrow $ Equation (REF ) or () or () (depending on $\\varepsilon $ ) $\\sigma \\in E$ /* This step modifies $F$ !",
"*/ Replace any restriction term $t$ in $F$ by AddConstraints$(t, \\lbrace \\sigma \\rbrace )$/* See Algo.",
"REF */ $\\sigma \\in A$ /* This step modifies $F$ !",
"*/ Replace any restriction term $t$ in $F$ by AddMandatory$(t, \\sigma )$ $F$ ($($$\\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n],\\gamma $$($) a rewriting of $\\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n] (\\gamma )$ as a union of restriction terms using Equation (REF ).",
"ComputeEqnForRestriction$({\\hat{\\mathcal {C}}}^{\\varepsilon },E,A)$ The pattern containment constraints are propagated by AddMandatory, in a very similar fashion to the pattern avoidance constraints propagated by AddConstraints.",
"To compute $t(\\gamma )$ for $\\gamma $ a permutation and $t = \\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n]$ a restriction term, we first compute all embeddings of $\\gamma $ into $\\pi $ .",
"In this case, a permutation belongs to $t(\\gamma )$ if and only if at least one embedding is satisfied.",
"Hence, any restriction term $t = \\pi [{\\mathcal {S}}_1, \\dots , {\\mathcal {S}}_n](\\gamma )$ rewrites as a (possibly ambiguous) union as follows: $\\textstyle \\bigcup _{i=1}^{\\ell } \\pi [{\\mathcal {S}}_{1}(\\gamma _{\\alpha _{i}(1)}),{\\mathcal {S}}_{2}(\\gamma _{\\alpha _{i}(2)}),\\ldots ,{\\mathcal {S}}_{n}(\\gamma _{\\alpha _{i}(n)})], $ where the $(\\alpha _{i})_{i \\in \\lbrace 1, \\ldots , \\ell \\rbrace }$ are all the embeddings of $\\gamma $ in $\\pi $ and if $\\gamma _{\\alpha _{i}(j)}=0$ , then ${\\mathcal {S}}_{j}(\\gamma _{\\alpha _{i}(j)}) = {\\mathcal {S}}_j$ .",
"For instance, for $t = 2413[{\\mathcal {S}}_{1},{\\mathcal {S}}_{2},{\\mathcal {S}}_{3},{\\mathcal {S}}_{4}]$ and $\\gamma = 3214$ , there are 9 embeddings of $\\gamma $ into 2413, and the embedding $2413\\lbrace 321,1,0,0\\rbrace $ contributes to the above union with the term $2413[{\\mathcal {S}}_{1}(321),{\\mathcal {S}}_{2}(1),{\\mathcal {S}}_{3},{\\mathcal {S}}_{4}]$ .",
"Notice that although the unions of Equation REF may be ambiguous, they will be transformed into disjoint unions by the outer loop of Algorithm REF .",
"Finally, the algorithm produces an unambiguous system which is the result of a finite number of iterations of computing equations followed by their disambiguation."
],
[
"Conclusion",
"We provide an algorithm to compute a combinatorial specification for a permutation class $Av(B)$ , when its basis $B$ and the set of its simple permutations are finite and given as input.",
"The complexity of this algorithm is however still to analyse.",
"In particular, we observe a combinatorial explosion of the number of equations in the system obtained, that needs to be quantified.",
"Combined with existing algorithms, our procedure provides a full algorithmic chain from the basis (when finite) of a permutation class t̏o a specification for This procedure may fail to compute its result, when c̏ontains an infinite number of simple permutations, this condition being tested algorithmically.",
"This procedure has two natural algorithmic continuations.",
"First, with the dictionnary of [13], the constructors in the specification of $\\mathcal {C}$ can be directly translated into operators on the generating function $C(z)$ of $\\mathcal {C}$ , turning the specification into a system of (possibly implicit) equations defining $C(z)$ .",
"Notice that, using the inclusion-exclusion principle as in [1], a system defining $C(z)$ could also be obtained from an ambiguous system describing $\\mathcal {C}$ .",
"Second, the specification can be translated directly into a Boltzmann uniform random sampler of permutations in $\\mathcal {C}$ , in the same fashion as the above dictionnary (see [11]).",
"This second translation is possible only from an unambiguous system: indeed, whereas adapted when considering enumeration sequences, the inclusion-exclusion principle does not apply when working on the combinatorial objects themselves.",
"When generating permutations with a Boltzmann sampler, complexity is measured w.r.t.",
"the size of the permutation produced (and is linear if we allow a small variation on the size of the output permutation; quadratic otherwise) and not at all w.r.t.",
"the number of equations in the specification.",
"In our context, this dependency is of course relevant, and opens a new direction in the study of Boltzmann random samplers.",
"With a complete implementation of the algorithmic chain from $B$ to the specification and the Boltzmann sampler, one should be able to test conjectures on and study permutation classes.",
"One direction would be to somehow measure the randomness of permutations in a given class, by comparing random permutations with random permutations in a class, or random permutations in two different classes, w.r.t.",
"well-known statistics on permutations.",
"Another perspective would be to use the specifications obtained to compute or estimate the growth rates of permutation classes, to provide improvements on the known bounds on these growth rates.",
"We could also explore the possible use the computed specifications to provide more efficient algorithms to test membership of a permutation to a class.",
"However, a weekness of our procedure that we must acknowledge is that it fails to be completely general.",
"Although the method is generic and algorithmic, the classes that are fully handled by the algorithmic process are those containing a finite number of simple permutations.",
"By [1], such classes have finite basis (which is a restriction we imposed already), but they also have an algebraic generating function.",
"Of course, this is not the case for every permutation class.",
"We may wonder how restrictive this framework is, depending on which problems are studied.",
"First, does it often happen that a permutation class contains finitely many simple permutations?",
"To properly express what often means, a probability distribution on permutation classes should be defined, which is a direction of research yet to be explored.",
"Second, we may want to describe some problems (maybe like the distribution of some statistics) for which algebraic permutation classes are representative of all permutation classes.",
"To enlarge the framework of application of our algorithm, we could explore the possibility of extending it to permutation classes that contain an infinite number of simple permutations, but that are finitely described (like the family of oscillations of [10] for instance).",
"With such an improvement, more classes would enter our framework, but it would be hard to leave the algebraic case.",
"This is however a promising direction for the construction of Boltzmann random samplers for such permutation classes."
]
] | 1204.0797 |
[
[
"Growing transverse oscillations of a multistranded loop observed by\n SDO/AIA"
],
[
"Abstract The first evidence of transverse oscillations of a multistranded loop with growing amplitudes and internal coupling observed by the Atomspheric Imaging Assembly (AIA) onboard the Solar Dynamics Observatory (SDO) is presented.",
"The loop oscillation event occurred on 2011 March 8, triggered by a CME.",
"The multiwavelength analysis reveals the presence of multithermal strands in the oscillating loop, whose dynamic behaviors are temperature-dependent, showing differences in their oscillation amplitudes, phases and emission evolution.",
"The physical parameters of growing oscillations of two strands in 171 A are measured and the 3-D loop geometry is determined using STEREO-A/EUVI data.",
"These strands have very similar frequencies, and between two 193 A strands a quarter-period phase delay sets up.",
"These features suggest the coupling between kink oscillations of neighboring strands and the interpretation by the collective kink mode as predicted by some models.",
"However, the temperature dependence of the multistarnded loop oscillations was not studied previously and needs further investigation.",
"The transverse loop oscillations are associated with intensity and loop width variations.",
"We suggest that the amplitude-growing kink oscillations may be a result of continuous non-periodic driving by magnetic deformation of the CME, which deposits energy into the loop system at a rate faster than its loss."
],
[
"Introduction",
"Transverse coronal loop oscillations have been extensively studied in both observation and theory [2], [30], [23].",
"Observations from TRACE and STEREO/EUVI show that these oscillations are triggered by a flare or a coronal mass ejection (CME) [4], [2].",
"They have been interpreted as fast standing magnetohydrodynamic (MHD) kink modes [3], [15].",
"Transverse loop oscillations are often observed with a rapid decay within several periods [15], [4], [36].",
"Sometimes the undamped oscillations are observed [4], [5].",
"It has been suggested that the expected damping is balanced by amplification due to cooling [24], [25], [26].",
"Moreover, transverse oscillations are observed not only in single loops but also in a bundle of loops [32], [19], [33], [5].",
"Recent theories have shown that the global kink mode still exists in models with multiple strands, but its transverse dynamics are influenced by the internal fine structure due to the coupling and phase mixing of neighboring strands in properties such as the frequency and damping [17], [18], [11], [12], [13], [29], [31].",
"These studies have significantly contributed to the progress of coronal seismology, a diagnostic tool to probe the physical parameters in the corona [16].",
"Here we present the first example of transverse oscillations of a multistranded loop observed by the Atmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SDO), showing the evidence for growing amplitudes and the internal coupling."
],
[
"Observations",
"The oscillation event occurred on 2011 March 8, 19:40$-$ 20:40 UT in AR 11165 on the limb, observed with SDO/AIA.",
"An M1.5 GOES-class flare associated with a CME and a surge were also observed during this time.",
"The AIA records continuous images of the full Sun with 1.5$^{^{\\prime \\prime }}$ resolution and 12 s cadence [9].",
"This flare-CME event was first studied by [28] using AIA, STEREO-A and RHESSI data.",
"We present the analysis of loop oscillations using data from four AIA bands, 171, 193, 211, and 304 Å, as well as STEREO-A/EUVI data.",
"Figure: Observations of the transverse loop oscillation eventon 2011 March 08, with SDO/AIA in (a) 193 Å and (b) 304 Å bands.",
"A small narrow box shows a cutused for stack plots.",
"The dashed lines outline the oscillating loop seen in 193 Å.",
"The solid line showsthe solar radial direction passing the loop's footpoint.",
"(c) and (d): Simultaneous observationsof the oscillating loop with AIA 171 Å and STEREO-A/EUVI 171 Å at 20:06:00 UT.",
"In (d), the red curveis the best fit of a circular loop model to the oscillating loop (outlined with pluses).The white line shows the limb as seen from SDO with the visible disk on its left.",
"The symbol ofasterisk marks the footpoint of the jet.",
"In (c) the lines and the symbol have the samemeanings as in (d) but for the SDO view.",
"[See animations in the online journal]"
],
[
"Results",
"The oscillating loop is visible in AIA 171, 193 and 211 Å bands (Figure REF ).",
"The loop plane is almost parallel to the line-of-sight.",
"In the 304 Å, a surge was observed passing by the loop, and could have triggered its oscillations (Figure REF (b)).",
"The surge began at 19:55 UT with the velocity of 170$-$ 230 km s$^{-1}$ , and the ejected material fell back on the Sun at 20:30 UT at about 80 km s$^{-1}$ .",
"However, the STEREO-A observations viewing the AR at the disk center show that this surge was not directly related to the loop oscillation.",
"Figure REF shows the flux evolution of the four bands along a cut at the loop's apex.",
"The loop started to oscillate at about 19:40 UT with a rapid drift towards the north (flare source).",
"The start time of oscillations is consistent with the CME acceleration time [28].",
"The oscillations became evident during the period of the surge.",
"In this period, the loop in 171 Å gradually split into two strands manifesting the unusual oscillations with growing amplitudes.",
"The loop in 193 and 211 Å, composed of several close strands, shows the oscillations with no clear change in amplitude.",
"The difference of these strands in spatial distribution and temporal evolution indicates that the loop consists of the multithermal structure.",
"The lower 171 Å strand disappeared after the surge, while a dimming formed in 193 and 211 Å.",
"This may suggest that parts of the loop erupted.",
"The upper branch of the loop dimmed slowly in 171 Å, while remained bright in 193 and 211 Å, suggesting possible heating.",
"We examine the phase relation between the oscillations of different strands.",
"Figure REF (a) shows the oscillations of two 171 Å strands almost in phase (see also Figure REF (a)).",
"Figure REF (b) shows the oscillations of two 193 Å strands initially in phase, but becoming shifted by a quarter-period after two periods.",
"This behavior can be clearly seen from a comparison of the intensity evolution at two locations (with a linear drift of 1 km s$^{-1}$ ) near the displacement maxima of the strands (Figure REF (d)).",
"For the upper strand, the oscillation period is estimated to be 258$\\pm $ 66 s for the initial phase, and 216$\\pm $ 15 s for the later phase, while for the lower strand, the period is 294$\\pm $ 22 s for the initial phase, and 216$\\pm $ 83 s for the later phase, where the periods are measured from the average of time intervals of the intensity peaks and the errors the standard deviation.",
"This evolution suggests the setup of collective oscillation between the two neighboring strands, which has a slightly shorter period than before the coupling.",
"Figure REF (c) compares the flux evolution in three bands at the location (Y1) near the upper strands, where Y1=20$^{^{\\prime \\prime }}$ in 171 Å and Y1=23$^{^{\\prime \\prime }}$ in 193 and 211 Å considering their slight offsets in position.",
"It shows that the upper strands seen in the three bands oscillated with similar periods and nearly in phase.",
"The oscillations of these strands with certain phase shifts and the similar frequencies suggest collective dynamics of a multistranded loop system [11], [12].",
"Figure: Time-distance maps along a cut at the loop apexas shown in Figure (a) (averaged over the narrow width)in four bands, (a) 304 Å, (b) 171 Å, (c) 193 Å, and (d) 211 Å.",
"A box marks the timeperiod of interest.",
"The vertical dashed lines indicate the start time of transverseloop oscillations.We measure the time variation of oscillation amplitudes for two 171 Å strands by locating the cross-sectional flux maxima using a double Gaussian best fit with a parabolic background.",
"Figure REF (b) shows an example for the fitting of emission profile across the loop.",
"The measured loop displacements, FWHM width, and cross-sectional peak flux as a function of time are shown in Figures REF (a), (c), and (d), respectively.",
"The upper strand has the average FWHM diameter of 4.2$\\pm $ 0.4 Mm, and the lower strand has that of 3.3$\\pm $ 0.7 Mm.",
"By fitting the displacement oscillations with an amplitude-growing sine function with a parabolic drift given by, $a(t)=A\\,{\\rm sin}(\\frac{2\\pi {(t-t_0)}}{P}+\\phi ){\\rm e}^{\\frac{t-t_0}{\\tau _g}}+a_0+a_1(t-t_0)+a_2(t-t_0)^2,$ we determine the parameters of the oscillation: amplitude ($A$ ), period ($P$ ), phase ($\\phi $ ), and amplitude growth time scale ($\\tau _g$ ), where $t_0$ is the start time of analyzed oscillations.",
"The measured parameters are shown on the plots and listed in Table .",
"The growing oscillations of the two strands started and ended almost simultaneously, lasting over four periods.",
"The lower strand has higher growth rate of the oscillation amplitude than the upper one.",
"We estimate the increase in their amplitudes by $A_2/A_1$ =3.8 for the lower strand, and $A_2/A_1$ =2.3 for the upper strand using $A_2/A_1$ =$e^{\\Delta {t}/\\tau _g}$ with the life time $\\Delta {t}$ =1020 s. lccccccc Physical parameters of the amplitude-growing oscillations in the 171 Å banda 0pt Loop $P$ $A$ $\\tau _g$ $\\phi $ $A_2/A_1$ $d$ $L$ (s) (Mm) (s) ($^{\\circ }$ ) (Mm) (Mm) upper 230 0.254 1248 $-$ 62 2.3 4.2$\\pm $ 0.4 77 lower 233 0.269 759 $-$ 43 3.8 3.3$\\pm $ 0.7 77 a$P$ –oscillation period, $A$ –amplitude at the start time, $\\tau _g$ –amplitude growth time, $\\phi $ -phase, $A_2/A_1$ –ratio of amplitudes at the end and start time of analyzed oscillations, $d$ –loop FWHM diameter, and $L$ –loop length.",
"The displacement oscillations are found in association with intensity and loop width fluctuations.",
"Figures REF (e) and (f) show comparisons between the relative displacement, cross-sectional peak flux, and loop width time variations, where a 290 s smoothed trend for all parameters has been subtracted, the relative displacements are normalized to a scale of 7 Mm, and the relative peak flux and loop width are normalized to their smoothed trend.",
"An inphase relationship is found between the loop width and intensity fluctuations (with relative amplitudes of $\\sim $ 5%$-$ 15%) for both strands.",
"The phase relationship between displacement and intensity oscillations is different for the two strands, being in-phase for the upper strand and a quarter-period shift for the lower strand.",
"Figure: Time-slice diagrams of (a) 171 Å and (b) 193 Å flux of theoscillating loop (in negative color).",
"The vertical dashed line marks the timefor Figure (b).",
"(c) Time profiles of the 171, 193and 211 Å relative intensities at position Y1 (marked in (a)).",
"(d) Time profiles of the 193 Å relative intensities at two positions Y1 and Y2 (marked in (b)), where for a clear comparisonthe light curve for Y2 (solid black line) is plotted as its negative.",
"In (c) and (d), therelative intensities (with the background subtracted) are normalized tothe smoothed background trend.To determine the trigger of the oscillation and measure the loop geometry, we analyze the STEREO-A observations (Figure REF (d)).",
"A jet which corresponds to the surge in AIA 304 Å band can be easily identified, whereas the direct identification of the oscillating loop in STEREO/EUVI images is not obvious due to the low (75 s) cadence.",
"We use the following procedure to locate the oscillating loop.",
"First we speculate that a faint loop (outlined in Figure REF (d)) is the target based on its evolutionary features.",
"The movies (available in online version of this letter) show that this loop was apparently shrinking in STEREO/EUVI, suggesting a correspondence to the change in inclination of the oscillating loop in SDO/AIA.",
"In addition, this STEREO loop dimmed simultaneously as the AIA loop (at about 20:40 UT).",
"Next we model this STEREO loop with a 3D arc and map it onto the AIA view to compare with the observation.",
"The method is similar to that used by [4].",
"A circular loop model is made by optimizing two free parameters, $h_0$ and $\\theta $ , where $h_0$ is the height of the circular loop center above the solar surface and $\\theta $ is the inclination angle of the loop plane to vertical.",
"The fact that the best fit loop model, when mapped to the AIA view, matches the observation confirms the initial conjecture.",
"The calculated loop parameters are $h_0$ =18 Mm, $\\theta $ =24$^{\\circ }$ , the curvature radius $r$ =59 Mm, and the loop length $L$ =212 Mm.",
"Taking the period $P$ =230 s, we estimate the phase speed of the oscillations in the fundamental mode, $V_p$ =2L/P=1840 km s$^{-1}$ .",
"For the kink mode [21], we obtain the Alfvén speed $V_A$ =1360 km s$^{-1}$ if assuming the phase speed ($V_p$ ) equal to the kink speed ($C_k$ ) and the loop density contrast of 10, and estimate the average magnetic field in the loop, $B$ =6$-$ 20 G, for typical coronal loop densities ($10^8-10^9$ cm$^{-3}$ ).",
"The STEREO-A observations suggest that the loop oscillation was triggered by the CME but not the surge/jet.",
"The EUVI 171, 195 and 284 Å bands observed the eruption of a large flux rope at 19:40 UT, which appeared as a CME at 21:12 UT in SOHO/LASCO C2.",
"In 171 Å a bright ribbon appeared at about 20:00 UT, and extended towards the left footpoint of the oscillating loop, followed by the formation of a dimming region (Figure REF (d), and movies in online version).",
"The dimmings observed in the EUV and/or soft X-ray range were interpreted as the coronal plasma evacuation at the footpoints of a magnetic flux rope when it rapidly opens or expands [27], [35].",
"The presence of a bright ribbon at the boundary of the extending dimming region has not been reported previously in literature.",
"We suggest that it could be caused by interaction (via local reconnection) between the expanding magnetic fields of the CME and the ambient closed magnetic loops.",
"[8] found the temperature-dependent strong upflows (up to $\\sim $ 150 km s$^{-1}$ ) at the boundary of the dimming region with Hinode/EIS, supporting this suggestion.",
"Therefore, continuous magnetic interaction by the CME may drive the loop oscillation with growing amplitudes, and also lead to the heating (e.g.",
"by hot outflows) and partial eruption of the oscillating loop, as observed.",
"This scenario is also supported by a coincidence of the flux rope eruption with the excitation of the loop oscillations in time (at $\\sim $ 19:40 UT).",
"Figure: (a) The displacement oscillations of two loop strands in 171 Å,and the best fits (red lines) with Equation ().",
"The red dashed lines are the parabolic fitto a drift.",
"(b) Loop cross-sectional flux profile at 20:17:00 UT, and the double Gaussian fit(red solid line), where the dotted and dashed lines show the fitted components forthe lower and upper strands, and the dot-dashed line for the fitted background.",
"(c) Loop FWHM width, and (d) cross-sectional peak flux of the upper (solid line) and lower(dashed line) strands as a function of time.",
"(e) and (f): The normalized relative variationsof displacement (black line), cross-sectional peak flux (red line), and loop FWHM width(green line) for the upper and lower strands, respectively."
],
[
"Summary and Discussion",
"The SDO/AIA observations of transverse loop oscillations analyzed have revealed several interesting new features.",
"The loop consists of multithermal strands, whose dynamical behaviors are temperature-dependent.",
"In the 171 Å band, two strands show in-phase oscillations with growing amplitudes and a separating drift.",
"Their displacement oscillations are associated with the intensity variations.",
"In the 193 Å band, two close strands show the oscillations with no clear amplitude change and a quarter-period phase delay has developed between them after a time of about two periods.",
"The oscillations of these strands have very similar periods.",
"The flare-excited transverse loop oscillations observed by TRACE have been interpreted as eigenmodes (mainly the fundamental kink mode) [15], [4].",
"These oscillations typically show the strong damping, which has been suggested due to resonant absorption or wave leakage [22], [23].",
"The unusual growing oscillations reported here suggest that they may be forced kink oscillations with continuous energy input at a rate faster than the damping.",
"The STEREO-A observations suggest that continuous interaction from the erupted flux rope in a CME may play the role of the external driver.",
"A theoretical study by [6] showed that a harmonic driver typically excites a mixture of standing kink modes harmonics (with both the driver's and the natural periods).",
"The oscillations analyzed here show mostly a single frequency.",
"This feature does not agree with the harmonic driver, and suggests the excitation by a continuous non-harmonic driver.",
"However, we notice that the timing of the amplitude-increasing oscillations is coincident with both the CME dimming formation near the loop footpoint and the emergence of the surge.",
"Although the flux rope eruption as the driver of the oscillations is the preferred interpretation as discussed in the last section, the possibility that the surge also could play a role in reinforcing the oscillations cannot be entirely excluded.",
"Recently, quasi-periodic fast mode magnetosonic waves with a propagation speed of more than 2000 km s$^{-1}$ and a total duration about 30 minutes were discovered by SDO/AIA [10], [20].",
"Whether such waves were produced by the surge and could have driven the amplification of oscillations needs further investigations observationally and theoretically.",
"Some studies have suggested that the loop cooling can strongly affect the kink oscillations [1], [14].",
"Recently, an undamped kink oscillation event was observed by SDO/AIA [5], and the lack of damping has been attributed to the cooling which can amplify the oscillation [24], [25], [26].",
"Note that for an unrealistic assumption, [14] obtained that cooling causes the damping of kink oscillations [24].",
"Our discussion below based on the models developed by [24] suggests that the growing oscillations reported here cannot be explained by changes of the loop temperature with time.",
"For the sake of present discussion, we assume that the decrease of the loop intensity in the 171 Å band is due to the cooling (although our analysis above suggests heating of the loop strands).",
"Considering the role of wave damping, the measured amplitude growth time ($\\tau _g$ ) should be the upper limit of the amplification time ($t_{amp}$ ) due to the cooling.",
"From the measured loop height ($h$ =77 Mm), we obtain the parameter, $\\kappa $ =$h/H_0\\approx 2$ , where $H_0$ is the atmospheric scale height for the initial loop plasma temperature of about 1 MK.",
"For the loop model with $\\kappa $ =2 and $\\chi \\approx $ 0.1 (the ratio of the loop external and internal plasma densities), we obtain $t_{amp}\\approx 4t_{cool}$ from the dependence of the oscillation amplitude on time for the model of stratified loop with constant temperature of external plasma [24], [25], where $t_{cool}$ is the loop cooling time.",
"Since $t_{amp}<\\tau _g$ , we obtain $t_{cool}/P<$ 1.4 and 0.8 for the upper and lower threads, respectively.",
"This means that the cooling must be very fast with the characteristic cooling time less than the oscillation period.",
"For such a rapid cooling the oscillating loop should become completely invisible in the 171 Å band after two periods, and the oscillation should show a dramatic ($>50\\%$ ) decrease in period over the lifetime [14], [24].",
"However, neither theoretical predictions are consistent with present observations.",
"For the same reason the undamped oscillations observed in the 193 Å band are also impossible to interpret by the cooling effect when the typical finite damping rate by resonant absorption is considered [26].",
"This disagreement supports our suggestion that the wave energy in the loop is supplied continuously during the oscillations in our case, in contrast with the initial impulsive excitation suggested by the typical damping scenario of resonant absorption.",
"The above discussions are based on the properties of monolithic tube models, whereas in our case the oscillating loop consists of multithermal strands, thus the interpretation of their dynamic behaviors may need to consider the properties of coupled multi-stranded loop models [18], [12].",
"Our observations show kink oscillations of several loop strands with similar frequencies and phase shift of in-phase or quarter-period, providing the evidence for a collective kink mode [12].",
"[11] showed the simultaneous excitation of several collective normal modes can lead to a $\\pi $ /2 phase difference between two neighboring loops and the beating of the system in some cases.",
"The setup of a quarter-period phase shift between two 193 Å strands may belong to such a case.",
"We notice that after the coupling the two 193 Å strands oscillate with a period slightly shorter (by $\\sim $ 20%) than before (assumed to be the kink-mode period of the individual loops).",
"This feature also agrees with the model-prediction [11] when the two loops are very close (with the similar separation as observed).",
"It is unclear whether the amplitude-growing oscillations of the 171 Å strands are due to the particular combination of collective normal modes excited in certain condition as no beating behavior was found as predicted [11], [13].",
"In addition, our observations show the evident temperature dependence of multistanded loop oscillations, which were not studied in previous models.",
"The association of loop displacement oscillations with intensity and loop width variations is found in this study.",
"However, the positive correlation between the loop width and intensity variations suggests that the loop width variations may be observational artifacts due to the line-of-sight intensity variations.",
"Assuming the mass conservation in the loop, an anti-correlation between them (ie., $\\Delta {I}/I\\sim -3\\Delta {d}/d$ , where $I$ is the loop intensity, and $d$ the loop diameter) is predicted theoretically [5].",
"The associated intensity oscillations could be due to variations in the line-of-sight column depth of the oscillating loop as suggested in some previous studies [7], [33], [34], [36], but this conjecture needs a forward modeling to confirm.",
"The authors are grateful to Drs.",
"Jaume Terradas and Manuel Luna for their valuable comments.",
"The work of TW was supported by NASA grants NNX08AE44G, NNX10AN10G, and NNX12AB34G.",
"LO acknowledges to support from NASA grants NNX09AG10G, NNX10AN10G, and NNX12AB34G."
]
] | 1204.1376 |
[
[
"On relativistic motion of a pair of particles having opposite signs of\n masses"
],
[
"Abstract (abbreviated) In this note we consider, in a weak-field limit, a relativistic linear motion of two particles with opposite signs of masses having a small difference between their absolute values $m_{1,2}=\\pm (\\mu\\pm \\Delta \\mu) $, $\\mu > 0$, $|\\Delta \\mu | \\ll \\mu$ and a small difference between their velocities.",
"Assuming that the weak-field limit holds and the dynamical system is conservative an elementary treatment of the problem based on the laws of energy and momentum conservation shows that the system can be accelerated indefinitely, or attain very large asymptotic values of the Lorentz factor $\\gamma$.",
"The system experiences indefinite acceleration when its energy-momentum vector is null and the mass difference $\\Delta \\mu \\le 0$.",
"When modulus of the square of the norm of the energy-momentum vector, $|N^2|$, is sufficiently small the system can be accelerated to very large $\\gamma \\propto |N^2|^{-1}$.",
"It is stressed that when only leading terms in the ratio of a characteristic gravitational radius to the distance between the particles are retained our elementary analysis leads to equations of motion equivalent to those derived from relativistic weak-field equations of motion of Havas and Goldberg 1962.",
"Thus, in the weak-field approximation, it is possible to bring the system to the state with extremely high values of $\\gamma$.",
"The positive energy carried by the particle with positive mass may be conveyed to other physical bodies say, by intercepting this particle with a target.",
"Suppose that there is a process of production of such pairs and the particles with positive mass are intercepted while the negative mass particles are expelled from the region of space occupied by physical bodies of interest.",
"This scheme could provide a persistent transfer of positive energy to the bodies, which may be classified as a 'Perpetuum Motion of Third Kind'."
],
[
"Introduction",
"Bondi 1957 [1] pointed out that in the Newtonian approximation two particles with opposite signs of masses at rest with respect to each other accelerate indefinitely in an inertial frame.",
"This process is allowed by the laws of conservation since the kinetic energy and angular momentum of such a system are conserved being exactly zero while the potential energy depends only on relative distance between the particles.",
"In the same paper he generalised this result by finding an appropriate static accelerated solution in General Relativity and discovered that a uniformly accelerated pair of particles with opposite signs of masses must have a mass difference determined by the fact that constant in time particle accelerations must be different to keep them static with respect to each other.",
"It is trivial to show that in the Newtonian approximation (see the next Section ) when the two particles with opposite signs of masses have a relative velocity, its value is approximately conserved.",
"As a result, the acceleration period is finite and the pair as a whole being initially at rest gains a finite value of velocity.",
"We also show that when initial relative velocity of the particles is sufficiently small the pair can be accelerated to a relativistic speed.",
"In the next Section we consider the problem in the relativistic setting and generalise Bondi's analysis considering pairs of particles with opposite masses and small difference between their absolute values: $m_{1,2}=\\pm (\\mu \\pm \\Delta \\mu ) $ , $\\mu > 0$ , $|\\Delta \\mu | \\ll \\mu $ having an initial relative velocity $v_{in}$ in a fixed lab frame where the pair as a whole is initially at rest.",
"We assume that gravitational interaction is weak and, therefore, $G\\mu /(c^2 D_{in})\\ll 1$ , where $D_{in}$ is an initial separation distance between the particles.",
"Also, for simplicity, in the relativistic treatment, it is assumed that the orbital angular momentum of the system is equal to zero and the motion is linear.",
"We analyse this situation by elementary means.",
"The equations of motion are obtained from the laws of energy and momentum conservation.",
"It is assumed that energy and momentum of the system in a Lorentz frame instantaneously comoving with the motion of the pair are given by the Newtonian expressions and that they form time and spacial components of a local four-vector.",
"Then this energy-momentum vector is projected onto the lab frame.",
"Since energy and momentum in the lab frame are conserved under the assumption that gravitational radiation from the system is insignificant we get two equations of first order in time fully describing dynamics of the system.",
"We also show how to derive an equivalent pair of second order equations considering the Newton's law of gravity in a frame accelerating with the particles.",
"It is shown that the pair as a whole always have a positive acceleration with its asymptotic value being either zero or a nonzero constant depending on initial conditions.",
"The relative distance between the particles can either have a turning point or increase monotonically.",
"The system accelerates indefinitely when the mass difference $\\Delta \\mu \\le 0$ and the norm of the energy-momentum vector $N=\\sqrt{(2\\Delta \\mu c^2 +{G\\mu ^{2}\\over D_{in}})^{2}-\\mu ^2v_{in}^2}=0$ , and, accordingly, the energy-momentum vector is null.",
"In this case the relative distance increases monotonically.",
"When $N^2$ is sufficiently small, for the initial conditions corresponding to the monotonic behaviour of the relative distance the acceleration period is finite but the asymptotic value of the Lorentz gamma factor is large being proportional to $|N^{2}|^{-1}$ .",
"Such pairs can play a role in realisation of a hypothetical effect, which we called 'Perpetuum Motion of Third Kind' [2], hereafter PMT.",
"In its most general formulation this effect is a possibility of a persistent energy transfer from a subsystem having negative energy to a subsystem with positive energy, in classical theories where negative energy subsystems are possible.",
"Indeed, the positive mass particle can, in principle, be used to transfer positive energy to other physical bodies after the pair has been accelerated to high values of the Lorentz factor.",
"Iterating this process as many times as we need we can extract as much positive energy as we wish.",
"Note, however, that it is not the only 'working model' of PMT, and that, in principal, in order to make PMT we need neither systems with negative rest mass nor gravitational interactions.",
"As is shown in [2] it suffices to have a medium violating the weak energy condition with certain additional properties and mere hydrodynamical interaction 'to construct a PMT'.",
"Additionally, we comment on several statements of paper [3], where the Kepler problem for a binary with opposite signs of masses has been considered which may, in our opinion, lead to misunderstanding of the problem."
],
[
"Newtonian treatment of the problem",
"At first let us consider the problem in the Newtonian approximation where mutual gravitational accelerations acting on the particles of masses $m_1$ and $m_2$ are given by the conventional expressions: $\\ddot{\\bf r}_1=-{Gm_2\\over |{\\bf D}|^3}{\\bf D}, \\quad \\ddot{\\bf r}_2={Gm_1\\over |{\\bf D}|^3}{\\bf D}, $ where ${\\bf r}_i$ are position vectors of particles with indices $i=1,2$ and ${\\bf D}={\\bf r}_{1}-{\\bf r}_2$ .",
"Setting $\\mu \\equiv Gm_1=-Gm_2$ we obtain from (REF ) $\\dot{\\bf V}= {\\mu \\over |{\\bf D}|^3}{\\bf D}, \\quad \\dot{\\bf v} =0, $ where ${\\bf V}\\equiv {1\\over 2}(\\dot{\\bf r}_1+ \\dot{\\bf r}_2)$ and ${\\bf v}\\equiv \\dot{\\bf D}$ .",
"From equation (REF ) it follows that when ${\\bf v}=0$ at some moment of time it remains zero in the course of evolution of the system.",
"Thus, in this case the relative separation distance ${\\bf D} $ does not change during the evolution and the system ever accelerates as a whole, with the acceleration vector ${\\bf a}\\equiv \\dot{\\bf V}= {\\mu \\over |{\\bf D}|^3}{\\bf D}$ being constant.",
"The laws of conservation are nonetheless respected since the kinetic energy and momentum of the system are precisely zero while the potential energy depends only on the relative separation distanceNote that it is easy to show that the same motion can be realised in a system containing $N$ particles provided that the total mass of the system $M=\\sum _{i=1,n} m_i =0$ and positions of the particles are chosen in a special way.",
"Say, for a system containing three particles their relative positions must form an equilateral triangle..",
"When ${\\bf v}(t=0)\\equiv {\\bf v}_{in}\\ne 0$ the absolute value of the relative distance changes with time.",
"Accordingly, the absolute value of the acceleration changes as well and eventually decays provided that $({\\bf D}_{in} \\cdot {\\bf v}_{in}) \\ne - |{\\bf v}_{in}||{\\bf D}_{in}|$Clearly, when $({\\bf D}_{in}\\cdot {\\bf v}_{in}) = - |{\\bf v}_{in}||{\\bf D}_{in}|$ the particles collide., where ${\\bf D}_{in}\\equiv {\\bf D}(t=0)$ .",
"We have $ {\\bf D}={\\bf v}_{in}t + {\\bf D}_{in}, $ and, thus, integrating equation (REF ) we obtain $|{\\bf V}(t)|={1\\over \\sqrt{1-\\alpha ^2}}{\\mu \\over D_{in} v_{in}}\\sqrt{2(1-{\\epsilon ^{1/2}+\\alpha \\tau \\over \\Delta })}, $ where we use the dimensionless time $\\tau =\\sqrt{{\\mu \\over D_{in}^3}}t$ , $D_{in}=|{\\bf D}_{in}|$ , $v_{in}=|{\\bf v}_{in}|$ , $\\epsilon ={\\mu \\over D_{in} v_{in}^2}$ , $\\alpha =({\\bf v}_{in}\\cdot {\\bf D}_{in})/(v_{in}D_{in})$ , and $\\Delta =\\sqrt{\\epsilon +\\tau ^2+2\\alpha \\epsilon ^{1/2}\\tau }$ .",
"Note that when the system moves along a straight line with increasing value of $|{\\bf D}|$ , and, accordingly, $\\alpha =1 $ equation (REF ) yields $|{\\bf V}(t)|={\\mu \\over D_{in} v_{in}}{\\tau \\over \\epsilon ^{1/2}+\\tau }.",
"$ In the limit $\\tau \\rightarrow \\infty $ we get from equations (REF ) and (REF ) $V_{\\infty }\\equiv |{\\bf V}(\\tau \\rightarrow \\infty )|=\\sqrt{{2\\over 1 +\\alpha }}{\\mu \\over D_{in} v_{in}}.",
"$ It follows from (REF ) that when $v_{in} < v_{crit}= \\sqrt{{2\\over 1 +\\alpha }}{\\mu \\over D_{in} c}, $ the asymptotic value of velocity of the system, $V_{\\infty }$ , formally exceeds the speed of light, $c$ .",
"Clearly, a relativistic approach to the problem is to be used in this situation."
],
[
"Derivation of dynamical equations",
"In order to keep our study as simple as possible let us consider in the relativistic case only the motion along a straight line with increasing value of ${\\bf D}$ ($\\alpha = 1$ ).",
"Additionally, in this Section we use the natural units setting the speed of light and the gravitational constant to unity.",
"However, unlike the Newtonian case, here we would like to consider particles having a small mass difference: $m_{1,2}=\\mu \\pm \\Delta \\mu $ , where it is assumed below that $\\mu > 0$ and $|\\Delta \\mu | \\ll \\mu $ .",
"It is useful to introduce two local frames and the respective coordinate systems 1) a fixed lab frame with global Lorentzian coordinates $(x,t)$ and 2) a local Lorentzian frame instantaneously comoving with the motion of the point $R(t)={1\\over 2}(x_{1}(t)+x_{2}(t))$ , where $x_1(t)$ and $x_{2}(t)$ are positions of the particles in the lab frame, with associated Lorentzian coordinates $(x^{com},t^{com})$ .",
"Is is assumed that at some particular moment of time $t=t_*$ coordinates of the event $(t_*, R(t=t_*))$ in the comoving coordinate system are equal to $(\\tau , 0)$ , where $\\tau $ is the proper time associated with the world line $(t, R(t))$ .",
"Hereafter, the world line $(t, R(t))$ is referred to as \"the reference world line\".",
"When $t^{com}=\\tau $ the positions of particles are given by $x^{com}_{1,2}(t^{com})$ , their velocities are $v^{com}_{1,2}={d\\over dt^{com}}x^{com}_{1,2}$ .",
"Let us also introduce the relative position and velocity in the comoving coordinate system $D=x_1^{com}-x_2^{com}$ , $v^{com}={d\\over dt^{com}}D$ .",
"Without loss of generality we assume hereafter $D^{com} > 0$ .",
"When the relative separation remains sufficiently small along the reference world line we have approximately $x^{com}_{2}=-x^{com}_{1}$ .",
"In the global coordinates at the time slice $t=t_*$ the velocity of motion of the system as a whole is given by $V={1\\over 2}({d\\over dt}x_1+{d\\over dt}x_2)(t=t_*)$ while the relative position and velocity of the relative motion are $D_{lab}=x_1(t_*)-x_2(t_*)$ and $v={d\\over dt}D$ .",
"Introducing the Lorentz gamma factor $\\gamma ={1\\over \\sqrt{1-V^{2}}}$ associated with the reference world line we may write in the limit of small separations $D(t^{com})=\\gamma D_{lab}, \\quad {dt\\over dt^{com}}=\\gamma ,$ and, accordingly, $v^{com}=\\gamma {d\\over dt}(\\gamma D_{lab})=\\gamma ^{2}v+\\gamma {d\\gamma \\over dt}D_{lab}.",
"$ Supposing below that, on one hand, the relative distance $D \\gg \\mu $ , and, therefore, a weak-field approximation holds and, on the other hand, it is not too large for the local Lorentzian coordinates to be adequate and, respectively, equations (REF -REF ) to be valid we can use the Newtonian expression for the energy, $E_c$ , and momentum, $P_c$ , of the system in the comoving frame at the time $t^{com}=\\tau $ $E_c=2\\Delta \\mu +{\\mu ^2\\over D}, \\quad P_c=\\mu \\dot{D}, $ where dot stands for differentiation w.r.t.",
"the proper time $\\tau $ .",
"In the same limit $E_c$ and $P_c$ represent time and spacial components of a local four vector, and, therefore, they values in the lab frame, $E$ and $P$ , respectively, can be obtained from (REF ) by the standard Lorentz transformation.",
"We have $E=\\gamma (2\\Delta \\mu +{\\mu ^2\\over D} + V\\mu \\dot{D}), \\quad P=\\gamma ( \\mu \\dot{D} + V(2\\Delta \\mu +{\\mu ^2\\over D})),$ where it is assumed that the velocity of the systems as a whole, $V$ , is a function of the proper time $\\tau $ .",
"Since energy and momentum in the lab frame are obviously conserved, equations (REF ) fully describe the dynamics of our system.",
"They should be solved subject to the condition that the system is initially at rest with respect to the lab frame: when $\\tau =0$ we have $V=0$ and $ E=E_{in}=2\\Delta \\mu +{\\mu ^2\\over D_{in}},\\quad P=P_{in}= \\mu v_{in}, $ where $D_{in}$ and $v_{in}$ are initial separation distance and relative velocity, respectively.",
"It is assumed below that $v_{in} > 0$ .",
"Although our derivation of dynamical equations (REF ) may look somewhat heuristic it is worth mentioning that when terms next to the leading order in $\\mu $ are discarded they can be derived from the precise weak-field equations of reference [4] in the limit of small separations and $|\\Delta \\mu | \\ll \\mu $ .",
"It is convenient to transform equations (REF ) to another form using their linear combination $E-VP$ and calculating square of the norm of the energy-momentum vector, $N^{2}=E^2-P^2$ .",
"We get $E-VP=\\gamma ^{-1}(2\\Delta \\mu +{\\mu ^2\\over D}) $ and $N^{2}= (2\\Delta \\mu +{\\mu ^2\\over D})^{2}-\\mu ^{2}(\\dot{D})^2.$ We also obviously have $N^{2}=(2\\Delta \\mu +{\\mu ^2\\over D_{in}})^{2}-\\mu ^{2}v_{in}^2$ .",
"Note that contrary to the usual situation the energy-momentum vector can be null, time-like or space-like, depending on initial conditions.",
"Equations (REF ) are first order integrals of two second order in time dynamical equations.",
"One of these equations can be obtained from (REF ) by differentiating this equation over $\\tau $ with the result $\\ddot{D}= -{2\\Delta \\mu \\over D^{2}}-{\\mu ^2\\over D^3}, $ and the second one by differentiating either of equations (REF ) and using (REF ): $\\gamma ^{2}\\dot{V}={\\mu \\over D^2} $ Equations (REF ) and (REF ) can be obtained from other independent qualitative arguments.",
"The derivation of the second order dynamical equations, which relate dynamical variables with different values of time coordinates is not, however, convenient in the local Lorentzian coordinates introduced above since these coordinates are defined with respect to some particular event on the reference world line and, therefore, the definition is different for different events along this world line.",
"It is much more convenient to use a coordinate system, where the proper time $\\tau $ plays the role of coordinate time.",
"For that let us consider another, the so-called local Fermi-Walker coordinate system $(\\tau , y)$ , see e.g.",
"[5], where the proper time $\\tau $ is the coordinate time and the unit vector in the spacial direction $y$ is always perpendicular to the four velocity along the reference world line.",
"The coordinates of the reference world line in this coordinate system are simply $(\\tau , 0)$ .",
"At the time slice $t^{com}=\\tau $ the local Lorentz coordinates and the Fermi-Walker coordinates coincide: $x^{com}_{1,2}=y_{1,2}$ , but the Fermi-Walker coordinate system is accelerating with respect to the local Lorentz coordinate system with an acceleration $g(y)$ .",
"Clearly, $g(y=0)$ must coincide with modulus of four acceleration of the reference world line with respect to the lab frame.",
"The equations of motion in the Fermi-Walker coordinates are assumed to be determined by the Newton's law (REF ) with added acceleration term $-g$ , which accounts for the fact that this system is not inertial: $\\ddot{y}_{1,2}={\\mu \\mp \\Delta \\mu \\over D^{2}_{FW}}-g,$ where $D_{FW}=y_1-y_2$ and we take into account that the acceleration term depends on the coordinate $y$ , e.g.",
"[5]: $g=a+a^2y$ .",
"For the average distance $Y=(y_1+y_2)/2$ to be at rest $Y(\\tau )=0$ the acceleration term $a$ must be balanced by the gravity term ${\\mu \\over D^{2}_{FW}}$ .",
"We get $a={\\mu \\over D^{2}_{FW}}.",
"$ Taking into account that in the lab frame the spacial coordinate of four acceleration is related to $a$ as $a^x=\\gamma a$ we have $\\dot{U}^x=\\gamma ^3\\dot{V}=\\gamma a, \\quad \\dot{V}= {\\mu \\over \\gamma ^2 D^{2}_{FW}}.",
"$ It is clear that this coincides with equation (REF ).",
"The dynamical equation for the relative distance $D_{FW}$ directly follows from (REF ) and (REF ): $\\ddot{D}_{FW}=-{2\\Delta \\mu \\over D_{FW}^{2}}-a^2D_{FW}=-{2\\Delta \\mu \\over D_{FW}^{2}}-{\\mu ^2\\over D_{FW}^{3}}.$ It coincides with (REF ).",
"The last term on the right hand side of (REF ) is due to the non-uniform acceleration force appearing in the Fermi-Walker coordinates.",
"Because it is $\\propto \\mu ^{2}$ , technically, it is a post-Newtonian term.",
"Since we consider the gravitational force in the Newtonian approximation in (REF ) it is important to check whether or not post-Newtonian corrections to the gravitational force are comparable with the acceleration term in (REF ).",
"In fact, as is described in standard handbooks, e.g.",
"[6], the post-Newtonian corrections are either proportional to $\\Delta \\mu $ or $\\dot{y}_{1,2}$ .",
"The mass difference and velocities are assumed to be small and therefore, the terms in (REF ) arising from the post-Newtonian corrections appear to be small compared to the terms taken into account.",
"From (REF ) it follows that when the mass difference is negative and $D_{FW}=2|\\Delta \\mu |$ the particles are at rest with respect to each other.",
"In this case the Fermi-Walker coordinate system locally coincide with the Rindler one and the particles accelerate indefinitely.",
"Thus, unlike the Newtonian case considered in the previous Section the particles accelerating indefinitely being at rest with respect to each other must have the small mass difference.",
"This effect was first noted by Bondi 1957 [1].",
"It is obviously due to the non-uniform character of the acceleration term."
],
[
"Solution of dynamical equations",
"Since equation (REF ) contains only $V$ and $D$ it can be used to express $V$ in terms of $D$ $V={EP\\over E_c^2+P^{2}}(1\\mp {E_c\\over EP}\\sqrt{E_{c}^{2}-N^{2}}),$ where $E_{c}$ is expressed through $D$ in equation (REF ), $E$ and $P$ are given in equation (REF ).",
"As we discussed above we assume that at the initial moment of time $t=\\tau =0$ we have $V=0$ .",
"That means that initially we have to choose the sign $(-)$ in (REF ).",
"However, under certain conditions discussed below the direction of motion of the particles relative to each other and, accordingly, $\\dot{D}$ , changes sign.",
"At the turning point $\\dot{D}=0$ we have $N^{2}=E_{c}^2$ .",
"Since velocity $V$ must grow monotonically according to (REF ) we must take the sign $(+)$ in (REF ) after the turning point.",
"On the other hand equation (REF ) contains only $D$ and its derivative with respect to the time $\\tau $ , and, therefore, it can be integrated to obtain the dependence of $D$ on time.",
"Explicitly we have $\\int ^{D}_{D_{min}}{xdx \\over \\sqrt{R(x)}}=\\tau /\\mu , $ where $R(x)=(\\mu ^2+2\\Delta \\mu x)^{2}-N^{2}x^{2}.$ The integral in (REF ) can be evaluated by a standard substitution to give an explicit relation between $\\tau $ and $D$ .",
"However, the final expressions are rather cumbersome and we do not show them here.",
"Instead, in general, we analyse qualitatively solutions to (REF ) based on analogy between this equation and an equation describing a motion of a particle in a potential well.",
"For that we bring (REF ) to a standard form ${{\\dot{D}}^{2}\\over 2}+ U(D)={\\cal E}, \\quad U(D)=-{2\\Delta \\mu \\over D}-{\\mu ^{2}\\over 2D^2},$ where ${\\cal E}={4\\Delta \\mu ^{2}-N^{2}\\over 2\\mu ^{2}}={v_{in}^{2}\\over 2}-{2\\Delta \\mu \\over D_{in}}-{\\mu ^{2}\\over 2D_{in}^2}.$ Introducing natural units $\\tilde{U} = {\\mu ^2\\over \\Delta \\mu ^{2}}U$ and $\\tilde{D} = {|\\Delta \\mu |\\over \\mu ^2}D$ we can express $\\tilde{U} $ in terms of $\\tilde{D}$ in a very simple form: $\\tilde{U} =\\mp {2\\over \\tilde{D}}-{1\\over 2{\\tilde{D}}^2}$ , where the sign $-$ ($+$ ) corresponds to $\\Delta \\mu > 0$ ($\\Delta \\mu < 0$ ).",
"The dependence $\\tilde{U}(\\tilde{D})$ is shown in Fig.",
"1.",
"Figure: The dependence of the potential UU on the spacialcoordinate DD.",
"The solid curve corresponds to the case Δμ>0\\Delta \\mu > 0 while the dashed one to the case Δμ<0\\Delta \\mu < 0 .At first let us consider in detail an important case of zero norm of the energy-momentum vector, $N^{2}=0$ , and set, accordingly, $P=E$ .",
"A simple analysis of equation (REF ) shows that in this case there are no turning points, the relative separation $D$ grows with time and the value of $V=1$ can be achieved in the asymptotic limit $\\tau \\rightarrow \\infty $ .",
"Therefore, in this case the system may accelerate indefinitely.",
"When $N^{2}=0$ equation (REF ) simplifies to $V={E^{2}-E_c|E_c|\\over E^2+E_c^2}, \\quad \\gamma ={(E^2+E_c|E_c|)\\over 2EE_c} $ and from equation (REF ) we get $\\tau ={1\\over 4\\Delta \\mu ^2}(2\\Delta \\mu (D-D_{min})-\\mu ^{2}\\log ({\\mu ^2 + 2\\Delta \\mu D\\over \\mu ^2 +2\\Delta \\mu D_{min}})).$ From equation (REF ) it follows that when $E_c > 0$ the indefinite acceleration is possible only if $E_c \\rightarrow 0$ when $\\tau \\rightarrow \\infty $ and from the expression for $E_c$ (REF ) it is seen that the mass difference $\\Delta \\mu $ must be negative for that.",
"We consider below only this case in detail.",
"When $|\\Delta \\mu |\\ne 0$ $E_c\\rightarrow 0$ provided that $D\\rightarrow D_{crit}=\\mu ^2/(2|\\Delta \\mu |)$ .",
"Equation (REF ) tells that the logarithm on the right hand side diverges when $D\\rightarrow D_{crit}$ .",
"That means that this limit does correspond to the limit $\\tau \\rightarrow \\infty $ .",
"Let us estimate the dependence of the Lorentz factor $\\gamma $ on time in this case.",
"To do so, we introduce a new variable $\\Delta =D_{crit}-D$ and substitute it to (REF ) assuming that it is small.",
"We get $\\tau \\approx {\\mu ^{2}\\over 4\\Delta \\mu ^2}\\log ({\\mu ^{2}-2|\\Delta \\mu |D_{in}\\over 2|\\Delta \\mu |\\Delta }), $ and, substituting this result into equation (REF ) we have $\\gamma \\approx {\\mu ^2\\over 4|\\Delta \\mu |D_{in}}\\exp {4\\Delta \\mu ^2\\over \\mu ^2}\\tau .$ Equation (REF ) tells that when $D\\approx D_{crit}$ acceleration is exponentially fast.",
"The degenerate case $\\Delta \\mu =0 $ must be analysed separately.",
"In this case from (REF ) we have $\\tau ={1\\over 2\\mu ^2}(D^{2}-D_{min}^{2}), $ and the distance $D$ increases indefinitely with time.",
"From equation (REF ) we obtain $\\gamma \\approx {\\mu \\over 2D_{min}}\\sqrt{2\\tau }.",
"$ Now let us turn to the general case $N^{2} \\ne 0$ .",
"Setting $\\dot{D}=0$ in (REF ) we get a general equation for the turning points $D_{1,2}={\\Delta \\mu \\over {\\cal E}}(-1\\pm \\sqrt{1-{\\mu ^{2}{\\cal E}\\over 2\\Delta \\mu ^2}})={\\Delta \\mu \\over {\\cal E}}(-1\\pm {\\sqrt{N^2}\\over 2\\Delta \\mu }).",
"$ Equation (REF ) tells that the turning points exist only when $N^{2} > 0$ .",
"Their number depends on signs of ${\\cal E}$ and $\\Delta \\mu $ .",
"When $\\Delta \\mu > 0$ the potential $U(D)$ is negative, see Fig.",
"1, and therefore, the relative motion is finite for ${\\cal E} < 0$ with one turning point Let us remember that we consider only positive values of $D$ .",
"$D_{1}={\\Delta \\mu \\over |{\\cal E}|}(1+{N\\over 2\\Delta \\mu }).$ In the opposite case ${\\cal E} > 0$ , and, accordingly, $N <2\\Delta \\mu $ , the motion is unbound and the relative distance $D$ grows indefinitely with time.",
"When $\\Delta \\mu < 0$ the potential $U(D)$ acquires positive values for $D > {\\mu ^{2}\\over 4|\\Delta \\mu |}$ , see Fig.",
"1.",
"It tends to zero when $D \\rightarrow \\infty $ and has a maximum at $D=D_{crit}$ .",
"Note that from the condition $U(D_{crit})={\\cal E}={2\\Delta \\mu ^2\\over \\mu ^2}$ we get $N^{2}=0$ there.",
"The character of the relative motion depends on whether ${\\cal E}$ is negative, belongs to the interval $0 < {\\cal E} < {2\\Delta \\mu ^2\\over \\mu ^2}$ corresponding to $0 < N < 2|\\Delta \\mu |$ , or ${\\cal E} > {2\\Delta \\mu ^2\\over \\mu ^2}$ and, accordingly, $N^{2} <0$ .",
"When the energy ${\\cal E}$ is negative the motion is bound with one turning point $D_{1}={|\\Delta \\mu |\\over |{\\cal E}|}(-1+ {N\\over 2\\Delta \\mu }).$ In the intermediate region $0 < {\\cal E} < {2\\Delta \\mu ^2\\over \\mu ^2}$ there are two turning points $D_{\\pm }={|\\Delta \\mu | \\over {\\cal E}}(1\\pm {N\\over 2\\Delta \\mu }).$ When $D_{in} < D_{-}$ the motion is bound while for $D_{in} >D_{+}$ $D$ grows indefinitely.",
"Finally, when $N^{2} < 0$ the motion is always unbound.",
"When the motion is bound the velocity $\\dot{D}$ changes sign after the turning point and in this case we should use the sign $(+)$ in (REF ).",
"Taking into account that $D$ is decreasing after the turning point and that $E_{c} \\propto D^{-1}$ we see from (REF ) that the velocity $V\\rightarrow 1$ .",
"The particles tend to collide.",
"However, our assumption that $D \\gg \\mu $ breaks down in this case and we cannot describe the motion at scales $D\\sim \\mu $ within the framework of our formalism.",
"Note that we consider in this study only pairs of particles with strictly zero angular momentum.",
"In the situation when the particles have a small but nonzero angular moment they would miss each other and after certain moment of time the distance $D$ would become negative.",
"In this case the analysis of this paper can be repeated without any major change for negative values of $D$ and one would conclude that for such parameters of motion there is another symmetric turning point at negative values of $D$ .",
"Thus, the relative motion of a pair of particles with small but nonzero orbital momentum would be periodic much similar to the case of ordinary particles with positive masses.",
"Now let us consider the case of the unbound motion and estimate the maximal value of the Lorentz gamma factor the system can reach.",
"As follows from our previous discussion when $N^2\\ne 0$ the distance $D$ grows indefinitely.",
"That means that the energy in the comoving frame, $E_{c}$ , must tend asymptotically to $2\\Delta \\mu $ .",
"Note that when $\\Delta \\mu < 0$ the asymptotic value of $E_c$ is negative.",
"We have from (REF ) setting $E_{c}=2\\Delta \\mu $ there $V={1\\over 4\\Delta \\mu ^2+E^{2}-N^2}(E\\sqrt{E^2-N^2}-2\\Delta \\mu \\sqrt{4\\Delta \\mu ^2 -N^2}).$ Equation (REF ) tells that when $\\Delta \\mu > 0$ the last term in the brackets is negative and the asymptotic value of velocity is smaller than 1.",
"Large values of $V$ can be achieved in the opposite case $\\Delta \\mu < 0$ assuming $|N^2| \\ll \\Delta \\mu ^2 $ .",
"In this case we expand expressions in (REF ) in the Taylor series in $|N^2|/\\Delta \\mu ^2 $ to obtain $V=1-{N^{4}\\over 32\\Delta \\mu ^2 E^2}, $ and, accordingly $\\gamma \\approx {1\\over \\sqrt{2(1-V)}}={4|\\Delta \\mu E|\\over |N^2|}.$ Equation (REF ) tells that for fixed values of $E$ and $\\Delta \\mu < 0$ the gamma factor can be made arbitrary large by choosing arbitrary small values of $|N^{2}|$ .",
"This conclusion is in agreement with our previous finding that the system accelerates indefinitely when $N^2=0$ ."
],
[
"Methodological comments",
"Here I would like to make comments on several methodological issues related to the problem.",
"1) At first glance the fact that the 'average' position of the pair $(x_1+x_2)/2$ always grows with time may seem to be in contradiction with the law of conservation of the centre of mass of the system.",
"This contradiction is resolved by observation that for the system containing particles of opposite masses position of the centre of mass, $R$ , is determined by a difference of positions of particular particles.",
"Say, in the Newtonian approximation we have $R=m_1x_1+m_2x_2=(\\mu +\\Delta \\mu )x_1-(\\mu -\\Delta \\mu )x_2$ .",
"In the relativistic case the situation is analogous for systems with $N^2 > 0$ .",
"In the opposite case the notion of centre of masses is ill defined.",
"Indeed, introducing the velocity of a coordinate system, where the centre of mass is at rest in a standard way as $V_{cm}=P/E$ [6] we see that when $N^{2}=0$ $V_{cm}=1$ and when $N^{2} < 0 $ $V_{cm}$ formally exceeds the speed of light.",
"It is obvious that the notion of the centre of mass is redundant in both cases.",
"2) In Introduction of their paper the authors of [3] claim that the conception of PMT put forward by the author of this note is related to the problem of indefinite acceleration of two gravitationally interacting particles.",
"This statement needs, in my opinion, a clarification.",
"First, let me note that as it is discussed above even in the case when only a finite acceleration of the particles is attained PMT is still possible in a situation where production of such pairs is provided by some physical mechanism.",
"Second, the conception of PMT, in general, does not rely on gravitation interactions at all.",
"In particular, in paper [2] I consider a model where there is a continuous flow of positive energy from some spacial regions having negative energy to other regions with positive energy provided by hydrodynamical effects.",
"In this model the space-time is assumed to be flat and gravitational interactions are absent.",
"Moreover, in order to construct a PMT it is not necessary to invoke objects having negative rest masses.",
"It is enough to consider a medium with positive comoving energy density violating the weak energy condition [2].",
"Additionally, there are ways of constructing PMT, where gravitational interaction plays a totally different role, say, transferring the energy from a non-stationary system having negative mass to gravitational waves, as for example in the model of a rotating relativistic string connected by two negative mass monopoles [2], [7].",
"The effects related to the dynamics of free negative mass particles are clearly irrelevant to such systems.",
"3) The authors of [3] claim that it is impossible to obtain, in principal, an indefinite acceleration of the system containing two particles with opposite signs of masses.",
"One may think that this clearly contradicts to the Bondi's result [1] and the results of this paper.",
"The conundrum is resolved by observation that the authors of [3] consider only relative motions while Bondi's analysis as well the analysis in this note also deal with the motion of the pair of particles as a whole with respect to an inertial frame."
],
[
"Conclusions",
"In this note we show by elementary means that in the weak limit approximation a pair of particles having opposite values of masses can be accelerated indefinitely provided that the energy-momentum vector characterising the system is null.",
"The system can also be accelerated to arbitrary large Lorentz factors when the mass difference $\\Delta \\mu < 0$ and the norm of the energy-momentum vector is sufficiently small.",
"Assuming that there is a process of production of such pairs and that the positive mass particles are intercepted with a target while the negative mass particles fly away it is possible to transfer to the target any desired amount of energy.",
"In a more natural situation one can also consider a theory where the positive and negative mass particles interact differently with a normal matter.",
"A general situation of this kind where there is a persistent transfer of energy from a subsystem having negative or almost zero energy (like this pair ) to a subsystem with positive energy was dubbed by us 'Perpetuum Motion of Third Kind' (PMT) [2].",
"Note, however, that it is just a classical analog of the well known instability of a quantum system with a number of negative energy states unbound from below.",
"The question of whether the existence of PMT or ever accelerating pairs of particles is a paradox depends, in our opinion, on definition of what paradox is.",
"On one hand, for example, Bonnor 1989 states 'I regard the runaway (or self-accelerating ) motion ... so preposterous that I prefer to rule it out by supposing that inertial mass is all positive or all negative' [8].",
"Clearly, existence of PMT can also be classified as a kind of runaway.",
"On the other hand, no laws of physics are broken in such systems.",
"We believe that the existence of runaways of these kinds is dangerous for theories where they present.",
"To exemplify, an indefinite concentration of energy of different signs in spatially separated regions could lead to a highly inhomogeneous space-time hardly compatible with presence of any life.",
"Therefore, such theories should be ruled out though some additional study of them in General Relativity may be of some interest.",
"Since in our approximation only linear metric perturbations and one next-to-the-leading order term determined by the acceleration of the pair as a whole are taken into account, it is interesting to estimate what kind of corrections can be obtained by considering other higher order terms quadratic in metric perturbations?",
"For a non-relativistic motion with $V \\ll c$ for this purpose one can use the well known Einstein-Infeld-Hoffmann equations of motion (e.g.",
"[6]).",
"In this way it is convenient to consider particles with a large mass difference as well as systems with non-zero angular momentum.",
"There are, however, many corrections, which are absent in such a treatment, notably the emission of gravitational waves.",
"Therefore, a self-consistent relativistic treatment of the problem in the next to the weak field approximation must be based on the second-order formalism of Havas and Goldberg 1962.",
"Such an approach is left for a possible future work.",
"Although in this paper we consider only particles with no internal structure our analysis may also be valid for a pair of extended objects with total energies of opposite signs provided that they have a sufficiently large separation distance and that their relative velocities are sufficiently small.",
"For example, Deser and Pirani [9] considered the behaviour of systems with all possible inertial/gravitational mass signs and noted that a pair of geons having opposite signs of their total energies would behave as a pair of point particles in the corresponding limit.",
"It is also interesting to point out that the notion of 'Perpetuum Motion of Third Kind' was introduced in the context of thermodynamical systems having negative temperatures, where one can withdraw heat from a negative temperature reservoir and convert it completely to work, see e. g. [10], p. 176.",
"Since thermodynamical systems with negatives masses of their components should have negative temperatures (e.g.",
"[11]) there is a link between thermodynamical properties of such systems and the ones discussed in this paper.",
"In particular, a runaway process occurring in a thermodynamical system having two subsystems containing particles with opposite signs of masses has been discussed in e.g.",
"[12].",
"It has been mentioned that this process is analogous to the self-acceleration of a pair of particles with opposites signs of their masses.",
"I am grateful to S. Deser, J. F. Gonza'lez Herna'ndez, I. D. Novikov, K. A. Postnov and V. N. Strokov for useful comments.",
"This work was supported in part by the Dynasty Foundation, in part by ”Research and Research/Teaching staff of Innovative Russia” for 2009 - 2013 years (State Contract No.",
"P 1336 on 2 September 2009) and by RFBR grant 11-02-00244-a."
]
] | 1204.1269 |
[
[
"Longitudinal and transverse velocity scaling exponents from merging of\n the Vortex filament and Multifractal models"
],
[
"Abstract We suggest a simple explanation of the difference between transverse and longitudinal scaling exponents observed in experiments and simulations.",
"Based on the Vortex filament model and Multifractal conjecture, we calculate both scaling exponents for any n without any fitting parameters and ESS anzatz.",
"The results are in very good agreement with the data of simulations."
],
[
"Longitudinal and transverse velocity scaling exponents from merging of the Vortex filament and Multifractal models.",
"K. P. ZybinElectronic address: [email protected] V. A. Sirota P.N.Lebedev Physical Institute of RAS, 119991, Leninskij pr.53, Moscow, Russia We suggest a simple explanation of the difference between transverse and longitudinal scaling exponents observed in experiments and simulations.",
"Based on the Vortex filament model and Multifractal conjecture, we calculate both scaling exponents for any $n$ without any fitting parameters and ESS anzatz.",
"The results are in very good agreement with the data of simulations.",
"Numerous direct numerical simulations (DNS) of hydrodynamical turbulent flow, as well as high-resolution experiments, have been performed during last twenty years [1].",
"The positive result of these investigations is that numerical and experimental approaches demonstrate a very good agreement.",
"In particular, velocity statistics calculated from DNS practically coincides with that obtained from experiments [1].",
"This justifies the accuracy of both kinds of results.",
"But there still is a lack of physical understanding of the processes occuring in a turbulent flow and contributing to statistics.",
"Which of the infinite number of solutions to the NSE are responsible for the observed intermittent behavior, and why?",
"Okamoto et al.",
"[2] claimed that the most part of the flow's helicity and, in some sense, the most part of information about the flow are contained in the same small part of the liquid volume spread more or less uniformly throughout the liquid.",
"These structures are stable, their lifetime exceeds many times the largest eddy turnover time in the flow.",
"This seems us to be a very important step.",
"Probably, these elongated regions of high vorticity are just the structures that determine the small-scale statistics of a turbulent flow.",
"Thus, it would be useful to elaborate some physical understanding of these objects.",
"Such an attempt was performed in [3], [4], [5], [6], [7], [8], [9], [10].",
"The main idea is that in the regions where vorticity is high (vortex filaments), the vorticity itself stabilizes the motion.",
"So, despite the stochastic large-scale forces that give energy to a filament, the motion inside the filament has an essential non-stochastic component.",
"This may be the reason of stability of filaments.",
"The random large-scale forces then act, on average, as a stretching force.",
"Stretching (not breaking!)",
"of vortices is the mechanism that provides the observed energy flux from larger to smaller scales, and observed statistical properties of turbulence.",
"This is the main difference of our approach from the Kolmogorov's approach developed in the K41 theory [11].",
"The relation between longitudinal and transversal Euler velocity structure functions is one of the problems that have had no answer up to now.",
"These functions are defined by $ S_n^{\\parallel }(l)=\\left< \\left| \\Delta {\\bf v}\\cdot \\frac{\\bf l}{l}\\right| ^n\\right>\\ , \\quad S_n^{\\perp }(l)=\\left< \\left| \\Delta {\\bf v} \\times \\frac{\\bf l}{l}\\right|^n\\right>$ Here $\\Delta {\\bf v} = {\\bf v}({\\bf r}+{\\bf l}) -{\\bf v}({\\bf r}) $ is velocity difference between two near points separated by ${\\bf l}$ , and the average is taken over all pairs of points separated by given $l$ .",
"Experiments and DNS show that inside the inertial range the structure functions obey scaling laws, $ S_n^{\\perp }(l)\\propto l^{\\zeta _n^{\\perp }} \\ , \\quad S_n^{\\parallel }(l)\\propto l^{\\zeta _n^{\\parallel }} \\ .$ The scaling exponents $\\zeta _n^{\\perp }$ , $\\zeta _n^{\\parallel }$ are believed to be independent on conditions of the experiment.",
"They are intermittent, i.e., $\\zeta _n^{\\perp }/n$ and $\\zeta _n^{\\parallel }/n$ are decreasing functions of $n$ .",
"The question whether the two scaling exponents coincide in isotropic and homogeneous turbulence or not, is open.",
"There is an exact statement that $\\zeta _n^{\\parallel }=\\zeta _n^{\\perp }$ for $n=2$ and 3 [11].",
"On the other hand, modern experiments [12], [13] and numerical simulations [14], [15], [16], [17] show some significant difference between $\\zeta _n^{\\parallel }$ and $\\zeta _n^{\\perp }$ at higher $n$ .",
"But the proponents of the equality argue that the difference may result from finite Reynolds number effects [18], [19], [20] or anisotropy [21].",
"The aim of the paper is to analyze the question by means of the Vortex filament model.",
"We show that the divergence of the scaling exponents can be understood and calculated from general physical considerations using the tools provided by the Vortex filament model and the well-known Multifractal (MF) conjecture.",
"Before we consider the problem from the viewpoint of the Vortex filament model, we note that isotropic and homogeneous medium does not necessarily imply $\\zeta _n^{\\perp }=\\zeta _n^{\\parallel }$ .",
"For a simple illustration, consider a gas of tops, each rotating around its own axis, the axes directions distributed randomly.",
"Although locally a strong asymmetry could be found near each top, the whole picture remains isotropic.",
"The Vortex filament model provides analogous situation that might be in real turbulence.",
"In [10] we suggested to join our model of vortex filaments with the MF model to get velocity scaling exponents.",
"We now remind briefly some ideas of the two models and the results of [10].",
"Then we proceed to the difference between the longitudinal and transverse scaling exponents.",
"The MF model implies that the determinative contribution to velocity structure functions is given by the solutions (regions) where $\\Delta v(l) = | {\\bf v }({\\bf r} + {\\bf l}) -{\\bf v}({\\bf r}) | \\sim l^{h ({\\bf r}) }$ So, to calculate structure functions, it is enough to consider only a set of scaling solutions that can be numbered by $h$ .",
"This property is called local scale invariance.",
"The Large Fluctuations Theorem states that the probability of measured velocity difference $\\Delta v (l)$ to have the scaling $h$ is a power-law function of $l$ : $ \\texttt {P} \\propto l^{3-D(h)} $ Knowing $D(h)$ , one could in principle calculate all structure functions: $\\langle \\Delta v^n \\rangle =\\int \\delta v^n \\texttt {P} dh =\\int l^{nh} l^{3-D(h)} d\\mu (h)$ Here $d\\mu (h)$ is the measure that is responsible for relative weights of different values $h$ .",
"In the limit $l\\rightarrow 0$ , only the smallest exponent contributes to the integral; it then follows $\\zeta _n= \\min \\limits _h \\left( nh+3-D(h)\\right)$ Without loss of generality, one can treat $D(h)$ as a concave function.",
"As follows from (REF ), the point $h_0$ where $D(h)$ reaches its maximum corresponds to $n=0$ : the requirement $\\zeta _0=0$ leads to $D(h_0)=3$ .",
"The definitional domain of $D$ is restricted by some $h_{min}$ and $h_{max}$ .",
"Since we are interested in positive values $n$ , we will hereafter restrict ourselves by $h\\le h_0$ .",
"What about $h_{min}$ , from the condition [11] $\\zeta ^{\\prime \\prime }_n<0$ and the finiteness of the Mach number (or from the absence of singularities of $\\Delta v $ as $l\\rightarrow 0$ ) it follows $ \\zeta ^{\\prime }_n \\ge 0$ for any $n$ .",
"The minimum in (REF ) is reached at $h_n=\\zeta ^{\\prime }_n$ .",
"Since negative values of the derivative are forbidden, we get $h_{min}\\ge 0$ .",
"The behavior of $D(h)$ near $h_{min}$ determines the behavior of $\\zeta _n$ at $n\\simeq n_* = D^{\\prime }(h_{min}) \\ .$ For larger $n$ , the minimum in (REF ) is reached at the boundary $h=h_{min}$ .",
"Thus, the behavior of $\\zeta _n$ at very large $n$ depends only on the value of $h_{min}$ : the asymptotic behavior of $\\zeta _n$ for all non-zero $h_{min}$ is $\\left.",
"\\zeta _n \\right|_{n>n_*} = n h_{min} +3- D(h_{min}) \\sim _{n\\rightarrow \\infty } n h_{min}$ If $h_{min}=0$ , the possible growth of $\\zeta _n$ at $n>n_*$ is steeper than linear; if $D(h_{min})$ is finite, we get $\\left.",
"\\zeta _n \\right| _{n>n_*} = const $ .",
"We now proceed to the Vortex filament model.",
"In [3] we derived the equation $\\ddot{\\omega }_i = \\rho _{ij}\\omega _j \\ , \\quad \\rho _{ij}=-\\nabla _i \\nabla _j p$ Here ${\\omega }$ is the vorticity of the flow, ${\\omega }=\\nabla \\times {\\bf v}$ ; .",
"$\\rho _{ij}$ is the pressure hessian.",
"This equation is the direct consequence of the inviscid limit of the NSE and describes the evolution of vorticity along the trajectory of a liquid particle.",
"It was for the first time obtained in [22].",
"The main assumption of the theory is that inside vortex filaments the 'longitudinal' (in relation to ${\\omega }$ ) part of the hessian doesn't depend on the local vorticity, and is determined by the large-scale component of a flow.$^1$ [1] An indirect confirmation of this statement is presented in [23], where the maximal eigenvalue of the pressure hessian is shown to be orthogonal to vorticity.",
"This makes (REF ) a linear stochastic equation in relation to ${\\omega }$ .",
"In [3] we wrote the corresponding equation for probability density function (PDF), $f({\\omega }, \\dot{{\\omega }}, t)$ .",
"Solving the equation and integrating over all angles and over $\\dot{\\omega }$ , we found the intermediate asymptotic solution for the PDF at $\\omega \\rightarrow \\infty $ : $ P(\\omega ) \\propto 1 / \\omega ^4 $ In [10] we used it to derive the condition $\\min \\limits _h (3h+2-D(h)) =0$ for the function $D(h)$ .",
"As follows from (REF ), this condition is just equivalent to $\\zeta _3 =1$ which is the well-known Kolmogorov's '4/5 law'.",
"The MF model is a dimensional theory, so it does not distinguish longitudinal and transverse scaling exponents.",
"Also the condition (REF ) is valid for both of them.",
"To describe the difference, it is natural to use two different functions $D(h)$ for $\\zeta _n^{\\perp }$ and $\\zeta _n^{\\parallel }$ [17].",
"But what is the relation between them, why does the difference appear?",
"In [4], [5] we introduced a simple model of an axially symmetric vortex filament: $ v_r= a(t) r\\,, \\qquad v_z = b(t) z \\,,\\qquad v_{\\phi }=\\omega (t) r$ From the incompressibility condition and the Euler equation it then follows $\\begin{array}{l}2a+b=0\\,,\\qquad \\dot{a} +a^2 -\\omega ^2 = -P_1(t) \\ ,\\\\\\dot{\\omega } +2 a \\omega =0\\,,\\qquad \\dot{b} +b^2 =-P_2(t) \\ ,\\\\ p =\\frac{1}{2} P_1(t) r^2 + \\frac{1}{2} P_2(t) z^2\\end{array}$ Taking the second derivative of $\\omega $ , we get $ \\ddot{\\omega } = -P_2(t)\\omega \\ ,$ which corresponds to (REF ).",
"In accordance with the assumption discussed above, let $P_2(t)$ be a random function independent of $\\omega $ .",
"Then all moments of $\\omega (t)$ increase exponentially as a function of time, $a\\sim \\dot{\\omega } / \\omega \\ll \\omega $ for large $t$ .",
"It is easy to calculate that in this model $S^{\\perp }_n \\propto \\langle \\omega ^n \\rangle l^n $ , $S^{\\parallel }_n \\propto \\langle a^n \\rangle l^n $ .",
"In [8] we discussed the connection between $t$ and $l$ and showed that $t \\propto - \\ln l$ .",
"Hence, in this model $S^{\\perp }_n \\gg S^{\\parallel }_n$ , $\\zeta _n^{\\perp } < \\zeta _n^{\\parallel }$ .",
"The equality of scaling exponents can be restored if we take the other branch of the same filament into account, assuming that the filament is closed and 'turns back' at very large $|z|$ .",
"Then there are two filaments with equal and opposite vorticities.",
"The second branch produces a perturbation of velocity $\\delta v \\sim \\alpha \\omega $ , where the coefficient $\\alpha $ is inversely proportional to the distance between the two branches.",
"(This is just analogous to an axial electric current producing magnetic field in vacuum; one can write the corresponding solution to the Euler equation by developing the perturbation as series in $\\alpha $ .)",
"The perturbation violates the axial symmetry, so the scaling exponents become equal.",
"However, the pre-exponent of $S^{\\parallel }_n$ is much smaller than that of $S^{\\perp }_n$ if $\\alpha $ is small.",
"The value of $\\alpha $ increases as we approach the turnover point, and the two branches become nearer.",
"This gives us a hint that the regions where a filament is strongly curved may make a small contribution to $S^{\\perp }_n$ but contribute much to $S^{\\parallel }_n$ .",
"This effect is stronger for large $n$ , since $S^{\\parallel }_n\\sim \\alpha ^n S^{\\perp }_n$ .",
"So, for infinitely large $n$ we can expect that transverse structure functions are dominated by extremely stretched-out and very thin, roughly axially symmetric filaments with very high vorticities, while for longitudinal structure functions one needs extremely curved parts of these filaments.",
"The discussed example is one of numerous solutions to the Euler equation that are presented in a turbulent flow.",
"The observed exponents $\\zeta _n^{\\perp }$ and $\\zeta _n^{\\parallel }$ are produced by contributions of many filaments.",
"Different filaments (and different parts of them) make contributions to different $n$ (or, in terms of MF theory, to different $h$ ).",
"This picture is very complicated, but, thanks to the MF model, we don't need to know it in details.",
"Approximating $D(h)$ by a second-order polynomial [10] and making use of (REF ), we only have to fix one more point of the curve.",
"For this purpose, we consider the minimal possible $h=h_{\\min }$ .",
"This corresponds to $n\\rightarrow \\infty $ .",
"In [10] we have shown that $h_{\\min }=0$ : values $h<0$ are forbidden by the condition $d \\zeta _n / d n \\ge 0$ (see p.2).",
"The existence of $h=0$ is proved by the possibility of a cylindric filament with rotating velocity independent of $r$ : ${\\bf v} = [{\\bf e}_z, {\\bf r}/r] $ One can check that in this extreme case, indeed, $\\delta v (l) \\sim l^0$ .",
"The velocity (REF ) satisfies the Euler equation with pressure logarithmically divergent; this means that for any positive $h$ , there exists a corresponding velocity distribution with converging pressure.",
"Calculation of the correlators directly from (REF ) gives under the limit $n\\rightarrow \\infty $ : $\\langle \\left| \\Delta v \\times {\\bf l}/l \\right| ^n \\rangle \\propto \\frac{2^n}{n} \\, l^2\\,,\\quad \\langle \\left| \\Delta v \\cdot {\\bf l}/l \\right| ^n \\rangle \\propto n^{-5/2} l^2$ The proportionality to $l^2$ is caused by the axial geometry of the filament (integrating $r dr$ ).",
"This corresponds to the definition of $D$ given in, e.g., [11]: the probability that at least one of a pair of points would get inside the filament is proportional to $l^2$ .",
"Thus, we get $D^{\\perp }(0)=1$ To find the difference between $S^{\\perp }_n$ and $S^{\\parallel }_n$ , we now remind that, at infinitely large $n$ , $S^{\\perp }_n$ are contributed mostly by 'cylindric' parts of filaments, while $S^{\\parallel }_n$ are dominated by point-like regions where filaments are bent very strongly.",
"Actually, the behavior of the pre-exponents in (REF ) shows that the contribution of the 'extreme' filament to $S_n^{\\perp }$ increases as $n \\rightarrow \\infty $ .",
"To the opposite, its contribution to $S_n^{\\parallel }$ for large $n$ (i.e., small $h$ ) is very small.",
"There must be other solutions to the Euler equation to determine the behavior of $D^{\\parallel }(h)$ for small $h$ and, equivalently, to make the most contribution to $S_n^{\\parallel }$ for large $n$ .",
"These solutions correspond to 'strongly curved' extreme filament.",
"To satisfy $h=0$ , velocity must be independent of $r$ .",
"It may, for example, take the form ${\\bf v} = (v_r(\\theta ), v_{\\theta }(\\theta ), 0)$ .",
"Such a solution exists but it cannot be written analytically.",
"At $\\theta =0$ , the radial velocity diverges weakly: $v_{\\theta } \\propto -\\theta \\sqrt{\\ln \\,(1/\\theta )}\\,, v_r \\propto \\sqrt{\\ln \\,(1/\\theta )}$ ; the pressure divergence is $p\\propto \\ln r$ , just as in (REF ).",
"Since $\\delta v \\sim l^0$ in the case, and averaging includes $r^2 dr$ , the correlator is proportional to $l^3$ .",
"This corresponds to $D^{\\parallel }(0)=0$ The difference between the boundary values (REF ) and (REF ) determines the difference between the functions $D_{\\perp }$ and $D^{\\parallel }$ .",
"We seek the solution $D(h)$ in the simplest form $D(h) = 3- b(h-h_0)^2$ and use (REF ) and (REF ),(REF ) to find the two unknown parameters.",
"We obtain two equations: $\\begin{array}{ll}h^{\\perp }_0 - \\frac{3}{8} (h^{\\perp }_0)^2 = \\frac{1}{3} \\ , \\ b^{\\perp }=2/(h^{\\perp }_0)^2\\quad & \\mbox{for} \\ D^{\\perp }\\\\h^{\\parallel }_0 - \\frac{1}{4} (h^{\\parallel }_0)^2 = \\frac{1}{3} \\ , \\ b^{ \\parallel }=3/(h^{\\parallel }_0)^2\\quad & \\mbox{for} \\ D^{\\parallel }\\end{array}$ Each of the equations has two roots.",
"The bigger roots are non-physical, because in this case the curve $\\zeta _n$ would be constant already at $n \\ge n_*=D^{\\prime }_h(0)=2b h_0 \\sim 2$ , and (REF ) would not hold.",
"For the second roots, in accordance with (REF ), we have $ \\zeta _n = n h_0 - n^2 /4b \\ , \\qquad n \\le n_* = 2 b h_0$ Substituting the values of $h_0$ ,$b$ , we get the scaling exponents: $ \\begin{array}{ll}\\zeta _n^{\\parallel } = & \\left\\lbrace \\begin{array}{lr}0.367 n - 1.12\\cdot 10^{-2} n^2 \\ , & \\quad n \\le 16.3 \\ ; \\\\3 \\ , & \\quad n>16.3 \\ .\\end{array} \\right.\\\\ \\zeta _n^{\\perp } = & \\left\\lbrace \\begin{array}{lr}0.391 n - 1.91 \\cdot 10^{-2} n^2 \\ , &\\quad n \\le 10.2 \\ ; \\\\2 \\ , & \\quad n>10.2 \\ .\\end{array} \\right.\\end{array}$ In Fig.",
"1 we compare this theoretical prediction with experimental data by [16] and [15].",
"We see that the result of our simple model is very close to the experimental results and lies inside the error bars of the experiments.",
"Figure: Euler longitudinal (upper branch) and transverse (lower branch) scalingexponents: the results of DNS from (◯\\bigcirc , □\\square )and (△\\bigtriangleup .",
"▽\\bigtriangledown ), and the prediction of the theory() (lines).Our simple model has one difficulty: the two parabolas plotted in Fig.",
"1 coincide at $n=0$ and $n=3$ , hence they do not coincide at $n=2$ .",
"This contradicts to the exact theoretical statement that $\\zeta _2 = \\chi _2$ .",
"However, this difficulty is caused by postulating the simplest parabolic shape for $D(h)$ .",
"It can easily be solved by adding one more parameter and assuming $D(h)$ to be a cubic polynomial.",
"Because of very small divergence ($1.6 \\cdot 10^{-2}$ ) between $\\zeta _2$ and $\\chi _2$ in Fig.",
"1, the coefficient by the eldest order would be very small ($\\sim 10^{-4}$ ).",
"It would change very slightly (unnoticeable for an eye) the lines presented in Fig. 1.",
"The only thing they may change significantly is the rate of approaching the constant at large (but still intermediate) $n$ (from 10 to 15, approximately).",
"But this range of $n$ is, anyway, badly described by the lowest-order polynomials: adding more degrees of freedom with very small coefficients, though unimportant for smaller $n$ , would change the solutions for these $n$ .",
"However, the changes cannot be very big, since the exponents are still restricted by the values 2 and 3, respectively.",
"One more comment is that, knowind $D(h)$ , one can use the MF model to calculate, e.g., the PDF of velocity gradients or accelerations.",
"In [1] it is shown that once $D(h)$ fits $\\zeta _n$ well, it would also fit well the other quantities.",
"Thus, we propose an explanation of the difference between $\\zeta _n^{\\parallel }$ and $\\zeta _n^{\\perp }$ based on the difference between the filaments that contribute to the two structure functions: roughly speaking, this is the difference between axially symmetric and strongly curved ones.",
"This allows to find the values of $\\zeta _n$ for very large $n$ , and merging of the Vortex filament and Multifractal theories gives the whole functions.",
"The obtained solutions (REF )-(REF ) fit very well the observed scaling exponents $\\zeta _n^{\\parallel }$ and $\\zeta _n^{\\perp }$ , and we hope that they reveal the nature of the difference between longitudinal and transverse structure functions.",
"We thank Prof.A.V.",
"Gurevich for his kind interest to our work.",
"The work was partially supported by the RAS Program 'Fundamental Problems of Nonlinear Dynamics'."
]
] | 1204.1465 |
[
[
"Periodic Rigidity on a Variable Torus Using Inductive Constructions"
],
[
"Abstract In this paper we prove a recursive characterisation of generic rigidity for frameworks periodic with respect to a partially variable lattice.",
"We follow the approach of modelling periodic frameworks as frameworks on a torus and use the language of gain graphs for the finite counterpart of a periodic graph.",
"In this setting we employ variants of the Henneberg operations used frequently in rigidity theory."
],
[
"Introduction",
"Given an embedding of a graph into Euclidean space as a collection of stiff bars and flexible joints (a framework), when is it possible to continuously deform the structure into a non-congruent position without breaking connectivity or changing the bar lengths (is the framework rigid or flexible)?",
"This is the fundamental question in rigidity theory.",
"The subject has many obvious applications, for example, in molecular biology, structural engineering and computer aided design [7].",
"Typically the question is $NP$ -hard [1] but for generic embeddings in the plane there is a complete combinatorial description.",
"Theorem 1.1 (Henneberg [10], Laman [12], Maxwell [15]) Let $G=(V,E)$ and let $p$ be a generic embedding into $\\mathbb {R}^2$ .",
"Then the following are equivalent: the framework $(G,p)$ is generically minimally rigid, $G$ satisfies $|E|=2|V|-3$ and $|E^{\\prime }|\\le 2|V^{\\prime }|-3$ for every $G^{\\prime }=(V^{\\prime },E^{\\prime })\\subset G$ , $G$ can be constructed from a single edge by recursively adding vertices of degree 2 and by removing an edge and adding vertices of degree 3 adjacent to the ends of the old edge.",
"We will assume some basic familiarity with the rigidity of finite frameworks, those unfamiliar may wish to consult [6], [7] or [26].",
"The operations defined in (3) will be referred to as Henneberg operations, [10].",
"Recently a lot of attention has been paid to finding analogues of Theorem REF for infinite graphs, specifically for periodic graphs (those with a finite quotient in an appropriate sense).",
"Particularly [19] and [20] developed the approach of considering periodic frameworks as frameworks embedded on a torus.",
"Indeed in [20] a characterisation was given for the case where the lattice (or torus) is fixed.",
"The purpose of this paper is to prove the following result concerning the case when one of the lattice vectors is allowed to change.",
"Theorem 1.2 A labelled graph is generically minimally rigid on the partially variable torus if and only if it can be derived from a single loop by gain-preserving Henneberg operations.",
"The gain-preserving Henneberg operations are extensions of the recursive moves in Theorem REF (3) to include labelled edges, and will be formally defined in Section .",
"One of the intricacies of the theorem is that we are working with a class of graphs, the $P(2,1)$ -graphs (defined in Section ), that fall in between $(2,1)$ -tight graphs and $(2,2)$ -circuits.",
"It is, in part, for this reason that we do not adopt a matroid theoretic approach in this paper.",
"Throughout the paper, we focus on a particular type of variable torus, namely one which is variable in the $x$ -direction only.",
"In fact, the results are much more general, and apply to frameworks on a torus which is variable in the $y$ -direction only, and frameworks on a torus which has a variable angle between two fixed-length generators.",
"We may also apply this result for a full characterisation of the generic rigidity of frieze-type patterns, that is, frameworks which are periodic in one direction only (interpreted as frameworks on a variable cylinder).",
"These variations are discussed in Section .",
"The study of periodic frameworks is a topic which has experienced a surge of interest over the past decade [3], [4], [14], [18], [9].",
"This work has been motivated in part by questions arising in materials science about the structural properties of zeolites, a type of mineral with a repetitive, micro-porous structure [22].",
"Furthermore, there may be physical meaning associated with certain restrictions of the fully variable torus.",
"It has been suggested that the time scales of atomic movement are significantly different from those of lattice deformation [23].",
"In this paper we take an inductive approach to the problem of characterizing the generic rigidity of periodic frameworks.",
"That is, we define a collection of local graph-theoretic moves which characterize the class of generically rigid periodic frameworks on a partially variable torus.",
"The inductive method has the advantage of being easy to state and understand.",
"Furthermore, while finding an inductive construction for a particular graph does not in general make for fast algorithms, once we have such an inductive sequence, it offers an immediate certificate of the rigidity of that framework."
],
[
"Results in Context",
"The basic theory of periodic frameworks has been well formalized by Borcea and Streinu [4].",
"The approach we use here is based on the presentation appearing in [19].",
"In that paper, the links between the approach of [4] and the present methodology are outlined in detail.",
"In [20], Ross proved an inductive characterisation of the generic rigidity of two-dimensional periodic frameworks on a fixed torus, that is, a torus with no variability.",
"The methods used here build on those results.",
"In [14] Malestein and Theran proved a characterisation of generic minimal rigidity of two-dimensional frameworks on the fully variable torus (three degrees of freedom).",
"They obtain the result of Ross as a restriction of their more general theorem.",
"However, their methods differ significantly from ours, in that they do not use an inductive characterisation.",
"Indeed giving an inductive construction for the relevant class of graphs on the fully variable torus is an intriguing open problem."
],
[
"Outline of Paper",
"In Section we recall the basic theory of periodic frameworks as frameworks on a torus.",
"The following two sections state the relevant rigidity results for the partially variable torus, Maxwell-type necessary conditions and Henneberg constructions preserving rigidity of frameworks.",
"In Section we prove some preliminary graph theory results, including an inductive construction of $P(2,1)$ -graphs that may be of independent interest.",
"The main body of the paper is contained in Section where we present a case by case analysis showing that the appropriate gains are preserved by the construction operations.",
"This allows us to prove our main theoretical result, Theorem REF , and hence to complete the proof of Theorem REF .",
"We describe some extensions of the work in Section .",
"The final section concludes the paper with some discussion of further work."
],
[
"Background",
"A periodic framework in the plane is a locally finite infinite graph which is symmetric with respect to the free action of $\\mathbb {Z}^2$ .",
"Such a framework has a finite number of vertex and edge orbits under the action of $\\mathbb {Z}^2$ .",
"Full definitions and details can be found in the work of Borcea and Streinu, [3], [4].",
"The approach taken here is to consider periodic frameworks as orbit frameworks on a torus, as in [19], [20]."
],
[
"Periodic Orbit Frameworks on the Variable Torus $\\mathcal {T}_x^2$",
"Let $\\mathcal {T}_x^2 = \\mathbb {R}^2 / {L_x\\mathbb {Z}^2}$ , where $L_x = L_x(t) = \\left(\\begin{array}{cc}x(t) & 0 \\\\y_1 & y_2\\end{array}\\right), \\ y_1, y_2 \\in \\mathbb {R}.$ We call $\\mathcal {T}_x^2$ the $x$ -variable torus, and the matrix $L_x = L_x(t)$ is the lattice matrix.",
"Similarly, let $L_0 = \\left(\\begin{array}{cc}x & 0 \\\\y_1 & y_2\\end{array}\\right), \\ x, y_1, y_2 \\in \\mathbb {R}$ be the fixed lattice matrix, and we call the quotient space $\\mathcal {T}_0^2 = \\mathbb {R}^2 / {L_0\\mathbb {Z}^2}$ the fixed torus.",
"For a graph $G$ , we will use the notation $V(G)$ and $E(G)$ to refer to the vertex and edge sets of $G$ , if not explicitly named.",
"A periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ on $\\mathcal {T}_x^2$ consists of a labelled, directed multigraph $\\langle G, \\mathbf {\\rangle }$ together with a position $p$ of the vertices $V(G)$ on the variable torus $\\mathcal {T}_x^2$ .",
"$\\langle G, \\mathbf {\\rangle }$ is called a gain graph [28], [8], and it encodes the way in which the graph $G$ is “wrapped\" around the torus.",
"Specifically, $\\langle G, \\mathbf {\\rangle }$ is composed of a directed multigraph $G=(V, E)$ , and a labelling $m: E^+ \\rightarrow \\mathbb {Z}^2$ .",
"The labelling of the edges is invertible, meaning that we may write $e=\\lbrace v_i, v_j; m_e\\rbrace = \\lbrace v_j, v_i; -m_e\\rbrace .$ From the periodic orbit graph we may define the derived periodic graph $G^m$ , which has vertex set $V^m = V \\times \\mathbb {Z}^2$ , and edge set $E^m = E \\times \\mathbb {Z}^2$ .",
"If $e = \\lbrace v_i, v_j; m_e\\rbrace \\in E\\langle G, \\mathbf {\\rangle }$ , then the edge $(e, z) \\in E^m, z \\in \\mathbb {Z}^2$ connects the vertices $(v_i, z)$ and $(v_j, z+m_e) \\in V^m$ .",
"In this way, the periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ is a kind of “recipe\" for the infinite derived graph $G^m$ (see Figure REF ).",
"Furthermore, the automorphism group of $G^m$ contains $\\mathbb {Z}^2$ .",
"Figure: The periodic orbit graph 〈G,〉\\langle G, \\mathbf {\\rangle } and its corresponding derived graph G m G^m.In a similar fashion we use the periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ to define an infinite periodic framework, the derived periodic framework $(\\langle G^m, L_x \\rangle , p^m)$ , where $p^m: V^m \\rightarrow \\mathbb {Z}^2$ is given by $p^m(v, z) = p(v) + zL_x, \\ \\textrm {where}\\ v \\in V, z \\in \\mathbb {Z}^2.$ An infinitesimal motion of $(\\langle G, \\mathbf {\\rangle },p)$ on $\\mathcal {T}_x^2$ is an element $(u, u_x) \\in \\mathbb {R}^{2|V| +1}$ , where $u: V \\rightarrow \\mathbb {R}^2, \\ \\textrm {and} \\ u_x:x(t) \\rightarrow \\mathbb {R},$ such that $(p_i - p_j - m_e L_x) \\cdot (u_i - u_j - (u_x(x), 0)) = 0 \\ \\textrm {for all} \\ \\lbrace v_i, v_j; m_e\\rbrace \\in E\\langle G, \\mathbf {\\rangle }.$ If $u_x = 0$ , and $u: V \\rightarrow z, z \\in \\mathbb {R}^d$ (i.e.",
"$u$ is a translation), then we say that $(u, u_x)$ is a trivial motion of $(\\langle G, \\mathbf {\\rangle },p)$ on $\\mathcal {T}_x^2$ .",
"If the only infinitesimal motions of a framework $(\\langle G, \\mathbf {\\rangle },p)$ on $\\mathcal {T}_x^2$ are trivial, then we say that $(\\langle G, \\mathbf {\\rangle },p)$ is infinitesimally rigid on $\\mathcal {T}_x^2$ .",
"Any infinitesimal motion $(u, u_x)$ of $(\\langle G, \\mathbf {\\rangle },p)$ on the $x$ -variable torus $\\mathcal {T}_x^2$ for which $u_x = 0$ is also an infinitesimal motion of $(\\langle G, \\mathbf {\\rangle },p)$ on the fixed torus $\\mathcal {T}_0^2$ .",
"The definitions of trivial motions and infinitesimal rigidity on $\\mathcal {T}_0^2$ are the same as for $\\mathcal {T}_x^2$ ."
],
[
"The Rigidity Matrix $\\mathbf {R}_x$",
"The rigidity matrix $\\mathbf {R}_x$ permits us to simultaneously solve the equations (REF ) for the space of infinitesimal motions of $(\\langle G, \\mathbf {\\rangle },p)$ .",
"It is an $|E| \\times (2|V| + 1)$ matrix with one row corresponding to each edge, two columns corresponding to each vertex, and a single column corresponding to the variable lattice element $x(t)$ .",
"The row corresponding to the edge $\\lbrace v_i, v_j; m_e\\rbrace $ is as follows: $ {\\begin{bordermatrix } && i & & j & & x(t) \\\\& 0 \\cdots 0 & p_i - (p_j+m_eL_x) & 0 \\cdots 0 & (p_j+m_eL_x) - p_i & 0 \\cdots 0 & (m_e)_x[p_i - (p_j+m_eL_x)]_x\\\\\\end{bordermatrix }}, $ where the entries under $i$ and $j$ are actually 2-tuples.",
"By $(m_e)_X$ we mean the $x$ -component of $m_e \\in \\mathbb {Z}^2$ .",
"The kernel of this matrix is the space of infinitesimal motions of $(\\langle G, \\mathbf {\\rangle },p)$ on $\\mathcal {T}_x^2$ , and we may write $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)\\cdot (u, u_x)^T = 0,$ where $(u, u_x) \\in \\mathbb {R}^{2|V|+1}$ is as described above.",
"A framework on $\\mathcal {T}_x^2$ always has a two-dimensional space of trivial infinitesimal motions, generated by the unit translations.",
"It follows that the kernel of the rigidity matrix always has dimension at least 2.",
"Furthermore, since a framework is infinitesimally rigid on $\\mathcal {T}_x^2$ if and only if the only infinitesimal motions are trivial (i.e.",
"are translations), we have the following result: Theorem 2.1 A periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ is infinitesimally rigid on the $x$ -variable torus $\\mathcal {T}_x^2$ if and only if the rigidity matrix $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)$ has rank $2|V| - 1$ ."
],
[
"The $T$ -gain Procedure on {{formula:b56cb088-d021-4129-88b7-adeef0060bd3}}",
"In [19] Ross described the $T$ -gain procedure, and showed that the rigidity matrices corresponding to two $T$ -gain equivalent periodic orbit frameworks have the same rank.",
"We now extend this to the variable torus case.",
"See Figure REF for an example.",
"The net gain on a cycle in a periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ is the sum of the gains on the edges of an oriented cycle of $G$ , where the gains are appropriately multiplied by $\\pm 1$ , depending on the direction of traversal.",
"The T-gain procedure is a procedure that can be used to easily identify the net gains on the cycles of a periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ .",
"As we will soon see, the rigidity of frameworks on $\\mathcal {T}_x^2$ is generically characterized by the net gains on the cycles of the periodic orbit graph (Theorem REF ).",
"The $T$ -gain procedure will thus be an essential proof technique which we use to show the necessity of the conditions in our main result (Proposition REF ).",
"The $T$ -gain procedure is defined in [8] for general gain (voltage) graphs.",
"Figure: A gain graph 〈G,〉\\langle G, \\mathbf {\\rangle } in (a), with identified tree TT (in red), root uu, and TT-potentials in (b).",
"The resulting TT-gain graph 〈G, T 〉\\langle G, \\mathbf {_}T \\rangle is shown in (c).$T$ -gain Procedure Select an arbitrary spanning tree $T$ of $G$ , and choose a vertex $u$ to be the root vertex.",
"For every vertex $v$ in $G$ , there is a unique path in the tree $T$ from the root $u$ to $v$ .",
"Denote the net gain along that path by $\\mathbf {(}v, T)$ , and we call this the $T$ -potential of $v$ .",
"Compute the $T$ -potential of every vertex $v$ of $G$ .",
"Let $e$ be a plus-directed edge of $G$ with initial vertex $v$ and terminal vertex $w$ .",
"Define the $T$ -gain of $e$ , $\\mathbf {_}T(e)$ to be $\\mathbf {_}T(e) = \\mathbf {(}v, T) + \\mathbf {(}e) - \\mathbf {(}w, T).$ Compute the $T$ -gain of every edge in $G$ .",
"Note that the $T$ -gain of every edge of the spanning tree will be zero.",
"Theorem 2.2 Let $(\\langle G, \\mathbf {\\rangle },p)$ be a periodic orbit framework on $\\mathcal {T}_x^2$ .",
"Then ${\\rm rank}\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)= {\\rm rank}\\mathbf {R}_x(\\langle G, \\mathbf {_}T \\rangle , p^{\\prime })$ , where $p^{\\prime }:V \\rightarrow \\mathbb {R}^2$ is given by $p^{\\prime }_i = p_i + \\mathbf {_}T(v_i)$ .",
"Suppose that a set of rows is dependent in $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)$ .",
"Then there exists a vector of scalars, say $\\omega = [\\begin{array}{ccc} \\omega _1 & \\cdots & \\omega _{|E|} \\end{array}]$ such that $\\omega \\cdot \\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)= 0.$ For each vertex $v_i \\in V$ the column sum of $\\mathbf {R}_x((\\langle G, \\mathbf {\\rangle },p))$ becomes $\\sum _{ e_{\\alpha } \\in E_+} \\omega _{e_{\\alpha }} (p_i - (p_j+m_{e_{\\alpha }})) +\\sum _{ e_{\\beta } \\in E_-} \\omega _{e_{\\beta }} (p_i - (p_k-m_{e_{\\beta }})) = 0,$ where $E_+$ and $E_-$ are the edges directed out from and into vertex $i$ repectively.",
"In [19], it was demonstrated that (REF ) is equivalent to the following: $& \\sum _{ e_{\\alpha } \\in E_+} \\omega _e \\Big (p_i+ \\mathbf {_}T(v_i) - (p_j + \\mathbf {_}T(v_j)) -\\mathbf {_}T(e)\\Big ) + \\nonumber \\\\& \\hspace{72.26999pt} \\sum _{ e_{\\beta } \\in E_-} \\omega _e \\Big (p_i+ \\mathbf {_}T(v_i) - (p_j + \\mathbf {_}T(v_j)) +\\mathbf {_}T(e)\\Big ) = 0$ which is the column sum of the column of $\\mathbf {R}_x(\\langle G, \\mathbf {_}T \\rangle , p^{\\prime })$ corresponding to the vertex $v_i$ .",
"Since we are working with the variable torus, we have one additional column corresponding to the flexibility of the $x$ -direction.",
"We will show that if there exists a vector of scalars $\\omega = [\\begin{array}{ccc} \\omega _1 & \\cdots & \\omega _{|E|} \\end{array}]$ such that $\\omega \\cdot \\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)= 0,$ then $\\omega \\cdot \\mathbf {R}_x(\\langle G, \\mathbf {_}T \\rangle ,p)= 0$ too.",
"Since the first $2|V|$ columns are exactly as in the fixed torus case, we need only show this holds for the new column.",
"Consider the column sum corresponding to the columns of the lattice elements in $\\mathbf {R}_x(\\langle G, \\mathbf {_}T \\rangle , p^{\\prime })$ : $\\sum _{e \\in E} \\omega _e \\Big (m_T(e)\\Big [(p_i+ \\mathbf {_}T(v_i)) - (p_j + \\mathbf {_}T(v_j)) -\\mathbf {_}T(e)\\Big ]\\Big )_x.$ Recall that $\\mathbf {_}T(e) = \\mathbf {_}T(v_i) + \\mathbf {(}e) - \\mathbf {_}T(v_j)$ , where $m_T(v_i)$ represents the $T$ -potential of the vertex $v_i$ (the $T$ -potential of a vertex $v_i$ is the net gain on the directed path along $T$ from the root vertex).",
"Expanding (REF ), we obtain $\\sum _{e \\in E} \\omega _e \\Big ( m(e)\\big [\\cdots \\big ] + m_T(v_i)\\big [\\cdots \\big ] - m_T(v_j)\\big [\\cdots \\big ]\\Big )_x,$ where $ \\big [\\cdots \\big ] = \\big [(p_i+ \\mathbf {_}T(v_i)) - (p_j + \\mathbf {_}T(v_j)) -\\mathbf {_}T(e)\\big ]$ .",
"We know that $&\\sum _{e \\in E} \\omega _e \\Big ( m(e)\\Big [(p_i+ \\mathbf {_}T(v_i)) - (p_j + \\mathbf {_}T(v_j)) -\\mathbf {_}T(e)\\Big ]\\Big )_x\\\\= & \\sum _{e \\in E} \\omega _e \\Big ( m(e)\\Big [p_i - p_j -\\mathbf {(}e)\\Big ]\\Big )_x \\\\= & 0, \\ \\ \\ \\textrm {since} \\ \\omega \\cdot \\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)= 0.$ Now note that $\\mathbf {_}T(v_i)$ and $\\mathbf {_}T(v_j)$ have one of $|V|$ different values.",
"Grouping (REF ) according to these values, we obtain $\\sum _{i=1}^{|V|} (m_T(v_i))_x\\Big [\\sum _{j: (i,j) \\in E} \\omega _e(p_i+ \\mathbf {_}T(v_i) - (p_j + \\mathbf {_}T(v_j)) -\\mathbf {_}T(e)) \\Big ]_x,$ where each edge is counted exactly twice, once for its initial vertex and once for its terminal vertex, with sign depending on the orientation of the edge.",
"But by (REF ), the sum inside the square brackets is zero, since it represents the column sum at any vertex.",
"Hence (REF ) is also zero.",
"The same argument also works in reverse, which proves the claim."
],
[
"Periodic Orbit Frameworks on the Fixed Torus $\\mathcal {T}_0^2$",
"From the rigidity matrix for the $x$ -variable torus, we can obtain the rigidity matrix for frameworks on the fixed torus, simply by striking out the column corresponding to $x(t)$ .",
"We are left with an $|E| \\times 2|V|$ matrix $\\mathbf {R}_0$ , and a periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ is infinitesimally rigid on $\\mathcal {T}_0^2$ if and only if the rank of $\\mathbf {R}_0(\\langle G, \\mathbf {\\rangle },p)$ is $2|V| - 2$ [20].",
"Frameworks on the fixed torus are the subject of the papers [20], [19], and we record only the main result.",
"For brevity throughout we use the following terminology.",
"Let $G=(V, E)$ be a graph.",
"We say that $G$ is $(k, \\ell )$ -sparse if all subgraphs $G^{\\prime } = (V^{\\prime }, E^{\\prime })$ of $G$ satisfy $|E^{\\prime }| = k|V^{\\prime }| - \\ell $ .",
"If in addition, $G$ satisfies $|E| = k|V| - \\ell $ , we say that $G$ is $(k, \\ell )$ -tight.",
"Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit graph, where $G$ is $(2,2)$ -tight.",
"We say that the gain assignment $m: E^+ \\rightarrow \\mathbb {R}^2$ is constructive if every subgraph $G^{\\prime } \\subset G$ with exactly $|E^{\\prime }| = 2|V^{\\prime }| - 2$ edges contains some cycle with non-trivial net gain.",
"For example, the periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ pictured in Figure REF (a) has a constructive gain assignment.",
"Note further that the $T$ -gain procedure preserves the net gains on cycles, and therefore the graph $\\langle G, \\mathbf {_}T \\rangle $ pictured in (c) also has a constructive gain assignment.",
"Theorem 2.3 The periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ is minimally rigid on the fixed torus $\\mathcal {T}_0^2$ if and only if $G$ is $(2,2)$ -tight, and $m$ is a constructive gain assignment."
],
[
"1-Dimensional Frameworks",
"The basic ideas of the rigidity of finite graphs on the line can be found in [26] or [7].",
"The key result is that a graph $G$ is rigid as a 1-dimensional framework if and only if it is connected.",
"1-dimensional periodic frameworks were discussed in [18] and [14].",
"We give a brief summary.",
"Just as we map 2-periodic frameworks onto the torus, we may view 1-periodic frameworks as graphs on a circle.",
"Such graphs may be on a circle of fixed circumference $x$ (the fixed circle), or they may be on a circle that is allowed to change circumference $x(t)$ (the flexible circle).",
"We denote the fixed circle by $\\mathcal {T}_0^1$ , and the flexible circle by 1.",
"In either case, a 1-periodic orbit framework is the pair $(\\langle G, \\mathbf {\\rangle },p)$ , with $m: E \\rightarrow \\mathbb {Z}$ , and $p:V \\rightarrow [0, x)$ , where $x$ is either a fixed element of $\\mathbb {R}$ for the fixed circle, or $x = x(t)$ is a continuous function of time for the flexible circle.",
"We assume further that $p$ maps the endpoints of any edge to distinct locations in $[0, x)$ , thereby avoiding edges of length zero.",
"Consider a periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ on the fixed circle.",
"For consistency with the notation used in the remainder of this paper, let $L_0$ be the $1 \\times 1$ matrix $[x]$ , where the circumference of the fixed circle is $x$ .",
"The $|E| \\times |V|$ rigidity matrix in this case will have one row corresponding to each edge $\\lbrace i, j; m\\rbrace $ : $ {\\begin{bordermatrix } && i & & j & \\\\& 0 \\cdots 0 & p_i - (p_j+mL_0) & 0 \\cdots 0 & (p_j+mL_0) - p_i & 0 \\cdots 0 & \\\\\\end{bordermatrix }},$ where $p_i, p_j \\in \\mathbb {R}$ , and $m \\in \\mathbb {Z}$ .",
"Since there is always a 1-dimensional space of trivial infinitesimal motions generated by the vector $(1, \\dots , 1)^T$ , the rigidity matrix has maximum rank $|V|-1$ .",
"Proposition 2.4 The periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ is (infinitesimally) rigid on $\\mathcal {T}_0^1$ if and only if $G$ is connected.",
"If we allow the radius of the circle to change size, in addition to connectivity, we now require the graph to “wrap\" in a non-trivial fashion around the circle.",
"That is, $\\langle G, \\mathbf {\\rangle }$ must contain a constructive cycle.",
"The rigidity matrix now has an extra column corresponding to $x(t)$ , with entry (for the edge $\\lbrace i,j; m\\rbrace $ ) given by $m(p_i - (p_j + mL))$ .",
"One way to see the necessity of a constructive cycle is to perform the $T$ -gain procedure on the edges of a periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ with $|E| = |V|$ .",
"If no cycle is constructive, the column corresponding to $x(t)$ will be identically zero.",
"Proposition 2.5 The periodic orbit framework $(\\langle G, \\mathbf {\\rangle },p)$ is (infinitesimally) rigid on 1 if and only if $G$ is connected, and $G$ contains a constructive cycle."
],
[
"Necessary Conditions for Rigidity on $\\mathcal {T}_x^2$",
"Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit framework where $G$ is $P(2,1)$ .",
"We say $m$ is $\\mathcal {T}_x^2$ -constructive if every $(2,2)$ -subgraph is constructive (i.e.",
"every $(2,2)$ -subgraph contains some cycle with non-trivial net gain) and every $(2,1)$ -subgraph is $x$ -constructive (i.e.",
"contains some cycle with non-trivial net gain in the $x$ -direction.)",
"In [13] it was shown that the class of $(k,\\ell )$ -tight graphs forms a matroid for natural numbers $0\\le \\ell <2k$ .",
"Thus we say that a $(k,\\ell -1)$ -circuit is a $(k,\\ell )$ -tight graph in which deleting any edge gives a $(k,\\ell -1)$ -tight graph.",
"Crucial to us will be the following weaker definition.",
"We will say that $G$ is a $P(2,1)$ -graph if $G$ is $(2,1)$ -tight and there exists $e \\in E$ such that $G-e$ is $(2,2)$ -tight.",
"The following proposition provides necessary conditions for minimal rigidity on $\\mathcal {T}_x^2$ .",
"Proposition 3.1 Let $(\\langle G, \\mathbf {\\rangle },p)$ be a periodic orbit framework.",
"If $(\\langle G, \\mathbf {\\rangle },p)$ is minimally rigid on $\\mathcal {T}_x^2$ , then $G$ is $P(2,1)$ and $m$ is $\\mathcal {T}_x^2$ -constructive.",
"That $G$ must be $P(2,1)$ for $(\\langle G, \\mathbf {\\rangle },p)$ to be rigid follows from Theorem REF and Lemma REF .",
"Let $H$ be the unique $(2,2)$ -circuit.",
"To see that $\\langle G, \\mathbf {\\rangle }$ is $\\mathcal {T}_x^2$ -constructive, we show that $H$ is $x$ -constructive, and that every $(2,2)$ -tight subgraph of $G$ obtained by deleting a single edge of $H$ is constructive.",
"To see that $H$ is $x$ -constructive, suppose toward a contradiction that $\\langle G, \\mathbf {\\rangle }$ contains no $x$ -constructive cycle.",
"Applying the $T$ -gain procedure to any tree in $G$ will produce a $T$ -gain assignment $m_T$ , where all $x$ -coordinates are zero.",
"Then all of the entries of the single lattice column of the rigidity matrix will be zero.",
"Hence we effectively have $2|V|-1$ edges in the fixed torus rigidity matrix, which has maximum rank $2|V| - 2$ , a contradiction.",
"Since $G$ is $P(2,1)$ , deleting any edge $e$ from $H$ results in a $(2,2)$ -subgraph of $G$ .",
"Since every such subgraph must correspond to a set of linearly independent rows in the rigidity matrix, Proposition REF implies that the gain assignment $m$ restricted to this subgraph must be constructive.",
"When we move from the fixed torus to the variable torus, we add columns to the rigidity matrix.",
"As a result, it seems possible that edges that were dependent on the fixed torus become independent on the variable torus.",
"When $d=2$ , and our dependent subgraphs are of size $2|V|-2$ , this is not the case.",
"Proposition 3.2 Let $\\langle G, \\mathbf {\\rangle }$ be a $(2,2)$ -tight periodic orbit graph.",
"If $\\langle G, \\mathbf {\\rangle }$ is dependent on $\\mathcal {T}_0^2$ then $\\langle G, \\mathbf {\\rangle }$ is also dependent on $x^2$ .",
"Suppose that $\\langle G, \\mathbf {\\rangle }$ is dependent on $\\mathcal {T}_0^2$ .",
"Then there is some subgraph $\\langle G^{\\prime }, \\mathbf {^{\\prime }} \\rangle \\subseteq \\langle G, \\mathbf {\\rangle }$ with $|E^{\\prime }| = 2|V^{\\prime }| - 2$ , and no constructive cycle.",
"Therefore, all gains on this subgraph are $T$ -gain equivalent to $(0,0)$ .",
"Then the entries in the lattice column of the rigidity matrix corresponding to these edges will be zero, since $(m_T)_x=0$ for all $e$ , and therefore the edges continue to be dependent on $x^2$ .",
"A map-graph is a graph in which each connected component has exactly one cycle.",
"By a result of Whiteley [25], $G=(V,E)$ is a $(k, \\ell )$ -tight graph if and only if $E$ is the edge-disjoint union of $\\ell $ spanning trees and $k-\\ell $ spanning map-graphs.",
"Note that each map-graph need not be connected.",
"This is in contrast to the situation for minimally rigid periodic orbit frameworks on $\\mathcal {T}_x^2$ : Theorem 3.3 Let $(\\langle G, \\mathbf {\\rangle },p)$ be a minimally rigid framework on the variable torus $\\mathcal {T}_x^2$ .",
"Then the edges of $\\langle G, \\mathbf {\\rangle }$ admit a decomposition into one spanning tree and one connected spanning map-graph.",
"This proof is similar to the proof of Theorem 2.18 in [24].",
"Let $(\\langle G, \\mathbf {\\rangle },p)$ be a minimally rigid framework on $\\mathcal {T}_x^2$ .",
"The rigidity matrix, $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)$ has rank $2|V|-1$ , and dimension $(2|V|-1) \\times (2|V| + 1)$ , with $2|V|$ columns corresponding to the vertices, and one column corresponding to the flexibility of the lattice.",
"Adding the two rows $\\left(\\begin{array}{cccccc}1 & 0 & 0 & \\cdots & 0 & 0\\end{array}\\right) ,\\left(\\begin{array}{cccccc}0 & 1 & 0 &\\cdots & 0 & 0\\end{array}\\right)$ has the effect of eliminating the 2-dimensional space of infinitesimal translations.",
"This “tie down\" is described in [24], and is equivalent to pinning one vertex on the torus.",
"The resulting square matrix has $2|V| +1$ independent rows, and hence a non-zero determinant.",
"Reorder the columns of $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)$ by coordinates, with the single lattice column grouped with the first coordinates.",
"Regard the determinant as a Laplace decomposition where the terms are products of the determinants of one square block $M_i$ with dimension $(|V| +1) \\times (|V|+1)$ , and one square block $N_i$ with dimension $|V| \\times |V|$ .",
"That is, $\\det \\mathbf {R}_x(\\langle G, \\mathbf {\\rangle },p)= \\sum M_i N_i$ .",
"The block $M_i$ contains all of the entries from the columns of the first coordinates, and the $N_i$ block contains the second coordinates.",
"Each block contains a single tie-down row.",
"There must be at least one nonzero product $M_kN_k$ .",
"By the Laplace decomposition, the rows used in $M_k$ and $N_k$ form disjoint subgraphs, and $M_k$ or $N_k$ will each contain one of the tie-down rows.",
"The $|V|$ rows of $M_k$ that are not tie-down rows have rank $|V|$ , and this submatrix corresponds to the rigidity matrix of a 1-dimensional graph on the flexible circle (the periodic line).",
"By Proposition REF , we know that such a graph must be connected, and must contain a constructive cycle.",
"Hence the $|V|$ edges of the $M_k$ block form a spanning connected map-graph.",
"Similarly, the $|V|-1$ edges of the $N_k$ block that are not tie-down edges correspond to the rigidity matrix of a graph on the fixed circle.",
"Since the block $N_k$ has rank $|V|$ , the edges that are not tie-down edges are independent on the fixed circle, and hence by Proposition REF the graph is connected.",
"Therefore the edges of the $N_k$ block form a spanning tree of $G$ .",
"Lemma 3.4 Suppose $G$ has $|E| = 2|V| - 1$ .",
"If the edges of $G$ admit a decomposition into one (edge-disjoint) spanning tree and one connected spanning map-graph, then $G$ is a $P(2,1)$ -graph and $G$ contains a unique $(2,2)$ -circuit.",
"In light of Theorem REF , the lemma is simply a re-statement of Lemma REF ."
],
[
"H1, H2 Preserve Rigidity on $\\mathcal {T}_x^2$",
"We will use $d(v)$ to denote the degree of the vertex $v$ and $d_G(v)$ when the context of the graph is not clear.",
"$N(v)$ denotes the set of neighbours of $v$ .",
"As is common in the literature we will refer to the following construction moves as Henneberg operations: add a vertex $v_0$ with $d(v_0)=2$ and $N(v_0)=\\lbrace v_1, v_2\\rbrace $ , $v_1 \\ne v_2$ , add a vertex $v_0$ with $d(v_0)=2$ and $N(v_0)=\\lbrace v_1\\rbrace $ , remove an edge $v_1v_2$ , $v_1\\ne v_2$ , and add a vertex $v_0$ with $d(v_0)=3$ and $N(v_0)=\\lbrace v_1, v_2, v_3\\rbrace $ for some $v_3 \\in V$ , remove an edge $v_1v_2$ , $v_1\\ne v_2$ , and add a vertex $v_0$ with $d(v_0)=3$ and $N(v_0)=\\lbrace v_1, v_2\\rbrace $ with one edge connecting $v_0$ with $v_1$ , and two edges connecting vertices $v_0$ and $v_2$ .",
"Figure: The H1H1 moves (periodic vertex addition).",
"The large circular region represents a generically rigid periodic orbit graph on 𝒯 x 2 \\mathcal {T}_x^2.Figure: The H2H2 moves (periodic edge split).",
"The gain m e m_e on the edge connecting 1 and 2 is preserved through this split.More strongly, when the moves are applied to a periodic orbit framework $\\langle G, \\mathbf {\\rangle }$ with changes to the gains as illustrated in Figures REF and REF , we say that these operations are gain-preserving Henneberg operations.",
"It was shown in [20] using linear algebra techniques that gain-preserving Henneberg operations preserve the maximality of the rank of the rigidity matrix, echoing the situation for finite frameworks.",
"The corresponding results for the variable torus $\\mathcal {T}_x^2$ can be proven entirely similarly; we leave the details to the reader.",
"Proposition 4.1 Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit and let $\\langle G^{\\prime },m^{\\prime } \\rangle $ be the result of a gain-preserving $H1$ operation on $\\langle G, \\mathbf {\\rangle }$ .",
"Let $p$ be generic and let $p^{\\prime }=(p,p_{n+1})$ be chosen generically with respect to $p$ .",
"Then the rows of $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle }, p)$ are linearly independent if and only if the rows of $\\mathbf {R}_x(\\langle G^{\\prime },m^{\\prime }\\rangle ,p^{\\prime })$ are linearly independent.",
"Proposition 4.2 Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit and let $\\langle G^{\\prime },m^{\\prime } \\rangle $ be the result of a gain-preserving $H2$ operation on $\\langle G, \\mathbf {\\rangle }$ .",
"Let $p$ be generic and let $p^{\\prime }=(p,p_{n+1})$ be chosen generically with respect to $p$ .",
"If the rows of $\\mathbf {R}_x(\\langle G, \\mathbf {\\rangle }, p)$ are linearly independent then the rows of $\\mathbf {R}_x(\\langle G^{\\prime },m^{\\prime }\\rangle ,p^{\\prime })$ are linearly independent."
],
[
"$P(2,1)$ Graphs",
"In this section we consider in detail the structure of $P(2,1)$ -graphs.",
"For $X\\subset V$ , $G[X]$ denotes the subgraph induced by $X$ .",
"For two subsets $J,K \\subset V$ , $d(J,K)$ denotes the number of edges in $E$ with one end-vertex in $G[J]$ and one in $G[K]$ .",
"For a subset $X\\subset V$ let $i(X)$ denote the number of edges in the subgraph induced by $X$ .",
"Observe first that a $P(2,1)$ -graph can have at most one loop and a $(2,2)$ -circuit can have a loop if and only if the graph is a single loop."
],
[
"Critical Sets",
"Let $G=(V,E)$ be a $P(2,1)$ -graph and let $X\\subset V$ .",
"We will say $X$ is over-critical if $i(X)=2|X|-1$ , critical if $i(X)=2|X|-2$ and semi-critical if $i(X)=2|X|-3$ .",
"In each case if there is a degree 3 vertex $v\\in V$ with $N(v)=\\lbrace x,y,z\\rbrace $ and $X$ contains $x,y$ but not $z,v$ then we say that $X$ is over-v-critical, v-critical or semi-v-critical respectively.",
"To simplify the arguments in Section we record some basic facts about $P(2,1)$ -graphs.",
"In the next lemma, minimal means having the least number of vertices.",
"Lemma 5.1 Let $G=(V,E)$ be a $P(2,1)$ -graph.",
"Then there is a unique minimal over-critical set $X\\subset V$ and $G[X]$ is a $(2,2)$ -circuit.",
"If $G$ is a $(2,2)$ -circuit, then $X=V$ .",
"Otherwise by definition there exists $X\\subsetneq V$ with $i(X)=2|X|-1$ .",
"Choose the minimal over-critical set $X^-\\subset X$ .",
"This induces a $(2,2)$ -circuit.",
"For uniqueness suppose $J\\subsetneq V$ such that $J\\ne X^-$ is over-critical.",
"If $J$ and $X^-$ are disjoint we contradict the definition of a $P(2,1)$ -graph.",
"Thus $i(X^- \\cup J)+i(X^-\\cap J)=i(X^-)+i(J)+d(X^--J,J-X^-)=2|X^-|-1+2|J|-1+d(X^--J,J-X^-)$ .",
"This implies that $d(X^--J,J-X^-)=0$ , $i(X^- \\cup J)=2|X^-\\cup J|-1$ and $i(X^-\\cap J)=2|X^-\\cap J)-1$ contradicting the minimality of $X^-$ .",
"Lemma 5.2 Let $G=(V,E)$ be a $P(2,1)$ -graph.",
"Then $G$ is 2-edge-connected.",
"If $X\\subset V$ is critical then $G[X]$ is connected.",
"If $X\\subset V$ is semi-critical then either $G[X]$ is connected or $X$ has two connnected components $A,B$ such that $A$ is over-critical and $B$ is critical.",
"In each case Lemma REF will imply at most one of $A,B$ is over-critical.",
"For (1) suppose $V=A\\cup B$ for $A,B \\subset V$ with $A\\cap B=\\emptyset $ and $d(A,B)=1$ .",
"Now $2|V|-1= |E|&=&i(A)+i(B)+1 \\\\ &\\le & 2|A|-1+2|B|-2+1\\\\ &=&2|V|-2,$ a contradiction.",
"For (2) suppose $V=A\\cup B$ for $A,B \\subset V$ with $A\\cap B=\\emptyset $ and $d(A,B)=0$ .",
"Now $2|X|-2= i(X)&=&i(A)+i(B) \\\\ &\\le & 2|A|-1+2|B|-2\\\\ &=&2|X|-3,$ a contradiction.",
"For (3) suppose $G[X]$ is not connected.",
"Then the proof is entirely similar to (2).",
"However $P(2,1)$ -graphs need not be 2-connected and need not be 3-edge-connected.",
"Lemma 5.3 Let $G=(V,E)$ be a $P(2,1)$ -graph with unique $(2,2)$ -circuit $G[X]$ and let $v\\in V$ have $N(v)=\\lbrace x,y,z\\rbrace $ .",
"Let $x,y,z\\in X$ .",
"Then $v\\in X$ and there is no over-v-critical set.",
"If $Y$ is v-critical on $x,y$ but not $z$ with no over-critical subset and $Z$ is semi v-critical on $x,z$ but not $y$ with no critical subset then if $|Y\\cap Z|>1$ then $Y\\cup Z$ is critical and $Y\\cap Z$ is semi-critical.",
"if $|Y\\cap Z|=1$ then $Y\\cup Z$ is semi-critical and $Y\\cap Z$ is critical.",
"First $v \\in X$ otherwise $i(X\\cup v)=2|X\\cup v|$ and any over-v-critical set contradicts Lemma REF .",
"For (2) $Y\\cap Z \\subset Z$ so $i(Y\\cap Z)\\le 2|Y\\cap Z|-3$ .",
"Since $ i(Y\\cup Z)+i(Y\\cap Z)=2|Y\\cup Z|+2|Y\\cap Z|-5+d(Y,Z)$ and $Y\\cup Z$ contains $x,y,z$ but not $v$ we deduce that $Y\\cup Z$ is critical and $Y\\cap Z$ is semi-critical.",
"(b) is entirely similar.",
"Lemma 5.4 Let $G=(V,E)$ be a $P(2,1)$ -graph with unique $(2,2)$ -circuit $G[X]$ and let $v\\in V$ have $N(v)=\\lbrace x,y,z\\rbrace $ .",
"Let $x,y \\in X, z \\in V-X$ .",
"Then $v \\in V-X$ and $X$ is the unique over-v-critical set, there is no v-critical set containing $z$ , if $Y\\subset V$ contains $x,z$ but not $y,v$ and $|X\\cap Y|>1$ then $G[Y]$ is not $(2,3)$ -tight.",
"First $v \\in V-X$ contradicts Lemma REF and $X$ is unique by Lemma REF and the definition of a $(2,2)$ -circuit.",
"For (2) suppose there is such a v-critical set $Z$ .",
"$ i(X\\cup Z)+i(X\\cap Z)&=&i(X)+i(Z)+d(X-Z,Z-X)\\\\ &\\le & 2|X|-1+2|Z|-2+d(X-Z,Z-X).",
"$ Since $i(X\\cap Z)\\le 2|X\\cap Z|-2$ we have $d(X-Z,Z-X)=0$ and $i(X\\cup Z)=2|X\\cup Z|-1$ but $i(X\\cup Z\\cup v)=2|X\\cup Z\\cup v|$ , a contradiction.",
"Finally, since $|X\\cap Y|>1$ we have $i(X\\cap Y)\\le 2|X\\cap Y|-3$ as $X\\cap Y\\subset Y$ .",
"Thus, similarly to before, $X\\cup Y$ is over v-critical, and adding back $v$ gives a contradiction.",
"Lemma 5.5 Let $G=(V,E)$ be a $P(2,1)$ -graph with unique $(2,2)$ -circuit $G[X]$ and let $v\\in V$ have $N(v)=\\lbrace x,y,z\\rbrace $ .",
"Let $y,z \\in V-X$ .",
"Then $v \\in V-X$ , there is at most one over-v-critical set $Y$ .",
"Moreover if there is such a $Y$ then there is no v-critical set and at most one semi-v-critical set.",
"First $v \\in V-X$ otherwise $i(X-v)>2|X-v|-1$ .",
"Let $Y$ be over-v-critical containing $x,z$ but not $y$ .",
"If $W$ is over-v-critical on $x,y$ but not $z$ then $i(Y\\cup W\\cup v)\\ge 2|Y\\cup W\\cup v|$ .",
"Now suppose $Z$ is v-critical on $x,y$ but not $z$ .",
"Then $i(Y\\cap Z)\\le 2|Y\\cap Z|-2$ so $i(Y\\cup Z)=2|Y\\cup Z|-1$ and adding back $v$ creates a contradiction.",
"Finally suppose $Z$ is semi-v-critical on $x,y$ but not $z$ .",
"Then $Y\\cup Z$ is critical.",
"Consider $W\\subset V$ containing $y,z$ but not $x,v$ .",
"$i(Y\\cap W)\\le 2|Y\\cap W|-2$ and $i(Z\\cap W)\\le 2|Z\\cap W|-2$ .",
"Therefore $i(Y\\cup Z\\cup W)\\le 2|Y\\cup Z \\cup W|-4$ .",
"Lemma 5.6 Let $G=(V,E)$ be a $P(2,1)$ -graph containing a unique $(2,2)$ -circuit $G[X]$ , with a degree 3 vertex $v$ with $N(v)=\\lbrace x,y,z\\rbrace $ such that $x,y \\in X, z \\notin X$ .",
"Let $Y_{xz}\\subset V$ contain $x,z$ but not $y,v$ and $Y_{yz}\\subset V$ contain $y,z$ but not $x,v$ .",
"Then at most one of $Y_{xz}$ and $Y_{yz}$ is semi-critical.",
"Suppose $Y_{yz}$ is semi-critical.",
"Firstly if $i(Y_{xz})<2|Y_{xz}|-3$ then $i(X\\cap Y_{xz})\\ge 2|X\\cap Y_{xz}|- 2$ by Lemma REF .",
"Thus $i(X\\cup Y_{xz})<2|X\\cup Y_{xz}|-2$ so adding back $v$ and its 3 edges contradicts the definition of a $P(2,1)$ -graph.",
"Suppose $i(Y_{xz})=2|Y_{xz}|-3$ .",
"$i(X\\cup Y_{xz})+i(X\\cap Y_{xz})&=&i(X)+i(Y_{xz})+d(X-Y_{xz},Y_{xz}-X)\\\\ &=&4+d(X-Y_{xz},Y_{xz}-X).$ If $i(X\\cup Y_{xz})=2|X\\cup Y_{xz}|-1$ add $v$ and its 3 edges for a contradiction.",
"If $i(X\\cap Y_{xz})=2|X\\cap Y_{xz}|-1$ then we contradict $G[X]$ being a $(2,2)$ -circuit.",
"Thus $i(X\\cup Y_{xz})=2|X\\cup Y_{xz}|-2$ and $i(X\\cap Y_{xz})=2|X\\cap Y_{xz}|-2$ .",
"Since $X\\cap Y_{xz} \\subset Y_{xz}$ we know $|X\\cap Y_{xz}|=1$ .",
"Similarly we derive that $|X\\cap Y_{yz}|=1$ .",
"Now $i(Y_{xz}\\cup Y_{yz})+i(Y_{xz}\\cap Y_{yz})=6+d(Y_{xz}-Y_{yz},Y_{yz}-Y_{xz}).$ $i(Y_{xz}\\cap Y_{yz})>2|Y_{xz}\\cap Y_{yz}|-1$ by Lemma REF and $i(Y_{xz}\\cup Y_{yz})>2|Y_{xz}\\cup Y_{yz}|-1$ otherwise adding $v$ and its 3 edges gives a contradiction.",
"Hence $i(Y_{xz}\\cup Y_{yz})=2|Y_{xz}\\cup Y_{yz}|-s$ where $s \\in \\lbrace 2,3,4\\rbrace $ .",
"$i(X\\cup (Y_{xz}\\cup Y_{yz}))=1+s-4 +d(X-(Y_{xz}\\cup Y_{yz}),(Y_{xz}\\cup Y_{yz})-X)\\le 1.$ Thus adding $v$ and its 3 edges violates the definition of a $P(2,1)$ -graph."
],
[
"Henneberg Operations on $P(2,1)$ -graphs",
"A vertex $v$ in a $P(2,1)$ -graph is admissible if there is some inverse Henneberg operation removing $v$ that results in a $P(2,1)$ -graph.",
"For brevity we will use $K_i^j$ to denote the complete graph on $i$ vertices with $j$ copies of each edge.",
"Note when $i=1$ then $j$ denotes the number of loops on that single vertex.",
"The next lemma is useful in extending the standard arguments showing the inverse moves preserve $(2,l)$ -tightness to show that the inverse moves preserve the $P(2,1)$ condition.",
"Lemma 5.7 Let $G=(V,E)$ be a $P(2,1)$ -graph not equal to $K_2^3$ with no degree 2 vertex.",
"Then there exists a degree 3 vertex $v$ and an edge $e$ not incident to $v$ such that $G-e$ is $(2,2)$ -tight.",
"This is trivial if $G$ is a $(2,2)$ -circuit.",
"Suppose $G$ is not a $(2,2)$ -circuit then there exists an over-critical set $K \\subsetneq V$ .",
"Let $J=V-K$ .",
"Now $i(V)=i(J)+i(K)+d(J,K)$ so $i(J)=d(J,K)$ .",
"Also $i(J)=d(J,K)\\ge 2$ by Lemma REF (1).",
"Suppose $d_G(v)\\ge 4$ for all $v \\in J$ .",
"There are at least $4|J|+2$ vertex/edge incidences which implies $i(J)+d(J-K,K-J) \\ge 2|J|+1$ but $ 2|V|-1=i(V)= i(K)+i(J)+d(J-K,K-J)\\ge 2|K|+2|J|$ so there is a vertex $v_3 \\in J$ with $d_G(v_3)=3$ .",
"Now from the definition of a $P(2,1)$ -graph there is some edge in $G[K]$ that gives the result.",
"The following lemma allows us to apply the inverse moves easily in the case that the $P(2,1)$ -graph $G$ happens to be a $(2,2)$ -circuit.",
"Lemma 5.8 Let $G=(V,E)$ be a graph.",
"If $G$ is a $(2,2)$ -circuit then $G$ contains a degree 3 vertex $v$ , and for any such $v$ , $G-v$ is $(2,2)$ -tight.",
"Note that the lemma is not in general true for $P(2,1)$ -graphs and that the converse fails for $(2,2)$ -circuits but is true by definition for $P(2,1)$ -graphs.",
"Let $G$ be a $(2,2)$ -circuit.",
"Since $|E|=2|V|-1$ we have $\\sum _{i=1}^{|V|}(4-d(i))=2$ .",
"Therefore there exists $v \\in V$ with $d(v) \\le 3$ .",
"$(2,2)$ -circuits cannot contain vertices of degree $\\le 2$ ; suppose $u$ was such a vertex then $i(V- u)=2|V- u|-1$ edges contradicting the definition of a $(2,2)$ -circuit.",
"Thus $G$ contains a degree 3 vertex $v$ .",
"Clearly $V \\setminus v$ is critical since we can think of the operation of deleting a degree 3 vertex as the composition of an edge deletion and an inverse 1a move.",
"Lemma 5.9 Let $G$ be a $(2,2)$ -circuit.",
"Then either $G=K_2^3$ or there is an inverse 2a or 2b move on any degree 3 vertex that results in a $P(2,1)$ -graph.",
"Let $v \\in V$ have $d(v)=3$ .",
"$V-v$ is critical by Lemma REF .",
"If $N(v)=\\lbrace a\\rbrace $ then $v \\in K_2^3$ .",
"If not then adding any non-edge between the neighbours of $v$ creates a $P(2,1)$ -graph.",
"Note the stronger statement that every degree 3 vertex in a $(2,2)$ -circuit is admissible is false, for similar considerations see [2] and [16].",
"In the following lemma, recall it is well known that if $G$ is $(2,\\ell )$ -tight ($\\ell =2,1$ ) then there is an inverse 2a or 2b move that results in a $(2,\\ell )$ -tight graph.",
"Hence we need only concern ourselves with the additional subgraph condition.",
"Lemma 5.10 Let $G=(V,E)$ be a $P(2,1)$ -graph containing a vertex $v$ with $d(v)=3$ and $|N(v)|>1$ .",
"Then $v$ is admissible.",
"By Lemma REF we may assume $G$ is not a $(2,2)$ -circuit.",
"Thus by Lemma REF $G$ contains a unique $(2,2)$ -circuit $G[X]$ .",
"By Lemma REF we find a vertex $v$ in $V-X$ with $d(v)=3$ .",
"For any edge $e$ in $G[X]$ , $G-e$ is $(2,2)$ -tight and $G-e-v+xy$ for some $x,y \\in N(v)$ is $(2,2)$ -tight.",
"Therefore $G-v+xy$ is a $P(2,1)$ -graph.",
"For brevity we use the terminology leaf for a degree 1 vertex.",
"We can now collect together the results in this section into our main result about $P(2,1)$ -graphs.",
"$(1) \\Leftrightarrow (3)$ is a slight refinement of a result of Whiteley.",
"Theorem 5.11 Let $G=(V,E)$ .",
"The following are equivalent: $G$ is a $P(2,1)$ -graph, $G$ can be constructed from $K_1^1$ or $K_2^3$ by 1a, 1b, 2a and 2b moves, $G$ is the edge disjoint union of a spanning tree $T$ and a connected map graph $M$ .",
"First we prove $(1) \\Leftrightarrow (2)$ .",
"It is easy to see that any of these four moves applied to an arbitrary $P(2,1)$ -graph results in a $P(2,1)$ graph.",
"Since $K_1^1$ and $K_2^3$ are $P(2,1)$ -graphs it follows that any graph constructed from a sequence of these moves is a $P(2,1)$ -graph.",
"The converse follows from the above sequence of results by induction on $|V|$ .",
"Suppose $G$ is a $P(2,1)$ -graph containing a loop.",
"Then this loop is the unique $(2,2)$ -circuit within $G$ and Lemma REF guarantees an inverse move.",
"Suppose now that $G$ is loopless.",
"Then either $G$ is a $(2,2)$ -circuit in which case apply Lemma REF or $G$ contains a unique $(2,2)$ -circuit and apply Lemma REF .",
"The result follows from Lemma REF .",
"$(3) \\Rightarrow (1)$ follows since $|E|=|E(T)|+|E(M)|$ , $|E(T)|=|V|-1$ , $E(T^{\\prime })|\\le |V(T^{\\prime })|-1$ for any subgraph $T^{\\prime }$ of $T$ , $|E(M)|=|V|$ and $|E(M^{\\prime })|\\le |V(M^{\\prime })|$ for any subgraph $M^{\\prime }$ of $M$ .",
"Let $G^{\\prime }$ be formed from $G$ by one of the four construction moves.",
"$(2) \\Rightarrow (3)$ follows by showing that in each case if $G$ satisfies $(3)$ then so does $G^{\\prime }$ .",
"This is trivial in each case.",
"Particularly in the 1a and 1b moves the new vertex is a leaf in the tree and in the connected map graph.",
"In the 2a and 2b moves removing $xy$ the new edges $xv,yv$ go in whichever of $T$ or $M$ contained $xy$ and the remaining edge goes in the other."
],
[
"$G$ is {{formula:99372e94-f82b-4be1-ba8d-abe207a9ccdf}} and {{formula:5f056720-f44e-40df-b4b6-7a15b2519d3d}} -constructive {{formula:dfb70127-dbdf-4cd9-947d-960392032e2a}} H1, H2",
"Let $\\langle G, \\mathbf {\\rangle }$ be a $\\mathcal {T}_x^2$ orbit graph, where $G$ is $P(2,1)$ and $m$ is $\\mathcal {T}_x^2$ -constructive.",
"We say that a vertex $v \\in V$ is a circuit vertex if it is contained within the minimal $(2,1)$ -tight subgraph of $G$ (this subgraph is a $(2,2)$ -circuit).",
"Recall the Henneberg operations defined in Section .",
"As we will see there is an infinite but controllable class of periodic orbit frameworks which are $P(2,1)$ -graphs and $\\mathcal {T}_x$ -constructive, yet for which the Henneberg operations are insufficient, see Figure REF .",
"For these graphs we must introduce an additional $H2$ Henneberg type move, see Figure REF for the definition.",
"In the following theorem, our main result, by Henneberg operation we mean $H1$ or $H2$ move.",
"Theorem 6.1 ($\\mathcal {T}_x^2$ Henneberg Theorem) Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit framework.",
"$G$ is a $P(2,1)$ -graph and $m$ is $\\mathcal {T}_x$ -constructive if and only if $G$ can be generated from a single loop by gain-preserving Henneberg operations.",
"Given Theorem REF we may now prove Theorem REF .",
"[Proof of Theorem REF ] The necessity of the construction operations follows from Proposition REF and the fact that each of the construction operations takes a $\\mathcal {T}_x^2$ -constructive $P(2,1)$ -graph to a $\\mathcal {T}_x^2$ -constructive $P(2,1)$ -graph.",
"By Theorem REF , Propositions REF and REF show that periodic rigidity is preserved by the Henneberg operations.",
"Thus the sufficiency follows from Theorem REF .",
"The remainder of this section will prove Theorem REF ."
],
[
"Paths, Cycles and Gains",
"Lemma 6.2 Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit framework where $G=(V,E)$ is a $P(2,1)$ graph with unique minimal over-critical set $X\\subset V$ and $m$ is $\\mathcal {T}_x$ -constructive.",
"Let $v_0\\in V$ have $N(v_0)=\\lbrace v_1,v_2,v_3\\rbrace \\subset X$ .",
"Let $G_1^1$ be $(2,2)$ -tight containing $v_1,v_2$ but not $v_0,v_3$ with no $x$ -constructive cycle and all paths from $v_1$ to $v_2$ have $x$ -gain $(m_2-m_1)_x$ .",
"Similarly let $G_2^1$ be $(2,2)$ -tight containing $v_2,v_3$ but not $v_0,v_1$ with no $x$ -constructive cycle and all paths from $v_1$ to $v_2$ have $x$ -gain $(m_3-m_2)_x$ .",
"Then all paths from $v_1$ to $v_3$ have $x$ -gain $(m_3-m_1)_x$ .",
"Let $P$ be a path from $v_1$ to $v_3$ .",
"The path must pass through $G_1^1 \\cap G_2^1$ , say through the vertex $u \\in V_1^1 \\cap V_2^1$ .",
"See Figure REF .",
"In the simplest case, the path $v_1 \\rightarrow u$ lies completely within $G_1^1$ , and the path $u \\rightarrow v_3$ lies completely within $G_2^1$ .",
"Since the graph $G_1^1 \\cap G_2^1$ is connected, there exists a path $P^{\\prime } \\in G_1^1 \\cap G_2^1$ from $u \\rightarrow v_2$ .",
"Suppose the gains on the path are as follows: $v_1 \\xrightarrow{} u \\xrightarrow{} v_3$ $v_1 \\xrightarrow{} u \\xrightarrow{} v_2 \\xrightarrow{} u \\xrightarrow{} v_3.$ But it is now clear that $(m_a + m_c)_x = (m_2 - m_1)_x, \\textrm { and \\ } (m_b - m_c)_x = (m_3 - m_2)_x.$ Summing the two equations, we find that $m_a + m_b = m_3 - m_1$ , as desired.",
"Figure: Proof of Lemma .",
"Both G 1 1 G_1^1 and G 2 1 G_2^1 are (2,2)-tight.Now consider the case that the path $P$ is as follows: $v_1 \\rightarrow u_1 \\rightarrow u_2 \\rightarrow \\cdots \\rightarrow u_n \\rightarrow v_3,$ where $v_1 \\rightarrow u_1$ is contained within $G_1^1$ , the path $u_1 \\rightarrow u2 \\in G_2^1$ , and the remaining segments alternate between $G_1^1$ and $G_2^1$ , with the final path $u_n \\rightarrow v_3 \\in G_2^1$ .",
"Note first that $v_2 \\rightarrow u_1 \\rightarrow u_2 \\rightarrow v_2$ is completely contained within $G_2^1$ , and therefore it must have trivial net $x$ -gain.",
"Similarly, the net $x$ -gain on any closed path $\\underbrace{v_2 \\rightarrow u_i \\rightarrow u_{i+1} \\rightarrow v_2}_{0} $ is trivial.",
"Then the net $x$ -gain on the path from $v_1$ to $v_3$ is as follows: $v_1 \\rightarrow u_1 \\rightarrow u_2 \\rightarrow \\cdots \\rightarrow u_n \\rightarrow v_3$ $\\underbrace{v_1 \\rightarrow u_1 \\rightarrow v_2}_{m_2-m_1} \\rightarrow u_1 \\rightarrow u_2 \\rightarrow v_2 \\rightarrow \\cdots v_2 \\rightarrow u_{n-2} \\rightarrow u_{n-1} \\rightarrow \\underbrace{v_2 \\rightarrow u_n \\rightarrow v_3}_{m_3-m_2},$ and all the paths in the middle contribute nothing to the net $x$ -gain by (REF ), which completes the proof.",
"Note that the only graph theoretical consideration in the proof was the fact that $G_1^1 \\cap G_2^1$ is connected.",
"Thus, in view of the results in Subsection REF , the argument adapts easily to each of the other cases we will consider."
],
[
"Two Distinct Neighbours",
"In approaching this case, we need to pay special attention to a particular class of graphs which we call the bunny ears class.",
"It consists of a three-valent vertex $v_0$ , which is adjacent only to the vertices $v_1$ , and $v_2$ .",
"The vertices $v_1$ and $v_2$ are both members of two edge-disjoint $(2,3)$ -tight subgraphs, which comprise the rest of the graph (see Figure REF (a)).",
"The simplest example, which gives the class its name, consists of only five edges, as shown in Figure REF (b).",
"Figure: The bunny ears class of graphs (a).",
"The smallest example, where G 1 2 G_1^2 and G 2 2 G_2^2 consist of single edges is shown in (b).",
"Note that attempting to delete either vertex 0 or 1 in a reverse H2H2 move will produce a dependent (2,1)(2,1)-tight subgraph.",
"We define a Henneberg 2c move to deal with this case, see Figure .For this class of graphs only, we introduce one additional Henneberg move (Henneberg 2c), which is essentially an edge split on a loop edge.",
"We will use the reverse move: deleting a three-valent vertex and adding a loop edge.",
"See Figure REF .",
"Figure: The reverse Henneberg 2c move on the bunny ears class of graphs (a).",
"The reverse Henneberg 2c move on the smallest example (b).Proposition 6.3 Let $\\langle G, \\mathbf {\\rangle }$ be a periodic orbit framework where $G=(V,E)$ is a $P(2,1)$ graph, and $m$ is $\\mathcal {T}_x$ -constructive.",
"Let $v_0$ be a three-valent vertex adjacent to two distinct vertices $v_1$ and $v_2$ , and suppose that $v_0,v_1, v_2$ are circuit vertices.",
"Let the edges adjacent to $v_0$ be $\\lbrace v_0, v_1; m_0\\rbrace , \\lbrace v_0, v_2; m_1\\rbrace , \\lbrace v_0, v_2; m_2\\rbrace .$ Then $v_0$ is admissible unless $v_0, v_1, v_2$ are part of the bunny ears class of graphs, in which case we add the (loop) edge $\\lbrace v_2, v_2; m_1 - m_2\\rbrace .$ The candidate edges for a reverse $H2$ move deleting $v_0$ are $\\lbrace v_1, v_2; m_0+m_1\\rbrace , \\lbrace v_1, v_2; m_0 + m_2\\rbrace .$ Toward a contradiction, suppose we cannot add either edge.",
"Then there are subgraphs of $G$ which prevent the addition of these edges.",
"Note first that there is no $(2,1)$ -tight subgraph of $G$ containing $v_1$ and $v_2$ but not the vertex $v_0$ , since this would contradict the fact that $G$ is $(2,1)$ -tight.",
"If we cannot add the edge $\\lbrace v_1, v_2; m_0 +m_1\\rbrace $ , then we must have one of two scenarios: There is a subgraph $G_1^1 \\subset G$ which contains $v_1, v_2$ but not $v_0$ , and satisfying $i(V_1^1) = 2|V_1^1|-2$ that will not have an $x$ -constructive gain assignment with the addition of the candidate edge $\\lbrace v_1, v_2; m_0 +m_1\\rbrace $ .",
"That is, $G_1^1$ is $(2,2)$ -tight, contains no $x$ -constructive cycles, and all paths from $v_1$ to $v_2$ have $x$ -gain $(m_0+m_1)_x$ .",
"Note that $G_1^1$ cannot contain a $(2,1)$ subgraph, since that subgraph would be $x$ -constructive by hypothesis.",
"There is a $(2,3)$ -tight subgraph $G_1^2 \\subset G$ which contains $v_1, v_2$ but not $v_0$ , that will not have a constructive gain assignment with the addition of $\\lbrace v_1, v_2; m_0 +m_1\\rbrace $ .",
"That is, $G_1^2$ contains no constructive cycles, and all paths from $v_1$ to $v_2$ have net gain $m_0+m_1$ .",
"Similarly, if we cannot add the edge $\\lbrace v_1, v_2; m_0 +m_2\\rbrace $ , then we have the two scenarios: $G_2^1$ is a $(2,2)$ -tight subgraph containing $v_1, v_2$ but not $v_0$ , and with all paths from $v_1$ to $v_2$ having $x$ -gain $(m_0 + m_2)_x$ .",
"$G_2^2$ is a $(2,3)$ -tight subgraph containing $v_1, v_2$ but not $v_0$ , and with all paths from $v_1$ to $v_2$ having net gain $m_0 + m_2$ .",
"The subscripts of the subgraphs correspond to which edge we are trying to put in, and the superscripts tell us how many edges we can add before creating an overbraced framework.",
"If we cannot put in either edge, then we have three cases, corresponding to the possible pairs of the four subgraphs above: $G_1^1$ and $G_2^1$ $G_1^1$ and $G_2^2$ (and by symmetry $G_1^2$ and $G_2^1$ ) $G_1^2$ and $G_2^2$ Case (1) $V_1^1 \\cap V_2^1$ and $V_1^1 \\cup V_2^1$ are both critical, otherwise adding back the vertex $v_0$ and its three adjacent edges provides a contradiction.",
"Further $G_1^1 \\cap G_2^1$ is connected by Lemma REF (2).",
"Therefore, any path from $v_1$ to $v_2$ in $G_1^1 \\cap G_2^1$ is also a path in both $G_1^1$ and $G_2^1$ .",
"Hence all paths from $v_1$ to $v_2$ have $x$ -gain $(m_0+m_1)_x = (m_0+m_2)_x$ , and therefore $(m_1)_x=(m_2)_x$ .",
"Now let $G^*$ be the graph formed from $G_1^1 \\cap G_2^1$ together with $v_0$ and the three adjacent edges.",
"$G^*$ satisfies $i(V^*) = 2|V^*| - 1$ , and it is not $x$ -constructive, a contradiction.",
"Case (2) Since $G_1^1$ is $(2,2)$ -tight and $G_2^2$ is $(2,3)$ tight, the intersection $G_1^1\\cap G_2^2$ satisfies $i(V_1^1 \\cap V_2^2) = 2|V_1^1 \\cap V_2^2| - \\ell , \\textrm {\\ where } \\ell \\in \\lbrace 2,3\\rbrace .$ If $\\ell = 2$ , then $V_1^1 \\cap V_2^2$ is critical but $G_2^2$ is $(2,3)$ -tight, a contradiction.",
"Therefore $\\ell = 3$ , and $G_1^1 \\cap G_2^2$ is connected by Lemma REF (3).",
"All paths from $v_1$ to $v_2$ have $x$ -gain $(m_0+m_1)_x$ , and net gain $m_0+m_2$ .",
"As in the previous case, $(m_1)_x = (m_2)_x$ .",
"Now consider the union $V_1^1 \\cup V_2^2$ , which is critical.",
"Let $G^*$ be the $(2,1)$ -tight graph created by adding the vertex $v_0$ and its three adjacent edges to $G_1^1 \\cup G_2^2$ .",
"As a subgraph of $G$ , the gain assignment on $G^*$ must be $x$ -constructive.",
"But all paths from $v_1$ to $v_2$ have $x$ -gain $(m_0+m_1)_x = (m_0+m_2)_x$ , and the gain assignment on $G^*$ is not $x$ -constructive, a contradiction.",
"Case (3) $G_1^2 \\cap G_2^2$ satisfies $i(V_1^1 \\cap V_2^2) = 2|V_1^1 \\cap V_2^2| - \\ell , \\textrm {\\ where } \\ell \\in \\lbrace 2,3, 4\\rbrace .$ By the same argument as before $\\ell \\ne 2$ .",
"Suppose $\\ell = 3$ .",
"Lemma REF (3) implies the intersection $G_1^2\\cap G_2^2$ is connected, and all paths in $G_1^2 \\cap G_2^2$ have net gain $m_0 + m_1 = m_0 + m_2$ , which implies that $m_1 = m_2$ .",
"Letting $G^*$ be the graph created from $G_1^2 \\cap G_2^2$ by adding the vertex $v_0$ and its three adjacent edges, we find that $G^*$ is a $(2,2)$ -tight subgraph of $G$ which is not constructive, a contradiction.",
"Thus $\\ell = 4$ .",
"When $|V_1^2 \\cap V_2^2| > 2$ , the intersection is connected, and we again obtain $m_1 = m_2$ .",
"As in Case (2), considering the graph $G_1^2 \\cup G_2^2$ provides a contradiction.",
"Finally we consider the case when $|V_1^2 \\cap V_2^2| = 2$ , in which case the intersection $E_1^2 \\cap E_2^2$ is empty.",
"This corresponds to the bunny ears class of graphs.",
"Then the graph $G^*$ formed from adding $v_0$ and its three adjacent edges to $G_1^2 \\cap G_2^2$ is $(2,1)$ -tight, and moreover it must be the minimal $(2,1)$ -tight subgraph (it does not contain any $(2,1)$ -tight subgraphs).",
"Therefore, there are no loops in $G$ (since loops always form the minimal $(2,1)$ -tight subgraph), and we may add the loop edge $\\lbrace v_2, v_2; m_1-m_2\\rbrace $ .",
"Now observe that if at most one of $v_1,v_2$ lies in the circuit (and hence $v_0$ does not) then we may proceed exactly as for the fixed torus.",
"This is clearly the case if $v_1,v_2$ are not in the circuit so suppose $v_1$ is.",
"There are two cases, (1) there are two copies of $v_1v_0$ and (2) there are two copies of $v_2v_0$ .",
"In (1) the inverse Henneberg 2c operation creates a subgraph which is not $(2,1)$ -sparse (the circuit with a loop added to $v_1$ ) and in (2) the inverse Henneberg 2c operation creates a second circuit contrary to uniqueness.",
"Hence no possible inverse Henneberg operation alters the existing circuit."
],
[
"Three Distinct Neighbours",
"Proposition 6.4 Let $\\langle G, \\mathbf {\\rangle }$ be periodic orbit framework where $G$ is a $P(2,1)$ graph, and $m$ is $\\mathcal {T}_x$ -constructive.",
"Let $v_0$ be a three-valent circuit vertex adjacent to three distinct vertices $v_1, v_2, v_3$ , all of which are in the circuit.",
"Let the three edges adjacent to $v_0$ be given by $\\lbrace v_0, v_i; m_i\\rbrace $ .",
"Then $v_0$ is admissible.",
"We proceed in a similar fashion to the proof of Proposition REF .",
"The three candidate edges are $\\lbrace v_1, v_2; m_2-m_1\\rbrace , \\lbrace v_2, v_3; m_3 - m_2\\rbrace , \\lbrace v_3, v_1; m_1-m_3\\rbrace .$ Suppose we cannot add any of these.",
"We will consider four distinct cases.",
"First we describe some notation.",
"If we cannot add the edge $\\lbrace v_1, v_2; m_2-m_1\\rbrace $ , then we must have one of two scenarios: There is a subgraph $G_1^1 \\subset G$ which contains $v_1, v_2$ but not $v_0, v_3$ , such that $V_1^1$ is critical that will not have an $x$ -constructive gain assignment with the addition of the candidate edge $\\lbrace v_1, v_2; m_2-m_1\\rbrace $ .",
"That is, $G_1^1$ is $(2,2)$ -tight, contains no $x$ -constructive cycles, and all paths from $v_1$ to $v_2$ have $x$ -gain $(m_2-m_1)_x$ .",
"Note that $G_1^1$ cannot contain a $(2,1)$ subgraph, since that subgraph would be $x$ -constructive by hypothesis.",
"Furthermore, $G_1^1$ must be constructive.",
"There is a $(2,3)$ -tight subgraph $G_1^2 \\subset G$ which contains $v_1, v_2$ but not $v_0, v_3$ , that will not have a constructive gain assignment with the addition of $\\lbrace v_1, v_2; m_2-m_1\\rbrace $ .",
"That is, $G_1^2$ contains no constructive cycles, and all paths from $v_1$ to $v_2$ have net gain $m_0+m_1$ .",
"We remark that any $(2,1)$ -tight subgraph containing $v_1$ and $v_2$ must be the whole circuit, so this situation does not arise.",
"Similarly, if we cannot add the edge $\\lbrace v_2, v_3; m_3 - m_2\\rbrace $ , then we have the two scenarios: $G_2^1$ is a $(2,2)$ -tight subgraph containing $v_2, v_3$ but not $v_0, v_1$ , and with all paths from $v_2$ to $v_3$ having $x$ -gain $(m_3 - m_2)_x$ .",
"$G_2^2$ is a $(2,3)$ -tight subgraph containing $v_2, v_3$ but not $v_0, v_1$ , and with all paths from $v_2$ to $v_3$ having net gain $m_3 - m_2$ .",
"Finally if we cannot add the edge $\\lbrace v_3, v_1; m_1-m_3\\rbrace $ , then we have the analogous subgraphs $G_3^1$ is a $(2,2)$ -tight subgraph containing $v_1, v_3$ but not $v_0, v_2$ , and with all paths from $v_3$ to $v_1$ having $x$ -gain $(m_1 - m_3)_x$ .",
"$G_3^2$ is a $(2,3)$ -tight subgraph containing $v_1, v_3$ but not $v_0, v_2$ , and with all paths from $v_3$ to $v_1$ having net gain $m_1 - m_3$ .",
"We consider triples of subgraphs which prevent the addition of any edge, and we have four cases.",
"$G_1^1, G_2^1, G_3^1$ $G_1^1, G_2^1, G_3^2$ $G_1^1, G_2^2, G_3^2$ $G_1^2, G_2^2, G_3^2$ The other combinations follow by symmetry.",
"In fact we will show that we can treat cases (1) and (2) together.",
"Cases (1), (2) Consider the intersection of $G_1^1$ and $G_2^1$ .",
"We show that these two subgraphs cannot co-exist, which eliminates these first two cases.",
"First note that $i(V_1^1 \\cap V_2^1) \\le 2|V_1^1 \\cap V_2^1| - 2$ as a subgraph of the $(2,2)$ -tight graphs $G_1^1, G_2^1$ .",
"In addition $|E_1^1 \\cup E_2^1| \\le 2|V_1^1 \\cup V_2^1| - 2$ , since $v_1, v_2, v_3 \\in V_1^1 \\cup V_2^1$ , and the addition of $v_0$ cannot create an overbraced subgraph.",
"Together these facts mean that we have equality in both cases.",
"In addition, the intersection graph is connected.",
"Now consider the graph $G_1^1 \\cup G_2^1$ .",
"Lemma REF implies all paths from $v_1$ to $v_3$ have $x$ -gain $(m_3 - m_1)_x$ .",
"We also claim that $G_1^1 \\cup G_2^2$ contains no $x$ -constructive cycles.",
"This can be proved using a similar argument.",
"Continuing with Cases (1) and (2), let $G^*$ be the graph formed by appending the vertex $v_0$ and its three adjacent edges to $G_1^1 \\cap G_2^1$ .",
"Since $G_1^1 \\cap G_2^1$ is $(2,2)$ -tight, $G^*$ is a $(2, 1)$ -tight subgraph of $G$ .",
"As such, it must be $x$ -constructive.",
"But $G_1^1 \\cap G_2^1$ does not contain any $x$ -constructive cycles, and adding $v_0$ does not create any new constructive cycles, by our claim.",
"This is a contradiction, and therefore $G_1^1$ and $G_2^1$ cannot both exist.",
"Case (3) Consider the intersection $G_1^1 \\cap G_2^2$ .",
"Lemma REF (2) gives two cases.",
"Case A.",
"$G_1^1 \\cup G_2^2$ is $(2,2)$ -tight, and $G_1^1 \\cap G_2^2$ is $(2,3)$ -tight (and hence connected).",
"We may now argue along the same lines as Cases (1) and (2).",
"That is, $G_1^1 \\cup G_2^2$ is not $x$ -constructive, and nor is the graph $G^*$ formed by appending $v_0$ and its three adjacent edges to $G_1^1 \\cup G_2^2$ , which is the contradiction, since $G^*$ is a $(2, 1)$ -tight subgraph of $G$ .",
"Case B.",
"$ G_1^1 \\cup G_2^2$ is semi-critical.",
"Let $G_4 = G_1^1 \\cup G_2^2$ and consider $G_4 \\cap G_3^2$ .",
"Since $V_4$ is semi-critical, and $G_3^2$ is $(2,3)$ -tight, we know that $ i(V_4 \\cap V_3^2) + i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| + 2|V_4 \\cap V_3^2| - 6.",
"$ Since $v_1, v_2, v_3 \\in V_4 \\cup V_3^2$ , but $v_0$ is not, it must be the case that $2|V_4 \\cup V_3^2|-4 \\le i(V_4 \\cup V_3^2) \\le 2|V_4 \\cup V_3^2| - 2.",
"$ Hence there are 3 cases to analyse.",
"Case I) If we have equality, $i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| - 2$ , then $ i(V_4 \\cap V_3^2) = 2|V_4 \\cap V_3^2| - 4$ by (REF ).",
"In this case, the induced subgraph on the vertices $V_4 \\cup V_3^2$ contains no more than $i(V_4 \\cup V_3^2)$ edges, since otherwise we would have an overbraced subgraph.",
"Figure: Proof of Proposition , Case 3,B,I.",
"G 4 =G 1 1 ∪G 2 2 G_4 = G_1^1 \\cup G_2^2, and the intersection G 4 ∩G 3 2 G_4 \\cap G_3^2 is disconnected into at least two components, including H 1 H_1 and H 2 H_2.",
"In Case 3,B,II, the intersection is connected, in which case it consists of a single edge linking H 1 H_1 and H 2 H_2.It follows that the intersection, $G_4 \\cap G_3^2$ is disconnected, since $v_2 \\notin G_4 \\cap G_3^2$ (Figure REF ).",
"Say the component that lies completely within $G_1^1$ is called $H_1$ , and the component lying completely within $G_2^2$ is called $H_2$ .",
"If $V(H_1)$ and $V(H_2)$ both contain more than one vertex, then we immediately obtain a contradiction, with the following argument.",
"Since $|E(H_1)| \\le 2|V(H_1)| - 2$ , and $|E(H_2)| \\le 2|V(H_2)|-3$ , then $i(V_4 \\cap V_3^2) = |E(H_1)| + |E(H_2)| \\le 2(|V(H_1)| + |V(H_2)|) - 5.$ But this contradicts (REF ).",
"Therefore, it must be the case that one or both of $H_1, H_2$ consists only of a single vertex.",
"Suppose $|V(H_2)| = 1$ , and therefore $|E(H_2)| = 0$ .",
"Then by (REF ), we find that $|E(H_1)| = 2|V(H_1)| - 2$ (and may therefore be a single vertex as well, but need not be).",
"Note all paths from $v_1$ to $v_3$ must pass through the vertex $v_2$ .",
"Thus $G_4 \\cup G_3^2$ is a $(2,2)$ -tight subgraph such that all paths from $v_i$ to $v_j$ have gain $m_j - m_i$ for $1\\le i <j\\le 3$ .",
"Let $G^*$ be the graph formed from $G_4 \\cap G_3^2$ by adding the vertex $v_0$ and its three adjacent edges.",
"Then $G^*$ has $i(V^*) = 2|V^*| -1$ , but no $x$ -constructive cycles.",
"This is a contradiction, since we assumed that $G$ has no such subgraphs.",
"Case II) If it is the case that $i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| - 3$ , then $ i(V_4 \\cap V_3^2) = 2|V_4 \\cap V_3^2| - 3$ by (REF ).",
"The intersection, $G_4 \\cap G_3^2$ must be connected, since otherwise we would have overbraced subgraphs of $G_1^1$ or $G_2^2$ .",
"In particular, there is exactly one edge, say $e$ , which connects $H_1 \\subset G_1^1 \\cap G_3^2$ with $H_2 \\subset G_2^2 \\cap G_3^2$ (see Figure REF ).",
"Moreover, since $|E(H_1)\\le 2|V(H_1)|-2$ , it follows that $H_2$ consists of a single vertex, namely $v_3$ .",
"$H_1$ is then $(2,2)$ -tight.",
"We claim that the graph $G_4 \\cup \\lbrace e\\rbrace $ is a $(2,2)$ -tight subgraph of $G$ which contains no $x$ -constructive cycles, and all paths from $v_i$ to $v_j$ have gain $m_j - m_i$ for $1\\le i <j\\le 3$ .",
"These facts follow from similar arguments to those used in Cases 1 and 2.",
"For example, any path from $v_1$ to $v_2$ that does not lie completely within $G_1^1$ must pass through the new edge $e$ .",
"But then $e$ joins $H_1 \\subset G_1^1$ at some vertex $u$ which is also contained within $G_3^2$ .",
"Since $H_1$ is connected, there is a path from $u$ to $v_1$ within $H_1$ .",
"The path $v_3 \\rightarrow u \\rightarrow v_1$ lies within $G_3^2$ , and therefore the net gain on this path is $m_1 - m_3$ .",
"The path from $v_2$ to $v_3$ is therefore: $v_2 \\rightarrow u \\rightarrow v_3$ $\\underbrace{v_2 \\rightarrow u \\rightarrow }_{(m_1 - m_2)_x} v_1 \\underbrace{\\rightarrow u \\rightarrow v_3}_{(m_3-m_1)_x}, $ and hence the net $x$ -gain from $v_2$ to $v_3$ on any path that goes through the edge $e$ is $(m_3 - m_2)_x$ , which proves (a).",
"Similar arguments apply to show (b) and (c).",
"Case III) If it is the case that $i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| - 4$ , then $ i(V_4 \\cap V_3^2) = 2|V_4 \\cap V_3^2| - 2$ by (REF ).",
"As before we find that $H_1 \\subset G_1^1 \\cap G_3^2$ is $(2,2)$ -tight, and $H_2 \\subset G_2^2 \\cap G_3^2$ consists of a single vertex.",
"In addition, $G_4 \\cap G_3^2$ contains two additional edges, say $e$ and $f$ , as a consequence of (REF ).",
"But then $G_4 \\cup \\lbrace e, f\\rbrace $ is a $(2,1)$ -tight subgraph containing $v_1, v_2, v_3$ but not $v_0$ , a contradiction Case (4) Consider the intersection of $G_1^2$ and $G_2^2$ .",
"From the previous arguments, we have the following cases: $|V_1^2 \\cap V_2^2| > 1$ , and $i(V_1^2 \\cap V_2^2) \\le 2|V_1^2 \\cap V_2^2| - 3$ , or $|V_1^2 \\cap V_2^2| = 1$ , and $i(V_2^1 \\cap V_2^2) = 0$ .",
"Case A.",
"We know that $G_1^2 \\cap G_2^2$ has $i(V_1^2 \\cap E_2^2) \\le 2|V_1^2 \\cap V_2^2| - 3, \\textrm { and }$ $i(V_1^2 \\cup E_2^2) \\ge 2|V_1^2 \\cup V_2^2| - 3.$ However, since the union does not contain any constructive cycles (as the union of two graphs without constructive cycles by the arguments of Cases 1 and 2), in fact these inequalities are both tight.",
"Let $G^*$ be the graph formed from $G_1^2 \\cup G_2^2$ together with $v_0$ and its three adjacent edges.",
"Then $|E^*| = 2|V^*| - 2$ , and must therefore be constructive as a subgraph of $G$ .",
"But, by the arguments of Cases (1) – (2), we find that all paths from $v_1$ to $v_3$ have net gain $m_3 - m_1$ , and therefore, $G^*$ contains no constructive cycles, a contradiction.",
"Case B.",
"Now suppose that $|V_1^2 \\cap V_2^2| = 1$ (i.e.",
"$V_1^2 \\cap V_2^2 = \\lbrace v_2\\rbrace $ ), and therefore $i(V_1^2 \\cup V_2^2) = 2|V_1^2 \\cup V_1^2| - 4$ .",
"Let $G_4 = G_1^2 \\cup G_2^2$ and consider $G_4 \\cap G_3^2$ .",
"Since $G_4$ has $|E_4| = 2|V_4| - 4$ , and $G_3^2$ is $(2,3)$ -tight, we know that $ i(V_4 \\cap V_3^2) + i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| + 2|V_4 \\cap V_3^2| - 7.",
"$ Since $v_1, v_2, v_3 \\in V_4 \\cup V_3^2$ , but $v_0$ is not, it must be the case that $i(V_4 \\cup V_3^2) \\le 2|V_4 \\cup V_3^2| - 2.",
"$ Here we again have three cases, depending on the number of edges in $G_4 \\cup G_3^2$ .",
"Case I) If $i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| - 2$ , then $i(V_4 \\cap V_3^2) \\le 2|V_4 \\cap V_3^2| - 5$ .",
"The intersection is therefore disconnected, and must consist of a single vertex $H_1 \\subset G_1^2$ , where $H_1 = \\lbrace v_1\\rbrace $ , and a $(2,3)$ -tight subgraph $H_2 \\subset G_3^2$ (i.e.",
"$H_2$ is not a singleton).",
"But then $G_4 \\cup G_3^2$ is a $(2,2)$ -tight subgraph of $G$ with no constructive cycles, a contradiction.",
"Case II) If $i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| - 3$ , then $i(V_4 \\cap V_3^2) \\le 2|V_4 \\cap V_3^2| - 4$ .",
"The intersection may be connected or disconnected, with $H_1 \\subset G_1^2$ , and $H_2 \\subset G_3^2$ being either single vertices, or $(2,3)$ -tight subgraphs.",
"In any case, let $G^*$ be the subgraph of $G$ consisting of $G_4 \\cup G_3^2$ together with $v_0$ and its three adjacent edges.",
"Then $G^*$ is a $(2,2)$ -tight subgraph of $G$ with no constructive cycles, a contradiction.",
"Case III) If $i(V_4 \\cup V_3^2) = 2|V_4 \\cup V_3^2| - 4$ , then $i(V_4 \\cap V_3^2) \\le 2|V_4 \\cap V_3^2| - 3$ .",
"The intersection must be connected, and therefore $E(V_4 \\cap V_3^2)$ contains more edges than $E_4 \\cap E_3^2$ .",
"There are at most two induced edges, otherwise $G_4$ together with the three induced edges would form a $(2,1)$ -tight subgraph containing $v_1, v_2, v_3$ , a contradiction.",
"If there are exactly two induced edges, then $G_4$ together with the induced edges $\\lbrace e, f\\rbrace $ is a $(2,2)$ -tight subgraph of $G$ .",
"By arguments similar to those in Case 3, this is a subgraph with no constructive cycles, a contradiction.",
"If there is exactly one induced edge, then $G_4$ together with the induced edge $\\lbrace e\\rbrace $ is a $(2,3)$ -tight subgraph with no constructive cycles, and all paths from $v_i$ to $v_j$ have gain $m_j - m_i$ for $1\\le i <j\\le 3$ .",
"Letting $G^*$ be the graph formed from $G_4 \\cup \\lbrace e\\rbrace $ by adding $v_0$ and its three adjacent edges.",
"Then $G^*$ is a $(2,2)$ -tight subgraph of $G$ with no constructive cycles, a contradiction.",
"When two of the vertices lie in the circuit: Proposition 6.5 Let $\\langle G, \\mathbf {\\rangle }$ be periodic orbit framework where $G$ is a $P(2,1)$ graph, and $m$ is $\\mathcal {T}_x$ -constructive.",
"Let $v_0$ be a three-valent vertex adjacent to three distinct vertices $v_1, v_2, v_3$ , where the edges adjacent to $v_0$ are $\\lbrace v_0, v_i; m_i\\rbrace $ .",
"Suppose that $v_1, v_2$ are circuit vertices, but $v_0$ and $v_3$ lie outside the circuit.",
"Then $v_0$ is admissible.",
"The three candidate edges are $\\lbrace v_1, v_2; m_2-m_1\\rbrace , \\lbrace v_2, v_3; m_3 - m_2\\rbrace , \\lbrace v_3, v_1; m_1-m_3\\rbrace .$ Suppose $G_1^0$ is a $(2,1)$ -tight subgraph of $G$ containing $v_1, v_2$ , but not $v_0$ or $v_3$ .",
"If $G_2^1$ or $G_3^1$ exist we contradict Lemma REF (2).",
"Similarly if $G_2^2$ exists and intersects the circuit in more than one vertex we contradict Lemma REF (3).",
"Finally if $|V_1^0\\cap V_2^2|=1$ then we contradict Lemma REF .",
"In other words, if two of the vertices $v_1, v_2, v_3$ lie in the circuit (or any $(2,1)$ -tight subgraph containing the circuit), then at least one of the other two candidate edges may always be added.",
"Proposition 6.6 Let $\\langle G, \\mathbf {\\rangle }$ be periodic orbit framework where $G=(V,E)$ is a $P(2,1)$ graph with unique over-critical set $X$ , and $m$ is $\\mathcal {T}_x$ -constructive.",
"Let $v_0\\in V$ have $N(v_0)=\\lbrace v_1,v_2,v_3\\rbrace $ and suppose $|X\\cap N(v_0)|\\le 1$ .",
"Then $v_0$ is admissible.",
"Lemma REF reduces this to a fixed torus problem, thus the result follows from Theorem REF .",
"We now reach the stated goal of this section.",
"[Proof of Theorem REF ] One direction follows from the definitions of the gain-preserving Henneberg operations.",
"The converse follows from Propositions REF , REF , REF and REF ."
],
[
"Extensions",
"Up to this point, we have focused on frameworks which are on a variable torus, where the variability is in the $x$ -direction only.",
"It is immediate that we can apply this result to frameworks which are variable in the $y$ -direction only, simply by demanding that our framework be $\\mathcal {T}_y$ -constructive.",
"We now outline several other variations on the torus with one degree of freedom."
],
[
"Fixed Area and Angle",
"One variation of the variable torus is the torus whose area remains fixed, as does the angle between the two generators, which we denote ${vol}^2$ .",
"It is generated by the lattice matrix $L_{vol}(t) = \\left(\\begin{array}{cc}x(t) & 0 \\\\0 & kx(t)\\end{array}\\right).$ In this case the angle is constrained to remain fixed at $\\pi /2$ .",
"Certainly, if a framework is generically rigid on $\\mathcal {T}_x^2$ or $y^2$ , then it is generically rigid on ${vol}^2$ too.",
"Of course, the natural way to understand the fixed area setting is to consider frameworks on the torus with two degrees of freedom, which we do not address here."
],
[
"Flexible Angle",
"Let ${\\theta }^2$ be the torus generated by the lattice matrix $L_{\\theta } = \\left(\\begin{array}{cc}1 & 0 \\\\\\cos \\theta & \\sin \\theta \\end{array}\\right).$ This is the torus that has generators with fixed lengths, and a variable angle between them.",
"Then the rigidity matrix for frameworks on ${\\theta }^2$ has a column corresponding to the variable $\\theta (t)$ .",
"Instead of requiring an $x$ -constructive cycle, the necessary condition for rigidity can be seen to be that the critical subgraph contains a cycle with net gain $(m_1, m_2)$ satisfying $m_1m_2 \\ne 0$ .",
"The other requirements for rigidity are as for frameworks on $x^2$ ."
],
[
"Frameworks on a Variable Cylinder",
"As a direct consequence of Theorem REF we obtain a characterisation of the generic rigidity of frameworks which are periodic in one direction only.",
"That is, Theorem REF provides necessary and sufficient conditions for the rigidity of frameworks on the cylinder with variable circumference, where the cylinder is a “flat cylinder.\"",
"That is, we are not considering frameworks supported on surfaces in three dimensional space as in [17].",
"See Figure REF for an example.",
"Such frameworks are similar to frieze patterns, although we assume that such patterns exhibit only translational symmetry (in one direction), and do not have any of the other symmetries of frieze patterns.",
"Figure: A framework which is periodic in one direction only (a), and its gain graph (b) which is labelled by elements of ℤ\\mathbb {Z}.Let $\\langle G, \\mathbf {\\rangle }$ be a gain graph with $m: E \\rightarrow \\mathbb {Z}$ .",
"For each edge $e = \\lbrace i,j;m_e\\rbrace $ , the integer $m_e$ now represents the number of times the edge $e$ “wraps\" around the cylinder.",
"Let $p:V \\rightarrow \\mathbb {R}^2 / (\\mathbb {Z} \\times Id)$ .",
"That is, for $v \\in V$ , $p(v) \\in [0,1) \\times \\mathbb {R}$ .",
"Let $ [0,1) \\times \\mathbb {R}$ , and we call this the variable cylinder.",
"Let $\\langle G, \\mathbf {\\rangle }$ be a gain graph with gain assignments from $\\mathbb {Z}$ .",
"Let $\\hat{m}: E \\rightarrow \\mathbb {Z}^2$ be the gain assignment on $G$ given by $\\hat{m}(e) = (m(e), 0)$ .",
"Since the rigidity matrix for the cylinder and the rigidity matrix for $\\mathcal {T}_x^2$ are identical (each has exactly one lattice column), we have the following proposition.",
"Proposition 7.1 A periodic orbit graph $\\langle G, \\mathbf {\\rangle }$ is generically rigid on the variable cylinder $ if and only if $ G, m $ is generically rigid on $ x2$.$"
],
[
"Fully Variable Torus",
"Generic minimal rigidity on the fully variable torus 2 has been characterised by Malestein and Theran [14]; though their proof is non-inductive.",
"There is a significant new challenge to providing such a constructive characterisation as the underlying graphs can have minimum degree 4.",
"This suggests a consideration of degree 4 Henneberg type operations such as $X$ and $V$ -replacement, however these operations are already known to be problematic for 3-dimensional rigidity, [27].",
"For periodic frameworks the main challenges are the large number of cases and the fact that the variants of $V$ -replacement do not necessarily preserve the relevant counting conditions."
],
[
"Global Periodic Rigidity",
"The global rigidity of frameworks in Euclidean space is well studied.",
"As in the case of rigidity there is a celebrated combinatorial characterisation in the plane, see [11], but no such characterisation is known in higher dimensions.",
"The result in the plane relied on inductive constructions of circuits in the plane rigidity matroid [2] and crucially of 3-connected, redundantly rigid graphs [11].",
"It is tempting then, as a first step, to take the inductive constructions provided here and in [20] and try to provide related constructions for the circuits in the fixed (or flexible) torus rigidity matroid.",
"As far as we know this has not yet been addressed."
],
[
"Periodic Body-Bar Frameworks",
"Recently, a characterisation of the generic rigidity of periodic body-bar frameworks on the fixed torus has been developed, when $d \\le 3$ [21].",
"The body-bar setting is somewhat different from the present study, since we are no longer working within a combinatorial subclass of the full inductive class (as we are with $P(2,1)$ graphs in the class of all $(2,1)$ -tight graphs).",
"For this reason, it is possible to use existing inductive characterisations of the relevant combinatorial structures [5].",
"However, it may be possible to extend those results to the partially variable torus, which may require a more subtle approach, as in the present paper.",
"Acknowledgements.",
"Much of this research was carried out while both authors were at the Fields Institute, University of Toronto.",
"We enjoyed several stimulating conversations with Justin Malestein and Louis Theran on the broad topic of inductive constructions for periodic and symmetric frameworks."
],
[
"Fully Variable Torus",
"Generic minimal rigidity on the fully variable torus 2 has been characterised by Malestein and Theran [14]; though their proof is non-inductive.",
"There is a significant new challenge to providing such a constructive characterisation as the underlying graphs can have minimum degree 4.",
"This suggests a consideration of degree 4 Henneberg type operations such as $X$ and $V$ -replacement, however these operations are already known to be problematic for 3-dimensional rigidity, [27].",
"For periodic frameworks the main challenges are the large number of cases and the fact that the variants of $V$ -replacement do not necessarily preserve the relevant counting conditions."
],
[
"Global Periodic Rigidity",
"The global rigidity of frameworks in Euclidean space is well studied.",
"As in the case of rigidity there is a celebrated combinatorial characterisation in the plane, see [11], but no such characterisation is known in higher dimensions.",
"The result in the plane relied on inductive constructions of circuits in the plane rigidity matroid [2] and crucially of 3-connected, redundantly rigid graphs [11].",
"It is tempting then, as a first step, to take the inductive constructions provided here and in [20] and try to provide related constructions for the circuits in the fixed (or flexible) torus rigidity matroid.",
"As far as we know this has not yet been addressed."
],
[
"Periodic Body-Bar Frameworks",
"Recently, a characterisation of the generic rigidity of periodic body-bar frameworks on the fixed torus has been developed, when $d \\le 3$ [21].",
"The body-bar setting is somewhat different from the present study, since we are no longer working within a combinatorial subclass of the full inductive class (as we are with $P(2,1)$ graphs in the class of all $(2,1)$ -tight graphs).",
"For this reason, it is possible to use existing inductive characterisations of the relevant combinatorial structures [5].",
"However, it may be possible to extend those results to the partially variable torus, which may require a more subtle approach, as in the present paper.",
"Acknowledgements.",
"Much of this research was carried out while both authors were at the Fields Institute, University of Toronto.",
"We enjoyed several stimulating conversations with Justin Malestein and Louis Theran on the broad topic of inductive constructions for periodic and symmetric frameworks."
]
] | 1204.1349 |
[
[
"Ricci focusing, shearing, and the expansion rate in an almost\n homogeneous Universe"
],
[
"Abstract The Universe is inhomogeneous, and yet it seems to be incredibly well-characterised by a homogeneous relativistic model.",
"One of the current challenges is to accurately characterise the properties of such a model.",
"In this paper we explore how inhomogeneities may affect the overall optical properties of the Universe by quantifying how they can bias the redshift-distance relation in a number of toy models that mimic the real Universe.",
"The models that we explore are statistically homogeneous on large scales.",
"We find that the effect of inhomogeneities is of order of a few percent, which can be quite important in precise estimation of cosmological parameters.",
"We discuss what lessons can be learned to help us tackle a more realistic inhomogeneous universe."
],
[
"Introduction",
"Large scale redshift surveys of galaxies reveal a hierarchy of structure that extends out to hundreds of Megaparsecs, yet a homogeneous cosmological model lies at the core of our understanding of how the Universe evolves.",
"Such an assumption works incredibly well, as can be seen from current analyses of the Cosmic Microwave Background (CMB).",
"The linearly perturbed Friedmann–Lemaître–Robertson–Walker (FLRW) model can fit observations of the CMB with exquisite precision [1].",
"Nevertheless we cannot avoid the fact that the Universe is inhomogeneous, especially on small scales.",
"Indeed, as a first approximation, space looks empty, with nuggets of mass and energy clustered together to form an intricate cosmic web.",
"It is quite possible that the smooth description of space emerges as the average behaviour of a fundamentally inhomogeneous universe (cf.",
"[2], [3], [4], [5]).",
"This is an operation that we are familiar with closer to home, when we describe a gas or liquid through their macroscopic properties instead of through a detailed atomistic description.",
"We would like to be able to do the same for the Universe, i.e.",
"to infer the smooth, large scale (macroscopic) properties of space time from cosmological observations without having to worry about the discreteness of its constituents on small scales.",
"We attempt to reconcile such an observation with a homogeneous universe by saying that our local, granular point of view is merely the small scale limit of the description given by the $\\Lambda $ -Cold Dark Matter ($\\Lambda $ CDM) cosmology.",
"We then assume that on large scales, $\\Lambda $ CDM is accurately described by a smooth cosmological fluid which on small scales fragments into particle dark matter and forms halos.",
"Some have expressed strong concerns about the validity of such an approach (cf.",
"[6], [7], [8]), especially because it is not clear whether the inhomogeneities on small scales will radically affect the large scale dynamics of the Universe (see [9] – [27], but see also [28], [29], [30] for counterexamples).",
"There are a number of interesting and important issues that must be addressed when tackling this problem.",
"In this paper we wish to focus on one: how inhomogeneities can affect the inferred optical properties of the Universe.",
"This problem has been looked at before, throughout the 1960s [31] – [37] and 1970s [38] – [40], and more recently [41] – [75], with a variety of different assumptions and results.",
"In this paper we look at how different effects can modify the redshift-distance relation, ($z$ , $D_A$ ).",
"One might expect, a priori, that studying the effect of inhomogeneities on ($z$ , $D_A$ ) is pointless: in 1976, Weinberg [40] claimed that the overall ($z$ , $D_A$ ) for an inhomogeneous Universe should, on the whole, match that of the average homogeneous Universe.",
"Views are somewhat divided on the validity of Weinberg's argument and its generality, with cosmologists tending to accept it wholesale, while some relativists are slightly more sceptical and question its assumptions (cf.",
"[76]).",
"In this paper we will be completely agnostic.",
"Our approach is based entirely on the Sachs equations.",
"Within the framework of these equations, we study different aspects of the redshift-distance relation.",
"This is a complementary procedure to what is usually doneThe most commonly-implemented procedure to account for the effect of inhomogeneities on light propagation is the magnification matrix approach.",
"In almost all cases, one further assumes that light propagates on unperturbed geodesics, and thus the redshift (to a source located at some fixed comoving distance) is calculated as in the background homogeneous models, while inhomogeneities only affect the magnification/demagnification (for theoretical analysis see [42], [52], [72], or for practical implementation in ray tracing codes see [49], [50], [73]).. Our approach will be to use a number of different approximate models: high resolution N-body simulations, halo models, and a variety of swiss cheese models with Lemaître-Tolman-Bondi inhomogeneities.",
"While what we find is not, as yet, definitive, it does shed light on how important the effects of inhomogeneities might be.",
"The paper is structured as follows.",
"In Section we present the Sachs equations and their various limits.",
"In Section we look at the effect of Ricci focusing, in Section we study the impact of shear, and in Section we attempt to include local fluctuations in the expansion rate.",
"In each of these sections, we need to consider different models for inhomogeneities.",
"In Section we discuss our results."
],
[
"Distance redshift relation",
"The Sachs optical equations describe the evolution of optical quantities, the expansion of the null bundle $\\theta $ and its shear $\\sigma $ [31] $\\frac{{\\rm d} \\theta }{{\\rm d} s} + \\theta ^2 + |\\sigma |^2 =- \\frac{1}{2} R_{\\alpha \\beta } k^{\\alpha } k^{\\beta }, \\\\\\frac{{\\rm d} \\sigma }{{\\rm d} s} + 2 \\theta \\sigma =C_{\\alpha \\beta \\mu \\nu } \\epsilon ^{*\\alpha }k^{\\beta } \\epsilon ^{*\\mu } k^{\\nu }.",
"$ where $R_{\\alpha \\beta }$ and $C_{\\alpha \\beta \\mu \\nu }$ are respectively the Ricci and Weyl tensors, $k^\\mu $ is the null vector and $\\epsilon ^\\mu $ is perpendicular to $k^\\mu $ and is confined to a surface tangent to a wave front.",
"In the gravitational lensing nomenclature, `shear' normally refers to the image distortion $\\gamma $ .",
"These two quantities are related but are not the same, and in order to avoid any confusion, we will always refer to $\\sigma $ as to the shearing (for a detailed set of equations that relates these two quantities, see [69]).",
"The rate of change of the distance depends on the expansion rate of the light bundle in such a way that the angular diameter distance, $D_A$ , satisfies $\\frac{{\\rm d} }{{\\rm d} s} \\ln D_A = \\theta .$ Recall that the luminosity distance is given by $D_L\\equiv (1+z)^2D_A$ [77], [78].",
"Rewriting equation (REF ), we find that $\\frac{{\\rm d^2} D_A}{{\\rm d} s^2} = - ( |\\sigma |^2 + \\frac{1}{2} R_{\\alpha \\beta } k^{\\alpha } k^{\\beta }) D_A.",
"$ We need a set of initial conditions to solve for ($z$ , $D_A$ ), but more importantly we also need: (1) the Ricci curvature $R_{\\alpha \\beta }$ ; (2) the Weyl curvature $C_{\\alpha \\beta \\mu \\nu }$ ; (3) the transformation from the affine parameter $s$ to the redshift $z$ .",
"Using the Einstein equations ($R_{\\alpha \\beta } - R g_{\\alpha \\beta }/2 = \\kappa T_{\\alpha \\beta } + \\Lambda g_{\\alpha \\beta }$ ) (where $\\kappa = 8 \\pi G$ and $G$ is the gravitational constant), with the energy momentum tensor of a perfect fluid, one has that $R_{\\alpha \\beta } k^{\\alpha } k^{\\beta } = \\kappa ( \\rho + p) (u_{\\alpha } k^\\alpha )^2.$ In comoving coordinates, using the definition of redshift $1+z \\equiv (u_{\\alpha } k^\\alpha )_e/ (u_{\\alpha } k^\\alpha )_o$ (where the subscripts $e$ and $o$ refer to the instants of emission and observation, respectively), one obtains $R_{\\alpha \\beta } k^{\\alpha } k^{\\beta } =\\kappa (\\rho + p) (1+z)^2,$ where we have used the freedom of the affine re-parametrizationThe affine parametrization is conserved with respect to linear transformations, $s \\rightarrow a s +b$ [79].",
"to set $(u_{\\alpha } k^\\alpha )_o = 1$ .",
"For the case of dust (pressureless matter), the Ricci focusing depends only on matter density along the past null cone.",
"From now on, we focus only on this case (dark matter can be treated as being pressureless, and for the cosmological constant we have $\\rho _\\Lambda + p_\\Lambda = 0$ ), i.e.",
"we assume that the Ricci focusing is given by $R_{\\alpha \\beta } k^{\\alpha } k^{\\beta } = \\rho (1+z)^2.$ Unfortunately, there is no `easy' trick for estimating the Weyl focusing, and so to calculate this factor one has to solve for the null geodesics.",
"In fact, solving null geodesics is also important for linking $s$ with $z$ , which can be done by using the redshift formula together with the null geodesic equations.",
"In comoving coordinates, we have $1+z = (u_{\\alpha } k^\\alpha )_e/ (u_{\\alpha } k^\\alpha )_o= k^0_e = \\frac{{\\rm d} t}{{\\rm d} s},$ which allows us to link $s$ with $z$ .",
"In a homogeneous space time we have $&& \\frac{{\\rm d} z}{{\\rm d} s} = (1+z)^2 H, \\nonumber \\\\&& \\sigma = 0, \\nonumber \\\\&& R_{\\alpha \\beta } k^{\\alpha } k^{\\beta } =\\rho _0 (1+z)^5,$ where $H$ is the Hubble rate, and one recovers the textbook redshift-distance relation.",
"If one assumes that $R_{\\alpha \\beta } k^{\\alpha } k^{\\beta } = \\alpha \\rho _0 (1+z)^5,$ where $\\alpha = const$ , the Dyer-Roeder formula is recovered [38], [39]."
],
[
"The role of Ricci focusing",
"We will first try to quantify the effect of Ricci focusing.",
"To do so, we need to solve (REF ) in terms of $\\rho (z)$ , $\\sigma (z)$ and $s(z)$ .",
"In this section, we assume that $\\sigma (z)$ and $s(z)$ are exactly the same as in the background FLRW model [see the first two terms in (REF )].",
"We wish to study `realistic' density profiles along the line of sight, and to do so we use density fields from the Millennium simulation [80], [81].",
"The Millennium simulation is an N-body simulation of the concordance cosmology.",
"It consists of 10,077,696,000 particles of mass $8.6 \\times 10^8 M_\\odot h^{-1}$ within a cube of volume $(500 h^{-1}$ Mpc$)^3$ .",
"The simulation was performed using the GADGET-2 code [82].",
"In our calculations, we use the Millennium MField, which is the dark matter density field put on a $256^3$ grid, smoothed with Gaussian kernels $\\rho = \\sum _i m_i W$ , where $W \\sim \\exp [ x^2/\\sigma ^2].",
"$ of size: 1.25 Mpc, 2.5 Mpc, 5 Mpc, and 10 Mpc, labelled $g_{1.25}$ , $g_{2.5}$ , $g_5$ , and $g_{10}$ respectively.",
"We randomly place an observer in the Millennium box, and then calculate $10^6$ different lines of sight (each in a different direction from the observer, and for each line of sight we randomly select a different observer).",
"We assume periodic boundary conditions, so that when the light ray exits the Millennium box, it enters the other side of the box with entry angles the same as the exit angles (to enforce periodic boundary conditions).",
"Since the MField consists of density maps on discreet time slices, to get $\\rho $ at any required instant we interpolate between different time slices.",
"We calculate the light propagation out to $z=1.6$ , solve Eq.",
"(REF ) and then write the distance as a deviation from the expected value in the background model (in our case the $\\Lambda $ CDM model), $D_{A}(z)= \\bar{D}_{A} ( 1 + \\Delta ).$ Results in terms of the probability distribution function (PDF) of $\\Delta $ are presented in Fig.",
"REF .",
"We find that the larger the smoothing radius, the smaller the variance: for $g_{10}$ the standard deviation is $5.18 \\times 10^{-3}$ (uppermost solid curve in Fig.",
"REF ); for $g_{5}$ the standard deviation is $7.28 \\times 10^{-3}$ (the solid curve second from the top in Fig.",
"REF ); for $g_{2.5}$ the standard deviation is $9.52 \\times 10^{-3}$ (the solid curve third from the top in Fig.",
"REF ); and for $g_{1.25}$ the standard deviation is $1.21 \\times 10^{-2}$ .",
"It is interesting to compare these results with the ones obtained using the weak lensing formula (in the Born approximation), $\\Delta _{WL} =- \\frac{3 }{2} H_0^2 \\Omega _m \\int \\limits _0^{\\chi _e} {\\rm d} \\chi \\frac{ \\chi _e - \\chi }{\\chi _e} \\chi a^{-1} \\delta (\\chi ),$ where $\\chi $ is the comoving coordinate, $d \\chi = dz/H(z)$ and we use the same $\\delta (z)$ as before.",
"The results are plotted with dotted lines in Fig.",
"REF .",
"As shown, the PDFs are very similar to the ones obtained within the Ricci focusing regime.",
"Finally, we calculate the standard deviation by squaring the expression above (where the mean is zero by construction) and replacing $\\delta ^2$ with the matter power spectrum [83], $ \\sigma ^2_{\\Delta } =\\frac{9 }{4} \\Omega _m^2 H_0^4 \\int \\limits _0^{\\chi _e} {\\rm d} \\chi \\left[ (1+z) \\frac{ \\chi _e - \\chi }{\\chi _e} \\chi \\right]^2\\int \\limits _0^{\\infty } {\\rm d} k ~ k \\frac{P(k,z)}{2\\pi }.$ In this case, the standard deviation of the distance correction isHere, in order to compare with the Millennium simulation, we use the cosmological parameters that were used in the Millennium simulation.",
"If the WMAP7 set of cosmological parameters is used instead, then $\\sigma _{\\Delta } =1.57 \\times 10^{-2} $ .",
"$1.45 \\times 10^{-2}$ (the dashed curve in Fig.",
"REF is the Gaussian PDF with mean of zero and standard deviation of $1.45 \\times 10^{-2}$ ).",
"The above results clearly reveal a pattern.",
"First of all, the smaller the smoothing radius, the larger the variance, although even with the 1.25 Mpc smoothing radius we get a smaller standard deviation than was obtained within the framework of linear approximation.",
"Also, note that the maximum of the PDF lies on the demagnification side (i.e.",
"$\\Delta >0$ ), so a random object is most likely to be dimmer than on average.",
"Nevertheless, in all cases the mean is almost zero: $1.20 \\times 10^{-4}$ , $1.38 \\times 10^{-4}$ , $1.29 \\times 10^{-4}$ , $1.44 \\times 10^{-4}$ , for $g_{1.25}$ , $g_{2.5}$ , $g_{5}$ , and $g_{10}$ respectivelyNote that, for each different line of sight, we chose a different observer location.",
"If we located an observer inside some deep inhomogeneity instead, such as a void of $\\delta _{1.25} = -0.84$ , $\\delta _{2.5} = -0.78$ , $\\delta _{5} = -0.66$ , $\\delta _{10} = -0.43$ , then the mean would be $2.54 \\times 10^{-4}$ , $2.58 \\times 10^{-4}$ , $2.46 \\times 10^{-4}$ , $2.07 \\times 10^{-4}$ .",
"This suggest that the effect of local inhomogeneity on the distance (to $z=1.6$ in our case) is of order $10^{-4}$ .",
"In this paper we try to avoid the `local' effects introduced by nearby inhomogeneities, so that we have a more clear insight to light propagation effects alone.",
"For papers that study the local bias see [84]–[89].",
"Figure: PDF of Δ\\Delta .Solid curves (from the top) are the resultsobtained by solving Eq.",
"within the Ricci focusing regime,for the Millennium maps g 10 g_{10}, g 5 g_{5}, g 2.5 g_{2.5}, and g 1.25 g_{1.25}.The dotted lines are the resultsobtained from () for the same density fields.The dashed line is the Gaussian PDF withthe variance calculated from Eq.",
"().Figure: The thick solid line is the PDF obtained in the full halo model.The dashed line is a variation of the halo model, where we switched off the Weyl focusing.",
"The dotted lines (from the top)are halo models with Δ SO \\Delta _{SO} equalto 80 and 40 respectively (see text for explanation).",
"For comparison, wealso present the results when the g 1.25 g_{1.25} Millennium map is used(dot-dashed line)."
],
[
"The role of smoothing and Weyl focusing",
"We now extend our analysis to investigate the role of smoothing and the effect of Weyl focusing on the optical properties of the Universe.",
"When dealing with the Weyl focusing, it is not straightforward to link density fields with the Weyl curvature.",
"An ideal situation would be to have an exact generic solution of the Einstein field equations, and use that to model the evolution and effects of the Weyl curvature.",
"Since there is no such a solution, in practise we must always employ some approximations.",
"Here we will evaluate the Weyl focusing within the halo model.",
"Simply put, the halo model considered here consists of a universe filled with different halos of masses above $10^{10} h^{-1} M_\\odot $ , while the rest of the mass density (in objects of masses below this threshold) is assumed to be distributed homogeneously.",
"The halo model we use closely follows the construction outlined in detail in Ref.",
"[64].",
"Each halo is described by the Navarro-Frenk-White (NFW) density profile [90], $\\rho (r)= \\rho _m(z) \\frac{\\delta _c}{(r/R)(1+r/R)^2},$ where $R$ is the radius of the halo, defined as $R = \\left( \\frac{M}{(4/3) \\pi \\rho _m(z) \\Delta _{SO}} \\right)^{1/3},$ with $\\rho _m(z) = \\rho _0 (1+z)^3$ , and $\\Delta _{SO} = 180$ (the halo is defined as an overdensity of mass $M$ , whose constant density contrast is 180 with respect to the mean matter density, cf.",
"[91], [64]).",
"Integrating $\\rho (r)$ , and using the expression above, one finds that $ \\delta _c = \\frac{\\Delta _{SO}}{3} \\frac{c^3}{\\ln (1+c) - c/(1+c)}, $ where $c$ is the concentration parameter, whose evolution given in [92] as $c = 10.14 [ M /(2 \\times 10^{12} h^{-1} M_\\odot )]^{-0.081} (1+z)^{-1.01}.$ The number density of halos at a given instant in time is given by the halo mass function $ {\\rm d} n = \\frac{\\rho _m(z)}{M} f {\\rm d} \\sigma ^{-1}, $ with $f$ and $\\sigma $ given by [91] $&& f(\\sigma ,z) = 0.301 \\exp [ - | \\ln \\sigma ^{-1} +0.67|^{3.82}| ], \\nonumber \\\\&& \\sigma (M,z) = \\frac{{\\cal G}}{2\\pi ^2} \\int \\limits _0^\\infty k^2 P(k) W^2(k,M) {\\rm d} k,$ where ${\\cal G}(z)$ is the growth factor (this is usually written as $D$ , but we use $ {\\cal G}$ here in order to avoid confusion with distance).",
"The halo model allows us to address the problem of smoothing — instead of dealing with a continuous density field, we now have a discrete set of dark matter halos — although one should keep in mind that, given the characteristics described above, this is not really a completely `discrete' model.",
"It also allows us to take into account the effect of the shearing, by assuming that the Weyl focusing for a particular halo, $i$ , is calculated using the Lemaître–Tolman (LT) model [93], [94] (an exact general relativistic model of an spherically symmetric, inhomogeneous, non-stationary space time) [47], [48] $\\mathcal {C}_i\\simeq C_{\\alpha \\beta \\mu \\nu } \\epsilon ^{*\\alpha } k^{\\beta } \\epsilon ^{*\\mu } k^{\\nu } = \\frac{1}{2} \\frac{b^2}{R^2} \\left( \\rho - \\bar{\\rho } \\right),$ where $b$ is the impact parameter, $R$ is the areal radius, $\\rho = 4 \\pi \\frac{G}{c^2} \\frac{ M^{\\prime }}{R^2 R^{\\prime }}, \\quad {\\rm and~} \\quad \\bar{\\rho } = 4 \\pi \\frac{G}{c^2} \\frac{ 3M}{R^3} .\\nonumber $ We apply the weak field approximations (cf.",
"Ref.",
"[33]) and assume that the total Weyl focusing is a sum over all contributors, $ {\\mathcal {C}} = \\sum _i {\\mathcal {C}}_i$ .",
"Note that, outside the halo, $\\mathcal {C}_i \\sim b^2/R^{5}$ , and so only halos that are close to the light ray contribute most significantly.",
"We again solve Eq.",
"(REF ) for $10^6$ different lines of sight.",
"The Ricci and Weyl focusing are calculated using the halo model, while $s(z)$ is assumed to be the same as in the FLRW background model [see the first term of (REF )].",
"The resulting distribution for $\\Delta $ is presented in Fig.",
"REF (thick solid curve).",
"As expected, the variance is much larger than when the smoothed Millennium maps were used.",
"Also, the maximum is shifted more towards positive values of $\\Delta $ .",
"The mean, as before, is still negligible, at $8\\times 10^{-4}$ .",
"For comparison, and to gain a better understanding of the different factors, we now consider rather unphysical variations of the halo model.",
"The first variation is when we neglect the shearing – when solving (REF ), we put $\\sigma =0$ .",
"The result of this is shown as the dashed line in Fig.",
"REF .",
"As can be seen, it is significantly shifted to the magnification side (positive $\\Delta $ ) – the mean is $1.02 \\times 10^{-2}$ .",
"This shows that, when considering the halo model, the role of the shearing should not be neglected when calculating the distance correction.",
"To better understand the relation between the halo model and the smoothed Millennium density fields, we can study two more unconventional modifications of the halo model.",
"In order to reduce the level of discreteness of the halo model, we decrease the value of the parameter $\\Delta _{SO}$ , which results in larger halos and a lower amplitude of the halo density profiles.",
"In the first case we reduce $\\Delta _{SO}$ to 80, which means that the radius of the halo increases by roughly $30\\%$ .",
"In the second case, we reduce $\\Delta _{SO}$ to 40, which results in a $65\\%$ increase in the halo's radiusFor example, a halo of mass $10^{12} M_\\odot $ and $\\Delta _{SO} = 180$ has $R \\approx 330$ kpc [this follows from (REF )], for $\\Delta _{SO} = 80$ and $\\Delta _{SO} =40$ , $R \\approx 430$ kpc and $R \\approx 545$ kpc respectively.. We plot these results as dotted lines in Fig.",
"REF .",
"As can be seen, when we increase the halo radii (and hence decrease the degree of discreteness), we approach the results obtained when using the smoothed Millennium maps.",
"So far, the results reveal the same pattern: a skewed distribution with a maximum at the demagnification side ($\\Delta _{max}>0$ ).",
"This can easily be understood within the framework of the weak lensing formula (REF ): when light rays pass through the large-scale structure, they are more likely to propagate through voids than through more dense, compact structures.",
"Therefore, it is more likely to have $\\delta <0$ along the line of sight and, as follows from (REF ), $\\delta < 0 \\Rightarrow \\Delta >0$ .",
"This is clearly visible when we use the smoothed Millennium maps.",
"If the smoothing scale is sufficiently large, the resulting density field approaches a Gaussian field (all non-linearities are washed away, and we recover symmetry between overdense and underdense regions), and so the maximum approaches zero (for pure Gaussian fluctuations $\\Delta _{max}=0$ ).",
"In the halo model, we deal with halos that are very compact and occupy a relatively small volume, which makes propagation through voids even more likely to occur.",
"We therefore see a shift of $\\Delta _{max}$ towards higher demagnification (see Fig.",
"REF ).",
"Occasionally, light rays pass close to a halo, and then both the Ricci and Weyl focusing magnify the light bundle.",
"Thus, on average, the total balance is recovered, and the mean of $\\Delta $ is almost zero (in the lensing framework, as seen in (REF ), due to matter conservation it is exactly zero).",
"Here we find a mean of order $10^{-4}$ .",
"A small deviation from the expected value in the FLRW model is expected.",
"For example the analogue of the Integrated Sachs-Wolfe effect will affect the redshift, and thus $(z,D_A)$ .",
"Also, the local environment plays a role – note that our observer is located inside an inhomogeneity (i.e.",
"not within a homogeneous region) – see footnote on page 6.",
"The same pattern has also been reported in previous studies – see for example in [42], and in ray tracing though N-body simulations.",
"In fact, our halo model provides very good agreement with the ray tracing through the Millennium simulation, cf.",
"[49], [50].",
"It is also in almost perfect agreement with the PDF generated using the turboGL code[59], [64]The turboGL code uses the halo model to calculate the PDF of the weak lensing convergence.",
"The effect of the shear is neglected in turboGL.",
"However, if the zero-shear PDFs are compared, they are in agreement.",
"Note that, as seen in Fig.",
"REF , the zero-shear PDF, apart from a small change in its amplitude, looks like the translation of the nonzero-shear PDF..",
"The results are therefore consistent with the most commonly-implemented procedures, where the effects of inhomogeneities on light propagation are calculated using the magnification matrix approach.",
"In almost all cases, one further assumes that light propagates on unperturbed geodesics, and so the redshift (to a source located at some fixed comoving distance) is calculated as in the background homogeneous models, while inhomogeneities only affect the magnification/demagnification.",
"So far, we have been doing the same (that is, we assumed that $s(z)$ is the same as in the background FLRW model).",
"We therefore examine the effect of changing $s(z)$ in the next section."
],
[
"The role of the non-uniform expansion rate",
"The last effect that remains to be examined is related to $s(z)$ .",
"So far we have been applying the FLRW formula, but as shown in (REF ), to calculate it correctly we need to solve null geodesics.",
"We now address this point by studying light propagation within the LT Swiss Cheese model.",
"The advantage of the LT model is that it is an exact model, and so no approximations are required when modelling the light propagation, evolution of perturbations, or the Weyl curvature.",
"The disadvantage is that each particular inhomogeneity must be spherically symmetric, and we do not have the freedom to mimic the Millennium density profile.",
"Furthermore, we cannot describe virialised objects such as massive matter halos (we will come back to this point later on).",
"Figure: Upper left: Density along a random line of sightthrough the g 1.25 g_{1.25} density field of the Millennium simulation.Upper right:Density along a random line of sightthrough the halo model.Lower left:`Mild' Swiss Cheese model.Lower right:Density along a random line of sightthrough the highly non-linear Swiss Cheese model(solid line – 'u'-shaped pattern) and the contrast of the expansion rateδ H =H/H ¯-1\\delta _H = H/\\bar{H} - 1 (dotted line – note that in regions where δ\\delta is ofhigh amplitude, δ H \\delta _H is negative).Figure: PDF of Δ\\Delta in various models.The central dotted line is the PDF obtained within the `mild' Swiss Cheese model.The PDF of the highly non-linear LT Swiss Cheese modelis given by the solid line.",
"The dashed line is also the PDF obtainedin the non-linear LT Swiss Cheese model, but when we neglectthe effect of collapse on the redshift relation (see text for details).For comparison, the PDF for the g 10 g_{10} density map(from Fig.",
")is shown as the dashed-dotted line.Let us first consider a `mild' Swiss Cheese model, i.e.",
"one without large density fluctuations.",
"Each inhomogeneity has a radius of 20 Mpc, beyond which the system becomes homogeneous.",
"The density contrast inside the inhomogeneity is generated using the log-normal PDF$P(\\delta ) = \\frac{1}{\\sqrt{2 \\pi \\sigma _{nl}^2} }\\exp \\left[ - \\frac{ (\\ln (1+\\delta ) + \\sigma _{nl}^2/2)^2 }{2\\sigma _{nl}^2} \\right] \\frac{1}{1+\\delta }$ , where $\\sigma _{nl}^2 = \\ln [ 1 + \\sigma _R^2]$ , and $\\sigma _R^2 = \\frac{1}{2\\pi ^2} \\int \\limits _0^\\infty {\\rm d} k {\\cal P}(k) W^2(kR) k^2$ ., with $\\sigma _R = 0.96$ (i.e.",
"$R = 10$ Mpc).",
"An example of a random line of sight through this model is presented in the panel to the lower left of Fig.",
"REF .",
"Within this model, we solve the null geodesic equations using (REF ) to obtain $s(z)$ .",
"The Weyl focusing is calculated from (REF ) and the distance is calculated by solving (REF ).",
"In order to avoid effects of local structures, we place the observer and the source within the homogeneous regions.",
"The resulting PDF of $\\Delta $ for a source at $z=1.6$ is presented in Fig.",
"REF (dotted line).",
"The mean is $ 5.75 \\times 10^{-4}$ .",
"Qualitatively, the results are very similar to the ones obtained before: the PDF is skewed, with the maximum at the positive side of $\\Delta $ .",
"Quantitatively, however, the variance is much smaller than in previous cases.",
"This is because of the smaller amplitude of inhomogeneities along the line of sight, which can be seen by comparing the upper left (Millennium) and upper right (halo model) panels of Fig.",
"REF to the lower left panel of Fig.",
"REF (the `mild' Swiss Cheese case).",
"To bring us closer to the Millennium case, but in such a way as to include shearing and a modified $s(z)$ relation, we now consider a more `realistic' model, with large deep voids (with, for example, a present day radius of 14 Mpc and a density contrast of $-0.83$ ).",
"These voids have highly non-linear walls which, due to the high density contrast, are at the collapsing stage, clearly visible in the lower right panel of Fig.",
"REF (dotted line).",
"The PDF obtained within this model is presented in Fig.",
"REF (solid line).",
"The mean is $ -1.01 \\times 10^{-3}$ .",
"The first qualitative difference is that the maximum is now at negative $\\Delta $ (magnification).",
"Secondly, due to large fluctuations in the density field, the variance is larger, and is comparable with the $g_{10}$ Millennium density field (dotted-dashed line in Fig.",
"REF ).",
"The surprising result is that the maximum is on the magnification side ($\\Delta _{max}<0$ ).",
"Such a phenomenon has not been reported before.",
"Let us closely examine what could lead to such an unexpected behaviour.",
"Three working hypotheses can be put forward to explain this: (1) the effect of the Weyl focusing, (2) the problem with collapsing regions, (3) an artefact of the symmetry of the density contrast.",
"With regards to the shearing, we have seen in the previous section, when comparing the halo model with and without the Weyl focusing, that if $\\sigma \\ne 0$ , the PDF shifts to lower values of $\\Delta $ .",
"However, as seen from Fig.",
"REF , the role of the Weyl focusing for our Swiss Cheese models is small (cf.",
"[70] and [72] for a similar conclusion regarding the role of the Weyl focusing in this type of model).",
"Essentially, if the density contrast is not high (unlike in the halo model; compare the upper right panel of Fig.",
"REF with the lower right panel), the role of the Weyl focusing is small (compare the set of 3 solid lines with the set of 3 dashed or dotted lines in Fig.",
"REF ).",
"We therefore conclude that the shear cannot be responsible for the shift of $\\Delta _{max}$ from demagnification to the magnification side.",
"This can be confirmed by re-running the same model with $\\sigma $ set to zero – the resulting PDF is exactly the same as in the case of $\\sigma \\ne 0$ (solid line in Fig.",
"REF ).",
"Our second hypothesis involves considering the role of collapsing regions.",
"In this case, $H<0$ , which may strongly affect $s(z)$ .",
"In the real Universe, whenever the density field has a high amplitude, cosmic structures are virialised, and hence they are not collapsing.",
"Unfortunately, there is no rotation within the LT model, which could prevent the collapse.",
"Therefore, as seen in the lower right panel of Fig.",
"REF , the expansion is negative within the wall.",
"In order to `correct' the model for the virialisation effect, whenever the expansion rate is negative, we assume that the structure is already virialised and so the redshift does not change within this region.",
"That is, we assume that $ {\\rm d} k^0 / {\\rm d} s = 0$ in these regions, whereas in all other regions we use the exact formula, i.e.",
"$ {\\rm d} k^0 / {\\rm d} s = - R \\dot{R} (k^3)^2 - R^{\\prime }(\\dot{R}^{\\prime }/(1+2E)) (k^1)^2 $ .",
"We should exercise caution when analysing these results, as this `correction' was done by hand.",
"The fact that some regions are virialised, and hence $ {\\rm d} k^0 / {\\rm d} s = 0$ , is not a consequence of a model, and so there is clearly an issue of self-consistency.",
"The resulting PDF is presented using a dashed line in Fig.",
"REF , where we find that the maximum of the curve is shifted even further towards negative values of $\\Delta $ .",
"The mean is almost twice as large as before, $ -1.91 \\times 10^{-3}$ .",
"Clearly this does not explain why $\\Delta _{max} <0$ , and in fact it makes things even `worse'.",
"Our final hypothesis concerns the symmetry of the density perturbations.",
"We can already see that this cannot be the cause, as $\\Delta _{max} >0$ for the `mild' Swiss Cheese model.",
"Nevertheless, we return to the Millennium smoothed maps and perform further tests.",
"We proceed as before, with the only difference being that, when solving $s(z)$ , we no longer assume a uniform expansion rate as in (REF ), but instead assume that $\\frac{{\\rm d} z}{{\\rm d} s} = (1+z)^2 \\bar{H} (1+\\delta _H),$ where $\\bar{H}$ is the expansion rate of the background model.",
"In general $\\bar{H} \\delta _H = \\frac{1}{3} \\left({\\Theta - \\bar{\\Theta }}\\right)+ \\sigma _{ab} e^a e^b $ where $\\Theta $ is the matter scalar of the expansion and $\\sigma _{ab}$ is the shear of the matter velocity field.",
"We further work under the assumption of zero shear and so we only consider fluctuations in the expansion field, which in the linear regime are given by $\\delta _H = -\\frac{1}{3} f \\delta .$ Here, $f$ is the growth function, $f = \\dot{{\\cal G}}/({\\cal G}H)$ .",
"This method has thus 2 limitations, and works well as long as (1) the density contrast is not too high and (2) the matter shear is negligible.",
"We proceed further with this toy model, as it will bring more insight into results obtained within the exact LT model.",
"Naively, one can think of this as follows: because $\\langle {\\delta }\\rangle = 0 = \\langle {\\delta _H}\\rangle $ , we should expect (qualitatively) the same results as before, except for a potentially larger variance due to the presence of both $\\delta $ and $\\delta _H$ .",
"Our results are presented in Fig.",
"REF .",
"First of all, the variance changes only slightly.",
"Secondly, and most importantly, we find that $\\Delta _{max}<0$ .",
"In the $g_{10}$ case (with 10 Mpc smoothing scale), the results are strongly comparable with our non-linear Swiss Cheese model; for comparison we present its PDF in the upper left panel of Fig.",
"REF (dot-dashed line).",
"Also, the mean of $\\Delta $ is of the same order of magnitude as in the Swiss Cheese model, i.e.",
"$\\sim 10^{-3}$ — see Table REF .",
"Finally, as before, we try to `correct' the model for the effects of virialization.",
"As seen from (REF ), if $ \\delta > \\delta _T = \\frac{3}{f}, $ then such a region collapses.",
"In the $g_{1.25}$ case, almost $1.5\\%$ of regions have a density contrast above this threshold (see Table REF ).",
"As before, we set $ {\\rm if~} \\delta > \\delta _T \\quad \\Rightarrow \\quad \\delta _H = -1.$ In this case, if we calculate the average $\\delta _H$ , due to the fact that we neglect the most negative contributions, the average is no longer zero, and in the case of $g_{1.25}$ the average $\\delta _H$ is approximately $0.0211$ (it can only be zero if $\\delta _H < -1$ ).",
"The resulting PDFs are presented in Fig.",
"REF (dashed lines).",
"As in the case of the non-linear Swiss Cheese model before, the PDF shifts towards even more negative values of $\\Delta $ .",
"For detailed values of the means of the PDFs, see Table REF .",
"Table: The mean and standard deviation of Δ\\Delta for a source at z=1.6z=1.6.Table: Middle column: The percentage of regions within the Millennium simulationwhose density contrast is higher than the threshold above which δ H <-1\\delta _H < -1Right column: The average expansion rate aftersetting δ H =-1\\delta _H = -1 wherever δ H <-1\\delta _H<-1 .Proceeding further with the idea of a variable expansion rate and its impact on $s(z)$ , we also recalculate our results for the halo models that were considered previously.",
"Since halos are virialised, their interiors are set to $\\delta _H = -1$ .",
"The resulting PDF is presented in the right panel of Fig.",
"REF .",
"As was the case for the Millennium maps, a shift towards magnification is evident.",
"This should be compared with the mild Swiss Cheese model, or with the models which assumed a uniform expansion rate, where the PDF peaked on the positive side of $\\Delta $ .",
"Figure: The Weyl focusing (i.e.",
"the ratio of the shearing to present daymatter distribution σ 2 /ρ 0 \\sigma ^2/\\rho _0) in the different models that were considered.From bottom:the three dotted lines are 3 examples of the shearingin the mild LT Swiss Cheese model.",
"The shearingat z=1z=1 is of order 10 -6 -10 -4 10^{-6}-10^{-4}.Above those, the next 3 (dashed) linesare 3 examples of shearing in the highly non-linear Swiss Cheese model.Here, at z=1z=1, the shearing is of order 10 -2 10^{-2}.The next 3 (solid) lines show shearingin 3 examples of the halo model.",
"At z=1z=1,the shearing is of order 10 -1 10^{-1}.For comparison, the uppermost dash-dotted lineis (1+z) 3 (1+z)^3 (the evolution of matter density).Figure: Upper left: Millennium map g 10 g_{10} with δ H =0\\delta _H =0(dotted line, cf.",
"Fig.",
"),δ H ≠0\\delta _H \\ne 0 (solid line),and δ H ≥-1\\delta _H \\ge -1 (dashed line, in this case indistinguishable from the solid line).For comparison, the PDF of the non-linear Swiss Cheese model is shownas the dash-dotted line.Upper right:Millennium map g 5 g_{5} with δ H =0\\delta _H =0(dotted line),δ H ≠0\\delta _H \\ne 0 (solid line),and δ H ≥-1\\delta _H \\ge -1 (dashed line, hardly distinguishable from the solid line).Lower left:Millennium map g 2.5 g_{2.5} with δ H =0\\delta _H =0(dotted line),δ H ≠0\\delta _H \\ne 0 (solid line),and δ H ≥-1\\delta _H \\ge -1 (dashed line).Lower right: Millennium map g 1.25 g_{1.25} with δ H =0\\delta _H =0(dotted line),δ H ≠0\\delta _H \\ne 0 (solid line),and δ H ≥-1\\delta _H \\ge -1 (dashed line).Figure: PDF of Δ\\Delta .Solid line: mean.",
"Dotted line: solution of ().Dashed line: when () is used together with the assumptionthat z ˜=z\\tilde{z} = z.",
"Left: halo model, Right: g 1.25 g_{1.25} Millennium map.Figure: Average of the fluctuation of the expansion rateat constant zz.",
"Solid lineg 1.25 g_{1.25} Millennium map, dashed line for the halo model.In Secs.",
"and we assumed that $s(z)$ was the same as in the FLRW background model.",
"As a result, we found that $\\langle {\\Delta }\\rangle \\approx 10^{-4}$ .",
"For the Swiss Cheese model, we had variations in $s(z)$ with regards to the background, but still found that $\\langle {\\Delta }\\rangle \\approx 10^{-3}$ .",
"It was only when we started to `correct' our models for virialization that we found deviations of order of $10^{-2}$ (for the halo and $g_{1.25}$ models).",
"We now want to follow this up further, and test how the average of the distance follows the background's $s(z)$ .",
"Note that when we `force' some regions not to collapse (by setting $\\delta _H = -1$ whenever $\\delta _H < -1$ ), we change the average expansion history (see Table REF and Fig.",
"REF ), and thus also $s(z)$ .",
"This leads us to question: if we recover the average $s(z)$ correctly, will we obtain the correct mean of the distance, without the need for a detailed calculation of the Ricci and Weyl focusing?",
"We study this issue within the halo and $g_{1.25}$ models, as they have the largest deviation of $\\langle {\\Delta }\\rangle $ .",
"The PDFs obtained within these models are presented in Fig.",
"REF .",
"The mean of $\\Delta $ is shown as a vertical solid line.",
"The average of the expansion rate fluctuations calculated on a slice of constant $z$ is presented in Fig.",
"REF .",
"Before proceeding further, we must comment on one important aspect of our computations.",
"Cosmologists tend to use the redshift as a proxy for time, so they express the evolution of different fields in terms of the redshift, for example: $\\rho (s) \\Leftrightarrow \\rho (\\tilde{z}) = \\rho _0 (1+\\tilde{z})^3, \\nonumber \\\\H(s) \\Leftrightarrow H(\\tilde{z}) = H_0 \\sqrt{\\Omega _m (1+\\tilde{z})^3 + \\Omega _k (1+\\tilde{z})^2 + \\Omega _\\Lambda }.$ Only in the case of perfect homogeneity do we have that $\\tilde{z}$ (the proxy for time, i.e.",
"the redshift of a photon at a given time in a perfect homogeneous model) coincides with $z$ (the actual shift of the photon's frequency).",
"Thus, in solving the Sachs equations, one must solve the following: $&& \\frac{{\\rm d^2} D_A}{{\\rm d} s^2} = - \\frac{1}{2} \\langle {\\rho }\\rangle (1+z)^2 = \\rho _0 (1+\\tilde{z})^3 (1+z)^2 \\nonumber \\\\&& \\frac{{\\rm d} z}{{\\rm d} s} = (1+z)^2 \\langle {H}\\rangle = (1+z)^2 H(\\tilde{z}) (1+\\delta _H(z)) \\nonumber \\\\&& \\frac{{\\rm d} \\tilde{z}}{{\\rm d} s} = (1+\\tilde{z})^2 H(\\tilde{z}).$ We solve these, and plot the results as dotted lines in Fig.",
"REF .",
"As can be seen, the difference between the actual value, and that obtained from the above set of equations, is of order of $10^{-3}$ , (as before, where we set $s(z)$ to its value in the background $\\Lambda $ CDM model).",
"For comparison, we solve the above equation with $ z = \\tilde{z} \\quad \\Rightarrow \\quad R_{\\alpha \\beta } k^\\alpha k^\\beta = \\rho _0 (1+z)^5.",
"$ This time, the result (given by the dashed line) is significantly displaced to the left.",
"The reason for the shift towards higher magnification is that if $\\langle {\\delta _H}\\rangle > 0$ then, as follows from (REF ), we have that $z > \\tilde{z}$ , and so if we replace $\\tilde{z}$ with $z$ , we increase the Ricci focusing, hence higher magnification.",
"This simple exercise shows that if one is interested in having an average $D_A(z)$ without having to worry about the Ricci or Weyl focusing, one can simply use the average $s(z)$ of one's model.",
"Caution must be taken, though, when studying models in which $\\rho $ is not proportional to $(1+z)^3$ .",
"Such models include, for example, Gpc-scale inhomogeneous LT modelsIn fact the results of this paper provide a fresh view into different configurations of the LT model, which consist of giant voids [95], [96] and giant humps [97].",
"Within the giant voids $\\rho $ increases outwards, which results in the decrease of the expansion rate along the past null cone, hence the dimming of the supernova within the giant voids is caused by the change of the $s(z)$ relation.",
"Whereas for the giant hump models $\\rho $ decreases outwards making the Ricci focusing less efficient, and hence causing additional dimming.",
"and some backreaction models.",
"In this section we have tested all 3 factors that affect the relation between $z$ and $D_A$ via the Sachs equation: $\\rho (z)$ , $\\sigma (z)$ , and $s(z)$ .",
"In order to model them in a self-consistent manner, we used the LT Swiss Cheese model.",
"In all cases, we found that the distance does not change, on average, by more than $10^{-3}$ (at $z = 1.6$ ).",
"This result is fairly consistent with what has been found by other groups who have employed exact models, and ensured the proper randomization of light raysNote that for models where the distance correction is calculated using the Born approximation, have by the construction $\\langle {\\Delta }\\rangle =0$ [52], [59], [64], [72] (cf.",
"[47], [48], [56], [70], [74], [75]).",
"Our main motivation has been to follow, as closely as possible, the standard procedures to test self-consistency and the impact of inhomogeneities on the optical properties of the Universe within models that follow the FLRW evolution.",
"Nevertheless, it is not completely obvious that one should recover the FLRW results for these models, unless one completely agrees with Weinberg's reasoning.",
"If one does take Weinberg's results at face value, then our $10^{-3}$ results should be considered as a big deviation, about two orders of magnitude larger than what is allowed by the analogue of the Integrated Sachs Wolfe effect (which should be of order $10^{-5}$ ).",
"From one point of view, our results have turned out to be quite surprising.",
"In the case of large fluctuations in the density and expansion fields, the mode of $\\Delta $ shifts to the magnification side.",
"Such a behaviour has not been reported before.",
"The effect is only visible if the degree of inhomogeneity is large – for small amplitude Swiss Cheese models, we recover the standard results.",
"This can be explained using the same `voids argument' that we used in the case of the weak lensing approximation – since the structures in the Universe form a cosmic web, it is more likely for photons to propagate through voids (hence the position of the mode is associated with the amount of void regions).",
"As seen from (REF ), the weak lensing regime is only sensitive to density fluctuations, and so the mode is positive.",
"If we introduce fluctuations in the expansion rate, then the redshift decreases, as seen from (REF ).",
"Hence an object located at the same distance has a higher redshift, and thus gives the impression of being magnified compared to the homogeneous caseThe same fact can also be visualised in terms of the slope of the relation $(z,D_A)$ – when the redshift increases faster than in the homogeneous case, the slope decreases.",
"See, for example, Fig.",
"4 of [74], which shows the change of the slope in the relation $(z,D_A)$ , depending on fluctuations in the density and expansion rate..",
"This is the reason why the mode is now on the magnification side.",
"A valid point, and one worth further investigation, is related to the fact that, due to the virialisation of high density regions, the average expansion rate may deviate from the background value.",
"This was already pointed out in [98], [99] in the context of backreaction models, although as shown in [100], [101], the presence of matter shear may decrease the backreaction effect, leaving the evolution of the average quantities relatively unchanged.",
"Nevertheless, in such cases, the 3D volume average on surfaces of constant time may not coincide with the average on slices of constant $z$ .",
"As a result, as we have shown above, we would still expect a change in the redshift-distance relation.",
"However, we reiterate that care should be taken in interpreting these results – changing $H$ by hand may not lead to self consistent results (although keeping the expansion rate uniform may also turn out to be inconsistent).",
"To test this phenomenon more closely, we require exact models that allow for virialisation.",
"The LT models are definitely not suitable for such a study."
],
[
"Conclusions",
"The overarching question that we wish to address is: does the FLRW model, which correctly describes the evolution of the background, correctly recover (on average) the distance-redshift relation?",
"Our results, while not entirely conclusive, are suggestive.",
"We have explored a suite of models which are inhomogeneous, but close to the FLRW.",
"We find that, if the background model correctly describes the evolution of the average expansion rate and the average matter density, then indeed the homogeneous model is a good approximation.",
"Our results show that Ricci focusing alone introduces deviations from the average value.",
"The PDF is skewed and the maximum is on the demagnification side (a random object will most likely be dimmer than the average).",
"The change of the mean is negligible, which means that, on average, distances coincide with the expected distance in the background model.",
"The deviation from the average (for a particular line of sight) depends crucially on the smoothing scale and, unsurprisingly, the smaller the smoothing scale, the larger the deviation.",
"We find that Weyl focusing is important for objects with very high density contrast, and yet even in this case, the bias in the mean of $\\Delta $ is negligibly small.",
"We also find that the presence of large fluctuations in the expansion rate also affects the final results: there is a slight shift of the maximum of the PDF towards the magnification side (a random object will most likely be brighter than the average), and the variance changes.",
"In particular, we find that if the average expansion rate deviates from the background expansion rate, the distance (on average) changes by a few percent.",
"In the real Universe, such a situation may occur in the late non-linear stage of evolution, where large voids expand much faster than the background, while virialised over-dense regions do not expand at all (for these fluctuations to cancel each other, at some stage the presence of collapsing regions is required).",
"In summary, we have studied the various biases that cosmological inhomogeneities can introduce into the inferred optical properties of the Universe by considering a range of toy models.",
"There are, of course, limitations to the accuracy and self-consistency of these models, and how well they mimic the real Universe.",
"We find that the presence of inhomogeneities (around a fixed background) changes distances by at most a few percent, although the average is closely related to the background model.",
"The only possible deviation of the average of the distance from the distance predicted by the background model is in situations where the background model does not correctly describe the evolution of the average quantities.",
"The lessons we have learned can be summarized in the form of a message to our readers.",
"For cosmologists, we would like to point out that, with the increasing precision of cosmological observations, a change of few percent might soon become important for accurate estimation of cosmological parameters (cf.",
"[63], [67]).",
"In particular, the PDFs are skewed and hence, for a limited number of observational data, there might be a bias due to the fact that the mode does not coincide with the mean, or median.",
"For relativists, we would argue that, to study the effect of inhomogeneities on observations, it is first important to understand how much inhomogeneities affect the evolution of the background model.",
"Such a study should focus on $\\rho (z)$ and $H(z)$ , and especially on potential deviations from (REF ), as this has the biggest impact on the distance-redshift relation (see, for example, Fig.",
"REF for a few percent change in the case where $\\rho (z) \\nsim (1+z)^3$ (cf.",
"[68]).",
"Thus, pinning down $\\rho (z)$ and $H(z)$ will also provide the average of $D_A(z)$ , without the need to account for the Weyl focusing or to model complex aspects of the Ricci focusing).",
"We acknowledge discussion with Timothy Clifton, Sabino Mataresse, and Chris Clarkson.",
"We are particularly grateful to Phil Bull for a detailed reading of the manuscript and extensive discussions.",
"This research of was supported by the European Union Seventh Framework Programme under the Marie Curie Fellowship, grant no.",
"PIEF-GA-2009-252950, STFC, BIPAC, and the Oxford Martin School."
]
] | 1204.0909 |
[
[
"Fast Calculation of Calendar Time-, Age- and Duration Dependent Time at\n Risk in the Lexis Space"
],
[
"Abstract In epidemiology, the person-years method is broadly used to estimate the incidence rates of health related events.",
"This needs determination of time at risk stratified by period, age and sometimes by duration of disease or exposition.",
"The article describes a fast method for calculating the time at risk in two- or three-dimensional Lexis diagrams based on Siddon's algorithm."
],
[
"Lexis diagram and person-years method",
"In epidemiology, oftentimes relevant events or outcomes simultaneously depend on different time scales: age of the subjects, calendar time and duration of an irreversible disease.",
"In event history analysis, [5], a useful concept is the Lexis diagram, which is a co-ordinate system with axes calendar time $t$ (abscissa) and age $a$ (ordinate).",
"The calendar time dimension sometimes is referred to as period.",
"Each subject is represented by a line segment from time and age at entry to time and age at exit.",
"Entry and exit may be birth and death, respectively, or entry and exit in a epidemiological study or trial.",
"There are excellent and extensive introductions about the theory of Lexis diagrams (see for example [3], [4], [1] and references therein), which allows to be short here.",
"When it comes to irreversible diseases, the commonly used two-dimensional Lexis diagram with axes in time and age direction may be generalized to a three-dimensional co-ordinate system with disease duration $d$ represented by the applicate (z-axis).",
"If a subject does not get the disease during life time, the life line remains in the time-age-plane parallel to the line bisecting abscissa and ordinate.",
"With other words, the life line for the time without disease points in the $(1,1,0)$ direction (where the triple $(t,a,d)$ denotes the co-ordinates in time, age and duration direction, respectively).",
"However if at a certain point in time $E$ the disease is diagnosed, the life line changes its direction, henceforth pointing to $(1,1,1)$ .",
"The situation is illustrated in Figure REF .",
"The life lines of two subjects are shown in the three-dimensional Lexis space.",
"At time of birth (denoted $B_n, ~n=1,2$ ) both subjects are disease-free; both life lines go to the $(1,1,0)$ direction.",
"The first subject gets the disease at $E$ , and henceforth the life line is parallel to $(1,1,1)$ until death at $D_1$ .",
"The second subject remains without the disease for the whole life, which ends at $D_2$ .",
"Figure: Three-dimensional Lexis diagram with two life lines.",
"Abscissa,ordinate and applicate represent calendar time tt, age aa and duration dd, respectively.The life lines start and end at birth B n B_n and death D n ,n=1,2.D_n, ~n=1,2.The first subject gets the disease at EE.",
"Then, the life line changes its direction.The second subject does not get the disease, the corresponding life line remains inthe tt-aa-plane.In order to measure the frequency of events in a population, such as onset of a chronic disease, the person-years methods records the number of people who are affected and the time elapsed before the event occurs.",
"The person-years incidence rate $\\lambda $ is estimated by $\\lambda = \\frac{e}{m},$ where $e$ is is the number of events and $m$ is the number of person-years at risk [7].",
"Calendar time and age often are important determinants for occurrence of events and have to be taken into account.",
"This usually is achieved by dividing the subjects' time spent in the study into calendar time and age groups.",
"Let $e_{ij}$ be the number of events taking place while subjects are in time and age group $(i,j).$ Furthermore, let $m_{ij}$ be the total time at risk spent in this group, then Equation (REF ) becomes $\\lambda _{ij} = \\tfrac{e_{ij}}{m_{ij}}.$ In the planar Lexis diagram it is clear, how the time at risk $m_{ij}$ can be obtained.",
"Let the time and age group $(i, j)$ be defined by Cartesian product $S_{ij} := [t_{i-1}, t_{i}) \\times [a_{j-1}, a_{j})$ .",
"Each subject whose life line intersects with the rectangle $S_{ij}$ contributes by its time at risk in $S_{ij}$ .",
"To be precise, $m_{ij}$ is the sum of all the subjects' times at risk spent in $S_{ij}$ : $m_{ij} = \\sum _{n=1}^N \\ell ^{(n)}_{ij},$ where $\\ell ^{(n)}_{ij}$ is the time at risk of subject $n, ~n=1, \\dots , N,$ in the rectangle $S_{ij}.$ Again, these ideas can be generalized to three-dimensional case: Then, $e_{ijk}$ is the number of events taking place in the rectangular hexahedron $S_{ijk} := [t_{i-1}, t_{i}) \\times [a_{j-1}, a_{j}) \\times [d_{k-1}, d_{k}).$ For the times at risk $m_{ijk}$ it holds $m_{ijk} = \\sum _{n=1}^N \\ell ^{(n)}_{ijk},$ where $\\ell ^{(n)}_{ijk}$ is the time at risk of subject $n$ in volume element $S_{ijk}.$ Given a certain study population of size $N$ , the question arises how the subjects' contributions $\\ell ^{(n)}_{ijk}$ to the overall time at risk $m_{ijk}$ spent in $S_{ijk}$ can be calculated.",
"Since $N$ may be large (up to several thousand), some attention should be paid to computation time.",
"The solution is straightforward by noting that the question is very similar to the problem of following a radiological path through a voxel grid in tomography or raytracing in computer graphics.",
"For both fields, tomography and raytracing, there is an ongoing research effort to efficiently discretize continuous lines (radiological paths or rays of light).",
"This article has been inspired by the seminal work of Siddon, [6]."
],
[
"Intersecting life lines with voxels in the Lexis diagram",
"Since the algorithm presented in this section is motivated from the field of computer tomography, some of the terminology is useful.",
"Typically one of the sets $S_{ijk}$ resulting from a partition of a rectangular hexahedron (right cuboid) into congruent volume elements, is called a voxel.",
"The six faces of each voxel are subsets of two adjacent planes parallel either to the $t$ -$a$ -plane, $a$ -$d$ -plane or $t$ -$d$ -plane.",
"Hence, the voxel space comes along with a set of equidistant, parallel planes which are perpendicular to the abscissa, ordinate or applicate and which are defined by the union of all voxel faces.",
"These planes play a crucial role in the algorithm.",
"In this article all voxels $S_{ijk}$ are considered to be cubical, with all edges having the length $t_r, ~t_r > 0:$ $S_{ijk} := \\left[ t_r \\cdot \\left(i-1 \\right), ~t_r \\cdot i \\right) \\times \\left[ t_r \\cdot \\left(j-1 \\right), ~t_r \\cdot j \\right) \\times \\left[ t_r \\cdot \\left(k-1 \\right), ~t_r \\cdot k \\right).$ These voxels form a grid where the life lines of all subjects in the study are sorted into.",
"As a consequence of cubical voxels, the temporal resolution with respect to calendar time, age and duration is the same.",
"However, generalization to partitions usings rectangular voxels with height, length and depth being different is easily possible.",
"The main idea for calculating the $\\ell ^{(n)}_{ijk}$ in the life line ${\\cal L}_n$ of subject $n$ starting at entry point $B_n := (t^{(n)}_0, a^{(n)}_0, d^{(n)}_0)$ , ending at exit point $D_n := (t^{(n)}_1, a^{(n)}_1, d^{(n)}_1)$ , is the parameterization in the form ${\\cal L}_n: ~B_n + \\alpha \\cdot (D_n - B_n), ~\\alpha \\in [0, 1].$ Note, that $t^{(n)}_1 - t^{(n)}_0 = a^{(n)}_1 - a^{(n)}_0 = d^{(n)}_1 - d^{(n)}_0 =:\\Delta t^{(n)}.$ Using this parameterization, all parameters $\\alpha ^{(n)} \\in [0, 1]$ are calculated where an intersection with a voxel face takes place.",
"Since the voxels are arranged in a regular grid, intersecting one of the voxel faces is equivalent with intersecting one of the $t$ -$a$ -, $a$ -$d$ - or $t$ -$d$ -planes formed by the union of all voxel faces mentioned above.",
"Hence, we calculate the intersections with these planes.",
"Let us start with the $a$ -$d$ -planes (perpendicular to the $t$ -axis): all those $\\alpha ^{(n)}_t$ where an intersection with an $a$ -$d$ -plane occurs are given by $\\alpha ^{(n)}_t(u) = \\frac{u \\cdot t_r - (t^{(n)}_0 ~\\% ~t_r)}{\\Delta t^{(n)}}, ~u = 1, \\dots , U^{(n)},$ where $\\%$ is the modulo-operator and $U^{(n)}$ denotes the number of intersected $a$ -$d$ -planes: $U^{(n)} = \\left\\lfloor {t^{(n)}_1}{t_r} \\right\\rfloor - \\left\\lfloor {t^{(n)}_0}{t_r} \\right\\rfloor .$ Similar formulas hold for those $\\alpha ^{(n)}_a(v), ~v=1, \\dots , V^{(n)},$ and $\\alpha ^{(n)}_d(w), ~w=1, \\dots , W^{(n)},$ where ${\\cal L}_n$ intersects the $t$ -$d$ - or $t$ -$a$ -planes, respectively.",
"Now define the set $A_n := & & \\lbrace \\alpha ^{(n)}_t(u)~\\vert ~u = 1, \\dots , ~U^{(n)} \\rbrace \\nonumber \\\\&\\cup & \\lbrace \\alpha ^{(n)}_a(v)~\\vert ~v = 1, \\dots , ~V^{(n)} \\rbrace \\\\&\\cup & \\lbrace \\alpha ^{(n)}_d(w)~\\vert ~w = 1, \\dots , W^{(n)} \\rbrace \\nonumber ,$ which contains those $\\alpha ^{(n)} \\in [0, 1]$ where an intersection occurs.",
"Note that the three sets on the right-hand side of Equation (REF ) are not necessarily disjoint.",
"Multiple values occur if an intersection happens to be on an edge or vertex of a voxel.",
"Let $A^\\star _n := A_n \\cup \\lbrace 0, ~1\\rbrace $ be ordered ascendingly $A^\\star _n = \\lbrace \\alpha ^{(n)}(p) ~\\vert ~p = 1, \\dots , P^{(n)} \\rbrace $ with $0 = \\alpha ^{(n)}(1) < \\dots < \\alpha ^{(n)}(P^{(n)}) = 1.$ For calculating the $\\ell ^{(n)}$ and the associated voxel indices $i, j, k,$ we have following algorithm: For each subject $n, ~n=1, \\dots , N,$ calculate the set $A^\\star _n$ as above and sort the elements $\\alpha ^{(n)}(p), ~p = 1, \\dots , P^{(n)},$ in ascending order.",
"For $p= 1, \\dots , P^{(n)}$ set $(i_p, j_p, k_p) := \\left\\lfloor \\frac{B_n + \\alpha ^{(n)}(p) \\cdot (D_n - B_n)}{t_r} \\right\\rfloor ,$ where the division is taken componentwise.",
"Then, calculate $\\ell ^{(n)}_{i_p j_p k_p} = \\left( \\alpha ^{(n)}(p+1) -\\alpha ^{(n)}(p) \\right) \\cdot \\Delta t^{(n)}, ~p = 1, \\dots , P^{(n)} - 1.$ For each voxel $(i, j, k)$ summing up all the times $\\ell ^{(n)}_{ijk}, ~n=1, \\dots , N,$ by Equation (REF ) yields the time at risk $m_{ijk}$ in period-, age- and duration class $(i, j, k),$ which can be used in the person-years method.",
"The idea of calculating the intersection points with the voxel faces goes back to Robert L. Siddon.",
"The algorithm proposed by Siddon has been developed for raytracing in tomography, where several millions of paths have to be computed to form a radiological image.",
"While the implementation provided with this article is not tuned for efficiency, remarkable speeding up is possible [2].",
"The execution of 500 runs of an R (The R Foundation for Statistical Computing) implementation of the algorithm with the data of 200 patients (equivalent to a total of $10^5$ patients) on a 2.6 GHz personal computer takes 92 seconds.",
"The (simulated) patient data set is described in more detail in the next section."
],
[
"Examples",
"An implemtentation of the method in this article has been tested with simulated patient data.",
"A study population of 200 subjects with entry age 55 to 85 born between 0 and 15 (TUs) suffering from a chronic disease for 3 to 15 TUs at is the time of entry is assumed to have the mortality rate $m(a, d) = \\exp (-10 + 0.1 \\cdot a) \\cdot (1 + 0.1\\cdot d).$ Exit from the study is assumed to be only due to death (no censoring).",
"The aim is to unfold the mortality rate from these patient data.",
"For setting up the data set, following code has been used: for(patNr in 1:200){ thisPatInAge <- runif(1, 55, 80) thisPatBirth <- runif(1, 0, 15) thisPatDuration <- runif(1, 3, 15) F <- fct_F(thisPatInAge, thisPatDuration) thisPatDeathAge <- round( which.min(runif(1) > F) + thisPatInAge + (runif(1) - 0.5), 3) patMatrix[patNr, ] <- c(thisPatBirth, thisPatInAge, thisPatDuration, thisPatDeathAge) } For the simulation of the age of death, inverse transform sampling is used: (which.min(runif(1) > F)).",
"Therefor the cumulative distribution function $F(t ~\\vert ~a_0, d_0)$ for someone who enters the study at age $a_0$ and having got the disease for $d_0$ TUs is calculated in the function fct_F: $F(t ~\\vert ~a_0, d_0) = 1 - \\exp \\left( - \\int _0^t m(a_0 + \\tau , d_0+\\tau ) \\mathrm {d} \\tau \\right).$ If the alorithm is applied to this data set with $t_r = 5,$ it is possible to estimate the mortality rate $m(a, d)$ by the person-years method.",
"The result is shown in Figure REF .",
"Figure: Age- and duration specific mortality as estimated by the person-yearsmethod."
],
[
"Conclusion",
"This article is about an extension of the person-years when period-, age- and duration-effects occur.",
"The method to calculate the time at risk is based on raytracing techniques used in tomography and provides a fast way to follow the individual life lines of subjects in the Lexis diagram.",
"The algorithm is able to treat two cases calculate the time at risk for newly incident cases, and calculate the time at risk, where the time elapsed after an event (e.g.",
"duration since onset of a disease) is relevant.",
"In the first case, the life lines are located in the $t$ -$a$ -plane, in the second the life lines are pointing to the $(1,1,1)$ direction.",
"The later case is important, when the duration is a covariable.",
"This might be the case in late sequalae or mortality after having got a disease.",
"Similarly, by interpreting the applicate axis (z-axis) as duration of exposure to a risk factor, the time at risk depending on period, age and duration of exposure can be calculated.",
"In large populations or register data ($\\log _{10} N > 5$ , times at risk are usually estimated by a formula going back to Sverdrup, [1].",
"It is noted that using the method described in this article, estimation is no longer necessary, because given entry and exit times of the subjects, calculation of times at risk is possible at feasible computational expense."
]
] | 1204.0798 |
[
[
"An explicit bijection between semistandard tableaux and non-elliptic\n sl_3 webs"
],
[
"Abstract The sl_3 spider is a diagrammatic category used to study the representation theory of the quantum group U_q(sl_3).",
"The morphisms in this category are generated by a basis of non-elliptic webs.",
"Khovanov- Kuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux.",
"They establish this bijection via a recursive growth algorithm.",
"Recently, Tymoczko gave a simple version of this bijection in the case that the tableaux are standard and used it to study rotation and joins of webs.",
"We build on Tymoczko's bijection to give a simple and explicit algorithm for constructing all non-elliptic sl_3 webs."
],
[
"Introduction",
"The $sl_3$ spider, introduced by Kuperberg [10] and subsequently studied by many others [8], [9], [11], [12], is a diagrammatic, braided monoidal category encoding the representation theory of $U_q(sl_3)$ .",
"The objects in this category, called sign strings, are finite words in the alphabet $\\lbrace +,-\\rbrace $ including the empty word.",
"The morphisms are $\\mathbb {Z}[q, q^{-1}]$ - linear combinations of certain graphs called webs.",
"See Figure REF for an example of a web.",
"The objects in the spider can be thought of as tensor products of the two dual 3-dimensional irreducible representations $V^{+}$ and $V^{-}$ of $U_q(sl_3)$ , and the morphisms can be thought of as intertwining maps between tensor products of these representations [10].",
"Spider categories for other Lie types have been defined.",
"See for instance [2], [11], [12].",
"Figure: A web in Hom(++++,+)\\textup {Hom}(++++,+).Webs in the $sl_3$ spider are oriented trivalent graphs drawn in a rectangular region with boundary points lying on the top and bottom edges of that region.",
"Edges incident on the boundary points have orientations compatible with the source and target sign strings.",
"We read webs from bottom to top.",
"All vertices are either sources or sinks.",
"Webs are also subject to Relations REF , REF , and REF below which are often referred to as the circle, bigon, and square relations respectively.",
"A web with no bigons, squares, or circles is called non-elliptic or irreducible.",
"Every web is a linear combination of non-elliptic webs.",
"We follow the normalization conventions found in Khovanov's work on $sl_3$ link homology [7].",
"$ \\raisebox {1pt}{\\begin{tikzpicture}[baseline=0cm, scale=0.5][style=thick,->] (0,0) arc (0:180:1cm);[style=thick](-2,0) arc (180:360:1cm);\\end{tikzpicture}} = [3]_q = q^2 + 1 + q^{-2}$ $\\raisebox {-5pt}{\\begin{tikzpicture}[baseline=0cm, scale=0.5][style=thick, -<] (0,1) -- (0,1.5);[style=thick](0,1.5)--(0,2);[style=thick](0,-1)--(0,-.5);[style=thick, ->](0,0)--(0,-.5);[style=thick,->](0,0) .. controls (-.5,.1) and (-.5,.35).. (-.5,.5);[style=thick](-.5,.5) .. controls (-.5,.65) and (-.5,.9).. (0,1);[style=thick,->](0,0) .. controls (.5,.1) and (.5,.35).. (.5,.5);[style=thick](.5,.5) .. controls (.5,.65) and (.5,.9).. (0,1);[radius=.08, fill=black](0,0)circle;[radius=.08, fill=black](0,1)circle;\\end{tikzpicture}}= [2]_q \\; \\;\\raisebox {-33pt}{\\begin{tikzpicture}[baseline=0cm, scale=0.5][style=thick, -<] (0,1) -- (0,2.5);[style=thick](0,2.5)--(0,4);\\end{tikzpicture}} = (q+q^{-1}) \\;\\; \\raisebox {-33pt}{\\begin{tikzpicture}[baseline=0cm, scale=0.5][style=thick, -<] (0,1) -- (0,2.5);[style=thick](0,2.5)--(0,4);\\end{tikzpicture}}$ $\\raisebox {-18pt}{\\begin{tikzpicture}[baseline=0cm, scale=0.5][style=thick, -<] (0,1) -- (0,1.5);[style=thick](0,1.5)--(0,2);[style=thick,->](0,2)--(.5,2);[style=thick](.5,2)--(1,2);[style=thick, ->](1,1)--(1,1.5);[style=thick](1,1.5)--(1,2);[style=thick, ->](1,1)--(.5,1);[style=thick](.5,1)--(0,1);[style=thick,-<](1,2) -- (1.25,2.25);[style=thick](1.25,2.25)--(1.5,2.5);[style=thick,-<](0,1) -- (-.25, .75);[style=thick](-.25, .75)--(-.5, .5);[style=thick,->](1,1) -- (1.25,.75);[style=thick](1.25,.75)--(1.5,.5);[style=thick,->](0,2) -- (-.25,2.25);[style=thick](-.25,2.25)--(-.5,2.5);[radius=.08, fill=black](0,1)circle;[radius=.08, fill=black](0,2)circle;[radius=.08, fill=black](1,2)circle;[radius=.08, fill=black](1,1)circle;\\end{tikzpicture}}$ = [baseline=0cm, scale=0.9] [xshift=1.5cm, style=thick,-<](-.25,0) .. controls (-.5,.2) and (-.5,.5).. (-.5,.5); [xshift=1.5cm, style=thick](-.5,.5) .. controls (-.5,.45) and (-.5,.9).. (-.25,1); [style=thick,->](.25,0) .. controls (.5,.2) and (.5,.5).. (.5,.5); [style=thick](.5,.5) .. controls (.5,.45) and (.5,.9).. (.25,1); + [baseline=0cm, scale=0.9, rotate=90] [xshift=1.5cm, style=thick,](-.25,0) .. controls (-.5,.2) and (-.5,.5).. (-.5,.5); [xshift=1.5cm, style=thick](-.5,.5) .. controls (-.5,.45) and (-.5,.9).. (-.25,1); [xshift=1.5cm,style=thick](-.5,.5)–(-.6,.4); [xshift=1.5cm,style=thick](-.5,.5)–(-.4,.4); [style=thick,](.25,0) .. controls (.5,.2) and (.5,.5).. (.5,.5); [style=thick](.5,.5) .. controls (.5,.45) and (.5,.9).. (.25,1); [style=thick](.5,.5)–(.6,.6); [style=thick](.5,.5)–(.4,.6); Given a sign string $s$ , construct the dual string $s^*$ of $s$ by reversing the order of $s$ and then replacing each $+$ with a $-$ and each $-$ with a $+$ .",
"This is really just a diagrammatic version of the statement that, for quantum group representations, $(V\\otimes W)^* \\cong W^* \\otimes V^*$ .",
"Let $\\textup {Inv}(V)$ be the space of invariant tensors of $V$ where $V$ is a tensor product of irreducible representations of $U_q(sl_3)$ .",
"Since $\\textup {Hom}(V,W) \\cong \\textup {Inv}(V^*, W)$ , it is enough to study $sl_3$ webs of the form $\\textup {Hom}(s, \\emptyset )$ .",
"Let $s=s_1 \\ldots s_n$ be a sign string.",
"The dimension of $\\textup {Inv}(V^{s_1}\\otimes \\cdots \\otimes V^{s_n})$ is the number of lattice paths in the dominant Weyl chamber from the origin to itself satisfying some additional condition coming from the string [8].",
"These dominant lattice paths for $s$ correspond to certain words in the alphabet $\\lbrace -1, 0, +1\\rbrace $ .",
"Khovanov-Kuperberg give a recursive growth algorithm which produces a non-elliptic web from a given lattice path word.",
"This growth algorithm establishes a bijection between dominant lattice paths and webs with inverse coming from a depth map on webs [8].",
"Recall that a semistandard Young tableau is a filling of a Young diagram which strictly increases in columns and weakly increases in rows.",
"The dimension of the invariant tensor space is reformulated by Petersen-Pylyavskyy-Rhoades using the language of semistandard tableaux [13].",
"Proposition 1 Let $s$ be a sign string of length $3n$ with $k$ minuses and $3n-k$ pluses.",
"The number of non-elliptic webs in $Hom(s, \\emptyset )$ is equal to the number of semistandard tableaux of shape $(3, 3, \\ldots , 3, 3) \\vdash 3n$ filled with $\\lbrace 1^2, \\ldots , k^2, k+1, \\ldots , 3n-k\\rbrace $ .",
"Tymoczko recently gave an explicit bijection between webs in $Hom(+++ \\ldots +++, \\emptyset )$ and standard tableaux [20].",
"This is accomplished by constructing an intermediate object called an $m$ -diagram which can then be modified slightly to produce a non-elliptic web.",
"Tymoczko shows that this straightforward procedure provides a concrete realization of the growth algorithm bijection of Khovanov-Kuperberg.",
"Building on the $m$ -diagram algorithm, this paper provides a simple bijection between all non-elliptic webs and a certain subset of semistandard Young tableaux.",
"We begin by recalling Tymoczko's $m$ -diagram algorithm and then describe the generalized bijection providing many examples.",
"We conclude with two theorems about rotation and join of webs that generalize results of Petersen-Pylyavskyy-Rhoades and Tymoczko to all $sl_3$ webs [13], [20].",
"An interesting potential application of this bijection is in the study of Spaltenstein varieties.",
"Combinatorial data from $sl_2$ webs has been used to describe the representation theory and topological structure of Springer varieties, certain flag varietyies used to construct irreducible representations of the symmetric group [3], [4], [6], [15], [14], [16], [19].",
"Spaltenstein varieties are a generalization of Springer varieties using partial flags.",
"Just as the components of Springer varieties are indexed by standard tableaux, the components of Spaltenstein varieties are indexed by semistandard tableaux.",
"Recent work of Brundan-Ostrik and Schäfer show strong evidence that the combinatorics of the more general class of webs studied here should aid in the study of three-row Spaltenstein varieties [1], [18]."
],
[
"Acknowledgements",
"We are very grateful to Julianna Tymoczko for suggesting this project and for many enlightening conversations.",
"Thanks also to Matt Housley for helpful discussions about $\\tau $ sets.",
"We also wanted to acknowledge Dongho Moon who has recently obtained similar results."
],
[
"Tymoczko's m-diagram algorithm",
"Let $n\\in \\mathbb {N}$ and consider the partition $(n,n,n) \\vdash 3n$ .",
"Let $T$ be a standard tableau of shape $(n,n,n)$ .",
"The bijection in this section is between tableaux $T$ and webs with $3n$ source vertices.",
"Note that the number of standard fillings of shape $(n,n,n)$ is the same as the number of standard fillings of the shape $(3, 3, \\ldots , 3, 3)$ , so this can also be thought of as a bijection with tableaux of that shape.",
"Given a tableau $T$ , the Tymoczko $m$ -diagram algorithm constructs the $m$ -diagram $m_T$ as follows [20].",
"Draw a horizontal line with $3n$ equally spaced dots labeled from left to right with the numbers $1, \\ldots , 3n$ .",
"This line forms the lower boundary for the diagram, and all arcs will lie above this line.",
"Starting with the smallest number $j$ on the second row, draw a semi-circular arc connecting $j$ to its nearest unoccupied neighbor $i$ to the left.",
"The arcs $(i,j)$ are the left arcs in the $m$ -diagram.",
"Starting with the smallest number $k$ on the bottom row, draw a semi-circular arc connecting $k$ to its nearest neighbor $j$ to the left that does not already have an arc coming to it from the left.",
"The arcs $(j,k)$ are the right arcs of the $m$ -diagram.",
"The collection of left arcs is nonintersecting as is the collection of right arcs, but left arcs can intersect right arcs.",
"Figure REF has an example of an $m$ -diagram.",
"Figure: The mm-diagram for a tableau.From an $m$ -diagram $m_T$ for $T$ there is a straightforward procedure for transforming $m_T$ into a non-elliptic web $w_T$ [20].",
"At each boundary vertex where two semi-circular arcs meet, replace the portion of the diagram in a small neighborhood of the vertex with a `Y` shape as shown in Figure REF .",
"Orient all arcs away from the boundary so that the branching point of each `Y` becomes a source.",
"Finally replace any 4-valent intersection point of a left arc and a right arc with a pair of trivalent vertices as shown in Figure REF .",
"There is a unique way to do this preserving orientation of incoming arcs.",
"Figure: Modifying the middle vertex of an mm.Figure: Replacing a 4-valent vertex with trivalent vertices.For each face of a web $w$ , define its depth to be the minimal number of times a path from the given face to the unbounded region must intersect $w$ .",
"An example is shown in Figure REF .",
"Depths of adjacent faces differ by at most one.",
"Figure: The depth map for a webLet $F_{i,L}$ be the face immediately to the left of the edge incident on boundary vertex $i$ , and let $F_{i,R}$ be the face immediately to the right of the edge incident on $i$ .",
"The following algorithm constructs a standard tableaux $T_w$ from a non-elliptic web $w$ with boundary $3n$ sources.",
"In fact this process is inverse to Tymoczko's web bijection in the sense that $T_{w_T} = T$ [20].",
"If the depth of $F_{i,L}$ is less than the depth of $F_{i,R}$ , put $i$ in the top row of $T_w$ .",
"If the depths of $F_{i,L}$ and $F_{i,R}$ are the same, put $i$ in the middle row of $T_w$ .",
"If the depth of $F_{i,L}$ is greater than the depth of $F_{i,R}$ , put $i$ in the bottom row of $T_w$ .",
"Let $\\tau (T)$ be the set of all pairs $(i, i+1)$ such that $i$ occurs in a row above $i+1$ in $T$ .",
"The terminology of $\\tau $ comes from the work of Vogan on primitive spectra of semi simple Lie algebras [21].",
"The $\\tau $ set is also often called the descent set of a tableau.",
"Lemma REF appears in an upcoming paper of the author with Housley and Tymoczko where we study the symmetric group action on $sl_3$ webs with $3n$ sources [5].",
"It is the key idea in establishing a bijection between semistandard tableaux and webs.",
"Lemma 1 Given a standard tableau $T$ and its associated web $w_T$ , if $(i,i+1)\\in \\tau (T)$ then boundary vertices $i$ and $i+1$ are connected to the same internal vertex in $w_T$ .",
"Say $(i,i+1)\\in \\tau (T)$ .",
"Since $T$ has three rows, there are three possibilities: $i$ is in the top row, and $i+1$ is in the middle row.",
"$i$ is in the middle row, and $i+1$ is in the bottom row.",
"$i$ is in the top row, and $i+1$ is in the bottom row.",
"In the first case, the boundary vertices $i$ and $i+1$ must be connected by the left arc of an $m$ .",
"If this were not the case, then the $m$ -diagram for $w_T$ would have two left arcs crossing, which cannot happen.",
"Since $i$ and $i+1$ are adjacent and connected by the arc of an $m$ , they will be connected to the same internal vertex in $w_T$ .",
"The second case is completely analogous to the first except that $i$ and $i+1$ are connected by the right arc of an $m$ .",
"This once again means that they connect to the same internal vertex in $w_T$ .",
"In the third case $i$ is at the far left of an $m$ , and $i+1$ is at the far right of an $m$ .",
"Since there are no external vertices between them, they must cross exactly in the manner shown in Figure REF .",
"This means that in $w_T$ , vertices $i$ and $i+1$ will connect to the same internal vertex.",
"Figure: Case 3: Vertices ii and i+1i+1 in the mm-diagram and web."
],
[
"A bijection between semistandard tableaux and webs",
"Given a semistandard tableau $T$ , the content $\\lambda $ of $T$ is the composition where $\\lambda _i$ is the number of occurrences of $i$ in $T$ .",
"Given a sign string $s=s_1 \\cdots s_n$ with $k$ minuses define the content of $s$ to be the composition $\\lambda _s = (\\lambda _{s,1}, \\ldots , \\lambda _{s,{3n-k}})$ where $\\lambda _{s,i} = \\left\\lbrace \\begin{array}{lr}1 & \\textup {if } s_i=+,\\\\2 & \\textup {if } s_i=-.\\end{array}\\right.$ This section studies semistandard fillings of $(3,\\ldots , 3) \\vdash 3n$ of content $\\lambda _s$ which we will refer to as fillings of content $s$ .",
"For example, the first tableau in Figure REF has content $s=--+++++$ .",
"Note that the number of fillings of content $s$ depends only on the number of pluses and minuses and not on the order in which these symbols appear.",
"The bijection described in this section works for all sign strings.",
"For ease of notation, we provide explicit instructions for the case that $s = - \\cdots - + \\cdots +$ .",
"The general case is similar.",
"An example is given at the end of Section .",
"Let $s = - \\cdots - + \\cdots +$ be a sign string consisting of $k$ minuses followed by $3n-k$ pluses.",
"Let $T_s$ be a filling of $(3,\\ldots , 3) \\vdash 3n$ of content $s$ .",
"From $T_s$ , construct a standard tableau $\\widetilde{T_s}$ by replacing each repeated pair $i, i$ with the numbers $2i-1, 2i$ such that $2i-1$ is to the left of $2i$ ; for $i>k$ , replace $i$ with $i+k$ .",
"Figure REF has an example.",
"For any tableau $T$ , write $T^{\\prime }$ for its conjugate.",
"Figure: Obtaining a standard from a semistandard and taking its conjugate.Lemma 2 Let $T_s$ be a tableau of content $s$ .",
"Then $(2i-1,2i)\\in \\tau (\\widetilde{T_s}^{\\prime })$ for all $1\\le i\\le k$ .",
"Since $T_s$ is semistandard, the repeated pairs $i,i$ can never be in the same column of $T_s$ .",
"To construct $\\widetilde{T_s}$ the leftmost instance of $i$ is replaced with $2i-1$ , and the rightmost instance is replaced with $2i$ .",
"This means that $2i-1$ will always lie in a row above $2i$ in the conjugate tableau $\\widetilde{T_s}^{\\prime }$ .",
"Let $w_{\\widetilde{T_s}^{\\prime }}$ be the non-elliptic web constructed from $\\widetilde{T_s}^{\\prime }$ using the $m$ -diagram algorithm in Section .",
"Let $w_{T_s}$ be the web formed by contracting the first $2k$ boundary edges of $w_{\\widetilde{T_s}^{\\prime }}$ as shown in Figure REF .",
"Figure: Contracting boundary edges to produce a sink vertex.Lemma 3 The sign string associated to the boundary of $w_{T_s}$ is $s$ .",
"Since the construction of $w_{T_s}$ leaves the last $3n-k$ boundary vertices undisturbed, it is clear that they will be sources since they were sources in $w_{\\widetilde{T_s}^{\\prime }}$ .",
"Since $(2i-1, 2i)\\in \\tau (\\widetilde{T_s}^{\\prime })$ for all $1\\le i\\le k$ , vertices $2i-1$ and $2i$ are connected to the same internal vertex.",
"When we contract the boundary edges incident on vertices $2i-1$ and $2i$ , the internal vertex they share becomes a new boundary vertex.",
"This vertex is a sink.",
"Thus, each pair of vertices $2i-1, 2i$ is replaced with a single sink vertex, and the boundary of $w_{T_s}$ consists of $k$ sinks followed by $3n-k$ sources as desired.",
"Lemma 4 The web $w_{T_s}$ is non-elliptic.",
"Any closed face of $w_{T_s}$ is also a closed face of $w_{\\widetilde{T_s}^{\\prime }}$ .",
"Since $w_{\\widetilde{T_s}^{\\prime }}$ is non-elliptic, it follows that $w_{T_s}$ is also non-elliptic.",
"Lemma 5 Given two different fillings $T_{s,1}$ and $T_{s,2}$ of content $s$ , the webs $w_{T_{s,1}}$ and $w_{T_{s,2}}$ are distinct.",
"If $T_{s,1}$ and $T_{s,2}$ differ on some repeated number $i$ where $1\\le i \\le k$ then the standard tableaux $\\widetilde{T_{s,1}}$ and $\\widetilde{T_{s,2}}$ will have at least one of the pair $2i-1, 2i$ in different positions.",
"If $T_{s,1}$ and $T_{s,2}$ differ on some unrepeated number $i$ where $k<i\\le 3n-k$ then the number $i+k$ will be in different positions in $\\widetilde{T_{s,1}}$ and $\\widetilde{T_{s,2}}$ .",
"In either case $\\widetilde{T_{s,1}}$ and $\\widetilde{T_{s,2}}$ are distinct.",
"Since $\\widetilde{T_{s,1}}$ and $\\widetilde{T_{s,2}}$ must be distinct, it follows that $w_{\\widetilde{T_{s,1}}^{\\prime }}$ and $w_{\\widetilde{T_{s,2}}^{\\prime }}$ are distinct webs with $3n$ sources.",
"The portions of $w_{\\widetilde{T_{s,1}}^{\\prime }}$ and $w_{\\widetilde{T_{s,2}}^{\\prime }}$ that are contracted to form $w_{T_{s,1}}$ and $w_{T_{s,2}}$ are identical according to Lemma REF .",
"Therefore $w_{\\widetilde{T_{s,1}}^{\\prime }}$ and $w_{\\widetilde{T_{s,2}}^{\\prime }}$ must differ away from the first $2k$ boundary edges which means that $w_{T_{s,1}}$ and $w_{T_{s,2}}$ must be distinct as well.",
"Theorem 1 The map sending fillings $T_s$ of content $s$ to webs $w_{T_s}$ with boundary $s$ is a bijection.",
"The previous lemmas show that this map sends distinct semistandard tableaux of content $s$ to distinct webs with boundary $s$ .",
"Since these two sets are in bijection, it follows that this map gives a bijective correspondence.",
"Given a web $w$ with boundary $s$ , construct a tableau ${T_{s_w}}$ of content $s$ as follows.",
"If the depth of $F_{i,L}$ is less than the depth of $F_{i,R}$ put the pair $i,i$ in the first and second column if vertex $i$ of $w$ is a sink and $i$ in the first column if it is a source.",
"If $F_{i,L}$ has the same depth as $F_{i,R}$ then put the pair $i,i$ in the first and third column if $i$ is a sink and $i$ in the second column if it is a source.",
"If the depth of $F_{i,L}$ is greater than $F_{i,R}$ then put the pair $i,i$ in the second and third column if $i$ is a sink and $i$ in the third column if it is a source.",
"By construction this process is inverse to the algorithm given above for building a web from a semistandard tableau."
],
[
"Examples",
"Consider the sign string $s = ++-++-+$ .",
"The composition $\\lambda _s$ in this case is $\\lambda _s = (1, 1, 2, 1, 1, 2, 1)$ .",
"Then the following tableau $T_s$ which is a filling using the numbers $\\lbrace 1, 2, 3^2, 4, 5, 6^2, 7\\rbrace $ is said to have content $s$ .",
"We construct the two standard tableaux $\\widetilde{T_s}$ and $\\widetilde{T_s}^{\\prime }$ using the natural generalization of the algorithm from the previous section.",
"In particular, we replace the ordered set $\\lbrace 1, 2, 3, 3, 4, 5, 6, 6, 7\\rbrace $ with the ordered set $\\lbrace 1,2, 3, 4, 5, 6, 7, 8, 9\\rbrace $ always placing the smaller number farthest left when replacing a repeated pair.",
"$T_s = \\raisebox {-15pt}{(134,256,367)} \\hspace{18.06749pt} \\longrightarrow \\hspace{18.06749pt} \\widetilde{T_s} = \\raisebox {-15pt}{(145,268,379)} \\hspace{18.06749pt} \\longrightarrow \\hspace{18.06749pt} \\widetilde{T_s}^{\\prime } = \\raisebox {-15pt}{(123,467,589)}$ From the tableau $\\widetilde{T_s}^{\\prime }$ we get the $m$ -diagram and the web shown in Figure REF .",
"After contracting the edges incident on vertices 3, 4, 7, and 8 we get the web with boundary $s$ shown in Figure REF .",
"Figure: The mm-diagram and web corresponding to T s ˜ ' \\widetilde{T_s}^{\\prime }.Figure: The web of content ss corresponding to T s T_s.We conclude this section with some additional examples.",
"Figures REF ,REF , and REF construct all webs corresponding to the sign strings $---, --++,$ and $-++++$ respectively.",
"Figure: Constructing the bijection for sign string s=---s=---.Figure: Constructing the bijection for sign string s=--++s=--++.Figure: Constructing the bijection for sign string s=-++++s=-++++."
],
[
"Applications: Rotation and join of webs",
"When the sign string of a web is all pluses and the corresponding tableau is standard, Petersen-Pylyavskyy-Rhoades prove that rotation of webs corresponds to jeu-de-taquin promotion [13].",
"Tymoczko uses the $m$ -diagram algorithm to give a simplified proof of this fact [20].",
"There is also a notion of the join of two webs.",
"Tymoczko proves that join can be understood as another move on standard tableaux called a shuffle [20].",
"In this section, we prove that jeu-de-taquin promotion and shuffle of semistandard tableaux correspond to rotation and join of all non-elliptic $sl_3$ webs."
],
[
"Rotation and jeu-de-taquin promotion",
"Jeu-de-taquin promotion is a process on semistandard Young tableaux whereby a box (or subset of boxes) is removed, and the tableau is rearranged to form a new filling of the same shape.",
"Figure REF has an example.",
"Say that $T$ is a semistandard tableaux filled with at least one of each of the numbers $1, \\ldots , \\ell $ .",
"Jeu-de-taquin promotion on $T$ produces a new tableau $jdt(T)$ as follows.",
"Begin by removing the entry 1 from the top left corner of $T$ .",
"Say that $a$ is below and $b$ is to the right of the removed box.",
"If $a\\le b$ , slide $a$ upwards into the empty position.",
"Otherwise, slide $b$ left into the empty position.",
"Continue this process until the empty box has no entries to its right or below.",
"If 1 appears multiple times in $T$ , repeat the first three steps until all 1's have been erased.",
"Decrement all entries by 1, and replace each empty box with $\\ell $ .",
"A proof of the following Lemma can be found in Sagan's book [17].",
"Lemma 6 The jeu-de-taquin process described above is well-defined on semistandard tableaux.",
"Figure: Promotion on a semistandard tableau.Lemma 7 Let $T_s$ be a semistandard tableau of content $s = s_1\\cdots s_n$ .",
"Then $\\widetilde{jdt(T_s)}^{\\prime } = jdt(\\widetilde{T_s}^{\\prime })$ if $s_1$ is a plus, and $\\widetilde{jdt(T_s)}^{\\prime } = jdt\\left(jdt\\left(\\widetilde{T_s}^{\\prime }\\right)\\right)$ if $s_1$ is a minus.",
"When the numbers below and to the right of an empty box are equal, jeu-de-taquin promotion chooses to move the box below into the empty position.",
"Since we construct the standard tableau $\\widetilde{T_s}$ from $T_s$ by replacing the leftmost instance of a repeated entry with a smaller number than the rightmost instance, it follows that $\\widetilde{jdt(T_s)} = jdt(\\widetilde{T_s})$ when $s_1$ is a plus.",
"If $s_1$ is a minus, then the numbers 1 and 2 must be promoted in the standard tableau to correspond to the promotion of a repeated entry of 1 in the semistandard tableau.",
"Thus $\\widetilde{jdt(T_s)} = jdt\\left(jdt\\left(\\widetilde{T_s}\\right)\\right)$ in the case that $s_2$ is a minus.",
"The lemma follows from the fact that jeu-de-taquin promotion commutes with the process of conjugation in standard tableaux.",
"We have been considering webs with boundary lying on a horizontal line, but webs are often also viewed in a disk with univalent vertices on the boundary circle [10], [13].",
"Boundary vertices are enumerated counterclockwise with respect to some base point.",
"To obtain a web with linear boundary, the circle bounding the disk is split open at the base point.",
"The notion of web rotation is more natural when viewed from the disk perspective.",
"Figure REF has an example of rotation on a web with linear boundary.",
"Theorem 2 Jeu-de-taquin promotion of semistandard tableaux corresponds to rotation of webs.",
"Jeu-de-taquin promotion in standard tableaux corresponds to rotation of webs with boundary $3n$ sources.",
"Since $\\widetilde{T_s}^{\\prime }$ is a standard tableau, the web $w_{jdt(\\widetilde{T_s}^{\\prime })}$ is a rotation of the web $w_{\\widetilde{T_s}^{\\prime }}$ .",
"It follows from Lemma REF that the web $w_{\\widetilde{jdt(T_s)}^{\\prime }}$ is also a rotation of $w_{\\widetilde{T_s}^{\\prime }}$ .",
"Therefore the web obtained by rotation of $w_{T_s}$ is the same as the web $w_{jdt(T_s)}$ .",
"Figure: Rotation of a web."
],
[
"Join and shuffling",
"Tymoczko defines the notion of a shuffle of two standard tableaux and proves that the web associated to the shuffle is the join of the webs corresponding to those tableaux [20].",
"The join of two webs is the insertion of one into the other between some designated pair of vertices.",
"The definition of shuffle has a natural extension to semistandard tableaux.",
"Let $T$ be a semistandard tableau filled with at least one of each of the numbers $1,\\ldots , \\ell _1$ and $T^{\\prime }$ be a semistandard tableau filled with at least one of each of the numbers $1,\\ldots , \\ell _2$ .",
"Let $i\\le \\ell _1$ .",
"The shuffle of $T^{\\prime }$ into $T$ at $i$ is denoted by $T^{\\prime }\\stackrel{i}{\\mapsto } T$ and defined by the following process.",
"For each instance of $j = 1, \\ldots , i$ in $T$ , put $j$ in the same column of $T^{\\prime }\\stackrel{i}{\\mapsto } T$ as in $T$ .",
"For each instance of $j = 1, \\ldots , \\ell _2$ in $T^{\\prime }$ , put $j+i$ in the same column of $T^{\\prime }\\stackrel{i}{\\mapsto } T$ as in $T^{\\prime }$ .",
"For each instance of $j=i+1, \\ldots , \\ell _1$ in $T$ , put $j+\\ell _2$ in the same column of $T^{\\prime }\\stackrel{i}{\\mapsto } T$ as in $T$ .",
"Figure REF compares the shuffling of tableaux to the join of webs.",
"The entries of $T^{\\prime }$ that have been shuffled into $T$ appear in bold.",
"Figure: Joining two webs.In the case where $T$ has content $s = s_1 \\cdots s_{\\ell _{1}}$ and $T^{\\prime }$ has content $t = t_1 \\cdots t_{\\ell _{2}}$ , the web $T^{\\prime }\\stackrel{i}{\\mapsto } T$ will be semistandard with content $s_1 \\cdots s_i t_1 \\cdots t_{\\ell _{2}} s_{i+1} \\cdots s_{\\ell _{1}}$ .",
"The following theorem, illustrated in Figure REF , is a straightforward generalization of Tymoczko's result [20].",
"Theorem 3 Shuffling of semistandard tableaux corresponds to the join of webs.",
"Specifically, shuffling $T^{\\prime }$ into $T$ at $i$ corresponds to inserting the web $w_{T^{\\prime }}$ into the web $w_T$ between vertices $i$ and $i+1$ of $T$ ."
]
] | 1204.1037 |
[
[
"Optimal Channel Efficiency in a Sensory Network"
],
[
"Abstract We show that the entropy of the distribution of avalanche lifetimes in the Kinouchi-Copelli model always achieves a maximum jointly with the dynamic range.",
"This is noteworthy and nontrivial because while the dynamic range is an equilibrium average measure of the sensibility of a sensory system to a stimulus, the entropy of relaxation times is a purely dynamical quantity, independent of the stimulus rate, that can be interpreted as the efficiency of the network seen as a communication channel.",
"The newly found optimization occurs for all topologies we tested, even when the distribution of avalanche lifetimes itself is not a power-law and when the entropy of the size distribution of avalanches is not concomitantly maximized, strongly suggesting that dynamical rules allowing a proper temporal matching of the states of the interacting neurons is the key for achieving good performance in information processing, rather than increasing the number of available units."
],
[
"pdftitle = Optimal Channel Efficiency in a Sensory Network, pdfauthor = Thiago S. Mosqueiro and Leonardo P. Maia, pdfkeywords = neurodynamics, optimal dynamic range, avalanches, entropy, colorlinks = true, linkcolor = blue, anchorcolor = blue, citecolor = blue, urlcolor = blue Optimal Channel Efficiency in a Sensory Network Thiago S. Mosqueiro [email protected] Leonardo P. Maia [email protected] Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970 São Carlos, SP, Brasil We show that the entropy of the distribution of avalanche lifetimes in the Kinouchi-Copelli model always achieves a maximum jointly with the dynamic range.",
"This is noteworthy and nontrivial because while the dynamic range is an equilibrium average measure of the sensibility of a sensory system to a stimulus, the entropy of relaxation times is a purely dynamical quantity, independent of the stimulus rate, that can be interpreted as the efficiency of the network seen as a communication channel.",
"The newly found optimization occurs for all topologies we tested, even when the distribution of avalanche lifetimes itself is not a power law and when the entropy of the size distribution of avalanches is not concomitantly maximized, strongly suggesting that dynamical rules allowing a proper temporal matching of the states of the interacting neurons is the key for achieving good performance in information processing, rather than increasing the number of available units.",
"neurodynamics, critical phenomena, avalanches 64.60.av, 89.70.Cf, 87.85.dq In the end of the last century appeared the first claims on the criticality hypothesis, stating that some biological systems could evolve towards the edge of chaos [1], [2], [3], [4], [5], [6], the critical surface separating two phases in an abstract space of parameters.",
"The heuristic justification for this hypothesis is that adaptation of biological systems would be guided by selective pressures favoring the optimization of some key attributes, e.g.",
"the capacity of sensing environments.",
"No matter how appealing was the proposal, earlier work [1], [2], [3] relied only on simulations and the lack of experimental validation combined with some criticisms [7] turned down the theory for some time.",
"However, captivating researchers on brain dynamics, the idea acquired a wholly new motivation [8].",
"Indeed not only new theoretical evidence of increased computational performance at criticality showed up recently [9], [10], [11], [12] but mainly the observation of power-law behavior of neuronal avalanches in cortical networks both in vitro [13], [14] and in vivo [15], [16], [17] constituted stronger-than-ever evidence of the relevance of the edge of chaos to the operation of biological systems.",
"There was some debate regarding the proper characterization of the recorded power laws [18], [19] and whether criticality would be the sole explanation for the observed scale invariance but this time the criticality hypothesis is standing up criticisms [5], [6].",
"Remarkably, some experiments have explicitly revealed maximized quantities as information transmission [13], , information capacity , synchronizability and dynamic range in cortical networks at a critical condition.",
"The dynamic range is a sensibility measure associated to the high-slope region of a tuning curve (also response function, the stimuli-response relationship characterizing a sensory network), where nearby stimuli can be most easily discriminated since small changes in stimulus lead to high changes in the firing response.",
"Theoretical work [11] indicated that the dynamic range should be maximized in sensory systems when the topology of the network was set in a specific condition, critical for the signal propagation among interacting excitable neurons.",
"A beautiful marriage of theoretical prediction with experimental confirmation happened when an optimal dynamic range was found in cortex slice cultures with a proper balance between excitatory and inhibitory interactions achieved through pharmacological manipulation.",
"In this study, we employ simulations to show that, when the Kinouchi-Copelli (KC) model [11] is tuned at the edge of chaos, the Shannon entropy of its avalanche lifetime statistics (hereafter information efficiency) is always jointly maximized with the dynamic range, both in the original [11] random graph topology and alternatives.",
"We note that the information capacity (the entropy of avalanche size distribution) does not always exhibit such a critical optimization.",
"Indeed, we believe that information efficiency rules the behavior of the dynamic range and outweighs by far the relevance of information capacity in determining the information processing properties of a sensory network.",
"Two previous works discussed critical optimization of entropies of avalanches sizes.",
"The first one was a theoretical study of measures of information propagation in Boolean networks and the second one was the discovery of optimal information capacity in critical cortical networks .",
"None of them have raised the possibility that the information efficiency could be maximized instead of capacity.",
"The model.— We start giving a concise description of the KC model [11]: each of the $N$ neurons is a cellular automaton that can be at states 0 (quiescent/excitable), 1 (excited) or $2,\\cdots ,m-1$ (refractory states).",
"The neurons are arranged as a weighted undirected graph with mean degree $K$ .",
"The sequential transitions $1 \\rightarrow 2$ , $2 \\rightarrow 3$ , ..., $m-2 \\rightarrow m-1$ and $m-1 \\rightarrow 0$ are deterministic.",
"On the other hand, the transition $0 \\rightarrow 1$ happens for neuron $k$ if either (i) a spark of any of its excited neighbors $j$ reaches it, with probability $p_{kj}$ drawn from an uniform distribution in $[0,\\mathrm {p_{max}}]$ , or (ii) if $k$ gets an external stimulus, modeled by a Poisson process with rate $r$ , resulting in an excitation probability $\\lambda = 1 - \\exp \\left( r \\Delta t \\right)$ at each time interval $\\Delta t$ .",
"All other transitions are forbidden.",
"By setting $\\mathrm {p_{max}} = 2\\sigma /K$ , the mean number of excitations an excited neuron could generate in one time step if all its neighbors were quiescent is $\\sigma $ , namely, the average branching ratio.",
"Given the fraction of excited nodes $\\rho _t$ at time $t$ , the proper psychophysical response of the system is the average activity $F = T^{-1} \\sum _{t=1}^T \\rho _t$ .",
"Consequently, the response is a function $F=F(r)$ of the stimulus rate $r$ .",
"The dynamic range $\\Delta $ is defined in decibels as $\\Delta = 10 \\log (r_{0.9}/r_{0.1})$ , where $F(r_x)=F_{\\mathrm {min}} + x [ F_{\\mathrm {max}} - F_{\\mathrm {min}}]$ , $x \\in [0,1]$ , $F_{\\mathrm {max}}=F(\\infty )$ is the satured response and $F_{\\mathrm {min}}=F(r\\rightarrow 0)$ is the spontaneous activity.",
"Kinouchi and Copelli showed that self-sustained activity is possible if $\\sigma $ is greater than $\\sigma _c=1$ [11], so that $F_{\\mathrm {min}}$ plays the role of the order parameter in a phase transition in the neural activity with $\\sigma $ as a control parameter.",
"They have also found a critical optimization for $\\Delta $ .",
"In [11] the authors studied only the Erdös-Rényi topology (ERT) with a fixed number $NK/2$ of connections and focused on characterizing the maximization of the dynamic range, as did further works on alternative topologies , , .",
"They did not dwell on exploring the bursts of activity (avalanches) generated by their model, although stating that critical networks exhibit both large variance of avalanche lifetimes and a power-law distribution for avalanche sizes with the classical exponent $-3/2$ , .",
"In this work, we study the avalanches exhibited by the KC model implemented on both the ERT and the Barabási-Albert topology (BAT).",
"Unless explicitly stated otherwise, the simulations were performed with $N = 10^5$ and $K = 10$ .",
"Given a randomly generated representant of a topology, with chosen average connectivity $K$ and average branching ratio $\\sigma $ , we randomly choose a neuron of the network to be initially excited while all others are quiescent and record both the number $s$ of neurons that get excited due to that single spark and the number $t$ of consecutive generations the network remained active.",
"We repeat this procedure a large number of times in order to get the distributions $\\lbrace p_s\\rbrace $ and $\\lbrace p_t\\rbrace $ of the size and the lifetime of an avalanche, respectively.",
"In this setting, there is no role at all for a stimulus rate.",
"Size and lifetime distributions.— Fig.",
"REF (a) illustrates the critical (at $\\sigma _c=1$ ) emergence of power-law scaling in the bursts of activity, $p_s \\sim s^{-1.5}$ and $p_t \\sim t^{-1.9}$ , for the ERT (exponents estimated with standard techniques , ).",
"Since the ERT allows the propagation of almost independent branches of activity, this behavior is perfectly compatible with the predictions from the theory of branching processes , .",
"Indeed, we will describe elsewhere how that formalism predicts the solid lines in the bottom of Fig.",
"REF (a).",
"In Fig.",
"REF (b) we exhibit the avalanche distributions in a BAT.",
"It is harder to estimate these distributions and there is much uncertainty in the size distribution, but the bottom of Fig.",
"REF (b) shows clearly the presence of a bump in the lifetime distribution for $\\sigma =0.4$ ($p_s \\sim s^{-2}$ and $p_t \\sim t^{-2.9}$ in the first decades) and strongly suggests power-law behavior for slightly smaller values of $\\sigma $ .",
"A naïve analysis based on the criterium of a power law as a signature of critical behavior would favor the latter against the former, but we remark that the observation of power laws in the KC model in this topology demands subsampling to a 2-10% level (not shown, but see also [16], ) and below we will present results supporting the “bumpy” curve as the critical one.",
"We start discussing the relation among the dynamic range and the entropies of the distributions just described.",
"Figure: Distributions of (top) burst size and (bottom) activity lifetimes for (a) Erdös-Rényi and (b) Barabási-Albert topologies.",
"In (a), power laws emerge for σ c =1\\sigma _c = 1 and the solid lines in the bottom result from the theory of branching process (see text).",
"In (b), especially in the bottom, huge avalanches become more frequent when σ=0.4\\sigma = 0.4.",
"We have evidence that this is the critical condition, despite the lack of a power law (see text).Information efficiency.— The Shannon entropy $H$ of a distribution $\\lbrace p_n\\rbrace $ , $H (\\lbrace p_n\\rbrace ) = - \\sum _{n} p_n \\log p_n,$ is a standard measure of the uncertainty of a stochastic observable and so it is quite obvious it should be applied to the analysis of avalanches.",
"We will set $n$ as either $s$ or $t$ to indicate which distribution we are talking about.",
"In Fig.",
"REF , we jointly illustrate the behavior of the dynamic range $\\Delta $ and of the entropies as functions of $\\sigma $ , given $K$ and a topology.",
"It was not surprising to observe in Fig.",
"REF (a) $H (\\lbrace p_s\\rbrace )$ getting maximized jointly with $\\Delta $ in the ERT, since two previous studies , , in different contexts, reported such an effect for the information capacity.",
"Moreover, it seems natural to associate the dynamic range to that entropies (as measures of the flexibility of the system in dealing with signals), so that heuristically $H (\\lbrace p_t\\rbrace )$ should exhibit critical optimization following $\\Delta $ .",
"Thus, since the critical optimization of $\\Delta $ in BAT's has already been stablished , , the lack of a corresponding peak for $H (\\lbrace p_s\\rbrace )$ in Fig.",
"REF (b) came as a complete surprise.",
"Later we will discuss how that happens while $H (\\lbrace p_t\\rbrace )$ and $\\Delta $ keep getting optimized jointly, this time at $\\sigma =0.4$ .",
"Is this optimization really critical?",
"As did the authors of , we claim so invoking as first evidence the plot of the order parameter $F_{\\mathrm {min}}$ of the KC model against the control parameter $\\sigma $ in the inset of Fig.",
"REF (b).",
"However, our results suggest $\\sigma _c=0.4$ , while $\\sigma _c \\approx 0.5$ was estimated in .",
"Despite the much smaller networks simulated in that work, such a divergence demands an even more stringent analysis regarding the critical nature of the position of the peaks in this topology, namely, a tentative of data collapse into a scaling function.",
"We emphasize we are not going to take the abscissa of the peak as the location of a critical point a priori, since there is no support for such a procedure.",
"Quite the opposite, in , for instance, stochasticity makes $H(\\lbrace p_s\\rbrace )$ exhibit a peak away from criticality.",
"We also remark that (i) the deviation of $\\sigma _c$ from 1 in alternative topologies has been recently explained in terms of a spectral analysis , and nowadays is not surprising at all, (ii) there is no critical optimization for BAT's grown node by node (i.e., the information efficiency and the dynamic range keep behaving jointly even in such a “charmless” scenery) and (iii) whatever be the critical point, it is very clear from Fig.",
"REF (b), mainly from the plot of the information efficiency $H (\\lbrace p_t\\rbrace )$ , that the joint optimization of $H (\\lbrace p_t\\rbrace )$ and $\\Delta $ does not happen at the values of $\\sigma $ resembling power-law behavior in Fig.",
"REF (b).",
"Figure: Dynamic range (Δ\\Delta ) and entropies of avalanche size and lifetime distributions as functions of the average branching ratio σ\\sigma in (a) Erdös-Rényi and (b) Barabási-Albert topologies.",
"Concomitant critical optimization emerges clearly in (a), but only the lifetime entropy gets optimized jointly with Δ\\Delta in (b).",
"The green dashed lines are guides for the eyes.",
"Inset: F min F_{\\mathrm {min}} vs. σ\\sigma , clearly indicating the critical value σ c ≈0.4\\sigma _c \\approx 0.4.Critical optimization.— Despite the inset in Fig.",
"REF (b) being a classical signature of a phase transition, the aforementioned displacement of the critical point prompted us to perform a scaling analysis in order to confidently assess the issue of criticality.",
"Let $\\rho _N (t)$ be the complementary cumulative distribution function (CDF) of the lifetime of an avalanche when there are $N$ neurons in the KC model (i.e., $\\rho _N (t)$ is the probability of the duration of an avalanche surviving for at least $t$ given a network with $N$ units).",
"Given a topology and a value of $\\sigma $ thought to be a critical point, we have looked for exponents $\\gamma $ and $D$ able to make all the transformed distributions $x^{\\gamma } \\rho _N (x)$ collapse into a single universal curve $\\mathcal {F}$ when plotted against $x/N^D$ , regardless the value of $N$ .",
"Fig.",
"REF illustrates the success of such endeavour, considering the distributions obtained with $\\sigma _c=1$ in the ERT and $\\sigma _c=0.4$ in the BAT.",
"Figure: Instances of (top) CDF's for several system sizes and (bottom) their collapse into a scaling function for both (a) Erdös-Rényi and (b) Barabási-Albert topologies.",
"To collapse the CDF's at the proposed critical conditions, we have fit γ ER =0.9\\gamma _{ER} = 0.9, D ER =0.5D_{ER} = 0.5, γ BA =1.25\\gamma _{BA} = 1.25 and D BA =1D_{BA} = 1 (see text).",
"The distributions in (b) closely resemble the ones presented recently by Dehghani et al .There are two more observations supporting our claim of criticality in the absence of pure power laws in the avalanches (data not shown).",
"Both preliminary results on the collapse of avalanche shapes, as advocated by and implemented in , and an analysis of the extinction probability (a branching-process-like study to be discussed elsewhere) point to $\\sigma _c=0.4$ in the BAT.",
"We also report complementary observations regarding data not shown in this paper.",
"In contrast with other contexts [9], , our results are qualitatively robust to changes in the distribution of edge weights (even for constant weigths).",
"Most alternative topologies , leading to more realistic degree distributions are qualitatively equivalent to the BAT and exhibit even more pronounced bumps.",
"In the uncorrelated version of the configuration model , bumps are still present in the avalanche distributions but both information capacity and efficiency exhibit critical optimization.",
"This behavior fits in the phenomenology we describe here, but also suggests that the existence of degree correlations in the BAT may be the reason why $H (\\lbrace p_s\\rbrace )$ does not maximize in that topology.",
"Discussion.— Neuroscientists regularly employ the mutual information as a measure of the statistical dependence between stimulus and response, taking into account both specificity (change in stimulus implying change in response) and fidelity (low variability given a stimulus) of responses.",
"Notwithstanding the relevance of such studies, indispensable for the comprehension of neural coding, we remark our aim in this paper is the study of intrinsic, stimulus-free, behavior of sensory systems rather than of patterns of response variability.",
"The KC model [11] revealed that the stimulus-dependant measure $\\Delta $ is optimized precisely at $\\sigma _c$ , a critical point of the self-sustained activity $F_{\\mathrm {min}}$ , that does not depend at all on the nature of the stimulus.",
"If not a theoretical artifact, that phenomenology suggests that generally the behavior of actual excitable networks (e.g., sensory systems) could be strongly determined by its stimulus-free properties.",
"Accordingly, an optimal dynamic range was observed experimentally at a condition determined in the absence of stimuli and there is further robust evidence that stimulus-evoked activity is strongly dependent on the spontaneous ($r=0$ ) firing patterns in the cortex (see references in ).",
"Therefore, despite it may seem quite plausible per se that the sensibility measured by the dynamic range gets optimal when signals do not either fade out fastly (subcritical) or frenetically superimpose themselves (supercritical), there is enough motivation for scrutiny on the microscopic mechanisms leading to improvements in information processing capabilities even in stimulus-free conditions.",
"We firmly believe to have discovered one such mechanism.",
"It seems fairly intuitive to expect that a sensory system be more flexible if it disposes of greater variability of the duration of the bursts of neural activity it must process.",
"Indeed the support of the avalanche lifetime distribution (the interval of lifetimes with positive probabilities) achieves a maximum jointly with the dynamic range, while noncritical distributions hold for no longer than two decades (see Fig.",
"REF ).",
"We have chosen the information efficiency as the proper measure of that effect because it takes into account not only the magnitude of the reportoire of lifetimes but also a balanced utilization As an entropy, the information efficiency gets higher the more spread is the avalanche lifetime distribution on its support.",
"Also on this issue, we remark that the breakdown of power-law scaling in the critical condition occurs by increased concentration of probability mass in the tails of the distributions instead of throwing away large and durable excursions, see Fig.",
"REF .",
"of that resource for information transmission (from the olfactory bulb to the cortex, for instance).",
"It is not evident why such arguments are always valid for the information efficiency but not for the capacity.",
"We speculate that an explanation must rely on the relationship between the microstructure of the network (motifs) and the dynamics of the excitable units.",
"Progress beyond the initial studies on the synchronizability properties of the KC model , , will be probably achieved by deciphering such relationship.",
"As a rule of thumb, greater values of the clustering coefficient should lead to stronger deviations from the $-3/2$ law.",
"It is also worth mentioning that it has already been suggested that “the lifetime distributions of neuronal avalanches may carry rich information about the local cortical circuit structure” and may exhibit consistent deviations from power-law scaling, while the size distribution would be much more well-behaved.",
"Anyway, despite our praise of the intrinsic activity, it is definitively worth studying stimuli-dependant features of excitable networks by information-theoretic tools like mutual information and transfer entropy.",
"The notion of criticality without power laws may have wide implications in the interpretation of observations of neuronal avalanches.",
"Recent experiments exhibiting critical optimization , , have described the phase transitions in terms of a control parameter $\\kappa $ resembling $\\sigma $ but based on the tacit assumption that neuronal avalanches are pure power laws.",
"Further investigations are necessary to reveal eventual consequences of the breakdown of that hypothesis.",
"Likewise, Ref.",
"employed robust statistical techniques to analyze neuronal avalanches in vivo and stand up against critical dynamics.",
"However, the CDF's they present are very similar to Fig.",
"REF (b), so that probably their data is ruling out power-law scaling, but not criticality.",
"Finally, distributions pretty much like the ones in Fig.",
"REF (b) have been recently observed in high-resolution experiments in vitro and the bumps were no obstacle for a remarkable data collapse constituting very compelling evidence of critical behavior in brain dynamics.",
"Summarizing, we studied the avalanches in Kinouchi-Copelli model in a first attempt to figure out detailed mechanisms of information transmission in cortical networks.",
"We discovered that, in a critical point, the entropy of avalanche lifetime statistics (information efficiency) is always maximized jointly with the dynamic range, an important measure of information transmission extracted from the psychophysical tuning curves.",
"Our findings fit in the discussions regarding the role of criticality in information processing [5], [6] and the relationship of long bursts of activity with the dynamic range , specially because they suggest critical behavior without pure scale invariance.",
"Acknowledgements.",
"We acknowledge the contribution of an anonymous referee that brought reference to our attention and gave suggestions that significantly improved the paper.",
"Thiago S. Mosqueiro acknowledges CAPES for financial support.",
"The work of Leonardo P. Maia was supported by FAPESP grant No.",
"2010/20446-5."
]
] | 1204.0751 |
[
[
"Froissart Bounds for Amplitudes and Cross Sections at High Energies"
],
[
"Abstract High-energy behavior of total cross sections is discussed in experiment and theory.",
"Origin and meaning of the Froissart bounds are described and explained.",
"Violation of the familiar log-squared bound appears to not violate unitarity (contrary to the common opinion), but correspond to rapid high-energy increase of the amplitude in nonphysical regions."
],
[
"english FROISSART BOUNDS FOR AMPLITUDES AND CROSS SECTIONS AT HIGH ENERGIES Based on the lecture presented at the XLV Winter School of PNPI (March 2011) and on the seminar talk at the Ruhr University Bochum (October 2011).",
"The Russian version is published in “Nuclear and particle physics, Theoretical physics (Proceedings of the XLV Winter School of PNPI, Feb. 28 - March 5, 2011)”, ed.",
"V.T.Kim, PNPI, St.Petersburg, 2011, pp.20-26.",
"Ya.",
"I. Azimov (PNPI) A b s t r a c t High-energy behavior of total cross sections is discussed in experiment and theory.",
"Origin and meaning of the Froissart bounds are described and explained.",
"Violation of the familiar log-squared bound appears to not violate unitarity (contrary to the common opinion), but correspond to rapid high-energy increase of the amplitude in nonphysical regions.",
"The elementary particle physics (or, the same, high energy physics) is considered as a separate branch of physics since 1956, when the Rochester University, USA, organized the Conference on High Energy Physics (since then the “Rochester” Conferences gathered once a year in different cities and countries; after 1964 they are biennials called “International Conferences on High Energy Physics”).",
"But, sure, elementary particles and their interactions had been investigated even before 1956.",
"It became clear as early as in '30-ies that particles have interactions of several different kinds.",
"And it was discovered in '40-ies that strong interactions with increasing energy provide increasing multiple meson production.",
"In other words, the role of inelastic processes grows with growing energy in collisions of strong-interacting particles (they are called “hadrons” since 1962, according to suggestion of L.B.",
"Okun).",
"To 1960, the idea had been formed that scattering of hadrons at very high energies should be similar to the classical diffraction of light on a black (completely absorbing) disc of a finite radius.",
"If this were true, the total interaction cross sections at very high energies should be asymptotically constant, and the elastic cross sections should be a fixed part of the total ones.",
"Angular distribution (or distribution in the momentum transfer $-t$ ) should look as the diffraction peak $\\sim \\exp (bt)$ with a constant slope $b$ , which is proportional to the radius squared of the disc.",
"Experimental data of those years (in the then available energy interval) seemed to agree with such expectations.",
"However, in 1961 there appeared two theoretical papers which cast doubts on applicability of such a simple picture.",
"One of them was presented by V.N.",
"Gribov [1].",
"It showed that the classical diffraction is incompatible with the analytical properties of hadron amplitudes when combined with the cross-channel unitarity condition.",
"This result has become an impetus to construct the Reggeon theory, according to which the diffraction peak changes (shrinks) with increasing energy, even if the total cross section is asymptotically constant.",
"The other paper was published by Marcel Froissart [2].",
"Froissart (he is, by the way, a member of the old noble French family) began his work with the hypothesis that total cross sections of hadron interactions may infinitely grow with energy (though no experimental evidence for such possibility had been seen to that time).",
"Then he applied the unitarity condition together with the analyticity of an elastic amplitude, as expressed by the dispersion relations with a finite number of subtractions.",
"Based on such, seemingly very “soft”, conditions (nearly from nothing) Froissart was able to receive quite tangible restriction for a possible energy growth rate of the forward (backward) scattering amplitude, and even stronger restrictions for the fixed angle nonforward (nonbackward) scattering.",
"Since the unitarity condition (the optical theorem) relates the forward elastic amplitude with the total interaction cross section, it appeared that the total cross sections might not grow faster than the logarithm squared of the energy.",
"This result, known as “the Froissart theorem”, has become one of key points when constructing theoretical models for high-energy strong interactions.",
"Moreover, it became a sincere belief for public opinion of the high-energy physics community, that violation of the Froissart theorem would mean violation of unitarity.",
"In the years after 1961, our knowledge of strong interactions has been significantly expanded to higher energies.",
"The following experimental facts have been definitely established.",
"The diffraction peaks shrink indeed with growing energy; their slopes in respect to the momentum transfer grow at least as the logarithm of energy.",
"The total cross sections, as is clear now, indeed increase with energy.",
"Existing data for different hadrons agree with the hypothesis that the total cross sections asymptotically grow as $\\ln ^2 s~$ ($s$ is the c.m.s.",
"energy squared).",
"The diffraction peaks shrink indeed with growing energy; their slopes in respect to the momentum transfer grow at least as the logarithm of energy.",
"The total cross sections, as is clear now, indeed increase with energy.",
"Existing data for different hadrons agree with the hypothesis that the total cross sections asymptotically grow as $\\ln ^2 s~$ ($s$ is the c.m.s.",
"energy squared).",
"Most advanced in the energy scale are investigations of nucleon-(anti)nucleon interactions, especially if one adds data from cosmic ray studies.",
"Existing values for the total $pp$ and $p\\bar{p}$ cross sections may be quite satisfactorily described by the curves shown in Fig.1 (taken from Ref.[3]).",
"Their high-energy behavior is proportional to the log-squared energy.",
"However, the accelerator energy interval available in the pre-LHC era is rather narrow in the logarithmic scale, while data extracted from cosmic ray experiments have great uncertainties.",
"As a result, significant ambiguities may (and do) appear in the description of the data.",
"In particular, possible are “heretic” descriptions, which contradict to the canonically understood Froissart result.",
"For example, Fig.2 (taken from Ref.",
"[4]) shows such a description of the total cross sections which corresponds to the power increase with energy as $s^{\\delta }$ , though with a small exponent $\\delta \\approx 0.08$ .",
"Thus, experiments have not allowed yet to reach a definite conclusion, whether the log-squared energy asymptotics is true or not.",
"LHC extends the accelerator energies to the values which have been available earlier, but only in cosmic rays.",
"Meanwhile, the accelerator measurements Figure: Fit for all acceleratot dataon the total pppp and pp ¯p\\bar{p} cross sectionsavailable before LHC .",
"The curvesasymptotically grow as a power of energy.are much more precise.",
"Therefore, one can hope that the LHC data, especially at its maximal energy (not reached yet), may be able to clarify the situation.",
"It is interesting (and useful), however, to examine also the theoretical basis of the Froissart theorem.",
"This was just the aim of the paper [5], which revises derivation of the theorem and meaning of its results.",
"The paper may be easily reached either in the journal or as the arXiv e-print, so it is not necessary to present here all its calculations and formulas.",
"Instead, it is sufficient to describe the main results and conclusions of the paper.",
"A necessary physical input for the Froissart theorem is, of course, unitarity.",
"It works in two ways: on one side, the scattering-channel unitarity restricts elastic partial-wave amplitudes; on the other side, the cross-channel unitarity relates positions for scattering angle singularities of the elastic amplitude with the mass spectrum in the cross channel.",
"Another physical input is the absence of massless particles.",
"It guaranties the absence of angle singularities both inside the physical region and on its edges.",
"A necessary mathematical base for the Froissart theorem is provided by properties of the Legendre functions.",
"Especially important appears the behavior of $P_l(z)$ at $l\\rightarrow +\\infty $ .",
"The infinite point in the $l$ -plane is an essential singularity for the Legendre functions.",
"As a result, their asymptotic forms at large positive $l$ are sharply different in the three cases: inside the $z$ -interval $(-1, +1)$ , at its edges (i.e., at $z=\\pm 1$ ), and outside this interval, though the points $z=\\pm 1$ are not singular for $P_l(z)$ with physical (integer positive) values of $l$ .",
"On one side, therefore, discontinuities become possible (and arise indeed) between high-energy asymptotics of an elastic amplitude in the three configurations: inside the physical region of angles, at its boundary (i.e., for the forward or backward scattering), at nonphysical (complex) angles (it is worth to emphasize that transitions between those three configurations do not touch any singularities of the amplitude).",
"On the other side, due to properties of $P_l(z)$ , the rate of high-energy increase of the amplitude is much more moderate for physical angles than for nonphysical ones.",
"Such sharp moderateness of the amplitudes in physical configurations is just the true meaning of the Froissart theorem.",
"All those results do nor fix, however, any particular asymptotic expression for the total cross sections.",
"To obtain the familiar “canonical” restriction of the form $\\ln ^2 s$ , one should add the hypothesis that in every nonphysical configuration (even including arbitrary nonphysical angles) the amplitude cannot grow with energy faster than some finite power of energy.",
"The Froissart paper [2] “hides” this hypothesis in dispersion relations with a finite number of subtractions.",
"Note that no physical or mathematical justifications have been ever suggested for such an asymptotic hypothesis.",
"Moreover, the observed linearity of Regge trajectories provides phenomenological arguments against the power boundedness (more detailed motivation see in Ref.[5]).",
"In a general case, the upper bound for the total cross section may grow with energy approximately as the squared logarithm of the fastest asymptotics of the amplitude in nonphysical configurations.",
"The more exact asymptotic expressions for the Legendre functions, used in Ref.",
"[5], allowed to strengthen the original Froissart inequalities [2] for physical amplitudes (and cross sections).",
"For example, even if the amplitude is bounded by a finite power of energy in any nonphysical configurations, the corresponding total cross section still cannot grow as $\\ln ^2(s/s_0)$ with a fixed scale $s_0$ (as is usually stated).",
"Instead, the scale $s_0$ itself should grow logarithmically with energy, reducing the growth rate for the total cross section.",
"Increase of a total cross section faster than the log-squared energy does not mean violation of unitarity and is not forbidden by any general principles, contrary to a widespread opinion.",
"It is interesting that neither dispersion relations, nor any particular properties of interactions were needed in the analysis of Ref.[5].",
"The strong interactions, as an object to apply Froissart restrictions, are marked out only by the fact of absence of massless hadrons (as compared, say, to the electrodynamics with its massless photon).",
"A necessary physical input for the Froissart theorem is, of course, unitarity.",
"It works in two ways: on one side, the scattering-channel unitarity restricts elastic partial-wave amplitudes; on the other side, the cross-channel unitarity relates positions for scattering angle singularities of the elastic amplitude with the mass spectrum in the cross channel.",
"Another physical input is the absence of massless particles.",
"It guaranties the absence of angle singularities both inside the physical region and on its edges.",
"A necessary mathematical base for the Froissart theorem is provided by properties of the Legendre functions.",
"Especially important appears the behavior of $P_l(z)$ at $l\\rightarrow +\\infty $ .",
"The infinite point in the $l$ -plane is an essential singularity for the Legendre functions.",
"As a result, their asymptotic forms at large positive $l$ are sharply different in the three cases: inside the $z$ -interval $(-1, +1)$ , at its edges (i.e., at $z=\\pm 1$ ), and outside this interval, though the points $z=\\pm 1$ are not singular for $P_l(z)$ with physical (integer positive) values of $l$ .",
"On one side, therefore, discontinuities become possible (and arise indeed) between high-energy asymptotics of an elastic amplitude in the three configurations: inside the physical region of angles, at its boundary (i.e., for the forward or backward scattering), at nonphysical (complex) angles (it is worth to emphasize that transitions between those three configurations do not touch any singularities of the amplitude).",
"On the other side, due to properties of $P_l(z)$ , the rate of high-energy increase of the amplitude is much more moderate for physical angles than for nonphysical ones.",
"Such sharp moderateness of the amplitudes in physical configurations is just the true meaning of the Froissart theorem.",
"All those results do nor fix, however, any particular asymptotic expression for the total cross sections.",
"To obtain the familiar “canonical” restriction of the form $\\ln ^2 s$ , one should add the hypothesis that in every nonphysical configuration (even including arbitrary nonphysical angles) the amplitude cannot grow with energy faster than some finite power of energy.",
"The Froissart paper [2] “hides” this hypothesis in dispersion relations with a finite number of subtractions.",
"Note that no physical or mathematical justifications have been ever suggested for such an asymptotic hypothesis.",
"Moreover, the observed linearity of Regge trajectories provides phenomenological arguments against the power boundedness (more detailed motivation see in Ref.[5]).",
"In a general case, the upper bound for the total cross section may grow with energy approximately as the squared logarithm of the fastest asymptotics of the amplitude in nonphysical configurations.",
"The more exact asymptotic expressions for the Legendre functions, used in Ref.",
"[5], allowed to strengthen the original Froissart inequalities [2] for physical amplitudes (and cross sections).",
"For example, even if the amplitude is bounded by a finite power of energy in any nonphysical configurations, the corresponding total cross section still cannot grow as $\\ln ^2(s/s_0)$ with a fixed scale $s_0$ (as is usually stated).",
"Instead, the scale $s_0$ itself should grow logarithmically with energy, reducing the growth rate for the total cross section.",
"Increase of a total cross section faster than the log-squared energy does not mean violation of unitarity and is not forbidden by any general principles, contrary to a widespread opinion.",
"It is interesting that neither dispersion relations, nor any particular properties of interactions were needed in the analysis of Ref.[5].",
"The strong interactions, as an object to apply Froissart restrictions, are marked out only by the fact of absence of massless hadrons (as compared, say, to the electrodynamics with its massless photon).",
"LHC has begun to contribute into the problem of the increasing total cross sections.",
"The recent analysis of accelerator data for $pp$ and $p\\bar{p}$ scattering [6] assumed the asymptotic behavior of their total cross sections in the form $(\\ln s)^\\alpha $ , the exponent $\\alpha $ being a free parameter.",
"The earlier data agree with the “canonical” value $\\alpha =2 $ .",
"However, addition of the first LHC data [7] appears to provide small but statistically meaningful excess $\\alpha >2$ [6].",
"Approach of Ref.",
"[5] enables one to investigate the high-energy asymptotics not only at a fixed scattering angle (as in Ref.",
"[2]), but also at a fixed momentum transfer.",
"This allows to study asymptotics of the diffraction peak slope as well.",
"As appears, if the total cross section increases with energy, then the diffraction slope should increase at the same rate or even faster.",
"In the saturation regime, when the total cross section grows with the maximal possible rate, its ratio to the slope should stay constant or even decrease [5].",
"Such expectation was in agreement with the pre-LHC accelerator data, but LHC seems to violate it [8].",
"This means that the present increase of the total cross sections is not saturated yet, and when going to even higher energies we may encounter some unexpected features.",
"The present work was partly supported by the Russian State grant RSGSS-65751.2010.2."
]
] | 1204.0984 |
[
[
"Noether current from the surface term of gravitational action, Virasoro\n algebra and horizon entropy"
],
[
"Abstract We describe a simple way of obtaining horizon entropy using the approach based on the Virasoro algebra and central charge.",
"We show that the Virasoro algebra defined by the Noether currents corresponding to the surface term of gravitational action, for the diffeomorphisms which leave the horizon structure unaltered, has a central extension that directly leads to the horizon entropy.",
"In this approach there are no ambiguities in the calculation of the central charge.",
"We explain why this approach is physically well motivated and could provide greater insight into the nature of horizon entropy."
],
[
"Acknowledgement",
"The research of TP is partially supported by the JC Bose research grant of DST India."
],
[
"Derivation of noether current",
"Consider a general Lagrangian which is a total derivative of a vector field so that the resulting action has only a surface contribution.",
"Then the Lagrangian density can be expressed as $\\sqrt{g}L = \\sqrt{g}\\nabla _aA^a~,$ where $L$ is a scalar.",
"Under a diffeomorphism $x^a\\rightarrow x^a+\\xi ^a$ the left hand side changes by: $\\delta _{\\xi }\\Big (\\sqrt{g}L\\Big ) \\equiv £_{\\xi } \\Big (\\sqrt{g}L\\Big ) = \\sqrt{g}\\nabla _a\\Big (L\\xi ^a\\Big )~.$ On the other hand, the variation of the right hand side of (REF ) is given by: $&&\\delta _{\\xi }\\Big (\\sqrt{g}\\nabla _aA^a\\Big ) = £_{\\xi }\\Big [\\partial _a\\Big (\\sqrt{g}A^a\\Big )\\Big ]\\nonumber \\\\&& = \\partial _a\\Big [A^a£_\\xi \\sqrt{g} + \\sqrt{g}£_\\xi A^a\\Big ]\\nonumber \\\\&& = \\sqrt{g}\\nabla _a\\Big [\\nabla _b\\Big (A^a\\xi ^b\\Big ) - A^b\\nabla _b\\xi ^a\\Big ]$ Equating (REF ) and (REF ) we obtain the conservation law $\\nabla _a J^a =0$ for the Noether current: $J^a[\\xi ] = L\\xi ^a - \\nabla _b(A^a\\xi ^b) + A^b\\nabla _b\\xi ^a~,$ which after using (REF ) reduces to the following form: $J^a[\\xi ] = \\nabla _bJ^{ab}[\\xi ] = \\nabla _b\\Big [\\xi ^aA^b-\\xi ^bA^a\\Big ]~.$ This was the result used in the paper."
]
] | 1204.1422 |
[
[
"A polynomial time algorithm for computing the HNF of a module over the\n integers of a number field"
],
[
"Abstract We present a variation of the modular algorithm for computing the Hermite Normal Form of an $\\OK$-module presented by Cohen, where $\\OK$ is the ring of integers of a number field K. The modular strategy was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature.",
"In this paper, we provide a new method to prevent the coefficient explosion and we rigorously assess its complexity with respect to the size of the input and the invariants of the field K."
],
[
"Introduction",
"The construction of a good basis of an $\\mathcal {O}_{K}$ -module, where $K$ is a number field and $\\mathcal {O}_{K}$ its ring of integers, has recently received a growing interest from the cryptographic community.",
"Indeed, $\\mathcal {O}_{K}$ -modules occur in lattice-based cryptography [8], [9], [10], [13], [14], where cryptosystems rely on the difficulty to find a short element of a module, or solving the closest vector problem.",
"The computation of a good basis is crucial for solving these problems, and most of the algorithms for computing a reduced basis of a $\\mathbb {Z}$ -lattice have an equivalent for $\\mathcal {O}_{K}$ -modules.",
"However, applying the available tools over $\\mathbb {Z}$ to $\\mathcal {O}_{K}$ -modules would result in the loss of of their structure.",
"The computation of a Hermite Normal Form (HNF)-basis was generalized to $\\mathcal {O}_{K}$ -modules by Cohen [2].",
"His algorithm returns a basis that enjoys similar properties as the HNF of a $\\mathbb {Z}$ -module.",
"A modular version of this algorithm is conjectured to run in polynomial time, although this statement is not proven (see last remark of [2]).",
"In addition, Fieker and Stehlé's recent algorithm for computing a sized-reduced basis relies on the conjectured possibility to compute an HNF-basis for an $\\mathcal {O}_{K}$ -module in polynomial time [5].",
"This allows a polynomial time equivalent of the LLL algorithm preserving the structure of $\\mathcal {O}_{K}$ -module.",
"In this paper, we adress the problem of the polynomiality of the computation of an HNF basis for an $\\mathcal {O}_{K}$ -module by presenting a modified version of Cohen's algorithm [2].",
"We thus assure the validity of the LLL algorithm for $\\mathcal {O}_{K}$ -modules of Fieker and Stehlé [5] which has applications in lattice-based cryptography, as well as in representations of matrix groups [4] and in automorphism algebras of Abelian varieties.",
"In addition, our HNF algorithm allows to compute a basis for the intersection of $\\mathcal {O}_{K}$ modules, which has applications in list decoding codes based on number fields (see [6] for their description)."
],
[
"Our contribution",
"We present in this paper the first polynomial time algorithm for computing an HNF basis of an $\\mathcal {O}_{K}$ -module based on the modular approach of Cohen [2].",
"We rigorously adress its correctness and derive bounds on its run time with respect to the size of the input, the dimension of the module and the invariants of the field."
],
[
"Generalities on number fields",
"Let $K$ be a number field of degree $d$ .",
"It has $r_1\\le d$ real embeddings $(\\sigma _i)_{i\\le r_1}$ and $2r_2$ complex embeddings $(\\sigma _i)_{r_1 < i \\le 2r_2}$ (coming as $r_2$ pairs of conjugates).",
"The field $K$ is isomorphic to $\\mathcal {O}_{K}\\otimes \\mathbb {Q}$ where $\\mathcal {O}_{K}$ denotes the ring of integers of $K$ .",
"We can embed $K$ in $K_\\mathbb {R}:= K\\otimes \\mathbb {R}\\simeq \\mathbb {R}^{r_1}\\times {r_2}, $ and extend the $\\sigma _i$ 's to $K_\\mathbb {R}$ .",
"Let $T_2$ be the Hermitian form on $K_\\mathbb {R}$ defined by $T_2(x,x^{\\prime }) := \\sum _i \\sigma _i(x)\\overline{\\sigma _i}(x^{\\prime }),$ and let $\\Vert x\\Vert := \\sqrt{T_2(x,x)}$ be the corresponding $L_2$ -norm.",
"Let $(\\alpha _i)_{i\\le d}$ such that $\\mathcal {O}_{K}= \\oplus _i \\mathbb {Z}\\alpha _i$ , then the discriminant of $K$ is given by $\\Delta _K = \\det ^2(T_2(\\alpha _i,\\alpha _j))$ .",
"The norm of an element $x\\in K$ is defined by $\\mathcal {N}(x) = \\prod _i|\\sigma _i(x)|$ .",
"To represent $\\mathcal {O}_{K}$ -modules, we rely on a generalization of the notion of ideal, namely the fractional ideals of $\\mathcal {O}_{K}$ .",
"They can be defined as finitely generated $\\mathbb {Z}$ -modules of $K$ .",
"When a fractional ideal is contained in $\\mathcal {O}_{K}$ , we refer to it as an integral ideal, which is in fact an ideal of $\\mathcal {O}_{K}$ .",
"Otherwise, for every fractional ideal $I$ of $\\mathcal {O}_{K}$ , there exists $r\\in \\mathbb {Z}_{>0}$ such that $rI$ is integral.",
"The sum and product of two fractional ideals of $\\mathcal {O}_{K}$ is given by IJ = { i1j1 + + iljllN, i1,ilI, j1,jlJ} I + J = { i + jiI , jJ}.",
"The fractional ideals of $\\mathcal {O}_{K}$ are invertible, that is for every fractional ideal $I$ , there exists $I^{-1}:= \\lbrace x\\in K\\mid xI\\subseteq \\mathcal {O}_{K}\\rbrace $ such that $II^{-1} = \\mathcal {O}_{K}$ .",
"The set of fractional ideals is equipped with a norm function defined by $\\mathcal {N}(I) = \\det (M^I)/\\det (\\mathcal {O}_{K})$ where the rows of $M^I$ are a $\\mathbb {Z}$ -basis of $I$ .",
"The norm of ideals is multiplicative, and in the case of an integral ideal, we have $\\mathcal {N}(I) = |\\mathcal {O}_{K}/ I|$ .",
"Also note that the norm of $x\\in K$ is precisely the norm of the principal ideal $(x) = x\\mathcal {O}_{K}$ .",
"Algorithms for ideal arithmetic in polynomial time are described in Section ."
],
[
"The HNF",
"Let $M\\subseteq K^l$ be a finitely generated $\\mathcal {O}_{K}$ -module.",
"As in [2], we say that $[(a_i),(\\mathfrak {a}_i)]_{i\\le n}$ , where $a_i\\in K$ and $\\mathfrak {a}_i$ is a fractional ideal, is a pseudo-basis for $M$ if $M = \\mathfrak {a}_1a_1\\oplus \\cdots \\oplus \\mathfrak {a}_na_n.$ Note that a pseudo-basis is not unique, and the main result of [5] is precisely to compute a pseudo-basis of short elements.",
"If the sum is not direct, we call $[(a_i),(\\mathfrak {a}_i)]_{i\\le n}$ a pseudo-generating set for $M$ .",
"Once a pseudo-generating set $[(a_i),(\\mathfrak {a}_i)]_{i\\le n}$ for $M$ is known, we can associate a pseudo-matrix $A = (A,I)$ to $M$ , where $A\\in K^{n\\times l}$ and $I = (\\mathfrak {a}_i)_{i\\le n}$ is a list of $n$ fractional ideals such that $M = \\mathfrak {a}_1 A_1 + \\cdots +\\mathfrak {a}_n A_n,$ where $A_i\\in K^l$ is the $i$ -th row of $A$ .",
"We can construct a pseudo-basis from a pseudo-generating set by using the Hermite normal form (HNF) over Dedekind domains (see [2]).",
"Note that this canonical form is also refered to as the pseudo-HNF in [2].",
"In this paper we simply call it HNF, but we implicitly refer to the standard HNF over $\\mathbb {Z}$ when dealing with an integer matrix.",
"Assume $A$ is of rank $l$ (in particular $n\\ge l$ ), then there exists an $n\\times n$ matrix $U = (u_{i,j})$ and $n$ non-zero ideals $\\mathfrak {b}_1, \\cdots , \\mathfrak {b}_n$ satisfying $\\forall i,j, u_{i,j}\\in \\mathfrak {b}_i^{-1}\\mathfrak {a}_j$ .",
"$\\mathfrak {a}= \\det (U)\\mathfrak {b}$ for $\\mathfrak {a}= \\prod _i\\mathfrak {a}_i$ and $\\mathfrak {b}= \\prod _i \\mathfrak {b}_i$ .",
"The matrix $UA$ is of the form $ UA = \\left(\\begin{BMAT}(@)[0.5pt,2cm,2cm]{c}{c.c}\\begin{BMAT}(e)[1pt,1cm,1cm]{cccc}{cccc}1 & 0 & \\hdots & 0 \\\\\\vdots & 1 & \\ddots & \\vdots \\\\\\vdots & \\vdots & \\ddots & 0 \\\\* & * & \\hdots & 1\\\\\\end{BMAT} \\\\\\begin{BMAT}[0.5pt,1cm,1cm]{c}{c}(0)\\end{BMAT}\\end{BMAT}\\right).$ $M = \\mathfrak {b}_{1}\\omega _1\\oplus \\cdots \\oplus \\mathfrak {b}_{l}\\omega _l$ where $\\omega _1,\\cdots \\omega _l$ are the first $l$ rows of $UA$ .",
"In general, the algorithm of [2] for computing the HNF of a pseudo-matrix takes exponential time, but as in the integer case, there exists a modular one which is polynomial in the dimensions of $A$ , the degree of $K$ , and the bit size of the modulus.",
"Note that in the case of a pseudo matrix representing an $\\mathcal {O}_{K}$ -module $M$ , the modulus is an integral multiple of the determinantal ideal $\\mathfrak {g}(M)$ , which is generated by all the ideals of the form $\\det _{i_1,\\cdots ,i_l}(A)\\cdot \\mathfrak {a}_{i_1}\\cdots \\mathfrak {a}_{i_l},$ where $\\det _{i_1,\\cdots ,i_l}(A)$ is the determinant of the $l\\times l$ minor consisting of the last $l$ columns of rows of indices $i_1,\\cdots ,i_l$ .",
"The determinantal ideal is a rather involved structure, except in the case $l = n$ .",
"In applications, the modulus is frequently known.",
"In the rest of the paper, we restrict ourselves to the case of an $n\\times n$ matrix $A$ of rank $n$ .",
"One can immediatly derive polynonmial time algorithms for the rectangular case, and for the case of a singular matrix $A$ ."
],
[
"Notion of size",
"To ensure that our algorithm for computing an HNF basis of an $\\mathcal {O}_{K}$ -module runs in polynomial time, we need a notion of size that bounds the bit size required to represent ideals and field elements.",
"An ideal $I\\subseteq \\mathcal {O}_{K}$ is given by the matrix $M^I\\in \\mathbb {Z}^{d\\times d}$ of its basis expressed in an integral basis $\\omega _1,\\cdots ,\\omega _d$ of $\\mathcal {O}_{K}$ .",
"If the matrix is in Hermite Normal Form, the size required to store it is therefore bounded by $d^2\\max _{i,j}\\left(\\log (|M^I_{i,j}|)\\right)$ , where $\\log (x)$ is the base 2 logarithm of $x$ .",
"In the meantime, every coefficient of $M^I$ is bounded by $|\\det (M^I)|=\\mathcal {N}(I)$ (see [3]).",
"Thus, we define the size of an ideal as $S(I):= d^2\\log (\\mathcal {N}(I)).$ If $\\mathfrak {a}= (1/k)I$ is a fractional ideal of $K$ , where $I\\subseteq \\mathcal {O}_{K}$ and $k\\in \\mathbb {Z}_{>0}$ is minimal, then the natural generalization of the notion of size is $S(\\mathfrak {a}) := \\log (k) + S(I),$ where $\\log (k)$ is the base 2 logarithm of $k$ .",
"We also define the size of elements of $K$ .",
"If $x\\in \\mathcal {O}_{K}$ can be written as $x = \\sum _{i\\le d}x_i\\omega _i$ , where $x_i\\in \\mathbb {Z}$ , then we define its size by $S(x) := d\\log (\\max _i|x_i|).$ It can be generalized to elements $y\\in K$ by writing $y = x/k$ where $x\\in \\mathcal {O}_{K}$ and $k$ is a minimal positive integer, and by setting $S(y) := \\log (k) + S(x).$ In the litterature, the size of elements of $K$ is often expressed with $\\Vert x\\Vert $ .",
"These two notions are in fact related.",
"Proposition 1 Let $x\\in \\mathcal {O}_{K}$ , the size of $x$ and its $T_2$ -norm satisfy (x)O( S(x)d + d2 + |K|) S(x)O( d( d+ (x))).",
"In appendix So, for all $x\\in \\mathcal {O}_{K}$ , $S(x) = O\\left(\\log \\left( \\Vert x\\Vert \\right)\\right)$ , and $\\log \\left( \\Vert x\\Vert \\right) = O(S(x))$ , where the constants are polynomial in $d$ and $\\log |\\Delta _K|$ .",
"Corollary 1 Let $x,y\\in \\mathcal {O}_{K}$ , their size satisfies $S(xy)\\le \\tilde{O}\\left( d^3 + d\\log |\\Delta _K| + S(x) + S(y)\\right).$"
],
[
"Cost model",
"We assume that the module $M$ satifies $M\\subseteq \\mathcal {O}_{K}^n$ and that $\\mathcal {O}_{K}$ is given by an LLL-reduced integral basis $\\omega _1,\\cdots ,\\omega _d$ such that $\\omega _1 = 1$ .",
"The computation of such a basis can be done by using [1] to produce a good integral basis for $\\mathcal {O}_{K}$ and then reducing it with the LLL algorithm [7].",
"In this section, we evaluate the complexity of the basic operations performed during our algorithm.",
"We rely on standard number theoretic algorithms.",
"We multiply two integers of bit size $h$ in time $\\mathcal {M}(h) \\le O\\left( h \\log (h)\\log (\\log (h))\\right)$ using Schönhage-Strassen algorithm, while the addition of such integers is in $O(h)$ , their division has complexity bounded by $O(\\mathcal {M}(h))$ , and the Euclidiean algorihm that provides their GCD has complexity $O(\\log (h)\\mathcal {M}(h))$ (see [11]).",
"In the following, we also refer to two standard linear algebra algorithms, namely the HNF computation over the integers due to Storjohann [15] in complexity $\\left(nmr^{\\omega -1}\\log |A|\\right)^{1+o(1)}$ and Dixon's $p$ -adic algorithm for solving linear systems in $\\left(n^{\\omega } \\log |A| \\right)^{1 + o(1)},$ where $A\\in \\mathbb {Z}^{m\\times n}$ has rank $r$ and has its entries bounded by $|A|$ , and where $3\\ge \\omega \\ge 2$ is the exponent of the complexity of matrix multiplication.",
"We need to perform additions, multiplications and inversions of elements of $K$ , as well as of fractional ideals.",
"There is no reference on the complexity of these operations, although many implementations can be found.",
"We adress this problem in the rest of this section.",
"We use $\\tilde{O}$ to denote the complexity were all the logarithmic factors are omitted.",
"Elements $x$ of $K$ are represented as quotients of an element of $\\mathcal {O}_{K}$ and a positive denominator.",
"We add them naively while their multiplication is done by using a precomputed table of the $\\omega _i\\omega _j$ for $i,j\\le d$ .",
"Proposition 2 Let $\\alpha ,\\beta \\in K$ such that $S(\\alpha ),S(\\beta )\\le B$ , then the following holds: $\\alpha + \\beta $ can be computed in $\\tilde{O}(dB)$ $\\alpha \\beta $ can be computed in $\\tilde{O}\\left(d^2(B + d^3 + d\\log |\\Delta _K|)\\right)$ $\\frac{1}{\\alpha }$ can be computed in $\\tilde{O}\\left(d^{\\omega -1}(B + d^3 + d\\log |\\Delta _K|)\\right)$ , Adding $\\alpha $ and $\\beta $ is straightforward.",
"Multiplying them is done by storing a precomputed multiplication table for the $\\omega _i\\omega _j$ .",
"Finally, inverting $\\alpha $ boils down to solving a linear system in the coefficients of $\\frac{1}{\\alpha }$ .",
"More details are given in appendix.",
"Ideals of $\\mathcal {O}_{K}$ are given by their HNF representation with respect to the integral basis $\\omega _1,\\cdots ,\\omega _d$ of $\\mathcal {O}_{K}$ .",
"It consists of the HNF of the matrix representing the $d$ generators of their $\\mathbb {Z}$ basis as rows.",
"Operations on this matrix yield the addition, multiplication and inverse of an integral ideal.",
"The corresponding operations on fractional ideals are trivialy deduced by taking care of the denominator.",
"Proposition 3 Let $\\mathfrak {a}$ and $\\mathfrak {b}$ be fractional ideals of $K$ such that $S(\\mathfrak {a}),S(\\mathfrak {b})\\le B$ , then the following holds: $\\mathfrak {a}+ \\mathfrak {b}$ can be computed in $\\tilde{O}(d^{\\omega +1}B)$ , $\\mathfrak {a}\\mathfrak {b}$ can be computed in $\\tilde{O}(d^3(d^4 + d^2\\log |\\Delta _K| + B))$ , $1/\\mathfrak {a}$ can be computed in $\\tilde{O}\\left( d^{2\\omega }(d^4 + d^2\\log |\\Delta _K| + B)\\right)$ .",
"The addition of integral ideals $\\mathfrak {a}$ and $\\mathfrak {b}$ given by their HNF matrix $A$ and $B$ is given by the HNF of $\\left(\\frac{A}{B}\\right)$ .",
"To multiply them, one has to compute the HNF of the matrix whose $d^2$ rows represent $\\gamma _i\\delta _j$ where $(\\gamma _i)_{i\\le d}$ is an integral basis for $\\mathfrak {a}$ and $(\\delta _i)_{i\\le d}$ is an integral basis for $\\mathfrak {b}$ .",
"Finally, following the approach of [3], inverting $\\mathfrak {a}$ boils down to solving a $d^2\\times (d+d^2)$ linear system.",
"More details are given in appendix.",
"Note that the reason why the dependency in $B$ in the complexity of the addition of fractional ideals is slightly more than in the complexity of the multiplication is the way we deal with the denominators.",
"In the case of integral ideals, the addition would be in $\\tilde{O}(d^{\\omega -1}B)$ .",
"The last operation that needs to be performed during our HNF algorithm is the multiplication between an element of $K$ and a fractional ideal.",
"Proposition 4 Let $\\alpha \\in K$ , a fractional ideal $\\mathfrak {a}\\subseteq K$ and $B_1,B_2$ such that $S(\\mathfrak {a})\\le B_1$ and $S(\\alpha )\\le B_2$ , then $\\alpha \\mathfrak {a}$ can be computed in expected time bounded by $\\tilde{O}\\left(d^{\\omega }\\left(d^3 + d\\log |\\Delta _K| + \\frac{B_1}{d} + B_2 \\right)\\right).$ If $\\gamma _1,\\cdots ,\\gamma _d$ is an integral basis for $\\mathfrak {a}\\subseteq \\mathcal {O}_{K}$ , then $(\\alpha \\gamma _i)_{i\\le d}$ is one for $(\\alpha )\\mathfrak {a}$ .",
"The HNF of the matrix representing these elements leads to the desired result.",
"More details are given in appendix."
],
[
"The normalization",
"The normalization is the key difference between our approach and the one of Cohen [2].",
"It is the strategy that prevents the coefficient swell by calculating a pseudo-basis for which the ideals are integral with size bounded by the field's invariants.",
"Given a one-dimensional $\\mathcal {O}_{K}$ -module $\\mathfrak {a}A\\subseteq \\mathcal {O}_{K}^n$ where $\\mathfrak {a}$ is a fractional ideal of $K$ , and $A\\in K^n$ , we find $b\\in K$ such that the size taken to represent our module as $(b\\mathfrak {a})(A/b)$ is reasonably bounded.",
"Indeed, any non trivial module can be represented by elements of arbitrary large size, which would cause a significant slow-down in our algorithm.",
"The first step to our normalization is to make sure that $\\mathfrak {a}$ is integral.",
"This allows us to bound the denominator of the coefficients of the matrix when manipulating its rows during the HNF algorithm.",
"If $k\\in \\mathbb {Z}$ is the denominator of $\\mathfrak {a}$ , then replacing $\\mathfrak {a}$ by $k\\mathfrak {a}$ and $A$ by $A/k$ increases the size needed to represent our module via the growth of all the denominators of the coefficients of $A\\in K^n$ .",
"Thus, after this operation, the size of each coefficient $a_i$ of $A$ is bounded by $S(a_i) + S(\\mathfrak {a})$ .",
"We can now assume that our one-dimensional module is of the form $\\mathfrak {a}A$ where $\\mathfrak {a}\\subseteq \\mathcal {O}_{K}$ and $A\\in K^n$ at the price of a slight growth of its size.",
"The next step of normalization is to express our module as $\\mathfrak {a}^{\\prime } A^{\\prime }$ where $A^{\\prime }\\in K^n$ and $\\mathfrak {a}^{\\prime }\\subseteq \\mathcal {O}_{K}$ such that $\\mathcal {N}(\\mathfrak {a}^{\\prime })$ only depends on invariants of the field.",
"To do this, we invert $\\mathfrak {a}$ and write it as $\\mathfrak {a}^{-1} = \\frac{1}{k}\\mathfrak {b},$ where $k\\in \\mathbb {Z}_{>0}$ and $\\mathfrak {b}\\subseteq \\mathcal {O}_{K}$ .",
"As $\\mathcal {N}(\\mathfrak {a})\\in \\mathfrak {a}$ , we have $\\mathcal {N}(\\mathfrak {a})\\mathfrak {a}^{-1}\\subseteq \\mathcal {O}_{K}$ and thus $k\\le \\mathcal {N}(\\mathfrak {a})$ .",
"Therefore, $\\mathcal {N}(\\mathfrak {b}) \\le \\frac{\\mathcal {N}(k)}{\\mathcal {N}(\\mathfrak {a})}\\le \\frac{k^{d}}{\\mathcal {N}(\\mathfrak {a})} \\le \\mathcal {N}(\\mathfrak {a})^{d-1}.$ Then we use the LLL algorithm to find an element $\\alpha \\in \\mathfrak {b}$ such that $\\Vert \\alpha \\Vert \\le d^{1/2}2^{d/2}|\\Delta _K|^{1/2d}\\mathcal {N}(\\mathfrak {b})^{1/d}.$ Our reduced ideal is $\\mathfrak {a}^{\\prime } := \\left(\\frac{\\alpha }{k}\\right)\\mathfrak {a}\\subseteq \\mathfrak {a}^{-1}\\mathfrak {a}= \\mathcal {O}_{K}.$ The integrality of $\\mathfrak {a}^{\\prime }$ comes from the definition of $\\mathfrak {b}^{-1}$ and the fact that $\\alpha \\in \\mathfrak {b}$ .",
"From the arithmetic-geometric mean, we know that $\\mathcal {N}(\\alpha )\\le \\frac{\\Vert \\alpha \\Vert ^d}{d^d}$ , therefore $\\mathcal {N}(\\alpha )\\le 2^{d^2/2}\\sqrt{|\\Delta _K|}\\mathcal {N}(\\mathfrak {b}),$ and the norm of the reduced ideal can be bounded by $\\mathcal {N}(\\mathfrak {a}^{\\prime })\\le 2^{d^2/2}\\sqrt{|\\Delta _K|}$ .",
"On the other hand, we set $A^{\\prime } := (k/\\alpha )A$ , which induces a growth of the coefficients $a_i$ of $A$ .",
"Indeed, each $a_i$ is multiplied by $(k/\\alpha )$ .",
"Proposition 5 The size of the normalized module $\\mathfrak {a}^{\\prime }A^{\\prime }$ of $\\mathfrak {a}A\\subseteq K^n$ satisfies S(a'i) O( d3 + d|K| + S(a) + S(ai)) S(a') O( d3 + d|K|) From Corollary REF we know that $S\\left(\\frac{a_ik}{\\alpha }\\right) \\le \\tilde{O}\\left( d^3 + d\\log |\\Delta _K| + \\frac{S(\\mathfrak {a})}{d} + S(a_i) + S\\left(\\frac{1}{\\alpha }\\right)\\right)$ In addition, if $\\frac{1}{\\alpha } = \\frac{x}{k^{\\prime }}$ where $x\\in \\mathcal {O}_{K}$ and $k^{\\prime }\\in \\mathbb {Z}_{>0}$ , then $S\\left(\\frac{1}{\\alpha }\\right)\\le \\tilde{O}\\left( \\log (k^{\\prime }) + d(d+\\log \\Vert x\\Vert )\\right).$ On the one hand, we have k' N()2d2/2|K|N(a)d-1, and on the other hand, we need to bound $\\Vert x\\Vert $ .",
"We notice that since $\\mathcal {N}(\\alpha )\\in \\mathbb {Q}$ , $\\forall j\\le d$ , $\\mathcal {N}(\\alpha ) = \\alpha \\beta = \\sigma _j(\\alpha \\beta )$ .",
"We also know that $\\forall j$ , $|\\sigma _j(\\alpha )|\\le \\Vert \\alpha \\Vert $ .",
"Therefore, $\\forall j\\le d, \\ |\\sigma _j(x)| = \\frac{\\mathcal {N}(\\alpha )}{|\\sigma _j(\\alpha )|}= \\prod _{i\\ne j}|\\sigma _i(\\alpha )|\\le \\Vert \\alpha \\Vert ^{d-1}.$ Therefore $\\Vert x\\Vert \\le \\sqrt{d}\\Vert \\alpha \\Vert ^{d-1}$ , and thus $S\\left(\\frac{1}{\\alpha }\\right)\\le \\tilde{O}\\left( d^3 + d\\log |\\Delta _K| + S(\\mathfrak {a})\\right).$ Our normalization, summarized in Algorithm , was performed at the price of a reasonable growth in the size of the object we manipulate.",
"Let us now evaluate its complexity.",
"[ht] Normalization of a one-dimensional module [1] $A \\in K^{n}$ , fractional ideal $\\mathfrak {a}$ of $K$ .",
"$A^{\\prime } \\in K^{n}$ , $\\mathfrak {a}^{\\prime }\\subseteq \\mathcal {O}_{K}$ such that $\\mathcal {N}(\\mathfrak {a}^{\\prime })\\le 2^{d^2/2}\\sqrt{|\\Delta _K|}$ and $\\mathfrak {a}A = \\mathfrak {a}^{\\prime } A^{\\prime }$ .",
"$\\mathfrak {a}\\leftarrow k_0\\mathfrak {a}$ , $A \\leftarrow A/k_0$ where $k_0$ is the denominator of $\\mathfrak {a}$ .",
"$\\mathfrak {b}\\leftarrow k \\mathfrak {a}^{-1}$ where $k$ is the denominator of $\\mathfrak {a}^{-1}$ .",
"Let $\\alpha $ be the first element of an LLL-reduced basis of $\\mathfrak {b}$ .",
"$\\mathfrak {a}^{\\prime }\\leftarrow \\left(\\frac{\\alpha }{k}\\right) \\mathfrak {a}$ , $A^{\\prime } \\leftarrow \\left(\\frac{k}{\\alpha }\\right)A$ .",
"$\\mathfrak {a}^{\\prime }$ , $A^{\\prime }$ .",
"Proposition 6 Let $B_1,B_2$ such that $S(\\mathfrak {a})\\le B_1$ and $\\forall i, S(a_i)\\le B_2$ , then the complexity of Algorithm is bounded by $\\tilde{O}\\left(nd^2(d^3 + B_1 + B_2 + d\\log |\\Delta _K|)\\right).$ The inversion of $\\mathfrak {a}$ is performed in time $\\tilde{O}\\left( d^{2\\omega }(d^4 + d^2\\log |\\Delta _K| + B_1)\\right),$ by using Proposition REF .",
"Then, the LLL-reduction of the basis of $\\mathfrak {b}$ is done by the $L^2$ algorithm of Stehlé and Nguyen [12] in expected time bounded by $\\tilde{O}\\left(d^3\\left(d + \\frac{S(\\mathfrak {b})}{d^2}\\right)\\frac{S(\\mathfrak {b})}{d^2}d\\right)\\le \\tilde{O}\\left(d^2S(\\mathfrak {a})(d^2+S(\\mathfrak {a}))\\right).$ Then, computing $(\\alpha /k)\\mathfrak {a}$ is the multiplication of the ideal $\\mathfrak {a}$ by the element $\\alpha /k$ which satisfies $S(\\alpha /k)\\le \\tilde{O}\\left( d^2 + \\log |\\Delta _K| + S(\\mathfrak {a})/d\\right).$ This takes $\\tilde{O}\\left(d^{\\omega -1}(S(\\mathfrak {a}) + d^4 + d^2\\log |\\Delta _K|)\\right)$ .",
"Finally, computing $k(1/\\alpha )A$ consists of inverting $\\alpha $ with $S(\\alpha )\\le \\tilde{O}(d^3 + \\log |\\Delta _K| + B_1/d)$ , which takes $\\tilde{O}\\left(d^{\\omega -1}(d^3 + B_1/d + d\\log |\\Delta _K|)\\right),$ and performing $n+1$ multiplications between elements of size bounded by $\\tilde{O}(d^3 + B_1 + B_2 + d\\log |\\Delta _K|)$ , which is done in time $\\tilde{O}\\left(nd^2(d^3 + B_1 + B_2 + d\\log |\\Delta _K|)\\right).$ The result follows from the combination of the above expected times and from the fact that $2\\le \\omega \\le 3$ ."
],
[
"Reduction modulo a fractional ideal",
"To achieve a polynomial complexity for our HNF algorithm, we reduce some elements of $K$ modulo ideals whose norm can be reasonably bounded.",
"We show in this section how to bound the norm of a reduced element with respect to the norm of the ideal and invariants of $K$ .",
"Let $\\mathfrak {a}$ be a fractional ideal of $K$ , and $x\\in K$ .",
"Our goal is to find $\\overline{x} \\in K$ such that $\\Vert \\overline{x} \\Vert $ is bounded, and that $x - \\overline{x} \\in \\mathfrak {a}$ .",
"The reduction algorithm consists of finding an LLL-reduced basis $r_1,\\cdots , r_d$ of $\\mathfrak {a}$ and to decompose $x = \\sum _{i\\le d } x_i r_i.$ Then, we define $\\overline{x} := x - \\sum _{i\\le d } \\lfloor x_i \\rceil r_i.$ Proposition 7 Let $x\\in K$ and $\\mathfrak {a}$ be a fractional ideal of $K$ , then Algorithm returns $\\overline{x}$ such that $x - \\overline{x}\\in \\mathfrak {a}$ and $\\Vert \\overline{x}\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}.$ In appendix [ht] Reduction modulo a fractional ideal [1] $\\alpha \\in K$ , fractional ideal $\\mathfrak {a}$ of $K$ .",
"$\\overline{\\alpha }\\in K$ such that $\\alpha - \\overline{\\alpha } \\in \\mathfrak {a}$ and $\\Vert \\overline{\\alpha }\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}$ .",
"$\\Vert \\alpha \\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}$ or $\\alpha = 1$ $\\alpha $ .",
"Compute an LLL-reduced basis $(r_i)_{i\\le d}$ of $\\mathfrak {a}$ .",
"Decompose $\\alpha = \\sum _{i\\le d} x_i r_i$ .",
"$\\overline{\\alpha } \\leftarrow \\alpha - \\sum _{i\\le d } \\lfloor x_i \\rceil r_i.$ $\\overline{\\alpha }$ .",
"Proposition 8 Let $B_1,B_2$ such that $S(\\mathfrak {a})\\le B_1$ and $S(\\alpha )\\le B_2$ , then the complexity of Algorithm is bounded by $\\tilde{O}\\left( B_1(d^3 + B_1) + d^{\\omega - 1}B_2 + d^{\\omega +2} \\right)$ To compute the LLL-reduced basis of $\\mathfrak {a}$ , we LLL-reduce the integral ideal $k\\mathfrak {a}$ where $k\\in \\mathbb {Z}_{>0}$ is the denominator of $\\mathfrak {a}$ .",
"Then, we express $x$ with respect to the basis of $k\\mathfrak {a}$ where $x\\in \\mathcal {O}_{K}$ satifies $\\alpha = x/a$ for $a\\in \\mathbb {Z}_{>0}$ .",
"Then we divide by the respective denominator at the extra cost of $d$ multiplications.",
"Using the $L^2$ algorithm of Stehlé and Nguyen [12] yields the reduced basis of $k\\mathfrak {a}$ in expected time bounded by $\\tilde{O}\\left( d^3\\left( d + \\frac{S(\\mathfrak {a})}{d^2}\\right) \\frac{S(\\mathfrak {a})}{d^2}d\\right)\\le \\tilde{O}\\left( S(\\mathfrak {a})(d^3 + S(\\mathfrak {a}))\\right).$ Then, expressing $x$ with respect to the reduced basis of $k\\mathfrak {a}$ costs $\\tilde{O}\\left( d^{\\omega }\\left(\\frac{S(\\mathfrak {a})}{d^2} + d\\log |\\Delta _K| +d^2 + \\frac{S(x)}{d}\\right)\\right).$ Finally, the subtraction and the division by the denominators are in $\\tilde{O}\\left(d\\frac{S(\\alpha )}{d}\\right).$"
],
[
"Modular HNF Algorithm",
"Let $M \\subseteq \\mathcal {O}_{K}^n$ be an $\\mathcal {O}_{K}$ -module.",
"We use a variant of the modular version of [2] which ensures that the current pseudo-basis $[\\mathfrak {a}_i,A_i]_{i\\le n}$ of the module satisfies $\\mathfrak {a}\\subseteq \\mathcal {O}_{K}$ at every step of the algorithm.",
"This extra feature allows us to bound the denominator of coefficients of the matrix whose rows we manipulate.",
"Algorithm computes the HNF modulo the determinantal ideal $\\mathfrak {g}$ , and Algorithm recovers an actual HNF for $M$ .",
"In this section, we discuss the differences between Algorithms and and their equivalent in [2].",
"After the original normalization, all the ideals are integral.",
"As $M\\subseteq \\mathcal {O}_{K}^n$ , we immediatly deduce that the ideal $\\mathfrak {d}$ created at Step 6 of Algorithm is integral as well.",
"In addition, from the definition of the inverse of an ideal we also have that $\\frac{b_{i,i}\\mathfrak {b}_ib_{i,j}\\mathfrak {b}_j}{b_{i,j}\\mathfrak {b}_j + b_{i,i}\\mathfrak {b}_i}\\subseteq \\mathcal {O}_{K},$ which allows us to conclude that the update of $(\\mathfrak {b}_i,\\mathfrak {b}_j)$ performed at Step 9 of Algorithm preserves the fact that our ideals are integral.",
"[ht] HNF of a full-rank square pseudo-matrix modulo $\\mathfrak {g}$ [1] $A \\in K^{n\\times n}$ , $\\mathfrak {a}_1,\\cdots ,\\mathfrak {a}_n$ , $\\mathfrak {g}$ .",
"pseudo-HNF $B$ , $\\mathfrak {b}_1,\\cdots ,\\mathfrak {b}_n$ modulo $\\mathfrak {g}$ .",
"$B\\leftarrow A$ , $\\mathfrak {b}_i\\leftarrow \\mathfrak {a}_i$ , $j\\leftarrow n$ .",
"Normalize $[(B_i),(\\mathfrak {b}_i)]_{i\\le n}$ with Algorithm $j\\ge 1$ $i \\leftarrow j-1$ .",
"$i\\ge 1$ $\\mathfrak {d}\\leftarrow b_{i,j}\\mathfrak {b}_i + b_{j,j}\\mathfrak {b}_j$ Find $u\\in \\mathfrak {b}_i\\mathfrak {d}^{-1}$ and $v\\in \\mathfrak {b}_j\\mathfrak {d}^{-1}$ such that $b_{i,j}u+ b_{j,j}v = 1$ with [2].",
"$(B_i,B_j)\\leftarrow (b_{j,j}B_i-b_{i,j}B_j,uB_i + vB_j)$ .",
"$(\\mathfrak {b}_i,\\mathfrak {b}_j)\\leftarrow (b_{i,j}\\mathfrak {b}_ib_{j,j}\\mathfrak {b}_j\\mathfrak {d}^{-1},\\mathfrak {d})$ .",
"Normalize $\\mathfrak {b}_i,B_i$ with Algorithm .",
"Reduce $B_i$ modulo $\\mathfrak {g}\\mathfrak {b}_i^{-1}$ and $B_j$ modulo $\\mathfrak {g}\\mathfrak {b}_j^{-1}$ with Algorithm .",
"$i\\leftarrow i-1$ .",
"$j\\leftarrow j-1$ .",
"$(\\mathfrak {b}_i)_{i\\le n}$ , $B$ .",
"The normalization and reduction at Step 10-11 allow us to keep the size of the $B_i$ and of the $\\mathfrak {b}_i$ reasonably bounded by invariants of $K$ and the dimension of the module.",
"By doing so, we give away some information about the module $M$ .",
"However, algorithm allows us to recover $M$ , as we state in Proposition REF .",
"Proposition 9 The $\\mathcal {O}_{K}$ -module defined by the pseudo-basis $[(W_i),(\\mathfrak {c}_i)]$ obtained by applying Algorithm to the HNF of $M$ modulo $\\mathfrak {g}(M)$ satisfies $\\mathfrak {c}_1 W_1 + \\cdots + \\mathfrak {c}_n W_n = M.$ The proof of this statement essentially follows its equivalent for matrices over the integers.",
"It consists of showing that $W:= \\sum _i\\mathfrak {c}_i$ and $M:= \\sum _i\\mathfrak {a}_iA_i$ have the same determinantal ideal and that $W\\subseteq A$ , and then showing that this implies that $W = M$ .",
"A more complete proof is given in appendix.",
"[ht] Eucledian reconstruction of the HNF [1] $B \\in K^{n\\times n}$ , $\\mathfrak {b}_1,\\cdots ,\\mathfrak {b}_n$ output of Algorithm modulo $\\mathfrak {g}$ for $M\\subseteq \\mathcal {O}_{K}^n$ .",
"An HNF $W$ ,$\\mathfrak {c}_1,\\cdots ,\\mathfrak {c}_n$ for $M$ .",
"$j\\leftarrow n$ , $\\mathfrak {g}_j\\leftarrow \\mathfrak {g}$ .",
"$j\\ge 1$ $\\mathfrak {c}_j\\leftarrow \\mathfrak {b}_j + \\mathfrak {g}_j$ .",
"Find $u\\in \\mathfrak {b}_j\\mathfrak {d}^{-1}$ and $v\\in \\mathfrak {g}\\mathfrak {c}^{-1}_j$ such that $u + v = 1$ .",
"$W_j\\leftarrow uB_j\\bmod \\unknown.",
"\\mathfrak {g}\\mathfrak {c}^{-1}_j$ .",
"$\\mathfrak {g}_j \\leftarrow \\mathfrak {g}_j\\mathfrak {c}^{-1}_j$ .",
"$j\\leftarrow j-1$ .",
"$W,(\\mathfrak {c}_i)_{i\\le n}$ ."
],
[
"Complexity of the modular HNF",
"Let us assume that we are able to compute the determinantal ideal $\\mathfrak {g}$ of our module $M$ in polynomial time with respect to the bit size of the invariants of the field and of $S(\\mathfrak {g})$ .",
"We discuss the computation of $\\mathfrak {g}$ in Section .",
"In this section, we show that Algorithm and Algorithm are polynomial wih respect to the same parameters.",
"This result is analogous to the case of integers matrices.",
"Indeed, the only thing we need to verify is that the size of the elements remains reasonably bounded during the algorithm.",
"In Algorithm , the coefficient explosion is prevented by the modular reduction of Step 11.",
"It ensures that $\\forall i_1, i_2<j, \\ \\Vert b_{i_1,i_2}\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {g}\\mathfrak {b}_{i_1}^{-1})^{1/d}\\sqrt{|\\Delta _K|}.$ This is not enough to prevent the explosion since $b_{i_1,i_2}$ might not be integral.",
"Therefore, there is a minimal $k\\in \\mathbb {Z}_{>0}$ such that $kb_{i_1,i_2}\\in \\mathcal {O}_{K}$ , which we need to bound to ensure that $S(b_{i_1,i_2})$ remains bounded as well.",
"We know that $b_{i,j}\\mathfrak {b}_i\\subseteq \\mathcal {O}_{K}$ , and that $\\mathfrak {b}_i$ is integral.",
"Thus, $\\mathcal {N}(k)\\mid \\mathcal {N}(\\mathfrak {b}_{i_1})$ , which in turns implies that $k\\le \\mathcal {N}(\\mathfrak {b}_{i_1})$ .",
"As on the other hand, the normalization of Step 10 ensures that $\\mathcal {N}(\\mathfrak {b}_{i_1})\\le 2^{d^2/2}\\sqrt{|\\Delta _K|}$ , we conclude that after Step 11, S(bi1,i2)O( d2 + d|K| + S(g)d2).",
"In Algorithm , we last manipulate $B_j$ and $\\mathfrak {b}_j$ when the index $j$ is the pivot.",
"In that case, we cannot use the normalization to bound the size since we require that $b_{j,j} = 1$ .",
"However we reduce $B_j$ modulo $\\mathfrak {g}\\mathfrak {b}_j$ , which means that $\\forall i\\le j,\\ \\Vert b_{i,j}\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {g}\\mathfrak {b}_i^{-1})^{1/d}\\sqrt{|\\Delta _K|}.$ In addition, the arithmetic-geometric tells us that $\\Vert b_{j,j}\\Vert \\ge \\sqrt{d}\\mathcal {N}(b_{i,j})^{1/d}$ , which in turn implies that $\\forall i\\le j,\\ \\mathcal {N}(b_{i,j}\\mathfrak {b}_i)\\le d^d 2^{d^2/2} \\mathcal {N}(\\mathfrak {g})^d |\\Delta _K|^{d/2}.$ As we know that $\\mathcal {N}(b_{i,j}\\mathfrak {b}_i + b_{j,j}\\mathfrak {b}_j) \\le \\max \\left(\\mathcal {N}(b_{i,j}\\mathfrak {b}_i),\\mathcal {N}(b_{j,j}\\mathfrak {b}_j)\\right),$ we therefore know that after Step 9 $\\mathcal {N}(\\mathfrak {b}_j)\\le d^d 2^{d^2/2} \\mathcal {N}(\\mathfrak {g})^d |\\Delta _K|^{d/2},$ which allows us to bound the size of the denominators in the $j$ -th row the same way we did for the rows of index $i_1<j$ : $\\forall i\\le j, \\ S(b_{i,j})\\le \\tilde{O}\\left( d^2 + d\\log |\\Delta _K| + \\frac{S(\\mathfrak {g})}{d^2}\\right).$ Proposition 10 The complexity of Algorithm is in $\\tilde{O}\\left( n^3d^2\\left(d^3 + d^2\\log |\\Delta _K| + S(\\mathfrak {g})\\right)^2\\right).$ Steps 6 to 11 of Algorithm are repeated $O(n^2)$ times.",
"Let us analyze their complexity.",
"First, at Step 6 we have S(bi,j) O(d3 + |K| + S(g)d2) S(bi) O(d3 + |K|) so from Proposition REF , computing $b_{i,j}\\mathfrak {b}_i$ takes $\\tilde{O}\\left( d^{\\omega -2}(d^5 + d^{3}\\log |\\Delta _K| + S(\\mathfrak {g}))\\right).$ Then, from Proposition REF and (REF ), $S(b_{i,j}\\mathfrak {b}_i)\\le \\tilde{O}\\left(d^4 + dS(\\mathfrak {g}) + d^3\\log |\\Delta _K|\\right),$ and computing $\\mathfrak {d}$ costs $\\tilde{O}\\left( d^{\\omega + 2}(d^3 + d^2\\log |\\Delta _K| + S(\\mathfrak {g}))\\right).$ As $S(\\mathfrak {d})\\le S(b_{i,j}\\mathfrak {b}_i)$ , computing $\\mathfrak {d}^{-1}$ takes $\\tilde{O}\\left( d^{2\\omega +1}\\left( d^3 + d^2\\log |\\Delta _K| + S(\\mathfrak {g}) \\right)\\right).$ From [3], this is done by solving a linear system on a matrix $D$ satisfying $\\log |D|\\le \\tilde{O}\\left( d^2 + \\log |\\Delta _K| + \\frac{S(\\mathfrak {g})}{d^2}\\right),$ and the coefficients of the HNF matrix of $\\mathfrak {d}$ are those of a matrix $M$ satisfying $\\log |\\det (M)|\\le \\tilde{O}\\left(d^2\\log |D|\\right)$ .",
"Therefore, we have $S(\\mathfrak {d}^{-1})\\le d^2\\log |\\det (M)| \\le \\tilde{O}\\left( d^2\\left( d^4 + d^2\\log |\\Delta _K| + S(\\mathfrak {g})\\right)\\right).$ As $S(\\mathfrak {b}_i),S(\\mathfrak {b}_j)\\le \\tilde{O}(d^3 + \\log |\\Delta _K|)$ , computing $\\mathfrak {b}_i\\mathfrak {d}^{-1}$ and $\\mathfrak {b}_j\\mathfrak {d}^{-1}$ takes $\\tilde{O}\\left( d^{5}\\left( d^4 + d^2\\log |\\Delta _K| + S(\\mathfrak {g})\\right)\\right).$ Then, from [2], computing $u$ and $v$ is done by finding $u^{\\prime }\\in b_{i,j}\\mathfrak {b}_i\\mathfrak {d}^{-1}$ and $v^{\\prime }\\in b_{j,j}\\mathfrak {b}_j\\mathfrak {d}^{-1}$ such that $u^{\\prime }+v^{\\prime }=1$ and returning $u:= u^{\\prime }/b_{i,j}$ and $v:= v^{\\prime }/b_{j,j}$ .",
"Let $I_i := b_{i,j}\\mathfrak {b}_i\\mathfrak {d}^{-1}\\subseteq \\mathcal {O}_{K}$ .",
"Then, from [2] computing $u^{\\prime },v^{\\prime }$ is done at the cost of an HNF computation of a $2d\\times d$ matrix whose entries have their size bounded by $\\log (\\mathcal {N}(I_j))$ .",
"This cost is in $\\tilde{O}\\left( d^{\\omega }(d^3 + d^2\\log |\\Delta _K| + S(\\mathfrak {g}))\\right).$ In addition, $S(u^{\\prime }),S(v^{\\prime })\\le \\tilde{O}(d^4 +d^3\\log |\\Delta _K| + dS(\\mathfrak {g}))$ .",
"Then, by using the same methdods as in the proof of Proposition REF , we know that $S\\left(\\frac{1}{b_{i,j}}\\right)\\le \\tilde{O}\\left( d^3 + \\frac{S(\\mathfrak {g})}{d} + d^2\\log |\\Delta _K|\\right)$ while Proposition REF ensures us that inverting $b_{i,j}$ is done in $\\tilde{O}\\left( d^{\\omega -1}\\left( d^3 + d\\log |\\Delta _K| + \\frac{S(\\mathfrak {g})}{d^2}\\right) \\right).$ Then, calculating $u^{\\prime }/b_{i,j}$ and $v^{\\prime }/b_{j,j}$ is done in time bounded by $\\tilde{O}\\left( d^2 \\left( d^4 + d^3\\log |\\Delta | + dS(\\mathfrak {g})\\right) \\right),$ and by Corollary REF , we know that $S(u),S(v)\\le \\tilde{O}\\left(d^4 + d^3\\log |\\Delta _K| + dS(\\mathfrak {g})\\right).$ Then, from Proposition REF and (), the expected time for Step 8 is bounded by $\\tilde{O}\\left( nd^2(d^4 + d^3\\log |\\Delta _K| + dS(\\mathfrak {g}))\\right).$ In addition, after Step 8, we have $S(b_{i,j})\\le \\tilde{O}\\left( d^4 + d^3\\log |\\Delta _K| + dS(\\mathfrak {g})\\right).$ Then, from Proposition REF and the bounds on $S(b_{i,j}\\mathfrak {b}_i)$ and $S(\\mathfrak {d}^{-1})$ computed above, Step 9 takes $\\tilde{O}\\left( d^{5}\\left( d^4 + d^2\\log |\\Delta _K| + S(\\mathfrak {g})\\right)\\right).$ By using Proposition REF , we bound the time taken by Step 10 by $\\tilde{O}\\left( nd^3(d^3 + d^2\\log |\\Delta _K| + S(\\mathfrak {g}))\\right),$ Finally, from the bound on $S(b_{i,j})$ after Step 8 and Proposition REF , Step 11 takes $\\tilde{O}\\left( nd^2\\left(d^3 + d^2\\log |\\Delta _K| + S(\\mathfrak {g})\\right)^2\\right).$ The Euclidian reconstruction of Algorithm can be seen as another pivot operation between the two one-dimensional $\\mathcal {O}_{K}$ -modules $\\mathfrak {b}_j B_j$ and $\\mathfrak {g}_je_j$ for each $j\\le n$ .",
"We can therefore bound the entries of $W$ by the same method as for Step 6-11 of Algorithm , we the extra observation $\\mathcal {N}(\\mathfrak {g}_j)\\le \\mathcal {N}(\\mathfrak {g}).$ Therefore, we showed that we could bound the size of the objects that are manipulated throughout the algorithm by values that are polynomial in terms of $n$ , $d$ , $S(\\mathfrak {g})$ and $\\log (|\\Delta _K|)$ , and that the complexity of the HNF algorithm was polynomial in these parameters."
],
[
"Computing the modulus",
"Let us assume that $A\\in \\mathcal {O}_{K}^{n\\times n}$ .",
"If it is not the case, then we need to multiply by the common denominator $k$ of the entries of $A$ and return $\\det (kA)/k^n$ .",
"In this section, we describe how to compute $\\mathfrak {g}$ in polynomial time with respect to $n$ , $d$ , $\\log |\\Delta _K|$ and the size of the entries of $A$ .",
"The idea is to compute $\\det (A)\\mod {(}p)$ for a sufficiently large prime number $p$ .",
"In practice, one might prefer to compute $\\det (A)\\mod {(}p_i)$ for several prime numbers $p_1,\\cdots ,p_l$ and recombine the values via the chinese remainder theorem, but for the sake of simplicity, we only describe that procedure for a single prime.",
"Once $\\det (A)$ is computed in polynomial time, we return $\\mathfrak {g}= \\det (A)\\cdot \\mathfrak {a}_1\\cdots \\mathfrak {a}_n.$ The first step consists of evaluating how large $p$ should be to ensure that we recover $\\det (A)$ uniquely.",
"As $p\\omega _1,\\cdots ,p\\omega _d$ is an integral basis for $(p)$ , it suffices that $p\\ge \\max _i|a_i|$ where $\\det (A) = \\sum _i a_i \\omega _i$ .",
"As $\\max _i|a_i|\\le 2^{3d/2}\\Vert \\det (A)\\Vert $ , it suffices to bound $\\Vert \\det (A)\\Vert $ .",
"We first compute an upper bound on $|\\sigma (\\det (A))|$ for the $d$ complex embeddings $\\sigma $ of $K$ via Hadamard's inequality and then we deduce a bound on $\\Vert \\det (A)\\Vert $ .",
"Let $\\sigma : K\\rightarrow \\mathcal {C}$ , we know from Hadamard's inequality that $|\\sigma (\\det (A))|\\le B^n n^{n/2},$ where $B$ is a bound on $\\sigma (a_{i,j})$ .",
"Such a bound can be derived from the size of the coefficient of $A$ by using $\\forall x, \\ \\forall i\\ |\\sigma _i(x)| \\le \\left(\\max _j|x_j|\\right)d^{3/2}2^{d^2/2}\\sqrt{|\\Delta _K|}.$ This way, we see that $B := 2^{\\max _{i,j}\\left(S(a_{i,j})\\right)}d^{3/2}2^{d^2/2}\\sqrt{|\\Delta _K|}$ suffices.",
"Then, our bound on $\\Vert \\det (A)\\Vert $ is simply $\\Vert \\det (A)\\Vert \\le \\sqrt{n} 2^{\\max _{i,j}\\left(S(a_{i,j})\\right)}d^{3/2}2^{d^2/2}\\sqrt{|\\Delta _K|}.$ [ht] Computation of $\\det (A)$ [1] $A\\in \\mathcal {O}_{K}^{n\\times n}$ , $B > \\max _{i,j}\\left(S(a_{i,j})\\right)$ $\\det (A)$ .",
"Let $p\\ge \\sqrt{n} 2^{B}d^{3/2}2^{d^2/2}\\sqrt{|\\Delta _K|}$ be a prime.",
"$\\mathfrak {p}_i\\mid (p)$ Compute $\\det (A)\\bmod \\unknown.",
"\\mathfrak {p}_i$ .",
"Recover $\\det (A)\\bmod (p)$ via successive applications of Algorithm $\\det (A)$ .",
"To reconstruct $\\det (A)\\mod {(}p)$ from $\\det (A)\\bmod \\unknown.",
"\\mathfrak {a}_i$ for $i\\le d$ , let us consider the simpler case of the reconstruction modulo two coprime ideals $\\mathfrak {a},\\mathfrak {b}$ of $\\mathcal {O}_{K}$ .",
"Let $M_\\mathfrak {a}$ and $M_\\mathfrak {b}$ be the matrices representing the $\\mathbb {Z}$ basis of $\\mathfrak {a}$ and $\\mathfrak {b}$ in the integral basis $(\\omega _i)_{i\\le d}$ of $\\mathcal {O}_{K}$ , and let $x,y,w\\in \\mathcal {O}_{K}$ such that x = ya x = wb.",
"We wish to compute $z\\in \\mathcal {O}_{K}$ such that $x = z\\bmod \\unknown.",
"\\mathfrak {a}\\mathfrak {b}$ .",
"As in [2], we can derive $a\\in \\mathfrak {a},b\\in \\mathfrak {b}$ such that $a + b = 1$ from the HNF of $\\left(\\frac{M_\\mathfrak {a}}{M_\\mathfrak {b}}\\right)$ .",
"Then, a solution to our CRT recomposition is given by $z := wa + yb.$ [ht] CRT recomposition [1] $\\mathfrak {a},\\mathfrak {b}\\subseteq \\mathcal {O}_{K}$ , $x,y,w\\in \\mathcal {O}_{K}$ such that $x = y\\bmod \\unknown.",
"\\mathfrak {a}$ and $x = w \\bmod \\unknown.",
"\\mathfrak {b}$ .",
"$z\\in \\mathcal {O}_{K}$ such that $x = z\\bmod \\unknown.",
"\\mathfrak {a}\\mathfrak {b}$ .",
"Compute $a\\in \\mathfrak {a},b\\in \\mathfrak {b}$ such that $a + b = 1$ .",
"$z$ .",
"Proposition 11 Let $B>\\max _{i,j}\\left(S(a_{i,j})\\right)$ , then the complexity of Algorithm is bounded by $\\tilde{O}\\left( n^3d^7(d^2 + B + \\log |\\Delta _K|)^2\\right).$ For each $\\mathfrak {p}_i$ , the computation of $\\det (A)\\bmod \\unknown.",
"\\mathfrak {p}_i$ consists of $n^3$ multiplications of reduced elements modulo $\\mathfrak {p}_i$ followed by a reduction modulo $\\mathfrak {p}_i$ .",
"Given our choice of $p$ , we have $\\log \\mathcal {N}(\\mathfrak {p}_i)\\le \\tilde{O}\\left( d(B + d^2 + \\log |\\Delta _K|)\\right).$ Therefore, the size of the elements $x\\in \\mathcal {O}_{K}$ involved in these multiplications satisfies $S(x)\\le \\tilde{O}\\left( d^2(d^2 + \\log |\\Delta _K| + B)\\right).$ The cost of the multiplications is in $\\tilde{O}\\left( d^4(d^2 + B + \\log |\\Delta _K|)\\right),$ while the mdular reductions cost $\\tilde{O}\\left( d^6(d^2 + B + \\log |\\Delta _K|)^2\\right).$ The time to reconstruct $\\det (A)\\bmod (p)$ corresponds to the computation of $n^2$ Hermite forms of $d^2\\times d$ integer matrices $M$ such that $\\log |M|\\le \\log (\\mathcal {N}(\\mathfrak {p}_i))$ .",
"This takes $\\tilde{O}\\left( n^2d^{\\omega +3}(d^2 + B + \\log |\\Delta _K|)^2\\right).$"
],
[
"Conclusion",
"We described a polynomial time algorithm for computing the HNF basis of an $\\mathcal {O}_{K}$ -module.",
"Our strategy relies on the one of Cohen [2] who had conjectured that his modular algorithm was polynomial.",
"The crucial difference between our algorithm and the one of [2] is the normalization which allows us to prove the complexity to be polynomial.",
"Without it, we cannot bound the denominator of the coefficients of the matrix when we recombine rows, even if they are reduced modulo the determinantal ideal.",
"We provided a rigorous proof of the complexity of our method with respect to the dimension of the module, the size of the input and the invariants of the field.",
"Our algorithm is the first polynomial time method for computing the HNF of an $\\mathcal {O}_{K}$ -module.",
"This result is significant since other applications rely on the possibility of computing the HNF of an $\\mathcal {O}_{K}$ -module in polynomial time.",
"In particular, Fieker and Stehlé [5] made this assumption in the analysis of their LLL algorithms for $\\mathcal {O}_{K}$ -modules.",
"Our result has natural ramifications in cryptography through the LLL algorithm of Fieker and Stehlé [5], but it can also be used for list-decoding number field codes.",
"Detailed proofs of statements notion of size Proposition 1 Let $x\\in \\mathcal {O}_{K}$ , the size of $x$ and its $T_2$ -norm satisfy (x)O( S(x)d + d2 + |K|) S(x)O( d( d+ (x))).",
"Let us show how $S(x)$ and $\\log (\\Vert x\\Vert )$ are related.",
"First, we can assume [5] that we choose an LLL-reduced integral basis $\\omega _1,\\cdots , \\omega _d$ of $\\mathcal {O}_{K}$ satisfying $\\max _i\\Vert \\omega _i\\Vert \\le \\sqrt{d}2^{d^2/2}\\sqrt{|\\Delta _K|}.$ Then, we have id, |x|i = |i(x)|= | jd |xj| i(j)| d(i |xi|) j (i |xi|) d3/22d2/2|K|.",
"Therefore, $\\log \\left(\\Vert x\\Vert \\right)\\le S(x)+ d\\log \\left(d^{3/2}2^{d^2/2}\\sqrt{|\\Delta _K|}\\right)$ .",
"On the other hand, we know from [5] that for our choice of an integral basis of $\\mathcal {O}_{K}$ , we have $\\forall x\\in \\mathcal {O}_{K}, \\ S(x)\\le d\\log \\left(2^{3d/2}\\Vert x\\Vert \\right).$ Cost model Proposition 2 Let $\\alpha ,\\beta \\in K$ such that $S(\\alpha ),S(\\beta )\\le B$ , then the following holds: $\\alpha + \\beta $ can be computed in $\\tilde{O}(dB)$ $\\alpha \\beta $ can be computed in $\\tilde{O}\\left(d^2(B + d^3 + d\\log |\\Delta _K|)\\right)$ $\\frac{1}{\\alpha }$ can be computed in $\\tilde{O}\\left(d^{\\omega -1}(B + d^3 + d\\log |\\Delta _K|)\\right)$ , where $\\tilde{O}$ denotes the complexity whithout the logarithmic factors.",
"Let $x,y\\in \\mathcal {O}_{K}$ and $a,b\\in \\mathbb {Z}_{>0}$ such that $\\alpha = x/a$ and $\\beta = y/b$ .",
"The first step of computing $\\alpha + \\beta $ consists of reducing them to the same denominator.",
"This takes a time bounded by $\\tilde{O}(dB)$ .",
"Then the addition of the numerators takes $\\tilde{O}(dB)$ , as well as and the simplification by the GCD of the denominator and the $d$ coefficients.",
"For $i,j,k\\le d$ , let $a^{(k)}_{i,j}$ be such that $\\omega _i\\omega _j = \\sum _{k\\le d}a^{(k)}_{i,j}\\omega _k$ .",
"From [5], we know that $\\forall i,\\Vert \\omega _i\\Vert \\le \\sqrt{d}2^{d^2/2}\\sqrt{|\\Delta _K|}$ , and thus $\\forall i,j\\Vert \\omega _i\\omega _j\\Vert \\le \\Vert \\omega _i \\Vert \\Vert \\omega _j\\Vert \\le d2^{d^2}|\\Delta _K|.$ Therefore, from Proposition REF , we have $\\forall i,j,k, \\log \\left(|a^{(k)}_{i,j}|\\right)\\le \\tilde{O}(d^2 + \\log |\\Delta _K|).$ Then, if $x = \\sum _{i\\le d}b_i\\omega _i$ and $y = \\sum _{j\\le d}c_j\\omega _j$ , we first need to compute $b_ic_j$ for every $i,j\\le d$ , which takes time $d^2\\mathcal {M}(B/d)$ .",
"Then, we compute $(b_ic_j)a^{(k)}_{i,j}$ for $i,j,k\\le d$ , which takes $\\tilde{O}(d^3\\mathcal {M}(2B/d + d^2 +\\log |\\Delta _K|)$ .",
"Then for each $k\\le d$ , we compute $\\sum _{i,j}b_ic_ja^{(k)}_{i,j}$ , which is in $\\tilde{O}(d(B/d + d^2 +\\log |\\Delta _K|))$ .",
"Finally, the multiplication of the denominators is in $\\mathcal {M}(B)$ , and the simplification of the numerator and denominator takes $\\tilde{O}(d\\mathcal {M}(B/d + d^2 + \\log |\\Delta _K|)).$ To invert $x = \\sum _i b_i \\omega _i$ , we first define $A := (d_{j,k})_{j,k\\le d}$ by $d_{j,k}:= \\sum _ib_ia^{(k)}_{i,j}$ , and notice that $\\forall i,\\ x\\omega _i = \\sum _ib_i\\left( \\sum _{k\\le d}a^{(k)}_{i,j}\\omega _k\\right) = \\sum _{k\\le d} d_{j,k}\\omega _k.$ Inverting $x$ boils down to finding $x_1,\\cdots ,x_d\\in \\mathbb {Q}$ such that $\\sum _i xx_i\\omega _i = 1$ .",
"It can be achieved by solving $XA = (1,0,\\cdots ,0).$ We derive the complexity of this step by noticing that $\\log |A|\\le 2^{B/d + d^2 +3d/2}d|\\Delta _K|$ .",
"From Hadamard's inequality, we know that the numerator and the denominator of $x_i$ are bounded by $d^{d/2}|A|^{d}\\le 2^{d^3 +3d^2/2 + B}d^{3d/2}|\\Delta _K|^d.$ Multiplying all numerators by $a$ where $\\alpha = x/a$ costs $\\tilde{O}\\left( d\\mathcal {M}(d^3 + B +d\\log (|\\Delta _K|)\\right),$ while reducing the $ax_i$ to the same denominator and simplifying the expression can be done in $\\tilde{O}\\left(d(d^3 + B + d\\log (|\\Delta _K|))\\right).$ As $\\omega \\ge 2$ , the complexity of the inversion is in fact dominated by the resolution of the linear system.",
"Proposition 3 Let $\\mathfrak {a}$ and $\\mathfrak {b}$ be fractional ideals of $K$ such that $S(\\mathfrak {a}),S(\\mathfrak {b})\\le B$ , then the following holds: $\\mathfrak {a}+ \\mathfrak {b}$ can be computed in $\\tilde{O}(d^{\\omega +1}B)$ , $\\mathfrak {a}\\mathfrak {b}$ can be computed in $\\tilde{O}(d^3(d^4 + d^2\\log |\\Delta _K| + B))$ , $1/\\mathfrak {a}$ can be computed in $\\tilde{O}\\left( d^{2\\omega }(d^4 + d^2\\log |\\Delta _K| + B)\\right)$ .",
"Let $A,C\\in \\mathbb {Z}^{d\\times d}$ in HNF form and $a,c\\in \\mathbb {Z}_{>0}$ such that $\\mathfrak {a}= \\frac{1}{a}\\left(\\sum _{i\\le d}\\mathbb {Z}A_i\\right)$ and $\\mathfrak {b}= \\frac{1}{c}\\left(\\sum _{i\\le d}\\mathbb {Z}C_i\\right)$ , where $A_i$ denotes the $i$ -th row of $A$ .",
"Adding $\\mathfrak {a}$ and $\\mathfrak {b}$ is done by computing the HNF of $\\left(\\frac{cA}{aC}\\right)$ and reducing the denominator.",
"The complexity is bounded by the one of the HNF which is in $\\tilde{O}(d^{\\omega +1}B)$ since $\\log |cA|,\\log |aC|\\le B + B/d^2$ .",
"Let $\\gamma _1,\\cdots ,\\gamma _d$ and $\\delta _1,\\cdots ,\\delta _d$ be integral elements such that a= 1a( Z1 + + Zd ) b= 1b( Z1 + + Zd ) for $a,b\\in \\mathbb {Z}_{>0}$ .",
"We first compute $\\gamma _i\\delta _j$ , which takes $\\tilde{O}\\left( d^3( S(\\mathfrak {a}) + d^4 + d^2\\log |\\Delta _K|)\\right).$ Their size satisfies $S(\\gamma _i\\gamma _j)\\le \\tilde{O}\\left( d^3 + d\\log |\\Delta _K| + \\frac{S(\\mathfrak {a})}{d}\\right)$ .",
"Then, we compute the HNF basis of the $\\mathbb {Z}$ -module generated by the $\\gamma _i\\delta _j$ , which costs $\\tilde{O}\\left( d^\\omega ( d^4 + d^2\\log |\\Delta _K| + S(\\mathfrak {a}))\\right),$ and we finally perform $d^2$ gcd reduction involving the product of the denominators which is bounded by $\\tilde{O}(B)$ .",
"Finally, we know from [3] that finding the inverse of $\\mathfrak {a}$ consists of calculating a basis of the nullspace of a matrix $D\\in \\mathbb {Z}^{(d^2+d)\\times d^2}$ satisfying $\\log |D|\\le \\tilde{O}(d^2 + \\log |\\Delta _K| +B/d^2)$ , and returning the HNF of its left $d\\times d$ minor $U$ .",
"By using [15], we find such a nullspace $M\\in \\mathbb {Z}^{d\\times d^2}$ satisfying $|M|\\le d(\\sqrt{d}|D|)^{2d}$ in expected time bounded by $\\tilde{O}\\left( d^{2+2\\omega }\\log |D|\\right)\\le \\tilde{O}\\left( d^{2\\omega }(d^4 + d^2\\log |\\Delta _K| + B)\\right).$ The HNF of $U$ has complexity bounded by $\\tilde{O}(d^{\\omega + 1}\\log |M|)\\le \\tilde{O}(d^{2 + \\omega }\\log |D|)$ .",
"Proposition 4 Let $\\alpha \\in K$ , a fractional ideal $\\mathfrak {a}\\subseteq K$ and $B_1,B_2$ such that $S(\\mathfrak {a})\\le B_1$ and $S(\\alpha )\\le B_2$ , then $\\alpha \\mathfrak {a}$ can be computed in expected time bounded by $\\tilde{O}\\left(d^{\\omega }\\left(d^3 + d\\log |\\Delta _K| + \\frac{B_1}{d} + B_2 \\right)\\right).$ Let $x\\in \\mathcal {O}_{K}$ and $a\\in \\mathbb {Z}_{>0}$ such that $\\alpha = x/a$ and let $k\\in \\mathbb {Z}_{>0}$ and $\\gamma _1,\\cdots ,\\gamma _d$ be an HNF basis for $\\mathfrak {a}$ .",
"Then, $(x\\gamma _i)_{i\\le d}$ is a $\\mathbb {Z}$ -basis for $(x)\\mathfrak {a}$ .",
"We perform $d$ multiplications $x\\gamma _i$ where $S(\\gamma _i)\\le B_1/d$ and $S(x)\\le B_2$ .",
"This costs $\\tilde{O}\\left( d^3\\left( \\frac{B_1}{d} + B_2 + d^3 + d\\log |\\Delta _K|\\right) \\right).$ Then, from Corollary REF , we know that $S(x\\gamma _i)\\le \\tilde{O}\\left(d^3 + d\\log |\\Delta _K| + S(x) + S(\\gamma _i)\\right).$ Therefore, computing the HNF of the resulting matrix of entries bounded by $S(x\\gamma _i)/d$ takes $\\tilde{O}\\left(S(x\\gamma _i) d^{\\omega }\\right)\\le \\tilde{O}\\left(d^{\\omega }\\left(d^3 + d\\log |\\Delta _K| + S(x) + S(\\gamma _i)\\right)\\right).$ Finally, we multiply the denominators and reduce them by successive GCD computations in time $\\tilde{O}(dS(x\\gamma _i)).$ Reduction modulo a fractional ideal Proposition 7 Let $x\\in K$ and $\\mathfrak {a}$ be a fractional ideal of $K$ , then Algorithm returns $\\overline{x}$ such that $x - \\overline{x}\\in \\mathfrak {a}$ and $\\Vert \\overline{x}\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}.$ The LLL [7] algorithm allows us to compute a basis $(r_j)_{j\\le d}$ for $I$ that satisfies $\\Vert r_j\\Vert \\le 2^{d/2}\\sqrt{d}\\mathcal {N}(I)^{1/d}\\sqrt{|\\Delta _K|}.$ The same holds for a fractional ideal $\\mathfrak {a}$ of $K$ by multiplying the above relation by the denominator of $\\mathfrak {a}$ .",
"Then, as $\\lfloor x_j \\rceil r_j\\le 1$ , we see that $\\Vert \\overline{x}\\Vert \\le d\\max _j\\Vert r_j\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}.$ The HNF At the end of Algorithm , we obtain a pseudo-basis $[(B_i)_{i\\le n},(\\mathfrak {b}_i)_{i\\le n}]$ such that $\\forall i\\le n\\ \\mathfrak {b}_iB_i \\subseteq M + \\mathfrak {g}e_i,$ where $e_i := (0,0,\\cdots ,1,0,\\cdots ,0)$ is the $i$ -th vector of the canonical basis of $K^n$ .",
"However, the determinant of $i\\times i$ minors is preserved modulo $\\mathfrak {g}$ .",
"Let $M_i\\subseteq \\mathcal {O}_{K}^{n-i}$ be the $\\mathcal {O}_{K}$ -module defined by $\\mathfrak {a}_1(a_{1,n-i},\\cdots , a_{1,n}) + \\cdots + \\mathfrak {a}_n(a_{n,n-i},\\cdots ,a_{n,n}),$ and $\\mathfrak {g}(M_i)$ its determinantal ideal.",
"The operations performed at Step 6 to 10 in Algorithm preserve $\\mathfrak {g}(M_i)$ while after Step 11, our pseudo-basis $[(B_i)_{i\\le n},(\\mathfrak {b}_i)_{i\\le n}]$ only defines a module $M^{\\prime }\\subseteq \\mathcal {O}_{K}^n$ satisfying $\\mathfrak {g}(M^{\\prime }_i) + \\mathfrak {g}= \\mathfrak {g}(M_i) + \\mathfrak {g}.$ This property is the equivalent of the integer case when the HNF is taken modulo a multiple $D$ of the determinant of the lattice.",
"To recover the ideals $\\mathfrak {c}_i$ of a pseudo-HNF of $M$ , we first notice that i,g(Mi') + g= g(Mi)+g = cn-icn + g = cn-icn + c1cn = cn-icn.",
"On the other hand, $\\mathfrak {g}(M^{\\prime }_i) + \\mathfrak {g}= \\mathfrak {b}_{n-i}\\cdots \\mathfrak {b}_n + \\mathfrak {g}$ .",
"Thus, we have $\\forall i,\\ \\mathfrak {b}_{n-i}\\cdots \\mathfrak {b}_n +\\mathfrak {g}= \\mathfrak {c}_{n-i}\\cdots \\mathfrak {c}_n,$ which allows us to recursively recover the $\\mathfrak {c}_i$ from the $(\\mathfrak {b}_j)_{j\\ge i}$ and $\\mathfrak {g}$ .",
"Indeed, as in the integer case, it boils down to taking $\\mathfrak {c}_i = \\frac{\\mathfrak {g}}{\\prod _{j > i}\\mathfrak {c}_j} + \\mathfrak {b}_i.$ To do so, we keep track of $\\mathfrak {g}_i := \\frac{\\mathfrak {g}}{\\prod _{j > i}\\mathfrak {c}_j}$ throughout Algorithm that reconstructs the actual pseudo-HNF from its modular version given by Algorithm .",
"At each step we set $\\mathfrak {c}_i\\leftarrow \\mathfrak {b}_i + \\mathfrak {g}_{i}.$ This replacement of the ideals in the pseudo-basis defining our module impacts the corresponding vectors in $K^n$ as well.",
"In particular, we require that the diagonal elements all be 1.",
"Do ensure thus, we find $u\\in \\mathfrak {b}_i\\mathfrak {c}^{-1}_i,\\ v\\in \\mathfrak {g}_{i}\\mathfrak {c}^{-1}_i$ such that $u + v = 1$ which implies that $\\mathfrak {c}_i(uB_i + ve_i)\\subseteq \\mathfrak {b}_iB_i + \\mathfrak {g}_ie_i,$ where the $i$ -th coefficient of $uB_i + ve_i\\in K^n$ is 1 and the coefficient of index $j>i$ in $uB_i + ve_i$ are 0.",
"Then we set $W_i\\leftarrow uB_i\\bmod \\unknown.",
"\\mathfrak {g}_i\\mathfrak {c}^{-1}_i,$ and observe that $\\sum _i \\mathfrak {c}_iW_i \\subseteq M$ .",
"These $\\mathcal {O}_{K}$ -modules have the same determinantal ideal, and as in the integer case, we can prove that it is sufficient to ensure that they are equal.",
"$uB_i + ve_i = W_i + d_i$ where the coefficients of $d_i\\in \\left(\\mathfrak {g}_i/\\mathfrak {c}_i\\right)^n$ of index $j > i$ are 0.",
"The vector $d_i$ satisfies $\\mathfrak {c}_id_i\\subseteq \\mathfrak {g}_id^{\\prime }_i$ where $d^{\\prime }_i\\in \\mathcal {O}_{K}^n$ with coefficients $j>i$ equal to 0.",
"This allows us to state that $\\mathfrak {c}_i W_i \\subseteq \\mathfrak {b}_iB_i + \\mathfrak {g}_ie_i + \\mathfrak {c}_id_i \\subseteq M + \\mathfrak {g}_ie_i + \\mathfrak {g}_id^{\\prime }_i \\subseteq M + \\mathfrak {g}_iD_i,$ where the coefficients of $D_i\\in \\mathcal {O}_{K}^n$ of index $j>i$ equal 0.",
"We now want to prove that $\\mathfrak {c}_iW_i\\subseteq M$ .",
"To do this, we prove that $\\mathfrak {g}_iD_i\\subseteq M$ .",
"Lemma 1 Let $M = \\mathfrak {a}_1A_1 + \\cdots \\mathfrak {a}_nA_n\\in \\mathcal {O}_{K}^n$ , then we have $\\mathfrak {g}(M)\\mathcal {O}_{K}^n \\subseteq M$ We can prove by induction that if $[(B_i),(\\mathfrak {b}_i)]$ is a pseudo-HNF basis of $M$ , then $\\forall i,\\ \\mathfrak {g}_1\\cdots \\mathfrak {g}_i e_i \\subseteq M,$ where $e_i$ is the $i$ -th vector of the canonical basis of $\\mathcal {O}_{K}^n$ .",
"Our statement immediatly follows.",
"We now consider the intersection $N_i$ of our module $M\\subseteq \\mathcal {O}_{K}^n$ with $\\mathcal {O}_{K}^i$ .",
"Note that with the previous definitions, we have in particular $M = N_i \\oplus M_i$ .",
"Lemma 2 Let $i\\le n$ and $D\\in \\mathcal {O}_{K}^n$ a vector whose entries of index $j>i$ are 0.",
"Then we have $\\mathfrak {g}_i D \\subseteq M.$ From Lemma REF , we know that $\\mathfrak {g}_i\\mathcal {O}_{K}^i \\subseteq N_i$ .",
"If $D_i\\in \\mathcal {O}_{K}^i$ is the first $i$ coordinates of $D$ , then $\\mathfrak {g}_i D_i\\subseteq N_i$ , and as the last $n-i$ coordinates of $D$ are 0, we have $\\mathfrak {g}_i D \\subseteq M.$ The module generated by the pseudo-basis $[(W_i),(\\mathfrak {c}_i)]$ computed by Algorithm is a subset of $M$ .",
"We ensured that its determinantal ideal $\\prod _i\\mathfrak {c}_i$ equals the determinantal ideal $\\mathfrak {g}$ of $M$ .",
"Let us prove that it is sufficient to ensure that $\\mathfrak {c}_1 W_1 + \\cdots + \\mathfrak {c}_n W_n = M.$ Lemma 3 Let $M = \\sum _{i\\le n} \\mathfrak {a}_i A_i$ and $M^{\\prime } = \\sum _{i\\le n} \\mathfrak {b}_i B_i$ two $n$ -dimensional $\\mathcal {O}_{K}$ -modules such that $M^{\\prime }\\subseteq M$ and $\\mathfrak {g}(M^{\\prime }) = \\mathfrak {g}(M)$ .",
"Then necessarily $M = M^{\\prime }.$ Let $[(W_i),(\\mathfrak {c}_i)]$ be a pseudo-HNF for $M$ , and $[(W^{\\prime }_i),(\\mathfrak {c}^{\\prime }_i)]$ a pseudo-HNF for $M^{\\prime }$ .",
"By assumption, we have $\\prod _i\\mathfrak {c}_i= \\prod _i\\mathfrak {c}^{\\prime }_i$ , and $M^{\\prime }\\subseteq M$ .",
"As both matrices $W$ and $W^{\\prime }$ have a lower triangular shape, it is clear that $\\forall i, \\ \\sum _{j\\le i}\\mathfrak {c}^{\\prime }_jW^{\\prime }_j \\subseteq \\sum _{j\\le i} \\mathfrak {c}_jW_j.$ As the diagonal coefficients of both $W$ and $W^{\\prime }$ are 1, we see by looking at the inclusion in the coefficient $i$ of (REF ) that $\\mathfrak {c}^{\\prime }_i\\subseteq \\mathfrak {c}_i$ .",
"Then as $\\mathfrak {g}(M) = \\mathfrak {g}(M^{\\prime })$ , we have $\\forall i \\mathfrak {c}_i = \\mathfrak {c}^{\\prime }_i.$ Now let us prove by induction that $\\forall i,\\ \\mathfrak {c}_iW_i \\subseteq \\mathfrak {c}_1W^{\\prime }_1 + \\cdots + \\mathfrak {c}_iW^{\\prime }_i.$ This assertion is clear for $i=1$ since $W_1 = W^{\\prime }_1 = e_1$ .",
"Then, assuming (REF ) for $1,\\cdots ,i-1$ , we first use the fact that $\\mathfrak {c}_iW^{\\prime }_i \\subseteq \\mathfrak {c}_1W_1 + \\cdots + \\mathfrak {c}_i W_i.$ In other words, $\\forall c^{\\prime }_i\\in \\mathfrak {c}_i$ , $\\exists (c_1,\\cdots ,c_i)\\in \\mathfrak {c}_1\\times \\cdots \\times \\mathfrak {c}_i$ such that $c^{\\prime }_i(w^{\\prime }_{i,1},\\cdots ,w^{\\prime }_{i,i-1},1) = \\left(\\sum _{1\\le j\\le i} c_jw_{j,1} , \\cdots , c_{i}w_{i,i-1} + c_{i-1} , c_i\\right).$ In particular, $c_i = c^{\\prime }_i$ , which allows us to state that $\\forall c_i\\in \\mathfrak {c}_i$ , $\\exists (c_1,\\cdots ,c_{i-1})\\in \\mathfrak {c}_1\\times \\cdots \\times \\mathfrak {c}_{i-1}$ such that ci wi,i-1 = ci-1 + ci w'i,i-1 ci wi,i-2 = ci-2 + ci-1 wi-1,i-2 + ciw'i,i-2 $\\vdots $ = $\\vdots $ ci wi,1 = c1 + + ci-1 wi-1,1 + ciw'i,1.",
"This shows that $\\mathfrak {c}_iW_i \\subseteq \\mathfrak {c}_1W_1 + \\cdots + \\mathfrak {c}_{i-1}W_{i-1} + \\mathfrak {c}_iW^{\\prime }_i,$ and since we have $\\forall j<i,\\ \\mathfrak {c}_jW_i\\subseteq \\sum _{j<i}\\mathfrak {c}_jW^{\\prime }_j$ , we obtain the desired result.",
"Lemma REF is a generalization of the standard result on $\\mathbb {Z}$ -modules stating that if $L^{\\prime }\\subseteq L$ and $\\det (L) = \\det (L^{\\prime })$ , then $L = L^{\\prime }$ .",
"Although implied in [2], Lemma REF is not stated, nor proved in the litterature.",
"Yet, it is essential to ensure the validity of Algorithm .",
"Proposition 9 The $\\mathcal {O}_{K}$ -module defined by the pseudo-basis $[(W_i),(\\mathfrak {c}_i)]$ obtained by applying lgorithm to the pseudo-HNF of $M$ modulo $\\mathfrak {g}(M)$ satisfies $\\mathfrak {c}_1 W_1 + \\cdots + \\mathfrak {c}_n W_n = M.$"
],
[
"notion of size",
"Proposition 1 Let $x\\in \\mathcal {O}_{K}$ , the size of $x$ and its $T_2$ -norm satisfy (x)O( S(x)d + d2 + |K|) S(x)O( d( d+ (x))).",
"Let us show how $S(x)$ and $\\log (\\Vert x\\Vert )$ are related.",
"First, we can assume [5] that we choose an LLL-reduced integral basis $\\omega _1,\\cdots , \\omega _d$ of $\\mathcal {O}_{K}$ satisfying $\\max _i\\Vert \\omega _i\\Vert \\le \\sqrt{d}2^{d^2/2}\\sqrt{|\\Delta _K|}.$ Then, we have id, |x|i = |i(x)|= | jd |xj| i(j)| d(i |xi|) j (i |xi|) d3/22d2/2|K|.",
"Therefore, $\\log \\left(\\Vert x\\Vert \\right)\\le S(x)+ d\\log \\left(d^{3/2}2^{d^2/2}\\sqrt{|\\Delta _K|}\\right)$ .",
"On the other hand, we know from [5] that for our choice of an integral basis of $\\mathcal {O}_{K}$ , we have $\\forall x\\in \\mathcal {O}_{K}, \\ S(x)\\le d\\log \\left(2^{3d/2}\\Vert x\\Vert \\right).$"
],
[
"Cost model",
"Proposition 2 Let $\\alpha ,\\beta \\in K$ such that $S(\\alpha ),S(\\beta )\\le B$ , then the following holds: $\\alpha + \\beta $ can be computed in $\\tilde{O}(dB)$ $\\alpha \\beta $ can be computed in $\\tilde{O}\\left(d^2(B + d^3 + d\\log |\\Delta _K|)\\right)$ $\\frac{1}{\\alpha }$ can be computed in $\\tilde{O}\\left(d^{\\omega -1}(B + d^3 + d\\log |\\Delta _K|)\\right)$ , where $\\tilde{O}$ denotes the complexity whithout the logarithmic factors.",
"Let $x,y\\in \\mathcal {O}_{K}$ and $a,b\\in \\mathbb {Z}_{>0}$ such that $\\alpha = x/a$ and $\\beta = y/b$ .",
"The first step of computing $\\alpha + \\beta $ consists of reducing them to the same denominator.",
"This takes a time bounded by $\\tilde{O}(dB)$ .",
"Then the addition of the numerators takes $\\tilde{O}(dB)$ , as well as and the simplification by the GCD of the denominator and the $d$ coefficients.",
"For $i,j,k\\le d$ , let $a^{(k)}_{i,j}$ be such that $\\omega _i\\omega _j = \\sum _{k\\le d}a^{(k)}_{i,j}\\omega _k$ .",
"From [5], we know that $\\forall i,\\Vert \\omega _i\\Vert \\le \\sqrt{d}2^{d^2/2}\\sqrt{|\\Delta _K|}$ , and thus $\\forall i,j\\Vert \\omega _i\\omega _j\\Vert \\le \\Vert \\omega _i \\Vert \\Vert \\omega _j\\Vert \\le d2^{d^2}|\\Delta _K|.$ Therefore, from Proposition REF , we have $\\forall i,j,k, \\log \\left(|a^{(k)}_{i,j}|\\right)\\le \\tilde{O}(d^2 + \\log |\\Delta _K|).$ Then, if $x = \\sum _{i\\le d}b_i\\omega _i$ and $y = \\sum _{j\\le d}c_j\\omega _j$ , we first need to compute $b_ic_j$ for every $i,j\\le d$ , which takes time $d^2\\mathcal {M}(B/d)$ .",
"Then, we compute $(b_ic_j)a^{(k)}_{i,j}$ for $i,j,k\\le d$ , which takes $\\tilde{O}(d^3\\mathcal {M}(2B/d + d^2 +\\log |\\Delta _K|)$ .",
"Then for each $k\\le d$ , we compute $\\sum _{i,j}b_ic_ja^{(k)}_{i,j}$ , which is in $\\tilde{O}(d(B/d + d^2 +\\log |\\Delta _K|))$ .",
"Finally, the multiplication of the denominators is in $\\mathcal {M}(B)$ , and the simplification of the numerator and denominator takes $\\tilde{O}(d\\mathcal {M}(B/d + d^2 + \\log |\\Delta _K|)).$ To invert $x = \\sum _i b_i \\omega _i$ , we first define $A := (d_{j,k})_{j,k\\le d}$ by $d_{j,k}:= \\sum _ib_ia^{(k)}_{i,j}$ , and notice that $\\forall i,\\ x\\omega _i = \\sum _ib_i\\left( \\sum _{k\\le d}a^{(k)}_{i,j}\\omega _k\\right) = \\sum _{k\\le d} d_{j,k}\\omega _k.$ Inverting $x$ boils down to finding $x_1,\\cdots ,x_d\\in \\mathbb {Q}$ such that $\\sum _i xx_i\\omega _i = 1$ .",
"It can be achieved by solving $XA = (1,0,\\cdots ,0).$ We derive the complexity of this step by noticing that $\\log |A|\\le 2^{B/d + d^2 +3d/2}d|\\Delta _K|$ .",
"From Hadamard's inequality, we know that the numerator and the denominator of $x_i$ are bounded by $d^{d/2}|A|^{d}\\le 2^{d^3 +3d^2/2 + B}d^{3d/2}|\\Delta _K|^d.$ Multiplying all numerators by $a$ where $\\alpha = x/a$ costs $\\tilde{O}\\left( d\\mathcal {M}(d^3 + B +d\\log (|\\Delta _K|)\\right),$ while reducing the $ax_i$ to the same denominator and simplifying the expression can be done in $\\tilde{O}\\left(d(d^3 + B + d\\log (|\\Delta _K|))\\right).$ As $\\omega \\ge 2$ , the complexity of the inversion is in fact dominated by the resolution of the linear system.",
"Proposition 3 Let $\\mathfrak {a}$ and $\\mathfrak {b}$ be fractional ideals of $K$ such that $S(\\mathfrak {a}),S(\\mathfrak {b})\\le B$ , then the following holds: $\\mathfrak {a}+ \\mathfrak {b}$ can be computed in $\\tilde{O}(d^{\\omega +1}B)$ , $\\mathfrak {a}\\mathfrak {b}$ can be computed in $\\tilde{O}(d^3(d^4 + d^2\\log |\\Delta _K| + B))$ , $1/\\mathfrak {a}$ can be computed in $\\tilde{O}\\left( d^{2\\omega }(d^4 + d^2\\log |\\Delta _K| + B)\\right)$ .",
"Let $A,C\\in \\mathbb {Z}^{d\\times d}$ in HNF form and $a,c\\in \\mathbb {Z}_{>0}$ such that $\\mathfrak {a}= \\frac{1}{a}\\left(\\sum _{i\\le d}\\mathbb {Z}A_i\\right)$ and $\\mathfrak {b}= \\frac{1}{c}\\left(\\sum _{i\\le d}\\mathbb {Z}C_i\\right)$ , where $A_i$ denotes the $i$ -th row of $A$ .",
"Adding $\\mathfrak {a}$ and $\\mathfrak {b}$ is done by computing the HNF of $\\left(\\frac{cA}{aC}\\right)$ and reducing the denominator.",
"The complexity is bounded by the one of the HNF which is in $\\tilde{O}(d^{\\omega +1}B)$ since $\\log |cA|,\\log |aC|\\le B + B/d^2$ .",
"Let $\\gamma _1,\\cdots ,\\gamma _d$ and $\\delta _1,\\cdots ,\\delta _d$ be integral elements such that a= 1a( Z1 + + Zd ) b= 1b( Z1 + + Zd ) for $a,b\\in \\mathbb {Z}_{>0}$ .",
"We first compute $\\gamma _i\\delta _j$ , which takes $\\tilde{O}\\left( d^3( S(\\mathfrak {a}) + d^4 + d^2\\log |\\Delta _K|)\\right).$ Their size satisfies $S(\\gamma _i\\gamma _j)\\le \\tilde{O}\\left( d^3 + d\\log |\\Delta _K| + \\frac{S(\\mathfrak {a})}{d}\\right)$ .",
"Then, we compute the HNF basis of the $\\mathbb {Z}$ -module generated by the $\\gamma _i\\delta _j$ , which costs $\\tilde{O}\\left( d^\\omega ( d^4 + d^2\\log |\\Delta _K| + S(\\mathfrak {a}))\\right),$ and we finally perform $d^2$ gcd reduction involving the product of the denominators which is bounded by $\\tilde{O}(B)$ .",
"Finally, we know from [3] that finding the inverse of $\\mathfrak {a}$ consists of calculating a basis of the nullspace of a matrix $D\\in \\mathbb {Z}^{(d^2+d)\\times d^2}$ satisfying $\\log |D|\\le \\tilde{O}(d^2 + \\log |\\Delta _K| +B/d^2)$ , and returning the HNF of its left $d\\times d$ minor $U$ .",
"By using [15], we find such a nullspace $M\\in \\mathbb {Z}^{d\\times d^2}$ satisfying $|M|\\le d(\\sqrt{d}|D|)^{2d}$ in expected time bounded by $\\tilde{O}\\left( d^{2+2\\omega }\\log |D|\\right)\\le \\tilde{O}\\left( d^{2\\omega }(d^4 + d^2\\log |\\Delta _K| + B)\\right).$ The HNF of $U$ has complexity bounded by $\\tilde{O}(d^{\\omega + 1}\\log |M|)\\le \\tilde{O}(d^{2 + \\omega }\\log |D|)$ .",
"Proposition 4 Let $\\alpha \\in K$ , a fractional ideal $\\mathfrak {a}\\subseteq K$ and $B_1,B_2$ such that $S(\\mathfrak {a})\\le B_1$ and $S(\\alpha )\\le B_2$ , then $\\alpha \\mathfrak {a}$ can be computed in expected time bounded by $\\tilde{O}\\left(d^{\\omega }\\left(d^3 + d\\log |\\Delta _K| + \\frac{B_1}{d} + B_2 \\right)\\right).$ Let $x\\in \\mathcal {O}_{K}$ and $a\\in \\mathbb {Z}_{>0}$ such that $\\alpha = x/a$ and let $k\\in \\mathbb {Z}_{>0}$ and $\\gamma _1,\\cdots ,\\gamma _d$ be an HNF basis for $\\mathfrak {a}$ .",
"Then, $(x\\gamma _i)_{i\\le d}$ is a $\\mathbb {Z}$ -basis for $(x)\\mathfrak {a}$ .",
"We perform $d$ multiplications $x\\gamma _i$ where $S(\\gamma _i)\\le B_1/d$ and $S(x)\\le B_2$ .",
"This costs $\\tilde{O}\\left( d^3\\left( \\frac{B_1}{d} + B_2 + d^3 + d\\log |\\Delta _K|\\right) \\right).$ Then, from Corollary REF , we know that $S(x\\gamma _i)\\le \\tilde{O}\\left(d^3 + d\\log |\\Delta _K| + S(x) + S(\\gamma _i)\\right).$ Therefore, computing the HNF of the resulting matrix of entries bounded by $S(x\\gamma _i)/d$ takes $\\tilde{O}\\left(S(x\\gamma _i) d^{\\omega }\\right)\\le \\tilde{O}\\left(d^{\\omega }\\left(d^3 + d\\log |\\Delta _K| + S(x) + S(\\gamma _i)\\right)\\right).$ Finally, we multiply the denominators and reduce them by successive GCD computations in time $\\tilde{O}(dS(x\\gamma _i)).$"
],
[
"Reduction modulo a fractional ideal",
"Proposition 7 Let $x\\in K$ and $\\mathfrak {a}$ be a fractional ideal of $K$ , then Algorithm returns $\\overline{x}$ such that $x - \\overline{x}\\in \\mathfrak {a}$ and $\\Vert \\overline{x}\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}.$ The LLL [7] algorithm allows us to compute a basis $(r_j)_{j\\le d}$ for $I$ that satisfies $\\Vert r_j\\Vert \\le 2^{d/2}\\sqrt{d}\\mathcal {N}(I)^{1/d}\\sqrt{|\\Delta _K|}.$ The same holds for a fractional ideal $\\mathfrak {a}$ of $K$ by multiplying the above relation by the denominator of $\\mathfrak {a}$ .",
"Then, as $\\lfloor x_j \\rceil r_j\\le 1$ , we see that $\\Vert \\overline{x}\\Vert \\le d\\max _j\\Vert r_j\\Vert \\le d^{3/2}2^{d/2}\\mathcal {N}(\\mathfrak {a})^{1/d}\\sqrt{|\\Delta _K|}.$"
],
[
"The HNF",
"At the end of Algorithm , we obtain a pseudo-basis $[(B_i)_{i\\le n},(\\mathfrak {b}_i)_{i\\le n}]$ such that $\\forall i\\le n\\ \\mathfrak {b}_iB_i \\subseteq M + \\mathfrak {g}e_i,$ where $e_i := (0,0,\\cdots ,1,0,\\cdots ,0)$ is the $i$ -th vector of the canonical basis of $K^n$ .",
"However, the determinant of $i\\times i$ minors is preserved modulo $\\mathfrak {g}$ .",
"Let $M_i\\subseteq \\mathcal {O}_{K}^{n-i}$ be the $\\mathcal {O}_{K}$ -module defined by $\\mathfrak {a}_1(a_{1,n-i},\\cdots , a_{1,n}) + \\cdots + \\mathfrak {a}_n(a_{n,n-i},\\cdots ,a_{n,n}),$ and $\\mathfrak {g}(M_i)$ its determinantal ideal.",
"The operations performed at Step 6 to 10 in Algorithm preserve $\\mathfrak {g}(M_i)$ while after Step 11, our pseudo-basis $[(B_i)_{i\\le n},(\\mathfrak {b}_i)_{i\\le n}]$ only defines a module $M^{\\prime }\\subseteq \\mathcal {O}_{K}^n$ satisfying $\\mathfrak {g}(M^{\\prime }_i) + \\mathfrak {g}= \\mathfrak {g}(M_i) + \\mathfrak {g}.$ This property is the equivalent of the integer case when the HNF is taken modulo a multiple $D$ of the determinant of the lattice.",
"To recover the ideals $\\mathfrak {c}_i$ of a pseudo-HNF of $M$ , we first notice that i,g(Mi') + g= g(Mi)+g = cn-icn + g = cn-icn + c1cn = cn-icn.",
"On the other hand, $\\mathfrak {g}(M^{\\prime }_i) + \\mathfrak {g}= \\mathfrak {b}_{n-i}\\cdots \\mathfrak {b}_n + \\mathfrak {g}$ .",
"Thus, we have $\\forall i,\\ \\mathfrak {b}_{n-i}\\cdots \\mathfrak {b}_n +\\mathfrak {g}= \\mathfrak {c}_{n-i}\\cdots \\mathfrak {c}_n,$ which allows us to recursively recover the $\\mathfrak {c}_i$ from the $(\\mathfrak {b}_j)_{j\\ge i}$ and $\\mathfrak {g}$ .",
"Indeed, as in the integer case, it boils down to taking $\\mathfrak {c}_i = \\frac{\\mathfrak {g}}{\\prod _{j > i}\\mathfrak {c}_j} + \\mathfrak {b}_i.$ To do so, we keep track of $\\mathfrak {g}_i := \\frac{\\mathfrak {g}}{\\prod _{j > i}\\mathfrak {c}_j}$ throughout Algorithm that reconstructs the actual pseudo-HNF from its modular version given by Algorithm .",
"At each step we set $\\mathfrak {c}_i\\leftarrow \\mathfrak {b}_i + \\mathfrak {g}_{i}.$ This replacement of the ideals in the pseudo-basis defining our module impacts the corresponding vectors in $K^n$ as well.",
"In particular, we require that the diagonal elements all be 1.",
"Do ensure thus, we find $u\\in \\mathfrak {b}_i\\mathfrak {c}^{-1}_i,\\ v\\in \\mathfrak {g}_{i}\\mathfrak {c}^{-1}_i$ such that $u + v = 1$ which implies that $\\mathfrak {c}_i(uB_i + ve_i)\\subseteq \\mathfrak {b}_iB_i + \\mathfrak {g}_ie_i,$ where the $i$ -th coefficient of $uB_i + ve_i\\in K^n$ is 1 and the coefficient of index $j>i$ in $uB_i + ve_i$ are 0.",
"Then we set $W_i\\leftarrow uB_i\\bmod \\unknown.",
"\\mathfrak {g}_i\\mathfrak {c}^{-1}_i,$ and observe that $\\sum _i \\mathfrak {c}_iW_i \\subseteq M$ .",
"These $\\mathcal {O}_{K}$ -modules have the same determinantal ideal, and as in the integer case, we can prove that it is sufficient to ensure that they are equal.",
"$uB_i + ve_i = W_i + d_i$ where the coefficients of $d_i\\in \\left(\\mathfrak {g}_i/\\mathfrak {c}_i\\right)^n$ of index $j > i$ are 0.",
"The vector $d_i$ satisfies $\\mathfrak {c}_id_i\\subseteq \\mathfrak {g}_id^{\\prime }_i$ where $d^{\\prime }_i\\in \\mathcal {O}_{K}^n$ with coefficients $j>i$ equal to 0.",
"This allows us to state that $\\mathfrak {c}_i W_i \\subseteq \\mathfrak {b}_iB_i + \\mathfrak {g}_ie_i + \\mathfrak {c}_id_i \\subseteq M + \\mathfrak {g}_ie_i + \\mathfrak {g}_id^{\\prime }_i \\subseteq M + \\mathfrak {g}_iD_i,$ where the coefficients of $D_i\\in \\mathcal {O}_{K}^n$ of index $j>i$ equal 0.",
"We now want to prove that $\\mathfrak {c}_iW_i\\subseteq M$ .",
"To do this, we prove that $\\mathfrak {g}_iD_i\\subseteq M$ .",
"Lemma 1 Let $M = \\mathfrak {a}_1A_1 + \\cdots \\mathfrak {a}_nA_n\\in \\mathcal {O}_{K}^n$ , then we have $\\mathfrak {g}(M)\\mathcal {O}_{K}^n \\subseteq M$ We can prove by induction that if $[(B_i),(\\mathfrak {b}_i)]$ is a pseudo-HNF basis of $M$ , then $\\forall i,\\ \\mathfrak {g}_1\\cdots \\mathfrak {g}_i e_i \\subseteq M,$ where $e_i$ is the $i$ -th vector of the canonical basis of $\\mathcal {O}_{K}^n$ .",
"Our statement immediatly follows.",
"We now consider the intersection $N_i$ of our module $M\\subseteq \\mathcal {O}_{K}^n$ with $\\mathcal {O}_{K}^i$ .",
"Note that with the previous definitions, we have in particular $M = N_i \\oplus M_i$ .",
"Lemma 2 Let $i\\le n$ and $D\\in \\mathcal {O}_{K}^n$ a vector whose entries of index $j>i$ are 0.",
"Then we have $\\mathfrak {g}_i D \\subseteq M.$ From Lemma REF , we know that $\\mathfrak {g}_i\\mathcal {O}_{K}^i \\subseteq N_i$ .",
"If $D_i\\in \\mathcal {O}_{K}^i$ is the first $i$ coordinates of $D$ , then $\\mathfrak {g}_i D_i\\subseteq N_i$ , and as the last $n-i$ coordinates of $D$ are 0, we have $\\mathfrak {g}_i D \\subseteq M.$ The module generated by the pseudo-basis $[(W_i),(\\mathfrak {c}_i)]$ computed by Algorithm is a subset of $M$ .",
"We ensured that its determinantal ideal $\\prod _i\\mathfrak {c}_i$ equals the determinantal ideal $\\mathfrak {g}$ of $M$ .",
"Let us prove that it is sufficient to ensure that $\\mathfrak {c}_1 W_1 + \\cdots + \\mathfrak {c}_n W_n = M.$ Lemma 3 Let $M = \\sum _{i\\le n} \\mathfrak {a}_i A_i$ and $M^{\\prime } = \\sum _{i\\le n} \\mathfrak {b}_i B_i$ two $n$ -dimensional $\\mathcal {O}_{K}$ -modules such that $M^{\\prime }\\subseteq M$ and $\\mathfrak {g}(M^{\\prime }) = \\mathfrak {g}(M)$ .",
"Then necessarily $M = M^{\\prime }.$ Let $[(W_i),(\\mathfrak {c}_i)]$ be a pseudo-HNF for $M$ , and $[(W^{\\prime }_i),(\\mathfrak {c}^{\\prime }_i)]$ a pseudo-HNF for $M^{\\prime }$ .",
"By assumption, we have $\\prod _i\\mathfrak {c}_i= \\prod _i\\mathfrak {c}^{\\prime }_i$ , and $M^{\\prime }\\subseteq M$ .",
"As both matrices $W$ and $W^{\\prime }$ have a lower triangular shape, it is clear that $\\forall i, \\ \\sum _{j\\le i}\\mathfrak {c}^{\\prime }_jW^{\\prime }_j \\subseteq \\sum _{j\\le i} \\mathfrak {c}_jW_j.$ As the diagonal coefficients of both $W$ and $W^{\\prime }$ are 1, we see by looking at the inclusion in the coefficient $i$ of (REF ) that $\\mathfrak {c}^{\\prime }_i\\subseteq \\mathfrak {c}_i$ .",
"Then as $\\mathfrak {g}(M) = \\mathfrak {g}(M^{\\prime })$ , we have $\\forall i \\mathfrak {c}_i = \\mathfrak {c}^{\\prime }_i.$ Now let us prove by induction that $\\forall i,\\ \\mathfrak {c}_iW_i \\subseteq \\mathfrak {c}_1W^{\\prime }_1 + \\cdots + \\mathfrak {c}_iW^{\\prime }_i.$ This assertion is clear for $i=1$ since $W_1 = W^{\\prime }_1 = e_1$ .",
"Then, assuming (REF ) for $1,\\cdots ,i-1$ , we first use the fact that $\\mathfrak {c}_iW^{\\prime }_i \\subseteq \\mathfrak {c}_1W_1 + \\cdots + \\mathfrak {c}_i W_i.$ In other words, $\\forall c^{\\prime }_i\\in \\mathfrak {c}_i$ , $\\exists (c_1,\\cdots ,c_i)\\in \\mathfrak {c}_1\\times \\cdots \\times \\mathfrak {c}_i$ such that $c^{\\prime }_i(w^{\\prime }_{i,1},\\cdots ,w^{\\prime }_{i,i-1},1) = \\left(\\sum _{1\\le j\\le i} c_jw_{j,1} , \\cdots , c_{i}w_{i,i-1} + c_{i-1} , c_i\\right).$ In particular, $c_i = c^{\\prime }_i$ , which allows us to state that $\\forall c_i\\in \\mathfrak {c}_i$ , $\\exists (c_1,\\cdots ,c_{i-1})\\in \\mathfrak {c}_1\\times \\cdots \\times \\mathfrak {c}_{i-1}$ such that ci wi,i-1 = ci-1 + ci w'i,i-1 ci wi,i-2 = ci-2 + ci-1 wi-1,i-2 + ciw'i,i-2 $\\vdots $ = $\\vdots $ ci wi,1 = c1 + + ci-1 wi-1,1 + ciw'i,1.",
"This shows that $\\mathfrak {c}_iW_i \\subseteq \\mathfrak {c}_1W_1 + \\cdots + \\mathfrak {c}_{i-1}W_{i-1} + \\mathfrak {c}_iW^{\\prime }_i,$ and since we have $\\forall j<i,\\ \\mathfrak {c}_jW_i\\subseteq \\sum _{j<i}\\mathfrak {c}_jW^{\\prime }_j$ , we obtain the desired result.",
"Lemma REF is a generalization of the standard result on $\\mathbb {Z}$ -modules stating that if $L^{\\prime }\\subseteq L$ and $\\det (L) = \\det (L^{\\prime })$ , then $L = L^{\\prime }$ .",
"Although implied in [2], Lemma REF is not stated, nor proved in the litterature.",
"Yet, it is essential to ensure the validity of Algorithm .",
"Proposition 9 The $\\mathcal {O}_{K}$ -module defined by the pseudo-basis $[(W_i),(\\mathfrak {c}_i)]$ obtained by applying lgorithm to the pseudo-HNF of $M$ modulo $\\mathfrak {g}(M)$ satisfies $\\mathfrak {c}_1 W_1 + \\cdots + \\mathfrak {c}_n W_n = M.$"
]
] | 1204.1298 |
[
[
"Large deviations for fractional Poisson processes"
],
[
"Abstract We prove large deviation principles for two versions of fractional Poisson processes.",
"Firstly we consider the main version which is a renewal process; we also present large deviation estimates for the ruin probabilities of an insurance model with constant premium rate, i.i.d.",
"light tail claim sizes, and a fractional Poisson claim number process.",
"We conclude with the alternative version where all the random variables are weighted Poisson distributed.",
"Keywords: Mittag Leffler function; renewal process; random time cha"
],
[
"Introduction",
"The theory of large deviations gives an asymptotic computation of small probabilities on exponential scale (we refer to [11] for this topic).",
"The aim of this paper is to prove some large deviation results for fractional Poisson processes.",
"To the best of our knowledge, these techniques have not been applied so far to the fractional Poisson process.",
"The study of fractional versions of the usual renewal processes has recently received an increasing interest, starting from the paper by [34].",
"In [17] the so-called fractal Poisson process is introduced (by means of non-standard analysis) and proved to have independent increments.",
"In analogy with the fractional Brownian motion, in [39]-[40] is proposed a process constructed as a stochastic integral with respect to the Poisson measure.",
"A different approach has been followed by other authors, in the mainstream of the fractional diffusions, in the sense of extending well-known differential equations by introducing fractional-order derivatives with respect to time: the relaxation equation (see e.g.",
"[28]), the heat and wave equations (see e.g.",
"[14]-[23]-[24]) as well as the telegraph equation (see e.g.",
"[29]) and the higher-order heat-type equations (see e.g.",
"[2]).",
"In this context the solution of the fractional generalization of the Kolmogorov-Feller equation, together with the distribution of the waiting time for the corresponding process, is derived in [19].",
"Many other aspects of this type of fractional Poisson processes have been analyzed: a probabilistic representation of the fractional Poisson process of order $\\nu $ as a composition of a standard Poisson process with a random time given by a fractional diffusion is given in [3] (note that this has some analogy with other results holding for compositions of different processes; see e.g.",
"[5]-[30]) and, for $\\nu =1/2$ , the time argument reduces to the absolute value of a Brownian motion; in [27] it is proved that we have the same one-dimensional distributions of a standard Poisson process time-changed via an inverse stable subordinator; in [33] it is given a full characterization of the fractional Poisson process in terms of its multidimensional distributions.",
"Other aspects of the fractional Poisson process are analyzed in [25]-[26]-[36].",
"We also recall other references with different approaches.",
"Applications based on fractional Poisson processes can be found in [37] (in the field of the transport of charge carriers in semiconductors) and in [20] (in the field the fractional quantum mechanics); an inference problem is studied in [6]; a version of space-fractional Poisson process where the state probabilities are governed by equations with a fractional difference operator found in time series analysis is presented in [31].",
"The outline of the paper is the following.",
"We start with some preliminaries in section .",
"In section we consider the main version which is a slight generalization of the renewal process in [3]-[4].",
"We give results for the empirical means of the i.i.d.",
"holding times and for the normalized counting processes; furthermore we study an insurance model with fractional Poisson claim number process.",
"In section we present large deviation results for an alternative version, which is the first version presented in section 4 in [3] with a suitable deterministic time-change.",
"In such a case we have a weighted Poisson process, i.e.",
"all the random variables are weighted Poisson distributed with the same weights (namely the weights do not depend on $t$ ).",
"In the literature a weighted Poisson process is defined in [1] and examples of weighted Poisson distributions can be found in [9] and [10]."
],
[
"Preliminaries on large deviations.",
"We start by recalling some basic definitions (see [11], pages 4-5).",
"Given a topological space $\\mathcal {Z}$ , we say that a family of $\\mathcal {Z}$ -valued random variables $\\lbrace Z_t:t>0\\rbrace $ satisfies the large deviation principle (LDP from now on) with rate function $I$ if: the function $I:\\mathcal {Z}\\rightarrow [0,\\infty ]$ is lower semi-continuous; the upper bound $\\limsup _{t\\rightarrow \\infty }\\frac{1}{t}\\log P(Z_t\\in C)\\le -\\inf _{x\\in C}I(x)$ holds for all closed sets $C$ ; the lower bound $\\liminf _{t\\rightarrow \\infty }\\frac{1}{t}\\log P(Z_t\\in G)\\ge -\\inf _{x\\in G}I(x)$ holds for all open sets $G$ .",
"The above definition can be given also for a sequence of $\\mathcal {Z}$ -valued random variables $\\lbrace Z_n:n\\ge 1\\rbrace $ (we mean the discrete parameter denoted by $n$ in place of the continuous parameter $t$ ).",
"Moreover a rate function is said to be good if all its level sets $\\lbrace \\lbrace x\\in \\mathcal {Z}:I(x)\\le \\eta \\rbrace :\\eta \\ge 0\\rbrace $ are compact.",
"Throughout this paper we often refer to the well-known Gärtner Ellis Theorem (see e.g.",
"Theorem 2.3.6 in [11]).",
"Furthermore we always set $0\\log 0=0$ and $0\\log \\frac{0}{0}=0$ ."
],
[
"Preliminaries on (generalized) Mittag Leffler function.",
"The Mittag Leffler function is defined by $E_{\\alpha ,\\beta }(x):=\\sum _{r\\ge 0}\\frac{x^r}{\\Gamma (\\alpha r+\\beta )}$ (see e.g.",
"[32], page 17).",
"We recall that, if we write $a_t\\sim b_t$ to mean that $\\frac{a_t}{b_t}\\rightarrow 1$ as $t\\rightarrow \\infty $ , we have $E_{\\nu ,\\beta }(z)\\sim \\frac{1}{\\nu }z^{(1-\\beta )/\\nu }e^{z^{1/\\nu }}\\ \\mathrm {as}\\ z\\rightarrow \\infty $ (see e.g.",
"eq.",
"(1.8.27) in [18]).",
"Finally we recall that the generalized Mittag Leffler function is defined by $E_{\\alpha ,\\beta }^\\gamma (x):=\\sum _{r\\ge 0}\\frac{(\\gamma )_rx^r}{r!\\Gamma (\\alpha r+\\beta )},$ where $(\\gamma )_r=1$ is the Pochammer symbol defined by $(\\gamma )_r:=\\left\\lbrace \\begin{array}{ll}1&\\ \\mathrm {if}\\ r=0\\\\\\gamma (\\gamma +1)\\cdots (\\gamma +r-1)&\\ \\mathrm {if}\\ r\\in \\lbrace 1,2,3,\\ldots \\rbrace \\end{array}\\right.$ (see e.g.",
"eq.",
"(1.9.1) in [18]); note that $E_{\\alpha ,\\beta }^1$ coincides with $E_{\\alpha ,\\beta }$ ."
],
[
"Results for the main version (renewal process)",
"Throughout this section we consider a class of fractional Poisson processes defined as renewal processes.",
"More precisely, for $\\nu \\in (0,1]$ and $h,\\lambda >0$ , we consider $\\lbrace M_{\\nu ,h,\\lambda }(t):t\\ge 0\\rbrace $ defined by $M_{\\nu ,h,\\lambda }(t):=\\sum _{n\\ge 1}1_{T_1+\\cdots +T_n\\le t},$ where the holding times $\\lbrace T_n:n\\ge 1\\rbrace $ are i.i.d.",
"with generalized Mittag Leffler distribution, i.e.",
"with continuous density $f_{\\nu ,h,\\lambda }$ defined by $f_{\\nu ,h,\\lambda }(t)=\\lambda ^h t^{\\nu h-1}E_{\\nu ,\\nu h}^h(-\\lambda t^\\nu )1_{(0,\\infty )}(t).$ We remark that, if we set $h=1$ , we recover the same process in [3]-[4] (see eq.",
"(2.16) in [4]; see also eq.",
"(4.14) in [3]).",
"Moreover $f_{\\nu ,k,\\lambda }$ coincides with eq.",
"(2.19) in [4], where $k$ is integer.",
"Finally we have $f_{1,h,\\lambda }(t)=\\frac{\\lambda ^h}{\\Gamma (h)}t^{h-1}e^{-\\lambda t}1_{(0,\\infty )}(t)$ which is a Gamma density; thus we obtain the classical case with exponentially distributed holding times for $(\\nu ,h)=(1,1)$ .",
"Now, in view of what follows, it is useful to recall that $\\kappa _{\\nu ,h,\\lambda }(\\theta ):=\\log \\mathbb {E}[e^{\\theta T_1}]=\\left\\lbrace \\begin{array}{ll}h\\log \\frac{\\lambda }{\\lambda +(-\\theta )^\\nu }&\\ \\mathrm {if}\\ \\theta \\le 0\\\\\\infty &\\ \\mathrm {if}\\ \\theta >0\\end{array}\\right.\\ \\mathrm {for}\\ \\nu \\in (0,1),$ and, for $\\nu =1$ , $\\kappa _{1,h,\\lambda }(\\theta ):=\\log \\mathbb {E}[e^{\\theta T_1}]=\\left\\lbrace \\begin{array}{ll}h\\log \\frac{\\lambda }{\\lambda -\\theta }&\\ \\mathrm {if}\\ \\theta <\\lambda \\\\\\infty &\\ \\mathrm {if}\\ \\theta \\ge \\lambda .\\end{array}\\right.$ We conclude with the outline of this section.",
"We start with the LDPs for $\\lbrace \\bar{T}_n:n\\ge 1\\rbrace $ , where $\\bar{T}_n:=\\frac{T_1+\\cdots +T_n}{n}$ for all $n\\ge 1$ , and for $\\left\\lbrace \\frac{M_{\\nu ,h,\\lambda }(t)}{t}:t>0\\right\\rbrace $ ; moreover, for the second LDP, we discuss the possible application of Gärtner Ellis Theorem.",
"In particular we study in detail the fractional case $\\nu =\\frac{1}{2}$ , for which we can provide explicit expressions for the rate functions, and we recover the LDP for $\\left\\lbrace \\frac{M_{\\nu ,1,\\lambda }(t)}{t}:t>0\\right\\rbrace $ (concerning the case $h=1$ ) by taking into account that it can be expressed in terms of a classical Poisson process computed at an independent random time given by a reflecting Brownian motion with variance parameter 2.",
"Finally we present some results for the ruin probabilities concerning an insurance model with a fractional Poisson claim number process."
],
[
"The basic LDPs",
"We start with two LDPs which can be easily proved: the first one (Proposition REF ) concerns $\\lbrace \\bar{T}_n:n\\ge 1\\rbrace $ , and it is a particular case of Cramér Theorem (see e.g.",
"Theorem 2.2.3 in [11]); the second one (Proposition REF ) concerns $\\left\\lbrace \\frac{M_{\\nu ,h,\\lambda }(t)}{t}:t>0\\right\\rbrace $ , and its proof is based on the combination of the first result and known results in the literature for nondecreasing processes and their inverses (here we refer to [12] which treats this kind of problem in a wide generality, allowing non-linear scaling functions; more precisely we refer to Theorem 1.1(i) in [12] with $u,v,w$ as the identity function because $I_{\\nu ,h,\\lambda }^{(T)}$ has no peaks with the unique base $x=\\infty $ if $\\nu \\in (0,1)$ , and $x=\\frac{h}{\\lambda }$ if $\\nu =1$ ).",
"Proposition 3.1 The sequence $\\lbrace \\bar{T}_n:n\\ge 1\\rbrace $ satisfies the LDP with rate function $I_{\\nu ,h,\\lambda }^{(T)}$ defined by $I_{\\nu ,h,\\lambda }^{(T)}(x):=\\sup _{\\theta \\in \\mathbb {R}}\\lbrace \\theta x-\\kappa _{\\nu ,h,\\lambda }(\\theta )\\rbrace $ .",
"In particular, for $\\nu =1$ , we have $I_{1,h,\\lambda }^{(T)}(x)=\\left\\lbrace \\begin{array}{ll}h\\left(\\frac{\\lambda x}{h}-1-\\log \\frac{\\lambda x}{h}\\right)&\\ if\\ x>0\\\\\\infty &\\ if\\ x\\le 0,\\end{array}\\right.$ and it is a good rate function.",
"For $\\nu \\in (0,1)$ we have: $I_{\\nu ,h,\\lambda }^{(T)}(x)=\\infty $ for $x\\le 0$ , $I_{\\nu ,h,\\lambda }^{(T)}(x)$ is decreasing on $(0,\\infty )$ , $\\lim _{x\\downarrow 0}I_{\\nu ,h,\\lambda }^{(T)}(x)=\\infty $ , $\\lim _{x\\rightarrow \\infty }I_{\\nu ,h,\\lambda }^{(T)}(x)=0$ , the rate function $I_{\\nu ,h,\\lambda }^{(T)}$ is not good.",
"We remark that, for all $\\nu \\in (0,1]$ and for all $h>0$ , we have $\\kappa _{\\nu ,h,\\lambda }(\\theta )=h\\kappa _{\\nu ,1,\\lambda }(\\theta )$ for all $\\theta \\in \\mathbb {R}$ , and therefore $I_{\\nu ,h,\\lambda }^{(T)}(x)=hI_{\\nu ,1,\\lambda }^{(T)}(\\frac{x}{h})$ for all $x\\in \\mathbb {R}$ .",
"Proposition 3.2 The family $\\left\\lbrace \\frac{M_{\\nu ,h,\\lambda }(t)}{t}:t>0\\right\\rbrace $ satisfies the LDP with good rate function $I_{\\nu ,h,\\lambda }^{(M)}$ defined by $I_{\\nu ,h,\\lambda }^{(M)}(x):=\\left\\lbrace \\begin{array}{ll}xI_{\\nu ,h,\\lambda }^{(T)}(1/x)&\\ if\\ x>0\\\\\\lambda 1_{\\nu =1}&\\ if\\ x=0\\\\\\infty &\\ if\\ x<0.\\end{array}\\right.$ In particular, for $\\nu =1$ , we have $I_{1,h,\\lambda }^{(M)}(x)=\\left\\lbrace \\begin{array}{ll}hx\\log \\frac{hx}{\\lambda }-hx+\\lambda &\\ if\\ x\\ge 0\\\\\\infty &\\ if\\ x<0.\\end{array}\\right.$ For $\\nu \\in (0,1)$ we have: $I_{\\nu ,h,\\lambda }^{(M)}(x)=\\infty $ for $x<0$ , $I_{\\nu ,h,\\lambda }^{(M)}(x)$ is increasing on $[0,\\infty )$ , $\\lim _{x\\downarrow 0}I_{\\nu ,h,\\lambda }^{(M)}(x)=I_{\\nu ,h,\\lambda }^{(M)}(0)=0$ , $\\lim _{x\\rightarrow \\infty }I_{\\nu ,h,\\lambda }^{(M)}(x)=\\infty $ ."
],
[
"A discussion on Gärtner Ellis Theorem for the proof of Proposition ",
"It is well-known that the rate function $I_{\\nu ,h,\\lambda }^{(T)}$ is convex on $(0,\\infty )$ .",
"Moreover $I_{\\nu ,h,\\lambda }^{(M)}$ is also convex on $(0,\\infty )$ ; in fact, for $x_1,x_2\\in (0,\\infty )$ and $\\gamma \\in [0,1]$ , we have $I_{\\nu ,h,\\lambda }^{(M)}(\\gamma x_1+(1-\\gamma )x_2)=&(\\gamma x_1+(1-\\gamma )x_2)I_{\\nu ,h,\\lambda }^{(T)}\\left(\\frac{1}{\\gamma x_1+(1-\\gamma )x_2}\\right)\\\\=&(\\gamma x_1+(1-\\gamma )x_2)I_{\\nu ,h,\\lambda }^{(T)}\\left(\\frac{\\gamma x_1}{\\gamma x_1+(1-\\gamma )x_2}\\cdot \\frac{1}{x_1}+\\frac{(1-\\gamma )x_2}{\\gamma x_1+(1-\\gamma )x_2}\\cdot \\frac{1}{x_2}\\right)$ and, by the convexity of $I_{\\nu ,h,\\lambda }^{(T)}$ , we get $I_{\\nu ,h,\\lambda }^{(M)}(\\gamma x_1+(1-\\gamma )x_2)\\le &\\gamma x_1I_{\\nu ,h,\\lambda }^{(T)}\\left(\\frac{1}{x_1}\\right)+(1-\\gamma )x_2I_{\\nu ,h,\\lambda }^{(T)}\\left(\\frac{1}{x_2}\\right)\\\\\\le &\\gamma I_{\\nu ,h,\\lambda }^{(M)}(x_1)+(1-\\gamma )I_{\\nu ,h,\\lambda }^{(M)}(x_2).$ Then one could try to prove the LDP in Proposition REF with an application of Gärtner Ellis Theorem.",
"If this was possible, we should have $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\mathbb {E}\\left[e^{\\theta M_{\\nu ,h,\\lambda }(t)}\\right]=\\left\\lbrace \\begin{array}{ll}\\lambda (e^{\\theta /h}-1)&\\ \\mathrm {if}\\ \\nu =1\\\\(\\lambda (e^{\\theta /h}-1))^{1/\\nu }1_{\\theta \\ge 0}&\\ \\mathrm {if}\\ \\nu \\in (0,1)\\end{array}\\right.=:\\Lambda _{\\nu ,h,\\lambda }(\\theta )\\ (\\mbox{for all}\\ \\theta \\in \\mathbb {R})$ and, since the function $\\theta \\mapsto \\Lambda _{\\nu ,h,\\lambda }(\\theta )$ satisfies the hypotheses of Gärtner Ellis Theorem in both cases $\\nu =1$ and $\\nu \\in (0,1)$ , we should get the LDP with rate function $\\Lambda _{\\nu ,h,\\lambda }^*$ defined by $\\Lambda _{\\nu ,h,\\lambda }^*(x)=\\sup _{\\theta \\in \\mathbb {R}}\\left\\lbrace \\theta x-\\Lambda _{\\nu ,h,\\lambda }(\\theta )\\right\\rbrace \\ (\\mbox{for all}\\ x\\in \\mathbb {R})$ because $\\Lambda _{\\nu ,h,\\lambda }^*$ coincides with the rate function $I_{\\nu ,h,\\lambda }^{(M)}$ in Proposition REF .",
"However we can have some difficulties with this approach because we cannot have an expression of the moment generating function $\\mathbb {E}\\left[e^{\\theta M_{\\nu ,h,\\lambda }(t)}\\right]$ .",
"For completeness we also remark that, in both the cases $\\nu =1$ and $\\nu \\in (0,1)$ , the function $\\theta \\mapsto \\Lambda _{\\nu ,h,\\lambda }(\\theta )$ above meets eq.",
"(12)-(13) in [15], i.e.",
": $\\Lambda _{\\nu ,h,\\lambda }(\\theta )=-\\kappa _{\\nu ,h,\\lambda }^{-1}(-\\theta )\\ \\mathrm {for}\\ \\left\\lbrace \\begin{array}{ll}\\theta \\in \\mathbb {R}&\\ \\mathrm {if}\\ \\nu =1\\\\\\theta \\ge 0&\\ \\mathrm {if}\\ \\nu \\in (0,1);\\end{array}\\right.$ $\\kappa _{\\nu ,h,\\lambda }(\\theta )=-\\Lambda _{\\nu ,h,\\lambda }^{-1}(-\\theta )\\ \\mathrm {for}\\ \\left\\lbrace \\begin{array}{ll}\\theta <\\lambda &\\ \\mathrm {if}\\ \\nu =1\\\\\\theta \\le 0&\\ \\mathrm {if}\\ \\nu \\in (0,1).\\end{array}\\right.$"
],
[
"Some remarks on the fractional case $\\nu =\\frac{1}{2}$ .",
"We can provide explicit formulas for the rate functions presented above.",
"By Proposition REF , we have $I_{\\frac{1}{2},h,\\lambda }^{(T)}(x)=&\\sup _{\\theta \\le 0}\\left\\lbrace \\theta x-h\\log \\left(\\frac{\\lambda }{\\lambda +(-\\theta )^{\\frac{1}{2}}}\\right)\\right\\rbrace =\\left\\lbrace \\theta x-h\\log \\left(\\frac{\\lambda }{\\lambda +(-\\theta )^{\\frac{1}{2}}}\\right)\\right\\rbrace _{\\theta =-\\left(-\\frac{\\lambda }{2}+\\frac{1}{2}\\sqrt{\\lambda ^2+\\frac{2h}{x}}\\right)^2}\\\\=&-\\left(-\\frac{\\lambda }{2}+\\frac{1}{2}\\sqrt{\\lambda ^2+\\frac{2h}{x}}\\right)^2x-h\\log \\left(\\frac{\\lambda }{\\lambda -\\frac{\\lambda }{2}+\\frac{1}{2}\\sqrt{\\lambda ^2+\\frac{2h}{x}}}\\right)\\\\=&-x\\left(\\frac{1}{2}\\sqrt{\\lambda ^2+\\frac{2h}{x}}-\\frac{\\lambda }{2}\\right)^2+h\\log \\left(\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+\\frac{2h}{\\lambda ^2x}}\\right)\\ (\\mbox{for all}\\ x>0);$ thus, by Proposition REF , we have $I_{\\frac{1}{2},h,\\lambda }^{(M)}(x)=xI_{\\frac{1}{2},h,\\lambda }^{(T)}(1/x)=hx\\log \\left(\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+\\frac{2hx}{\\lambda ^2}}\\right)-\\left(\\frac{1}{2}\\sqrt{\\lambda ^2+2hx}-\\frac{\\lambda }{2}\\right)^2\\ (\\mbox{for all}\\ x>0).$ We remark that $I_{\\frac{1}{2},h,\\lambda }^{(M)}$ in (REF ) meets $\\Lambda _{\\frac{1}{2},h,\\lambda }^*$ in (REF ) presented in the above discussion on the proof of the LDP in Proposition REF with an application of Gärtner Ellis Theorem: the case $x\\le 0$ is trivial (the details are omitted) and, for $x>0$ , we have $\\sup _{\\theta \\ge 0}\\left\\lbrace \\theta x-(\\lambda (e^{\\theta /h}-1))^2\\right\\rbrace =&\\left\\lbrace \\theta x-(\\lambda (e^{\\theta /h}-1))^2\\right\\rbrace _{\\theta =h\\log \\left(\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+\\frac{2hx}{\\lambda ^2}}\\right)}\\\\=&h\\log \\left(\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+\\frac{2hx}{\\lambda ^2}}\\right)x-\\lambda ^2\\left(\\frac{1}{2}\\sqrt{1+\\frac{2hx}{\\lambda ^2}}-\\frac{1}{2}\\right)^2\\\\=&hx\\log \\left(\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+\\frac{2hx}{\\lambda ^2}}\\right)-\\left(\\frac{1}{2}\\sqrt{\\lambda ^2+2hx}-\\frac{\\lambda }{2}\\right)^2=I_{\\frac{1}{2},h,\\lambda }^{(M)}(x).$"
],
[
"An alternative proof of Proposition ",
"The starting point consists of the following representation in the literature (see Remark 2.1 in [3]): for each fixed $t>0$ , $M_{\\frac{1}{2},1,\\lambda }(t)$ is distributed as $N_\\lambda (|B(2t)|)$ where $\\left\\lbrace \\begin{array}{ll}\\lbrace N_\\lambda (t):t\\ge 0\\rbrace \\ \\mbox{and}\\ \\lbrace B(t):t\\ge 0\\rbrace \\ \\mbox{are independent},\\\\\\lbrace N_\\lambda (t):t\\ge 0\\rbrace \\ \\mbox{is a classical Poisson process, i.e.",
"it is distributed as}\\ \\lbrace M_{1,1,\\lambda }(t):t\\ge 0\\rbrace ,\\\\\\mbox{and}\\ \\lbrace B(t):t\\ge 0\\rbrace \\ \\mbox{is a standard Brownian motion}\\end{array}\\right.$ (note that $\\lbrace N_\\lambda (|B(2t)|):t\\ge 0\\rbrace $ does not represent a version of $\\lbrace M_{\\frac{1}{2},1,\\lambda }(t):t\\ge 0\\rbrace $ the process $\\lbrace N_\\lambda (|B(2t)|):t\\ge 0\\rbrace $ is nondecreasing with respect to $t$ ).",
"We start with the following two statements.",
"The family of random variables $\\left\\lbrace \\frac{|B(2t)|}{t}:t>0\\right\\rbrace $ satisfies the LDP with good rate function $J$ defined by $J(y):=\\left\\lbrace \\begin{array}{ll}\\frac{y^2}{4}&\\ \\mathrm {if}\\ y\\ge 0\\\\\\infty &\\ \\mathrm {if}\\ y<0.\\end{array}\\right.$ Sketch of the proof.",
"Firstly $\\left\\lbrace \\frac{B(2t)}{t}:t>0\\right\\rbrace $ satisfies the LDP by an easy application of Gärtner Ellis Theorem with the good rate function $H$ defined by $H(x)=\\frac{x^2}{4}$ ; then the required LDP holds by applying the contraction principle (see e.g.",
"Theorem 4.2.1 in [11]) and noting that $J(y)=\\inf \\lbrace H(x):|x|=y\\rbrace $ for all $y\\in \\mathbb {R}$ .",
"If $y_t\\rightarrow y$ as $t\\rightarrow \\infty $ , then $\\left\\lbrace \\frac{N_\\lambda (y_tt)}{t}:t>0\\right\\rbrace $ satisfies the LDP with rate function $K(\\cdot |y)$ defined by $K(x|y):=\\sup _{\\theta \\in \\mathbb {R}}(\\theta x-\\lambda y(e^\\theta -1))=\\left\\lbrace \\begin{array}{ll}\\left\\lbrace \\begin{array}{ll}x\\log \\left(\\frac{x}{\\lambda y}\\right)-x+\\lambda y&\\ \\mathrm {if}\\ x\\ge 0\\\\\\infty &\\ \\mathrm {if}\\ x<0,\\end{array}\\right.&\\ \\mathrm {if}\\ y>0,\\\\\\Delta _0(x)&\\ \\mathrm {if}\\ y=0,\\end{array}\\right.$ where $\\Delta _0$ is the function defined by $\\Delta _0(x):=\\left\\lbrace \\begin{array}{ll}0&\\ \\mathrm {if}\\ x=0\\\\\\infty &\\ \\mathrm {if}\\ x\\ne 0.\\end{array}\\right.$ Moreover the function $(x,y)\\mapsto K(x|y)$ is lower semi-continuous on $[0,\\infty )\\times [0,\\infty )$ .",
"Sketch of the proof.",
"The LDP can be proved by an easy application of Gärtner Ellis Theorem; moreover $(x,y)\\mapsto K(x|y)$ is a lower semi-continuous function on $[0,\\infty )\\times [0,\\infty )$ because, if $(x_n,y_n)\\rightarrow (x,y)$ , for all $\\theta \\in \\mathbb {R}$ we have $\\liminf _{n\\rightarrow \\infty }K(x_n|y_n)\\ge \\liminf _{n\\rightarrow \\infty }(\\theta x_n-\\lambda y_n(e^\\theta -1))=\\theta x-\\lambda y(e^\\theta -1)$ and we conclude by taking the supremum with respect to $\\theta \\in \\mathbb {R}$ .",
"In conclusion, by Theorem 2.3 in [7] (namely we mean the LDP for marginal distributions), the family of random variables $\\left\\lbrace \\frac{N_\\lambda (|B(2t)|)}{t}:t>0\\right\\rbrace $ satisfies the LDP with rate function $J_{\\frac{1}{2},1,\\lambda }^{(M)}$ (say) defined by $J_{\\frac{1}{2},1,\\lambda }^{(M)}(x):=\\inf \\lbrace K(x|y)+J(y):y\\ge 0\\rbrace .$ Finally we show that $J_{\\frac{1}{2},1,\\lambda }^{(M)}$ coincides with $I_{\\frac{1}{2},1,\\lambda }^{(M)}$ .",
"The equality $J_{\\frac{1}{2},1,\\lambda }^{(M)}(x)=I_{\\frac{1}{2},1,\\lambda }^{(M)}(x)$ can be easily checked if $x<0$ (because $K(x|y)=\\infty $ for all $y\\ge 0$ ) and if $x=0$ (because $J_{\\frac{1}{2},1,\\lambda }^{(M)}(0)=\\inf \\left\\lbrace \\lambda y+\\frac{y^2}{4}:y\\ge 0\\right\\rbrace =0$ since the infimum is attained at $y=0$ ).",
"If $x>0$ we have $J_{\\frac{1}{2},1,\\lambda }^{(M)}(x)=\\inf \\left\\lbrace x\\log \\left(\\frac{x}{\\lambda y}\\right)-x+\\lambda y+\\frac{y^2}{4}:y>0\\right\\rbrace ;$ then one can easily check that the infimum is attained $y=\\sqrt{\\lambda ^2+2x}-\\lambda $ , and we obtain $J_{\\frac{1}{2},1,\\lambda }^{(M)}(x)=&x\\log \\left(\\frac{x}{\\lambda (\\sqrt{\\lambda ^2+2x}-\\lambda )}\\right)-x+\\lambda \\left(\\sqrt{\\lambda ^2+2x}-\\lambda \\right)+\\frac{\\left(\\sqrt{\\lambda ^2+2x}-\\lambda \\right)^2}{4}\\\\=&x\\log \\left(\\frac{x(\\sqrt{\\lambda ^2+2x}+\\lambda )}{2\\lambda x}\\right)-x+\\lambda \\left(\\sqrt{\\lambda ^2+2x}-\\lambda \\right)+\\left(\\frac{1}{2}\\sqrt{\\lambda ^2+2x}-\\frac{\\lambda }{2}\\right)^2\\\\=&x\\log \\left(\\frac{1}{2}+\\frac{1}{2}\\sqrt{1+\\frac{2x}{\\lambda ^2}}\\right)-x+\\lambda \\left(\\sqrt{\\lambda ^2+2x}-\\lambda \\right)+\\left(\\frac{1}{2}\\sqrt{\\lambda ^2+2x}-\\frac{\\lambda }{2}\\right)^2;$ finally, by (REF ) with $h=1$ , we get $J_{\\frac{1}{2},1,\\lambda }^{(M)}(x)=&I_{\\frac{1}{2},1,\\lambda }^{(M)}(x)-x+\\lambda \\left(\\sqrt{\\lambda ^2+2x}-\\lambda \\right)+2\\left(\\frac{1}{2}\\sqrt{\\lambda ^2+2x}-\\frac{\\lambda }{2}\\right)^2\\\\=&I_{\\frac{1}{2},1,\\lambda }^{(M)}(x)-x+\\lambda \\sqrt{\\lambda ^2+2x}-\\lambda ^2+\\frac{\\lambda ^2+2x}{2}+\\frac{\\lambda ^2}{2}-\\lambda \\sqrt{\\lambda ^2+2x}=I_{\\frac{1}{2},1,\\lambda }^{(M)}(x).$"
],
[
"An insurance model with fractional Poisson claim number process",
"In this subsection we study the ruin probability $\\Psi (u):=P(\\lbrace \\exists t\\ge 0:R(t)<0\\rbrace )$ concerning the insurance model $R(t):=u+ct-\\sum _{k=1}^{M_{\\nu ,h,\\lambda }(t)}U_k,$ where (we refer to the terminology for eq.",
"(5.1.14) in [35]) $\\lbrace R(t):t\\ge 0\\rbrace $ is the reserve process, $u>0$ is the initial capital of the company, $c>0$ is the premium rate and $\\lbrace U_k:k\\ge 1\\rbrace $ are the claim sizes assumed to be i.i.d.",
"positive random variables and independent of the claim number process $\\lbrace M_{\\nu ,h,\\lambda }(t):t\\ge 0\\rbrace $ defined by (REF ).",
"Here we consider a slightly different notation for the holding times, which will be denoted by $\\lbrace T_n^{(\\nu )}:n\\ge 1\\rbrace $ instead of $\\lbrace T_n:n\\ge 1\\rbrace $ .",
"We always consider a fractional claim number process, i.e.",
"we assume that $\\nu \\in (0,1)$ and $h>0$ .",
"We recall that, if $(\\nu ,h)=(1,1)$ , the claim number process is a homogeneous Poisson process and we have the compound Poisson model (see e.g.",
"section 5.3 in [35]; see also the Cramér-Lundberg model in section 1.1 in [13]).",
"It is easy to check that the ruin probability $\\Psi (u)$ coincides with a level crossing probability for the random walk $\\lbrace \\sum _{k=1}^n(U_k-cT_k^{(\\nu )}):n\\ge 1\\rbrace $ , i.e.",
"$\\Psi (u)=P\\left(\\left\\lbrace \\exists n\\ge 1:\\sum _{k=1}^n(U_k-cT_k^{(\\nu )})>u\\right\\rbrace \\right);$ this happens because the ruin can occur only at the time epochs of the claims.",
"Furthermore it is known that, if we consider the case $\\nu =1$ , the ruin problem is non-trivial (i.e.",
"$\\Psi (u)\\in (0,1)$ ) if $c$ is large enough to have $\\mathbb {E}[U_1-cT_1^{(1)}]<0$ , i.e.",
"if the net profit condition $c>\\frac{\\lambda }{h}\\mathbb {E}[U_1]$ holds (note that, for $h=1$ , this meets eq.",
"(5.3.2) in [35], or eq.",
"(1.7) in [13] (page 26)).",
"On the contrary, for the fractional case $\\nu \\in (0,1)$ considered here, the ruin problem is non trivial for any $c>0$ because we have $\\mathbb {E}[U_1-cT_1^{(\\nu )}]=-\\infty $ .",
"Here we present two results which can be derived from straightforward applications of Theorems 1-2 in [21] for the random walk $\\lbrace \\sum _{k=1}^n(U_k-cT_k^{(\\nu )}):n\\ge 1\\rbrace $ , respectively.",
"Thus we need to consider the function $\\tilde{\\kappa }_\\nu $ defined by $\\tilde{\\kappa }_\\nu (\\theta ):=\\log \\mathbb {E}[e^{\\theta U_1}]+\\log \\mathbb {E}[e^{-c\\theta T_1^{(\\nu )}}]=\\log \\mathbb {E}[e^{\\theta U_1}]+\\kappa _{\\nu ,h,\\lambda }(-c\\theta ),$ and the following condition: $\\mathbf {(C1)}$ : there exists $w_{\\nu ,h,\\lambda }\\in (0,\\infty )\\cap \\lbrace \\theta \\in \\mathbb {R}:\\tilde{\\kappa }_\\nu (\\theta )<\\infty \\rbrace ^\\circ $ such that $\\tilde{\\kappa }_\\nu (w_{\\nu ,h,\\lambda })=0$ .",
"We start with the first result which provides an asymptotic estimate of $\\Psi (u)$ in the fashion of large deviations.",
"Proposition 3.3 Assume that $\\mathbf {(C1)}$ holds.",
"Then we have $\\lim _{u\\rightarrow \\infty }\\frac{1}{u}\\log \\Psi (u)=-w_{\\nu ,h,\\lambda }$ .",
"Note that, if $\\nu _1<\\nu _2$ (with $\\nu _1,\\nu _2\\in (0,1)$ ), then $w_{\\nu _1,h,\\lambda }<w_{\\nu _2,h,\\lambda }$ ; this can be checked noting that, by (REF ) and the definition of $\\kappa _{\\nu ,h,\\lambda }$ in (REF ), $\\tilde{\\kappa }_{\\nu _1}(\\theta )>\\tilde{\\kappa }_{\\nu _2}(\\theta )$ for $\\theta >0$ .",
"Thus the smaller is the value $\\nu $ , the more dangerous is the situation (i.e.",
"the more slowly the ruin probabilities decay as $u\\rightarrow \\infty $ ).",
"The second result gives an optimal importance sampling distribution for the estimation of $\\Psi (u)$ by simulation for large values of $u$ .",
"We need some further preliminaries.",
"Let $P_U\\otimes P_T$ be the common law for the random variables $\\lbrace (U_n,T_n^{(\\nu )}):n\\ge 1\\rbrace $ .",
"Moreover, for each $\\theta $ such that $\\tilde{\\kappa }_\\nu (\\theta )<\\infty $ , let $P_U^\\theta \\otimes P_T^\\theta $ be the absolutely continuous law with density $\\frac{d(P_U^\\theta \\otimes P_T^\\theta )}{d(P_U\\otimes P_T)}(x,t)=\\frac{dP_U^\\theta }{dP_U}(x)\\frac{dP_T^\\theta }{dP_T}(t)$ where $\\frac{dP_U^\\theta }{dP_U}(x)=e^{\\theta x-\\log \\mathbb {E}[e^{\\theta U_1}]}$ and $\\frac{dP_T^\\theta }{dP_T}(t)=e^{-c\\theta t-\\log \\mathbb {E}[e^{-c\\theta T_1^{(\\nu )}}]}=e^{-c\\theta t-\\kappa _{\\nu ,h,\\lambda }(-c\\theta )}$ .",
"Here we think to have i.i.d.",
"random variables $\\lbrace (U_n,T_n^{(\\nu )}):n\\ge 1\\rbrace $ whose common law is $P_U^\\theta \\otimes P_T^\\theta $ (for some $\\theta $ ); thus, in particular, each one of the random variables $\\lbrace (U_n,T_n^{(\\nu )}):n\\ge 1\\rbrace $ has independent components as happens under the original law $P_U\\otimes P_T$ (i.e.",
"$P_U^0\\otimes P_T^0$ ) of the random variables.",
"Proposition 3.4 Assume that $\\mathbf {(C1)}$ holds.",
"Then, for $\\theta =w_{\\nu ,h,\\lambda }$ , $P_U^\\theta \\otimes P_T^\\theta $ is an optimal importance sampling distribution for the estimation of $\\Psi (u)$ by simulation for large values of $u$ .",
"Note that the exponential change of measure $P_U^\\theta \\otimes P_T^\\theta $ presented above can be considered also for $\\nu =1$ .",
"Then we have the two following situations.",
"If $\\nu =1$ , for $\\theta >-\\frac{\\lambda }{c}$ we have $dP_T^\\theta (t)=\\frac{e^{-c\\theta t}\\frac{\\lambda ^h}{\\Gamma (h)}t^{h-1}e^{-\\lambda t}1_{(0,\\infty )}(t)dt}{\\int _0^\\infty e^{-c\\theta y}\\frac{\\lambda ^h}{\\Gamma (h)}y^{h-1}e^{-\\lambda y}dy}=\\frac{(c\\theta +\\lambda )^h}{\\Gamma (h)}t^{h-1}e^{-(c\\theta +\\lambda )t}1_{(0,\\infty )}(t)dt;$ thus $\\lbrace P_T^\\theta :\\theta >-\\frac{\\lambda }{c}\\rbrace $ are all Gamma distributions.",
"If $\\nu \\in (0,1)$ , for $\\theta \\ge 0$ we have $dP_T^\\theta (t)=\\frac{e^{-c\\theta t}\\lambda ^h t^{\\nu h-1}E_{\\nu ,\\nu h}^h(-\\lambda t^\\nu )1_{(0,\\infty )}(t)dt}{\\int _0^\\infty e^{-c\\theta y}\\lambda ^h y^{\\nu h-1}E_{\\nu ,\\nu h}^h(-\\lambda y^\\nu )dy}=e^{-c\\theta t}(\\lambda +(c\\theta )^\\nu )^ht^{\\nu h-1}E_{\\nu ,\\nu h}^h(-\\lambda t^\\nu )1_{(0,\\infty )}(t)dt;$ thus $\\lbrace P_T^\\theta :\\theta >0\\rbrace $ are not generalized Mittag Leffler distributions as it is $P_T^0$ because the equality $e^{-c\\theta t}E_{\\nu ,\\nu h}^h(-\\lambda t^\\nu )=E_{\\nu ,\\nu h}^h(-(\\lambda +(c\\theta )^\\nu )t^\\nu )$ holds if and only if $\\theta =0$ (on the contrary the equality always holds if $\\nu =1$ )."
],
[
"Results for the alternative version (weighted Poisson laws)",
"In this section we consider an alternative version of the fractional Poisson process $\\lbrace A_{\\nu ,\\lambda }(t):t\\ge 0\\rbrace $ which is the first version presented in section 4 in [3] with $t^\\nu $ in place of $t$ : $P(A_{\\nu ,\\lambda }(t)=k)=\\frac{(\\lambda t^\\nu )^k}{\\Gamma (\\nu k+1)}\\frac{1}{E_{\\nu ,1}(\\lambda t^\\nu )}\\ \\mbox{for all}\\ k\\in \\mathbb {N}^*:=\\lbrace 0,1,2,3,\\ldots \\rbrace .$ We remark that each random variable $A_{\\nu ,\\lambda }(t)$ has a particular weighted Poisson distribution (we refer to the terminology in [16], page 90; see also the references cited therein), and the weights do not depend on $t$ .",
"More precisely, for each fixed $t$ , the discrete density of $A_{\\nu ,\\lambda }(t)$ is $q_w(k):=\\frac{w(k)q(k)}{\\sum _{j\\ge 0}w(j)q(j)}\\ \\mbox{for all}\\ k\\in \\mathbb {N}^*,$ where the density $\\lbrace q(k):k\\in \\mathbb {N}^*\\rbrace $ and the weights $\\lbrace w(k):k\\in \\mathbb {N}^*\\rbrace $ are defined by $q(k):=\\frac{(\\lambda t^\\nu )^k}{k!",
"}e^{-\\lambda t^\\nu }$ (the classical Poisson density with mean $\\lambda t^\\nu $ ) and $w(k):=\\frac{k!",
"}{\\Gamma (\\nu k+1)}$ , respectively.",
"In this section we prove the LDP for $\\left\\lbrace \\frac{A_{\\nu ,\\lambda }(t)}{t}:t>0\\right\\rbrace $ and we provide a formula (see eq.",
"(REF ) below) for the rate function in terms of a suitable limit of normalized relative entropies (see eq.",
"(REF ) below).",
"This has some analogy with a recent result for stationary Gaussian processes (see section 2 in [22]); moreover it is well-known (see e.g.",
"the discussion in [38]) that the rate functions are often expressed in terms of the relative entropy.",
"Proposition 4.1 For $\\nu \\in (0,1]$ , $\\left\\lbrace \\frac{A_{\\nu ,\\lambda }(t)}{t}:t>0\\right\\rbrace $ satisfies the LDP with good rate function $I_{\\nu ,\\lambda }^{(A)}$ defined by $I_{\\nu ,\\lambda }^{(A)}(x):=\\left\\lbrace \\begin{array}{ll}\\nu x\\log \\frac{\\nu x}{\\lambda ^{1/\\nu }}-\\nu x+\\lambda ^{1/\\nu }&\\ if\\ x\\ge 0\\\\\\infty &\\ if\\ x<0.\\end{array}\\right.$ Remark.",
"For each fixed $t\\ge 0$ , $A_{1,\\lambda }(t)$ is distributed as $M_{1,1,\\lambda }(t)$ in (REF ).",
"Thus, if $\\nu =1$ , we recover Proposition REF with $h=1$ and, actually, one can check that $I_{1,\\lambda }^{(A)}$ coincides with $I_{1,1,\\lambda }^{(M)}$ .",
"Proof.",
"Firstly we can immediately check that $\\mathbb {E}\\left[e^{\\theta A_{\\nu ,\\lambda }(t)}\\right]=\\frac{E_{\\nu ,1}(e^\\theta \\lambda t^\\nu )}{E_{\\nu ,1}(\\lambda t^\\nu )}$ for all $\\theta \\in \\mathbb {R}$ ; note that $\\mathbb {E}\\left[e^{\\theta A_{\\nu ,\\lambda }(t)}\\right]=m(e^\\theta )$ , where $m(\\cdot )$ is the probability generating function in eq.",
"(4.4) in [3] (with $t^\\nu $ in place of $t$ ).",
"Therefore, by using (REF ), we can check the limit $\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\log \\mathbb {E}\\left[e^{\\theta A_{\\nu ,\\lambda }(t)}\\right]=\\lambda ^{1/\\nu }(e^{\\theta /\\nu }-1).$ Then, by Gärtner Ellis Theorem, the LDP holds with good rate function $I_{\\nu ,\\lambda }^{(A)}$ defined by $I_{\\nu ,\\lambda }^{(A)}(x):=\\sup _{\\theta \\in \\mathbb {R}}\\lbrace \\theta x-\\lambda ^{1/\\nu }(e^{\\theta /\\nu }-1)\\rbrace $ which coincides with the rate function in the statement (we omit the details).",
"$\\Box $ In view of what follows we recall the definition and some properties of the relative entropy (see e.g.",
"section 2.3 in [8]).",
"Given two probability measures $Q_1$ and $Q_2$ on the same measurable space $(\\Omega ,\\mathcal {B}(\\Omega ))$ , we write $Q_1\\ll Q_2$ to mean that $Q_1$ is absolutely continuous with respect to $Q_2$ and, in such a case, the density will be denoted by $\\frac{dQ_1}{dQ_2}$ .",
"Then the relative entropy of $Q_1$ with respect to $Q_2$ is defined by $H(Q_1|Q_2)=\\left\\lbrace \\begin{array}{ll}\\int _\\Omega \\log (\\frac{dQ_1}{dQ_2}(\\omega ))Q_1(d\\omega )&\\ \\mathrm {if}\\ Q_1\\ll Q_2\\\\\\infty &\\ \\mathrm {otherwise}.\\end{array}\\right.$ It is known that $H(Q_1|Q_2)$ is nonnegative and it is equal to zero if and only if $Q_1=Q_2$ .",
"Now, in view of what follows, let $Q_{\\nu ,\\lambda ,t}$ be the law of $A_{\\nu ,\\lambda }(t)$ ; here we also allow the case $\\lambda =0$ , and $Q_{\\nu ,0,t}$ is the law of the constant random variable equal to 0.",
"Then, if we consider the following limit of normalized relative entropies $\\mathcal {H}_\\nu (\\lambda _1|\\lambda _2):=\\lim _{t\\rightarrow \\infty }\\frac{1}{t}H(Q_{\\nu ,\\lambda _1,t}|Q_{\\nu ,\\lambda _2,t})$ (for $\\nu \\in (0,1]$ and $\\lambda _1,\\lambda _2\\ge 0$ ), we have $I_{\\nu ,\\lambda }^{(A)}(x)=\\mathcal {H}_\\nu ((\\nu x)^\\nu |\\lambda )\\ \\mbox{for all}\\ x\\ge 0$ as an immediate consequence of the following result.",
"Proposition 4.2 For $\\nu \\in (0,1]$ and $\\lambda _1,\\lambda _2\\ge 0$ , we have $\\mathcal {H}_\\nu (\\lambda _1|\\lambda _2)=\\lambda _1^{1/\\nu }\\log \\frac{\\lambda _1^{1/\\nu }}{\\lambda _2^{1/\\nu }}-\\lambda _1^{1/\\nu }+\\lambda _2^{1/\\nu }$ .",
"Proof.",
"We start assuming that $\\lambda _1,\\lambda _2>0$ .",
"We have the following chain of equalities where, for the latter equality, we take into account eq.",
"(4.6) in [3] (with $t^\\nu $ in place of $t$ ) for the expected value $\\sum _{k=0}^\\infty kP(A_{\\nu ,\\lambda _1}(t)=k)$ : $\\frac{1}{t}H(Q_{\\nu ,\\lambda _1,t}|Q_{\\nu ,\\lambda _2,t})=&\\frac{1}{t}\\sum _{k=0}^\\infty P(A_{\\nu ,\\lambda _1}(t)=k)\\log \\left(\\frac{P(A_{\\nu ,\\lambda _1}(t)=k)}{P(A_{\\nu ,\\lambda _2}(t)=k)}\\right)\\\\=&\\frac{1}{t}\\sum _{k=0}^\\infty P(A_{\\nu ,\\lambda _1}(t)=k)\\log \\left(\\frac{\\lambda _1^k}{\\lambda _2^k}\\frac{E_{\\nu ,1}(\\lambda _2 t^\\nu )}{E_{\\nu ,1}(\\lambda _1t^\\nu )}\\right)\\\\=&\\frac{1}{t}\\log \\frac{\\lambda _1}{\\lambda _2}\\sum _{k=0}^\\infty kP(A_{\\nu ,\\lambda _1}(t)=k)+\\frac{1}{t}\\log \\left(\\frac{E_{\\nu ,1}(\\lambda _2t^\\nu )}{E_{\\nu ,1}(\\lambda _1 t^\\nu )}\\right)\\\\=&\\frac{1}{t}\\frac{\\lambda _1t^\\nu }{\\nu }\\frac{E_{\\nu ,\\nu }(\\lambda _1t^\\nu )}{E_{\\nu ,1}(\\lambda _1t^\\nu )}\\cdot \\log \\frac{\\lambda _1}{\\lambda _2}+\\frac{1}{t}\\log \\left(\\frac{E_{\\nu ,1}(\\lambda _2t^\\nu )}{E_{\\nu ,1}(\\lambda _1t^\\nu )}\\right).$ Then, by using (REF ), the limit in (REF ) holds with $\\mathcal {H}_\\nu (\\lambda _1|\\lambda _2)=\\frac{\\lambda _1^{1/\\nu }}{\\nu }\\log \\frac{\\lambda _1}{\\lambda _2}+\\lambda _2^{1/\\nu }-\\lambda _1^{1/\\nu }=\\lambda _1^{1/\\nu }\\log \\frac{\\lambda _1^{1/\\nu }}{\\lambda _2^{1/\\nu }}-\\lambda _1^{1/\\nu }+\\lambda _2^{1/\\nu }.$ Thus the proof of the proposition is complete when $\\lambda _1,\\lambda _2>0$ , and now we give some details for the other cases.",
"If $\\lambda _1=0$ and $\\lambda _2>0$ , we can consider this procedure, but the above sum reduces to the first addendum (the one with $k=0$ ) and we have $\\mathcal {H}_\\nu (\\lambda _1|\\lambda _2)=\\lambda _2^{1/\\nu }$ .",
"If $\\lambda _2=0$ , we have $\\mathcal {H}_\\nu (\\lambda _1|0)=\\left\\lbrace \\begin{array}{ll}0&\\ \\mathrm {if}\\ \\lambda _1=0\\\\\\infty &\\ \\mathrm {if}\\ \\lambda _1>0\\end{array}\\right.$ because, for all $t>0$ , we trivially have $H(Q_{\\nu ,0,t}|Q_{\\nu ,0,t})=0$ and, if $\\lambda _1>0$ , $H(Q_{\\nu ,\\lambda _1,t}|Q_{\\nu ,0,t})=\\infty $ .",
"$\\Box $ Finally we remark that $\\frac{1}{t}H(Q_{1,\\lambda _1,t}|Q_{1,\\lambda _2,t})&=\\frac{1}{t}\\sum _{k=0}^\\infty P(A_{1,\\lambda _1}(t)=k)\\log \\left(\\frac{\\lambda _1^k}{\\lambda _2^k}\\frac{E_{1,1}(\\lambda _2 t)}{E_{1,1}(\\lambda _1 t)}\\right)\\\\&=\\frac{1}{t}\\log \\frac{\\lambda _1}{\\lambda _2}\\sum _{k=0}^\\infty k\\frac{(\\lambda _1t)^k}{k!",
"}e^{-\\lambda _1t}+\\frac{1}{t}\\log \\left(e^{(\\lambda _2-\\lambda _1)t}\\right)\\\\&=\\lambda _1\\log \\frac{\\lambda _1}{\\lambda _2}-\\lambda _1+\\lambda _2=H(Q_{1,\\lambda _1,1}|Q_{1,\\lambda _2,1})$ does not depend on $t>0$ , and therefore coincides with $\\mathcal {H}_1(\\lambda _1|\\lambda _2)$ ."
]
] | 1204.1446 |
[
[
"Bunching and anti-bunching of localised particles in disordered media"
],
[
"Abstract We consider pairs of non-interacting quantum particles transmitted through a disordered medium, with emphasis on the role of their quantum statistics.",
"It is shown that particle-number correlations measured in transmission are strikingly sensitive to the quantum nature of the particles when they undergo Anderson localisation, due to bosonic bunching and fermionic anti-bunching in the scattering channels of the medium.",
"The case of distinguishable particles is also discussed."
],
[
"Model",
"We consider a source producing pairs of monochromatic particles with identical wave number $k_0$ .",
"These pairs propagate through a medium containing heterogeneities randomly arranged in space and characterised by a scattering mean free path $\\ell $ , see fig.",
"REF .",
"The medium has length $L$ and cross-section $A$ , and is assumed to be connected to perfect leads (not shown in fig.",
"REF ).",
"In such a geometry the incoming and outgoing wave fronts can be decomposed on a discrete set of $N\\simeq k_0^2A$ transverse modes, each of them corresponding to a value of the transverse component of the wave vector compatible with the boundary conditions.",
"The statistical properties of transport observables are independent of the transverse shape of the medium in the limit $N\\gg 1$ that we assume from here on.",
"Particles' coincidences between two modes $k\\ne k^\\prime $ are detected in transmission.",
"Such a single-mode detection setup requires a detector area smaller than the typical size $R^2/N$ of a speckle spot, with $R$ the distance between sample and detector.",
"To quantify the coincident count statistics, we quantise the field before and after the disordered region, using the basis of the transverse modes [12], and define the normalised correlation function $C = \\frac{\\overline{\\langle :\\!",
"\\hat{n}_k \\hat{n}_{k^{\\prime }}\\!\\!",
": \\rangle }}{\\overline{\\langle \\hat{n}_k \\rangle } \\times \\overline{\\langle \\hat{n}_{k^{\\prime }} \\rangle }} ,$ where $\\hat{n}_{k}$ denotes the particle-number operator in the outgoing mode $k$ , $\\langle \\ldots \\rangle $ the quantum expectation value, $:\\,\\,:$ the normal ordering of operators, and the overbar the average over a statistical ensemble of disordered samples.",
"Since we are here primarily interested in the quantum statistics of the particles (fermions, bosons or distinguishable particles), we remain general in the following and do not specify how the two-particle state is initially distributed over the incoming modes.",
"In the spirit of [14], we thus write the initial density matrix in the general form $\\hat{\\varrho } = \\frac{1}{2}\\!\\sum _{i,j,i^\\prime ,j^\\prime } \\!",
"w_{i j i^\\prime j^\\prime }\\, \\hat{a}^{\\dagger }_{i} \\hat{a}^{\\dagger }_{j} \\vert 0 \\rangle \\langle 0 \\vert \\hat{a}_{i^\\prime } \\hat{a}_{j^\\prime },$ where the operator $\\hat{a}^{\\dagger }_{i}$ creates a particle in the transverse mode labeled by index $i$ , and all sums run from 1 to $N$ .",
"Creation and annihilation operators fulfill the commutation relations $\\hat{a}^{\\dagger }_{i}\\hat{a}^{\\dagger }_{j}=\\pm \\hat{a}_{j}^{\\dagger }\\hat{a}_i^{\\dagger }$ , $\\hat{a}_{i}\\hat{a}_{j}=\\pm \\hat{a}_{j}\\hat{a}_i$ and $\\hat{a}^{\\dagger }_{i}\\hat{a}_{j}=\\delta _{ij}\\pm \\hat{a}_{j}^{\\dagger }\\hat{a}_i$ , with $+$ for bosons and $-$ for fermions.",
"Due to these relations, the tensor $w_{iji^\\prime j^\\prime }$ can be chosen (anti-) symmetric under the exchange of the first and last two indices for (fermions) bosons.",
"The prefactor $1/2$ in eq.",
"(REF ) guarantees the state normalization, $\\text{tr}\\hat{\\rho }=\\sum _{i,j}w_{ijij}=1$ .",
"Figure: A source produces pairs of particles characterised by a two-particle density matrix ρ ^\\hat{\\rho }.",
"These pairs are scattered from a disordered medium with length LL, mean free path ℓ\\ell , and supporting NN transverse modes.",
"The coincidence rate between two outgoing modes kk and k ' k^\\prime is analysed in transmission.From the initial density matrix $\\hat{\\rho }$ , the mean particle number in some outgoing mode $k$ is defined as $\\langle \\hat{n}_k \\rangle =\\text{tr}[\\hat{\\rho }\\,\\hat{c}_k^\\dagger (\\underline{a}^\\dagger )\\hat{c}_k(\\underline{a})],$ where $\\hat{c}_k(\\underline{a})=\\sum _i t_{ki}\\hat{a}_i$ and $\\hat{c}^\\dagger _k(\\underline{a}^\\dagger )=\\sum _i t_{ki}^*\\hat{a}_i^\\dagger $ are the input-output relations connecting the annihilation (creation) operator $\\hat{c}_k$ ($\\hat{c}_k^\\dagger $ ) of the outgoing mode $k$ to the set of operators $\\hat{a}_i$ ($\\hat{a}_i^\\dagger $ ) of the incoming modes $i$ through the (random) transmission matrix $t$ of the disordered medium.",
"Substituting these relations into eq.",
"(REF ), we readily obtain $\\langle \\hat{n}_k \\rangle =2\\sum _{i,j,i^\\prime }w_{iji^\\prime j}t_{ki}t^*_{ki^\\prime }.$ Similarly, from the definition $\\langle :\\!\\hat{n}_k\\hat{n}_{k^\\prime }\\!\\!",
": \\rangle =\\text{tr}[\\hat{\\rho }\\,\\hat{c}_k^\\dagger (\\underline{a}^\\dagger )\\hat{c}_{k^\\prime }^\\dagger (\\underline{a}^\\dagger )\\hat{c}_k(\\underline{a})\\hat{c}_{k^\\prime }(\\underline{a})]$ of the particle-number correlation, we obtain $\\langle :\\!\\hat{n}_k\\hat{n}_{k^\\prime }\\!\\!",
": \\rangle =2\\!\\sum _{i,j,i^\\prime ,j^\\prime }w_{iji^\\prime j^\\prime }t_{ki}t^*_{k^\\prime j}t^*_{ki^\\prime }t_{k^\\prime j^\\prime }.$ Note that at this stage eqs.",
"(REF ) and (REF ) hold for fermions and bosons, provided the tensor $w_{iji^\\prime j^\\prime }$ is properly (anti-) symmetrised.",
"The next step in the derivation of the correlation function (REF ) consists in performing the ensemble averages.",
"In eqs.",
"(REF ) and (REF ), statistical properties of the disordered medium are encoded in the transmission matrix elements $t_{ki}$ .",
"In the limit $N\\ell /L\\equiv g\\gg 1$ , which corresponds to a regime where particles are multiply scattered according to a diffusion process, these elements are normally distributed over the unitary group [19].",
"This property was notably used in [14] to calculate the statistical distribution of $\\langle :\\!\\hat{n}_k\\hat{n}_{k^\\prime }\\!\\!",
": \\rangle $ for photons.",
"Here however, we aim at studying the behaviour of $C$ for arbitrary values of the parameter $g$ .",
"For this purpose we make use of the unitarity and symmetry of the transmission matrix, which allows us to employ the polar decomposition [19], [20] $t_{k i} = \\sum _{a } u_{k a} \\sqrt{\\tau _a} v_{a i},$ where $u$ and $v$ are random unitary matrices and $\\tau _1, \\dots , \\tau _N$ are the eigenvalues of $t$ .",
"Eq.",
"(REF ) is at the basis of the macroscopic description of disordered media [19].",
"Physically, the matrix $v$ first distributes a field amplitude incident in mode $i$ among the scattering channels of the medium, each having its transmission coefficient $\\tau _a$ (between 0 and 1).",
"The matrix $u$ finally recombines the transmitted field amplitudes in the outgoing mode $k$ .",
"For a medium with length much larger than its width, i.e in the limit $L\\gg \\sqrt{A}$ , ensemble averaging can be carried out in two steps: first over the matrices $u$ and $v$ , which are uniformly distributed over the unitary group (isotropy assumption), and second over the eigenvalues $\\tau _1, \\dots , \\tau _N$ , whose joint distribution follows the so-called DMPK equation [20].",
"Substituting the decomposition (REF ) into eqs.",
"(REF ) and (REF ), and performing the average over unitary matrices, we obtain $C = \\frac{1}{2} \\frac{\\sum _{a,b}\\overline{ \\tau _a\\tau _b}\\pm \\sum _a\\overline{\\tau _a^2}}{\\left(\\sum _a\\overline{\\tau _a}\\right)^2},$ where the positive sign refers to bosons and the negative sign to fermions.",
"Eq.",
"(REF ) is the first important result of this Letter.",
"It shows that, when $L\\gg \\sqrt{A}$ , the correlation function $C$ is independent of how the incident state is distributed over the incident modes of the disordered medium, and only depends on the statistical correlations between scattering channels [14].",
"This property originates from the fact that when deriving eq.",
"(REF ), we assumed each field amplitude incoming in a given mode to be equally distributed among the scattering channels of the medium (isotropy assumption).",
"Another important comment concerns the prefactor $1/2$ in eq.",
"(REF ), also found in [15], [16].",
"For the particular case of a two-photon Fock state with Fano factor $F=0$ , this prefactor was shown to express itself as $1+(F-1)/2$ [10].",
"Its value $1/2<1$ therefore signals the sub-Poissonian statistics, here of an arbitrary incident two-particle state.",
"We now comment on the structure of the mean correlation [numerator of eq.",
"(REF )].",
"The latter consists of two terms.",
"The first one, $\\sum _{a,b}\\overline{ \\tau _a\\tau _b}$ , corresponds to the sum of all possible correlations between scattering channels.",
"The second one, $\\pm \\sum _{a}\\overline{\\tau _a^2}$ , corresponds to the sum of all possible autocorrelations of scattering channels and is affected by the specific statistics of quantum particles: fermions cannot both propagate in the same channel, therefore this sum appears with a minus sign, and cancels the autocorrelation contributions in the first term (anti-bunching effect).",
"On the other hand, bosons tend to bunch in the scattering channels, as manifested by the positive sign in front of the second term.",
"These bunching and anti-bunching effects have important consequences on the magnitude of the correlation function $C$ , as we now show."
],
[
"Numerics for bosons and fermions",
"Eq.",
"(REF ) is valid for any value of the parameter $g=N\\ell /L$ , provided $L\\gg \\sqrt{A}$ .",
"In order to clarify how bunching and anti-bunching effects would show up concretely in an experiment accessing the correlation function (REF ), we evaluated $C$ numerically as a function of $L/\\ell $ , for a fixed number of modes, $N =10$ .",
"For this purpose we computed each of the averages $\\sum _a\\overline{\\tau _a^2}$ , $\\sum _{a,b}\\overline{ \\tau _a\\tau _b}$ and $\\sum _a\\overline{\\tau _a}$ from the DMPK equation, by means of a Markov chain Monte Carlo approach [22].",
"Such a numerical procedure was already used in the context of the conductance distribution of metallic conductors [23] and of intensity correlations of classical light in disordered media [24].",
"Figure: Correlation function () for bosons (dots, blue online) and fermions (squares, red online), plotted as functions of L/ℓL/\\ell , for N=10N=10.",
"Diamonds (orange online) are the contribution ∑ a τ a 2 ¯/∑ a τ a ¯ 2 \\sum _a\\overline{\\tau _a^2}/\\sum _a\\overline{\\tau _a}^2 to C bosons C^\\text{bosons}.",
"Dashed horizontal lines are the analytical prediction (), and dashed curves are given by eqs.",
"() and ().",
"Vertical dotted lines indicate the crossovers L∼ℓL\\sim \\ell and L∼NℓL\\sim N\\ell from ballistic to diffusion and from diffusion to localisation, respectively.At fixed $N$ , the study of $C$ as a function of $L/\\ell $ allows us to probe all possible regimes of transport in the disorder, from quasi-ballistic ($L\\lesssim \\ell $ ) to diffusive propagation ($\\ell \\ll L\\ll N\\ell $ ) and eventually Anderson localisation ($L\\gg N\\ell $ ).",
"The results can be seen in fig.",
"REF , for bosons (dots, blue online) and fermions (squares, red online).",
"In the quasi-ballistic and diffusive regimes, $C$ is very close to $1/2$ for bosons and fermions, i.e.",
"the correlation is hardly sensitive to the quantum statistics of the particles (the small difference observed in fig.",
"REF is discussed in the next section).",
"This picture dramatically changes as $L$ increases, at the onset of localisation ($L\\sim N\\ell $ ): the bosonic correlation function starts growing, while the fermionic counterpart decreases.",
"Far in the localisation regime ($L\\gg N\\ell $ ), $C\\gg 1$ for bosons and $C\\ll 1$ for fermions.",
"These radically different behaviours can be interpreted in the framework of the theory of active transmission channels [21], [25] by rewriting eq.",
"(REF ) as $C^\\text{bosons}= (1/2)\\times \\left(\\sum _{a\\ne b}\\overline{ \\tau _a\\tau _b}+2\\sum _a\\overline{\\tau _a^2}\\right)/\\left(\\sum _a\\overline{\\tau _a}\\right)^2$ and $C^\\text{fermions}= (1/2)\\times \\left(\\sum _{a\\ne b}\\overline{ \\tau _a\\tau _b}\\right)/\\left(\\sum _a\\overline{\\tau _a}\\right)^2$ .",
"As the length $L$ increases, more and more scattering channels become closed, $i.e.$ their transmission coefficient becomes exponentially small.",
"Eventually, in the localisation regime, only one channel $a_0$ retains a fairly large transmission coefficient.",
"In this limit, we thus have $\\left(\\sum _a\\overline{\\tau _a}\\right)^2\\simeq \\overline{\\tau _{a_0}}^2$ , $\\sum _a\\overline{\\tau _a^2}\\simeq \\overline{\\tau _{a_0}^2}$ and $\\sum _{a\\ne b}\\overline{ \\tau _a\\tau _b}\\simeq \\overline{ \\tau _{a_0}\\tau _{b_0}}$ , where $b_0$ denotes a closed channel.",
"Since the correlation between a closed and the open channel is typically very small compared to $\\overline{\\tau _{a_0}}^2$ , we have $C^\\text{fermions}\\ll 1$ .",
"On the other hand, we have also $\\overline{\\tau _{a_0}\\tau _{b_0}}\\ll \\overline{\\tau _{a_0}^2}$ , such that $C^\\text{bosons}\\simeq (1/2)\\times \\overline{\\tau _{a_0}^2}/\\overline{\\tau _{a_0}}^2$ .",
"This ratio is very large because transmission fluctuations well exceed the mean transmission in the localisation regime [19].",
"To get a better picture of the weight of each term appearing in $C^\\text{bosons}$ and $C^\\text{fermions}$ , in the different regimes of transport, we also show in fig.",
"REF the contribution of channel autocorrelations to $C^\\text{bosons}$ , $\\sum _a\\overline{\\tau _a^2}/\\sum _a\\overline{\\tau _a}^2$ (diamonds, orange online).",
"It is instructive to compare the results in fig.",
"REF for bosons (dots) with those obtained by Ott et al., fig.",
"4 of Ref.",
"[15], for the particular case of a two-photon Fock state.",
"Our correlation function is insensitive to the form of the incident state, unlike in [15], where the correlation function is constant for two photons incident in the same mode whereas it increases and saturates for two photons incident in two different modes.",
"This difference relies on the different quantity appearing in the denominator of the correlation function of [15], $\\overline{\\langle n_k \\rangle \\langle n_k^\\prime \\rangle }$ , instead of $\\overline{\\langle n_k \\rangle }\\times \\overline{\\langle n_k^\\prime \\rangle }$ in our case, eq.",
"(REF )."
],
[
"Analytical results",
"In order to support our numerical observations, we now calculate explicitly $C$ in the quasi-ballistic, diffusive and localisation regimes.",
"First, when $L\\lesssim \\ell $ , we have $\\tau _a\\simeq 1\\, \\forall a$ , such that $C\\simeq \\frac{1}{2} \\frac{N^2\\pm N}{N^2}=\\frac{1}{2}\\left(1\\pm \\frac{1}{N}\\right).$ Eq.",
"(REF ) is shown in fig.",
"REF for bosons ($+$ ) and fermions ($-$ ) as dashed horizontal lines.",
"The $1/N$ difference between the two types of particles is visible in the figure since we considered a finite value of $N$ for the numerics.",
"In practice however, $N\\gg 1$ and this difference is negligible.",
"In the diffusive regime $\\ell \\ll L\\ll N\\ell $ , averages over transmission eigenvalues can be evaluated by making use of the method of moments introduced in [26].",
"The result is a perturbation expansion of $C$ in powers of the parameter $1/g=L/(N\\ell )\\ll 1$ : $C= \\frac{1}{2} \\left[1\\pm \\frac{2}{3g}+\\mathcal {O} \\left(\\frac{1}{g^2}\\right)\\right].$ In the regime of diffusive transport, bosonic bunching ($+$ ) and fermionic anti-bunching ($-$ ) in the scattering channels are thus visible, but remain small as long as $g\\gg 1$ .",
"Finally, far in the localisation regime $L\\gg N\\ell $ , $C$ can be calculated from the probability distribution of the transmission eigenvalues [19].",
"A straightforward calculation then yields: $C^\\text{bosons} = \\frac{1}{3} \\sqrt{\\frac{\\pi L}{2\\xi }} \\exp \\left( \\frac{L}{2 \\xi } \\right), \\\\C^\\text{fermions} = \\frac{1}{2} \\sqrt{\\frac{\\pi L}{2 \\xi }} \\exp \\left( - \\frac{3 L}{2\\xi } \\right), $ where we have explicitly introduced the localisation length $\\xi =N\\ell $ .",
"Eqs.",
"(REF ) and () are consistent with the discussion above, namely $C^\\text{fermions}\\ll 1\\ll C^\\text{bosons}$ .",
"Analytical predictions (REF ) and () are shown in fig.",
"REF as dashed curves, and agree well with our Monte Carlo simulations."
],
[
"Distinguishable particles",
"To complete our discussion, we finally consider the scattering problem of a pair of distinguishable particles.",
"For this purpose, we equip them with an additional spin degree of freedom, indicated by the symbols $\\uparrow $ and $\\downarrow $ .",
"With this strategy, we consider an incident state of the form $\\hat{\\rho } =\\sum _{i,j,i^\\prime ,j^\\prime } \\!",
"w_{i j i^\\prime j^\\prime }\\, \\hat{a}^{\\dagger }_{i\\uparrow } \\hat{a}^{\\dagger }_{j\\downarrow } \\vert 0 \\rangle \\langle 0 \\vert \\hat{a}_{i^\\prime \\uparrow } \\hat{a}_{j^\\prime \\downarrow }.$ In comparison with eq.",
"(REF ), note the missing prefactor $1/2$ .",
"This originates from the new commutation relation involving the two particles which have now different spins, $\\hat{a}_{k\\uparrow }\\hat{a}_{k\\downarrow }^\\dagger =\\hat{a}_{k\\downarrow }^\\dagger \\hat{a}_{k\\uparrow }$ .",
"Normalization of the state ($\\ref {eq:statedist}$ ) again reads $\\text{tr}\\hat{\\rho }=\\sum _{i,j}w_{ijij}=1$ .",
"The formalism developed above for indistinguishable particles remains essentially the same for distinguishable particles, but care has to be taken in defining the particle-number and correlation operators.",
"Indeed, since one detects particles on output of the medium irrespective of their spin, these operators must now be symmetrised with respect to the two particles, i.e $\\hat{n}_k=\\hat{c}_{k\\uparrow }^\\dagger \\hat{c}_{k\\uparrow }+\\hat{c}_{k\\downarrow }^\\dagger \\hat{c}_{k\\downarrow }$ for the correlation operator in the outgoing mode $k$ , and $:\\!\\hat{n}_k\\hat{n}_{k^\\prime }\\!\\!",
":=\\hat{c}_{k\\uparrow }^\\dagger \\hat{c}_{k^\\prime \\downarrow }^\\dagger \\hat{c}_{k\\uparrow }\\hat{c}_{k^\\prime \\downarrow }+\\hat{c}_{k\\downarrow }^\\dagger \\hat{c}_{k^\\prime \\uparrow }^\\dagger \\hat{c}_{k\\downarrow }\\hat{c}_{k^\\prime \\uparrow }.$ for the coincidence rate operator between the outgoing modes $k$ and $k^\\prime $ .",
"The input-output relations now read $\\hat{c}_{k\\uparrow (\\downarrow )}=\\sum _i t_{ki}^{\\uparrow (\\downarrow )}\\hat{a}_{i\\uparrow (\\downarrow )}$ and $\\hat{c}^\\dagger _{k\\uparrow (\\downarrow )}=\\sum _i t_{ki}^{*\\uparrow (\\downarrow )}\\hat{a}_{i\\uparrow (\\downarrow )}^\\dagger $ (note the spin labeling also in the transmission matrix elements).",
"Quantum expectation values for $\\hat{n}_k$ and $:\\!\\hat{n}_k\\hat{n}_{k^\\prime }\\!\\!",
":$ are now given by $\\langle \\hat{n}_k \\rangle =\\sum _{iji^\\prime }w_{iji^\\prime j} t_{ki}^{\\uparrow }t_{ki^\\prime }^{*\\uparrow }+\\sum _{ijj^\\prime }w_{ijij^\\prime } t_{kj}^{\\downarrow }t_{kj^\\prime }^{*\\downarrow },$ and $\\langle :\\!\\hat{n}_k\\hat{n}_{k^\\prime }\\!\\!",
": \\rangle =\\sum _{iji^\\prime i^\\prime }w_{iji^\\prime j^\\prime }\\left(t_{ki}^{\\uparrow }t_{ki^\\prime }^{*\\uparrow }t_{k^\\prime j}^{\\downarrow }t_{k^\\prime j^\\prime }^{*\\downarrow }+t_{k^\\prime i}^{\\uparrow }t_{k^\\prime i^\\prime }^{*\\uparrow }t_{k j}^{\\downarrow }t_{k j^\\prime }^{*\\downarrow }\\right).$ Note that in eq.",
"(REF ), the first (second) term on the right-hand side is nothing but the contribution of the particle with spin up (down) to the detected mean particle number in mode $k$ .",
"Similarly, the two terms on the right-hand side of eq.",
"(REF ) correspond to the two possible coincidence events, i.e.",
"particle with spin up detected in mode $k$ and particle with spin down detected in mode $k^\\prime $ for the first term, and vice versa for the second one.",
"Finally, we again decompose transmission coefficients according to $t_{k i}^{\\uparrow (\\downarrow )} = \\sum _{a = 1}^N u_{k a}^{\\uparrow (\\downarrow )} \\sqrt{\\tau _a} v_{a i}^{\\uparrow (\\downarrow )}$ and carry out the averages over the matrices $u^{\\uparrow ,\\downarrow }$ and $v^{\\uparrow ,\\downarrow }$ .",
"This yields: $C^\\text{disting} = \\frac{1}{2} \\frac{\\sum _{a,b}\\overline{ \\tau _a\\tau _b}}{\\left(\\sum _a\\overline{\\tau _a}\\right)^2}.$ Comparing with eq.",
"(REF ), we see that the bunching/anti-bunching term has disappeared for distinguishable particles, as expected.",
"We computed $C^\\text{disting}$ numerically, using the same Monte Carlo approach as above to carry out the remaining averages over transmission eigenvalues.",
"The results are shown in fig.",
"REF as a function of $L/\\ell $ (triangles, green online).",
"For comparison we also show $C^\\text{bosons}$ (dots, blue online).",
"Figure: Correlation function for bosons (dots, blue online) and distinguishable particles (triangles, green online), plotted as functions of L/ℓL/\\ell for N=10N=10.",
"The upper and lower dashed curves are the analytical predictions () and () respectively, and the vertical dotted line indicates the crossover L∼NℓL\\sim N\\ell from diffusion to localisation.As for bosons, the correlation function of distinguishable particles increases with $L/\\ell $ .",
"This is expected since at the onset of localisation, one scattering channel $a_0$ dominates transport and $\\sum _{a,b}\\overline{\\tau _a\\tau _b}\\simeq \\overline{\\tau _{a_0}^2}\\gg (\\sum _{a}\\overline{\\tau _a})^2\\simeq \\overline{\\tau _{a_0}}^2$ .",
"Again, simple analytical results can be obtained in the quasi-ballistic regime of transport, where $C^\\text{disting}=1/2$ , and in the diffusion regime, where $C^\\text{disting}=(1/2)\\times \\left[1+2/(15g^2)+\\mathcal {O}(1/g^3)\\right]$ .",
"On the other hand, far in the localisation regime we find $C^\\text{disting}=\\dfrac{1}{6}\\sqrt{\\dfrac{\\pi L}{2\\xi }}\\exp \\left(\\dfrac{L}{2\\xi }\\right).$ This result is shown in fig.",
"REF (lower dashed curve), and agrees well with the numerical calculations.",
"It also confirms that the bunching effect of bosons is only fully visible in the localisation regime where $\\frac{C^\\text{bosons}}{C^\\text{disting}} \\stackrel{L \\gg N\\ell }{\\longrightarrow } \\; 2,$ as also demonstrated by the numerical points in fig.",
"REF ."
],
[
"Conclusion",
"We have shown that particle-number correlations measured on output of a disordered medium are extremely sensitive to the quantum statistics of localised particles.",
"This phenomenon reflects the bosonic bunching and fermionic anti-bunching in the scattering channels of the medium.",
"An interesting extension of this work would be to analyse the scattering problem of many-particle quantum states.",
"We thank M. C. Tichy for useful discussions and S. E. Skipetrov for his comments on the manuscript.",
"N. C. acknowledges financial support from the Alexander von Humboldt Foundation."
]
] | 1204.1214 |
[
[
"Circumbinary MHD Accretion into Inspiraling Binary Black Holes"
],
[
"Abstract As 2 black holes bound to each other in a close binary approach merger their inspiral time becomes shorter than the characteristic inflow time of surrounding orbiting matter.",
"Using an innovative technique in which we represent the changing spacetime in the region occupied by the orbiting matter with a 2.5PN approximation and the binary orbital evolution with 3.5PN, we have simulated the MHD evolution of a circumbinary disk surrounding an equal-mass non-spinning binary.",
"Prior to the beginning of the inspiral, the structure of the circumbinary disk is predicted well by extrapolation from Newtonian results.",
"The binary opens a low-density gap whose radius is roughly two binary separations, and matter piles up at the outer edge of this gap as inflow is retarded by torques exerted by the binary; nonetheless, the accretion rate is diminished relative to its value at larger radius by only about a factor of 2.",
"During inspiral, the inner edge of the disk at first moves inward in coordination with the shrinking binary, but as the orbital evolution accelerates, the rate at which the inner edge moves toward smaller radii falls behind the rate of binary compression.",
"In this stage, the rate of angular momentum transfer from the binary to the disk slows substantially, but the net accretion rate decreases by only 10-20%.",
"When the binary separation is tens of gravitational radii, the rest-mass efficiency of disk radiation is a few percent, suggesting that supermassive binary black holes in galactic nuclei could be very luminous at this stage of their evolution.",
"If the luminosity were optically thin, it would be modulated at a frequency that is a beat between the orbital frequency of the disk's surface density maximum and the binary orbital frequency.",
"However, a disk with sufficient surface density to be luminous should also be optically thick; as a result, the periodic modulation may be suppressed."
],
[
"Introduction",
"There is now excellent evidence that every galaxy with a bulge contains a supermassive black hole at its center [34].",
"In addition, the prevailing theory of galaxy formation posits that today's massive galaxies were assembled from smaller pieces, as dark-matter haloes of progressively greater size merged [23], [10].",
"If massive black holes were already present in those progenitors, they would bring their black holes with them into the new combined galaxy, creating an opportunity for the black holes to merge.",
"Such an event would be very exciting to detect for many reasons: It would reveal the presence of supermassive black holes early in the life of galaxies.",
"It would shed important light on the growth of the strong correlations between nuclear black hole mass and galaxy structure [34].",
"Most of all, it would provide a concrete example of one of general relativity's most spectacular predictions and possibly also allow a test of the validity of general relativity in a truly strong-field regime.",
"An extremely large amount of energy is very rapidly released in a binary black hole (BBH) merger event, almost all of it through gravitational radiation [several percent of the black hole masses in a timescale of $\\sim (M_{\\rm BBH}/M_\\odot )\\times 493\\,\\mu $ s].",
"Gravitational radiation may be strong enough to eject the final remnant from its host galaxy, with recoil velocities or “kicks” up to $\\approx 10^3$ km/s predicted by numerical relativity simulations [5], [20], [21], [32], [37], [49], [6], [36], [55], [56], [57].",
"Unfortunately, a gravitational wave observatory with adequate sensitivity in the appropriate frequency range is still well in the future, whether it operates by direct detection or through pulsar timing [42], [94].",
"On the other hand, even if only a small part of the energy is deposited in nearby gas, the associated photon signals might be much more readily seen with instruments operating today.",
"Because the energy given to the gas comes from work done by gravitational forces, one would expect, on the basis of the Equivalence Principle, that the total energy added to the gas would be proportional to its mass.",
"If most of this added energy is dissipated into heat (local irregularities are likely to drive shocks), the total energy radiated in photons would then be similarly proportional to the gas mass [51].",
"The question is, therefore: “How much mass would one expect in the neighborhood of a black hole merger?\"",
"Even if a BBH were supplied with mass at a rate characteristic of high luminosity quasars ($\\sim 10 M_{\\odot }$ yr$^{-1}$ ), several effects may severely reduce how much gas remains close to the binary.",
"Torques exerted by the binary on the inflowing gas may hold back the inflow, preventing much of it from approaching closer than a few times the binary separation $a$ [76], [58].",
"As the binary compresses, whether by interactions with passing stars and external gas or by gravitational radiation, the gas follows, but is held off at a distance of at least $\\simeq 2a$ .",
"Toward the end of the binary's evolution, gravitational radiation losses grow rapidly and dominate the orbital shrinkage.",
"Ultimately, the orbit shrinks on a timescale shorter than the characteristic accretion inflow time and the BBH is expected to decouple from the disk.",
"After such decoupling, there would not be enough time for much disk mass to catch up with the black holes before they merge [59].",
"Thus, for a given external supply rate, the amount of gas available to be heated in a merger is determined by a competition between the internal stresses that drive inflow and a pair of dynamical mechanisms that tend to keep gas at “arms-length\" from the merging black holes.",
"Until recently, efforts to quantify these effects have relied almost entirely on the phenomenological Shakura-Sunyaev $\\alpha $ -disk model to describe internal stresses [59], [58], [54], [88], in which the vertically-integrated and time- and azimuthally-averaged internal stress is supposed to be a factor $\\alpha $ times the similarly integrated and averaged pressure; the only exceptions were studies focusing on binaries with large mass ratios between primary and secondary, a limit primarily relevant to planet formation and to extreme-mass ratio inspiral sources [96], [70], [63], [46], [99].",
"Moreover, with the exception of [29] and [15], which assumed these stresses were negligible, all these calculations also assumed Newtonian dynamics.",
"However, there is strong reason to think that the actual mechanism of these stresses is magnetohydrodynamic (MHD) turbulence, stirred by the magnetorotational instability (MRI) [8].",
"In contrast to ordinary, isotropic turbulence, orbital shear makes this turbulence highly anisotropic, so that there is a non-zero correlation between the radial and azimuthal components of the magnetic field; this correlation creates the stress.",
"MHD calculations are therefore required, at the very least, to establish the appropriate scale of the stresses and the approximate magnitude of $\\alpha $ .",
"In addition, although the $\\alpha $ -model may give a reasonable description of time-averaged behavior well inside the body of an accretion flow, it is particularly ill-suited for predicting dynamical behavior on shorter timescales [40] and at disk edges [52], [66].",
"Because the key issues in how much gas reaches a merging BBH depend, of course, on the time-dependent behavior of gas near and within the inner edge of a disk, explicit calculation of the MHD turbulence is also required for an accurate treatment of the time- and spatial-dependence of the internal stress.",
"In this paper, we present the first simulation of a circumbinary accretion disk around a binary black hole system during the epoch in which the binary's inspiral time grows shorter than the inflow time through the disk.",
"Generically, this period occurs not long before the binary's final merger.",
"Our physics treatment includes fully relativistic MHD.",
"This study differs from that of [82], who presented similar MHD simulations, but concentrated on the Newtonian regime, when the black holes are very widely separated.",
"Moreover, their Newtonian treatment did not allow for the black holes to inspiral, a reasonable assumption when the semi-major axis is hundreds of thousands of gravitational radii, but a terrible assumption in the late inspiral.",
"We here focus on BBHs with separations of $\\sim (10$ –$20)\\,r_g$ , where $r_g \\equiv GM/c^2$ and $M$ is the total mass of the binary (we adopt geometric units with $G=c=1$ for the remainder of this paper).",
"The spacetime associated with the BBH orbital dynamics is described through a vacuum post-Newtonian (PN) approximation (see the review paper of [12] and references therein), where we neglect the back reaction of the disk on the BBH dynamics.",
"Our work also contrasts with that of [31], who employed full numerical relativity to compute the spacetime in which an initially uniform gas distribution with an externally-imposed uniform magnetic field evolved during the last 3 orbits before merger.",
"As we will describe below, the PN approximation is adequate to describe the spacetime evolution for our needs.",
"The PN scheme is a method to describe approximately the dynamics of physical systems in which motions are slow compared to the speed of light and gravitational fields are weak.",
"That is, one solves the Einstein field equations perturbatively, expanding in $(v/c)^2 \\ll 1$ and $r_g/r = GM/(rc^2) \\ll 1$ Here $v$ , $M$ and $r$ are the characteristic velocity, mass and size or separation of the system.",
"This approximation has been remarkably effective in describing the perihelion precession of Mercury [27], and the gravitational-wave loss from binary systems, such as the Hulse-Taylor pulsar, PSR B1913+16 (see e.g.",
"[93], [95]).",
"PN theory also plays a key role in the construction of the gravitational-waveform templates [78] for inspiraling compact objects currently used in the search for gravitational waves by laser-interferometric observatories.",
"PN theory has also been recently interfaced with numerical relativity simulations to serve as initial data for the modeling of BBH mergers [91], [17], [101], [100], [97], [44], [19], [43], [45], [62].",
"In all cases, the PN approximation is developed to sufficiently high perturbative order that the error contained in the approximation is much smaller than that associated with either the data in hand (in the case of binary pulsars) or the data expected (in the case of direct gravitational wave detection).",
"Using this PN-approximated description of the spacetime, we first evolve the BBH at a fixed initial separation $a_0=20 M$ to allow the accretion disk to relax to a quasi-steady state.",
"To study the effect of orbital shrinkage on the accretion disk, we then follow the binary inspiral until it reaches a separation $a = 8M$ , beyond which the PN approximation ceases to be sufficiently accurate for our purposes.",
"In a separate simulation, we kept the binary's separation fixed at $20 M$ and continued to evolve until $\\simeq 76000M$ to study the secular dynamics of the quasi-steady state, and, by contrasting with the first simulation, highlight the special effects induced by the inspiral.",
"Our findings can be summarized as follows.",
"The mass at $r \\simeq 2.5a$ builds steadily throughout the quasi-steady state, but much of it eventually concentrates in a distinct “lump”.",
"At smaller radii, a gap is cleared as torques and forces exerted by the binary either sweep matter inward or fling it outward.",
"Much of the small amount of mass in this gap is found in a pair of streams emanating from the inner edge of the disk and curving inward toward each black hole.",
"These streams carry nearly half of the mass accreting through the bulk of the disk to the inner boundary of the problem volume at $r=0.75a_0$ .",
"As the binary starts to shrink, the inner edge of the disk at first moves inward following the orbital evolution of the binary, but eventually cannot keep up, as the orbital shrinkage grows faster.",
"Nevertheless, a significant amount of mass still follows the binary's inspiral within the gap region.",
"We find that the final accretion rate in the inspiral stage is about $70\\%$ of the corresponding rate in the steady-state stage.",
"The luminosity of the disk is proportional to its surface density.",
"If the accretion rate fed to the disk were comparable to that of ordinary AGN, the surface density, and therefore the luminosity, of such a circumbinary disk could approach AGN level.",
"Most strikingly, the luminosity should be modulated periodically at a frequency determined by the binary orbital frequency and the binary mass ratio—$1.46$ times the binary orbital frequency in the case of equal masses.",
"However, the amplitude of modulation may be reduced by the large optical depth of the disk if the surface density is large enough to generate a sizable luminosity.",
"This paper is organized as follows.",
"In Section we describe the construction of the dynamical BBH spacetime.",
"In Section , we report the details of the MHD simulations of the disk.",
"In Section and Section , we present the results from these simulations and interpret them in the light of previous work and from the point of view of potential observational signatures.",
"Finally, in Section , we summarize our principal conclusions."
],
[
"Binary Black-Hole Spacetime",
"While solutions of the Einstein equations for single black holes were discovered as early as 1916 [80], no exact closed-form solution to the two-body problem exists and one generally needs to solve the Einstein equation numerically.",
"With the breakthroughs in numerical relativity [75], [18], [7], [79], it is now possible to perform stable and accurate full numerical simulations of BBHs in vacuum for a wide variety of mass ratios and spins parameters.",
"However, because the Einstein equations can be thought of as modified wave equations, with wave speeds of $c$ , the Courant conditionThe Courant condition is a stability condition relating the timestep $dt$ to the spatial resolution $h$ .",
"The timestep is limited by $dt < h / v$ , where $v$ is the fastest propagation speed in the system of interest.",
"greatly limits the timestep size, making full numerical simulations impractical when the characteristic MHD speeds are significantly smaller than $c$ .",
"On the other hand, if an approximate, but accurate, spacetime is given, the Courant condition is set by the MHD speeds, allowing for a much larger timestep.",
"Fortunately, analytic perturbative techniques have been successfully developed to tackle the spacetime problem both in the regime where the black holes are not too close, as well as in the close limit regime where the spacetime can be treated as a perturbed single black hole.",
"In this paper, we use the PN approach to model the spacetime of an inspiraling binary system prior to merger, neglecting the effect of the disk on the evolution of the black holes, from orbital separations of $20M$ down to $\\sim 8M$ , roughly where the standard PN approximation becomes inaccurate for our purposes.",
"Using this PN-approximated solution, we then solve for the relativistic MHD evolution of the circumbinary accretion disk.",
"We stop the PN evolution at $r = 8 M$ , but one can in principle continue the simulations beyond this regime.",
"To do so, one could use a snapshot of the PN metric and MHD data at that radius as initial conditions to then carry out a fully non-linear GRMHD evolution, using numerical relativity techniques to solve the coupled GRMHD Einstein system of equations (which will be the subject of an upcoming paper).",
"We perform two simulations: (i) RunSS keeps the semi-major axis of the binary artificially fixed at $20M$ ; (ii) RunIn starts from a snapshot of RunSS at $t=40000 \\, M$ (or after $\\simeq 70$ orbits) and then lets the black holes inspiral at the PN-theory prescribed rate down to a separation of $\\sim 8M$ .",
"RunSS is used to study the secular evolution of the accretion disk at fixed binary separation, while RunIn is used to investigate how the diminishing separation alters this secular evolution.",
"We describe our PN approach to model the spacetime metric below."
],
[
"The post-Newtonian Approximation",
"The PN approximation is based on a perturbative expansion of all fields, assuming slow motion $v/c \\ll 1$ and weak fields $GM/(r c^2) \\ll 1$Note that we have explicitly re-introduced $c$ and $G$ in this section in order to discuss the PN approximation..",
"These assumptions allow us to search for solutions that can be expressed as a divergent asymptotic series about a flat Minkowski background spacetime.",
"These perturbations obey differential equations determined by the PN-expanded Einstein field equations.",
"One then solves such equations perturbatively and iteratively to construct an approximate solution.",
"The two body problem in the slow-motion/weak-field limit is better understood by classifying the spacetime into different regions, where different assumptions hold and different approximations can be used (see e.g.",
"[90], [2], [3], [101], [100], [97], [43] for a review).",
"Here we concentrate on the near zone, which is the region sufficiently far from the horizons that the weak-field approximation of PN is valid, but less than a reduced gravitational wave wavelength ${\\mathchoice{\\displaystyle \\lambda }{\\hbox{}}{\\hspace{0.0pt}\\vrule width.5height.1pt depth.1pt\\hss }{\\box }0}$$\\textstyle \\lambda $$\\scriptstyle \\lambda $$\\scriptscriptstyle \\lambda $$ away from the center ofmass of the system, so that retardation effects can be treatedperturbatively.",
"We note that in the {\\emph {far zone}}, i.e.\\ theradiation zone where retardation effects can no longer be treatedperturbatively, a multipolar post-Minkowskian expansion can be used rather than aPN one.",
"Very close to each black hole (i.e.\\ in the {\\emph {innerzone}}),perturbedSchwarzschild solutions are used (which can be extended to includespin by using perturbed Kerr solutions).In this paper, the binary black hole metric will be approximated withonly the near zone solution.$ The PN approximation, of course, has its limits: binary systems eventually become so closely separated that a slow-motion/weak-field description is inappropriate.",
"For example, consider the simple case of a test particle spiraling into a non-spinning (Schwarzschild) black hole in a quasi-circular orbit.",
"Eventually, the particle will reach the innermost stable circular orbit, at which point its orbital velocity $v/c \\sim 0.41$ .",
"Clearly, such a velocity is not much less than unity, and thus, the PN approximation need not be an accurate description of the relativistic orbital dynamics.",
"Similarly, when comparable-mass BBHs inspiral, they eventually reach a separation at which the PN approximation is a bad predictor of the dynamics, since the small-velocity/weak-field assumptions are violated.",
"The determination of the formal region of validity of the PN approximation is crucial, but it can only be assessed when one possesses a more accurate, perhaps numerical, description of the orbital dynamics.",
"This is indeed the case when considering extreme mass-ratio inspirals (EMRIs), consisting of a stellar-mass compact object spiraling into a supermassive black hole.",
"When considering EMRIs, one can model the spacetime through black hole perturbation theory, i.e., by decomposing the full metric as that of the supermassive black hole plus a perturbation induced by the stellar-mass compact object, without assuming slow motions (see e.g.",
"Section 4 of [41] for a recent review, [60], [77], [9], [74] for related topics).",
"To leading-order, the orbital dynamics are then described by geodesics of the small object in the spacetime of the supermassive black hole.",
"The orbital motions are slowly perturbed by the radiation reaction due to the emission of gravitational waves.",
"The comparison of black hole perturbation theory and PN theory predictions has allowed for the construction of different measures to estimate the PN region of validity.",
"When considering the reduction in signal-to-noise ratio induced by filtering an “exact\" black-hole perturbation theory gravitational wave with a 4PNThis was extended to 5.5PN order in the erratum and addendum of [73].",
"order filter, [73] found that PN theory is sufficiently accurate provided $v \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 0.2$.When considering the 5.5 and 4PN order predictions for the loss of the binary^{\\prime }s binding energy for non-spinningand spinning background black holes respectively, relative to an ``exact\" black hole perturbation theory prediction, \\cite {Yunes:2008tw} and\\cite {Zhang:2011vh} found that the former is accurate provided $ v $\\sim $$<$ 0.29$, whichcorresponds to an orbital separation of $ a $\\sim $$>$ 11 M$.",
"The differencebetween these estimates is due to the different measures used and the different order of the PNapproximation employed\\footnote {It is not surprising that beyond 3PN order the region of validityof the PN approximation shrinks.",
"This is a property of divergent asymptotic series, whose behaviorin the context of PN theory was analyzed by \\cite {Yunes:2008tw} and\\cite {Zhang:2011vh}.",
"}.$ The region of validity of the PN approximation for comparable-mass binaries has not been as well studied.",
"This is because black hole perturbation theory is not applicable here, and one must rely on full numerical relativistic simulations.",
"Currently, state-of-the-art simulations can only model the last few tens of orbits prior to merger, while the determination of the formal region of validity would require knowledge of at least the last thousand orbits.",
"Nonetheless, there exist analytical arguments suggesting that the region of validity in the comparable-mass case is larger than in the EMRI case (i.e., the PN expansion is valid for even larger velocities) [85], [13], [61].",
"Moreover, the NINJA (Numerical INJection Analysis)-2 project [1], a collaboration between numerical relativists and gravitational wave data analysts, has established that certain 3PN order gravitational waveforms are sufficiently accurate for use as templates provided $v \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 0.33$ ($ a $\\sim $$>$ 8 M$).$"
],
[
"Near-Zone PN Evolution",
"The PN order of a given term is determined by the exponent of the perturbation parameter contained in that term.",
"In the near zone, the PN expansion of a BBH spacetime metric is a series expansion in the orbital velocity $v/c \\ll 1$ and the field strength $G M_{A}/(r_{A} c^{2})$ , where $M_{A}$ and $r_{A}$ are the masses of the $A^\\mathrm {th}$ particle and the distances from the $A^\\mathrm {th}$ particle to a field point respectively.",
"Here, we may consider $1/c$ as the PN parameter which goes to zero in the Newtonian limit $c \\rightarrow \\infty $ .",
"Notice that by the virial theorem $v^2/c^2 = {\\cal {O}}[GM/(a \\, c^2)]$ .",
"A term proportional to $(1/c)^n$ beyond the Newtonian (leading-order) expression is said to be of $(n/2)^\\mathrm {th}$ PN order.",
"The near zone metric will be described here by a resummed PN expression.",
"One begins with the 2.5PN expansion of the metric for non-spinning point particles in a quasi-circular orbit in harmonic coordinates, given for example in [14].",
"Such a metric, however, describes black holes as point-particles, which is why one then applies a “background resummation\", as in [100], [97], [43].",
"This resummation is intended to improve the strong-field behavior of the metric close to each point-particle, i.e., it recovers the horizon of each individual black hole.",
"The metric can then be formally written as $ g_{\\mu \\nu }(t,\\,\\vec{x}) =g_{\\mu \\nu }[\\vec{x};\\,\\vec{y}_{A}(t),\\,\\vec{v}_{A}(t)] \\,, $ where $\\vec{x}$ is a spatial vector from the binary's center of mass to a field point, while $\\vec{y}_{A}(t)$ and $\\vec{v}_{A}(t)$ are the particle's spatial location and 3-velocity with $A=(1,\\,2)$ .",
"This metric depends on the mass of each individual black hole, but also on the binary orbital evolution $\\lbrace \\vec{y}_{A}(t),\\,\\vec{v}_{A}(t)\\rbrace $ that must be prescribed separately.",
"We will use Greek letters (e.g., $\\mu ,\\,\\nu ,\\,\\lambda ,\\,\\kappa $ ) to represent spacetime indices $\\left[0,\\,1,\\,2,\\,3\\right]$ , and Roman letters (e.g., $i,\\,j,\\,k,\\,l$ ) to represent spatial indices $\\left[1,\\,2,\\,3\\right]$ .",
"The orbital evolution can also be prescribed within the PN approximation.",
"We simplify the analysis by considering only quasi-circular orbits.",
"This simplification is justified because gravitational wave emission tends to circularize binaries very efficiently, as demonstrated in the weak [72] and strong field regimes [87], [39], [38].",
"We will here use a 3.5PN expansion for the orbital phase evolution $\\phi (t)$ , as given for example by Equation (234) of [12], which depends on the quantity $\\Theta = \\nu ( t_c -t ) / (5 M) $ , where $t$ is time, $t_c$ is the time of coalescence, $M$ is the total mass and $\\nu = M_1 M_2/M^2$ is the symmetric mass ratio.",
"The PN orbital frequency is calculated from $\\Omega _{\\rm bin}=d\\phi /dt$ in this paper.",
"Although waveforms can be fully characterized by harmonics of the orbital phase, the latter depend on the orbital trajectories.",
"One may model these in harmonic coordinates via $ y_{1}^{i}(t) &=& \\frac{M_{2}}{M} a(t) \\left[\\cos {\\phi (t)},\\,\\sin {\\phi (t)},\\, 0 \\right]\\,,\\\\y_{2}^{i}(t) &=& -\\frac{M_{1}}{M} a(t) \\left[\\cos {\\phi (t)},\\,\\sin {\\phi (t)},\\, 0 \\right]\\,, $ where $a(t)=|\\vec{y}_{1}(t)-\\vec{y}_{2}(t)|$ is the orbital separation as a function of time.",
"This separation can be calculated via the balance law (see e.g., [12]), which states that the local rate of change of the binary's orbital binding energy is exactly balanced by the gravitational wave luminosity carried out to future null infinity and into black hole horizons, namely $ \\frac{dE_{\\rm Orb}}{dt} = - \\cal {L} \\,.",
"$ The rate of change of the orbital separation is then given by $ \\frac{da}{dt} = - \\left(\\frac{dE_{\\rm Orb}}{da}\\right)^{-1} {\\cal L} \\,.",
"$ Assuming the initial condition $a(t=0)=a_0$ , we then find $ t = t_c -\\int _0^{a} da^{\\prime } \\left(\\frac{dE_{\\rm Orb}}{da^{\\prime }}\\right) \\; {\\cal L}^{-1}\\,, $ where the integrand is expanded in a Taylor PN series and the coalescence time $t_c$ is defined by $ t_c = \\int _0^{a_0} da \\left(\\frac{dE_{\\rm Orb}}{da}\\right) \\; {\\cal L}^{-1} \\,.",
"$ In this paper, we only use the part of ${\\cal {L}}$ that is carried to future null infinity and, although Equations (REF ,REF ) are typically Taylor expanded to evaluate $t(a)$ , here they are inverted through Newton-Raphson minimization to yield $a(t)$ .",
"Given the above analysis, we can now calculate the time of coalescence and the number of orbits in each of the simulations carried out.",
"As already discussed, RunSS is artificially kept at a fixed semi-major axis, so $a(t)=a_0$ for all times, and thus, formally $t_{c} = \\infty $ .",
"On the other hand, RunIn keeps $a(t)$ fixed to $a(t) = a_0=20M$ for $t < t_\\mathrm {shrink} \\equiv 40000 M$ , after which it is allowed to evolve according to the PN equations of motion.",
"The time of coalescence for this run can be computed by inverting Equation (REF ) to obtain $t_c \\sim 14000M$ , although to leading order it is approximately described by [72] $ t_c \\sim \\frac{5}{256 \\nu }\\left(\\frac{a_0}{M}\\right)^4 M= \\frac{5}{256}\\left(\\frac{a_0}{M}\\right)^4 \\frac{(1+q)^2}{q} M\\,, $ where $q = M_2/M_1$ is the binary mass ratio.",
"A BBH clearly takes longer to merge for systems that start at larger initial separations and that possess extreme mass ratios.",
"Obviously, $t_{c}$ is always defined as the length of time to coalesce, when the binary is allowed to inspiral.",
"The total simulation time of RunIn is then $t_{c} + t_\\mathrm {shrink} \\sim 54000 M$ .",
"Figure: The orbital motion of one of the black holes in the binary from RunIn .",
"Its trajectory startsfrom an initial separation of a(0)=20Ma(0)=20M and stops at a(t f )≃8Ma(t_f)\\simeq 8M.Its black hole companion is located at a parity-symmetric pointacross the origin (track not drawn in the figure for the sake of clarity).Figure: The evolution of the orbital separation with respect to time, a(t)a(t).We turn on the gravitational radiation reaction at t=40000Mt=40000M.Figure: The number of orbits as a function of time.",
"(Black) RunSS .",
"(Grey) RunIn , inwhich the first part from t=0t=0 to 40000M40000M shows the circular orbit without the radiation reaction.We have plotted a few diagnostics to get a sense of the evolution of the binary system in RunIn .",
"Figure REF shows the orbital evolution of the binary in the $x$ -$y$ plane, after it is allowed to inspiral.",
"Figure REF plots the orbital separation as a function of time.",
"Observe that initially the semi-major axis is artificially kept fixed, while after $t>t_\\mathrm {shrink}$ it is allowed to decrease due to gravitational radiation reaction.",
"Figure REF plots the number of orbits $N_{\\rm orbits}$ traced by the binary system as a function of time, which is given by $ N_{\\rm orbits} =\\left\\lbrace \\begin{array}{l}{\\displaystyle \\frac{1}{2\\pi }} \\,\\Omega _{\\rm bin,0}\\,t\\ ({\\rm if} \\ t < t_{\\rm shrink}) \\,,\\\\[2ex]{\\displaystyle \\frac{1}{2\\pi }} \\,\\left[\\Omega _{\\rm bin,0}\\,t_{\\rm shrink} + \\phi (t-t_{\\rm shrink})\\right]\\ ({\\rm if} \\ t \\ge t_{\\rm shrink}) \\,.\\end{array}\\right.",
"$ We have here defined $\\Omega _{\\rm bin,0}=\\Omega _{\\rm bin}(r=a_0)$ to be the (constant) PN orbital frequency at a fixed semi-major axis, and we have set $\\phi (0)=0$ for the PN phase evolution.",
"The number of orbits is obviously a piece-wise function since when $t < t_\\mathrm {shrink}$ , $N_{\\rm orbits}$ increases linearly, as the binary is artificially kept at fixed $a_{0}$ , while when $t > t_\\mathrm {shrink}$ , $N_{\\rm orbits}$ can be approximated to leading order by $ N_{\\rm orbits} \\sim \\frac{1}{64 \\pi \\nu }\\left(\\frac{a_0}{M}\\right)^{5/2}= \\frac{1}{64 \\pi }\\left(\\frac{a_0}{M}\\right)^{5/2} \\frac{(1+q)^2}{q}\\,.",
"$ Therefore, for RunIn there are approximately 100 total orbits, while for RunSS there are approximately 127 orbits."
],
[
"Simulation Details",
"Like accretion disks around single black holes, circumbinary accretion flows are well described by the ideal MHD equations of motion (EOM) in the curved spacetime of only the black hole or holes.",
"We therefore neglect the matter's contribution to spacetime curvature and the accumulation of mass and momentum by the black holes from gas accretion.",
"Many codes have been written to simulate the single black hole case (e.g., [47], [25], [30], [64], [4], [48], [89], [65]), while only the equations of electrodynamics [67], force-free MHD [69], [68] and nonmagnetized hydrodynamics (e.g., [16], [28], [29], [15]) have been solved in the relativistic circumbinary setting.",
"Unfortunately, these latter simulations employ methods, like block-structured adaptive mesh refinement (AMR) in Cartesian coordinates, that typically lead to poor conservation of fluid angular momentum and excessive dissipation at refinement boundaries.",
"These two effects alter the disk's angular momentum transport mechanism and thermodynamics in a nontrivial way.",
"Furthermore, they require the solution of the Einstein equations, which—in turn—imposes a significant computational burden.",
"In order to avoid these problems, we take an alternate route and solve the MHD EOM using a code designed for single black hole systems: Harm3d [65].",
"Fortunately, Harm3d was written to be almost independent of coordinate system or choice of spacetime, so modifying it to handle non-axisymmetric, time-dependent spacetimes was straightforward.",
"In fact, the only differences between the algorithm described in [65] and here are that the metric (and its affine connection or gravitational source terms) needs to be updated every sub-step of the second-order Runge-Kutta time-integration procedureNote that many other technical changes were made that do not affect the algorithm, but do affect the runtime efficiency and design of the code.. Below, we describe the equations solved, initial data setup and other details of the disk evolution."
],
[
"MHD Evolution",
"Since we assume that the gas does not self-gravitate and alter the spacetime dynamics, we need only solve the GRMHD equations on a specified background spacetime, $g_{\\mu \\nu }(x^\\lambda )$ , where $\\left\\lbrace x^\\lambda \\right\\rbrace $ represents a set of general spacetime coordinates.",
"The EOM originate from the local conservation of baryon number density, the local conservation of energy, and the induction equations from Maxwell's equations (please see [65] for more details).",
"They take the form of a set of conservation laws: $ {\\partial }_t {\\bf U}\\left({{\\mathbf {P}}}\\right) =-{\\partial }_i {\\bf F}^i\\left({{\\mathbf {P}}}\\right) + \\mathbf {S}\\left({{\\mathbf {P}}}\\right) \\, $ where ${\\bf U}$ is a vector of “conserved” variables, ${\\bf F}^i$ are the fluxes, and $\\mathbf {S}$ is a vector of source terms.",
"Explicitly, these are $ {\\bf U}\\left({{\\mathbf {P}}}\\right) = \\sqrt{-g} \\left[ \\rho u^t ,\\, {T^t}_t+ \\rho u^t ,\\, {T^t}_j ,\\, B^k \\right]^T $ $ {\\bf F}^i\\left({{\\mathbf {P}}}\\right) = \\sqrt{-g} \\left[ \\rho u^i ,\\, {T^i}_t + \\rho u^i ,\\, {T^i}_j ,\\,\\left(b^i u^k - b^k u^i\\right) \\right]^T $ $ \\mathbf {S}\\left({{\\mathbf {P}}}\\right) = \\sqrt{-g}\\left[ 0 ,\\,{T^\\kappa }_\\lambda {\\Gamma ^\\lambda }_{t \\kappa } - \\mathcal {F}_t ,\\,{T^\\kappa }_\\lambda {\\Gamma ^\\lambda }_{j \\kappa } - \\mathcal {F}_j ,\\,0 \\right]^T \\, $ where $g$ is the determinant of the metric, ${\\Gamma ^\\lambda }_{\\mu \\kappa }$ is the metric's affine connection, $B^i = {^{^*}\\!\\!F}^{it}/\\sqrt{4\\pi }$ is our magnetic field (proportional to the field measured by observers traveling normal to the spacelike hypersurface), ${^{^*}\\!\\!F}^{\\mu \\nu }$ is the Maxwell tensor, $u^\\mu $ is the fluid's 4-velocity, $b^\\mu = \\frac{1}{u^t} \\left({\\delta ^\\mu }_{\\nu } + u^\\mu u_\\nu \\right) B^\\nu $ is the magnetic 4-vector or the magnetic field projected into the fluid's co-moving frame, and $W = u^t / \\sqrt{-g^{tt}}$ is the fluid's Lorentz function.",
"The MHD stress-energy tensor, $T_{\\mu \\nu }$ , is defined as $ T_{\\mu \\nu } = \\left( \\rho h + 2 p_{m} \\right) u_\\mu u_\\nu + \\left( p + p_{m}\\right) g_{\\mu \\nu } - b_\\mu b_\\nu $ where $p_m = b^\\mu b_\\mu / 2$ is the magnetic pressure, $p$ is the gas pressure, $\\rho $ is the rest-mass density, $h = 1 + \\epsilon + p/\\rho $ is the specific enthalpy, and $\\epsilon $ is the specific internal energy, We evolve the quantity $( \\rho u^t + {T^t}_t )$ instead of ${T^t}_t$ in order to reduce the magnitude of the internal energy's numerical error [30].",
"Note that the terms proportional to ${\\Gamma ^{\\lambda }}_{t\\kappa }$ and ${\\Gamma ^{\\lambda }}_{\\phi \\kappa }$ in the source no longer vanish as the metric is now dependent on time and azimuthal coordinate, $\\phi $ .",
"Also, note that we add a negative source term $\\left(-\\mathcal {F}_\\mu \\right)$ to the local energy conservation equation to model energy/momentum loss from radiative cooling; please see Section REF for more details.",
"The MHD evolution is facilitated by calculating and using so-called primitive variables: the rest-mass density ($\\rho $ ), the internal energy density ($u = \\rho \\epsilon $ ), the velocities relative to the observer moving normal to the spacelike hypersurface, $\\tilde{u}^i = u^i - u^t g^{ti} / g^{tt}$ .",
"The magnetic field $B^i$ is considered both a primitive and a conserved variable.",
"We employ piecewise parabolic reconstruction of the primitive variables for calculating the local Lax-Friedrichs flux at each cell interface [30].",
"We use a 3-dimensional version of the FluxCT to impose the solenoidal constraint, $\\partial _i \\sqrt{-g} B^i = 0$ [92].",
"The EMFs (electromotive forces) are calculated midway along each cell edge using piecewise parabolic interpolation of the fluxes from the induction equation.",
"A second-order accurate Runge-Kutta method is used to integrate the EOM using the method of lines once the numerical fluxes are found.",
"The primitive variables are found from the conserved variables using the “2D” scheme of [64].",
"A conservation equation for the entropy density is evolved and used to replace the total energy equation of the 2D method whenever the plasma becomes too magnetically dominated, or—specifically—when $\\rho \\epsilon < 0.02 p_{m}$ ; this procedure helps us avoid numerical instabilities and negative pressures from developing.",
"Please see [65] for more details.",
"The MHD evolution is performed in the same way as in single black hole cases except that the metric is evaluatedWe remind the reader that the metric is known in closed form, requiring only direct evaluation except for the Newton-Raphson iteration to find the current time's binary separation.",
"at the present sub-step's time before the MHD fields are updated.",
"The metric is required in many facets of the update procedure.",
"For example, it is used in calculating the 4-velocity from the primitive velocities, source terms and geometric factors in the EOM, and for deriving the primitive variables from the conserved variables.",
"The affine connection is calculated via finite differencing the PN metric to evaluate $ {\\Gamma ^\\mu }_{\\nu \\kappa } = \\frac{1}{2} g^{\\mu \\sigma } \\left( \\partial _\\nu g_{\\kappa \\sigma } + \\partial _\\kappa g_{\\nu \\sigma } - \\partial _\\sigma g_{\\nu \\kappa } \\right)\\quad .",
"$ The spatial finite differences use fourth-order centered stencils away from the physical boundaries, and backward/forward stencils adjacent to the physical boundaries.",
"Since the metric is evaluated and stored at the cell centers and faces, but the connection is only evaluated at the centers, fourth-order stencils require only three cells' worth of data to compute.",
"The time derivatives are second-order accurate, but use a time spacing $10^{-3}$ times that used in the MHD integration.",
"This means that additional evaluations of the metric are made at advanced and retarded times at the cell centers to calculate the time derivatives for each connection evaluation.",
"We have verified that the truncation error from the time derivatives is smaller than that from the spatial derivatives.",
"Also, the connection's spatial finite differencing is one order more accurate than that of the MHD procedure, implying it is not the primary source of error in the calculation.",
"Please see Appendix for a discussion on our resolution tests."
],
[
"Initial Conditions",
"In this project, we avoid evolving the gas in the neighborhood of the black holes, choosing instead to focus on establishing reasonable prior conditions for the gas that ultimately feeds the BBH.",
"We therefore excise a spherical domain, which includes the binary, from our calculation.",
"It is common to begin with initial conditions devoid of large transient artifacts.",
"This is often done by starting from a torus of material in equilibrium (via pressure and rotational support) about the central gravitating source (e.g., a black hole) [22], [26].",
"Unfortunately, such tori will not be near equilibrium in our spacetime as it is $\\left(t,\\phi \\right)$ -dependent.",
"Plus, the equations describing their structure assume that the metric has the same form (i.e.",
"share the same zero-valued elements) as the Kerr metric in Boyer-Lindquist coordinates.",
"We resolve these issues in the following way.",
"First, since we hold the binary at fixed separation for several orbits, the spacetime initially has a helical Killing symmetry, with Killing vector $\\mathcal {K}^a = \\Omega _\\mathrm {bin} \\left(\\partial _\\phi \\right)^a + \\left(\\partial _t \\right)^a$ .",
"In other words, the spacetime is invariant in a frame rotating with the binary, while the separation is held constant.",
"Since the torus will lie a few $a_0$ away from the binary, its dynamical response time—comparable to its orbital period—will be longer than the binary period, implying that a torus near equilibrium in this helically-symmetric spacetime will also be near equilibrium in its time average.",
"Due to its helical symmetry, its time average is also its azimuthal average.",
"We therefore start with a torus in equilibrium in a $\\left(t,\\phi \\right)$ -independent spacetime, $\\hat{g}_{\\mu \\nu }$ , found by averaging over $\\phi $ : $ \\hat{g}_{\\mu \\nu } = \\frac{ \\int g_{\\mu \\nu } \\sqrt{g_{\\phi \\phi }} \\, d\\phi }{ \\int \\sqrt{g_{\\phi \\phi }} \\, d\\phi }\\quad .",
"$ We have verified that the same components that are zero-valued in Boyer-Lindquist coordinates are consistent with zero to within our PN-order accuracy in the $\\hat{g}_{\\mu \\nu }$ metric.",
"This means we can employ a similar torus solution method as described in [22].",
"A description of our modifications to the procedure—including the generalization to our $\\phi $ -averaged spacetime—is provided in Appendix .",
"Note that we now ensure that the equilibrium solution is found iteratively to greater precision, instead of the approximate method described in [26] which has been used in prior work of the authors in single black hole disk evolutions [65], [66] and by others studying the hydrodynamic circumbinary case [29].",
"We find that our procedure produces initial tori that are much closer to equilibrium than the approximate scheme.",
"Please see Appendix for more details.",
"Previous studies have shown that a gap develops near $2.5a$ for equal mass binaries [58], [82].",
"We aim to study how this gap develops, so we choose to start material outside this radius.",
"We therefore set up a disk with inner edge located at $r_\\mathrm {in}= 3 a_0$ and pressure maximum located at $r_{p}= 5 a_0$ ; from prior experience, $r_{p}$ is approximately the radius at which the disk transitions from accreting to decreting since matter must shed its angular momentum to fluid elements further out in order to accrete.",
"These outer elements gain angular momentum and form a time-averaged decretion flow away from the central potential.",
"We will therefore focus on $r < r_{p}= 5 a_0$ in our analyses.",
"The initial disk extends to $r_\\mathrm {out}\\simeq 12 a_0$ , is isentropic with $p/\\rho ^\\Gamma = 0.01$ , and is tuned to have an aspect ratio of $H/r=0.1$ at $r=r_{p}$ , where $H$ is the density scale height defined as the first moment of the rest-mass density with respect to distance from the midplane: $ H \\equiv \\frac{\\langle \\rho \\sqrt{g_{\\theta \\theta }} \\, |\\theta - \\pi /2| \\rangle }{\\langle \\rho \\rangle } \\quad , $ and where the $\\langle X \\rangle $ denotes the average over origin-centered spheres: $ \\langle X \\rangle \\equiv \\frac{\\int X \\, \\sqrt{-g} \\, d\\theta d\\phi }{\\int \\sqrt{-g} \\, d\\theta d\\phi } \\quad .",
"$ More information about the initial torus and its solution method is given in Appendix .",
"In the disk, we add random, cell-scale noise to the the internal energy, $u$ , in order to hasten the development of turbulence; the random noise is evenly distributed over the range $\\pm 5\\times 10^{-3}$ .",
"Once the torus is in place on the grid, a surrounding nonmagnetized atmosphere is added as our numerical scheme requires us to maintain positive values of $\\rho $ and $p$ .",
"The atmosphere is initially static, $u^i=0$ , and in approximate pressure equilibrium: $\\rho _\\mathrm {atm} = 1 \\times 10^{-7} \\rho _\\mathrm {max} \\left(r/M\\right)^{-3/2}$ , $u_\\mathrm {atm} = 3.3\\times 10^{-6} \\, u_\\mathrm {max} \\left(r/M\\right)^{-5/2}$ , where $\\rho _\\mathrm {max}$ and $u_\\mathrm {max}$ are—respectively—the initial maxima of $\\rho $ and $u$ .",
"We note that when either $\\rho $ or $u$ are found to go below, respectively, $\\rho _\\mathrm {atm}$ or $u_\\mathrm {atm}$ , they are set to those atmosphere values without any modification to the magnetic field or fluid velocity; this happens very rarely once the disk's turbulence saturates.",
"The magnetic field is initialized as a set of dipolar loops that follow density contours in the disk's interior.",
"We set the azimuthal component of the vector potential and differentiate it to yield $B^i$ ; $B^\\phi (t=0) = 0$ in our configuration.",
"The vector potential component is $ A^\\phi = A^\\phi _0 \\, \\max \\left[ \\left(\\rho - \\frac{1}{4}\\rho _\\mathrm {max} \\right) , 0 \\right] \\quad .",
"$ The magnitude of the field, $A^\\phi _0$ , is set such that the ratio of the disk's total internal energy to its total magnetic energy is 100."
],
[
"Grid, Boundary Conditions, and Parameters",
"The domain on which the MHD EOM are solved is a uniformly discretized space of spatial coordinates $\\left\\lbrace x^{\\left(i\\right)}\\right\\rbrace $ that are isomorphic to spherical coordinates $\\left\\lbrace r,\\theta ,\\phi \\right\\rbrace $ : $ r(x^{\\left(1\\right)}) = M e^{x^{\\left(1\\right)}} \\quad , $ $ \\theta (x^{\\left(2\\right)}) =\\frac{\\pi }{2} \\left[ 1 + \\left(1-\\xi \\right) \\left(2 x^{\\left(2\\right)}- 1 \\right)+ \\left( \\xi - \\frac{2 \\theta _c}{\\pi } \\right) \\left( 2 x^{\\left(2\\right)}- 1 \\right)^n \\right]\\quad , $ and $\\phi = x^{\\left(3\\right)}$ .",
"We set $n=9$ , $\\xi =0.87$ , and $\\theta _c=0.2$ .",
"The logarithmic radial coordinates are such that the radial cell extents are smaller at smaller radii in order to resolve smaller scale features of the accretion flow there.",
"The $x^{\\left(2\\right)}\\leftrightarrow \\theta $ mapping concentrates more cells near the plane of the disk and the binary's orbit, the equator of our coordinate system.",
"Let each grid cell in our numerical domain be labeled by three spatial indices that each cover $[0,N^{\\left(n\\right)}-1]$ , where $\\left\\lbrace N^{\\left(n\\right)}\\right\\rbrace $ are the number of cell divisions along each dimension.",
"A cell with indices $\\left(i,j,k\\right)$ is located at $\\left(x^{\\left(1\\right)}_{i}, x^{\\left(2\\right)}_{j}, x^{\\left(3\\right)}_{k} \\right)$ , where $x^{\\left(n\\right)}_{j} = x^{\\left(n\\right)}_{b} + \\left(j + \\frac{1}{2} \\right) \\Delta x^{\\left(n\\right)}$ .",
"The grid we used is completely specified by: $x^{\\left(1\\right)}_{b} = \\ln \\left(r_\\mathrm {min}/M\\right)$ , $\\Delta x^{\\left(1\\right)}= \\ln \\left(r_\\mathrm {max}/r_\\mathrm {min}\\right)/N^{\\left(1\\right)}$ , $r_\\mathrm {min}=15M$ , $r_\\mathrm {max}=260M$ , $N^{\\left(1\\right)}= 300$ , $x^{\\left(2\\right)}_{b} = 0$ , $\\Delta x^{\\left(2\\right)}= 1/N^{\\left(2\\right)}$ , $N^{\\left(2\\right)}= 160$ , $x^{\\left(3\\right)}_{b} = 0$ , $\\Delta x^{\\left(3\\right)}= 2\\pi /N^{\\left(3\\right)}$ , $N^{\\left(3\\right)}= 400$ .",
"We chose our resolution and grid extent based upon a number of criteria.",
"First, our $\\theta $ and $\\phi $ resolutions were set in order to adequately resolve the MRI based on guidelines of [35] and [86].",
"The radial resolution was chosen to resolve the spiral density waves—generated by the binary's time-varying tidal field—by several radial zones.",
"We find that our grid adequately resolves the MRI, as measured by the criteria of [35] and [86], throughout the domain of interest (i.e.",
"$r < 5a_0$ ) $\\forall t$ .",
"Please see Appendix for a quantitative description of these resolution criteria and for a demonstration of how well we resolve the MRI.",
"The radial extent of the grid was inspired by [82] and the limits of our near-zone PN metric.",
"Since the near-zone PN metric that we use is only valid at distances more than $10 M_i$ from the black hole with mass $M_i$ [100], [101], [43], then—in the equal mass case considered here—we have $r_\\mathrm {min}\\ge 10 \\frac{M}{2} + \\max (a(t))/2 = 5M + a_0/2 = 15M$ .",
"[82] found that an inner radial boundary located at $r_\\mathrm {min}\\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ 1.1 a$ was sufficientlyfar away from the gap and deep in the potential as to not significantly alter the development and evolution of the surface density peak at the edge ofthe gap.",
"These two constraints justify our choice of $ rmin= 15M = 0.75 a0$ and suggest that our inner boundary conditionmay begin affecting the gap^{\\prime }s evolution when$ a(t) $\\sim $$<$ rmin/1.1 13.6M 0.68 a0$, which occurs after approximately $ t=51235M$ in \\hbox{RunIn\\ }.",
"We set $ rmax=260M=13a0$ toencompass the initial torus.$ All cells are advanced in time with the same time increment ($\\Delta x^0 = \\Delta t$ ), which itself changes in time; $\\Delta x^0$ is set to $0.45 \\Delta t_\\mathrm {min}$ , where $\\Delta t_\\mathrm {min}$ is the shortest cell crossing time of any MHD wave over the entire domain.",
"Boundary conditions were imposed through assignment of primitive variables and $B^i$ in ghost zones.",
"Outflow boundary conditions are imposed at $r=r_\\mathrm {min}$ and $r=r_\\mathrm {max}$ which amounts to extrapolating the primitive variables at $0^\\mathrm {th}$ -order into the ghost zones.",
"Additionally, $u^r$ is set to zero—and $\\tilde{u}^i$ recalculated—whenever it points into the domain at $r=r_\\mathrm {max}$ .",
"We note that we attempted to implement a similar condition on $u^r$ at $r=r_\\mathrm {min}$ , but found it to be unstable during the earliest part of the simulation.",
"Even though it was successfully used in [58] and [82], we found that this condition was inconsistent with the tendency of negative radial pressure gradients developing ahead of each black hole as it moved around its orbit.",
"This pressure gradient moves small amounts of material onto the grid, elevating the density just above the floor ahead of the black holes.",
"Even without the special condition on $u^r$ at $r_\\mathrm {min}$ and just using $0^\\mathrm {th}$ -order extrapolation of the primitive variables there, we observe insignificant amounts ($\\ll 1\\%$ of the total) of positive mass flux into the domain there and a nearly flat $\\dot{M}(r,t)$ profile over $r_\\mathrm {min}< r < 2 a$ with no noticeable artifacts near $r_\\mathrm {min}$ (e.g., Figure REF )."
],
[
"Thermodynamics",
"Depending on internal properties of the gas (e.g., density, accretion rate), the disk may or may not be optically thin, geometrically thin, or have a constant aspect ratio $H/r$ .",
"As these disk characteristics are sensitive to the assumed initial conditions and thermodynamics of the system, we therefore must select which kind of disk to model a priori and verify the consistency of our assumptions a posteriori.",
"We chose to model a disk with intermediate thickness ($H/r=0.1$ ) in order to both address the fact that binary's torque will likely heat the gas efficiently and our expectation that the disk will be dense, optically thick and radiating efficiently.",
"We chose the ideal-gas “$\\Gamma $ -law” equation of state to close the MHD EOM: $P = \\left(\\Gamma - 1\\right) \\rho \\epsilon $ .",
"We set $\\Gamma = 5/3$ , which reasonably well describes the behavior of a plasma whose specific thermal energy is smaller than an electron's rest-mass energy (i.e.",
"it is not relativistically hot).",
"The gas is cooled to a target entropy, the initial entropy of the disk, at a rate equal to $ \\mathcal {L}_c = \\frac{\\rho \\epsilon }{T_\\mathrm {cool}} \\left( \\frac{\\Delta S}{S_0} + \\left|\\frac{\\Delta S}{S_0}\\right| \\right) \\quad , $ where $\\Delta S \\equiv S - S_{0}$ and $T_\\mathrm {cool} = 2 \\pi \\left(r/M\\right)^{3/2}$ is the cooling time, which we approximate as the Newtonian period of a circular equatorial orbit at radius $r$ .",
"Our procedure is similar to those used by [65] and [71].",
"The term in the parentheses acts as a switch ensuring that $\\mathcal {L}_c \\ge 0$ always, and is zero when the local entropy, $S=p/\\rho ^\\Gamma $ , is below the target entropy, $S_0 = 0.01$ , which is the constant value used in the initial data's torus.",
"Hence, the cooling function should release any heat generated through dissipation since the initial state.",
"We do not cool unbound material—i.e.",
"fluid elements that satisfy $\\left( \\rho h + 2p_{m} \\right)u_t < - \\rho $ —since we do not want to include cooling that results from application of density or pressure floors.",
"Since $\\mathcal {L}_c$ is the cooling rate in the local fluid frame, its implementation in the EOM must be expressed in the coordinate frame: $ \\mathcal {F}_\\mu = \\mathcal {L}_c u_\\mu \\quad .",
"$ Another advantage of the cooling function is that it provides us with a proxy for bolometric emissivity that is consistent with the disk's thermodynamics—unlike a posteriori estimates of synchrotron and/or bremsstrahlung luminosity that have typically been made in numerical relativity simulations (e.g., [16], [28], [29]).",
"We will use $\\mathcal {L}_c$ to make predictions of the total luminosity from circumbinary disks.",
"These predictions are made by integrating $\\mathcal {L}_c$ over the domain in the coordinate frame; we expect to verify their accuracy using full GR ray-tracing in future work."
],
[
"Approximate Steady State",
"At the beginning of both simulations, orbital shear transforms part of the radial component of the magnetic field to toroidal, creating a laminar Maxwell stress.",
"Meanwhile, in the same region, the magnetorotational instability grows, its amplitude exponentially growing on the local dynamical timescale, $\\simeq 500M$ at the initial inner edge of the disk, $r = 60M$ .",
"The turbulence in the inner disk reaches nonlinear saturation at $t \\simeq 10000M$ .",
"Under the combined influence of the initial laminar and later turbulent Maxwell stress, matter flows inward (see Figure REF ).",
"Figure: Color contours of logΣ(r)\\log \\Sigma (r) as a function of time.",
"The scale is shownin the color bar.",
"The black dashed curve shows 2a(t)2a(t).",
"(Left) RunIn .",
"(Right) RunSS .Soon after $t \\simeq 10000M$ , the inward flow begins to pile up at $r \\simeq 50M$ , between two and three times the binary separation (the dashed line in both panels of Figure REF marks the location of $2a(t)$ in order to guide the eye).",
"We define the surface density $\\Sigma $ as $\\Sigma (r,\\phi ) \\equiv \\int \\, d\\theta \\sqrt{-g} \\rho / \\sqrt{g_{\\phi \\phi }(\\theta =\\pi /2)};$ when we quote it as $\\Sigma (r)$ , that denotes an azimuthal average of equation (REF ).",
"In later discussion, we will sometimes normalize the surface density to $\\Sigma _0$ , the maximum surface density in the initial condition; in code-units $\\Sigma _0 =0.0956$ .",
"In RunSS , $\\Sigma (r \\sim 2a)$ grows steadily for the duration of the simulation, but after $t \\simeq 20000M$ , the logarithmic rate of growth (i.e., $d\\ln \\Sigma (r)/dt$ ) gradually becomes slower and slower.",
"Because a number of azimuthally-averaged properties like $\\Sigma (r)$ all become steadier after $t=40000M$ , we call the period from then until the end of RunSS the “quasi-steady epoch\".",
"For the same reason, we began the binary orbital evolution of RunIn at that time.",
"Figure: Σ(r/a)\\Sigma (r/a) every 1000M1000M in time from t=30000Mt = 30000M to the end of thesimulation.",
"Time increases from violet color to red.",
"The dotted curve shows theinitial condition.",
"The dashed curve shows the average of the colored curves.",
"(Left) RunIn , where the time span extends to 53000M53000M.",
"Notethat in this simulation aa decreases after t=40000Mt=40000M, so that a fixed value ofr/ar/a corresponds to a progressively smaller radial coordinate after that time.",
"(Right) RunSS , where the time span extends to 76000M76000M.",
"The binary separationis fixed throughout this simulation.Once this quasi-steady state is reached, $\\Sigma (r)$ rises sharply from the inner boundary at $r=16M$ to $r\\simeq 50M$ , initially $\\propto r^{2.5}$ , but at late times in RunSS , $\\propto \\exp (3r/a)$ (Figure REF ).",
"At first, the azimuthally-averaged surface density profile forms a relatively flat plateau at radii greater than $50M \\simeq 2.5a$ , but by $t=30000M$ , a distinct local maximum appears at $r \\simeq 50M$ and persists for the remainder of the simulation.",
"This maximum is noticeably asymmetric in the sense that $|d\\Sigma /dr|$ is always considerably smaller in the disk body (i.e., $r > 2.5a$ ) than in the gap region inside $r=2a$ (Figure REF ).",
"This behavior resembles closely what has previously been seen in the Newtonian regime (e.g., [58], [82]).",
"By construction, the behavior of RunIn is identical to that of RunSS up to $t=40000M$ , when the binary inspiral was begun.",
"In fact, at large radius, the behavior of the surface density profile in RunIn continues to be very similar to that of RunSS even after the binary begins to shrink.",
"Near the surface density peak and at smaller radii, however, things change.",
"In RunIn , the location of the peak moves inward as the binary becomes smaller, and the slope of the disk's inner edge becomes noticeably shallower as the inspiral accelerates.",
"Comparing the curve of the dashed line in the RunIn panel of Figure REF to the curving edge of the colors denoting higher surface density, one can see that the location of the disk's inner edge follows the evolution of the binary until shortly before the end of the simulation.",
"Figure: Color contours of surface density in units of Σ 0 \\Sigma _0 as a function of radius and azimuthalangle in RunSS at four different times in two different scales: (Left) Logarithmiccolor scale emphasizing the streams from the disk toward the binary members.",
"(Right) Linearcolor scale emphasizing the growth of asymmetry in the inner disk.",
"In both panels, the timesshown are t=40000Mt=40000M (upper-left), t=51963Mt=51963M (upper-right),t=63926Mt=63926M (lower-left), and t=75890Mt=75890M (lower-right).However, speaking in terms of azimuthally-averaged surface density obscures an important aspect of circumbinary disks: near and inside their inner edges, their structure is generically far from axisymmetric.",
"In Figure REF , we show $\\Sigma (r,\\phi )$ at $t = 40000M$ .",
"As mentioned previously, at radii smaller than $\\simeq 2a \\simeq 40M$ , there is relatively little matter.",
"The reason this gap forms is that, unlike a time-steady, axisymmetric potential, the time-dependent quadrupolar potential of the binary does not conserve either the energy or the angular momentum of test-particles.",
"Consequently, closed orbits do not exist, and torques driven by the binary can rapidly expel some matter to the outside, while matter on other trajectories can be forced inward [82].",
"As a result, even though the rate at which matter enters the gap is comparable to the outer-disk accretion rate, at any given time, relatively little matter can be found in the region within $\\simeq 2a$ , and the matter that is present follows trajectories with little resemblance to stationary circular orbits.",
"Instead, a pair of streams leave the inner edge of the disk and curve inward toward each member of the binary.",
"Part of their flow gains enough angular momentum to return to the disk, but part crosses the inner simulation boundary, traveling toward the domain of the binary.",
"In addition, several tens of binary orbits after matter begins to pile up at $r \\simeq 2.5a$ , a distinct “lump\" (as [82] called it) forms in the region of the surface density peak.",
"The density contrast between this lump and adjacent regions grows steadily in time.",
"Thus, despite the relatively slow variation of azimuthally-averaged disk properties during the period we call “quasi-steady\", the “lump\" continues to evolve secularly.",
"Although RunSS was not continued long enough to see this effect, [82] found that the eccentricity of the lump's orbit also grows slowly."
],
[
"Accretion rate, internal stresses, and angular momentum budget",
"It is also useful to characterize the global dynamics of circumbinary disks in terms of the radial dependence of the net mass flow, i.e., the accretion rate as a function of radius.",
"We show in Figure REF how this quantity slowly evolved during the quasi-steady epoch of RunSS by dividing the time from $30000M$ until the end of the simulation at $76000M$ into four segments and averaging over each one separately.",
"The accretion rate is constant as a function of radius only inside the gap region, at most times increasing gradually outside $r \\simeq 2a$ .",
"During the first part of this period, the accretion rate rises steadily to radii beyond $5a$ , but after $t \\simeq 50000M$ , the accretion rate in the outer disk gradually falls.",
"At the end of the simulation, $\\dot{M}(r)$ is actually about a factor of 2 greater at $r \\simeq 3a$ than anywhere else.",
"Averaging over the entire quasi-steady epoch, the rate at which mass passes through the inner boundary is a bit less than half the accretion rate at $r=5a$ .",
"Although the first analytic theories of circumbinary disks [76] assumed that no accretion would pass the inner edge of such a disk, Newtonian simulations, both purely hydrodynamic [58] and MHD [82], have generally seen leakage fractions of a few tens of percent; our fraction is thus only somewhat greater than previously found.",
"Figure: Log 10 _{10} of the azimuthally-averaged plasma β\\beta parameter at four times during RunSS .As mentioned earlier, Maxwell stresses due to correlations induced in MHD turbulence by orbital shear dominate angular momentum transport within accretion disks.",
"Because the ratio of Maxwell stress to magnetic pressure, $2\\langle B^{(r)} B_{(\\phi )} \\rangle /\\langle B^2\\rangle $ , is fixed [35] at $\\simeq 0.3$ –0.4 in a point-mass potential (here the notation $X^{(\\mu )}$ denotes the magnitude of the $\\mu $ -component of four-vector X projected into the fluid frame), the stress is linearly proportional to the magnetic pressure.",
"A useful measure of the strength of magnetic effects is therefore the plasma $\\beta \\equiv \\langle p \\rangle /\\langle B^2\\rangle $ .",
"In most previous accretion disk simulations, this quantity is $\\sim 100$ in the midplane and drops to $\\sim O(1)$ a few scale-heights out of the plane.",
"We show its dependence on position in the poloidal plane at several times during the quasi-steady epoch of RunSS in Figure REF ; to be more precise, we show the ratio of the time- and azimuthally-averaged gas pressure to the similarly averaged magnetic pressure.",
"As that figure illustrates, the level of magnetization is rather larger than usual (i.e., $\\beta $ is smaller than usual), but gradually diminishes over time.",
"At $t=30000M$ , $\\beta \\simeq 1$ in the midplane at $r \\sim 3$ –$5a$ and $\\simeq 3$ in the region of the surface density peak ($r \\sim 2$ –$3a$ ); by the end of the simulation, it is $\\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 10$ in the disk body for the whole range $ 2a < r < 5a$ and reaches as much as$ 30$ in the lump.$ Ever since the work of [81], it has been popular to measure the vertically-integrated, azimuthally-averaged, and time-averaged internal disk stress in units of the similarly integrated and averaged pressure.",
"In order to avoid unphysical pressures found in the unbound regions, we computed the stresses and pressures only in bound material.",
"Outside $r \\sim 4a$ , where the disk resembles an ordinary accretion disk, we find that the Maxwell stress alone has magnitude $\\simeq 0.3$ –0.5 in these units.",
"This is roughly 3–5 times larger than the stress levels found in general relativistic simulations of MHD flows in the Kerr metric [52].",
"In the gap region, the ratio of Maxwell stress to pressure rises about a factor of 2, while the Reynolds stress in the gap rises dramatically (as also found by [82]).",
"These large Reynolds stresses are entirely due to the strong binary torque, which pushes part of the inflowing streams back out to the disk with additional angular momentum.",
"An overview of angular momentum flow in the system can be gleaned from Figure REF , in which we show the radial derivatives of the time-averaged angular momentum fluxes integrated on shells, i.e., the time-averaged torques due to the several mechanisms acting.",
"Several important points stand out in this figure.",
"The first is that the binary torques are delivered primarily in the gap region $a \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ r $\\sim $$<$ 2a$.",
"The torque density $ dT/dr$ peaks at$ r 1.45a$, and the region surrounding that peak dominates the integral over all radii.Moreover, all these torques are positive in net,but they are locally negative both at small radii ($ r $\\sim $$<$ a$) and at large ($ r $\\sim $$>$ 1.9a$).Thus, most of the angular momentum the binary gives the disk is delivered in the gap, wherethe gas density is very much lower than in the disk proper.",
"This point has previously beenemphasized by \\cite {Shi11}.",
"Second, that angular momentum is conveyed to the disk properby fluid flows, i.e., Reynolds stresses.",
"That is why the Reynolds stress is large and positivefrom $ r 1.8a$ to $ r 2.5a$.",
"Outside those regions, Maxwell stress, which alwaysacts so as to remove angular momentum from the gas and carry it outward, dominates the internalstresses.",
"Finally, the net angular momentum change at any given radius is generally positive inthe inner disk because matter continues to pile up between $ r 2a$ and $ r 5a$throughout the simulation.$ Figure: Radial derivatives of the angular momentum flux due to shell-integrated Maxwell stressin the coordinate frame (red), the angular momentum flux due to shell-integrated Reynolds stressin the coordinate frame (green), and advected angular momentum (gold).",
"Torque densities perunit radius due to the actual binary potential and radiation losses are shown by blue andcyan curves, respectively.",
"The net rate of change of angular momentum ∂ r ∂ t J\\partial _r \\partial _t J(solid black).",
"All quantities are time-averaged over the quasi-steady epoch in RunSS ."
],
[
"Disk thickness",
"We close this section by commenting on the disk thickness $H/r$ [defined in Equation (REF )], a parameter that will play an important role during the period when the binary orbit evolves.",
"Our initial data and cooling function were chosen so as to keep $H$ roughly constant over time at a fixed ratio to the local radius: $H/r \\simeq 0.1$ .",
"However, although the gas temperature stayed very close to the target entropy at all radii $r > 2a$ , and the ratio $H/r$ did stay nearly independent of radius, its value first rose to $\\simeq 0.15$ and then fell slightly (to $\\simeq 0.12$ by the end of the simulation).",
"The departure from the prediction of simple hydrostatic equilibrium was proportional to how much the magnetic pressure contributed to support against the vertical component of gravity."
],
[
"Binary Separation Evolution",
"At $t=40000M$ in RunIn , we began to evolve the binary orbit, letting it compress as gravitational radiation removes its orbital energy.",
"The rate of orbital evolution is extremely sensitive to separation: $\\dot{a}/a \\propto a^{-4}$ when $a/r_g \\gg 1$ .",
"Consequently, even at the relatively small initial separation assumed here ($a_0 = 20M$ ), orbital evolution is comparatively slow at first.",
"However, it accelerates dramatically after $t \\simeq 50000M$ .",
"By the end of RunIn ($t = 54000 M$ ), $\\dot{a}/a$ is quite rapid, and $a \\simeq 8M$ , small enough to make our PN expansion problematic.",
"Figure: Color contours of logΣ(r/a(t))\\log \\Sigma (r/a(t)).",
"The scaleis shown in the color bar.",
"(Left) RunIn .",
"(Right) RunSS .Figure: Mass enclosed within several sample radii as functions of time in RunIn .",
"Frombottom to top, the radii are r/a=1r/a = 1, 1.5, 2, 3, and 4.",
"(Left) For fixed a=a 0 a = a_0.",
"(Right) For time-dependent a(t)a(t).While the binary orbit changes relatively slowly, the inner edge of the disk moves inward in pace with the change in the binary separation, staying close to $\\simeq 2a(t)$ (as shown in Figures REF and REF ) until $t \\simeq 50000M$ .",
"However, as the orbital evolution becomes more rapid (after $t \\simeq 50000M$ ), although the inner edge of the disk continues to move inward in terms of absolute distance (Figure REF ), it begins to recede in terms of $r/a(t)$ (Figure REF ).",
"At the end of the simulation, the disk edge has moved in to $\\simeq 20M$ , but that is $\\simeq 2.5a$ .",
"Simultaneous with this evolution, the slope of the inner edge also becomes gentler (Figure REF ).",
"In other words, the contrast between the surface density in the disk body and in the gap weakens, particularly when considering the outer part of the gap.",
"As shown by the RunSS panel in Figure REF , none of this adjustment (in $r/a(t)$ terms) occurs without binary evolution.",
"Another view of this process may be seen in Figure REF .",
"In that figure, we see the way matter accumulates in the inner disk over time, at first during the quasi-steady epoch and later during the binary orbital evolution of RunIn .",
"The left-hand panel shows what happens when referred to an absolute radius scale.",
"When the binary begins to shrink, the quantity of matter found at small radii grows abruptly, particularly in the original gap region: the amount of mass inside $r = 40M$ almost doubles, and the mass inside $r=20M$ increases by a factor of 5 during the period of binary orbital evolution.",
"The right-hand panel shows the same events from a different point of view.",
"In this figure, we see that the mass enclosed within small multiples of $a(t)$ declines rapidly as the binary's shrinkage accelerates.",
"For larger multiples (e.g., $3a$ and $4a$ ), the mass enclosed continues to rise for a while after binary orbital evolution, but eventually drops once the compression becomes rapid.",
"In particular, the mass within the gap region (i.e., $r < 2a(t)$ ) falls by roughly a factor of 40 during the period of orbital evolution, although this ratio is in fact a bit ill-defined because $2a(t)$ is almost at the simulation's inner boundary by the end of the simulation.",
"Figure: Accretion rate through the inner boundary of the simulation as a function of time.",
"(Left) RunIn .",
"(Right) RunSS .The accretion rate behaves differently.",
"It falls (see Figure REF ) from $\\simeq 30000M$ –$40000M$ , even before the binary begins to compress.",
"Without binary orbital evolution (RunSS ), it levels out from $\\simeq 40000M$ –$50000M$ , before declining more gradually from $\\simeq 50000M$ until the end of RunSS at $\\simeq 76000M$ .",
"In RunIn , the onset of binary evolution at $t=40000M$ leads to a continuing decrease in the rate at which mass flows through the inner boundary that levels out only after $\\simeq 50000M$ .",
"Although the accretion rates in the simulations with and without binary orbital evolution decline at different times and at different rates, the final accretion rate in RunIn , when the binary separation has shrunk to $8M$ , is only 20–$30\\%$ less than at the same time in RunSS .",
"Another consequence of the changing relationship between disk material and the binary is a diminution in the integrated torque when the binary compresses (Figure REF ).",
"During the initial slow stages of energy loss due to gravitational wave emission, the binary continues to exert nearly as much torque on the disk as in RunSS , in which the binary orbit does not change at all.",
"However, once the orbital shrinkage begins to accelerate, the torque plummets; at the end of RunIn , it has fallen to $\\simeq 1/5$ of the value at that time in RunSS .",
"The greater part of this diminution in torque is due to the fact that at this stage in the binary's evolution, its separation diminishes so rapidly that the region between $a$ and $2a$ , where most of the torque is expressed, moves inward faster than the matter can follow.",
"There is consequently much less matter on which these torques can be exerted.",
"The connection between available matter and torque is shown clearly in the right-hand panel of Figure REF , in which one can easily see that for nearly the entire inspiral the torque density at the location of its maximum ($r=1.45a(t)$ ) is almost exactly proportional to the surface density there.",
"However, there is also a smaller part due to an artifact of the simulation.",
"Its inner boundary lies at $r_{\\rm min} = 15M$ .",
"As soon as $a(t)$ becomes smaller than $15M$ , part of the region in which the binary torque is applied is no longer in the problem volume, so we cannot calculate any torque occurring there.",
"As shown in the right-hand panel of Figure REF , this effect becomes significant at $t \\simeq 5.2 \\times 10^4 M$ , when $a(t) \\simeq 13M$ .",
"By the end of RunIn , $a \\simeq 8M$ , so that nearly the entire region where the torque is exerted ($a \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ r $\\sim $$<$ 2a$) has left theproblem volume.",
"At that point, even if there were significant matter there, our calculationcan neither say what its mass is nor what torque it feels.$ Figure: (Left) Integrated torque as a function of time in RunSS (black) andRunIn (grey).",
"Total torque is shown by the solid curves; the dotted curve showstorque in RunSS including only the radial range r>r min a 0 /a 2 (t)r > r_\\mathrm {min}a_0 / a_{2}(t),where a 2 (t)a_{2}(t) is the orbital separation as a function of time in RunIn .",
"(Right) Surface density (solid) and torque density at its peak, i.e., at r=1.45a(t)r=1.45a(t)(dashed) in RunSS (black) and RunIn (gray)."
],
[
"EM Luminosity: Magnitude, Modulation",
"We define the (coordinate frame) cooling rate per unit radius of the disk by $\\frac{dL}{dr} = \\int \\, \\sqrt{-g} \\, d\\theta \\, d\\phi \\mathcal {L}_c u_t .$ During the approximate stationary state, it is best described in terms of two separate regimes.",
"As shown in Figure REF , at large radius ($r \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 2a$), it is very well described by a power-law,$ dL/d(r/a0) 5 10-4 (r/a0)-2 0 a0$.At around $ r 2a$, the cooling rate per unit radius reaches a local maximumand declines inward.",
"This distinction neatly corresponds to two different mechanisms forgenerating the requisite heat: the dissipation of MHD turbulence associated with massaccretion (at large radius) and the dissipation of fluid kinetic energy given to therelatively small amount of gas in the gap by the binary torques (at small radius).In fact, this identification is confirmed semi-quantitatively.",
"In time-steadyaccretion, the luminosity per unit radius is $ (3/2)M c4/[(r/rg)2 GM]$ at radiiwhere the local orbital angular momentum per unit mass is large compared to thenet angular momentum flux per unit mass.",
"Our disk is never in inflow equilibrium,and this expression is not exact when $ M$ is a function of radius.",
"Nonetheless,taking it as an estimator, it predicts\\begin{equation}\\frac{dL}{dr/a_0} = 4 \\times 10^{-4} (\\dot{M}/0.01) (r/a_0)^{-2} \\Sigma _0 a_0.\\end{equation}As Figure~\\ref {fig:accrate_radius} shows, the mean accretion rate in code unitsat $ r=2a$ in \\hbox{RunSS\\ }was $ 0.01$, while $ M$ at larger radii istypically similar or perhaps a factor of two greater.",
"Thus, this prediction ofthe luminosity profile on the basis of the time-averaged accretion rate andexpectations derived from time-steady accretion onto a solitary mass quiteaccurately matches the actual luminosity profile seen in the simulation.$ Figure: Luminosity per unit radius averaged over the quasi-steady epoch in RunSS .The dashed line shows a logarithmic slope of -2.Figure: Luminosity as a function of time.",
"(Grey) RunIn .",
"(Black) RunSS .",
"We note that the vertical axis'range does not include zero in order to accentuate the curves' fluctuations.Integrated over radius, the total luminosity reaches a peak $\\hat{L} \\simeq 5.5 \\times 10^{-3}$ at $t \\simeq 33000M$ (Figure REF ), where $\\hat{L}$ is the integrated luminosity in units of $GM \\Sigma _0 c$ .",
"After reaching this peak, $\\hat{L}$ falls slowly, reaching $\\simeq 3 \\times 10^{-3}$ at $t \\simeq 76000M$ in RunSS ; averaged over the entire quasi-steady period in this simulation, it is $3.8\\times 10^{-3}$ .",
"The light output from RunIn remains very close to that in RunSS until the binary orbital evolution becomes rapid at $t\\simeq 50000M$ .",
"After that time, it falls more sharply, so that by the time at which RunIn stops, $\\hat{L} \\simeq 2.7 \\times 10^{-3}$ ; this is, however, still 2/3 the luminosity in RunSS at the same time.",
"As the binary shrinks, the radial distribution of the luminosity changes in parallel, with the peak in surface brightness moving inward.",
"We attribute the gradual decline in luminosity to the gradual decline in accretion rate.",
"The sharp drop in the final stages of binary orbital shrinkage is due to the interaction of a boundary effect with genuine dynamics.",
"As shown by [82], gas streams flow inward from the inner edge of a quasi-steady circumbinary disk to radii $\\simeq 1.2a$ , where they can be strongly torqued and some of their material flung back outward toward the disk.",
"The outward-moving matter shocks against the disk proper at a radius near that of the surface density peak, and the heat dissipated in these shocks contributes significantly to the luminosity.",
"When the binary shrinks, this mechanism is weakened for two reasons.",
"The inner boundary of our simulation ($r = 0.8a_0$ ) eventually becomes larger than $1.2a(t)$ ; when it does, matter is no longer thrown outward by binary torques.",
"At the same time, however, it is possible that the retreat of the disk's inner edge when measured in terms of $a(t)$ might also lead to weaker inward streams.",
"The fact that the energy deposited by binary torques is ultimately radiated in the disk proper leads to a method of estimating the relative contributions to the total luminosity coming from accretion and binary torques.",
"For that reason, and also because the accretion rate diminishes as the region of the surface density peak is approached from larger radius, it is a reasonable approximation to suppose that most of the luminosity from the region of the surface density peak inward has its source in the binary torques.",
"We can therefore estimate the work done by the torques by bounding it between $L(r<2a_0)$ and $L(r<3a_0)$ .",
"On this basis, accretion would account for $\\simeq 1/2$ –$3/4$ of the total (i.e., $\\hat{L} \\simeq 1.8$ –$2.9 \\times 10^{-3}$ ) and the binary torque for $\\simeq 1/4$ –$1/2$ ($\\hat{L}\\simeq 0.9$ –$2 \\times 10^{-3}$ ).",
"The rest-mass efficiency of this luminosity is comparable to the rest-mass efficiency due to accretion that goes all the way to the black hole.",
"Measured in terms of the time-dependent luminosity relative to the time-averaged accretion rate through the inner boundary, the efficiency in RunSS falls from a peak $\\simeq 0.06$ achieved for $20000M \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$<$}}$ t $\\sim $$<$ 45000M$ to $ 0.03$ at the end of this simulation.There are several reasons that this efficiency is so great even though the potential at$ r=50M$ is an order of magnitude shallower than the potential at the innermost stablecircular orbit (the ``ISCO\").",
"One is that the accretionrate in the circumbinary disk is roughly twice the accretion rate through the innerboundary, so the local accretion dissipation in the disk is boosted by that samefactor of two relative to the rate at which mass passes the inner boundary.",
"Anotheris that in a conventional disk around a single black hole the dissipation rate in theregion just outside the ISCO is depressed relative to larger radii because some ofthe potential energy released is transported outward by the inter-ring stresses.",
"In theNovikov-Thorne model (in which the stresses are assumed to vanish at the ISCO),almost 40\\% of the total luminosity is released outside $ r=40M$ when the blackhole has no spin.",
"This fraction is smaller when the spin is greater, and may befurther reduced to the degree that the net angular momentum flux is smaller \\cite {KHH05}.Lastly, of course, additional energy is deposited in the disk by the work done bythe binary torques.$ Translating the peak cooling rate into physical units gives $L_{\\rm disk} \\simeq 2.4 \\times 10^{40} (\\hat{L}/10^{-3}) M_6 \\tau _0\\hbox{~erg/s}.$ Here $\\tau _0$ is the Thomson optical depth through a disk of surface density $\\Sigma _0$ and $\\hat{L}$ is the luminosity in code units, i.e., 3–$5\\times 10^{-3}$ .",
"In Eddington units, this becomes $L_{\\rm disk}/L_E \\simeq 1.7 \\times 10^{-4} (\\hat{L}/10^{-3}) \\tau _0$ .",
"Thus, for such a system to be readily observable at cosmological distances, it will be necessary both for the disk to be optically thick to Thomson scattering and for the mass of the binary to be relatively large.",
"As a gauge of what might reasonably be expected, we note that in a steady-state accretion disk around a solitary black hole, the optical depth of the disk at $r/r_g = 20$ would be $\\sim 2 \\times 10^3 (\\alpha /0.1)^{-1} (\\eta /\\dot{m})$ , where $\\eta $ is the usual rest-mass efficiency and $\\dot{m}$ is the accretion rate in Eddington units.",
"With this disk surface density, the luminosity would approach that of a typical AGN when $M_6$ is at least $\\sim 1$ .",
"If this light were radiated thermally, the corresponding effective temperature would be $T_{\\rm eff} \\simeq 4 \\times 10^4 (\\hat{L}/10^{-3})^{1/4} M_6^{-1/4} \\tau _0^{1/4}\\hbox{~K},$ where we have assumed that the radiating area is $2\\pi (2a)^2$ .",
"Thus, it would emerge primarily in the ultraviolet for fiducial values of black hole mass and optical depth.",
"Figure: Fourier power spectrum of the luminosity radiated during the quasi-steadyepoch of RunSS .",
"The vertical lines represent: the orbital frequency atthe surface density maximum (dashes) and the peak in the spectrum (dots).The luminosity (assumed to be optically thin) exhibits a noticeable modulation as a function of time, with peak-to-trough contrast of $\\simeq 5\\%$ .",
"Its Fourier power spectrum shows a strong, sharp peak at a frequency $1.47\\Omega _{\\rm bin}$ (see Figure REF ) and a weaker peak at $0.26\\Omega _{\\rm bin}$ .",
"The latter is the orbital frequency at the radius of the surface density maximum, $\\simeq 2.4a$ ; because the lump is located at this radius, we call this frequency $\\Omega _{\\rm lump}$ .",
"The former we identify with the rate at which the lump approaches the orbital phase of a member of the binary, $2(\\Omega _{\\rm bin} - \\Omega _{\\rm lump}) = 1.46\\Omega _{\\rm bin}$ .",
"When the lump draws near one of the black holes, a new stream forms, falls inward, and is split into two pieces, one of which gains angular momentum, sweeps back out to the disk, and ultimately shocks against the disk gas.",
"It is this process, whose frequency is $1.46\\Omega _{\\rm bin}$ , that modulates the light curve.",
"If the binary mass ratio were far from unity, we expect that the modulation frequency would fall to $\\simeq \\Omega _{\\rm bin} - \\Omega _{\\rm lump}$ ."
],
[
"Comparison to Newtonian MHD",
"In many respects, the behavior we found in this post-Newtonian regime resembles what was previously found in the Newtonian limit [82].",
"There is very good agreement in the shapes of their azimuthally-averaged surface density profiles, with any contrasts attributable to their somewhat different initial conditions.",
"In both cases, during the quasi-steady epoch the surface density at the disk's inner edge rises $\\propto \\exp (3r/a)$ , reaches a maximum at $r \\simeq 2.5a$ , and then declines to larger radii.",
"At early times in both, there is a pair of streams leading from the disk edge to the inner boundary, which they typically reach at an orbital phase slightly ahead of the nearest member of the binary.",
"Both also develop a strong $m=1$ asymmetry (a “lump\") in the surface density at $r \\simeq 2.5a$ at late times.",
"This asymmetry ultimately causes, in both the Newtonian and post-Newtonian simulations, a single stream in the gap to become dominant.",
"Almost the only contrast in this regard is that the orbit of the lump developed a growing eccentricity in the Newtonian case, but not in RunSS .",
"The level of magnetization is likewise qualitatively similar: the mean plasma $\\beta $ in the Newtonian case fell from $\\sim 1$ at $\\simeq 6a$ to $\\simeq 0.3$ at $r \\simeq 2a$ , while the value (averaged over the quasi-steady epoch in RunSS ) in our simulations was $\\simeq 1.5$ at $r=6a$ , grew to $\\simeq 2.5$ at the surface density peak, and then decreased inward.",
"The magnetic stress-to-pressure ratio $\\alpha $ in the disk body follows the same pattern of close resemblance.",
"It was $\\simeq 0.3$ in the disk body in the Newtonian case, and $\\simeq 0.2$ in RunSS .",
"In the gap, the similarity was more qualitative than quantitative: in both cases, it rose steeply into the gap, but reached only $\\simeq 0.7$ at $r \\simeq a$ in the PN simulation, whereas it climbed to $\\simeq 10$ in the Newtonian one.",
"Most strikingly, the luminosity estimated by [82] scales extremely well to the PN case.",
"[82] could not directly compute the luminosity because they assumed an isothermal equation of state.",
"However, they argued that the work done by the binary torques would be delivered to the disk and ultimately dissipated there into heat.",
"Rewriting in our units their value for the rate at which the torques did work on the gas gives a luminosity of $0.018 GM\\Sigma _p c(a/r_g)^{-1/2}$ , where $\\Sigma _p$ is the surface density at the maximum; for our separation ($a=20M)$ and our surface density at the maximum ($\\simeq 0.55$ averaged over the quasi-steady epoch in RunSS ), that becomes $2.2 \\times 10^{-3} GM\\Sigma _0 c$ .",
"This prediction agrees well with the upper end of our estimated range for the binary torque share of the luminosity."
],
[
"Comparison to Analytic Estimates of Binary Runaway",
"[59] predicted that at some point well before the merger, the BBH should begin compressing so fast by gravitational radiation that internal stresses within the disk would not allow it to move inward rapidly enough to stay near the binary.",
"At the order of magnitude level, this breakaway point would be expected to come when the gravitational radiation time $t_{\\rm gr} = \\frac{5}{64}\\left(\\frac{a}{M}\\right)^4 \\frac{(1+q)^2}{q} M$ becomes shorter than the characteristic disk inflow time $t_{\\rm in} = \\alpha ^{-1}(H/r)^{-2}(d\\ln \\Sigma /d\\ln r)^{-1}\\Omega ^{-1}= \\alpha ^{-1}(H/r)^{-2}(d\\ln \\Sigma /d\\ln r)^{-1}(r/r_g)^{3/2}M.$ In these equations $\\Omega $ is the local disk orbital frequency.",
"The logarithmic derivative of the surface density enters because spreading of the inner edge is more rapid when it is especially sharp.",
"With a typical estimate of the stress level, $\\alpha \\sim 0.01$ , the binary separation at which $t_{\\rm gr}$ and $t_{\\rm in}$ would match, and the disk and binary might decouple is $a_{\\rm dec} = 70 (d\\ln \\Sigma /d\\ln r)^{-2/5}\\left(\\frac{H/r}{0.15}\\right)^{-4/5} M,$ where the fiducial radius at which the inflow time was computed is $r_* = 2a$ , and we set $q=1$ .",
"Indeed, it was this sort of estimate that led us to choose the initial conditions for our simulation.",
"However, scaling to the actual parameters of our simulation leads to a considerably smaller predicted value, $a_{\\rm dec} \\simeq 10 [(d\\ln \\Sigma /d\\ln r)/6]^{-2/5}\\left(\\frac{\\alpha }{0.2}\\right)^{-2/5}\\left(\\frac{H/r}{0.15}\\right)^{-4/5}M ,$ which is much closer to what is found in RunIn .",
"Thus, the substantially stronger magnetic stresses than predicted by usual $\\alpha $ -based estimates lead to decoupling at a much smaller binary separation.",
"Nonetheless, in the end the inward motion of the disk is limited by angular momentum transport, so once the magnitude of those stresses are known, $a_{\\rm dec}$ can be estimated quite well by this means.",
"On the other hand, the meaning of the term “binary runaway\" should also be made more nuanced.",
"As we have seen, the accretion rate through the simulation inner boundary decreases as the binary shrinks, but almost the same decrease in accretion rate occurs when the binary does not shrink.",
"Moreover, the continuing advance of the disk during the period of orbital evolution brought matter rapidly inward.",
"The amount of matter within $30M$ rose by about a factor of 4 while the binary shrank, so that almost as much matter could be found within that radius as had been within $40M$ at the beginning of the binary orbital evolution.",
"Thus, acceleration of binary orbital evolution does lead to a state in which the surface density at $r < 3a$ is smaller than would be expected if the orbital evolution were slower, and this diminution in the mass close to the binary does lead to consequences such as sharply diminished torque (as discussed in Section REF ) and luminosity (see Section REF ).",
"On the other hand, neither the torque nor the luminosity falls by as much as an order of magnitude because the decoupling of binary and disk matter is not complete.",
"Most notably, accretion continues at a rate only tens of percent lower than in the absence of inspiral.",
"Continuing accretion into the binary orbital region has a particularly interesting consequence.",
"The material in those streams should, just as happens when the binary orbital evolution is slower, be captured into orbit around one or the other of the members of the binary.",
"It will then settle into two smaller disks, one around each black hole.",
"In the conditions of our simulation, the inflow time in the individual black hole disks might be only slightly shorter than the merger time because decoupling occurs when the binary separation is not a great deal larger than the ISCO, even if both black holes spin rapidly.",
"If the circumbinary disk were cooler than in our simulation, so that $a_{\\rm dec} \\gg M$ , the inflow time in the smaller disks would be shorter than that in the circumbinary disk by a sizable ratio: $ \\sim 3 \\times 10^{-4}$ at decoupling if the scale heights in the inner disks and the outer disk are the same, and even smaller at later times.",
"In either case, there could be interesting hydrodynamic interaction between the two smaller disks as the binary compresses."
],
[
"Different Disk Thermal States",
"In terms of an ultimate comparison to observations, a larger question is posed by the disk's thermal state.",
"As just shown, the binary separation at decoupling scales as $(H/r)^{-4/5}$ , so the disk's internal pressure ($H \\propto c_s \\propto (p/\\rho )^{1/2}$ ) can influence $a_{\\rm dec}$ .",
"For reasons of numerical convenience, we chose parameters yielding a relatively thick disk.",
"Although the factors controlling the saturation of magneto-rotational turbulence are still not well understood, a scaling with disk pressure remains plausible.",
"If the effective sound speed of the gas were lower, decoupling might occur at rather larger binary separation, well outside the domain of relativistic orbits.",
"Several factors can influence the actual equation of state of the disk.",
"In ordinary AGN, local heating due to accretion can make the disk radiation-dominated inside $r \\simeq 100M$ when $\\dot{m} \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 0.3$ and the central mass $ M > 106 M$ \\cite {krolik99}.Larger central masses lead to radiation dominance even when $ m$ is smaller.",
"Whenthe disk surrounds a binary, the local heating should be similar at radii $ r $\\sim $$>$ 2a$,as shown by Figure~\\ref {fig:dlumdr}; the diminished accretion inside $ 4a$ iscompensated by dissipation of the work done by the binary torques and delivered to the disk.In those circumstances, the disk scale height for $ r $\\sim $$>$ 2a$ is independentof radius, giving an aspect ratio $ H/r (3/2) (m/) (r/rg)-1$, where$ m$ is now the {\\it local} accretion rate in Eddington units.",
"Thus, our aspect ratioof $ 0.15$ would correspond to a nominal $ m 0.4(r/40rg)$; that is,this $ m(r)$ is the mass accretion rate at $ r$ that would, if it reached the blackhole in a flow with $ = 0.1$, produce that fraction of an Eddington luminosity.$ Given the relatively large leakage fraction through the inner edge of the circumbinary disk, the accretion rate onto the two black holes will in general be smaller than the accretion rate in the disk by only a factor of a few.",
"They might therefore generate a sizable luminosity with a spectrum not too different from that of a generic AGN.",
"Because the density in the gap is considerably smaller than in the disk, this luminosity may irradiate the disk, particularly if it is relatively thick.",
"The inner edge of the disk could be heated by this means, as well.",
"Thus, there remains considerable uncertainty in the thickness profile of a circumbinary disk surrounding a relatively compact binary black hole.",
"The particular thickness we have simulated lies within that range of uncertainty, but may not be generic."
],
[
"Distinctive EM Signals",
"Given a sufficient external mass supply rate, the luminosity from the circumbinary disk alone could be great enough to be detected, even from a cosmological distance (Section REF ).",
"However, until the binary separation becomes as small as that considered here, the luminosity from matter accreting onto the two individual black holes would dominate the circumbinary luminosity by a large ratio.",
"In many respects, a binary black hole with $a \\gg 10M$ should strongly resemble a conventional AGN.",
"However, when the separation is as small as the $\\sim 20M$ scale studied in our simulations, contrasts with ordinary AGN continua might occur due both to the additional heating near the inner disk and the gap from $\\sim 2.5a$ down to $\\sim a/3$ in the range of radii in which a thermal disk exists.",
"The supplementary heating due to the binary torques will increase the luminosity of the portion of the disk near $r \\simeq 2a$ , but the temperature in this region is smaller than the hottest part of the accretion flow by a ratio $\\sim (1.5r_{\\rm ISCO}/2a)^{3/4}$ , where the factor 1.5 multiplying the ISCO radius is meant to account approximately for the displacement of the temperature maximum from the ISCO.",
"Consequently, the additional luminosity will appear at rather longer wavelengths than the peak of the thermal continuum.",
"On the other hand, radiation from the gap region will be very different from what might be expected from a conventional disk in those radii.",
"The dissipation rate is much smaller because the motions are laminar, not turbulent; moreover, whatever light does issue from that region is unlikely to be effectively thermalized, and would therefore emerge at considerably shorter wavelengths.",
"Thus, the luminosity at wavelengths intermediate between those characteristic of the innermost part of the disk and those characteristic of $r \\simeq 2a$ would be significantly suppressed.",
"The periodic modulation in the heating rate of the circumbinary disk that we have found might make its emission easier to isolate.",
"The key question governing that “might\" is how effectively optical depth in the disk blurs the modulation.",
"Our cooling function, which operates at a characteristic rate $\\sim \\Omega (r)$ filters out variations on timescales $\\ll \\Omega ^{-1}$ , but optical depth would impose a rather more severe upper bound on the maximum effective frequency of variation.",
"According to [53], a constant-density sphere of radius $R$ and optical depth $\\tau $ suppresses the amplitude of a periodic signal of angular frequency $\\omega $ injected at its center by a factor $\\simeq 3 c/(R\\tau \\omega )$ when $\\omega R \\tau /c \\gg 1$ .",
"To apply this estimator, we suppose that a stratified disk segment with scale height $H$ can be approximated by a homogeneous sphere of radius $R=H$ .",
"For binary separation $a$ , the relevant orbital radius is $\\simeq 2.5a$ and the signal frequency $\\omega = 1.46\\Omega _{\\rm bin} \\simeq 1.5(GM/a^3)^{1/2}$ for total binary mass $M$ .",
"As shown in Figure REF , the surface density in the lump grows to be $\\simeq \\Sigma _0$ , so we take $\\tau = \\tau _0$ .",
"The suppression factor is then $\\simeq 0.024 (\\tau _0/1000)^{-1} (a/20r_g)^{1/2} (H/0.15r)^{-1}$ .",
"In other words, when the relevant region of the disk is optically thick, the luminosity in the modulated component is independent of surface density, so that its fractional modulation decreases as the luminosity increases.",
"As we have discussed in section REF , there is also considerable uncertainty in $H/r$ , even given a disk surface density.",
"If the disk thickness and optical depth were determined by the considerations of conventional time-steady accretion flows around single black holes, the characteristic cooling time $\\tau H/c$ can be identified with $(\\alpha \\Omega )^{-1}$ .",
"The fluctuation suppression factor could then be estimated by $\\simeq 3 \\alpha \\Omega /\\omega $ .",
"Here the relevant orbital frequency is $\\Omega (2.4a) = 0.26\\Omega _{\\rm bin}$ , while $0.74\\Omega _{\\rm bin} \\le \\omega \\le 1.46\\Omega _{\\rm bin}$ (the upper limit applies in the equal-mass case, the lower limit when the masses are very unequal).",
"Thus, the suppression factor estimated in this way is $\\simeq 0.1(\\alpha /0.2)$ for equal black hole masses, rising to double that in the limit of very unequal masses.",
"However, this estimate is made uncertain by the fact that the disk in the vicinity of the surface density peak is certainly not in a state of inflow equilibrium.",
"In addition, just as for the other estimate, the association of the periodic modulation with the lump means that any estimate based on assumptions of axisymmetry likely underestimates the local optical depth.",
"Both estimates suggest that the modulation will be suppressed by at least a factor of several, but both are also subject to considerable uncertainty, making the actual outcome unclear.",
"It is worth pointing out that in the event the modulation is detectable, the period of the modulation would allow an estimate of the binary orbital period.",
"When the binary mass ratio is unity, the binary orbital frequency is 0.68 times the frequency of the modulation; as the mass ratio departs from unity, the binary orbital frequency should rise toward $\\simeq 1.36$ times the modulation frequency.",
"Finally, we remark that our predictions of EM signals from circumbinary disks around merging black holes are complementary to those previously made [15] in two ways.",
"In the previous work, the period of EM emission began when the binary separation shrank to $8M$ ; that is when our calculation ends.",
"In addition, that effort expressly excluded the disk proper, which they defined as $r \\ge 16M$ ; in our work, that is the location of the overwhelming majority of the emission.",
"Our effort also differs from previous work in this area in that we explicitly include radiation losses in the gas's energy equation and also tie the rate of radiation directly to the instantaneous local thermodynamic state of the gas (albeit in only a formal way).",
"In addition, our discussion of the observability of periodic modulation in the lightcurve takes into account possible suppression of the variation due to optical depth in the source."
],
[
"Summary",
"By describing the binary black hole spacetime at separations of tens of gravitational radii through the PN approximation, we have been able to simulate many orbits of fluid motion around such a system in fully relativistic MHD.",
"In so doing, we have demonstrated that the qualitative properties of circumbinary disks in such a regime are well described by an extrapolation from their properties in the Newtonian limit: Matter piles up at $\\simeq 2.5a$ , while smaller radii are largely cleared of mass; nonetheless, accretion continues through the inner gap, albeit reduced by a factor of a few from the rate at which it is supplied at larger radii.",
"At the same time, however, we have also investigated the initial stages of strongly relativistic behavior in the form of the disk's response to binary orbital evolution by gravitational wave emission.",
"By carrying the disk's evolution through the transition from the epoch in which its characteristic inflow time is short compared to the binary evolution timescale all the way to the epoch in which the binary evolves much faster than the disk, we have established the time at which the binary “runs away\" from the disk, and more importantly, the degree to which it does so.",
"This decoupling causes a drop in both the torque the binary exerts on the disk and in the disk luminosity.",
"However, a sizable fraction of the accretion rate at large radius continues to makes its way to the binary throughout this period.",
"The binary separation at which this decoupling occurs is rather smaller than commonly estimated, largely because the internal stresses produced by MHD turbulence in the disk are considerably greater than typical applications of the $\\alpha $ -model had guessed.",
"The actual value of $a_{\\rm dec}$ is sensitive to the disk's thermodynamics to the degree that the absolute level of the internal stresses are proportional to the disk's internal pressure.",
"Because accretion continues, luminosity released when the accreting gas reaches the black holes may illuminate the disk and heat it.",
"This sort of feedback has the potential to keep the disk's inflow rate high, self-consistently sustaining the accretion rate.",
"Given the sort of accretion rates associated with AGN, the inner regions of circumbinary disks around binary black holes with separations of tens of gravitational radii can be almost as bright as AGN, although there may be identifying alterations in the shapes of their optical/UV continua.",
"We have also shown that the work done on streams passing from the inner edge of the circumbinary disk through the evacuated gap around the binary is carried back to the disk and dissipated there.",
"Because the disk generically develops a non-axisymmetric density distribution at $\\simeq 2.5$ binary separations, the dissipation rate is modulated periodically with $\\sim 5\\%$ fractional amplitude.",
"In the right circumstances, this modulation might be detectable, although optical depth in the disk is likely to diminish its fractional amplitude, particularly when the accretion rate is high enough to make the system luminous.",
"If this modulation can be detected, its period would provide an estimator of the binary's orbital frequency with factor of 2 accuracy.",
"A number of our results may be sensitive to the particular parameters chosen, most importantly equal masses in the binary, spinless black holes, and perfect alignment between the orientation of the binary's orbital angular momentum and the disk's angular momentum.",
"Moreover, these choices can interact: for example, black hole spins oblique to the gas orbital plane can induce changes in that plane.",
"Future work exploring a variety of choices for these parameters may reveal additional effects.",
"This work was supported by several NSF grants.",
"S.C.N, M.C, and Y.Z.",
"received support from AST-1028087 and J.H.K.",
"from AST-1028111.",
"M.C., B.C.M, H.N, Y.Z.",
"also acknowledge partial support from PHY-0929114, PHY-0969855, PHY-0903782, OCI-0832606 and DRL-1136221.",
"NY acknowledges support from NSF grant PHY-1114374 and NASA grant NNX11AI49G, under sub-award 00001944.",
"We thank C. Lousto (RIT), John Hawley (UVa), K. Sorathia (JHU), J. Schnittman (Goddard) and Jiming Shi (UC Berkeley) for valuable discussions on the manuscript.",
"Computational resources were provided by the Ranger system at the Texas Advance Computing Center (Teragrid allocation TG-PHY060027N), which is supported in part by the NSF, and by NewHorizons at Rochester Institute of Technology, which was supported by NSF grant No.",
"PHY-0722703, DMS-0820923 and AST-1028087."
],
[
"Hydrostationary Torus Solutions in General Axisymmetric Spacetimes",
"Here, we describe a method for calculating axisymmetric non-magnetized gas distributions supported by pressure gradients and rotation within an axisymmetric spacetime.",
"We will assume a general spacetime with Killing vectors $\\left( \\partial / \\partial \\phi \\right)^a$ and $\\left( \\partial / \\partial t \\right)^a$ that can be expressed in the simple form (in coordinates similar to spherical Boyer-Lindquist coordinates): $ g_{\\mu \\nu } = \\left[ \\begin{array}{cccc}g_{t t} & 0 &0 &g_{t \\phi } \\\\[0.25cm]0 & g_{r r} & 0 & 0 \\\\[0.25cm]0 & 0 & g_{\\theta \\theta } & 0 \\\\[0.25cm]g_{t \\phi } & 0 & 0 & g_{\\phi \\phi } \\\\[0.25cm]\\end{array} \\right] \\quad , $ which means that the inverse metric is $ g^{\\mu \\nu } = \\left[ \\begin{array}{cccc}-\\frac{g_{\\phi \\phi }}{A} & 0 &0 & \\frac{g_{t \\phi }}{A} \\\\[0.25cm]0 & \\frac{1}{g_{r r}} & 0 & 0 \\\\[0.25cm]0 & 0 & \\frac{1}{g_{\\theta \\theta }} & 0 \\\\[0.25cm]\\frac{g_{t \\phi }}{A} & 0 & 0 & -\\frac{g_{t t}}{A} \\\\[0.25cm]\\end{array} \\right] \\,.",
"$ where $A = g_{t \\phi }^2 - g_{t t} \\, g_{\\phi \\phi }$ .",
"We have verified that the $\\phi $ -average of our PN spacetime, $\\hat{g}_{\\mu \\nu }$ , has this form to within the accuracy of our PN procedure.",
"The initial state of the simulation consists of matter in axisymmetric hydrostatic equilibrium with a specific angular momentum profile, $\\ell $ .",
"We start from the discussion of [26], which is based on [22] and other citations mentioned therein.",
"The disk is centered about the equator of the black hole's spin; we will eventually assume that it is initially isentropic.",
"The time-independent and axisymmetric Euler-Lagrange equations reduce, essentially, to $ \\frac{\\partial _i h}{h} + \\frac{1}{2} u^2_t \\partial _i u_t^{-2} - \\frac{\\Omega }{1 - \\ell \\Omega } \\partial _i \\ell = 0\\,, $ where the angular frequency—$\\Omega = u^\\phi /u^t$ —is not a simple function of the specific angular momentum—$\\ell = -u_\\phi /u_t$ .",
"The 4-velocity, $u^\\mu $ , in our symmetry has zero components: $u^r = u^\\theta = u_r = u_\\theta = 0$ .",
"One can show, from the normalization condition $u_\\mu u^\\mu = -1$ , that $ u_t = - \\left[ -g^{tt} + 2 \\ell \\,g^{t\\phi } - \\ell ^2 g^{\\phi \\phi } \\right]^{-1/2}\\,, $ and $ \\Omega = \\frac{ g^{t\\phi } - \\ell \\,g^{\\phi \\phi } }{ g^{tt} - \\ell \\,g^{t\\phi } }\\,.",
"$ The solutions assume that $ \\Omega = \\eta \\,\\lambda ^{-q}\\,, $ where $\\eta $ and $q$ are yet to be determined parameters, and $\\lambda $ is defined by $ \\lambda ^2 &=& \\frac{\\ell }{\\Omega }\\nonumber \\\\&=& \\ell \\,\\frac{g^{tt} - \\ell \\,g^{t\\phi }}{g^{t\\phi } - \\ell \\,g^{\\phi \\phi }}\\,.",
"$ We can eliminate $\\Omega $ from this system by combining equations (REF ) and (REF ) to yield a non-linear algebraic equation for $\\ell =\\ell (r,\\theta )$ in terms of the metric: $ R(\\ell ) = g^{t\\phi } \\left[ \\ell ^2 + \\lambda ^2(\\ell ) \\right] - g^{tt} \\ell - g^{\\phi \\phi } \\ell \\,\\lambda ^2(\\ell )= 0\\,, $ where $ \\ell &=& \\Omega \\,\\lambda ^2\\nonumber \\\\&=& \\eta \\,\\lambda ^{2-q}\\,, $ or $ \\lambda = \\left( \\frac{\\ell }{\\eta } \\right)^{1/\\left(2-q\\right)}\\,.",
"$ Also, we can show that $ \\Omega &=& \\eta ^{-2/\\left(q-2\\right)} \\ \\ell ^{q/\\left(q-2\\right)}\\nonumber \\\\& \\equiv & k \\,\\ell ^\\zeta \\,, $ where $k = \\eta ^{-2/\\left(q-2\\right)}$ and $\\zeta = q/\\left(q-2\\right)$ .",
"Typically, one “solves” equation (REF ) by approximating $\\lambda ^2$ with its Schwarzschild value: $\\lambda ^2 \\simeq -g^{tt}/g^{\\phi \\phi }$ [26], [65], [29].",
"For our $\\phi $ -averaged spacetimes, the Schwarzschild approximation is not so goodWhen using the approximate method, we found the disk to undergo a low frequency breathing mode that dominated the early evolution of the disk..",
"Therefore, we need a better solution.",
"We can solve this equation to roundoff precision by using a Newton-Raphson scheme.",
"To do so, we will need to know $\\partial R(\\ell ) / \\partial \\ell $ : $ \\frac{ \\partial R }{ \\partial \\ell } =g^{t\\phi } \\left[ 2 \\,\\ell + \\frac{ \\partial \\lambda ^2 }{ \\partial \\ell } \\right] - g^{tt}- g^{\\phi \\phi } \\left[ \\lambda ^2 + \\ell \\,\\frac{ \\partial \\lambda ^2 }{ \\partial \\ell } \\right]\\,, $ where $\\partial \\lambda ^2/\\partial \\ell =2 \\lambda ^2/[(2-q)\\ell ]$ .",
"Let us come back to equation (REF ).",
"A solution to this equation yields our disk solution.",
"The solution process involves integrating it from the inner disk's edge — located at $r_\\mathrm {in}$ — to a point within the disk: $ \\int ^h_{h_\\mathrm {in}} \\frac{d h}{h}= - \\frac{1}{2} \\int ^{u_t}_{u_{t \\mathrm {in}}} \\frac{ d (u_t)^{-2} }{ (u_t)^{-2} }+ \\int ^{\\ell }_{\\ell _\\mathrm {in}} \\frac{k \\,\\ell ^\\zeta }{ 1 - k \\,\\ell ^{\\zeta + 1}} d\\ell \\,.",
"$ With the boundary condition, $h_\\mathrm {in} = h(r_\\mathrm {in},\\theta =\\pi /2) = 1$ , one can solve this integral equation for $h=h(r,\\theta )$ : $ h = \\frac{ u_{t \\mathrm {in}} f(\\ell _\\mathrm {in}) }{ u_{t}(r,\\theta ) f(\\ell (r,\\theta )) }\\,, $ where $u_t$ is given by equation (REF ), $\\ell $ is found using Newton-Raphson on equation (REF ), $\\ell _\\mathrm {in} = \\ell (r_\\mathrm {in},\\pi /2)$ is a boundary value, and $f(\\ell ) \\equiv \\left| 1 - k \\, \\ell ^{\\zeta +1} \\right|^{1/\\left(\\zeta +1\\right)}$ .",
"We want a distribution that resembles a torus, which has a pressure maximum at some radius $r_{p}$ , and has finite extent.",
"The parameters $\\left\\lbrace \\ell _\\mathrm {in}, q, \\eta \\right\\rbrace $ determine whether we get such a solution.",
"We would like to replace one of the degrees of freedom with $r_{p}$ , however, there is not a closed-form solution for $r_{p}$ in terms of any of the original parameters.",
"We know that the fluid attains the Keplerian angular momentum at the pressure maximum ($r_{p}$ ) as the pressure gradient must be zero there.",
"It means that $\\ell _\\mathrm {in}$ should be super-Keplerian at the inner edge ($r_\\mathrm {in}$ ), and $\\ell _\\mathrm {out}$ should be sub-Keplerian at the outer edge ($r_\\mathrm {out}$ ).",
"We therefore know that $\\ell _\\mathrm {in} > \\ell _K(r_\\mathrm {in})$ , $\\ell _\\mathrm {out} < \\ell _K(r_\\mathrm {out})$ , and $\\ell _p = \\ell _K(r_{p})$ , where $\\ell _K$ is the Keplerian specific angular momentum (see the next section below).",
"However, we only have two free parameters, and now have three constraints (if we specify all $\\ell _\\mathrm {in}, \\ell _\\mathrm {out}$ and $\\ell _p$ ).",
"It may be possible to change equation (REF ) to look like: $ \\ell = \\eta \\left( \\lambda - \\lambda _0 \\right)^{2-q}\\,, $ and then find $\\lambda _0$ with this third constraint.",
"Using these three constraints, however, does not yield a closed-form solution for $q$ , $\\eta $ , and $\\lambda _0$ .",
"Therefore, another Newton-Raphson procedure would be required.",
"Hence, we relax the constraint on $\\ell _\\mathrm {out} < \\ell _K(r_\\mathrm {out})$ , and let $r_\\mathrm {out}$ be a result of our procedure.",
"Using $\\lambda _0=0$ , we find that $ q &=& 2 - \\frac{ \\log {\\left(\\ell _\\mathrm {in}/\\ell _p\\right)} }{ \\log {\\left( \\lambda _\\mathrm {in} / \\lambda _p \\right)} }\\,,\\\\\\eta &=& \\frac{ \\ell _p }{ \\lambda _p^{2-q} } \\,, $ where $\\lambda _p = \\lambda (\\ell _p, r_{p})$ and $\\lambda _\\mathrm {in} = \\lambda (\\ell _\\mathrm {in}, r_\\mathrm {in})$ given by equation (REF ).",
"We follow the solution process based on one described in [22], and is the following: Chose values of $(r_\\mathrm {in}, r_{p}, \\ell _\\mathrm {in})$ , and derive $q$ and $\\eta $ using equations (REF ) and (); Calculate $\\lambda $ (and then $\\ell $ ) at a new $(r,\\theta )$ via Newton-Raphson (using equations (REF ) and (REF )) close to the previous location, so that the old location's value can be used as a seed to the Newton iteration to successfully find a solution; Calculate $u_t$ via equation (REF ) then $h$ at $(r,\\theta )$ via equation (REF ); The solution process progresses throughout $(r,\\theta )$ space until the boundary of the disk is found, where $h(r,\\theta )=1$ .",
"The path we take starts at $(r,\\theta )=(r_\\mathrm {in},\\pi /2)$ , moves in increasing $r$ along the $\\theta =\\pi /2$ line, and we test to see if $h$ is increasing.",
"If $h$ is not increasing at first, then we stop and try a different set of parameters.",
"If $h$ is increasing at first, we then proceed until the outer edge of the disk is reached; this radius is denoted as $r_\\mathrm {out}$ .",
"Then, $\\forall r \\in \\left[r_\\mathrm {in}, r_\\mathrm {out}\\right]$ , we start from the $\\theta =\\pi /2$ solution and proceed backward and forward in $\\theta $ along constant $r$ to find $h(r,\\theta )$ .",
"The initial torus solution used herein is parameterized by $\\ell _\\mathrm {in}=8.743$ , $\\eta = 1.961$ , $q = 1.642$ , $r_\\mathrm {in}= 3 a_0 = 60M$ , $r_{p}= 5 a_0 = 100M$ , $r_\\mathrm {out}= 11.75 a_0 = 235M$ .",
"Figure: Change in total energy relative from its initial value versus time of non-magnetized tori from different runs.The first run used the so-called “approximate method” and was evolved in a PN BBH spacetime (solid); the second run usedour new and more accurate procedure, but used the same spacetime as the first (dashes);the last used the same disk from the second run, but was evolved in the φ\\phi -average of the other runs' spacetime (dots).",
"Thecurve for the last run (dots) oscillates with amplitude ∼10 -8 \\sim 10^{-8}, which is why it appears consistent with zero in this figure.All the disks share the same parameters: r in =2a 0 r_\\mathrm {in}= 2 a_0, r p =4a 0 r_{p}= 4 a_0, a 0 =30Ma_0 = 30 M."
],
[
"General Keplerian Velocity",
"The stationary torus solution described in Section requires the Keplerian, or circular equatorial geodesic orbits, of the spacetime.",
"Since we do not calculate $\\hat{g}_{\\mu \\nu }$ in closed form, we require equations for these orbits based on a generalized metric of the form Equation (REF ).",
"Here, we state the equations governing the Keplerian orbits.",
"Keplerian orbits in our spacetime have 4-velocity, $u^\\mu = [u^t, 0, 0, u^\\phi ] = u^t [1, 0, 0, \\Omega _K]$ .",
"$\\Omega _K$ is found from the $r$ -component of the geodesic equation, $du^r/d\\tau = - {\\Gamma ^r}_{\\mu \\nu } u^\\mu u^\\nu = 0$ , which ultimately yields $ \\Omega _{K \\, \\pm } &=& \\frac{1}{ {\\Gamma ^r}_{\\phi \\phi } } \\left[- {\\Gamma ^r}_{t \\phi }\\pm \\sqrt{\\left( {\\Gamma ^r}_{t \\phi } \\right)^2- {\\Gamma ^r}_{\\phi \\phi } {\\Gamma ^r}_{t t}}\\right]\\nonumber \\\\&=& - \\frac{1}{ \\partial _r g_{\\phi \\phi } } \\left[\\partial _r g_{t \\phi }\\pm \\sqrt{\\left( \\partial _r g_{t \\phi } \\right)^2- \\left(\\partial _r g_{\\phi \\phi }\\right) \\left(\\partial _r g_{t t}\\right)}\\right]\\,.",
"$ $\\Omega _{K \\, -}$ and $\\Omega _{K \\, +}$ are the prograde angular velocity and retrograde angular velocity, respectively.",
"The 4-velocity components are found by the normalization condition: $ u^t &=& [ g_{t t} + 2 \\,\\Omega \\,g_{t \\phi } + \\Omega ^2 g_{\\phi \\phi } ]^{-1/2}\\,,\\\\u^\\phi &=& \\Omega \\,u^t\\,.",
"$ The Keplerian specific angular momentum, $\\ell _K$ , is found by the relation between $\\ell $ and $\\Omega $ : $ \\ell = - \\frac{ g_{t \\phi } + \\Omega \\,g_{\\phi \\phi } }{ g_{t t} + \\Omega \\,g_{t \\phi } }\\,.",
"$"
],
[
"Resolution Requirements",
"Our grid resolution was chosen to adequately resolve the MRI, to resolve the spiral density waves generated by the binary's potential, and to involve cells that are nearly cubical.",
"We discuss each choice in turn.",
"Many recent studies have explored the resolution dependence of global MHD accretion disk simulations [35], [86], [83].",
"[35] found that many global properties of the disk nearly asymptote with increasing resolution once the following criteria are satisfied: $ N^{\\left(z\\right)} \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 16 (100)1/2 (z)1/2 (Q(z)10) , $ N^{\\left(3\\right)} \\mathrel {\\hbox{\\unknown.",
"{\\hbox{$\\sim $}}}\\hbox{$>$}}$ 790 (0.1H/r) (10)1/2 (Q(3)25) , where $N^{\\left(z\\right)}$ is the number of cells per scale height, $H$ , $Q^{\\left(z\\right)} > 10$ is the recommended quality factor of the simulation, $\\beta _z \\equiv \\langle p \\rangle / \\langle \\left|\\sqrt{g_{zz}} B^z\\right|^2 \\rangle $ .",
"We use spherical coordinates, so $N^{\\left(\\theta \\right)}$ is needed instead: $ N^{\\left(z\\right)} = \\frac{H}{\\Delta z} = \\frac{H}{r \\Delta \\theta } = \\frac{H/r}{\\Delta \\theta } \\equiv N_{H/r}\\quad , $ where $N_{H/r}$ is the number of cells in the poloidal direction per scale height.",
"We see from prior simulations (e.g., [66]) that $\\beta \\simeq 10$ and $\\beta _z/\\beta \\simeq 50$ are reasonable for a disk in its asymptotic steady state, suggesting that $N_{H/r} > 36$ .",
"The initial condition values of $\\beta \\simeq 100$ and $\\beta _z/\\beta \\simeq 1$ , however, yield a weaker constraint ($N_{H/r} > 16$ ) on the resolution.",
"Thus, we setup a grid such that $N_{H/r} \\simeq 36$ with $H/r=0.1$ , our simulation's scale height.",
"This is satisfied by the $x^{\\left(2\\right)}$ discretization described in Section REF .",
"We also note that our condition satisfies the recommendation of $N_{H/r} > 32$ by [86].",
"The more severe constraint is on the azimuthal symmetry.",
"Both [35] and [86] suggest that past simulations under-resolved the azimuthal direction and that one should cover the full azimuthal range $\\phi \\in [0,2\\pi ]$ instead of assuming quarter- or half-circle symmetry.",
"Since $\\Delta \\phi $ limits the time step size, we were only able to afford $N^{\\left(3\\right)} = 400$ as anything larger was impractical given our computational resources at the time.",
"We were optimistic with this resolution, however, since the thinnest run of [66] failed to satisfy Equation () yet still resolved the MRI with $Q^{\\left(3\\right)} > 25$ throughout most of the disk's body.",
"We demonstrate how well RunIn and RunSS resolve the MRI in Figures REF - REF , where we show mass-weighted averages of the $Q^{\\left(2\\right)}$ and $Q^{\\left(3\\right)}$ MRI quality factors: $ Q^{\\left(i\\right)} = \\frac{2 \\pi \\left|b^i\\right|}{ \\Delta x^{\\left(i\\right)} \\, \\Omega _K(r) \\, \\sqrt{\\rho h + 2 p_{m} } }\\quad .",
"$ The averages were made over $x^{\\left(2\\right)}$ in the following way: $ \\langle Q^{\\left(i\\right)} \\rangle _{\\rho } \\equiv \\frac{\\int _0^{1} Q^{\\left(i\\right)} \\rho \\, \\sqrt{-g} \\, dx^{\\left(2\\right)}}{\\int _0^{1} \\rho \\, \\sqrt{-g} \\, dx^{\\left(2\\right)}}\\quad .",
"$ A mass-weighting is used to calculate $\\langle Q^i \\rangle _{\\rho }$ in order to bias the integral over the turbulent portion of the disk (the disk's bulk) rather than the laminar regions (e.g., corona, funnel).",
"We find that the $Q^{\\left(z\\right)}$ constraint, i.e.",
"$\\langle Q^{\\left(2\\right)} \\rangle _{\\rho } > 10$ , is satisfied for all times and regions in either RunIn or RunSS except for the densest parts of the lump at late times in RunSS .",
"Similarly, the $Q^{\\left(3\\right)}$ constraint, i.e.",
"$\\langle Q^{\\left(3\\right)} \\rangle _{\\rho } > 25$ , is satisfied for all times and regions in either RunIn or RunSS except for in the lump at late times in RunSS .",
"We further note that $\\langle B^{r} B_{\\phi } \\rangle /\\langle p_m \\rangle \\simeq 0.3-0.35$ when averaged over the quasi-steady period of RunIn and RunSS ; this level is consistent with the asymptotic value found in resolution studies about point masses [11], [33], [24], [84], [35], [86], [83].",
"Figure: 〈Q 2 〉 ρ \\langle Q^{\\left(2\\right)} \\rangle _{\\rho } from RunIn (top row) and RunSS (second row) at late times in each simulation.The times of each snapshot are specified in the upper-right corner of each frame in units of MM.",
"The verticaland horizontal axes are in units of a 0 =20Ma_0=20M.We note that the t=40000Mt=40000M snapshot is shared by RunIn and RunSS .",
"The color map used to make the snapshots isgiven in the bottom row.Figure: Same as in Figure , but for 〈Q 3 〉 ρ \\langle Q^{\\left(3\\right)} \\rangle _{\\rho }.We also aim to resolve the spiral density waves generated by the binary's time-dependent quadrupolar potential.",
"This means that we need about $\\sim 10$ cells per wavelength of the sound wave generated by the binary, $\\lambda _d = 2 \\,\\pi \\,c_s/\\Omega _\\mathrm {bin}$ , where $c_s = \\left( H/r \\right) r \\, \\Omega _K$ is the speed of sound.",
"We use the Newtonian approximates $\\Omega _\\mathrm {bin}\\simeq a^{-3/2}$ and $\\Omega _K \\simeq r^{-3/2}$ to simplify $\\lambda _d$ : $\\lambda _d \\simeq 2 \\pi \\left(H/r\\right) r \\left(a/r\\right)^{3/2}$ .",
"We want to resolve $\\lambda _d$ in $\\phi $ and $r$ out to $r_{p}$ , which means that we want to satisfy another quality condition: $\\lambda _d / \\Delta r , \\lambda _d / r \\Delta \\phi \\simeq Q_d$ , where $Q_d$ will be the target number of cells per spiral density wavelength.",
"Using the grid specifications described in Section REF , it is easy to show that the resolution constraints become: $ N^{\\left(1\\right)}\\simeq 305 \\left(\\frac{r_{p}/a_0}{5}\\right)^{3/2}\\left(\\frac{0.1}{H/r}\\right)\\left(\\frac{\\ln \\left(r_\\mathrm {max}/r_\\mathrm {min}\\right)}{\\ln \\left(13/0.75\\right)}\\right)\\left(\\frac{Q_d}{6}\\right)\\quad , $ $ N^{\\left(3\\right)}\\simeq 671 \\left(\\frac{0.1}{H/r}\\right) \\left(\\frac{r_{p}/a_0}{5} \\right)^{3/2} \\left(\\frac{Q_d}{6}\\right)\\quad .",
"$ We find that the spiral density wave criterion is stricter than the MRI criterion when $ Q_d \\left(\\frac{r}{a}\\right)^{3/2} > \\beta ^{1/2} Q^{\\left(3\\right)}\\,.",
"$ Again, we did not satisfy the constraint on azimuthal resolution with our choice of $N^{\\left(3\\right)}= 400$ .",
"In this case, we were reassured by evidence found by [82] that found that the spiral density waves were extended over a large azimuthal extent and required far fewer cells than expected in that direction to resolve.",
"In practice, we find that spiral density waves are short-lived as they propagate through the turbulent, shear flow of the disk; they are hardly ever seen in snapshots of intrinsic quantities past $r\\simeq 3a_0$ .",
"Since gridscale dissipation scales with the ratio of the cell extent to characteristic length scales of the physical quantities, cells that are oblate may effectively lead to anisotropic dissipation.",
"Because physical dissipation mechanisms are isotropic, this effect could lead to unphysical artifacts.",
"For this reason, we attempt to make the cells within the bulk of the disk—where most of the dissipation occurs—as isotropic as possible.",
"Our runs use grids with $\\Delta r : r \\Delta \\theta : r \\Delta \\phi : \\simeq 3 : 1 : 3$ as measured in the $\\theta = \\pi / 2$ plane.",
"[86] suggested that $\\Delta r = r \\Delta \\phi $ and $\\Delta \\phi / \\Delta \\theta \\le 2$ ; we satisfy the former, and violate the latter by a slim margin: $\\Delta \\phi $ should be just $2/3$ times the size we use.",
"We note that the poloidal extent increases off the equator, so that the cells become more cubical at larger $\\left|\\theta -\\pi /2\\right|$ ."
],
[
"Mass, Energy, and Angular Momentum Budgets",
"Space and time gradients of the accretion flow's extensive quantities, mass ($M$ ), energy ($E$ ) and angular momentum ($J$ ) are fundamental for understanding the disk's evolution and structure.",
"In stationary spacetimes, $M$ , $E$ , and $J$ are all conserved.",
"In our $\\left(t,\\phi \\right)$ -dependent spacetime, $E$ and $J$ are no longer strictly conserved.",
"We describe here how several functions used throughout the paper are derived from the evolution equations of mass, energy and angular momentum."
],
[
"Angular Momentum",
"We begin this section with the angular momentum equation because of its import to accretion physics.",
"We follow the notation and derivation procedure outlined in [29].",
"An extensive quantity, $J$ , is the integral over the spatial volume of the time component of the its associated current, $j^\\mu $ : $J = \\int j^t \\, \\sqrt{-g} \\, dV$ , where $dV$ is the spatial volume component in the spacelike hypersurface (e.g., $dr d\\theta d\\phi $ ).",
"We are interested in the azimuthal component of the momentum as that is the dominant component of the gas' and the binary's momenta.",
"We therefore recognize that $j^\\mu = {T^\\mu }_\\nu \\phi ^\\nu $ , and $\\phi ^\\nu = \\left(\\partial _\\phi \\right)^\\nu = \\partial x^\\nu /\\partial \\phi = [0,0,0,1]$ in spherical coordinates, which is what we use.",
"We wish to calculate $d^2 J/dt dr$ .",
"If $J$ is locally conserved perfectly, $\\nabla _\\mu j^\\mu = 0$ .",
"In our case it will not be conserved exactly, and exploring the radial gradient of its volume integral will help us understand how MHD stresses and the binary's gravitational torque compete over the run of the flow.",
"This quantity is: $ \\frac{ d }{ d r } \\int \\left( \\nabla _\\mu j^\\mu \\right) \\, \\sqrt{-g} \\, dr d\\theta d\\phi & = & \\partial _r \\int \\left( \\partial _\\mu \\sqrt{-g} \\, j^\\mu \\right) \\, dr d\\theta d\\phi \\nonumber \\\\& = & \\partial _r \\partial _t J + \\partial _r \\partial _i \\int j^i \\, \\sqrt{-g}\\, dr d\\theta d\\phi \\nonumber \\\\& = & \\partial _r \\partial _t J+ \\partial _r \\int {T^r}_\\phi \\,\\sqrt{-g} \\, d\\theta d\\phi \\,, $ where the last equality results from the fact that $j^\\theta $ is zero on the axis, and $j^\\phi (\\phi =0) = j^\\phi (\\phi =2\\pi )$ .",
"On the other hand, we know from the stress-energy EOM—$\\nabla _\\mu {T^\\mu }_\\nu = -\\mathcal {F}_\\nu $ —that $ \\frac{ d }{ d r } \\int \\left( \\nabla _\\mu j^\\mu \\right) \\, \\sqrt{-g} \\, dr d\\theta d\\phi & = & \\int \\left( \\nabla _\\mu j^\\mu \\right) \\, \\sqrt{-g} \\, d\\theta d\\phi \\nonumber \\\\& = & \\frac{ d T }{ d r } - \\int \\mathcal {F}_\\phi \\, \\sqrt{-g} \\, d\\theta d\\phi \\,, $ where the torque density, $dT/dr$ , can be expressed as $ \\frac{ d T }{ d r } = \\int {T^\\mu }_\\nu {\\Gamma ^\\nu }_{\\mu \\phi } \\, \\sqrt{-g} \\, d\\theta d\\phi \\ = \\ \\frac{1}{2} \\int T^{\\mu \\nu } \\partial _\\phi g_{\\mu \\nu } \\, \\sqrt{-g} \\, d\\theta d\\phi \\,.",
"$ We remind the reader that $\\mathcal {F}_\\nu $ is the radiative cooling flux (see Section for details).",
"Therefore, equating the two equations (REF ) and (REF ), we have $ \\begin{array}{ccccccccccc}\\partial _r \\partial _t J & = &\\frac{ d T }{ d r } & - & \\left\\lbrace \\mathcal {F}_\\phi \\right\\rbrace & - & \\partial _r \\left\\lbrace {T^r}_\\phi \\right\\rbrace &&&&\\\\& = &\\frac{ d T }{ d r } &- &\\left\\lbrace \\mathcal {F}_\\phi \\right\\rbrace & - &\\partial _r \\left\\lbrace {M^r}_\\phi \\right\\rbrace & - &\\partial _r \\left\\lbrace {R^r}_\\phi \\right\\rbrace & - &\\partial _r \\left\\lbrace {A^r}_\\phi \\right\\rbrace \\,,\\end{array} $ where we have used here the shorthand $ \\left\\lbrace X \\right\\rbrace \\equiv \\int \\sqrt{-g} \\, X d\\theta \\, d\\phi \\ = \\langle X \\rangle \\int \\sqrt{-g} \\, d\\theta \\, d\\phi \\,.",
"$ Also, ${M^r}_\\phi $ , ${R^r}_\\phi $ , and ${A^r}_\\phi $ are—respectively—the Maxwell (MHD) stress, Reynolds stress, and advected flux of angular momentum.",
"We note that ${M^\\mu }_\\nu = 2p_m u^\\mu u_\\nu + p_m {\\delta ^\\mu }_\\nu - b^\\mu b_\\nu $ is the EM part of ${T^\\mu }_\\nu $ , while $\\left( {R^\\mu }_\\nu + {A^\\mu }_\\nu \\right) = {{T_H}^\\mu }_\\nu = \\rho h u^\\mu u_\\nu + p {\\delta ^\\mu }_\\nu $ is the hydrodynamic part.",
"The Reynolds stress alone is more complicated to calculate as we have to find the perturbation from the mean flow: $ {R^r}_\\phi = \\rho h \\, \\delta u^r \\, \\delta u_\\phi \\,, $ where $ \\delta u^\\mu \\equiv u^\\mu - \\left\\lbrace \\rho u^\\mu \\right\\rbrace / \\left\\lbrace \\rho \\right\\rbrace \\,.",
"$ We note that we include the enthalpy as it technically contributes to the stress; its contribution is insignificant, however, for our relatively cool flow.",
"The quantities $\\left\\lbrace {R^\\mu }_{\\nu } \\right\\rbrace $ and $\\left\\lbrace {A^\\mu }_{\\nu } \\right\\rbrace $ are not calculated during the simulation, but found approximately from other shell-integrated quantities we do calculate; $\\left\\lbrace {M^\\mu }_{\\nu } \\right\\rbrace $ , $\\left\\lbrace \\mathcal {F}_\\mu \\right\\rbrace $ , and $dT/dr$ are calculated as stated above during the run.",
"One can easily show from equations (REF ) and (REF ) that $ \\left\\lbrace {R^r}_\\phi \\right\\rbrace \\ = \\ \\left\\lbrace \\rho h \\, \\delta u^r \\, \\delta u_\\phi \\right\\rbrace \\ \\simeq \\ \\left\\lbrace {{T_H}^r}_\\phi \\right\\rbrace - \\left\\lbrace {A^r}_\\phi \\right\\rbrace $ where ${{T_H}^r}_\\phi $ is the hydrodynamic part of ${T^r}_\\phi $ , and $\\left\\lbrace {A^r}_\\phi \\right\\rbrace $ is calculated approximately as $ \\left\\lbrace {A^r}_\\phi \\right\\rbrace \\simeq \\frac{\\left\\lbrace \\rho \\ell \\right\\rbrace \\left\\lbrace \\rho h u^r \\right\\rbrace }{\\left\\lbrace \\rho \\right\\rbrace }\\,.",
"$ Here, $\\ell = - u_\\phi / u_t$ as its defined in Appendix .",
"The approximations used to find equations (REF -REF ) include: 1) $h \\simeq 1$ , and 2) $u_t \\simeq -1$ .",
"We have demonstrated that these assumptions are valid to the few percent level in the bound portion of the flow for our simulations described in this paper."
],
[
"Energy",
"Torques and stresses do work on the gas, transporting angular momentum.",
"This work can be dissipated in the disk, changing its internal energy, which is eventually radiated away in part.",
"Here we calculate the partitions in which the energy can move into; this calculation is nearly identical to that for $d^2 J/dtdr$ in Appendix REF The current associated with $E$ is $e^\\mu = {T^\\mu }_\\nu t^\\nu $ , where $t^\\mu $ is the 4-vector along time coordinate, $t^\\mu = \\left[1,0,0,0\\right]$ .",
"They are related by $E = \\int e^t \\sqrt{-g} \\,dV$ .",
"Just as with $j^\\mu $ , the divergence of $e^\\mu $ is not exactly zero, because of the time-dependent spacetime.",
"Using a similar analysis as before, we get $ \\partial _r \\partial _t E \\ = \\ dW/dr \\ - \\ \\left\\lbrace \\mathcal {F}_t \\right\\rbrace \\ - \\ \\partial _r \\left\\lbrace {T^r}_t \\right\\rbrace \\nonumber $ where $dW/dr = \\left\\lbrace \\frac{1}{2} T^{\\mu \\nu } \\partial _t g_{\\mu \\nu } \\right\\rbrace $ is the work done by the spacetime on the matter."
],
[
"Mass Accretion Rate",
"The current $j^\\mu = \\rho u^\\mu $ is associated with the conserved quantity $M$ , so we have $M = \\int \\rho u^t \\sqrt{-g} \\,dV$ , and $ \\frac{ d M }{ d t }\\ = \\ - \\int \\rho u^r \\sqrt{-g} \\,d\\theta d\\phi \\,.",
"$ by using a similar technique to obtain Equation (REF )."
]
] | 1204.1073 |
[
[
"Predictions for the CO emission of galaxies from a coupled simulation of\n galaxy formation and photon dominated regions"
],
[
"Abstract We combine the galaxy formation model GALFORM with the Photon Dominated Region code UCL_PDR to study the emission from the rotational transitions of 12CO (CO) in galaxies from z=0 to z=6 in the Lambda CDM framework.",
"GALFORM is used to predict the molecular (H2) and atomic hydrogen (HI) gas contents of galaxies using the pressure-based empirical star formation relation of Blitz & Rosolowsky.",
"From the predicted H2 mass and the conditions in the interstellar medium, we estimate the CO emission in the rotational transitions 1-0 to 10-9 by applying the UCL_PDR model to each galaxy.",
"We find that deviations from the Milky-Way CO-H2 conversion factor come mainly from variations in metallicity, and in the average gas and star formation rate surface densities.",
"In the local universe, the model predicts a CO(1-0) luminosity function (LF), CO-to-total infrared (IR) luminosity ratios for multiple CO lines and a CO spectral line energy distribution (SLED) which are in good agreement with observations of luminous and ultra-luminous IR galaxies.",
"At high redshifts, the predicted CO SLED of the brightest IR galaxies reproduces the shape and normalization of the observed CO SLED.",
"The model predicts little evolution in the CO-to-IR luminosity ratio for different CO transitions, in good agreement with observations up to z~5.",
"We use this new hybrid model to explore the potential of using colour selected samples of high-redshift star-forming galaxies to characterise the evolution of the cold gas mass in galaxies through observations with the Atacama Large Millimeter Array."
],
[
"Introduction",
"The connection between molecular gas and star formation (SF) is a fundamental but poorly understood problem in galaxy formation.",
"Observations have shown that the star formation rate (SFR) correlates with the abundance of cold, dense gas in galaxies, suggesting that molecular gas is needed to form stars.",
"A variety of observational evidence supports this conclusion, such as the correlation between the surface densities of SFR and $^{12}\\rm CO$ (hereafter CO) emission and between the CO and infrared (IR) luminosities (e.g.",
"[107]; [13]).",
"The CO luminosity traces dense gas in the interstellar medium (ISM), which is dominated by molecular hydrogen (H$_2$ ).",
"The IR luminosity approximates the total luminosity emitted by interstellar dust, which, in media that are optically thick to ultraviolet (UV) radiation, is expected to correlate closely with the SFR in star-forming galaxies.",
"In the local Universe, high-quality, spatially resolved CO data show a tight and close to linear correlation between the surface density of the SFR and the surface density of CO emission, that extends over several orders of magnitude and in very different environments: from low-metallicity, atomic-dominated gas to high-metallicity, molecular-dominated gas (e.g.",
"[123]; [72]; Bigiel at al., 2008, 2011; [102]; [96]).",
"This suggests that SF proceeds in a similar way in these different environments.",
"Support for this also comes from the IR-CO luminosity relation, with high-redshift submillimeter galaxies (SMGs) and quasi-stellar objects (QSOs) falling on a similar relation to luminous IR galaxies (LIRGs) and ultra-luminous IR galaxies (ULIRGs) in the local Universe (e.g.",
"[106]; [103]; [53]; [112]; [40]; [11]; [5]; [111]; [48]; [99]; [32]; [47]; [56]; [22]).",
"[5] studied the correlation between different CO transitions and IR luminosity and found that this correlation holds even up to $z\\approx 6$ .",
"The CO emission from galaxies is generally assumed to be a good indicator of molecular gas content.",
"However, to infer the underlying H$_2$ mass from CO luminosity it is necessary to address how well CO traces H$_2$ mass.",
"This relation is usually parametrised by the conversion factor, $X_{\\rm CO}$ , which is the ratio between the H$_2$ column density and the integrated CO line intensity.",
"Large efforts have been made observationally to determine the value of $X_{\\rm CO(1-0)}$ for the CO$(1-0)$ transition, and it has been inferred directly in a few galaxies, mainly through virial estimates and measurements of dust column density.",
"Typical estimates for normal spiral galaxies lie in the range $X_{\\rm CO(1-0)} \\approx (2-3.5) \\times 10^{20} \\rm cm^{-2} / K\\,km\\,s^{-1}$ (e.g.",
"[125]; [19]; [15]; [17]).",
"However, systematic variations in the value of $X_{\\rm CO(1-0)}$ have been inferred in galaxies whose ISM conditions differ considerably from those in normal spiral galaxies, favouring a larger $X_{\\rm CO(1-0)}$ in low-metallicity galaxies and a smaller $X_{\\rm CO(1-0)}$ in starburst galaxies (e.g.",
"[70], 2011; [76]; see [107] for a review).",
"Theoretically, most studies of $X_{\\rm CO}$ are based on Photon Dominated Region (PDR; e.g.",
"[8]) or large velocity gradient (LVG; e.g.",
"[120]) models.",
"Such models have been shown to be an excellent theoretical tool, reproducing the emission of various chemical species coming from regions where the CO emission dominates (i.e.",
"in giant molecular clouds, GMCs, where most of the gas is in the atomic or molecular phase, with kinetic temperatures typically below 100 K, and densities ranging from $10^3\\,\\rm cm^{-3}$ to $10^5\\,\\rm cm^{-3}$ ).",
"These models have shown that $X_{\\rm CO}$ can vary considerably with some of the physical conditions in the ISM, such as gas metallicity, interstellar far-UV (FUV) radiation field and column density of gas and dust (e.g.",
"[9]; Bayet et al.",
"2012, in prep.).",
"Recently, large efforts have been devoted to measuring the CO emission in high-redshift galaxies.",
"These observations have mainly been carried out for higher CO rotational transitions (e.g.",
"[111]; [99]; [48]; [32]; [47]).",
"Thus, in order to estimate molecular gas masses from these observations, a connection to the CO$(1-0)$ luminosity is needed, as expressed through the $\\rm CO(J\\rightarrow J-1)/\\rm CO(1-0)$ luminosity ratio.",
"The latter depends upon the excitation of the CO lines and the conditions in the ISM, and is therefore uncertain.",
"The study of several CO transitions, as well as other molecular species, has revealed a wide range of ISM properties that drive large differences in the excitation levels of CO lines in different galaxy types.",
"Through comparisons with PDR and LVG models, a broad distinction has been made between the ISM in normal star-forming galaxies, in starburst-like galaxies, and most recently, the ISM excited by radiation from active galactic nuclei (AGN), suggesting large differences in gas temperature (e.g.",
"[122]; [79]; [116]; [121]; [115]; [7]).",
"[56] show that large uncertainties are introduced in the study of the CO-to-IR luminosity ratio of galaxies when including measurements which rely on an assumed $\\rm CO(J\\rightarrow J-1)/CO(1-0)$ luminosity ratio.",
"In recent years, efforts have also been made to develop a theoretical framework in a cosmological context to understand the relation between cold, dense gas, SF and other galaxy properties (e.g.",
"[94]; [49]; Narayanan et al.",
"2009, 2012; [23]).",
"In particular, a new generation of semi-analytical models of galaxy formation have implemented improved recipes for SF which use more physical descriptions of the ISM of galaxies ([38]; [46]; [31]; [67]).",
"This has allowed a major step forward in the understanding of a wide range of galaxy properties, including gas and stellar contents and their scaling relations (e.g.",
"[66]; [57]).",
"Lagos et al.",
"(2011a) presented simple predictions for CO emission, based on assuming a constant conversion factor between CO luminosity and H$_2$ mass, and successfully recovered the $L_{\\rm CO(1-0)}/L_{\\rm IR}$ ratio in normal and starburst galaxies from $z=0$ to $z=6$ .",
"Despite this progress, a crucial step in the comparison between observations and theoretical predictions is missing: a physically motivated CO-H$_2$ conversion factor, $X_{\\rm CO}$ .",
"Hydrodynamical cosmological simulations have successfully included the formation of CO, as well as H$_2$ (e.g.",
"[93]), but their high-computational cost does not allow a large number of galaxies spanning a wide range of properties to be simulated to assess the origin of statistical relations such as that between the CO and IR luminosities.",
"[88] presented a simple phenomenological model to calculate the luminosities of different CO transitions, based on a calculation of the ISM temperature depending on the surface density of SFR and the AGN bolometric luminosity, under the assumption of local thermodynamic equilibrium.",
"However, this modelling introduces several extra free parameters which, in most cases, are not well constrained by observations.",
"In this paper we propose a theoretical framework to statistically study the connection between CO emission, SF and H$_2$ mass based on a novel approach which combines a state-of-the-art semi-analytic model of galaxy formation with a single gas phase PDR model of the ISM which outputs the chemistry of the cold ISM.",
"From this hybrid model we estimate the CO emission in different transitions using the predicted molecular content, gas metallicity, UV and X-ray radiation fields in the ISM of galaxies, attempting to include as much of the physics determining $X_{\\rm CO}$ as possible.",
"The underlying assumption is that all molecular gas is locked up in GMCs.",
"Although inferences from observations indicate that galaxies have some diffuse H$_2$ in the outer parts of GMCs that is not traced by the CO emission (e.g.",
"[97]; [52]), it has been suggested theoretically that this gas represents a constant correction of $\\approx 0.3$ over a large range of media conditions [121].",
"We therefore do not attempt to model this diffuse component in this paper and focus on the inner part of the PDRs exclusively, where there is CO emission.",
"We show in this paper that by coupling a PDR model with the predictions of a galaxy formation model, we are able to explain the observed CO luminosity in several CO transitions and its dependence on IR luminosity.",
"The theoretical framework presented in this paper will help the interpretation of CO observations with the current and next generation of millimeter telescopes, such as the Atacama Large Millimeter Arrayhttp://www.almaobservatory.org/ (ALMA), the Large Millimeter Telescopehttp://www.lmtgtm.org/ (LMT) and the new configuration of the Plateu de Bureau Interferometerhttp://www.iram-institute.org/EN (PdBI).",
"These instruments will produce an unprecedented amount of data, helping to statistically assess the cold gas components of the ISM in both local and high-redshift galaxies.",
"This paper is organised as follows.",
"In $§2$ we present the galaxy formation and the PDR models used and describe how we couple the two codes to predict the CO emission in galaxies.",
"$§3$ presents the predicted CO$(1-0)$ emission of galaxies in the local universe, and its relation to other galaxy properties, and compares with available observations.",
"$§4$ is devoted to the study of the emission of multiple CO lines in the local and high-redshift universe, i.e.",
"the CO spectral line energy distribution (SLED), how the CO emission relates to the IR luminosity and how this depends on selected physical ingredients used in the model.",
"In $§5$ we analyse the assumptions of the PDR model and how these affect the predictions for the CO luminosity.",
"In $§6$ , we focus on the ALMA science case for measuring the cold gas content of high-redshift star-forming galaxies to illustrate the predictive power of the model.",
"We discuss our results and present our conclusions in $§6$ .",
"In Appendix , we describe how we convert the CO luminosity to the different units used in this paper and how we estimate it from the H$_2$ mass and $X_{\\rm CO}$ ."
],
[
"Modelling the CO emission of galaxies",
"We study the $\\rm CO$ emission from the $(1-0)$ to the $(10-9)$ rotational transitions, and its relation to other galaxy properties, using a modified version of the GALFORM semi-analytical model of galaxy formation described by Lagos et al.",
"(2011a, 2011b) in combination with the Photon Dominated Region code, UCL$_{-}$ PDR of [7].",
"In this section we describe the galaxy formation model and the physical processes included in it in $§2.1$ , the UCL$_{-}$ PDR model and its main parameters in $§2.2$ , give details about how we couple these two models to estimate the CO emission of galaxies in $§2.3$ , and briefly describe how the CO conversion factor depends on galaxy properties predicted by GALFORM in $§2.4$ ."
],
[
"The galaxy formation model",
"The GALFORM model [28] takes into account the main physical processes that shape the formation and evolution of galaxies.",
"These are: (i) the collapse and merging of dark matter (DM) halos, (ii) the shock-heating and radiative cooling of gas inside DM halos, leading to the formation of galactic disks, (iii) quiescent star formation (SF) in galaxy disks, (iv) feedback from supernovae (SNe), from AGN and from photo-ionization of the intergalactic medium (IGM), (v) chemical enrichment of stars and gas, and (vi) galaxy mergers driven by dynamical friction within common DM halos, which can trigger bursts of SF and lead to the formation of spheroids (for a review of these ingredients see [3]; [10]).",
"Galaxy luminosities are computed from the predicted star formation and chemical enrichment histories using a stellar population synthesis model.",
"Dust extinction at different wavelengths is calculated self-consistently from the gas and metal contents of each galaxy and the predicted scale lengths of the disk and bulge components using a radiative transfer model (see [28] and [64]).",
"[67] improved the treatment of SF in quiescent disks, (iii) in the above list, which allowed more general SF laws to be used in the model.",
"GALFORM uses the formation histories of DM halos as a starting point to model galaxy formation (see [28]).",
"In this paper we use halo merger trees extracted from the Millennium N-body simulation [108], which assumes the following cosmological parameters: $\\Omega _{\\rm m}=\\Omega _{\\rm DM}+\\Omega _{\\rm baryons}=0.25$ (with a baryon fraction of $0.18$ ), $\\Omega _{\\Lambda }=0.75$ , $\\sigma _{8}=0.9$ and $h=0.73$ .",
"The resolution of the $N$ -body simulation corresponds to a minimum halo mass of $1.72 \\times 10^{10} h^{-1} M_{\\odot }$ .",
"This is sufficient to resolve the halos that contain most of the H$_2$ in the universe at $z<8$ [66].",
"[67] studied three SF laws, (i) the empirical SF law of [58], (ii) the empirical SF law of [16] and (iii) the theoretical SF law of [62].",
"Here we follow Lagos et al.",
"(2011a, hereafter L11), who adopted the empirical SF law of [16] as their preferred model.",
"The main successes of the L11 model include the reproduction of the optical and near-infrared luminosity functions (LF), the $z=0$ atomic hydrogen (HI) mass function (MF), the global density evolution of HI at $z<3.5$ , and scaling relations between HI, H$_2$ , stellar mass and galaxy morphology in the local Uuniverse.",
"The [16] empirical SF law has the form $\\Sigma _{\\rm SFR} = \\nu _{\\rm SF} \\,\\rm f_{\\rm mol} \\, \\Sigma _{\\rm gas},$ where $\\Sigma _{\\rm SFR}$ and $\\Sigma _{\\rm gas}$ are the surface densities of SFR and the total cold gas mass, respectively, $\\nu _{\\rm SF}$ is the inverse of the SF timescale for the molecular gas and $\\rm f_{\\rm mol}=\\Sigma _{\\rm mol}/\\Sigma _{\\rm gas}$ is the molecular to total gas mass surface density ratio.",
"The molecular and total gas contents include the contribution from helium, while HI and H$_2$ masses only include hydrogen (helium accounts for 26% of the overall cold gas mass).",
"The ratio $\\rm f_{\\rm mol}$ is assumed to depend on the internal hydrostatic pressure of the disk as $\\Sigma _{\\rm H_2}/\\Sigma _{\\rm HI}=\\rm f_{\\rm mol}/(f_{\\rm mol}-1)=(P_{\\rm ext}/P_{0})^{\\alpha }$ [16].",
"The parameters values we use for $\\nu _{\\rm SF}$ , $\\rm P_{0}$ and $\\alpha $ are the best fits to observations of spirals and dwarf galaxies, $\\nu _{\\rm SF}=0.5\\, \\rm Gyr^{-1}$ , $\\alpha =0.92$ and $\\rm log(P_{0}/k_{\\rm B} [\\rm cm^{-3} K])=4.54$ ([16]; [72]; [14]; [96]).",
"[75] explain the relation between the $\\Sigma _{\\rm H_2}/\\Sigma _{\\rm HI}$ ratio and the midplane pressure being the result of an underlying and more fundamental relation between these two quantities and the local density in normal spiral galaxies (see also [93]).",
"In this paper, however, we adopt the empirical relation to avoid fine-tuning of the parameters associated with it (see [67]).",
"For starbursts the situation is less clear.",
"Observational uncertainties, such as the conversion factor between CO and H$_2$ in starbursts, and the intrinsic compactness of star-forming regions, have not allowed a clear characterisation of the SF law (e.g.",
"[58]; [48]; [29]).",
"Theoretically, it has been suggested that the SF law in starbursts is different from that in normal star-forming galaxies: the relation between $\\Sigma _{\\rm H_2}/\\Sigma _{\\rm HI}$ and gas pressure is expected to be different in environments of very high gas densities typical of starbursts ([94]; [93]; [75]), where the ISM is predicted to always be dominated by H$_2$ independently of the exact gas pressure.",
"For these reasons we choose to apply the Blitz & Rosolowsky SF law only during quiescent SF (fuelled by the accretion of cooled gas onto galactic disks) and retain the original SF prescription for starbursts (see [28] and L11 for details).",
"In the latter, the SF timescale is taken to be proportional to the bulge dynamical timescale above a minimum floor value (which is a model parameter) and involves the whole ISM gas content in the starburst, giving $\\rm SFR = \\it M_{\\rm gas}/\\tau _{\\rm SF}$ (see [51] and [65] for details), with $\\tau _{\\rm SF}=\\rm max(\\tau _{min},f_{\\rm dyn}\\tau _{\\rm dyn}).$ Here we adopt $\\tau _{\\rm min}=100\\, \\rm Myr$ and $f_{\\rm dyn}=50$ .",
"This parameter choice helps to reproduce the observed rest-frame UV (1500Å) luminosity function from $z\\approx 3$ to $z\\approx 6$ (see [4]; [64]).",
"In [66] these parameters were set to $\\tau _{\\rm min}=5\\, \\rm Myr$ and $f_{\\rm dyn}=2$ , inherited from the parameter choice in [25].",
"However, the modification of these two parameters does not have any relevant influence on the results presented previously in Lagos et al.",
"(2011a,b), but mainly affects the UV luminosity of very high redshift galaxies through the dust production during starbursts.",
"In order to estimate the CO emission in starbursts, we assume here that the cold gas content is fully molecular, $f_{\\rm mol} = 1$ .",
"Note that this is similar to assuming that the Blitz & Rosolowsky relation between the midplane pressure and the $\\Sigma _{\\rm H_2}/\\Sigma _{\\rm HI}$ ratio holds in starbursts, given that large gas and stellar densities lead to $f_{\\rm mol} \\approx 1$ .",
"Throughout the paper we will refer to galaxies as `starburst galaxies' if their total SFR is dominated by the starburst mode, $\\rm SFR_{\\rm starburst}>SFR_{\\rm quiescent}$ , while the rest of the galaxies will be refered to as `quiescent galaxies'.",
"The three properties predicted by GALFORM which we use as inputs for the PDR model are (1) the ISM metallicity, $Z_{\\rm gas}$ , (2) the average internal FUV radiation field, $G_{\\rm UV}$ , and (3) the average internal hard X-ray radiation field, $F_{\\rm X}$ .",
"In this subsection we describe how these three properties are estimated and compare with observations in the case of $Z_{\\rm gas}$ .",
"Figure: Top-panel: Gas metallicity vs. dust extincted BB-band absolute magnitude ofgalaxies at z=0z=0 in the L11 model.The solid line and the dark shaded area show the median andthe 10 to 90 percentile range of model galaxies in the L11 model withan equivalent width of the Hβ\\beta line EW (Hβ)>1.5\\rm EW(H\\beta )>1.5Å.The lighter colourshaded region shows the 10 to 90 percentile range of all model galaxies withSFR>0>0.The best fit and standard deviation of the observed gas metallicity-luminosity relationsare shown as dot-dashed lines witherrorbars (; ; ).Circles and errorbars show the median and 2σ2\\sigma range on theobservational estimates of , using their correction to convert thegg-band luminosity-metallicity relation to the B-B-band.Note that the observational datacorrespond to ametallicity inferred from emission lines coming from the central parts of galaxies(i.e.",
"emission within the fiber, which is of a diameter2 arcsec in the case of 2dFGRS in , 3 arcsec in the case ofSDSS in , and 2 and 3.53.5 arcsecs in the smaller surveys of and , respectively).We also show the median relation shifted by -0.3-0.3 dex (dashed line) to illustratethe possible systematic error in the strong line method (see text for details).Bottom-panel: Gas metallicity vs. IR luminosityfor the L11 model.",
"Here we include all model galaxies.For reference, the solar metallicity value reported by is shownas a horizontal dotted line in both panels.",
"$Z_{\\rm gas}$ .",
"In GALFORM, $Z_{\\rm gas}$ corresponds to the total mass fraction in metals in the ISM, and is calculated by assuming instantaneous recycling.",
"$Z_{\\rm gas}$ is the result of the non-linear interplay between the existing metal content in the ISM, the metal content of the incoming cooled gas, which originated in the hot halo, and the metals ejected by dying stars [28].",
"The top-panel of Fig.",
"REF shows the gas metallicity as a function of the $B-$ band luminosity for galaxies in the model compared to the observational results of [30], [81], [68] and [113].",
"A correction factor needs to be applied to the observations to convert from the inferred abundance of oxygen relative to hydrogen, $\\rm O/H$ , to $Z_{\\rm gas}$ .",
"We use the solar metallicity ratios reported by [2], $\\rm O/H_{\\odot }=4.57\\times 10^{-4}$ and $Z_{\\odot }=0.0122$ .",
"This choice of the value of solar abundance is to keep consistency with the solar abundance assumed in the UCL$_{-}$ PDR model.",
"In the case of the [113] data, we applied the conversion suggested by these authors to derive a $B$ -band luminosity-metallicity relation from their $g$ -band relation.",
"The luminosity-metallicity relations estimated by [113] and [68] used large area redshift surveys (the Sloan Digital Sky Survey and the 2 Degree Field Galaxy Redshift survey, respectively).",
"In the case of [30] and [81], results were based on smaller samples of star-forming galaxies which were followed up in spectroscopy.",
"The [30] $B$ -band luminosity-metallicity relation was derived from a sample of UV-selected galaxies, which includes a few higher redshift galaxies with $z>0.15$ .",
"The observational results shown in Fig.",
"REF use abundance indicators based on emission lines to calculate oxygen abundances.",
"To estimate oxygen abundances, [30], [81] and [68] use an empirical relation between oxygen abundance and the $R_{23}$ abundance ratio, where $R_{23}=\\rm ([OII]+[OIII])/H\\beta $ , which is often called the `strong-line' technique.",
"[113], on the other hand, fit all prominent emission lines with a model designed to describe integrated galaxy spectra, which includes HII regions and diffuse ionized gas.",
"[59] compared the `strong-line' technique with abundances inferred from electron temperature measurements in a sample of HII regions with very high resolution spectroscopy, and argued that the empirical `strong-line' method systematically results in larger abundances by approximately a factor of 2 due to uncertainties in the nebular models used in calibration.",
"In order to illustrate this possible systematic error, we show as a dashed line the relation of [81] data shifted by $-0.3$ dex.",
"The L11 model predicts a lower normalization of the luminosity-metallicity relation than implied by observations, but with a similar slope.",
"When selecting star-forming galaxies in the model by their H$\\beta $ equivalent width, this discrepancy increases due to the tighter gas metallicity-luminosity relation predicted for these galaxies.",
"The correction suggested by [59] to remove the systematic introduced by the `strong-line' technique reduces the discrepancy between the observed and predicted median relations to a factor 2, well within the typical dispersion of observational data (see errorbars for [113] in Fig.",
"REF ).",
"Another caveat in the comparison between observations and the model predictions is the fact that the observational data are inferred from the emission lying within a spectrograph fiber (typically of 3 arcsecs or less in diameter), which typically covers only the central parts of the galaxy, and therefore are not global mass-weighted metallicities.",
"Galaxies show metallicity gradients, with the central parts being more metal rich than the outskirts (e.g.",
"[92]; [37]).",
"The differences in metallicity between centers and the outer regions of galaxies can be as large as a factor $2-3$ in early-type galaxies and $3-10$ in late-type galaxies [54].",
"Thus, one would expect metallicities inferred from the fibers to be upper limits for the mass-weighted ISM metallicity.",
"Given this caveat, the model predictions are in reasonable agreement with the observations.",
"Throughout the paper we make extensive comparisons between the CO and total IR luminosity (see Appendix for a description of how we calculate the IR luminosity in GALFORM).",
"We plot in the bottom-panel of Fig.",
"REF , the gas metallicity as a function of the IR luminosity.",
"For $L_{\\rm IR}<10^{10}\\, L_{\\odot }$ , the gas metallicity increases with $L_{\\rm IR}$ , but tends to a constant value at higher luminosities.",
"As we show later in the text, the dependence of the CO emission on IR luminosity is influenced by the gas metallicity ($§3$ and $§4$ ), as it alters both luminosities.",
"The bottom panel of Fig.",
"REF will therefore help in the interpretation of the results later in this paper.",
"$G_{\\rm UV}$ .",
"The average internal UV radiation field, $G_{\\rm UV}$ , corresponds to a local radiation field that depends on the transmission of UV photons from star-forming regions and their propagation through the diffuse ISM.",
"The exact transmitted fraction of UV radiation depends on the local conditions in the ISM, such as the optical depth, the ratio of gas in the diffuse ISM and in GMCs, etc (see [64]).",
"Whilst $G_{\\rm UV}$ is a local property, we make a rough estimate by considering two simple approximations which are based on global galaxy properties.",
"The first scaling is motivated by the close relation between UV luminosity and SFR [64], so that in an optically thin slab, the average UV flux scales approximately as $\\langle I_{\\rm UV}\\rangle \\propto \\Sigma _{\\rm UV}\\propto \\Sigma _{\\rm SFR}$ .",
"This is expected if the UV radiation field in the wavelength range considered here ($\\lambda =900-2100$ Å) is dominated by radiation from OB stars.",
"We therefore assume that $G_{\\rm UV}$ is related to the surface density of SFR by $\\frac{G_{\\rm UV}}{G_{0}}=\\left(\\frac{\\Sigma _{\\rm SFR}}{\\Sigma ^0_{\\rm SFR}}\\right)^{\\gamma }.$ Here we take $\\Sigma _{\\rm SFR}=\\rm SFR/2\\, \\pi \\rm r^2_{50}$ , where $\\rm r_{50}$ corresponds to the half-mass radius, either of the disk or the bulge, depending on where the SF is taking place (in the disk for quiescent SF and in the bulge for starbursts).",
"We set $\\gamma =1$ so that $G_{\\rm UV}$ increases by the same factor as $\\Sigma _{\\rm SFR}$ .",
"However, values of $\\gamma =0.5-2$ do not change the predictions of the model significantly.",
"For the normalisation, we choose $\\Sigma ^0_{\\rm SFR}=10^{-3}\\, M_{\\odot }\\, \\rm yr^{-1}\\, kpc^{-2}$ , so that $G_{\\rm UV}=G_{0}=1.6\\times 10^{-3}\\, \\rm erg\\, cm^{-2}\\, s^{-1}$ for the solar neighborhood [18].",
"A dependence of $G_{\\rm UV}$ solely on $\\Sigma _{\\rm SFR}$ assumes that an increase in the local UV radiation field takes place if a galaxy forms stars at a higher rate per unit area, but does not take into account the transparency of the gas.",
"To attempt to account for this, we consider an alternative scaling in which we include in a simple way the average optical depth of the ISM in the description of $G_{\\rm UV}$ .",
"In a slab, the transmission probability of UV photons, $\\beta _{\\rm UV}$ , scales with the optical depth, $\\tau _{\\rm UV}$ , so that $\\beta _{\\rm UV}\\sim (1-e^{-\\tau _{\\rm UV}})/\\tau _{\\rm UV}$ .",
"The optical depth, on the other hand, depends on the gas metallicity and column density of atoms as $\\tau _{\\rm UV}\\propto Z_{\\rm gas}\\, N_{\\rm H}$ .",
"In optically thick gas ($\\tau _{\\rm UV}\\gg 1$ ), $\\beta _{\\rm UV}\\sim \\tau ^{-1}_{\\rm UV}$ .",
"By assuming that the average local UV field depends on the average emitted UV field ($\\langle I_{\\rm UV}\\rangle \\propto \\Sigma _{\\rm SFR}$ ) times an average UV transmission factor, we get the scaling $\\frac{G_{\\rm UV}}{G_{0}}=\\left(\\frac{\\Sigma _{\\rm SFR}/\\Sigma ^0_{\\rm SFR}}{[Z_{\\rm gas}/Z_{\\odot }]\\, [\\Sigma _{\\rm gas}/\\Sigma ^0_{\\rm gas}]}\\right)^{\\gamma {\\prime }}.$ We set $\\gamma {\\prime }=1$ , but as with Eq.",
"REF , varying the exponent in the range $\\gamma {\\prime }=0.5-2$ has little impact on the model predictions (see $§4$ ).",
"We use the solar neighborhood value, $\\Sigma ^0_{\\rm gas}=10\\, M_{\\odot }\\, \\rm pc^{-2}$ [27].",
"The parametrization of Eq.",
"REF has been shown to explain the higher $G_{\\rm UV}$ in the Small Magellanic Cloud compared to the Milky-Way, which is needed to explain the low molecular-to-atomic hydrogen ratios [17].",
"We test these two parametrizations of $G_{\\rm UV}$ against broader observations in $§3$ and $§4$ .",
"$F_{\\rm X}$ .",
"In GALFORM we model the growth and emission by supermassive black holes (SMBHs) which drive AGN in galaxies [41].",
"[42] estimate the emission from accreting SMBHs over a wide wavelength range, from hard X-rays to radio wavelengths.",
"The SMBH modelling of Fanidakis et al.",
"includes an estimate of the efficiency of energy production by accretion onto the black hole, taking into account the value of the black hole spin, which is followed through all the gas accretion episodes and mergers with other black holes.",
"[42] show that the model can successfully explain the LF of AGN and quasars and its time evolution at different wavelengths.",
"In this work we use this SMBH modelling to take into account the heating of the ISM by the presence of an AGN in the galaxy, which has been shown to be important in the hard X-ray energy window [79].",
"The emission of AGN in hard X-rays ($2-10$ KeV), $L_{\\rm X}$ , is calculated in Fanidakis et al.",
"using the bolometric luminosity of the AGN and the bolometric corrections presented in [78].",
"We estimate the average hard X-ray flux, $F_{\\rm X}$ , at the half-mass radius of the bulge, $F_{\\rm X}=\\frac{L_{\\rm X}}{4\\pi \\, r^2_{\\rm 50}}.$"
],
[
"The ",
"The UCL$_{-}$ PDR code attempts to fully describe the chemical and thermal evolution of molecular clouds under different conditions in the surrounding ISM as quantified by: the far-UV (FUV) radiation background, the cosmic ray background, the volume number density of hydrogen, the average dust optical depth and the gas metallicity (see Bell et al.",
"2006; 2007 for a detailed description).",
"More recently, [5] and [7] explored these parameters for the ISM in external galaxies.",
"We use the code released by [7], in which additional cooling mechanisms were included, such as $^{13}\\rm C^{16}O$ , $\\rm ^{12}C^{18}O$ , CS and OH.",
"[7] showed that these coolants are important when dealing with galaxies which are forming stars at high rates.",
"The UCL$_{-}$ PDR code is a time-dependent model which treats a cloud as a one-dimensional, semi-infinite slab illuminated from one side by FUV photons.",
"Molecular gas described by the PDR code correspond to clouds having a single gas phase, with a single gas volume density, although gradients in temperature and chemical composition depend on the optical depth.",
"The radiative transfer equations are solved and the thermal balance between heating and cooling mechanisms is calculated leading to the determination of the gradients of kinetic temperature, chemical composition and emission line strength across the slab (i.e.",
"as a function of optical extinction in the visible, $A_{\\rm V}$ ).",
"The gas kinetic temperature at which this balance is achieved will be referred to throughout the paper as the typical kinetic temperature of molecular clouds in galaxies in the model, $T_{\\rm K}$ .",
"The starting point in the model is to assume that hydrogen is mostly molecular and that other species are atomic.",
"The model follows the relative abundance of 131 species, including atoms and molecules, using a network of more than $1,700$ chemical reactions (see [6]; [7]).",
"The initial element abundances, dust-to-gas ratio and H$_2$ formation rate are assumed to scale linearly with the metallicity of the gas.",
"The physical mechanisms included in the UCL$_{-}$ PDR code are (i) H$_2$ formation on dust grain surfaces, (ii) H$_2$ photodissociation by FUV radiation (which we define as the integrated emission for the wavelength range $\\lambda =900-2100$ Å), (iii) H$_2$ UV fluorescence, (iv) the photoelectric effect from silicate grains and polycyclic aromatic hydrocarbons (PAHs), (v) C II recombination and (vi) interaction of low-energy cosmic-rays (CRs) with the gas, which boosts the temperature of the gas.",
"The latter results in stronger CO emission from high order rotational transitions that resembles the observed CO emission from galaxies which host AGN.",
"The CO spectral line energy distribution (commonly referred to as the CO ladder or SLED) therefore can be obtained for a wide range of parameters included in the UCL$_{-}$ PDR code.",
"Given that the ISM of galaxies is not resolved in GALFORM, we assume the following fiducial properties for GMCs.",
"We adopted a gas density of $n_{\\rm H}=10^4\\, \\rm cm^{-3}$ , where each model was run for $10^{6}~\\rm yr$ .",
"Note that $n_{\\rm H}$ represents the total number of hydrogen nuclei.",
"The value of $n_{\\rm H}$ adopted is similar to the assumption used previously in GALFORM for GMCs (i.e.",
"$n_{\\rm H}=7\\times 10^3\\, \\rm cm^{-3}$ ; [51]; [64]), which in turn is motivated by the assumptions used in the GRASIL code [105], which calculates the reprocessing of stellar radiation by dust.",
"The parameters above correspond to star-forming gas which is likely to be opaque to radiation.",
"Note that this dense gas phase typically has $A_{V}$ in the range $3-8$ mag.",
"We choose to focus on dense gas of $A_{V}=8$ mag to obtain a $X_{\\rm CO(1-0)}$ for the local neighbourhood properties consistent with observational results.",
"We expect this approximation to be accurate particularly in gas-rich galaxies, which to first order have a larger proportion of gas in this dense phase with respect to the total gas reservoir compared to more passive galaxies.",
"This is simply because of the energetics and the dynamics involved in highly star-forming regions, which typically increase both density and temperature, leading to a more dense, opaque and fragmented medium (see Bayet 2008, 2009 for more details).",
"Note, however, that the assumption of a lower $A_{V}$ for lower gas surface density galaxies, $\\Sigma _{\\rm gas}<10^7 M_{\\odot }\\, \\rm kpc^{-2}$ , would not affect the results shown in this paper, for example simply moving galaxies along the fainter part of the CO luminosity function, without modifying its bright-end.",
"[122] suggested that a minimum density $n_{\\rm H,min} \\propto G_{\\rm {UV}}$ is necessary to obtain pressure balance between the warm and cold neutral media in the ISM.",
"All the models shown in Table REF fulfill this condition, with $n_{\\rm H}>n_{\\rm H,min}$ .",
"However, we test the effect of assuming that $n_{\\rm H}$ scales with the minimum density of [122], which leads to $n_{\\rm H} \\propto G_{\\rm {UV}}$ .",
"With this in mind, we ran four more models with $n_{\\rm H}$ varying in such a way that the $n_{\\rm H}/G_{\\rm {UV}}$ ratio is left invariant, in addition to the PDR models run using $n_{\\rm H}=10^4\\, \\rm cm^{-3}$ .",
"We analyse the CO luminosities prediced by this set of models in $§5$ .",
"The output of the UCL$_{-}$ PDR code includes the conversion factor, $X_{\\rm CO(J \\rightarrow J-1)}$ , between the intensity of a particular CO rotational transition and the column number density of H$_2$ molecules, $X_{\\rm CO(J \\rightarrow J-1)}=\\frac{N_{\\rm H_2}}{I_{\\rm CO(J\\rightarrow J-1)}},$ where $N_{\\rm H_2}$ is the H$_2$ column density and $I_{\\rm CO}$ is the integrated CO line intensity (see Appendix ).",
"This conversion factor, which is the one we are interested in here, depends on the conditions in the ISM.",
"Table: Conversion factors from CO(1-0)(1-0)-H 2 _2 (6), CO(3-2)(3-2)-H 2 _2 (7) and CO(7-6)(7-6)-H 2 _2 (8) in unitsof 10 20 cm -2 (K km s -1 ) -1 10^{20}\\, \\rm cm^{-2}\\, (K\\,km\\,s^{-1})^{-1}, and the kinetictemperature of the gas (9)for galaxieswith different ISM conditions: (1) FUV radiation background, G UV G_{\\rm UV}, in units of G 0 =1.6×10 -3 erg cm -2 s -1 G_{0}= 1.6\\times 10^{-3} \\,\\rm erg\\, cm^{-2} s^{-1}, (2) gas metallicity, Z gas Z_{\\rm gas}, in units of Z ⊙ =0.0122Z_{\\odot }=0.0122,(3) hard X-ray flux, F X F_{\\rm X},in units of erg s -1 cm -2 \\rm erg\\, s^{-1}\\, cm^{-2} and (4)total number of hydrogen nuclei, n H n_{\\rm H}, in units of cm -3 {\\rm cm}^{-3} (; ).We present, for the first time, the conversion factors for different transitions predicted by the UCL$_{-}$ PDR model.",
"The output is listed in Table REF for 41 different combinations of input properties of the ISM, where 37 models use $n_{\\rm H}=10^4\\, {\\rm cm}^{-3}$ , and 4 have variable $n_{\\rm H}$ , chosen so that $n_{\\rm H}/G_{\\rm UV}$ is constant.",
"These values are intended to span the range of possibilities in the galaxy population as a whole, ranging from low metallicity dwarf galaxies to metal rich starbursts.",
"These models consider UV radiation field strengths of 1, 10, 100 and 1000 times the value in our local neighbourhood ($G_{0}=1.6\\times 10^{-3} \\,\\rm erg\\, cm^{-2} s^{-1}$ ), gas metallicities ranging from $Z_{\\rm gas}=0.01-2\\, Z_{\\odot }$ and a flux (in the hard X-rays window) of $0.01$ , $0.1$ and $1\\, \\rm erg\\, s^{-1}\\, cm^{-2}$ (where X-rays are used as a proxy for cosmic rays; see [90]; [80]; [7]).",
"The UCL$_{-}$ PDR model inputs the cosmic rays ionization rate instead of hard X-rays flux.",
"We assume a direct proportionality between the cosmic ray ionization rate and hard X-rays flux following the studies of [80] and [7], where $F_{\\rm X}/F_{0}=\\zeta _{\\rm CR}/\\zeta _{0}$ , with $F_{0}=0.01\\, \\rm erg\\, s^{-1}\\, cm^{-2}$ and $\\zeta _{0}=5\\times 10^{-17}\\, \\rm s^{-1}$ .",
"We only list the CO-H$_2$ conversion parameters of three CO transitions in Table REF .",
"However, the UCL$_{-}$ PDR model was run to output all CO transitions from $1-0$ to $10-9$ , which we use to construct CO SLEDs in $§4$ and $§6$ .",
"A comprehensive analysis and results of the PDR model listed in Table REF will be presented in Bayet et al.",
"(2012, in prep.).",
"From Table REF it is possible to see that the general dependence of $X_{\\rm CO}$ on the three properties $Z_{\\rm gas}$ , $G_{\\rm UV}$ and $F_{\\rm X}$ , depends on the transition considered.",
"For instance, $X_{\\rm CO(1-0)}$ increases as the gas metallicity decreases, but its dependence on $G_{\\rm UV}$ and $F_{\\rm X}$ depends on the gas metallicity: for solar or supersolar metallicities, $X_{\\rm CO(1-0)}$ tends to decrease with increasing $G_{\\rm UV}$ and $F_{\\rm X}$ , given that the higher temperatures increase the CO$(1-0)$ emission.",
"However, for very subsolar gas metallicities, these trends are the opposite: $X_{\\rm CO(1-0)}$ tends to increase with increasing $G_{\\rm UV}$ and $F_{\\rm X}$ .",
"In this case this is due to the effect of the high radiation fields and the lack of an effective CO self-shielding, which destroys CO molecules.",
"In the case of higher CO transitions, for example CO$(7-6)$ , $X_{\\rm CO(7-6)}$ generally increases with decreasing kinetic temperature, however this is not strictly the case in every set of parameters.",
"These general trends will help explain the relations presented in $§3-4$ and $§6$ .",
"At very low metallicities, $Z_{\\rm g}\\approx 0.01 Z_{\\odot }$ , the CO lines become optically thin in some of the cases, e.g.",
"in those models where there is a high UV and X-ray flux illuminating the molecular clouds.",
"This represents a limitation of the PDR model given the uncertainties in the opacity effect on the CO lines.",
"However, such galaxies are extremely rare in our model after selecting galaxies with $L_{\\rm IR}>10^9\\, L_{\\odot }$ , which are those we use to study the CO SLED at redshifts $z>0$ (e.g.",
"only $0.05$ % of galaxies with $L_{\\rm IR}>10^9\\, L_{\\odot }$ at $z=6$ have $Z_{\\rm g}\\le 0.01 Z_{\\odot }$ ).",
"This is not the case for very faint IR galaxies.",
"We find that in the subsample of galaxies with $L_{\\rm IR}>10^7\\, L_{\\odot }$ , more than 10% of galaxies have $Z_{\\rm g}\\le 0.01 Z_{\\odot }$ at $z>1.5$ ."
],
[
"Coupling the ",
"We use the properties $Z_{\\rm gas}$ , $G_{\\rm UV}$ and $F_{\\rm X}$ as inputs to the UCL$_{-}$ PDR model.",
"For each galaxy, we calculate the $X_{\\rm CO}$ conversion factors for several CO transitions, and use the molecular mass of the galaxy to estimate the CO luminosity of these transitions (see Appendix ).",
"We use the models from Table REF to find the $X_{\\rm CO}$ conversion factors and the gas kinetic temperature of molecular clouds, $T_{\\rm K}$ , for each galaxy according to its ISM properties.",
"Given that $Z_{\\rm gas}$ , $G_{\\rm UV}$ and $F_{\\rm X}$ are discretely sampled, we interpolate over the entries of Table REF on a logarithmic scale in each parameter.",
"Throughout the paper we will refer to the coupled code as the GALFORM+UCL$_{-}$ PDR model.",
"Galaxies in GALFORM can have SF taking place simultaneously in the disk and the bulge, corresponding to the quiescent and starburst SF modes, respectively.",
"The gas reservoirs of these two modes are different and we estimate the CO luminosity of the two phases independently.",
"This can be particularly important at high-redshift, where the galaxy merger rate is higher and where galaxies are more prone to have dynamically unstable disks, which can lead to starbursts.",
"For instance, [36] showed that two gas phases, a diffuse and a dense phase, are necessary to describe the CO spectral line energy distribution of the $z=2.3$ galaxy SMM J2135-0102 (see also [22]), illustrating the importance of allowing for the possibility of concurrent quiescent and burst episodes of SF in the modelling of CO emission."
],
[
"The dependence of the CO-H$_2$ conversion factor on galaxy properties in ",
"In order to illustrate how much the $X_{\\rm CO}$ conversion factor varies with galaxy properties, in this subsection we focus on the predictions of the CO luminosity-to-molecular mass ratio in the GALFORM+UCL$_{-}$ PDR model in the case where $G_{\\rm UV}$ depends on $Z_{\\rm gas}$ , $\\Sigma _{\\rm SFR}$ and $\\Sigma _{\\rm gas}$ (see Eq.",
"REF ).",
"Fig.",
"REF shows the $L_{\\rm CO}/M_{\\rm H_2}$ ratio as a function of the average ISM gas surface density, $\\Sigma _{\\rm gas}$ , at three different redshifts, and for three different CO transitions.",
"In the case of CO$(1-0)$ , we show for reference the standard conversion factors typically adopted in the literature for starburst, normal spiral and dwarf galaxies as horizontal lines.",
"At $z=0$ there is, on average, a positive correlation between $L_{\\rm CO}/M_{\\rm H_2}$ and $\\Sigma _{\\rm gas}$ , with $L_{\\rm CO}/M_{\\rm H_2} \\propto \\Sigma ^{0.15}_{\\rm gas}$ , regardless of the CO transition.",
"At higher redshifts, the relation between the $L_{\\rm CO}/M_{\\rm H_2}$ ratio and $\\Sigma _{\\rm gas}$ flattens, mainly due to the lower gas metallicities of galaxies.",
"These trends are similar for all the CO transitions.",
"In the case of the CO$(1-0)$ at $z=0$ , galaxies of low $\\Sigma _{\\rm gas}$ have $X_{\\rm CO}$ closer to the value measured in dwarf galaxies moving to values closer to starburst galaxies at very high $\\Sigma _{\\rm gas}$ .",
"In terms of stellar mass, galaxies with $M_{\\rm stellar}\\approx 7\\times 10^{10} M_{\\odot }$ , close to the Milky-Way stellar mass, have on average $X_{\\rm CO(1-0)}\\approx 2-3\\times 10^{20}\\, \\rm cm^{-2}\\, (K\\,km\\,s^{-1})^{-1}$ , in agreement with the measurements of the solar neighbourhood.",
"For relatively massive galaxies, the model predicts that the $L_{\\rm CO}/M_{\\rm H_2}$ ratio evolves only weakly with redshift at $z\\lesssim 2$ , which explains the similarity between the $X_{\\rm CO(1-0)}$ measured by [32] in normal star-forming galaxies at $z\\approx 1.5$ and the value for local spiral galaxies.",
"The coupled GALFORM+UCL$_{-}$ PDR model thus predicts a dependence of $X_{\\rm CO}$ on galaxy properties which broadly agrees with observations in the local Universe and explains the few observations of high-redshift galaxies."
],
[
"The CO(1-0) emission of galaxies in the local universe",
"In the local Universe, the CO$(1-0)$ emission of galaxies has been studied extensively in different environments with large samples of galaxies (e.g.",
"[60]; [107] [21]; [101]; [74]).",
"In this section we compare our predictions for the CO$(1-0)$ emission of galaxies at $z=0$ and how this relates to their IR luminosity, with available observations."
],
[
"The CO(1-0) luminosity function",
"In this subsection we focus on the CO$(1-0)$ LF and how the predictions depend on the assumptions and the physics of the model.",
"Figure: The z=0z=0 CO (1-0)\\rm CO(1-0) luminosity function predicted by GALFORM+UCL - _{-}PDR model.Observational estimates of for samples of galaxies selected in the BB-band (triangles) and at 60μ60\\,\\mu m (filled circles),are also shown.",
"The predictions of the models are shown forthe GALFORM+UCL - _{-}PDR model when (i) G UV =G UV (Σ SFR )G_{\\rm UV}=G_{\\rm UV}(\\Sigma _{\\rm SFR}) (Eq.",
";dashed line),(ii) G UV =G UV (Σ SFR ,Z gas ,Σ gas )G_{\\rm UV}=G_{\\rm UV}(\\Sigma _{\\rm SFR}, Z_{\\rm gas}, \\Sigma _{\\rm gas}) (Eq.",
"; solid line),and (iii) using G UV G_{\\rm UV} as in (ii) but assuming AGN do not contribute to heat the ISM (dot-dashed line).For reference, we also show the predictions of the GALFORM model withoutthe PDR coupling, assumingtwo constant X CO X_{\\rm CO} factors,X CO (1-0) =(2,0.8)×10 20 cm -2 (K km s -1 ) -1 \\rm X_{\\rm CO(1-0)}=(2,0.8)\\times 10^{20}\\, \\rm cm^{-2}\\,(K\\,km\\,s^{-1})^{-1}for quiescent and starburst galaxies, respectively (dotted line).Fig.",
"REF shows the $\\rm CO(1-0)$ LF at $z=0$ for the hybrid GALFORM+UCL$_{-}$ PDR model, for the two parametrizations of $G_{\\rm UV}$ : (i) $G_{\\rm UV}(\\Sigma _{\\rm SFR})$ (Eq.",
"REF ) and (ii) $G_{\\rm UV}(\\Sigma _{\\rm SFR}, Z_{\\rm gas}, \\Sigma _{\\rm gas})$ (Eq.",
"REF ).",
"We also show the latter model, (ii), without the inclusion of AGN as an ISM heating source.",
"For reference, we also show the predictions of the GALFORM model without the processing of the PDR model, in the simplistic case where we assume two constant conversion factors, $X_{\\rm CO(1-0)}=2\\times 10^{20}\\, \\rm cm^{-2}\\, (K\\,km\\,s^{-1})^{-1}$ for quiescent galaxies and $X_{\\rm CO(1-0)}=0.8\\times 10^{20}\\, \\rm cm^{-2}\\, (K\\,km\\,s^{-1})^{-1}$ for starbursts.",
"Observational estimates of the $\\rm CO(1-0)$ LF made using both, $B$ -band and $60\\,\\mu $ m selected samples, are plotted as symbols [60].",
"Differences between the predictions of the model using different assumptions about $G_{\\rm UV}$ become evident at CO luminosities brighter than the break in the CO LF (i.e.",
"$\\rm log(\\it L_{\\rm CO}/\\rm Jy\\, km/s\\, Mpc^{2})\\approx 6.7$ ).",
"The model assuming a dependence of $G_{\\rm UV}$ solely on $\\Sigma _{\\rm SFR}$ predicts a larger number density of bright galaxies due to the higher $G_{\\rm UV}$ values in galaxies with large molecular mass and high SFRs.",
"We show later that the kinetic gas temperatures of the GALFORM+UCL$_{-}$ PDR model, when assuming a dependence of $G_{\\rm UV}$ solely on $\\Sigma _{\\rm SFR}$ , are very high and also translate into unrealistic emission from higher order CO transitions.",
"The values of $G_{\\rm UV}$ are smaller when including the dependence on the optical depth, $\\tau _{\\rm UV}$ , given that the increase in $\\Sigma _{\\rm SFR}$ is compensated by an increase in $\\tau _{\\rm UV}$ , which brings $G_{\\rm UV}$ down.",
"This model predicts a LF which is closer to and in reasonable agreement with the observations.",
"When AGN are not included as a heating mechanism, the model predictions for the CO$(1-0)$ LF are not affected, indicating that lower CO transitions are not sensitive to the presence of AGN.",
"However, as we show later (Fig.",
"REF ), the emission in high CO transitions is very sensitive to the presence of an AGN.",
"The GALFORM model without the PDR (i.e.",
"using two ad hoc constant values of $X_{\\rm CO}$ for starburst and quiescent galaxies) gives a LF closer to the observed number density of bright galaxies.",
"This happens because galaxies in the bright-end of the CO LF mainly correspond to quiescent, gas-rich galaxies, whose $G_{\\rm UV}>G_{0}$ , driving lower $X_{\\rm CO}$ in the GALFORM+UCL$_{-}$ PDR model compared to the value typically assumed for quiescent galaxies ($X_{\\rm CO}=2\\times 10^{20}\\, \\rm cm^{-2}\\, (K\\,km\\,s^{-1})^{-1}$ ).",
"The predictions of the GALFORM+UCL$_{-}$ PDR model using the form $G_{\\rm UV}(\\Sigma _{\\rm SFR}, Z_{\\rm gas}, \\Sigma _{\\rm gas})$ (Eq REF ), give a reasonable match to the observational data from [60].",
"We remind the reader that the model has not been tuned to reproduce the CO LF.",
"However, it is important to bear in mind that the CO LF from [60] is not based on a blind CO survey, but instead on galaxy samples selected using $60\\,\\mu \\rm m$ or $B$ -band fluxes.",
"These criteria might bias the LF towards galaxies with large amounts of dust or large recent SF."
],
[
"The CO-to-Infrared luminosity ratio",
"Fig.",
"REF shows the $L_{\\rm CO(1-0)}/L_{\\rm IR}$ ratio as a function of $L_{\\rm IR}$ .",
"Lines show the median of the GALFORM+UCL$_{-}$ PDR model predictions when using $G_{\\rm UV}(\\Sigma _{\\rm SFR})$ (Eq.",
"REF ; dashed line) and $G_{\\rm UV}(\\Sigma _{\\rm SFR},Z_{\\rm gas},\\Sigma _{\\rm gas})$ (Eq.",
"REF ; solid line).",
"Errorbars show the 10 and 90 percentiles of the distributions.",
"We also show, for reference, the molecular mass-to-IR luminosity ratio, $M_{\\rm mol}/L_{\\rm IR}$ , predicted by the L11 model and shifted by an arbitrary factor of 7 dex (dotted line).",
"Symbols show an observational compilation of local LIRGs and ULIRGs.",
"The model predicts that the $L_{\\rm CO(1-0)}/L_{\\rm IR}$ ratio decreases as the IR luminosity increases.",
"This trend is primarily driven by the dependence of the molecular mass-to-IR luminosity ratio on the IR luminosity which has the same form (dotted line).",
"The trend of decreasing $M_{\\rm mol}/L_{\\rm IR}$ with increasing IR luminosity is driven by the gas metallicity-IR luminosity relation.",
"Gas metallicity declines as the IR luminosity decreases (bottom panel of Fig.",
"REF ), which results in lower dust-to-total gas mass ratios and therefore lower IR luminosities for a given SFR.",
"The molecular mass is not affected by gas metallicity directly since it depends on the hydrostatic pressure of the disk (see $§2.1$ ).",
"Note that this effect affects galaxies with $L_{\\rm IR}<5\\times 10^{10}\\, L_{\\odot }$ , given that the gas metallicity-IR luminosity relation flattens above this IR luminosity (bottom panel of Fig.",
"REF ).",
"The distributions of $L_{\\rm CO(1-0)}/L_{\\rm IR}$ predicted by the model extend to very low $L_{\\rm CO(1-0)}/L_{\\rm IR}$ ratios (as shown by the errorbars in Fig.",
"REF ).",
"This is due to satellite galaxies in groups and clusters, which tend to have lower molecular mass-to-IR luminosity ratios, but that are relatively bright in IR due to their high gas metallicities (solar or supersolar), and therefore, large dust-to-gas mass ratios.",
"At $5\\times 10^{10}\\, L_{\\odot }<L_{\\rm IR}<5\\times 10^{11}\\, L_{\\odot }$ , the $L_{\\rm CO(1-0)}/L_{\\rm IR}$ -$L_{\\rm IR}$ relation tends to flatten.",
"This is due to a transition from galaxies dominated by quiescent SF to starburst galaxies, and the two different SF laws assumed in the model (see $§2.1$ ).",
"The SF law determines how fast the cold gas is converted into stars, thus playing a key role in determining the molecular reservoir at a given time.",
"In starburst galaxies, the SF timescale depends on the dynamical timescale of the bulge component with a floor (Eq.",
"REF ).",
"Starburst galaxies, which largely contribute to the number density at $L_{\\rm IR}>5\\times 10^{10}\\, L_{\\odot }$ , have similar SF timescales given their similar properties in stellar mass and size, therefore resulting in, similar molecular-to-SFR ratios, except for the brightest ones with $L_{\\rm IR}> 5\\times 10^{11}\\, L_{\\odot }$ .",
"This effects dominates the behaviour of the $L_{\\rm CO(1-0)}/L_{\\rm IR}$ ratio, with a second order contribution from variations in $X_{\\rm CO}$ , which tend to be small at these high gas surface densities (see Fig.",
"REF ).",
"This prediction of the model explains what has been observed in local and a few high-redshift galaxies: variations in the CO-to-IR luminosity ratio are of the same order as the variations of molecular mass-to-IR luminosity ratios, as inferred from the dust emission ([71]; [76]).",
"For the brightest galaxies, $L_{\\rm IR}> 5\\times 10^{11}\\, L_{\\odot }$ , the SF timescale decreases rapidly with increasing IR luminosity and consequently, the molecular mass-to-IR luminosity ratio also decreases.",
"We conclude that the GALFORM+UCL$_{-}$ PDR model is able to explain the observed CO$(1-0)$ emission of galaxies in the local Universe and its relation to the IR luminosity."
],
[
"The CO emission of galaxies in multiple transitions",
"We now focus on the predictions of the GALFORM+UCL$_{-}$ PDR model for the CO emission of galaxies in multiple CO lines in the local and high-redshift universe.",
"We focus on the CO LF of galaxies, the relation between the CO and IR luminosity, and the CO SLED.",
"In contrast to the case of the CO$(1-0)$ transition, the available observational data for higher CO transitions are scarce and limited to individual objects, instead of homogeneous samples of galaxies.",
"To carry out the fairest comparison possible at present, we select model galaxies in order to sample similar IR luminosity distributions to those in the observational catalogues."
],
[
"The luminosity function of multiple CO lines",
"The top row of Fig.",
"REF shows the $\\rm CO(1-0)$ , $\\rm CO(3-2)$ and $\\rm CO(7-6)$ LFs at $z=0$ for the GALFORM+UCL$_{-}$ PDR model using the scalings of $G_{\\rm UV}(\\Sigma _{\\rm SFR})$ (Eq.",
"REF ) and $G_{\\rm UV}(\\Sigma _{\\rm SFR},Z_{\\rm gas},\\Sigma _{\\rm gas})$ (Eq.",
"REF ).",
"The contributions from X-ray bright AGNs, with $L_{\\rm X}>10^{44}\\, \\rm s^{-1}\\, erg$ , and from quiescent and starburst galaxies with $L_{\\rm X}<10^{44}\\, \\rm s^{-1}\\, erg$ , are shown separately only for the model using $G_{\\rm UV}(\\Sigma _{\\rm SFR},Z_{\\rm gas},\\Sigma _{\\rm gas})$ .",
"Note that X-ray bright AGNs can correspond to both quiescent and starburst galaxies.",
"The observational results for the $\\rm CO(1-0)$ LF from [60] at $z=0$ and from [1] and [32] at $z=2$ are also plotted in the top and bottom left-hand panels, respectively.",
"The model using $G_{\\rm UV}(\\Sigma _{\\rm SFR})$ predicts a higher number density of bright galaxies for the three CO transitions shown in Fig.",
"REF due to the fact that with this assumption, galaxies typically have a higher value of $G_{\\rm UV}$ than in the parametrisation of Eq.",
"REF , which leads to lower values of $X_{\\rm CO}$ .",
"The offset in the bright-end between the model predictions when using $G_{\\rm UV}(\\Sigma _{\\rm SFR})$ and $G_{\\rm UV}(\\Sigma _{\\rm SFR},Z_{\\rm gas},\\Sigma _{\\rm gas})$ increases for higher CO transitions, since $J>4$ CO transitions are more sensitive to changes in kinetic temperature, and therefore in $G_{\\rm UV}$ .",
"For $G_{\\rm UV}(\\Sigma _{\\rm SFR})$ , galaxies are on average predicted to be very bright in the CO$(7-6)$ transition.",
"As we show later, this model predicts an average CO$(7-6)$ luminosity brighter than observed for local LIRGs (see Fig.",
"REF in $§4.2.1$ ), but still consistent with the observations within the errorbars.",
"Quiescent galaxies in the model are responsible for shaping the faint end of the CO LF, regardless of the CO transition.",
"Starburst galaxies make a very small contribution to the CO LFs at $z=0$ , given their low number density.",
"Galaxies which host a X-ray bright AGN are an important contributor to the bright-end of the CO LF, along with normal star-forming galaxies, regardless of the transition.",
"In the case of the CO$(1-0)$ and CO$(3-2)$ transitions, this is not due to the presence of the AGN in these galaxies, but instead to the large molecular gas reservoir, the typically high gas metallicities, and the high SFRs, which on average produce higher $G_{\\rm UV}$ , and therefore more CO luminosity per molecular mass.",
"The powerful AGN is therefore a consequence of the large gas reservoirs, which fuel large accretion rates, along with a massive central black hole, and has only a secondary effect through increasing the kinetic temperature, that is not enough to produce a visible effect on the low CO transitions.",
"However, the CO$(7-6)$ transition is slightly more sensitive to variations in the kinetic temperature of the gas, as we show in $§4.2$ .",
"The contribution from bright AGN and quiescent galaxies to the bright-end of the CO LF at $z=0$ is very similar, regardless of the CO transition.",
"This is due to the X-ray luminosity threshold chosen to select AGN bright galaxies, $L_{\\rm XR}>10^{44}\\, \\rm erg\\, s^{-1}$ , which takes out most of quiescent galaxies in the bright-end of the CO LF, which have hard X-ray luminosities in the range $10^{43}<L_{\\rm XR}/\\rm erg\\, s^{-1}<10^{44}$ .",
"The bottom row of Fig.",
"REF is the same as the top row but shows the LFs at $z=2$ .",
"In this case we only show the predictions of the GALFORM+UCL$_{-}$ PDR model in the $G_{\\rm UV}$ approximation of Eq.",
"REF .",
"To illustrate evolution between $z=0$ and $z=2$ , the dotted straight lines show the number density of galaxies at $z=0$ with a luminosity of $10^7\\, \\rm Jy\\, km/s\\, Mpc^2$ in the different CO transitions.",
"Bright CO galaxies are more common at $z=2$ , which is reflected in the higher number density of galaxies with $L_{\\rm CO,V}> 10^{7} \\rm Jy\\, km/s\\, Mpc^{2}$ compared to $z=0$ .",
"Bright AGNs, which are more common and brighter at $z=2$ than at $z=0$ , are responsible for most of the evolution in the bright-end of the CO LF with redshift, with a less important contribution from quiescent and starburst galaxies that host fainter AGN.",
"In the faint-end, there is an significant increase in the number density of galaxies, driven by the evolution of quiescent galaxies.",
"In general, the LF for higher CO transitions shows stronger evolution with redshift than it does for lower CO transitions, again indicating that the higher CO transitions are more sensitive to variations in $G_{\\rm UV}$ and $F_{\\rm X}$ .",
"From an observational point of view, measuring CO luminosity ratios, such as the CO$(7-6)$ -to-CO$(1-0)$ ratio, is promising for constraining the average physical state of the molecular gas.",
"However, in terms of estimating the total molecular mass in galaxies, lower CO transitions are more useful, given their lower sensitivity to changes in the conditions in the ISM in galaxies.",
"Our predictions for the CO$(1-0)$ at $z=2$ agree very well with the observed number density of bright CO$(1-0)$ galaxies reported by [32] and [1].",
"However, the uncertainty in the inferred space density displayed by the observations at $z=2$ is large, suggesting that further observations, desirably from CO blind surveys, are necessary to put better constraints in the CO LF.",
"Our predictions for the LF show that intermediate CO transitions are brighter in units of the velocity-integrated CO luminosity than lower and higher order CO transitions.",
"This trend is similar to the predictions of [88], who used a completely different approach, which relied on estimating a gas temperature based on the SFR surface density or AGN bolometric luminosity under local thermodynamic equilibrium (i.e.",
"a single gas phase).",
"However, Obreschkow et al.",
"predict a significant decrease in the number density of faint CO galaxies as the upper level $\\rm J$ increases, behaviour that is not seen in our model.",
"A possible explanation for this is that their model assumes local thermodynamic equilibrium, for which high order CO transitions would be thermalised.",
"This, in addition to the parameters Obreschkow et al.",
"use to estimate $L_{\\rm CO}$ , can lead to much lower CO emission in high-J transitions compared to that in our model.",
"For example at $T_{\\rm K}=10$ K, and using the equations and parameters given in Obreschkow et al., a ratio of $L_{\\rm CO(7-6),V}/L_{\\rm CO(1-0),V}\\approx 10^{-3}$ is obtained, while our model predicts $L_{\\rm CO(7-6),V}/L_{\\rm CO(1-0),V}\\approx 0.1$ for the same temperature.",
"Our approach does not require any of these assumptions given that the PDR model is designed to represent much more accurately the excitation state of GMCs.",
"In general, the GALFORM+UCL$_{-}$ PDR model predicts a higher number density of bright galaxies at high-redshifts, a trend which is slightly more pronounced for the higher CO transitions.",
"For low CO transitions, the main driver of this effect is the higher number density of galaxies with large molecular gas reservoirs at high redshift (see [66]).",
"For high CO transitions what makes the effect stronger is the higher average kinetic temperatures of the gas in molecular clouds at high redshifts (see Fig.",
"REF ).",
"Figure: CO(J→J-1)(J\\rightarrow J-1) to IR luminosity ratio as a function of upper level J at z=0z=0.Grey solid bars show the range of observed ratios reported by for 70 LIRGs at z≤0.1z\\le 0.1.",
"Horizontal segments show the median in the observed datafor two IR luminosity bins, L IR /L ⊙ <10 11 L_{\\rm IR}/L_{\\odot }<10^{11} (dark red with errorbars as dashed lines)and L IR /L ⊙ >10 11 L_{\\rm IR}/L_{\\odot }>10^{11} (blue with errorbars as solid lines).We show as symbols the predictions of a sample of model galaxies randomly chosen to have the same IRluminosity distribution as the Papadopoulos et al.",
"sample.Symbols and errorbars correspond to the median and 10 and 90 percentiles of thepredictions for the GALFORM+UCL - _{-}PDR model using theG UV G_{\\rm UV} parametrisations of Eqs.",
"(squares) and (large stars).For reference we also show for the bright IR luminosity binthe predictions of the model using G UV G_{\\rm UV} from Eq.",
"when AGN are not considered as an ISM heating mechanism (small stars).Figure: Top panel: The velocity-integrated luminosity normalized to the CO(1-0)(1-0) luminosityas a function of the upper quantum level of the CO rotational transition, JJ,predicted by the GALFORM+UCL - _{-}PDR model at z=0z=0 for galaxieswith IR luminosities in different ranges, as labelled, andusing the G UV G_{\\rm UV} parametrisation of Eqs. .",
"Solid lines and errorbars show themedians and 10 and 90 percentile ranges of the distributions.Dashed lines show individual galaxies from the observational sample, followingthe same colour code as the model galaxies.Bottom panel: Brightness temperature luminosity, L CO ' L^{\\prime }_{\\rm CO},as a function of JJ for galaxies with IR luminosities in different ranges, as labelled.",
"Note that L CO (1-0) ' L^{\\prime }_{\\rm CO(1-0)} monotonically increases with IR luminosity."
],
[
"The CO-to-IR luminosity ratio and the CO SLED",
"In this section we study the CO-to-IR luminosity ratio for multiple CO lines and compare to observational data in the local and high redshift Universe."
],
[
"The CO-to-IR luminosity ratio and CO SLED in the local universe",
"Observations have shown that emission from multiple CO transitions can help to constrain the state of the ISM in galaxies through the comparison with the predictions of PDR and LVG models.",
"In the local Universe, around 100 galaxies have been observed in more than one CO transition.",
"However, the one caveat to bear in mind is the selection of these samples, as they are built from studies of individual galaxies and are, therefore, inevitably biased towards bright galaxies.",
"To try to match the composition of the observational sample, we select galaxies from the model with the same distribution of IR luminosities as in the observed samples.",
"In this section we present this comparison, which allows us to study whether or not the model predicts galaxies that reproduce the observed CO ladder.",
"We compare the model predictions with the observational catalogue presented in [91].",
"This sample comprises 70 LIRGs and ULIRGs at $z\\le 0.1$ which have the emission of several CO transitions measured, as well as other molecular species.",
"The IR luminosities of these galaxies cover the range $10^{10}-5\\times 10^{12}\\, L_{\\odot }$ .",
"We randomly select galaxies in the model at $z=0$ to have the same distribution of IR luminosities as the sample of [91].",
"Fig.",
"REF shows the predicted CO-to-IR luminosity ratio for different transitions compared to the observational data, in two bins of IR luminosity, $10^{10}<L_{\\rm IR}/L_{\\odot }<10^{11}$ and $10^{11}<L_{\\rm IR}/L_{\\odot }<5\\times 10^{12}$ .",
"In the case of the observations, gray bands show the whole range of observed CO-to-IR luminosity ratios, while the horizontal segments show the medians of the bright (blue) and faint (dark red) IR luminosity bins.",
"In the case of the model, we show the medians and 10 and 90 percentiles of the distributions as symbols and errorbars, respectively, where dark red and blue symbols corresponds to the low and high luminosity bins, respectively.",
"Model predictions are presented for the two parametrisations of $G_{\\rm UV}$ discussed in $§3.1.1$ (see Eqs.",
"REF -REF ).",
"We also show for the bright IR luminosity bin, the CO-to-IR luminosity ratios for the model using $G_{\\rm UV}(\\Sigma _{\\rm SFR},Z_{\\rm gas},\\Sigma _{\\rm gas})$ (Eq.",
"REF ) and assuming no heating of the ISM by AGN (small stars).",
"The model predicts $L_{\\rm CO}/L_{\\rm IR}$ ratios which are well within the observed ranges.",
"For higher CO transitions, the model predicts broader distributions of the $L_{\\rm CO}/L_{\\rm IR}$ ratio than at low CO transitions, independent of the $G_{\\rm UV}$ parametrisation.",
"The model also predicts that galaxies in the bright IR luminosity bin have slightly lower $L_{\\rm CO}/L_{\\rm IR}$ ratios compared to galaxies in the faint IR luminosity bin.",
"The two parametrisations of $G_{\\rm UV}$ predict $L_{\\rm CO}/L_{\\rm IR}$ ratios that are only slightly offset, except for the highest CO transitions, $\\rm J>5$ , where the predictions differ by up to $\\approx 0.5$ dex.",
"At $\\rm J>5$ , the model in which $G_{\\rm UV}$ depends on the average UV optical depth, $G_{\\rm UV}(\\Sigma _{\\rm SFR}, Z_{\\rm gas},\\Sigma _{\\rm gas})$ , predicts on average $L_{\\rm CO}/L_{\\rm IR}$ ratios in better agreement with the observations than those predicted by the $G_{\\rm UV}$ depending solely on $\\Sigma _{\\rm SFR}$ .",
"This is due to the fact that galaxies with very high $\\Sigma _{\\rm SFR}$ , which drives high UV production, also tend to have high $Z_{\\rm gas}\\, \\Sigma _{\\rm gas}$ , decreasing the UV ionizing background if the average UV optical depth is considered.",
"For lower CO transitions, the difference between the predictions of the model when $G_{\\rm UV}$ is estimated as in Eqs.",
"REF and REF , becomes more evident for galaxies that are bright in CO, which affects the bright end of the CO luminosity function, as discussed in $§3.1$ .",
"Galaxies in the fainter IR luminosity bin correspond primarily to normal star-forming galaxies with $G_{\\rm UV}/G_{0}\\approx 1-10$ , while galaxies in the brighter IR luminosity bin are a mixture of normal star-forming and starburst galaxies.",
"The range of $G_{\\rm UV}$ in these galaxies varies significantly.",
"Starburst galaxies usually have larger $G_{\\rm UV}$ in the range $G_{\\rm UV}/G_{0}\\approx 10-10^3$ .",
"Faint IR galaxies in the model, with $L_{\\rm IR}<10^{9}\\, L_{\\odot }$ , can also correspond to passive galaxies, whose UV ionizing background is very small, $G_{\\rm UV}/G_{0}\\approx 0.01-1$ .",
"The variation in $G_{\\rm UV}$ within the faint and bright IR luminosity bins has a direct consequence on the range of gas kinetic temperatures displayed by galaxies in each bin.",
"Galaxies in the faint IR bin have $T_{\\rm K}\\approx 10-20\\, \\rm K$ , while galaxies in the bright IR bin have $T_{\\rm K}\\approx 10-60\\, \\rm K$ .",
"The presence of an AGN also has an effect on the kinetic temperature of the gas, and therefore on the CO emission of galaxies, as seen from the small stars in Fig.",
"REF .",
"When assuming that the AGN does not heat the ISM of galaxies, galaxies appear to have lower CO-to-IR ratios for $J>6$ transitions by a factor $\\approx 1.7$ , while lower transitions are largely unaffected.",
"This indicates again that high CO transitions are useful to constrain the effect of AGN in heating the ISM.",
"Fig.",
"REF shows the CO SLED in units of velocity-integrated CO luminosity, $L_{\\rm CO,V}$ (top panel), and brightness temperature luminosity, $L^{\\prime }_{\\rm CO}$ (bottom panel), for $z=0$ galaxies with IR luminosities in different luminosity bins, as labelled.",
"Individual galaxies from the [91] observational sample of LIRGs are shown as dashed lines in the top panel of Fig.",
"REF .",
"A typical way to show the CO SLED is velocity-integrated luminosity normalised by $L_{\\rm CO(1-0),V}$ , given that this way the SLED shows a peak, which indicates the degree of excitation: the higher the J of the peak, the higher the gas kinetic temperature of molecular clouds, which typically indicates more SF and/or AGN activity (e.g.",
"[119]).",
"When the CO SLED is shown in brightness temperature luminosity there is no clear peak (bottom panel of Fig.",
"REF ) .",
"This happens because $L_{\\rm CO,V}$ and $L^{\\prime }_{\\rm CO}$ have different dependencies on $J$ (see Appendix ).",
"The GALFORM+UCL$_{-}$ PDR model predicts a peak in the CO SLED at $\\rm J=4$ for galaxies with $L_{\\rm IR}\\lesssim 10^{11}\\, L_{\\odot }$ and at $\\rm J=5$ for galaxies with $L_{\\rm IR}\\gtrsim 10^{11}\\, L_{\\odot }$ , due to the starburst nature of the latter.",
"We find that the lowest IR luminosity bin, $10^{9}<L_{\\rm IR}/L_{\\odot }<1.7\\times 10^{9}$ , shows a peak at higher J values, closer to starburst galaxies.",
"This is due to the lower gas metallicities of these galaxies which increases $G_{\\rm UV}$ and $T_{\\rm K}$ .",
"Our predictions agree with the observed peaks of LIRGs (dashed lines).",
"However, we remind the reader that the LIRG catalogue of [91] is not a statistically complete sample.",
"Further observations are needed to construct volume-limited samples of galaxies with CO measurements in order to better constrain the physics of the ISM."
],
[
"Redshift evolution of the CO-to-IR luminosity ratio and the\nCO SLED at high redshift",
"At high redshifts, observational data on the CO emission from galaxies is based on studies of individual galaxies, typically LIRGs, ULIRGs, QSOs and SMGs.",
"This has allowed the characterisation of the CO-to-IR luminosity ratio for bright normal star-forming and starburst galaxies.",
"In this section we compare these observations with the predictions of the GALFORM+UCL$_{-}$ PDR model.",
"Fig.",
"REF shows the redshift evolution of the $L_{\\rm CO}/L_{\\rm IR}$ luminosity ratio for model galaxies for four different CO transitions, in different bins of IR luminosity.",
"Quiescent and starburst galaxies are shown separately in the top and bottom rows, respectively.",
"We also show a large compilation of observational results of local and high redshift normal star-forming galaxies, local LIRGs, local ULIRGs, high redshift colour selected galaxies, SMGs, and local and high redshift QSOs, and plot them in the panel corresponding to the CO transition that was studied in each case.",
"We plot observed CO-to-IR luminosity ratios in the top rows of Fig.",
"REF if observed galaxies correspond to normal star-forming galaxies or LIRGs, or in the bottom row if they are classified as starburst galaxies (ULIRGs, SMGs or QSOs).",
"We warn the reader that most of the observational data do not directly measure total IR luminosity, but infer it from either the emission in mid-IR or sub-millimeter bands, such as $24\\mu $ m or $850\\mu $ m, or an observationally inferred SFR.",
"Thus, the comparison between the model predictions and the observations in Fig.",
"REF has to be done with care.",
"There is a weak trend of lower $L_{\\rm CO}/L_{\\rm IR}$ ratios as $L_{\\rm IR}$ increases in both galaxy types as was shown at $z=0$ in Fig.",
"REF .",
"In the case of quiescent galaxies, this trend is driven by the gas metallicity: IR faint galaxies have lower metallicities, which, on average, decrease the dust opacity and the corresponding IR luminosity, producing higher $L_{\\rm CO}/L_{\\rm IR}$ ratios.",
"In the case of starburst galaxies, the main driver of the decreasing $L_{\\rm CO}/L_{\\rm IR}$ ratio with increasing $L_{\\rm IR}$ is the accompanying decrease in molecular mass for a given SFR due to the dependence of the SF law on the dynamical timescale of the bulge in starbursts.",
"This reduces the SF timescale in the most massive and brightest galaxies.",
"Observations shown in the panels corresponding to quiescent galaxies at intermediate and high redshifts show galaxies selected through different methods: [47] measured the CO$(1-0)$ emission in a 24$\\mu $ m-selected sample at $z\\approx 0.4$ of galaxies infalling into a rich galaxy cluster, [32] measured the CO$(2-1)$ emission in a colour-selected sample of galaxies (BzK; see $§6.1$ ), and [111] and [48] measured CO$(3-2)$ in a sample of normal star-forming galaxies located on the star forming sequence of the SFR$-M_{\\rm stellar}$ plane.",
"In the case of [32], [111] and [48], IR luminosities are inferred from the SFR, which in turn is estimated from the rest-frame UV and mid-IR emission with an uncertainty of a factor $\\approx 2$ .",
"The conversion between SFR and IR luminosity used in these works corresponds to the local Universe relation calibrated for solar metallicity.",
"High-redshift galaxies tend to have lower metallicities (e.g.",
"[77]; [69]), for which the use of the local Universe relation could possibly lead to an overestimate of the IR luminosity, and therefore, an underestimate of the $L_{\\rm CO}/L_{\\rm IR}$ ratio.",
"Given this caveat, the apparent discrepancy of $\\approx 0.3$ dex between the model predictions and the high redshift observations does not seem to be critical.",
"Accurate IR luminosity measurements for high redshift galaxies are needed to better assess how the model predictions compare with the observations.",
"In the case of starbursts, observations correspond to the brightest galaxies observed in the local and high redshift Universe.",
"IR luminosities for these galaxies are usually inferred from far-IR or sub-mm bands, e.g.",
"$850\\, \\mu $ m, and they are predicted to have gas metallicities close to solar, for which uncertainties in the IR luminosity are expected to be less important than in normal star-forming galaxies.",
"These bright galaxies should be compared to the model predictions for the brightest IR galaxies.",
"The model predicts a mean $L_{\\rm CO}/L_{\\rm IR}$ ratio and its evolution in good agreement with observations.",
"These galaxies in the model are predicted to have gas kinetic temperatures of $\\approx 50$ K (see red lines in Fig.",
"REF ).",
"Fig.",
"REF is similar to Fig.",
"REF , but shows the CO SLED at $z=2$ .",
"The redshift is chosen to match the median of the SMG observational compilation also shown in Fig.",
"REF .",
"This catalogue comprises 50 SMGs observed in various CO transitions with IR luminosities in the range $L_{\\rm IR}\\approx 10^{12}-4\\times 10^{13}\\, L_{\\odot }$ .",
"In order to infer a typical CO SLED of SMGs, we scale the CO luminosities in the SMG observational catalogue to the median IR luminosity of the sample, $\\langle L_{\\rm IR}\\rangle \\approx 8\\times 10^{12}\\, L_{\\odot }$ .",
"We do this by assuming that the $L^{\\prime }_{\\rm CO}/L_{\\rm IR}$ ratio for a given source is conserved.",
"Thus, the $L^{\\prime }_{\\rm CO}$ plotted in Fig.",
"REF corresponds to the observed CO luminosity scaled by a factor $L_{\\rm IR}/\\langle L_{\\rm IR}\\rangle $ .",
"These scaled observations are shown as symbols in the bottom panel of Fig.",
"REF .",
"With the aims of performing a fair comparison to the model predictions, we select galaxies in the model to have the same IR luminosity distribution as the observational sample and then scale their CO luminosities following the same procedure as with the observations.",
"This is shown as the black solid line in Fig.",
"REF .",
"The CO lines with the best statistics in the observational sample are the CO$(1-0)$ , CO$(2-1)$ , CO$(3-2)$ and CO$(4-3)$ .",
"The latter three correspond to the ones the model matches the best.",
"In the case of the CO$(1-0)$ , there is a slight discrepancy between the model and the observations, but still consistent with the dispersion predicted by the model.",
"At higher-J values, there are only two observations, one in agreement and the other one slightly above the model predictions.",
"However, the low number statistics prevents us from determining how representative these points are of the general SMG population.",
"The model predicts the peak of the CO SLED occurs, on average, at $J=5$ for these very luminous IR galaxies.",
"For the general galaxy population, the model predicts that the brightest IR galaxies have slightly flatter CO SLEDs than fainter IR counterparts.",
"Differences in the CO SLEDs of faint- and bright-IR galaxies at $z=2$ are predicted to be smaller than for $z=0$ galaxies.",
"In other words, at a fixed IR luminosity, high redshift galaxies tend to have shallower CO SLEDs compared to their $z=0$ counterparts.",
"This is due to a tendency of increasing average gas kinetic temperature in molecular clouds with increasing redshift (see Fig.",
"REF ), driven by the systematically lower metallicities and higher SFR surface densities of high redshift galaxies."
],
[
"Kinetic temperature evolution",
"The variation of the $L_{\\rm CO}/L_{\\rm IR}$ luminosity ratio with IR luminosity differs between quiescent and starburst galaxies, and is related to variations in the gas kinetic temperature, which depend on IR luminosity and redshift.",
"Fig.",
"REF shows the gas kinetic temperature of molecular clouds for quiescent (blue lines) and starburst (red lines) gas phases for different IR luminosity bins, as a function of redshift.",
"For reference, we also show the evolution of the temperature of the Cosmic Microwave Background (CMB) with redshift.",
"Note that some galaxies undergoing quiescent SF appear to have kinetic temperatures below the CMB temperature at $z\\gtrsim 4.5$ .",
"For these galaxies, extra heating from the CMB needs to be included in the UCL$_{-}$ PDR model to describe the thermal and chemical state of these galaxies.",
"This represents a limitation of the current GALFORM+UCL$_{-}$ PDR model.",
"However, this only becomes relevant at very high redshift and for quiescent galaxies.",
"The relation between the kinetic temperature of the gas and IR luminosity is primarily dominated by gas metallicity in quiescent galaxies and by the UV radiation field in starburst galaxies.",
"In the case of quiescent galaxies, the gas metallicity increases as the IR luminosity increases, and therefore the gas cools more efficiently, decreasing $T_{\\rm K}$ as $L_{\\rm IR}$ increases.",
"This is true only in this quiescent regime given that $G_{\\rm UV}$ only varies around $1-10\\times G_{0}$ .",
"In the case of starburst galaxies, as the IR luminosity increases so does the UV radiation field, $G_{\\rm UV}$ , which boosts the kinetic temperature of the gas, driving a positive relation between $T_{\\rm K}$ and $L_{\\rm IR}$ .",
"In general, starburst galaxies tend to have higher $T_{\\rm K}$ than quiescent galaxies.",
"The GALFORM+UCL$_{-}$ PDR model predicts that both cool and a warm ISM phases should be present in the high redshift universe, particularly in relatively bright galaxies (but not exclusively in the brightest ones), given that a large fraction of galaxies in the model at high redshift have SF taking place simultaneously in both the disk and bulge components."
],
[
"Assessing the robustness of the model predictions",
"We analyse in this section how the predictions of the coupled model presented in $§3$ and $§4$ depend on the assumptions made in the PDR model.",
"We focus on (i) the effect of metallicity, and (ii) the effect of a varying hydrogen number density (as opposed to the fixed density adopted previously).",
"For a detailed analysis on how other assumptions in the PDR modelling affect the results, e.g.",
"the assumed geometry, see [100].",
"We have shown that gas metallicity has an important effect on the predicted CO luminosity and SLED, particularly for relatively IR-faint galaxies.",
"These variations with metallicities have been extensively analysed in PDR and LVG models, such as in [122], [8] and [119].",
"However, comparisons between observations and PDR or LVG models with the aim of inferring average GMC properties tend to ignore the metallicity effect by assuming that the metallicity is fixed at solar or super-solar (e.g.",
"[55]; [36]; [83]).",
"In order to assess how much our predictions change if we ignore changes in metallicity we perform the same calculations as in $§4$ but ignore the metallicity information in GALFORM.",
"We therefore select the subset of the PDR models shown in Table REF that have $Z_{\\rm g}=1\\, Z_{\\odot }$ and calculate the $X_{\\rm CO}$ for each galaxy from that subset, regardless of its actual metallicity.",
"Figure: Logarithm of the median ratio between the predicted CO luminosity in the PDR model variants andthe standard GALFORM+UCL - _{-}PDR model as a functionof redshiftfor two alternative PDR models.Quiescent and starburst galaxies are shown in the left-hand and right-hand columns, respectively,and for four CO transitions,CO(1-0)(1-0) (top panels), CO(3-2)(3-2) (middle-top panels), CO(5-4(5-4) (middle-bottom panels) and CO(7-6)(7-6)(bottom panels), and two IR luminosities ranges (blue and red, as labelled), which are different in the rightand left-hand columns to enhance differences.The assumptions made in these two alternative PDR models are: (i) constant n H =10 4 cm -3 n_{\\rm H}=10^4\\, {\\rm cm}^{-3} and metallicity,Z g =Z ⊙ Z_{\\rm g}=Z_{\\odot } (dashed lines), and (ii) varying Z g Z_{\\rm g} and n H n_{\\rm H}(solid lines).Fig.",
"REF shows the ratio between the predicted CO luminosity in the standard GALFORM+UCL$_{-}$ PDR model and the variant with a fixed gas metallicity, for 4 CO transitions and two IR luminosity ranges.",
"Low CO transitions ($J\\le 4$ ) are only slightly affected by this change in the PDR models, with differences of less than a factor 3.",
"IR bright galaxies show the least variation in CO luminosity with respect to the standard model due to their already high gas metallicities, which tend to be close to solar or super-solar.",
"The CO luminosities of fainter galaxies in the IR are more affected given that they show larger gas metallicity differences, as Fig.",
"REF shows.",
"As we move to higher CO transitions, differences with respect to the standard model increase to up to a factor $\\approx 10$ .",
"This is driven by the generally larger variation of the population level of high CO transitions with cloud properties, as we described in $§4$ .",
"These results indicate that to assume a gas metallicity for observed galaxies might lead to a misinterpretation of the data, particularly when analysing high CO transitions.",
"This effect has also been previously seen in detailed ISM hydro-dynamical simulations, such as in [43].",
"Throughout the paper we have so far assumed that GMCs are characterised by a constant hydrogen density of $n_{\\rm H}=10^4\\, {\\rm cm}^{-3}$ , which, we have shown, allows us to explain the observed CO luminosities of local and high-redshift galaxies.",
"However, it is interesting to study the impact of allowing the hydrogen number density, $n_{\\rm H}$ , to vary.",
"This is because simulations and theoretical models suggest that a minimum density of hydrogen in GMCs is required to assure pressure equilibrium between a thermally supported warm medium and a turbulence supported cold neutral medium ([122]; [61]).",
"This minimum density depends on the UV flux, hard X-ray flux and metallicity as described by [122], $n_{\\rm H,min}\\propto \\frac{G_{\\rm UV}}{1+3.1(G_{\\rm UV}Z_{\\rm g} F_{\\rm X})^{0.37}},$ where $n_{\\rm H,min}$ , $G_{\\rm UV}$ , $Z_{\\rm g}$ and $F_{\\rm X}$ are in units of ${\\rm cm^{-3}}$ , $G_{0}$ , $Z_{\\odot }$ and $10 \\rm erg\\, s^{-1}\\, cm^{-2}$ (following our conversion between $F_{\\rm X}$ and $\\zeta _{0}$ described in $§2.2$ ).",
"We explore the effect of assuming that $n_{\\rm H}\\propto n_{\\rm H,min}\\propto G_{\\rm UV}$ on the predictions presented in $§3$ and $§4$ .",
"For this we select a subset of PDR models from Table REF , so that $n_{\\rm H}= 10^3 {\\rm cm}^{-3}\\, (G_{\\rm UV}/G_{\\rm 0})$ .",
"We repeat the analysis of $§4$ using this subset of PDR models.",
"The ratio between the predicted CO luminosity in the standard GALFORM+UCL$_{-}$ PDR and the model using a variable $n_{\\rm H}$ is shown in Fig.",
"REF as solid lines.",
"Figure: Kinetic temperature of the gas in molecular clouds for quiescent (blue curves) and starburst(red curves) gas phases in the ISM of galaxies as a function of redshift and for IR luminosities in the range10 10 -10 11 L ⊙ 10^{10}-10^{11}\\, L_{\\odot } for quiescent gas and 10 11 -10 12 L ⊙ 10^{11}-10^{12}\\, L_{\\odot } for starburst gas.Lines show the median of the predicted T K T_{\\rm K}, for the two models of Fig.",
", in addition to thestandard GALFORM+UCL - _{-}PDR model, which assumes n H =10 4 cm -3 n_{\\rm H}=10^4\\, {\\rm cm}^{-3} and varyingmetallicity, as labelled.",
"For reference, the Cosmic Microwave Background temperature is shown as a thick grey solid line.",
"For the same line style, starbursts have higher T K T_{\\rm K} than quiescent gas in galaxies.Allowing $n_{\\rm H}$ to vary has a small impact on the predicted CO luminosities at low CO transitions typically less than a factor of 3, and gives slightly smaller differences with respect to the standard GALFORM+UCL$_{-}$ PDR model than the model which uses a $Z_{\\rm g}=Z_{\\odot }$ and $n_{\\rm H}=10^4\\, {\\rm cm}^{-3}$ (dashed lies in Fig.",
"REF ).",
"This suggests that including the gas metallicity information from GALFORM in the PDR has a similar impact on the predicted CO luminosities than the assumption of a constant $n_{\\rm H}=10^4\\, {\\rm cm}^{-3}$ .",
"This again supports our interpretation of the major role that metallicity plays in determining $X_{\\rm CO}$ (Bayet et al.",
"2012, in prep.).",
"When moving to high CO transitions, deviations from the CO luminosities predicted by the model with variable $n_{\\rm H}$ become more important.",
"There is a tendency to produce fainter (brighter) CO emission from high CO transitions at low (high) redshifts with respect to the standard model.",
"This is because in the standard model there is a clear increase of temperature with redshift for both, quiescent and starburst gas, that is not as obvious in the case of the quiescent gas in the subset of PDR models in which $n_{\\rm H}$ is varied (Fig.",
"REF ).",
"We have argued that metallicity plays an important role in determining the average transmission of UV photons in galaxies and that therefore affects the incident UV flux.",
"This translates into low metallicity galaxies having higher temperatures.",
"Fig.",
"REF shows the kinetic temperature evolution for the standard GALFORM+UCL$_{-}$ PDR model and the two subsets of PDR models we describe above for two ranges of IR luminosities.",
"In the subset of PDR models where the metallicity is fixed, there is no visible evolution of $T_{\\rm K}$ with redshift for quiescent gas.",
"This supports our conclusion that the ISM metallicity evolution is a main driver of the increasing $T_{\\rm K}$ with redshift in quiescent gas in the standard GALFORM+UCL$_{-}$ PDR model ($§4.2.3$ ).",
"In the PDR subset of models where $n_{\\rm H}$ varies, the trend between $T_{\\rm K}$ and $z$ for the quiescent gas is only weakly recoveredi at $z<2.5$ .",
"In the case of starburst gas, the main driver of the trend of increasing $T_{\\rm K}$ with redshift, is the increasing $G_{\\rm UV}$ with redshift, that is only weakly affected by metallicity (see $§4.2.3$ ).",
"Thus, the subset of PDRs with fixed metallicity recovers the trend of the standard GALFORM+UCL$_{-}$ PDR model.",
"In the case of the subset of PDRs with variable $n_{\\rm H}$ , the trend is lost due to the more efficient cooling in the PDR due to the higher $n_{\\rm H}$ .",
"The small deviations in the emission of low CO transitions introduced by different assumptions in the PDR modelling suggest that the predictions presented in this paper for these transitions are robust under these changes.",
"However, high CO transitions are more sensitive to the assumptions in the PDR modelling.",
"The small number of observations available in these transitions does not so far allow us to distinguish these different possibilities.",
"More data on these high CO transitions are needed, particularly if they cover a wide redshift range.",
"Homogeneity in the observed samples, even though it is desired, it is not essential given that our modelling permits the prediction of a plethora of galaxy properties, which allows us to select galaxies to have similar properties to the observed ones."
],
[
"Predictive power of the ",
"We have shown that the predictions of the GALFORM+UCL$_{-}$ PDR model for CO emission are in good agreement with observations of galaxies in the local and high redshift Universe.",
"Consequently, this coupled model is a powerful theoretical tool to study the observability of molecular lines in different types of galaxies and can therefore contribute to the development of science cases for the new generation of millimeter telescopes.",
"In this section we focus on star-forming galaxies at high redshift, selected through two different techniques based on broad band colours: (i) BzK colour selection [33], which can be used to select star-forming galaxies in the redshift range $1.4\\lesssim z\\lesssim 2.5$ , and (ii) the Lyman-break technique which is used to select star-forming galaxies at $z\\sim 3-10$ .",
"In this section, we study the observability of these star-forming galaxies with ALMA.",
"We use AB magnitudes throughout this section."
],
[
"BzK galaxies",
"The BzK colour selection has shown to be efficient at selecting galaxies around $z\\sim 2$ [33].",
"BzK-selected galaxies have been used to study the build-up of the stellar mass, the SF history of the Universe and properties of star-forming and passive galaxies at the peak of SF activity (e.g.",
"Daddi et al.",
"2005, 2007; [73]).",
"The BzK criterion is based on observer frame magnitudes in the $B$ , $z$ and $K$ bands.",
"Star-forming galaxies, also referred to as sBzK, are selected as those whose BzK colour index BzK$=(z-K)_{\\rm AB}-(B-z)_{\\rm AB}>-0.2$ .",
"The model predicts that the BzK criterion is a very efficient way to select star-forming galaxies; with only a small contamination of $\\approx 10$ % of galaxies outside the redshift range $1.4<z<2.5$ .",
"Furthermore, for a given limit $m_{K}$ , there is a positive correlation between the BzK colour index and the SFR of galaxies, although with a large dispersion ([82]).",
"Large BzK colour indices (BzK$>1.5$ ) correspond almost exclusively to highly star-forming galaxies, SFR$\\gtrsim 10\\, M_{\\odot }\\, \\rm yr^{-1}$ .",
"We take the galaxy population predicted by the GALFORM model at $z=2$ and select a sample of sBzK galaxies based on their BzK colour indices and apparent K-band magnitudes: BzK$>-0.2$ and $m_{\\rm K}<24$ .",
"The latter cut corresponds roughly to the deepest K-band surveys to date ([12]).",
"We randomly select four galaxies from this $z=2$ BzK sample in bins of BzK colour index, and list selected properties in Table REF .",
"In Table REF , the CO line velocity width, $\\sigma ^{\\rm los}_{\\rm CO}$ , corresponds to the line-of-sight circular velocity and the velocity dispersion in the case of the disk and the bulge components, respectively (considering a random inclination).",
"We choose to focus on the CO$(1-0)$ , CO$(3-2)$ and CO$(6-5)$ emission of these galaxies, which fall into band 1, 3 and 6 of ALMA, respectively, but note that other CO lines also fall into the ALMA bands at this redshift.",
"We used the software CASASpecifically, we use the ALMA OST software developed by the ALMA regional centre in the UK, http://almaost.jb.man.ac.uk/., which is part of the observational tools associated with ALMA, to simulate observations of model galaxies, by including instrumental and atmospheric effects, such as the convolution with the primary beam and the sky noise.",
"We calculate the integration timesIntegration times were calculated using the ALMA sensitivity calculator https://almascience.nrao.edu/call-for-proposals/ sensitivity-calculator.",
"necessary to obtain a theoretical root mean square sensitivity of at least 5 times lower than the peak CO flux with the full ALMA configuration (50 antennae) under average water vapor conditions ($\\approx 1.2-1.5 \\,\\rm mm$ of column density).",
"The peak CO flux corresponds to $s_{\\nu }=S_{\\rm CO,V}/\\sigma _{\\rm los}$ .",
"Input parameters used in the CASA software and integration times are listed in Table REF for both CO emission lines considered.",
"Table: Properties of the four Lyman-break galaxies at z=3z=3 plotted in Fig :row (1) rest-frame, extincted UV magnitude, columns (2)-(7) are as in Table ,(8) velocity-integrated CO line flux of the CO(3-2)(3-2), the (9) CO(5-4)(5-4),and (10) the CO(6-5)(6-5) emission lines,(11), (12) and (13) show the integration times to get adetection at the sensitivity indicated in theparentheses in band 3, 4 and 5 of ALMA for the CO(3-2)(3-2), CO(5-4)(5-4) and CO(6-5)(6-5) emission lines,respectively.",
"We also list ν c \\nu _{\\rm c}, Δν\\Delta \\nu and RRused to simulate observations.Figure: Four Lyman-break galaxies at z=3z=3.Top panels: the CO(3-2)(3-2) flux maps in logarithmic units of mJy / pixel \\rm mJy/pixel.",
"The horizontal line shows1 arcsec.Bottom panels: the observed CO(3-2)(3-2) flux maps in declination vs. right ascension.Flux is in units of mJy / beam \\rm mJy/beam.Maps corresponds to the outputs of the CASA software,after convolving the original map with the primary beam andincluding an atmospheric model for the background noise.",
"Ellipses in the bottom-left cornerindicate the beam size and shape and the cross indicatesa 1×11\\times 1 arcsec 2 ^2.",
"The flux scale is shown at the top of each figure.Some relevant properties of these galaxies are listed in Table ,along with the integration timeused to generate the CASA maps of the LBGs.Fig.",
"REF shows two of the four BzK galaxies listed in Table REF .",
"For each of these galaxies we show mock images with perfect angular resolution and no noise (top panels) and the simulated observations (bottom panels) for the CO$(3-2)$ and CO$(6-5)$ lines assuming random inclinations and position angles.",
"The beam of the instrumentThe beam corresponds to the FWHM of a two-dimensional gaussian fitted to the central lobe of the Point Spread Function.",
"is plotted in the panels showing the simulated images.",
"Note that the angular extents of the output images for the CO$(3-2)$ and CO$(6-5)$ observations are different: crosses in the panels show for reference $1\\times 1$ arcsec$^2$ .",
"This happens because the lower frequencies are observed with lower angular resolution than the higher frequencies.",
"From the integration times calculated here, it is clear that ALMA can obtain $5 \\sigma $ detections in relatively short integration times only in some of these galaxies.",
"Even though sBzKs of Table REF have velocity-integrated CO fluxes that are large, these galaxies are sufficiently big so that the peak flux in some cases is faint enough to need large integration times.",
"Note that, if the required $5\\sigma $ detection is in the integrated flux instead of peak flux, the integration times are generally reduced to $\\tau _{\\rm int}<1$ hour.",
"This indicates that the sBzK selection could be an effective way of constructing a parent catalogue of galaxies to follow-up with ALMAThe ALMA basic specifications are described in https://almascience.nrao.edu/about-alma/full-alma.",
"However, spatially resolving the ISM of these high redshift galaxies will be a very difficult task because (1) high order CO lines can be observed at better angular resolution, but are at the same time fainter and therefore need much longer exposures (see Fig.",
"REF ), and (2) the decreasing galaxy size with increasing redshift predicted by GALFORM [64] and observed by several authors (e.g.",
"[24]; [89]), imply that galaxies at these high redshifts are intrinsically smaller than their local universe counterparts, and therefore even more difficult to resolve.",
"In the examples of Fig.",
"REF , it is possible to observe more than one CO emission line in reasonable integration times, which could help to constrain the excitation levels of the cold ISM in these galaxies.",
"Note that observationally, targeting of sBzK galaxies to study CO has been done by [32] for 5 very bright galaxies using the PdBI and integration times $>10$ hours per source.",
"Their CO emission is $\\approx 2-3$ larger than our brightest example, BzK+gal4, which requires an integration time in ALMA of less than 20 minutes in CO$(3-2)$ , indicating again that ALMA will be able to detect CO routinely in these galaxies, even with the relatively modest amount of emission from the CO$(3-2)$ transition."
],
[
"Lyman-break galaxies",
"Lyman-break galaxies (LBGs) are star-forming galaxies which are identified through the Lyman-break feature in their spectral energy distributions.",
"This feature is produced by absorption by neutral hydrogen in the atmospheres of massive stars, in the ISM of the galaxy and in the intergalactic medium ([110]; [109]).",
"Colour selection of these galaxies has been shown to be very efficient and has allowed the statistical assessment of their properties (such as the rest-frame UV LF and the size-luminosity relation; [109]; [24]).",
"LBGs are of great interest as a tracer of the galaxy population at high redshift (see [64]).",
"These galaxies are at even higher redshifts than BzK galaxies, and are therefore key to probing the evolution in the ISM of galaxies at early epochs.",
"We show in the following subsection examples of LBGs at $z=3$ and $z=6$ ."
],
[
"Lyman-break galaxies at $z=3$",
"Fig.",
"REF shows four LBGs at $z=3$ from the GALFORM+UCL$_{-}$ PDR model.",
"In the top panels we show mock images of the CO$(3-2)$ emission of the model LBGs, and in the bottom panels we show the simulated ALMA observations.",
"The intrinsic properties of these four galaxies are listed in Table REF , along with other relevant information, as discussed below.",
"Here, the rest-frame UV luminosity includes dust extinction.",
"We estimate integration times as in $§6.1$ for imaging the CO$(3-2)$ , CO$(5-4)$ and CO$(6-5)$ transitions, modifying the inputs accordingly (e.g.",
"$\\nu _{\\rm c}$ , $\\Delta \\nu $ , resolution).",
"Integration times are also listed in Table REF for the three CO emission lines.",
"Note that the CO$(6-5)$ transition in three of the four LBGs shown here needs integration times larger than a day to obtain a $5\\sigma $ detection.",
"These cases are not suitable for observation, but it is interesting to see how long an integration would be needed to be to get a minimum signal for a detection in such a high order CO transition.",
"However, in the four LBGs it would be possible to observe more than one CO transition line, which would allow the physical conditions in the ISM in these galaxies to be constrained.",
"LBGs were randomly chosen from the full sample of LBGs at $z=3$ in the GALFORM model, in bins of UV rest-frame luminosity.",
"The break in the UV LF at $z=3$ is at $M^{*}_{\\rm UV}-5\\rm log(h)\\approx -20.3\\,$ [98], so the LBGs in Fig.",
"REF have a UV luminosities covering a large range around $L^{*}_{\\rm UV}$ .",
"In terms of the $M_{\\rm stellar}-\\rm SFR$ plane (see [67]), the four LBGs in Table REF lie on the so-called `main' sequence.",
"The integration times we calculate for these galaxies indicate that imaging $z=3$ LBGs will be an easy task for ALMA, detecting CO$(3-2)$ in integrations shorter than few hours per source.",
"Therefore LBG selection should provide a promising way of constructing a parent galaxy catalogue to follow up using ALMA.",
"Note that imaging of the CO$(3-2)$ line in these LBGs at $z=3$ is easier than in the BzKs at $z=2$ shown before.",
"This happens because LBGs are predicted to typically have smaller $\\sigma _{\\rm los}$ than BzKs due to their lower baryonic content.",
"The GALFORM model predicts a weak correlation between the molecular mass and the UV luminosity, $M_{\\rm mol}\\propto L^{0.5}_{\\rm UV}$ , while the SFR and the gas metallicity have stronger correlations with the UV luminosity.",
"Thus, most of the differences in the CO emission between LBGs in Fig.",
"REF result from the different ISM conditions (e.g.",
"gas metallicities, $\\Sigma _{\\rm SFR}$ ), rather than molecular gas mass.",
"In the case of LBG+gal4, the CO$(5-4)$ flux is larger than that of CO$(3-2)$ .",
"This happens because this LBG is undergoing a bright starburst, which leads to a much more excited ISM.",
"Its CO SLED peaks at higher J and falls slowly as J increases, compared to the other LBGs shown here.",
"These differences in the excitation levels of CO have a big impact on the observability of LBGs in the high-order CO transitions ($J>5$ ), producing large variations in the integration times needed to get a $\\gtrsim 5\\sigma $ detection.",
"However, it is important to remark that, in this model, starburst galaxies constitute only $\\approx 10$ % of the galaxies with $M_{\\rm UV}\\rm -5log(h)< -18$ at this redshift, even though their number density is much higher compared to low redshifts."
],
[
"Lyman-break galaxies at $z=6$",
"We select four Lyman-break model galaxies at $z=6$ in terms of their UV luminosity.",
"Given the intrinsically faint CO emission of these galaxies, we estimate the integration times for these $z=6$ LBGs assuming good weather conditions (i.e.",
"a water vapour column density of $0.5$ mm), unlike BzKs and $z=3$ LBGs, for which we assumed average weather conditions.",
"Note that only two LBGs out of the four shown in Table REF require an integration time to detect the CO$(2-1)$ emission line of $\\tau _{\\rm int}<1$ day.",
"We show one of these two `observable' LBGs in Fig REF .",
"At this very high redshift, the detection of any CO emission line will be a challenging task even for ALMA.",
"Predictions from hydrodynamic simulations of $z=6$ Lyman-$\\alpha $ emitters reach a similar conclusion.",
"[114], using a constant $X_{\\rm CO(1-0)}$ and assuming local thermodynamic equilibrium to estimate the luminosities of higher CO transitions showed that only the brightest Lyman-$\\alpha $ emitters at $z=6$ would be suitable for observation in the full ALMA configuration in under $<10$ hours of integration time.",
"A possible solution for the study of these very high redshift galaxies is CO intensity mapping using instruments which are designed primarily to detect atomic hydrogen, such as the South-African SKA pathfinder, MeerKAThttp://www.ska.ac.za/meerkat/, and in the future, the Square Kilometer Arrayhttp://www.skatelescope.org/ (SKA).",
"At such high redshifts these telescopes will also cover the redshifted frequencies of low-J CO transitions.",
"Given their larger field-of-view compared to ALMA, it is possible to collect the molecular emission of all galaxies in a solid angle and to isolate the emission from a narrow redshift range by cross-correlating emission maps of different molecules.",
"From this, it is possible to construct the emission line power spectrum and its evolution, inferring valuable information, such as the total molecular content from galaxies that are too faint to be detected individually (Visbal et al.",
"2010, 2011; see [95] for a review)."
],
[
"Discussion and conclusions",
"We have presented a new theoretical tool to study the CO emission of galaxies and its connection to other galaxy properties.",
"One of the aims of this work is to expand the predictive power of galaxy formation models.",
"Previously, there was no connection between the conditions in the ISM in model galaxies and their CO emission.",
"The CO emission from a galaxy was obtained from the predicted mass of molecular hydrogen essentially by adopting an ad-hoc $X_{\\rm {CO}}$ .",
"In this new hybrid model, the value of $X_{\\rm {CO}}$ is computed by the PDR model after taking as inputs selected predicted galaxy properties.",
"A lack of resolution and the use of simplifying assumptions (which make the calculation tractable) means that we use proxy properties to describe the conditions in the ISM.",
"At the end of this exercise, the number of testable predictions which can be used to reduce the model parameter space has been considerably increased (e.g.",
"CO luminosity functions, CO-to-IR luminosity ratios and CO SLEDs).",
"The hybrid model presented in this work combines the galaxy formation model GALFORM with the Photon Dominated Region code UCL$_{-}$ PDR, which calculates the chemistry of the cold ISM.",
"We use state-of-the-art models: the [66] galaxy formation model, which includes a calculation of the H$_2$ abundance in the ISM of galaxies and self-consistently estimates the instantaneous SFR from the H$_2$ surface density, and the [7] PDR model, which models the thermal and chemical states of the ISM in galaxies.",
"The combined code uses the molecular gas mass of galaxies and their average ISM properties as predicted by GALFORM as inputs to the UCL$_{-}$ PDR model, to estimate the CO emission in several CO transitions for each galaxy.",
"The average ISM properties required from GALFORM are the gas metallicity, and the average UV and X-ray radiation fields within galaxies.",
"The gas metallicity and the X-ray luminosity from AGN are calculated directly in GALFORM.",
"We use a phenomenological approach to estimate the UV radiation field by assuming a semi-infinite slab and a relation between the UV intensity and the SFR surface density in galaxies, with and without a correction for the average attenuation of UV photons.",
"Given that the GALFORM model does not produce detailed radial profiles of galaxies, the combined GALFORM+UCL$_{-}$ PDR model focuses on the interpretation of global CO luminosities and their relation to other galaxy properties.",
"We show that this hybrid model is able to explain a wide range of the available CO observations of galaxies from $z=0$ to $z=6$ , including LIRGs, ULIRGs, high redshift normal star-forming galaxies, SMGs and QSOs.",
"Our main conclusion are: (i) The GALFORM+UCL$_{-}$ PDR model predicts a $z=0$ CO$(1-0)$ luminosity function and CO$(1-0)$ -to-IR luminosity relation in good agreement with observations (e.g.",
"[60]; [107]).",
"The model favours the inclusion of the attenuation of UV photons due to dust extinction in the estimate of the internal UV radiation field.",
"(ii) Starburst galaxies have lower CO$(1-0)$ /IR luminosity ratios than normal star-forming galaxies, which leads to an anti-correlation between CO$(1-0)$ /IR luminosity ratio and IR luminosity.",
"This is due in part to the different SF laws in bursts compared to quiescent SF.",
"(iii) The GALFORM+UCL$_{-}$ PDR model predicts that the CO-to-IR luminosity ratio evolves weakly with redshift, regardless of the CO transition, and in agreement with local and high redshift observational data.",
"(iv) We find that the model is able to explain the shape and normalization of the CO SLEDs for local Universe LIRGs and high redshift SMGs.",
"The model predicts a peak in their CO SLEDs, on average, at $J=4$ and $J=5$ , respectively.",
"The model predicts that the peak shifts to higher-J values with increasing IR luminosity.",
"At a fixed IR luminosity, high redshift galaxies are predicted to have ISMs with higher gas kinetic temperature than low redshift galaxies, a result driven by the lower metallicities and higher SFR surface densities in such objects.",
"The presence of an AGN affects the emission of high CO transitions in galaxies, with galaxies with AGN showing larger CO$(\\rm J\\rightarrow J-1)$ /IR luminosity ratios at $J>6$ than galaxies without AGN.",
"The model predicts that observations of these high-J CO transitions should provide useful constraints on the heating effects of AGN on the ISM of galaxies.",
"We have shown that, despite its simplicity, this exploratory hybrid model is able to explain the observed CO emission of a wide range of galaxy types at low and high redshifts without the need for further tuning.",
"This is the first time that a galaxy formation model has been able to successfully reproduce such a wide range of observations of CO along with other galaxy properties.",
"This hybrid model can be used to predict the observability of galaxies with the current and upcoming generation of millimeter telescopes, such as LMT, GMT, PdBI and ALMA.",
"This is also applies to radio telescopes which can be used to study molecular emission of high redshift galaxies, such as MeerKAT, and further into the future, the SKA.",
"As an example of the diagnostic power of the GALFORM+UCL$_{-}$ PDR model, we study the observability of high redshift star-forming galaxies with ALMA, which is one of its key science goals.",
"In particular, we focus on colour-selected BzK galaxies at $z=2$ and LBGs at $z=3$ and $z=6$ .",
"We use the ALMA OST software to simulate observations of the GALFORM+UCL$_{-}$ PDR model galaxies.",
"For the first time, we present the expected CO fluxes and the integration times needed to obtain a 5$\\sigma $ detections in the full ALMA configuration.",
"We find that ALMA should be able to observe star-forming galaxies in low-J CO transitions routinely up to $z\\approx 3$ , with integration times of less than a few hours per source, and in a large fraction of the samples, in under 1 hour.",
"However, for star-forming galaxies at $z=6$ , this will be a much more difficult task, given their lower gas masses and metallicities, which lead to lower CO luminosities.",
"For these galaxies, future radio telescopes offer a promising alternative of intensity mapping of molecular emission lines, from which it is possible to learn about the molecular content of faint galaxies.",
"Therefore, colour selection of galaxies should be an effective way to construct parent samples for follow up with ALMA.",
"Further observational data on the CO SLEDs of galaxies and how these relate to other galaxy properties will be key to constraining the physical mechanisms included in the model, and determine whether our model is sufficient to explain the observations (particularly of high-redshift galaxies), or whether an improved (more general) calculation is needed, or indeed if further physical processes have to be considered.",
"The physics included and the simplifications made in this work seem to be good enough to explain current observations of CO.",
"The GALFORM+UCL$_{-}$ PDR model will facilitate the interpretation of observations which aim to studying the evolution of the mass of molecular gas in galaxies and assist the planning of science cases for the new generation of millimeter telescopes, and lays the foundation for a new generation of theoretical models."
],
[
"Acknowledgements",
"We thank Serena Viti, Ian Smail, Mark Swinbank, Linda Tacconi, Karin Sandstrom and Desika Narayanan for useful discussion and comments on this work.",
"We thank the anonymous referee for helpful suggestions that improved this work.",
"We thank the ALMA OT team, and particularly Adam Avinson, Justo González and Rodrigo Tobar, for helpful remarks on the ALMA OT and the CASA software.",
"CL gratefully acknowledges a STFC Gemini studentship.",
"EB acknowledges the rolling grants `Astrophysics at Oxford' PP/EE/E001114/1 and ST/H504862/1 from the UK Research Councils and the John Fell OUP Research fund, ref 092/267.",
"JEG is supported by a Banting Fellowship, administered by the Natural Science and Engineering Council of Canada.",
"The authors benefited from a visit of EB to Durham University supported by a STFC visitor grant at Durham.",
"Calculations for this paper were mainly performed on the ICC Cosmology Machine, which is part of the DiRAC Facility jointly funded by STFC, the Large Facilities Capital Fund of BIS, and Durham University."
],
[
"The CO line and IR luminosity",
"In this appendix we explain in more detail how we calculate CO luminosities in the different units used in the paper and also the total IR luminosity.",
"We express the CO luminosity in three different ways: (i) as a line luminosity, $L_{\\rm CO}$ , typically expressed in solar luminosities, (ii) as a velocity-integrated CO luminosity, $L_{\\rm CO,V}$ , which is typically expressed in units of $\\rm Jy\\, km\\, s^{-1}\\, Mpc^{-2}$ , and (iii) as a brightness temperature luminosity, $L^{\\prime }_{\\rm CO}$ , in units of $\\rm K\\, km\\, s^{-1}\\, pc^{-2}$ .",
"We estimate these luminosities from the molecular hydrogen mass and the value of $X_{\\rm CO}$ for each galaxy.",
"The spectral energy distribution of a source is characterised by the monochromatic luminosity, $l_{\\nu }(\\nu _{\\rm rest})$ , where $\\nu _{\\rm rest}$ is the rest-frame frequency.",
"The total luminosity of the emission line we are interested in is simply the integral of $l_{\\nu }(\\nu _{\\rm rest})$ over the frequency width of the line, $L_{\\rm CO}=\\int l_{\\nu }(\\nu _{\\rm rest})\\, {\\rm d} \\nu _{\\rm rest}.$ The units of $L_{\\rm CO}$ are proportional to $\\rm erg\\, s^{-1}$ .",
"Observationally, what is measured is the monochromatic flux, $s_{\\nu }(\\nu _{\\rm obs})$ , where $\\nu _{\\rm obs}=\\nu _{\\rm rest}/(1+z)$ is the observed frequency.",
"The flux density of this emitter is simply the frequency-integrated flux, $S=\\int s_{\\nu }(\\nu _{\\rm obs}){\\rm d}\\nu _{\\rm obs}$ .",
"The frequency-integrated flux can be calculated from the intrinsic luminosity, which is what we predict in the GALFORM+UCL$_{-}$ PDR model, and the luminosity distance, $D_{\\rm L}$ , $S_{\\rm CO}=\\frac{L_{\\rm CO}}{4\\pi D^2_{\\rm L}}.$ A widely used luminosity in radio observations is the velocity-integrated CO luminosity, $L_{\\rm CO,V}$ .",
"This corresponds to the integral of the monochromatic luminosity over velocity $L_{\\rm CO,V}=\\int l_{\\nu }(\\nu _{\\rm rest})\\, {\\rm d} V=\\frac{c}{\\nu _{\\rm rest}}\\, L_{\\rm CO}.$ Here $c$ is the speed of light and ${\\rm d} V$ is the differential velocity, which is related to $\\nu _{\\rm rest}$ and $\\nu _{\\rm obs}$ as ${\\rm d} V=c\\, ({\\rm d} \\nu _{\\rm rest}/\\nu _{\\rm rest})=c\\, ({\\rm d} \\nu _{\\rm obs}/\\nu _{\\rm obs})$ .",
"Observationally, the velocity-integrated luminosity is calculated from the velocity-integrated flux, $S_{\\rm CO,V}=\\int s_{\\nu }(\\nu _{\\rm obs}) {\\rm d} V$ .",
"We can estimate the observable quantity, $S_{\\rm CO,V}$ , from our predicted $L_{\\rm CO,V}$ as, $S_{\\rm CO,V} &=& (1+z)\\, \\frac{L_{\\rm CO,V}}{4\\pi \\,D^2_{\\rm L}}.$ The third widely used luminosity in radio observations is the brightness temperature luminosity.",
"The definition of the rest-frame brightness temperature of an emitting source is $T_{\\rm B}(\\nu _{\\rm obs})=\\frac{c^2}{2\\, k_{\\rm B}}\\, \\frac{s_{\\nu }(\\nu _{\\rm obs})\\, (1+z)}{\\nu ^{2}_{\\rm obs}\\, \\Omega }.$ Here $k_{\\rm B}$ is Boltzmann's constant and $\\Omega $ is the solid angle subtended by the source.",
"The brightness temperature in Eq.",
"REF is an intrinsic quantity, given that the factor $(1+z)$ converts the brightness temperature from the observer-frame to the rest-frame.",
"In the regime of low frequencies (the Rayleigh-Jeans regime), such for as the rotational transitions of CO, in an optically thick medium and with thermalised CO transitions, the brightness temperature corresponds to the true temperature of the gas.",
"The integrated CO line intensity is defined as the velocity-integrated brightness temperature, $I_{\\rm CO}=\\int T_{\\rm B}(\\nu _{\\rm obs})\\, {\\rm d} V$ .",
"The brightness temperature luminosity is then defined as $L^{\\prime }_{\\rm CO}=I_{\\rm CO}\\, \\Omega \\, D^2_{\\rm A},$ where $D_{\\rm A}=D_{\\rm L}(1+z)^2$ is the angular diameter distance and therefore $\\Omega \\, D^2_{\\rm A}$ is the area of the source.",
"From Eqs.",
"REF , REF and REF it is possible to relate $L^{\\prime }_{\\rm CO}$ to $L_{\\rm CO}$ , $L^{\\prime }_{\\rm CO}=\\frac{c^3}{8\\pi k_{\\rm B}\\, \\nu ^3_{\\rm rest}}\\, L_{\\rm CO}.$ By definition, the relation between $L^{\\prime }_{\\rm CO}$ and the molecular hydrogen mass is parametrised by the factor $\\alpha _{\\rm CO}$ , $L^{\\prime }_{\\rm CO}=\\frac{M_{\\rm H_2}}{\\alpha _{\\rm CO}}.$ Note that here we define $\\alpha _{\\rm CO}$ in terms of molecular hydrogen mass, as done for example by [111] and [48].",
"However, other authors define $\\alpha _{\\rm CO}$ in terms of the total molecular gas mass (e.g.",
"[107]).",
"These two definitions differ by a factor $X_{\\rm H}$ , the hydrogen mass fraction.",
"In Eq.",
"6 in $§2.2$ we introduced the relation between $I_{\\rm CO}$ and the molecular hydrogen column density $N_{\\rm H_2}$ , $X_{\\rm CO}=N_{\\rm H_2}/I_{\\rm CO}$ .",
"Given that $M_{\\rm H_2}=m_{\\rm H_2}\\, N_{\\rm H_2}\\, \\Omega \\, D^2_{\\rm A}$ , where $m_{\\rm H_2}$ is the mass of a hydrogen molecule, the relation between $\\alpha _{\\rm CO}$ and $X_{\\rm CO}$ is simply $\\alpha _{\\rm CO}=m_{\\rm H_2}\\, X_{\\rm CO}.$ We can therefore estimate the brightness temperature CO luminosity introduced above from the molecular hydrogen mass, calculated in GALFORM, and the $X_{\\rm CO}$ conversion factor calculated in the $\\tt UCL_{-}PDR$ model as, $L^{\\prime }_{\\rm CO}&=&\\frac{M_{\\rm H_2}}{m_{\\rm H_2}\\, X_{\\rm CO}}.$ $L_{\\rm CO}$ and $L_{\\rm CO,V}$ are also estimated from $M_{\\rm H_2}$ and $X_{\\rm CO}$ using Eqs REF , REF , REF and REF .",
"For a more extended review of all the conversions between units and from CO luminosity to molecular mass, see Appendices A and B in [88].",
"To facilitate the comparison with observations, we use $L_{\\rm CO,V}$ to construct the CO luminosity function and $L_{\\rm CO}$ to compare against IR luminosity.",
"To construct CO flux density maps in $§5$ , we use the above relations to determine the velocity-integrated line flux, $S_{\\rm V}$ , from $M_{\\rm H_2}$ and $X_{\\rm CO}$ .",
"Throughout this paper we make extensive comparisons between the CO and IR luminosities.",
"In GALFORM, we define the total IR luminosity to be an integral over the rest-frame wavelength range 8$-$ 1000 $\\mu $ m, which approximates the total luminosity emitted by interstellar dust, free from contamination by starlight.",
"To estimate the IR luminosity, we use the method described in [64] and [50] (see also [63], in prep.",
"), which uses a physical model for the dust extinction at each wavelength to calculate the total amount of stellar radiation absorbed by dust in each galaxy, which is then equal to its total IR luminosity.",
"The dust model assumes a two-phase interstellar medium, with star-forming clouds embedded in a diffuse medium.",
"The total mass of dust is predicted by GALFORM self-consistently from the cold gas mass and metallicity, assuming a dust-to-gas ratio which is proportional to the gas metallicity, while the radius of the diffuse dust component is assumed to be equal to that of the star-forming component, which corresponds to the disk or the bulge half-mass radius depending on whether the galaxy is a quiescent disk or a starburst, respectively.",
"This dust model successfully explains the Lyman-break galaxy LF up to $z\\sim 10$ (see [64])."
]
] | 1204.0795 |
[
[
"Bayesian Centroid Estimation for Motif Discovery"
],
[
"Abstract Biological sequences may contain patterns that are signal important biomolecular functions; a classical example is regulation of gene expression by transcription factors that bind to specific patterns in genomic promoter regions.",
"In motif discovery we are given a set of sequences that share a common motif and aim to identify not only the motif composition, but also the binding sites in each sequence of the set.",
"We present a Bayesian model that is an extended version of the model adopted by the Gibbs motif sampler, and propose a new centroid estimator that arises from a refined and meaningful loss function for binding site inference.",
"We discuss the main advantages of centroid estimation for motif discovery, including computational convenience, and how its principled derivation offers further insights about the posterior distribution of binding site configurations.",
"We also illustrate, using simulated and real datasets, that the centroid estimator can differ from the maximum a posteriori estimator."
],
[
"Introduction",
"In motif discovery we are given a set of sequences that share a common motif and aim to identify the motif composition—the frequency of symbols for each position in the pattern—and the positions in each sequence where the motifs are.",
"It is assumed that the motifs are significantly different, in composition, from sequence background.",
"This problem has gained attention and relevance in the past 25 years mainly due to biological applications; a classical example is regulation of gene expression by transcription factors that bind to specific motifs in genomic promoter regions [17], [10], [25].",
"For this reason, we refer to the positions where the motifs are realized in the sequences as “binding sites”.",
"Due to its importance, hundreds of procedures have been proposed for motif discovery [11], [30].",
"While some approaches seek to characterize motifs and their binding sites using dictionary methods that capture over-representation of words as evidence [23], [21], it is common to represent motif compositions by a position weight matrix [26] and specify a parametric model where sequences are generated conditionally on motif and background compositions and binding sites.",
"Binding sites can then be regarded as missing data; parameters for the compositions can be estimated using expectation-maximization [6] in a frequentist setup, as in MEME [1], or assigned a prior distribution in a Bayesian setup [12], [19].",
"Following the Bayesian model from [15], we assume that there is only one motif of fixed length $L$ and that sequences are generated conditionally independently according to a product multinomial model given binding site positions and motif and background compositions.",
"Thus, for an alphabet $\\mathcal {S}$ , we define $\\theta _0 = (\\theta _{0,s})_{s \\in \\mathcal {S}}$ as background probabilities of generating each letter in $\\mathcal {S}$ and, for each position $i = 1, \\ldots , L$ in the motif, $\\theta _i = (\\theta _{i,s})_{s \\in \\mathcal {S}}$ as the probabilities of generating each letter at the $i$ -th position in the motif.",
"To simplify the notation we denote $\\Theta = (\\theta _0, \\theta _1, \\ldots , \\theta _L)$ .",
"As in [15], we set a conjugate Dirichlet prior for $\\Theta $ .",
"Product multinomial and product Dirichlet models are justified as a good working, first approximation based on position independence.",
"There are many extensions to this model that consider DNA strand complementarity [24], a more informative Markov structure for the background composition [16], and an explicit representation of the number of binding sites per sequence [28].",
"However, since we will be discussing a new inferential procedure, we adopt an extended model that yields a feasible computational method while still retaining a realistic interpretation and allows us to focus the discussion on the proposed estimator.",
"Motif discovery is considered a hard problem since motifs are usually short relative to sequence length and have a composition that might be hard to distinguish from background (see, for instance, [11].)",
"It is then imperative to rely on more refined, informative estimation methods that better glean information from the posterior distribution of binding site configurations.",
"Discrete inferential methods with this goal have recently been proposed, including the median probability model of [2] and the centroid estimator [7], [5].",
"Centroid estimation, in particular, has been successfully used for motif discovery [29], including models that account for sequence conservation [20].",
"In this paper we present a Bayesian model for motif discovery on multiple sequences with multiple possible binding sites and formalize a new flavor of inference based on centroid estimation.",
"As we will argue, the proposed estimator offers a good representative of the posterior space of binding site configurations; moreover, as a by-product of its derivation, we obtain informative summaries of the distribution of posterior mass.",
"We start the discussion by addressing a simple case when there is only one sequence and we accept only one binding site; next we extend the presentation to include multiple binding sites; then, we treat the full case when $\\Theta $ is random, in a fully Bayesian approach.",
"Finally, we offer some concluding remarks and directions for future work in the last section."
],
[
"One sequence, one binding site",
"Suppose we observe a sequence $R$ , $|R| \\doteq n$ , and wish to infer the location of the only binding site $Y$ , $Y \\in \\lbrace 1, \\ldots , n-L+1\\rbrace $ .",
"Setting a non-informative prior on $Y$ , ${\\mathbb {P}}(Y) = (n - L + 1)^{-1}$ , we have the posterior: ${\\mathbb {P}}(Y {\\,|\\,}R, \\Theta ) =\\frac{{\\mathbb {P}}(R {\\,|\\,}Y, \\Theta ) {\\mathbb {P}}(Y {\\,|\\,}\\Theta )}{\\sum _{\\widetilde{Y}=1}^{n-L+1} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta ) {\\mathbb {P}}(\\widetilde{Y}{\\,|\\,}\\Theta )}= \\frac{{\\mathbb {P}}(R {\\,|\\,}Y, \\Theta )}{\\sum _{\\widetilde{Y}=1}^{n-L+1} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )}.$ The likelihood, as previously stated, follows a product multinomial distribution given $Y$ : ${\\mathbb {P}}(R {\\,|\\,}Y, \\Theta ) = \\prod _{s \\in \\mathcal {S}}\\prod _{j \\in BG} \\theta _{0,s}^{I(R_j = s)}\\prod _{j=1}^L \\theta _{j,s}^{I(R_{Y-j+1} = s)},$ where $j \\in BG$ means position $j$ in background.",
"One traditional estimator is the MAP estimator, $\\widehat{Y}_M = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}=1,\\ldots ,n-L+1}{\\mathbb {P}}(\\widetilde{Y} {\\,|\\,}R, \\Theta ),$ but we argue for an estimator that accounts for differences in positions when comparing binding site configurations.",
"Using Bayesian decision theory [3] we look for an estimator that minimizes, on average, a more refined loss function $H$ : $\\widehat{Y}_C = \\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}=1,\\ldots ,n-L+1}{\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\big [ H(\\widetilde{Y}, Y) \\big ].$ We adopt a generalized Hamming loss $H$ , $H(\\widetilde{Y}, Y) = \\sum _{i=1}^n h(l_i(\\widetilde{Y}), l_i(Y)),$ where $l_i(Y)$ returns the “state” of position $i$ : if $i$ is a background position, $l_i(Y) = 0$ , otherwise $l_i(Y) = Y - i + 1$ , that is, $l_i(Y)$ returns the position in the motif.",
"Loss function $H$ compares configurations position-wise according to $h$ , which in turn compares states.",
"One option for $h$ when $\\Theta $ is known is a probability distance, the symmetric Kullback-Leibler distance, $h(i,j) = D_{KL}(\\theta _i \\,||\\, \\theta _j) + D_{KL}(\\theta _j \\,||\\, \\theta _i)= \\sum _{s \\in \\mathcal {S}}\\theta _{i,s} \\log \\frac{\\theta _{i,s}}{\\theta _{j,s}}+ \\theta _{j,s} \\log \\frac{\\theta _{j,s}}{\\theta _{i,s}},$ for $i,j = 0, 1, \\ldots , L$ .",
"It is, however, not common to have such an informed loss function.",
"An alternative metric arises by simply allowing $\\theta _{j,s} \\doteq \\theta _s \\ne \\theta _{0,s}$ for all $s \\in \\mathcal {S}$ and $j = 1, \\ldots , L$ in the motif.",
"In this case, if $m(i) \\doteq I(i > 0)$ indicates if state $i$ is a motif state, $h(i,j) = h(m(i), m(j)) = I(m(i) \\ne m(j)) \\Bigg [\\sum _{s \\in \\mathcal {S}}\\theta _s \\log \\frac{\\theta _s}{\\theta _{0,s}}+ \\theta _{0,s} \\log \\frac{\\theta _{0,s}}{\\theta _s} \\Bigg ].$ Since we are ultimately concerned with the argument of a minimum, as per Equation REF , we can define the loss function up to a shift and (positive) scale.",
"Thus, for our inferential purposes it suffices to define $h(i, j) = I(m(i) \\ne m(j))$ to obtain a loss $H$ that accounts for overlap in binding sites.",
"Such metric is commonly adopted to measure binding site level accuracy, as in the performance coefficients in [22], [11], [30].",
"From now on we will be focusing on this minimally informed loss function.",
"Estimator $\\widehat{Y}_C$ is a generalized centroid estimator; for instance, if $h$ is a common zero-one loss, $h(i,j) = I(i \\ne j)$ , $H$ corresponds to Hamming loss, and thus $\\widehat{Y}_C$ is the regular centroid estimator [7], [5].",
"As [5] argue, centroid estimators more effectively represent the space since they are closer to posterior means; in contrast, it can be shown that $\\widehat{Y}_M$ arises from a zero-one loss function which yields the posterior mode [4].",
"Let us now derive more specific expressions for $H$ and $\\widehat{Y}_C$ .",
"We first notice that if $|\\widetilde{Y}-Y| \\ge L$ then the binding sites do not overlap and so $H(\\widetilde{Y}, Y) = 2 \\sum _{j=1}^L h(j,0) \\doteq H^*$ , the null overlap distance between two configurations.",
"Alternatively, when $|\\widetilde{Y}-Y| < L$ then $H(\\widetilde{Y}, Y) =\\sum _{j=1}^{|\\widetilde{Y}-Y|} h(j, 0)+ \\sum _{j=L - |\\widetilde{Y}-Y| + 1}^L h(j, 0)+ \\sum _{j=1}^{L-|\\widetilde{Y}-Y|} h(j, j+|\\widetilde{Y}-Y|),$ since the common backgrounds in $\\widetilde{Y}$ and $Y$ do not affect $H(\\widetilde{Y}, Y)$ , the first two terms above account for the left and right “tails” where binding sites in one sequence are matched with background in the other sequence, and the last term accounts for the overlap in binding sites.",
"We also note that $H(\\widetilde{Y}, Y)$ is actually a function of $|\\widetilde{Y}-Y|$ .",
"Instead of a loss function we can also define our estimator in terms of a gain function $G(\\widetilde{Y}, Y)\\doteq 1 - H(\\widetilde{Y}, Y) / H^*$ .",
"Note that $0 \\le G(\\widetilde{Y}, Y) \\le 1$ ; in particular, when $|\\widetilde{Y}-Y| \\ge L$ there is no gain, $G(\\widetilde{Y}, Y) = 0$ , and if $\\widetilde{Y}=Y$ we have $G(\\widetilde{Y}, Y) =1$ .",
"As a consequence, we can simply write $G(\\widetilde{Y}, Y) =I(|\\widetilde{Y}-Y|<L) (1 - H(\\widetilde{Y}, Y)/H^*)$ with $H$ from Equation REF .",
"Noting that $G$ , like $H$ , is also a function of $|\\widetilde{Y}-Y|$ , we obtain the following characterization: Theorem 1 The centroid estimator $\\widehat{Y}_C$ is $\\widehat{Y}_C = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y} = 1, \\ldots , n - L + 1}G(\\widetilde{Y}, \\cdot ) * {\\mathbb {P}}(\\cdot {\\,|\\,}R, \\Theta ),$ a convolution between $G$ and the posterior distribution on $Y$ .",
"The result follows directly from the definition in Equation REF : $\\begin{split}\\widehat{Y}_C & = \\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y} = 1, \\ldots , n - L + 1}{\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\big [ H(\\widetilde{Y}, Y) \\big ] \\\\& = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y} = 1, \\ldots , n - L + 1}{\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\big [I(|\\widetilde{Y}-Y|<L) (1 - H(\\widetilde{Y}, Y)/H^*)\\big ] \\\\& = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y} = 1, \\ldots , n - L + 1}\\sum _{Y = \\max \\lbrace 1, \\widetilde{Y}-L+1\\rbrace }^{\\min \\lbrace n-L+1, \\widetilde{Y}+L-1\\rbrace }G(\\widetilde{Y}, Y) {\\mathbb {P}}(Y {\\,|\\,}R, \\Theta ) \\\\& = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y} = 1, \\ldots , n - L + 1}G(\\widetilde{Y}, \\cdot ) * {\\mathbb {P}}(\\cdot {\\,|\\,}R, \\Theta ),\\end{split}$ as required.",
"When contrasted to $\\widehat{Y}_M$ we can see the effect of having a higher resolution loss function: $\\widehat{Y}_C$ gathers probability support from nearby, relative to $H$ , binding site configurations instead of just picking the most likely configuration.",
"The following example should give us some insight into this new estimator.",
"Example 1 Consider the following sequence of length $n=200$ from the nucleotide alphabet $\\mathcal {S} = \\lbrace $A, C, G, T$\\rbrace $ , [boxwidth=auto] 10 20 30 40 50 | | | | | GCCACTTTCGGGCCCGTGTCTAACGCACCACGGGCTACGTGACGGTGTGG CTCTATACTGACGACGTGAACCAAGCTTTACTGAAGGACTTGCTGTTCCC CGACCCATTTCCTGCCAGAACCTCTGACCAGTGTCTAGGGCTATCGCCCG TGATGTCTCATGGCGACGCGCGAGGCGGTTGCTCGCCTCACTCCGTTCTG and a motif of length $L=6$ with parameters $\\Theta $ given by Table REF .",
"Table: Background and motif compositions: background is assumed to beCG-rich, while the motif represents a canonical palindromic E-box,CACGTG .Figure REF shows the conditional marginal posterior ${\\mathbb {P}}(Y {\\,|\\,}R,\\Theta )$ and the convolution $G * {\\mathbb {P}}(\\cdot {\\,|\\,}R, \\Theta )$ used to obtain the centroid $\\widehat{Y}_C = 36$ , binding at the subsequence TACGTG, close to the consensual motif.",
"Note that since $\\Theta $ is very informative the posterior profile has clear peaks and in this case $\\widehat{Y}_c =\\widehat{Y}_M$ , the two estimators coincide.",
"Figure: Conditional marginal probability distribution ℙ(Y|R,Θ){\\mathbb {P}}(Y {\\,|\\,}R,\\Theta ) in solid line and convolution G*ℙ(·|R,Θ)G * {\\mathbb {P}}(\\cdot {\\,|\\,}R, \\Theta ) indotted line.",
"The black thick line close to the axis marks the binding sitecorresponding to the centroid Y ^ C \\widehat{Y}_C."
],
[
"One sequence, multiple binding sites",
"We now allow for multiple binding sites by defining $Y = \\lbrace Y_k\\rbrace $ as the collection of binding sites $Y_k$ .",
"The likelihood is similar, but accounts for the multiple binding sites: ${\\mathbb {P}}(R {\\,|\\,}Y,\\Theta ) =\\prod _{s \\in \\mathcal {S}}\\prod _{i \\in BG} \\theta _{0,s}^{I(R_i=s)}\\prod _{k=1}^{|Y|} \\prod _{i=1}^L \\theta _{i,s}^{I(R_{Y_k+i-1}=s)}.$ Given the “entropic” effect of possibly having many binding sites, we need to adopt a better prior for $Y$ that takes into account the number of possible configurations for the binding sites.",
"So, instead of naively electing ${\\mathbb {P}}(Y)\\propto 1$ , we can explore a hierarchical structure: if $c(Y) = |Y|$ , the number of binding sites in $Y$ , we note that ${\\mathbb {P}}(Y) = {\\mathbb {P}}(Y, c(Y)) = {\\mathbb {P}}(Y {\\,|\\,}c(Y)) {\\mathbb {P}}(c(Y))$ and set ${\\mathbb {P}}(Y {\\,|\\,}c(Y)) \\propto 1$ and ${\\mathbb {P}}(c(Y)) \\propto 1$ to obtain ${\\mathbb {P}}(Y) = {\\mathbb {P}}(Y {\\,|\\,}c(Y)) {\\mathbb {P}}(c(Y)) = \\binom{n - c(Y)(L-1)}{c(Y)}^{-1} \\cdot \\frac{1}{C},$ where $C \\doteq \\lfloor n/L \\rfloor $ is the maximum number of binding sites in $R$ .",
"Another, possibly more familiar, approach is to adopt a Markov chain with two states, background and motif, where the probability of transitioning to background, either from background or motif, and of starting at background is $p$ .",
"In this case we keep ${\\mathbb {P}}(Y {\\,|\\,}c(Y))$ as before, but now ${\\mathbb {P}}(c(Y)) \\propto \\binom{n - c(Y)(L-1)}{c(Y)} p^{n-c(Y)L} (1-p)^{c(Y)},$ since there needs to be $c(Y)$ transitions to the motif state.",
"This prior structure offers more flexibility through $p$ : we can further set a hyperprior distribution on $p$ , or specify it directly based on the expected number $b$ of binding sites in the sequence; if $n$ is large compared to $b$ , as usual, then $p$ should be close to one, $c(Y)$ is approximately Poisson with mean $n(1-p)$ and thus $p \\doteq 1 - b/n$ becomes a good candidate.",
"The posterior is then $\\begin{split}{\\mathbb {P}}(Y {\\,|\\,}R, \\Theta ) & = \\frac{{\\mathbb {P}}(R, Y {\\,|\\,}\\Theta )}{\\sum _{\\widetilde{Y}} {\\mathbb {P}}(R, \\widetilde{Y} {\\,|\\,}\\Theta )} \\\\& =\\underbrace{\\frac{{\\mathbb {P}}(R, Y {\\,|\\,}\\Theta )}{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c(Y)}{\\mathbb {P}}(R, \\widetilde{Y} {\\,|\\,}\\Theta )}}_{{\\mathbb {P}}(Y {\\,|\\,}c(Y), R, \\Theta )}\\cdot \\underbrace{\\frac{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c(Y)} {\\mathbb {P}}(R, \\widetilde{Y} {\\,|\\,}\\Theta )}{\\sum _{c=0}^C \\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c}{\\mathbb {P}}(R,\\widetilde{Y} {\\,|\\,}\\Theta )}}_{{\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta )}.\\end{split}$ By the structure of our prior it follows that $\\begin{split}{\\mathbb {P}}(Y {\\,|\\,}c(Y), R, \\Theta ) & =\\frac{{\\mathbb {P}}(R {\\,|\\,}Y, \\Theta ) {\\mathbb {P}}(Y)}{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c(Y)} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta ){\\mathbb {P}}(\\widetilde{Y})} \\\\& = \\frac{{\\mathbb {P}}(R {\\,|\\,}Y, \\Theta )}{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c(Y)} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )},\\end{split}$ and $\\begin{split}{\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta ) & =\\frac{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c(Y)} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta ){\\mathbb {P}}(\\widetilde{Y})}{\\sum _{c=0}^C \\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c}{\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta ) {\\mathbb {P}}(\\widetilde{Y})} \\\\& =\\frac{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c(Y)} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta ){\\mathbb {P}}(\\widetilde{Y} {\\,|\\,}c(\\widetilde{Y})) {\\mathbb {P}}(c(\\widetilde{Y}))}{\\sum _{c=0}^C \\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c}{\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta ) {\\mathbb {P}}(\\widetilde{Y} {\\,|\\,}c(\\widetilde{Y})){\\mathbb {P}}(c(\\widetilde{Y}))}.\\end{split}$ This decomposition suggests a good approach to sampling from ${\\mathbb {P}}(Y {\\,|\\,}R,\\Theta )$ : we first sample $c(Y)$ according to ${\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta )$ and then sample $Y$ given the number of binding sites, according to ${\\mathbb {P}}(Y {\\,|\\,}c(Y), R, \\Theta )$ .",
"As we will see next, we need to work more to obtain a centroid estimator for the binding sites: we need to establish a hierarchical inferential structure by first finding centroids for $c(Y) = 1, \\ldots , C$ and then proceed to estimate a global centroid.",
"To this end we find ${\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta )$ and then compute marginal posteriors ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y), R, \\Theta )$ ."
],
[
"Marginal posterior on $c(Y)$",
"From Equations REF and REF we observe that we need to compute $\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )$ up to a constant to find both conditional posteriors of $c(Y)$ and $Y$ and thus the posterior ${\\mathbb {P}}(Y {\\,|\\,}R, \\Theta )$ .",
"Let us now denote by $R_{i:j}$ the subsequence of $R$ from positions $i$ to $j$ and by $Y_{i:j}$ the binding sites in $Y$ between $i$ and $j$ —that is, all $Y_k$ such that $i \\le Y_k \\le j - L + 1$ .",
"If we then define forward sums $F_{c,j} \\doteq \\frac{\\sum _{\\widetilde{Y}_{1:j}: c(\\widetilde{Y}_{1:j}) = c}{\\mathbb {P}}(R_{1:j} {\\,|\\,}\\widetilde{Y}_{1:j}, \\Theta )}{\\prod _{i=1}^j \\prod _{s \\in \\mathcal {S}} \\theta _{0,s}^{I(R_i=s)}}$ we have that $\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y},\\Theta ) \\propto F_{c,n}$ .",
"To further simplify the notation, let us define $\\lambda (j; \\Theta ) =\\prod _{i=1}^L \\prod _{s \\in \\mathcal {S}}\\Bigg (\\frac{\\theta _{i,s}}{\\theta _{0,s}}\\Bigg )^{I(R_{j-1+i}=s)},$ the composition ratio between motif and background for a binding site starting at $j$ .",
"The forward sums $F_{c,j}$ can be computed recursively, $F_{c,j} = F_{c,j-1} + F_{c-1,j-L} \\lambda (j-L+1; \\Theta ),$ by considering two options for the tail of the sequence: either having a background position—and hence the first summand above—or by having a binding site on the last $L$ positions—and thus requiring the second summand.",
"Thus, we have ${\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta ) =\\frac{F_{c(Y),n} \\binom{n-c(Y)(L-1)}{c(Y)}^{-1} {\\mathbb {P}}(c(Y))}{\\sum _{c=0}^C F_{c,n} \\binom{n-c(L-1)}{c}^{-1}{\\mathbb {P}}(c(Y)=c)},$ which yields a straightforward way to sample the posterior $c(Y)$ conditional on $\\Theta $ ."
],
[
"Marginal posterior on $Y_k$ given {{formula:ea1757b7-4edd-4ba8-8c01-749b2e761086}}",
"To compute ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y), R, \\Theta )$ we now need backward sums.",
"We can define them analogously to the forward sums: $B_{c,j} \\doteq \\frac{\\sum _{\\widetilde{Y}_{j:n}: c(\\widetilde{Y}_{j:n}) = c}{\\mathbb {P}}(R_{j:n} {\\,|\\,}\\widetilde{Y}_{j:n}, \\Theta )}{\\prod _{i=j}^n \\prod _{s \\in \\mathcal {S}} \\theta _{0,s}^{I(R_i=s)}},$ and hence $\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y},\\Theta ) \\propto B_{c,1}$ , as expected.",
"Moreover, by a similar argument to the previous subsection, we also have that the backward sums are recursive: $B_{c,j} = B_{c,j+1} + B_{c-1,j+L} \\lambda (j; \\Theta ).$ Having forward and backward sums enable us to readily compute the marginal posterior on $Y_k$ conditional on $c(Y)$ : since $\\begin{split}{\\mathbb {P}}(Y_k {\\,|\\,}c(Y)=c, R, \\Theta ) & =\\sum _{Y_1, \\ldots , Y_{k-1}, Y_{k+1}, \\ldots , Y_c}{\\mathbb {P}}(Y {\\,|\\,}c(Y)=c, R, \\Theta ) \\\\& = \\sum _{Y_1, \\ldots , Y_{k-1}, Y_{k+1}, \\ldots , Y_c}\\frac{{\\mathbb {P}}(R {\\,|\\,}Y, \\Theta )}{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )},\\end{split}$ and $\\sum _{Y_1, \\ldots , Y_{k-1}, Y_{k+1}, \\ldots , Y_c}{\\mathbb {P}}(R {\\,|\\,}Y, \\Theta ) =\\sum _{Y_1, \\ldots , Y_{k-1}}{\\mathbb {P}}(R_{1:Y_k-1} {\\,|\\,}Y_{1:Y_k-1}, \\Theta ) \\\\\\cdot {\\mathbb {P}}(R_{Y_k:Y_k+L-1} {\\,|\\,}Y_{Y_k:Y_k+L-1}, \\Theta )\\cdot \\sum _{Y_{k+1}, \\ldots , Y_c}{\\mathbb {P}}(R_{Y_k+L:n} {\\,|\\,}Y_{Y_k+L:n}, \\Theta ),$ and thus ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y)=c, R, \\Theta ) =\\frac{F_{k-1, Y_k-1} \\lambda (Y_k; \\Theta ) B_{c-k, Y_k+L}}{\\sum _{\\widetilde{Y}_k = (k-1)L}^{n-(c-k+1)L+1}F_{k-1, \\widetilde{Y}_k-1} \\lambda (\\widetilde{Y}_k; \\Theta ) B_{c-k, \\widetilde{Y}_k+L}}.$ Note that $\\frac{\\sum _{\\widetilde{Y}:c(\\widetilde{Y})=c} {\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )}{\\prod _{i=1}^n \\prod _{s \\in \\mathcal {S}} \\theta _{0,s}^{I(R_i = s)}} =F_{c,n} = B_{c,1} \\\\= \\sum _{\\widetilde{Y}_k = (k-1)L}^{n-(c-k+1)L+1}F_{k-1, \\widetilde{Y}_k-1} \\lambda (\\widetilde{Y}_k; \\Theta ) B_{c-k, \\widetilde{Y}_k+L},$ for $k = 1, \\ldots , c$ .",
"Before discussing posterior inference we summarize the results of this section in Algorithm REF .",
"[htpb] Computes ${\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta )$ and ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y), R, \\Theta )$ for $k = 1, \\ldots , c(Y)$ .",
"(Initialize) Set $F_{0,0} = B_{0,n+1} = F_{0,j} = B_{0,j} = 1$ for $j = 1, \\ldots , n$ ; for $c = 1, \\ldots , C$ , set $F_{c,j} =0$ when $j < cL$ and $B_{c,j} = 0$ when $j > n - cL + 1$ .",
"(Compute forward sums) For $c = 1, \\ldots , C$ and $j = cL+1,\\ldots , n$ do: set $F_{c,j}$ as in Equation REF , $F_{c,j} = F_{c,j-1} + F_{c-1,j-L} \\lambda (j-L+1; \\Theta )$ (Compute ${\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta )$ ) For $c = 0, \\ldots , C$ do: compute marginal posterior $c(Y)$ as in Equation REF , ${\\mathbb {P}}(c(Y) = c {\\,|\\,}R, \\Theta ) =\\frac{F_{c,n} \\binom{n-c(L-1)}{c}^{-1} {\\mathbb {P}}(c(Y)=c)}{\\sum _{\\widetilde{c}=0}^C F_{\\widetilde{c},n}\\binom{n-\\widetilde{c}(L-1)}{\\widetilde{c}}^{-1} {\\mathbb {P}}(c(Y)=\\widetilde{c})}$ (Compute backward sums) For $c = 1, \\ldots , C$ and $j = n-cL, \\ldots , 1$ do: set $B_{c,j}$ as in Equation REF , $B_{c,j} = B_{c,j+1} + B_{c-1,j+L} \\lambda (j; \\Theta )$ (Compute ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y), R, \\Theta )$ ) For $c = 1, \\ldots ,C$ , $k = 1, \\ldots , c$ , and $Y_k = (k-1)L+1, \\ldots , n-(c-k+1)L+1$ do: compute marginal posterior $Y_k$ given $c(Y)$ as in Equation REF , ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y)=c, R, \\Theta ) =F_{k-1, Y_k-1} \\lambda (Y_k; \\Theta ) B_{c-k, Y_k+L} / F_{c,n}$"
],
[
"Posterior Inference",
"In contrast to the one binding site case from last section, posterior inference is more difficult since comparing configurations with different number of binding sites is not amenable to a systematic approach.",
"Our first approximation is to consider local estimators for each group of configurations with a fixed number of binding sites and then appeal to a triangle inequality: $H(Y, \\widehat{Y}) \\le H(Y, \\widehat{Y}_c) + H(\\widehat{Y}_c, \\widehat{Y}),$ where $Y$ is a configuration with $c$ binding sites, $\\widehat{Y}_c$ is the constrained estimator for all configurations with $c$ binding sites, and $\\widehat{Y}$ is the (overall) centroid estimator.",
"Recall that for the centroid estimator we wish to find $\\widetilde{Y}$ that minimizes ${\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\big [ H(\\widetilde{Y}, Y) \\big ] =\\sum _{c = 0}^C \\sum _{Y : c(Y) = c}H(\\widetilde{Y}, Y) {\\mathbb {P}}(Y {\\,|\\,}R, \\Theta ).$ Using the triangle inequality for each group we then have ${\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\big [ H(\\widetilde{Y}, Y) \\big ] \\le \\sum _{c = 0}^C \\sum _{Y : c(Y) = c}\\big [ H(\\widetilde{Y}, \\widetilde{Y}_c) + H(\\widetilde{Y}_c, Y) \\big ]{\\mathbb {P}}(Y {\\,|\\,}R, \\Theta ) \\\\= \\sum _{c=0}^C \\Bigg [H(\\widetilde{Y}, \\widetilde{Y}_c)+ \\sum _{Y:c(Y)=c} H(\\widetilde{Y}_c, Y) {\\mathbb {P}}(Y|c(Y)=c, R, \\Theta )\\Bigg ] {\\mathbb {P}}(c(Y) = c {\\,|\\,}R, \\Theta ),$ where $\\widetilde{Y}_c$ is an arbitrary point in $\\lbrace Y:c(Y)=c\\rbrace $ .",
"Our task is now to find an estimator—let us still call it centroid—that minimizes the right-hand bound in Equation REF above.",
"This goal suggests a two-step strategy: For each number of binding sites, $c = 1, \\ldots , C$ , find the local centroids $\\widehat{Y}_c =\\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y}) = c}{\\mathbb {E}}_{Y {\\,|\\,}c(Y)=c, R, \\Theta } \\big [ H(\\widetilde{Y}, Y) \\big ]$ as the $\\widetilde{Y}_c$ in Equation REF .",
"Find the global centroid given the local centroids $\\lbrace \\widehat{Y}_c\\rbrace _{c=1}^C$ , $\\widehat{Y} = \\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}}{\\mathbb {E}}_{c(Y) {\\,|\\,}R, \\Theta } \\big [ H(\\widehat{Y}_{c(Y)}, \\widetilde{Y}) \\big ].$ We note that this strategy does not guarantee that the bound is minimized; the main goal here is computational convenience.",
"Let us tackle each step of this heuristic next."
],
[
"Local centroids",
"Even when the number of binding sites is fixed, minimizing the conditional posterior expectation of $H(\\widetilde{Y}, Y)$ can be challenging: we would still have to consider for each candidate configuration $\\widetilde{Y}$ the posterior probability of configurations with all binding sites to the left of the first binding site in $\\widetilde{Y}$ , in-between binding sites in $\\widetilde{Y}$ , and so on.",
"We adopt another approximation and decide to minimize a paired Hamming loss $H_A$ where binding site positions are matched according to their order: $H_A(\\widetilde{Y}, Y) = \\sum _{k=1}^{c(Y)} H_1(\\widetilde{Y}_k, Y_k),$ where $H_1(\\widetilde{Y}_k, Y_k)$ is Hamming loss when comparing sequences with only one binding site at $\\widetilde{Y}_k$ and $Y_k$ , respectively, that is, $H_1(\\widetilde{Y}_k, Y_k) = 2\\max \\lbrace |\\widetilde{Y}_k - Y_k|, L\\rbrace $ .",
"From the definition we have that $H_A$ upper bounds $H$ : $H_A(\\widetilde{Y}, Y)\\ge H(\\widetilde{Y}, Y)$ .",
"As a bad approximation example, if $\\widetilde{Y}_k =Y_{k+1}$ for $k = 1, \\ldots , c(Y)-1$ then $H_A(\\widetilde{Y}, Y) = c(Y) L$ , since each pair of binding sites $\\widetilde{Y}_k$ and $Y_k$ does not overlap, while $H(\\widetilde{Y}, Y) = 2L$ since only $Y_1$ and $\\widetilde{Y}_{c(Y)}$ are in disagreement with background.",
"The next result adapts Theorem REF to yield the paired local centroids.",
"Lemma 2 If ${\\mathbb {P}}_k(\\cdot {\\,|\\,}c(Y)=c, R, \\Theta )$ is the marginal conditional posterior on $Y_k$ then the paired local centroids are $\\widehat{Y}_c = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=c}\\sum _{k=1}^cG(\\widetilde{Y}_k, \\cdot ) * {\\mathbb {P}}_k(\\cdot {\\,|\\,}c(Y)=c, R, \\Theta )$ In the same spirit of Theorem REF , we use the conditional estimator in Equation REF with the paired loss $H_A$ : $\\begin{split}\\widehat{Y}_c & =\\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=c}{\\mathbb {E}}_{Y {\\,|\\,}c(Y)=c, R, \\Theta } \\big [ H_A(\\widetilde{Y}, Y) \\big ] \\\\& =\\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=c}\\sum _{Y:c(Y)=c} \\sum _{k=1}^c H_1(\\widetilde{Y}_k, Y_k){\\mathbb {P}}(Y {\\,|\\,}c(Y)=c, R, \\Theta ) \\\\& =\\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=c}\\sum _{k=1}^c \\sum _{Y_k=(k-1)L+1}^{n-(c-k+1)L+1} H_1(\\widetilde{Y}_k, Y_k){\\mathbb {P}}(Y_k {\\,|\\,}c(Y)=c, R, \\Theta ) \\\\& =\\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=c}\\sum _{k=1}^c\\sum _{Y_k=\\max \\lbrace (k-1)L+1, \\widetilde{Y}_k-L\\rbrace }^{\\min \\lbrace n-(c-k+1)L+1, \\widetilde{Y}_k+L\\rbrace }G(\\widetilde{Y}_k, Y_k){\\mathbb {P}}(Y_k {\\,|\\,}c(Y)=c, R, \\Theta ) \\\\& =\\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=c}\\sum _{k=1}^cG(\\widetilde{Y}_k, \\cdot ) * {\\mathbb {P}}_k(\\cdot {\\,|\\,}c(Y)=c, R, \\Theta ),\\end{split}$ and the result follows.",
"We can spot in Lemma REF the familiar convolutions, but now with the marginal posteriors ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y), R, \\Theta )$ and in a more restricted range.",
"We have a nice characterization, but we still have to optimize a sum to obtain the local centroids; to this end we explore the same recursive structure that allowed us to compute forward and backward sums.",
"Let us define $f(\\widetilde{Y}_k) \\doteq G(\\widetilde{Y}_k, \\cdot ) * {\\mathbb {P}}_k(\\cdot {\\,|\\,}c(Y)=c, R,\\Theta )$ as the convolution against the marginal posterior on $Y_k$ ; then we should have $\\max _{\\widetilde{Y}:c(\\widetilde{Y})=c} \\sum _{k=1}^c f(\\widetilde{Y}_k)= \\max _{\\widetilde{Y}_c = (c-1)L+1, \\ldots , n-cL+1}\\Bigg [ f(\\widetilde{Y}_c)+ \\max _{\\widetilde{Y}_1, \\ldots , \\widetilde{Y}_{c-1}} \\sum _{k=1}^{c-1} f(\\widetilde{Y}_k)\\Bigg ].$ This important observation allows us to obtain $\\widehat{Y}_c$ using the dynamic programming approach listed in Algorithm REF , as Theorem REF formalizes.",
"[htbp] Find $\\widehat{Y}_c$ using dynamic programming.",
"Construct partial maxima and backtrack pointers: Set $m_1(\\widetilde{Y}_1) = f(\\widetilde{Y}_1)$ for $\\widetilde{Y}_1 = 1, \\ldots , n-cL+1$ .",
"For $k = 2, \\ldots , c$ and $\\widetilde{Y}_k = (k-1)L + 1, \\ldots , n-(c-k+1)L + 1$ do: set backtrack pointers $A_{k-1}(\\widetilde{Y}_k) =\\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}_{k-1} = (k-2)L+1, \\ldots , \\widetilde{Y}_k-L}m_{k-1}(\\widetilde{Y}_{k-1}).$ and set partial sum maximum $m_k$ as $m_k(\\widetilde{Y}_k) =f(\\widetilde{Y}_k) + m_{k-1} \\Big (A_{k-1}(\\widetilde{Y}_k)\\Big ).$ Reconstruct centroid $\\widehat{Y}_c$ using backtrack pointers: Set last binding site position: $\\widehat{Y}_{c,c} = \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}_c = (c-1)L+1, \\ldots , n-L+1}m_c(\\widetilde{Y}_c).$ Note that, by construction, $\\max _{\\widetilde{Y}:c(\\widetilde{Y})=c} \\sum _{k=1}^c f(\\widetilde{Y}_k) =m_c(\\widehat{Y}_{c,c})$ .",
"For $k = c, \\ldots , 2$ do: recover the remainder of $\\widehat{Y}_c$ by setting $\\widehat{Y}_{c,k-1} = A_{k-1}(\\widehat{Y}_{c,k})$ .",
"Theorem 3 Algorithm REF correctly identifies the paired local centroids $\\widehat{Y}_c =\\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y}) = c}{\\mathbb {E}}_{Y {\\,|\\,}c(Y)=c, R, \\Theta } \\big [ H_A(\\widetilde{Y}, Y) \\big ].$ From Lemma REF we know that $\\widehat{Y}_c$ is the argument of $\\max _{\\widetilde{Y}:c(\\widetilde{Y})=c} \\sum _{k=1}^c f(\\widetilde{Y}_k)$ .",
"The key device in Algorithm REF is to exploit the recursion in Equation REF to define $m_1(\\widetilde{Y}_1) = f(\\widetilde{Y}_1)$ and $m_k(\\widetilde{Y}_k) =f(\\widetilde{Y}_k) + \\max _{\\widetilde{Y}_{k-1} = (k-2)L+1, \\ldots , \\widetilde{Y}_k-L}m_{k-1}(\\widetilde{Y}_{k-1}),$ for $k > 1$ , to store partial sum maxima.",
"Now it follows that $\\max _{\\widetilde{Y}:c(\\widetilde{Y})=c} \\sum _{k=1}^c f(\\widetilde{Y}_k)= \\max _{\\widetilde{Y}_c = (c-1)L+1, \\ldots , n-cL+1} m_c(\\widetilde{Y}_c),$ and so Step 3 must be correct.",
"The correctness of Step 4 relies on the right specification of $m$ in Steps 1 and 2; but these steps are a straightforward application of Equation REF using the definition of $m_1$ and a formulation of Equation REF based on the backtrack pointers $A$ , and so the algorithm is correct.",
"We note that the paired local centroids minimize an expected posterior upper bound $H_A$ on the loss $H$ , and so the actual local centroid might not be attained.",
"We expect, however, that for common cases in which the motif coverage $c(Y) L$ is much smaller than $n$ that the bound is tight since $H_A$ approximates $H$ well and thus the two local centroids often coincide."
],
[
"Global centroid",
"While the local centroids already convey information about the distribution of posterior mass in the space of binding site configurations, the end goal of the analysis is a point estimate that is, in itself, a good representative of the space.",
"Following the strategy we outlined in the beginning of this section, we can further summarize the information in the local centroids by identifying a configuration $\\widehat{Y}$ that minimizes the expected conditional Hamming loss, as in Equation REF .",
"This approach, however, entails the same difficulties as defining the centroid based on all points in the space, and it is thus not treatable by a systematic approach—we are now just restricting the configurations to the local centroids.",
"The global centroid can be defined by direct enumeration of all possible configurations while keeping the minimizer of the expected conditional posterior loss, but this “brute-force” approach considers an exponential number of solutions.",
"A simple heuristic is to restrict the global centroid to be one of the local centroids, $\\widehat{Y} = \\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y} \\in \\lbrace \\widehat{Y}_c\\rbrace _{c=0}^C}{\\mathbb {E}}_{c(Y) {\\,|\\,}R, \\Theta } \\big [ H(\\widehat{Y}_{c(Y)}, \\widetilde{Y}) \\big ].$ Another alternative is to just take as global centroid the local centroid of the modal number of binding sites, $\\widehat{Y}=\\widehat{Y}_{c^*}$ , where $c^* \\doteq \\mathop {\\rm arg\\,max}\\limits _{c=0, \\ldots , C} {\\mathbb {P}}(c(Y)=c{\\,|\\,}R, \\Theta )$ .",
"From now on we adopt the global centroid in Equation REF for simplicity and, again, computational expediency.",
"Before we continue to our next example, let us remark that a constrained, on the number of binding sites, global centroid might be more computationally feasible since we are restricting the space of available configurations.",
"For instance, consider the 1-global centroid, $\\widehat{Y}_o \\doteq \\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y}) = 1}{\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\Big [ H(\\widetilde{Y}, Y) \\Big ].$ As when defining local centroids, we can approximate $\\widehat{Y}_o$ using a paired loss, and since $\\begin{split}{\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\Big [ H_A(\\widetilde{Y}, Y) \\Big ] & =\\sum _{c=0}^C \\sum _{Y:c(Y)=c} \\sum _{k=1}^cH_1(\\widetilde{Y}, Y_k) {\\mathbb {P}}(Y {\\,|\\,}R, \\Theta ) \\\\& =\\sum _{i=1}^n \\sum _{c=0}^C \\sum _{Y:c(Y)=c} \\sum _{k=1}^cH_1(\\widetilde{Y}, i) {\\mathbb {P}}(Y_k = i {\\,|\\,}R, \\Theta ) \\\\& =\\sum _{i=1}^n H_1(\\widetilde{Y}, i)\\sum _{c=0}^C \\sum _{Y:c(Y)=c} \\sum _{k=1}^c{\\mathbb {P}}(Y_k = i {\\,|\\,}R, \\Theta ) \\\\& =\\sum _{i=1}^n H_1(\\widetilde{Y}, i) P_c(i {\\,|\\,}R, \\Theta ),\\end{split}$ where $P_c(i {\\,|\\,}R, \\Theta ) \\doteq \\sum _{c=1}^C \\sum _{Y : c(Y)=c}\\sum _{k=1}^c {\\mathbb {P}}(Y_k = i {\\,|\\,}R, \\Theta ),$ we have that $\\widehat{Y}_o =\\mathop {\\rm arg\\,min}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=1}{\\mathbb {E}}_{Y {\\,|\\,}R, \\Theta } \\Big [ H_A(\\widetilde{Y}, Y) \\Big ]= \\mathop {\\rm arg\\,max}\\limits _{\\widetilde{Y}:c(\\widetilde{Y})=1}G(\\widetilde{Y}, \\cdot ) * P_c(\\cdot {\\,|\\,}R, \\Theta ).$ It is important to note that while the restriction of one binding site might seem artificial, the derivation of $\\widehat{Y}_o$ is helpful in recognizing sequence regions that are likely to host binding sites.",
"In fact, since $P_c$ captures the posterior probability of having a binding site starting at each position, and considering the overlap gain $G$ , the convolution of $G$ and $P_c$ highlights positions that have higher posterior probability of being covered by a binding site.",
"Example 2 We revisit the same sequence from Example REF , but now allow for at most $C = \\lfloor n / L \\rfloor = 33$ binding sites, and adopt the prior given in Equation REF with $b = 3$ and thus $p = 1 - b/n = 0.985$ .",
"Using Algorithm REF we are able to compute the conditional marginal posteriors ${\\mathbb {P}}(c(Y) {\\,|\\,}R, \\Theta )$ and ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y), R,\\Theta )$ for $k = 1, \\ldots , c(Y)$ .",
"These posterior distributions yield the local centroids—according to Algorithm REF —and the global centroid from Equation REF .",
"In Table REF we list the marginal posterior ${\\mathbb {P}}(c(Y) = c {\\,|\\,}R, \\Theta )$ up to the smallest $c$ such that ${\\mathbb {P}}(c(Y) \\le c {\\,|\\,}R, \\Theta ) > 0.95$ , along with the local centroids; the global centroid $\\widehat{Y}_C$ is highlighted.",
"Interestingly, the global centroid coincides with the local centroid from the modal number of binding sites.",
"Table: Centroids and marginal posterior distribution of number of bindingsites.",
"The global centroid and the modal number of binding sites arehighlighted in bold.In Figure REF we display the posterior probabilities of binding site coverage $P_c$ from Equation REF , along with the convolutions that are needed to define the 1-global centroid $\\widehat{Y}_o = 36$ .",
"As can be seen, position 36 has a lot of support, being present in all the local centroids listed in Table REF ; in fact, the probability of a binding site starting at position 36 is greater than $50\\%$ .",
"Figure: Posterior binding site coverage P c P_c in solid line and convolutionG*P c G * P_c in dotted line.",
"Local centroids are listed below in gray; the globalcentroid is in black.While $P_c$ can provide us guidance for which positions are likely to start a binding site, using $P_c$ to define local centroids can be misleading.",
"For instance, we could expect that the local centroid with three binding sites—the modal number of binding sites—would be, following a decreasing order on $P_c$ , 36, 63, and 147.",
"However, if we examine the marginal posteriors ${\\mathbb {P}}(Y_k {\\,|\\,}c(Y)=3, R, \\Theta )$ in Figure REF we realize that position 13 is favored over position 63 because, if $F_k \\doteq G * {\\mathbb {P}}_k(\\cdot {\\,|\\,}c(Y)=3, R, \\Theta )$ , $F_1(13) + F_2(36) > F_1(36) + F_2(63)$ .",
"Figure: Marginal posterior distributionsℙ(Y k |c(Y)=3,R,Θ){\\mathbb {P}}(Y_k {\\,|\\,}c(Y) = 3, R, \\Theta ) in solid line and convolutions G*ℙ(·|c(Y),R,Θ)G *{\\mathbb {P}}(\\cdot {\\,|\\,}c(Y), R, \\Theta ) in dotted line.",
"The local centroid isdisplayed at the bottom."
],
[
"Multiple sequences, multiple binding sites per sequence, random\nmotif",
"We are ready to address our model in broader generality: the dataset now comprises $m$ sequences, $R = \\lbrace R_i\\rbrace _{i=1}^m$ , and thus binding site configurations are also indexed by sequence, $Y = \\lbrace Y_i\\rbrace _{i=1}^m$ .",
"As before, we have that $Y$ is independent of motif parameters $\\Theta $ , but we further assume that sequences and configurations are conditionally independent given $\\Theta $ : ${\\mathbb {P}}(R, Y {\\,|\\,}\\Theta ) = \\prod _{i=1}^m {\\mathbb {P}}(R_i, Y_i {\\,|\\,}\\Theta )= \\prod _{i=1}^m {\\mathbb {P}}(R_i {\\,|\\,}Y_i,\\Theta ) {\\mathbb {P}}(Y_i).$ Given $\\Theta $ we would be able to apply the methods discussed this far to each sequence separately: compute forward and backward sums to obtain marginal posterior probabilities for each $Y_i$ and then find local centroids and the $i$ -th global centroid.",
"We will, however, assume that $\\Theta $ is random, $\\theta _j \\sim \\text{Dir}(\\alpha _j), \\quad j = 0, 1, \\ldots , L,$ independently, and we thus wish to also conduct inference on the background and motif compositions.",
"This assumption, albeit more realistic, complicates matters, since the marginal unconditioned posterior distributions of $Y$ and $\\Theta $ are not readily available; we are now required to estimate them before obtaining centroid estimates.",
"To this end, we present next a Gibbs sampler [8], [14] that draws $Y_i$ for each sequence given $\\Theta $ and then samples $\\Theta $ conditional on the binding site configurations $Y$ , similar to the approach in [15]."
],
[
"Sampling $\\Theta $ given {{formula:ee5d39da-eadf-4bcf-a5da-810ee99ab33d}} and {{formula:34447652-2824-42ff-bba0-6f1e36e971f6}}",
"Since the prior on $\\Theta $ is conjugate, we should be able to sample $\\Theta $ exactly from a Dirichlet distribution.",
"From Equations REF and REF we have $\\begin{split}{\\mathbb {P}}(\\theta _0 {\\,|\\,}Y, R) & \\propto \\Bigg [ \\prod _{i=1}^m \\prod _{s \\in \\mathcal {S}} \\prod _{j \\in BG_i}\\theta _{0,s}^{I(R_{ij}=s)} \\Bigg ]\\Bigg [ \\prod _{s \\in \\mathcal {S}} \\theta _{0,s}^{\\alpha _{0,s}-1} \\Bigg ] \\\\& =\\prod _{s \\in \\mathcal {S}}\\theta _{0,s}^{\\sum _{i=1}^m \\sum _{j \\in BG_i} I(R_{ij}=s) + \\alpha _{0,s}-1},\\end{split}$ and so $\\theta _0 {\\,|\\,}Y, R \\sim \\text{Dir}(N_0(Y, R) + \\alpha _0)$ , where $N_0(Y, R) = \\lbrace N_{0,s}\\rbrace _{s \\in \\mathcal {S}}$ and $N_{0,s} = \\sum _{i=1}^m \\sum _{j \\in BG_i} I(R_{ij}=s)$ is the number of background positions across all sequences that have symbol $s$ .",
"Similarly, for the $j$ -th position in the motif, $\\begin{split}{\\mathbb {P}}(\\theta _j {\\,|\\,}Y, R) & \\propto \\Bigg [ \\prod _{i=1}^m \\prod _{s \\in \\mathcal {S}} \\prod _{k=1}^{|Y_i|}\\theta _{j,s}^{I(R_{i,Y_{ik}+j-1}=s)} \\Bigg ]\\Bigg [ \\prod _{s \\in \\mathcal {S}} \\theta _{j,s}^{\\alpha _{j,s}-1} \\Bigg ] \\\\& =\\prod _{s \\in \\mathcal {S}}\\theta _{0,s}^{\\sum _{i=1}^m \\sum _{j \\in BG_i} I(R_{ij}=s) + \\alpha _{0,s}-1},\\end{split}$ and thus $\\theta _j {\\,|\\,}Y, R \\sim \\text{Dir}(N_j(Y, R) + \\alpha _j)$ , with $N_j(Y, R) = \\lbrace N_{j,s}\\rbrace _{s \\in \\mathcal {S}}$ and $N_{j,s} = \\sum _{i=1}^m \\sum _{k=1}^{|Y_i|} I(R_{i,Y_{ik} + j - 1} = s)$ is the number of motif $j$ -th positions across all sequences and binding sites that have symbol $s$ ."
],
[
"Sampling $Y_i$ given {{formula:6ad0a3d0-b309-443f-ab78-854ff8f36dae}} and {{formula:bf1f59e7-c85f-4f02-b153-689156410393}}",
"Each configuration $Y_i$ for the $i$ -th sequence is conditionally independent given $\\Theta $ , so we can devise a sampling procedure that can be applied to each sequence in turn.",
"To simplify the notation, let us drop the sequence index in what follows, that is, $Y_i$ is $Y$ , $R_i$ is $R$ , and so on.",
"We will be following a similar approach to Sections REF and REF , but instead of summing to obtain marginal distributions we will be sampling exactly.",
"To sample from the conditional posterior on $Y$ , we first sample $c(Y)=c$ according to Equation REF and then proceed to sample $Y$ from its last, $c$ -th binding site up to its first binding site.",
"For this reason, this strategy is commonly referred to as “stochastic backtracking”, since it can be regarded as a stochastic version of Step 4 in Algorithms REF and REF .",
"Sampling $Y$ is similar to the predictive update step in [15], which, on its turn, is based on a stochastic variation of expectation-maximization where missing data is imputed [27]; however, here we exploit a hierarchical structure on $c(Y)$ and do not use the collapsing technique of [13].",
"Exploiting the conditional independence of the sequence configurations and Equation REF the last binding site can be sampled using $\\begin{split}{\\mathbb {P}}(Y_c {\\,|\\,}c(Y), R, \\Theta ) & =\\frac{\\sum _{Y_1, \\ldots , Y_{c-1}} {\\mathbb {P}}(R {\\,|\\,}Y, \\Theta )}{\\sum _{\\widetilde{Y}_c} \\sum _{\\widetilde{Y}_1, \\ldots , \\widetilde{Y}_{c-1}}{\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )} \\\\& =\\frac{F_{c-1,Y_c-1} \\lambda (Y_c; \\Theta )}{\\sum _{\\widetilde{Y}_c = (c-1)L + 1}^{n-L+1}F_{c-1,\\widetilde{Y}_c-1} \\lambda (\\widetilde{Y}_c; \\Theta )}.\\end{split}$ To sample the (intermediate) $j$ -th binding site we use a similar expression: $\\begin{split}{\\mathbb {P}}(Y_j {\\,|\\,}Y_{j+1}, \\ldots , Y_c, c(Y), R, \\Theta ) & =\\frac{{\\mathbb {P}}(Y_j, \\ldots , Y_c, c(Y), R, \\Theta )}{\\sum _{\\widetilde{Y}_j} {\\mathbb {P}}(\\widetilde{Y}_j, \\ldots , \\widetilde{Y}_c, c(Y), R, \\Theta )} \\\\& =\\frac{\\sum _{Y_1, \\ldots , Y_{j-1}} {\\mathbb {P}}(R {\\,|\\,}Y, \\Theta )}{\\sum _{\\widetilde{Y}_j} \\sum _{\\widetilde{Y}_1, \\ldots , \\widetilde{Y}_{j-1}}{\\mathbb {P}}(R {\\,|\\,}\\widetilde{Y}, \\Theta )} \\\\& =\\frac{F_{j-1, Y_j-1} \\lambda (Y_j; \\Theta )}{\\sum _{\\widetilde{Y}_j=(j-1)L+1}^{Y_{j+1}-L}F_{j-1, \\widetilde{Y}_j-1} \\lambda (\\widetilde{Y}_j; \\Theta )}.\\end{split}$ By making the convention that $Y_{c+1}=|R|+1$ we can reduce Equation REF to Equation REF .",
"Moreover, note that Equation REF implies that ${\\mathbb {P}}(Y_j {\\,|\\,}Y_{j+1}, \\ldots , Y_c, c(Y), R, \\Theta ) ={\\mathbb {P}}(Y_j {\\,|\\,}Y_{j+1}, c(Y), R, \\Theta ),$ as expected.",
"We summarize the whole procedure in Algorithm REF .",
"Note how Steps 1.1 to 1.3 are analogous to Steps 1 to 3 in Algorithm REF , and how Step 1.4 is an stochastic version of Step 4 in Algorithm REF : as previously stated, we are now sampling backwards instead of summing backwards.",
"To obtain the centroids we follow the procedure described in Section REF , but adopting Monte Carlo estimates of the marginal posterior distributions, for $i = 1, \\ldots , m$ , $\\begin{split}\\widehat{{\\mathbb {P}}}(c(Y_i) = c {\\,|\\,}R) & \\approx \\frac{1}{T} \\sum _{t=1}^T I \\big ( c(Y_i^{(t)}) = c \\big ), \\\\\\widehat{{\\mathbb {P}}}(Y_{ik} = j {\\,|\\,}c(Y_i) = c, R) & \\approx \\frac{\\sum _{t=1}^T I \\big ( Y_{ik}^{(t)} = j \\big )I\\big ( c(Y_i^{(t)}) = c \\big )}{\\sum _{t=1}^T I\\big ( c(Y_i^{(t)}) = c \\big )},\\quad k = 1, \\ldots , c, \\\\\\end{split}$ where $T$ is the number of samples.",
"[htpb] Gibbs sampler for ${\\mathbb {P}}(Y, \\Theta {\\,|\\,}R)$ .",
"Set $\\Theta ^{(0)}$ arbitrarily.",
"For $t = 1, \\ldots $ (until convergence) do: (Sample $Y {\\,|\\,}\\Theta , R$ ) For each sequence $i = 1, \\ldots ,m$ , do: let $n = |R_i|$ , $C = \\lfloor n/L \\rfloor $ and sample $Y_i {\\,|\\,}R_i, \\Theta $ .",
"(Initialize) Set $F_{0,j} = 1$ for $j = 0, 1, \\ldots , n$ and for $c = 1, \\ldots , C$ set $F_{c,j} = 0$ when $j < cL$ .",
"(Compute forward sums) For $c = 1, \\ldots , C$ and $j = cL+1,\\ldots , n$ do: set $F_{c,j}$ as in Equation REF , $F_{c,j} = F_{c,j-1} + F_{c-1,j-L} \\lambda _i(j-L+1; \\Theta ^{(t-1)}),$ where $\\lambda _i$ uses $R_i$ .",
"(Sample $c(Y_i^{(t)}) {\\,|\\,}R_i, \\Theta ^{(t-1)}$ ) For $c = 0,\\ldots , C$ do: compute marginal posterior $c(Y_i)$ as in Equation REF when applied to the $i$ -th sequence, ${\\mathbb {P}}(c(Y_i) = c {\\,|\\,}R_i, \\Theta ^{(t-1)}) =\\frac{F_{c,n} \\binom{n-c(L-1)}{c}^{-1} {\\mathbb {P}}(c(Y_i) = c)}{\\sum _{\\widetilde{c}=0}^C F_{\\widetilde{c},n}\\binom{n-\\widetilde{c}(L-1)}{\\widetilde{c}}^{-1} {\\mathbb {P}}(c(Y_i) = \\widetilde{c})}$ and sample $c^{(t)} \\doteq c(Y_i^{(t)})$ according to ${\\mathbb {P}}(c(Y_i) = c {\\,|\\,}R_i, \\Theta ^{(t-1)})$ .",
"(Sample $Y_i^{(t)} {\\,|\\,}c(Y_i^{(t)})=c^{(t)}, R_i,\\Theta ^{(t-1)}$ ) For $k = c^{(t)}, \\ldots , 1$ do: sample $Y_{ik}^{(t)}$ proportional to $F_{k-1, Y_{ik}^{(t)} - 1} \\lambda _i(Y_{ik}^{(t)}; \\Theta ^{(t-1)})$ as in Equation REF , ${\\mathbb {P}}(Y_{ik}^{(t)} {\\,|\\,}Y_{i,k+1}^{(t)},c(Y_i^{(t)}) = c^{(t)}, R_i, \\Theta ^{(t-1)}) = \\\\= \\frac{F_{k-1, Y_{ik}^{(t)}-1} \\lambda _i(Y_{ik}^{(t)}; \\Theta ^{(t-1)})}{\\sum _{\\widetilde{Y}_k=(k-1)L+1}^{Y_{i,k+1}^{(t)}-L}F_{k-1, \\widetilde{Y}_k-1} \\lambda _i(\\widetilde{Y}_k; \\Theta ^{(t-1)})}$ (Sample $\\Theta {\\,|\\,}Y, R$ ) For $j = 0, \\ldots , L$ compute $N_j(Y^{(t)}, R)$ and then sample $\\theta _j^{(t)} {\\,|\\,}Y^{(t)}, R \\sim \\text{Dir}(N_j(Y^{(t)}, R) + \\alpha _j)$ .",
"Example 3 For the random motif version of Example REF we simulate $m = 20$ sequences of same length $n = 200$ using $\\Theta $ from Table REF and the prior for $Y_i$ , $i = 1, \\ldots , m$ , from Equation REF with $p = 1 - 1/n = 0.995$ .",
"We continue focusing on the inference of binding site configurations in the same sequence from previous examples, which is the first sequence in the simulated dataset.",
"We assume a non-informative prior on $\\Theta $ by setting $\\alpha _{j,s} = 1$ for $s \\in \\mathcal {S}$ and $j = 0, \\ldots , L$ ; the prior on each sequence $Y_i$ is the same prior from Example REF with $p =0.985$ .",
"Algorithm REF is run for $10,\\!000$ iterations to guarantee convergence (diagnostics not shown.)",
"The marginal posterior distribution of $\\Theta $ can be assessed in Figure REF .",
"Since most positions in the sequences are background sequences $\\theta _0$ has very small posterior variances.",
"Also note that the canonical palindromic E-box motif, with consensus CACGTG, is recovered.",
"Figure: Boxplots of MCMC samples for Θ\\Theta (outliers are not shown.",
")The procedure is now similar to what we presented in Example REF ; the main difference is that the marginal posterior distributions are estimated from the MCMC samples.",
"Table REF lists the estimated marginal posterior distribution of the number of binding sites, the local and global centroids.",
"The global centroid does not coincide with the local centroid for the modal number of binding sites.",
"Moreover, the local centroids here are different from the (conditional) local centroids in Example REF , most likely due to the randomness of $\\Theta $ being taken into account.",
"Table: Centroids and estimated marginal posterior distribution of number ofbinding sites.",
"The global centroid and the modal number of binding sites arehighlighted in bold.Figure REF displays the estimated $P_c$ , $G * P_c$ , and the centroids.",
"We see that compared to Example REF some posterior mass has shifted to positions 29 and to the group of positions 166, 167, and 168.",
"Here we clearly see the advantage of a centroid estimator: $G * P_c$ , and later $G * {\\mathbb {P}}_k(\\cdot {\\,|\\,}R)$ , gathers evidence of motif binding from nearby positions, yielding a better summary—according to our choice of loss function—of the distribution of posterior mass.",
"Figure: Estimated posterior binding site coverage P c P_c in solid line andconvolution G*P c G * P_c in dotted line.",
"Local centroids are listed below ingray; the global centroid is in black.The selection of position 167 in the second local centroid $\\widehat{Y}_2$ might seem puzzling since the peaks at positions 36, 63, and 147 hold higher coverage probabilities.",
"Checking $\\widehat{{\\mathbb {P}}}(Y_k {\\,|\\,}R)$ in Figure REF helps dismiss any doubts: most of the support for these positions come from configurations with higher number of binding sites, as evidenced by the respective local centroids, but these configurations hold relatively low posterior mass.",
"When $c(Y)=2$ , the prior on $Y_{2,2}$ assigns more posterior probability to higher positions, close to the end of the sequence, simply because there are more configurations for $Y_{2,2}$ on these positions.",
"It is also important to notice that while none of the positions in the cluster 166–168 has higher marginal posterior mass than positions 63 and 147, the convolution $G * \\widehat{{\\mathbb {P}}}_2(\\cdot {\\,|\\,}R)$ is maximized at position 167, that is, the cluster when taken together has more support from the data, as weighted by $G$ .",
"Figure: Estimated marginal posterior distributionsℙ(Y k |c(Y)=2,R,Θ){\\mathbb {P}}(Y_k {\\,|\\,}c(Y) = 2, R, \\Theta ) in solid line and convolutionsG*ℙ(·|c(Y),R,Θ)G * {\\mathbb {P}}(\\cdot {\\,|\\,}c(Y), R, \\Theta ) in dotted line.",
"The local centroid isdisplayed at the bottom.Example 4 We end this section with an example from the real-world dataset in [30], sequence set yst02r.",
"The dataset contains $m = 4$ sequences each with $n = 500$ letters.",
"We set $L=16$ and adopt a non-informative prior on $\\Theta $ , as in the previous example, and the prior on each $Y_i$ , for the $i$ -th sequence, from Equation REF with $b= 3$ per thousand positions, so $p = 1 - 3 / 1000 = 0.997$ .",
"As in the previous example, $10,\\!000$ iterations suffice to reach convergence.",
"Let us focus on the second sequence.",
"Figure REF pictures the binding site coverage probabilities, along with the local centroids.",
"The global centroid $\\widehat{Y}_C = \\lbrace 85,105,169\\rbrace $ contains three binding sites, and it is also the local centroid for the modal number of binding sites, with $\\widehat{{\\mathbb {P}}}(c(Y)=3 {\\,|\\,}R) = 0.32$ .",
"Since most of the posterior mass in concentrated in configurations with $c(Y) = 3$ , the posterior profiles $\\widehat{{\\mathbb {P}}}(Y_k {\\,|\\,}c(Y)=3, R)$ are similar to $P_c$ and are thus omitted.",
"Figure: Estimated posterior binding site coverage P c P_c in solid line andconvolution G*P c G * P_c in dotted line for real-world dataset, second sequence.Local centroids are listed below in gray; the global centroid is in black.From the MCMC samples we can produce the MAP estimate $\\widehat{Y}_M = \\lbrace 86, 105, 174\\rbrace $ as the configuration with highest frequency among the samples: $\\widehat{{\\mathbb {P}}}(\\widehat{Y}_M {\\,|\\,}R) = 0.032$ .",
"In fact, we can estimate the posterior probability of each sampled binding site configuration and then, using classic multidimensional scaling [9], visualize the estimated posterior distribution in Figure REF .",
"It is interesting to note that the null configuration—that is, without binding sites—is also very likely with posterior probability $0.024$ .",
"In contrast, the global centroid has very small posterior probability, close to $0.001$ ; it sits, however, closer to configurations with high posterior mass, including the local centroids with one, two, and four binding sites.",
"Figure: Estimated posterior distribution of configurations YY based on MCMCsamples and projected using multidimensional scaling.",
"The colors codeconfigurations with different number of binding sites.",
"Bold points mark localcentroids, while a square (bold) point highlights the global centroid.To better assess how the centroid estimator is closer to a mean than a mode estimator, we plot the estimated posterior distribution of the generalized loss function $H$ centered at both $\\widehat{Y}_C$ and $\\widehat{Y}_M$ in Figure REF .",
"Since ${\\mathbb {E}}_{Y {\\,|\\,}R}[H(\\widehat{Y}_M,Y)] = 42.40$ and ${\\mathbb {E}}_{Y {\\,|\\,}R}[H(\\widehat{Y}_C,Y)] = 40.22$ , we see that the binding sites in the centroid configuration are, on average, overlapping two extra positions with the binding sites in all the configurations when compared to the MAP estimate's binding sites.",
"Both estimates are fairly similar, but the centroid reminds us that placing the third binding site at position 169, instead of 174, yields an unlikely configuration, but with a higher chance of overlapping with binding sites in positions 160–175 that have high posterior probability.",
"In the context of Figures REF and REF , the centroid places itself between two clusters that concentrate posterior mass: one with configurations $Y$ such that $25 \\le H(\\widehat{Y}_C, Y) \\le 40$ and another with configurations further away, satisfying $40 \\le H(\\widehat{Y}_C, Y) \\le 50$ .",
"Figure: Estimated posterior distribution of loss function centered atY ˜\\widetilde{Y} for the MAP (Y ˜=Y ^ M \\widetilde{Y} = \\widehat{Y}_M) and centroid (Y ˜=Y ^ C \\widetilde{Y} =\\widehat{Y}_C) estimates."
],
[
"Discussion",
"In this paper we have presented a Bayesian approach, similar to the Gibbs motif sampler in [12], [15], that jointly models motif and background compositions and binding site locations in a set of sequences.",
"More importantly, we discuss and formalize an inferential procedure based on the centroid estimator proposed by [5].",
"As in any Bayesian analysis, we wish to evaluate features of interest in a model based on their posterior distribution; however, if we are required to pick a representative configuration, a point in the parameter space, then a principled approach is to elect a loss function and conduct formal statistical decision analysis.",
"In this sense, by exploring a more refined loss function that depends on position-wise comparisons between sequence states—background or motif positions—we are able to identify a better representative of the posterior space of binding site configurations.",
"As pointed out in [5], the centroid estimator better accounts for the distribution of posterior mass; it is more similar to a median than to a mode, and can thus offer better predictive resolution than the MAP estimator [2].",
"When applied to motif discovery, the centroid estimator captures information in the vicinity of binding site positions through a convolution in marginal posterior distributions of binding sites.",
"Given the combinatorial number of possible configurations in the parameter space it is not feasible to identify the centroid estimate through enumeration or even a systematic approach.",
"Yet, we devise an approximative scheme that efficiently optimizes an upper bound on the posterior expected loss and thus provides a related centroid.",
"Despite its heuristic nature, the proposed method has another advantage besides computational convenience: it allows for an informative depiction of the posterior distribution on binding site configurations.",
"First, when defining the local centroids, we are able to assess the contributions from each binding site through their marginal posterior distributions conditional on the number of binding sites, and, in particular, through the convolution of these marginal profiles with the gain filter; secondly, when finding the global centroid we explore the marginal posterior distribution on the number of binding sites.",
"Moreover, other representations might be helpful in understanding the distribution of posterior mass, as in the use of $P_c$ (in Equation REF ) to pinpoint the 1-global centroid and measure the overall support of the configurations to a binding site at some specific position in the sequence.",
"These comments are in the spirit of an estimator being also a communicator of the posterior space and the particular choice of prior distribution [3].",
"It is important to note that even when the model is accurate, a poor inference might fail in recovering relevant features of the space.",
"In Example 2, the MAP estimate is the null configuration, while the centroid indicates three binding sites that represent a group of configurations that jointly pool significant posterior mass.",
"It is also common that the posterior distribution is too complex to be reasonably captured by a single representative; in this case the expected posterior loss could also be used to partition the space and further define additional representatives as conditional estimates on each subspace.",
"This is a direction of work that warrants interest and that we intend to follow next.",
"Further improvements can be obtained by specifying a more complex model that accounts, for example, for higher order Markov chains with more states for the background, as in [24], [16], phylogenetic profiles [20], structural information [31], a variable motif length, or dependency among motif positions.",
"As pointed out by [11], motif discovery using sequence only is well known for low signal-to-noise ratio; future extensions would also incorporate other data sources, such as gene expression or ChIP-Seq data, to increase the signal-to-noise ratio."
],
[
"Acknowledgements",
"The author would like to thank Antonio Gomes for the helpful discussions and comments in the text."
]
] | 1204.1571 |
[
[
"Electron-nuclei spin dynamics in II-VI semiconductor quantum dots"
],
[
"Abstract We report on the dynamics of optically induced nuclear spin polarization in individual CdTe/ZnTe quantum dots loaded with one electron by modulation doping.",
"The fine structure of the hot trion (charged exciton $X^-$ with an electron in the $P$-shell) is identified in photoluminescence excitation spectra.",
"A negative polarisation rate of the photoluminescence, optical pumping of the resident electron and the built-up of dynamic nuclear spin polarisation (DNSP) are observed in time-resolved optical pumping experiments when the quantum dot is excited at higher energy than the hot trion triplet state.",
"The time and magnetic field dependence of the polarisation rate of the $X^-$ emission allows to probe the dynamics of formation of the DNSP in the optical pumping regime.",
"We demonstrate using time-resolved measurements that the creation of a DNSP at B=0T efficiently prevents longitudinal spin relaxation of the electron caused by fluctuations of the nuclear spin bath.",
"The DNSP is built in the microsecond range at high excitation intensity.",
"A relaxation time of the DNSP in about 10 microseconds is observed at $B=0T$ and significantly increases under a magnetic field of a few milli-Tesla.",
"We discuss mechanisms responsible for the fast initialisation and relaxation of the diluted nuclear spins in this system."
],
[
"Introduction",
"The confinement of single electrons in semiconductor quantum dots (QDs) and the control of their spin has been motivated by perspectives of using the electron spin as the ultimate solid state system to store and process quantum information.",
"In the commonly studied III-V semiconductor QDs, the hyperfine interaction of the electron with the fluctuating nuclear spins limits the time scale on which an electron spin can be manipulated at low magnetic field.",
"It has been proposed that a full polarization of the nuclei could cancel the decoherence of the electron induced by the fluctuating hyperfine field [1].",
"Alternatively, the decoherence created by nuclei could be circumvented by using isotopically purified II-VI materials [2] since Zn, Cd, Mg, O, Se and Te all have dominant isotopes without nuclear spins.",
"However, as highlighted in reference 3 and 4, the interaction between a confined electron in a II-VI QD and the low density of nuclear spins I=1/2 in a QD volume ten to one hundred times smaller than InAs/GaAs QDs leads to some spin dynamics which is fundamentally different from the one observed in III-V systems [3], [4].",
"Due to the small QD size and low density of nuclear spins, the electron-nuclei dynamics in II-VI QDs is ruled by a large Knight field and significant nuclear spin fluctuations despite a small Overhauser field.",
"Consequently, the nuclei-induced spin decoherence of the electron is also an issue in II-VI QDs.",
"However, the built-up of a dynamic nuclear spin polarisation (DNSP) at B=0T, can be much faster than the relaxation induced by the dipole interaction between nuclear spins allowing the creation of a strong non-equilibrium DNSP [3].",
"Under these conditions, decoherence of the electron should be efficiently suppressed.",
"Experimental study of the electron-nuclear dynamics in II-VI QDs are few.",
"Providing a thorough experimental study of this system at a single dot level is the aim of this work.",
"In this paper, we report on the dynamics of coupled electron and nuclear spins polarization in individual CdTe/ZnTe QDs with a resident electron introduced by aluminium modulation doping.",
"Here, in contrast to gated structures where the number of resident charges is controlled by an applied voltage, we use the characteristic spectral feature of the charged exciton triplet state observed in photoluminescence excitation spectra (PLE) to identify QDs containing a single resident electron.",
"Non-resonant circularly polarized excitation of the negatively charged exciton has been shown to lead to a polarisation of the nuclear spins in both III-V [6] and II-VI QDs [4] and will be used throughout this study to build-up and probe DNSP.",
"In II-VI QDs, the Overhauser shift is much smaller than the photoluminescence (PL) linewidth and cannot be observed directly: The nuclear field is detected through the polarization rate of the resident electron which is controlled by the nuclear spin fluctuations (NSF).",
"We give a description of the studied structures and experimental techniques in section II of this paper.",
"In section III, we discuss nuclear spin-polarisation in II-VI QDs, and derive orders of magnitude for the effective hyperfine fields encountered in this system.",
"Experimental evidences of $X^-$ triplet states and details about the mechanism of optical spin injection and build-up of negative circular polarization in singly charged CdTe/ZnTe QDs are presented in section IV.",
"In section V we describe how the polarization of nuclear spins influences spin dynamics of the confined electron.",
"The dynamics of nuclear spin polarization is considered in section VI.",
"Finally, in section VII we present and discuss results of coupled electron/nuclei spin decay in the absence of optical excitation."
],
[
"Sample and experiment",
"The sample used in this study is grown on a ZnTe substrate and contains CdTe/ZnTe QDs.",
"A 6.5- monolayer-thick CdTe layer is deposited at $280\\,^{\\circ }\\mathrm {C}$ by atomic layer epitaxy on a ZnTe barrier grown by molecular beam epitaxy at $360\\,^{\\circ }\\mathrm {C}$ .",
"The dots are formed by the high Tellurium deposition process described in reference 7 and protected by a 100-nm-thick ZnTe top barrier [7].",
"A 20 nm thick Al doped ZnTe layer is introduced 30 nm above the QDs leading to an average negative charging of the QDs.",
"The height of the QDs core is about 2-3 nanometers and their diameter is 10 to 20 nm.",
"We estimate an average QD volume of about 250 $nm^3$ containing $\\approx 8000$ nuclei, 1200 of which carry a spin 1/2.",
"Optical addressing of individual QDs containing a single electron is achieved using micro-spectroscopy techniques.",
"A high refractive index hemispherical solid immersion lens is mounted on the bare surface of the sample to enhance the spatial resolution and the collection efficiency of single-dot emission in a low-temperature ($T=5K$ ) scanning optical microscope [8].",
"Despite the quite large QD density ($\\approx 10^{10}cm^{-2}$ ) and the large number of dots in the focal spot area, single QD transitions can be identified by their spectral signatures.",
"A weak magnetic field of a few tens of milli-Tesla can be applied in Voigt or Faraday configuration using permanent magnets.",
"To investigate the mechanisms of spin injection, the QDs are excited with energy tunable picosecond ($\\approx 2ps$ ) laser pulses from a frequency doubled optical parametric oscillator with a repetition time of 13ns.",
"A delay line can be used to divide a single pulse into one co and one cross-polarized pulses.",
"Time-resolved experiments to observe slower dynamics (optical pumping of electron and nuclei) are performed using a modulated tunable continuous wave (CW) dye laser.",
"Laser pulses of controllable duration and polarisation are created using Acousto-Optic Modulators or an Electro-Optic modulator with rise times of about 10ns.",
"The collected PL is dispersed by a $1m$ double monochromator before being detected by a CCD camera or a fast avalanche photodiode in conjunction with a time-correlated photon-counting unit with an overall time resolution of about 50 ps.",
"In CW experiments, the polarisation rate of the PL is measured using a birefringent prism to separate the $\\sigma +$ from the $\\sigma -$ component and to detect them at the same time on different areas of the CCD camera."
],
[
"Nuclear spin polarization in II-VI quantum dots",
"In a singly charged QD under the injection of spin polarized electrons, a nuclear spin polarization builds up by integration over many mutual spin flip-flops of the confined electrons and the lattice nuclei.",
"This nuclear magnetic field modifies the coherent electron spin dynamics and consequently the average polarization of the PL of the $X^-$ .",
"The knowledge of the nuclear spin polarization can then be used to estimate the resident electron spin polarization.",
"In this section, we want to estimate the order of magnitude of the nuclear spin polarization that can build-up in a II-VI QD and its influence on the spin dynamics of a confined electron.",
"The dominant contribution to the coupling between the confined electron and the nuclear spins originates from a Fermi contact hyperfine interaction.",
"This interaction can be written as [23]: $ H_{hf}=\\nu _0\\sum _iA^I_i|\\psi (R_i)|^2(I^i_z\\sigma _z+\\frac{I^i_+\\sigma _-+I^i_-\\sigma _+}{2})$ where R$_i$ is the position of the nuclei $i$ with spin I$^i$ and hyperfine interaction constant A$^I_i$ .",
"$\\sigma $ and I$^i$ are the spin operators of the electron and nuclei respectively.",
"$\\nu _0$ is the volume of the unitary cell containing Z=2 nuclei (one Cd and one Te).",
"This Hamiltonian can be decomposed in a static part affecting the energy of the electron and nuclear spins and a dynamical part proportional to $(I^i_+\\sigma _-+I^i_-\\sigma _+)$ , allowing for the transfer of angular momentum between the electron and nuclear spin system.",
"The static part of the hyperfine interaction leads to the notion of effective magnetic field, either seen by the electron due to the spin polarized nuclei (Overhauser field $\\overrightarrow{B}_N$ ), or by a nucleus at position R$_i$ due to a spin polarized electron (Knight field $\\overrightarrow{B}_e^i$ ).",
"These fields are defined by the electron-nuclei interaction energy: $H_{hf}=g_e\\mu _B\\overrightarrow{\\sigma }.\\overrightarrow{B}_N=-\\sum _i\\mu _I^i\\overrightarrow{I}^i.\\overrightarrow{B}_e^i$ where $g_e$ is the Lande factor of the electron and $\\mu _I^i$ the magneton of nucleus $i$ with spin $I^i$ defined by $\\mu _I^i=\\hbar \\gamma _I^iI$ with $\\gamma _I^i$ the gyromagnetic ratio of the nucleus $i$ ."
],
[
"Overhauser field in a CdTe/ZnTe quantum dot",
"The maximum Overhauser field resulting from a complete polarization of the nuclei, $B_N^{max}$ , is defined by intrinsic parameters characterizing the material and the hyperfine interaction inside the material [5], [18].",
"The Overhauser field can be written as: $B_N=\\frac{\\nu _0}{g_e\\mu _B}\\sum _i^{N_I}A^I_i|\\psi (R_i)|^2\\langle I_z^i\\rangle $ where the sum runs over N$_I$ , the number of nuclei carrying a spin I.",
"When all the Cd and Te nuclear spins are polarized and if we assume a homogeneous electron wave function $\\psi (R)=\\sqrt{2/(\\nu _0N_L)}$ , with N$_L$ the total number of nuclei in the QD, the nuclear field reads: $B_N^{max}=\\frac{1}{g_e\\mu _B}(I^{Cd}A^{Cd}p^{Cd}+I^{Te}A^{Te}p^{Te})$ The nuclear spin of Cd and Te are $I^{Cd}$ =$I^{Te}$ =1/2 and their corresponding abundance $p^{I}=N_I/N_L$ are $p^{Cd}=0.25$ and $p^{Te}=0.08$ (see table REF ).",
"Taking the hyperfine coupling constants $A^{Cd}\\approx $ -31$\\mu $ eV, $A^{Te}\\approx $ -45$\\mu $ eV from reference 5 and an average electron Lande factor $g_e\\approx -0.5$ [19], [20], we obtain $B_N^{max}\\approx $ 200mT.",
"Table: Isotopic abundance, nuclear spin II and magnetonof the nucleus μ I \\mu _I for Cd and Te alloys .μ I \\mu _I is given in unit of the nuclear magneton μ N \\mu _N.The Overhauser field really obtained in a QD under optical pumping, $B_N$ , is proportional to the average nuclear spin polarization $\\langle I_z\\rangle $ and reaches $B_N^{max}$ when $\\langle I_z\\rangle $ =1/2.",
"As the electron Lande factor in CdTe/ZnTe QDs is negative and the hyperfine constants are negative, the sign of $B_N$ is fixed by the sign of $\\langle I_z\\rangle $ which is given by the average electron spin polarization [22], [5] $\\langle S_z \\rangle $ along the QD growth axis $z$ ($\\langle S_z\\rangle $ =1/2 for fully polarized spin up electrons).",
"In the present study, the resident electron is pumped down ($\\langle S_z\\rangle \\le $ 0) for a $\\sigma $ + excitation (spin $|\\downarrow \\rangle $ electron) in the presence of a positive external magnetic field.",
"Thus, a $\\sigma $ + excitation leads to an Overhauser field, $B_N$ , antiparallel to the applied magnetic field ($B_N<0$ )."
],
[
"Knight field in a CdTe/ZnTe quantum dot",
"At zero external magnetic field, the formation of a nuclear spin polarization is only possible if the effective field induced by the electron spin on the nuclei exceeds the local field B$_l$ created by the nuclear dipole-dipole interaction [22].",
"The Knight field is inhomogeneous across the nuclear ensemble because the electron wave function is not constant across the QD.",
"In the core of the QD where the Knight field is the strongest, the spin diffusion induced by the nuclear dipole-dipole coupling is suppressed and it is there that the nuclear spins may become polarized.",
"The magnitude of the time averaged Knight field for a nucleus with a hyperfine constant A$^I$ at the position R$_i$ is given by $B_e^i=-\\nu _0\\frac{A^I}{\\mu _I}|\\psi (R_i)|^2(\\langle S_z\\rangle f_e)$ where $f_e$ is the probability that the dot is occupied by an electron.",
"Considering a constant electron/nuclei overlap (homogeneous wave function $\\psi (R)=\\sqrt{2/(\\nu _0N_L)}$ ) one can obtain [23], [24] the maximum Knight field for nuclei with a hyperfine constant A$^I$ : $ B_e^{max}=-\\frac{A^I}{N_L\\mu _I}$ With $\\mu _{Cd}\\approx -0.6$ and a total number of nuclei in the QD assumed to be $N_L\\approx 8 \\times 10^3$ , $B_e^{max}\\approx $ 100 mT is derived for Cd nuclei.",
"A Knight field of 10mT is typical for InAs/GaAs QDs [25].",
"The difference is a consequence of the smaller QD size in II-VI materials.",
"However, B$_e$ follows the distribution of the electron wave function in the dot leading to a nuclear site-dependent field varying across the dot and in optical pumping experiments, only a weighted-averaged value of the Knight field can be accessible."
],
[
"Dynamic nuclear spin polarization",
"Under circularly polarized CW excitation, the rate equation describing the nuclear spin polarization $I_z$ in a singly negatively charged QD can be written as [27]: $ \\frac{\\partial I_z}{\\partial t}=\\frac{1}{T_{1e}}(\\frac{4}{3}I(I+1)S_0-I_z)-\\frac{1}{T_{dd}}I_z-\\frac{1}{T_{r}}I_z$ The last term on the right side of the equation accounts for any spin relaxation mechanism except the dipole-dipole interaction between nuclei which is described by the relaxation time $T_{dd}$ .",
"The first term corresponds to the transfer of angular momentum between the spin of the electron and the nuclear spin bath, with $S_0$ the polarization rate of the injected electrons and T$_{1e}$ the probability of an electron/nuclei spin \"flip-flop\" given by [26]: $\\frac{1}{T_{1e}}=\\frac{1}{T_{1e}^0}\\frac{1}{1+(\\Delta E_{eZ}/\\hbar )^2\\tau _e^2}$ Here $\\tau _e$ is the correlation time of the electron/nuclear spin interaction [27].",
"$\\Delta E_{eZ}=g_e\\mu _B(B_{ext}+B_N)$ is the electron Zeeman splitting which depends on the external magnetic field, B$_{ext}$ , and the effective nuclear field B$_N$ .",
"This term provides a feedback process mechanism between the spin transfer rate and the nuclear polarization leading to the enhancement of flip-flop processes when B$_N$ reduces the electron Zeeman splitting.",
"This feedback is responsible for the bi-stability in the nuclear spin polarization observed under magnetic field in InAs/GaAs QDs [6], [26], [18].",
"$T_{1e}^0$ , is given by: $\\frac{1}{T_{1e}^0}=f_e\\tau _e(\\frac{E_{hf}}{\\hbar })^2$ with $E_{hf}=\\nu _0A_i^I|\\psi (R_i)|^2$ the interaction energy between the electron and nuclear spin I at position $R_i$ .",
"$T_{1e}^0$ corresponds to the nuclear spin relaxation induced by the electron at zero electron splitting.",
"For a homogeneous electron wave function ($\\psi (R)=\\sqrt{2/(\\nu _0N_L)}$ ), we obtain $1/T_{1e}^0=f_e\\tau _e(2A^I_i/(N_L\\hbar ))^2$ .",
"The contribution of the dipole-dipole interaction to the relaxation process is given, in the presence of an external magnetic field, B$_{ext}$ , by: $\\frac{1}{T_{dd}}=\\frac{1}{T_{dd}^0}\\frac{B_l^2}{(B_{ext}+B_e)^2+B_l^2}$ where B$_l$ is the local field describing the nuclear spin-spin interaction and T$_{dd}^0$ the characteristic time of this interaction at zero field [28], [29], [30].",
"This formula describes the acceleration of the nuclear spin relaxation when an applied magnetic field compensates the Knight field.",
"Figure: Nuclear spin polarization obtained from equation() for a single family of nuclei with an averagehyperfine coupling A av _{av}=-40μ\\mu eV and abundancep av _{av}=16% and with the parameters: f e _e=0.4,S 0 S_0=-0.4, T dd 0 _{dd}^0=5μ\\mu s, B l _l=2.5mT, dd=2.5nm,ξ\\xi =5nm, τ e \\tau _e=5ns, T r _{r}=5ms.",
"The inset shows theapproximation of the Gaussian electron density withξ\\xi =5nm and L z _z=2.5nm by a step function used in thenumerical calculation.The magnetic field dependence of the nuclear spin polarization for a single family of nuclei with an average hyperfine coupling A$_{av}$ =-40$\\mu $ eV and abundance p$_{av}$ =16% obtained by a numerical determination of the steady state of the rate equation (REF ) is presented in Fig.",
"REF .",
"In this model, the electron wave function is described by a Gaussian function in the QD plane and a constant function in the $z$ direction: $\\psi (\\rho ,z)=\\frac{1}{\\sqrt{L_z}}\\frac{1}{\\xi \\sqrt{\\pi }}e^{-\\frac{\\rho ^2}{2\\xi ^2}}$ where $L_z$ is the thickness of the QD and $\\xi $ the lateral extension of the electron wave function.",
"For the numerical calculation, the Gaussian wave function is approximated by a 7 steps function (inset of Fig.REF ) and the 7 coupled differential equations corresponding to the 7 families of nuclear spins (i.e.",
"with different Knight fields) are solved simultaneously.",
"The acceleration of the dipole-dipole interaction between the nuclear spins when the external field compensate the Knight field is responsible for the decrease of the nuclear spin polarisation observed at low negative magnetic field (between $B_z\\approx -50mT$ and $B_z\\approx -100mT$ ).",
"The position of this minimum depends on the average electron spin polarization and is a measurement of the mean value of the Knight field.",
"The nuclear spin polarization does not drop to zero because of the inhomogeneity of the Knight field: the condition B$_{tot}=0$ is satisfied only for a small number of nuclei at any given B$_{ext}$ .",
"Provided B$_e$ $\\gg $ B$_l$ , the majority of nuclei experience negligible change in depolarization.",
"The feedback process occurring when the Overhauser field compensates the applied magnetic field leads to a strong increase of the nuclear spin polarization at low positive magnetic field ($B_z\\approx 50mT$ ).",
"This resonant effect is enhanced by a long electron correlation time $\\tau _e$ , i.e.",
"a weak broadening of the electron transition.",
"This time of free coherent electron-nuclei precession is likely to be controlled in our experimental condition (i.e.",
"chemically doped QDs under CW excitation) by the non-resonant optical injection of an electron-hole pair: we chose $\\tau _e=5ns$ in the calculation presented in Fig.REF .",
"The important variations of the nuclear spin polarization observed in Fig.",
"REF for a small varying magnetic field around B$_z$ =0T are expected to significantly influence the spin dynamics of the resident electron.",
"In particular, an increase of the relaxation rate of the electron spin should be observed when the external magnetic field compensates the Overhauser field increasing the influence of the fluctuating nuclear field."
],
[
"Polarized fine structure of the excited state of the charged exciton",
"In order to prepare the spin state of a resident carrier in a singly charged QD, spin polarized electron-hole pairs are injected through circularly polarized photo-excitation of an excited state of the QD.",
"Low power PLE spectra on a singly negatively charged QD presented in Fig.",
"REF reveals intense absorption resonances for X$^-$ with a strong polarization dependence.",
"In general we find three distinctive features in these excitation spectra.",
"The first is a set of lower energy resonances that are strongly co-polarized with the excitation laser.",
"These transitions can be assigned to nominally forbidden transitions involving states with two s-shell electrons and an excited or delocalized hole.",
"These transitions are particularly well observed in CdTe/ZnTe structures because of the weak valence band offset.",
"Figure: (a) PL and PLE spectra resolved in circularpolarisation under σ+\\sigma + CW excitation.",
"The inset is azoom on the polarized doublet in the PLE spectra, thepolarisation rate is also displayed.",
"(b) Energy levels ofthe negative trion states.",
"The electrons triplet state issplit by the electron-hole exchange energy Δ eh \\Delta _{eh}.The electrons singlet state, also part of the p-shell isnot represented here.",
"Left scheme: when exciting withσ+\\sigma + light on the triplet stateS=1,S z =-1\\left|S=1,S_z=-1\\right\\rangle , photon absorption occursonly if the resident electron is down.",
"As demonstrated byM.E.",
"Ware , during the excited trionrelaxation, an electron-hole flip-flop process allowed byanisotropic exchange interactions results in a σ-\\sigma -PL.",
"Right scheme: when exciting with σ+\\sigma + light onthe triplet state S=1,S z =0\\left|S=1,S_z=0\\right\\rangle , photonabsorption occurs only if the resident electron is up.",
"Fastrelaxation from this state leads to σ+\\sigma +PL.The second feature is a higher energy resonance that displays a fine structure doublet well resolved in circular polarization.",
"As presented in Fig.",
"REF (a), the PLE exhibits a strongly co- and then cross-polarized resonance as the laser energy increases around 2110 meV.",
"As proposed by M.E.",
"Ware et al.",
"[9] we can assign this doublet to the direct excitation of the two bright triplet states (see Fig.",
"REF (b)) of the excited negatively charged exciton ($X^{-*}$ ).",
"$X^{-*}$ consists of an electron-hole pair in the $S$ -shell and an electron in the $P$ -shell.",
"This doublet is a characteristic signature of the presence of a single electron in the QD ground state.",
"We have found a triplet splitting $\\Delta _{eh}$ around 400 $\\mu $ eV changing from dot to dot.",
"This is higher than the values found in InAs QDs, in agreement with the stronger exchange interaction in our II-VI QD system.",
"For an excitation above the $X^{-*}$ triplet states, a series of excited states and an absorption background with a significant negative circular polarisation rate are observed.",
"As we will discuss in the next section, this condition of excitation can be used to perform an optical pumping of the resident electron spin."
],
[
"Kinetics of the degree of circular polarization",
"As a probe of the resident electron spin orientation, we will use the amplitude of the negative circular polarization of the charged QDs [10], [11], [13].",
"In the case of $X^-$ , the circular polarization of the emitted light reflects both the spin of the resident electron before the absorption of a photon and the spin of the hole before emission.",
"Negative polarization of X$^-$ implies that the hole spin has flipped prior to recombination and that a spin flipped hole contributes to the X$^-$ formation with a higher probability than a non-flipped hole.",
"This process also leads to an optical pumping of the resident electron spin.",
"Figure: (a) PL and PLE spectra of a singly charged quantumdot.",
"(b) Time-evolution of the co-polarized PL (red line),the cross-polarized (blue line) PL, and the degree ofcircular polarisation (black line) under pulsed excitationfor four different excitation energies (E 1 E_{1} toE 4 E_{4}).",
"A clear negative polarization rate isobserved when the excitation energy is higher than thetriplet states of the excited charged exciton X -* ^{-*}(E 2 E_{2}).The kinetics of the degree of circular polarisation observed in time resolved PL experiments reflects the mechanisms of spin injection.",
"The time dependence of the polarisation rate $\\rho _c$ under quasi-resonant pulsed excitation ($\\approx $ 2ps) is displayed in Fig.",
"REF .",
"As we discussed for the PLE spectra of the QD presented in Fig.",
"REF , the lowest excited state (E$_{1}$ in Fig.",
"REF ) is fully co-polarized.",
"It corresponds to the negatively charged state with two electrons in the $S$ -shell and a hole in a higher shell.",
"It is spin selective as the spin of the two electrons in the $S$ -shell have to be opposite: under $\\sigma +$ excitation, an absorption only occurs if the resident electron is $|\\uparrow \\rangle $ .",
"The fast initial decay (in the range of 100 ps) of the polarization under excitation on E$_{1}$ is attributed to a relaxation of the hole spin on the high energy shell while the slower decay (in the range of 5 ns) is attributed to relaxation of the hole spin when it is in the $S$ -shell.",
"Resonant $\\sigma +$ excitation on the charged exciton triplet states (E$_{2}$ in Fig.",
"REF ) results in the creation of an excited trion $\\Uparrow _P\\downarrow _P\\uparrow _S$ or $\\Uparrow _P\\downarrow _P\\downarrow _S$ depending on the spin of the resident electron (see Fig.",
"REF (b)).",
"Under excitation on E$_{2}$ we observe a fast decay of the initial positive polarization rate: the polarization rate becomes negative in a few tens of $ps$ .",
"This evolution reflects the spin dynamics of $X^{-*}$ .",
"The excited state $\\Uparrow _P\\downarrow _P\\uparrow _S$ can relax quickly to the ground trion state while for the state $\\Uparrow _P\\downarrow _P\\downarrow _S$ , relaxation to the ground state is forbidden until an electron-hole flip-flop occurs through anisotropic exchange interaction [12].",
"Therefore, relaxation from $X^{-*}$ results in a positive polarization rate at short delays and negative at longer delays.",
"For higher excitation energies (E$_{3}$ and E$_{4}$ in Fig.",
"REF ) a major part of the decay of the polarization rate takes place within the first 200 ps.",
"The polarization rate becomes quickly negative and approches a value of about -30$\\%$ .",
"It further decreases at longer time delay ($\\approx 1ns$ ) and reaches a steady state value lower than -50$\\%$ .",
"To understand the three regimes in the dynamics of the polarization rate, we have to consider 0D-2D cross-transitions where the electron is injected in the dot and the hole in the wetting layer.",
"Such transitions are particularly important in our system presenting a weak valence band offset.",
"The spin of the hole is randomize before its capture by the QD whereas the electron spin is conserved (spin $|\\downarrow \\rangle $ for $\\sigma +$ excitation).",
"The captured bright excitons (i.e.",
"without hole spin flip) or dark excitons (i.e.",
"after a hole spin flip) relax to form the hot trion $X^{-*}$ .",
"For an unpolarized resident electron, four possible channels are then possible for the relaxation of the hot trion formed by a $\\sigma +$ excitation: $\\uparrow _s\\downarrow _p\\Uparrow $ $\\longmapsto $ $\\uparrow _s\\downarrow _s\\Uparrow $ $\\longmapsto $ $\\sigma +$ $and$ $\\uparrow _s$ $\\uparrow _s\\downarrow _p\\Downarrow $ $\\longmapsto $ $\\uparrow _s\\downarrow _s\\Downarrow $ $\\longmapsto $ $\\sigma -$ $and$ $\\downarrow _s$ $\\downarrow _s\\downarrow _p\\Uparrow $ $\\stackrel{\\delta _a}{\\longmapsto }$ $\\downarrow _s\\uparrow _s\\Downarrow $ $\\longmapsto $ $\\sigma -$ $and$ $\\downarrow _s$ $\\downarrow _s\\downarrow _p\\Downarrow $ $\\stackrel{\\tau _h}{\\longmapsto }$ $\\downarrow _s\\downarrow _p\\Uparrow $ $\\stackrel{\\delta _a}{\\longmapsto }$ $\\uparrow _s\\downarrow _s\\Downarrow $ $\\longmapsto $ $\\sigma -$ $and$ $\\downarrow _s$ where $\\delta _a$ is an anisotropic electron-hole exchange interaction term responsible for the flip-flop of the electron-hole pair in the QD excited state and $\\tau _h$ the spin flip time of a hole in the $S$ -shell.",
"The realization of (1) and (2) is proportional to the probability for the resident electron to be $|\\uparrow \\rangle $ and does not require any spin flip of the hot trion: they take place in the $ps$ range.",
"(3) and (4) depend on the probability of having the resident electron $|\\downarrow \\rangle $ and involve spin flips of the hot trion.",
"These last two channels ((3) and (4)) lead to the appearance of a negative circular polarization rate with two time scales, one in the tens of $ps$ range governed by $\\delta _a$ and one in the $ns$ range governed by $\\tau _h$ .",
"If the spin relaxation rate of the resident electron is longer than the optical excitation rate, cumulative effects lead to the optical orientation of the electron spin.",
"At this stage, we have to notice that the observation of negative circular polarization does not necessarily mean that the resident electron is polarized but a variation of its polarization will cause a change in the negative circular polarization rate."
],
[
"Orientation of the spin of the electron",
"The presence of optical pumping of the resident electron is confirmed by the power dependence of the negative circular polarization rate obtained under CW excitation (Fig.",
"REF (a)).",
"As the pump power intensity is increased, we observe a rapid growth of the negative circular polarization with a saturation at about -55$\\%$ .",
"This reveals the progressive orientation of the resident electron spin by the exciting beam.",
"The remaining polarization rate observed at low excitation power or in a weak transverse magnetic field is attributed to the different processes of carrier relaxation discussed in section IV.",
"In the optical pumping regime (i.e.",
"without transverse magnetic field and at large excitation intensity), the measurement of the negative circular polarization gives an estimate of the degree of the electron spin polarization.",
"The electron spin memory can be significantly erased by a weak magnetic field (see Fig.",
"REF (b)) applied in the plane of the QD.",
"At B$_x$$\\approx 0.1T$ , all the contribution of the electron spin polarization to the negative circular polarization rate has disappeared.",
"Despite the weak transverse component of the hole g-factor, a further increase of the transverse magnetic field can induce a precession of the confined hole spin during the lifetime of the negatively charged exciton.",
"At high field, this precession depolarizes the hole spin and finishes to destroy the average negative circular polarisation of the $X^-$ .",
"This effect is observed in Fig.",
"REF (b) as an oscillation of the polarization rate for a transverse field larger than 0.1T.",
"This oscillation corresponds to the first period of precession of a spin polarized hole injected at $t=0ns$ .",
"The decrease of the polarization rate at long time delay corresponds to the late recombination of spin flipped holes stored as dark excitons in the triplet state of X$^{-*}$ .",
"Figure: (a) Evolution of the degree of circularpolarisation as a function of CW excitation power.",
"Emptycircles are zero field measurements, filled circles aremeasurements in a transverse magnetic field B x _x=0.13T.",
"(b) Time-evolution of the degree of circular polarisationunder pulsed (2ps) quasi-resonant excitation for differenttransverse magnetic fields.",
"(c) Time-evolution of thedegree of circular polarisation with a two-pulse-excitationsequence: the pulses are separated by 2ns.",
"(i) is obtained with cross-polarized pulses atzero field.",
"(ii) is obtained with co-polarized pulses atzero field.",
"(iii) is obtained with co-polarized pulses anda transverse magnetic field B x _x=0.13T.The dynamics of the pumping and relaxation of the resident electron spin can be estimated through the time evolution of the polarization rate when the QD is excited by a sequence of two circularly polarized picosecond laser pulses.",
"The results of the experiments using equal intensities for the two pulses are shown in Fig.",
"REF (c).",
"When the QD is excited with circularly co-polarized pulses (ii), a large average negative circular polarization is observed for both PL pulses.",
"However, in the case of excitation by cross-polarized beams (i), the average PL polarization vanishes.",
"These results directly demonstrate that the spin orientation created by the first pulse affects the polarization of the PL excited by the second one.",
"It means that after recombination of the electron-hole pair, the information about the polarization of the excitation is stored in the orientation of the resident electron spin.",
"In addition, we notice that the polarization rate is identical for the two pulses in the excitation sequence.",
"As these pulses are separated in time by either $2ns$ or $11ns$ this suggest that the relaxation time of the electron exceeds by far the laser pulses repetition rate ($\\approx 13ns$ ).",
"Consequently, the resident electron can be fully depolarized by a weak transverse magnetic field.",
"A significant decrease of the negative polarisation rate is observed for both pulses in a transverse field B$_x$ =0.13T (iii) confirming the influence of the optical pumping of the electron spin on the negative polarization rate."
],
[
"Dynamics of the electron spin orientation",
"In the presence of optical pumping, the degree of negative circular polarization of $X^-$ reflects the spin polarisation of the resident electron.",
"The dynamics of its optical orientation can then be revealed by the observation of the negative polarization rate under modulated circularly polarized excitation.",
"As presented in Fig.",
"REF (a), the negative polarisation strongly depends on the modulation frequency: an increase of the degree of circular polarisation when the modulation frequency is decreased is observed at B=0T.",
"This modulation frequency dependence is canceled by a magnetic field of B$_z$ =0.16T applied along the QD growth axis.",
"As already observed in InAs QDs, this behavior is a fingerprint of the coupling of the electron to a fluctuating nuclear field[17].",
"The QD contains a finite number $N_I$ of nuclei carrying a spin, which means that statistically, the number of spins parallel and antiparallel in any given direction differs by a value $\\sqrt{N_I/3}$ .",
"The result is an effective magnetic field B$_f$ , oriented in a random direction.",
"This field will induce a precession of the spin of the electron for every B$_{f}$ not aligned along the QD growth axis $z$ .",
"B$_f$ can be estimated, for a CdTe QD.",
"Assuming a homogeneous envelope-function for the electron ($\\psi (R)=\\sqrt{2/(\\nu _0N_L)}$ ), B$_{f}$ is given by [28]: $B_f^2=\\frac{2}{(g_e\\mu _B\\sqrt{N_L})^2}(I_{Cd}(I_{Cd}+1)A_{Cd}^2p_{Cd}\\\\ \\nonumber +I_{Te}(I_{Te}+1)A_{Te}^2 p_{Te})$ For our estimation of $N_L=8000$ one obtain $B_f\\approx 12mT$ .",
"The electron spin precession frequency in the frozen nuclear spin fluctuation B$_f$ $\\approx $ 12mT is $\\approx $ 80MHz.",
"This frequency can be smaller than the rate of optical injection of the spin polarized carriers at high excitation intensity allowing an optical pumping of the electron spin even in the presence of nuclear spin fluctuations.",
"At high modulation frequency of the polarisation and low excitation intensity, a dynamic nuclear spin polarisation does not have time to build-up.",
"Over time scales less than 1 $\\mu s$ , the electron is exposed to a snapshot of B$_f$ where the nuclear spin configuration remains frozen.",
"In the absence of an external magnetic field, only this internal field B=B$_f$ acts on the electron.",
"For a randomly oriented nuclear spin system, the electron spin polarization quickly decays to 1/3 of its initial value due to the frozen nuclear field [14].",
"This decay is not a real relaxation process as the electron coherently evolves in a frozen nuclear spin configuration.",
"On an averaged measurement, a fast decay of the electron polarization on a characteristic timescale $t\\approx h/(g_e\\mu _BB_f)$ is expected[14].",
"In the absence of nuclear spin polarization, the influence of the fluctuating nuclear field can be suppressed by applying an external magnetic field.",
"For sufficiently large external fields, the nuclear spin fluctuations does not contribute significantly to the total field, B$_{tot}$ =B$_{ext}$ +B$_f$ , and the electron-spin polarization is preserved.",
"At small modulation frequencies of the polarization or under CW excitation, nuclei can be dynamically oriented through flip-flop with the spin polarized resident electron.",
"This nuclei orientation leads to the formation of an Overhauser field, B$_N$ along the $z$ axis, which may be much larger than the in-plane component of the fluctuating field (B$_f$ ).",
"The electron now precesses around a nuclear field whose $z$ component dominates: the result is an increase of the average electron spin polarization compared to the case of a totally randomly oriented nuclear spin system.",
"The effect of the Overhauser field is similar to an applied magnetic field along the $z$ axis allowing an optical orientation of the electron spin even at low excitation power.",
"This influence of the nuclear spin fluctuations in the optical pumping of the electron is confirmed by the magnetic field dependence of the time resolved polarisation rate obtained at high modulation frequency (Fig.",
"REF (b)).",
"The increase of the polarization rate and the appearance of a transient with an applied external magnetic field along $z$ reflects an increase of the optical pumping efficiency of the electron [3].",
"This optical pumping, which takes place in a few tens of $ns$ , is promoted by the presence of the external field which can dominate the fluctuations of the Overhauser field [15].",
"This short timescale component in the dynamics of the polarization of the spin of the electron becomes faster with the increase of the optical generation rate of spin polarized carriers and reach the $ns$ range (Fig.",
"REF (c)).",
"Figure: (a) Evolution of the NCP with the frequency ofthe σ+\\sigma +/σ-\\sigma - modulation of the excitation laserat zero magnetic field and at a field B z _z=0.16T appliedalong the QD growth axis zz.",
"Time-evolution of theσ+\\sigma + PL excited alternatively with σ+\\sigma + orσ-\\sigma - light for different magnetic fields applied inFaraday geometry (b) and for different excitationintensities and a fixed magnetic field B z _z=0.19T (c).",
"(d)Time-evolution of the σ+\\sigma + PL excited alternativelywith σ+\\sigma + and σ-\\sigma - light trains.",
"The excitationsequence are displayed above the spectrum.",
"The inset is azoom on the fast transient at short timedelay.Similarly to the application of an external magnetic field, the build-up of a DNSP favors the electron spin polarization.",
"This is confirmed by the following experiment: in Fig.",
"REF (d), the $\\sigma +$ PL has been time-resolved using the two different excitation sequences displayed on each spectrum.",
"In the sequence (ii), the excitation pulses are of equal length and power, and are short enough to prevent the creation of DNSP.",
"In sequence (i), the difference of pulses length allows the creation of DNSP.",
"The measurements show two striking differences.",
"First, the average circular polarisation, given by the difference of the PL intensity obtained under $\\sigma -$ and $\\sigma +$ excitation, is higher in (i) than in (ii).",
"Second, the PL of (i) exhibits a fast PL transient at short delay reflecting an optical pumping of the electron spin (detail of this transient is shown in the inset of Fig.",
"REF (d)).",
"These two features demonstrate that the Overhauser field created in (i) is strong enough to block the longitudinal decay of the electron spin by the fluctuating nuclear field."
],
[
"Built-up of the nuclear spin polarization",
"Direct evidence of the build-up of a DNSP can be observed using sequences of pulses of long duration (tens of $\\mu $ s).",
"As displayed in Fig.",
"REF (a), the $\\sigma +$ PL recorded under $\\sigma +$ excitation presents first a fast transient with a drop of the intensity due to the orientation of the resident electron spin (a zoom of the transient at short delay is presented in Fig.",
"REF (b)).",
"Then, a slower transient is observed: the $\\sigma +$ PL increases during a few $\\mu $ s reflecting a decrease of the absolute value of the negative polarisation (i.e.",
"of the spin polarization of the resident electron) before it decreases again.",
"This evolution has been predicted by M. Petrov et al.",
"[16] and results from a destruction of the Overhauser field created at the end of the $\\sigma -$ excitation pulse, and a build-up of an Overhauser field in the opposite direction under $\\sigma +$ excitation.",
"During this process, the amplitude of the Overhauser field becomes zero and the electron spin is strongly affected by the nuclear spin fluctuations B$_f$ .",
"Figure: (a) Time-evolution of the σ+\\sigma + PL excitedalternatively with σ+\\sigma +/σ-\\sigma - light trains.",
"Theexcitation sequence is displayed above the spectrum.",
"(b)Zoom on the transient corresponding to the optical pumpingof the electron.",
"The time-evolution of the σ-\\sigma - PLrecorded in the same conditions of excitation is alsodisplayed.",
"(c) Dependence of the the degree of circularpolarisation of the QD PL on the frequency of theσ+\\sigma +/σ-\\sigma - modulation of the light for differentexcitation power.",
"(d) Time-evolution of the σ+\\sigma + PLexcited alternatively with σ+\\sigma +/σ-\\sigma - light fordifferent excitation power.As presented in Fig.",
"REF (d), the speed of the destruction and build-up of the nuclear polarization strongly increases with the increase of the excitation power.",
"Simultaneously, the average negative polarisation of the $X^-$ , given by the intensity difference of the PL obtained under $\\sigma +$ and $\\sigma -$ excitation, increases.",
"This effect is also directly observed in the modulation frequency dependence of the polarization rate displayed in Fig.",
"REF (c).",
"The modulation frequency required to suppress the nuclear spin polarisation (i.e.",
"to decrease the absolute value of the polarization) increases with the excitation intensity.",
"At low excitation intensity, a formation time of the nuclear spin polarization of about $50\\mu s$ can be estimated from the modulation frequency dependence of the negative circular polarization [17].",
"This is three order of magnitude faster than in InAs/GaAs QDs where a pumping time of the nuclei around $10 ms$ has been reported [18].",
"At hight excitation power, the pumping rate of the nuclei becomes faster than the polarization modulation frequency and a stable negative polarization of about -50$\\%$ is obtained (Fig.",
"REF (c)).",
"An increase of the value of the negative polarization with the excitation intensity is also observed suggesting an increase of the average nuclear spin polarization and Overhauser field with the excitation intensity."
],
[
"Magnetic field dependence of the nuclear spin polarisation",
"A typical magnetic field dependence of the polarization rate of a singly charged QD under CW circularly polarized excitation, in the optical pumping regime, is presented in Fig.",
"REF : Fig.",
"REF (a) focuses on the Faraday geometry while the Voigt geometry and the influence of the modulation frequency of the polarization of the excitation beam are presented in Fig.",
"REF (b).",
"The asymmetry of the response in the Faraday geometry is a fingerprint of the presence of a nuclear spin polarization induced by the helicity of the excitation beam.",
"Figure: (a) Magnetic field dependence in Faradayconfiguration of the negative polarization rate underσ+\\sigma + CW excitation.",
"(b) Magnetic field dependence ofthe negative polarization rate, on a different QD, inVoight (left) and Faraday (right) configuration underσ+\\sigma +/σ-\\sigma - modulated excitation at low(f mod _{mod}=200Hz) and high (f mod _{mod}=175kHz) modulationfrequency.A striking feature of this magnetic field dependence is the small increase of the absolute value of the negative polarization around B=0T.",
"This is in opposition to what is usually observed in III-V semiconductor QDs where a decrease of the electron spin polarisation occurs at weak magnetic field because of the dominant contribution of the fluctuating nuclear field B$_f$ .",
"As presented in curve (i) of Fig.",
"REF (b), a standard increase of polarisation with magnetic field is restored in the absence of nuclear polarisation (i.e.",
"under polarization modulated excitation).",
"This shows that the increase of the electron polarization around $B=0T$ observed under CW excitation is linked to the DNSP: the nuclear spin fluctuations are strongly suppressed by the build-up of a large Overhauser field.",
"The experiment presented in Fig.",
"REF (a) was carried out under CW $\\sigma +$ excitation, pumping the resident electron down.",
"This leads to an average polarization of nuclei with $\\left\\langle I_z\\right\\rangle <0$ and an Overhauser field $B_N\\le 0$ .",
"For $B_z\\ge 0$ , we observe around 50mT an increase of the circular polarization rate of $10\\%$ which reflects a depolarization of the resident electron.",
"This behavior is attributed to the compensation of the Overhauser field by the external Faraday field ($B_{z}=-B_N$ ).",
"As the electron precesses around the total field $B_{tot}=B_{z}+B_N+B_f$ , the electron dynamics is then governed by the nuclear spin fluctuations $B_f$ , resulting in a depolarization of the resident electron.",
"Considering the left panel of Fig.",
"REF (a) (corresponding to $B_z\\le 0$ ), we observe at $B=0T$ a maximum in the electron polarization, then a decrease of about $5\\%$ in the first 25 mT followed by a small increase at larger fields.",
"This extremum around $B_z=-25mT$ also reflects a depolarization of the resident electron spin.",
"This depolarization is attributed to a compensation of the Knight field by the external magnetic field.",
"The nuclear field is then close to zero and the electron dynamics is ruled by the sum of $B_{z}\\approx 25mT$ and the nuclear spin fluctuations $B_f$ .",
"The effect of $B_f$ is not negligible at $25mT$ .",
"As observed in the experiment of Fig.",
"REF (b), in the absence of DNSP, the polarization of the PL continuously increases as the Faraday magnetic field is increased from 0 to $60mT$ .",
"This influence of the Knight and Overhauser fields is further-confirmed by the following experiment: We present on the right panel of Fig.",
"REF (b) a measurement, were we have studied the polarization rate as a function of the magnetic field $B_z$ under modulated excitation.",
"The light is modulated $\\sigma +/\\sigma -$ at two different rates: (i) 175 kHz ($\\approx 3\\mu s$ of $\\sigma +$ exc., then $\\approx 3\\mu s$ of $\\sigma -$ exc.",
"of equal intensity) and (ii) 200 Hz ($\\approx 2500\\mu s$ for a given polarization).",
"Hence, at low power of excitation, DNSP is achieved in (ii) and not in (i).",
"The detection is done on an APD synchronized with the modulation, and for a fixed circularly polarized detection.",
"We measure a polarization rate by varying the excitation polarization and not the detection.",
"Hence, this polarization rate is an average of the polarization rates measured in Fig.",
"REF (a) for $B_z$ and $-B_z$ .",
"The magnetic field dependence for (ii) is consistent with the one observed in the CW regime, with an evolution ruled by the competition between the $B_z$ , $B_N$ , $B_e$ and $B_f$ .",
"On the other hand the magnetic field dependence (i) is only controlled by the competition between $B_z$ and $B_f$ .",
"For sufficiently large external fields, the nuclear spin fluctuations $B_f$ do not contribute to the total field and the electron-spin polarization does not decay.",
"The width at half maximum is $25\\pm 5mT$ .",
"This gives an order of magnitude of the fluctuating Overhauser field.",
"Figure: Excitation power dependence of the negativepolarization rate under magnetic field.",
"(a) Faradayconfiguration under σ\\sigma + CW excitation.",
"(b) Voigtconfiguration.The magnetic field dependence of the polarization rate under CW excitation has been performed for $B_z>0$ for different excitation powers (Fig.",
"REF (a)).",
"With increasing power, the minimum in the electron polarization is shifted to higher magnetic field, evidencing an increase of the polarization of the nuclei and of the resulting Overhauser field.",
"At high excitation intensity, a significant portion of the nuclei are polarized and the minimum of electron polarization is observed around $B_z=-B_N=100mT$ which would correspond to $50\\%$ of the maximum Overhauser field.",
"However, the parameters of II-VI QDs used to estimate this maximum field are not known with precision and this percentage is subject to caution."
],
[
"Electron-nuclear spin system in a transverse\nmagnetic field",
"A transverse magnetic field dependence of the polarization rate of X$^-$ is presented in the left panel of Fig.",
"REF (b).",
"As the transverse magnetic field is increased, we observe in the absence of DNSP (black curve, corresponding to fast $\\sigma +/\\sigma -$ modulation), a progressive decrease of the negative polarization rate over the first $80mT$ .",
"For a spin polarized electron and in the absence of nuclear spin polarization, two processes can contribute to the observed Hanle depolarization of X$^-$ .",
"The first is a depolarization of the resident electron governed by the transverse relaxation time of the spin of the electron, $T_2$ , in an unpolarized nuclear spin bath (standard Hanle depolarization).",
"This $T_2$ should give rise to a half width of the Hanle curve $B_{1/2}=B_f\\approx 25mT$ , deduced from the Faraday measurement in Fig.",
"REF (b).",
"The second mechanism is a precession of the hole during the charged exciton life-time.",
"This process is expected to play a role above $50mT$ as we have seen in the time resolved polarization rates presented in Fig.",
"REF (b).",
"It is not possible to discriminate between the two mechanisms as, because of the weak polarization of the electron, they are both responsible of a small decrease of few $\\%$ of the circular polarization rate.",
"More interesting is the comparison with the data where a DNSP is created (red curve on the left panel of Fig.",
"REF (b)).",
"In this later case, a fast decrease of the negative polarization rate is observed when increasing the transverse field from 0 to $10mT$ .",
"The half width at half maximum of the depolarization curve is $\\approx 5mT$ .",
"This efficient depolarization of the resident electron is due to the precession of the coupled electron-nuclei system [22], [31], [29].",
"After this fast depolarization of the electron, the negative polarization rate reaches the value observed in the absence of DNSP (i.e.",
"under fast $\\sigma +$ /$\\sigma -$ modulated excitation, black curve).",
"This is also observed in the modulation frequency dependence of the polarization rate presented in Fig.",
"REF (a).",
"We observe that the depolarization curve in transverse magnetic field in the presence of nuclear spin polarization (Fig.",
"REF (a)) strongly depends on the excitation power and deviates from a Lorentzian shape at high excitation power.",
"For the later case, the negative polarization rate seems to be weakly affected by the first few $mT$ of transverse magnetic field, and then decreases abruptly.",
"The width of such depolarization curve is typically 50mT in a singly charged InAs/GaAs QDs in the presence of nuclear spin polarization [32].",
"In InAs QDs, an influence of the magnetic anisotropy of the nuclei produced by the in-plane strain is also observed in the transverse magnetic field dependence of the Overhauser field [33], [34].",
"This cannot be the case in CdTe QDs as the nuclear spins I=1/2 for Cd and Te.",
"Figure: Variation of the amplitude (c) and characteristictime (d) of the DNSP transient under σ+\\sigma + excitation(an exemple is presented in (a)) as a function of theexcitation intensity at B z _z=0mT with a constant dark timeτ dark =50μs\\tau _{dark}=50\\mu s. (b) Variation of the amplitude ofthe DNSP transient with a magnetic field applied along theQD growth axis.The power dependence observed in our system could arise as the electron is pumped faster than the precession in the transverse applied ($\\tau _{press}^e=4ns$ at $B_x=5mT$ ).",
"At hight excitation intensity, the resident electron is replaced by an injected spin polarized electron faster than the precession in the transverse magnetic field.",
"The depolarization of the electron spin is then given by the power dependent Hanle curve: $S_{z}(\\Omega )=\\frac{S_z(0)}{(1+(\\Omega \\tau )^2)}$ where $\\Omega =g_e\\mu _BB/\\hbar $ and $1/\\tau =1/\\tau _p+1/\\tau _s$ with $1/\\tau _p$ the pumping rate and $1/\\tau _s$ the electron spin relaxation rate.",
"However, this power broadening does not explain the slight deviation form the Lorentzian shape we observed at high excitation intensity in (Fig.",
"REF (b)).",
"The creation of DNSP could also be faster than the nuclei precession ($\\tau _{press}^N=5\\mu s$ at $B_x=5mT$ ).",
"The decrease of the electron polarization in a transverse magnetic field can then be influenced by the decrease of the steady state nuclear field.",
"As a result of this decrease, the total in-plane component of the magnetic field which controls the electron precession increases more slowly than the external field B$_x$ : the precession of the electron would be efficiently blocked by the Overhauser field, and the electron polarization would be conserved.",
"Such a scenario would be specific to II-VI quantum dots, where the build-up of DNSP is fast enough to block the precession of nuclei.",
"This point requires further investigation."
],
[
"Dynamics of the nuclear spin polarization",
"In order to analyze quantitatively the build-up time and the characteristic amplitude of the polarization transient induced by the DNSP, we perform a time-resolved measurement using a $100\\mu s$ pulse of a $\\sigma +$ helicity, followed by a $50\\mu s$ dark time during which the DNSP relaxes partially (quantitative analysis of this relaxation will be done in the next section and is indeed found to occur on a time-scale shorter than $50\\mu s$ for $B_z<5mT$ ).",
"This experimental configuration enables us to fit the observed DNSP transient by an exponential variation, permitting to extract a characteristic rate $1/\\tau $ and amplitude $\\Delta I$ (Fig.",
"REF (a)).",
"Figure: (a) Magnetic field dependence of the destructionand build-up of the nuclear spin polarisation observedunder σ+\\sigma +/σ-\\sigma - modulated excitation (left:detection σ-\\sigma -, right: detection σ+\\sigma +).",
"Adetailed vue of the transients is presented in (b).",
"Theinset shows the magnetic field dependence of the transientamplitude (Δ\\Delta I) and position(τ 0 \\tau _0).We observe in Fig.",
"REF (d), a linear increase of the pumping rate with the excitation power.",
"While the build up of DNSP takes a few $ms$ in III-V materials at $B=0T$ [18], it occurs in the $\\mu s$ range in our case.",
"This results from the strong localization of the electron in II-VI quantum dots: the built-up rate of DNSP scales as[38] $\\left|\\Psi _e\\right|^4$ so that we typically expect $\\tau _{II-VI}/\\tau _{III-V}\\approx 8000^2/(10^5)^2\\approx 5.10^{-3}$ .",
"The amplitude of the pumping transient ($\\Delta I/I$ ) presented in Fig.",
"REF (c) increases linearly at low excitation power, reaches a maximum and decreases at high power.",
"The increase is attributed to an increase of the nuclear spin polarization.",
"The reduction at high excitation power likely comes from a decoupling of the dynamics of the electron spin from the fluctuating nuclear spin.",
"As we have already seen in Fig.",
"REF (c), at high excitation intensity, optical pumping of the electron spin becomes faster than the precession in the fluctuating field of the nuclear spin B$_f$ and the measured polarisation rate of the $X^-$ becomes less sensitive to the polarization of the nuclei.",
"A similar decrease of the amplitude of the transient is observed under a magnetic field of a few mT applied along the QD growth axis.",
"The magnetic field dependence, of $\\Delta I/I$ is shown in Fig.",
"REF (b).",
"We observe an important decrease of $\\Delta I/I$ as soon as a few $mT$ are applied along the QD growth axis $z$ .",
"This fast decrease mainly comes from the increase of the relaxation time of the DNSP under magnetic field (this increase of the relaxation time is evidenced in Fig.",
"REF and will be further discussed): As the nuclear spin polarization does not fully relax during the dark time, the amplitude of the pumping transient decreases.",
"This increase of the relaxation time explains the general shape at fields lower than a few mT.",
"At larger fields, a decrease also occurs when the static magnetic field exceed the fluctuating nuclear field.",
"The magnetic field dependence of $\\Delta I/I$ presents an asymmetry as the magnetic field is reversed.",
"Similarly to the calculated asymmetry presented in Fig.",
"REF and the observed asymmetry presented in Fig.",
"REF (a), this is the signature of the creation of an effective internal field with well defined direction.",
"The faster drop of $\\Delta I/I$ in a positive magnetic field comes from the increase of the influence of the fluctuating nuclear field B$_f$ when the external magnetic field compensates the Overhauser field.",
"Such behavior has already been observed on ensemble of negatively charged CdSe/ZnSe QDs [4].",
"Our study of the DNSP build-up time scales was complemented by adding a magnetic field in the Faraday configuration in the time resolved pumping experiments.",
"Under $\\sigma +$ /$\\sigma -$ modulated excitation, each switching of the polarization results in an instantaneous (on the time-scale of the nuclear spin dynamics) change in the electron spin polarization followed by a slow evolution due to the re-polarization of the nuclei.",
"This re-polarization process is responsible for the minimum observed in the time evolution of the negative polarization rate.",
"Under a magnetic field, an asymmetry between the cases of $\\sigma +$ and $\\sigma -$ excitation is observed in the dynamics of the coupled electron-nuclei spin system (Fig.",
"REF ).",
"Under $\\sigma +$ excitation, as expected, the application of a magnetic field along $z$ progressively decreases the influence of the nuclear spin fluctuations on the electron-spin dynamics and the minimum in the electron polarization rate vanishes.",
"The behavior of the electron polarization is different under $\\sigma -$ excitation: we observe an acceleration with the increase of B$_z$ of the destruction of the DNSP at the beginning of the $\\sigma -$ pulse.",
"This is illustrated in the inset of Fig.",
"REF (b): The position of the minimum of polarization, $\\tau _0$ , linearly shift from $\\tau _0\\approx 8\\mu s$ at $B_z=0mT$ to $\\tau _0\\approx 0\\mu s$ at $B_z\\approx 30mT$ .",
"At the end of the $\\sigma +$ light train, the polarized light has created a nuclear field B$_N^{\\sigma +}$ anti-parallel to B$_{z}$ .",
"At some time after switching to $\\sigma -$ excitation, B$_N$ decreases and approach -B$_{z}$ .",
"At this point, the non-linear feed-back process in the electron-nuclei \"flip-flops\" starts and accelerates the depolarization until the DNSP vanishes.",
"Simultaneously, the absolute value of the negative polarization reaches a minimum.",
"Then the nuclei are re-polarized by the $\\sigma -$ excitation until B$_N$ reaches B$_N^{\\sigma -}$ parallel to B$_{z}$ .",
"Consequently, as observed in Fig.",
"REF (b), under $\\sigma -$ excitation the destruction of the DNSP is expected to be faster than its buildup.",
"However, it is not clear why such a magnetic field dependent acceleration is not observed during the build-up of the DNSP at the end of the transient in $\\sigma +$ polarization when B$_N^{\\sigma +}$ reaches -B$_{z}$ .",
"To fully understand this behavior, a complete model of the coherent dynamics of coupled electron and nuclear spins in a weak Faraday magnetic field should be developed.",
"Such model at zero field has already shown that the minimum of $\\langle S_z\\rangle $ can apparently be shifted from the point $\\langle I_z\\rangle $ =0 [16]."
],
[
"Nuclear spin polarization decay",
"In order to investigate the variation with magnetic field of the relaxation time in the dark of the DNSP, we follow the protocol shown in Fig.REF .",
"For a given magnetic field, we prepare a DNSP and measure after a time $\\tau _{dark}$ the amplitude of the transient, corresponding to the partial relaxation of the nuclear polarization.",
"As $\\tau _{dark}$ is increased, this amplitude saturates, demonstrating the full relaxation.",
"The variation of the amplitude of the transient with $\\tau _{dark}$ is used to estimate the relaxation time of the DNSP at a given magnetic field.",
"Figure: (a) Evolution of the DNSP transient, under amagnetic field B z _z=9.3mT, with the variation of the darktime introduced between circularly polarized light trainsof constant length.",
"(b) Magnetic field dependence of thenuclear spin relaxation time.",
"The inset show the evolutionof amplitude of the DNSP transient with the dark time forB z _z=9.3mT.The evolution of this relaxation time is presented in Fig.REF (b).",
"It ranges from $14\\mu s$ at $B=0T$ to $170\\mu s$ at $B=16mT$ .",
"The relaxation rate is one order of magnitude faster than the one observed by Feng et al.",
"on ensemble of CdSe/ZnSe QDs[3], and the one expected from nuclear dipole-dipole interactions.",
"Furthermore, in InAs/GaAs Schottky structure the decay in electron charged dots occurred in a millisecond time scale [18] while nuclear spin lifetime in an empty dot has been shown to exceed 1 hour.",
"[35].",
"The magnetic field dependence of the DNSP relaxation presents a significant increase of the decay time over the first few $mT$ .",
"It has been demonstrated that a magnetic field of $1mT$ efficiently inhibits nuclear dipole-dipole interactions in III-V materials [18].",
"Since this interaction is expected to be smaller in our system with diluted nuclear spins, we can definitely rule out dipole-dipole interaction as a major cause of DNSP relaxation.",
"Co-tunneling to the close-by reservoir could be responsible for this depolarization.",
"Via hyperfine-mediated flip-flop, the randomization of the electron spin creates an efficient relaxation of the nuclei.",
"Following Merkulov et.",
"al.",
"[38], this relaxation time is given by: $T_{1e}^{-1}=\\frac{2\\left\\langle \\omega ^2\\right\\rangle \\left\\Vert s\\right\\Vert ^2\\tau _c}{3[1+(\\Omega \\tau _e)^2]}$ In this expression, $\\omega $ is the precession frequency of the nuclei in the Knight field, $\\tau _e$ is the correlation time of the electron (in the dark), $\\Omega $ is the precession frequency of the electron in the Overhauser field.",
"At last, $\\left\\Vert s\\right\\Vert ^2$ is equal to $s(s+1)=3/4$ .",
"The fastest relaxation we expect from this process can be estimated taking $\\Omega =\\Omega _{fluc}=2\\pi /2ns^{-1}$ and $\\tau _c=10ns$ .",
"We obtain $T_{1e}\\approx 200\\mu s$ which is not fast enough.",
"Therefore, we are tempted to conclude that co-tunneling alone cannot explain the observed dynamics.",
"Another mechanism to consider is the depolarization resulting from an electron-mediated nuclear dipole-dipole interaction.",
"This results in exchange constants between the nuclei which typically scale as $A^2/(N^2\\epsilon _z)$ .",
"The resulting rate of nuclear-spin depolarization is $T^{-1}_{ind}\\approx A^2/(N^{3/2}\\hbar \\epsilon _z)$ , where $\\epsilon _z$ is the Zeeman splitting of the electron.",
"This mechanism could explain a depolarization of the nuclei on a $\\mu s$ scale [36].",
"However, this expression gives only a minor bound to the relaxation time because the inhomogeneity of the Knight field can strongly inhibit this decay [39], [40].",
"A magnetic field along the $z$ axis is expected to affect this process, progressively decoupling the nuclei from the indirect coupling created by the electron, as observed in our experiments in the first few $mT$ (Fig.REF ).",
"The electron-induced nuclear depolarization was demonstrated in [18] in which the $ms$ relaxation was completely suppressed using a voltage pulse on a Schottky diode in order to remove the resident electron."
],
[
"conclusion",
"In summary, we have evidenced in the PLE spectra of a negatively charged CdTe QD the polarized fine structure of the triplet states of the charged exciton.",
"We have studied, using PL decay measurements, the dynamics of the injection of spin polarized photo-carriers as a function of the energy of the injection.",
"We have shown that the injection above the triplet states of the charged exciton can be use to pump the resident electron on a time-scale of $10-100ns$ and to create a dynamic nuclear spin polarization.",
"At $B=0T$ , the creation of the dynamic nuclear spin polarization can be as fast as a few $\\mu s$ , and the decay of the nuclear polarization, attributed to an electron mediated relaxation, is $\\approx 10\\mu s$ .",
"The measured dynamics are $\\approx 10^3$ faster than the ones observed in III/V QDs at $B=0T$ .",
"The relaxation time of the coupled electron-nuclei system is increased by one order of magnitude under a magnetic field of 5mT.",
"The magnetic-field dependence of the PL polarization rate revealed that the nuclear spin fluctuations are the dominant process in the dephasing of the resident electron.",
"We proved that this dephasing is efficiently suppressed by a large dynamic nuclear spin polarization at $B=0T$ .",
"This work is supported by the French ANR contract QuAMOS and EU ITN project Spin-Optronics."
]
] | 1204.1469 |
[
[
"Fluctuation Relation for Heat"
],
[
"Abstract We present a fluctuation relation for heat dissipation in a nonequilibrium system.",
"A nonequilibrium work is known to obey the fluctuation theorem in any time interval $t$.",
"A heat, which differs from a work by an energy change, is shown to satisfy a modified fluctuation relation.",
"Modification is brought by correlation between a heat and an energy change during nonequilibrium processes whose effect may not be negligible even in the $t\\to\\infty$ limit.",
"The fluctuation relation is derived for overdamped Langevin equation systems, and tested in a linear diffusion system."
],
[
"Fluctuation Relation for Heat Jae Dong Noh Department of Physics, University of Seoul, Seoul 130-743, Republic of Korea School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea Jong-Min Park Department of Physics, University of Seoul, Seoul 130-743, Republic of Korea We present a fluctuation relation for heat dissipation in a nonequilibrium system.",
"A nonequilibrium work is known to obey the fluctuation theorem in any time interval $t$ .",
"A heat, which differs from a work by an energy change, is shown to satisfy a modified fluctuation relation.",
"Modification is brought by correlation between a heat and an energy change during nonequilibrium processes whose effect may not be negligible even in the $t\\rightarrow \\infty $ limit.",
"The fluctuation relation is derived for overdamped Langevin equation systems, and tested in a linear diffusion system.",
"05.70.Ln, 05.40.-a,02.50.-r,05.10.Gg Fluctuations of thermodynamic quantities of nonequilibrium systems obey a universal relation referred to as fluctuation theorem (FT) [1], [4], [2], [3], [5], [6], [8], [7], [9].",
"Discovery of the FT leads to a great advance in nonequilibrium statistical mechanics.",
"Based on the FT, one can generalize the fluctuation dissipation relation to nonequilibrium systems [10], [11], [12] and figure out fluctuations observed in experimental small-sized systems [13], [14], [15].",
"The FT for a quantity ${\\mathcal {R}}$ over a time interval $t$ takes the form $\\langle e^{-{\\mathcal {R}}}\\rangle = 1$ , where the average $\\langle \\cdot \\rangle $ is taken over a probability distribution for an initial state and over all time trajectories.",
"Some quantities further satisfy the FT in the form $P_r(R)/\\tilde{P}_r (-R) = e^R$ where $P_r(R) = \\langle \\delta ({\\mathcal {R}}-R)\\rangle $ is a probability density function (PDF) for a nonequilibrium process and $\\tilde{P}_r(R)$ for a corresponding reverse process.",
"The latter is called the detailed FT and implies the former called the integral FT.",
"Consider a system being in thermal equilibrium with a heat reservoir.",
"We will set the temperature and the Boltzmann constant to unity.",
"The system is driven into a nonequilibrium state if one adds a nonconservative force or applies a time-dependent perturbation.",
"Then, there exist nonzero net flows of a nonequilibrium work ${\\mathcal {W}}$ into the system and a heat dissipation ${\\mathcal {Q}}$ into the reservoir.",
"It is well established that the work ${\\mathcal {W}}$ over a time interval $t$ obeys the FT [2], [6].",
"In addition, the total entropy change $\\Delta {\\mathcal {S}}_{tot} = \\Delta {\\mathcal {S}}_{sys}+\\Delta {\\mathcal {S}}_{res}$ with the system (reservoir) entropy ${\\mathcal {S}}_{sys}~({\\mathcal {S}}_{res})$ satisfies the integral FT for an arbitrary initial state, and even the detailed FT for a steady state initial condition [7].",
"Thermodynamic quantities are measurable experimentally from time trajectories in classical systems [16], while their experimental measurability in quantum systems is still an open issue [17].",
"Fluctuations of heat ${\\mathcal {Q}}$ , or entropy production $\\Delta {\\mathcal {S}}_{res}={\\mathcal {Q}}/T$ , has also been attracting much interest [18], [19], [20], [21], [22], [23], [25], [24], [26].",
"Note that a heat differs from a work by an energy change $\\Delta {\\mathcal {E}}$ as ${\\mathcal {Q}}= {\\mathcal {W}}- \\Delta {\\mathcal {E}}$ .",
"When $t$ becomes large, the system will reach a steady state with constant work and heat production rates on average.",
"Hence one may expect the FT for heat in the large $t$ limit where an energy change can be negligible (${\\mathcal {Q}}\\simeq {\\mathcal {W}}\\gg \\Delta {\\mathcal {E}}$ ).",
"In fact, the FT for the heat production rate $({\\mathcal {Q}}/t)$ is derived formally in the $t\\rightarrow \\infty $ limit [3], [4].",
"On the other hand, some model studies demonstrate the FT for heat [25] or failure of the FT in the $t\\rightarrow \\infty $ limit [19], [23], [24].",
"So, it is interesting to understand how and why the FT is violated for finite $t$ and whether it is restored in the large $t$ limit [22], [21].",
"In this Letter, we present a fluctuation relation for heat, given in Eq.",
"(REF ).",
"For any process, fluctuations are constrained by the energy conservation ${\\mathcal {Q}}= {\\mathcal {W}}- \\Delta {\\mathcal {E}}$ .",
"So a correlation between thermodynamic quantities plays an important role in characterizing the heat fluctuation.",
"We find that the heat distribution satisfies a modified fluctuation relation that differs from the ordinary FT by a factor reflecting such a correlation.",
"The fluctuation relation is confirmed for a linear diffusion system analytically and numerically.",
"Our work provides an insight into origin for failure of the FT for heat for finite-$t$ interval and possibly for infinite-$t$ interval.",
"We consider a dynamical system described by an overdamped Langevin equation $\\frac{d{q}(t)}{dt} = {f}({q}(t)) + {\\xi }(t)$ where ${q}=(q_1,q_2,\\cdots , q_d)^T$ is a configuration vector, ${f}({q}) = (f_1({q}),f_2({q}),\\cdots ,f_d({q}))^T$ is a force, and ${\\xi } = (\\xi _1,\\xi _2,\\cdots ,\\xi _d)^T$ is a white noise with $\\langle \\xi _i(t) \\rangle = 0 \\ , \\ \\langle \\xi _i (t) \\xi _j (t^{\\prime }) \\rangle = 2\\delta _{ij} \\delta (t-t^{\\prime })$ A damping coefficient and a noise strength are set to unity by rescaling $t$ and ${q}$ properly.",
"The force can be decomposed as ${f}= {f}_c +{f}_{nc}$ , where ${f}_c({q}) = -{\\nabla }_{{q}}\\Phi ({q})$ is a conservative force with a scalar potential energy function $\\Phi ({q})$ and ${f}_{nc}({q})$ is a nonconservative force.",
"In this work, we focus on systems with a time-independent potential.",
"We assume that the system is in thermal equilibrium following the Boltzmann distribution $P_{eq}({q}) \\propto e^{-\\Phi ({q})}$ initially at $t=0$ .",
"Then it evolves into a nonequilibrium state due to the nonconservative force.",
"When the system follows a path ${q}(\\tau )$ for a time interval $0\\le \\tau \\le t$ , a nonequilibrium work done by the nonconservative force, a heat dissipation, and an energy change are given by functionals $\\mathcal {W}[{q}(\\tau )] = \\int _0^t d\\tau \\dot{q}(\\tau ) \\cdot {f}_{nc}({q}(\\tau ))$ , $\\mathcal {Q}[{q}(\\tau )] = \\int _0^t d\\tau \\dot{q}(\\tau ) \\cdot {f}({q}(\\tau ))$ , and $\\Delta {\\mathcal {E}}[{q}(\\tau )] =\\Phi ({q}(t))-\\Phi ({q}(0))$ , respectively [27].",
"They satisfy the energy conservation $\\Delta \\mathcal {E} = \\mathcal {W} - \\mathcal {Q}$ .",
"Among these, the PDF for work $P_w(W) \\equiv \\langle \\delta (\\mathcal {W}[{q}(\\tau )]-W) \\rangle $ satisfies the FT [6] $\\frac{P_w(W)}{P_w(-W)} = e^{W} \\ .$ Recently, it was found that joint probabilities for thermodynamic quantities also satisfy similar fluctuation relations [28].",
"Let $\\lbrace \\mathcal {A}_i[{q}(\\tau )]\\rbrace $ be a set of functionals whose sum is equal to the work ${\\mathcal {W}}[{q}(\\tau )] = \\sum _i \\mathcal {A}_i[{q}(\\tau )]$ and $\\mathcal {A}_i[\\bar{{q}}(\\tau )] = -\\mathcal {A}_i[{q}(\\tau )]$ where $\\bar{{q}}(\\tau ) = {q}(t-\\tau )$ is a time-reversed path.",
"Then, it was found that the joint PDF $P(\\lbrace A_i\\rbrace ) \\equiv \\langle \\prod _i\\delta (\\mathcal {A}_i[{q}]-A_i)\\rangle $ satisfies [28] $\\frac{P(\\lbrace A_i\\rbrace )}{P(\\lbrace -A_i\\rbrace )} = e^{\\sum _i A_i} \\ .$ Those relations reproduce Eq.",
"(REF ), and provide more detailed informations on nonequilibrium fluctuations [29].",
"We apply the formalism to the study of heat fluctuations.",
"Consider the joint PDF $P_{h,e}(Q,\\Delta E) \\equiv \\langle \\delta ({\\mathcal {Q}}[{q}]-Q)\\delta (\\Delta {\\mathcal {E}}[{q}]-\\Delta E)\\rangle $ .",
"Since ${\\mathcal {W}}= {\\mathcal {Q}}+ \\Delta {\\mathcal {E}}$ , it follows from Eq.",
"(REF ) that $P_{h,e}(Q,\\Delta E) = e^{Q+\\Delta E} P_{h,e}(-Q,-\\Delta E) \\ ,$ which will be referred to as a generalized FT (GFT).",
"One may consider another joint PDF $P_{w,e}(W,\\Delta E) \\equiv \\langle \\delta ({\\mathcal {W}}[{q}]-W)\\delta (\\Delta {\\mathcal {E}}[{q}]-\\Delta E)\\rangle $ .",
"They are related as $P_{h,e}(Q,\\Delta E) = P_{w,e}(Q + \\Delta E,\\Delta E) \\ .$ So the GFT for $P_{w,e}$ takes a slightly different form as $P_{w,e}(W,\\Delta E) = e^{W} P_{w,e}(-W,-\\Delta E) \\ .$ The PDF $P_h(Q) \\equiv \\langle \\delta ({\\mathcal {Q}}[{q}]-Q)\\rangle $ for heat is reduced from $P_{h,e}(Q,\\Delta E)$ .",
"Integrating both sides of Eq.",
"(REF ) over $(\\Delta E)$ , we obtain a fluctuation relation for the heat: $\\frac{P_h(Q)}{P_h(-Q)} = e^{Q} / \\Psi (Q) \\ ,$ where $\\Psi (Q) \\equiv \\int d(\\Delta E)\\ e^{-\\Delta E} P_{e|h}(\\Delta E| Q) \\ .$ Note that $P_{e|h}(\\Delta E|Q) = P_{h,e}(Q,\\Delta E)/P_h(Q)$ denotes a conditional probability for an energy change $\\Delta E$ to a given value of heat dissipation $Q$ .",
"A reciprocity relation $\\Psi (-Q) = \\Psi (Q)^{-1}$ was used in Eq.",
"(REF ).",
"The integral version is obtained from Eq.",
"(REF ) or Eq.",
"(REF ).",
"It is given by $\\left\\langle e^{-{\\mathcal {Q}}[{q}(\\tau )]} \\right\\rangle = \\left\\langle e^{-\\Delta {\\mathcal {E}}[{q}(\\tau )]} \\right\\rangle \\ .$ The detailed FT for heat is modified by the factor $\\Psi (Q)$ .",
"The original FT requires that $\\Psi (Q)=1$ for all $Q$ .",
"However one, in general, expects a correlation between $Q$ and $\\Delta E$ .",
"Such a correlation leads to a $Q$ -dependence in $\\Psi (Q)$ , hence invalidates the detailed FT for finite $t$ .",
"The fluctuation relations can be rewritten in terms of moment generating functions ${\\mathcal {G}}_{\\hat{w}}(\\lambda ) \\equiv \\langle e^{-\\lambda {\\mathcal {W}}}\\rangle $ , ${\\mathcal {G}}_{\\hat{h}}(\\eta ) \\equiv \\langle e^{-\\eta {\\mathcal {Q}}}\\rangle $ , ${\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\lambda ,\\kappa ) \\equiv \\langle e^{-\\lambda {\\mathcal {W}}- \\kappa \\Delta {\\mathcal {E}}}\\rangle $ , and ${\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) \\equiv \\langle e^{-\\eta {\\mathcal {Q}}- \\kappa \\Delta {\\mathcal {E}}}\\rangle $ .",
"All of them are not independent, but are deduced from a single one, e.g., ${\\mathcal {G}}_{\\hat{w},\\hat{e}}$ : Equation (REF ) yields that ${\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) ={\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\eta ,\\kappa -\\eta )$ , and ${\\mathcal {G}}_{\\hat{w}}(\\lambda ) = {\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\lambda ,\\kappa =0)$ and ${\\mathcal {G}}_{\\hat{h}}(\\eta ) = {\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa =0)$ .",
"The GFTs in Eqs.",
"(REF ) and (REF ) are equivalent to ${\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\lambda ,\\kappa ) &=&{\\mathcal {G}}_{\\hat{w},\\hat{e}}(1-\\lambda ,-\\kappa ) \\ , \\\\{\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) &=&{\\mathcal {G}}_{\\hat{h},\\hat{e}}(1-\\eta ,1-\\kappa ) \\ .$ Setting $\\kappa =0$ in Eq.",
"(REF ), one recovers the FT for work, ${\\mathcal {G}}_{\\hat{w}}(\\lambda ) = {\\mathcal {G}}_{\\hat{w}} (1-\\lambda )$ .",
"The fluctuation relation for heat cannot be written in a simple form with generating functions.",
"Instead, the modification factor $\\Psi (Q)$ can be written as $\\Psi (Q) = {\\mathcal {G}}_{h,\\hat{e}}(Q,\\kappa =1) / {\\mathcal {G}}_{h,\\hat{e}}(Q,\\kappa =0)$ with ${\\mathcal {G}}_{h,\\hat{e}}(Q,\\kappa ) \\equiv \\int d(\\Delta E)e^{-\\kappa (\\Delta E)} P_{h,e}(Q,\\Delta E)$ .",
"In the $t\\rightarrow \\infty $ limit, the FT is formulated in terms of the large deviation function (LDF) [4].",
"For the heat distribution, it is defined as $e_{h}(q) \\equiv \\lim _{t\\rightarrow \\infty } -\\frac{1}{t}\\ln P_h(Q=qt) \\ .$ Then, Eq.",
"(REF ) yields that $e_{h}(-q) - e_h(q) = q + \\psi (q)$ where $\\psi (q) = \\lim _{t\\rightarrow \\infty } -\\frac{1}{t} \\ln \\Psi (Q=qt) \\ .$ We can further simplify it by introducing a LDF $e_{{h},\\hat{e}}(q,\\kappa ) = \\lim _{t\\rightarrow \\infty }-\\frac{1}{t}\\ln {\\mathcal {G}}_{{h},\\hat{e}}(Q=qt,\\kappa ) \\ ,$ which is obtained from the Legendre transformation $e_{h,\\hat{e}}(q,\\kappa ) = \\max _{\\eta } \\lbrace e_{\\hat{h},\\hat{e}}(\\eta ,\\kappa )-q\\eta \\rbrace $ of a LDF $e_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) = \\lim _{t\\rightarrow \\infty } - \\frac{1}{t} \\ln {\\mathcal {G}}_{h,e}(\\eta ,\\kappa )$ .",
"Combining these, we obtain that $\\psi (q) = e_{h,\\hat{e}}(q,\\kappa =1) - e_{h,\\hat{e}}(q,\\kappa =0) \\ .$ This is a central quantity that determines whether the FT holds for heat in the $t\\rightarrow \\infty $ limit.",
"We apply the formalism to a $d=2$ dimensional linear diffusion system where the force is given by ${f}({q}) = - \\mathsf {F}\\cdot {q}$ with a force matrix $\\mathsf {F} = \\left(\\begin{array}{cc} 1 & \\varepsilon \\\\ -\\varepsilon & 1 \\end{array} \\right) \\ .$ This model is a specific case of a general linear diffusion system studied in Ref.",
"[27], where one can find closed form solutions for various distribution functions.",
"The purpose of this study is to confirm the fluctuation relations in Eqs.",
"(REF ), (REF ), and (REF ) explicitly, and to understand the effect of the correlation on the fluctuation relation.",
"The force matrix is decomposed into the symmetric part ${\\mathsf {F}}_s =({\\mathsf {F}}+{\\mathsf {F}}^T)/2 = {\\mathsf {I}}$ and the anti-symmetric part ${\\mathsf {F}}_a = ({\\mathsf {F}}-{\\mathsf {F}}^T)/2$ .",
"Then, the conservative force, the energy function, and the nonconservative force are given by ${f}_c({q}) = - {q}$ , $\\Phi ({q}) = {q}^T \\cdot {\\mathsf {F}}_s \\cdot {q}/2 = \\frac{1}{2} |{q}|^2$ , and ${f}_{nc} = -{\\mathsf {F}}_a \\cdot {q} = \\varepsilon (-q_2, q_1)^T$ , respectively.",
"So the model describes a particle trapped in an isotropic harmonic potential $\\Phi $ and driven by a swirling force ${f}_{nc}$ .",
"The parameter $\\varepsilon $ represents a strength of the driving force.",
"The linear diffusion system was studied in Ref.",
"[27] using a path-integral formalism.",
"We extend the formalism to obtain the joint probability distributions.",
"The algebra is straightforward but rather lengthy.",
"So we present the explicit expression for the moment generation function ${\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\lambda ,\\kappa )$ without derivation.",
"Details will be published elsewhere [30].",
"It is given by ${\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\lambda ,\\kappa ) = {\\mathcal {F}}(\\lambda ,\\kappa ) \\ ,$ where ${\\mathcal {F}}(x,y) \\equiv \\frac{e^t}{\\cosh ({\\Omega (x)}t)+\\frac{(1-4y^2)+\\Omega (x)^2}{2{\\Omega (x)}}\\sinh ({\\Omega (x)}t)}$ and $\\Omega (x) \\equiv \\sqrt{1 - 4\\varepsilon ^2 x(x-1)} \\ .$ Note that $\\Omega (x) = \\Omega {(1-x)}$ .",
"Hence, ${\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\lambda ,\\kappa )$ satisfies the GFT in Eq.",
"(REF ).",
"Figure: Large deviation functions for the generation function in (a) and thePDF in (b) for heat at ε=1/(23)\\varepsilon =1/(2\\sqrt{3}) are drawn with solid lines.Dashed lines are the plot of (Ω(η)-1\\Omega (\\eta )-1) in (a) and its Legendretransformation in (b).The generating function for $P_h(Q)$ is given by ${\\mathcal {G}}_{\\hat{h}}(\\eta ) = {\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa =0) ={\\mathcal {F}}(\\eta ,-\\eta ) \\ .$ It does not obey the FT (${\\mathcal {G}}_{\\hat{h}}(\\eta )\\ne {\\mathcal {G}}_{\\hat{h}}(1-\\eta )$ ) for finite $t$ .",
"The corresponding LDF is given by $e_{\\hat{h}}(\\eta _-\\le \\eta \\le \\eta _+)= \\Omega (\\eta ) - 1 \\ .$ We remark that the limit should be taken carefully.",
"The function ${\\mathcal {F}}(x,y)$ has a pole singularity at $(\\Omega (x)-1)^2 =4y^2$ in the $t\\rightarrow \\infty $ limit.",
"Hence, the LDF is well-defined only within the interval $\\eta _- \\le \\eta \\le \\eta _+$ where $\\eta _+=1$ and $\\eta _- =\\frac{1}{2}-\\frac{1}{2}\\sqrt{1+1/\\varepsilon ^2}$ for $\\varepsilon ^2>1/3$ and $\\eta _- =(\\varepsilon ^2-1)/(\\varepsilon ^2+1)$ for $\\varepsilon ^2\\le 1/3$ .",
"Equation (REF ) is valid only within the interval, while $e_{\\hat{h}}(\\eta )=-\\infty $ otherwise.",
"The Legendre transformation $e_h(q) = \\max _\\eta \\lbrace e_{\\hat{h}}(\\eta ) - q \\eta \\rbrace $ yields that $e_{h}(q) = \\left\\lbrace \\begin{array}{ll}-\\eta _+ q + \\Omega (\\eta _+) - 1 & ,\\ q \\le q_+ \\\\ [1mm]-\\eta _- q + \\Omega (\\eta _-) - 1 & ,\\ q \\ge q_- \\\\ [1mm]\\sqrt{\\frac{(1+\\varepsilon ^2)(q^2+4\\varepsilon ^2)}{4\\varepsilon ^2}}- \\frac{q}{2}-1 & , \\mbox{ otherwise}\\end{array}\\right.$ where $q_{\\pm } = \\left.",
"{d e_{\\hat{h}}}/{d\\eta }\\right|_{\\eta =\\eta _\\pm }$ .",
"The linear branches indicate exponential tails in $P_h(Q)$ [19].",
"Figure REF (a) shows the LDF at $\\varepsilon =1/(2\\sqrt{3})$ .",
"The function $[\\Omega (\\eta )-1]$ is drawn with a dashed line, while $e_{\\hat{h}}(\\eta )$ is drawn with a solid line.",
"The Legendre transformation $e_h(q)$ is plotted in Fig.",
"REF (b) with a solid line.",
"The Legendre transformation of $[\\Omega (\\eta )-1]$ is also drawn with a dashed line.",
"They deviate from each other at $q_{\\pm }$ .",
"In order to test the FT, we plot $e_h(-q)-e_h(q)$ (solid line) in Fig.",
"REF (a).",
"It does not coincide with the dashed straight line representing $e_h(-q)-e_h(q)=q$ for large $q$ , which shows that heat does not obey the FT.",
"It is worthy to compare our result with that of Ref. [19].",
"In both cases, the FT appears to be valid for small values of $q$ , specifically within the domain $|q| \\le |q_+|$ in our study.",
"The value of $|e_h(-q)-e_h(q)|$ saturates to a constant for large $|q|$ in Ref. [19].",
"It contrasts with the linear increase when $|q|>|q_-|$ in our case.",
"It suggests that the heat fluctuations do not exhibit a universal behavior [19], [24].",
"Figure: (a) Deviation from the FT for heat at ϵ=1/(23)\\epsilon =1/(2\\sqrt{3}).",
"(b) Validity domain for the LDF e h ^,e ^ (η,κ)e_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ).Our theory predicts that $\\psi (q)$ describes the deviation from the FT. We now evaluate $\\psi (q)$ explicitly to confirm the proposed relation in Eq.",
"(REF ).",
"Using Eq.",
"(REF ) and ${\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) ={\\mathcal {G}}_{\\hat{w},\\hat{e}}(\\eta ,\\kappa -\\eta )$ , one has ${\\mathcal {G}}_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) = {\\mathcal {F}}(\\eta ,\\kappa -\\eta )$ .",
"So the LDF is given by $e_{\\hat{h},\\hat{e}}(\\eta ,\\kappa ) = \\Omega (\\eta ) - 1 \\ .$ It appears to be independent of $\\kappa $ .",
"However, due to the singularity of ${\\mathcal {F}}$ in Eq.",
"(REF ), the LDF is well defined only within the region $4(\\kappa -\\eta )^2 \\le (\\Omega (\\eta )-1)^2$ , i.e., $\\eta -\\frac{1+\\Omega (\\eta )}{2} \\le \\kappa \\le \\eta + \\frac{1+\\Omega (\\eta )}{2} \\ .$ Accordingly, the LDF $e_{\\hat{h},\\hat{e}}(\\eta ,\\kappa )$ has a $\\kappa $ dependence.",
"The domain is drawn in Fig.",
"REF (b).",
"Now we need perform the Legendre transformation of Eq.",
"(REF ) at $\\kappa =0$ and 1.",
"When $\\kappa =0$ , $\\eta $ is restricted to the interval $\\eta _- \\le \\eta \\le \\eta _+$ , and $e_{h,\\hat{e}}(q,\\kappa =0)$ becomes equal to $e_h(q)$ given in Eq.",
"(REF ).",
"When $\\kappa =1$ , the validity region is shifted to $\\eta _-^{\\prime }\\le \\eta \\le \\eta _+^{\\prime }$ (see Fig.",
"REF (b)).",
"So, $e_{h,\\hat{e}}(q,\\kappa =1)$ is given by the function in Eq.",
"(REF ) with $\\eta _\\pm $ and $q_{\\pm }$ being replaced with $\\eta _{\\pm }^{\\prime }$ and $q_{\\pm }^{\\prime }=de_{\\hat{h}}/d\\eta |_{\\eta =\\eta _\\pm ^{\\prime }}$ , respectively.",
"Notice the symmetry $\\Omega (\\eta ) = \\Omega (1-\\eta )$ .",
"It yields that $\\eta _{\\pm }^{\\prime } = 1-\\eta _{\\mp }$ and $q_{\\pm }^{\\prime } = - q_{\\mp }$ .",
"Inserting these into Eq.",
"(REF ), one can find that $e_{h,\\hat{e}}(q,\\kappa =1) = e_h(-q)-q$ .",
"This completes the proof that $\\psi (q) = e_h(-q)-e_h(q)-q$ .",
"We also test validity of the relation (REF ) at finite $t$ .",
"We have solved Eq.",
"(REF ) with $\\varepsilon =1/\\sqrt{3}$ numerically $10^7$ times up to $t=0.5$ to measure various PDFs and $\\Psi (Q)$ .",
"In Fig.",
"REF (a), $P_w(W)$ and $e^W P_w(-W)$ are compared, which confirms the FT for work.",
"In Fig.",
"REF (b) $P_h(Q)$ displays a disagreement with $e^Q P_h(-Q)$ but matches perfectly with $e^Q P_h(-Q)/\\Psi (Q)$ .",
"This is a numerical verification of the relation in Eq.",
"(REF ).",
"The joint PDF $P_{w,e}(W,\\Delta E)$ shown in Fig.",
"REF (c) is symmetric under inversion $\\Delta E\\rightarrow -\\Delta E$ .",
"So the energy may increase or decrease equally likely irrespective of the amount of work.",
"It explains the reason why the heat distribution is wider than the work distribution as shown in Fig.",
"REF .",
"One can find an anti-correlation between ${\\mathcal {Q}}$ and $\\Delta {\\mathcal {E}}$ in Fig.",
"REF (d).",
"Due to the correlation, $\\Psi (Q)= \\langle e^{-\\Delta E}\\rangle _Q \\ne 1$ .",
"Figure: (Color online)(a) Semi-log plots of P w (W)P_w(W) and e W P w (-W)e^W P_w(-W).",
"(b) Semi-log plots of P h (Q)P_h(Q), e Q P h (-Q)e^Q P_h(-Q), and e Q P h (-Q)/Ψ(Q)e^Q P_h(-Q) / \\Psi (Q).Density plots of lnP w,e (W,ΔE)\\ln P_{w,e}(W,\\Delta E) in (c) andlnP h,e (Q,Δ)\\ln P_{h,e}(Q,\\Delta ) in (d).In summary, we have derived the fluctuation relation for heat in Eq.",
"(REF ) using the GFT in Eq.",
"(REF ) for the joint PDF.",
"The heat distribution does not obey the same type of the fluctuation relation as the work distribution does.",
"The modification is given by the factor $\\Psi (Q)\\equiv \\langle e^{-\\Delta E}\\rangle _Q$ that depends on the correlation between heat and energy change.",
"The modified fluctuation relation for the heat has been tested analytically and numerically for a linear diffusion system.",
"Our result shows that the FT for heat is not valid in general for finite $t$ .",
"The model studies in this work and in Ref.",
"[19] show explicitly that the FT is violated even for the LDF in the $t\\rightarrow \\infty $ limit.",
"Nevertheless, it still remains as an open question whether there is a criterion for the FT in terms of the LDF.",
"A sufficient condition is readily obtained from our result.",
"Suppose that the energy function is strictly bounded as $E_0<\\Phi ({q})<E_1$ with finite $E_{0,1}$ [5], [3], [23].",
"Then, $e^{-(E_{1}-E_{0})}< \\Psi (Q)<e^{E_{1}-E_{0}}$ , hence $\\psi (q)=0$ and the FT holds.",
"Hopefully, our formalism may yield a more strict condition for the FT. Future works are necessary in order to understand implication of the proposed fluctuation relation and to generalize it for systems with a time-dependent perturbation or systems in contact with many reservoirs.",
"Experimental studies in small-sized systems [16] are also necessary in order to characterize nonequilibrium fluctuations of heat.",
"This work was supported by Mid-career Researcher Program through NRF Grant No.",
"2011-0017982 funded by the Ministry of Education, Science, and Technology of Korea.",
"We thank Hyunggyu Park and Chulan Kwon for stimulating discussions."
]
] | 1204.1004 |
[
[
"Extracting Geospatial Preferences Using Relational Neighbors"
],
[
"Abstract With the increasing popularity of location-based social media applications and devices that automatically tag generated content with locations, large repositories of collaborative geo-referenced data are appearing on-line.",
"Efficiently extracting user preferences from these data to determine what information to recommend is challenging because of the sheer volume of data as well as the frequency of updates.",
"Traditional recommender systems focus on the interplay between users and items, but ignore contextual parameters such as location.",
"In this paper we take a geospatial approach to determine locational preferences and similarities between users.",
"We propose to capture the geographic context of user preferences for items using a relational graph, through which we are able to derive many new and state-of-the-art recommendation algorithms, including combinations of them, requiring changes only in the definition of the edge weights.",
"Furthermore, we discuss several solutions for cold-start scenarios.",
"Finally, we conduct experiments using two real-world datasets and provide empirical evidence that many of the proposed algorithms outperform existing location-aware recommender algorithms."
],
[
"Introduction",
"With the affordable prices of GPS-enabled mobile devices and the success of social networks, location-based social media has become increasingly popular in recent years.",
"Users can upload content, e.g., photos, videos, and text, and annotate that content with geographical identification metadata, typically known as geotags.",
"Geotags act as geographic indexes helping users to organize and retrieve location-specific information.",
"Foursquarehttp://foursquare.com/, as an example, is a location-based service where users endorse and share tips about visited points of interest (POI).",
"It reached 725 thousand registered users and 22 million check-ins (i.e.",
"endorsed POIs) in 2010http://mashable.com/2010/03/29/foursquare-growth-numbers/.",
"Recommender systems (RS) are among the best known techniques for helping users filter out and discover relevant information in large data sets.",
"In the typical scenario, RS algorithms exploit user-item matrices representing user preferences for items, e.g., the rating history of purchased books in Amazon, with the aim of recommending the items most likely to be relevant to the user.",
"While most of the RS work to date has ignored the locations where users demonstrated interest for an item, there are many scenarios in which the geographic context of an item has a direct influence on the preferences of the user for that item.",
"Mao et al.",
"[21], for example, showed that Foursquare users prefer POIs that are nearby the POIs they already visited in the past, while [13] showed that Panoramio users who took pictures in nearby locations in the past, tend to share similar preferences of geographic regions in the future.",
"Efficient extraction and representation of location-specific user preferences are thus essential to decide on what item to recommend.",
"Location-aware recommender systems suggest relevant geotagged items for a given user within a declared geographic area.",
"Relevance here can assume different notions, depending on the geographic constraints imposed by the user.",
"For example, a user may be interested in objects nearby his previous, current, or future location within a given radius.",
"For example, a first time user visiting the Stanford University campus in Palo Alto, US, might be interested to know what is worth visiting inside the campus, while a second time visitor may want to know what else is worth visiting.",
"Each of these scenarios can lead to various definitions of user preference, hence it is important to know which definition works best for each scenario.",
"The literature concerning location-aware recommender system is still sparse, where the methods are ad-hoc and can not be easily changed to meet the different recommendation scenarios outlined above (e.g.",
"[13], [18], [6]).",
"In this paper we propose a relational graph for capturing the geographic context of users that suits all the aforementioned recommendation scenarios.",
"We introduce several strategies to represent location-specific user preferences in the graph and show how to derive many recommendation algorithms, including ensembles of them, by only changing the definition of the edge weights.",
"Our contributions are as follows: We propose a new model for geotagged data that is able to capture both the geographic context of users and their preferences for objects within the declared geographic context.",
"We introduce new similarity measures that take into account the spatial decisions of users.",
"We propose a recommendation algorithm based on a relational neighbor graph, that derive many recommendation algorithms and ensembles of them, as special cases, by only requiring changes in the definition of the edge weights.",
"Finally, we conduct experiments using two datasets in various recommendation scenarios, including cold-start ones, and provide empirical evidence that many of the proposed algorithms outperform existing location-aware recommender algorithms.",
"The rest of the paper is organized as follows.",
"Section introduces the problem setting.",
"Section presents our relational graph representation of the data, weighting schemes for capturing user geographic preferences, and a recommendation algorithm based on relational neighbors.",
"Section presents the experimental setting and evaluation.",
"Section describes related work, and Section concludes the paper and discusses future work."
],
[
"Problem Setting",
"The recommendation scenario is as follows.",
"A user specifies the geographic region of interest, e.g., a city or a region in that city, and the recommender engine suggests items within the declared geographic region that are likely to be relevant to the user.",
"So, let $U$ be the set of users, $G$ the set of regions denoting geographic contexts, and $I$ the set of geotagged items.",
"Notice that what is meant by a region is application dependent since regions can assume different geographic shapes, such as point coordinates, circles, lines, and polygons.",
"In this paper we only consider implicit feedback dataAlthough our framework can trivially support explicit feedback as well., i.e., the set $S\\subseteq U\\times G\\times I$ of ternary relations between users, geographic contexts, and geotagged items.",
"The task is then to find a prediction scoring function $\\hat{s}:U\\times G\\times I\\rightarrow \\mathbb {R}$ that predicts a preference score for items within certain geographic regions, given a target user.",
"Now, for a given user $u\\in U$ , and a given geographic context $g\\in G$ , the topN recommendations can be computed by $\\textit {topN}(u,g):=\\operatornamewithlimits{argmax}_{i\\in I_g}^n \\hat{s}(u,g,i)$ where $n$ denotes the topN items to be recommended and $I_g$ the set of items within the geographic context $g$ .",
"For convenience, we also define $I_{u,g}:=S\\cap (\\lbrace u\\rbrace \\times \\lbrace g\\rbrace \\times I)$ as the set of items of user $u\\in U$ in a given geographic context $g\\in G$ ."
],
[
"A Relational Approach",
"Relational classification refers to an active area of machine learning where classifiers usually consider, additionally to the typical attribute-value data of objects, relational information.",
"A scientific paper, for example, can be connected to another paper that has been written by the same author or because they share common citations.",
"It has been shown that in many classification problems, relational classifiers outperform purely attribute-based classifiers [5], [10], [16].",
"In particular, Macskassy and Provost [11] showed that simple relational neighbor-based techniques, besides requiring low computational costs, perform competitively to, and in some cases even outperforms, more complex relational methods such as Probabilistic Relational Models and Relational Probability Trees.",
"The basic idea is that the classification of a target instance solely depends on the class labels of related instances of the same type.",
"Since geotagging data is inherently relational, we propose to capture the geographic context of users and items by a relational graph, which, as a side effect, gives us many tools from relational classification that can be directly applied to the location-aware recommendation problem.",
"In order to easily use neighborhood-based classification methods, similarly to [11], [16], we adopt a homogenous view of the relations in the data.",
"In a homogeneous view we have only one entity type, such that, there is a set of target entities $x\\in X$ and relations $R\\subseteq X\\times X$ between them.",
"However, as we saw in Section , geotagging data forms a set $S$ of ternary relations between three different types of entities.",
"Therefore, we first need to convert these ternary relations into the desired homogeneous relations.",
"We do this as follows relational classification is possible with heterogenous node types as well as has been demostrated in [9] and [14] but the added complexity defeats our goal of fast model building and evaluation.."
],
[
"Graph Definition",
"First let $V:=\\lbrace (u,g)\\,|\\,\\exists i\\in I\\,:\\,(u,g,i)\\in S\\rbrace $ denote the set of all distinct user/geographic context combinations in $S$ .",
"We now propose to interpret $S$ as a graph $\\mathbb {G}:=(V,E)$ , where $V$ is the set of vertices and $E\\subseteq V\\times V$ the set of edges.",
"We assume that there is an edge between two vertices if they share the same geographic context, i.e., $\\lbrace (v,v^{\\prime })\\in E \\,|\\, g_v = g_{v^{\\prime }}\\rbrace $ For convenience, let $g_v:=g$ and $u_v:=u$ denote the geographic context and the user of node $v\\in V$ respectively.",
"In other words, we assume that users who share the same geographic region are related to some extent.",
"Now suppose that John is a first time visitor in Rio de Janeiro, Brazil, and wants to know from other people who already have been in Rio which places are worth visiting there.",
"To continue our example, we denote the pair John/Rio by a colored node in the graph of Figure REF , i.e., the target node for which we want to compute recommendations.",
"John is denoted as $u_1$ and Rio as $g_1$ respectively, and the other nodes connected to it contain the users who already have been in Rio.",
"For computing recommendations, we just need to go through the items of the neighbors and define some selection criterion on which items to recommend.",
"This idea assumes that entities related to each other, in this case users sharing the same geographic context, are similar and tend to select the same items.",
"Notice that the strength of the similarity depends on the size of the geographic region being considered.",
"If the declared geographic context is a small region, say the Copacabana beach in Rio de Janeiro, then it is more likely that users within this area will be more strongly related to each other than users sharing larger geographic contexts, such as the whole country of Brazil.",
"We can alleviate these effects by defining appropriate weights to each relation, as we will see next.",
"Figure: Example of relational graph for capturing geographic context."
],
[
"Weighting Schemes",
"In this section we present several weighting strategies for the edges of the relational graph.",
"And as we will see in the next section, each weighting scheme leads to a specific recommendation algorithm, without, however, the need to change the overall algorithm."
],
[
"Uniform Weighting",
"Here the same weight value is assigned to each edge (1 in our case), denoting that each neighbor is of the same importance to the target user.",
"Therefore, the weight $w$ for any edge $(v,^{\\prime }v)\\in E$ , is defined as $w^{\\text{uni}}(v,v^{\\prime }):=1$ .",
"The computational cost for assigning uniform weights to the relations of a target node $v$ is in the order of $O(|N_v|)$ , i.e., a linear scan in the neighbors of $v$ , here denoted by $N_v$ ."
],
[
"Correlation Weighting",
"For two nodes $(v,v^{\\prime })\\in E$ , we can represent $v$ and $v^{\\prime }$ as profile vectors, where each component of the vector is a geotagged item and the values denote the preference of a user for an item, i.e., $\\vec{m}_v:=(i_1,...,i_{|I|})$ We can either construct this vector solely based on the items of the geographic context of interest, or we could consider the items of all geographic contexts.",
"The assumption is that users who have selected the same items within the same geographic contexts are more similar than users who did not.",
"If a user declared interest for some geographic context for which he has not yet selected any item, it will not be possible to compute any correlation similarity with other users, unless we build the profile vectors considering the items of other geographic contexts.",
"The edge weight between two nodes is finally computed by applying a correlation metric between the desired nodes' profile vectors, which in our case, is the cosine similarity: $w^{\\text{cor}}(v,v^{\\prime }):=\\frac{\\langle \\vec{m}_v,\\vec{m}_{v^{\\prime }} \\rangle }{\\Vert \\vec{m}_v\\Vert \\Vert \\vec{m}_{v^{\\prime }}\\Vert }$ The computational cost for assigning cosine similarities to the relations of a target node $v$ is in the order of $O(Z\\cdot |N_v|)$ since we need to compute $|N_v|$ similarities, each requiring $Z$ operations."
],
[
"Geographic Similarity",
"We can use the geographic distances between users' items to define the strength of their relation.",
"The assumption is that users who select items nearby the items of other users should be closely related.",
"Therefore, for defining the weight between two nodes $(v,v^{\\prime })\\in E$ through geographic similarity, we first calculate the geographic centroids of the set of items of users $u_v$ and $u_{v^{\\prime }}$ .",
"For computing the geographic centroid of a given user $u\\in U$ , we sum up the coordinates of each geotagged item of user $u$ within the declared geographic context $g$ , i.e., the items in the set $I_{u,g}$ , and divide the resulting sum by the number of items in the set $\\frac{1}{|I_{u,g}|}\\displaystyle \\sum _{i\\in I_{u,g}} p(i)\\\\$ where $p(i)$ returns the latitude/longitude coordinate used to geotag item $i\\in I$ .",
"Now, the geographic distance between $u(v)$ and $u(v^{\\prime })$ centroids is given by any of the many existing functions for calculating geodetic distances between latitude/longitude coordinates, e.g.",
"the Haversine formula.",
"Here we denote such a function by $d(x,y)$ where $x$ and $y$ are two coordinates.",
"Finally, we only need to turn the distance into a similarity and bound it to the range $[0,1]$ , which is done as follows $w^{\\text{geo}}(v,v^{\\prime }) := 1-\\frac{d\\left(c_{u_{v}},c_{u_{v^{\\prime }}}\\right)}{d_{\\text{max}}}$ where $d_{max}$ is the maximal possible distance between any points in the region of interest and $c_u$ is the geographic centroid of the set of items of user $u$ .",
"In other words, when the distance between the items of two users is small, their similarity tend to 1, and is 0 when the distance equals $d_{\\text{max}}$ .",
"Figure REF illustrates this idea.",
"Noticed that for this to work, we are assuming that users tend to form geographic clusters among the selected items.",
"This was empirically observed in [21] by showing that Foursquare users tend to check-in to POIs that are nearby the POIs they have already visited.",
"The computational cost of this weighting scheme is in the same order as the correlation weighting above.",
"Figure: Example of geographic similarity.",
"The items of users u 1 u_1, u 2 u_2, u 3 u_3 are represented by triangles, circles and rectangles respectively.",
"Centroids are represented by “X” and distances by dashed lines.",
"Thus, u 3 u_3 would be regarded as more similar to u 1 u_1 than u 2 u_2."
],
[
"Partonomy-Based Similarity",
"The different granularities between geographic contexts can be represented by a hierarchical partonomy, i.e., a graph $\\mathbb {G}:=(G,E)$ where vertices are represented by geographic contexts and each edge $e\\in E$ represents a part-of relation between two geographic regions [13].",
"For example, a POI is a part of a city, a city is a part of a state, a state is a part of a country, etc.",
"Matyas et al.",
"[13] proposed several similarity measures for capturing user similarities with respect to specific levels of a weighted geographic partonomy.",
"The idea is that if two users are not found to be similar in a lower level of the hierarchy, they may, eventually, be found to be similar in a higher level.",
"For example, even if two users did not visit the same places in some city, they still can be considered to be similar since they have visited the same city.",
"We can use these measures, together with a partonomy, to weigh the relations in our relational graph.",
"Here we just present the best performing measure reported in [13].",
"The similarity of two users $u,u^{\\prime }\\in U$ with respect to node $g$ is calculated as follows $\\textit {sim}^{\\textit {inf}}(u,u^{\\prime },g):=\\frac{\\sum _{(c\\in A_g\\cap B_g)}\\textit {information}(c)}{\\sum _{(c\\in A_g\\cup B_g)}\\textit {information}(c)}$ where $A_g$ and $B_g$ denote the sets of children nodes of $g$ in which user $u$ and $u^{\\prime }$ selected items respectively.",
"The similarity is weighted by the function information, which can be seen as the inverse of the popularity of a node.",
"The assumption is that users sharing less popular nodes are more similar than users sharing more popular ones [13].",
"Now, for computing the similarity of two users $u,u^{\\prime }\\in U$ with respect to a certain partonomy layer $l$ , the following formula is used $\\textit {sim}^{\\textit {two-layer}}(u,u^{\\prime }):=\\frac{\\sum _{(g\\in G_l)}\\textit {sim}^{\\textit {inf}}(u,u^{\\prime },g)\\cdot w^{\\textit {node}}(g)}{\\sum _{(g\\in G_l)}w(g)}$ where $G_l$ is the set of nodes in the $l$ -th layer of the partonomy.",
"This measure was named two-layer similarity in [13] because when computing the similarity of two users in a certain layer, the measure uses the layer immediately below.",
"Returning the discussion to the weighting of the edges in our relational graph, we can now define the weight of a certain edge $(v,v^{\\prime })\\in E$ in terms of the two-layer similarity, i.e., $w(v,v^{\\prime }):=\\textit {sim}^{\\textit {two-layer}}(u_v,u_{v^{\\prime }})$ This similarity can be quite expensive to compute, given all the necessary steps described above for weighting the partonomy.",
"Even assuming that the weighted partonomy is given, the complexity for computing the weight between $v$ and all its neighbors is still higher than for the other similarities.",
"The computational cost is in the order of $O(|N_v||G_l||C_l|\\cdot Z)$ where $C_l$ is the set of children nodes of the $l$ -th layer, since for each neighbor, we need to go through all the locations $g$ in the $l$ -th layer of the partonomy, for each location go through all its children, and finally for each child we need to perform $Z$ operations."
],
[
"Recommendation Algorithm",
"Now, we have all the components for deriving a location-aware recommendation procedure.",
"Algorithm REF describes the overall recommendation process.",
"It receives as input a weighted graph $\\mathbb {G}$ , a user/geographic context pair denoted by $v$ , for which we want to generate recommendations, and the number $n$ of recommendations to be returned.",
"The algorithm iterates through the neighbors of $v$ (line 3), denoted by $N_v$ , and for each neighbor, it iterates through the items within the geographic context of interest (line 4) and accumulates weights in the array scores, that is indexed by items.",
"Finally, it sorts the scores in descending order of weights and presents the top-$n$ geotagged items that the target user has not already selected.",
"Note that if we define all weights to 1, we end up recommending the most popular items within the geographic context of interest.",
"Or if we decide to weigh edges according to the similarities presented in Section REF , we end up with many flavors of collaborative filtering-based algorithms.",
"Assuming that the weighted relational graph is given, the complexity of this algorithm only depends on the computation of a weighted sum of geotagged items, which means $|N_v|$ passes in the set of items $I$ .",
"Hence, the complexity is given by $O\\left(|I||N_v|\\right)$ .",
"Graph-based Location Recommendations [1] Input: $\\mathbb {G}(V,E)$ , $v\\in V$ , $n$ Output: list of topN recommendations $v^{\\prime } \\in N_v$ $i \\in I_{u_{v^{\\prime }},g_{v^{\\prime }}}$ $\\textit {scores}[i]\\leftarrow \\textit {scores}[i] + w(v,v^{\\prime })$ $\\text{topN}\\leftarrow \\displaystyle \\operatornamewithlimits{argmax}_{i\\in I\\setminus I_{u_{v},g_{v}}}^n \\textit {scores}[i] $"
],
[
"Experiments",
"In this section we describe our two datasets, geotagged photos from Panoramio, and print jobs from the HP ePrint Mobile Print Location service; the evaluation protocol adopted; and the results for each dataset.",
"We considered three different recommendation scenarios.",
"In the first scenario, we hide all the geotagged items of each test user in a geographic context, and use the remaining data for trying to predict the removed items.",
"This corresponds to a cold-start scenario where a user has not selected any item in the geographic context of interest.",
"We will refer to this scenario as leave-all-out.",
"In the second scenario, we remove some geotagged items for each test user, 4 photos in Panoramio and 1 printer provider in ePrint, and use the remaining data for predicting the removed items.",
"This scenario represents those users who already selected some items in a given location but want to know what other items are worth selecting in this location.",
"We will refer to this scenario as leave-some-out In the third scenario, we have a mix of both scenarios, i.e., some fraction of the users are first time users and the other fraction already have selected some items in the location of interest.",
"This corresponds to a more realistic scenario, and to the best of our knowledge, this is the first time location-aware algorithms are evaluated in this kind of scenario.",
"We will refer to this scenario as leave-some/all-out"
],
[
"Panoramio Experiments",
"Panoramio is a photo-sharing website from Google where users can upload, geotag, and retrieve photos of landmarks.",
"Each photo in Panoramio is georeferenced using latitude and longitude information.",
"Similarly to [13], we assume that if a user takes a picture in a specific location, then he has some interest in that location.",
"As geographic context of interest we have chosen the city of Rio de Janeiro, which is one of the top touristic places in Brazil, so, it has a large set of photos."
],
[
"Data Collection and Preparation",
"In order to retrieve the set of photos taken in Rio, we did a spatial search for photos inside the bounding box of Rio using the data access API of Panoramio.",
"We then iterated through each photo in the result set, and retrieved the users who took these photos.",
"For each of these users, we then retrieved the other locations where they took photos, as well as the users and photos in those locations.",
"Then we removed the cities with too little activity from users who also visited Rio.",
"Approximately the top three cities in all the crawled states were kept in the evaluation set.",
"We used the gazetteer of HP Gloehttp://www.hpgloe.com/ to obtain the place names.",
"In order to use the two-layer-similarity weighting scheme presented in Section REF , we built a geographic partonomy, and similarly to [13], we worked with three countries (Brazil, Chile, and USA) as high level nodes, states and cities as intermediary nodes, and geographic clusters as leaf nodes.",
"Our data set contains 35,920 photos (4,906 from Rio) and 7,048 users in total.",
"In Panoramio the geographic items are represented by latitude and longitude coordinates, hence, it is very difficult for two users to take a picture in the exact same location.",
"Therefore, we adopted the same approach as in [13], where the authors used geographic clusters to represent geographic items.",
"So, instead of recommending individual point coordinates, we recommend regions where users may be interested in taking photos.",
"For computing the clusters, we used the DBSCAN [7] algorithm with the following parameters: $\\textit {MaxRadius}=1$ Km and $\\textit {MinPoints}=3$ .",
"The $\\textit {MaxRadius}$ parameter was set to 1 Km, as we assume that a radius of 1 Km from a given photo is sufficient to establish a geographic similarity between photos, e.g., photos taken at the University of Stanford campus.",
"We tested to set $\\textit {MaxRadius}=500$ meters without any significant differences in performance, so we do not show those results here.",
"Also, the minimum number of points to form a cluster was set to 3, in order to establish popularity of a given POI.",
"We computed 1,187 clusters, 221 of which are in Rio."
],
[
"Evaluation Protocol",
"For testing the algorithms, we considered the dense part of the data, i.e., only the users who have taken at least 5 photos in Rio (see Table REF ).",
"For the leave-some-out and leave-some/all-out scenarios, we generated 5 random splits of training/test sets and took the average precision and recall on top-10 recommendation lists over all splits.",
"We have a hit every time a hidden photo is found to belong to some of the recommended geographic clusters.",
"Whenever CF is not able to fill the recommendation list up to 10, we fill up the list with the most popular items that are not already in the recommendation list.",
"Table: Panoramio data for Rio de Janeiro"
],
[
"Algorithms",
"We have used several weighting schemes for our relational graph, which resulted in the following recommendation algorithms: Most popular (MP): Recommends the most popular geographic clusters in Rio.",
"For doing that, we just apply Uniform Weighting (see Section REF ) to the graph in Algorithm REF .",
"Intra-Cluster (IC): Uses the geographic similarity defined in Section REF for defining intra-cluster similarities between users.",
"The idea is as follows.",
"The weight of any edge $(v,v^{\\prime })\\in E$ is now calculated by summing up $w^{\\text{geo}}(v,v^{\\prime })$ for all clusters where $u_v$ and $u_{v^{\\prime }}$ have photos, and dividing the resulting sum by the total number of clusters.",
"After that, we again just need to plug the weighted graph into Algorithm REF for computing recommendations.",
"Collaborative Filtering (CF): This is the geographic version of the classic user-based collaborative filtering algorithm.",
"We use the correlation weighting scheme defined in Section REF where profile vectors' components are the geographic clusters where users took pictures, within and out of the context of interest.",
"Two-Layer (TL): This is the algorithm originally introduced in [13].",
"We weigh the relational graph with the two-layer similarity and plug it into the recommendation algorithm.",
"Correlation + Two-Layer (CF-TL): We found empirically that TL works best for cold-start scenarios, while CF outperforms the other algorithms for non cold-start scenarios (see Figures REF and REF ).",
"This gave us the insight to propose a weighting combination strategy where we weigh each relation differently according to the case presented.",
"If the target user is a cold-start user, we weigh his relations with the two-layer similarity, if not, we use the correlation weighting.",
"The flexibility for combining different weighting schemes in such an easy way is one of the main advantages of our approach."
],
[
"Results",
"Figure REF depicts the results for the leave-some-out scenario.",
"Notice that when there is enough data available, CF outperforms all the other methods.",
"The Intra-Cluster recommender, although worse than CF and TL, is better than MP in all cases.",
"This indicates that the geographic similarity indeed is able to capture some preferences of the user, under the assumption that users tend to like the items of other users that are nearby the items they have selected in the past.",
"Figure REF shows the results for the leave-all-out.",
"Notice that TL is the winner in this scenario.",
"This is in line with the results of [13], where they showed that geographic partonomies can help to improve the recommendations in cold-start scenarios.",
"Notice that since we remove all the photos of all test users in the context of interest, we just have one possible split of training/test, and thus cannot compute standard deviations and plot error bars.",
"Figure: Evaluation on leave-some-out scenario.Figure: Evaluation on leave-all-out scenario.In Figure REF we show the results of the leave-some/all-out scenario when 50% of the test users are cold-start users.",
"In this case, the combination method CF-TL is slightly better, both in precision and recall, than the other methods.",
"But when 70% of ther users are cold-start users (see Figure REF ) the superiority of CF-TL becomes more evident.",
"We also evaluated this scenario when 30% of the users are cold-start users, but since there was no significant differences in performance in comparison to 50% of cold-start users, we do not show those results here.",
"Figure: Evaluation on the leave-some/all-out scenario when half of the users are cold-start users.Figure: Evaluation on the leave-some/all-out scenario when 70% of the users are cold-start users."
],
[
"HP ePrint Experiments",
"The HP ePrint Mobile Print Location (MPL) http://www.hp.com/go/eprintmobile service allows smartphone users to print documents, photos, web pages and emails directly to public print providers, such as business centers at hotels and airports, and dedicated copy and print service stores.",
"Users can query for providers near their current location and then submit a print job upon which a pickup code is generated.",
"This code can then be used in the store to obtain the printout."
],
[
"Data Preparation",
"Our dataset contains a 5-month trace of all print jobs sent through the service.",
"To protect business sensitive information the users as well as providers were anonymized.",
"Furthermore, the volume was obfuscated using a bootstrap inspired sampling algorithm that takes uniform random samples with replacement of print jobs from the original dataset and replays them with the original timestamp randomly modified up to one week from the original job submission.",
"We sampled 100k print jobs in this way, and then extracted all the entries that occurred in Manhattan (about 2,580).",
"Each entry comprises the user id of the user submitting the print job, the provider id of the provider receiving the print job, the timestamp of the print job and the latitude and longitude of the print provider location."
],
[
"Evaluation Protocol",
"Here the geographic items are print providers and in contrast to Panoramio, the POIs are well defined, and therefore we do not need to compute geographic clusters.",
"We chose Manhattan as the geographic context of interest because of its large volume of printing activity.",
"Here we used the leave-one-out protocol, i.e., for each user we randomly removed one of the items and used the rest for computing recommendations.",
"We considered all the users who have used at least one print provider, i.e., all users in Manhattan, for our test set.",
"due to the nature of the data set, we only evaluate the leave-some/all-out scenario, where some users are cold-start users and others not.",
"Furthermore, we ignore users for which the recommendation list is empty.",
"Since we only hide one photo, we do not compute precision, because this would be the same as recall up to a multiplicative constant.",
"Thus, we only compute the average recall over ten random splits of training/test.",
"Again, due to the nature of this dataset we only evaluated the MP and CF algorithms for top-10 recommendations.",
"Whenever CF is not able to fill the recommendation list up to 10, we fill up the list with the most popular items that are not already in the recommended list."
],
[
"Results",
"Figure REF shows the results for MP and CF in ePrint.",
"The error bars represent one standard deviation.",
"We can see that the most popular item algorithm outperforms the collaborative filtering algorithm consistently, in particular for long recommendation lists.",
"Given the relative short period this service has been on the market it is dangerous to draw too many conclusions from this behavior, but there are two lessons to be learned from this result.",
"First, we could easily adapt our algorithm to generate useful recommendations for a use case vastly different than the Panoramio application, by just simply applying a different weighting scheme.",
"Second, a very simple popularity-based recommendation engine can do very well, and deploying sophisticated collaborative filtering techniques may not always be the best choice.",
"Figure: MP vs. CF in HP ePrint."
],
[
"Related Work",
"Related work falls into three broad categories: relational classification and similarity measures, location-aware methods, and methods focusing on improving the computational scalability of the model building and execution phases."
],
[
"Relational Classification and Similarity Measures",
"Relational classification has been applied to areas where entities are linked in an explicit manner, like hypertext documents, such that the class of a target instance only depends on the class of its related instances [16].",
"In [12] we presented a relational neighbor classification approach in the ECML/PKDD Discovery Challenge 2009, which was about tag recommendations.",
"Similarly to that work, in this paper we formulate the recommendation problem in a relational neighbor framework where geographic contexts link entities represented by user/location pairs.",
"In [13] a new similarity measure for computing location recommendations based on a non-overlapping hierarchical taxonomy of locations is presented.",
"The key idea is that co-activity in locations can be better captured if you zoom out to larger and larger locations.",
"Similar to our work they use a panoramio data set to evaluate their algorithm.",
"However, their model assumes a cold-start user is making the query, i.e.",
"a user who has no trace in the geographic context of the query.",
"The reliance on a place taxonomy is also more restricting than our general model.",
"Jeh and Widom define SimRank in [9].",
"The SimRank similarity measure is as the name suggests highly influenced by the more famous PageRank [1] algorithm.",
"The general idea is that “two objects are similar if they are referenced by similar objects” [9].",
"This definition recursively propagates similarity through a relational graph to leverage structural context in addition to the more traditional use of object content and attribute information.",
"In our work akin to the model used in [13], we also have a recursive, relation-based definition of similarity but in the form of hierarchical geographic contexts.",
"For faster computation we simplify the similarity model to not be recursive between leaf-nodes within the same geographic context in our graph.",
"Our evaluations show that this can be done without any loss of accuracy if the user who requested the recommendation already has an activity trace within the geographic context for which the recommendation is sought.",
"Minkov and Cohen [14] also explore relational properties among objects to compute similarity scores.",
"They focus on the problem of effectively searching in graphs comprised of interrelated objects of various types.",
"They propose effective random-walk based procedures to evaluate the Personal PageRank measure [1], a measure that takes into account the scope of the query in addition to the well-known random surfer assumption of the original PageRank algorithm."
],
[
"Location-aware Methods",
"In [21] a standard user-based collaborative filtering approach is extended with a novel mechanism of geographic influence based on a statistical model of how likely a user is to check-in to two places based on their distance.",
"The geographic model is a distributional assumption (power law) fitted to real data sets from two popular check-in services.",
"A number of studies have looked at timestamped GPS traces to predict future locations within very restricted geographic regions [4], [23], [3].",
"Markov models and tensor factorization models are fit to the data and non-personalized predictions of the most likely next location or the most likely activity given a location and time are produced.",
"There is a lot of novel work on automatically detecting geographic context in these papers but the general approach cannot be replicated easily in our scenario since we do not have the same luxury of rich traces as we mainly focus on implicit feedback as input.",
"Relying on GPS traces is furthermore a privacy concern and has scalability and power consumption implications.",
"We produce more personalized recommendations given user-user similarities as opposed to just looking at the most popular or most frequent behavior.",
"Furthermore large Markov models and tensor factorization algorithms tend to be very costly to compute, which would therefore need to be done in an off-line setting whereas we also target real-time recommendations.",
"Bayesian networks have also been studied to model and learn patterns in location, time and weather contexts for individual users in [15].",
"However, compared to our model these models tend to be complex and require expert human knowledge to construct, and furthermore they are not tractable.",
"[2] applies a center of mass model to detect and recommend locations and POIs.",
"This work was before the check-in systems era so now we could just more easily query the check-in services for this information.",
"In [8] user-user based CF with location pre-filtering is employed in an explicit voting scenario.",
"The cold start problem is solved by generating random recommendations using pseudo users.",
"We address the problem by incorporating out-of geo context similarities for in-context recommendations which is less ad-hoc.",
"In our previous work in [19] we studied popularity inferred recommendations based on location, friends and tag prefiltering using both explicit and implicit feedback.",
"In this work we extend that model to provide personalized recommendations similar to the work in [18] but with a new model that incorporates location, and distance metrics directly in the evaluation graph as opposed to relying on ad-hoc and costly pre and post filtering.",
"The GeoFolk system [20] was designed to take both geographic context and text features into account for various information retrieval tasks such as tag recommendation, content classification and clustering.",
"Experiments show that combining both textual and geographic relevance leads to more accurate results than using the two factors in isolation.",
"Although our methods and use case targets are quite different from this work, the empirical evidence of the influence geographic context has on information retrieval is promising and serves as motivation for our work."
],
[
"Computational Scalability",
"Popular and accurate recommender system methods such as those based on matrix factorization can incure very high model building as well as execution overhead, in particular as more contextual factors beyond users and items are taken into account.",
"As a result there have been many attempts at improving the computational scalability of pre-existing methods.",
"In [18], [22], [6] the general issue of complex and high-latency model building and execution for location-based recommendations is addressed.",
"To achieve real-time performance [18] and [6] pre-filter based on location and [22] pre-filters based on friends to reduce the complexity of the models.",
"As opposed to pre- or post-filtering context we make contextual paramaters an inherent part of our graph model to allow interesting combinations of various types of context in an efficient way.",
"In [17] a fast context-aware recommendation algorithm is proposed that maintains the features of state-of-the-art multi-tensor matrix factorization while bringing down the complexity of the previously known algorithms from exponential to linear growth in problem size.",
"The main idea is to solve the least-squares optimization problem for each model parameter separately.",
"Our method in contrast achieves scalability by not utilizing any complex matrix factorization."
],
[
"Conclusions and Future Work",
"In this paper we introduced a relational graph for capturing the geographic preferences of users with the purpose of generating personalized recommendations in services with geotagged content.",
"We also presented several weighting schemes for representing different types of user preferences in the proposed graph.",
"Furthermore, we propose a recommendation algorithm template that is sufficiently generic to derive many traditional and new location-aware recommendation algorithms, including combinations of them, by only requiring changes in the definition of the edge weights.",
"Assuming the graph is given, the algorithm requires modest computational effort since it runs linearly in the number of neighbors.",
"We have tested the proposed algorithms with two real-world datasets, geotagged photos from Panoramio and print jobs from the HP ePrint Mobile Print Location service, and showed how our model easily suits many different recommendation scenarios.",
"We also gained insights about which notion of similarity works best for a set of scenarios.",
"In cold-start scenarios a geographic partonomy seems to be a good alternative, whereas when there is enough data available the plain location-aware collaborative filtering algorithm yields the best result.",
"In response to this finding, we proposed to combine a partonomy-based similarity measure with the cosine similarity by weighting individual relations in the graph according to the type of the user, i.e., cold-start versus non cold-start.",
"By doing this, we achieved better recall and precision in particular in scenarios where there are many cold-start users.",
"As future work, we plan to incorporate temporal aspects in the model, such that the items to be recommended match the temporal context of the user.",
"For example, it may not make as much sense to recommend ski resorts in New York during the summer as it would to make the same recommendation during the peak winter season.",
"Another natural extension of our work would be to assign the weights in our graph based on the strength of the social ties between the users, e.g.",
"based on their declared or implied social networks.",
"Finally, we plan to investigate machine learning approaches for learning optimal weights based on the location-aware recommendation task at hand."
],
[
"Acknowledgements",
"This work was supported by a cooperation with Hewlett-Packard Brasil Ltda.",
"using incentives of Brazilian Informatics Law (Law No.",
"8.2.48 of 1991).",
"We would also like to thank Christina Aperjis, Sitaram Asur and Mao Ye for insightful comments on our work."
]
] | 1204.1528 |
[
[
"Normal Hyperbolicity and Unbounded Critical Manifolds"
],
[
"Abstract This work is motivated by mathematical questions arising in differential equation models for autocatalytic reactions.",
"In particular, this paper answers an open question posed by Guckenheimer and Scheper [SIAM J. Appl.",
"Dyn.",
"Syst.",
"10-1 (2011), pp.",
"92-128] and provides a more general theoretical approach to parts of the work by Gucwa and Szmolyan [Discr.",
"Cont.",
"Dyn.",
"Sys.-S. 2-4 (2009), pp.",
"783-806].",
"We extend the local theory of singularities in fast-slow polynomial vector fields to classes of unbounded manifolds which lose normal hyperbolicity due to an alignment of the tangent and normal bundles.",
"A projective transformation is used to localize the unbounded problem.",
"Then the blow-up method is employed to characterize the loss of normal hyperbolicity for the transformed slow manifolds.",
"Our analysis yields a rigorous scaling law for all unbounded manifolds which exhibit a power-law decay for the alignment with a fast subsystem domain.",
"Furthermore, the proof also provides a technical extension of the blow-up method itself by augmenting the analysis with an optimality criterion for the blow-up exponents."
],
[
"Introduction",
"The motivation of this work are several models of singularly perturbed differential equations arising in applications.",
"In [20] Merkin et al.",
"propose to study a prototypical autocatalytic system given by four reactions $P\\rightarrow Y,\\quad Y\\rightarrow X,\\quad Y+2X\\rightarrow 3X, \\quad X\\rightarrow Z$ where $X,Y$ are the two main reactants, $P$ is a constant 'pool'-chemical and $Z$ is the product.",
"Then it can be shown, using standard mass-action kinetics and non-dimensionalization [23], that (REF ) leads to a two-dimensional system of ordinary differential equations (ODEs) given by $\\begin{array}{rcl}\\epsilon \\dot{x}&=& yx^2+y-x,\\\\\\dot{y}&=& \\mu -yx^2-y,\\\\\\end{array}$ where $\\dot{~}$ denotes derivative with respect to time, $x,y$ are dimensionless concentrations associated to $X,Y$ respectively and $\\epsilon ,\\mu $ are parameters; we note that the parameter $\\epsilon $ is defined by the ratio of reaction rates for $Y\\rightarrow X$ and $X\\rightarrow C$ [23].",
"Obviously it is natural to assume that the concentrations are non-negative $x,y\\in \\mathbb {R}^+_0:=\\lbrace w\\in \\mathbb {R}:w\\ge 0\\rbrace $ .",
"Furthermore, note carefully that the nonlinear term arises due to the autocatalytic reaction part $Y+2X\\rightarrow 3X$ .",
"It has been proven in [9] that the 2D-autocatalator (REF ) can exhibit an attracting relaxation-oscillation periodic orbit for certain ranges of the parameters; see also Figure REF (a).",
"In [23] Petrov et al.",
"generalized (REF ) by including a further reactant $P\\rightarrow Y,\\quad P+Z\\rightarrow Y+Z,\\quad Y\\rightarrow X,\\quad X\\rightarrow Z,\\quad Y+2X\\rightarrow 3X,\\quad Z\\rightarrow W.$ As before, it is straightforward [23] to derive from (REF ) the ODEs $\\begin{array}{rcl}\\epsilon \\dot{x}&=& yx^2+y-x,\\\\\\dot{y}&=& \\mu (\\kappa +z)-yx^2-y,\\\\\\dot{z}&=& x-z,\\\\\\end{array}$ where $(\\mu ,\\kappa ,\\epsilon )$ are parameters and $(x,y,z)\\in (\\mathbb {R}^+_0)^3$ are the concentrations.",
"Numerical studies [8], [21], [22] have shown that periodic, mixed-mode and chaotic oscillations exist for the 3D autocatalator (REF ).",
"Figure: (a) Numerical illustration for the 2D autocatalator with μ=1.1\\mu =1.1 and ϵ=0.01\\epsilon =0.01.",
"The thin line segments show the vector field.",
"The attracting periodic relaxation-type periodic orbit (thick black curve) is also shown.",
"(b) The critical manifold 𝒞 0 {\\mathcal {C}}_0 (gray) starts to align with the fast subsystem domains {y=const.",
"}\\lbrace y=\\text{const.",
"}\\rbrace (dashed) as x→∞x\\rightarrow \\infty .For both models (REF ), (REF ) it is often assumed that the ratio of time scales $\\epsilon >0$ is a sufficiently small parameter; we shall also denote this assumption by $0<\\epsilon \\ll 1$ .",
"In this case it follows that both autocatalator models are fast-slow (or singularly perturbed) ODEs.",
"We use the notation $p$ to denote a point in the phase space of concentrations for the autocatalator models i.e.",
"$p=(x,y)\\in (\\mathbb {R}_0^+)^2$ or $p=(x,y,z)\\in (\\mathbb {R}_0^+)^3$ .",
"The $x$ -nullcline, or critical manifold, for both models is given by ${\\mathcal {C}}_0=\\lbrace p:yx^2+y-x=0\\rbrace =\\left\\lbrace p:y=\\frac{x}{1+x^2}\\right\\rbrace .$ Note that ${\\mathcal {C}}_0$ is an unbounded smooth manifold.",
"The angle between the tangent spaces $T_{p}{\\mathcal {C}}_0$ and the hyperplanes $\\lbrace p:y=\\text{const.",
"}\\rbrace $ decays to zero as $x\\rightarrow +\\infty $ ; see Figure REF (b).",
"This alignment, which is expected to imply a loss of normal hyperbolicity in the system, already causes substantial difficulties in the rigorous analysis of the dynamics of the (2D) autocatalator model [9].",
"The global return mechanism induced by the unbounded part of ${\\mathcal {C}}_0$ also plays a key role for the complex oscillatory patterns of the 3D autocatalator [8] which have been observed in numerical simulations.",
"In a completely different context a model with similar properties to the autocatalator was proposed by Rankin et al.",
"[24] as a caricature system for effects in aircraft ground dynamics $\\left\\lbrace \\begin{array}{rcl}\\epsilon \\dot{x}&=& y+(x-\\mu )\\exp \\left(\\frac{x}{\\kappa }\\right),\\\\\\dot{y}&=& \\nu -x,\\\\\\end{array}\\right.$ where $(\\mu ,\\kappa ,\\nu ,\\epsilon )$ are parameters, $0<\\epsilon \\ll 1$ and ${\\mathcal {C}}_0=\\lbrace y=(\\mu -x)\\exp \\left(x/\\kappa \\right)\\rbrace $ also aligns with $\\lbrace y=0\\rbrace $ as $x\\rightarrow -\\infty $ if $\\kappa >0$ or as $x\\rightarrow +\\infty $ if $\\kappa <0$ .",
"Motivated by these examples, it is desirable to build a general theory of fast-slow systems with unbounded critical manifolds.",
"The main contributions of this work are: We study a general class of critical manifolds which may have an arbitrary power-law decay for the alignment with a fast subsystem domain.",
"This includes the autocatalytic critical manifolds as special cases and answers open questions arising from various numerical studies.",
"Using the blow-up method we give a rigorous proof when normal hyperbolicity for a perturbation of the critical manifold fails.",
"The relevant scaling law turns out to be given by $(x,y)=\\left({\\mathcal {O}}(\\epsilon ^{-1/(s+1)}),{\\mathcal {O}}(\\epsilon ^{s/(s+1)})\\right),\\qquad \\text{as $\\epsilon \\rightarrow 0$}$ where $s$ is a power-law exponent computable from the critical manifold.",
"This scaling has immediate consequences for the asymptotics of oscillatory patterns.",
"On a technical level we contribute to a further development of the blow-up method by augmenting it with an 'optimality-criterion' of blow-up coefficients which have to be chosen in the analysis.",
"The paper is structured as follows: In Section we introduce the notation and the required background from fast-slow systems.",
"In Section a formal asymptotic argument is given to illustrate the important scaling properties and the system is localized via a projective transformation.",
"The local system is desingularized via blow-up in Section .",
"The charts and transition functions between the two relevant charts are calculated.",
"The dynamics in each chart are analyzed in Section and respectively.",
"The optimality criterion for the blow-up coefficients is proven in Section .",
"The main scaling result is summarized in Section and its implications are discussed."
],
[
"Background and Notation",
"In this paper we restrict our attention to the analysis of family of planar fast-slow systems given by $\\begin{array}{rcrcr}\\frac{dx}{dt}&=&x^{\\prime }&=& f(x,y),\\\\\\frac{dy}{dt}&=&y^{\\prime }&=& \\epsilon g(x,y),\\\\\\end{array}$ where $(x,y)\\in \\mathbb {R}^2$ , $0<\\epsilon \\ll 1$ and $f:\\mathbb {R}^2\\rightarrow \\mathbb {R}$ , $g:\\mathbb {R}^2\\rightarrow \\mathbb {R}$ are assumed to be sufficiently smooth.",
"Remark: By restricting to planar systems the (3D) autocatalator does not immediately fit within the theory we develop.",
"However, we expect that center manifold techniques, similar to the generalization for folded singularities from three to arbitrary finite-dimensional fast-slow systems [29], can be applied.",
"We are going to choose particular forms of $f,g$ below but introduce some general terminology beforehand; for more detailed reviews/introductions to fast-slow systems see [12], [13], [2].",
"The critical set of (REF ) is given by ${\\mathcal {C}}_0:=\\left\\lbrace (x,y)\\in \\mathbb {R}^2:f(x,y)=0\\right\\rbrace .$ We are going to assume that ${\\mathcal {C}}_0$ defines a smooth manifold.",
"${\\mathcal {C}}_0$ is called normally hyperbolic at a point $(x^*,y^*)\\in {\\mathcal {C}}_0$ if $\\partial _xf(x^*,y^*)\\ne 0$ .",
"We will assume that ${\\mathcal {C}}_0$ is normally hyperbolic at every point in $\\mathbb {R}^2$ but ${\\mathcal {C}}_0$ will be unbounded.",
"Let us point out that there has been a lot of work on the geometric theory of planar fast-slow systems recently by De Maesschalck and Dumortier, see e.g.",
"[17], [18], [19], particularly in the context of local singularities, periodic orbits and Liénard systems.",
"Remark: The general definition of normal hyperbolicity of an invariant manifold [10], [5], [30] requires the splitting of tangent and normal dynamics.",
"Recall that a compact manifold ${\\mathcal {M}}\\subset \\mathbb {R}^N$ is normally hyperbolic for a vector field $F:\\mathbb {R}^N\\rightarrow \\mathbb {R}^N$ if there exists a continuous splitting of the tangent bundle $T_{{\\mathcal {M}}}\\mathbb {R}^n=N^u\\oplus T{\\mathcal {M}}\\oplus N^s$ where the linearization $DF$ expands $N^u$ and contracts $N^s$ more sharply than $T{\\mathcal {M}}$ .",
"In $\\mathbb {R}^2$ , we obviously just need one of the normal bundles if $\\dim ({\\mathcal {M}})=1$ .",
"It is important to point out that we do restrict ourselves to perturbations of normally hyperbolic manifolds in the class of fast-slow vector fields of the form (REF ) being interested in loss of normal hyperbolicity of invariant manifolds within this class of perturbed vector fields where the perturbation is given by $\\epsilon g(x,y)$ .",
"The implicit function theorem yields that ${\\mathcal {C}}_0$ is locally a graph.",
"In our case, the parametrization will be global so that ${\\mathcal {C}}_0=\\lbrace (x,y)\\in \\mathbb {R}^2:x=h(y)\\rbrace .$ for some smooth function $h:\\mathbb {R}\\rightarrow \\mathbb {R}$ .",
"The singular limit $\\epsilon \\rightarrow 0$ of (REF ) defines a differential algebraic equation, also called the fast subsystem $\\begin{array}{rcl}x^{\\prime }&=& f(x,y),\\\\y^{\\prime }&=& 0.\\\\\\end{array}$ Changing to the slow time scale $\\tau =\\epsilon t$ in (REF ) and taking the singular limit $\\epsilon =0$ yields the slow subsystem defined on ${\\mathcal {C}}_0$ $\\begin{array}{rcrcl}0&=&0&=& f(x,y),\\\\\\frac{dy}{d\\tau }&=&\\dot{y}&=& g(x,y).\\\\\\end{array}$ The slow subsystem can be written as $\\dot{y}=g(h(y),y)$ .",
"If $h$ is invertible, the critical manifold can also be defined via ${\\mathcal {C}}_0=\\lbrace (x,y)\\in \\mathbb {R}^2:y=h^{-1}(x)=:c(x)\\rbrace $ .",
"Implicit differentiation with respect to $\\tau $ yields $\\dot{y}=c^{\\prime }(x)\\dot{x}$ so that the slow subsystem is $c^{\\prime }(x)\\dot{x}= g(x,c(x)).$ Fenichel's Theorem [6], [12] implies that that any compact submanifold ${\\mathcal {M}}_0\\subset {\\mathcal {C}}_0$ persists as a slow manifold ${\\mathcal {M}}_\\epsilon $ for $0<\\epsilon \\ll 1$ .",
"Furthermore, ${\\mathcal {M}}_\\epsilon $ is locally invariant and diffeomorphic to ${\\mathcal {M}}_0$ .",
"The flow on ${\\mathcal {M}}_\\epsilon $ converges to the slow flow as $\\epsilon \\rightarrow 0$ .",
"For any fixed small $\\epsilon >0$ we can define ${\\mathcal {M}}_\\epsilon $ and then extend it under the flow of (REF ) but this extension may no longer normally hyperbolic as an invariant manifold for the full system (REF ).",
"To understand this effect we propose to study the family of model systems given by $f(x,y,\\epsilon )=1-x^sy\\qquad \\text{for $s\\in \\mathbb {N}$}$ and $g(x,y)=\\mu $ for some $\\mu \\ne 0$ .",
"The choice for the fast vector field is motivated by the asymptotic expansion of ${\\mathcal {C}}_0$ for the autocatalator models (REF ),(REF ) as $x\\rightarrow \\infty $ .",
"In fact, returning to the chemical interpretation the general model class corresponds to autocatalytic reaction steps of the form $Y+(s+1)X\\rightarrow (s+2)X$ .",
"The smooth unbounded critical manifold for (REF ) is given by ${\\mathcal {C}}_0:=\\left\\lbrace (x,y)\\in \\mathbb {R}^2:y=\\frac{1}{x^s}=c(x)\\right\\rbrace .$ As $|x|\\rightarrow \\infty $ the tangent space $T_{(x,y)}{\\mathcal {C}}_0$ starts to align with the $x$ -axis since $c(x)\\rightarrow 0$ and $c^{\\prime }(x)\\rightarrow 0$ as $|x|\\rightarrow \\infty $ .",
"By reflection symmetry we restrict our focus to the positive part ${\\mathcal {C}}_0\\cap (\\mathbb {R}^+_0)^2$ from now on i.e.",
"the intersection with the non-negative quadrant will be understood.",
"Note that the model class (REF ) covers already a wide variety of power-law decay rates.",
"Since we are only interested in the unbounded part of ${\\mathcal {C}}_0$ we shall restrict the dynamics to ${\\mathcal {H}}^\\sigma :=\\lbrace (x,y)\\in \\mathbb {R}^2:x>\\sigma \\rbrace $ for a suitable fixed $\\sigma >0$ with $\\sigma ={\\mathcal {O}}(1)$ as $\\epsilon \\rightarrow 0$ ; see also Figure REF (a).",
"Since $\\partial _xf(x^*,y^*)=-s(x^*)^{s-1}y^*\\ne 0$ for $(x^*,y^*)\\in {\\mathcal {C}}_0$ it follows that any compact submanifold ${\\mathcal {M}}_0\\subset {\\mathcal {C}}_0$ is normally hyperbolic and perturbs, by Fenichel's Theorem, to a normally hyperbolic slow manifold for $\\epsilon >0$ sufficiently small.",
"However, for fixed small $\\epsilon >0$ we do not know yet when the extension of ${\\mathcal {M}}_\\epsilon $ under the flow starts to deviate from the critical manifold approximation."
],
[
"Asymptotic Expansion and Projective Transformation",
"A good intuitive understanding of the problem is gained by considering one of the simplest possible systems in the model class (REF ) given by $s=1$ and $\\mu =1$ so that $\\begin{array}{rcl}\\epsilon \\dot{x}&=& 1-yx,\\\\\\dot{y}&=& 1.\\\\\\end{array}$ From the viewpoint of formal asymptotics we can simply calculate a slow manifold ${\\mathcal {M}}_\\epsilon $ by making the ansatz $x(y)=x_0(y)+\\epsilon x_1(y)+\\epsilon ^2 x_2(y)+\\cdots .$ Inserting (REF ) into (REF ) and collecting terms of different orders in $\\epsilon $ yields $\\begin{array}{rrcl}{\\mathcal {O}}(1):\\qquad & 0&=&1-yx_0(y),\\\\{\\mathcal {O}}(\\epsilon ^k):\\qquad & \\frac{dx_{k-1}}{dy}&=&-yx_k(y),\\\\\\end{array}$ for $k\\in \\mathbb {N}$ .",
"A direct calculation using (REF ) gives the formal asymptotics of the slow manifold $x(y)=\\frac{1}{y}-\\sum _{k=1}^K(2k-1)\\frac{\\epsilon ^k}{y^{2k+1}}+{\\mathcal {O}}(\\epsilon ^{K+1})\\qquad \\text{as $\\epsilon \\rightarrow 0$}.$ The series (REF ) fails to be asymptotic when $1/y\\sim \\epsilon ^k/y^{2k+1}$ as $\\epsilon \\rightarrow 0$ or $y={\\mathcal {O}}(\\sqrt{\\epsilon })$ .",
"Since the critical manifold of (REF ) is given as the graph of $y=1/x$ we expect that normal hyperbolicity of the associated slow manifold breaks down for $x={\\mathcal {O}}(\\epsilon ^{-1/2})$ .",
"The example (REF ) suggests that, depending on the asymptotics of the critical manifold as $x\\rightarrow \\infty $ , there is an $\\epsilon $ -dependent scaling law that governs within which regime we can use the critical manifold approximation provided by Fenichel's Theorem.",
"To give a rigorous proof of the asymptotic scaling we would like to desingularize the problem.",
"In [9] the autocatalator is considered and phase space is re-scaled by an $\\epsilon $ -dependent transformation; see also further remarks in Section .",
"Here we use a different approach by first completely localizing the problem; see also [28].",
"Consider a projective coordinate transformation $\\rho :{\\mathcal {H}}^1\\rightarrow (\\mathbb {R}^2-{\\mathcal {H}}^1)\\qquad \\rho (x,y)=\\left(\\frac{1}{v},y\\right)$ where $\\sigma =1$ is chosen for computational convenience for the domain ${\\mathcal {H}}^\\sigma $ .",
"All the following results are easily checked to be independent of a fixed chosen $\\sigma >0$ independent of $\\epsilon $ ; see also Figure REF (b).",
"A direct calculation yields the following result: Lemma 3.1 The map (REF ) yields a transformed vector field defined on $(\\mathbb {R}^2-{\\mathcal {H}}^1)$ and given by $\\begin{array}{rcl}\\epsilon \\dot{v}&=& \\left(yv^{2-s}-v^{2}\\right),\\\\\\dot{y}&=& \\mu .\\end{array}$ Figure: (a) Critical manifolds () for s=1s=1 (solid gray) and s=2s=2 (dashed black).",
"(b) Critical manifolds after the transformation ().",
"Recall that we always restrict the dynamics to the non-negative quadrant (ℝ 0 + ) 2 (\\mathbb {R}^+_0)^2.The vector field (REF ) can be desingularized, similar to the desingularization for folded singularities [27], via multiplication by $v^s$ and a time re-scaling which leads to $\\begin{array}{rcl}\\epsilon \\dot{v}&=& v^2\\left(y-v^{s}\\right),\\\\\\dot{y}&=& \\mu v^{s}.\\end{array}$ Note that the transformation reverses time on orbits for $v<0$ if $s$ is odd but that we always restrict to the non-negative quadrant which implies that we do not have to consider this issue here.",
"After the transformation the original critical manifold ${\\mathcal {C}}_0=\\lbrace y=1/x^s\\rbrace $ is given by $\\rho ({\\mathcal {C}}_0)=\\left\\lbrace (v,y)\\in \\mathbb {R}^2-{\\mathcal {H}}^1:y=v^{s}\\right\\rbrace =:{\\mathcal {S}}_0.$ Observe that ${\\mathcal {S}}_0$ is normally hyperbolic except at the degenerate singularity $(v,y)=(0,0)$ .",
"The key point of the projective transformation $\\rho $ is that it localized the approach towards a singular point to calculate the breakdown of normal hyperbolicity; in fact, it is not the singularity arising from infinity we are interested in but the scaling of the associated slow manifold ${\\mathcal {S}}_\\epsilon $ in a sufficiently small ball excluding $(v,y)=(0,0)$ .",
"Note that the case $s=1$ yields the classical transcritical structure for ${\\mathcal {S}}_0$ with $f(v,y)=v(y-v)$ ; see [15], [26] and references therein.",
"To desingularize (REF ) one can try to apply a blow-up transformation.",
"The blow-up method [3] was introduced into fast-slow systems theory in the work of Dumortier and Roussarie [4].",
"For additional background on the blow-up method for fast-slow local singularities see [16]."
],
[
"Desingularization via Blow-Up",
"Re-writing (REF ) on the fast time scale and augmenting the system by $\\epsilon ^{\\prime }=0$ leads to the vector field $\\begin{array}{rcl}v^{\\prime }&=& v^2\\left(y-v^{s}\\right),\\\\y^{\\prime }&=& \\epsilon \\mu v^{s},\\\\\\epsilon ^{\\prime }&=&0,\\end{array}$ which we denote by $X$ .",
"Consider the manifold $\\bar{{\\mathcal {B}}}:={\\mathcal {S}}^2\\times [0,r_0]$ for some $r_0>0$ where ${\\mathcal {S}}^2\\subset \\mathbb {R}^3$ denotes the unit sphere.",
"Denote the coordinates on $\\bar{{\\mathcal {B}}}$ by $(\\bar{v},\\bar{y},\\bar{\\epsilon },\\bar{r})$ where $(\\bar{v},\\bar{y},\\bar{\\epsilon })\\in {\\mathcal {S}}^2$ .",
"A general weighted blow-up transformation $\\Phi :\\bar{{\\mathcal {B}}}\\rightarrow \\mathbb {R}^3$ is then given by $v=\\bar{r}^{\\alpha _v}\\bar{v},\\qquad y=\\bar{r}^{\\alpha _y}\\bar{y},\\qquad \\epsilon =\\bar{r}^{\\alpha _\\epsilon }\\bar{\\epsilon },$ where we choose the coefficients as $(\\alpha _v,\\alpha _y,\\alpha _\\epsilon )=(1,s,s+1).$ The choice of $(\\alpha _v,\\alpha _y,\\alpha _\\epsilon )$ can be motivated by using the Newton polygon [3], [1].",
"Via $\\Phi $ a vector field $\\bar{X}$ is induced on the space $\\bar{{\\mathcal {B}}}$ by requiring $\\Phi _*(\\bar{X})=X$ where $\\Phi _*$ is the usual push-forward.",
"Recall that we restrict to the manifold ${\\mathcal {C}}_0$ in the positive quadrant.",
"Let ${\\mathcal {B}}_{\\bar{y}}:=\\bar{{\\mathcal {B}}}\\cap \\lbrace \\bar{y}>0\\rbrace $ and ${\\mathcal {B}}_{\\bar{\\epsilon }}:=\\bar{{\\mathcal {B}}}\\cap \\lbrace \\bar{\\epsilon }>0\\rbrace $ and consider two charts $\\kappa _1:{\\mathcal {B}}_{\\bar{y}}\\rightarrow \\mathbb {R}^3 \\qquad \\text{and}\\qquad \\kappa _2:{\\mathcal {B}}_{\\bar{\\epsilon }}\\rightarrow \\mathbb {R}^3$ where $(v_1,r_1,\\epsilon _1)$ and $(v_2,y_2,r_2)$ denote the respective coordinates for ${\\mathcal {B}}_{\\bar{y}}$ and ${\\mathcal {B}}_{\\bar{\\epsilon }}$ ; see also [16] for the basic geometry of a blow-up in $\\mathbb {R}^3$ .",
"Slightly tedious, but straightforward, calculations yield the following result: Lemma 4.1 The maps $\\kappa _1$ and $\\kappa _2$ are given by $\\begin{array}{lll}v_1=\\bar{v}\\bar{y}^{-1/s},\\quad & r_1=\\bar{r}\\bar{y}^{1/s}, \\quad & \\epsilon _1=\\bar{\\epsilon }\\bar{y}^{-(s+1)/s},\\\\v_2=\\bar{v}\\bar{\\epsilon }^{-1/(s+1)},\\quad & y_2=\\bar{y}\\bar{\\epsilon }^{-s/(s+1)},\\quad & r_2=\\bar{r}\\bar{\\epsilon }^{1/(s+1)}.\\\\\\end{array}$ The transitions functions $\\kappa _{12}$ and $\\kappa _{21}$ between charts are $\\begin{array}{lll}v_2=v_1\\epsilon _1^{-1/(s+1)}, \\quad & y_2=\\epsilon _1^{-s/(s+1)},\\quad & r_2=r_1\\epsilon _1^{1/(s+1)},\\\\v_1=v_2y_2^{-1/s},\\qquad & r_1=r_2y_2^{1/s},\\quad & \\epsilon _1=y_2^{-(s+1)/s}.\\end{array}$ Lemma 4.2 The desingularized vector field in $\\kappa _1$ is given by $\\begin{array}{lcl}v_1^{\\prime }&=&sv_1^2(1-v_1^s)-\\mu \\epsilon _1 v_1^{s+1},\\\\r_1^{\\prime }&=&\\mu r_1\\epsilon _1v_1^s,\\\\\\epsilon _1^{\\prime }&=&-(s+1)\\mu \\epsilon _1^2v_1^s,\\\\\\end{array}$ The desingularized vector field in $\\kappa _2$ is given by $\\begin{array}{lcl}v_2^{\\prime }&=&v_2^2(y_2-v_1^s),\\\\y_2^{\\prime }&=&\\mu v_2^s.\\\\\\end{array}$ (of Lemma REF ) The required calculations follow by direct differentiation, for example, one has $r_1^{\\prime }=(\\mu /s) r_1^{s+2}\\epsilon _1v_1^s\\qquad \\text{and} \\qquad v^{\\prime }=r_1^{\\prime }v_1+r_1v_1^{\\prime }$ which imply upon algebraic manipulation that $v_1^{\\prime }=r_1^{s+1}\\left(v_1^2(1-v_1^s)-\\frac{\\mu }{s}\\epsilon _1 v_1^{s+1}\\right).$ Similarly, one obtains the equation for $\\epsilon _1$ given by $\\epsilon _1^{\\prime }=-\\frac{(s+1)}{s}\\mu r^{s+1}\\epsilon _1^2v_1^s.$ A division by the common factor $r_1^{s+1}$ and a time re-scaling yield (REF ).",
"In $\\kappa _2$ the transformation is just a re-scaling $(v,y)=\\left(\\epsilon ^{1/(s+1)}v_2,\\epsilon ^{s/(s+1)}y_2\\right)$ and a time rescaling $t\\rightarrow t/\\epsilon $ gives (REF )."
],
[
"Dynamics in the First Chart",
"We can continue the critical manifold ${\\mathcal {S}}_0=\\lbrace (v,y)\\in (\\mathbb {R}^2)^+:y=v^s\\rbrace $ into the chart $\\kappa _1$ .",
"Using the definition of the blow-up map and Lemma (REF ) imply the relevant continuation result: Lemma 5.1 $(\\kappa _1\\circ \\Phi ^{-1})({\\mathcal {S}}_0)=\\lbrace (v_1,y_1)\\in (\\mathbb {R}^2)^+:v_1=1\\rbrace =:{\\mathcal {S}}_{1,a}$ .",
"Analyzing the flow (REF ) gives more information about ${\\mathcal {S}}_{1,a}$ .",
"The vector field (REF ) has two invariant subspaces $\\epsilon _1=0$ and $r_1=0$ .",
"In the first subspace the flow is $\\begin{array}{lcl}v_1^{\\prime }&=&sv_1^2(1-v_1^s),\\\\r_1^{\\prime }&=&0,\\\\\\end{array}$ which has a line of equilibrium points ${\\mathcal {S}}_{1,a}$ which are attracting in the $v_1$ -direction.",
"There is also a line of equilibria at $v_1=0$ which we do not have to consider.",
"In the second invariant subspace the flow is $\\begin{array}{lcl}v_1^{\\prime }&=&sv_1^2(1-v_1^s)-\\mu \\epsilon _1 v_1^{s+1},\\\\\\epsilon _1^{\\prime }&=&-(s+1)\\mu \\epsilon _1^2v_1^s.\\\\\\end{array}$ Linearizing around $(v_1,\\epsilon _1)=(1,0)$ provides the important local dynamics: Lemma 5.2 The vector field (REF ) has a center-stable equilibrium at $(v_1,\\epsilon _1)=(1,0)=:p_{1,a}$ with eigenvalues 0 and $-s^2$ and associated eigenvectors $(-\\mu /s^2,1)^T$ and $(1,0)^T$ .",
"Lemma 5.3 Locally near $p_{a,1}$ the equilibrium $p_{1,a}$ has a one-dimensional center manifold ${\\mathcal {N}}_{1,a}=\\lbrace v_1=1-\\epsilon _1\\frac{\\mu }{s}+c_{11}\\epsilon _1^2+{\\mathcal {O}}(\\epsilon _1^3):=n_{1,a}(\\epsilon _1)+{\\mathcal {O}}(\\epsilon _1^3))\\rbrace $ where $c_{11}=-(1+s+s^2)\\mu ^2/(2s^3)$ .",
"(of Lemma REF ) Existence is guaranteed by the center manifold theorem [7].",
"Using a translation $V_1=v_1-1$ moves $p_{1,a}$ to the origin.",
"A further linear coordinate change $\\left(\\begin{array}{c}V_1 \\\\ \\epsilon _1\\\\ \\end{array}\\right)=Mz=\\left(\\begin{array}{cc}-\\mu /s^2 & 1 \\\\ 1 & 0\\\\ \\end{array}\\right)\\left(\\begin{array}{c}z_1 \\\\ z_2\\\\ \\end{array}\\right)$ transforms (REF ) into $\\begin{array}{lcrll}z_1^{\\prime }&=& 0 &-\\mu (1+s)z_1^2&+{\\mathcal {O}}(3),\\\\z_2^{\\prime }&=& -s^2 &-\\mu ^2\\left(1+\\frac{1}{2s}+\\frac{1}{2s^2}\\right)z_1^2+2\\mu z_1z_2-\\left(\\frac{3s^2}{2}+\\frac{s^3}{2}\\right)z_2^2&+{\\mathcal {O}}(3),\\\\\\end{array}$ where ${\\mathcal {O}}(3)={\\mathcal {O}}(z_1^3,z_1^2z_2,z_1z_2^2,z_2^3)$ .",
"Making the ansatz $z_2=c_{11}z_1^2+{\\mathcal {O}}(z_1^3)$ and substituting into the invariance equation [7] for the center manifold at $z=(0.0)$ implies the condition $0=\\left(c_{11}s+\\mu ^2+\\frac{\\mu ^2}{2s}+\\frac{\\mu ^2}{2s^2}\\right)z_1^2+{\\mathcal {O}}(z_1^3).$ Therefore, $c_{11}=-(1+s+s^2)\\mu ^2/(2s^3)$ and transforming back to $(v_1,\\epsilon _1)$ yields the result.",
"As before, other equilibria of (REF ) will not be of relevance here.",
"Lemma REF and the center manifold theorem [7] imply the next result: Lemma 5.4 There exists a center-stable manifold ${\\mathcal {M}}_{1,a}$ for (REF ) at $p_{1,a}$ containing ${\\mathcal {S}}_{1,a}$ and ${\\mathcal {N}}_{1,a}$ .",
"Furthermore, ${\\mathcal {M}}_{1,a}$ is locally given as a graph of a map $y_1=h_{1,a}(r_1,\\epsilon _1)$ .",
"From the flow on the center manifold we see that there are two cases based upon the sign of $\\mu $ .",
"If $\\mu >0$ then trajectories in the center manifold flow away from the sphere ${\\mathcal {S}}^2$ while for $\\mu <0$ , trajectories in ${\\mathcal {M}}_{1,a}$ flow from the chart $\\kappa _1$ onto the sphere.",
"We are only going to deal with the case $\\mu <0$ from now on.",
"The case $\\mu >0$ can be obtained from $\\mu <0$ by a time reversal of the the original problem (REF ).",
"Figure REF (a) provides a sketch of the results and the notation for the relevant dynamical objects.",
"Figure: (a) Sketch of the dynamics in chart κ 1 \\kappa _1.",
"(b) Sketch of the dynamics in chart κ 2 \\kappa _2"
],
[
"Dynamics in the Second Chart",
"Now the critical manifold in the chart $\\kappa _1$ has to be continued into the chart $\\kappa _2$ .",
"This transition is studied using the continuation ${\\mathcal {M}}_{2,a}$ of ${\\mathcal {M}}_{1,a}$ under the flow in $\\kappa _2$ ; see also Figure REF (b).",
"Lemma 6.1 The curve ${\\mathcal {N}}_{1,a}\\subset {\\mathcal {M}}_{1,a}$ transforms as $\\kappa _{12}({\\mathcal {N}}_{1,a})=\\left\\lbrace (v_2,y_2)\\in \\mathbb {R}^2:v_2=1-\\frac{\\mu }{s y_2}+\\frac{c_{11}}{y_2^{(2s+1)/s}}+{\\mathcal {O}}\\left(y_2^{\\frac{-3s-2}{s}}\\right)\\text{ as $y_2\\rightarrow +\\infty $}\\right\\rbrace .$ We introduce the notation ${\\mathcal {N}}_{2,a}:=\\kappa _{12}({\\mathcal {N}}_{1,a})$ and note that ${\\mathcal {N}}_{2,a}$ can be extended under the flow (REF ) in chart $\\kappa _2$ for initial conditions with $v_2\\approx 1$ and arbitrarily large $y_2>0$ .",
"To emphasize that this extension is a trajectory we are going to denote it by $n_{2,a}(t)=(n_{v_2,a}(t),n_{y_2,a}(t))^T\\in \\mathbb {R}^2$ .",
"A case distinction based upon the value of $s$ will be necessary.",
"For $s=2k$ with $k\\in \\mathbb {N}$ the differential equation is $\\begin{array}{lcl}v_2^{\\prime }&=&v_2^2(y_2-v_2^{2k}),\\\\y_2^{\\prime }&=&\\mu v^{2k}_2,\\\\\\end{array}$ so that $y_2^{\\prime }<0$ and $v_2^{\\prime }<0$ for $y_2<v_2^{2k}$ imply that $n_{v_2,a}(t)\\rightarrow 0$ for $t\\rightarrow +\\infty $ .",
"The variational equation is defined via the time-dependent matrix $A(t)&:=&\\left.D_{(v_2,y_2)}\\left(\\begin{array}{c}v_2^2(y_2-v_2^{2k}) \\\\ \\mu v^{2k} \\\\\\end{array}\\right)\\right|_{n_{2,a}(t)}\\\\&=&\\left(\\begin{array}{cc}2n_{y_2,a}(t)n_{v_2,a}(t)-(2k+2)n_{v_2,a}(t)^{2k+1} & 1 \\\\ 2k \\mu ~n_{v_2,a}(t)^{2k-1} & 0 \\\\\\end{array}\\right),$ which shows that the variational equation becomes asymptotically autonomous in forward time [25] and $A(t)$ approaches a double zero eigenvalue when $t\\rightarrow +\\infty $ .",
"Hence ${\\mathcal {M}}_{2,a}$ is not everywhere normally hyperbolic in $\\kappa _2$ .",
"A possible alternative argument for $k>1$ , leading to the same conclusion, involves desingularizing (REF ) using division by $v_2^2$ and observing that the resulting vector field is parallel on the $y_2$ -axis with no flow in the $y_2$ -component.",
"However, with this argument the case $k=0$ is special but can be treated using standard results about the resulting Ricatti equation which also appears in the blow-up analysis of the fold point [14].",
"For $s=2k+1$ and $k\\in \\mathbb {N}_0$ the vector field is $\\begin{array}{lcl}v_2^{\\prime }&=&v_2^2(y_2-v_2^{2k+1}),\\\\y_2^{\\prime }&=&\\mu v^{2k+1}_2.\\\\\\end{array}$ Note that (REF ) does not have the same nice monotonicity properties as before.",
"To show the convergence towards $v_2=0$ define $V(v_2,y_2):=\\frac{1}{2k}v_2^{2k}+\\frac{1}{2|\\mu |}y_2^{2}\\quad \\Rightarrow \\quad \\frac{dV}{dt}=v_2^{2k-1}v_2^{\\prime }+\\frac{1}{|\\mu |}y_2y_2^{\\prime }=-v_2^{2(2k+1)}<0$ showing that $V$ is a Lyapunov function [11] which implies the required convergence.",
"Computing the variational equation yields $A(t)&:=&\\left.D_{(v_2,y_2)}\\left(\\begin{array}{c}v_2^2(y_2-v_2^{2k+1}) \\\\ \\mu v_2^{2k+1} \\\\\\end{array}\\right)\\right|_{n_{2,a}(t)}\\\\&=&\\left(\\begin{array}{cc}2n_{v_2,a}(t)n_{y_2,a}(t)-(2k+3)n_{v_2,a}(t)^{2k+2} & n_{v_2,a}(t)^2 \\\\ \\mu (2k+1)n_{v_2,a}(t)^{2k} & 0 \\\\\\end{array}\\right).$ which shows that the variational equation becomes asymptotically autonomous in forward time and $A(t)$ approaches a double zero eigenvalue when $t\\rightarrow +\\infty $ .",
"As before, one could also use a suitable division by a power of $v_2$ and observe that the resulting vector field is parallel on the $y_2$ -axis.",
"Lemma 6.2 ${\\mathcal {M}}_{2,a}$ is not everywhere a normally hyperbolic manifold inside $\\kappa _2$ .",
"Lemma REF is just a local restatement of the alignment property of the original slow manifold ${\\mathcal {C}}_\\epsilon $ with the fast subsystem domains.",
"This confirms the conjectured loss of normal hyperbolicity as $x\\rightarrow +\\infty $ .",
"The previous results combine to give the following statement which is a partial version of our main result.",
"Proposition 6.3 Consider the family of planar fast-slow systems $\\begin{array}{rcl}x^{\\prime }&=& 1-x^sy,\\\\y^{\\prime }&=& \\epsilon \\mu .\\\\\\end{array}$ for $s\\in \\mathbb {N}$ , $\\mu \\ne 0$ and $\\epsilon >0$ sufficiently small.",
"Let ${\\mathcal {M}}_0\\subset {\\mathcal {C}}_0$ be a compact submanifold of the critical manifold.",
"Then ${\\mathcal {M}}_\\epsilon $ extends to a normally hyperbolic manifold of (REF ) up to a domain of size $(x,y)=\\left({\\mathcal {O}}(\\epsilon ^{-1/(s+1)}),{\\mathcal {O}}(\\epsilon ^{s/(s+1)})\\right),\\qquad \\text{as $\\epsilon \\rightarrow 0$.",
"}$ Under the blow-up map (REF )-(REF ) the manifold ${\\mathcal {M}}_\\epsilon $ cannot be extended to a normally hyperbolic slow manifold to a subset of a larger domain.",
"We work in projective coordinates via $\\rho $ .",
"We can restrict to pieces of the critical manifold lying in the positive quadrant without loss of generality.",
"${\\mathcal {M}}_\\epsilon $ is obtained from ${\\mathcal {M}}_0$ by Fenichel's Theorem since $\\epsilon >0$ is sufficiently small.",
"Using the blow-up transformation and Lemma REF , it follows that ${\\mathcal {M}}_\\epsilon $ extends to a normally hyperbolic manifold in the chart $\\kappa _1$ .",
"By Lemma REF the extension is not normally hyperbolic in the chart $\\kappa _2$ .",
"By Lemma REF and Lemma REF it follows that the neighborhood of $(v,y)=(0,0)$ scales as $(v,y)=\\left(\\epsilon ^{1/(s+1)}v_2,\\epsilon ^{s/(s+1)}y_2\\right).$ By applying a blow-down transformation it follows that ${\\mathcal {M}}_\\epsilon $ is normally hyperbolic up to a scaling region given by (REF ) and not normally hyperbolic inside some larger domain.",
"Via $\\rho ^{-1}$ we get $(x,y)=\\left({\\mathcal {O}}(\\epsilon ^{-1/(s+1)}),{\\mathcal {O}}(\\epsilon ^{s/(s+1)})\\right)$ as $\\epsilon \\rightarrow 0$ in original coordinates so that the result follows."
],
[
"Smaller Regions",
"To prove the final result, we aim to strengthen Proposition REF as it claims that the slow manifold extends and then normal hyperbolicity breaks down for the scaling (REF ) using the blow-up (REF )-(REF ).",
"A priori, we could have chosen different exponents for the blow-up which could have allowed us to extend the slow manifold even further.",
"Here we show that the exponents in (REF ) are indeed optimal in the sense that for any pair $(\\alpha _1,\\alpha _2)$ , $\\alpha _i\\ge 0$ with $0<\\alpha _1+\\alpha _2$ the manifold ${\\mathcal {C}}_\\epsilon $ is not normally hyperbolic in the larger region $(x,y)=\\left({\\mathcal {O}}(\\epsilon ^{-(1+\\alpha _1)/(s+1)}),{\\mathcal {O}}(\\epsilon ^{(s+\\alpha _2)/(s+1)})\\right),\\qquad \\text{as $\\epsilon \\rightarrow 0$.",
"}$ Note that it suffices to assume that $0<\\alpha _1+\\alpha _2\\ll 1$ so that the region is slightly larger.",
"Consider the modified blow-up $v=\\bar{r}^{1+\\alpha _1}\\bar{v},\\qquad y=\\bar{r}^{s+\\alpha _2}\\bar{y},\\qquad \\epsilon =\\bar{r}^{s+1}\\bar{\\epsilon }.$ Observe that in the chart $\\kappa _2$ the blow-up (REF ) reduces to the re-scaling $v=\\epsilon ^{(1+\\alpha _1)/(s+1)}v_2,\\qquad y=\\epsilon ^{(s+\\alpha _2)/(s+1)}y_2$ which yields a strictly smaller scaling region than before.",
"Lemma 7.1 In the chart $\\kappa _1$ the vector field is $\\begin{array}{lcl}r_1^{\\prime }&=&\\frac{\\mu }{s+\\alpha _2}r_1^{2-\\alpha _2+s+s\\alpha _1}\\epsilon _1v_1^s,\\\\v_1^{\\prime }&=& \\frac{-\\mu (1+\\alpha _1)}{s+\\alpha _2}r_1^{1-\\alpha _2+s+s\\alpha _1}\\epsilon _1v_1^{s+1}+r_1^{1+\\alpha _1+s}v_1^2(r_1^{\\alpha _2}-r_1^{s\\alpha _1}v_1^s),\\\\\\epsilon _1&=& \\frac{-(s+1)\\mu }{s+\\alpha _2}r_1^{1-\\alpha _2+s+s\\alpha _1}\\epsilon _1^2v_1^s,\\\\\\end{array}$ Proposition 7.2 The extension of the manifold ${\\mathcal {S}}_\\epsilon $ in the chart $\\kappa _1$ for the blow-up (REF ) is not normally hyperbolic as $r_1\\rightarrow 0$ .",
"Suppose first that $s\\alpha _1-\\alpha _2\\le 0$ .",
"Since $0<\\alpha _1+\\alpha _2\\ll 1$ we can desingularize the vector field (REF ) by $r_1^{1-\\alpha _2+s+s\\alpha _1}$ as $1-\\alpha _2+s+s\\alpha _1>0$ .",
"Considering the invariant subspace defined by $\\epsilon _1=0$ of the desingularized vector field yields the equation $v_1^{\\prime }= r_1^{\\alpha _1+\\alpha _2-s\\alpha _1}v_1^2(r_1^{\\alpha _2}-r_1^{s\\alpha _1}v_1^s)$ which has a line of equilibria given by $v_1=r_1^{(\\alpha _2-s\\alpha _1)/s}$ corresponding to the critical manifold ${\\mathcal {S}}_0$ .",
"Linearizing (REF ) around this line gives the variational equation $V^{\\prime }=r_1^{\\alpha _1+\\alpha _2-s\\alpha _1}(2r_1^{(\\alpha _2-s\\alpha _1)/s}r_1^{\\alpha _2}-(s+1)r_1^{s\\alpha _1}r_1^{(\\alpha _2-s\\alpha _1)(s+1)/s})V$ which reduces to $V^{\\prime }=0\\cdot V$ as $r_1\\rightarrow 0$ .",
"Hence, in this case, the extension of ${\\mathcal {S}}_\\epsilon $ is not normally hyperbolic in all of $\\kappa _1$ .",
"In the case $s\\alpha _1-\\alpha _2>0$ we can desingularize by $r_1^{\\alpha ^*}$ where $\\alpha ^*=1+s+\\min (\\alpha _1+\\alpha _2,s\\alpha _1-\\alpha _2)$ in which case again a multiplicative factor containing $r_1$ appears in the variational equation.",
"The main point of the previous proof is that we cannot remove the $r_1$ -dependence in the the chart $\\kappa _1$ which means that there is no center manifold at the point $p_{1,a}$ extending up to the sphere ${\\mathcal {S}}^2$ , as previously for the blow-up with $\\alpha _1=0=\\alpha _2$ ."
],
[
"The Main Result and Conclusions",
"Finally, we can combine the previous result to obtain the main result about loss of normal hyperbolicity and scaling.",
"Recall again that we always restrict the dynamics to $(\\mathbb {R}_0^+)^2\\cap {\\mathcal {H}}^1$ which is a subset in the non-negative quadrant bounded away from $x=0$ .",
"Theorem 8.1 Consider the family of planar fast-slow systems $\\begin{array}{rcl}x^{\\prime }&=& 1-x^sy,\\\\y^{\\prime }&=& \\epsilon \\mu .\\\\\\end{array}$ for $s\\in \\mathbb {N}$ , $\\mu \\ne 0$ and $\\epsilon >0$ sufficiently small.",
"Let ${\\mathcal {M}}_0\\subset {\\mathcal {C}}_0$ be a compact submanifold of the critical manifold.",
"Then ${\\mathcal {M}}_\\epsilon $ extends to a normally hyperbolic manifold of (REF ) up to a domain of size $(x,y)=\\left({\\mathcal {O}}(\\epsilon ^{-1/(s+1)}),{\\mathcal {O}}(\\epsilon ^{s/(s+1)})\\right),\\qquad \\text{as $\\epsilon \\rightarrow 0$.",
"}$ under the flow of (REF ).",
"${\\mathcal {M}}_\\epsilon $ is not normally hyperbolic for any larger domain i.e.",
"there does not exist a normally hyperbolic invariant manifold $\\tilde{{\\mathcal {M}}}_\\epsilon $ in a domain strictly larger than (REF ) as $\\epsilon \\rightarrow 0$ such that ${\\mathcal {M}}_\\epsilon $ is strictly contained in $\\tilde{{\\mathcal {M}}}_\\epsilon $ .",
"Apply Proposition REF and Proposition REF .",
"Theorem REF immediately applies to the 2D autocatalator.",
"It is very interesting to point out the relation between the approach by Gucwa and Szmolyan [9] and the general Theorem REF .",
"The scaling used in [9] to capture the large relaxation-type oscillation for the autocatalator is $x=\\tilde{x}/\\epsilon $ .",
"Theorem REF states that the slow manifold loses normal hyperbolicity when $x={\\mathcal {O}}(\\epsilon ^{-1/2})$ .",
"This scaling certainly does not exclude the possibility of global crossings of the critical manifold when $x={\\mathcal {O}}(\\epsilon ^{-1})$ .",
"In particular, the relaxation loop shown in Figure REF (a) jumps near a fold point of the critical manifold 'towards infinity' but it has a travel time so that it only turns around for $x={\\mathcal {O}}(\\epsilon ^{-1})$ .",
"This yields one possible explanation why two consecutive blow-ups are needed in [9] as the scaling $x=\\tilde{x}/\\epsilon $ compresses two important scaling regions for the global return simultaneously.",
"For the model (REF ) by Rankin et al.",
"the decay rate of the critical manifold is exponential and hence faster than any polynomial.",
"In this context, Theorem REF implies that a compact submanifold of the critical manifold will loose normal hyperbolicity for arbitrarily large choices of $s\\in \\mathbb {N}$ in (REF ).",
"Hence the scaling is expected to be $x,y={\\mathcal {O}}(1)$ , or at least close to this scaling for all practical purposes, so that normal hyperbolicity is lost in a very small region independent of $\\epsilon $ .",
"As an important conclusion from Theorem REF one may also understand the behaviour of large amplitude oscillations (LAOs) induced by a global return mechanism with the asymptotic dynamics given by (REF ) i.e.",
"those LAOs which cross the critical manifold.",
"For example, given a general autocatalytic reaction mechanism $Y+(s+1)X\\rightarrow (s+2)X$ then the lower bound for the crossing is ${\\mathcal {O}}(\\epsilon ^{-1/(s+1)})$ for the fast component amplitude.",
"Since $s$ is usually known and $\\epsilon $ can be computed from the reaction rates the result has quite general applicability.",
"Acknowledgments: I would like to thank the European Commission (EC/REA) for support by a Marie-Curie International Re-integration Grant.",
"I also acknowledge support via an APART fellowship of the Austrian Academy of Sciences (ÖAW)."
]
] | 1204.0947 |
[
[
"Mixed-state quantum transport in correlated spin networks"
],
[
"Abstract Quantum spin networks can be used to transport information between separated registers in a quantum information processor.",
"To find a practical implementation, the strict requirements of ideal models for perfect state transfer need to be relaxed, allowing for complex coupling topologies and general initial states.",
"Here we analyze transport in complex quantum spin networks in the maximally mixed state and derive explicit conditions that should be satisfied by propagators for perfect state transport.",
"Using a description of the transport process as a quantum walk over the network, we show that it is necessary to phase correlate the transport processes occurring along all the possible paths in the network.",
"We provide a Hamiltonian that achieves this correlation, and use it in a constructive method to derive engineered couplings for perfect transport in complicated network topologies."
],
[
"Introduction",
"In the quest toward a scalable quantum computer [1], a promising model comprises distributed computing units connected by passive wires that transmit quantum information [2], [3], [4], [5], [6].",
"This architecture would provide several advantages, since the wires require no or limited control, easing the fabrication requirements and improving their isolation from the environment.",
"For a simpler integration in a solid-state architecture, the wires can be composed of spins.",
"Following seminal work by Bose [7], which showed that spin chains enable transporting quantum states between the ends of the chain, the dynamics of quantum state transfer has been widely studied (see Ref.",
"[8] for a review) and protocols for improving the fidelity by coupling engineering [9], [10], [11], [12], [13], [14], [15], [16], dual-rail topologies [17], active control on the chain spins [18] or on the end spins only [19], [20], [21], [22] have been proposed.",
"Recently these studies have been extended to mixed state spin chains [23], [24], [25], [26], [27], which are more easily obtained in high-temperature laboratory settings – making them important protagonists in practical quantum computing.",
"A further challenge to experimental implementation of quantum transport is the lack of chains with the desired coupling strengths, since coupling engineering is limited by fabrication constraints and by the presence of long-range interactions.",
"These challenges highlight the need for a systematic study of mixed state transport in quantum systems beyond chains, including more complex network topologies.",
"These topologies reflect more closely actual experimental conditions as well as systems occurring in nature.",
"For example, there is remarkable recent evidence [28], [29] that coherent quantum transport may be the underlying reason for the high efficiency (of over 99%) of photosynthetic energy transfer [30].",
"To derive explicit conditions for perfect transport we quantify geometric constraints on the unitary propagator that drives transport in an arbitrary network.",
"As a consequence, we find that transferring some mixed states in a network in general requires fewer conditions than pure state transfer.",
"Transport conditions for pure states have been previously quantified [31], however, our method – relying on decomposing the propagator in orthogonal spaces – is fundamentally different and more suitable for mixed states.",
"Perfect transport occurs when the bulk of the network acts like a lens to focus transport to its ends.",
"To make this physical picture more concrete, we describe mixed state transport as a continuous quantum walk over the network [32], [33], [34] that progressively populates its nodes.",
"Through this formalism, we derive constructive conditions on the Hamiltonian that results in the correlation of transport processes through different possible paths in the network.",
"The correlation of transport processes leads to their constructive interference at the position of the two end-spins, giving perfect transport.",
"While similar walk models have been applied to coherent transfer before (see [35] for a review), our work provides their first extension to transport involving mixed states.",
"The insight gained by describing quantum transport as correlated quantum walks can be used to construct larger networks where perfect transport is possible.",
"Here we show a strategy to achieve this goal by engineering the coupling strengths between different nodes of the network to construct weighted spin networks that support perfect mixed state transport.",
"Feder [36] had considered a similar problem for pure states by mapping the quantum walk of $N$ spinor bosons to a single particle; this has been extended in more recent work [37], [38], [39].",
"Here we find far more relaxed weighting requirements for mixed state transport, thanks in part to a fermionic instead of bosonic mapping.",
"In turn, this could ease the fabrication requirements for coupling engineering.",
"The paper is organized as follows.",
"In Sec.",
"we define the problem of mixed state transport.",
"Sec.",
"provides the geometric conditions on the propagator for perfect transport in arbitrary networks.",
"We finally present in Sec.",
"the quantum walk formalism, which allows the correlation of transport processes over different paths and the construction of families of weighted networks that support perfect transfer."
],
[
"Transport in mixed-state networks",
"Consider an $N$ -spin network ${\\cal N}$ , whose vertices (nodes) ${\\cal V}$ represent spins and whose edges ${\\cal E}=\\lbrace \\alpha _{ij}\\rbrace $ describe the couplings between spins $i$ and $j$ (see Fig.",
"REF ).",
"The system dynamics is governed by the Hamiltonian $H=\\sum _{i<j}\\alpha _{ij}H_{ij}$ , where $H_{ij}$ is the operator form of the interaction.",
"In the most general case, a spin of ${\\cal N}$ may be coupled to several others, for instance, in a dipolar coupled network, $\\alpha _{ij}\\sim 1/r_{ij}^3$ is a function of the distance between the spins in the network.",
"We assume that we can identify two nodes, labeled 1 and $N$ , that we can (partially) control and read out, independently from the “bulk” of the network, and thus act as the “end” spins between which transport will occur.",
"The rest of the spins in the network can at most be manipulated by collective control.",
"This also imposes restrictions on the network initialization [24], [25], [26].",
"To relax the requirements for the network preparation, we assume to work in the infinite-temperature limit [25] – a physical setting easily achievable for many experimental systems – where the bulk spins are in the maximally mixed state, $\\rho \\propto $ .",
"We will then consider the transport of a slight excess polarization from node 1 to node $N$ .",
"The initial state is $\\rho _i \\sim (+\\delta Z_1)$ , where $Z_1$ is the Pauli matrix acting on spin 1 and $\\delta \\ll 1$ denotes the polarization excess.",
"Since only the traceless part of the density matrix evolves in time, we will monitor the transport from $\\rho ^{\\Delta }_i = Z_1$ to a desired final state $\\rho ^{\\Delta }_f=Z_N$ .",
"The fidelity of the transport process is then defined as $F(t)={\\rm Tr}(\\rho ^{\\Delta }(t)Z_N)/{\\rm Tr}(Z^{\\dag }_1Z_1)$ , with $\\rho ^{\\Delta }(t)=U(t)\\rho ^{\\Delta }_iU^\\dag (t)$ being the evolved state.",
"The polarization behaves like a wave-packet traveling over the network ${\\cal N}$ [40], [41].",
"In most cases, the Hamiltonian $H$ drives a rapidly dispersive evolution, where the wave-packet quickly spreads out into many-body correlations among the nodes of ${\\cal N}$ , from which it cannot be recovered [34].",
"This is for example the case of evolution under the naturally occurring dipolar Hamiltonian, which induces a fast-decay of the spin polarization as measured in solid-state NMR, even if many-body correlations can be detected at longer times [42].",
"In order to drive a dispersionless transport, thus ensuring perfect fidelity, the network Hamiltonian should satisfy very specific conditions.",
"In this paper we will investigate these conditions by answering the questions: (i) What are the possible operator forms of the Hamiltonian $H_{ij}$ for dispersionless transport?",
"(ii) What are the coupling topologies and (iii) strengths $\\alpha _{ij}$ that support perfect transport?",
"Figure: (Color online) Transport in a spin network.",
"The network edgesrepresent the interaction among spins (nodes).The pink (dark) nodes represent the end spins between which transport should occur.",
"The shadedregion comprising yellow (light) nodes is the bulk network, H B H_B.",
"Aninitial polarization packet (hatched) is prepared on spin 1 and allowed to propagate onthe network through various possible paths(arrows).",
"For perfect transport, the polarization refocuses at spin NN."
],
[
"Fidelity of mixed-state transport",
"The condition for perfect transport, $F=1$ , can be expressed in a compact form by using the product-operator (PO) basis [43].",
"For the $N$ -spin network system there are $2^{2N}$ basis elements, $&&\\textbf {B} =\\textbf {B}_1\\otimes \\textbf {B}_{\\textrm {bulk}}\\otimes \\textbf {B}_N=\\lbrace ,X_{1},Y_{1},Z_{1},X_{2},\\dots , X_{1}X_{2},\\nonumber \\\\&&\\dots ,Z_{1}Z_{2},X_{1}X_{2}X_{3},\\dots ,\\dots ,Z_{1}Z_{2}\\cdots Z_{N-1}Z_{N}\\rbrace ,$ where $\\textbf {B}_{1,N}$ and $\\textbf {B}_{\\textrm {bulk}}$ are the basis for the end and the bulk spins, respectively.",
"Using the PO basis, the propagator $U(t)=e^{-iHt}$ can be represented by a vector $\\left|{U}\\right\\rangle =[c_{B_1},c_{B_2},\\dots ,c_{B_{4^N}}]^T$ in the $2^{2N}$ dimensional Hilbert-Schmidt (HS) operator space spanned by $\\textbf {B}$ [44]: $U(t) = \\textstyle \\sum _i c_{B_i}(t)B_i ,\\quad {\\rm with}\\quad c_{B_i}=\\displaystyle \\frac{\\textrm {Tr}(B_i^{\\dagger }U)}{\\textrm {Tr}(B_i^{\\dagger } B_ { i })}\\nonumber $ From an initial state with a polarization excess on spin 1, $\\rho _i^\\Delta =Z_1$ , the system evolves to $\\rho ^{\\Delta }_f=U\\rho ^{\\Delta }_iU^\\dag =\\textstyle \\sum _{i,j}c_{B_i}c_{B_j}^{\\ast }B_iZ_1B_j\\:,$ yielding the transport fidelity to spin $N$ $F&\\!=\\!&\\frac{1}{{\\rm Tr}(Z^{\\dag }_1Z_1)}\\textrm {Tr}\\left[\\textstyle \\sum _{i,j}c^{\\ast }_{B_i}c_{B_j}B_iZ_1B_jZ_N\\right]\\!\\!=\\!\\textstyle \\sum _ic_{B_i}c_{B_j}^{\\ast },\\nonumber \\\\&& {\\rm with}\\ B_j= \\pm Z_1Z_NB_i, \\quad {\\rm for}\\quad [B_i,Z_N]_{\\mp }=0.$ The last equation follows from the property that all elements of $\\textbf {B}$ , except $$ , are traceless.",
"The fidelity derived in Eq.",
"(REF ) has a simple form in the HS space.",
"Note that in this operator space, the product $B_iU$ is a linear transformation, $\\textbf {T}:\\left|{U}\\right\\rangle \\rightarrow \\hat{P}_{B_i}\\left|{U}\\right\\rangle $ , where $\\hat{P}_{B_i}$ is a permutation matrix corresponding to the action of $B_i$ [44] (here and in the following we denote operators in the HS space by a hat).",
"The unitarity of $U$ yields the conditions: $\\Vert {U}\\Vert =\\langle {U}\\vert {U}\\rangle =1\\qquad \\left\\langle {U}\\right|\\hat{P}_{B_i}\\left|{U}\\right\\rangle =0\\: ,\\: B_i\\ne .$ Let us partition the HS space in two subspaces $\\cal G$ , $\\widetilde{\\cal G}$ , spanned by the basis $G$ and $\\widetilde{G}$ , ${ G} = \\textbf {B}_1\\otimes \\textbf {B}_{\\rm bulk}\\otimes \\lbrace ,Z_N\\rbrace ,\\quad \\widetilde{G}= \\textbf {B}_1\\otimes \\textbf {B}_{\\rm bulk}\\otimes \\lbrace X_N,Y_N\\rbrace ,$ and note that all elements of ${G}$ commute with $Z_N$ , while all elements of $\\widetilde{G}$ anti-commute with $Z_N$ .",
"We will label by superscripts ${{\\cal G}}$ and ${\\widetilde{\\cal G}}$ the projections of operators in these subspaces.",
"Using this partition, we can simplify the expression for the fidelity of Eq.",
"(REF ) to obtain $F=\\left\\langle {U}\\right|\\hat{P}_{Z_1Z_N}\\hat{P}_{R}\\left|{U}\\right\\rangle \\:,$ where $\\hat{P}_{R}$ is a reflection about ${\\cal G}$ and $\\hat{P}_{Z_1Z_N}$ is block-diagonal in the $\\lbrace {G},\\widetilde{{G}}\\rbrace $ basis (since $Z_1Z_N\\in {\\cal G}$ ): $\\hat{P}_{R}=\\left[\\begin{array}{cc}^{\\cal G} & 0\\\\0 & -^{\\widetilde{\\cal G}}\\\\\\end{array}\\right],\\quad \\hat{P}_{Z_1Z_N}=\\left[\\begin{array}{cc}\\hat{P}_{Z_1Z_N}^{\\cal G} & 0\\\\0 & \\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}\\\\\\end{array}\\right]$ Rewriting the fidelity as the inner product between two vectors, $F=\\langle {(\\hat{P}_{Z_1Z_N}U)}\\vert {\\hat{P}_{R}{U}}\\rangle $ , provides a simple geometric interpretation of the perfect transport condition, as shown in Fig.",
"REF .",
"The vector $\\hat{P}_{Z_1Z_N}\\left|{U}\\right\\rangle $ should be parallel to $\\hat{P}_{R}\\left|{U}\\right\\rangle $ , which can be obtained if $\\hat{P}_{Z_1Z_N}$ rotates $\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle $ by an angle $\\pi $ , while leaving $\\left|{U^{{\\cal G}}}\\right\\rangle $ unaffected.",
"Alternatively, since $\\hat{P}_{Z_1Z_N}$ just describes a $\\pi -$ rotation of the vector $\\left|{U}\\right\\rangle $ about the $Z_1Z_N$ axis, for perfect transport the rotation-reflection operation $\\hat{S}=\\hat{P}_{Z_1Z_N}\\hat{P}_{R}$ should be a symmetry operation for $\\left|{U}\\right\\rangle $ .",
"Figure: (Color online) Geometric interpretation of the condition for maximumZ 1 →Z N Z_1\\rightarrow Z_N transport fidelity.",
"The unitary UU is representedas a vector U\\left|{U}\\right\\rangle (red thick arrow) in the HS space, with componentsU 𝒢 \\left|{U^{\\cal G}}\\right\\rangle and U 𝒢 ˜ \\left|{U^{\\widetilde{\\cal G}}}\\right\\rangle in the subspaces 𝒢{\\cal G} (represented by an axis) and 𝒢 ˜\\widetilde{\\cal G} (represented by theshaded grey plane).",
"P ^ R U\\hat{P}_R\\left|{U}\\right\\rangle (blue dashed arrow) is the reflection ofU\\left|{U}\\right\\rangle about the 𝒢{\\cal G} axis.",
"Maximumfidelity occurs only when the P ^ Z 1 Z N \\hat{P}_{Z_1Z_N} causes a π\\pi -rotation ofU 𝒢 ˜ \\left|{U^{\\widetilde{\\cal G}}}\\right\\rangle .From Eq.",
"(REF ) we have $\\left\\langle {U}\\right|\\hat{P}_{Z_1Z_N}\\left|{U}\\right\\rangle =0$ and using Eq.",
"(REF ) we can derive explicit conditions to be satisfied by the propagator to achieve perfect transport, $F=1$ $\\left\\langle {U^{\\cal G}}\\right|\\hat{P}_{Z_1Z_N}^{\\cal G}\\left|{U^{\\cal G}}\\right\\rangle +\\left\\langle {U^{\\widetilde{{\\cal G}}}}\\right|\\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle &=&0\\\\\\left\\langle {U^{\\cal G}}\\right|\\hat{P}_{Z_1Z_N}^{\\cal G}\\left|{U^{\\cal G}}\\right\\rangle -\\left\\langle {U^{\\widetilde{{\\cal G}}}}\\right|\\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle &=& 1$ that simplify to $\\left\\langle {U^{\\cal G}}\\right|\\hat{P}_{Z_1Z_N}^{\\cal G}\\left|{U^{\\cal G}}\\right\\rangle =-\\left\\langle {U^{\\widetilde{{\\cal G}}}}\\right|\\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle =\\frac{1}{2}\\:.$ When is this equation satisfied?",
"By symmetry, it happens when $\\Vert {U}^{\\cal G}\\Vert =\\Vert {U}^{\\widetilde{{\\cal G}}}\\Vert =1/2$ , and $\\left|{U^{\\cal G}}\\right\\rangle $ and $\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle $ are (up to a phase) eigenvectors of $\\hat{P}_{Z_1Z_N}^{\\cal G}$ and $\\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}$ with eigenvalues $\\pm 1$ respectively: $\\hat{P}_{Z_1Z_N}^{\\cal G}\\left|{U^{\\cal G}}\\right\\rangle =+\\left|{U^{\\cal G}}\\right\\rangle ;\\quad \\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}\\left|{U^{\\widetilde{\\cal G}}}\\right\\rangle =-\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle $ To enable perfect transport, $\\left|{U}\\right\\rangle $ must thus have an equal projection on the two subspaces ${\\cal G}$ and $\\widetilde{{\\cal G}}$ , as shown geometrically in Fig.",
"REF .",
"Also, intuitively from the symmetry operation $\\hat{S}$ , all components of $\\left|{U}\\right\\rangle $ lying on the plane ${\\cal G}$ should be rotationally symmetric with respect to $Z_1Z_N$ , while components of $\\left|{U}\\right\\rangle $ lying on the plane $\\widetilde{\\cal G}$ should have reflection symmetry about $Z_1Z_N$ .",
"Note that Eq.",
"(REF ) imposes fairly weak constraints on the transport unitaries, as opposed to the constraints for pure state transport [31], [45].",
"In particular Eq.",
"(REF ) provides no explicit constraint on the bulk of the network.",
"For example, the two propagators, $\\begin{array}{l}U_1=B_{\\rm bulk}(\\pm Z_1Z_N)+ B_{\\rm bulk}^{^{\\prime }}(X_1X_N \\pm Y_1Y_N),\\\\U_2=B_{\\rm bulk}(\\pm Z_1Z_N)+ B_{\\rm bulk}^{^{\\prime }}(X_1Y_N \\mp Y_1X_N),\\end{array}$ with $B_{\\rm bulk}$ and $B_{\\rm bulk}^{^{\\prime }}$ arbitrary equi-norm operators ($\\in {\\rm span}\\lbrace \\bf {B}_{\\rm bulk}\\rbrace $ ) acting on the bulk, support perfect transport.",
"Other propagators can be obtained thanks to an invariance property that we present in the next section.",
"More generally, in Appendix we explicitly provide a prescription to construct classes of unitaries for perfect mixed state transport."
],
[
"Invariance of transport Hamiltonians",
"The fidelity $F$ in Eq.",
"(REF ) is invariant under a transformation $U^{^{\\prime }}=VU$ , where $V$ is unitary and commutes with $\\hat{S}$ , that is, $[\\hat{V},\\hat{P}_{Z_1Z_N}\\hat{P}_{R}] =0.$ This invariance can be used to construct Hamiltonians that support perfect transport starting from known ones.",
"Consider an Hamiltonian $H$ that generates the transport evolution $U=\\exp (-iHt)$ .",
"Then the transport driven by $H$ is identical to that generated by the Hamiltonian $H^{^{\\prime }} =V^{\\dagger }HV$ , where $V$ satisfies Eq.",
"(REF ).",
"Ref.",
"[46] proved similar symmetry requirements for Hamiltonians that transport pure states; here, however, we derived these Hamiltonian properties just from the geometric conditions on $U$ .",
"Ref.",
"[47] treated a similar problem, defining classes of Hamiltonians that perform the same action on a state of interest.",
"A special case of this result was used in [23] to study transport in a mixed-state spin chain driven either by the nearest-neighbor coupling isotropic XY Hamiltonian, HXY=iiTii+1+ with Tij+=(Si+Sj- + Si-Sj+) or double-quantum (DQ) Hamiltonian $H_{\\rm DQ}=\\textstyle \\sum _{i}\\alpha _{i}D_{ii+1}^{+}\\text{ with }D_{ij}^{+}=(S_i^{+}S_j^{+} +S_i^{-}S_j^{-}),$ with $S_j^{\\pm }=\\frac{1}{2}(X_j \\pm iY_j)$ .",
"The unitary operator relating the two Hamiltonians, $V =\\prod _{k^{^{\\prime }}}X_{k^{^{\\prime }}}$ , where the product $k^{^{\\prime }}$ extends over all even or odd spins, does indeed satisfy Eq.",
"(REF )."
],
[
"Quantum information transport via mixed state networks",
"The requirements for perfect transport (Eq.",
"(REF )) can be easily generalized to the transport between any two elements of $\\textbf {B}$ , say from ${\\cal I}$ to ${\\cal F}$ .",
"One has simply to appropriately construct the subspaces ${\\cal G}$ and $\\widetilde{\\cal G}$ and the corresponding permutation operator $\\hat{P}_{{\\cal IF}}$ .",
"One could further consider under which conditions this transport (for example, $X_1\\rightarrow X_N$ ) can occur simultaneously with the $Z_1\\rightarrow Z_N$ transport already considered.",
"More generally, the simultaneous transfer of operators forming a basis for $\\bf {B_1}$ would enable the transport of quantum information [25], [10] via a mixed-state network.",
"The unitary $U$ should now not only be symmetric under $\\hat{S}$ , but should also under a similar operator derived for $X_1X_N$ .",
"The requirements on $U$ thus become more stringent and only a special case of the propagators constructed in Eq.",
"(REF ) in Appendix is allowed, $U=B_{\\rm bulk}(\\pm Z_1Z_N + X_1X_N \\pm Y_1Y_N)\\:.$ This is exactly a SWAP operation (up to a phase) between the end-spins, which can also lead to a transfer of arbitrary pure states between 1 and $N$ .",
"Therefore, we find that perfect transport of non-commuting mixed states between the end-spins also allows transport in pure state networks.",
"We note that quantum information could be encoded in multi-spin states [48], [25] that satisfy proper symmetry conditions and thus do not impose additional conditions on the transport propagators."
],
[
"Which Hamiltonians support mixed state transport?",
"It would be interesting to determine which Hamiltonians can generate propagators $\\left|{U(t)}\\right\\rangle = \\exp (-i\\hat{H}t)\\left|{}\\right\\rangle $ for perfect transport.",
"Unfortunately, deriving requirements for the Hamiltonian from the conditions on the unitaries is non-trivial; however, as we show below, one can still extract useful information.",
"A general Hamiltonian can be decomposed as $H = H^{{\\cal G}} + H^{\\widetilde{{\\cal G}}}$ , where $H^{{\\cal G},\\widetilde{{\\cal G}}}$ lie in the subspaces ${\\cal G}$ and $\\widetilde{{\\cal G}}$ , respectively.",
"We cannot set $H=H^{{\\cal G}}$ since the Hamiltonian does need to have a component that is non-commuting with the target operator ($Z_N$ ) in order to drive the transport.",
"If $H=H^{\\widetilde{\\cal G}}$ , odd powers of $H$ are in $\\widetilde{{\\cal G}}$ , while even powers of $H$ belong to ${{\\cal G}}$ .",
"Then the propagator has contributions from $\\left|{U^{\\cal G}}\\right\\rangle $ and $\\left|{U^{\\widetilde{\\cal G}}}\\right\\rangle $ with $\\left|{U^{{\\cal G}}}\\right\\rangle &=& \\left|{}\\right\\rangle +\\frac{(it)^2}{2!",
"}\\hat{H}\\left|{H}\\right\\rangle + \\frac{(it)^4}{4!",
"}\\hat{H}^3\\left|{H}\\right\\rangle +\\cdots \\nonumber \\\\\\left|{U^{\\widetilde{{\\cal G}}}}\\right\\rangle &=& it\\left|{H}\\right\\rangle +\\frac{(it)^3}{3!",
"}\\hat{H}^2\\left|{H}\\right\\rangle + \\cdots $ We can demonstrate that in this case the Hamiltonian must satisfy two conditions to drive perfect transport.",
"First, the “vector” form of the Hamiltonian must be an eigenstate of $\\hat{P}_{Z_1Z_N}$ , $\\hat{P}_{Z_1Z_N}\\left|{H}\\right\\rangle =-\\left|{H}\\right\\rangle $ , which ensures that the second equation in (REF ) is trivially satisfied, as $\\hat{P}_{Z_1Z_N}\\left|{U^{\\widetilde{\\cal G}}}\\right\\rangle = -\\left|{U^{\\widetilde{\\cal G}}}\\right\\rangle $ .",
"Second, since we have $\\hat{P}_{Z_1Z_N}\\left|{U^{{\\cal G}}}\\right\\rangle = \\left|{Z_1Z_N}\\right\\rangle +\\frac{(it)^2}{2!",
"}\\hat{H}\\left|{H}\\right\\rangle + \\frac{(it)^4}{4!",
"}\\hat{H}^3\\left|{H}\\right\\rangle +\\cdots ,$ the first equation in (REF ) implies that $H^{2n}\\!=\\!\\frac{1}{2}(\\!-\\!",
"Z_1Z_N)$ for any $n$ .",
"These conditions are for example satisfied by the XY-like Hamiltonian, $H=B_{\\rm bulk}T_{1N}^{+}$ , where $B_{\\rm bulk}$ is any operator acting on the bulk and $T_{1N}^{\\pm }=(S_1^{+}S_N^{-} \\pm S_1^{-}S_N^{+})$ .",
"In this case, at $t=\\pi /4$ , all conditions in Eq.",
"(REF ) are satisfied and perfect transport is achieved.",
"Indeed the XY Hamiltonian has been widely studied for quantum transport [7], [9] and it is interesting that we could derive its transport properties solely by the symmetry conditions on the propagator.",
"An Hamiltonian $H=H^{\\widetilde{\\cal G}}$ with support only in $\\widetilde{\\cal G}$ is however a very restrictive case as it refers to the situation where all nodes of the network are connected to $N$ .",
"Hamiltonians with support in both subspaces are more experimentally relevant, as they correspond to a common physical situation, where the ends of the network are separated in space and direct interaction between them is zero or too weak.",
"In the following, we will consider this more general situation, although restricting the study to XY Hamiltonians in order to derive conditions for perfect transport.",
"In the following, we will consider the network ${\\cal N}$ to consist of spins that are coupled by XY-like interaction, $\\lbrace T_{ij}^{+}\\rbrace $ .",
"We focus on this interaction since it has been shown that with appropriate engineered coupling strengths, $\\alpha _{ij}\\propto \\sqrt{i(N-i)}\\delta _{j,i+1}$ , the XY-Hamiltonian can support perfect transport in linear spin chains (see e.g.",
"[11], [12], [10], [49]).",
"Thanks to the invariance property described in Sec.",
"REF , this analysis applies to a much broader class of Hamiltonians, in particular to the DQ Hamiltonian.",
"We assume that the end spins of ${\\cal N}$ are not directly coupled, thus transport needs to be mediated by the bulk of the network.",
"The simplest such topology is a $\\Lambda $ -type configuration where the end spins are coupled to a single spin in the bulk.",
"The Hamiltonian $\\Lambda _j= (T_{1j}^{+} + T_{jN}^{+})/\\sqrt{2}$ , where $j$ is a spin in the bulk, is enough to drive this transport.",
"In this case, $(\\Lambda _j^2)^{n}\\!=\\!\\Lambda _j^2\\ \\forall \\,n$ , and hence the propagator is $U=\\exp (-iHt)=+[\\cos (t)-1] H^2 -i \\sin (t) H,$ where $H^2 = 1/2[(T_{1j}^{+})^2 + (T_{jN}^{+})^2 + T_{1N}^{+}]$ .",
"Since $(- 2H^2)$ has the form of $U_3$ in Eq.",
"(REF ), setting $t=\\pi $ ensures $U=U_3$ , yielding perfect transport.",
"This is an expected result, since this simple lambda-network is just a 3-spin linear chain.",
"This result can be extended to longer chains, as long as engineered couplings ensure that the resulting Hamiltonian is mirror-symmetric [45], [10], [16].",
"A different situation arises when there is more than one transport path possible, that is, the end spins are coupled to more than one spin in the bulk with an Hamiltonian $H=\\sum _{j\\in \\rm {bulk}}\\alpha _j\\Lambda _j$ .",
"For example, Fig.",
"REF depicts a network similar to the one considered in [26], [50] where there are three $\\Lambda $ paths between the end-spins.",
"Even if each path individually supports perfect transport, evolution along different paths may not be correlated, leading to destructive interference reducing the fidelity (see Fig.",
"REF ).",
"Figure: (Color online) (a) Λ\\Lambda -type network with threeΛ\\Lambda paths between the bulk and end-spins and equal coupling strength α\\alpha .",
"(b) Transportfidelity as a function of normalized time for the Λ\\Lambda -network coupled by theXY-Hamiltonian T ij + T_{ij}^{+} (blue dashed) and the modified XY-Hamiltonian T ˜ ij + \\widetilde{T}_{ij}^{+}(red solid).",
"In the latter case, correlated quantum walks lead to perfecttransport.Perfect fidelity can be achieved only if different paths can be collapsed into a single “effective” one that supports perfect transport (Fig.",
"REF ).",
"This strategy not only allows us to determine if an Hamiltonian can support perfect transport, but it also gives a recipe to build allowed Hamiltonians, by combining simpler networks known to support perfect transport into more complex ones.",
"To this end, we use the fact that linear chains enable perfect transport with appropriate engineered couplings.",
"Our first step will then to give conditions under which two chains of the same length (with end-spins in common) can be combined.",
"To obtain these conditions, we describe the evolution of the spin polarization as a quantum walk over the operators in the network [33], [34].",
"This description reveals the need to correlate the parallel paths over the network, in order to achieve a constructive refocusing of the polarization at the other end of the network.",
"We then generalize the conditions by a recursive construction to quite general networks."
],
[
"Transport as a quantum walk over ${\\cal N}$",
"We describe the transport evolution as a quantum walk over the network, which progressively populates operators in the HS space.",
"This process of progressively populating different parts of the HS space upon continuous time evolution under the Hamiltonian can be considered as a quantum walk [33], [34], [51].",
"We first expand the transport fidelity $F(t)$ (Eq.",
"REF ) in a time series, $F(t) &=&\\left\\langle U_0\\left|\\hat{P}_{Z_1Z_N}\\hat{P}_{R}\\right|U_0\\rangle - it\\langle U_0\\left|[\\hat{H},\\hat{P}_{Z_1Z_N}\\hat{P}_{R}]\\right|U_0\\right\\rangle \\nonumber \\\\&+&\\frac{i^2t^2}{2!",
"}\\left\\langle U_0\\left|\\left[\\hat{H},\\left[\\hat{H},\\hat{P}_{Z_1Z_N}\\hat{P}_{R}\\right]\\right]\\right|U_0\\right\\rangle + \\cdots $ with $|U_0\\rangle =\\left|{}\\right\\rangle $ .",
"Defining the nested commutators, ${\\cal C}_0 = \\hat{P}_{Z_1Z_N}\\hat{P}_{R} \\ ; \\ {\\cal C}_n =[\\hat{H},{\\cal C}_{n-1}] \\:,$ Eq.",
"(REF ) takes the form $F(t)=\\sum _{n=0}^{\\infty } \\frac{(it)^n}{n!",
"}\\langle {\\cal C}_n\\rangle ,$ where the expectation value is taken with respect to $\\left|{{U}_0}\\right\\rangle $ .",
"A large part of the Hamiltonian commutes with $\\hat{P}_{Z_1Z_N}\\hat{P}_{R}$ and can be neglected.",
"We can isolate the non-commuting part by defining the operator $\\hat{A}$ via the relationship $[\\hat{H},\\hat{P}_{Z_1Z_N}\\hat{P}_{R}] =\\hat{A}\\hat{P}_{Z_1Z_N}\\hat{P}_{R}\\:.$ The operator $\\hat{A}$ and its nested commutators ${\\cal C}_n^{A}=[\\hat{H},{\\cal C}_{n-1}^{A}] $ (with ${\\cal C}_0^{A}=\\hat{A}$ ) have a simple graphical construction.",
"The commutation relations, $\\left[{T}_{ij}^{+},{T}_{jk}^{\\pm }\\right] =-Z_j{T}_{ik}^{\\mp };\\qquad \\left[{T}_{ij } ^ { +},{T}_{kl}^{+}\\right] =0\\,,$ (see Fig.",
"REF ) can be used to provide a simple prescription to graphically determine the flip-flop terms in ${\\cal C}_n^{A}$ .",
"Figure: (Color online) Graphical representation of commutatorsfor (a) the XY and (b) the modified XYHamiltonian.",
"The commutator between two top legs (blue) of the directed graph isthe third edge (red).",
"In case of the XY Hamiltonian, the commutator is conditionedon node jj.For any two edges, one in ${\\cal C}_{n-1}^{A}$ and one in $H$ , that share a common node, ${\\cal C}_n^{A}$ contains the edge required to complete the triangle between them.",
"Thus, each higher order in the commutation expansion creates a link between nodes in the network, progressively populating it.",
"We will refer to the operators ${\\cal C}_n^{A}$ as quantum walk operators, since as we show below, the nested commutators ${\\cal C}_n$ in Eq.",
"(REF ) can be built exclusively out of them.",
"Consider the network of Fig.",
"REF (a), with coupling strengths $\\alpha _{ij}=1$ : $\\hat{A}$ contains only the edges of ${\\cal N}$ that connect to node 1, as represented by the red lines in Fig.",
"REF (b).",
"Figure: (Color online) (a) A six spin network with two pathsP 1 P_1 and P 2 P_2 between the end spins.",
"(b)-(f) representgraphically the successive orders of the quantum walk operators𝒞 n A {\\cal C}_{n}^{A}: A red line linking twonodes indicates that there is a flip-flop term T ij ± T_{ij}^{\\pm } between them, while path-dependent prefactors are not depicted.Once the walk hascovered the entire network, successive ordersin 𝒞 n A {\\cal C}_{n}^{A} reproduce 𝒞 3 A {\\cal C}_{3}^{A} and 𝒞 4 A {\\cal C}_{4}^{A}.",
"Theexplicit expressions for the commutators are shown in Table .Fig.",
"REF (c-d) represent the higher order commutators, with a red line linking two nodes denoting a term $T_{ij}^{\\pm }$ between them.",
"We note that the graphical construction only predicts the presence of a flip-flop term $T_{ij}^{\\pm }$ linking two nodes in the commutator ${\\cal C}_n^{A}$ , while the explicit forms of the commutators is generally more complex, as shown in Table REF , with additional appropriate weights for arbitrary coupling strengths $\\alpha _{ij}$ .",
"Still, as we now show, only the $T_{ij}^{\\pm }$ terms are important to determine the fidelity, and the presence of a $T_{1N}^{\\pm }$ term in the graphical series is an indication that transport can occur between the end-nodes.",
"The commutators ${\\cal C}_n$ can indeed be written in terms of the ${\\cal C}_n^{A}$ nested commutators, ${\\cal C}_{n}= \\sum _{k=0}^{n-1}\\dbinom{n-1}{k}{\\cal C}_{n-1-k}^{A}{\\cal C}_{k},$ yielding an expression for the fidelity containing only products of the nested commutators ${\\cal C}_{n}^{A}$ : ${\\cal C}_{n}= \\sum _{k_1=0}^{n-1}\\sum _{k_2=0}^{k_1-1}\\cdots \\sum _{k_n=0}^{k_{n-1}-1}\\dbinom{n-1}{k_1}\\dbinom{k_1-1}{k_2}\\cdots \\dbinom{k_{n-1}-1}{k_n}{\\cal C}_{n-k_1-1}^{A}{\\cal C}_{k_1-k_2-1}^{A}\\cdots {\\cal C}_{k_{n-1}-k_n-1}^{A}\\hat{P}_{Z_1Z_N}\\hat{P}_{R}\\:.$ For a commutator ${\\cal C}_n$ to yield a non-zero contribution to the fidelity, the product of the operators ${\\cal C}_k^{A}$ should be proportional to $Z_1Z_N$ , that is, it should evaluate to even powers of $T_{1N}^{\\pm }$ .",
"Hence very few terms appearing in Eq.",
"(REF ) actually contribute to the transfer fidelity $F$ .",
"Table: Nested commutators 𝒞 n A {\\cal C}_{n}^A (walk operators) corresponding to thegraphs in Fig.",
"if the edges represent the XY Hamiltonian or themodified XY Hamiltonian.",
"In the first case, note the presence of path dependentZ j Z_jprefactors, which are absent if the modified XY Hamiltonian isused.",
"This allows for the correlation of transport through parallel paths.The geometric construction of ${\\cal C}_{n}^{A}$ only yields the XY operators contained in each commutator, but it does not reflect the appearance of prefactors $\\propto Z_j$ (due to the commutator in Eq.",
"(REF )) that are explicitly written out in Table REF .",
"Thus, the geometric construction gives a necessary condition for transport, but not a sufficient one.",
"The operators ${\\cal C}_{n}^{A}$ describe the sum of walks over different paths: for example in ${\\cal C}_{2}^{A}$ , $Z_2Z_3T_{1N}^{+}$ can be interpreted as the information packet reaching node $N$ through path $P_1$ in Fig.",
"REF (a), while $Z_4Z_5T_{1N}^{+}$ represents propagation via path $P_2$ .",
"These two terms could in principle contribute to the fidelity, as they contain $T_{1N}^{+}$ .",
"However, the additional path dependent factors $\\prod _k Z_k$ lead to a loss of fidelity.",
"Transport through different paths yield different $\\prod _k Z_k$ factors, resulting into a destructive “interference” effect.",
"Note also that since different paths are weighted by different correlation factors, they cannot be canceled through some external control to recover the fidelity.",
"In the following section we show how a modified Hamiltonian can remove this path-conditioning and thus drive perfect transport.",
"Note that the path-dependent factors are as well unimportant in the case of pure states, provided the states reside in the same excitation manifold [52], [53], [25].",
"Figure: (Color online) Operators appearing in the quantum walkof a network consisting of two Λ\\Lambda paths,1→2→N1\\rightarrow 2\\rightarrow N and 1→3→N1\\rightarrow 3\\rightarrow N. In panel (a), where the transport is driven by the XY-Hamiltonian, the two paths through spins2 and 3 are different, as they traverse a different set of operators, and are thus depicted in two separated grey panels.",
"In panel(b), where we consider the modified XY Hamiltonian, both walks go through a commonset of operators.",
"The previously separate walks are bridged by the operators inthe red box, making both walks indistinguishable and hence correlated."
],
[
"Correlating quantum walks over ${\\cal N}$ : Modified XY-Hamiltonian",
"To remove the path-conditioning one should modify the Hamiltonian so that the $Z_j$ term in the commutator Eq.",
"(REF ) disappears.",
"This can be done via a modified XY-Hamiltonian $\\widetilde{T}_{ij}^{\\pm }=T_{ij}^{\\pm }\\prod _{i<u<j}Z_u$ since it satisfies this condition: $\\left[\\widetilde{T}_{ij}^{+},\\widetilde{T}_{jk}^{\\pm }\\right] =-\\widetilde{T}_{ik} ^{\\mp }\\:;\\quad \\left[\\widetilde{T}_{ij}^{+},\\widetilde{T}_{kl}^{+}\\right] =0$ These operators now depend on the number of nodes between $i$ and $j$ , thus introducing a metric in the spin-space that distinguishes paths between the two nodes $i$ and $j$ .",
"Note that when the network $\\cal N$ is a simple linear chain with nearest-neighbor couplings the modified Hamiltonian $\\widetilde{T}_{ij}^{+}$ is equivalent to the bare XY Hamiltonian.",
"The modification in Eq.",
"(REF ) of the XY-Hamiltonian could also be seen as mapping the spin system into a set of non-interacting fermions [54], [55] via a Jordan-Wigner transformation [56], [57], since $C_i=\\prod _{u<i}Z_uS^{+}_{i}$ are operators that satisfy the usual fermionic anti-commutation relationships.",
"When these modified operators are employed in the network ${\\cal N}$ of Fig.",
"REF (a), the two paths $P_1$ and $P_2$ in Fig.",
"REF (a) are indistinguishable or, equivalently, they become perfectly correlated (see Table REF ).",
"In effect, the modified XY Hamiltonian drives the quantum walks over different paths through a common set of operators of B.",
"This is shown in Fig.",
"REF for a simple $\\Lambda $ -network consisting of two $\\Lambda $ paths.",
"The graphic construction used to calculate the transport over the network in Fig.",
"REF remains unchanged, except that now the red lines between two nodes denote modified flip-flops $\\widetilde{T}_{ij}^{\\pm }$ between them.",
"Crucially there are no path dependent prefactors and symmetric nodes in each path become equivalent in each of the operators ${\\cal C}_{n}^{A}$ .",
"It is then possible to collapse different paths into a single effective one, until a complex network ${\\cal N}$ is collapsed into a linear chain.",
"This is depicted in Fig.",
"REF (a).",
"We can express this result more formally, by defining collapsed XY operators, where we denote in parenthesis equivalent nodes in two parallel paths: $\\widetilde{T}_{i(j,k)}^{\\pm } =\\frac{1}{\\sqrt{\\gamma _{ij}^2 + \\gamma _{ik}^2}}\\left(\\gamma _{ij}\\widetilde{T}_{ij}^{\\pm } +\\gamma _{ik}\\widetilde{T}_{ik}^{\\pm }\\right)\\:,$ where $\\gamma _{ij}$ and $\\gamma _{ik}$ are arbitrary parameters, $0<\\gamma _{ij},\\gamma _{ik}<1$ (see also Appendix ).",
"Remarkably, these operators satisfy the same path-independent commutation relations as in Eq.",
"(REF ) $\\left[\\widetilde{T}_{i(j,k)}^{+},\\widetilde{T}_{(j,k)\\ell }^{\\pm }\\right] =-\\widetilde{T}_{i\\ell }^{\\mp }\\:,$ thus showing that intermediate equivalent nodes can be neglected in higher order commutators.",
"In addition the nested commutators ${\\cal C}_n^A$ , and the graphical method to construct them (Fig.",
"REF ), remain invariant when substituting the modified XY operator with the collapsed operators $\\widetilde{T}_{i(j,k)}^{\\pm }$ .",
"Using the collapsed operators, the network of Fig.",
"REF (a) can thus be reduced to a simpler linear chain (Fig.",
"REF ).",
"Analogous arguments for path-collapsing were presented in [58], and have been applied before to some classes of graphs [9], [38].",
"In the following we show that path-equivalence could be constructed even for more complex network topologies, since, as we described, path collapsing can be derived just from the commutation relationships between the edges of the network."
],
[
"Engineered spin networks",
"The path collapsing described in the previous section provides a constructive way to build networks, with appropriate coupling geometries and strengths, that achieve perfect transport.",
"Alternatively, given a certain network geometry, the method determines all the possible coupling strength distributions that leave its transport fidelity unchanged.",
"For example, starting from a linear chain, any node can be substituted by two equivalent nodes, thus giving rise to two equivalent paths.",
"Then, within the subspace of the equivalent nodes, the couplings can be set using Eq.",
"(REF ) with arbitrary weights $\\gamma $ , thus giving much flexibility in the final allowed network.",
"The engineered network corresponding to Fig.",
"REF (a) is represented in Fig.",
"REF (a), where equivalent nodes from $P_1$ are weighted by $\\gamma $ , while those from $P_2$ are weighted by $\\sqrt{1-\\gamma ^2}$ .",
"A more complex example is shown in Fig.",
"REF (b), where the network is built combining the networks in Fig.",
"REF (a) and Fig.",
"REF (a).",
"It consists of three paths and can be collapsed into a 4-spin linear chain.",
"The couplings shown lead to perfect $Z_1\\rightarrow Z_N$ transport for arbitrary path weights $\\gamma _1$ and $\\gamma _2$ , with $0<\\gamma _1,\\gamma _2<1$ , as shown in Fig.",
"REF (c).",
"The network engineering scheme can be recursively integrated to construct larger and more complicated network topologies (see for example Fig.",
"REF ).",
"Similar weighted networks have been considered before for bosons [36].",
"The engineered couplings derived by mapping quantum walks of $N$ spinor bosons to the walk of a single particle are however much more restricted than what we found here via the mapping of spins to non-interacting fermions.",
"Figure: (Color online) A complex network topology that can beengineered for perfect transport with the modified XY Hamiltonian.Collapsing the equivalent nodes along the lines ℓ 2 -ℓ 5 \\ell _2-\\ell _5leads tothree equivalent paths that can be suitably engineered for perfect transport."
],
[
"Conclusions and Outlook",
"Experimental implementation of quantum information transport requires relaxing many of the assumptions made in ideal schemes.",
"In this paper we analyzed a physical situation that is closer to experimental settings – information transport in mixed-state spin networks with complex topologies.",
"We first derived general conditions on propagators that allow perfect transport in these mixed-state spin networks.",
"We used the conditions on the propagators to show that there exist classes of symmetry transformations on the Hamiltonians driving the transport for which the transport fidelity is invariant.",
"We also showed that the propagator conditions also imply that transporting some mixed states requires fewer control requirements than pure state transport, an added advantage to using mixed-state channels in quantum information architectures.",
"In order to study quantum transfer in complex spin networks, we described the dynamics as a continuous quantum walk over the possible paths offered by the network.",
"This description provided a graphical construction to predict the system evolution, which highlighted the need of correlating the transport processes occurring along different paths of the network to obtain perfect transport.",
"We thus introduced a modified XY-Hamiltonian, based on Jordan-Wigner fermionization, that achieves correlation among paths by establishing a metric for the quantum walks occurring on the network.",
"Conversely, the graphical construction could be as well used to study the generation from the usual XY-Hamiltonian of states of interest in measurement-based quantum computation [59].",
"Finally, the quantum-walk picture and the graphical construction lead us to define a constructive method to build complex networks from simpler ones, with appropriate coupling geometries and strengths, that achieve perfect transport.",
"We thus found that there is considerable freedom in the choice of topology and interaction strength that still allows perfect transport in complex networks.",
"While the requirement of a well-defined network topology could be further relaxed [60], the precise construction proposed in this paper would provide faster transport and the freedom in the coupling distributions could make these networks implementable in experimental systems."
],
[
"Acknowledgment",
"This work was partially funded by NSF under grant DMG-1005926."
],
[
"Constructing perfect transport unitaries",
"Here we show how the conditions specified in Eq.",
"(REF ) could be used to construct perfect transport propagators.",
"Our motivation for this is to demonstrate that the conditions of Eq.",
"(REF ) are very weak, in the sense that is possible to construct an infinite classes of unitaries that support $Z_1\\rightarrow Z_N$ transport.",
"Consider the matrix forms of $\\hat{P}_{Z_1Z_N}^{\\cal G}$ and $\\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}$ in the two-dimensional $\\lbrace 1,N\\rbrace $ subspace of ${\\cal G}$ and $\\widetilde{\\cal G}$ : ${\\cal G}\\!&\\!\\sim \\!&\\!\\Big \\lbrace \\lbrace ,Z_1Z_N\\rbrace ,\\lbrace Z_1 , Z_N\\rbrace ,\\lbrace X_1 ,Y_1Z_N\\rbrace ,\\lbrace Y_1 ,X_1Z_N\\rbrace \\Big \\rbrace \\nonumber \\\\\\widetilde{\\cal G}\\!&\\!\\sim \\!&\\!\\Big \\lbrace \\lbrace X_1X_N,Y_1Y_N\\rbrace ,\\lbrace X_1Y_N,Y_1X_N\\rbrace ,\\lbrace X_N,Z_1Y_N\\rbrace ,\\nonumber \\\\&&\\lbrace Y_N,Z_1X_N\\rbrace \\Big \\rbrace $ where the $\\sim $ refers to the restriction in the $\\lbrace 1,N\\rbrace $ subspace.",
"Then, for this order of basis, the matrix forms are block diagonal $\\hat{P}_{Z_1Z_N}^{\\cal G}&=&\\textrm {diag}([X, X,-Y,Y]) \\nonumber \\\\\\hat{P}_{Z_1Z_N}^{\\widetilde{\\cal G}}&=&\\textrm {diag}([-X,X,-Y,Y])$ where $X$ and $Y$ are the standard Pauli matrices, whose eigenvectors with eigenvalues $\\pm 1$ are respectively $[1,\\pm 1]^T$ and $[1,\\pm i]^T$ ; this imposes a restriction on $U$ .",
"If $B_{\\rm bulk},B_{\\rm bulk}^{^{\\prime }} \\in {\\rm span}\\lbrace \\bf {B}_{\\rm bulk}\\rbrace $ , one can explicitly list from Eq.",
"(REF ) possible forms of $U$ for perfect transport, $U_1&=&B_{\\rm bulk}(\\pm Z_1Z_N)+ B_{\\rm bulk}^{^{\\prime }}(X_1X_N \\pm Y_1Y_N),\\nonumber \\\\U_2&=&B_{\\rm bulk}(\\pm Z_1Z_N)+ B_{\\rm bulk}^{^{\\prime }}(X_1Y_N \\mp Y_1X_N),\\nonumber \\\\U_3&=& B_{\\rm bulk}(Z_1 \\pm Z_N)+B_{\\rm bulk}^{^{\\prime }}(X_1X_N \\pm Y_1Y_N),\\nonumber \\\\U_4&=&B_{\\rm bulk}(Z_1 \\pm Z_N)+ B_{\\rm bulk}^{^{\\prime }}(X_1Y_N \\mp Y_1X_N),\\nonumber \\\\U_5&=& B_{\\rm bulk}(X_1 \\pm iY_1Z_N)+B_{\\rm bulk}^{^{\\prime }}(X_N \\mp iZ_1Y_N),\\nonumber \\\\U_6&=& B_{\\rm bulk}(X_1 \\pm iY_1Z_N)+B_{\\rm bulk}^{^{\\prime }}( Y_N \\pm iZ_1X_N),\\nonumber \\\\U_7&=&B_{\\rm bulk}(Y_1 \\pm iX_1Z_N)+B_{\\rm bulk}^{^{\\prime }}( X_N\\pm iZ_1Y_N),\\nonumber \\\\U_8&=&B_{\\rm bulk}(Y_1 \\pm iX_1Z_N)+B_{\\rm bulk}^{^{\\prime }}( Y_N\\mp iZ_1X_N),\\nonumber \\\\$ Note that the bulk of the network specified by $B_{\\rm bulk}$ and $B_{\\rm bulk}^{^{\\prime }}$ can be any arbitrary operators with equal norms.",
"In fact, the invariance described in Sec.",
"REF could be used to show that the eight forms of $U$ in Eq.",
"(REF ) are equivalent to $U_1$ or $U_2$ .",
"Of course, one can combine the forms in Eq.",
"(REF ) to form other propagators that continue to support perfect transport.",
"Consider for example a propagator constructed out of $U_1$ and $U_2$ in Eq.",
"(REF ), with $B_{\\rm bulk},B_{\\rm bulk}^{^{\\prime }}=$ $U = \\lambda _1(1\\pm Z_1Z_N) + \\lambda _2(X_1X_N \\pm Y_1Y_N) + \\lambda _3(X_1Y_N\\mp Y_1X_N)\\nonumber $ where $\\lambda _j$ are coefficients to be determined.",
"Then, from Eq.",
"(REF ) we have $\\langle {U}\\vert {U}\\rangle =1 &\\Rightarrow & |\\lambda _1|^2+ |\\lambda _2|^2 + |\\lambda _3|^2 = 1\\nonumber \\\\\\left\\langle {U}\\right|\\hat{P}_{Z_1Z_N}\\left|{U}\\right\\rangle =0 &\\Rightarrow &|\\lambda _1|^2 = |\\lambda _2|^2 + |\\lambda _3|^2\\nonumber \\\\\\left\\langle {U}\\right|\\hat{P}_{Z_N}\\left|{U}\\right\\rangle =0 &\\Rightarrow &{\\rm Im}(\\lambda _2^{\\ast }\\lambda _3)=0$ Other conditions in Eq.",
"(REF ) are satisifed trivially.",
"Eq.",
"(REF ) can be solved exactly; for example $\\lambda _j=\\lbrace 1/\\sqrt{2},1/2,1/2\\rbrace $ is a solution.",
"Importantly however, if the $B_{\\rm bulk}^{^{\\prime }}$ 's were different from each other for $U_1$ and $U_2$ , the set of equations Eq.",
"(REF ) becomes far simpler.",
"In summary, achieving $Z_1\\rightarrow Z_N$ transport requires weak conditions on the propagator driving the transport.",
"This is as opposed to perfect pure state transport, that requires the propagators to be isomorphic to permutation operators [31] that are mirror symmetric [45] about the end spins of the network."
],
[
"Properties of flip-flop and double-quantum Hamiltonians",
"In this appendix, we present simple relations satisfied by the flip-flop (XY) operators that will be used in the main paper.",
"Note that the double-quantum (DQ) operators in Eq.",
"(REF ) follow analogous equations.",
"In what follows, distinct indices label distinct positions on the spin network unless otherwise specified.",
"We start with the definition of the operators $S$ and $E$ : $E_j^{\\pm }=\\frac{1}{2}(\\pm Z_j)\\:,\\qquad S_j^{\\pm }=\\frac{1}{2}(X_j \\pm iY_j).$ These operators satisfy the following product rules: $\\begin{array}{l}Z_jS_j^{\\pm }\\!=\\!\\pm S_j^{\\pm },\\quad (S_j^{\\pm })^2\\!=\\!E_j^{\\pm }E_j^{\\mp }=0,\\\\S_j^{\\pm }S_j^{\\mp }\\!=\\!",
"(E_j^{\\pm })^2=E_j^{\\pm }.\\end{array}$ We define the flip-flop operators $T_{ij}^{\\pm }$ and $L_{ij}^{\\pm }$ : $T_{ij}^{\\pm }=(S_i^{+}S_j^{-} \\pm S_i^{-}S_j^{+});\\quad L_{ij}^{\\pm }=(E_i^{+}E_j^{-} \\pm E_i^{-}E_j^{+})$ From the definition in Eq.",
"(REF ) it follows that $T_{ij}^{\\pm }=\\pm T_{ji}^{\\pm }\\:;\\:Z_jT_{ij}^{+}=T_{ij}^{-}.$ We have then the following product relations: $\\begin{array}{l}\\left(T_{ij}^{\\pm }\\right)^2=\\pm L_{ij}^{+}\\,,\\quad T_{ij}^{\\pm }L_{ij}^{+}=L_{ij}^{\\pm }T_{ij}^{+}=T_{ij}^{\\pm },\\\\\\left(L_{ij}^{\\pm }\\right)^2=L_{ij}^{+}\\,,\\quad T_{ij}^{\\pm }T_{jk}^{+}=\\frac{1}{2}\\left(T_{ik}^{\\pm }-Z_jT_{ik}^{\\mp }\\right)\\end{array}$ and the commutation relations: $\\begin{array}{ll}\\left[T_{ij}^{+},T_{jk}^{+}\\right] =-Z_jT_{ik}^{-}\\,,\\ &\\left[T_{ij}^{+},Z_jT_{ik}^{+}\\right] =T_{kj}^{-},\\\\\\left[T_{ij}^{+},Z_i\\right] = -2T_{ij}^{-}\\,,&\\left[T_{ij}^{-},Z_i\\right] = -2T_{ij}^{+}.\\end{array}$ We define the modified flip-flop operators $\\widetilde{T}_{ij}^{\\pm }$ , $\\widetilde{T}_{ij}^{\\pm }=T_{ij}^{\\pm }\\prod _{i<u<j}Z_u,$ obtained by multiplying the flip-flop operator in Eq.",
"(REF ) by a factor of $Z_u$ for all nodes between $i$ and $j$ .",
"The modified flip-flop operators follow especially simple commutation rules $\\left[\\widetilde{T}_{ij}^{+},\\widetilde{T}_{jk}^{\\pm }\\right] =-\\widetilde{T} _{ik}^{\\mp }\\:;\\:\\left[\\widetilde{T}_{ij}^{+},\\widetilde{T}_{k\\ell }^{\\pm }\\right] =0$ Note that crucially, these commutators only depend on the initial and final nodes ($i$ and $k$ ), and are independent of intermediate nodes.",
"In a physical analogy, the modified operators $\\widetilde{T}_{ij}^{+}$ behave as if they were path independent.",
"Thus, when considering two (or more) paths, we could omit any intermediate node, since it would not enter in the ensuing commutators.",
"We then denote $\\textit {equivalent}$ nodes in parenthesis –for example, $(j,k)$ means nodes $j$ and $k$ are equivalent– and define the collapsed operators: $\\widetilde{T}_{i(j,k)}^{\\pm } =\\frac{1}{\\sqrt{\\gamma _{ij}^2 + \\gamma _{ik}^2}}\\left(\\gamma _{ij}\\widetilde{T}_{ij}^{\\pm } +\\gamma _{ik}\\widetilde{T}_{ik}^{\\pm }\\right),$ where $\\gamma _{ij},\\ \\gamma _{ik}$ are arbitrary parameters, $0\\!<\\!\\gamma _{ij},\\!\\gamma _{ik}\\!<~\\!1$ .",
"The collapsed operators satisfy commutation relations similar to Eq.",
"(REF ): $\\begin{array}{l}\\left[\\widetilde{T}_{i(j,k)}^{+},\\widetilde{T}_{(j,k)\\ell }^{\\pm }\\right]\\!=\\!-\\widetilde{T}_{i\\ell }^{\\mp }\\,,\\ \\ \\left[\\widetilde{T}_{(j,k)i}^{+},\\widetilde{T}_{i\\ell }^{\\pm }\\right]\\!=\\!-\\widetilde{T}_{(j,k)\\ell }^{\\mp },\\\\\\displaystyle \\left[\\widetilde{T}_{(j,k)i}^{+},\\widetilde{T}_{i(m,n)}^{\\pm }\\right] =-\\widetilde{T}_{(j,k)(m,n)}^{\\mp }\\end{array}$ The collapsed operators in Eq.",
"(REF ) can be generalized.",
"If $\\mathbf {I}=(a_1,a_2,\\cdots ,a_m)$ and ${\\bf J}=(b_1,b_2,\\cdots ,b_n)$ denote two sets of equivalent nodes, we have the collapsed operator $\\widetilde{T}_{{\\bf IJ}}^{\\pm } =\\frac{1}{\\sqrt{\\sum _{i=1}^{m}\\sum _{j=1}^{n}\\gamma _{a_ib_j}^2}}\\sum _{i=1}^{m}\\sum _{j=1}^{n}\\gamma _{a_ib_j}\\widetilde{T}_{a_ib_j}^{\\pm }\\:,$ which satisfies the commutation relationships: $\\left[\\widetilde{T}_{\\mathbf {IJ}}^{+},\\widetilde{T}_{\\mathbf {JK}}^{\\pm }\\right] =-\\widetilde{T} _{\\mathbf {IK}}^{\\mp }\\:;\\:\\left[\\widetilde{T}_{\\mathbf {IJ}}^{+},\\widetilde{T}_{\\mathbf {KL}}^{\\pm }\\right]=0.$"
]
] | 1204.1363 |
[
[
"{\\Gamma}-species, quotients, and graph enumeration"
],
[
"Abstract The theory of {\\Gamma}-species is developed to allow species-theoretic study of quotient structures in a categorically rigorous fashion.",
"This new approach is then applied to two graph-enumeration problems which were previously unsolved in the unlabeled case-bipartite blocks and general k-trees."
],
[
"Introduction",
"Many of the most important historical problems in enumerative combinatorics have concerned the difficulty of passing from `labeled' to `unlabeled' structures.",
"In many cases, the algebra of generating functions has proved a powerful tool in analyzing such problems.",
"However, the general theory of the association between natural operations on classes of such structures and the algebra of their generating functions has been largely ad-hoc.",
"André Joyal's introduction of the theory of combinatorial species in [15] provided the groundwork to formalize and understand this connection.",
"A full, pedagogical exposition of the theory of species is available in [3], so we here present only an outline, largely tracking that text.",
"To begin, we wish to formalize the notion of a `construction' of a structure of some given class from a set of `labels', such as the construction of a graph from its vertex set or or that of a linear order from its elements.",
"The language of category theory will allow us capture this behavior succinctly yet with full generality: Let ${FinBij}$ be the category of finite sets with bijections and ${FinSet}$ be the category of finite sets with set maps.",
"Then a species is a functor $F: {FinBij} \\rightarrow {FinSet}$ .",
"For a set $A$ and a species $F$ , an element of $F {A}$ is an $F$ -structure on $A$.",
"Moreover, for a species $F$ and a bijection $\\phi : A \\rightarrow B$ , the bijection $F {\\phi }: F {A} \\rightarrow F {B}$ is the $F$ -transport of $\\phi $.",
"A species functor $F$ simply associates to each set $A$ another set $F {A}$ of its $F$ -structures; for example, for ${S}$ the species of permutations, we associate to some set $A$ the set ${S} {A} = \\operatorname{Bij} {A}$ of self-bijections (that is, permutations as maps) of $A$ .",
"This association of label set $A$ to the set $F {A}$ of all $F$ -structures over $A$ is fundamental throughout combinatorics, and functorality is simply the requirement that we may carry maps on the label set through the construction.",
"Let ${G}$ denote the species of simple graphs labeled at vertices.",
"Then, for any finite set $A$ of labels, $G {A}$ is the set of simple graphs with ${A}$ vertices labeled by the elements of $A$ .",
"For example, for label set $A = {3} = {1, 2, 3}$ , there are eight graphs in ${G} {A}$ , since there are $\\binom{3}{2} = 3$ possible edges and thus $2^{3} = 8$ ways to choose a subset of those edges: ${G} {{1, 2, 3}} = {\\begin{array}{c}\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};}\\end{aligned},\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (2);}\\end{aligned},\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(2) to (3);}\\end{aligned},\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (3);}\\end{aligned}, \\\\\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (2);(1) to (3);(2) to (3);}\\end{aligned},\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (3);(2) to (3);}\\end{aligned},\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (2);(1) to (3);}\\end{aligned},\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (2);(2) to (3);}\\end{aligned}\\end{array}}.$ The symmetric group 3 acts on the set 3 as permutations.",
"Consider the permutation ${(23)}$ that interchanges 2 and 3 in 3.",
"Then ${G} {{(23)}}$ is a permutation on the set ${G} {{1, 2, 3}}$ ; for example, ${G} {{(23)}} {\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (2);}\\end{aligned}} =\\begin{aligned}{ \\node [style=graphnode](1) at (90:1) {1};\\node [style=graphnode](2) at (210:1) {2};\\node [style=graphnode](3) at (330:1) {3};(1) to (3);}.\\end{aligned}.$ Since the image of a bijection under such a functor is necessarily itself a bijection, many authors instead simply define a species as a functor $F: {FinBij} \\rightarrow {FinBij}$ .",
"Our motivation for using this definition instead will become clear in s:quot.",
"Note that, having defined the species $F$ to be a functor, we have the following properties: for any two bijections $\\alpha : A \\rightarrow B$ and $\\beta : B \\rightarrow C$ , we have $F {\\alpha \\circ \\beta } = F {\\alpha } \\circ F {\\beta }$ , and for any set $A$ , we have $F {_{A}} = _{F {A}}$ .",
"Accordingly, we (generally) need not concern ourselves with the details of the set $A$ of labels we consider, so we will often restrict our attention to a canonical label set ${n} := {1, 2, \\dots , n}$ for each cardinality $n$ .",
"Moreover, the permutation group ${A}$ on any given set $A$ acts by self-bijections of $A$ and induces automorphisms of $F$ -structures for a given species $F$ .",
"The orbits of $F$ -structures on $A$ under the induced action of ${A}$ are then exactly the `unlabeled' structures of the class $F$ , such as unlabeled graphs.",
"Finally, we note that it is often natural to speak of maps between classes of combinatorial structures, and that these maps are sometimes combinatorially `natural'.",
"For example, we might wish to map the species of trees into the species of general graphs by embedding; to map the species of connected bicolored graphs to the species of connected bipartite graphs by forgetting some color information; or the species of graphs to the species of sets of connected graphs by identification.",
"These maps are all `natural' in the sense that they are explicitly structural and do not reference labels; thus, at least at a conceptual level, they are compatible with the motivating ideas of species.",
"We can formalize this notion in the language of categories: Let $F$ and $G$ be species.",
"A species map $\\phi $ is a natural transformation $\\phi : F \\rightarrow G$ — that is, an association to each set $A \\in {FinBij}$ of a set map $\\phi _{A} \\in {FinSet}$ such that the following diagram commutes: $\\begin{tikzpicture}[every node/.style={fill=white}](m) [matrix of math nodes, row sep=4em, column sep=4em]{F {A} & F {B} \\\\G {A} & G {B} \\\\};[->,font=\\scriptsize ](m-1-1) edge node {\\phi _{A}} (m-2-1)edge node {F {\\sigma }} (m-1-2)(m-2-1) edge node {G {\\sigma }} (m-2-2)(m-1-2) edge node {\\phi _{B}} (m-2-2);\\end{tikzpicture}$ We call the set map $\\phi _{A}$ the $A$ component of $\\phi $ or the component of $\\phi $ at $A$.",
"Such species maps may capture the idea that two species are essentially `the same' or that one `contains' or `sits inside' another.",
"Let $F$ and $G$ be species and $\\phi : F \\rightarrow G$ a species map between them.",
"In the case that the components $\\phi _{A}$ are all bijections, we say that $\\phi $ is a species isomorphism and that $F$ and $G$ are isomorphic.",
"In the case that the components $\\phi _{A}$ are all injections, we say that $\\phi $ is a species embedding and that $F$ embeds in $G$ (denoted $\\phi : F \\hookrightarrow G$ ).",
"Likewise, in the case that the components $\\phi _{A}$ are all surjections, we say that $\\phi $ is a species covering and that $F$ covers $G$ (denoted $\\phi : F \\twoheadrightarrow G$ ).",
"With the full power of the language of categories, we may make the following more general observation: Let ${Spc}$ denote the functor category of species; that is, define ${Spc} {FinSet}^{{FinBij}}$ , the category of functors from ${FinBij}$ to ${FinSet}$ .",
"Species maps as defined in def:specmap are natural transformations of these functors and thus are exactly the morphisms of ${Spc}$ .",
"It is a classical theorem of category theory (cf.",
"[17]) that the epi- and monomorphisms of a functor category are exactly those whose components are epi- and monomorphisms in the target category if the target category has pullbacks and pushouts.",
"Since ${FinSet}$ is such a category, species embeddings and species coverings are precisely the epi- and monomorphisms of the functor category ${Spc}$ .",
"Species isomorphisms are of course the categorical isomorphisms in ${Spc}$ .",
"In the case that $F$ and $G$ are isomorphic species, we will often simply write $F = G$ , since they are combinatorially equivalent; some authors instead use $F \\simeq G$ , reserving the notation of equality for the much stricter case that additionally requires that $F {A} = G {A}$ as sets for all $A$ .",
"The notions of species embedding and species covering are original to this work.",
"In the motivating examples from above: The species $\\mathfrak {a}$ of trees embeds in the species ${G}$ of graphs by the map which identifies each tree with itself as a graph, since any two distinct trees are distinct as graphs.",
"The species ${BC}$ of bicolored graphs covers the species ${BP}$ of bipartite graphs by the map which sends each bicolored graph to its underlying bipartite graph, since every bipartite graph has at least one bicoloring.",
"The species ${G}$ of graphs is isomorphic with the species ${E} {{G}^{{C}}}$ of sets of connected graphs by the map which identifies each graph with its set of connected components, since this decomposition exists uniquely."
],
[
"Cycle indices and species enumeration",
"In classical enumerative combinatorics, formal power series known as `generating functions' are used extensively for keeping track of enumerative data.",
"In this spirit, we now associate to each species a formal power series which counts structures with respect to their automorphisms, which will prove to be significantly more powerful: For a species $F$ , define its cycle index series to be the power series ${F}{p_{1}, p_{2}, \\dots } := \\sum _{n \\ge 0} \\frac{1}{n!}",
"\\big ( \\sum _{\\sigma \\in {n}} {F {\\sigma }} p_{1}^{\\sigma _{1}} p_{2}^{\\sigma _{2}} \\dots \\big ) = \\sum _{n \\ge 0} \\frac{1}{n!}",
"\\big ( \\sum _{\\sigma \\in {n}} {F {\\sigma }} p_{\\sigma } \\big )$ where ${F {\\sigma }} := {{s \\in F {A} : F {\\sigma } {s} = s}}$ , where $\\sigma _{i}$ is the number of $i$ -cycles of $\\sigma $ , and where $p_{i}$ are indeterminates.",
"(That is, ${F {\\sigma }}$ is the number of $F$ -structures fixed under the action of the transport of $\\sigma $ .)",
"We will make extensive use of the compressed notation $p_{\\sigma } = p_{1}^{\\sigma _{1}} p_{2}^{\\sigma _{2}} \\dots $ hereafter.",
"In fact, by functorality, ${F {\\sigma }}$ is a class functionThat is, the value of ${F {\\sigma }}$ will be constant on conjugacy classes of permutations, which we note are exactly the sets of permutations of fixed cycle type.",
"on permutations $\\sigma \\in {n}$ .",
"Accordingly, we can instead consider all permutations of a given cycle type at once.",
"It is a classical theorem that conjugacy classes of permutations in ${n}$ are indexed by partitions $\\lambda \\vdash n$ , which are defined as multisets of natural numbers whose sum is $n$ .",
"In particular, conjugacy classes are determined by their cycle type, the multiset of the lengths of the cycles, which may clearly be identified bijectively with partitions of $n$ .",
"For a given partition $\\lambda \\vdash n$ , there are $n!",
"/ z_{\\lambda }$ permutations of cycle type $\\lambda $ , where $z_{\\lambda } := \\prod _{i} i^{\\lambda _{i}} \\lambda _{i}!$ where $\\lambda _{i}$ denotes the multiplicity of $i$ in $\\lambda $ ..",
"Thus, we can instead write ${F}{p_{1}, p_{2}, \\dots } := \\sum _{n \\ge 0} \\sum _{\\lambda \\vdash n} {F {\\lambda }} \\frac{p_{1}^{\\lambda _{1}} p_{2}^{\\lambda _{2}} \\dots }{z_{\\lambda }} = \\sum _{n \\ge 0} \\sum _{\\lambda \\vdash n} {F {\\lambda }} \\frac{p_{\\lambda }}{z_{\\lambda }}$ for $F {\\lambda } := F {\\sigma }$ for some choice of a permutation $\\sigma $ of cycle type $\\lambda $ .",
"Again, we will make extensive use of the notation $p_{\\lambda } = p_{\\sigma }$ hereafter.",
"That the cycle index ${F}$ usefully characterizes the enumerative structure of the species $F$ may not be clear.",
"However, as the following theorems show, both labeled and unlabeled enumeration are immediately possible once the cycle index is in hand.",
"Recall that, for a given sequence $a = {a_{0}, a_{1}, a_{2}, \\dots }$ , the ordinary generating functionAlthough these are called `functions' for historical reasons, convergence of these formal power series is often not of immediate interest.",
"of $a$ is the formal power series $\\tilde{A} {x} = \\sum _{i = 0}^{\\infty } a_{i} x^{i}$ and the exponential generating function is the formal power series $A {x} = \\sum _{i = 1}^{\\infty } \\frac{1}{i!}",
"a_{i} x^{i}$ .",
"The scaling factor of $\\frac{1}{n!",
"}$ in the exponential generating function is convenient in many contexts; for example, it makes differentiation of the generating function a combinatorially-significant operation.",
"The cycle index of a species is then directly related to two important generating functions: The exponential generating function $F {x}$ of labeled $F$ -structures is given by $F {x} = {f}{x, 0, 0, \\dots }.$ The ordinary generating function $\\tilde{F} {x}$ of unlabeled $F$ -structures is given by $\\tilde{F} {x} = {F} [big]{x, x^{2}, x^{3}, \\dots }.$ Proofs of both theorems are found in [3].",
"In essence, eq:ciegf counts each labeled structure exactly once (as a fixed point of the trivial automorphism on ${n}$ ) with a factor of $1/n!$ , while eq:ciogf simply counts orbits à la Burnside's Lemma.",
"In cases where the unlabeled enumeration problem is interesting, it is generally more challenging than the labeled enumeration of the same structures, since the characterization of isomorphism in a species may be nontrivial to capture in a generating function.",
"If, however, we can calculate the complete cycle index for a species, both labeled and unlabeled enumerations immediately follow.",
"The use of $p_{i}$ for the variables instead of the more conventional $x_{i}$ alludes to the theory of symmetric functions, in which $p_{i}$ denotes the power-sum functions $p_{i} = \\sum _{j} x_{j}^{i}$ , which form an important basis for the ring $\\Lambda $ of symmetric functions.",
"When the $p_{i}$ are understood as symmetric functions rather than simply indeterminates, additional Pólya-theoretic enumerative information is exposed.",
"In particular, the symmetric function in $x$ -variables underlying a cycle index in $p$ -variables may be said to count partially-labeled structures of a given species, where the coefficient on a monomial $\\prod x_{i}^{\\alpha _{i}}$ counts structures with $\\alpha _{i}$ labels of each sort $i$ .",
"This serves to explain why the coefficients of powers of $p_{1} = \\sum _{i} x_{i}$ counts labeled structures (where the labels must all be distinct) and why the automorphism types of structures are enumerated by ${f}{x, x^{2}, x^{3}, \\cdots }$ , which allows clusters of labels to be the same.",
"Another application of the theory of symmetric functions to the cycle indices of species may be found in [7].",
"A more detailed exploration of the history of cycle index polynomials and their relationship to classical Pólya theory may be found in [18].",
"Of course, it is not always obvious how to calculate the cycle index of a species directly.",
"However, in cases where we can decompose a species as some combination of simpler ones, we can exploit these relationships algebraically to study the cycle indices, as we will see in the next section."
],
[
"Algebra of species",
"It is often natural to describe a species in terms of combinations of other, simpler species—for example, `a permutation is a set of cycles' or `a rooted tree is a single vertex together with a (possibly empty) set of rooted trees'.",
"Several combinatorial operations on species of structures are commonly used to represent these kinds of combinations; that they have direct analogues in the algebra of cycle indices is in some sense the conceptual justification of the theory.",
"In particular, for species $F$ and $G$ , we will define species $F + G$ , $F \\cdot G$ , $F \\circ G$ , ${F}$ , and $F^{\\prime }$ , and we will compute their cycle indices in terms of ${F}$ and ${G}$ .",
"In what follows, we will not say explicitly what the effects of a given species operation are on bijections when those effects are obvious (as is usually the case).",
"For two species $F$ and $G$ , define their sum to be the species $F + G$ given by $[big]{F + G} {A} = F {A} \\sqcup G {A}$ (where $\\sqcup $ denotes disjoint set union).",
"In other words, an ${F + G}$ -structure is an $F$ -structure or a $G$ -structure.",
"We use the disjoint union to avoid the complexities of dealing with cases where $F {A}$ and $G {A}$ overlap as sets.",
"For species $F$ and $G$ , the cycle index of their sum is ${F + G} = {F} + {G}.$ In the case that $F = G_{1} + G_{2}$ , we can simply invert the equation and write $F - G_{2} = G_{1}$ .",
"However, we may instead wish to study the species $F - G$ without first writing $F$ as a sum.",
"In the spirit of the definition of species addition, we wish to define the species subtraction $F - G$ as the species of $F$ -structures that `are not' $G$ -structures.",
"For slightly more generality, we may apply the notions of def:specmaptypes: For two species $F$ and $G$ with a species embedding $\\phi : G \\rightarrow F$ , define their difference with respect to $\\phi $ to be the species $F {\\phi } G$ given by $[big]{F {\\phi } G} {A} F {A} - \\phi {G {A}}$ .",
"When there is no ambiguity about the choice of embedding $\\phi $ , especially in the case that $G$ has a combinatorially natural embedding in $F$ , we may instead simply write $F - G$ and call this species their difference.",
"For example, for ${G}$ the species of graphs and $\\mathfrak {a}$ the species of trees with the natural embedding, we have that ${G} - \\mathfrak {a}$ is the species of graphs with cycles.",
"We note also that species addition is associative and commutative (up to species isomorphism), and furthermore the empty species ${0}: A \\mapsto \\varnothing $ is an additive identity, so species with addition form an abelian monoid.",
"This can be completed to create the abelian group of virtual species, in which the subtraction $F - G$ of arbitrary species is defined; the two definitions in fact agree where our definition applies.",
"We will not delve into the details of virtual species theory here, directing the reader instead to [3].",
"For two species $F$ and $G$ , define their product to be the species $F \\cdot G$ given by ${F \\cdot G} {A} = \\sum _{A = B \\sqcup C} F {B} \\times G {C}$ .",
"In other words, an ${F \\cdot G}$ -structure is a partition of $A$ into two sets $B$ and $C$ , an $F$ -structure on $B$ , and a $G$ -structure on $C$ .",
"This definition is partially motivated by the following result on cycle indices: For species $F$ and $G$ , the cycle index of their product is ${F \\cdot G} = {F} \\cdot {G}.$ Conceptually, the species product can be used to describe species that decompose uniquely into substructures of two specified species.",
"For example, a permutation on a set $A$ decomposes uniquely into a (possibly empty) set of fixed points and a derangement of their complement in $A$ .",
"Thus, ${S} = {E} \\cdot \\operatorname{Der}$ for ${S}$ the species of permutations, ${E}$ the species of sets, and $\\operatorname{Der}$ the species of derangements.",
"We note also that species multiplication is commutative (up to species isomorphism) and distributes over addition, so the class of species with addition and multiplication forms a commutative semiring, with the species ${1}: {\\left\\lbrace \\begin{array}{ll} \\varnothing \\mapsto {\\varnothing } \\\\ A \\ne \\varnothing \\mapsto \\varnothing \\end{array}\\right.",
"}$ as a multiplicative identity; if addition is completed as previously described, the class of virtual species with addition and multiplication forms a true commutative ring.",
"In addition, the question of which species can be decomposed as sums and products without resorting to virtual species is one of great interest; the notions of molecular and atomic species are directly derived from such decompositions, and represent the beginnings of the systematic study of the structure of the class of species as a whole.",
"Further details on this topic are presented in [3].",
"For two species $F$ and $G$ with $G {\\varnothing } = \\varnothing $ , define their composition to be the species $F \\circ G$ given by ${F \\circ G} {A} = \\prod _{\\pi \\in P {A}} {F {\\pi } \\times \\prod _{B \\in \\pi } G {B}}$ where $P {A}$ is the set of partitions of $A$ .",
"In other words, the composition $F \\circ G$ produces the species of $F$ -structures of collections of $G$ -structures.",
"The definition is, again, motivated by a correspondence with a certain operation on cycle indices: Let $f$ and $g$ be cycle indices.",
"Then the plethysm $f \\circ g$ is the cycle index $f \\circ g = f {g {p_{1}, p_{2}, p_{3}, \\dots }, g {p_{2}, p_{4}, p_{6}, \\dots }, \\dots },$ where $f {a, b, \\dots }$ denotes the cycle index $f$ with $a$ substituted for $p_{1}$ , $b$ substituted for $p_{2}$ , and so on.",
"This definition is inherited directly from the theory of symmetric functions in infinitely many variables, where our $p_{i}$ are basis elements, as previously discussed.",
"This operation on cycle indices then corresponds exactly to species composition: For species $F$ and $G$ with $G {\\varnothing } = \\varnothing $ , the cycle index of their plethysm is ${F \\circ G} = {F} \\circ {G}$ where $\\circ $ in the right-hand side is as in eq:cipleth.",
"Many combinatorial structures admit natural descriptions as compositions of species.",
"For example, every graph admits a unique decomposition as a (possibly empty) set of (nonempty) connected graphs, so we have the species identity ${G} = {E} \\circ {G}^{C}$ for ${G}$ the species of graphs and ${G}^{C}$ the species of nonempty connected graphs.",
"Diligent readers may observe that the requirement that $G {\\varnothing } = \\varnothing $ in def:speccomp is in fact logically vacuous, since the given construction would simply ignore the $\\varnothing $ -structures.",
"However, the formula in thm:speccompci fails to be well-defined for any ${G}$ with non-zero constant term (corresponding to species $G$ with nonempty $G {\\varnothing }$ ) unless ${F}$ has finite degree (corresponding to species $F$ with support only in finitely many degrees).",
"Consider the following example: Let ${E}$ denote the species of sets, ${E}_{3}$ its restriction to sets with three elements, 1 the species described above (which has one empty structure), and $X$ the species of singletons (which has one order-1 structure).",
"If ${E} {{1} + X}$ were well-defined, it would denote the species of `partially-labeled sets'.",
"However, for fixed cardinality $n$ , there is an ${E} {{1} + X}$ -structure on $n$ labels for each nonnegative $k$—specifically, the set ${n}$ together with $k$ unlabeled elements.",
"Thus, there would be infinitely many structures of each cardinality for this `species', so it is not in fact a species at all.",
"However, the situation for ${E}_{3} {{1} + X}$ is entirely different.",
"A structure in this species is a 3-set, some of whose elements are labeled.",
"There are only four possible such structures: ${*, *, *}$ , ${*, *, 1}$ , ${*, 1, 2}$ , and ${1, 2, 3}$ , where $*$ denotes an unlabeled element and integers denote labeled elements.",
"Moreover, by discarding the unlabeled elements, we can clearly see that ${E}_{3} {{1} + X} = \\sum _{i = 0}^{3} {E}_{i}$ .",
"In our setting, we will not use this alternative notion of composition, so we will not develop it formally here.",
"Several other binary operations on species are defined in the literature, including the Cartesian product $F \\times G$ , the functorial composition $F \\square G$ , and the inner plethysm $F \\boxtimes G$ of [24].",
"We will not use these here.",
"However, we do introduce two unary operations: ${F}$ and $F^{\\prime }$ .",
"For a species $F$ , define its species derivative to be the species ${F}$ given by ${F} {A} = F {A \\cup {*}}$ for $*$ an element chosen not in $A$ (say, the set $A$ itself).",
"It is important to note that the label $*$ of an ${F}$ -structure is distinguished from the other labels; the automorphisms of the species ${F}$ cannot interchange $*$ with another label.",
"Thus, species differentiation is appropriate for cases where we want to remove one `position' in a structure.",
"For example, for ${L}$ the species of linear orders and ${C}$ the species of cyclic orders, we have ${L} = {{C}}$ ; a cyclic order on the set $A \\cup {*}$ is naturally associated with the linear order on the set $A$ produced by removing $*$ .",
"Terming this operation `differentiation' is justified by its effect on cycle indices: For a species $F$ , the cycle index of its derivative is given by ${{F}}{p_{1}, p_{2}, \\dots } = \\frac{\\partial }{\\partial p_{1}} {F}{p_{1}, p_{2}, \\dots }.$ We note that we cannot in general recover ${F}$ from ${{F}}$ , since there may be terms in ${F}$ which have no $p_{1}$ -component (corresponding to $F$ -structures which have no automorphisms with fixed points).",
"Finally, we introduce a variant of the species derivative which allows us to label the distinguished element $*$ : For a species $F$ , define its pointed species to be the species ${F}$ given by ${F} {A} = F {A} \\times A$ (that is, pairs of the form ${f, a}$ where $f$ is an $F$ -structure on $A$ and $a \\in A$ ) with transport ${F} {\\sigma } {f, a} = {F {\\sigma } {f}, \\sigma {a}}$ .",
"We can also write ${F} {A} = X \\cdot {F}$ for $X$ the species of singletons.",
"In other words, an ${F} {A}$ -structure is an $F {A}$ -structure with a distinguished element taken from the set $A$ (as opposed to ${F}$ , where the distinguished element is new).",
"Thus, species pointing is appropriate for cases such as those of rooted trees: for $\\mathfrak {a}$ the species of trees and ${A}$ the species of rooted trees, we have ${A} = {\\mathfrak {a}}$ .",
"eq:specderivci leads directly to the following: For a species $F$ , the cycle index of its corresponding pointed species is given by ${{F}} = {X} \\cdot {{F}}.$ Note that, again, we cannot in general recover ${F}$ from ${{F}}$ , for the same reasons as in the case of ${{F}}$ ."
],
[
"Multisort species",
"A species $F$ as defined in def:species is a functor $F: {FinBij} \\rightarrow {FinSet}$ ; an $F$ -structure in $F {A}$ takes its labels from the set $A$ .",
"The tool-set so produced is adequate to describe many classes of combinatorial structures.",
"However, there is one particular structure type which it cannot effectively capture: the notion of distinct sorts of elements within a structure.",
"Perhaps the most natural example of this is the case of $k$ -colored graphs, where every vertex has one of $k$ colors with the requirement that no pair of adjacent vertices shares a color.",
"Automorphisms of such a graph must preserve the colorings of the vertices, which is not a natural restriction to impose in the calculation of the classical cycle index in eq:cycinddef.",
"We thus incorporate the notion of sorts directly into a new definition: For a fixed integer $k \\ge 1$ , define a $k$ -sort set to be an ordered $k$ -tuple of sets.",
"Say that a $k$ -sort set is finite if each component set is finite; in that case, its $k$ -sort cardinality is the ordered tuple of its components' set cardinalities.",
"Further, define a $k$ -sort function to be an ordered $k$ -tuple of set functions which acts componentwise on $k$ -sort sets.",
"For two $k$ -sort sets $U$ and $V$ , a $k$ -sort function $\\sigma $ is a $k$ -sort bijection if each component is a set bijection.",
"For $k$ -sort sets of cardinality ${c_{1}, c_{2}, \\dots , c_{k}}$ , denote by ${c_{1}, c_{2}, \\dots , c_{k}} = {c_{1}} \\times {c_{2}} \\times \\dots \\times {c_{k}}$ the $k$ -sort symmetric group, the elements of which are in natural bijection with $k$ -sort bijections from a $k$ -sort set to itself.",
"Finally, denote by ${FinBij}^{k}$ the category of finite $k$ -sort sets with $k$ -sort bijections.",
"We can then define an extension of species to the context of $k$ -sort sets: A $k$ -sort species $F$ is a functor $F: {FinBij}^{k} \\rightarrow {FinBij}$ which associates to each $k$ -sort set $U$ a set $F {U}$ of $k$ -sort $F$ -structures and to each $k$ -sort bijection $\\sigma : U \\rightarrow V$ a bijection $F {\\sigma }: F {U} \\rightarrow F {V}$ .",
"Functorality once again imposes naturality conditions on these associations.",
"Just as in the theory of ordinary species, to each multisort species is associated a power series, its cycle index, which carries essential combinatorial data about the automorphism structure of the species.",
"To keep track of the multiple sorts of labels, however, we require multiple sets of indeterminates.",
"Where in ordinary cycle indices we simply used $p_{i}$ for the $i$ th indeterminate, we now use $p_{i} {j}$ for the $i$ th indeterminate of the $j$ th sort.",
"In some contexts with small $k$ , we will denote our sorts with letters (saying, for example, that we have `$X$ labels' and `$Y$ labels'), in which case we will write $p_{i} {x}$ , $p_{i} {y}$ , and so forth.",
"In natural analogy to def:cycind, the formula for the cycle index of a $k$ -sort species $F$ is given by ${F}{p_{1} {1}, p_{2} {1}, \\dots ; p_{1} {2}, p_{2} {2}, \\dots ; \\dots ; p_{1} {k}, p_{2} {k}, \\dots } = \\\\\\sum _{{n \\ge 0 \\\\ a_{1} + a_{2} + \\dots + a_{k} = n}} \\frac{1}{a_{1}!",
"a_{2}!",
"\\dots a_{k}!}",
"\\sum _{\\sigma \\in {a_{1}, a_{2}, \\dots , a_{k}}} {F {\\sigma }} p^{\\sigma _{1}}_{{1}} p^{\\sigma _{2}}_{{2}} \\dots p^{\\sigma _{k}}_{{k}}.$ where by $p^{\\sigma _{i}}_{{i}}$ we denote the product $\\prod _{j} {p_{j} {i}}^{{\\sigma _{i}}_{j}}$ where ${\\sigma _{i}}_{j}$ is the number of $j$ -cycles of $\\sigma _{i}$ .",
"The operations of addition and multiplication extend to the multisort context naturally.",
"To make sense of differentiation and pointing, we need only specify a sort from which to draw the element or label which is marked; we then write $[X]{F}$ and $[X]{F}$ for the derivative and pointing respectively of $F$ `in the sort $X$ ', which is to say with its distinguished element drawn from that sort.",
"When $F$ is a 1-sort species and $G$ a $k$ -sort species, the construction of the $k$ -sort species $F \\circ G$ is natural; in other settings, we will not define a general notion of composition of multisort species."
],
[
"$\\Gamma $ -species and quotient species",
"It is frequently the case that interesting combinatorial problems admit elegant descriptions in terms of quotients of a class of structures $F$ under the action of a group $\\Gamma $ .",
"In some cases, this group action will be structural in the sense that it commutes with permutations of labels in the species $F$ , or, informally, that it is independent of the choice of labelings on each $F$ -structure.",
"In such a case, we may also say that $\\Gamma $ acts on `unlabeled structures' of the class $F$ .",
"Let ${G}$ denote the species of simple graphs.",
"Let the group 2 act on such graphs by letting the identity act trivially and letting the non-trivial element ${(12)}$ send each graph to its complement (that is, by replacing each edge with a non-edge and each non-edge with an edge).",
"This `complementation action' is structural in the sense described previously.",
"We note that a group action is structural is exactly the condition that each $\\gamma \\in \\Gamma $ acts by a species isomorphism $\\gamma : F \\rightarrow F$ in the sense of def:specmaptypes.",
"We now incorporate such species-compatible actions into a new definition: For $\\Gamma $ a group, a $\\Gamma $ -species $F$ is a combinatorial species $F$ together with an action of $\\Gamma $ on $F$ -structures by species isomorphisms.",
"Explicitly, for $F$ a $\\Gamma $ -species, the diagram $\\begin{tikzpicture}[every node/.style={fill=white}](m) [matrix of math nodes, row sep=4em, column sep=4em, text height=1.5ex, text depth=0.25ex]{A & F {A} & F {A} \\\\B & F {B} & F {B} \\\\};[->,font=\\scriptsize ](m-1-1) edge node {F} (m-1-2)edge node {\\sigma } (m-2-1)(m-1-2) edge node {\\gamma _{A}} (m-1-3)edge node {F {\\sigma }} (m-2-2)(m-1-3) edge node {F {\\sigma }} (m-2-3)(m-2-1) edge node {F} (m-2-2)(m-2-2) edge node {\\gamma _{B}} (m-2-3);\\end{tikzpicture}$ commutes for every $\\gamma \\in \\Gamma $ and every set bijection $\\sigma : A \\rightarrow B$ .",
"(Note that commutativity of the left square is required for $F$ to be a species at all.)",
"${G}$ is then a 2-species with the action described in ex:graphcomp.",
"For such a $\\Gamma $ -species, of course, it is then meaningful to pass to the quotient under the action by $\\Gamma $ : For $F$ a $\\Gamma $ -species, define ${F}{\\Gamma }$ , the quotient species of $F$ under the action of $\\Gamma $ , to be the species of $\\Gamma $ -orbits of $F$ -structures.",
"Consider ${G}$ as a 2-species in light of the action defined in ex:graphcomp.",
"The structures of the quotient species ${{G}}{{2}}$ are then pairs of complementary graphs.",
"We may choose to interpret each such pair as representing a 2-partition of the set of vertex pairs of the complete graph (that is, of edges of the complete graph).",
"More natural examples of quotient structures will present themselves in later chapters.",
"For each label set $A$ , let ${\\Gamma } {A}: F {A} \\rightarrow {F}{\\Gamma } {A}$ denote the map sending each $F$ -structure over $A$ to its quotient ${F}{\\Gamma }$ -structure over $A$ .",
"Then ${\\Gamma } {A}$ is an injection for each $A$ , and the requirement that $\\Gamma $ acts by natural transformations implies that the induced functor map ${\\Gamma }: F \\rightarrow {F}{\\Gamma }$ is a natural transformation.",
"Thus, the passage from $F$ to ${F}{\\Gamma }$ is a species cover in the sense of def:specmaptypes.",
"A brief exposition of the notion of quotient species may be found in [3], and a more thorough exposition (in French) in [4].",
"Our motivation, of course, is that combinatorial structures of a given class are often `naturally' identified with orbits of structures of another, larger class under the action of some group.",
"Our goal will be to compute the cycle index of the species ${F}{\\Gamma }$ in terms of that of $F$ and information about the $\\Gamma $ -action, so that enumerative data about the quotient species can be extracted.",
"As an intermediate step to the computation of the cycle index associated to this quotient species, we associate a cycle index to a $\\Gamma $ -species $F$ that keeps track of the needed data about the $\\Gamma $ -action.",
"For a $\\Gamma $ -species $F$ , define the $\\Gamma $ -cycle index ${\\Gamma }{F}$ as in [14]: for each $\\gamma \\in \\Gamma $ , let ${\\Gamma }{F}{\\gamma } = \\sum _{n \\ge 0} \\frac{1}{n!}",
"\\sum _{\\sigma \\in {n}} {\\gamma \\cdot F {\\sigma }} p_{\\sigma } $ with $p_{\\sigma }$ as in eq:cycinddef.",
"We will call such an object (formally a map from $\\Gamma $ to the ring ${Q} {{p_{1}, p_{2}, \\dots }}$ of symmetric functions with rational coefficients in the $p$ -basis) a $\\Gamma $ -cycle index even when it is not explicitly the $\\Gamma $ -cycle index of a $\\Gamma $ -species, and we will sometimes call ${\\Gamma }{F}{\\gamma }$ the “$\\gamma $ term of ${\\Gamma }{F}$ ”.",
"So the coefficients in the power series count the fixed points of the combined action of a permutation and the group element $\\gamma $ .",
"Note that, in particular, the classical (`ordinary') cycle index may be recovered as ${F} = {\\Gamma }{F}{e}$ for any $\\Gamma $ -species $F$ .",
"The algebraic relationships between ordinary species and their cycle indices generally extend without modification to the $\\Gamma $ -species context, as long as appropriate allowances are made.",
"The actions on cycle indices of $\\Gamma $ -species addition and multiplication are exactly as in the ordinary species case considered componentwise: For two $\\Gamma $ -species $F$ and $G$ , the $\\Gamma $ -cycle index of their sum $F + G$ is given by ${\\Gamma }{F + G}{\\gamma } = {\\Gamma }{F}{\\gamma } + {\\Gamma }{G}{\\gamma }$ and the $\\Gamma $ -cycle index of their product $F \\cdot G$ is given by ${\\Gamma }{F \\cdot G}{\\gamma } = {\\Gamma }{F}{\\gamma } \\cdot {\\Gamma }{G}{\\gamma }$ The action of composition, which in ordinary species corresponds to plethysm of cycle indices, can also be extended: For two $\\Gamma $ -species $F$ and $G$ , define their composition to be the $\\Gamma $ -species $F \\circ G$ with structures given by ${F \\circ G} {A} = \\prod _{\\pi \\in P {A}} {F {\\pi } \\times \\prod _{B \\in \\pi } G {B}}$ where $P {A}$ is the set of partitions of $A$ and where $\\gamma \\in \\Gamma $ acts on a ${F \\circ G}$ -structure by acting on the $F$ -structure and the $G$ -structures independently.",
"The requirement in def:gspecies that the action of $\\Gamma $ commutes with transport implies that this is well-defined.",
"Informally, for $\\Gamma $ -species $F$ and $G$ , we have defined the composition $F \\circ G$ to be the $\\Gamma $ -species of $F$ -structures of $G$ -structures, where $\\gamma \\in \\Gamma $ acts on an ${F \\circ G}$ -structure by acting independently on the $F$ -structure and each of its associated $G$ -structures.",
"A formula similar to that thm:speccompci requires a definition of the plethysm of $\\Gamma $ -symmetric functions, here taken from [14]: For two $\\Gamma $ -cycle indices $f$ and $g$ , their plethysm $f \\circ g$ is a $\\Gamma $ -cycle index defined by ${f \\circ g} {\\gamma } = f {\\gamma } {g {\\gamma } {p_{1}, p_{2}, p_{3}, \\dots }, g [big]{\\gamma ^{2}} {p_{2}, p_{4}, p_{6}, \\dots }, \\dots }.$ This definition of $\\Gamma $ -cycle index plethysm is then indeed the correct operation to pair with the composition of $\\Gamma $ -species: [Theorem 3.1, [14]] If $A$ and $B$ are $\\Gamma $ -species and $B {\\varnothing } = \\varnothing $ , then ${\\Gamma }{A \\circ B} = {\\Gamma }{A} \\circ {\\Gamma }{B}.$ Thus, $\\Gamma $ -species admit the same sorts of `nice' correspondences between structural descriptions (in terms of functorial algebra) and enumerative characterizations (in terms of cycle indices) that ordinary species do.",
"However, to make use of this theory for enumerative purposes, we also need to be able to pass from the $\\Gamma $ -cycle index of a $\\Gamma $ -species to the ordinary cycle index of its associated quotient species under the action of $\\Gamma $ .",
"This will allow us to adopt a useful strategy: if we can characterize some difficult-to-enumerate combinatorial structure as quotients of more accessible structures, we will be able to apply the full force of species theory to the enumeration of the prequotient structures, then pass to the quotient when it is convenient.",
"Exactly this approach will serve as the core of both of the following chapters.",
"Since we intend to enumerate orbits under a group action, we apply a generalization of Burnside's Lemma found in [7]: If $\\Gamma $ and $\\Delta $ are finite groups and $S$ a set with a ${\\Gamma \\times \\Delta }$ -action, for any $\\delta \\in \\Delta $ the number of $\\Gamma $ -orbits fixed by $\\delta $ is $\\frac{1}{{\\Gamma }} \\sum _{\\gamma \\in \\Gamma } {\\gamma , \\delta }$ .",
"Recall from eq:cycinddef that, to compute the cycle index of a species, we need to enumerate the fixed points of each $\\sigma \\in {n}$ .",
"However, to do this in the quotient species ${F}{\\Gamma }$ is by definition to count the fixed $\\Gamma $ -orbits of $\\sigma $ in $F$ under commuting actions of ${n}$ and $\\Gamma $ (that is, under an ${{n} \\times \\Gamma }$ -action).",
"Thus, lem:grouporbits implies the following: For a $\\Gamma $ -species $F$ , the ordinary cycle index of the quotient species ${F}{\\Gamma }$ is given by ${F / \\Gamma } = {\\Gamma }{F} \\frac{1}{{\\Gamma }} \\sum _{\\gamma \\in \\Gamma } {\\Gamma }{F}{\\gamma } = \\frac{1}{{\\Gamma }} \\sum _{{n \\ge 0 \\\\ \\sigma \\in {n} \\\\ \\gamma \\in \\Gamma }} \\frac{1}{n!}",
"{\\gamma \\cdot F {\\sigma }} p_{\\sigma }.$ where we define ${\\Gamma }{F} = \\frac{1}{{\\Gamma }} \\sum _{\\gamma \\in \\Gamma } {\\Gamma }{F}{\\gamma }$ for future convenience.",
"Note that this same result on cycle indices is implicit in [4].",
"With it, we can compute explicit enumerative data for a quotient species using cycle-index information of the original $\\Gamma $ -species with respect to the group action, as desired.",
"Recall from thm:ciegf,thm:ciogf that the exponential generating function $F {x}$ of labeled $F$ -structures and the ordinary generating function $\\tilde{F} {x}$ of unlabeled $F$ -structures may both be computed from the cycle index ${F}$ of an ordinary species $F$ by simple substitutions.",
"In the $\\Gamma $ -species context, we may perform similar substitutions to derive analogous generating functions.",
"The exponential generating function $F_{\\gamma } {x}$ of labeled $\\gamma $ -invariant $F$ -structures is $F_{\\gamma } {x} = {\\Gamma }{F}{\\gamma }{x, 0, 0, \\dots }.$ The ordinary generating function $\\tilde{F}_{\\gamma } {x}$ of unlabeled $\\gamma $ -invariant $F$ -structures is $\\tilde{F}_{\\gamma } {x} = {\\Gamma }{F}{\\gamma }{x, x^{2}, x^{3}, \\dots }.$ These theorems follow directly from eq:ciegf,eq:ciogf, thinking of $F_{\\gamma } {x}$ and $\\widetilde{F_{\\gamma } {x}}$ as enumerating the combinatorial class of $F$ -structures which are invariant under $\\gamma $ .",
"Note that the notion of `unlabeled $\\gamma $ -invariant $F$ -structures' is always well-defined precisely because def:gspecies requires that the action of $\\Gamma $ commutes with transport of structures.",
"From these results and thm:qsci, we can the conclude: The exponential generating function $F {x}$ of labeled ${F}{\\Gamma }$ -structures is $F {x} = \\frac{1}{{\\Gamma }} \\sum _{\\gamma \\in \\Gamma } F_{\\gamma } {x}.$ Similarly, The ordinary generating function $\\tilde{F} {x}$ of unlabeled ${F}{\\Gamma }$ -structures is $\\tilde{F} {x} = \\frac{1}{{\\Gamma }} \\sum _{\\gamma \\in \\Gamma } \\tilde{F}_{\\gamma } {x}.$ Note also that all of the above extends naturally into the multisort species context.",
"We will use this extensively in c:ktrees.",
"It also extends naturally to weighted contexts, but we will not apply this extension here."
],
[
"Introduction",
"We first apply the theory of quotient species to the enumeration of bipartite blocks.",
"A bicolored graph is a graph $\\Gamma $ each vertex of which has been assigned one of two colors (here, black and white) such that each edge connects vertices of different colors.",
"A bipartite graph (sometimes called bicolorable) is a graph $\\Gamma $ which admits such a coloring.",
"There is an extensive literature about bicolored and bipartite graphs, including enumerative results for bicolored graphs [12], bipartite graphs both allowing [8] and prohibiting [13] isolated points, and bipartite blocks [11].",
"However, this final enumeration was previously completed only in the labeled case.",
"By considering the problem in light of the theory of $\\Gamma $ -species, we develop a more systematic understanding of the structural relationships between these various classes of graphs, which allows us to enumerate all of them in both labeled and unlabeled settings.",
"Throughout this chapter, we denote by ${BC}$ the species of bicolored graphs and by ${BP}$ the species of bipartite graphs.",
"The prefix ${C}$ will indicate the connected analogue of such a species.",
"We are motivated by the graph-theoretic fact that each connected bipartite graph may be identified with exactly two bicolored graphs which are color-dual.",
"In other words, a connected bipartite graph is (by definition or by easy exercise, depending on your approach) an orbit of connected bicolored graphs under the action of 2 where the nontrivial element $\\tau $ reverses all vertex colors.",
"We will hereafter treat all the various species of bicolored graphs as 2-species with respect to this action and use the theory developed in s:quot to pass to bipartite graphs.",
"Although the theory of multisort species presented in s:mult is in general well-suited to the study of colored graphs, we will not need it here.",
"The restrictions that vertex colorings place on automorphisms of bicolored graphs are simple enough that we can deal with them directly."
],
[
"Bicolored graphs",
"We begin our investigation by directly computing the 2-cycle index for the species ${BC}$ of bicolored graphs with the color-reversing 2-action described previously.",
"We will then use various methods from the species algebra of c:species to pass to various other species."
],
[
"Computing ${{2}}{{BC}}{e}$",
"We construct the cycle index for the species ${BC}$ of bicolored graphs in the classical way, which in light of our 2-action will give ${{2}}{{BC}}{e}$ .",
"Recall the formula for the cycle index of a $\\Gamma $ -species in eq:gcycinddef: ${\\Gamma }{F}{\\gamma } = \\sum _{n \\ge 0} \\frac{1}{n!}",
"\\sum _{\\sigma \\in {n}} {\\gamma \\cdot F {\\sigma }} p_{\\sigma }.$ Thus, for each $n > 0$ and each permutation $\\pi \\in {n}$ , we must count bicolored graphs on ${n}$ for which $\\pi $ is a color-preserving automorphism.",
"To simplify some future calculations, we omit empty graphs and define ${BC} {\\varnothing } = \\varnothing $ .",
"We note that the number of such graphs in fact depends only on the cycle type $\\lambda \\vdash n$ of the permutation $\\pi $ , so we can use the cycle index formula in eq:cycinddefpart interpreted as a $\\Gamma $ -cycle index identity.",
"Fix some $n \\ge 0$ and let $\\lambda \\vdash n$ .",
"We wish to count bicolored graphs for which a chosen permutation $\\pi $ of cycle type $\\lambda $ is a color-preserving automorphism.",
"Each cycle of the permutation must correspond to a monochromatic subset of the vertices, so we may construct graphs by drawing bicolored edges into a given colored vertex set.",
"If we draw some particular bicolored edge, we must also draw every other edge in its orbit under $\\pi $ if $\\pi $ is to be an automorphism of the graph.",
"Moreover, every bicolored graph for which $\\pi $ is an automorphism may be constructed in this way Therefore, we direct our attention first to counting these edge orbits for a fixed coloring; we will then count colorings with respect to these results to get our total cycle index.",
"Consider an edge connecting two cycles of lengths $m$ and $n$ ; the length of its orbit under the permutation is ${m, n}$ , so the number of such orbits of edges between these two cycles is $mn / {m, n} = \\gcd {m, n}$ .",
"For an example in the case $m = 4, n = 2$ , see fig:exbcecycle.",
"The number of orbits for a fixed coloring is then $\\sum \\gcd {m, n}$ where the sum is over the multiset of all cycle lengths $m$ of white cycles and $n$ of black cycles in the permutation $\\pi $ .",
"We may then construct any possible graph fixed by our permutation by making a choice of a subset of these cycles to fill with edges, so the total number of such graphs is $\\prod 2^{\\gcd {m, n}}$ for a fixed coloring.",
"Figure: An edge ee (solid) between two cycles of lengths 4 and 2 in a permutation and that edge's orbit (dashed)We now turn our attention to the possible colorings of the graph which are compatible with a permutation of specified cycle type $\\lambda $ .",
"We split our partition into two subpartitions, writing $\\lambda = \\mu \\cup \\nu $ , where partitions are treated as multisets and $\\cup $ is the multiset union, and designate $\\mu $ to represent the white cycles and $\\nu $ the black.",
"Then the total number of graphs fixed by such a permutation with a specified decomposition is ${\\mu , \\nu } = \\prod _{{i \\in \\mu \\\\ j \\in \\nu }} 2^{\\gcd {i, j}}$ where the product is over the elements of $\\mu $ and $\\lambda $ taken as multisets.",
"However, since $\\mu $ and $\\nu $ represent white and black cycles respectively, it is important to distinguish which cycles of $\\lambda $ are taken into each.",
"The $\\lambda _{i}$ $i$ -cycles of $\\lambda $ can be distributed into $\\mu $ and $\\nu $ in $\\binom{\\lambda _{i}}{\\mu _{i}} = \\lambda _{i}!",
"/ {\\mu _{i}!",
"\\nu _{i}!",
"}$ ways, so in total there are $\\prod _{i} \\lambda _{i}!",
"/ {\\mu _{i}!",
"\\nu _{i}!}",
"= z_{\\lambda } / {z_{\\mu } z_{\\nu }}$ decompositions.",
"Thus, ${\\lambda } = \\frac{z_{\\lambda }}{z_{\\mu } z_{\\nu }} {\\mu , \\nu } = \\sum _{\\mu \\cup \\nu = \\lambda } \\frac{z_{\\lambda }}{z_{\\mu } z_{\\nu }} \\prod _{{i \\in \\mu \\\\ j \\in \\nu }} 2^{\\gcd {i, j}}.$ Therefore we conclude: ${{2}}{{BC}}{e} = \\sum _{n > 0} \\sum _{{\\mu , \\nu \\\\ \\mu \\cup \\nu \\vdash n}} \\frac{p_{\\mu \\cup \\nu }}{z_{\\mu } z_{\\nu }} \\prod _{i, j} 2^{\\gcd {\\mu _{i}, \\nu _{j}}}$ Explicit formulas for the generating function for unlabeled bicolored graphs were obtained in [12] using conventional Pólya-theoretic methods.",
"Conceptually, this enumeration in fact largely mirrors our own.",
"Harary uses the algebra of the classical cycle index of the `line groupThe line group of a graph is the group of permutations of edges induced by permutations of vertices.'",
"of the complete bicolored graph of which any given bicolored graph is a spanning subgraph.",
"He then enumerates orbits of edges under these groups using the Pólya enumeration theorem.",
"This is clearly analogous to our procedure, which enumerates the orbits of edges under each specific permutation of vertices."
],
[
"Calculating ${{2}}{{BC}}{\\tau }$",
"Recall that the nontrivial element of $\\tau \\in {2}$ acts on bicolored graphs by reversing all colors.",
"We again consider the cycles in the vertex set ${n}$ induced by a permutation $\\pi \\in {n}$ and use the partition $\\lambda $ corresponding to the cycle type of $\\pi $ for bookkeeping.",
"We then wish to count bicolored graphs on ${n}$ for which $\\tau \\cdot \\pi $ is an automorphism, which is to say that $\\pi $ itself is a color-reversing automorphism.",
"Once again, the number of bicolored graphs for which $\\pi $ is a color-reversing automorphism is in fact dependent only on the cycle type $\\lambda $ .",
"Each cycle of vertices must be color-alternating and hence of even length, so our partition $\\lambda $ must have only even parts.",
"Once this condition is satisfied, edges may be drawn either within a single cycle or between two cycles, and as before if we draw in any edge we must draw in its entire orbit under $\\pi $ (since $\\pi $ is to be an automorphism of the underlying graph).",
"Moreover, all graphs for which $\\pi $ is a color-reversing automorphism and with a fixed coloring may be constructed in this way, so it suffices to count such edge orbits and then consider how colorings may be assigned.",
"Consider a cycle of length $2n$ ; we hereafter describe such a cycle as having semilength $n$ .",
"There are exactly $n^{2}$ possible white-black edges in such a cycle.",
"If $n$ is odd, diametrically opposed vertices have opposite colors, so we can have an edge of length $l = n$ (in the sense of connecting two vertices which are $l$ steps apart in the cycle), and in such a case the orbit length is exactly $n$ and there is exactly one orbit.",
"See fig:exbctincycd for an example of this case.",
"However, if $n$ is odd but $l \\ne n$ , the orbit length is $2n$ , so the number of such orbits is $\\frac{n^{2} - n}{2n}$ .",
"Hence, the total number of orbits for $n$ odd is $\\frac{n^2 + n}{2n} = {\\frac{n}{2}}$ .",
"Similarly, if $n$ is even, all orbits are of length $2n$ , so the total number of orbits is $\\frac{n^{2}}{2n} = \\frac{n}{2} = {\\frac{n}{2}}$ also.",
"See fig:exbctincyce for an example of each of these cases.",
"Figure: Both types of intra-cycle edges and their orbits on a typical color-alternating 6-cycleNow consider an edge to be drawn between two cycles of semilengths $m$ and $n$ .",
"The total number of possible white-black edges is $2mn$ , each of which has an orbit length of ${2m, 2n} = 2 {m, n}$ .",
"Hence, the total number of orbits is $2mn / {2 {m, n}} = \\gcd {m, n}$ .",
"Figure: An edge ee and its orbit between color-alternating cycles of semilengths 2 and 1All together, then, the number of orbits for a fixed coloring of a permutation of cycle type $2 \\lambda $ (denoting the partition obtained by doubling every part of $\\lambda $ ) is $\\sum _{i} {\\frac{\\lambda _{i}}{2}} + \\sum _{i < j} \\gcd {\\lambda _{i}, \\lambda _{j}}$ .",
"All valid bicolored graphs for a fixed coloring for which $\\pi $ is a color-preserving automorphism may be obtained uniquely by making some choice of a subset of this collection of orbits, just as in ss:ecibc.",
"Thus, the total number of possible graphs for a given vertex coloring is $\\prod _{i} 2^{{\\frac{\\lambda _{i}}{2}}} \\prod _{i < j} 2^{\\gcd {\\lambda _{i}, \\lambda _{j}}},$ which we note is independent of the choice of coloring.",
"For a partition $2\\lambda $ with $l {\\lambda }$ cycles, there are then $2^{l {\\lambda }}$ colorings compatible with our requirement that each cycle is color-alternating, which we multiply by the previous to obtain the total number of graphs for all permutations $\\pi $ with cycle type $2 \\lambda $ .",
"Therefore we conclude: ${{2}}{{BC}}{\\tau } = \\sum _{{n > 0 \\\\ \\text{$n$ even}}} \\sum _{\\lambda \\vdash \\frac{n}{2}} 2^{l {\\lambda }} \\frac{p_{2 \\lambda }}{z_{2 \\lambda }} \\prod _{i} 2^{{\\frac{\\lambda _{i}}{2}}} \\prod _{i < j} 2^{\\gcd {\\lambda _{i}, \\lambda _{j}}}$"
],
[
"Connected bicolored graphs",
"As noted in the introduction of this section, we may pass from bicolored to bipartite graphs by taking a quotient under the color-reversing action of 2 only in the connected case.",
"Thus, we must pass from the species ${BC}$ to the species ${CBC}$ of connected bicolored graphs to continue.",
"It is a standard principle of graph enumeration that a graph may be decomposed uniquely into (and thus species-theoretically identified with) the set of its connected components.",
"We must, of course, require that the component structures are nonempty to ensure that the construction is well-defined, as discussed in s:specalg.",
"This same relationship holds in the case of bicolored graphs.",
"Thus, the species ${BC}$ of nonempty bicolored graphs is the composition of the species ${CBC}$ of nonempty connected bicolored graphs into the species ${E}^{+} = {E} - 1$ of nonempty sets: $ {BC} = {E}^{+} \\circ {CBC} $ Reversing the colors of a bicolored graph is done simply by reversing the colors of each of its connected components independently; thus, once we trivially extend the species ${E}^{+}$ to an 2-species by applying the trivial action, eq:bcdecomp holds as an identity of 2-species for the color-reversing 2-action described previously.",
"To use the decomposition in eq:bcdecomp to derive the 2-cycle index for ${CBC}$ , we must invert the 2-species composition into ${E}^{+}$ .",
"In the context of the theory of virtual species, this is possible; we write $:= {{E} - 1}^{{-1}}$ to denote this virtual species.",
"We can derive from [3] that its cycle index is ${} = \\sum _{k \\ge 1} \\frac{\\mu {k}}{k} \\log {1 + p_{k}}$ where $\\mu $ is the Möbius function.",
"We can then rewrite eq:bcdecomp as ${CBC} = \\circ {BC}$ It then follows immediately from thm:gspeccompci that $ {{2}}{{CBC}} = {} \\circ {{2}}{{BC}} $"
],
[
"Bipartite graphs",
"As we previously observed, connected bipartite graphs are naturally identified with orbits of connected bicolored graphs under the color-reversing action of 2.",
"Thus, ${CBP} = {{CBC}}{{2}}.$ By application of thm:qsci, we can then directly compute the cycle index of ${CBP}$ in terms of previous results: ${{CBP}} = {{2}}{{CBC}} = \\frac{1}{2} {{{2}}{{CBC}}{e} + {{2}}{{CBC}}{\\tau }}.$ Finally, to reach a result for the general bipartite case, we return to the graph-theoretic composition relationship previously considered in s:cbc: ${BP} = {E} \\circ {CBP}.$ This time, we need not invert the composition, so the cycle-index calculation is simple: ${{BP}} = {{E}} \\circ {{CBP}}.$ A generating function for labeled bipartite graphs was obtained first in [13] and later in [8]; the latter uses Pólya-theoretic methods to calculate the cycle index of what in modern terminology would be the species of edge-labeled complete bipartite graphs."
],
[
"Nonseparable graphs",
"We now turn our attention to the notions of block decomposition and nonseparable graphs.",
"A graph is said to be nonseparable if it is vertex-2-connected (that is, if there exists no vertex whose removal disconnects the graph); every connected graph then has a canonical `decomposition'Note that this decomposition does not actually partition the vertices, since many blocks may share a single cut-point, a detail which significantly complicates but does not entirely preclude species-theoretic analysis.",
"into maximal nonseparable subgraphs, often shortened to blocks.",
"In the spirit of our previous notation, we we will denote by ${NBP}$ the species of nonseparable bipartite graphs, our object of study.",
"The basic principles of block enumeration in terms of automorphisms and cycle indices of permutation groups were first identified and exploited in [22].",
"In [3], a theory relating a specified species $B$ of nonseparable graphs to the species $C_{B}$ of connected graphs whose blocks are in $B$ is developed using similar principles.",
"It is apparent that the class of nonseparable bipartite graphs is itself exactly the class of blocks that occur in block decompositions of connected bipartite graphs; hence, we apply that theory here to study the species ${NBP}$ .",
"From [3] we obtain ${NBP} = {CBP} {{CBP}^{\\bullet {-1}}} + X \\cdot {{NBP}} - X,$ where by [3] we have ${{NBP}} = {\\frac{X}{{CBP}^{\\bullet {-1}}}}.$ We have already calculated the cycle index for the species ${CBP}$ , so the calculation of the cycle index of ${NBP}$ is now simply a matter of algebraic expansion.",
"A generating function for labeled bipartite blocks was given in [11], where their analogue of eq:nbpexp for the labeled exponential generating function for blocks comes from [5].",
"However, we could locate no corresponding unlabeled enumeration in the literature.",
"The numbers of labeled and unlabeled nonseparable bipartite graphs for $n \\le 10$ as calculated using our method are given in tab:bpblocks."
],
[
"$k$ -trees",
"Trees and their generalizations have played an important role in the literature of combinatorial graph theory throughout its history.",
"The multi-dimensional generalization to so-called `$k$ -trees' has proved to be particularly fertile ground for both research problems and applications.",
"The class ${k}$ of $k$ -trees (for $k \\in {N}$ ) may be defined recursively: The complete graph on $k$ vertices ($K_{k}$ ) is a $k$ -tree, and any graph formed by adding a single vertex to a $k$ -tree and connecting that vertex by edges to some existing $k$ -clique (that is, induced $k$ -complete subgraph) of that $k$ -tree is a $k$ -tree.",
"The graph-theoretic notion of $k$ -trees was first introduced in 1968 in [10]; vertex-labeled $k$ -trees were quickly enumerated in the following year in both [19] and [2].",
"The special case $k=2$ has been especially thoroughly studied; enumerations are available in the literature for edge- and triangle-labeled 2-trees in [20], for plane 2-trees in [21], and for unlabeled 2-trees in [10] and [9].",
"In 2001, the theory of species was brought to bear on 2-trees in [6], resulting in more explicit formulas for the enumeration of unlabeled 2-trees.",
"An extensive literature on other properties of $k$ -trees and their applications has also emerged; Beineke and Pippert claim in [1] that “[t]here are now over 100 papers on various aspects of $k$ -trees”.",
"However, no general enumeration of unlabeled $k$ -trees appears in the literature to date.",
"To begin, we establish two definitions for substructures of $k$ -trees which we will use extensively in our analysis.",
"A hedron of a $k$ -tree is a ${k+1}$ -clique and a front is a $k$ -clique.",
"We will frequently describe $k$ -trees as assemblages of hedra attached along their fronts rather than using explicit graph-theoretic descriptions in terms of edges and vertices, keeping in mind that the structure of interest is graph-theoretic and not geometric.",
"The recursive addition of a single vertex and its connection by edges to an existing $k$ -clique in def:ktree is then interpreted as the attachment of a hedron to an existing one along some front, identifying the $k$ vertices they have in common.",
"The analogy to the recursive definition of conventional trees is clear, and in fact the class $\\mathfrak {a}$ of trees may be recovered by setting $k = 1$ .",
"For higher $k$ , the structures formed are still distinctively tree-like; for example, 2-trees are formed by gluing triangles together along their edges without forming loops of triangles (see fig:ex2tree), while 3-trees are formed by gluing tetrahedra together along their triangular faces without forming loops of tetrahedra.",
"Figure: A (vertex-labeled) 2-treeIn graph-theoretic contexts, it is conventional to label graphs on their vertices and possibly their edges.",
"However, for our purposes, it will be more convenient to label hedra and fronts.",
"Throughout, we will treat the species ${k}$ of $k$ -trees as a two-sort species, with $X$ -labels on the hedra and $Y$ -labels on their fronts; in diagrams, we will generally use capital letters for the hedron-labels and positive integers for the front-labels (see fig:exlab2tree).",
"Figure: A (hedron-and-front–labeled) 2-tree"
],
[
"The dissymmetry theorem for $k$ -trees",
"Studies of tree-like structures—especially those explicitly informed by the theory of species—often feature decompositions based on dissymmetry, which allow enumerations of unrooted structures to be recharacterized in terms of rooted structures.",
"For example, as seen in [3], the species $\\mathfrak {a}$ of trees and ${A} = {\\mathfrak {a}}$ of rooted trees are related by the equation ${A} + {E}_{2} {{A}} = \\mathfrak {a} + {A}^{2}$ where the proof hinges on a recursive structural decomposition of trees.",
"In this case, the species ${A}$ is relatively easy to characterize explicitly, so this equation serves to characterize the species $\\mathfrak {a}$ , which would be difficult to do directly.",
"A similar theorem holds for $k$ -trees.",
"The species ${k}$ and ${k}$ of $k$ -trees rooted at hedra and fronts respectively, ${k}$ of $k$ -trees rooted at a hedron with a designated front, and ${k}$ of unrooted $k$ -trees are related by the equation ${k} + {k} = {k} + {k}$ as an isomorphism of species.",
"We give a bijective, natural map from ${{k} + {k}}$ -structures on the left side to ${{k} + {k}}$ -structures on the right side.",
"Define a $k$ -path in a $k$ -tree to be a non-self-intersecting sequence of consecutively adjacent hedra and fronts, and define the length of a $k$ -path to be the total number of hedra and fronts along it.",
"Note that the ends of every maximal $k$ -path in a $k$ -tree are fronts.",
"It is easily verified, as in [16], that every $k$ -tree has a unique center clique (either a hedron or a front) which is the midpoint of every longest $k$ -path (or, equivalently, has the greatest $k$ -eccentricity, defined appropriately).",
"An ${{k} + {k}}$ -structure on the left-hand side of the equation is a $k$ -tree $T$ rooted at some clique $c$ , which is either a hedron or a front.",
"Suppose that $c$ is the center of $T$ .",
"We then map $T$ to its unrooted equivalent in ${k}$ on the right-hand side.",
"This map is a natural bijection from its preimage, the set of $k$ -trees rooted at their centers, to ${k}$ , the set of unrooted $k$ -trees.",
"Now suppose that the root clique $c$ of the $k$ -tree $T$ is not the center, which we denote $C$ .",
"Identify the clique $c^{\\prime }$ which is adjacent to $c$ along the $k$ -path from $c$ to $C$ .",
"We then map the $k$ -tree $T$ rooted at the clique $c$ to the same tree $T$ rooted at both $c$ and its neighbor $c^{\\prime }$ .",
"This map is also a natural bijection, in this case from the set of $k$ -trees rooted at vertices which are not their centers to the set ${k}$ of $k$ -trees rooted at an adjacent hedron-front pair.",
"The combination of these two maps then gives the desired isomorphism of species in eq:dissymk.",
"In general we will reformulate the dissymmetry theorem as follows: For the various forms of the species ${k}$ as above, we have ${k} = {k} + {k} - {k}.$ as an isomorphism of ordinary species.",
"This species subtraction is well-defined in the sense of def:specdif, since the species ${k}$ embeds in the species ${k} + {k}$ by the centering map described in the proof of thm:dissymk.",
"Essentially, eq:dissymkreform identifies each unrooted $k$ -tree with itself rooted at its center simplex.",
"thm:dissymk and the consequent eq:dissymkreform allow us to reframe enumerative questions about generic $k$ -trees in terms of questions about $k$ -trees rooted in various ways.",
"However, the rich internal symmetries of large cliques obstruct direct analysis of these rooted structures.",
"We need to break these symmetries to proceed."
],
[
"Symmetry-breaking",
"In the case of the species ${A} = {{1}}$ of rooted trees, we may obtain a simple recursive functional equation [3]: ${A} = X \\cdot {E} {{A}}.$ This completely characterizes the combinatorial structure of the class of trees.",
"However, in the more general case of $k$ -trees, no such simple relationship obtains; attached to a given hedron is a collection of sets of hedra (one such set per front), but simply specifying which fronts to attach to which does not fully specify the attachings, and the structure of that collection of sets is complex.",
"We will break this symmetry by adding additional structure which we can later remove using the theory of quotient species.",
"Let $h_{1}$ and $h_{2}$ be two hedra joined at a front $f$ , hereafter said to be adjacent.",
"Each other front of one of the hedra shares $k-1$ vertices with $f$ ; we say that two fronts $f_{1}$ of $h_{1}$ and $f_{2}$ of $h_{2}$ are mirror with respect to $f$ if these shared vertices are the same, or equivalently if $f_{1} \\cap f = f_{2} \\cap f$ .",
"Let $T$ be a coherently-oriented $k$ -tree with two hedra $h_{1}$ and $h_{2}$ joined at a front $f$ .",
"Then there is exactly one front of $h_{2}$ mirror to each front of $h_{1}$ with respect to their shared front $f$ .",
"Define an orientation of a hedron to be a cyclic ordering of the set of its fronts and an orientation of a $k$ -tree to be a choice of orientation for each of its hedra.",
"If two oriented hedra share a front, their orientations are compatible if they correspond under the mirror bijection.",
"Then an orientation of a $k$ -tree is coherent if every pair of adjacent hedra is compatibly-oriented.",
"See fig:exco2tree for an example.",
"Note that every $k$ -tree admits many coherent orientations—any one hedron of the $k$ -tree may be oriented freely, and a unique orientation of the whole $k$ -tree will result from each choice of such an orientation of one hedron.",
"We will denote by ${k}$ the species of coherently-oriented $k$ -trees.",
"By shifting from the general $k$ -tree setting to that of coherently-oriented $k$ -trees, we break the symmetry described above.",
"If we can now establish a group action on ${k}$ whose orbits are generic $k$ -trees we can use the theory of quotient species to extract the generic species ${k}$ .",
"First, however, we describe an encoding procedure which will make future work more convenient.",
"Figure: A coherently-oriented 2-tree"
],
[
"Bicolored tree encoding",
"Although $k$ -trees are graphs (and hence made up simply of edges and vertices), their structure is more conveniently described in terms of their simplicial structure of hedra and fronts.",
"Indeed, if each hedron has an orientation of its faces and we choose in advance which hedra to attach to which by what fronts, the requirement that the resulting $k$ -tree be coherently oriented is strong enough to characterize the attaching completely.",
"We thus pass from coherently-oriented $k$ -trees to a surrogate structure which exposes the salient features of this attaching structure more clearly—structured bicolored trees in the spirit of the $R, S$ -enriched bicolored trees of [3].",
"A ${{C}_{k+1}, {E}}$ -enriched bicolored tree is a bicolored tree each black vertex of which carries a ${C}_{k+1}$ -structure (that is, a cyclic ordering on $k+1$ elements) on its white neighbors.",
"(The ${E}$ -structure on the black neighbors of each white vertex is already implicit in the bicolored tree itself.)",
"For later convenience, we will sometimes call such objects $k$ -coding trees, and we will denote by ${k}$ the species of such $k$ -coding trees.",
"We now define a map $\\beta : {k} {n} \\rightarrow {k} {n}$ .",
"For a given coherently-oriented $k$ -tree $T$ with $n$ hedra: For every hedron of $T$ construct a black vertex and for every front a white vertex, assigning labels appropriately.",
"For every black-white vertex pair, construct a connecting edge if the white vertex represents a front of the hedron represented by the black vertex.",
"Finally, enrich the collection of neighbors of each black vertex with a ${C}_{k+1}$ -structure inherited directly from the orientation of the $k$ -tree $T$ .",
"The resulting object $\\beta {T}$ is clearly a $k$ -coding tree with $n$ black vertices.",
"We can recover a $T$ from $\\beta {T}$ by following the reverse procedure.",
"For an example, see fig:exbctree, which shows the 2-coding tree associated to the coherently-oriented 2-tree of fig:exco2tree.",
"Note that, for clarity, we have rendered the black vertices (corresponding to hedra) with squares.",
"Figure: A C k+1 ,E{{C}_{k+1}, {E}}-enriched bicolored tree encoding a coherently-oriented 2-treeThe map $\\beta $ induces an isomorphism of species ${k} \\simeq {k}$ .",
"It is clear that $\\beta $ sends each coherently-oriented $k$ -tree to a unique $k$ -coding tree, and that this map commutes with permutations on the label sets (and thus is categorically natural).",
"To show that $\\beta $ induces a species isomorphism, then, we need only show that $\\beta $ is a surjection onto ${k} {n}$ for each $n$ .",
"Throughout, we will say `$F$ and $G$ have contact of order $n$ ' when the restrictions $F_{\\le n}$ and $G_{\\le n}$ of the species $F$ and $G$ to label sets of cardinality at most $n$ are naturally isomorphic.",
"First, we note that there are exactly $k!$ coherently-oriented $k$ -trees with one hedron—one for each cyclic ordering of the $k+1$ front labels.",
"There are also $k!$ coding trees with one black vertex, and the encoding $\\beta $ is clearly a natural bijection between these two sets.",
"Thus, the species ${k}$ of coherently-oriented $k$ -trees and ${k}$ of $k$ -coding trees have contact of order 1.",
"Now, by way of induction, suppose ${k}$ and ${k}$ have contact of order $n \\ge 1$ .",
"Let $C$ be a $k$ -coding tree with $n+1$ black vertices.",
"Then let $C_{1}$ and $C_{2}$ be two distinct sub-$k$ -coding trees of $C$ , each obtained from $C$ by removing one black node which has only one white neighbor which is not a leaf.",
"Then, by hypothesis, there exist coherently-oriented $k$ -trees $T_{1}$ and $T_{2}$ with $n$ hedra such that $\\beta {T_{1}} = C_{1}$ and $\\beta {T_{2}} = C_{2}$ .",
"Moreover, $\\beta {T_{1} \\cap T_{2}} = \\beta {T_{1}} \\cap \\beta {T_{2}}$ , and this $k$ -coding tree has $n-1$ black vertices, so $T_{1} \\cap T_{2}$ has $n-1$ hedra.",
"Thus, $T = T_{1} \\cup T_{2}$ is a coherently-oriented $k$ -tree with $n+1$ black hedra, and $\\beta {T} = C$ as desired.",
"Thus, $\\beta ^{-1} {\\beta {T_{1}} \\cup \\beta {T_{2}}} = T_{1} \\cup T_{2} = T$ , and hence ${k}$ and ${k}$ have contact of order $n+1$ .",
"Thus, ${k}$ and ${k}$ are isomorphic as species; however, $k$ -coding trees are much simpler than coherently-oriented $k$ -trees as graphs.",
"Moreover, $k$ -coding trees are doubly-enriched bicolored trees as in [3], for which the authors of that text develop a system of functional equations which fully characterizes the cycle index of such a species.",
"We thus will proceed in the following sections with a study of the species ${k}$ , then lift our results to the $k$ -tree context."
],
[
"Functional decomposition of $k$ -coding trees",
"With the encoding $\\beta : {k} \\rightarrow {k}$ , we now have direct graph-theoretic access to the attaching structure of coherently-oriented $k$ -trees.",
"We therefore turn our attention to the $k$ -coding trees themselves to produce a recursive decomposition.",
"As with $k$ -trees, we will study rooted versions of the species ${k}$ of $k$ -coding trees first, then use dissymmetry to apply the results to unrooted enumeration.",
"Let ${k}$ denote the species of $k$ -coding trees rooted at black vertices, ${k}$ denote the species of $k$ -coding trees rooted at white vertices, and ${k}$ denote the species of $k$ -coding trees rooted at edges (that is, at adjacent black-white pairs).",
"By construction, a ${k}$ -structure consists of a single $X$ -label and a cyclically-ordered ${k+1}$ -set of ${k}$ -structures.",
"See fig:ctxconst for an example of this construction.",
"Figure: An example 4-structure, rooted at the XX-vertex.Similarly, a ${k}$ -structure essentially consists of a single $Y$ -label and a (possibly empty) set of ${k}$ -structures, but with some modification.",
"Every white neighbor of the black root of a ${k}$ -structure is labeled in the construction above, but the white parent of a ${k}$ -structure in this recursive decomposition is already labeled.",
"Thus, the structure around a black vertex which is a child of a white vertex consists of an $X$ label and a linearly-ordered $k$ -set of ${k}$ -structures.",
"Thus, a ${k}$ -structure consists of a $Y$ -label and a set of pairs of an $X$ label and an ${L}_{k}$ -structure of ${k}$ -structures.",
"We note here for conceptual consistency that in fact ${L}_{k} = {{C}}_{k+1}$ for ${L}$ the species of linear orders and ${C}$ the species of cyclic orders and that ${{E}} = {E}$ for ${E}$ the species of sets; readers familiar with the $R, S$ -enriched bicolored trees of [3] will recognize echoes of their decomposition in these facts.",
"Finally, a ${k}$ -structure is simply an $X \\cdot {L}_{k} [big]{{k}}$ -structure as described above (corresponding to the black vertex) together with a ${k}$ -structure (corresponding to the white vertex).",
"For reasons that will become clear later, we note that we can incorporate the root white vertex into the linear order by making it last, thus representing a ${k}$ -structure instead as an $X \\cdot {L}_{k+1} [big]{{k}}$ -structure.",
"See fig:ctxyconst for an example of this construction.",
"Figure: An example 4-structure, rooted at the XX-vertex and the thick edge adjoining it.The various species of rooted $k$ -coding trees are therefore related by a system of functional equations: For the (ordinary) species ${k}$ of $X$ -rooted $k$ -coding trees, ${k}$ of $Y$ -rooted $k$ -coding trees, and ${k}$ of edge-rooted $k$ -coding trees, we have the functional relationships k = X Ck+1 [big]k k = Y E X Lk [big]k k = k X Lk [big]k = X Lk+1 [big]k as isomorphisms of ordinary species.",
"However, a recursive characterization of the various ordinary species of $k$ -coding trees is insufficient to characterize the species of $k$ -trees itself, since $k$ -coding trees represent $k$ -trees with coherent orientations."
],
[
"Generic $k$ -trees",
"To remove the additional structure of coherent orientation imposed on $k$ -trees before their conversion to $k$ -coding trees, we now apply the theory of $\\Gamma $ -species developed in s:quot.",
"In [6], the orientation-reversing action of 2 on $_{{3}}$ is exploited to study 2-trees species-theoretically.",
"We might hope to develop an analogous group action under which general $k$ -trees are naturally identified with orbits of coherently-oriented $k$ -trees under an action of ${k}$ .",
"Unfortunately: For $k \\ge 3$ , no transitive action of any group on the set $_{{k+1}}$ of cyclic orders on ${k+1}$ commutes with the action of ${k+1}$ that permutes labels.",
"We represent the elements of $_{{k+1}}$ as cyclic permutations on the alphabet ${k+1}$ ; then the action of ${k+1}$ that permutes labels is exactly the conjugation action on these permutations.",
"Consider an action of a group $G$ on $_{{k+1}}$ that commutes with this conjugation action.",
"Then, for any $g \\in G$ and any $c \\in _{{k+1}}$ , we have that $g \\cdot c = g \\cdot c c c^{-1} = c {g \\cdot c} c^{-1}$ and so $c$ and $g \\cdot c$ commute.",
"Thus, $c$ commutes with every element of its orbit under the action of $G$ .",
"But, for $k \\ge 3$ , not all elements of $_{{k+1}}$ commute, so the action is not transitive.",
"We thus cannot hope to attack the coherent orientations of $k$ -trees by acting directly on the cyclic orderings of fronts.",
"Accordingly, we cannot simply apply the results of ss:codecomp to compute a $\\Gamma $ -species ${k}$ with respect to some hypothetical action of a group $\\Gamma $ whose orbits correspond to generic $k$ -trees.",
"Instead, we will use the additional structure on rooted coherently-oriented $k$ -trees; with rooting, the cyclic orders around black vertices are converted into linear orders, for which there is a natural action of ${k+1}$ ."
],
[
"Group actions on $k$ -coding trees",
"We have noted previously that every labeled $k$ -tree admits exactly $k!$ coherent orientations.",
"Thus, there are $k!$ distinct $k$ -coding trees associated to each labeled $k$ -tree, which differ only in the ${C}_{k+1}$ -structures on their black vertices.",
"Consider a rooted $k$ -coding tree $T$ and a black vertex $v$ which is not the root vertex.",
"Then one white neighbor of $v$ is the `parent' of $v$ (in the sense that it lies on the path from $v$ to the root).",
"We thus can convert the cyclic order on the $k+1$ white neighbors of $v$ to a linear order by choosing the parent white neighbor to be last.",
"There is a natural, transitive, label-independent action of ${k+1}$ on the set of such linear orders which induces an action on the cyclic orders from which the linear orders are derived.",
"However, only elements of ${k+1}$ which fix $k+1$ will respect the structure around the black vertex we have chosen, since its parent white vertex must remain last.",
"In addition, if we simply apply the action of some $\\sigma \\in {k+1}$ to the order on white neighbors of $v$ , we change the coherently-oriented $k$ -tree $\\beta ^{-1} {T}$ to which $T$ is associated in such a way that it no longer corresponds to the same unoriented $k$ -tree.",
"Let $t$ denote the unoriented $k$ -tree associated to $\\beta ^{-1} {T}$ ; then there exists a coherent orientation of $t$ which agrees with orientation around $v$ induced by $\\sigma $ .",
"The $k$ -coding tree $T^{\\prime }$ corresponding to this new coherent orientation has the same underlying bicolored tree as $T$ but possibly different orders around its black vertices.",
"If we think of the $k$ -coding tree $T^{\\prime }$ as the image of $T$ under a global action of $\\sigma $ , orbits under all of ${}$ will be precisely the classes of $k$ -coding trees corresponding to all coherent orientations of specified $k$ -trees, allowing us to study unoriented $k$ -trees as quotients.",
"The orientation of $T^{\\prime }$ will be that obtained by applying $\\sigma $ at $v$ and then recursively adjusting the other cyclic orders so that fronts which were mirror are made mirror again.",
"This will ensure that the combinatorial structure of the underlying $k$ -tree $t$ is preserved.",
"Therefore, when we apply some permutation $\\sigma \\in {k+1}$ to the white neighbors of a black vertex $v$ , we must also permute the cyclic orders of the descendant black vertices of $v$ .",
"In particular, the permutation $\\sigma ^{\\prime }$ which must be applied to some immediate black descendant $v^{\\prime }$ of $v$ is precisely the permutation on the linear order of white neighbors of $v^{\\prime }$ induced by passing over the mirror bijection from $v^{\\prime }$ to $v$ , applying $\\sigma $ , and then passing back.",
"We can express this procedure in formulaic terms: If a permutation $\\sigma \\in {k+1}$ is applied to linearized orientation of a black vertex $v$ in rooted $k$ -coding tree, the permutation which must be applied to the linearized orientation a child black vertex $v^{\\prime }$ which was attached to the $i$ th white child of $v$ (with respect to the linear ordering induced by the orientation) to preserve the mirror relation is $\\rho _{i} {\\sigma }$ , where $\\rho _{i}$ is the map given by $\\rho _{i} {\\sigma }: a \\mapsto \\sigma {i + a} - \\sigma {i}$ in which all sums and differences are reduced to their representatives modulo $k+1$ in ${1, 2, \\dots , k+1}$ .",
"Let $v^{\\prime }$ denote a black vertex which is attached to $v$ by the white vertex 1, which we suppose to be in position $i$ in the linear order induced by the original orientation of $v$ .",
"Let 2 denote the white child of $v^{\\prime }$ which is $a$ th in the linear order induced by the original orientation around $v^{\\prime }$ .",
"It is mirror to the white child 3 of $v$ which is ${i+a}$ th in the linear order induced by the original orientation around $v$ .",
"After the action of $\\sigma $ is applied, vertex 3 is $\\sigma {i+a}$ th in the new linear order around $v$ .",
"We require that 2 is still mirror to 3, so we must move it to position $\\sigma {i + a} - \\sigma {i}$ when we create a new linear order around $v^{\\prime }$ .",
"This completes the proof.",
"This procedure is depicted in fig:rhoapp.",
"Figure: Application of a permutation σ\\sigma to the orientation of a non-root black vertex vv.The vertices 2 and 3 are mirror in the original orientation (lower set of edges), as shown by the arrows μ\\mu , so we must preserve this mirror relation when we apply σ\\sigma .The permutation σ\\sigma moves 3 from the i+a{i+a}th place to the σi+a\\sigma {i+a}th, so ρ i σ\\rho _{i} {\\sigma } must carry 2 from the aath place to the σi+a-σi{\\sigma {i+a} - \\sigma {i}}th.As an aside, we note that, although the construction $\\rho $ depends on $k$ , the value of $k$ will be fixed in any given context, so we suppress it in the notation.",
"Any $\\sigma $ which is to be applied to a non-root black vertex $v$ must of course fix $k+1$ .",
"We let $\\Delta : {k} \\rightarrow {k+1}$ denote the obvious embedding; then the image of $\\Delta $ is exactly the set of $\\sigma \\in {k+1}$ which fix $k+1$ .",
"We then have an action of ${k}$ on non-root black vertices induced by $\\Delta $ .",
"(Equivalently, we can think of ${k}$ as the subgroup of ${k+1}$ of permutations fixing $k+1$ , but the explicit notation $\\Delta $ will be of use in later formulas.)",
"In light of obs:funcdecompct, we now wish to adapt these ideas into explicit ${k}$ - and ${k+1}$ -actions on ${k}$ , ${k}$ , and ${k}$ whose orbits correspond to the various coherent orientations of single underlying rooted $k$ -trees.",
"In the case of a $Y$ -rooted $k$ -coding tree $T$ , if we declare that $\\sigma \\in {k}$ acts on $T$ by acting directly (as $\\Delta {\\sigma }$ ) on each of the black vertices immediately adjacent to the root and then applying $\\rho $ -derived permutations recursively to their descendants, orbits behave as expected.",
"The same ${k}$ -action serves equally well for edge-rooted $k$ -coding trees, where (for purposes of applying the action of some $\\sigma $ ) we can simply ignore the black vertex in the root.",
"However, if we begin with an $X$ -rooted $k$ -coding tree, the cyclic ordering of the white neighbors of the root black vertex has no canonical choice of linearization.",
"If we make an arbitrary choice of one of the $k+1$ available linearizations, and thus convert to an edge-rooted $k$ -coding tree, the full ${k+1}$ -action defined previously can be applied directly to the root vertex.",
"The orbit under this action of some edge-rooted $k$ -coding tree $T$ with a choice of linearization at the root then includes all possible linearizations of the root orders of all possible $X$ -rooted $k$ -coding trees corresponding to the different coherent orientations of a single $k$ -coding tree."
],
[
"$k$ -trees as quotients",
"Since these actions are label-independent, we may now treat ${k}$ and ${k}$ as ${k}$ -species and ${k}$ as an ${k+1}$ -species.",
"The ${k}$ - and ${k+1}$ -actions on ${k}$ are compatible, but we will make explicit reference to ${k}$ as an ${k}$ - or ${k+1}$ -species whenever it is important and not completely clear from context which we mean.",
"As a result of the above results, we can then relate the rooted $\\Gamma $ -species forms of ${k}$ to the various ordinary species forms of generic rooted $k$ -trees in thm:dissymk: For the various rooted forms of the ordinary species ${k}$ as in thm:dissymk and the various rooted $\\Gamma $ -species forms of ${k}$ as in obs:funcdecompct as ${k}$ - and ${k+1}$ -species, we have k = kk k = kk k = kk+1 as isomorphisms of ordinary species, where ${k}$ is an ${k}$ -species in eq:axyquot and an ${k+1}$ -species in eq:axquot.",
"As a result, we have explicit characterizations of all the rooted components of the original dissymmetry theorem, thm:dissymk.",
"To compute the cycle indices of these components (and thus the cycle index of ${k}$ itself), we need only compute the cycle indices of the various rooted ${k}$ species, which we will do using a combination of the functional equations in eq:ctfunc and explicit consideration of automorphisms."
],
[
"$k$ -coding trees: {{formula:cdcee90f-dcfd-432f-bcbd-f766e2edfd64}} and {{formula:97824b7f-f3ee-41aa-85d6-268eea1e6a3d}}",
"cor:dissymkreform of the dissymmetry theorem for $k$ -trees has a direct analogue in terms of cycle indices: For the various forms of the species ${k}$ as in s:dissymk, we have ${{k}} = {{k}} + {{k}} - {{k}}.$ Thus, we need to calculate the cycle indices of the three rooted forms of ${k}$ .",
"From thm:arootquot and by thm:qsci we obtain: For the various forms of the species ${k}$ as in s:dissymk and the various ${k}$ -species and ${k+1}$ -species forms of ${k}$ as in ss:actct, we have k = kk = 1k!",
"k kk k = kk = 1k!",
"k kk k = k+1k = 1k+1!",
"k+1 k+1k We thus need only calculate the various $\\Gamma $ -cycle indices for the ${k}$ -species and ${k+1}$ -species forms of ${k}$ and ${k}$ to complete our enumeration of general $k$ -trees.",
"In obs:funcdecompct, the functional equations for the ordinary species ${k}$ and ${k}$ both include terms of the form ${L}_{k} \\circ {k}$ .",
"The plethysm of ordinary species does have a generalization to $\\Gamma $ -species, as given in def:gspeccomp, but it does not correctly describe the manner in which ${k}$ acts on linear orders of ${k}$ -structures in these recursive decompositions.",
"Recall from s:quot that, for two $\\Gamma $ -species $F$ and $G$ , an element $\\gamma \\in \\Gamma $ acts on an ${F \\circ G}$ -structure (colloquially, `an $F$ -structure of $G$ -structures') by acting on the $F$ -structure and on each of the $G$ -structures independently.",
"In our action of ${k}$ , however, the actions of $\\sigma $ on the descendant ${k}$ -structures are not independent—they depend on the position of the structure in the linear ordering around the parent black vertex.",
"In particular, if $\\sigma $ acts on some non-root black vertex, then $\\rho _{i} {\\sigma }$ acts on the white vertex in the $i$ th place, where in general $\\rho _{i} {\\sigma } \\ne \\sigma $ .",
"Thus, we consider automorphisms of these ${k}$ -structures directly.",
"First, we consider the component species $X \\cdot {L}_{k} [big]{{k}}$ .",
"Let $B$ be a structure of the species $X \\cdot {L}_{k} [big]{{k}}$ .",
"Let $W_{i}$ be the ${k}$ -structure in the $i$ th position in the linear order.",
"Then some $\\sigma \\in {k}$ acts as an automorphism of $B$ if and only if, for each $i \\in {k+1}$ , we have $\\Delta ^{-1} {\\rho _{i} {\\Delta \\sigma }} W_{i} \\cong W_{\\sigma {i}}$ .",
"Recall that the action of $\\sigma \\in {k}$ is in fact the action of $\\Delta \\sigma \\in {k+1}$ .",
"The $X$ -label on the black root of $B$ is not affected by the application of $\\Delta \\sigma $ , so no conditions on $\\sigma $ are necessary to accommodate it.",
"However, the ${L}_{k}$ -structure on the white children of the root is permuted by $\\Delta \\sigma $ , and we apply to each of the $W_{i}$ 's the action of $\\Delta ^{-1} {\\rho _{i} {\\Delta \\sigma }}$ .",
"Thus, $\\sigma $ is an automorphism of $B$ if and only if the combination of applying $\\Delta \\sigma $ to the linear order and $\\Delta ^{-1} {\\rho _{i} {\\Delta \\sigma }}$ to each $W_{i}$ is an automorphism.",
"Since $\\sigma $ `carries' each $W_{i}$ onto $W_{\\sigma {i}}$ , we must have that $\\Delta ^{-1} {\\rho _{i} {\\Delta \\sigma }} W_{i} \\cong W_{\\sigma {i}}$ , as claimed.",
"That this suffices is clear.",
"Consider a structure $T$ of the ${k}$ -species ${k}$ and an element $\\sigma \\in {k}$ .",
"As discussed in ss:codecomp, $T$ is composed of a $Y$ -label and a set of $X \\cdot {L}_{k} [big]{{k}}$ -structures.",
"The permutation $\\sigma $ acts trivially on $Y$ and ${E}$ and acts on each of the component $X \\cdot {L}_{k} [big]{{k}}$ -structures independently.",
"For each of these component structures, by lem:ctyinvar, we have that $\\sigma $ is an automorphism if and only if $\\Delta \\sigma $ carries each ${k}$ -structure to its $\\Delta ^{-1} {\\rho _{i} {\\Delta \\sigma }}$ -image.",
"Thus, when constructing $\\sigma $ -invariant $X \\cdot {L}_{k} [big]{{k}}$ -structures, we must construct for each cycle of $\\sigma $ a ${k}$ -structure which is invariant under the application of all the permutations $\\Delta ^{-1} {\\rho _{i} {\\Delta \\sigma }}$ which will be applied to it along the cycle.",
"For $c$ the chosen cycle of $\\sigma $ , this permutation is $\\Delta ^{-1} {\\prod _{i \\in c} \\rho _{i} {\\Delta \\sigma }}$ , where the product is taken over any chosen linearization of the cyclic order of the terms in the cycle.",
"Once a choice of such a ${k}$ -structure for each cycle of $\\sigma $ is made, we can simply insert the structures into the ${L}_{k}$ -structure to build the desired $\\sigma $ -invariant $X \\cdot {L}_{k} [big]{{k}}$ -structure.",
"Accordingly: The ${k}$ -cycle index for the species ${k}$ is characterized by the recursive functional equation ${{k}}{{k}}{\\sigma } = p_{1} {y} \\\\\\times {{E}} \\circ [Big]{p_{1} {x} \\cdot \\prod _{c \\in C {\\sigma }} {{k}}{{k}} [Big]{\\Delta ^{-1} \\prod _{i \\in c} \\rho _{i} {\\Delta \\sigma }} {p_{{c}} {x}, p_{2 {c}} {x}, \\dots ; p_{{c}} {y}, p_{2 {c}} {y}, \\dots }}.$ where $C {\\sigma }$ denotes the set of cycles of $\\sigma $ (as a $k$ -permutation) and the inner product is taken with respect to any choice of linearization of the cyclic order of the elements of $c$ .",
"The situation for the ${k+1}$ -species ${k}$ is almost identical.",
"Recall from ss:actct that $\\sigma \\in {k+1}$ acts on a ${k}$ -structure $T$ by applying $\\sigma $ directly to the linear order on the $k+1$ white neighbors of the root black vertex and applying $\\rho $ -variants of $\\sigma $ recursively to their descendants.",
"Thus, we once again need only require that, along each cycle of $\\sigma $ , the successive white-vertex structures are pairwise isomorphic under the action of the appropriate $\\rho _{i} {\\sigma }$ .",
"Thus, we again need only choose for each cycle $c \\in C {\\sigma }$ a ${k}$ -structure which is invariant under $\\prod _{i \\in c} \\rho _{i} {\\sigma }$ .",
"Accordingly: The ${k+1}$ -cycle index for the species ${k}$ is given by ${{k+1}}{{k}}{\\sigma } = p_{1} {x} \\\\\\times \\prod _{c \\in C {\\sigma }} {{k}}{{k}} [Big]{\\prod _{i \\in c} \\rho _{i} {\\sigma }} {p_{{c}} {x}, p_{2 {c}} {x}, \\dots , p_{{c}} {y}, p_{2 {c}} {y}, \\dots }.$ under the same conditions as thm:ctyfuncci.",
"Terms of the form $\\prod _{i \\in c} \\rho _{i} {\\sigma }$ appear in eq:ctyfuncci,eq:ctxyfuncci.",
"For the simplification of calculations, we note here a two useful results about these products.",
"First, we observe that certain $\\rho $ -maps preserve cycle structure: Let $\\sigma \\in {k}$ be a permutation of which $i \\in {k}$ is a fixed point and let $\\lambda $ be the map sending each permutation in ${k}$ to its cycle type as a partition of $k$ .",
"Then $\\lambda {\\rho _{i} {\\sigma }} = \\lambda {\\sigma }$ .",
"Suppose $i + a \\in {k}$ is in an $l$ -cycle of $\\sigma $ .",
"Then i j a = i j - 1 i + a - i = i j - 2 i + i + a - i - i = i j - 2 2 i + a - 2 i $\\vdots $ = j i + a - j i But the values of ${\\rho _{i} {\\sigma }}^{j} {a} = \\sigma ^{j} {i + a} - \\sigma ^{j} {i}$ are all distinct for $j \\le l$ , since $i + a$ is in an $l$ -cycle and $i$ is a fixed point of $\\sigma $ .",
"Furthermore, ${\\rho _{i} {\\sigma }}^{l} {a} = \\sigma ^{l} {i + a} = i+a$ .",
"Thus, $a$ is in an $l$ -cycle of $\\rho _{i} {\\sigma }$ .",
"This establishes a length-preserving bijection between cycles of $\\rho _{i} {\\sigma }$ and cycles of $\\sigma $ , so their cycle types are equal.",
"But then we note that the products in the above theorems are in fact permutations obtained by applying such $\\rho $ -maps: Let $\\sigma \\in {k}$ be a permutation with a cycle $c$ .",
"Then $\\lambda {\\prod _{i \\in c} \\rho _{i} {\\sigma }}$ is determined by $\\lambda {\\sigma }$ and ${c}$ .",
"Let $c = {c_{1}, c_{2}, \\dots , c_{{c}}}$ .",
"First, we calculate: i = 1c ci = cc ...c2 c1 = cc ...c2 a c1 + a - c1 = cc ...c3 a c2 + c1 + a - c1 - c2 = cc ...c3 a 2 c1 + a - 2 c1 $\\vdots $ = a c c1 + a - c c1 = c1 c. But $c_{1}$ is a fixed point of $\\sigma ^{{c}}$ , so by the result of lem:rhofp, this has the same cycle structure as $\\sigma ^{{c}}$ , which in turn is determined by $\\lambda {\\sigma }$ and ${c}$ as desired.",
"From this and the fact that the terms of $X$ -degree 1 in all ${{k}}{{k}}$ and ${{k}}{{k+1}}$ are equal (to $p_{1} {x} p_{1} {y}^{k+1}$ ), we can conclude that: ${{k}}{{k}}{\\sigma }$ and ${{k}}{{k+1}}{\\sigma }$ are class functions of $\\sigma $ (that is, they are constant over permutations of fixed cycle type).",
"This will simplify computational enumeration of $k$ -trees significantly, since the number of partitions of $k$ grows exponentially while the number of permutations of ${k}$ grows factorially."
],
[
"$k$ -trees: {{formula:8f4c430e-a670-4f30-b5fa-a5a29044d5e0}}",
"We now have all the pieces in hand to apply thm:dissymkci to compute the cycle index of the species ${k}$ of general $k$ -trees.",
"eq:dissymkci characterizes the cycle index of the generic $k$ -tree species ${k}$ in terms of the cycle indices of the rooted species ${k}$ , ${k}$ , and ${k}$ ; thm:arootquot gives the cycle indices of these three rooted species in terms of the $\\Gamma $ -cycle indices ${{k}}{{k}}$ , ${{k}}{{k}}$ , and ${{k+1}}{{k}}$ ; and, finally, thm:ctyfuncci,thm:ctxyfuncci give these $\\Gamma $ -cycle indices explicitly.",
"By tracing the formulas in eq:ctyfuncci,eq:ctxyfuncci back through this sequence of functional relationships, we can conclude: [Cycle index for the species of $k$ -trees] For $\\mathfrak {a}_{k}$ the species of general $k$ -trees, ${{k}}{{k}}$ as in eq:ctyfuncci, and ${{k+1}}{{k}}$ as in eq:ctxyfuncci we have: k = 1k+1!",
"k+1 k+1k + 1k!",
"k kk - 1k!",
"k kk = k+1k + kk - kk.",
"eq:akci in fact represents a recursive system of functional equations, since the formulas for the $\\Gamma $ -cycle indices of ${k}$ and ${k}$ are recursive.",
"Computational methods can yield explicit enumerative results.",
"However, a bit of care will allow us to reduce the computational complexity of this problem significantly."
],
[
"Unlabeled enumeration and the generating function $\\tilde{\\mathfrak {a}}_{k} {x}$",
"eq:akci in thm:akci gives a recursive formula for the cycle index of the ordinary species ${k}$ of $k$ -trees.",
"The number of unlabeled $k$ -trees with $n$ hedra is historically an open problem, but by application of thm:ciogf the ordinary generating function counting such structures can be extracted from the cycle index ${{k}}$ .",
"Actually computing terms of the cycle index in order to derive the coefficients of the generating function is, however, a computationally expensive process, since the cycle index is by construction a power series in two infinite sets of variables.",
"The computational process can be simplified significantly by taking advantage of the relatively straightforward combinatorial structure of the structural decomposition used to derive the recursive formulas for the cycle index.",
"Recall from thm:gciogf that, for a $\\Gamma $ -species $F$ , the ordinary generating function $\\tilde{F}_{\\gamma } {x}$ counting unlabeled $\\gamma $ -invariant $F$ -structures is given by $\\tilde{F} [big]{\\gamma } {x} = {\\Gamma }{F}{\\gamma }{x, x^2, x^3, \\dots }$ and that the ordinary generating function for counting unlabeled ${F}{\\Gamma }$ -structures is given by $\\tilde{F} {x} = \\frac{1}{{\\Gamma }} \\sum _{\\gamma \\in \\Gamma } \\tilde{F} [big]{\\gamma } {x}.$ These formula admits an obvious multisort extension, but we in fact wish to count $k$ -trees with respect to just one sort of label (the $X$ -labels on hedra), so we will not deal with multisort here.",
"Each of the two-sort cycle indices in this chapter can be converted to one-sort by substituting $y_{i} = 1$ for all $i$ .",
"For the rest of this section, we will deal directly with these one-sort versions of the cycle indices.",
"We begin by considering the explicit recursive functional equations in thm:ctyfuncci,thm:ctxyfuncci.",
"In each case, by the above, the ordinary generating function is exactly the result of substituting $p_{i} {x} = x^{i}$ into the given formula.",
"Thus, we have: For ${k}$ the ${k}$ -species of $Y$ -rooted $k$ -coding trees and ${k}$ the ${k+1}$ -species of edge-rooted $k$ -coding trees, the corresponding $\\Gamma $ -ordinary generating functions are given by k x = [Big] n 1 xnn c C n k [Big]-1 i c i [big]n xc and k x = x c C k [Big]i c i [big]xc.",
"where $\\widetilde{{k}}$ is an ${k}$ -generating function and $\\widetilde{{k}}$ is an ${k+1}$ -generating function.",
"However, as a consequence of thm:ctciclassfunc, we can simplify these expressions significantly: For ${k}$ the ${k}$ -species of $Y$ -rooted $k$ -coding trees and ${k}$ the ${k+1}$ -species of edge-rooted $k$ -coding trees, the corresponding $\\Gamma $ -ordinary generating functions are given by k x = [Big]n 1 xnn i n k [big]i [big]xi and k x = x i k [big]i [big]xi where $\\lambda ^{i}$ denotes the $i$ th `partition power' of $\\lambda $ — that is, if $\\sigma $ is any permutation of cycle type $\\lambda $ , then $\\lambda ^{i}$ denotes the cycle type of $\\sigma ^{i}$ — and where $f {\\lambda } {x}$ denotes the value of $f {\\sigma } {x}$ for every $\\sigma $ of cycle type $\\lambda $ .",
"As in thm:ctyfuncci, we have recursively-defined functional equations, but these are recursions of power series in a single variable, so computing their terms is much less computationally expensive.",
"Also, as an immediate consequence of thm:ctciclassfunc, we have that $\\widetilde{{k}}$ and $\\widetilde{{k}}$ are class functions of $\\sigma $ , so we can restrict our computational attention to cycle-distinct permutations.",
"Moreover, the cycle index of the species ${k}$ , as seen in eq:akci, is given simply in terms of quotients of the cycle indices of the two $\\Gamma $ -species ${k}$ and ${k}$ .",
"Thus, we also have: For ${k}$ the species of $k$ -trees and $\\widetilde{{k}}$ and $\\widetilde{{k}}$ as in thm:ctrhoogf, we have $\\tilde{\\mathfrak {a}}_{k} {x} = \\frac{1}{{k+1}!}",
"\\sum _{\\sigma \\in {k+1}} \\widetilde{{k}} {\\sigma } {x} + \\frac{1}{k!}",
"\\sum _{\\sigma \\in {k}} \\widetilde{{k}} {x} {\\sigma } - \\frac{1}{k!}",
"\\sum _{\\sigma \\in {k}} \\widetilde{{k}} {\\sigma } {x}.$ Again, as a consequence of thm:ctciclassfunc by way of cor:ctogf, we can instead write For ${k}$ the species of $k$ -trees and $\\widetilde{{k}}$ and $\\widetilde{{k}}$ as in cor:ctogf, we have $\\tilde{\\mathfrak {a}}_{k} {x} = \\sum _{\\lambda \\vdash k+1} \\frac{1}{z_{\\lambda }} \\widetilde{{k}} {\\lambda } {x} + \\sum _{\\lambda \\vdash k} \\frac{1}{z_{\\lambda }} \\widetilde{{k}} {\\lambda } {x} - \\sum _{\\lambda \\vdash k} \\frac{1}{z_{\\lambda }} \\widetilde{{k}} {\\lambda \\cup {1}} {x}.$ This direct characterization of the ordinary generating function of unlabeled $k$ -trees, while still recursive, is much simpler computationally than the characterization of the full cycle index in eq:akci.",
"For computation of the number of unlabeled $k$ -trees, it is therefore much preferred.",
"Classical methods for working with recursively-defined power series suffice to extract the coefficients quickly and efficiently.",
"The results of some such explicit calculations are presented in s:ktenum."
],
[
"Special-case behavior for small $k$",
"Many of the complexities of the preceding analysis apply only for $k$ sufficiently large.",
"We note here some simplifications that are possible when $k$ is small."
],
[
"Ordinary trees ($k = 1$ )",
"When $k = 1$ , an ${k}$ -structure is merely an ordinary tree with $X$ -labels on its edges and $Y$ -labels on its vertices.",
"There is no internal symmetry of the form that the actions of ${k}$ are intended to break.",
"The actions of 2 act on ordinary trees rooted at a directed edge, with the nontrivial element $\\tau \\in {2}$ acting by reversing this orientation.",
"The resulting decomposition from the dissymmetry theorem in thm:dissymk and the recursive functional equations of obs:funcdecompct then clearly reduce to the classical dissymmetry analysis of ordinary trees."
],
[
"2-trees",
"When $k=2$ , there is a nontrivial symmetry at fronts (edges); two triangles may be joined at an edge in two distinct ways.",
"The imposition of a coherent orientation on a 2-tree by directing one of its edges breaks this symmetry; the action of 2 by reversal of these orientations gives unoriented 2-trees as its orbits.",
"The defined action of 3 on edge-rooted oriented triangles is simply the classical action of the dihedral group $D_{6}$ on a triangle, and its orbits are unoriented, unrooted triangles.",
"We further note that $\\rho _{i}$ is the trivial map on 2 and that $\\rho _{i} {\\sigma } = {1\\ 2}$ for $\\sigma \\in {3}$ if and only if $\\sigma $ is an odd permutation, both regardless of $i$ .",
"We then have that: 22 = p1 y E [Big]p1 x c C 22epc x, p2 c x, ...; pc y, p2 c y, ... 32 = p1 x c C 22 [big]c pc x, p2 c x, ...; pc y, p2 c y, .... where, by abuse of notation, we let $\\rho $ represent any $\\rho _{i}$ .",
"By the previous, the argument $\\rho {\\sigma }^{{c}}$ in eq:ctxyfuncci2 is $\\tau $ if and only if $\\sigma $ is an odd permutation and $c$ is of odd length.",
"This analysis and the resulting formulas for the cycle index ${{2}}$ are essentially equivalent to those derived in [6]."
],
[
"Cycle indices of compositional inverse species",
"In s:nbp, our results included two references to the compositional inverse ${CBP}^{\\bullet {-1}}$ of the species ${CBP}^{\\bullet }$ .",
"Although we have not explored computational methods in depth here, the question of how to compute the cycle index of the compositional inverse of a specified species efficiently is worth some consideration.",
"Several methods are available, including one developed in [3] as part of the proof that arbitrary species have compositional inverses, but our preferred method is one of iterated substitution.",
"Suppose that $\\Psi $ is a species (with known cycle index) of the form $X + \\Psi _{2} + \\Psi _{3} + \\dots $ where $\\Psi _{i}$ is the restriction of $\\Psi $ to structures on sets of cardinality $i$ and that $\\Phi $ is the compositional inverse of $\\Psi $ .",
"Then $\\Psi \\circ \\Phi = X$ by definition, but by hypothesis $X = \\Psi \\circ \\Phi = \\Phi + \\Psi _{2} {\\Phi } + \\Psi _{3} {\\Phi } + \\dots $ also.",
"Thus $\\Phi = X - \\Psi _{2} {\\Phi } - \\Psi _{3} {\\Phi } - \\dots .$ This recursive equation is the key to our computational method.",
"To compute the cycle index of $\\Phi $ to degree 2, we begin with the approximation $\\Phi \\approx X$ and then substitute it into the first two terms of eq:compinv: $\\Phi \\approx X - \\Psi _{2} {X}$ and thus ${\\Phi } \\approx {X} - {\\Psi _{2}} \\circ {X}$ .",
"All terms of degree up to two in this approximation will be correct.",
"To compute the cycle index of $\\Phi $ to degree 3, we then take this new approximation $\\Phi \\approx X - \\Psi _{2} {X}$ and substitute it into the first three terms of eq:compinv.",
"This process can be iterated as many times as are needed; to determine all terms of degree up to $n$ correctly, we need only iterate $n$ times.",
"With appropriate optimizations (in particular, truncations), this method can run very quickly on a personal computer to reasonably high degrees; we were able to compute ${{CBP}^{\\bullet {-1}}}$ to degree sixteen in thirteen seconds."
],
[
"Bipartite blocks",
"With the tools developed in c:bpblocks, we can calculate the cycle indices of the species $\\mathcal {NBP}$ of nonseparable bipartite graphs to any finite degree we choose using computational methods.",
"This result can then be used to enumerate unlabeled bipartite blocks.",
"We have done so here using Sage 1.7.4 [23] and code listed in s:bpbcode.",
"The resulting values appear in tab:bpblocks.",
"Table: Enumerative data for unlabeled bipartite blocks with nn hedra"
],
[
"$k$ -trees",
"With the recursive functional equations for cycle indices of s:ktcycind, we can calculate the explicit cycle index for the species ${k}$ to any finite degree we choose using computational methods; this cycle index can then be used to enumerate both unlabeled and labeled (at fronts, hedra, or both) $k$ -trees up to a specified number $n$ of hedra (or, equivalently, $kn + 1$ fronts).",
"We have done so here for $k \\le 7$ and $n \\le 30$ using Sage 1.7.4 [23] using code available in s:ktcode.",
"The resulting values appear in tab:ktrees.",
"We note that both unlabeled and hedron-labeled enumerations of $k$ -trees stabilize: For $k \\ge n + 2$ , the numbers of unlabeled and hedron-labeled $k$ -trees are independent of $k$ .",
"We show that the species ${k}$ and ${k+1}$ have contact up to order $k+2$ by explicitly constructing a natural bijection.",
"We note that in a ${k+1}$ -tree with no more than $k+2$ hedra, there will exist at least one vertex which is common to all hedra.",
"For any $k$ -tree with no more than $k+2$ hedra, we can construct a ${k+1}$ -tree with the same number of hedra by adding a single vertex and connecting it by edges to every existing vertex; we can then pass labels up from the ${k+1}$ -cliques which are the hedra of the $k$ -tree to the ${k+2}$ -cliques which now sit over them.",
"The resulting graph will be a ${k+1}$ -tree whose ${k+1}$ -tree hedra are adjacent exactly when the $k$ -tree hedra they came from were adjacent.",
"Therefore, any two distinct $k$ -trees will pass to distinct ${k+1}$ -trees.",
"Similarly, for any ${k+1}$ -tree with no more than $k+2$ hedra, choose one of the vertices common to all the hedra and remove it, passing the labels of ${k+1}$ -tree hedra down to the $k$ -tree hedra constructed from them; again, adjacency of hedra is preserved.",
"This of course creates a $k$ -tree, and for distinct ${k+1}$ -trees the resulting $k$ -trees will be distinct.",
"Moreover, by symmetry the result is independent of the choice of common vertex, in the case there is more than one.",
"However, thus far we have neither determined a direct method for computing these stabilization numbers nor identified a straightforward combinatorial characterization of the structures they represent.",
"Table: Enumerative data for kk-trees with nn hedraTable: *Table: *Table: *Table: *Table: *Table: *"
],
[
"Code listing",
"Our results in c:bpblocks,c:ktrees provide a framework for enumerating bipartite blocks and general $k$ -trees.",
"However, there is significant work to be done adapting the theory into practical algorithms for computing the actual numbers of such structures.",
"Using the computer algebra system Sage 1.7.4 [23], we have done exactly this.",
"In each case, the script listed may be run with Sage by invoking > sage --python scriptname.py args on a computer with a functioning Sage installation.",
"Alternatively, each code snippet may be executed in the Sage `notebook' interface starting at the comment “MATH BEGINS HERE”; in this case, the final print... invocation should be replaced with one specifying the desired parameters."
],
[
"Bipartite blocks",
"The functional eq:nbpexp characterizes the cycle index of the species ${NBP}$ of bipartite blocks.",
"Python/Sage code to compute the coefficients of the ordinary generating function $\\widetilde{{NBP}} {x}$ of unlabeled bipartite blocks explicitly follows in lst:bpcode.",
"Specifically, the generating function may be computed to degree $n$ by invoking > sage --python bpblocks.py n on a computer with a functioning Sage installation.",
"[caption=Sage code to compute numbers of bipartite blocks (bpblocks.py), label=lst:bpcode, language=Python]python/bpblocks.py"
],
[
"$k$ -trees",
"The recursive functional equations in eq:ctyogf,eq:ctxyogf,eq:akogf characterize the ordinary generating function $\\tilde{\\mathfrak {a}}_{k} {x}$ for unlabeled general $k$ -trees.",
"Python/Sage code to compute the coefficients of this generating function explicitly follows in lst:ktcode.",
"Specifically, the generating function for unlabeled $k$ -trees may be computed to degree $n$ by invoking > sage --python ktrees.py k n on a computer with a functioning Sage installation.",
"This code uses the class-function optimization of thm:ctciclassfunc extensively; as a result, it is able to compute the number of $k$ -trees on up to $n$ hedra quickly even for relatively large $k$ and $n$ .",
"For example, the first thirty terms of the generating function for 8-trees in tab:8trees were computed on a modern desktop-class computer in approximately two minutes.",
"[caption=Sage code to compute numbers of $k$ -trees (ktrees.py), label=lst:ktcode, language=Python]python/ktrees.py"
]
] | 1204.1402 |
[
[
"Optimal Save-Then-Transmit Protocol for Energy Harvesting Wireless\n Transmitters"
],
[
"Abstract In this paper, the design of a wireless communication device relying exclusively on energy harvesting is considered.",
"Due to the inability of rechargeable energy sources to charge and discharge at the same time, a constraint we term the energy half-duplex constraint, two rechargeable energy storage devices (ESDs) are assumed so that at any given time, there is always one ESD being recharged.",
"The energy harvesting rate is assumed to be a random variable that is constant over the time interval of interest.",
"A save-then-transmit (ST) protocol is introduced, in which a fraction of time {\\rho} (dubbed the save-ratio) is devoted exclusively to energy harvesting, with the remaining fraction 1 - {\\rho} used for data transmission.",
"The ratio of the energy obtainable from an ESD to the energy harvested is termed the energy storage efficiency, {\\eta}.",
"We address the practical case of the secondary ESD being a battery with {\\eta} < 1, and the main ESD being a super-capacitor with {\\eta} = 1.",
"The optimal save-ratio that minimizes outage probability is derived, from which some useful design guidelines are drawn.",
"In addition, we compare the outage performance of random power supply to that of constant power supply over the Rayleigh fading channel.",
"The diversity order with random power is shown to be the same as that of constant power, but the performance gap can be large.",
"Furthermore, we extend the proposed ST protocol to wireless networks with multiple transmitters.",
"It is shown that the system-level outage performance is critically dependent on the relationship between the number of transmitters and the optimal save-ratio for single-channel outage minimization.",
"Numerical results are provided to validate our proposed study."
],
[
"Introduction",
"The operation of communication networks powered either largely or exclusively by renewable sources has become increasingly attractive, both due to the increased desire to reduce energy consumption in human activities at large, and due to necessity brought about by the concept of networking heterogeneous devices ranging from medical sensors on/in the human body to environment sensors in the wilderness [2], [1].",
"Sensor nodes are powered by batteries that often cannot be replaced because of the inaccessibility of the devices.",
"Therefore, once the battery of a sensor node is exhausted, the node dies.",
"Thus the potentially maintenance-free and virtually perpetual operation offered by energy harvesting, whereby energy is extracted from the environment, is appealing.",
"The availability of an inexhaustible but unreliable energy source changes a system designer's options considerably, compared to the conventional cases of an inexhaustible reliable energy source (powered by the grid), and an exhaustible reliable energy source (powered by batteries).",
"There has been recent research on understanding data packet scheduling with an energy harvesting transmitter that has a rechargeable battery, most of which employed a deterministic energy harvesting model.",
"In [3], the transmission time for a given amount of data was minimized through power control based on known energy arrivals over all time.",
"Structural properties of the optimum solution were then used to establish a fast search algorithm.",
"This work has been extended to battery limited cases in [4], battery imperfections in [5], [6], and the Gaussian relay channel in [7].",
"Energy harvesting with channel fading has been investigated in [8] and [9], wherein a water-filling energy allocation solution where the so-called water levels follow a staircase function was proved to be optimal.",
"In scenarios where multiple energy harvesting wireless devices interact with each other, the design needs to adopt a system-level approach [12], [13], [14].",
"In [13], the medium access control (MAC) protocols for single-hop wireless sensor networks, operated by energy harvesting capable devices, were designed and analyzed.",
"In [14], $N$ energy harvesting nodes with independent data and energy queues were considered, and the queue stability was analyzed under different MAC protocols.",
"An information theoretic analysis of energy harvesting communication systems has been provided in [15], [16].",
"In [15], the authors proved that the capacity of the AWGN channel with stochastic energy arrivals is equal to the capacity with an average power constraint equal to the average recharge rate.",
"This work has been extended in [16] to the fading Gaussian channels with perfect/no channel state information at the transmitter.",
"Due to the theoretical intractability of online power scheduling under the energy causality constraint (the cumulative energy consumed is not allowed to exceed the cumulative energy harvested at every point in time), most current research is focused on an offline strategy with deterministic channel and energy state information, which is not practical and can only provide an upper bound on system performance.",
"An earlier line of research considers the problem of energy management, with only causal energy state information, in communications satellites [10], which formulated the problem of maximizing a reward that is linear in the energy as a dynamic programming problem.",
"In [11], energy management policies which stabilize the data queue have been proposed for single-user communication under linear energy-rate approximations.",
"In this paper, we focus our study on the design of practical circuit model and transmission protocol for energy harvesting wireless transmitters.",
"To be more specific, we consider a wireless system with one transmitter and one receiver, with the transmitter using a save-then-transmit (ST) protocol (see Fig.",
"REF ) to deliver $Q$ bits within $T$ seconds, the duration of a transmission frame.",
"Because rechargeable energy storage devices (ESDs) cannot both charge and discharge simultaneously (the energy half-duplex constraint), an energy harvesting transmitter needs two ESDs, which we call the main ESD (MESD) and secondary ESD (SESD).",
"The transmitter draws power from the MESD for data transmission, over which time the SESD is connected to the energy source and charges up.",
"At the end of transmission for a frame, the SESD transfers its stored energy to the MESD.",
"A fraction $\\rho $ (called the save-ratio) of every frame interval is used exclusively for energy harvesting by the MESD.Note that the energy source can be connected only to either the SESD or the MESD, but not both.",
"The energy storage efficiency, denoted by $\\eta $ , of each ESD may not be 100 percent, and a fixed amount of power $P_c$ is assumed to be consumed by the transmitter hardware whenever it is powered up.",
"The frame interval $T$ is assumed to be small relative to the time constant of changes in the ESD charging rate (or energy arrival rate).",
"The energy arrival rate is therefore modeled as a random variable $X$ in Joules/second, which is assumed to be constant over a frame.",
"Figure: Save-Then-Transmit (ST) ProtocolUnder the above realistic conditions, we minimize the outage probability (to be defined in the next section) over $\\rho $ , when transmitting over a block fading channel with an arbitrary fading distribution.",
"In this work, we particularize to the case where the MESD is a high-efficiency super-capacitor with $\\eta = 1$ , and the SESD is a low-efficiency rechargeable battery with $0 \\le \\eta \\le 1$ .",
"Based on the outage analysis, we compare the performance between two system setups: the (new) case with random power supply versus the (conventional) case with constant power supply, over the Rayleigh fading channel.",
"It is shown that energy harvesting, which results in time-varying power availability in addition to the randomness of the fading channel, may severely degrade the outage performance.",
"To be concrete, we further consider exponentially distributed random power, and show that although the diversity order with exponential power is the same as that with constant power over the Rayleigh fading channel, the outage probability curve may only display the slope predicted by this diversity analysis at substantially higher SNRs.",
"Finally, we extend the ST protocol for the single-channel case to the general case of wireless network with multiple transmitters.",
"We propose a time division multiple access (TDMA) based ST (TDMA-ST) protocol to allocate orthogonal time slots to multiple transmitters that periodically report to a fusion center.",
"Specifically, we consider two types of source data at transmitters as follows: Independent Data: transmitters send independent data packets to the fusion center for independent decoding; Common Data: transmitters send identical data packets to the fusion center, where diversity combining is applied to decode the common data.",
"It is shown that for both cases if the number of transmitters $N$ is smaller than the reciprocal of the optimal transmit-ratio ($1-\\rho $ ) for the single-channel outage minimization, all transmitters can operate at their individual minimum outage probability.",
"However, as $N$ goes up and exceeds this threshold, the system-level outage performance behaves quite differently for the two types of source data.",
"The rest of this paper is organized as follows.",
"Section presents the system model.",
"Section considers finding the optimal save-ratio for outage minimization and analyzes its various properties.",
"Section compares the outage performance between fixed power and random power.",
"Section introduces the TDMA-ST protocol for the multi-transmitter case.",
"Section shows numerical results.",
"Finally, Section concludes the paper.",
"The block diagram of the system is given in Fig.",
"REF .",
"The energy harvested from the environmentWind, solar, geothermal, etc.",
"is first stored in either the MESD or the SESD at any given time, as indicated by switch $a$ , before it is used in data transmission.",
"The MESD powers the transmitter directly and usually has high power density, good recycle ability and high efficiency, e.g.",
"a super-capacitor [17].",
"Since the MESD cannot charge and discharge simultaneously, a SESD (e.g.",
"rechargeable battery) stores up harvested energy while the transmitter is on, and transfers all its stored energy to the MESD once the transmitter is off.",
"We assume in the rest of this paper that the SESD is a battery with an efficiency $\\eta $ ,In practice, the battery efficiency can vary from $60\\%$ to $99\\%$ , depending on different recharging technologies [18].",
"where $\\eta \\in [0, 1]$ .",
"This means that a fraction $\\eta $ of the energy transferred into the SESD during charging can be subsequently recovered during discharging.",
"The other $1-\\eta $ fraction of the energy is thus lost, due to e.g.",
"battery leakage and/or circuit on/off overhead.",
"The reason of choosing a single-throw switch (switch $a$ in Fig.",
"REF ) between the energy harvesting device (EHD) and ESDs is that splitting the harvested energy with a portion going to the SESD, when the transmitter does not draw energy from the MESD, is not energy efficient due to the SESD's lower efficiency.",
"Note that at the current stage of research, the optimal detailed structure of an energy harvesting transmitter is not completely known and there exist various models in the literature (see e.g.",
"[8], [6], [9]).",
"The proposed circuit model, given in Fig.",
"REF , provides one possible practical design.",
"Figure: Energy Harvesting Circuit ModelWe assume that $Q$ bits of data are generated and must be transmitted within a time slot of duration $T$ seconds (i.e., delay constrained).",
"In the proposed ST protocol, the save-ratio $\\rho $ is the reserved fraction of time for energy harvesting by the MESD within one transmission slot.",
"In other words, data delivery only takes place in the last $(1-\\rho )T$ seconds of each time slot, which results in an effective rate of $R_{\\mathrm {eff}} =\\frac{Q}{(1-\\rho )T}$ bits/sec.",
"We also allow for a constant power consumption of $P_c$ Watts by the transmitter hardware whenever it is powered on.",
"The combined influence of $\\rho $ , $\\eta $ and $P_c$ on outage probability is quantified in this work.",
"Assume a block-fading frequency-nonselective channel, where the channel is constant over the time slot $T$ .",
"Over any time slot, the baseband-equivalent channel output is given by $ y = h\\cdot x + n,$ where $x$ is the transmitted signal, $y$ is the received signal, and $n$ is i.i.d.",
"circularly symmetric complex Gaussian (CSCG) noise with zero mean and variance $\\sigma _n^2$ .",
"For any frame, the ST protocol (cf.",
"Fig.",
"REF ) is described as follows: During time interval $(0, \\rho T]$ , harvested energy accumulates in the MESD, which corresponds to the situation that switches $b$ , $c$ are open and $a$ connects to the MESD in Fig.",
"REF ; From time $\\rho T$ to $T$ , the transmitter is powered on for transmission with energy from the MESD.",
"Since the transmitter has no knowledge of the channel state, we assume that all the buffered energy in the MESD is used up (best-effort transmission) in each frame.",
"Since the MESD cannot charge and discharge at the same time, the SESD starts to store up harvested energy while the transmitter is on.",
"Referring to Fig.",
"REF , $c$ is closed, $b$ is open and $a$ switches to the SESD; At time $T$ , the transmitter completes the transmission and powers off.",
"The SESD transfers all its buffered energy to the MESD within a negligible charging time, at efficiency $\\eta $ .",
"In other words, $b$ is closed and switches $a$ and $c$ are open in Fig.",
"REF ."
],
[
"Outage Probability",
"It is clear that the energy harvesting rate $X$ is a non-negative random variable with finite support, i.e., $0 \\le X \\le P_H < \\infty $ , as the maximum amount of power one can extract from any source is finite.",
"Suppose $f_X(x)$ and $F_X(x)$ represent its probability density function (PDF) and cumulative distribution function (CDF), respectively.",
"According to the proposed ST protocol, the total buffered energy in the MESD at $t = \\rho T$ (the start of data transmission within a transmission slot) is given by $E_T = X\\left[\\rho +\\eta (1-\\rho )\\right]T.$ Denote $P = \\frac{E_T}{(1-\\rho )T} =X\\left[\\frac{\\rho }{1-\\rho }+\\eta \\right]$ as the average total power, which is constant over the entire transmission period, and $P_c$ as the circuit power (i.e.",
"the power consumed by the hardware during data transmission), again assumed constant.",
"The mutual information of the channel (REF ) conditioned on $X$ and the channel gain $h$ is (assuming $P > P_c$ ) $R_T = \\log _2\\left(1+\\frac{(P-P_c)|h|^2}{\\sigma _n^2}\\right) = \\log _2\\left(1+(P-P_c)\\Gamma \\right)$ where $\\Gamma = \\frac{|h|^2}{\\sigma _n^2}$ with PDF $f_{\\Gamma }(\\cdot )$ and CDF $F_{\\Gamma }(\\cdot )$ .",
"For a transmitter with energy harvesting capability and working under the ST protocol, the outage event is the union of two mutually exclusive events: Circuit Outage and Channel Outage.",
"Circuit outage occurs when the MESD has insufficient energy stored up at $t = \\rho T$ to even power on the hardware for the duration of transmission i.e.",
"$E_T < P_c (1-\\rho )T$ or equivalent $P < P_c$ .",
"Channel outage is defined as the MESD having sufficient stored energy but the channel realization does not support the required target rate $R_{\\mathrm {eff}} = \\frac{Q}{(1-\\rho )T}$ bits/s.",
"Recalling that $X \\in [0,P_H]$ , the probabilities of Circuit Outage and Channel Outage are therefore: $P_{out}^{circuit} & = \\mbox{Pr}\\left\\lbrace P < P_c\\right\\rbrace \\nonumber \\\\& = \\left\\lbrace \\begin{array}{cl} \\displaystyle F_X\\left[\\phi (\\cdot )\\right] & \\mbox{if } P_H > \\phi (\\cdot ) \\\\1 & \\mbox{otherwise}.",
"\\end{array}\\right.",
"\\\\P_{out}^{channel} & = \\mbox{Pr}\\left\\lbrace \\log _2\\left(1+(P-P_c)\\Gamma \\right) < R_{\\mathrm {eff}}, P > P_c \\right\\rbrace \\nonumber \\\\& = \\mbox{Pr}\\left\\lbrace \\Gamma < \\frac{2^{R_{\\mathrm {eff}}}-1}{P-P_c}, P > P_c \\right\\rbrace \\nonumber \\\\& = \\left\\lbrace \\begin{array}{cl} \\displaystyle \\int _{\\phi (\\cdot )}^{P_H}f_X(x)F_{\\Gamma }\\left[g(\\cdot )\\right]dx & \\mbox{if } P_H > \\phi (\\cdot ) \\\\0 & \\mbox{otherwise}.",
"\\end{array}\\right.",
"$ where $g(\\rho , \\eta , P_c) =\\frac{2^{\\frac{Q}{(1-\\rho )T}}-1}{x[\\frac{\\rho }{1-\\rho }+\\eta ]-P_c}$ and $\\phi (\\rho , \\eta , P_c) = \\frac{P_c}{\\frac{\\rho }{1-\\rho } +\\eta }$ .",
"Since Circuit Outage and Channel Outage are mutually exclusive, it follows that $P_{out} & = P_{out}^{circuit} + P_{out}^{channel} \\nonumber \\\\& = \\left\\lbrace \\begin{array}{cl} \\displaystyle F_X\\left[\\phi (\\cdot )\\right] +\\\\\\int _{\\phi (\\cdot )}^{P_H}f_X(x)F_{\\Gamma }\\left[g(\\cdot )\\right]dx & \\mbox{if } P_H > \\phi (\\cdot ) \\\\1 & \\mbox{otherwise}.",
"\\end{array}\\right.$ For convenience, we define $\\hat{P}_{out}(\\rho , \\eta , P_c) = F_X\\left[\\phi (\\cdot )\\right] + \\int _{\\phi (\\cdot )}^{P_H}f_X(x)F_{\\Gamma }\\left[g(\\cdot )\\right]dx$ where $\\hat{P}_{out}(\\rho , \\eta , P_c) < 1$ and $P_H > \\phi (\\cdot )$ .",
"Unlike the conventional definition of outage probability in a block fading channel, which is dependent only on the fading distribution and a fixed average transmit power constraint, in an energy harvesting system with block fading and the ST protocol, both transmit power and channel are random, and the resulting outage is thus a function of the save-ratio $\\rho $ , the battery efficiency $\\eta $ and the circuit power $P_c$ ."
],
[
"Outage Minimization",
"In this section, we design the save-ratio $\\rho $ for the ST protocol by solving the optimization problem $\\mathrm {(P1)}:~\\mathop {\\mathtt {min.",
"}}_{0 \\le \\rho \\le 1} &~~ P_{out} \\nonumber $ i.e.",
"minimize average outage performance $P_{out}$ in (REF ) over $\\rho $ , for any given $\\eta \\in [0, 1]$ and $P_c \\in [0, \\infty )$ .",
"Denote the optimal (minimum) outage probability as $P_{out}^*(\\eta , P_c)$ and the optimal save-ratio as $\\rho ^*(\\eta , P_c)$ .",
"Note that $\\rho \\nearrow 1$ represents transmission of a very short burst at the end of each frame, and the rest of each frame is reserved for MESD energy harvesting.",
"$\\rho =0$ is another special case, in which the energy consumed in frame $i$ was collected (by the SESD) entirely in frame $i-1$ .",
"(P1) can always be solved through numerical search, but it is challenging to give a closed-form solution for $\\rho ^*(\\eta , P_c)$ in terms of $P_c$ and $\\eta $ in general.",
"We will instead analyze how $\\rho ^*(\\eta , P_c)$ varies with $P_c$ and $\\eta $ and thereby get some insights in the rest of this section.",
"Proposition 3.1 $P_{out}(\\rho , \\eta , P_c)$ in (REF ) is a non-increasing function of battery efficiency $\\eta $ and a non-decreasing function of circuit power $P_c$ for $\\rho \\in [0,1)$ .",
"The optimal value of (P1) $P_{out}^*(\\eta , P_c)$ is strictly decreasing with $\\eta $ and strictly increasing with $P_c$ .",
"Please refer to Appendix .",
"The intuition of Proposition $\\ref {proposition:1}$ is clear: If $\\eta $ grows, the energy available to the transmitter can only grow or remain the same, whatever the values of $\\rho $ and $P_c$ , hence $P_{out}$ must be non-increasing with $\\eta $ ; if $P_c$ grows, the energy available for transmission decreases, leading to higher $P_{out}$ ."
],
[
"Ideal System: $\\eta = 1$ and {{formula:1ab493b0-f06f-4ce0-a08d-2b809c4d4fa7}}",
"Suppose that circuit power is negligible, i.e.",
"all the energy is spent on transmission, and the SESD has perfect energy-transfer efficiency.",
"The condition $P_H > P_c/(\\frac{\\rho }{1-\\rho } + \\eta )$ is always satisfied, and problem (P1) is simplified to $\\mathrm {(P2)}:~\\mathop {\\mathtt {min.",
"}}_{0 \\le \\rho \\le 1} &~~ \\int _0^{P_H}f_X(x)F_{\\Gamma }\\left[\\frac{(2^{\\frac{Q}{(1-\\rho )T}}-1)(1-\\rho )}{x}\\right]dx \\nonumber $ where the optimal value of (P2) is denoted as $P_{out}^*(1,0)$ , and the optimal save-ratio is denoted as $\\rho ^*(1,0)$ .",
"Lemma 3.1 The minimum outage probability when $\\eta = 1$ and $P_c = 0$ is given by $P_{out}^*(1,0) = \\int _0^{P_H}f_X(x)F_{\\Gamma }\\left[\\frac{2^{Q/T}-1}{x}\\right]dx$ and is achieved with the save-ratio $\\rho ^*(1,0) = 0$ .",
"Please refer to Appendix .",
"Lemma REF indicates that the optimal strategy for a transmitter that uses no power to operate its circuitry powered by two ESDs with 100 percent efficiency, is to transmit continuously.Except for the time needed in each slot to transfer energy from the SESD to the MESD, which we assume to be negligible.",
"This is not surprising because the SESD collects energy from the environment just as efficiently as the MESD does, and so idling the transmitter while the MESD harvests energy wastes transmission resources (time) while not reaping any gains (energy harvested).",
"However, we will see that this is only true when there is no circuit power and the battery efficiency is perfect."
],
[
"Inefficient Battery: $\\eta < 1$ and {{formula:c4c95df7-6672-4401-89d8-30fe7ef3a85b}}",
"When the SESD energy transfer efficiency $\\eta < 1$ and $P_c = 0$ , (P1) becomes $\\mathrm {(P3)}:~\\mathop {\\mathtt {min.",
"}}_{0 \\le \\rho \\le 1} &~~ \\int _0^{P_H}f_X(x)F_{\\Gamma }\\left[\\frac{(2^{\\frac{Q}{(1-\\rho )T}}-1)}{x(\\frac{\\rho }{1-\\rho }+\\eta )}\\right]dx \\nonumber $ where the optimal value of (P3) is denoted as $P_{out}^*(\\eta ,0)$ , and the optimal save-ratio is denoted as $\\rho ^*(\\eta ,0)$ .",
"Lemma 3.2 When SESD energy transfer efficiency $\\eta < 1$ and circuit power $P_c = 0$ , the optimal save-ratio $\\rho $ has the following properties.",
"A “phase transition” behavior: $\\left\\lbrace \\begin{array}{ll}\\rho ^*(\\eta , 0) = 0, ~~~~ & \\eta \\in \\left[\\frac{2^{\\frac{Q}{T}}-1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}}, 1 \\right) \\\\\\rho ^*(\\eta , 0) > 0, ~~~~ & \\eta \\in \\left[0, \\frac{2^{\\frac{Q}{T}}-1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}} \\right)\\end{array}\\right.$ $\\rho ^*(\\eta ,0)$ is a non-increasing function of $\\eta $ , for $0 \\le \\eta \\le 1$ .",
"Please refer to Appendix .",
"According to (REF ), if the SESD efficiency is above a threshold, the increased energy available to the transmitter if the MESD rather than the SESD collects energy over $[0,\\rho T]$ is not sufficient to overcome the extra energy required to transmit at the higher rate $R_{\\mathrm {eff}}$ over $(\\rho T, T]$ .",
"The result is that the optimal $\\rho $ is 0.",
"On the other hand, if $\\eta $ is below that threshold, then some amount of time should be spent harvesting energy using the higher-efficiency MESD even at the expense of losing transmission time.",
"Lemma REF quantifies precisely the interplay among $\\eta $ , $Q$ , $T$ and $\\rho $ .",
"We should note here that even though we consider the case of having two ESD's, by setting $\\eta = 0$ , we effectively remove the SESD and hence our analysis applies also to the single-ESD case.",
"According to (REF ), if we only have one ESD, the optimal save-ratio is $\\rho ^*(0,0)$ , which is always larger than 0.",
"This is intuitively sensible, because with only one ESD obeying the energy half-duplex constraint, it would be impossible to transmit all the time ($\\rho = 0$ ) because that would leave no time at all for energy harvesting."
],
[
"Non-Zero Circuit Power: $\\eta \\le 1$ , {{formula:44d702a1-a86c-4509-8d3b-2050ce4e3466}}",
"Non-zero circuit power $P_c$ leads to two mutually exclusive effects: (i) inability to power on the transmitter for the $(1-\\rho )T$ duration of transmission – this is when $P_H <\\phi (\\cdot )$ in (REF ); and (ii) higher outage probability if $P_H > \\phi (\\cdot )$ because some power is devoted to running the hardware.",
"Since $\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }$ decreases as $\\rho $ increases, its maximum value is $\\frac{P_c}{\\eta }$ .",
"Therefore, if $P_H > \\frac{P_c}{\\eta }$ , the transmitter would be able to recover enough energy (with non-zero probability) to power on the transmitter, i.e.",
"$\\rho \\in [0, 1)$ .",
"If $P_H \\le \\frac{P_c}{\\eta }$ , by condition $P_H \\le \\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }$ , save-ratio $\\rho $ is required to be larger than $\\frac{\\frac{P_c}{P_H}-\\eta }{1-\\eta +\\frac{P_c}{P_H}}$ .",
"In summary, If $P_c < P_H\\eta $ $P_{out} = \\hat{P}_{out}(\\rho , \\eta , P_c), ~~~~ \\forall \\rho \\in [0, 1) \\nonumber $ If $P_c \\ge P_H\\eta $ $P_{out} = \\left\\lbrace \\begin{array}{ll}1, ~~~~ & \\rho \\le \\frac{\\frac{P_c}{P_H}-\\eta }{1-\\eta +\\frac{P_c}{P_H}} \\\\\\hat{P}_{out}(\\rho , \\eta , P_c), ~~~~ & \\rho > \\frac{\\frac{P_c}{P_H}-\\eta }{1-\\eta +\\frac{P_c}{P_H}}\\end{array}\\right.$ If $P_c \\ge \\eta P_H$ , referring to (), we may conclude that $\\rho ^*(\\eta , P_c) >\\frac{\\frac{P_c}{P_H}-\\eta }{1-\\eta +\\frac{P_c}{P_H}}$ due to the need to offset circuit power consumption.",
"If $P_c < \\eta P_H$ , theoretically, the transmitter is able to recover enough energy (with non-zero probability for all $\\rho \\in [0, 1)$ ) to transmit.",
"Lemma 3.3 For an energy harvesting transmitter with battery efficiency $\\eta $ and non-zero circuit power $P_c$ , $\\eta - \\frac{P_c}{P_H} < \\frac{2^{\\frac{Q}{T}} - 1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}} \\;\\;\\Longrightarrow \\;\\;\\rho ^*(\\eta , P_c) > 0.$ Please refer to Appendix .",
"Intuitively, the smaller the circuit power, the more energy we can spend on transmission; the larger the battery efficiency is, the more energy we can recover from energy harvesting.",
"Small circuit power and high battery efficiency suggest continuous transmission ($\\rho ^*(\\eta , P_c) = 0$ ), which is consistent with our intuition.",
"According to Lemma REF , larger circuit power may be compensated by larger ESD efficiency (when the threshold for $\\eta $ is smaller than 1).",
"A non-zero save-ratio is only desired if there exists significant circuit power to be offset or substantial ESD inefficiency to be compensated.",
"The threshold depends on required transmission rate.",
"Remark 3.1 It is worth noticing that if we ignore the battery inefficiency or set $\\eta = 1$ , Lemma REF could be simplified as $P_c > \\frac{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T} - 2^{\\frac{Q}{T}} +1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}}P_H \\;\\;\\Longrightarrow \\;\\;\\rho ^*(1, P_c) > 0$ where only circuit power $P_c$ impacts the save-ratio.",
"Since the MESD and the SESD are equivalent ($\\eta =1$ ), harvesting energy using the MESD is not the reason for delaying transmission.",
"Instead, $\\rho ^* > 0$ when $P_c$ is so large that we should transmit over a shorter interval at a higher power, so that the actual transmission power minimizes $P_{out}$ .",
"Circuit power similarly determined the fundamental tradeoff between energy efficiency and spectral efficiency (data rate) in [19], in which it was shown that with additional circuit power making use of all available time for transmission is not the best strategy in terms of both energy and spectral efficiency.",
"In this paper, outage is minimized through utilizing available (random) energy efficiently, wherein circuit power causes a similar effect."
],
[
"Diversity Analysis",
"The outage performance of wireless transmission over fading channels at high SNR can be conveniently characterized by the so-called diversity order [20], which is the high-SNR slope of the outage probability determined from a SNR-outage plot in the log-log scale.",
"Mathematically, the diversity order is defined as $d = - \\lim _{\\bar{\\gamma } \\rightarrow \\infty } \\frac{\\log _{10}(P_{out})}{\\log _{10}(\\bar{\\gamma })}$ where $P_{out}$ is the outage probability and $\\bar{\\gamma }$ is the average SNR.",
"Diversity order under various fading channel conditions has been comprehensively analyzed in the literature (see e.g.",
"[20] and references therein).",
"Generally speaking, if the fading channel power distribution has an accumulated density near zero that can be approximated by a polynomial term, i.e., $\\mbox{Pr}\\left(|h|^2 \\le \\epsilon \\right) \\approx \\epsilon ^k$ , where $\\epsilon $ is an arbitrary small positive constant, then the constant $k$ indicates the diversity order of the fading channel.",
"For example, in the case of Rayleigh fading channel with $\\mbox{Pr}\\left(|h|^2 \\le \\epsilon \\right)\\approx \\epsilon $ , the diversity order is thus 1 according to (REF ).",
"However, the above diversity analysis is only applicable to conventional wireless systems in which the transmitter has a constant power supply.",
"Since energy harvesting results in random power availability in addition to fading channels, the PDF of the receiver SNR due to both random transmit power and random channel power may not necessarily be polynomially smooth at the origin (as we will show later).",
"As a result, the conventional diversity analysis with constant transmit power cannot be directly applied.",
"In this section, we will investigate the effect of random power on diversity analysis, as compared with the conventional constant-power case.",
"For clarity, in the rest of this section, we consider the ideal system with $\\eta = 1$ and $P_c = 0$ , and the Rayleigh fading channel with $\\mathbb {E}[\\Gamma ] =\\mathbb {E}[\\frac{|h|^2}{\\sigma _n^2}] =\\frac{\\sigma _{h}^2}{\\sigma _{n}^2} = \\lambda _{\\gamma }$ .",
"From (REF ) and (REF ), the outage probability when $\\eta = 1$ and $P_c = 0$ is given by $P_{out} = \\mbox{Pr}\\left\\lbrace \\log _2(1+P\\Gamma ) < \\frac{Q}{(1-\\rho )T} \\right\\rbrace .$ Based on Lemma REF , the minimum outage probability is achieved with the save-ratio $\\rho = 0$ .",
"Therefore, the outage probability is simplified asFor convenience, $P_{out}^{*}$ is used to represent $P_{out}^*(1,0)$ in the rest of this section.",
"$P_{out}^{*} = \\mbox{Pr}\\left\\lbrace P\\Gamma < C\\right\\rbrace = \\int _0^{\\infty }\\int _0^{\\frac{C}{P}}f_P(p)f_{\\Gamma }(\\gamma )d\\gamma dp$ where $C = 2^{\\frac{Q}{T}} - 1$ and the last equality comes from the assumption of Rayleigh fading channel so the $\\Gamma $ is exponential distributed.",
"It is worth noting that in this case with $\\eta = 1$ and $\\rho = 0$ , according to (REF ), the energy arrival rate $X$ and the average total power $P$ are identical and thus can be used interchangeably.",
"Clearly, the near-zero behavior of $P_{out}^{*}$ critically depends on the PDF of random power $f_P(p)$ , while intuitively we should expect that random power can only degrade the outage performance.",
"We choose to use the Gamma distribution to model the random power $P$ , because the Gamma distribution models many positive random variables (RVs) [21], [22].",
"The Gamma distribution is very general, including exponential, Rayleigh, and Chi-Square as special cases; furthermore, the PDF of any positive continuous RV can be properly approximated by the sum of Gamma PDFs.",
"Supposing that $P$ follows a Gamma distribution denoted by $P\\sim \\mathcal {G}(\\beta , \\lambda _p)$ , then its PDF is given by $f_P(p) = \\frac{p^{\\beta - 1}\\mbox{exp}\\left(-\\frac{p}{\\lambda _p}\\right)}{\\lambda _p^{\\beta }\\Gamma (\\beta )}U(p)$ where $U(\\cdot )$ is the unit step function, $\\Gamma (\\cdot )$ is the Gamma function, and $\\beta >0$ , $\\lambda _p>0$ are given parameters.",
"Referring to [23], which gives the distribution of the product of $m$ Gamma RVs, the outage probability in (REF ) can be computed as $P_{out}^{*} = \\frac{1}{\\Gamma (\\beta )}G_{13}^{21}\\left(\\frac{C\\lambda _{\\gamma }}{\\lambda _p}\\left|\\begin{array}{ll}1 \\\\1, \\beta , 0\\end{array}\\right.\\right)$ where $G(\\cdot )$ is the Meijer G-function [21].",
"Meijer G-function can in general only be numerically evaluated and does not give much insights about how random power affects the outage performance.",
"Next, we further assume that the random power $P$ is exponentially distributed (as a special case of Gamma distribution with $\\beta = 1$ ) to demonstrate the effect of random power.",
"Lemma 4.1 Suppose that $P$ is exponentially distributed with mean $\\lambda _p$ , the channel is Rayleigh fading with $\\mathbb {E}[\\Gamma ] =\\frac{\\sigma _{h}^2}{\\sigma _{n}^2} = \\lambda _{\\gamma }$ , and thus the average received SNR $\\bar{\\gamma } = \\lambda _p\\lambda _{\\gamma } =\\frac{\\lambda _p\\sigma _h^2}{\\sigma _n^2}$ .",
"The minimum outage probability $P_{out}^{*}$ , under an ideal system with $\\eta = 1$ and $P_c = 0$ , is given by $P_{out}^{*} = \\sum _{k=0}^{\\infty }\\frac{C^{k+1}}{(k!",
")^2(k+1)\\bar{\\gamma }^{k+1}}\\left[\\frac{1}{k+1} - \\ln \\frac{C}{\\bar{\\gamma }} + 2\\psi (k+1)\\right]$ where $\\psi (x)$ is the digamma function [24] and $\\ln \\left(\\cdot \\right)$ represents the natural logarithm.",
"Please refer to Appendix .",
"In the asymptotically high-SNRWe assume that high SNR is achieved via decreasing noise power $\\sigma _{n}^2$ , while fixing the average harvested energy.",
"regime, we can approximate $P_{out}^{*}$ by taking only the first term of () as $P_{out}^{*} & \\approx \\frac{C}{\\bar{\\gamma }}\\left(1 - \\ln {\\frac{C}{\\bar{\\gamma }}} + 2\\psi (1)\\right) \\approx \\frac{\\ln {\\bar{\\gamma }}}{\\bar{\\gamma }}.$ As observed, $P_{out}^{*}$ decays as $\\bar{\\gamma }^{-1}\\ln {(\\bar{\\gamma })}$ rather than $\\bar{\\gamma }^{-1}$ as in the conventional case with constant power, which indicates that the PDF of the receiver SNR is no longer polynomially smooth near the origin.",
"Hence, the slope of $P_{out}^{*}$ in the SNR-outage plot, or the diversity order, will converge much more slowly to $\\bar{\\gamma }^{-1}$ with SNR than in the constant-power case, suggesting that random energy arrival has a significant impact on the diversity performance.",
"More specifically, we obtain the diversity order in the case of exponentially distributed random power as $ d = -\\lim _{\\bar{\\gamma } \\rightarrow \\infty } \\frac{-\\log _{10}{\\bar{\\gamma }} +\\log _{10}{\\left(\\ln {\\bar{\\gamma }}\\right)}}{\\log _{10}{\\bar{\\gamma }}}= 1$ which is, in principle, the same as that over the Rayleigh fading channel with constant power.",
"We thus conclude that diversity order may not be as meaningful a metric of evaluating outage performance in the presence of random power, as in the conventional case of constant power."
],
[
"Multiple Transmitters",
"In this section, we extend the ST protocol for the single-channel case to the more practical case of multiple transmitters in a wireless network, and quantify the system-level outage performance as a function of the number of transmitters in the network."
],
[
"TDMA-ST",
"We consider a wireless network with $N$ transmitters, each of which needs to transmit $Q$ bits of data within a time frame of duration $T$ seconds to a common fusion center (FC).",
"It is assumed that each transmitter is powered by the same energy harvesting circuit model as shown in Fig.",
"REF , and transmits over the baseband-equivalent channel model given in ().",
"We also assume a homogeneous system setup, in which the channel gains, energy harvesting rates or additive noises for all transmitter-FC links are independent and identically distributed (i.i.d).",
"Figure: TDMA based ST (TDMA-ST)In order to allow multiple transmitters to communicate with the FC, we propose a TDMA based ST (TDMA-ST) protocol as follows (cf.",
"Fig.",
"REF ): Every frame is equally divided into $N$ orthogonal time slots with each slot equal to $\\frac{T}{N}$ seconds.",
"Assuming perfect time synchronization, each transmitter is assigned a different (periodically repeating) time slot for transmission, i.e., in each frame, transmitter $i$ is allocated the time slot $\\left[\\frac{(i-1)}{N}T, ~ \\frac{i}{N}T\\right)$ , $1 \\le i \\le N$ .",
"Assuming $\\rho _i = \\rho $ for all $i$ 's, each transmitter implements the ST protocol with the transmission time in each frame aligned to be within its assigned time slot; as a result, the maximum transmit-ratio, denoted by $1-\\rho $ , for each transmitter cannot exceed $1/N$ , which means that $\\rho \\ge 1-\\frac{1}{N}$ .",
"The protocol described above is illustrated in Fig.",
"REF .",
"Unlike the single-channel case where the transmitter can select any save-ratio $\\rho $ in the interval $0\\le \\rho \\le 1$ , in the case of TDMA-ST, $\\rho $ is further constrained by $\\rho \\ge 1- \\frac{1}{N}$ to ensure orthogonal transmissions by all transmitters.",
"Due to this limitation, each transmitter may not be able to work at its individual minimum outage probability unless the corresponding optimal save-ratio $\\rho ^*$ satisfies $\\rho ^* \\ge 1- \\frac{1}{N}$ or $N \\le \\frac{1}{1-\\rho ^*}$ .",
"In this case, ST protocol naturally extends to TDMA-ST with every transmitter operating at the optimal save-ratio $\\rho ^*$ .",
"However, if $N$ exceeds the threshold $\\frac{1}{1-\\rho ^*}$ , transmitters have to deviate from $\\rho ^*$ to maintain orthogonal transmissions.",
"Next, we evaluate the performance of TDMA-ST for two types of source data at transmitters: Independent Data and Common Data."
],
[
"Independent Data",
"First, consider the case where all transmitters send independent data packets to the FC in each frame, which are decoded separately at the FC.",
"Under the symmetric setup, for a given $\\rho $ , all transmitters should have the same average outage performance.",
"Consequently, the system-level outage performance in the case of independent data can be equivalently measured by that of the individual transmitter, i.e., $P_{out}^s = P_{out}(\\rho , \\eta , P_c).$ We can further investigate the following two cases: $N \\le \\frac{1}{1-\\rho ^*}$ In this case, the additional constraint due to TDMA, $\\rho ^* \\ge 1-\\frac{1}{N}$ , is satisfied.",
"Since $P_{out}^s$ is the same as that of the single-transmitter case, the system is optimized when all transmitters work at their individual minimum outage with save-ratio $\\rho ^*$ .",
"Thus, the minimum system outage probability is $P_{out}^{s*} = P_{out}^*(\\eta , P_c)$ .",
"$N > \\frac{1}{1-\\rho ^*}$ In this case, the TDMA constraint on $\\rho ^*$ is violated and thus we are not able to allocate all transmitters the save-ratio $\\rho ^*$ , which means that each transmitter has to deviate from its minimum outage point.",
"Since in this case $\\rho ^*< 1 -\\frac{1}{N}\\le \\rho $ , the best strategy for each transmitter is to choose $\\rho = 1 - \\frac{1}{N} $ .",
"Thus, $P_{out}^{s*} =P_{out}(1-\\frac{1}{N},\\eta , P_c)$ .",
"To summarize, the optimal strategy for each transmitter in the case of independent data is given by $\\rho = \\left\\lbrace \\begin{array}{ll}\\rho ^*, ~~~~ & N \\le \\frac{1}{1-\\rho ^*} \\\\1 - \\frac{1}{N}, ~~~~ & N > \\frac{1}{1-\\rho ^*}\\end{array}\\right.$ which implies that the number of transmitters should be kept below the reciprocal of the single-channel optimal transmit-ratio; otherwise, the system outage performance will degrade."
],
[
"Common Data",
"Next, consider the case where all transmitters send identical data packets in each frame to the FC, which applies diversity combining techniques to decode the common data.",
"For simplicity, we consider selection combining (SC) at the receiver, but similar results can be obtained for other diversity combining techniques [20].",
"With SC, the system outage probability is given by [20] $P_{out}^s = \\left(P_{out}(\\rho , \\eta , P_c)\\right)^{N}.$ Similarly to the case of independent data, we can get exactly the same result for the optimal transmit strategy given in (REF ) for the common-data case, with which the minimum system outage probability is obtained as $P_{out}^{s*} = \\left\\lbrace \\begin{array}{ll}\\left(P_{out}^*(\\eta , P_c)\\right)^{N}, ~~~~ & N \\le \\frac{1}{1-\\rho ^*} \\\\\\left(P_{out}(1 - \\frac{1}{N}, \\eta , P_c)\\right)^{N}, ~~~~ & N > \\frac{1}{1-\\rho ^*}\\end{array}\\right.$ From the above, it is evident that the system outage probability initially drops as $N$ increases, provided that $N \\le \\frac{1}{1-\\rho ^*}$ .",
"However, when $N> \\frac{1}{1-\\rho ^*}$ , it is not immediately clear whether the system outage increases or decreases with $N$ , since increasing $N$ improves the SC diversity, but also makes each transmitter deviate even further from its minimum outage save-ratio according to (REF )."
],
[
"Numerical Examples",
"In this section, we provide numerical examples to validate our claims.",
"We assume that the energy harvesting rate $X$ follows a uniform distribution (unless specified otherwise) within $[0, 100]$ (i.e., $P_H = 100$ J/s), and the channel is Rayleigh fading with exponentially distributed $\\Gamma $ with parameter $\\lambda =0.02$ .",
"We also assume the target transmission rate $R_{\\mathrm {req}} = \\frac{Q}{T} = 2$ bits/s.This is normalized to a bandwidth of 1 Hz, i.e.",
"$R_{\\mathrm {req}}$ is the spectral efficiency in bis/s/Hz.",
"Figure: Optimal save-ratio ρ * \\rho ^*Fig.",
"REF demonstrates how battery efficiency $\\eta $ and circuit power $P_c$ affect the optimal save-ratio $\\rho ^*$ for the single-channel case.",
"As observed, larger $P_c$ and smaller $\\eta $ result in larger $\\rho ^*$ , i.e.",
"shorter transmission time.",
"Since the increment is more substantial along $P_c$ axis, circuit power has a larger influence on the optimal save-ratio compared with battery efficiency.",
"$\\rho ^*(1,0) = 0$ verifies the result of Lemma REF for an ideal system, while $\\rho ^*(\\eta , 0)$ along the line $P_c = 0$ demonstrates the “phase transition” behavior stated in Lemma REF .",
"The transition point is observed to be $\\eta = 0.541$ , which can also be computed from ().",
"Figure: Optimal outage probability P out * P_{out}^*Fig.",
"REF shows the optimal (minimum) outage probability $P_{out}^*(\\eta , P_c)$ corresponding to $\\rho ^*$ in Fig.",
"REF .",
"Consistent with Proposition REF , $P_{out}^*(\\eta , P_c)$ is observed to be monotonically decreasing with battery efficiency $\\eta $ and monotonically increasing with circuit power $P_c$ .",
"Again, $P_c$ affects outage performance more significantly than $\\eta $ .",
"From Fig.",
"REF , we see that for a reasonable outage probability e.g.",
"below $0.05$ , $P_c$ has to be small and $\\eta $ has to be close to 1.",
"Our results can thus be used to find the feasible region in the $\\eta -P_c$ plane for a given allowable $P_{out}$ .",
"Figure: Outage performance comparison: P c P H =0.5\\frac{P_c}{P_H} = 0.5Figure: Outage performance comparison: η=0.8\\eta = 0.8Figs.",
"REF and REF compare the outage performance with versus without save-ratio optimization.",
"In Fig.",
"REF we fix the normalized circuit power $\\frac{P_c}{P_H} = 0.5$ , while in Fig.",
"REF we fix the battery efficiency $\\eta = 0.8$ .",
"We observe that optimizing the save-ratio can significantly improve the outage performance.",
"It is worth noting that $P_{out}$ has an approximately linear relationship with the normalized circuit power $\\frac{P_c}{P_H}$ as observed in Fig.",
"REF , which indicates that $P_c$ considerably affects the outage performance as stated previously.",
"Figure: Outage probability for an ideal (η=1\\eta = 1, P c =0P_c = 0) system with constant powerversus random powerFigure: Outage probability comparison for ideal (η=1\\eta = 1, P c =0P_c = 0) versus non-ideal (η=0.8\\eta = 0.8, P c =0.1*𝔼[P]P_c = 0.1*\\mathbb {E}[P])systemsIn Fig.",
"REF , the outage probability for an ideal system ($\\eta = 1$ , $P_c = 0$ ) is shown by numerically evaluating (REF ).",
"By fixing the mean value of $P$ as $\\mathbb {E}[P] = 50$ J/s and varying $\\beta $ for the Gamma distributed power from 1 to 5, the resulting outage performance is compared with the case of constant power.",
"As observed, the outage probability increases due to the existence of power randomness.",
"As $\\beta $ increases, the outage curve approaches the case of constant power.",
"In Fig.",
"REF , we also plot the outage probability for the ideal system with exponentially distributed power based on the approximation given in (REF ), as well as for a non-ideal system with the normalized circuit power $\\frac{P_c}{\\mathbb {E}[P]} = 0.1$ and battery efficiency $\\eta = 0.8$ .",
"In comparison with the constant-power case, for the case of ideal system we observe that the high-SNR slope or diversity order with random power clearly converges much slower with SNR, which is in accordance with our analysis in Section .",
"Furthermore, at $P_{out} = 10^{-3}$ , there is about 10 dB power penalty observed due to exponential random power, even with the same diversity order as the constant-power case.",
"It is also observed that there is a small rising part for the outage approximation given in (REF ), since this approximation is only valid for sufficiently high SNR values ($\\bar{\\gamma } > 10$ dB).",
"Finally, it is worth noting that the outage probability for the non-ideal system eventually saturates with SNR (regardless of how small the noise power is or how large the SNR is), which indicates that the diversity order is zero for any non-ideal system.",
"Figure: Outage performance of multiple transmitters under TDMA-STprotocol, with 1 1-ρ * =4.83\\frac{1}{1 - \\rho ^*} = 4.83Fig.",
"REF shows the outage performance for the case of multiple transmitters operating under the TDMA-ST protocol.",
"We set the normalized circuit power $\\frac{P_c}{P_H} = 0.5$ and the battery efficiency $\\eta = 0.9$ .",
"Then, the optimal save-ratio $\\rho ^*$ for single-transmitter outage minimization can be obtained as $0.7930$ by numerical search.",
"Therefore, the threshold value for $N$ in the optimal rule of assigning save-ratio values in (REF ) is $\\frac{1}{1 - \\rho ^*} = 4.83$ .",
"For the case of independent data, it is observed that when $N \\le 4$ , the system outage probability is constantly equal to the optimal single-transmitter outage probability $P_{out}^*(0.9,0.5P_H)$ ; however, as $N>4$ , the outage probability increases dramatically.",
"In contrast, for the case of common data, it is observed that the system outage probability decreases initially as $N$ increases, even after the threshold value and until $N = 7$ , beyond which it starts increasing.",
"This implies that there is an optimal decision on the number of transmitters to achieve the optimal outage performance."
],
[
"Conclusion",
"In this paper, we studied a wireless system under practical energy harvesting conditions.",
"Assuming a general model with non-ideal energy storage efficiency and transmit circuit power, we proposed a Save-then-Transmit (ST) protocol to optimize the system outage performance via finding the optimal save-ratio.",
"We characterized how the optimal save-ratio and the minimum outage probability vary with practical system parameters.",
"We compared the outage performance between random power and constant power under the assumption of Rayleigh fading channel.",
"It is shown that random power considerably degrades the outage performance.",
"Furthermore, we presented a TDMA-ST protocol for wireless networks with multiple transmitters.",
"In particular, two types of source data are examined: independent data and common data.",
"It is shown that if the number of transmitters is smaller than the reciprocal of the optimal transmit-ratio for single-transmitter outage minimization, each transmitter should work with its minimum outage save-ratio; however, when the number of transmitters exceeds this threshold, each transmitter has to deviate from its individual optimal operating point.",
"There are important problems that remain unaddressed in this paper and are worth investigating in the future.",
"For example, we may consider the effect of different configurations of battery/supercapacitor and MESD/SESD on the system performance.",
"It is also interesting to investigate the information-theoretic limits for the ST protocol in the case of multiple transmitters using more sophisticated multiple-access techniques other than the simple TDMA."
],
[
"Proof of Proposition ",
"According to the Fundamental Theorem of Calculus [26], we can derive the first derivative of $\\hat{P}_{out}(\\rho , \\eta ,P_c)$ in (REF ) with respect to $\\eta $ , $P_c$ and $\\rho $ as $\\frac{\\partial \\hat{P}_{out}}{\\partial \\beta } & = \\left(\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }\\right)^{^{\\prime }}f_x\\left(\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }\\right) \\nonumber \\\\& - \\left(\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }\\right)^{^{\\prime }}f_x\\left(\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }\\right)F_{\\Gamma }(\\infty ) \\nonumber \\\\& +\\int _{\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }}^{P_H}f_X(x)f_{\\Gamma }\\left[g(\\cdot )\\right]\\frac{\\partial g(\\cdot )}{\\partial \\beta }dx \\nonumber \\\\& = \\int _{\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }}^{P_H}f_X(x)f_{\\Gamma }\\left[g(\\cdot )\\right]\\frac{\\partial g(\\cdot )}{\\partial \\beta }dx$ where $\\beta $ could be $\\eta $ , $P_c$ or $\\rho $ and $g(\\rho , \\eta , P_c) = \\frac{2^{\\frac{Q}{(1-\\rho )T}}-1}{x[\\frac{\\rho }{1-\\rho }+\\eta ]-P_c}$ .",
"It is easy to verify that $\\frac{\\partial g(\\rho , \\eta , P_c)}{\\partial \\eta } < 0, \\forall \\eta \\in [0, 1]$ and $\\frac{\\partial g(\\rho , \\eta , P_c)}{\\partial P_c} > 0, \\forall P_c \\in [0, \\infty ]$ .",
"Therefore $\\hat{P}_{out}(\\rho , \\eta , P_c)$ is strictly decreasing with battery efficiency $\\eta $ and strictly increasing with circuit power $P_c$ .",
"Next, we are going to prove the monotonicity of $P_{out}$ and $P_{out}^*$ with battery efficiency $\\eta $ , where circuit power $P_c$ is treated as constant.",
"The condition $P_H > \\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }$ in (REF ) could be expressed in terms of battery efficiency: $\\eta > \\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho }$ , then $P_{out} = \\left\\lbrace \\begin{array}{ll}1, ~~~~ & \\eta \\le \\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho } \\\\\\hat{P}_{out}, ~~~~ & \\eta > \\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho }\\end{array}\\right..$ Consider the following two cases: Suppose $\\eta _1 < \\eta _2$ and $\\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho } > 0$ If $\\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho } < \\eta _1 < \\eta _2 $ , then $P_{out}(\\rho , \\eta _1, P_c) = \\hat{P}_{out}(\\rho , \\eta _1, P_c)$ and $P_{out}(\\rho , \\eta _2, P_c) = \\hat{P}_{out}(\\rho , \\eta _2, P_c)$ .",
"Since $\\hat{P}_{out}(\\rho , \\eta , P_c)$ is strictly decreasing with battery efficiency $\\eta $ , we have $P_{out}(\\rho , \\eta _1, P_c) > P_{out}(\\rho , \\eta _2, P_c).",
"\\nonumber $ If $\\eta _1 \\le \\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho } < \\eta _2$ , then $P_{out}(\\rho , \\eta _1, P_c) = 1$ and $P_{out}(\\rho , \\eta _2, P_c) = \\hat{P}_{out}(\\rho , \\eta _2, P_c)$ .",
"Therefore $P_{out}(\\rho , \\eta _1, P_c) = 1 > P_{out}(\\rho , \\eta _2, P_c).",
"\\nonumber $ If $\\eta _1 < \\eta _2 \\le \\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho }$ , then $P_{out}(\\rho , \\eta _1, P_c) = P_{out}(\\rho , \\eta _2, P_c) = 1$ , which means $P_{out}(\\rho , \\eta _1, P_c) = P_{out}(\\rho , \\eta _2, P_c).",
"\\nonumber $ Suppose $\\eta _1 < \\eta _2$ and $\\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho } \\le 0$ , we have $\\frac{P_c}{P_H} - \\frac{\\rho }{1-\\rho } \\le \\eta _1 < \\eta _2$ .",
"Then it could be easily verified that $P_{out}(\\rho , \\eta _1, P_c) > P_{out}(\\rho , \\eta _2, P_c).",
"\\nonumber $ Combining all the above cases, we can conclude that $P_{out}(\\rho ,\\eta , P_c)$ is a non-increasing function of battery efficiency $\\eta $ given any non-zero circuit power $P_c$ for $\\rho \\in [0, 1)$ .",
"Next, we proceed to prove the monotonicity of $P_{out}^*(\\eta ,P_c)$ .",
"Assuming $\\eta _1 < \\eta _2$ again, then we could argue that $\\frac{P_c}{P_H} - \\frac{\\rho _1^*}{1-\\rho _1^*} < \\eta _1$ and $\\frac{P_c}{P_H} - \\frac{\\rho _2^*}{1-\\rho _2^*} < \\eta _2$ , where $\\rho _1^*$ and $\\rho _2^*$ are the optimal save-ratio for $\\eta =\\eta _1$ and $\\eta =\\eta _2$ , respectively.",
"Therefore we only need to consider two cases: $\\max \\left\\lbrace \\frac{P_c}{P_H} -\\frac{\\rho _1^*}{1-\\rho _1^*}, \\frac{P_c}{P_H} -\\frac{\\rho _2^*}{1-\\rho _2^*}\\right\\rbrace < \\eta _1 < \\eta _2$ and $\\eta _1\\le \\max \\left\\lbrace \\frac{P_c}{P_H} - \\frac{\\rho _1^*}{1-\\rho _1^*},\\frac{P_c}{P_H} - \\frac{\\rho _2^*}{1-\\rho _2^*}\\right\\rbrace < \\eta _2$ .",
"From the arguments we have given for the proof of the monotonicity of $P_{out}$ we know that, under these two conditions we have $P_{out}(\\rho , \\eta _1, P_c) > P_{out}(\\rho , \\eta _2, P_c).",
"\\nonumber $ Therefore, $P_{out}^*(\\eta _1, P_c) > P_{out}(\\rho _1^*, \\eta _2, P_c) \\ge P_{out}^*(\\eta _2, P_c) \\nonumber $ which completes the proof of the monotonicity for $P_{out}^*(\\eta ,P_c)$ .",
"With similar arguments, we could get the results regarding circuit power $P_c$ .",
"Proposition REF is thus proved."
],
[
"Proof of Lemma ",
"Since $F_\\Gamma (\\cdot )$ is non-negative and non-decreasing, we have $a < b \\;\\;\\Rightarrow \\;\\;F_\\Gamma \\left(\\frac{a}{x}\\right) \\le F_\\Gamma \\left(\\frac{b}{x}\\right)$ for any $x \\in [0,P_H]$ .",
"Since $f_X(\\cdot )$ is non-negative, this leads to $a < b \\;\\;&\\Rightarrow \\;\\; \\int _0^{P_H} f_X(x)F_\\Gamma \\left(\\frac{a}{x}\\right) dx \\nonumber \\\\&\\le \\int _0^{P_H} f_X(x)F_\\Gamma \\left(\\frac{b}{x}\\right) dx.",
"\\nonumber $ Given the form of $P_{out}$ in Problem (P2), with $\\rho $ appearing only in the numerator of the argument of $F_\\Gamma (\\cdot )$ , we conclude that $P_{out}$ is a non-decreasing function of $g(\\rho ) = \\left(2^{\\frac{Q}{(1-\\rho )T}}-1\\right)(1-\\rho )$ .",
"Hence minimizing $g(\\rho )$ is equivalent to minimizing $P_{out}$ .",
"The first and second derivatives of $g(\\rho )$ are $g^{\\prime }(\\rho ) & = 2^{\\frac{Q}{(1-\\rho )T}}(\\ln 2)\\frac{Q}{(1-\\rho )T} - 2^{\\frac{Q}{(1-\\rho )T}} + 1 \\nonumber \\\\g^{\\prime \\prime }(\\rho ) & = 2^{\\frac{Q}{(1-\\rho )T}}(\\ln 2)^2\\frac{Q^2}{T^2(1-\\rho )^3} > 0 \\quad \\mbox{since $Q > 0$.}",
"\\nonumber $ Let $h(\\rho ) = g^{\\prime }(\\rho )$ .",
"From the second equation above, $h(\\rho )$ is an increasing function.",
"In the range $0 \\le \\rho \\le 1$ , $h(\\rho )$ is thus minimized at $\\rho = 0$ , i.e.",
"the minimum of $g^{\\prime }(\\rho )$ is $h(0)$ , given by $g^{\\prime }_{\\mathrm {min}} &=& 2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T} - 2^{\\frac{Q}{T}} + 1 \\\\&=& 2^{\\frac{Q}{T}}\\left(\\ln 2^{Q/T} - 1\\right) + 1 > 0$ for $Q > 0$ .",
"In other words, the smallest value that the gradient of $g(\\rho )$ can take in the range $0 \\le \\rho \\le 1$ for any feasible $Q$ is positive, which implies that $g(\\rho )$ is increasing and therefore minimized at $\\rho = 0$ , as claimed.",
"The proof of Lemma REF is thus completed."
],
[
"Proof of Lemma ",
"To prove Property 1, we observe that as noted in the proof of Lemma REF , $P_{out}$ is a monotonic function of $g(\\rho ) = \\frac{(2^{\\frac{Q}{(1-\\rho )T}}-1)}{(\\frac{\\rho }{1-\\rho }+\\eta )}$ in Problem (P3), hence minimizing $g(\\rho )$ leads to the same solution as minimizing $P_{out}$ .",
"The first derivative of $g(\\rho )$ is $g^{\\prime }(\\rho ) & = \\frac{2^{\\frac{Q}{(1-\\rho )T}}(\\ln 2)\\frac{Q}{(1-\\rho )T}[\\rho +\\eta (1-\\rho )] - 2^{\\frac{Q}{(1-\\rho )T}} + 1}{[\\rho +\\eta (1-\\rho )]^2} \\nonumber \\\\& = \\frac{u(\\rho )}{[\\rho +\\eta (1-\\rho )]^2}.",
"\\nonumber $ It is clear in the above that the sign of $g^{\\prime }(\\rho )$ is the same as that of $u(\\rho )$ .",
"Since $u(1) = +\\infty $ and $u(\\rho )$ is a differentiable function, if $u(0)$ is negative then there exists a value $\\rho _c \\in (0,1)$ such that $u(\\rho _c) = 0 = g^{\\prime }(\\rho _c)$ .",
"It is easily verified that $u^{\\prime }(\\rho ) > 0$ ; hence $\\rho _c$ is the unique optimal value of $\\rho $ in this case.",
"Conversely, if there exists an $\\rho _c$ such that $u(\\rho _c) = 0$ , then $u(0)$ must be negative.",
"Hence $u(0) < 0$ is a necessary and sufficient condition for the optimal $\\rho $ to lie in $(0,1)$ .",
"The condition $u(0) < 0$ translates into the following condition on $\\eta $ , which proves the first part of the lemma: $ u(0) < 0 \\;\\;& \\Rightarrow \\;\\; 2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}\\eta - 2^{\\frac{Q}{T}} + 1 < 0 \\nonumber \\\\& \\Rightarrow ~~ \\eta < \\frac{2^{\\frac{Q}{T}}-1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}}.$ To prove the second point, suppose $\\rho _1^*(\\eta _1,0)$ and $\\rho _2^*(\\eta _2,0)$ are optimal save-ratios of (P3) for SESD efficiencies $\\eta _1$ and $\\eta _2$ , where $\\eta _1 < \\eta _2$ .",
"Then, $u(\\rho _1^*, \\eta _1) = 0$ and $u(\\rho _2^*, \\eta _2) = 0$ .",
"Since $\\eta _1 < \\eta _2$ and $u(\\rho , \\eta )$ is an increasing function of $\\eta $ , we have $u(\\rho _1^*, \\eta _2) > 0$ .",
"Combining what we have that $u(\\rho , \\eta )$ is an increasing function of $\\rho $ , $u(\\rho _2^*, \\eta _2) = 0$ and $u(\\rho _1^*, \\eta _2) > 0$ , we may conclude $\\rho _2^*(\\eta _2,0) < \\rho _1^*(\\eta _1,0)$ .",
"Lemma REF is thus proved."
],
[
"Proof of Lemma ",
"According to the proof of Proposition REF , the first derivative of $\\hat{P}_{out}(\\rho , \\eta , P_c)$ with respect to $\\eta $ , $P_c$ and $\\rho $ is, $\\frac{\\partial \\hat{P}_{out}}{\\partial \\beta } = \\int _{\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }}^{P_H}f_X(x)f_{\\Gamma }\\left[g(\\cdot )\\right]\\frac{\\partial g(\\cdot )}{\\partial \\beta }dx \\nonumber $ where $\\beta $ could be $\\eta $ , $P_c$ or $\\rho $ and $g(\\rho , \\eta ,P_c) =\\frac{2^{\\frac{Q}{(1-\\rho )T}}-1}{x[\\frac{\\rho }{1-\\rho }+\\eta ]-P_c}$ .",
"Furthermore, we have $\\frac{\\partial g(\\rho )}{\\partial \\rho } & = \\frac{2^{\\frac{Q}{(1-\\rho )T}}(\\ln 2)\\frac{Q}{(1-\\rho )T}\\left[\\rho +\\eta (1-\\rho ) - (1-\\rho )\\frac{P_c}{x}\\right]}{x\\left[\\rho +\\eta (1-\\rho ) - (1-\\rho )\\frac{P_c}{x}\\right]^{2}} \\nonumber \\\\& - \\frac{2^{\\frac{Q}{(1-\\rho )T}} - 1}{x\\left[\\rho +\\eta (1-\\rho ) - (1-\\rho )\\frac{P_c}{x}\\right]^{2}} \\nonumber \\\\& = \\frac{v(\\rho )}{x\\left[\\rho +\\eta (1-\\rho ) - (1-\\rho )\\frac{P_c}{x}\\right]^{2}}.",
"\\nonumber $ With similar arguments about $u(\\rho )$ in the proof of Lemma REF , we claim that $v(0) < 0, \\forall x \\in (\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }, P_H]$ is a sufficient condition of having $\\rho ^*(\\eta , P_c) > 0$ while $P_c < \\eta P_H$ .",
"Since $v(0)$ is an increasing function of $x$ , the condition $v(0) < 0, \\forall x \\in (\\frac{P_c}{\\frac{\\rho }{1-\\rho }+\\eta }, P_H]$ translates into the following condition on $\\eta $ and $P_c$ $v(0) & = 2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}\\left(\\eta - \\frac{P_c}{x}\\right) - 2^{\\frac{Q}{T}} + 1 \\nonumber \\\\& < 2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}\\left(\\eta - \\frac{P_c}{P_H}\\right) - 2^{\\frac{Q}{T}} + 1 < 0 \\nonumber \\\\& \\Longrightarrow 0 < \\eta - \\frac{P_c}{P_H} < \\frac{2^{\\frac{Q}{T}} - 1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}}.",
"\\nonumber $ Combined with the fact that $\\rho ^*(\\eta , P_c) > \\frac{\\frac{P_c}{P_H}-\\eta }{1-\\eta +\\frac{P_c}{P_H}}$ when $P_c \\ge \\eta P_H$ , we may conclude $\\rho ^*(\\eta , P_c) > 0, ~~~~ \\eta - \\frac{P_c}{P_H} < \\frac{2^{\\frac{Q}{T}} - 1}{2^{\\frac{Q}{T}}(\\ln 2)\\frac{Q}{T}}.$ Lemma REF is thus proved."
],
[
"Proof of Lemma ",
"Let $Z = P\\Gamma $ , where $P$ and $\\Gamma $ are exponential random variables with mean $\\lambda _p$ and $\\lambda _{\\gamma }$ respectively.",
"Then the PDF of $Z$ could be derived as follows, $F_Z(z) & = \\mbox{Pr}\\left\\lbrace P\\Gamma \\le z \\right\\rbrace \\nonumber \\\\& = 1 - \\frac{1}{\\lambda _p}\\int _0^{\\infty }e^{-\\frac{z}{p\\lambda _{\\gamma }}}e^{-\\frac{p}{\\lambda _p}}dp \\nonumber \\\\& = 1 - 2\\sqrt{\\frac{z}{\\lambda _p\\lambda _{\\gamma }}}K_1\\left(2\\sqrt{\\frac{z}{\\lambda _p\\lambda _{\\gamma }}}\\right)$ where $K_1(x)$ is the first-order modified Bessel function of the second kind and the last equality is given by [25]: $\\int _0^{\\infty }\\exp \\left(-\\frac{\\beta }{4x} - \\gamma x\\right)dx = \\sqrt{\\frac{\\beta }{\\gamma }}K_1\\left(\\sqrt{\\beta \\gamma }\\right) \\nonumber $ where $\\Re (\\beta ) \\ge 0, \\Re (\\gamma ) \\ge 0$ .",
"Let $M =\\frac{1}{\\sqrt{\\lambda _p\\lambda _{\\gamma }}}$ .",
"Taking the derivative of $F(z)$ yields $f(z) & = M\\left\\lbrace -\\frac{1}{\\sqrt{z}}K_1\\left(2M\\sqrt{z}\\right) - 2\\sqrt{z}\\left(K_1\\left(2M\\sqrt{z}\\right)\\right)^{^{\\prime }}\\right\\rbrace \\nonumber \\\\& = M\\left\\lbrace -\\frac{1}{\\sqrt{z}}K_1\\left(2M\\sqrt{z}\\right)\\right.",
"\\nonumber \\\\& - \\left.2\\sqrt{z}\\left(-K_0(2M\\sqrt{z})-\\frac{1}{2M\\sqrt{z}}K_1(2M\\sqrt{z})\\right)\\frac{M}{\\sqrt{z}}\\right\\rbrace \\nonumber \\\\& = 2M^2K_0\\left(2M\\sqrt{z}\\right) \\nonumber \\\\& = \\frac{2}{\\lambda _p\\lambda _{\\gamma }}K_0\\left(2\\sqrt{\\frac{z}{\\lambda _p\\lambda _{\\gamma }}}\\right)$ where $\\frac{\\partial K_v(z)}{\\partial z} = - K_{v-1}(z) - \\frac{v}{z}K_v(z)$ .",
"Next, we characterize the outage probability using (REF ).",
"According to (), we have $P_{out}^{*} &= \\mbox{Pr}\\left[P\\Gamma < C\\right] \\nonumber \\\\& =\\int _{0}^{C}\\frac{2}{\\lambda _p\\lambda _{\\gamma }}K_0\\left(2\\sqrt{\\frac{z}{\\lambda _p\\lambda _{\\gamma }}}\\right)dz.$ Let $X = \\frac{z}{\\lambda _p\\lambda _{\\gamma }}$ and $D =\\frac{C}{\\lambda _p\\lambda _{\\gamma }}$ .",
"We then have $P_{out}^{*} = 2\\int _0^{D}K_0\\left(2\\sqrt{x}\\right)dx.$ Using the series presentation [25], we have $K_0(x) = -\\ln \\left(\\frac{x}{2}\\right)I_0(x)+ \\sum _{k=0}^{\\infty }\\frac{x^{2k}}{2^{2k}(k!",
")^2}\\psi (k+1)$ with the series expansion for the modified Bessel function given by $I_0(x) = \\sum _{k=0}^{\\infty }\\frac{x^{2k}}{2^{2k}(k!",
")^2}.$ (REF ) could be expanded as $P_{out}^{*} = \\sum _{k=0}^{\\infty }\\frac{2}{(k!",
")^2}\\left[-\\frac{1}{2}\\int _0^D x^k\\ln xdx + \\psi (k+1)\\int _0^{D}x^kdx\\right]$ where $\\psi (x) = \\frac{d}{dx}\\ln \\Gamma (x) = \\frac{\\Gamma (x)^{^{\\prime }}}{\\Gamma (x)}$ is the digamma function [24].",
"Since the two integrals in (REF ) could be evaluated as $\\int _0^Dx^kdx & = \\frac{D^{k+1}}{k+1} \\nonumber \\\\\\int _0^D x^k\\ln xdx & = \\left.x^{k+1}\\left(\\frac{\\ln x}{k+1} - \\frac{1}{(k+1)^2}\\right)\\right|_{x=0}^{x=D} \\nonumber \\\\& = D^{k+1}\\left(\\frac{\\ln D}{k+1} - \\frac{1}{(k+1)^2}\\right) \\nonumber $ where $\\lim _{x \\rightarrow 0} (x\\ln {x}) = 0$ .",
"Then we have $P_{out}^{*} =\\sum _{k=0}^{\\infty }\\frac{2}{(k!",
")^2}\\frac{D^{k+1}}{k+1}\\left[-\\frac{1}{2}\\left(\\ln D - \\frac{1}{k+1}\\right) + \\psi (k+1)\\right].$ Since $D = \\frac{C}{\\lambda _p\\lambda _{\\gamma }} =\\frac{C\\sigma _n^2}{\\lambda _p\\sigma _h^2} = \\frac{C}{\\bar{\\gamma }}$ , (REF ) follows.",
"Lemma REF is thus proved."
]
] | 1204.1240 |
[
[
"The $\\alpha$, $\\beta$ and $\\gamma$ parameterizations of CP violating CKM\n phase"
],
[
"Abstract The CKM matrix describing quark mixing with three generations can be parameterized by three mixing angles and one CP violating phase.",
"In most of the parameterizations, the CP violating phase chosen is not a directly measurable quantity and is parametrization dependent.",
"In this work, we propose to use experimentally measurable CP violating quantities, $\\alpha$, $\\beta$ or $\\gamma$ in the unitarity triangle as the phase in the CKM matrix, and construct explicit $\\alpha$, $\\beta$ and $\\gamma$ parameterizations.",
"Approximate Wolfenstein-like expressions are also suggested."
],
[
"Introduction",
"The mixing between different quarks is described by an unitary matrix in the charged current interaction of W-boson in the mass eigen-state of quarks, the Cabibbo [1]-Kobayashi-Maskawa [2](CKM) matrix $V_{\\rm {CKM}}$ , defined by $L = -{g\\over \\sqrt{2}} \\overline{U}_L \\gamma ^\\mu V_{\\rm CKM} D_LW^+_\\mu + H.C.\\;,$ where $U_L = (u_L,c_L,t_L,...)^T$ , $D_L = (d_L,s_L,b_L,...)^T$ .",
"For n-generations, $V =V_{\\rm CKM}$ is an $n\\times n$ unitary matrix.",
"With three generations, one can write $V_{\\rm CKM} = \\left( \\begin{array}{lll}V_{ud}&V_{us}&V_{ub}\\\\V_{cd}&V_{cs}&V_{cb}\\\\V_{td}&V_{ts}&V_{tb}\\end{array}\\right)\\;.$ A commonly used parametrization for mixing matrix with three generations of quark is given by [3], [4], $V_{PDG} = \\left(\\begin{array}{ccc}c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\\delta _{PDG}} \\\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\\delta _{PDG}} &c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\\delta _{PDG}} & s_{23}c_{13} \\\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\\delta _{PDG}} &-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta _{PDG}} & c_{23}c_{13}\\end{array}\\right),$ where $s_{ij}=\\sin \\theta _{ij}$ and $c_{ij}=\\cos \\theta _{ij}$ with $\\theta _{ij}$ being angles rotating in flavor space and $\\delta _{PDG}$ is the CP violating phase.",
"We refer this as the PDG parametrization.",
"There are a lot of experimental data on the mixing pattern of quarks.",
"Fitting available data, the mixing angles and CP violating phase are determined to be [5] $&&\\theta _{12}=13.015^\\circ \\pm 0.059^\\circ ,\\quad \\theta _{23}=2.376^\\circ \\pm 0.046^\\circ ,\\quad \\theta _{13}=0.207^\\circ \\pm 0.008^\\circ , \\nonumber \\\\&&\\delta _{PDG}=69.7^\\circ \\pm 3.1^\\circ .",
"$ From the above, we obtain the magnitude of the matrix elements as $ \\left(\\begin{array}{lll}0.9743\\pm 0.0002 & 0.2252\\pm 0.0010 &0.0036\\pm 0.0001 \\\\0.2251\\pm 0.0010 & 0.9735 \\pm 0.0002 & 0.0415\\pm 0.0008 \\\\0.0088\\pm 0.0003 & 0.0407\\pm 0.0008 & 0.99913\\pm 0.00003\\end{array} \\right)\\;.$ The angles can be viewed as rotations in flavor spaces.",
"But both the angles and the phase in the CKM matrix are not directly measurable quantities.",
"There are different ways to parameterize the mixing matrix.",
"In different parametrizations, the angles and phase are different.",
"To illustrate this point let us study the original KM parametrization [2], $V_{KM} = \\left( \\begin{array}{ccc} c_1& - s_1 c_3& -s_1 s_3\\\\s_1c_2&c_1c_2c_3 - s_2s_3 e^{i\\delta _{KM}}&c_1c_2s_3 + s_2c_3 e^{i\\delta _{KM}}\\\\s_1s_2&c_1s_2c_3 + c_2 s_3 e^{i\\delta _{KM}}& c_1s_2 s_3 - c_2c_3 e^{i\\delta _{KM}}\\end{array}\\right)\\;.$ Using the observed values for the mixing matrix, one would obtain $&&\\theta _1 = 13.016^\\circ \\pm 0.003^\\circ \\;,\\;\\;\\theta _2 = 2.229^\\circ \\pm 0.066^\\circ \\;,\\;\\;\\theta _3 =0.921^\\circ \\pm 0.036^\\circ \\;,$ and the central value of the CP violating phase angle is $\\delta _{KM} = 88.2^\\circ $ .",
"Figure: The unitarity triangle.We see that the angles and phases in the PDG and KM parameterizations are indeed very different.",
"The angles and phase are parametrization dependent.",
"It is interesting to see whether all quantities used to parameterize the mixing matrix can all have well defined physical meanings, that is, all are experimentally measurable quantities, as have been done for several other quantities related to mixing matrices [6], [7], [8], [9].",
"To this end we notice that the magnitudes of the CKM matrix elements are already experimentally measurable quantities, one can take them to parameterize the mixing matrix.",
"However, the information on CP violation is then hid in the matrix elements, a single magnitude of an element would not be able to be taken as the measure of CP violation.",
"Only a combination of several magnitude is able to signify the CP violation.",
"For example, $Am = 1- (|V_{td}|^2|V_{tb}|^2 + |V_{ud}|^2|V_{ub}|^2 -|V_{cd}|^2|V_{cb}|^2)^2/4 |V_{td}|^2|V_{tb}|^2|V_{ud}|^2|V_{ub}|^2$ is non-zero.",
"Or one needs to know the phase of several matrix elements such as $J\\sum _{n,m = 1}^3\\epsilon _{ikm}\\epsilon _{jln}=Im(V_{ij}V_{kl}V^*_{il}V^*_{kj})$ [6] is non-zero.",
"Experimentally there are several measurable phases which can signify CP violations.",
"The famous ones are the angles $\\alpha $ , $\\beta $ and $\\gamma $ in the unitarity triangle defined by the unitarity condition $V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V^*_{tb}=0$ In the complex plane, the above defines a triangle shown in Fig 1.",
"This triangle defines three phase angles $\\ {\\alpha } = {\\arg \\left(-{{V_{td}V_{tb}^*}\\over {V_{ud}V_{ub}^*}}\\right)}\\;,\\;\\;\\ {\\beta } = {\\arg \\left(-{{V_{cd}V_{cb}^*}\\over {V_{td}V_{tb}^*}}\\right)}\\;,\\;\\;\\ {\\gamma } = {\\arg \\left(-{{V_{ud}V_{ub}^*}\\over {V_{cd}V_{cb}^*}}\\right)}\\;.",
"$ CP violation dictates that the area of this triangle to be non-zero.",
"This implies that none of the angles $\\alpha $ , $\\beta $ and $\\gamma $ can be zero.",
"Experimentally these three angles have been measured directly [4], $\\alpha =(89.0^{+4.4}_{-4.2})^{\\circ } $ , $\\beta =(21.1\\pm 0.9)^{\\circ } $ and $\\gamma =(73^{+22}_{-25})^{\\circ } $ .",
"These numbers are consistent with that obtained using the numerical numbers in eq.",
"REF , $\\alpha ={88.14}^{\\circ } $ , $\\beta ={22.20}^{\\circ } $ and $\\gamma ={69.67}^{\\circ } $ .",
"Also the directly measured numbers are consistent with the SM prediction $\\alpha + \\beta +\\gamma = \\pi $ .",
"Notice that the values $\\alpha $ , $\\gamma $ are very close to the two phases $\\delta _{KM}$ , $\\delta _{PDG}$ , respectively.",
"We will see later that although they are close to each other, they are not exactly equal.",
"Since we know that one only needs one quantity to signify the existence of CP violation, $\\alpha $ , $\\beta $ and $\\gamma $ must be related.",
"In fact they are related to the parameter $J$ as [7] $J=|V_{td}||V_{tb}^*||V_{ud}||V_{ub}^*| sin\\alpha = |V_{td}||V_{tb}^*||V_{cd}||V_{cb}^*| sin\\beta = |V_{cd}||V_{cb}^*||V_{ud}||V_{ub}^*| sin\\gamma \\;.$ The above is twice of the triangle area in Fig 1.",
"The quantity $Am$ is equal to $\\sin ^2\\alpha $ .",
"To have a parametrization for $V_{CKM}$ in which all quantities are experimentally measurable ones, one can choose three modulus of $|V_{ij}|$ and one of the above CP violating parameters.",
"For the CP violating phase, it is clear that the three phase angles $\\alpha $ , $\\beta $ and $\\gamma $ are among the simplest and have clear geometric meaning.",
"We refer these as the $\\alpha $ , $\\beta $ and $\\gamma $ parameterizations.",
"In the following we discuss how the parametrization can be constructed and how they can be transformed from each other."
],
[
"The $\\alpha $ , {{formula:b25d4da8-d4e2-426a-8e42-01b0ee89106c}} and {{formula:32b2ab6b-8334-431b-9ac4-06b973ec32c6}} parametrizations",
"In the $\\alpha $ parametrization we take $\\alpha $ as the phase appearing in the CKM matrix along with three modulus of $V_{ij}$ .",
"From the definition of $\\alpha $ in eq.REF , we can have four ways in which only one of the $V_{ud, ub, td, tb}$ relevant to the definition of $\\alpha $ is complex and all others are real and positive, $&&\\alpha _1)\\;.",
"\\;\\;(|V_{ud}|, |V_{ub}|, |V_{td}|, -|V_{tb}|e^{-i\\alpha })\\;,\\nonumber \\\\&&\\alpha _2)\\;.",
"\\;\\;(|V_{ud}|, |V_{ub}|, -|V_{td}|e^{i\\alpha }, |V_{tb}|)\\;,\\nonumber \\\\&&\\alpha _3)\\;.",
"\\;\\;(|V_{ud}|, -|V_{ub}|e^{i\\alpha }, |V_{td}|, V_{tb}|)\\;,\\nonumber \\\\&&\\alpha _4)\\;.",
"\\;\\;(- |V_{ud}|e^{-i\\alpha }, |V_{ub}|, |V_{td}|, |V_{tb}|)\\;.$ In the above one can change the signs of the elements by defining quark phases.",
"Indicating the CKM matrix for the four cases by $V^{\\alpha _i}_{CKM}$ .",
"We have $V^{\\alpha _1}_{CKM} = \\left(\\begin{array}{ccc}|V_{ud}|& |V_{us}| & |V_{ub}|\\\\|V_{cd}|&- \\frac{(|V_{us}|^{2}-|V_{td}|^{2})|V_{ud}|+|V_{ub}||V_{td}||V_{tb}|e^{-i\\alpha }}{|V_{us}||V_{cd}|}&\\frac{|V_{td}||V_{tb}|e^{-i\\alpha }-|V_{ud}||V_{ub}|}{|V_{cd}|}\\\\|V_{td}| & \\frac{|V_{ub}||V_{tb}|e^{-i\\alpha }-|V_{ud}||V_{td}|}{|V_{us}|}& - |V_{tb}|e^{-i\\alpha }\\\\\\end{array}\\right)$ One can take $\\alpha $ , $|V_{ud}|$ , $|V_{us}|$ , $|V_{cd}|$ as the four independent variables to parameterize the CKM matrix.",
"The other elements can be expressed as functions of them with $|V_{td}|&=&\\sqrt{1-|V_{ud}|^{2}-|V_{cd}|^{2}},\\;\\;|V_{ub}|=\\sqrt{1-|V_{ud}|^{2}-|V_{us}|^{2}},\\nonumber \\\\|V_{tb}|&=&\\frac{|V_{td}||V_{ud}||V_{ub}|\\cos {\\alpha }}{1-|V_{ud}|^{2}}\\\\&+&\\sqrt{(\\frac{|V_{td}||V_{ud}||V_{ub}|\\cos {\\alpha }}{1-|V_{ud}|^{2}})^{2}-\\frac{|V_{cd}|^{2}(|V_{ub}|^{2}-1)+|V_{ud}|^{2}|V_{ub}|^{2}}{1-|V_{ud}|^{2}}}\\;.\\nonumber $ For the other three cases, $V_{CKM}$ are given by $&&V^{\\alpha _2}_{CKM} = \\left(\\begin{array}{ccc}|V_{ud}|& |V_{us}| & |V_{ub}|\\\\\\frac{ |V_{td}||V_{tb}|e^{i\\alpha }-|V_{ud}||V_{ub}|}{|V_{cb}|}&\\frac{(|V_{ud}|^{2}-|V_{cb}|^{2})|V_{ub}|-|V_{ud}||V_{td}||V_{tb}|e^{i\\alpha }}{|V_{us}||V_{cb}|}&|V_{cb}|\\\\-|V_{td}|e^{i\\alpha } & \\frac{|V_{ud}||V_{td}|e^{i\\alpha }-|V_{ub}||V_{tb}|}{|V_{us}|} & |V_{tb}|\\\\\\end{array}\\right)\\;,\\nonumber \\\\&&V^{\\alpha _3}_{CKM} = \\left(\\begin{array}{ccc}|V_{ud}|&- \\frac{|V_{ud}||V_{td}|-|V_{tb}||V_{ub}|e^{i\\alpha }}{|V_{ts}|} & - |V_{ub}|e^{i\\alpha }\\\\|V_{cd}|&\\frac{(|V_{tb}|^{2}-|V_{cd}|^{2})|V_{td}|-|V_{ub}||V_{ud}||V_{tb}|e^{i\\alpha }}{|V_{ts}||V_{cd}|}&\\frac{|V_{ud}||V_{ub}|e^{i\\alpha }-|V_{td}||V_{tb}|}{|V_{cd}|}\\\\|V_{td}| & |V_{ts}|& |V_{tb}|\\\\\\end{array}\\right)\\;,\\\\&&V^{\\alpha _4}_{CKM} = \\left(\\begin{array}{ccc}- |V_{ud}|e^{-i\\alpha }& - \\frac{|V_{ub}||V_{tb}|-|V_{td}||V_{ud}|e^{-i\\alpha }}{|V_{ts}|} & |V_{ub}|\\\\- \\frac{|V_{td}||V_{tb}|-|V_{ud}||V_{ub}|e^{-i\\alpha }}{|V_{cb}|}&-\\frac{(|V_{cb}|^{2}-|V_{td}|^{2})|V_{tb}|+|V_{ud}||V_{td}||V_{ub}|e^{-i\\alpha }}{|V_{ts}||V_{cb}|}&|V_{cb}|\\\\|V_{td}| & |V_{ts}| & |V_{tb}|\\\\\\end{array}\\right)\\;.\\nonumber $ Similar to case $\\alpha _1$ , one can choose the phase $\\alpha $ and three modulus of $V_{ij}$ as independent variables for the above three cases.",
"It is convenient to choose the parameter sets ($\\alpha $ , $|V_{us}|$ , $|V_{ub}|$ , $|V_{cb}|$ ), ($\\alpha $ , $|V_{ts}|$ , $|V_{cd}|$ , $|V_{td}|$ ) and ($\\alpha $ , $|V_{cb}|$ , $|V_{tb}|$ , $|V_{ts}|$ ) as independent variables for the above three cases, respectively.",
"In all the above four cases, the Jarlskog parameter $J$ is given by $J=|V_{ub}||V_{ud}||V_{td}||V_{tb}|\\sin {\\alpha }$ This is not surprising because the above four cases are equivalent.",
"To see the above four cases discussed are equivalent explicitly, let us demonstrate how one can transform case $\\alpha _1$ to case $\\alpha _2$ by redefining quark phases.",
"The different parameterizations are equivalent implies that by redefining quark phases, one can transform the different ways of parameterizations for $V_{CKM}^{\\alpha _i}$ from one to another, that is, ${V^{\\alpha _{i}}_{CKM}}=\\left(\\begin{array}{ccc}1 & 0 &0 \\\\0 &e^{im} &0 \\\\0&0 &e^{in}\\end{array}\\right){V^{\\alpha _{j}}_{CKM}}\\left(\\begin{array}{ccc}e^{ix} & 0 &0 \\\\0 &e^{iy} &0 \\\\0&0 &e^{iz}\\end{array}\\right),$ where $i$ and $j$ stand for various types of parametrization.",
"For example, transforming case $\\alpha _1$ to case $\\alpha _2$ becomes a mission of finding the different parameters such that $&&\\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad {V^{\\alpha _{2}}_{CKM}}=\\left(\\begin{array}{ccc}1 & 0 &0 \\\\0 &e^{im} &0 \\\\0&0 &e^{in}\\end{array}\\right){V^{\\alpha _{1}}_{CKM}}\\left(\\begin{array}{ccc}e^{ix} & 0 &0 \\\\0 &e^{iy} &0 \\\\0&0 &e^{iz}\\end{array}\\right)\\\\&=&\\left(\\begin{array}{ccc}|V_{ud}|e^{ix}& |V_{us}|e^{iy} & |V_{ub}|e^{iz}\\\\|V_{cd}|e^{i(m+x)}&- \\frac{(|V_{us}|^{2}-|V_{td}|^{2})|V_{ud}|e^{i(m+y)}+|V_{ub}||V_{td}||V_{tb}|e^{i(m+y-\\alpha )}}{|V_{us}||V_{cd}|}&\\frac{|V_{td}||V_{tb}|e^{i(m+z-\\alpha )}-|V_{ud}||V_{ub}|e^{i(m+z)}}{|V_{cd}|}\\\\|V_{td}|e^{i(n+x)} & \\frac{|V_{ub}||V_{tb}|e^{i(n+y-\\alpha )}-|V_{ud}||V_{td}|e^{i(n+y)}}{|V_{us}|}& - |V_{tb}|e^{i(n+z-\\alpha )}\\\\\\end{array}\\right).\\nonumber $ Comparing the coefficients, one obtains $x=y=z=0$ , $n=\\alpha +\\pi $ , and ${m}=\\arccos {{(|V_{td}||V_{tb}|)^2-[(|V_{ud}||V_{ub}|)^2+(|V_{cd}||V_{cb}|)^2]}\\over {2|V_{ud}||V_{ub}||V_{cd}||V_{cb}|}}\\;.$ Therefore, the transformation from case $\\alpha _1$ to case $\\alpha _2$ is achieved by, ${V^{\\alpha _{2}}_{CKM}}=\\left(\\begin{array}{ccc}1 & 0 &0 \\\\0 &e^{im} &0 \\\\0&0 &-e^{i\\alpha }\\end{array}\\right){V^{\\alpha _{1}}_{CKM}}\\;.$ Similarly one can transform the other two cases to case $\\alpha _2$ too.",
"We now discuss the $\\beta $ parametrization.",
"From eq.REF , we can have four choices for the location of the phase similar to the $\\alpha $ parametrization.",
"They are $&&\\beta _1)\\;.",
"\\;\\;(|V_{cd}|, |V_{cb}|, |V_{td}|, -|V_{tb}|e^{i\\beta })\\;,\\nonumber \\\\&&\\beta _2)\\;.",
"\\;\\;(|V_{cd}|, |V_{cb}|, -|V_{td}|e^{-i\\beta }, |V_{tb}|)\\;,\\nonumber \\\\&&\\beta _3)\\;.",
"\\;\\;(|V_{cd}|, -|V_{cb}|e^{-i\\beta }, |V_{td}|, V_{tb}|)\\;,\\nonumber \\\\&&\\beta _4)\\;.",
"\\;\\;(- |V_{cd}|e^{i\\beta }, |V_{cb}|, |V_{td}|, |V_{tb}|)\\;.$ In a similar way as for the four cases of $\\alpha _i$ , one can show that the above four cases are equivalent.",
"Detailed expressions for these four cases are given in the Appendix.",
"We will display the explicit form for case $\\beta _2$ .",
"We have $V^{\\beta _2}_{CKM} = \\left(\\begin{array}{ccc}\\frac{|V_{td}||V_{tb}|e^{-i\\beta }-|V_{cb}||V_{cd}|}{|V_{ub}|}& \\frac{(|V_{cd}|^{2}-|V_{ub}|^{2})|V_{cb}|-|V_{cd}||V_{td}||V_{tb}|e^{-i\\beta }}{|V_{cs}||V_{ub}|} &|V_{ub}|\\\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\\\-|V_{td}|e^{-i\\beta } & \\frac{|V_{cd}||V_{td}|e^{-i\\beta }-|V_{cb}||V_{tb}|}{|V_{cs}|} & |V_{tb}|\\\\\\end{array}\\right)\\;.$ For this case it is convenient to take $\\beta $ , $|V_{cs}|$ , $|V_{cb}|$ , $|V_{tb}|$ as independent variables.",
"The other quantities can be expressed as $|V_{cd}|&=&\\sqrt{1-|V_{cs}|^{2}-|V_{cb}|^{2}},|V_{ub}|=\\sqrt{1-|V_{cb}|^{2}-|V_{tb}|^{2}},\\\\|V_{td}|&=&\\frac{|V_{tb}||V_{cd}||V_{cb}|\\cos {\\beta }}{1-|V_{cb}|^{2}}\\\\&+&\\sqrt{(\\frac{|V_{tb}||V_{cd}||V_{cb}|\\cos {\\beta }}{1-|V_{cb}|^{2}})^{2}-\\frac{|V_{ub}|^{2}(|V_{cd}|^{2}-1)+|V_{cd}|^{2}|V_{cb}|^{2}}{1-|V_{cb}|^{2}}}\\;.$ The Jarlskog parameter $J$ is given by $J=|V_{cb}||V_{tb}||V_{cd}||V_{td}|\\sin {\\beta }.$ For $\\gamma $ parametrization, the four different cases are defined by $&&\\gamma _1)\\;.",
"\\;\\;(|V_{ud}|, |V_{ub}|, |V_{cd}|, -|V_{cb}|e^{i\\gamma })\\;,\\nonumber \\\\&&\\gamma _2)\\;.",
"\\;\\;(|V_{ud}|, |V_{ub}|, -|V_{cd}|e^{-i\\gamma }, |V_{cb}|)\\;,\\nonumber \\\\&&\\gamma _3)\\;.",
"\\;\\;(|V_{ud}|, -|V_{ub}|e^{-i\\gamma }, |V_{cd}|, V_{cb}|)\\;,\\nonumber \\\\&&\\gamma _4)\\;.",
"\\;\\;(- |V_{ud}|e^{i\\gamma }, |V_{ub}|, |V_{cd}|, |V_{cb}|)\\;.$ The detailed expressions for these four cases are given in Appendix.",
"We display the explicit form for case $\\gamma _3$ here.",
"We have $V^{\\gamma _3}_{CKM} = \\left(\\begin{array}{ccc}|V_{ud}|& -\\frac{|V_{ud}||V_{cd}|-|V_{ub}||V_{cb}|e^{-i\\gamma }}{|V_{cs}|} &-|V_{ub}|e^{-i\\gamma }\\\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\\\|V_{td}| & \\frac{(|V_{cb}|^{2}-|V_{td}|^{2})|V_{cd}|-|V_{cb}||V_{ud}||V_{ub}|e^{-i\\gamma }}{|V_{cs}||V_{td}|} & \\frac{|V_{ud}||V_{ub}|e^{-i\\gamma }-|V_{cd}||V_{cb}|}{|V_{td}|}\\\\\\end{array}\\right)\\;.$ Taking $\\gamma $ , $|V_{cd}|$ , $|V_{cs}|$ , $|V_{td}|$ as variables, the other quantities can be expressed as $|V_{ud}|&=&\\sqrt{1-|V_{cd}|^{2}-|V_{td}|^{2}},|V_{cb}|=\\sqrt{1-|V_{cd}|^{2}-|V_{cs}|^{2}},\\\\|V_{ub}|&=&\\frac{|V_{ud}||V_{cd}||V_{cb}|\\cos {\\gamma }}{1-|V_{cd}|^{2}}\\\\&-&\\sqrt{(\\frac{|V_{ud}||V_{cd}||V_{cb}|\\cos {\\gamma }}{1-|V_{cd}|^{2}})^{2}-\\frac{|V_{cs}|^{2}(|V_{ud}|^{2}-1)+|V_{ud}|^{2}|V_{cd}|^{2}}{1-|V_{cd}|^{2}}},$ and $J=|V_{ub}||V_{cb}||V_{ud}||V_{cd}|\\sin {\\gamma }.$"
],
[
"Relations between different parameterizations",
"We now show that the $\\alpha $ , $\\beta $ and $\\gamma $ parameterizations can also be transformed from each other.",
"The twelve parameterizations discussed before in the previous section are all equivalent.",
"We have already shown that the four parameterizations for $\\alpha $ or $\\beta $ or $\\gamma $ are equivalent ones, we will therefore only need to show that one of the $\\alpha $ parameterizations is equivalent to one of the $\\beta $ or $\\gamma $ parameterizations.",
"We now show that $V_{CKM}^{\\alpha _3}$ and $V_{CKM}^{\\beta _3}$ .",
"For these two parametrizations, the first column and the third row are already identical in these two parameterizations.",
"Using $|V_{ub}||V_{ud}|e^{i\\alpha }+|V_{cb}||V_{cd}|e^{-i\\beta }=|V_{td}||V_{tb}|.$ One can readily show that the 12, 13, 22 and 23 entries of $V^{\\alpha _3}_{CKM}$ and $V_{CKM}^{\\beta _3}$ are equal.",
"Therefore $V^{\\alpha _3}_{CKM}& = &\\left(\\begin{array}{ccc}|V_{ud}|&- \\frac{|V_{ud}||V_{td}|-|V_{tb}||V_{ub}|e^{i\\alpha }}{|V_{ts}|} & - |V_{ub}|e^{i\\alpha }\\\\|V_{cd}|&\\frac{(|V_{tb}|^{2}-|V_{cd}|^{2})|V_{td}|-|V_{ub}||V_{ud}||V_{tb}|e^{i\\alpha }}{|V_{ts}||V_{cd}|}&\\frac{|V_{ud}||V_{ub}|e^{i\\alpha }-|V_{td}||V_{tb}|}{|V_{cd}|}\\\\|V_{td}| & |V_{ts}|& |V_{tb}|\\\\\\end{array}\\right)\\\\&=&\\left(\\begin{array}{ccc}|V_{ud}|& -\\frac{(|V_{ud}|^{2}-|V_{tb}|^{2})|V_{td}|+|V_{tb}||V_{cd}||V_{cb}|e^{-i\\beta }}{|V_{ts}||V_{ud}|} & -\\frac{|V_{td}||V_{tb}|-|V_{cb}||V_{cd}|e^{-i\\beta }}{|V_{ud}|}\\\\|V_{cd}|&\\frac{|V_{cb}||V_{tb}|e^{-i\\beta }-|V_{cd}||V_{td}|}{|V_{ts}|}&-|V_{cb}|e^{-i\\beta }\\\\|V_{td}| & |V_{ts}| & |V_{tb}|\\\\\\end{array}\\right) = V^{\\beta _3}_{CKM}.\\nonumber $ Similarly, using $|V_{tb}||V_{td}|e^{i\\alpha }+|V_{cb}||V_{cd}|e^{-i\\gamma }=|V_{ud}||V_{ub}|,\\\\|V_{ud}||V_{ub}|e^{-i\\gamma }+|V_{tb}||V_{td}|e^{i\\beta }=|V_{cd}||V_{cb}|,$ we obtain $V_{CKM}^{\\alpha 2}=V_{CKM}^{\\gamma 2},V_{CKM}^{\\beta 1}=V_{CKM}^{\\gamma 3}.$ We therefore have shown explicitly that the $\\alpha $ , $\\beta $ and $\\gamma $ parameterizations are related and can be transformed from one to another.",
"Numerically, one finds that the approximate relations $\\delta _{KM} \\approx \\alpha $ and $\\delta _{PDG}\\approx \\gamma $ .",
"These can be understood easily by noticing the relations between them [8], [10], $&&\\alpha =\\arctan ({\\sin \\delta _{KM} \\over x_{\\alpha }-\\cos \\delta _{KM} }),\\;\\;\\;\\;x_{\\alpha } = {c_1s_2s_3\\over c_2c_3} = {|V_{ud}||V_{td}||V_{ub}|\\over |V_{cd}||V_{us}|}=0.0006.\\nonumber \\\\&&\\gamma =\\arctan ({\\sin \\delta _{PDG} \\over x_{\\gamma }+\\cos \\delta _{PDG} }),\\;\\;\\;\\;x_{\\gamma }= {c_{12}s_{23}s_{13}\\over s_{12} c_{23}} = {|V_{ud}||V_{cb}||V_{ub}|\\over |V_{tb}||V_{us}|} =0.0006.\\nonumber $ Therefore, $\\delta _{KM} + \\alpha $ is approximately $\\pi $ , since $\\alpha $ is close to $90^\\circ $ , $\\delta _{KM}$ must also be close to $90^\\circ $ and therefore $\\delta _{KM} \\approx \\alpha $ .",
"It is also clear that $\\delta _{PDG}$ is approximately equal to $\\gamma $ .",
"One may wonder if there is a parametrization where the phase is close to $\\beta $ .",
"We find indeed there are angle prametrizations in which the CP violating phase is close to $\\beta $ .",
"An example is provided by the parametrization $P4$ discussed in Ref.",
"[11] where ${V_{CKM}^{P4}}= \\left(\\begin{array}{ccc}c_{\\theta }c_{\\tau }& c_{\\theta }s_{\\sigma }s_{\\tau }+ s_{\\theta }c_{\\sigma }e^{-i\\varphi } & c_{\\theta }c_{\\sigma }s_{\\tau }-s_{\\theta }s_{\\sigma }e^{-i\\varphi }\\\\-s_{\\theta } c_{\\tau }&-s_\\theta s_\\sigma s_\\tau + c_\\theta c_\\sigma e^{-i\\varphi }&-s_{\\theta } c_{\\sigma } s_\\tau - c_\\theta s_\\sigma e^{-i\\varphi }\\\\-s_{\\tau } & s_{\\sigma }c_{\\tau } &c_{\\sigma }c_{\\tau }\\\\\\end{array}\\right).$ We have $&&\\beta =\\arctan ({\\sin \\varphi \\over x_{\\beta }+ \\cos \\varphi }),\\;\\;\\;\\;x_{\\beta } = {s_\\theta c_\\sigma s_\\tau \\over c_\\theta s_\\sigma } = {|V_{cd}||V_{tb}||V_{td}|\\over |V_{ud}||V_{ts}|} = 0.0497.$ One may also wonder if one can find a parametrization in which the CP violating angle is one of the $\\alpha $ , $\\beta $ and $\\gamma $ , and the other three quantities to parameterize the mixing matrix can be chosen to be three angles similar to those in the PDG or KM paramerizations.",
"We find that this is impossible.",
"It has been shown that there are nine different ways to parameterize the mixing matrix using one phase and three angles [11].",
"Explicit inspections find that none of the phases can be exactly, allowing sign differences or plus or minus a $\\pi $ , identified as one of the $\\alpha $ , $\\beta $ or $\\gamma $ .",
"The reason is that in the $\\alpha $ , $\\beta $ and $\\gamma $ parametrization we have introduced, one needs one of the elements in mixing matrix to be of the form $|V_{ij}|e^{\\pm i\\alpha }$ , $|V_{ij}|e^{\\pm i\\beta }$ , or $|V_{ij}|e^{\\pm i\\gamma }$ , and also another five real matrix elements which cannot be satisfied with only three angles."
],
[
"Wolfenstein-like Expansions",
"It has proven to be convenient to use approximate formula such as the Wolfenstein parametrization[12].",
"In the literature different approximate forms have been proposed[13], [14].",
"In this section, we discuss the Wolfenstein-like expansions in the $\\alpha $ , $\\beta $ and $\\gamma $ parameterizations demanding to use one of the $\\alpha $ , $\\beta $ and $\\gamma $ as one of the parameters.",
"For the $\\alpha _i)$ cases, we find it convenient to work with $\\alpha _1)$ case.",
"We use $|V_{us}| = \\lambda $ , $|V_{ub}| = a \\lambda ^3$ , $|V_{td}| = b\\lambda ^3$ and $\\alpha $ as parameters.",
"The numerical values of $\\lambda $ , $a$ and $b$ are determined to be $\\lambda = 0.2252\\pm 0.0010$ , $a= 0.3170\\pm 0.0130 $ , and $b= 0.7670\\pm 0.0250 $ .",
"To order $\\lambda ^3$ , we have $V_{CKM}^{\\alpha _{1}}\\approx \\left(\\begin{array}{ccc}{1-{1\\over 2}{\\lambda }^2}& \\lambda & a{\\lambda }^3\\\\\\lambda &{-1+{1\\over 2}{\\lambda }^2} &-(a-be^{-i\\alpha })\\lambda ^2\\\\b{\\lambda }^3 & (a{e}^{-i\\alpha }-b){\\lambda }^2 & -e^{-i\\alpha }\\\\\\end{array}\\right).$ One can further rotate the phase of c-quark by $\\pi $ and b-quark by $\\pi +\\alpha $ to obtain an expression where the diagonal entries are close to 1.",
"We obtain $V_{CKM}^{\\alpha _{1}}\\approx \\left(\\begin{array}{ccc}{1-{1\\over 2}{\\lambda }^2}& \\lambda & - a{\\lambda }^3 e^{i\\alpha }\\\\-\\lambda &{1-{1\\over 2}{\\lambda }^2} &-(ae^{i\\alpha }-b)\\lambda ^2\\\\b{\\lambda }^3 & (a{e}^{-i\\alpha }-b){\\lambda }^2 & 1\\\\\\end{array}\\right).$ The above expansion is equivalent to that discussed in Ref.[14].",
"The parameters $\\delta $ , $h$ and $f$ in Ref.",
"[14] are related to the above parameters by, $\\delta =-\\alpha $ , $f = b$ and $h = -a$ .",
"At more than $\\lambda ^3$ order, there are differences for our approximation and that proposed in Ref.",
"[14].",
"For $\\beta _i)$ cases, it is convenient to use $\\beta _1)$ for expansion.",
"Setting $|V_{cd}|= \\lambda $ , $|V_{td}| = b \\lambda ^3$ , and $|V_{cb}| = c \\lambda ^2$ with $\\lambda =0.2251\\pm 0.0010$ , and $b=0.7685\\pm 0.0250 $ $c= 0.8185\\pm 0.0176$ .",
"Rotating the b-quark field by a phase $\\pi -\\beta $ , we obtain to order $\\lambda ^3$ $V_{CKM}^{\\beta _{1}} \\approx \\left(\\begin{array}{ccc}{1-{1\\over 2}{\\lambda }^2}& -\\lambda & {\\lambda }^3(c e^{-i\\beta }-b)\\\\\\lambda & 1-{1\\over 2}{\\lambda }^2 & -c\\lambda ^2 e^{-i\\beta }\\\\b{\\lambda }^3 & c\\lambda ^2{e}^{i\\beta } & 1\\\\\\end{array}\\right).$ For the $\\gamma _i)$ cases, $\\gamma _4)$ case is convenient for expansion.",
"Setting $|V_{cd}|= \\lambda $ , $|V_{ub}|={a{\\lambda }^3}$ and $|V_{cb}|={c{\\lambda }^2}$ with $\\lambda = 0.2251\\pm 0.0010$ , $a= 0.3176\\pm 0.0130$ , and $c= 0.8185\\pm 0.0176$ .",
"Rotating d-quark by a phase $\\pi $ and u-quark by a phase $-\\gamma $ , we obtain to order $\\lambda ^3$ $V_{CKM}^{\\gamma _4} \\approx \\left(\\begin{array}{ccc}1-{1\\over 2}{\\lambda }^2& \\lambda & a{\\lambda }^3 e^{- i\\gamma }\\\\- \\lambda & 1-{1\\over 2}{\\lambda }^2 & c\\lambda ^2\\\\- {\\lambda }^3(ae^{i\\gamma }-c) & - c\\lambda ^2 & 1\\\\\\end{array}\\right).$ To $\\lambda ^3$ order, the above expansion is equivalent to the Wolfenstein parametrization[12].",
"The parameters $A$ , $\\rho $ and $\\eta $ in the Wolfenstein parametrization are related to $a$ , $c$ and $\\gamma $ by $c= A$ , $\\rho = a\\cos \\gamma /c$ and $\\eta = a\\sin \\gamma /c$ .",
"Again at more than $\\lambda ^3$ order, there are differences in these two approximations.",
"Among the three phase angles, $\\alpha $ , $\\beta $ and $\\gamma $ , the best measured one is $\\beta $ .",
"This makes the approximate expression $V^{\\beta _1}_{CKM}$ better than others."
],
[
"Conclusion",
"To conclude, we have proposed new parametrizations of the CKM matrix using the three measurable phase angles $\\alpha $ , $\\beta $ and $\\gamma $ in the unitarity triangle as the CP violating phase.",
"For each of the $\\alpha $ , $\\beta $ and $\\gamma $ parametrization there are four ways to parameterize the mixing matrix where one column and one row elements are all real.",
"We have shown explicitly that all these cases are equivalent.",
"We have studied relations of the $\\alpha $ , $\\beta $ and $\\gamma $ parametrizations with the usual three rotation angles in flavor space and one CP violating phase parametrizations.",
"We find that it is not possible to parameterize the CKM matrix using three angles and taking one of the $\\alpha $ , $\\beta $ and $\\gamma $ as the CP violating phase.",
"However, there are rotation angle parametrizations in which the CP violating are very close to one of the phase angles $\\alpha $ , $\\beta $ or $\\gamma $ .",
"We, however, emphasis that the $\\alpha $ , $\\beta $ and $\\gamma $ parametrizations we proposed have the advantage that all the elements used to describe the mixing matrix are physically measurable quantities, unlike the parametrizations of using rotation angles in flavor space and a phase whose values are parametrization dependent.",
"The $\\alpha $ , $\\beta $ and $\\gamma $ parametrizations are parametrization independent representation of the CKM matrix.",
"We also suggest new Wolfenstein-like paramterizations.",
"Acknowledgement This work was supported in part by NSC and NCTS of ROC, NNSF(grant No:11175115) and Shanghai science and technology commission (grant no: 11DZ2260700) of PRC."
],
[
"Expressions of $V_{CKM}$ for {{formula:d5a6e28c-ab68-4ada-bc07-1517f0e83ba3}} and {{formula:1f64b97f-c8b7-486e-be75-5a54b9911266}} cases",
"Expressions for the four $\\beta $ and four $\\gamma $ parameterizations.",
"$V_{CKM}^{\\beta _1} = \\left(\\begin{array}{ccc}|V_{ud}|& -{\\frac{(|V_{ud}|^{2}-|V_{cb}|^{2})|V_{cd}|+|V_{cb}||V_{td}||V_{tb}|e^{i\\beta }}{|V_{cs}||V_{ud}|}} &-{\\frac{|V_{cb}||V_{cd}|-|V_{td}||V_{tb}|e^{i\\beta }}{|V_{ud}|}}\\\\|V_{cd}|&|V_{cs}| &|V_{cb}|\\\\|V_{td}| & \\frac{|V_{cb}||V_{tb}|e^{i\\beta }-|V_{cd}||V_{td}|}{|V_{cs}|} & -|V_{tb}|e^{i\\beta }\\\\\\end{array}\\right)$ $V_{CKM}^{\\beta _2} = \\left(\\begin{array}{ccc}\\frac{|V_{td}||V_{tb}|e^{-i\\beta }-|V_{cb}||V_{cd}|}{|V_{ub}|}& \\frac{(|V_{cd}|^{2}-|V_{ub}|^{2})|V_{cb}|-|V_{cd}||V_{td}||V_{tb}|e^{-i\\beta }}{|V_{cs}||V_{ub}|} &|V_{ub}|\\\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\\\-|V_{td}|e^{-i\\beta } & \\frac{|V_{cd}||V_{td}|e^{-i\\beta }-|V_{cb}||V_{tb}|}{|V_{cs}|} & |V_{tb}|\\\\\\end{array}\\right)$ $V_{CKM}^{\\beta _3} = \\left(\\begin{array}{ccc}|V_{ud}|& -\\frac{(|V_{ud}|^{2}-|V_{tb}|^{2})|V_{td}|+|V_{tb}||V_{cd}||V_{cb}|e^{-i\\beta }}{|V_{ts}||V_{ud}|} & -\\frac{|V_{td}||V_{tb}|-|V_{cb}||V_{cd}|e^{-i\\beta }}{|V_{ud}|}\\\\|V_{cd}|&\\frac{|V_{cb}||V_{tb}|e^{-i\\beta }-|V_{cd}||V_{td}|}{|V_{ts}|}&-|V_{cb}|e^{-i\\beta }\\\\|V_{td}| & |V_{ts}| & |V_{tb}|\\\\\\end{array}\\right)$ $V_{CKM}^{\\beta _4} = \\left(\\begin{array}{ccc}-\\frac{|V_{td}||V_{tb}|-|V_{cb}||V_{cd}|e^{i\\beta }}{|V_{ub}|}& -\\frac{(|V_{ub}|^{2}-|V_{td}|^{2})|V_{tb}|+|V_{td}||V_{cb}||V_{cd}|e^{i\\beta }}{|V_{ts}||V_{ub}|} & |V_{ub}|\\\\-|V_{cd}|e^{i\\beta }&-\\frac{|V_{cb}||V_{tb}|-|V_{cd}||V_{td}|e^{i\\beta }}{|V_{ts}|}&|V_{cb}|\\\\|V_{td}| & |V_{ts}| & |V_{tb}|\\\\\\end{array}\\right)$ $V_{CKM}^{\\gamma _1} = \\left(\\begin{array}{ccc}|V_{ud}|& |V_{us}| &|V_{ub}|\\\\|V_{cd}|&-\\frac{|V_{ud}||V_{cd}|-|V_{ub}||V_{cb}|e^{i\\gamma }}{|V_{us}|}&-|V_{cb}|e^{i\\gamma }\\\\|V_{td}| & \\frac{(|V_{ub}|^{2}-|V_{td}|^{2})|V_{ud}|-|V_{ub}||V_{cd}||V_{cb}|e^{i\\gamma }}{|V_{us}||V_{td}|} & -\\frac{|V_{ud}||V_{ub}|-|V_{cd}||V_{cb}|e^{i\\gamma }}{|V_{td}|}\\\\\\end{array}\\right)$ $V_{CKM}^{\\gamma _2} = \\left(\\begin{array}{ccc}|V_{ud}|& |V_{us}| &|V_{ub}|\\\\-|V_{cd}|e^{-i\\gamma }&\\frac{|V_{ud}||V_{cd}|e^{-i\\gamma }-|V_{ub}||V_{cb}|}{|V_{us}|}&|V_{cb}|\\\\\\frac{|V_{cd}||V_{cb}|e^{-i\\gamma }-|V_{ud}||V_{ub}|}{|V_{tb}|} & \\frac{(|V_{ud}|^{2}-|V_{tb}|^{2})|V_{ub}|-|V_{ud}||V_{cd}||V_{cb}|e^{-i\\gamma }}{|V_{us}||V_{tb}|} &|V_{tb}| \\\\\\end{array}\\right)$ $V_{CKM}^{\\gamma _3} = \\left(\\begin{array}{ccc}|V_{ud}|& -\\frac{|V_{ud}||V_{cd}|-|V_{ub}||V_{cb}|e^{-i\\gamma }}{|V_{cs}|} &-|V_{ub}|e^{-i\\gamma }\\\\|V_{cd}|&|V_{cs}| &|V_{cb}|\\\\|V_{td}| & \\frac{(|V_{cb}|^{2}-|V_{td}|^{2})|V_{cd}|-|V_{cb}||V_{ud}||V_{ub}|e^{-i\\gamma }}{|V_{cs}||V_{td}|} & \\frac{|V_{ud}||V_{ub}|e^{-i\\gamma }-|V_{cd}||V_{cb}|}{|V_{td}|}\\\\\\end{array}\\right)$ $V_{CKM}^{\\gamma _4} = \\left(\\begin{array}{ccc}-|V_{ud}|e^{i\\gamma }& \\frac{|V_{ud}||V_{cd}|e^{i\\gamma }-|V_{ub}||V_{cb}|}{|V_{cs}|} &|V_{ub}|\\\\|V_{cd}|&|V_{cs}|&|V_{cb}|\\\\\\frac{|V_{ud}||V_{ub}|e^{i\\gamma }-|V_{cd}||V_{cb}|}{|V_{tb}|} & \\frac{(-|V_{tb}|^{2}+|V_{cd}|^{2})|V_{cb}|-|V_{cd}||V_{ud}||V_{ub}|e^{i\\gamma }}{|V_{cs}||V_{tb}|} & |V_{tb}|\\\\\\end{array}\\right)$"
]
] | 1204.1230 |
[
[
"A new Laplace operator in Finsler geometry and periodic orbits of Anosov\n flows"
],
[
"Abstract In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives.",
"This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami operator when the Finsler metric is Riemannian.",
"We compute explicit spectral data for some Katok-Ziller metrics.",
"When the Finsler metric is negatively curved, we show, thanks to a result of Ancona that the Martin boundary is H\\\"older-homeomorphic to the visual boundary.",
"This allow us to deduce the existence of harmonic measures and some ergodic preoperties.",
"In the second part of this dissertation, we study Anosov flows in 3-manifolds, with leaf-spaces homeomorphic to $\\mathbb{R}$.",
"When the manifold is hyperbolic, Thurston showed that the (un)stable foliations induces a regulating flow.",
"We use this second flow to study isotopy class of periodic orbits of the Anosov flow and existence of embedded cylinders."
],
[
"Foulon's dynamical formalism for Finsler geometry and some consequences",
"Dynamical Formalism"
],
[
"Definitions",
"Here we explain the formalism that we will use in this dissertation regarding Finsler geometry.",
"In that respect, we follow the work introduced by Patrick Foulon in [59]."
],
[
"Notations",
"In this dissertation, we only consider orientable manifolds.",
"For the readers' convenience, we will start with a list of notations that will be used throughout this text but defined later : If $M$ is a manifold, $TM$ is its tangent bundle and $T^{\\ast }M$ its cotangent bundle.",
"$\\mathring{T}M$ (resp.",
"$\\mathring{T}^{\\ast }M$ ) is the tangent (resp.",
"cotangent) bundle minus the zero section.",
"The bundles $HM$ and $H^{\\ast }M$ are the (co)-homogenized bundles.",
"$p_{|M} \\colon TM \\rightarrow M$ $\\hat{p}_{|M} \\colon T^{\\ast }M \\rightarrow M$ $r \\colon \\mathring{T}M \\rightarrow HM$ $\\hat{r} \\colon \\mathring{T}^{\\ast }M \\rightarrow H^{\\ast }M$ $\\pi \\colon HM \\rightarrow M$ $\\hat{\\pi } \\colon H^{\\ast }M \\rightarrow M$ $\\mathcal {L}_F \\colon \\mathring{T}M \\rightarrow \\mathring{T^{\\ast }}M$ the Legendre transform.",
"$\\ell _F \\colon HM \\rightarrow H^{\\ast }M$ the Legendre transform on the homogeneous bundles.",
"If $\\alpha $ is a $p$ -form on a manifold (where $p$ can be equal to 0), then $d\\alpha $ is the exterior derivative of $\\alpha $ .",
"Otherwise, if $f\\colon M \\rightarrow N$ is a map between manifolds, $df \\colon TM \\rightarrow TN$ is the differential of the map.",
"$VHM = \\ker d\\pi $ .",
"$L_Z$ stands for the Lie derivative of the vector field $Z$ .",
"In all of this dissertation, a $\\ast $ in superscript (resp.",
"subscript) of a map will mean the pull-back (resp.",
"the push-forward) of the following object."
],
[
"Finsler metric and geodesic flow",
"Let $M$ be a smooth manifold of dimension $n$ , there are several definitions of a Finsler metric in the literature.",
"We will use what is probably the most common and, unfortunately, the most restrictive of smooth, strongly convex Finsler structure : Definition 1.1.1 A smooth Finsler metric on $M$ is a continuous function $F \\colon TM \\rightarrow \\mathbb {R}^+$ that is : $C^{\\infty }$ except on the zero section, positively homogeneous, i.e., $F(x,\\lambda v)=\\lambda F(x,v)$ for any $\\lambda >0$ , positive-definite, i.e., $F(x,v)\\ge 0$ with equality iff $v=0$ , strongly convex, i.e., $ \\left(\\dfrac{\\partial ^2 F^2}{\\partial v_i \\partial v_j}\\right)_{i,j}$ is positive-definite.",
"If, for any $(x,v)\\in TM$ , $F(x, -v ) = F(x,v)$ , then we say that $F$ is reversible.",
"A Finsler metric can be thought of as a smooth family $\\lbrace I_x \\rbrace $ of strongly convex sets containing 0 in each tangent space $T_xM$ , or equivalently as a smooth family of Minkowski norms.",
"If the convex sets are always ellipsoids centered at 0 then the Finsler metric is called Riemannian.",
"Some examples of Finsler metrics include for instance smooth and strongly convex deformations of Riemannian metrics.",
"In [95], G. Randers introduced the following : take $g$ a Riemannian metric on $M$ , $\\theta $ a 1-form on $M$ and define $F := \\sqrt{g} + \\theta $ .",
"If $\\theta $ has norm less than 1, then $F$ is a Finsler metric (see [14]).",
"Randers metrics are an important particular example of Finsler structures : they are the simplest kind of non-Riemannian metrics, arise a lot in physics and have been widely studied.",
"Note that Randers metrics are never reversible.",
"Under our conditions, it can be shown that $F$ defines a distance on $M$ : for $x,y \\in M$ , $d(x,y) = \\inf _c \\int _0^1 F(c(t), \\dot{c}(t) ) dt ,$ where $c$ runs over all $C^1$ -by-part paths such that $c(0) = x$ and $c(1)=y$ .",
"Note that, for a non-reversible Finsler metric, the distance function will not be symmetric.",
"In the definition, the positive homogeneity condition is necessary to ensure that changing the parametrization of a curve does not change its length and the third point assures us that a constant path does not have positive length.",
"The last requirement is not as self-explanatory as the others, but asking for $F$ to be convex in the second variable implies that the length structure is lower semi-continuous which, in turns, implies that the length of a rectifiable path computed with $d$ is the same as its “integral length” computed with the formula above (see [29]).",
"Using this distance, we can define geodesics as curves that locally minimize the distance and hence a geodesic flow on the unit tangent bundle.",
"Note that our stronger assumption for convexity is necessary for our purpose, for instance to obtain geodesics as solutions of a second-order differential equation, or equivalently, so that the Hilbert form $A$ , defined below, is a contact form.",
"The right place to study dynamical objects (by this, we mean objects linked to the geodesic flow) seems to be the unit tangent bundle.",
"However, in order to study flows associated to different Finsler metrics without having to change the space, we study everything on the homogenized tangent bundle $HM = \\mathring{T}M / \\mathbb {R}^+_{\\ast }.$ We write $r \\colon \\mathring{T}M \\rightarrow HM$ and $\\pi \\colon HM \\rightarrow M$ for the canonical projections.",
"Remark that the fibers of $\\pi $ defines a canonical distribution on $THM$ , called the vertical distribution $VHM$ .",
"It is the set of vectors in $THM$ that are tangent to a fiber of $\\pi $ , or equivalently, $VHM:= \\text{Ker} \\; d\\pi $ ."
],
[
"Hilbert form",
"To a Finsler metric $F$ , we can canonically associate a 1-form $A$ on $HM$ , called the Hilbert form, in the following manner : for $(x,\\xi ) \\in HM$ and $Z \\in T_{(x,\\xi )}HM$ , choose $v \\in T_xM$ such that $r(x,v) = (x,\\xi )$ and set $A_{(x,\\xi )}(Z) := \\lim _{\\varepsilon \\rightarrow 0} \\frac{F\\left(x, v + \\varepsilon d\\pi (Z) \\right) - F\\left( x,v \\right)}{\\varepsilon }.$ The homogeneity of $F$ implies that the definition of $A$ does not depend on the choice of $v$ .",
"Note that we can also define the vertical derivative of $F$ : it is the 1-form $d_vF \\colon TM \\rightarrow T^{\\ast }TM$ such that, for any $(x,v) \\in TM$ and $Z \\in T_{(x,v)}TM$ , $d_vF_{(x,v)}(Z) := \\lim _{\\varepsilon \\rightarrow 0} \\frac{F\\left(x, v + \\varepsilon dp(Z) \\right) - F\\left( x,v \\right)}{\\varepsilon }.$ And we clearly have $d_vF = r^{\\ast } A.$ Theorem 1.1.2 (Hilbert, ..., Foulon [59]) The form $A$ is a contact form, i.e., $A \\wedge dA^{n-1}$ is a volume form on $M$ .",
"Furthermore, if $X$ denotes its Reeb field, then $X$ generates the geodesic flow for $F$ .",
"Recall that a Reeb field is uniquely determined by the following two equations : $\\left\\lbrace \\begin{aligned}A(X) &= 1 \\\\i_X dA &= 0 \\, .\\end{aligned}\\right.$ [Sketch of proof] Proving that $A$ is contact can be done in local coordinates, using that $F$ is strongly convex and 1-homogeneous, via the Euler formula.",
"To prove the affirmation about the Reeb field, we just have to remark that Equation REF is a very nice way to write the Euler-Lagrange equations.",
"Remark 1.1.3 This implies that the canonical volume $A \\wedge dA^{n-1}$ is invariant by the flow, i.e., $L_X\\left(A \\wedge dA^{n-1}\\right) = 0.$ The Reeb field $X$ is a second order differential equation, as defined in [59], i.e., Definition 1.1.4 A vector field $X$ on $HM$ is called a second order differential equation if the following diagram commutes ${THM [r]^{d\\pi } & TM [r]^{r} & HM [lld]^{\\operatorname{Id}} & \\\\HM [u]^{X}}$ There is an easy and very useful lemma about second order differential equations : Lemma 1.1.5 If $X_1,X_2$ are two second order differential equations on $HM$ , then there exists a function $m \\colon HM \\rightarrow \\mathbb {R}$ and a vertical vector field $Y \\in VHM$ such that $X_2 = m X_1 + Y.$ By definition, $r \\circ d\\pi \\circ X_1=r \\circ d\\pi \\circ X_2$ , so there exists $m \\colon HM \\rightarrow \\mathbb {R}$ such that $d\\pi \\circ X_1=m d\\pi \\circ X_2$ , so $X_1 - m X_2 \\in VHM$ ."
],
[
"Dynamical derivative and Jacobi endomorphism",
"Foulon defined many objects associated with a geodesic flow, hence generalizing some Riemannian concepts, like curvature, in a purely intrinsic way, without requiring any of the connections in Finsler geometry.",
"However, we do not wish to introduce those objects at length (or at all), because it is already done (both in French and English) in the following articles : [59], [60], [61], [48], [49], [50], [46].",
"Moreover we will not use that much of this machinery, at least directly.",
"In [48] in particular, the reader can find much about the link between Foulon's definitions and their Riemannian (or connection-obtained Finsler) equivalent.",
"In [46] or [45], Mickaël Crampon even generalizes these methods to Hilbert geometry (so to some Finsler metrics with very low regularity).",
"However, as we do need some results further on, we introduce the bare minimum.",
"There exists a $C^{\\infty }$ -linear operator $H_{X} \\colon VHM \\rightarrow THM$ , called the horizontal endomorphism such that, if we set $h_{X}HM:= H_{X}(VHM)$ , we have : $THM = VHM \\oplus h_{X}HM \\oplus \\mathbb {R}\\cdot X.$ We write $\\operatorname{Id}= p_{v} + p_{h} + p_X$ for the associated projections.",
"Note that this decomposition generalizes the classical vertical/horizontal decomposition in Riemannian geometry.",
"Let us also state the following easy fact : Lemma 1.1.6 Let $Z \\in T_{(x,\\xi )}HM$ such that $d\\pi (Z) = (x,v) \\in TM$ .",
"Let $\\lambda \\in \\mathbb {R}$ such that $Z = \\lambda X + h + Y$ , then $\\lambda \\le F(x,v), \\quad \\text{ with equality iff } r(x,\\xi ) = (x,v).$ As $A$ is zero on $H_X HM \\oplus VHM$ , we have $A(Z) = \\lambda $ .",
"Now, let $u \\in T_xM$ such that $r(x,u) = (x ,\\xi )$ , then $A(Z) &= \\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon } \\left( F(x, u + \\varepsilon d\\pi (Z)) - F(x,u) \\right) \\\\&\\le \\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon } \\left( F(x,u) + \\varepsilon F(x,v) - F(x,u) \\right),$ where the last line is obtained by convexity of $F$ , giving us that $\\lambda \\le F(x,v)$ .",
"The equality condition comes from the fact that $F$ is strongly, hence strictly, convex.",
"There exists a first-order differential operator $D_{X}$ defined on the space of $C^1$ vector fields on $HM$ , called the dynamical derivative associated with $X$ .",
"The splitting of $THM$ , namely $THM = VHM \\oplus h_{X}HM \\oplus \\mathbb {R}\\cdot X$ is invariant under $D_{X}$ and ,for any vector field $Y \\colon HM \\rightarrow VHM$ , $ H_{X}(Y) = -[X,Y] +D_{X}(Y) \\, .$ In the following, uppercase letters will refer to vector fields.",
"The space $HM$ comes naturally equipped with a Riemannian metric $g$ such that $g(X,X) = 1$ , the above splitting of $THM$ is orthogonal and, for any $y_1,y_2 \\in V_{(x,\\xi )}HM$ , if we choose $Y_1,Y_2$ any extensions of $y_1$ and $y_2$ to vertical vector fields, then $g(y_1,y_2) = dA ([X,Y_2],Y_1)\\, .$ Furthermore, $g$ is compatible with $D_{X}$ , in the following sense, $L_{X} \\left( g(Y_1,Y_2)\\right) = g(D_X Y_1, Y_2) + g(Y_1,D_X Y_2)$ .",
"The metric $g$ is called the vertical metric.",
"There exists a $C^{\\infty }$ -linear operator $R^{X} \\colon THM \\rightarrow THM$ , called the Jacobi endomorphism or curvature endomorphism.",
"It is defined by $ R^{X}(X)=0, \\quad R^{X}(Y) = p_v ([X,H_X (Y) ]) , \\quad R^{X}(h) = H_{X} (p_v [X,h]).$"
],
[
"Cartan's structure equations",
"Let $\\Sigma $ be a surface and $F$ a Finsler metric on it.",
"There is a way to generalize Cartan's structure equations to this Finsler setting using Foulon's formalism.",
"Foulon never published the proof, A. Reissmann did it in a not easily available preprint [100], so we provide his proof.",
"Note that Bryant [27], [28] gives the same result using a different presentation.",
"Proposition 1.1.7 (Cartan's structure equations) Let $(\\Sigma , F)$ be a Finsler surface.",
"Let $Y^{\\prime }$ be the unique vertical vector field such that $g(Y^{\\prime },Y^{\\prime }) =1$ and $h = H_{X}(Y^{\\prime }) \\in h_{X}HM$ .",
"If we write $k$ for the function on $H\\Sigma $ such that $R^X (Y^{\\prime }) = k Y^{\\prime }$ , then we have the following relations : $\\left\\lbrace \\begin{aligned}\\left[X,Y^{\\prime }\\right] &= -h \\\\\\left[X,h\\right] &= k Y^{\\prime } \\\\\\left[Y^{\\prime },h\\right] &= -X + a Y^{\\prime } +b h \\, ,\\end{aligned}\\right.$ where $a,b \\colon HM \\rightarrow \\mathbb {R}$ satisfy $L_X b =a$ and $L_X a + b k - L_{Y^{\\prime }} k =0 $ .",
"Note that $Y^{\\prime }$ is unique because we consider only orientable manifold.",
"Let start out by computing $\\left[X,Y^{\\prime }\\right] $ .",
"As $g(Y^{\\prime },Y^{\\prime }) = 1$ and $D_{X}$ is compatible with $g$ , we have $0 = L_{X} \\left(g(Y^{\\prime },Y^{\\prime }) \\right) = 2g (D_{X}Y^{\\prime }, Y^{\\prime }) \\, .$ Hence $D_{X}Y^{\\prime }$ and $Y^{\\prime }$ are orthogonal, but $D_{X}$ leaves $VH\\Sigma $ invariant, so ${D_{X}Y^{\\prime }=0}$ which yields, by Equation (REF ) : $h = -\\left[X,Y^{\\prime }\\right].$ Now let us compute the three projections of $\\left[X,h\\right]$ .",
"First recall that $\\ker A$ coincides with $h_{X}H\\Sigma \\oplus VH\\Sigma $ and that $A$ is invariant by $X$ .",
"So the equality $ L_{X} (A(h) ) = (L_{X} A ) (h) + A(\\left[X,h\\right] ) $ yields $A(\\left[X,h\\right] )=0$ , hence the projection of $\\left[X,h\\right]$ along $X$ is null.",
"By definition (see [59]), $D_{X} h = p_h ([X,h])$ and applying the same argument used on $Y^{\\prime }$ above shows that $D_{X} h=0$ .",
"And finally, again by definition, $ R^{X}(Y) = p_v ([X,H_X (Y) ])$ .",
"We are left with $\\left[Y^{\\prime },h\\right]$ .",
"By choice of $Y^{\\prime }$ , we have that $dA(\\left[X,Y^{\\prime }\\right],Y^{\\prime })=1$ , so $dA(Y^{\\prime },h)=1$ .",
"Now, $dA(Y^{\\prime },h) = L_{Y^{\\prime }} \\left(A(h) \\right) - L_{h} \\left(A(Y^{\\prime }) \\right) - A\\left( \\left[ Y^{\\prime },h \\right] \\right) = - A\\left( \\left[ Y^{\\prime },h \\right] \\right),$ so the projection of $\\left[ Y^{\\prime },h \\right]$ along $X$ is $-1$ and projecting on the horizontal and vertical distributions shows that there exist two real-valued functions $a$ and $b$ on $H\\Sigma $ such that $\\left[ Y^{\\prime },h \\right]= -X + a Y^{\\prime } + b h $ .",
"We finish by proving the assertions about the Lie derivatives of $a$ and $b$ .",
"They follow from the Bianchi identity.",
"Indeed, we have $\\left[X,\\left[Y^{\\prime },h\\right]\\right] &= \\left[X, -X +aY^{\\prime } + b h\\right] \\\\&= a\\left[X,Y^{\\prime }\\right] + \\left( L_{X} a\\right) Y^{\\prime } + b \\left[X,h\\right] + \\left( L_{X} b\\right) h \\\\&= -a h + \\left( L_{X} a\\right) Y^{\\prime } + b R^{X}(Y^{\\prime }) + \\left( L_{X} b\\right) h ,\\\\\\left[h, \\left[X,Y^{\\prime }\\right] \\right] &= 0 , \\\\\\left[Y^{\\prime }, \\left[h,X\\right]\\right] &= \\left[Y^{\\prime }, -R^X (Y^{\\prime })\\right] \\\\&= -\\left(L_{Y^{\\prime }} k \\right) (Y^{\\prime }) \\, ,$ So writing that $\\left[X,\\left[Y^{\\prime },h\\right]\\right] +\\left[h, \\left[X,Y^{\\prime }\\right] \\right] +\\left[Y^{\\prime }, \\left[h,X\\right]\\right] =0$ yields $\\left( -a + L_{X} b \\right) h + \\left( L_X a + b k - L_{Y^{\\prime }} k \\right) Y^{\\prime } = 0.$ This implies that $L_{X} b = a$ and $L_X a + b k - L_{Y^{\\prime }} k =0 $ .",
"Proposition 1.1.8 The function $b$ is identically zero if and only if $F$ is Riemannian.",
"Suppose that $b = 0$ .",
"Remark that, as $L_Xb = a$ , we immediately get that $a=0$ .",
"Denote by $\\chi _s$ the flow generated by $Y^{\\prime }$ , take $x\\in \\Sigma $ and choose a point $u_0 \\in H_x \\Sigma $ .",
"Let $v(s):= d\\pi \\left( X_{\\chi _s (u_0)} \\right)$ .",
"By construction, we have $F(v(s))=1$ and so $v$ is a parametrization of the unit circle $F^{-1}(1)$ at $x$ .",
"The goal is to show that $v$ parametrizes an Euclidean circle.",
"We start by computing $\\dot{v} = \\frac{dv}{ds}$ .",
"By definition, we have $\\left[Y^{\\prime },X \\right]_{\\chi _s(u_0)} = \\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon } \\left( d \\chi _{-\\varepsilon } \\left( X_{\\chi _{s + \\varepsilon } (u_0)} \\right) -X_{\\chi _{s} (u_0)} \\right),$ therefore, using that $\\pi \\circ \\chi _{-\\varepsilon } = \\pi $ , $d\\pi \\left( \\left[Y^{\\prime },X \\right]_{\\chi _s(u_0)} \\right) &= \\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon } \\left( d\\pi \\circ d \\chi _{-\\varepsilon } \\left( X_{\\chi _{s + \\varepsilon } (u_0)} \\right) - d\\pi \\left( X_{\\chi _{s} (u_0)} \\right)\\right) \\\\&= \\lim _{\\varepsilon \\rightarrow 0} \\frac{1}{\\varepsilon } \\left( v(s+\\varepsilon ) -v(s) \\right) \\\\&= \\dot{v}\\, .$ So, by Cartan's structure equations, we get that $\\dot{v} = d\\pi (h)$ .",
"Similar computations show that $\\ddot{v} = d\\pi \\left( \\left[Y^{\\prime },h\\right] \\right)$ .",
"Applying $d\\pi $ to the last of the structure equations shows that $ \\ddot{v} = -v + b \\dot{v}\\, .$ So assuming that $b$ is identically null gives us a differential equation for which every integral curves are Euclidean circles.",
"Hence, $F$ comes from a quadratic form.",
"Now assume that $F$ is Riemannian.",
"Recall (see [59]) that in that case, for any $u$ in $H_{x} \\Sigma $ , $d\\pi _u$ is an isometry from $\\left(\\mathbb {R}\\cdot X \\oplus h_X H\\Sigma \\right)_u$ equipped with the vertical metric $g_u$ onto $T_x \\Sigma $ .",
"Thus, by the above computations, $\\dot{v}$ is orthogonal to $v$ and $F(\\dot{v}) = 1$ .",
"Taking the derivative shows that $\\dot{v}$ is orthogonal to $\\ddot{v}$ , which, by Equation (REF ) yields that $b=0$ ."
],
[
"Cotangent space and Legendre transform",
"In this section, we will recall the construction of the Legendre transform associated to a Finsler metric and the dual Finsler metric.",
"The Legendre transform gives a natural diffeomorphism between the tangent and the cotangent bundles.",
"The construction is well-known in Finsler geometry and even better known in the Hamiltonian context as is the dual Finsler metric, but some of the results proved below might be less well-known.",
"We also try to give intrinsic proofs whenever we can.",
"Definition 1.2.1 We define the dual Finsler metric $F^{\\ast } \\colon T^{\\ast }M \\rightarrow \\mathbb {R}$ by setting, for $(x,p) \\in T^{\\ast }M$ , $ F^{\\ast }(x,p) = \\sup \\lbrace p(v) \\mid v\\in T_xM \\; \\text{\\rm such that } F(x,v)=1 \\rbrace .$ Remark 1.2.2 The function $F^{\\ast }$ is a Finsler co-metric, that is, it verifies the same conditions as in Definition REF but on the cotangent bundle.",
"Definition 1.2.3 The Legendre transform $\\mathcal {L}_F : TM \\rightarrow T^{\\ast }M$ associated with $F$ is defined by $L_F(x,0) = (x,0)$ and, for $(x,v) \\in \\mathring{T}M$ and $u \\in T_x^{\\ast }M$ , $\\mathcal {L}_F ( x,v) (u) := \\frac{1}{2} \\left.",
"\\frac{d}{dt} F^2(x,v + tu)\\right|_{t=0}.$ As $F^2$ is 2-homogeneous, we have that $\\mathcal {L}_F$ is 1-homogeneous, so we can project $\\mathcal {L}_F$ to the homogenized bundles.",
"Set $H^{\\ast }M := \\mathring{T}^{\\ast }M / \\mathbb {R}^+_{\\ast }$ and write $\\ell _F\\colon HM \\rightarrow H^{\\ast }M$ for the projection.",
"We can also construct $\\ell _F$ via the Hilbert form $A$ ; As $A$ is zero on ${VHM = \\ker d\\pi }$ , for any $(x,\\xi ) \\in HM$ , the linear function $(d\\pi _{(x,\\xi )})_{\\ast } \\left( A_{(x,\\xi )} \\right) \\colon TM \\rightarrow \\mathbb {R}$ is well defined and taking its class in $H^{\\ast }M$ gives $\\ell _F$ .",
"Remark that $\\mathcal {L}_F (x,v) = F(x,v) (d\\pi _{r(x,v)})_{\\ast } \\left( A_{r(x,v)} \\right) $ .",
"Considering directly $\\ell _F$ , instead of $\\mathcal {L}_F$ , can be quite helpful sometimes.",
"Proposition 1.2.4 We have the following properties : $F = F^{\\ast } \\circ \\mathcal {L}_F$ ; $\\ell _F$ is a diffeomorphism, $\\mathcal {L}_F$ is a bijection from $TM$ to $T^{\\ast }M$ and a diffeomorphism outside the zero section ; The following diagram commutes ${& \\mathring{T}^{\\ast }M [r]^{\\hat{r}} [ld]_{\\hat{p}} & H^{\\ast }M [rd]^{\\hat{\\pi }} & \\\\M & & & M \\\\& \\mathring{T}M [uu]^{\\mathcal {L}_F} [r]_r [ul]^{p} & HM [uu]_{\\ell _F} [ur]_{\\pi } &}$ We start with 1.",
"Let $(x,v) \\in TM$ , then $F^{\\ast } \\circ \\mathcal {L}_F (x,v) = \\sup \\lbrace \\mathcal {L}_F(x,v) (u) \\mid F(x,u)=1 \\rbrace $ .",
"Now, for any $u$ such that $F(x,u)=1$ , we choose $Z \\in T_{r(x,v)}HM$ such that $d\\pi (Z) = (x,u)$ and we have $\\mathcal {L}_F(x,v) (u) = F(x,v) A_{r(x,v)}(Z) \\le F(x,v) F(x,u) = F(x,v),$ with equality iff $r(x,v) = r(x,u)$ (by Lemma REF ).",
"This implies that $F^{\\ast } \\circ \\mathcal {L}_F (x,v) = F(x,v)$ .",
"For 2., several things are already clear ; The unique pre-image of 0 by $\\mathcal {L}_F$ is 0, $\\ell _F$ and $\\mathcal {L}_F$ are as smooth as $F$ outside the zero section and finally, $\\ell _F$ is bijective iff $\\mathcal {L}_F$ is, so we will prove injectivity or surjectivity for one of them and it will imply it for the other.",
"The injectivity is given by Lemma REF ; Indeed, suppose that there exist $\\xi _1,\\xi _2 \\in H_xM$ distinct and $\\mu \\in \\mathbb {R}^+$ such that $(d\\pi )_{\\ast } A_{(x,\\xi _1)} = \\mu (d\\pi )_{\\ast } A_{(x,\\xi _2)}$ .",
"Let $u_1,u_2 \\in T_xM$ such that $r(x,u_i) = (x,\\xi _i)$ and $F(x,u_i) =1$ , then, by Lemma REF , $1= (d\\pi )_{\\ast } A_{(x,\\xi _1)} (u_1) = \\mu (d\\pi )_{\\ast } A_{(x,\\xi _2)} (u_1) < \\mu ,$ and switching $u_1$ and $u_2$ gives $\\mu <1$ .",
"To prove surjectivity ; take $p \\in \\mathring{T}_x^{\\ast }M$ and set $\\mu = F^{\\ast }(x,p)$ .",
"Let $v_p \\in T_xM$ such that $F(x,v_p)=1$ and $p(v_p) = \\mu $ , i.e., $v_p$ is the vector realizing the supremum in Equation (REF ) and write $(x,\\xi _p) :=r(x,v_p)$ .",
"Let $h \\in \\left(h_{X}HM \\right)_{(x,\\xi _p)}$ and $w = d\\pi (h)$ , we claim that $p(w) =0$ .",
"Assume the claim for a moment and recall that $d\\pi _{(x,\\xi _p)}$ is a bijection from $\\left(h_{X}HM \\oplus \\mathbb {R}\\cdot X \\right)_{(x,\\xi _p)}$ to $T_x M$ .",
"So, for any $u \\in T_xM$ , there are $\\lambda $ and $h$ such that $u = \\lambda d\\pi (X(x,\\xi _p)) + d\\pi (h)$ , that is, $u = \\lambda v_p + w$ .",
"Therefore $p(u) = \\lambda p(v_p) = \\lambda \\mu = \\mu (d\\pi )_{\\ast } A_{(x,\\xi _p)}( \\lambda v_p + w) = \\mu (d\\pi )_{\\ast } A_{(x,\\xi _p)}(u)$ and the surjectivity would be proven.",
"So we just need to prove the claim.",
"Let $h \\in \\left(h_{X}HM \\right)_{(x,\\xi _p)}$ and set $c(t) := \\left(F(v_p + t d\\pi (h)) \\right)^{-1}\\left(v_p + t d\\pi (h)\\right)$ .",
"By definition of $v_p$ , $\\left.",
"\\frac{d}{dt} p\\left( c(t) \\right) \\right|_{t=0} = 0$ so $0 = \\left.",
"\\frac{d}{dt} p\\left( c(t) \\right) \\right|_{t=0} = p \\left( \\frac{d\\pi (h)}{F(x,v_p + t d\\pi (h))} - \\frac{A_{(x,\\xi _p)}(h) v_p }{F^2(x,v_p + t d\\pi (h))} \\right).$ As $A_{(x,\\xi _p)}(h) =0$ , we get that $p(d\\pi (h) ) =0$ , which proves our claim.",
"Now, writing $\\mathcal {L}_F$ in local coordinates, we can easily see that its Jacobian is given by $\\left(\\dfrac{\\partial ^2 F^2}{\\partial v_i \\partial v_j}\\right)_{i,j}$ and, as this matrix is non-degenerate, $\\mathcal {L}_F$ is a local diffeomorphism, hence a global one because it is bijective.",
"Finally, the fact that the diagram commutes is trivial.",
"Definition 1.2.5 We set $B:=\\left(\\ell _F^{-1}\\right)^{\\ast } A$ , it is a contact form and we write $X^{\\ast }$ for its Reeb field.",
"Lemma 1.2.6 The following diagram commutes ${TH^{\\ast }M [r]^{d\\hat{\\pi }} & TM [r]^{r} & HM \\\\H^{\\ast }M [u]_{X^{\\ast }} [urr]_{\\ell _{F}^{-1} } & &}$ Note also that the projection by $\\hat{p}$ of the integral curves of $X^{\\ast }$ are geodesics of the metric on $M$ .",
"As $B = \\left( \\ell _F^{-1}\\right)^{\\ast } A$ and $X^{\\ast }$ is the Reeb field of $B$ , we have ${X^{\\ast } = \\left( \\ell _F^{-1}\\right)_{\\ast } X}$ (just verify that $B \\left( \\left( \\ell _F^{-1}\\right)_{\\ast } X \\right)=1$ and $i_{\\left( \\ell _F^{-1}\\right)_{\\ast } X} dB =0$ ), and since $\\hat{\\pi }\\circ \\ell _F = \\pi $ , we get that the projections to $M$ of the integral curves of $X^{\\ast }$ are the same as those of $X$ , which proves the second claim.",
"As $X^{\\ast } = \\left( \\ell _F^{-1}\\right)_{\\ast } X$ , we can rewrite it as $X \\circ \\ell _F^{-1} = d\\ell _F^{-1} \\circ X^{\\ast }$ , and $X$ being a second order differential equation, we get that $\\ell _F^{-1} = r\\circ d\\pi \\circ X \\circ \\ell _F^{-1} = r\\circ d\\pi \\circ d\\ell _F^{-1} \\circ X^{\\ast } = r\\circ d\\hat{\\pi } \\circ X^{\\ast }.$ Lemma 1.2.7 Let $H \\colon T^{\\ast }M \\rightarrow \\mathbb {R}$ be a Finsler co-metric, then we can define $H^{\\ast }\\colon TM \\rightarrow \\mathbb {R}$ by $H^{\\ast }(x,v) = \\sup \\lbrace p(v) \\mid p\\in T_x^{\\ast }M \\; \\text{\\rm such that } H(x,p)=1 \\rbrace .$ The function $H^{\\ast }$ satisfies the following properties : $H^{\\ast }$ is a Finsler metric ; If $F$ is a Finsler metric, then $F^{\\ast \\ast } = F$ ; We define the Legendre transform associated with $H$ , $\\mathcal {L}_H \\colon T^{\\ast }M \\rightarrow TM$ by $\\mathcal {L}_H ( x,p_1) (p_2) := \\frac{1}{2} \\left.",
"\\frac{d}{dt} H^2(x,p_1 + tp_2)\\right|_{t=0},$ where we identified $TM$ and $T^{\\ast \\ast }M$ .",
"It is a diffeomorphism outside the zero section.",
"Furthermore, if $F$ is a Finsler metric then $\\mathcal {L}_{F^{\\ast }} \\circ \\mathcal {L}_F = \\operatorname{Id}_{TM}$ and $\\mathcal {L}_F \\circ \\mathcal {L}_{F^{\\ast }} = \\operatorname{Id}_{T^{\\ast }M}$ .",
"$H = H^{\\ast } \\circ \\mathcal {L}_H$ .",
"We do not prove the first point nor the third one, as we will not use them in this dissertation.",
"Note also that the proof of 4. is the same as in Proposition REF .",
"The only thing we will use later is 2., so we prove it.",
"Set $F^{\\prime }(x,v) := \\sup \\lbrace p(v) \\mid p\\in T_x^{\\ast }M \\; \\text{such that } F^{\\ast }(x,p)=1 \\rbrace $ , we want to show that $F^{\\prime } = F$ , by homogeneity, it suffices to show it on the $F$ -unit circle.",
"Let $v\\in T_xM$ such that $F(x,v)=1$ .",
"First, if we set $p:= \\mathcal {L}_F(x,v) \\in T_x^{\\ast }M$ then, by definition, we have $p(v) = F(x,v) = F^{\\ast }(x,p) = 1,$ so $F^{\\prime }(x,v) \\ge F(x,v)$ .",
"Now, if we take $p_0$ such that $F^{\\ast }(x,p_0)= 1$ and $p_0(v) = F^{\\prime }(x,v)$ , again by definition $F^{\\prime }(x,v) = p_0(v) \\le F^{\\ast }(x,p_0) = 1 = F(x,v).$ The following result was communicated to us by P. Foulon and might be already known to others.",
"However, we are not aware of the existence of a published proof and hence give one : Theorem 1.2.8 (Foulon [58]) Any Finsler metric on $M$ defines the same contact structure on $H^{\\ast }M$ , i.e., if $F$ is a Finsler metric on $M$ and ${B = \\left(\\ell _F^{-1} \\right)^{\\ast } A}$ , the distribution $\\ker B \\subset TH^{\\ast }M$ is independent of $F$ .",
"Furthermore, if we denote by $\\lambda $ the Liouville form on $T^{\\ast }M$ , we have $ \\hat{r}^{\\ast } B = \\frac{\\lambda }{F^{\\ast }},$ and $ \\hat{r}^{\\ast } B\\wedge dB^{n-1} = \\frac{\\lambda \\wedge d\\lambda ^{n-1} }{(F^{\\ast })^n}.$ We will start by showing Equation (REF ).",
"First recall the definition of the Liouville form : for any $\\in T^{\\ast }M$ , $\\lambda _{p} = p \\circ d\\hat{p}_{|M}$ , where $\\hat{p}_{|M} \\colon T^{\\ast }M \\rightarrow M$ is the base point projection.",
"In order to show that $\\hat{r}^{\\ast } B = \\frac{\\lambda }{F^{\\ast }}$ , we will prove that their pull-back by $\\mathcal {L}_F$ coincides.",
"On one hand, as $\\hat{r} \\circ \\mathcal {L}_F = r \\circ \\ell _F$ , we have $\\mathcal {L}_F^{\\ast } \\hat{r}^{\\ast } B = r^{\\ast } \\ell _F^{\\ast } B = r^{\\ast } A = d_vF \\, ,$ and on the other hand, $\\mathcal {L}_F^{\\ast } \\left( \\frac{\\lambda }{F^{\\ast }} \\right) = \\frac{\\mathcal {L}_F^{\\ast } \\lambda }{ F^{\\ast } \\circ \\mathcal {L}_F} = \\frac{\\mathcal {L}_F^{\\ast } \\lambda }{F} \\, .$ Now, let us compute $\\mathcal {L}_F^{\\ast } \\lambda $ : for $(x,v) \\in TM$ and $Z \\in T_{(x,v)} TM$ , $\\left( \\mathcal {L}_F^{\\ast } \\lambda \\right)_{(x,v)} (Z) &= \\lambda _{\\mathcal {L}_F(x,v)} \\left( d\\mathcal {L}_F (Z) \\right) \\\\&= \\mathcal {L}_F(x,v) \\circ d\\hat{p}_{|M} \\circ d\\mathcal {L}_F (Z) \\\\&= \\mathcal {L}_F(x,v) \\circ dp_{|M} (Z) \\\\&= \\frac{1}{2} \\frac{d}{dt} F^2\\left(x, v + t dp_{|M} (Z) \\right) \\\\&= F(x,v) d_vF_{(x,v)}(Z)\\, .$ And we proved Equation (REF ).",
"Once we have that, the uniqueness of the contact structure is trivial.",
"For the last equality, we have $\\hat{r}^{\\ast } dB = d \\hat{r}^{\\ast } B = \\frac{d\\lambda }{F^{\\ast }} - \\frac{\\lambda \\wedge dF^{\\ast } }{(F^{\\ast })^2}\\, .$ Therefore $\\hat{r}^{\\ast } dB^{n-1} = \\left(\\frac{d\\lambda }{F^{\\ast }}\\right)^{n-1} + \\lambda \\wedge \\left( \\text{Something} \\right)$ , so $\\hat{r}^{\\ast } B\\wedge dB^{n-1}= \\frac{\\lambda \\wedge d\\lambda ^{n-1}}{(F^{\\ast })^{n}} + \\lambda \\wedge \\lambda \\wedge \\left( \\text{Something} \\right) = \\frac{\\lambda \\wedge d\\lambda ^{n-1}}{(F^{\\ast })^{n}}\\, .$ Corollary 1.2.9 If $B$ and $B_1$ correspond to two Finsler metrics $F$ and $F_1$ , then there exists a function $f \\colon H^{\\ast }M \\rightarrow \\mathbb {R}$ such that $B_1 = f B$ .",
"By Equation (REF ), $\\hat{r}^{\\ast } B_1 = \\frac{F^{\\ast }}{F^{\\ast }_1} \\hat{r}^{\\ast } B,$ hence the result."
],
[
"Angle and volume in Finsler geometry",
"We will give here a definition of a solid angle in Finsler geometry, i.e., a volume form on each $H_{x}M$ naturally associated with the Finsler metric.",
"Even if the construction seems to be known, this angle does not appear as such in the literature, at least to the best of the knowledge of the author.",
"We will also construct a similar solid “co-angle”, i.e., a volume form on each $H_{x}^{\\ast }M$ .",
"Simultaneously to the angle, the construction gives a volume form on $M$ .",
"We will show that this volume is the Holmes-Thompson volume for Finsler geometry [73].",
"There are already several different angles defined in Finsler geometry.",
"We did not however try to compare known Finsler angles to this new one, making just a quick address to that problem in Section REF ."
],
[
"Construction",
"As we have seen in Section REF , the only truly canonical volume associated with $F$ is given by $A \\wedge dA^{n-1}$ .",
"We are going to split this volume on $HM$ into a volume on the base manifold $M$ and a volume on each fiber $H_xM$ in a canonical way.",
"From now on, we will assume that $M$ is oriented and will only consider volumes preserving the orientation.",
"The construction is done in two steps.",
"First, Lemma 1.3.1 Let $\\omega $ be a volume form on $M$ , then there exists an $(n-1)$ -form $\\alpha ^{\\omega }$ on $HM$ such that $\\alpha ^{\\omega } \\wedge \\pi ^{\\ast }\\omega = A\\wedge dA^{n-1}.$ Moreover, the value taken by $\\alpha ^{\\omega }$ on $VHM$ is uniquely determined.",
"The existence of some $\\alpha ^{\\omega }$ is straightforward, we just complete a $n$ -form $\\pi ^{\\ast }\\omega $ into a $(2n-1)$ -form $A \\wedge dA^{n-1}$ .",
"The uniqueness is given by the following.",
"Let $Y_1, \\dots , Y_{n-1} \\in VHM$ be $(n-1)$ linearly independent vertical vector fields.",
"Then $ Y_1, \\dots , Y_{n-1}$ , $ X$ , $\\left[X, Y_1 \\right], \\dots ,\\\\ \\left[X, Y_{n-1}\\right]$ are linearly independent (see [59]).",
"Therefore, we must have $\\alpha ^{\\omega } \\left(Y_1, \\dots , Y_{n-1} \\right) = \\frac{A\\wedge dA^{n-1} \\left( Y_1, \\dots , Y_{n-1}, X, \\left[X, Y_1 \\right], \\dots , \\left[X, Y_{n-1}\\right] \\right) }{\\pi ^{\\ast }\\omega \\left( X, \\left[X, Y_1 \\right], \\dots , \\left[X, Y_{n-1}\\right] \\right) }.$ Note that $\\alpha ^{\\omega }$ does give us a notion of solid angle, despite its non-uniqueness : Lemma 1.3.2 For any $\\omega $ , the integral $l^{\\omega }(x)= \\int _{H_x M} \\alpha ^{\\omega }$ does not depend on the choice of $\\alpha ^{\\omega }$ .",
"Follows from the fact that the forms $\\alpha ^{\\omega }$ are the same on $VHM$ .",
"Secondly, in order to have a reasonable notion of angle, we wish the volume of the fibers to be constant.",
"To coincide with the Riemannian case we take this constant to be the volume of a Euclidean sphere.",
"It turns out that this condition is realized for a unique volume form on $M$ , hence giving a really natural way to associate a pair angle/volume with a Finsler metric : Lemma 1.3.3 There exists a unique volume form $\\Omega ^F$ on $M$ such that, for all $ x \\in M$ , $l^{\\Omega ^F}(x)= \\text{vol}_{\\text{Eucl}} \\left( \\mathbb {S}^{n-1} \\right).$ Moreover, if $\\omega $ is any volume on $M$ , then $\\Omega ^F$ is given by $ \\Omega ^F = \\frac{l^{\\omega }}{\\text{vol}_{\\text{Eucl}} \\left( \\mathbb {S}^{n-1} \\right)} \\omega \\, .$ Before getting on to the proof, let us state the following remark which has some interest of its own and will be needed afterwards : Remark 1.3.4 If $\\omega ^{\\prime }$ is another volume form on $M$ , preserving the orientation, and $f : M \\rightarrow \\mathbb {R}_+^{\\ast }$ such that $\\omega ^{\\prime }= f \\omega $ .",
"Then, $\\alpha ^{f\\omega } \\wedge \\pi ^{\\ast } \\left( f\\omega \\right) = \\alpha ^{f\\omega } \\wedge f \\pi ^{\\ast } \\omega = A\\wedge dA^{n-1}.$ And so, for any $Y_1, \\dots , Y_n \\in VHM$ , $i_{Y_1}\\dots i_{Y_n} \\alpha ^{f\\omega }= \\frac{1}{f} i_{Y_1}\\dots i_{Y_n} \\alpha ^{\\omega }.$ [Proof of Lemma REF ] Let $c_n =\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left( \\mathbb {S}^{n-1} \\right)$ and $ \\Omega = \\frac{l^{\\omega }}{ c_n } \\omega $ , then the remark shows that, on $VHM$ , $\\alpha ^{\\Omega } = \\frac{c_n}{l^{\\omega }} \\alpha ^{\\omega }$ , which in turns yields $l^{\\Omega }(x) = \\int _{H_x M} \\alpha ^{\\Omega } = \\int _{H_x M} \\frac{c_n}{l^{\\omega }(x)} \\alpha ^{\\omega } = \\frac{c_n}{l^{\\omega }(x)} \\int _{H_x M} \\alpha ^{\\omega } = c_n.$ Uniqueness is also straightforward.",
"We can summarize the construction in the following : Proposition 1.3.5 There exists a unique volume form $\\Omega ^F$ on $M$ and an $(n-1)$ -form $\\alpha ^F$ on $HM$ , never zero on $VHM$ , such that $\\alpha ^{F} \\wedge \\pi ^{\\ast }\\Omega ^F = A\\wedge dA^{n-1},$ and, for all $x\\in M$ , $\\int _{H_xM} \\alpha ^F = \\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}(\\mathbb {S}^{n-1})\\, .$ Remark 1.3.6 Let us emphasize again that $\\alpha ^F$ is not unique, only its restriction to $VHM$ , which is what we need in order to have a solid angle.",
"We will see later (Section REF ) that $\\Omega ^F$ is in fact the well-known Holmes-Thompson volume."
],
[
"Coangle",
"Taking the Hamiltonian point of view, we can see that the volume $B \\wedge dB^{n-1}$ on $H^{\\ast }M$ is as canonical as $A \\wedge dA^{n-1}$ is, so we could have carried out the above construction on the homogenized cotangent bundle $H^{\\ast }M$ .",
"The exact same steps gives the following : Proposition 1.3.7 There exists a unique volume form $\\Omega ^{F^{\\ast }}$ on $M$ and an $(n-1)$ -form $\\beta ^{F^{\\ast }}$ on $H^{\\ast }M$ , never zero on $VH^{\\ast }M:= \\ker d\\hat{\\pi }$ , such that $\\beta ^{F^{\\ast }} \\wedge \\hat{\\pi }^{\\ast }\\Omega ^{F^{\\ast }} = B \\wedge dB^{n-1},$ and, for all $x\\in M$ , $\\int _{H^{\\ast }_xM} \\beta ^{F^{\\ast }} = \\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}(\\mathbb {S}^{n-1})\\, .$ Fortunately for our claim of “natural” angle and volume associated with $F$ , there is a relationship between our two constructions : Proposition 1.3.8 Recall that $\\ell _{F} \\colon HM \\rightarrow H^{\\ast }M$ denotes the Legendre transform.",
"Then $\\Omega ^F &= \\Omega ^{F^{\\ast }}, \\\\\\alpha ^F &= \\ell _{F}^{\\ast } \\beta ^{F^{\\ast }},$ where the second equality holds on $VHM$ .",
"In the sequel, when the metric is clear from the context, we will often forget the superscript when writing $\\alpha ^F$ , $\\beta ^{F^{\\ast }}$ or $\\Omega ^F$ .",
"First, note that $A \\wedge dA^{n-1}= \\ell _F^{\\ast }\\left(B \\wedge dB^{n-1}\\right) = \\ell _F^{\\ast }\\left( \\beta ^{F^{\\ast }} \\wedge \\hat{\\pi }^{\\ast }\\Omega ^{F^{\\ast }} \\right) = \\ell _F^{\\ast }\\left( \\beta ^{F^{\\ast }} \\right) \\wedge \\ell _F^{\\ast } \\hat{\\pi }^{\\ast } \\left( \\Omega ^{F^{\\ast }} \\right).$ As $\\hat{\\pi } \\circ \\ell _F = \\pi $ we have that $\\ell _F^{\\ast } \\hat{\\pi }^{\\ast } \\left( \\Omega ^{F^{\\ast }} \\right) = \\pi ^{\\ast }\\Omega ^{F^{\\ast }}$ , which yields $A \\wedge dA^{n-1}= \\ell _F^{\\ast } \\beta ^{F^{\\ast }} \\wedge \\pi ^{\\ast }\\Omega ^{F^{\\ast }}.$ It remains to show that the length of the fibers for $\\ell _F^{\\ast } \\beta ^{F^{\\ast }}$ are equal to $\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1} \\right)$ as the uniqueness part in Proposition REF would then prove the claim.",
"By the change of variables formula and the definition of $\\beta ^{F^{\\ast }}$ , we have for any $x\\in M$ $\\int _{H_x M} \\ell _F^{\\ast } \\beta ^{F^{\\ast }} = \\int _{\\ell _F(H_xM)} \\beta ^{F^{\\ast }} = \\int _{H^{\\ast }_xM} \\beta ^{F^{\\ast }} = \\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1} \\right).",
"$"
],
[
"Angle and conformal change",
"Here we will show two properties relating our angle and conformal change of Finsler metrics.",
"The first says that the angle and coangle are invariant under conformal change, which is natural.",
"The second is more surprising ; the coangle determines the conformal class of a Finsler metric, therefore a Finsler metric is uniquely determined by a coangle and a volume form on the manifold.",
"Proposition 1.3.9 Let $(M,F)$ be a Finsler manifold, $f \\colon M \\xrightarrow{} \\mathbb {R}$ , $F_f= e^{f} F$ , $\\alpha _f$ , $\\beta _f$ and $\\Omega _f$ the angle, co-angle and volume form of $F_f$ .",
"Then $\\alpha _f = \\alpha $ on $VHM$ , $\\beta _f = \\beta $ on $VH^{\\ast }M$ and $\\Omega _f = e^{nf} \\Omega $ .",
"Using the definition of the Hilbert form, we immediately have $A_f = e^{f} A$ , so $A_f \\wedge dA_f^{n-1} = e^{nf} A\\wedge dA^{n-1}.$ Let $\\omega $ be a volume form on $M$ .",
"Let $\\alpha _F^{\\omega }$ and $\\alpha _{F_f}^{\\omega }$ be the two $(n-1)$ -forms defined by $\\alpha _F^{\\omega }\\wedge \\pi ^{\\ast } \\omega = A\\wedge dA^{n-1}$ and $\\alpha _{F_f}^{\\omega }\\wedge \\pi ^{\\ast } \\omega = A_f \\wedge dA_f^{n-1}$ .",
"We have $\\alpha _{F_f}^{\\omega }\\wedge \\pi ^{\\ast } \\omega = e^{nf} \\alpha _F^{\\omega }\\wedge \\pi ^{\\ast } \\omega .$ From there we get that, for any $Y_1, \\dots , Y_{n-1} \\in VHM$ , $i_{Y_1}\\dots i_{Y_{n-1}} \\left( \\alpha _{F}^{\\omega }\\wedge \\pi ^{\\ast } \\omega \\right) &= \\alpha _{F}^{\\omega }\\left(Y_1, \\dots , Y_{n-1} \\right)\\pi ^{\\ast } \\omega \\, , \\\\i_{Y_1}\\dots i_{Y_{n-1}} \\left( \\alpha _{F_f}^{\\omega }\\wedge \\pi ^{\\ast } \\omega \\right) &= \\alpha _{F_f}^{\\omega }\\left(Y_1, \\dots , Y_{n-1} \\right)\\pi ^{\\ast } \\omega \\, .$ Therefore $\\alpha _{F_f}^{\\omega }\\left(Y_1, \\dots , Y_{n-1} \\right)\\pi ^{\\ast } \\omega = e^{nf} \\alpha _{F}^{\\omega }\\left(Y_1, \\dots , Y_{n-1} \\right)\\pi ^{\\ast } \\omega \\, ,$ which leads to $\\alpha _{F_f}^{\\omega }\\left(Y_1, \\dots , Y_{n-1} \\right) = e^{nf} \\alpha _{F}^{\\omega }\\left(Y_1, \\dots , Y_{n-1} \\right).$ And we deduce that, for any $x\\in M$ , $\\int _{H_xM} \\hspace{-5.69054pt} \\alpha _{F_f}^{\\omega } = e^{nf(x)} \\int _{H_xM} \\hspace{-5.69054pt} \\alpha _{F}^{\\omega }\\, ,$ The two volume forms $\\Omega $ and $\\Omega _f$ on $M$ associated with $F$ and $F_f$ are given by (see Equation (REF )) $\\Omega _f = \\frac{\\int _{\\scriptscriptstyle {H_xM}} \\hspace{-2.84526pt} \\alpha _{F_f}^{\\omega }}{c_n} \\omega ,\\; \\text{and} \\; \\, \\Omega = \\frac{\\int _{\\scriptscriptstyle {H_xM}} \\hspace{-2.84526pt} \\alpha _{F}^{\\omega }}{c_n} \\omega \\, ,$ which yields $\\Omega _f = e^{nf} \\Omega \\, .$ Using the definition of $\\alpha _f$ and Equation (REF ), we obtain $e^{nf} \\alpha _f \\wedge \\pi ^{\\ast } \\Omega = \\alpha _f \\wedge \\pi ^{\\ast } \\Omega _f = A_f\\wedge dA_f^{n-1} = e^{nf} A\\wedge dA^{n-1} =e^{nf} \\alpha \\wedge \\pi ^{\\ast }\\Omega \\, ,$ which in turns implies that, for any $Y_1, \\dots , Y_{n-1} \\in VHM$ , we have $\\alpha \\left(Y_1, \\dots , Y_{n-1}\\right) = \\alpha _f\\left(Y_1, \\dots , Y_{n-1}\\right).$ Applying the following lemma will prove the claim about the co-angles.",
"Lemma 1.3.10 If $F_f = e^{f} F$ , then $\\ell _{F} = \\ell _{F_f}$ .",
"Recall (see Section REF ) that for $(x,\\xi ) \\in HM$ , $\\ell _F(x,\\xi )$ is given by the class in $H^{\\ast }_x M$ of $ A_{(x,\\xi )}$ seen as an element of $T^{\\ast }_xM$ .",
"Now $A_f = e^f A$ , therefore, $ A_{(x,\\xi )}$ and $ \\left.A_f \\right._{(x,\\xi )}$ are in the same class.",
"Hence $\\ell _{F} = \\ell _{F_f}$ .",
"Proposition 1.3.11 Let $\\beta $ and $\\beta _1$ be coangles associated with two Finsler metrics $F$ and $F_1$ .",
"If $\\beta $ and $\\beta _1$ are equal on $VH^{\\ast }M$ , then there exists a positive function $f$ on $M$ such that $F_1 = f F$ .",
"By Equation (REF ), we have that $ B \\wedge dB^{n-1}= \\left(\\frac{F_1^{\\ast }}{F^{\\ast }}\\right)^n B_1 \\wedge dB_1^{n-1}$ , where $F_1^{\\ast }/ F^{\\ast }$ is seen as a function on $H^{\\ast }M$ , we call it $f$ for the moment (note that $f$ is positive).",
"Using the definition of coangle, we have that $\\beta \\wedge \\hat{\\pi }^{\\ast }\\Omega = f^n \\beta _1 \\wedge \\hat{\\pi }^{\\ast }\\Omega _1 = f^n \\beta \\wedge \\hat{\\pi }^{\\ast }\\Omega _1\\, .$ Now, applying $i_{Y_1} \\dots i_{Y_{n-1}}$ to both sides, for every $Y_1, \\dots , Y_{n-1} \\in VH^{\\ast }M$ , shows that we must have $\\hat{\\pi }^{\\ast }\\Omega = f^n \\hat{\\pi }^{\\ast }\\Omega _1$ , i.e., $f$ must be constant on the fibers.",
"Hence, we can see $f$ as a function on $M$ , and, as $f = \\frac{F_1^{\\ast }}{F^{\\ast }}$ , we get, $F_1^{\\ast }=f F^{\\ast }$ , with $f \\colon M \\rightarrow \\mathbb {R}$ .",
"So $F_1^{\\ast }$ and $F^{\\ast }$ differs by a conformal change.",
"From there, it is easy to deduce it for $F_1$ and $F$ .",
"Indeed, recall that $F_1^{\\ast \\ast } = F_1$ (see Lemma REF ), therefore $F_1(x,v) &= \\sup \\lbrace p(v) \\mid p\\in T_x^{\\ast }M \\; \\text{such that } F_1^{\\ast }(x,p)=1 \\rbrace \\\\&= \\sup \\lbrace p(v) \\mid p\\in T_x^{\\ast }M \\; \\text{such that } f(x) F^{\\ast }(x,p)=1 \\rbrace \\\\&= f(x) F(x,v)\\, .$ Corollary 1.3.12 A Finsler metric $F$ is uniquely determined by the two forms $\\beta ^F$ and $\\Omega ^F$ .",
"Note that this corollary gives an interesting characterization of a Finsler metric.",
"However, given two such forms, we did not try to give conditions implying that they come from a Finsler metric even so it is an interesting question.",
"Remark that it would seem natural to have the above proposition also true for angles (instead of coangles).",
"Unfortunately, we do not know if that is the case.",
"One thing is sure : it is not, as it might seem, a direct consequence of Proposition REF : If $\\alpha = \\alpha _1$ , we just have $\\beta _1 = \\left( \\ell _F \\circ \\ell _{F_1}^{-1} \\right)^{\\ast }(\\beta )$ , but we have not yet found a reason to believe that having the same angle would imply that the Legendre transforms are the same."
],
[
"Angles in dimension two",
"In dimension 2, a solid angle is the same thing as a traditional angle.",
"In this section, we quickly go over a few properties of our angle.",
"It is not aimed to be anything like a thorough study of the angle, more of a quick overlook of some questions that at once came to our mind.",
"First, we study the relationship between this angle and another that is traditionally used in Finsler geometry.",
"Then we state some easy properties of the rotation generated by this angle."
],
[
"A characterization of Riemannian surfaces",
"Recall that there is a canonical Riemannian metric $g$ on $VHM$ given by : $g (y_1,y_2) = dA\\left( \\left[X,Y_2\\right] , Y_1 \\right),$ where $Y_1$ and $Y_2$ are vertical vector fields such that $Y_i(x,v) = y_i$ .",
"This defines a distance function on each $H_x\\Sigma $ , i.e., an angle.",
"Note that this gives an angle in any dimension, not just a solid angle as our $\\alpha ^F$ .",
"Even though the presentation of this metric was given by P. Foulon, the angle that it defines was known well before him, because it turns out that this vertical metric is the same as the one obtained by considering $\\left(\\dfrac{\\partial ^2 F^2}{\\partial v_i \\partial v_j}\\right)_{i,j}$ .",
"Our goal in this section is to show that $\\alpha ^F$ is in general very different from the angle obtained via the vertical metric.",
"We will show even more : in dimension 2, if those angles coincides at a point, then the Finsler metric is Riemannian at that point.",
"Proposition 1.4.1 Let $Y$ be the vertical vector field such that $\\alpha ^F(Y)=1$ and $Y^{\\prime }\\in VH\\Sigma $ such that $g(Y^{\\prime },Y^{\\prime }) =1$ .",
"Let $c \\colon HM \\rightarrow \\mathbb {R}$ , such that $Y^{\\prime } = c Y$ .",
"If $c$ is constant on the fibers, then $F$ is Riemannian.",
"If $c$ is constant on the fibers, then $A\\wedge i_{Y^{\\prime }} dA =i_{Y^{\\prime }} (A\\wedge dA) = \\alpha (Y^{\\prime }) \\pi ^{\\ast } \\Omega = c \\pi ^{\\ast } \\Omega $ can be projected to $M$ .",
"Therefore $L_{Y^{\\prime }} \\left(A\\wedge i_{Y^{\\prime }} dA \\right)= 0$ , and a direct computation gives $L_{Y^{\\prime }} \\left(A\\wedge i_{Y^{\\prime }} dA \\right) = A\\wedge L_{Y^{\\prime }} \\left( i_{Y^{\\prime }} dA \\right)$ .",
"We deduce that $L_{Y^{\\prime }} \\left( i_{Y^{\\prime }} dA \\right)$ is null on $VHM \\oplus h_{X} HM$ .",
"In particular, if $h = H^X (Y^{\\prime })$ , then $L_{Y^{\\prime }} \\left( i_{Y^{\\prime }} dA \\right) (h) = 0$ .",
"Using Cartan's structure equations (Proposition REF ), we have $L_{Y^{\\prime }} \\left( i_{Y^{\\prime }} dA \\right) (h) &= i_{Y^{\\prime }}\\left(d i_{Y^{\\prime }} dA \\right) (h) \\\\&= d\\left( i_{Y^{\\prime }} dA \\right) (h, Y^{\\prime }) \\\\&= L_h \\left( i_{Y^{\\prime }} dA (Y^{\\prime })\\right) - L_{Y^{\\prime }} \\left( i_{Y^{\\prime }} dA (h)\\right) - i_{Y^{\\prime }}dA\\left( \\left[h,Y^{\\prime }\\right]\\right) \\\\&= - L_{Y^{\\prime }} \\left( L_{Y^{\\prime }} A(h) - L_h A(Y^{\\prime }) - A([H,Y^{\\prime }]) \\right) - dA\\left( \\left[h,Y^{\\prime }\\right], Y^{\\prime }\\right)\\\\&= - L_{Y^{\\prime }} \\left( - A(X - a Y^{\\prime }-b h) \\right) - dA\\left( X - a Y^{\\prime }-b h, Y^{\\prime } \\right) \\\\&= - L_{Y^{\\prime }} (1) - dA\\left(-b h, Y^{\\prime }\\right)\\\\&= - b dA\\left(\\left[X, Y^{\\prime }\\right], Y^{\\prime }\\right) \\\\&= -b\\, .$ So $b=0$ , which is equivalent to $F$ being Riemannian (by Proposition REF )."
],
[
"Rotations and reversibility",
"Proposition 1.4.2 Let $(\\Sigma ,F)$ be a Finsler 2-manifold.",
"There exists a unique vertical vector field $Y \\colon H\\Sigma \\rightarrow VH\\Sigma $ such that $\\alpha (Y)=1$ .",
"The one-parameter group $\\theta ^t$ generated by $Y$ is such that $\\forall (x,\\xi ) \\in H\\Sigma $ , $t\\in \\mathbb {R}$ , $\\left\\lbrace \\begin{aligned}\\pi \\left(\\theta ^t(x,\\xi )\\right) &= x \\\\\\theta ^{2\\pi }(x,\\xi ) &= (x,\\xi ) \\, .\\end{aligned}\\right.$ Take a non-degenerate vertical vector field $Y_1$ and set $Y := Y_1/\\alpha \\left(Y_1\\right)$ .",
"Uniqueness is due to the fact that $VH\\Sigma $ is 1-dimensional.",
"As $Y$ is a non-degenerate smooth vector field, it generates a one-parameter group $\\theta ^t$ , and the first part of (REF ) follows from the fact that $Y$ is vertical.",
"The second claim follows from the fact that the length of the fibers is taken to be $2\\pi $ .",
"Indeed, if we set $x \\in \\Sigma $ , for any $\\xi \\in H_x\\Sigma $ , $t \\mapsto \\theta ^{t} (x,\\xi )$ gives a parametrization of $H_x\\Sigma $ .",
"Let $T_{(x,\\xi )} \\in \\mathbb {R}^+$ be the period of $\\theta ^{t}(x,\\xi )$ .",
"It is easy to see that this period does not depend on $\\xi $ .",
"Indeed, if $\\xi _1,\\xi _2 \\in H_x\\Sigma $ , there exists $t<T_{(x,\\xi _1)}$ such that $\\theta ^{t}(x,\\xi _1) = (x,\\xi _2)$ .",
"Therefore $\\theta ^{T_{(x,\\xi _1)}}(x,\\xi _2) = \\theta ^{T_{(x,\\xi _1)}}\\left(\\theta ^{t}(x,\\xi _1) \\right) = \\theta ^{t+T_{(x,\\xi _1)}}(x,\\xi _1) = \\theta ^{t}(x,\\xi _1) = (x,\\xi _2),$ so $T_{(x,\\xi _1)}$ is also a period for $\\xi _2$ .",
"Now, as the length of the fibers is $2\\pi $ , we can see that the period must be $2\\pi $ .",
"Indeed $2\\pi = \\int _{H_x\\Sigma } \\hspace{-5.69054pt} \\alpha = \\int _{0}^{T_x} \\!\\alpha \\left( \\dot{\\theta }(t) \\right) dt = \\int _{0}^{T_x} \\alpha \\left(Y \\right) dt = \\int _{0}^{T_x} dt = T_x \\, .$ Therefore, for any $(x,\\xi )\\in H\\Sigma $ , $\\theta ^{2\\pi }\\left(x,\\xi \\right)= \\left(x,\\xi \\right)$ .",
"The first question that sprang to our mind about this rotation is its link with the reversibility of the metric.",
"We prove here something that seems very natural : if the metric is reversible, then rotating a vector by $\\pi $ gives its opposite.",
"Note that this is not a characterization of reversible metrics.",
"Indeed, as we will see in the proof, if we write $s \\colon HM \\rightarrow HM$ such that $s(x,\\xi ) = (x,-\\xi )$ , the only thing we need is that $s^{\\ast }\\alpha = \\alpha $ .",
"Proposition 1.4.3 Let $(\\Sigma ,F)$ be a Finsler 2-manifold.",
"If $F$ is reversible and $(x,\\xi ) \\in H\\Sigma $ , then $\\theta ^{\\pi }\\left(x,\\xi \\right) = \\left(x,-\\xi \\right).$ Let $ s \\colon {T\\Sigma \\rightarrow T\\Sigma } $ , $(x,v) \\mapsto (x,-v)$ be the local symmetry.",
"If $F$ is reversible, then $s^{\\ast } d_v F = - d_v F$ : for $(x,v)\\in T\\Sigma $ and $Z\\in T_{(x,v)}T\\Sigma $ , $s^{\\ast } \\left(d_v F\\right)_{(x,v)}(Z) &= d_v F_{(x,-v)}(ds\\, Z) \\\\&= \\lim _{\\epsilon \\rightarrow 0} \\frac{1}{\\epsilon }\\left( F(x, -v + \\epsilon d\\pi \\circ ds \\, Z) - F(x,-v)\\right) \\\\&= \\lim _{\\epsilon \\rightarrow 0} \\frac{1}{\\epsilon }\\left( F(x, v - \\epsilon d\\pi \\, Z) - F(x,v)\\right) \\\\&= - d_v F_{(x,v)}(Z).$ Now, if we also denote by $s$ the symmetry on $H\\Sigma $ , we have shown that $s^{\\ast } A = -A $ .",
"Therefore $s^{\\ast } \\left( A\\wedge dA\\right) = A\\wedge dA$ and, using the definition of the angle form, we obtain $s^{\\ast } \\alpha = \\alpha $ .",
"The map $s$ is a diffeomorphism of $H\\Sigma $ , so for $x\\in \\Sigma $ and $U \\subset H_x \\Sigma $ measurable, the change of variable formula gives $\\int _{U} s^{\\ast }\\alpha = \\int _{s(U)} \\alpha ,$ if $\\xi \\in H_x\\Sigma $ and $U$ is one interval from $\\xi $ to $-\\xi $ , then $H_x\\Sigma = U \\cup s\\left(U\\right)$ and $2\\pi = \\int _{H_x\\Sigma } \\alpha = \\int _{U} \\alpha + \\int _{s\\left(U\\right)} \\alpha = 2 \\int _{U} \\alpha .$ Recall that there exists $t_0$ such that $U = \\lbrace \\theta ^t(\\xi ) \\mid t\\in \\left[0,t_0 \\right] \\rbrace $ , so $\\pi = \\int _{U} \\alpha = \\int _0^{t_0} \\alpha _{\\theta ^t(\\xi )}\\left(\\dot{\\theta }^t(\\xi ) \\right) dt = \\int _0^{t_0} dt = t_0,$ therefore $-\\xi = \\theta ^{\\pi }\\left(\\xi \\right)$ ."
],
[
"$\\Omega ^F $ is the Holmes-Thompson volume",
"Contrarily to what happens in Riemannian geometry, there is no canonical volume in Finsler geometry.",
"We do not go into the reason for that as it is very well explained in [29].",
"There is however a certain number of “natural” volumes that have come up in convex geometry (see [29], [5]).",
"Among them the most studied ones are the Busemann-Hausdorff and the Holmes-Thompson volumes.",
"The Busemann-Hausdorff volume ([30]) is obtained by considering a Finsler manifold as a metric space and taking its Hausdorff measure, but Alvarez-Paiva and Berck [4] showed that, despite its naturality, it might not be the “good” one.",
"The Holmes-Thompson volume ([73]) comes from the symplectic structure of the cotangent bundle of a manifold.",
"Recall that, if $\\lambda $ denotes the Liouville form on $T^{\\ast }M$ , then $d\\lambda $ is the canonical symplectic form on $T^{\\ast }M$ and $\\left(d\\lambda \\right)^n / n!$ is the Liouville volume.",
"Now the Holmes-Thompson volume $\\text{Vol}_{HT}$ associated with a Finsler metric $F$ is defined, for $U$ a Borel set on $M$ and $B^{\\ast }U := \\lbrace (x,p) \\in T^{\\ast }M \\mid \\hat{p}_{|M} (x,p) \\in U, \\;\\; F^{\\ast }(x,p) <1 \\rbrace \\subset T^{\\ast }M$ , by $\\text{Vol}_{HT}(U) = \\frac{1}{\\epsilon _n} \\int _{B^{\\ast }U} \\frac{\\left(d\\lambda \\right)^n}{n!",
"},$ where $\\epsilon _n$ is the volume of the unit ball in the Euclidean space $\\mathbb {E}^n$ .",
"Remark that, if we denote by $S^{\\ast }M$ the cotangent sphere bundle, i.e., the subset of $T^{\\ast }M$ given by $(F^{\\ast })^{-1}(1)$ , then an application of Stokes Theorem shows that $\\text{Vol}_{HT}(U) = \\frac{1}{\\epsilon _n} \\int _{S^{\\ast }U} \\frac{\\lambda \\wedge \\left(d\\lambda \\right)^{n-1}}{n!",
"}.$ We can now prove that $\\Omega ^F$ corresponds to $\\text{Vol}_{HT}$ : For $U$ a Borel set in $M$ , an application of Fubini Theorem (see Lemma REF ) gives $\\int _{U} \\Omega ^F = \\frac{1}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{H^{\\ast }U} \\beta ^F \\wedge \\hat{\\pi }^{\\ast } \\Omega ^F = \\frac{1}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{H^{\\ast }U} B \\wedge dB^{n-1}.$ Now, by Theorem REF , we have that $\\hat{r}^{\\ast } B \\wedge dB^{n-1}= \\frac{1}{(F^{\\ast })^{n} }\\lambda \\wedge \\left(d\\lambda \\right)^{n-1}$ , and ${\\hat{r} \\colon \\mathring{T}^{\\ast }M \\rightarrow H^{\\ast }M }$ is a diffeomorphism when restricted to $S^{\\ast }M$ .",
"So, by the change of variable formula, we obtain $\\int _{H^{\\ast }U} B \\wedge dB^{n-1}= \\int _{\\hat{r} \\left(S^{\\ast }U \\right)} B \\wedge dB^{n-1}= \\int _{S^{\\ast }U} \\frac{\\lambda \\wedge \\left(d\\lambda \\right)^{n-1}}{(F^{\\ast })^{n} } = \\int _{S^{\\ast }U} \\lambda \\wedge \\left(d\\lambda \\right)^{n-1}.$ Therefore, $\\int _{U} \\Omega ^F = \\frac{1}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{S^{\\ast }U} \\lambda \\wedge \\left(d\\lambda \\right)^{n-1} = \\frac{n!",
"\\epsilon _n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\text{Vol}_{HT}(U) = (n-1)!",
"\\text{Vol}_{HT}(U).$ So we proved the following : Proposition 1.5.1 Let $F$ be a Finsler metric on an $n$ -manifold, then $\\Omega ^F = (n-1)!",
"\\text{Vol}_{HT}\\,.$ Before moving on, we wish to make a few remarks.",
"First, we could get rid of the constant by considering $A \\wedge dA^{n-1}/ (n-1)!$ instead of just $A \\wedge dA^{n-1}$ .",
"However, our aim was never to study this volume and so we felt that adding this constant would unnecessarily complicate matters.",
"Indeed, almost everything in this dissertation is invariant under a change of volume by a constant, the only exception being the actual computation of the volume of a Finsler manifold.",
"However, our only actual computation of volume is in dimension two.",
"Second, we must admit that, until very recently, we wrongly believed that $\\Omega ^F$ was different from the Holmes-Thompson volume.",
"Hence in this dissertation, we prove every claim we make about this volume even so they are probably classical results.",
"Finally, and this is a side note, it is interesting to point out that the Holmes-Thompson volume is very well-known, but that the naturally associated angle does not seem to have been studied, so we hope that we repair at least a little that injustice."
],
[
"A natural Finsler–Laplace operator",
"Finsler–Laplacian"
],
[
"Definition",
"In this section we will introduce our generalization of the operator of Laplace–Beltrami, and begin by recalling the different equivalent definitions in Riemannian geometry.",
"The “historic” Laplacian on $\\mathbb {R}^n$ is defined as $\\Delta = \\sum _i \\frac{\\partial ^2}{\\partial x_i^2}.$ So, given a Riemannian metric $g$ , at any point $p$ , we can choose normal coordinates and use the above expression.",
"For generic local coordinates $(x_1, \\dots , x_n)$ , where the metric reads $g = [g_{ij}]$ , the local expression becomes $\\Delta ^{LB} := \\frac{1}{\\sqrt{\\text{det} g}} \\frac{\\partial }{\\partial x_i}\\left( \\sqrt{\\text{det} g} g^{ij} \\frac{\\partial }{\\partial x_j}\\right).$ A possibly more convenient, coordinate-free expression for the Laplacian is $\\Delta ^{LB}f = \\text{div} \\left( \\nabla f \\right),$ where $\\nabla f$ is the gradient of $f$ with respect to $g$ and $\\text{div}$ is the divergence operator (see, for instance, [66]).",
"Finally, the Hodge–Laplace operator gives an expression for the Laplacian on differential forms (see [66] or [110]).",
"The study of the Laplace operator is of paramount importance as it exhibit deep links between its spectral data and the geometry of the manifold carrying it (for a proof of this claim, the reader can consult, for a start, [21], [22], [37] and then move on to the thousands of articles on this subject).",
"Hence, giving a generalization of this operator to the Finslerian context is of great importance, especially if we manage to construct one having the same kind of behavior.",
"There have already been several generalizations of the Laplace operator to Finslerian geometry [15], [36], [102], each starting from a different definition of the Laplace–Beltrami and obtaining different operators.",
"This can be seen either as a drawback or a new source of interest in Finsler geometry.",
"Indeed, it is not uncommon that, when we try to extend definitions that were equivalent, we often end up with different notions, yielding some new insight in the process.",
"And this is particularly true of Finsler geometry, as the history of Finslerian connections, for instance, proves.",
"Bao and Lackey [15] gave a generalization of the Hodge-star operator, allowing them to define a Finslerian Hodge–Laplace operator.",
"Shen [102] gave the very natural generalization of $\\text{div} \\left( \\nabla \\right)$ to Finsler metrics.",
"Indeed, the gradient of a function is just its derivative seen as an element of the tangent space, i.e., its pullback by the Legendre transform.",
"So clearly this is not just Riemannian.",
"But note however that, for Finsler metrics, the Legendre transform is in general not linear, so Shen's Laplacian is not linear.",
"Remark also that, to define a divergence, one needs to choose a volume form on the manifold and, as we have already mentioned, there is no canonical volume form in Finsler geometry.",
"Finally, Centore [36] did not use directly one of the above definitions, but the fact that harmonic functions satisfy the mean-value property and designed an operator in order to keep that property.",
"Here, once again, his definition relies on the choice of a volume form.",
"Our approach for a generalization relies on the first definition, as the sum of the second derivatives in orthonormal directions.",
"As there is no good notion of orthogonality in Finsler geometry, we consider instead the average of the second derivatives in every direction.",
"The average being taken with respect to the angle we introduced in Section REF .",
"More precisely, we introduce : Definition 2.1.1 Let $F$ be a Finsler metric on an $n$ -manifold $M$ .",
"We define the Finsler–Laplace operator, denoted $\\Delta ^F$ , as $\\Delta ^F f (x) = \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) }\\int _{H_xM} L_X ^2 (\\pi ^{\\ast } f ) \\alpha ^F,$ for every $x\\in M$ and every $f \\colon M \\rightarrow \\mathbb {R}$ (or $) such that the integral exists.$ As we will see in the next section, the constant $n/\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right)$ is chosen so that $\\Delta ^F$ is the Laplace–Beltrami operator when $F$ is Riemannian.",
"Remark 2.1.2 Note that we can define a Laplace-like operator in this fashion for any contact form on the homogenized bundle $HM$ , but we have not pursued the study of this more general kind.",
"It is already clear from the definition that $\\Delta ^F$ is a linear differential operator of order two.",
"It also verifes the following : Theorem 2.1.3 Let $F$ be a Finsler metric on $M$ , then $\\Delta ^F$ is a second-order differential operator, furthermore : (i) $\\Delta ^F$ is elliptic ; (ii) $\\Delta ^F$ is symmetric, i.e., for any $f,g \\in C^{\\infty }_0(M)$ , $\\int _M f\\Delta ^F g - g\\Delta ^F f \\; \\Omega ^F = 0\\,;$ (iii) $\\Delta ^F$ is unitarily equivalent to a Schrödinger operator ; (iv) $\\Delta ^F$ coincides with the Laplace–Beltrami operator when $F$ is Riemannian.",
"Remark that our definition of Laplace operator could be applied with any angle.",
"However, to obtain a symmetric operator, we fundamentally rely on the fact that $\\alpha ^F$ and $\\Omega ^F$ come from the volume $A \\wedge dA^{n-1}$ , which is invariant under the geodesic flow.",
"So, in order to get an operator satisfying the above conditions, the only choice we really made was to ask for the constancy with respect to $\\alpha ^F$ of the volume of each fiber.",
"We split the proof of the theorem into four parts presented in the next four sections.",
"Note that all the proofs are surprisingly simple, which is, in my opinion, an asset of this operator."
],
[
"The Riemannian case",
"We start by proving Theorem REF (iv).",
"Proposition 2.1.4 Let $g$ be a Riemannian metric on $M$ , $F = \\sqrt{g}$ , $\\Delta ^F$ the Finsler–Laplace operator and $\\Delta ^g$ the usual Laplace–Beltrami operator.",
"Then, $\\Delta ^F = \\Delta ^g.$ We will compute both operators in normal coordinates for $g$ and show that they coincide.",
"Let $p\\in M$ and $x_1, \\dots , x_n$ be normal coordinates around it.",
"Denote by $v_1, \\dots , v_n$ their canonical lift to $T_x M$ .",
"For $f \\colon M \\rightarrow \\mathbb {R}$ , the Laplace–Beltrami operator is $ \\Delta ^{g}f(p) = \\sum _i \\frac{\\partial ^2 f }{\\partial x_i^2} (p)\\, .",
"$ The first step to compute the Finsler–Laplace operator is to compute the Hilbert form $A$ and the geodesic flow $X$ .",
"In order to write $A$ , we identify $HM$ with $T^1M$ and coordinates on $H_pM$ are then given by the $v_i$ 's with the condition $\\sqrt{\\sum (v_i)^2 }=1$ .",
"The vertical derivative of $F$ at $p$ is $d_v F_p = \\left(v_i dx_i\\right) \\big /\\sqrt{\\sum (v_i)^2 }$ .",
"So $ A_p = v_i\\;dx^i$ and $dA_p = dv_i \\wedge dx^i $ .",
"Hence $X(p, \\cdot ) = v_i \\frac{\\partial }{\\partial x_i}$ .",
"Indeed, we just need to check that $A_p\\left(X_p\\right) = 1$ and $\\left(i_X dA \\right)_p = 0 $ , but both equalities follow from $\\sum (v_i)^2 =1$ .",
"Let $f \\colon M \\rightarrow \\mathbb {R}$ .",
"Then $L_X^2\\left(\\pi ^{\\ast }f \\right) (p,v) = v_i v_j \\frac{\\partial ^2 f}{\\partial x_i\\partial x_j} (p,v)\\, ,$ so that the Finsler–Laplace operator is $\\Delta ^F f(p) = \\frac{ n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\mathbb {S}^{n-1}} \\int _{H_pM} v_i v_j \\; \\alpha \\; \\frac{\\partial ^2 f}{\\partial x_i\\partial x_j}(p)\\, ,$ and the proof follows from the next two claims.",
"Claim 2.1.5 For all $i \\ne j$ , $\\int _{H_pM} v_i v_j \\, \\alpha =0 \\,.$ $H_pM$ is parametrized by $ H_pM = \\left\\lbrace (v_1, \\dots , v_n ) \\mid v_i \\in [-1, 1], \\sum (v_i)^2 =1 \\right\\rbrace .$ A parity argument then yields the desired result.",
"Claim 2.1.6 For any $1\\le i \\le n $ , $\\int _{H_pM} v_i^2 \\, \\alpha = \\frac{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\mathbb {S}^{n-1}}{ n} \\, .$ As the $v_i$ 's are symmetric by construction, we have that, for any $i\\ne j$ , $\\int _{H_pM} v_i^2 \\, \\alpha = \\int _{H_pM} v_j^2 \\, \\alpha \\, .$ So $n \\int _{H_pM} \\hspace{-5.12149pt} v_i^2 \\, \\alpha = \\sum _j \\int _{H_pM} \\hspace{-5.12149pt} v_j^2 \\, \\alpha = \\int _{H_pM} \\sum _j v_j^2 \\, \\alpha = \\int _{H_pM} \\hspace{-6.54413pt}1 \\, \\alpha = \\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\mathbb {S}^{n-1}.",
"$"
],
[
"Ellipticity",
"We now prove Theorem REF (i).",
"Proposition 2.1.7 The operator $\\Delta ^F \\colon C^{\\infty }(M) \\rightarrow C^{\\infty }(M) $ is elliptic.",
"The symbol $\\sigma ^F$ is given by $\\sigma ^F_x(\\xi _1,\\xi _2) = \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{H_xM} L_X(\\pi ^{\\ast } \\varphi _1) L_X(\\pi ^{\\ast }\\varphi _2)\\, \\alpha ^F$ for $\\xi _1,\\xi _2 \\in T^{\\ast }_x M$ , where $\\varphi _i \\in C^{\\infty }(M)$ such that $\\varphi _i(x)=0$ and $\\left.d\\varphi _i\\right._x = \\xi _i$ .",
"Remark 2.1.8 If we identify the unit tangent bundle $T^1 M$ with the homogenized tangent bundle $HM$ and write again $\\alpha ^F$ for the angle form on $T^1M$ , then the symbol is given by $\\sigma ^F_x(\\xi _1,\\xi _2) = \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{v\\in T^1_xM} \\xi _1(v) \\xi _2(v) \\, \\alpha ^F(v)$ for $\\xi _1,\\xi _2 \\in T^{\\ast }_x M$ .",
"The symbol of an elliptic second-order differential operator is a non-degenerate symmetric 2-tensor on the cotangent bundle, and therefore defines a Riemannian metric on $M$ .",
"This gives one more way to obtain a Riemannian metric from a Finsler one.",
"Let $\\Delta ^{\\sigma }$ be the Laplace–Beltrami operator associated with the symbol metric, then $\\Delta ^F -\\Delta ^{\\sigma }$ is a differential operator of first order, so is given by a vector field $Z$ on $M$ .",
"The Finsler–Laplace operator therefore is a Laplace–Beltrami operator together with some “drift” given by $Z$ .",
"We will later see that our operator is in fact characterized by its symbol and the symmetry condition (Section REF ).",
"To show that $\\Delta $ is elliptic at $p\\in M$ , it suffices to show that for each $\\varphi \\colon M \\rightarrow \\mathbb {R}$ such that $\\varphi (p) = 0$ and $d\\varphi |_p$ does not vanish, and for $u \\colon M \\rightarrow \\mathbb {R}^+$ we have $\\Delta ^F(\\varphi ^2 u) (p) > 0$ unless $u(p)=0$ .",
"We first compute $L^{2}_X \\left(\\pi ^{\\ast }\\varphi ^2 u\\right)$ : $L^{2}_X \\left(\\pi ^{\\ast }\\varphi ^2 u\\right) &= L_X \\left( 2 \\pi ^{\\ast }\\varphi u L_X\\left( \\pi ^{\\ast }\\varphi \\right) + \\pi ^{\\ast }\\varphi ^2 L_X \\left(\\pi ^{\\ast }u\\right) \\right), \\\\&= 2 \\pi ^{\\ast }u \\left(L_X\\left( \\pi ^{\\ast }\\varphi \\right)\\right)^2 + 2 \\pi ^{\\ast } \\varphi u L^{2}_X \\left(\\pi ^{\\ast }\\varphi \\right) \\\\& \\quad + 4 \\pi ^{\\ast }\\varphi L_X \\left(\\pi ^{\\ast }\\varphi \\right) L_X \\left(\\pi ^{\\ast }u\\right) + 2\\pi ^{\\ast } \\varphi ^2 L^{2}_X\\left( \\pi ^{\\ast }u\\right) .$ Evaluating at $\\xi \\in H_p M$ , we obtain $L^{2}_X \\left(\\pi ^{\\ast }\\varphi ^2 u\\right) \\left(\\xi \\right) = 2 u(p) \\left(L_X \\pi ^{\\ast }\\varphi \\right)^2 \\left(\\xi \\right).$ Therefore $\\Delta ^F(\\varphi ^2 u) (p) &= \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{H_p M} 2 u(p) \\left(L_X \\pi ^{\\ast }\\varphi \\right)^2 \\alpha ,\\\\&= \\frac{2 u(p)n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{H_p M} \\left(L_X \\pi ^{\\ast }\\varphi \\right)^2 \\alpha \\; > 0 \\, .",
"$"
],
[
"Symmetry",
"We have a hermitian product defined on the space of $C^{\\infty }$ complex functions with compact support on $M$ by $\\langle f,g \\rangle = \\int _M f(x)\\overline{g(x)} \\Omega ^F.$ So we can now prove Theorem REF (ii).",
"Proposition 2.1.9 The operator $\\Delta ^F$ is symmetric for $\\langle \\cdot , \\cdot \\rangle $ on $C^{\\infty }_0(M)$ , i.e., for any ${f,g\\in C^{\\infty }_0(M)}$ , we have $\\langle \\Delta ^F f , g \\rangle = \\langle f,\\Delta ^F g \\rangle .$ Remark 2.1.10 The proof of this result is remarkably simple due to our choice of angle form and volume.",
"Indeed, as $\\alpha \\wedge \\pi ^{\\ast } \\Omega $ is the canonical volume on $HM$ , it is invariant under the geodesic flow (i.e., $L_X(\\alpha \\wedge \\pi ^{\\ast } \\Omega ) =0$ ) which is the key to the computation.",
"In order to prove the proposition, we first need a Fubini-like result.",
"It is certainly known, but as it appears to us that it would take less time to do it than try to look for a reference in the literature, we provide the proof below.",
"Lemma 2.1.11 Let $f \\colon HM \\rightarrow \\ be a continuous, integrable function on $ HM$.",
"We have\\begin{equation}\\int _M \\left( \\int _{H_xM} f(x,\\cdot ) \\, \\alpha \\right) \\Omega = \\int _{HM} f \\; \\alpha \\wedge \\pi ^{\\ast }\\Omega \\, .\\end{equation}$ In the following, we will write $f_x \\colon H_x M \\rightarrow for $ f(x,)$.\\\\Let $ Ua $ be a trivializing open covering for $ : HM M$, i.e., there exists $ S$\\ such that, for every $ a$, there exists a homeomorphism$$\\varphi _a \\colon HU_a= \\pi ^{-1}\\left( U_a \\right) \\rightarrow U_a \\times S \\,.$$Moreover, for every $ x Ua$, $ a(x, ) HxM=Hx Ua S$ is also a homeomorphism.\\\\Let $ a $\\ be a partition of unity subordinated to $ Ua $.",
"We have\\begin{equation*}\\int _M \\left( \\int _{H_xM} f_x \\alpha \\right) \\Omega = \\sum _a \\int _{U_a} \\varphi _a(x) \\left( \\int _{H_xM} f_x \\alpha \\right) \\Omega \\,.\\end{equation*}Let $ xUa$.",
"We set $ ax := ( a(x,)-1 ) $.",
"It is a $ (n-1)$-form on $ S$, but, by definition of $ a$, $ ax$ does not depend on $ x$, just on $ a$, so we can write $ a :=ax$.",
"We have\\begin{equation*}\\int _{U_a} \\varphi _a(x) \\left( \\int _{H_xM} f_x \\alpha \\right)\\Omega = \\int _{U_a} \\varphi _a(x) \\left( \\int _S f_x \\alpha ^a \\right)\\Omega \\, .\\end{equation*}Then, applying Fubini^{\\prime }s Theorem (see, for instance, \\cite {BerGos}) gives\\begin{equation*}\\int _{U_a} \\varphi _a(x) \\left( \\int _S f_x \\alpha \\right)\\Omega = \\int _{U_a \\times S} \\varphi _a f \\alpha ^a\\wedge \\Omega \\, .\\end{equation*}Now, we have $ a a= $, hence we get\\begin{equation*}\\int _{U_a \\times S} \\varphi _a f \\alpha \\wedge \\Omega = \\int _{HU_a} \\varphi _a f \\alpha \\wedge \\pi ^{\\ast }\\Omega \\,.\\end{equation*}Summing over $ a$, we finally obtain{\\begin{@align*}{1}{-1}\\int _M \\left( \\int _{H_xM} f_x \\alpha \\right) \\Omega &= \\sum _a \\int _{U_a} \\varphi _a(x) \\left( \\int _{H_xM} f_x \\alpha \\right) \\Omega \\\\&= \\sum _a \\int _{HU_a} \\varphi _a f \\alpha \\wedge \\pi ^{\\ast }\\Omega \\\\&= \\int _{HM} \\left( \\sum _a \\varphi _a\\right) f \\alpha \\wedge \\pi ^{\\ast }\\Omega \\\\&= \\int _{HM} f \\alpha \\wedge \\pi ^{\\ast }\\Omega \\, .\\end{@align*}}$ We can now proceed with the [Proof of Proposition REF ] Let $f,g: M \\xrightarrow{} \\ and set $ cn := nvolEucl(Sn-1) $.",
"{\\begin{@align*}{1}{-1}\\langle \\Delta ^F f,g \\rangle &= \\int _M \\overline{g} \\Delta ^F f \\; \\Omega \\\\&= c_n \\int _M \\overline{g} \\left( \\int _{H_xM} L_X ^2 (\\pi ^{\\ast } f ) \\alpha \\right) \\Omega \\\\&= c_n \\int _M \\left( \\int _{H_xM} \\overline{\\pi ^{\\ast }g} L_X ^2 (\\pi ^{\\ast } f ) \\alpha \\right) \\Omega \\\\&= c_n \\int _{HM} \\overline{\\pi ^{\\ast }g} L_X ^2 (\\pi ^{\\ast } f ) \\; \\alpha \\wedge \\pi ^{\\ast }\\Omega \\, ,\\end{@align*}}where the last equality follows from the preceding lemma.",
"\\\\As $ = AdAn-1$, we can write$$\\langle \\Delta ^F f,g \\rangle = c_n \\int _{HM} \\overline{\\pi ^{\\ast }g} L_X ^2 (\\pi ^{\\ast } f ) \\; A\\wedge dA^{n-1}.$$Now\\begin{multline*}L_X \\left( \\overline{\\pi ^{\\ast }g} L_X (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} \\right) = \\overline{\\pi ^{\\ast }g} L_X^2 (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} \\\\+ L_X ( \\overline{\\pi ^{\\ast }g} ) L_X (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} + \\overline{\\pi ^{\\ast }g} L_X (\\pi ^{\\ast } f ) L_X (A \\wedge dA^{n-1}).\\end{multline*}The last part of the above equation vanishes because of (\\ref {eq:lxada}).",
"We also have\\begin{equation*}L_X \\left( \\overline{\\pi ^{\\ast }g} L_X (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} \\right) = d\\left( i_X \\overline{\\pi ^{\\ast }g} L_X (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} \\right).\\end{equation*}Hence\\begin{multline*}\\langle \\Delta ^F f,g \\rangle = \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) }\\Biggl [ \\int _{HM} d\\left( i_X \\overline{\\pi ^{\\ast }g} L_X (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} \\right) \\\\- L_X ( \\overline{\\pi ^{\\ast }g} ) L_X (\\pi ^{\\ast } f ) A\\wedge dA^{n-1} \\Biggr ].\\end{multline*}As $ M$\\ is closed, $ HM$\\ is closed and applying Stokes Theorem gives (\\ref {eq:green_formula}), thus proving the claim.$ In the proof we obtained a Finsler version of Green's formulas : Proposition 2.1.12 For any $f,g \\in C^{\\infty }(M)$ , we have $ \\langle \\Delta ^F f,g \\rangle = \\frac{-n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{HM} L_X ( \\overline{\\pi ^{\\ast }g} ) L_X (\\pi ^{\\ast } f ) \\, A\\wedge dA^{n-1}.$ Let $U$ be a submanifold of $M$ of the same dimension and with boundaries.",
"Then, for any $f \\in C^{\\infty }(U)$ , we have $\\int _U \\Delta ^F f \\; \\Omega ^F =\\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) }\\int _{\\partial HU } L_X (\\pi ^{\\ast } f ) dA^{n-1}.$"
],
[
"A characterization of $\\Delta ^F$",
"The following results were explained to us by Yves Colin de Verdière and are probably well known to many people.",
"However, they might not be known to everyone and are quite interesting, so we provide the proofs.",
"Lemma 2.1.13 Let $(M,g)$ be a closed Riemannian manifold and $\\omega $ a volume form on $M$ .",
"There exists a unique second-order differential operator $\\Delta _{g,\\omega }$ on $M$ with real coefficients such that its symbol is the dual metric $g^{\\star }$ , that is symmetric with respect to $\\omega $ and zero on constants.",
"If $a\\in C^{\\infty }(M)$ is such that $\\omega = a^2 v_g$ , where $v_g$ is the Riemannian volume, then, for $\\varphi \\in C^{\\infty }(M)$ , $\\Delta _{g,\\omega } \\varphi = \\Delta ^{g} \\varphi - \\frac{1}{a^2} \\langle \\nabla \\varphi , \\nabla a^2 \\rangle .$ Before getting on to the proof, this result deserves a few remarks.",
"We have seen above that, to a Finsler metric, we can associate a volume and a Riemannian metric via the symbol of the Finsler–Laplace operator.",
"This lemma tells us that conversely a volume together with a Riemannian metric give a Laplace-like operator.",
"This shows that we could have a wealth of Finsler–Laplace operators — just associate a Riemannian metric and a volume to a Finsler metric — but not all of them are natural.",
"As there are many more Finsler metrics than pairs (volume/Riemannian metric), this lemma shows that many Finsler metrics will share the same Finsler–Laplacian.",
"A related question raised by Yves Colin de Verdière was to determine the range of pairs (volume/Riemannian metric) that can be obtained from a Finsler metric.",
"We prove that we get everything in the case of surfaces (Proposition REF ), but we do not know the general answer.",
"The operators of the type $\\Delta _{g,\\omega }$ seem to have been introduced by Chavel and Feldman [38] and Davies [47].",
"They are called weighted Laplace operators and have been quite widely studied (see, for instance [71]).",
"Up to now we only considered our operators as acting on $C^{\\infty }(M)$ .",
"In the next section, for the purpose of spectral theory, we will start considering them as unbounded operators on $L^2(M,\\omega )$ .",
"By considering the Friedrich extension, the above result stays true replacing “symmetric” by “self-adjoint” (we recall the definitions in Appendix REF ).",
"It is evident from the definition of $\\Delta _{g,\\omega }$ that it is zero on constant functions, that its symbol is $g^{\\ast }$ and that for $\\varphi , \\psi \\in C^{\\infty }(M)$ , $\\int _M \\psi \\Delta _{g,\\omega }\\varphi \\; \\omega = \\int _{M} g^{\\ast }\\left(d\\varphi , d\\psi \\right) \\, \\omega = \\int _M \\varphi \\Delta _{g,\\omega }\\psi \\; \\omega $ Let us now prove uniqueness.",
"Let $\\Delta _1$ and $\\Delta _2$ be two second-order differential operators such that they vanish on constant functions and have the same symbol.",
"This implies that there exists a smooth vector field $Z$ on $M$ such that $\\Delta _1 - \\Delta _2 = L_Z$ .",
"Now, suppose that both operators are symmetric with respect to $\\omega $ .",
"We get, $\\int _M \\varphi L_Z \\psi -\\psi L_Z \\varphi \\, \\omega =0$ for any $\\varphi , \\psi \\in C^{\\infty }(M)$ .",
"And taking $\\psi = 1$ yields $\\int _M L_Z \\varphi \\, \\omega =0$ .",
"But, if $Z$ is not zero, it is easy to construct a function $\\varphi \\in C^{\\infty }(M)$ such that $L_Z \\varphi >0$ in any open set that does not contain a singular point of $Z$ .",
"So by continuity, $Z$ must vanish.",
"An important consequence of this lemma is that any symmetric, elliptic linear second order operator is unitarily equivalent to a Schrödinger operator.",
"Proposition 2.1.14 Let $\\Delta _{g,\\omega }$ , $v_g$ and $a$ be as above.",
"Define an operator ${U \\colon L^2\\left(M, \\omega \\right) \\rightarrow L^2\\left(M, v_g \\right)}$ by $Uf = af$ .",
"Then $U \\Delta _{g,\\omega } U^{-1} = \\Delta ^{g} + V $ is a Schrödinger operator with potential $V = a\\Delta _{g,\\omega } a^{-1}$ .",
"Remark 2.1.15 This fact shows that the spectral theory of our operator restricts to the theory of Schrödinger operators such that the infimum of the spectrum is zero.",
"It suffices to show that $U \\Delta _{g,\\omega } U^{-1} - V$ is symmetric with respect to $\\omega $ , vanishes on constant functions and has $g^{\\ast }$ for symbol, because then Lemma REF proves the claim.",
"It clearly vanishes on constant functions and the symmetry property is obvious by construction.",
"Let $x\\in M$ and $\\varphi \\in L^2\\left(M, v_g \\right)$ be such that $\\varphi (x) = 0$ and $d\\varphi _{x} \\ne 0$ .",
"We have $\\left(U \\Delta _{g,\\omega } U^{-1} - V\\right) \\varphi ^2 (x) = a\\Delta _{g,\\omega }(\\varphi ^2 a^{-1})(x) = \\Delta _{g,\\omega }(\\varphi ^2)(x)\\, .$ Therefore the symbol of $\\left(U \\Delta _{g,\\omega } U^{-1} - V\\right)$ is the same as that of $\\Delta _{g,\\omega }$ ."
],
[
"Relation to other Laplacians",
"We did not pursue the study of the comparison between this Finsler–Laplace operator and those introduced before by Bao and Lackey [15] and Centore [36].",
"However, we can make the following easy remark.",
"Suppose that $L$ is a second-order differential operator on a closed manifold, vanishing on constant functions and symmetric with respect to two volumes $\\Omega _1$ and $\\Omega _2$ , then $\\Omega _1$ is a constant multiple of $\\Omega _2$ .",
"Indeed, if we write $\\Omega _2 = f \\Omega _1$ , then for any function $g$ , $0 = \\int _M Lg \\; \\Omega _2 = \\int _M (Lg) f \\; \\Omega _1 = \\int _M g Lf \\; \\Omega _1 \\, ,$ hence $Lf = 0$ , so, if $M$ is closed, $f = \\text{cst}$ .",
"So an easy way to see that our Finsler–Laplace operator is different from Centore's is by remarking that his operator is symmetric with respect to the Busemann-Hausdorff volume and applying the above remark."
],
[
"Spectral theory",
"Most of the results of this section follow from the general theory of elliptic, symmetric operators on compact manifolds.",
"However, we felt that for the convenience of the reader, as well as for the interest of the results, it was worthwhile to give the proofs.",
"Hence we either reproduced or adapted the proofs to our special case."
],
[
"The space $H^1$",
"In order to deal with the spectral theory of our operator, we will stop seeing it as acting on $C^{\\infty }$ functions but as an unbounded operator on $L^2(M)$ .",
"We collected in Appendix REF the basic definitions and the main results that we need.",
"We start by defining a very useful functional space.",
"Let $C^{\\infty }_0(M)$ be the space of smooth functions with compact support on $M$ (so that, if $M$ is boundaryless, then the second condition is empty).",
"Consider the following inner product on $C^{\\infty }_0(M)$ $\\langle u,v \\rangle _{1} = \\int _M uv \\; \\Omega + \\int _{HM} L_X\\left(\\pi ^{\\ast }u \\right) L_X\\left(\\pi ^{\\ast }v \\right)\\; A \\wedge dA^{n-1}$ and denote by $\\Vert \\cdot \\Vert _1$ the associated norm.",
"Definition 2.2.1 We denote by $H^1(M)$ the completion of $C^{\\infty }_0 (M)$ with respect to the norm $\\Vert \\cdot \\Vert _{_{1}}$ .",
"Remark 2.2.2 Using the Riemannian metric given by the symbol of the Laplacian, we have : $\\langle u,v \\rangle _{1} = \\int _M uv \\; \\Omega + \\int _M \\nabla u \\nabla v \\; \\Omega \\, .$ Note also that we do not use the classical notations of $H^1(M)$ and $H^1_0(M)$ for the completion of respectively $C^{\\infty }(M)$ and $C^{\\infty }_0(M)$ .",
"But, as our main focus will later be closed manifolds, we did not feel it worth introducing two notations.",
"The space $H^1(M)$ is a Sobolev space and we have the following embedding result (see [90]) : Theorem 2.2.3 (Rellich–Kondrachov) If $M$ is compact with smooth boundary, then $H^1(M)$ is compactly embedded in $L^2(M)$ .",
"The Finsler–Laplace operator is an unbounded operator on $L^2(M)$ with domain in $H^1(M)$ ."
],
[
"Energy integral and Rayleigh quotients",
"Definition 2.2.4 For any function $u\\in H^1(M)$ , we define the energy of $u$ by $E(u) := \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{HM} \\left|L_X\\left(\\pi ^{\\ast }u \\right)\\right|^2 A \\wedge dA^{n-1}$ and the Rayleigh quotient by $R(u) := \\frac{E(u)}{\\int _M u^2\\, \\Omega }\\, .$ Remark 2.2.5 The energy as well as the Rayleigh quotient can also be defined using the cotangent setting, i.e.",
"for any $u\\in H^1(M)$ , we have $E(u) := \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } \\int _{H^{\\ast }M} \\left|L_{X^{\\ast }}\\left(\\hat{\\pi }^{\\ast }u \\right)\\right|^2 B \\wedge dB^{n-1},$ where $B = (\\ell _F^{-1})^{\\ast } A$ and $X^{\\ast }$ is its Reeb field (see Chapter REF ).",
"The Energy we defined is naturally linked to the Finsler–Laplace operator : Theorem 2.2.6 A function $u \\in H^1(M)$ is a minimum of the energy if and only if $u$ is harmonic, i.e., $\\Delta ^F(u) = 0$ .",
"Remark 2.2.7 When $M$ is closed, this just proves that harmonic functions are constant.",
"But this result stays true without restrictions on the manifold and is therefore fundamental when used on manifolds with boundary.",
"Let $u,v \\in H^1(M)$ we want to compute $\\frac{d}{dt} E(v+tu)$ .",
"Let $c_n = \\dfrac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) } $ , we have $E(v+tu) = c_n \\int _{HM} \\!",
"\\left(L_X \\pi ^{\\ast }v\\right)^2 +2 t L_X \\pi ^{\\ast }v L_X \\pi ^{\\ast }u +t^2 \\left(L_X \\pi ^{\\ast }u\\right)^2 \\; A \\wedge dA^{n-1},$ and therefore $\\frac{d}{dt}\\left(E(v+tu) \\right)_{|_{t=0}} &=& 2 c_n \\int _{HM} L_X \\pi ^{\\ast }v L_X \\pi ^{\\ast }u \\, A \\wedge dA^{n-1}.$ Applying the Finsler–Green formula (Proposition REF , note that $u \\in H^1(M)$ implies that $u|_{\\partial M} =0$ hence the Finsler–Green formula applies without modifications even when $M$ has a boundary), we obtain $\\frac{d}{dt}\\left(E(v+tu) \\right)_{|_{t=0}} = 2 \\int _{HM} u \\Delta ^F v \\, \\Omega ^F.$ So, if $v$ is harmonic, then it is a critical point of the energy, and (REF ) shows that it must be a minimum.",
"Conversely, if $v$ is a critical point, then, for any $u \\in H^1(M)$ , $\\langle \\Delta ^F v, u \\rangle = 0$ , which yields $\\Delta ^F v = 0$ ."
],
[
"Spectrum",
"Theorem 2.2.8 Let $M$ be a compact manifold and $F$ a Finsler metric on $M$ .",
"The set of eigenvalues of $-\\Delta ^F$ consists of an infinite, unbounded sequence of non-negative real numbers $ \\lambda _0 < \\lambda _1 < \\lambda _2 < \\dots $ .",
"Each eigenvalue has finite multiplicity and the eigenspaces corresponding to different eigenvalues are $L^2\\left(M, \\Omega \\right)$ -orthogonal.",
"The direct sum of the eigenspaces is dense in $L^2\\left(M, \\Omega \\right)$ for the $L^2$ -norm and dense in $C^k\\left(M\\right)$ for the uniform $C^k$ -topology.",
"For the convenience of the reader, we will give the adaptation to this special case of the two classical proofs of Riemannian geometry (see, for instance [21]).",
"The first one uses the whole machinery of the theory of unbounded operators, the second one uses the min-max principle and is, in my opinion, more agreeable.",
"Before starting the proof, we wish to recall one special characteristic of elliptic operators (see, for instance [90]) : Theorem 2.2.9 (Elliptic Regularity Theorem) Let $L$ be an elliptic operator on a smooth manifold $M$ .",
"If $s \\in H^1(M)$ is such that $Ls \\in C^{\\infty }\\left(M\\right)$ , then $s \\in C^{\\infty }\\left(M\\right)$ .",
"One classical problem of partial differential equations is, given a differential operator $L$ on $M$ and a smooth function $w$ , to find a smooth function $u$ such that $Lu = w$ and $u$ verifying some additional conditions on the boundary of $M$ if it exists.",
"Finding solutions to this problem is generally hard, but can sometimes be simplified by weakening the expected regularity of $u$ .",
"These solutions are called weak solutions.",
"Now, if the operator is elliptic, then finding weak solutions is sufficient due to the above regularity theorem !",
"This technique of finding weak solutions is applied in the proofs to follow.",
"(We did not wish to go into any detail about weak/classical solutions as it is not the main concern here and it can be found in many books on PDEs, see for instance [69], [104].)",
"[Proof of Theorem REF : first method] The operator $-\\Delta ^F$ is a positive, symmetric operator, so its Friedrich extension $\\mathfrak {F}$ exists (see Theorem REF ).",
"The operator $\\mathfrak {F}$ is closed, positive, and self-adjoint, so its spectrum is in $\\mathbb {R}^+$ .",
"Now take any $\\mu $ not in $\\mathbb {R}^+$ , by definition, the resolvent $R_{\\mu }:= \\left( \\mathfrak {F} -\\mu \\operatorname{Id}\\right)$ is a bounded operator from $L^2(M)$ onto the domain of $\\mathfrak {F}$ which is $H^1$ .",
"By Rellich-Kondrachov Theorem, the embedding $H^1(M)\\hookrightarrow L^2(M)$ is compact, so $R_{\\mu }$ is a compact operator.",
"Therefore, by the classical result on the spectrum of compact operators (see for instance [99]), we deduce that $\\mathfrak {F}$ has an infinite, unbounded, discrete spectrum.",
"Finally, as $-\\Delta ^F$ is elliptic, so is $\\mathfrak {F}$ , hence all its eigenfunctions are smooth (by the elliptic regularity theorem) and so belong to the domain of $-\\Delta ^F$ .",
"The following proof is classical for the Laplace–Beltrami operator (and stays true in a much wider setting, see [97]).",
"Here, we adapted the proof given in [21].",
"[Proof of Theorem REF : The Min-Max method] Let $\\mu _1= \\inf \\lbrace R(u)\\; : \\; u \\in H^1(M), \\; \\int _M |u|^2 \\Omega \\ne 0 \\rbrace .$ As $R(u) \\ge 0$ , $\\mu _1$ exist.",
"Using the Rellich-Kondrachov theorem, we show that from a sequence $(u_n) \\in H^1(M)$ such that $R(u_n)$ tends to $\\mu _1$ , we can extract a subsequence converging in $\\mathcal {L}^2(M)$ to a function in $H^1(M)$ .",
"Therefore, $\\mu _1$ is realized in $H^1(M)$ .",
"Define $E_1$ as all $v\\in H^1(M)$ such that $R(v)=\\mu _1$ or $v=0$ .",
"The second step of the proof is to show that $E_1$ is an eigenspace for $\\Delta ^F$ .",
"Claim 2.2.10 We have the following characterization : $\\left( v \\in E_1 \\right) \\quad \\Leftrightarrow \\quad \\left( \\forall u \\in H^1(M), \\; \\langle u , v \\rangle _{{1}} = \\left(\\frac{\\mu _1}{c_n} + 1 \\right) \\langle u , v \\rangle \\right).$ [Proof of Claim] The implication from right to left is trivial, just take $u=v$ , so we focus on the other implication.",
"Choose any $v \\in E_1$ , $u\\in H^1(M)$ and $t\\in \\mathbb {R}$ sufficiently small, we have $R\\left( v +tu\\right) \\ge R(v) = \\mu _1,$ so the derivative at $t=0$ of the function $t \\mapsto R\\left( v +tu\\right)$ is zero.",
"We denote $c_n := n \\left(\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) \\right)^{-1}$ , direct computation gives $R\\left(v +tu\\right) &= c_n \\frac{\\int _{HM} \\left(L_X v\\right)^2 + 2t L_Xv L_Xu + o(t) \\; A \\wedge dA^{n-1}}{\\int _M v^2 + 2t uv +o(t) \\; \\Omega } \\\\&= \\frac{c_n}{ \\int _M v^2 \\; \\Omega } \\left( \\int _{HM} \\left(L_X v\\right)^2 + 2t L_Xv L_Xu + o(t) \\; A \\wedge dA^{n-1}\\right) \\\\& \\times \\left(1- \\frac{1}{\\int _M v^2 \\; \\Omega } \\int _M 2tuv \\; \\Omega +o(t) \\right) \\\\&= R(v) + 2t \\frac{c_n}{ \\int _M v^2 \\; \\Omega } \\left(- \\frac{\\int _{HM} \\left(L_X v\\right)^2 \\; A \\wedge dA^{n-1}}{\\int _M v^2 \\; \\Omega } \\int _M uv \\; \\Omega \\right.",
"\\\\& + \\left.",
"\\int _{HM} L_Xv L_Xu \\; A \\wedge dA^{n-1}\\right) +o(t) \\\\&= R(v) + \\frac{2t}{\\int _M v^2 \\; \\Omega } \\left(- \\mu _1 \\langle u,v \\rangle + c_n \\left( \\langle u,v\\rangle _{_{1}} - \\langle u,v \\rangle \\right) \\right) + o(t) \\, .$ So writing that the term in $t$ is 0 yields the characterization (REF ).",
"The Claim REF shows first that $E_1$ is a vector space.",
"Furthermore, as the $\\Vert .",
"\\Vert _1$ -norm and the $\\Vert .",
"\\Vert $ -norm are proportional on $E_1$ , using once more the Rellich-Kondrachov theorem, it shows that the balls of $E_1$ are compact in $L^2(M)$ , so $E_1$ is finite dimensional.",
"Using the Finsler-Green formula (Proposition REF ), we can rewrite Equation (REF ) as $\\left( v \\in E_1 \\right) \\quad \\Leftrightarrow \\quad \\left( \\forall u \\in H^1(M), \\; \\langle \\Delta v , u \\rangle _0 = -\\mu _1 \\langle u,v\\rangle _0 \\right)\\, ,$ that is, an element of $E_1$ is a weak solution of the closed eigenvalue problem, for the eigenvalue $\\mu _1$ .",
"So the elliptic regularity theorem yields that $E_1$ is inside $C^{\\infty }(M)$ , and that all the weak solutions are in fact classical solutions.",
"To get the next eigenvalue, set $H_1$ (resp.",
"$L_1$ ) the orthogonal complement of $E^1$ with regard to $\\Vert .",
"\\Vert _1$ (resp.",
"$\\Vert .",
"\\Vert $ ) and we define $\\mu _2 = \\inf \\lbrace R(u) : u\\in H_1, u\\ne 0 \\rbrace .$ Claim REF shows that $H_1 \\subset L_1$ , and these spaces are closed in $H^1(M)$ and $L^2(M)$ respectively.",
"So, the inclusion $H_1 \\subset L_1$ is closed and we can apply the same arguments to show that $\\mu _2$ is attained, and that the space of functions $E_2$ that realizes this minimum is the eigenspace for $\\mu _2$ .",
"By definition, $\\mu _2 > \\mu _1$ .",
"Redoing the previous steps, we construct a sequence of eigenvalues $0 \\le \\mu _1 < \\mu _2 < \\mu _3 ...$ together with a sequence of associated, mutually orthogonal, finite dimensional subspaces of $H^1(M)$ , $E_1,E_2, \\dots $ .",
"The sequence is necessarily infinite because $H^1(M)$ is infinite dimensional and the $E_i$ are only finite dimensional.",
"To finish the proof of the first part of the theorem, we must show that the sequence $\\lbrace \\mu _i \\rbrace $ is unbounded.",
"If it was bounded by a real number $\\mu $ , we could take an infinite sequence $\\lbrace \\phi _i \\rbrace $ of orthonormal functions in $L^2(M)$ such that $R(\\phi _i) \\le \\mu $ for every $i$ (just take a $L^2$ -orthonormal basis of every $E_i$ 's), so $\\Vert \\phi _i \\Vert _1 \\le \\mu +1$ , and as the inclusion $H^1(M) \\hookrightarrow L^2(M)$ is compact we would obtain a subsequence of orthonormal functions which converges in $L^2(M)$ , which is absurd.",
"The advantage of the above proof is that it gives an actual expression for eigenvalues.",
"Let us summarize it in the following : Theorem 2.2.11 (Min-Max principle) If $M$ is a compact manifold, then the first eigenvalue of $-\\Delta ^F$ is given by $\\lambda _0 = \\inf \\left\\lbrace R(u) \\mid u \\in H^1(M) \\right\\rbrace ,$ and its eigenspace $E_0$ is the set of functions realizing the above infimum.",
"The following eigenvalues are given by $\\lambda _k= \\inf \\left\\lbrace R(u) \\; \\biggl | \\; u \\in \\bigcap _{i=1}^{k-1} E_i^{\\perp } \\right\\rbrace ,$ where their eigenspaces $E_k$ are given by the set of functions realizing the above infimum.",
"Remark 2.2.12 In particular, if $M$ is closed, the first non-zero eigenvalue is $\\lambda _1 = \\inf \\lbrace R(u) \\mid u \\in H^1(M), \\; \\int _M \\!\\!",
"u \\; \\Omega = 0 \\rbrace \\, .$ The Min-Max principle admits another formulation : Theorem 2.2.13 (Min-Max principle (bis)) Let $M$ be a compact manifold and $\\lambda _k$ the $k^{\\text{th}}$ -eigenvalue (counted with multiplicity) of $-\\Delta ^F$ , then $\\lambda _k = \\inf _{V_k} \\sup _{u \\in V_k} R(u),$ where the infimum is taken over all the $k$ -dimensional subspaces of $H^1(M)$ when $M$ is closed, and $k+1$ -dimensional when $M$ has a non-empty boundary.",
"The proof given in [21] applies without any modifications.",
"When $M$ is non-compact, we do not have such a nice spectra.",
"However, we still have the following : Proposition 2.2.14 The infimum of the essential spectrum of $-\\Delta ^F$ on a non-compact manifold is given by $\\lambda _1 = \\inf \\lbrace R(u) \\mid u \\in H^1(M) \\rbrace \\, .$ This result is a consequence of the Friedrich extension Theorem (see Theorem REF )."
],
[
"Behavior under conformal change",
"As for the Laplace–Beltrami operator, the Energy allows us to give a simple proof that the Laplacian is a conformal invariant only in dimension 2.",
"Theorem 2.3.1 Let $(\\Sigma ,F)$ be a Finsler surface, $f \\colon \\Sigma \\xrightarrow{} \\mathbb {R}$ and $F_f = e^f F$ .",
"Then, $\\Delta ^{F_f} = e^{-2f} \\Delta ^F.$ We first prove the following result : Proposition 2.3.2 Let $(M,F)$ be a Finsler manifold of dimension $n$ , $f \\colon M \\xrightarrow{} \\mathbb {R}$ and $F_f = e^f F$ .",
"Set $E_f$ the Energy associated with $F_f$ .",
"Then, for $u\\in H^1\\left(M\\right)$ $E_f(u) = c_n \\int _{HM} e^{(n-2)f} \\left(L_X \\pi ^{\\ast }u \\right)^2 \\, A \\wedge dA^{n-1},$ where $c_n = n \\left(\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right) \\right)^{-1}$ .",
"In particular, when $n=2$ the Energy is a conformal invariant.",
"The subscript $f$ indicates that we refer to the object associated with the Finsler metric $F_f$ .",
"The vector field $X_f$ is a second-order differential equation, so (see Lemma REF ) there exists a function $m \\colon HM \\rightarrow \\mathbb {R}$ and a vertical vector field $Y$ such that $X_f = m X + Y.$ We have already seen (in Section REF ) that $A_f = e^f A$ and that $A_f \\wedge dA_f^{n-1} = e^{nf} A \\wedge dA^{n-1}.$ Using $A_f\\left(X_f\\right)=1$ and that $VHM$ is in the kernel of $A$ , we have $1 = e^f A\\left(mX +Y\\right) = e^f m A\\left(X\\right) = e^f m\\,.$ Now, $E_f(u) &= c_n \\int _{HM} \\left(L_{X_f} \\pi ^{\\ast } u \\right)^2 \\, A_f \\wedge dA_f^{n-1}, \\\\&= c_n \\int _{HM} \\left(L_{m X + Y} \\pi ^{\\ast } u \\right)^2 e^{nf} \\, A \\wedge dA^{n-1}, \\\\&= c_n \\int _{HM} e^{nf}\\left(mL_{ X } \\pi ^{\\ast } u + L_{Y} \\pi ^{\\ast } u \\right)^2 \\, A \\wedge dA^{n-1}.$ As $u$ is a function on the base and $Y$ is a vertical vector field, $L_{Y} \\pi ^{\\ast } u =0$ .",
"So the preceding equation becomes $E_f(u) &= c_n \\int _{HM} e^{nf} m^2 \\left(L_{ X } \\pi ^{\\ast } u \\right)^2 \\, A \\wedge dA^{n-1}, \\\\&= c_n \\int _{HM} e^{(n-2)f} \\left(L_{ X } \\pi ^{\\ast } u \\right)^2 \\, A \\wedge dA^{n-1}.",
"$ [Proof of Theorem REF ] Let $u,v \\in H^1\\left(\\Sigma \\right)$ , we have already shown (Theorem REF ) that $\\frac{d}{dt}\\left(E(v+tu) \\right)_{|_{t=0}} = -2 \\int _{\\Sigma } u \\Delta ^{F} v \\; \\Omega $ .",
"The conformal invariance of the Energy yields : for $u,v\\in H^1 \\left(\\Sigma \\right)$ , $-2 \\int _{\\Sigma } u \\Delta ^{F} v \\; \\Omega = -2 \\int _{\\Sigma } u \\Delta ^{F_f} v \\; \\Omega _f = -2 \\int _{\\Sigma } e^{2f} u \\Delta ^{F_f} v \\; \\Omega \\,,$ where we used $\\Omega _f = e^{2f} \\Omega $ (see Equation (REF )) to obtain the last equality.",
"We can rewrite this last equality as : for $u,v\\in H^1\\left(\\Sigma \\right)$ , $\\langle \\left(\\Delta ^{F} - e^{2f} \\Delta ^{F_f} \\right)v , u \\rangle = 0,$ which yields the desired result."
],
[
"Unbounded operators and the Friedrich extension",
"We give here a quick presentation of unbounded operators and the results that we used above.",
"We refer to [76] or to the series of books [99], [96], [98], [97] for the details and the proofs."
],
[
"Some basic definitions",
"Let $H$ be a Hilbert space and $\\langle \\cdot , \\cdot \\rangle $ its scalar product.",
"Definition 2.4.1 An unbounded operator $L$ is a linear map from a dense linear subspace of $H$ , called the domain of $L$ and denoted by $D(L)$ , into $H$ .",
"The operator $L_1$ is an extension of $L$ if $D(L_1) \\supset D(L)$ and if $L_1 \\varphi = L \\varphi $ for any $\\varphi \\in D(L)$ .",
"Definition 2.4.2 An unbounded operator $L$ is called closed if its graph $\\Gamma (L)$ is closed in $H \\times H$ .",
"The graph of $L$ is the set of pairs $\\Gamma (L):= \\lbrace \\left( \\psi , L \\psi \\right) \\mid \\psi \\in D(L) ^rbrace \\subset H \\times H \\,.$ The operator $L$ is called closable if there exists a closed extension of $L$ .",
"Definition 2.4.3 Let $L$ be a closed unbounded operator on $H$ and $D(L)$ its domain.",
"A complex number $\\lambda $ is said to be in the resolvent set for $L$ if $L-\\lambda \\operatorname{Id}$ is a bijection from $D(L)$ to $H$ such that its inverse is bounded.",
"We denote the resolvent set of $L$ by $\\rho (L)$ , and $R_{\\lambda }(L) := \\left(L-\\lambda \\operatorname{Id}\\right)^{-1}$ is called the resolvent of $L$ .",
"The set $\\sigma (L):= \\rho (L)$ is the spectrum.",
"The point spectrum consists of those elements in the spectrum which are eigenvalues."
],
[
"The Friedrich extension",
"Definition 2.4.4 Let $L$ be an unbounded operator on $H$ .",
"Let $D(L^{\\ast })$ be the set of $\\varphi \\in H$ for which there exists a $\\psi \\in H$ such that $\\langle L \\rho , \\varphi \\rangle =\\langle \\rho , \\psi \\rangle , \\quad \\text{for } \\rho \\in D(L).$ For any such $\\varphi \\in D(L^{\\ast })$ , we set $L^{\\ast }\\varphi := \\psi $ .",
"$L^{\\ast }$ is called the adjoint of $L$ .",
"Note that we always have $D(L) \\subset D(L^{\\ast })$ , this will provide the difference between being symmetric and self-adjoint : Definition 2.4.5 The operator $L$ is called symmetric if $L^{\\ast }$ is an extension of $L$ , i.e., if for $\\varphi , \\psi \\in D(L)$ $\\langle L \\psi , \\varphi \\rangle =\\langle \\psi , L\\varphi \\rangle \\, .$ The operator $L$ is called self-adjoint if $L = L^{\\ast }$ , i.e., if $L$ is symmetric and $D(L)=D(L^{\\ast })$ .",
"There are many criteria to determine whether a symmetric operator is self-adjoint (see [99]) however we do not recall them as we will not use them directly.",
"Indeed, for the operator studied in this thesis there exists a self-adjoint extension called the Friedrich extension (see below Theorem REF ).",
"But, before stating that result, we will need some more definitions.",
"Definition 2.4.6 The operator $L$ is called semi-bounded if there exists a positive constant $C$ such that $\\langle L \\psi , \\psi \\rangle \\ge -C \\Vert \\psi \\Vert ^2$ for all $\\psi \\in D(L)$ .",
"When $C$ can be taken to be 0, then we say that $L$ is positive.",
"Definition 2.4.7 An unbounded quadratic form $q$ on $H$ is a bilinear form $q \\colon D(q) \\times D(q) \\rightarrow \\mathbb {R}$ , where $D(q)$ is a dense linear subspace of $H$ .",
"It is called symmetric if $q(\\varphi ,\\psi ) = q(\\psi ,\\varphi )$ for any $\\varphi ,\\psi \\in D(q)$ and semi-bounded if there exists a positive constant $C$ such that $q(\\psi ,\\psi ) \\ge -C \\Vert \\psi \\Vert ^2$ for all $\\psi \\in D(L)$ .",
"As we defined an extension of an unbounded operator, we define in the same manner an extension of a quadratic form.",
"To a symmetric operator $L$ , we can naturally associate a symmetric quadratic form, just by letting $q_L (\\varphi , \\psi ) = \\langle \\varphi , L \\psi \\rangle $ for any $\\varphi ,\\psi \\in D(L)$ .",
"For bounded operators, the Riesz Lemma gives a one-to-one correspondence, but this is not always true for unbounded quadratic forms.",
"Definition 2.4.8 Let $q$ be a semi-bounded quadratic form, $q(\\psi ,\\psi ) \\ge -C \\Vert \\psi \\Vert ^2$ for all $\\psi \\in D(L)$ .",
"It is called closed if $D(q)$ is complete under the norm : $\\Vert \\psi \\Vert _{C+1} = \\sqrt{q(\\psi ,\\psi ) + (C+1) \\Vert \\psi \\Vert ^2}.$ A semi-bounded quadratic form $q$ is called closable if there exists a closed extension $\\overline{q}$ of $q$ Theorem 2.4.9 (Friedrich extension) Let $L$ be a symmetric semi-bounded operator and let $q(\\varphi , \\psi ) := \\langle L \\psi , \\varphi \\rangle $ for $\\psi , \\varphi \\in D(L)$ .",
"Then $q$ is a closable quadratic form, its closure $\\hat{q}$ is the quadratic form associated with a self-adjoint operator $\\hat{L}$ .",
"The operator $\\hat{L}$ is the unique self-adjoint extension of $L$ with domain in $D(\\hat{q})$ .",
"Furthermore, the lower bound of its spectrum is the lower bound of $q$ .",
"See, for instance, [96] or [76]"
],
[
"Explicit representations and computations of spectra",
"Explicit representations and spectra In this chapter, we give explicit representations for our Finsler–Laplace operator.",
"We start with general Randers metrics on surfaces and then give explicit spectra for some Katok-Ziller metrics."
],
[
"Some generalities on Randers metrics",
"Among the classical examples of non-Riemannian Finsler metrics, the Randers metrics play an important role, arise naturally in physics ([95]) and have been widely studied.",
"Let us give the definition : Let $g$ be a Riemannian metric on $M$ and $\\theta $ a 1-form on $M$ , define, for $(x,v) \\in TM$ $F\\left(x,v\\right) = \\sqrt{g_x(v,v)} + \\theta _x(v).$ If the norm of $\\theta $ with respect to $g$ is strictly less than one, than $F$ is a Finsler metric (see [14]) and it is called a Randers metric.",
"Note that Randers metrics are never reversible if they are not Riemannian.",
"One advantage of Randers metrics in our case is that it is particularly easy to compute the volume and angle forms.",
"Proposition 3.1.1 Let denote by $\\Omega _0$ , $\\alpha _0$ and $X_0$ the volume form, angle form and geodesic flow associated with the Riemannian metric $g$ .",
"We have $\\Omega ^F &= \\Omega _0\\,, \\\\\\alpha ^F &= \\left(1 + \\pi ^{\\ast }\\theta (X_0) \\right) \\alpha _0 \\,.$ Remark 3.1.2 The above result holds in general whenever a Finsler metric $F$ differs from a reversible Finsler metric $F_0$ by a 1-form $\\theta $ .",
"The fact that the Holmes-Thompson volume for Randers metrics is equal to the Riemannian volume is already known (see for instance [39]) but maybe not very widely.",
"By definition of $A$ (see Chapter REF ), for any $\\xi \\in H_{x}M$ and ${Z \\in T_{\\xi }HM}$ , we have $A_{\\xi }(Z) &= \\lim _{\\varepsilon \\rightarrow 0} \\frac{F_0\\left(x,v + \\varepsilon d\\pi (Z)\\right) -F_0\\left(x,v \\right) + \\varepsilon \\theta _x\\left( d\\pi (Z) \\right)}{\\varepsilon } \\\\&= \\left.",
"A_0 \\right._{\\xi }(Z) + \\pi ^{\\ast }\\theta (Z).$ From now on we will write $\\theta $ instead of $\\pi ^{\\ast }\\theta $ as it will simplify notations and hopefully not lead to any confusion.",
"Using this notation, we have : $A = A_0 + \\theta $ and therefore $dA = dA_0 + d\\theta $ .",
"Note that $dA^{n-1} = dA_0^{n-1} + T$ where $T$ is a $(2n-2)$ -form.",
"So, as $d\\theta $ is a 2-form vanishing on $VHM$ , and for $Y_1, Y_2 \\in VHM$ , $i_{Y_1} i_{Y_2} dA_0 = 0$ , $T$ can be given at most $n-2$ vertical vectors, i.e., if $Y_1, \\dots , Y_{n-1} \\in VHM$ , then $i_{Y_1} \\dots i_{Y_{n-1}} T = 0$ .",
"Now this implies that the top-form $A\\wedge T$ vanishes, hence $A \\wedge dA^{n-1}= (A_0 + \\theta )\\wedge dA_0^{n-1}$ .",
"As $A \\wedge dA^{n-1}$ and $A_0 \\wedge dA_0^{n-1}$ are both volume forms, there exists a function $\\lambda $ such that $A \\wedge dA^{n-1}= \\lambda A_0 \\wedge dA_0^{n-1}$ .",
"We have $i_{X_0}(A \\wedge dA^{n-1}) = (1+\\theta (X_0)) dA_0^{n-1} = \\lambda dA_0^{n-1},$ therefore $\\lambda = 1+\\theta (X_0)$ .",
"Let $\\alpha ^{\\Omega _0}$ be defined by $\\alpha ^{\\Omega _0} \\wedge \\pi ^{\\ast }\\Omega _0 = A \\wedge dA^{n-1}$ (it exists by Lemma REF ).",
"We have $\\alpha ^{\\Omega _0} \\wedge \\pi ^{\\ast }\\Omega _0 = \\lambda A_0\\wedge dA_0^{n-1} = \\lambda \\alpha _0 \\wedge \\pi ^{\\ast }\\Omega _0 $ , hence $\\alpha ^{\\Omega _0}$ and $\\lambda \\alpha _0$ coincide on $VHM$ .",
"It is then immediate (see Section REF ) that $\\Omega = \\frac{\\int _{H_xM} (1+\\theta (X_0)) \\alpha _0}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right)} \\Omega _0\\, .$ As the metric $g$ is Riemannian, it is reversible, therefore $\\int _{H_xM} \\theta (X_0) \\, \\alpha _0$ must be zero.",
"Hence $\\Omega = \\Omega _0$ and $\\alpha = (1+\\theta (X_0)) \\alpha _0$ .",
"With our knowledge of the angle and volume forms for Randers metrics, we can give a more explicit expression of our Finsler–Laplace operator.",
"Indeed, recall that there exists a function $m:HM \\rightarrow \\mathbb {R}$ and a vertical vector field $Y_0$ such that $X =mX_0 +Y_0$ (because $X$ and $X_0$ are second-order differential equations, see Section REF ).",
"As $1 =A(X) = m A_0(X_0) + m\\theta (X_0)$ , we get that $m = (1 + \\theta (X_0))^{-1}$ .",
"Hence, we can rewrite the Finsler–Laplace operator as $ \\Delta ^F f(p) = \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}(\\mathbb {S}^{n-1})}\\left( \\int _{H_p M} \\frac{1}{1+\\theta (X_0)} L_{X_0}^2(\\pi ^{\\ast }f) \\: \\alpha _0 \\right.",
"\\\\\\left.",
"+ \\int _{H_p M} \\left(\\frac{-L_{X_0}\\left(\\theta (X_0)\\right)}{(1+\\theta (X_0))^2} + \\frac{-L_{Y_0}\\left( \\theta (X_0) \\right) }{(1+\\theta (X_0))^3} \\right) L_{X_0}(\\pi ^{\\ast }f) \\: \\alpha _0 \\right).$ We agree that this formula does not represent a great improvement compared to the definition.",
"However, note that the symbol of $\\Delta ^F$ is determined only by the first integral, so to obtain a coordinate expression, we can just compute the symbol and then use the formula in Lemma REF and the traditional Riemannian local expressions.",
"A particularly easy case arises from Randers metrics that are also Minkowski metrics (see Remark REF )."
],
[
"The symbol of the Laplacian on Randers surfaces",
"This part is unfortunately a quite heavy computational section, but our justification for doing so is two-fold.",
"First, we want to compute explicitly the symbol for any Randers metric on a surface to prove that it can be done by hand.",
"Furthermore, this expression indicates what kind of Riemannian metric we can expect to come from a Randers metric via the symbol of its Finsler-Laplacian.",
"Which brings us to our second goal.",
"Namely, to give an answer, in the case of 2-dimensional manifolds, to the question of which pairs (Riemannian metric, volume) can be obtained from a Finsler metric via the Finsler-Laplacian (see Section REF ).",
"Proposition 3.1.3 Let $F= \\sqrt{g} + \\theta $ be a Randers metric on a 2-dimensional manifold, $(x,y)$ be normal coordinates for $g$ at $p$ , $\\left(\\sigma _{ij}(p)\\right)$ be the symbol at $p$ of $\\Delta ^F$ .",
"We write $\\theta = \\theta _x dx + \\theta _y dy$ and $T:= \\frac{\\theta _x -i \\theta _y}{2} =|T|e^{i\\varphi }$ .",
"We have : $\\sigma _{11}(p) &= \\frac{1}{\\sqrt{1-\\Vert \\theta \\Vert ^2}}\\left( 1 + \\cos \\left(2\\varphi \\right) \\left( \\frac{1-\\sqrt{1-\\Vert \\theta \\Vert ^2}}{1+\\sqrt{1-\\Vert \\theta \\Vert ^2}}\\right) \\right) ,\\\\\\sigma _{22}(p) &= \\frac{1}{\\sqrt{1-\\Vert \\theta \\Vert ^2}} \\left( 1 - \\cos \\left(2\\varphi \\right) \\left(\\frac{1-\\sqrt{1-\\Vert \\theta \\Vert ^2}}{1+\\sqrt{1-\\Vert \\theta \\Vert ^2} } \\right) \\right) , \\\\\\sigma _{12}(p) &= \\frac{ \\sin \\left(2\\varphi \\right)}{\\sqrt{1-\\Vert \\theta \\Vert ^2}} \\left( \\frac{1-\\sqrt{1-\\Vert \\theta \\Vert ^2}}{1+\\sqrt{1-\\Vert \\theta \\Vert ^2}}\\right) ,$ where $\\Vert \\theta \\Vert $ is the norm of $\\theta $ with respect to the Riemannian metric $g$ .",
"Remark 3.1.4 One interest of this result is that it shows that for a given Riemannian metric $g$ , no two Randers metrics $F= \\sqrt{g} + \\theta $ gives the same symbol, hence the same Laplacian.",
"We can also obtain a full coordinate expression of the Laplacian for Randers surfaces by using the formula given in Lemma REF .",
"However, I doubt that it would yield much information in the general case, so I have not done it.",
"So now, given a couple $(g_{\\text{goal}},\\Omega _{\\text{goal}} )$ our aim is to give a condition for the existence of a Randers metric $F = \\sqrt{g} + \\theta $ such that $\\Omega ^F = \\Omega _{\\text{goal}}$ and, if we denote by $g_{\\sigma }$ the dual of the symbol of $\\Delta ^F$ , then $g_{\\sigma } = g_{\\text{goal}}$ .",
"We remark that the above local expression of the symbol already gives a condition on the volumes.",
"Corollary 3.1.5 Let $F= \\sqrt{g} + \\theta $ be a Randers metric on a 2-manifold, and $g_{\\sigma }$ the dual of the symbol of $\\Delta ^F$ .",
"Then $\\Omega ^{g_{\\sigma }} = \\sqrt{\\frac{\\sqrt{1-\\Vert \\theta \\Vert ^2}\\left(1 + \\sqrt{1-\\Vert \\theta \\Vert ^2} \\right)^2 }{4} } \\Omega ^F.$ Using this result, we see that the norm of $\\theta $ is uniquely determined by the quotient $\\Omega ^{g_{\\sigma }}/\\Omega ^F$ , and so, a first condition to get a positive answer to the above question is that the quotient $\\Omega ^{g_{\\text{goal}}} /\\Omega _{\\text{goal}} $ can be realized by the norm of a 1-form.",
"A trivial counter-example would be to take any $g_{\\text{goal}}$ on the 2-sphere and $\\Omega _{\\text{goal}}$ to be a constant multiple of $\\Omega ^{g_{\\text{goal}}}$ , as we cannot have a 1-form on the sphere with constant norm we get a negative answer to our question.",
"However, this question was spurred by the fact that $\\Delta ^F = \\Delta _{g,\\Omega }$ (where $\\Delta _{g,\\Omega }$ is defined in Lemma REF ) and the question was really “can we obtain every second-order, elliptic, symmetric operators that vanish on the constants from a Finsler–Laplace operator ?”.",
"But, again from Lemma REF , it is clear that $\\Delta _{g,\\Omega } = \\Delta _{g,K \\Omega }$ for any constant $K >0$ .",
"So we should not be too upset about not being able to get all the possible $\\Omega _{\\text{goal}}$ but just one in each constant multiple class.",
"So, in order to answer our question, we will have to reconstruct a Randers metric just from the information given by a Riemannian metric of the form obtained in Proposition REF .",
"We described the 1-form $\\theta $ by its norm and a certain angle $\\varphi $ , which depends on the normal coordinates we chose.",
"Now, suppose that on a contractible set $U$ , we are given two smooth functions $k\\colon U \\rightarrow \\mathbb {R}^+$ and $\\varphi \\colon U \\rightarrow \\left[0,2 \\pi \\right]$ , a Riemannian metric $g$ and a preferred choice of coordinates $(x,y)$ .",
"With this information, we can construct a 1-form $\\theta $ such that its norm for $g$ is $k$ and such that $\\varphi $ is the angle computed in the (unique) normal coordinates $(x^n,y^n)$ for $g$ such that $x^n$ and $x$ are collinear.",
"In other words, to reconstruct $\\theta $ from its norm $k$ and angle $\\varphi $ what we really need is a Riemannian metric and a vector field that is not zero when $k$ is not zero.",
"This remark is all we need to prove the following : Proposition 3.1.6 Let $g_1$ be a Riemannian metric on a 2-manifold $M$ and $\\Omega $ a volume form on $M$ .",
"Denote by $\\mu \\colon M\\rightarrow \\mathbb {R}_+^{\\ast }$ the function such that $\\Omega ^{g_1} = \\mu \\Omega $ .",
"Let $K:= \\sup \\mu $ , suppose that either $M$ is contractible and $K < \\infty $ or that $M$ is compact.",
"Then there exists a Randers metric $F$ such that $\\Omega = K \\Omega ^F$ and $g_1$ is the dual of the symbol of $\\Delta ^F$ .",
"Remark 3.1.7 The proof we give is entirely based on the coordinate expression we obtained before, but it would be much more interesting to have a coordinate-free proof in order to generalize it to any dimension.",
"We now get on to the proofs.",
"[Proof of Proposition REF ] Equation (REF ) shows that the symbol of $F$ at $p$ is obtained by computing $\\frac{1}{\\pi } \\int _{H_pM} \\frac{1}{1+\\theta \\left(X_0\\right)} L_{X_0}^2\\left(\\pi ^{\\ast }f\\right) \\alpha _0\\, .$ In normal coordinates $(x,y, \\theta )$ at $p$ on $HM$ , it is easy to check that $(X_0)_p &= \\cos \\theta \\frac{\\partial }{\\partial x}+ \\sin \\theta \\frac{\\partial }{\\partial y}\\, , \\\\(A_0)_p &= \\cos \\theta dx + \\sin \\theta dy \\, , \\\\(A_0 \\wedge dA_0)_p &= -d\\theta \\wedge dx\\wedge dy\\, .$ Therefore we can write, $\\alpha _0 = d\\theta $ , $\\theta \\left(X_0\\right) = \\theta _x \\cos \\theta + \\theta _y \\sin \\theta $ and $L_{X_0}^2\\left(\\pi ^{\\ast }f\\right) = \\cos ^2 \\theta \\frac{\\partial f}{\\partial ^2 x}+ \\sin ^2 \\theta \\frac{\\partial f}{\\partial ^2 y}+2\\cos \\theta \\sin \\theta \\frac{\\partial f}{\\partial x \\partial y}\\, .$ Hence the symbol at $p$ is $\\sigma _{11}(p) &= \\frac{1}{\\pi } \\int _0^{2\\pi } \\frac{\\cos ^2 \\theta }{1+\\theta _x \\cos \\theta + \\theta _y \\sin \\theta } d\\theta \\,, \\\\\\sigma _{22}(p) &= \\frac{1}{\\pi } \\int _0^{2\\pi } \\frac{\\sin ^2 \\theta }{1+\\theta _x \\cos \\theta + \\theta _y \\sin \\theta } d\\theta \\,, \\\\\\sigma _{12}(p) &= \\frac{1}{\\pi } \\int _0^{2\\pi } \\frac{\\cos \\theta \\sin \\theta }{1+\\theta _x \\cos \\theta + \\theta _y \\sin \\theta } d\\theta \\,.$ Unfortunately, a lack of knowledge on my part associated with Maple's unhelpfullness prevented me to give a computerized computation.",
"Therefore I give below the computations by-hand proving the proposition (and would strongly advise the reader to skip the next two pages).",
"Computation of $\\sigma _{22}$ : Recall that $T = \\frac{\\theta _x -i \\theta _y}{2}$ and let $R :=\\frac{\\theta _x + i \\theta _y}{2}$ and $z=e^{i\\theta }$ , we have $\\sigma _{22} = \\int _{S^1} \\frac{\\left( \\frac{1}{2i}\\left(z-z^{-1} \\right) \\right)^2}{1 + T z + R z^{-1} } \\frac{dz}{iz} = \\frac{i}{4} \\int _{S^1} \\frac{z^4 -2 z^2 + 1 }{T z^4 + z^3 + R z^2 } dz \\,.$ We are going to apply the Residue Theorem, hence we must found the zeros of the polynomial $T z^4 + z^3 + R z^2$ and compute the residues.",
"$T z^4 + z^3 + R z^2 = Tz^2 \\left(z-z^-\\right) \\left(z-z^+\\right),$ where $z^- = \\frac{-1 -\\sqrt{1- 4|T|^2} }{2T},\\quad z^+ = \\frac{-1 +\\sqrt{1- 4|T|^2} }{2T}\\,.$ As we had chosen the 1-form $\\theta $ with a norm strictly less than 1, we have $4|T|^2 = \\theta _x^2 +\\theta _y^2 <1$ , so $z^-$ and $z^+$ are well-defined reals.",
"Furthermore, as $\\theta $ is non-null, $\\left|z^-\\right| > 1$ , so $z^-$ is not inside the unit disc.",
"The poles of $\\frac{z^4 -2 z^2 + 1 }{T z^4 + z^3 + R z^2 }$ inside the unit disc are then 0 and $z^+$ .",
"As $\\frac{z^4 -2 z^2 + 1 }{T z^4 + z^3 + R z^2 } = \\frac{z^2}{T \\left(z-z^-\\right) \\left(z-z^+\\right)} - \\frac{2}{T \\left(z-z^-\\right) \\left(z-z^+\\right)} \\\\ + \\frac{1}{Tz^2 \\left(z-z^-\\right) \\left(z-z^+\\right)}\\,,$ we get $\\operatorname{Res}_{z^+}\\left( \\frac{z^2}{ \\left(z-z^-\\right) \\left(z-z^+\\right)} \\right) &= \\frac{\\left(z^+\\right)^2}{z^+ - z^-} \\,,\\\\\\operatorname{Res}_{z^+}\\left(\\frac{2}{ \\left(z-z^-\\right) \\left(z-z^+\\right)} \\right) &= \\frac{2}{z^+ - z^-}, \\\\\\operatorname{Res}_{z^+}\\left( \\frac{1}{z^2 \\left(z-z^-\\right) \\left(z-z^+\\right)} \\right) &= \\frac{1}{\\left(z^+\\right)^2\\left(z^+ - z^-\\right)}\\, , \\\\ \\operatorname{Res}_{0}\\left( \\frac{1}{z^2 \\left(z-z^-\\right) \\left(z-z^+\\right)} \\right) &= \\frac{z^+ + z^-}{\\left(z^+ - z^-\\right)^2}\\,.",
"$ Then, using the Residue Theorem, we obtain $\\int _{S^1} \\frac{z^2}{T \\left(z-z^-\\right) \\left(z-z^+\\right)} dz &= \\frac{2i\\pi }{T} \\frac{\\left(z^+\\right)^2}{z^+ - z^-}\\,, \\\\\\int _{S^1} -\\frac{2}{T \\left(z-z^-\\right) \\left(z-z^+\\right)} dz &= -\\frac{2i\\pi }{T} \\frac{\\left(z^+\\right)^2}{z^+ - z^-}\\,, \\\\\\int _{S^1} \\frac{1}{Tz^2 \\left(z-z^-\\right) \\left(z-z^+\\right)} dz &= \\frac{2i\\pi }{T} \\left( \\frac{1}{\\left(z^+\\right)^2\\left(z^+ - z^-\\right)} + \\frac{z^+ + z^-}{\\left(z^+ - z^-\\right)^2} \\right)\\,,$ and the sum of these three integrals gives $\\sigma _{22}$ $\\sigma _{22} = \\frac{2i^2 \\pi }{4 T} \\left( \\frac{\\left(z^+\\right)^2}{z^+ - z^-} - \\frac{2}{z^+ - z^-} + \\frac{1}{\\left(z^+\\right)^2\\left(z^+ - z^-\\right)} + \\frac{z^+ + z^-}{\\left(z^+ - z^-\\right)^2} \\right).$ To simplify a bit the above equation, note that $z^+ + z^- = -\\frac{1}{T}, \\;\\; z^+ z^- = \\frac{R}{T}, \\;\\; z^+ - z^- = \\frac{\\sqrt{1-|T|^2}}{T} \\;\\; \\text{and} \\left(z^+\\right)^2 = -\\left( \\frac{z^+ + R }{T} \\right).$ So $\\sigma _{22} &= -\\frac{ 1}{2 T}\\frac{1}{\\left(z^+ z^- \\right)^2} \\left( \\left( \\left( \\left(z^+\\right)^2-2 \\right)\\left(z^+ z^- \\right)^2 + \\left(\\frac{z^+ z^- }{z^+}\\right)^2 \\right) \\frac{1}{z^+ - z^-} +z^+ +z^- \\right) \\\\&= -\\frac{ 1}{2 T}\\frac{1}{\\left(z^+ z^- \\right)^2}\\frac{1}{z^+ - z^-} \\left( \\left(z^+\\right)^4\\left(z^-\\right)^2 - 2\\left(z^+ z^- \\right)^2 + \\left(z^+\\right)^2 \\right) \\\\&= -\\frac{ 1}{2 T}\\frac{1}{z^+ - z^-} \\left( \\left(z^+\\right)^2\\left(1+\\frac{1}{\\left(z^+ z^- \\right)^2} \\right) -2 \\right)\\\\&= -\\frac{ 1}{2 T} \\frac{T}{\\sqrt{1-|T|^2}} \\left( - \\left( \\frac{z^+ + R }{T} \\right) \\left(1+ \\frac{T^2}{R^2} \\right) -2 \\right) \\\\&= \\frac{1}{2\\sqrt{1-|T|^2}} \\left(2 + \\left(\\frac{z^+}{T} + \\frac{R}{T}\\right)\\left(1+ \\frac{T^2}{R^2} \\right) \\right).$ We define $\\varphi $ as the argument of $T$ so that $\\frac{z^+}{T} = \\frac{-1 + \\sqrt{1-|T|^2}}{2|T|^2e^{2i\\varphi }} \\quad \\text{and } \\frac{R}{T} = e^{-2i\\varphi }.$ Hence $\\sigma _{22} &= \\frac{1}{2\\sqrt{1-|T|^2}} \\left(2 + e^{-2i\\varphi } \\left(\\frac{-1 + \\sqrt{1-|T|^2} }{2|A|^2} +1 \\right)\\left( 1 + e^{4i\\varphi } \\right) \\right) \\\\&= \\frac{1}{\\sqrt{1-|T|^2}} + \\frac{1}{\\sqrt{1-|T|^2}} \\cos \\left(2\\varphi \\right) \\left( 1 + \\frac{-1+\\sqrt{1-4|T|^2}}{2|T|^2} \\right).$ Using that $\\Vert \\theta \\Vert ^2 =4|T|^2 = \\left( 1-\\sqrt{1-4|T|^2}\\right)\\left(1+\\sqrt{1-4|T|^2}\\right)$ gives Formula ().",
"Computation of $\\sigma _{11}$ : Using the above notations, we have $\\sigma _{11} = -\\frac{i}{4\\pi } \\int _{S^1} \\frac{z^4 + 2z^2 +1}{Tz^4 + z^3 + R z^2} dz \\, .$ We need to compute the residues at 0 and $z^+$ of $\\frac{z^2}{ \\left(z-z^-\\right) \\left(z-z^+\\right)}$ , $\\frac{2}{ \\left(z-z^-\\right) \\left(z-z^+\\right)}$ and $\\frac{1}{z^2 \\left(z-z^-\\right) \\left(z-z^+\\right)}$ .",
"The values of these residues are given by Equations (REF ) to ().",
"So, applying once again the Residue Theorem, we get $\\sigma _{11} = \\frac{-2i^2 }{4 T} \\left( \\frac{\\left(z^+\\right)^2}{z^+ - z^-} + \\frac{2}{z^+ - z^-} + \\frac{1}{\\left(z^+\\right)^2\\left(z^+ - z^-\\right)} + \\frac{z^+ + z^-}{\\left(z^+ - z^-\\right)^2} \\right).",
"\\nonumber $ Then, as above, we rewrite this formula to have something that behaves well when $|T|$ tends to 0.",
"By doing the same transformations as in the case of $\\sigma _{22}$ , we obtain $\\sigma _{11} = \\frac{ 1}{2 T}\\frac{1}{z^+ - z^-} \\left( \\left(z^+\\right)^2\\left(1+\\frac{1}{\\left(z^+ z^- \\right)^2} \\right) + 2 \\right).$ Simplifying once again, we get $\\sigma _{11} &= \\frac{ 1}{2 T} \\frac{T}{\\sqrt{1-|T|^2}} \\left( - \\left( \\frac{z^+ + R }{T} \\right) \\left(1+ \\frac{T^2}{R^2} \\right) + 2 \\right) \\\\&= \\frac{1}{2\\sqrt{1-|T|^2}} \\left(2 - \\left(\\frac{z^+}{T} + \\frac{R}{T}\\right)\\left(1+ \\frac{T^2}{R^2} \\right) \\right)\\\\&= \\frac{1}{2\\sqrt{1-|T|^2}} \\left(2 - e^{-2i\\varphi } \\left(\\frac{-1 + \\sqrt{1-|T|^2} }{2|A|^2} +1 \\right)\\left( 1 + e^{4i\\varphi } \\right) \\right) \\\\&= \\frac{1}{\\sqrt{1-|T|^2}} - \\frac{1}{2\\sqrt{1-|T|^2}} \\cos \\left(2\\varphi \\right) \\left( 2 + \\frac{-1+\\sqrt{1-4|T|^2}}{|T|^2} \\right),$ that we can rewrite as (REF ).",
"Computation of $\\sigma _{12}$ : The now usual transformations give $2\\sigma _{12} = -\\frac{1}{2 \\pi T} \\int _{S^1} \\frac{z^2}{\\left(z - z^+\\right)\\left(z - z^-\\right)} + \\frac{1}{z^2\\left(z - z^+\\right)\\left(z - z^-\\right)} dz \\,.$ The residues we need are given by Equations (REF ), () and (), so we obtain $2\\sigma _{12} = -\\frac{i}{ T} \\left(\\frac{\\left(z^+\\right)^2}{z^+ - z^-} - \\frac{1}{\\left(z^+\\right)^2\\left(z^+ - z^-\\right)} - \\frac{z^+ + z^-}{\\left(z^+ - z^-\\right)^2} \\right) \\\\ = -\\frac{i\\pi }{ T} \\frac{\\left(z^+\\right)^2 }{\\left(z^+ - z^-\\right)} \\left( 1 - \\frac{1}{\\left(z^+z^-\\right)^2} \\right).$ Simplifying in the same fashion as above yields $2\\sigma _{12} &= -\\frac{i}{ T} \\frac{T}{\\sqrt{1-4|T|^2}} \\frac{z^+ +R}{T}\\left(1- \\left(\\frac{T}{R} \\right)^2 \\right) \\\\&= -\\frac{i}{\\sqrt{1-4|T|^2}} \\left( \\frac{-1 + \\sqrt{1-4|T|^2} }{2 |T|^2 e^{2i\\varphi }} + e^{-2i\\varphi } \\right) \\left(1-e^{4i\\varphi }\\right) \\\\&= -\\frac{ \\sin \\left(2 \\varphi \\right)}{\\sqrt{1-4|T|^2}} \\left( 2 + \\frac{-1 + \\sqrt{1-4|T|^2} }{|T|^2} \\right).$ We gave a local expression of $\\sigma $ in normal coordinate for $g$ .",
"Now we generalize it to any local coordinates, and in the same stroke, we prove Corollary REF : Lemma 3.1.8 Let $F = \\sqrt{g} + \\theta $ and $g_{\\sigma }$ be the dual of the symbol of $\\Delta ^F$ .",
"Let $|g| := \\det g$ .",
"Then we have $|g_{\\sigma }| = \\frac{\\sqrt{1-\\Vert \\theta \\Vert ^2}\\left(1 + \\sqrt{1-\\Vert \\theta \\Vert ^2} \\right)^2 }{4} |g|\\, ,$ Now let $(x,y)$ be local coordinates on $M$ and write $g = \\left[ g_{ij} \\right]$ and $g_{\\sigma } = \\left[ g_{\\sigma ;ij} \\right]$ in this basis.",
"Then $g_{\\sigma ;11} &= \\frac{g_{11}}{4} \\left( \\left(1 + \\sqrt{1-\\Vert \\theta \\Vert ^2} \\right)^2 + \\Vert \\theta \\Vert ^2 \\cos (2\\varphi ) \\right) \\,,\\\\g_{\\sigma ;12} &= \\frac{g_{12} }{4} \\left(1 + \\sqrt{1-\\Vert \\theta \\Vert ^2} \\right)^2 + \\frac{1}{4} \\Vert \\theta \\Vert ^2 \\left( -g_{12}\\cos (2\\varphi ) + \\sqrt{|g|} \\sin (2\\varphi ) \\right) \\,,\\\\g_{\\sigma ;22} &= \\frac{g_{22}}{4} \\left(1 + \\sqrt{1-\\Vert \\theta \\Vert ^2} \\right)^2 + \\frac{1}{4} \\Vert \\theta \\Vert ^2 \\left(\\frac{ g_{11}g_{22} - 2 g_{12}^{2} }{g_{11}} \\cos (2\\varphi ) + 2\\frac{ g_{12} \\sqrt{|g|}}{g_{11}} \\sin (2\\varphi ) \\right)\\,,$ where $\\varphi $ is given in Proposition REF , computed in the normal coordinates for $g$ given by $\\left( \\dfrac{x}{\\sqrt{g_{11}}} , -\\dfrac{g_{12}}{\\sqrt{g_{11}} \\sqrt{|g|}} x + \\dfrac{\\sqrt{g_{11}} }{ \\sqrt{|g|}} y \\right)$ .",
"If $F = \\sqrt{g}+\\theta $ , then, according to Proposition REF , in normal coordinates at $p$ for $g$ , we have $|\\sigma | = \\frac{4}{b(1 + b)^2}\\,,$ where $b := \\sqrt{1-\\Vert \\theta \\Vert ^2}$ .",
"Hence, switching from normal coordinates to any coordinates gives the claim and therefore Corollary REF because it is well known that the volume form of a Riemannian metric is given by the square-root of its determinant.",
"Note that $\\left(x^n, y^n\\right) := \\left( \\dfrac{x}{\\sqrt{g_{11}}} , -\\dfrac{g_{12}}{\\sqrt{g_{11}} \\sqrt{|g|}} x + \\dfrac{\\sqrt{g_{11}} }{ \\sqrt{|g|}} y \\right)$ are normal coordinates for $g$ .",
"Let $T := \\left[ \\begin{array}{cc}\\sqrt{g_{11}} & \\frac{g_{12}}{\\sqrt{g_{11}}} \\\\0 & \\frac{\\sqrt{|g|}}{\\sqrt{g_{11}}}\\end{array} \\right].$ We have that $T\\left(x^n, y^n\\right) = (x,y)$ and $ {\\vphantom{T}}^{t}{T} T = [g_{ij}]$ .",
"Now, as $g_{\\sigma } = \\sigma ^{\\ast }$ , in the normal coordinate $\\left(x^n, y^n\\right)$ at $p$ , we have, using Proposition REF : $g_{\\sigma }(p) = \\frac{1}{4} \\left[ \\begin{array}{cc}(1+b)^2 - (1-b^2)\\cos (2\\varphi ) & (1-b^2)\\sin (2\\varphi ) \\\\(1-b^2)\\sin (2\\varphi ) & (1+b)^2 + (1-b^2)\\cos (2\\varphi )\\end{array} \\right]$ and ${\\vphantom{T}}^{t}{T} g_{\\sigma }(p) T$ gives $g_{\\sigma }$ in the $(x,y)$ coordinates.",
"[Proof of Proposition REF ] Let $g_1$ and $\\Omega $ be respectively a Riemannian metric and a volume form on a 2-manifold $M$ .",
"Let $\\mu \\colon M\\rightarrow \\mathbb {R}_{+}^{\\ast }$ be the function such that $\\Omega ^{g_1} = \\mu \\Omega $ , so that we can write $K:= \\sup _{x\\in M} \\mu (x)$ , by hypothesis $K$ is finite, and set $\\mu ^{\\prime }:= \\frac{\\mu }{K}$ .",
"We choose a smooth vector field $Z$ on $M$ such that it is non-zero on $\\mu ^{\\prime -1}\\lbrace 1 \\rbrace $ .",
"Such a vector field exists by hypothesis, indeed, either $M$ is contractible and we can take $Z$ to be never zero, or $M$ is supposed to be compact, which force $\\mu ^{\\prime -1}\\lbrace 1 \\rbrace $ to be non-empty.",
"Now our goal is to construct a Riemannian metric $g$ and a 1-form $\\theta $ such that $g_1$ is obtained from the Randers metric $F = \\sqrt{g} + \\theta $ via the symbol of the Finsler–Laplace operator and $\\Omega = \\frac{1}{K} \\Omega ^F$ .",
"We set $b \\colon M \\rightarrow \\left] 0, 1 \\right]$ to be the unique function such that $\\frac{b(1+b)^2}{4}= \\left.\\mu ^{\\prime }\\right.^2 .$ Such a $b$ exists, and is smooth, by definition of $\\mu ^{\\prime }$ .",
"At any point $z \\in \\mu ^{\\prime -1}\\lbrace 1 \\rbrace $ , we set $g(z) := g_1(z)$ and $\\theta _z = 0$ .",
"Now, take $z \\notin \\mu ^{\\prime -1}\\lbrace 1 \\rbrace $ and choose local coordinates $(x,y)$ around $z$ such that $Z(z)$ and $\\left.\\frac{\\partial }{\\partial x}\\right|_z$ are collinear (such coordinates exist as $\\mu ^{\\prime -1}\\lbrace 1 \\rbrace $ is closed).",
"In order to avoid too much subscript clutter, we write $g_1 = \\left[ \\begin{array}{cc}u& v\\\\v & w\\end{array} \\right]$ in the $(x,y)$ coordinates.",
"In those coordinates, we define $g(z)$ by $\\left\\lbrace \\begin{aligned}g_{11}(z) &:= \\frac{4u}{\\left(1 + b\\right)^2} \\,, \\\\g_{12}(z) &:= \\frac{4v -(1-b^2) \\frac{\\sqrt{uw-v^2}}{\\mu ^{\\prime }} }{\\left(1 + b\\right)^2} \\,, \\\\g_{22}(z) &:= \\frac{4w -2 (1-b^2) \\frac{\\sqrt{uw-v^2}}{\\mu ^{\\prime }} \\left(\\frac{4v -(1-b^2) \\frac{\\sqrt{uw-v^2}}{\\mu ^{\\prime }} }{4u} \\right) }{\\left(1 + b\\right)^2} \\,.\\end{aligned} \\right.$ Remark that it implies that $\\left\\lbrace \\begin{aligned}\\sqrt{uw-v^2} &= \\mu ^{\\prime } \\sqrt{|g|} \\,, \\\\4u &= g_{11}(z) \\left(1 + b\\right)^2 \\,,\\\\4v &= g_{12}(z) \\left(1 + b \\right)^2 + (1-b^2) \\sqrt{|g(z)|} \\,,\\\\4w &= g_{22}(z)\\left(1 + b \\right)^2 + 2 (1-b^2)\\frac{ g_{12}(z) \\sqrt{|g(z)|}}{g_{11}(z)} \\,,\\end{aligned} \\right.$ where again we wrote $|g| = g_{11}(z) g_{22}(z) - g_{12}(z)^2$ .",
"Only the first equation is not evident, but the computation can be done by hand or by plugging $g(z)$ in your favorite formal computation program.",
"By Lemma REF , the second system of equations shows that $g_1(z) = g_{\\sigma }(z)$ where $g_{\\sigma }$ is the dual symbol coming from the Randers metric $F_z = \\sqrt{g(z)} + \\theta _z$ , where $\\theta _z$ is defined by $\\Vert \\theta _z \\Vert ^2_{g} = 1- b(z)^2$ and $\\varphi (z) = \\pi /4$ , i.e., $\\theta _z$ is $\\mathcal {L}_g \\circ Z(z)$ rotated (for the metric $g$ ) by $\\pi /2$ .",
"Moreover, $\\Omega ^F = \\Omega ^{g}$ because $F$ is Randers, and $\\Omega ^{g_{\\sigma }} = \\sqrt{\\dfrac{b(1-b)^2}{4}} \\Omega ^g$ .",
"By our choice of $b$ , we have $\\mu ^{\\prime } \\Omega ^F = \\Omega ^{g_1} = K \\mu ^{\\prime } \\Omega ,$ so $\\Omega ^F = K \\Omega $ as wanted.",
"For the moment, given any point $z$ and local coordinates $(x,y)$ around $z$ (satisfying the above condition with respect to $Z$ ) we constructed a scalar product $g(z)$ on $T_zM$ and an element $\\theta _z$ of $T_z^{\\ast }M$ which verifies our conclusion.",
"But in order to be done, we still need to show two things : First, the definitions of $g(z)$ and $\\theta _z$ must be independent of the local coordinates we choose, and once we have that, it remains to be shown that everything is smooth.",
"Let us prove the independence of $g(z)$ from the coordinates $(x,y)$ : For any $z\\in \\mu ^{\\prime -1}\\lbrace 1 \\rbrace $ there exists only one local coordinate system $(x^n, y^n)$ that is normal for $g_1$ at $z$ and such that $Z(z)$ is collinear to $\\left.\\frac{\\partial }{\\partial x^n}\\right|_z$ .",
"In these coordinates, $g(z) := [g_{ij}^n(z)]$ is given by $\\left\\lbrace \\begin{aligned}g_{11}^{n}(z) &:= \\frac{4}{\\left(1 + b\\right)^2} \\,, \\\\g_{12}^{n}(z) &:= -\\frac{ (1-b^2) }{\\lambda \\left(1 + b\\right)^2} \\,, \\\\g_{22}^{n}(z) &:= \\frac{4 + 2 (1-b^2)^2 \\frac{1}{4\\lambda ^2}}{\\left(1 + b\\right)^2} \\,.\\end{aligned} \\right.",
".$ A transformation from $(x^n,y^n)$ to $(x,y)$ is given by $T_z := \\left[ \\begin{array}{cc}\\sqrt{u(z)} & \\frac{v(z)}{\\sqrt{u(z)}} \\\\0 & \\frac{\\sqrt{u(z)w(z) - v(z)^2}}{\\sqrt{u(z)}}\\end{array} \\right].",
".$ and to get the independence, we just need to verify that ${\\vphantom{T_z}}^{t}{T_z} \\left[ \\begin{array}{cc}g_{11}^{n}(z) & g_{12}^{n}(z) \\\\ g_{12}^{n}(z) & g_{22}^{n}(z)\\end{array} \\right]T_z = \\left[ \\begin{array}{cc}g_{11}(z) & g_{12}(z) \\\\ g_{12}(z) & g_{22}(z)\\end{array} \\right].$ This is easily done, even by hand.",
"So $g$ is well defined and as $\\theta $ just depends on $g$ , $b$ and $Z$ , it is also well-defined.",
"We are left with the smoothness issue, which is easy.",
"Indeed, $b$ is a smooth function because $\\mu ^{\\prime }$ is.",
"So $g$ is also smooth because of how we defined it, and finally as $\\theta $ depends just on $g$ , $b$ and $Z$ , all of which are smooth, it is smooth."
],
[
"Finsler–Laplacian and spectral data for Katok–Ziller metrics",
"Spectral data for KZ metrics When we started studying this Finsler–Laplace operator, one of our first goals was to show that it was “usable”, that is, that we could take purely Finslerian examples and compute the spectrum and the eigenfunctions.",
"However, computing spectral data is a daunting task even in the Riemannian case.",
"Indeed, past the three model spaces $\\mathbb {R}^n$ , $\\mathbb {S}^n$ and $\\mathbb {H}^n$ and some of their quotients, we do not know any full spectra of a Laplace–Beltrami operator.",
"So, in order to have any chance of success, we wanted to start with some equivalent of a Finsler model space, but as this does not exist, we settled for just constant flag curvature, preferably on surfaces.",
"Akbar-Zadeh [2] showed that any closed surface endowed with a Finsler metric of constant negative flag curvature is in fact Riemannian.",
"On the 2-sphere however, the Finsler case is richer, Bryant [27], [28] constructed a 2-parameter family of such metrics.",
"Katok previously [77] had constructed a 1-parameter family of Finsler metrics on the sphere which later turned out to be of constant flag curvature 1 (see [94]).",
"We chose to study Katok's examples for several reasons, the main being its dynamical interest.",
"Indeed, they were constructed to give examples of metrics on the 2-sphere with only a finite number of closed geodesics at a time when it was thought impossible.",
"Furthermore, the construction method was generalized by Ziller [111] and so we use these metrics to give examples of spectral data in the torus case.",
"Lastly, these metrics admits a quite agreeable explicit expression compared to the above mentioned examples due to Bryant (see Proposition REF and compare to Equation (12.7.4) on p.346 of [14]).",
"We recall the construction, in a slightly more general context than in [111], as well as some other properties below."
],
[
"Construction",
"Let $M$ be a closed manifold and $F_0$ a smooth Finsler metric on $M$ .",
"We suppose furthermore that $M$ admits a Killing field $V$ , i.e., $V$ is a vector field on $M$ such that the one-parameter group of diffeomorphisms that it generates are composed of isometries for $F_0$ .",
"Katok-Ziller examples are constructed in the Hamiltonian setting.",
"Recall that the Legendre transform $\\mathcal {L}_{F_0} \\colon TM \\rightarrow T^{\\ast }M$ associated with $F_0$ is a diffeomorphism outside the zero section and that $H_0 := F_0 \\circ \\mathcal {L}_0 ^{-1} : T^{\\ast }M \\rightarrow \\mathbb {R}$ is a Finsler co-metric (see Section REF ).",
"We consider $T^{\\ast }M$ as a symplectic manifold with canonical form $\\omega $ .",
"Any function $H \\colon T^{\\ast }M \\rightarrow \\mathbb {R}$ gives rise to a Hamiltonian vector field $X_H$ defined by $dH(y) = \\omega \\left(X_H , y \\right), \\; \\text{for all} \\; y\\in TT^{\\ast }M\\, .$ We define $H_1: T^{\\ast }M \\rightarrow \\mathbb {R}$ by $ H_1(x) := x(V)$ , and, for sufficiently small $\\varepsilon $ , we set $H_{\\varepsilon } := H_0 + \\varepsilon H_1 \\, .$ The function $H_{\\varepsilon }$ is smooth off the zero-section, homogeneous of degree one, and also strongly convex for sufficiently small $\\varepsilon $ .",
"Therefore, the Legendre transform ${\\mathcal {L}_{\\varepsilon } : T^{\\ast }M \\rightarrow TM}$ associated to $\\frac{1}{2} H_{\\varepsilon }^2$ is a global diffeomorphism, so we can state the Definition 3.2.1 The family of generalized Katok-Ziller metrics on $M$ associated to $F_0$ and $V$ is given by $F_{\\varepsilon } = H_{\\varepsilon } \\circ \\mathcal {L}_{\\varepsilon }^{-1}.$ Katok, in his first example, took $F_0$ to be the standard Riemannian metric on $\\mathbb {S}^n$ and showed that some of these metrics had only a finite number of closed geodesics.",
"Ziller [111] then showed that, for $\\varepsilon /\\pi $ irrational, there was in fact $n$ closed geodesics for $n$ even and $n-1$ for $n$ odd.",
"In general, Bangert and Long [13] showed that every non-reversible Finsler metric on $\\mathbb {S}^2$ has at least two closed geodesics.",
"The minimal number of closed geodesics is still unknown in higher dimension.",
"As we will use local coordinates formulas for the Katok-Ziller metrics on the torus and the sphere, we can state the general formula in local coordinates when $F_0$ is Riemannian.",
"The computation of these metrics in local coordinates is probably not new (see Rademacher [94] for the expression on the sphere) and was communicated to us in this more general form by Foulon.",
"Proposition 3.2.2 (Foulon [58]) Let $F_0 = \\sqrt{g}$ be a Riemannian metric on $M$ , $V$ a Killing field on $M$ , and $F_{\\varepsilon }$ the Katok-Ziller metric associated.",
"Then $F_{\\varepsilon }(x,\\xi ) = \\frac{1}{1- \\varepsilon ^2 g\\left(V,V\\right)} \\left[ \\sqrt{g\\left(\\xi ,\\xi \\right) \\left(1-\\varepsilon ^2 g\\left(V,V\\right) \\right) + \\varepsilon ^2 g\\left(V,\\xi \\right)^2} -\\varepsilon g\\left(V,\\xi \\right) \\right].$ Let $x\\in M$ , we choose the normal coordinates $\\left(\\xi _i\\right)$ on $T_xM$ , so that we have $F_0^2\\left(x,\\xi \\right) = \\sum \\xi _i^2$ .",
"We will write $p$ for an element of $T_x^{\\ast }M$ , and $p^i$ will be the associated coordinates.",
"As $F_0$ is Riemannian, we have $H_0 \\left(x,p\\right) = ||p|| = \\sqrt{\\sum (p^i)^2}.$ The function $H_{\\varepsilon }$ is then given by $H_{\\varepsilon }\\left(x,p\\right) &= H_0 \\left(x,p\\right) + \\varepsilon H_1 \\left(x,p\\right) \\\\&= ||p|| + \\varepsilon \\left<p | V\\right>,$ and, if we set $\\mathcal {L}_{\\varepsilon }= d_v\\left(\\frac{1}{2} H_{\\varepsilon }^2 \\right): T^{\\ast }M \\rightarrow TM,$ we have $F_{\\varepsilon }\\left(x, \\xi \\right) = H_{\\varepsilon } \\circ \\mathcal {L}_{\\varepsilon }^{-1} \\left(x, \\xi \\right) .$ In order to compute $ F_{\\varepsilon }$ , we will first compute $\\mathcal {L}_{\\varepsilon }$ .",
"Recall that $\\left(\\frac{\\partial }{\\partial x_i}\\right)$ represents a basis of $T_xM$ $\\mathcal {L}_{\\varepsilon }\\left(x,p\\right) &= \\frac{\\partial }{\\partial p^i}\\left(\\frac{1}{2} H_{\\varepsilon }^2 \\right) \\frac{\\partial }{\\partial x_i} \\\\&= \\frac{\\partial }{\\partial p^i}\\left[ \\frac{1}{2} ||p||^2 + \\varepsilon ||p|| \\left<p | V\\right> + \\frac{\\varepsilon ^2}{2} \\left<p | V\\right>^2 \\right] \\frac{\\partial }{\\partial x_i} \\\\&= \\left[ p^i + \\varepsilon \\left(\\frac{p^i}{||p||} \\left<p | V\\right> + \\varepsilon ||p|| V_i \\right) + \\varepsilon ^2 V_i \\left<p | V\\right> \\right] \\frac{\\partial }{\\partial x_i} \\\\&= \\left( p^i + \\varepsilon ||p|| V_i \\right) \\left( 1 + \\frac{\\varepsilon }{||p||} \\left< p | V\\right>\\right) \\frac{\\partial }{\\partial x_i} \\, .$ Using that $H_{\\varepsilon } \\left(x,p\\right) = F_{\\varepsilon }\\left( \\mathcal {L}_{\\varepsilon }\\left(x,p\\right) \\right)$ and setting $\\frac{p}{||p||} = u $ , we obtain $||p|| + \\varepsilon \\left<p | V\\right> &= F_{\\varepsilon }\\left(||p|| \\left( u^i + \\varepsilon V_i \\right) \\left( 1 + \\frac{\\varepsilon }{||p||} \\left< p | V\\right>\\right) \\frac{\\partial }{\\partial x_i} \\right) \\\\&= \\left(||p|| + \\varepsilon \\left<p | V\\right> \\right) F_{\\varepsilon }\\left( \\left( u^i + \\varepsilon V_i \\right) \\frac{\\partial }{\\partial x_i} \\right) ,$ hence $1 = F_{\\varepsilon }\\left( \\left( u^i + \\varepsilon V_i \\right) \\frac{\\partial }{\\partial x_i} \\right).$ Let $ \\xi = \\mathcal {L}_{\\varepsilon }\\left(x,p\\right)$ , we have obtained that : $\\xi &= F_{\\varepsilon }\\left(x,\\xi \\right) \\left( u + \\varepsilon V \\right),$ where this equality must be understood coordinate by coordinate.",
"Therefore, $\\left< u |V \\right> &= \\frac{1}{F_{\\varepsilon }\\left(x,\\xi \\right)} \\left< \\xi |V \\right> - \\varepsilon ||V||^2,$ and $||\\xi ||^2 &= F_{\\varepsilon }^2\\left(x,\\xi \\right) \\left[ ||u||^2 + 2 \\varepsilon \\left< u | V\\right> + \\varepsilon ^2 ||V||^2 \\right] \\\\&= F_{\\varepsilon }^2\\left(x,\\xi \\right)\\left[ 1 + 2 \\varepsilon \\frac{ \\left< \\xi |V \\right> }{F_{\\varepsilon }\\left(x,\\xi \\right)} -2\\varepsilon ^2 ||V||^2+ \\varepsilon ^2 ||V||^2 \\right] .$ And we are led to solve $F_{\\varepsilon }^2\\left(x,\\xi \\right) \\left(1 - \\varepsilon ^2 ||V||^2 \\right) + 2 \\varepsilon \\left< \\xi |V \\right> F_{\\varepsilon }\\left(x,\\xi \\right) - ||\\xi ||^2 = 0,$ which yields $F_{\\varepsilon }\\left(x,\\xi \\right) = \\frac{ - \\varepsilon \\left< \\xi |V \\right> + \\sqrt{\\varepsilon ^2 \\left< \\xi |V \\right>^2 + \\left(1 - \\varepsilon ^2 ||V||^2 \\right)||\\xi ||^2 }}{\\left(1 - \\varepsilon ^2 ||V||^2 \\right)}\\, .$ Before getting on to the examples, we want to point out the following property of the Katok-Ziller examples : Theorem 3.2.3 (Foulon [58]) The flag curvatures of the family of Katok-Ziller metrics are constant."
],
[
"On the Torus",
"Let $ be an $ n$-dimensional torus, $ g$ a flat metric on $ and $V$ a Killing field for $g$ .",
"Let $F_{\\varepsilon }$ be the associated Katok–Ziller metric, we have : Proposition 3.2.4 There exists a (unique) Riemannian metric $\\sigma _{\\varepsilon }$ such that $\\Delta ^{F_{\\varepsilon }} = \\Delta ^{\\sigma _{\\varepsilon }}$ .",
"This result follows from the fact that all objects involved are invariant by translations, hence independent of the point on the torus.",
"Indeed, $V$ is a Killing field on $ and so is translation invariant.",
"From Proposition \\ref {prop:expression_KZ}, we deduce that $ F$ is independent of the base point on the torus, which yields, via Equation (\\ref {eq:laplacien_pour_randers_general}) and Proposition \\ref {prop:volume_angle_Randers} that both the symbol $$ and the volume $ F$ are constant on $ .",
"Now, recall (see Lemma REF ) that ${\\Delta ^{F_{\\varepsilon }} = \\Delta ^{\\sigma } - \\frac{1}{a^2} \\langle \\nabla ^{\\sigma } \\varphi , \\nabla ^{\\sigma } a^2 \\rangle }$ , where $a$ is the function such that $\\Omega ^{F_{\\varepsilon }} = a^2 \\Omega ^{\\sigma }$ .",
"So as $a$ is constant, $\\nabla ^{\\sigma } a^2 = 0$ and we get the result.",
"Remark 3.2.5 The above result of course holds for any “Minkowski-Randers space” (i.e., a Randers metric depending only on the tangent vector and not on the base-point on the manifold) and we could therefore use what is known on the Riemannian spectrum on subsets of $\\mathbb {R}^n$ to obtain the Finsler–Laplace spectrum of these spaces.",
"We give below the actual computation of the Finsler–Laplace operator for Katok–Ziller metrics on the 2-torus to get an idea of how the Katok–Ziller transformation actually acts on the spectrum.",
"This could be obtained by doing the right change of variables in the formula of Proposition REF .",
"However, direct computations are not much longer and were already typed.",
"Therefore that is how we proceed."
],
[
"An example in dimension two",
"We set $ \\mathbb {R}^2 /\\mathbb {Z}^2$ , $(x,y)$ (global) coordinates on $\\ and $ ( x, y )$\\ local coordinates on $ Tp.",
"Let $\\varepsilon <1$ , the Katok-Ziller metric on $\\ associated with the standard metric and the Killing field $ V = x$, is given by$$F_{\\varepsilon }(x,y; \\xi _x, \\xi _y ) = \\frac{1}{1-\\varepsilon ^2} \\left( \\sqrt{\\xi _x^2 + (1-\\varepsilon ^2) \\xi _y^2} - \\varepsilon \\xi _x \\right),$$$ Theorem 3.2.6 The Laplace operator, in local coordinates, is given by $\\Delta ^{F_{\\varepsilon }} = \\frac{2 \\left(1-\\varepsilon ^2\\right)}{1 + \\sqrt{1- \\varepsilon ^2}} \\left( \\sqrt{1- \\varepsilon ^2} \\frac{\\partial ^2}{\\partial x^2}+ \\frac{\\partial ^2}{\\partial y^2}\\right)$ and the spectrum is the set of $\\lambda _{(p,q)}$ , $(p,q)\\in \\mathbb {Z}^2$ , given by $\\lambda _{(p,q)} = 4\\pi ^2 \\frac{2 \\left(1-\\varepsilon ^2\\right)}{1 + \\sqrt{1- \\varepsilon ^2}} \\left( \\sqrt{1-\\varepsilon ^2} p^2 + q^2 \\right).$ Recall that for any flat Riemannian torus, the Poisson formula gives a link between the eigenvalues of the Laplacian and the length of the periodic orbits.",
"In the case at hand we lose this relationship, as there is no a priori link between the length of the periodic geodesics for the Finsler metric and the length of the closed geodesics in the isospectral torus.",
"Vertical derivative and coordinate change.",
"In the local coordinates $\\left(x,y, \\xi _x, \\xi _y\\right)$ on $T we have\\begin{equation*}d_v F_{\\varepsilon }= \\frac{1}{1-\\varepsilon ^2} \\left( f_x dx + f_y dy \\right),\\end{equation*}where\\begin{equation*}f_x := \\frac{\\xi _x}{\\sqrt{\\xi _x^2 + (1-\\varepsilon ^2) \\xi _y^2}} - \\varepsilon \\quad \\text{and} \\quad f_y := \\frac{\\xi _y}{\\sqrt{\\xi _x^2 + (1-\\varepsilon ^2) \\xi _y^2}}\\, .\\end{equation*}We choose a local coordinate system $ (x,y,)$\\ on $ H where $\\theta $ is determined by $\\left\\lbrace \\begin{aligned}\\cos \\theta & = f_x + \\varepsilon \\\\\\sin \\theta & = \\frac{f_y}{\\sqrt{1-\\varepsilon ^2}}\\, .\\end{aligned}\\right.$ As the Hilbert form $A$ is the projection on $H\\ of the vertical derivative of $ F$, we have\\begin{equation*}A = \\frac{1}{1-\\varepsilon ^2} \\left( \\left(\\cos \\theta -\\varepsilon \\right) dx + \\sqrt{1-\\varepsilon ^2}\\sin \\theta dy \\right).\\end{equation*}$Liouville volume and angle form.",
"We have $dA & = \\frac{1}{1-\\varepsilon ^2} \\left( -\\sin \\theta d\\theta \\wedge dx + \\sqrt{1-\\varepsilon ^2} \\cos (\\theta )d\\theta \\wedge dy \\right) \\\\A \\wedge dA & = \\left(\\frac{1}{1-\\varepsilon ^2}\\right)^{\\frac{3}{2}} \\left(-1 + \\varepsilon \\cos \\theta \\right) d\\theta \\wedge dx \\wedge dy \\, .$ Therefore $ \\alpha = \\left(1 - \\varepsilon \\cos (\\theta )\\right) d\\theta $ .",
"Geodesic flow.",
"Let $X = X_x \\frac{\\partial }{\\partial x}+ X_y \\frac{\\partial }{\\partial y}+ X_{\\theta } \\frac{\\partial }{\\partial \\theta }$ be the geodesic flow, Equation (REF ) is equivalent to $\\left\\lbrace \\begin{aligned}X_{\\theta } &= 0 \\\\\\sin (\\theta )X_x - \\sqrt{1-\\varepsilon ^2} \\cos (\\theta )X_y &= 0 \\\\\\left(\\cos (\\theta )-\\varepsilon \\right) X_x + \\sqrt{1-\\varepsilon ^2} \\sin (\\theta )X_y &= 1-\\varepsilon ^2 \\,.\\end{aligned}\\right.$ Hence $X_x = \\frac{1-\\varepsilon ^2}{1- \\varepsilon \\cos (\\theta )} \\cos (\\theta )$ , $X_y = \\frac{\\sqrt{1-\\varepsilon ^2}}{1- \\varepsilon \\cos (\\theta )} \\sin (\\theta )$ and $X_{\\theta } = 0$ .",
"The Laplacian.",
"The second Lie derivative of $X$ is $L_X^2 = X_x^2 \\frac{\\partial ^2}{\\partial x^2}+ X_y^2 \\frac{\\partial ^2}{\\partial y^2}+ X_x X_y \\frac{\\partial ^2}{\\partial x \\partial y}$ .",
"So, for $p\\in S$ , $\\Delta ^{\\varepsilon } = \\frac{1}{\\pi }\\left( \\int _{H_pS} X_x^2 \\alpha \\frac{\\partial ^2}{\\partial x^2}+ \\int _{H_pS} X_y^2 \\alpha \\frac{\\partial ^2}{\\partial y^2}+ \\int _{H_pS} X_x X_y \\alpha \\frac{\\partial ^2}{\\partial x \\partial y}\\right).$ As $X_x$ and $X_y$ are of different parity (in $\\theta $ ), we have $\\int _{H_pS} X_x X_y \\alpha =0 $ .",
"Hence $\\Delta ^{\\varepsilon } = \\frac{1}{\\pi }\\left( \\int _{H_pS} X_x^2 \\alpha \\frac{\\partial ^2}{\\partial x^2}+ \\int _{H_pS} X_y^2 \\alpha \\frac{\\partial ^2}{\\partial y^2}\\right).$ Direct computations give $\\int _{H_pS} X_x^2 \\alpha &= 2 \\pi \\frac{\\left(1-\\varepsilon ^2\\right)^{\\frac{3}{2}}}{1 + \\sqrt{1- \\varepsilon ^2}}\\, ,\\\\\\int _{H_pS} X_y^2 \\alpha &= 2 \\pi \\frac{1-\\varepsilon ^2}{1 + \\sqrt{1- \\varepsilon ^2}} \\, .$ Therefore, the Finsler–Laplace operator is given by $\\Delta ^{\\varepsilon } = \\frac{2 \\left(1-\\varepsilon ^2\\right)^{\\frac{3}{2}}}{1 + \\sqrt{1- \\varepsilon ^2}} \\frac{\\partial ^2}{\\partial x^2}+ \\frac{2 \\left(1-\\varepsilon ^2 \\right)}{1 + \\sqrt{1- \\varepsilon ^2}} \\frac{\\partial ^2}{\\partial y^2}\\, .$ The spectrum.",
"To compute the spectrum we consider Fourier series of functions on $.\\\\Any function $ f C($\\ can be written as$$f(x,y) = \\sum _{(p,q)\\in \\mathbb {Z}^2} c_{(p,q)} e^{2i\\pi (px+qy)}$$and we are led to solve\\begin{equation}\\Delta ^{F_{\\varepsilon }} f + \\lambda f = \\sum _{(p,q)\\in \\mathbb {Z}^2} c_{(p,q)} \\left[ -4\\pi ^2 \\left(a p^2 + b q^2\\right) +\\lambda \\right] e^{2i\\pi (px+qy)} = 0 \\,,\\end{equation}where\\begin{equation*}a = \\frac{2 \\left(1-\\varepsilon ^2\\right)^{3/2}}{1 + \\sqrt{1- \\varepsilon ^2}} \\quad \\text{and}\\quad b = \\frac{2 \\left(1-\\varepsilon ^2\\right)}{1 + \\sqrt{1- \\varepsilon ^2}}\\, .\\end{equation*}$ Now, for any $(p,q) \\in \\mathbb {Z}^2$ , $\\lambda _{(p,q)} = 4\\pi ^2 \\frac{2 \\left(1-\\varepsilon ^2\\right)}{1 + \\sqrt{1- \\varepsilon ^2}} \\left( \\sqrt{1-\\varepsilon ^2} p^2 + q^2 \\right)$ is a solution to ()."
],
[
"On the 2-Sphere",
"Let $\\mathbb {S}^2 \\setminus \\lbrace \\text{N,S}\\rbrace = \\lbrace (\\phi ,\\theta ) \\mid \\phi \\in \\left]0,\\pi \\right[, \\; \\theta \\in \\left[0,2 \\pi \\right] \\rbrace $ be polar coordinates on the sphere minus the poles, and take $\\left(\\phi ,\\theta ; \\xi _{\\phi }, \\xi _{\\theta } \\right)$ the associated local coordinates on $T\\mathbb {S}^2\\setminus \\lbrace \\text{N,S}\\rbrace $ .",
"All the formulas afterwards can be extended by taking $\\phi =0$ and $\\phi =\\pi $ for the North and South poles.",
"The Katok–Ziller metrics associated with the standard metric and the Killing field $V = \\sin (\\phi )\\frac{\\partial }{\\partial \\theta }$ are given by $F_{\\varepsilon } \\left(\\phi ,\\theta ; \\xi _{\\phi }, \\xi _{\\theta } \\right) = \\frac{1}{1-\\varepsilon ^2\\sin ^2(\\phi )} \\left( \\sqrt{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)\\xi _{\\phi }^2 + \\sin ^2(\\phi )\\xi _{\\theta }^2} - \\varepsilon \\sin ^2(\\phi ) \\xi _{\\theta }\\right),$ Theorem 3.2.7 The Finsler–Laplace operator on $(\\mathbb {S}^2, F_{\\varepsilon })$ is given by $ \\Delta ^{F_{\\varepsilon }} = \\frac{2}{1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\Biggl [ \\frac{1}{\\sin ^2(\\phi )} \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{\\frac{3}{2}} \\frac{\\partial ^2}{\\partial \\theta ^2}\\\\+ \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right) \\frac{\\partial ^2 }{\\partial \\phi ^2}+\\frac{\\cos (\\phi )}{\\sin (\\phi )} \\left(\\varepsilon ^2\\sin ^2(\\phi )+ \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\right)\\frac{\\partial }{\\partial \\phi }\\Biggr ].$ By computing the Laplacian associated with the symbol metric, we can remark that this Laplacian is not a Riemannian Laplacian, contrarily to the flat torus case above.",
"Hence the question of it being isospectral to a Riemannian Laplacian becomes non-trivial, but we do not know the answer.",
"Recall that the spectrum of the Laplace–Beltrami operator on $\\mathbb {S}^2$ is the set ${\\lbrace -l(l+1) \\mid l \\in \\mathbb {N}\\rbrace }$ and that an eigenspace is spanned by functions $Y_{l}^{m}$ with $m\\in \\mathbb {Z}$ such that $-l\\le m \\le l$ .",
"These functions are called spherical harmonics and are defined by $Y_{l}^{m}\\left(\\phi , \\theta \\right) := e^{i m \\theta } P_l^m\\left(\\cos (\\phi )\\right),$ where $P_l^m$ is the associated Legendre polynomial.",
"We can see clearly from Formula (REF ) that, when $\\varepsilon $ tends to 0, we obtain the usual Laplace–Beltrami operator on $\\mathbb {S}^2$ , we will therefore look for eigenfunctions close to the spherical harmonics.",
"It turns out that the $Y_1^m$ are eigenfunctions of $\\Delta ^{F_{\\varepsilon }}$ for any $\\varepsilon $ , which yields : Corollary 3.2.8 The smallest non-zero eigenvalue of $-\\Delta ^{F_{\\varepsilon }}$ is $\\lambda _1 = 2 -2 \\varepsilon ^2 = \\frac{8 \\pi }{\\text{vol}_{\\Omega ^{F_{\\varepsilon }}}\\left(\\mathbb {S}^2\\right)} \\,.$ It is of multiplicity two and the eigenspace is generated by $Y_1^1$ and $Y_1^{-1}$ .",
"The fact that we have the above formula for $\\lambda _1$ is quite interesting ; first, it shows us that there does exist relationships between some geometrical data associated with a Finsler metric (here the volume) and the spectrum of the Finsler–Laplace operator.",
"Secondly, recall the following result : Theorem (Hersch [72]) For any Riemannian metric $g$ on $\\mathbb {S}^2$ , $\\lambda _1 \\le \\frac{8\\pi }{\\text{vol}_{g}\\left(\\mathbb {S}^2\\right)}\\, .$ Furthermore, the equality is realized only in the constant curvature case.",
"So the Katok–Ziller metrics on $\\mathbb {S}^2$ give us a continuous family of metrics realizing that equality !",
"We do not know however whether this is a Finslerian maximum or not.",
"Note that we also have $\\Delta ^{F_{\\varepsilon }} Y_1^0 = -2 Y_1^0$ .",
"However, the $Y_{l}^{m}$ with $l\\ge 2$ are no longer eigenfunctions of $\\Delta ^{F_{\\varepsilon }}$ .",
"This is probably related to the breaking of the symmetries that the Katok–Ziller metrics induce.",
"In the following, if $m$ happens to be greater than $l$ , we set $Y_{l}^{m}=0$ .",
"We denote by $\\langle \\cdot , \\cdot \\rangle $ the inner product on $L^2\\left(\\mathbb {S}^2\\right)$ defined by : $\\langle f,g \\rangle = \\int _0^{2 \\pi } \\int _0^{\\pi } f \\bar{g} \\sin (\\phi )d\\phi d\\theta \\, .$ Theorem 3.2.9 Let $f$ be an eigenfunction for $\\Delta ^{F_{\\varepsilon }}$ and $\\lambda $ its eigenvalue.",
"There exist unique numbers $l$ and $m$ in $\\mathbb {N}$ , $0\\le m \\le l$ , such that $f = a Y_{l}^{m}+ b Y_l^{-m} + g$ , where $g$ uniformly tends to 0 with $\\varepsilon $ , and $\\lambda = -l(l+1) + \\varepsilon ^2 \\Biggl [ \\frac{m^2}{2\\left(2l-1 \\right)} \\left( 2\\left(l+1\\right) + \\frac{3 l\\left(l-1\\right)}{ \\left(2l+3 \\right)} \\right) \\\\+\\frac{3l\\left(l-1\\right)}{2\\left(2l-1 \\right)}\\left( 1 + \\frac{l^2+l-1}{ \\left(2l+3 \\right)\\left(2l-1 \\right)} \\right) \\Biggr ] +o\\left(\\varepsilon ^2 \\right).$ Note that the Katok–Ziller transformation gets rid of most of the degeneracy of the spectrum.",
"If $\\varepsilon \\ne 0$ , the eigenvalues are at most of multiplicity two, and are of multiplicity $2l+1$ if $\\varepsilon $ is zero.",
"We can state even more on the multiplicity of eigenvalues.",
"Define $\\Psi \\colon \\mathbb {S}^2 \\rightarrow \\mathbb {S}^2 $ by $\\Psi (\\phi , \\theta ) := (\\pi - \\phi , -\\theta ) \\, .$ Theorem REF implies that $\\Delta ^{F_{\\varepsilon }}$ is stable by $\\Psi $ , i.e., for any $g$ , $\\left(\\Delta ^{F_{\\varepsilon }} g \\right) \\circ \\Psi = \\Delta ^{F_{\\varepsilon }} \\left( g \\circ \\Psi \\right)$ .",
"So, if $f$ is an eigenfunction for $\\lambda $ , then $f \\circ \\Psi $ also.",
"Therefore, either the subspace generated by $f$ is stable by $\\Psi $ or $\\lambda $ is of multiplicity at least (and hence exactly) two.",
"Remark 3.2.10 When $\\varepsilon >0$ , $F_{\\varepsilon }$ is not preserved by $\\Psi $ ."
],
[
"Proof of Theorem ",
"This proof follows the same lines as that of Theorem REF , the computations being more involved and a bit lengthy.",
"We just give the main steps.",
"Vertical derivative and change of coordinates.",
"Set $g_{\\varepsilon }\\left(\\phi ,\\theta ; \\xi _{\\phi }, \\xi _{\\theta } \\right) = \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)\\xi _{\\phi }^2 + \\sin ^2(\\phi )\\xi _{\\theta }^2$ .",
"We have $d_v F_{\\varepsilon } = \\frac{\\partial F_{\\varepsilon }}{\\partial \\xi _{\\phi }} d\\phi + \\frac{\\partial F_{\\varepsilon }}{\\partial \\xi _{\\theta }} d\\theta $ where $ \\dfrac{\\partial F_{\\varepsilon }}{\\partial \\xi _{\\phi }} = \\dfrac{\\xi _{\\phi }}{\\sqrt{g_{\\varepsilon }}}$ and $\\dfrac{\\partial F_{\\varepsilon }}{\\partial \\xi _{\\theta }} = \\dfrac{1}{1-\\varepsilon ^2\\sin ^2(\\phi )}\\left(\\dfrac{\\xi _{\\theta } \\sin ^2(\\phi )}{\\sqrt{g_{\\varepsilon }}} - \\varepsilon \\sin ^2(\\phi )\\right)$ .",
"From now on we consider the local coordinate $\\psi \\in \\left[0,2\\pi \\right]$ on $H_{(\\phi ,\\theta )}\\mathbb {S}^2$ defined by, $\\left\\lbrace \\begin{aligned}\\cos (\\psi )& = \\frac{\\xi _{\\theta } \\sin (\\phi )}{\\sqrt{g_{\\varepsilon }}} \\\\\\sin (\\psi )& = \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\frac{\\xi _{\\phi }}{\\sqrt{g_{\\varepsilon }}} \\,.\\end{aligned}\\right.$ Hilbert form and Liouville volume.",
"As in the above coordinates, we have $ \\frac{\\partial F_{\\varepsilon }}{\\partial \\xi _{\\phi }} = \\frac{\\sin (\\psi )}{\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\quad \\text{and} \\quad \\frac{\\partial F_{\\varepsilon }}{\\partial \\xi _{\\theta }} = \\frac{1}{1-\\varepsilon ^2\\sin ^2(\\phi )}\\left(\\sin (\\phi )\\cos (\\psi )- \\varepsilon \\sin ^2(\\phi )\\right),$ we deduce that the Hilbert form $A$ associated to $F_{\\varepsilon }$ is given by $A = \\frac{1}{1- \\varepsilon ^2\\sin ^2(\\phi )}\\left( f_1 d\\phi + f_2 d\\theta \\right),$ with $f_1 = \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\sin (\\psi )$ and $f_2 = \\sin (\\phi )\\cos (\\psi )- \\varepsilon \\sin ^2(\\phi )$ .",
"In order to simplify the computations, note that $f_1$ is odd in $\\psi $ , $f_2$ is even and they do not depend on $\\theta $ .",
"The exterior derivative of $A$ is given by $dA = \\frac{1}{1- \\varepsilon ^2\\sin ^2(\\phi )} \\left(\\frac{\\partial f_1 }{\\partial \\psi }d\\psi \\wedge d\\phi + \\frac{\\partial f_2 }{\\partial \\psi }d\\psi \\wedge d\\theta + f_3 d\\phi \\wedge d\\theta \\right).$ where $\\frac{\\partial f_1 }{\\partial \\psi }&= \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\cos (\\psi ), \\\\\\frac{\\partial f_2 }{\\partial \\psi }&= -\\sin (\\phi )\\sin (\\psi ), \\\\f_3 &= \\cos (\\phi )\\frac{\\cos (\\psi )-2 \\varepsilon \\sin (\\phi )+\\varepsilon ^2\\sin ^2(\\phi )\\cos (\\psi )}{1-\\varepsilon ^2\\sin ^2(\\phi )} \\, .$ Now, we have $A \\wedge dA = \\frac{1}{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^2} \\left( -f_1 \\frac{\\partial f_2 }{\\partial \\psi }+ f_2 \\frac{\\partial f_1 }{\\partial \\psi }\\right) d\\psi \\wedge d\\phi \\wedge d\\theta $ and $-f_1 \\frac{\\partial f_2 }{\\partial \\psi }+ f_2 \\frac{\\partial f_1 }{\\partial \\psi }= \\sin (\\phi )\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\left(1 - \\varepsilon \\sin (\\phi )\\cos (\\psi )\\right)$ .",
"Therefore $A \\wedge dA = \\frac{\\sin (\\phi )}{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{3/2}} \\left( 1- \\varepsilon \\sin (\\phi )\\cos (\\psi )\\right) d\\psi \\wedge d\\phi \\wedge d\\theta \\, .$ We can now use the construction of the angle form (see Section REF ).",
"Let $\\alpha ^{\\prime }$ be the 1-form associated to the volume $d\\phi \\wedge d\\theta $ on $\\mathbb {S}^2$ , on $VH\\mathbb {S}^2$ we have $\\alpha ^{\\prime } = -\\frac{1}{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^2} \\left(f_1 \\frac{\\partial f_2 }{\\partial \\psi }- f_2 \\frac{\\partial f_1 }{\\partial \\psi }\\right) d\\psi \\, .$ If we denote by $l(p)$ the length for $\\alpha ^{\\prime }$ of the fiber above a point $p\\in \\mathbb {S}^2$ we obtain $l(p) &= \\int _{H_p \\mathbb {S}^2} \\alpha ^{\\prime } \\\\&= -\\frac{1}{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^2} \\int _0^{2\\pi } \\left(f_1 \\frac{\\partial f_2 }{\\partial \\psi }- f_2 \\frac{\\partial f_1 }{\\partial \\psi }\\right) d\\psi \\\\&= \\frac{2\\pi }{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{\\frac{3}{2}}} \\sin (\\phi )\\,,$ where we used again that $f_1 \\frac{\\partial f_2 }{\\partial \\psi }- f_2 \\frac{\\partial f_1 }{\\partial \\psi }= \\sin (\\phi )\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\left(-1 + \\varepsilon \\sin (\\phi )\\cos (\\psi )\\right).$ As $ \\alpha = \\frac{2\\pi }{l(p)} \\alpha ^{\\prime }$ we obtain $\\alpha = \\left(1- \\varepsilon \\sin (\\phi )\\cos (\\psi )\\right) d\\psi .$ Geodesic flow.",
"Let $X = X_{\\psi }\\frac{\\partial }{\\partial \\psi }+ X_{\\theta }\\frac{\\partial }{\\partial \\theta }+ X_{\\phi }\\frac{\\partial }{\\partial \\phi }$ be the geodesic flow of $F_{\\varepsilon }$ .",
"As $X$ is the Reeb field of $A$ , we can use Equations (REF ) to determine $X$ .",
"We have $0 = i_X dA = \\frac{1}{1-\\varepsilon ^2\\sin ^2(\\phi )} \\Biggl ( \\left(- \\frac{\\partial f_1 }{\\partial \\psi }X_{\\phi }- \\frac{\\partial f_2 }{\\partial \\psi }X_{\\theta }\\right) d\\psi +\\left( X_{\\phi }f_3+ X_{\\psi }\\frac{\\partial f_2 }{\\partial \\psi }\\right) d\\theta \\\\+ \\left( - X_{\\theta }f_3 + X_{\\psi }\\frac{\\partial f_1 }{\\partial \\psi }\\right) d\\phi \\Biggr ),$ and $1= A(X) = \\frac{1}{1-\\varepsilon ^2\\sin ^2(\\phi )} \\left( f_1 X_{\\phi }+ f_2 X_{\\theta }\\right).$ The above equations give the system $\\left\\lbrace \\begin{aligned}\\frac{\\partial f_1 }{\\partial \\psi }X_{\\phi }+ \\frac{\\partial f_2 }{\\partial \\psi }X_{\\theta }&= 0 \\\\X_{\\phi }f_3+ X_{\\psi }\\frac{\\partial f_2 }{\\partial \\psi }&= 0 \\\\- X_{\\theta }f_3 + X_{\\psi }\\frac{\\partial f_1 }{\\partial \\psi }&= 0 \\\\f_1 X_{\\phi }+ f_2 X_{\\theta }&= 1-\\varepsilon ^2\\sin ^2(\\phi )\\end{aligned}\\right.$ which yields $X_{\\theta }&= \\frac{1-\\varepsilon ^2\\sin ^2(\\phi )}{\\sin (\\phi )} \\; \\frac{\\cos (\\psi )}{1-\\varepsilon \\sin (\\phi )\\cos (\\psi )} ,\\\\X_{\\phi }&= \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\; \\frac{\\sin (\\psi )}{1-\\varepsilon \\sin (\\phi )\\cos (\\psi )} ,\\\\X_{\\psi }&= \\frac{1}{\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\; \\frac{\\cos (\\phi )}{\\sin (\\phi )} \\; \\frac{\\cos (\\psi )-2\\varepsilon \\sin (\\phi )+ \\varepsilon ^2\\sin ^2(\\phi )\\cos (\\psi )}{1-\\varepsilon \\sin (\\phi )\\cos (\\psi )}.$ The Finsler–Laplace operator.",
"Let $f \\colon \\mathbb {S}^2 \\rightarrow \\mathbb {R}$ .",
"We start by computing $L_X^2 \\pi ^{\\ast }f$ .",
"As $\\frac{\\partial }{\\partial \\psi }\\left(\\pi ^{\\ast }f\\right) = 0 $ and $X$ does not depend on $\\theta $ , we get $L_X^2 \\pi ^{\\ast }f = X_{\\theta }^2 \\frac{\\partial ^2 f}{\\partial \\theta ^2} + X_{\\theta }X_{\\phi }\\frac{\\partial ^2 f}{\\partial \\phi \\partial \\theta } + X_{\\phi }X_{\\theta }\\frac{\\partial ^2 f}{\\partial \\theta \\partial \\phi } + X_{\\phi }\\frac{\\partial X_{\\theta }}{\\partial \\phi } \\frac{\\partial f}{\\partial \\theta } \\\\+ X_{\\phi }\\frac{\\partial X_{\\phi }}{\\partial \\phi } \\frac{\\partial f}{\\partial \\phi } + X_{\\phi }^2 \\frac{\\partial ^2 f}{\\partial \\phi ^2} + X_{\\psi }\\frac{\\partial X_{\\theta }}{\\partial \\psi } \\frac{\\partial f}{\\partial \\theta } + X_{\\psi }\\frac{\\partial X_{\\phi }}{\\partial \\psi } \\frac{\\partial f}{\\partial \\phi } \\, .$ Since we are only interested in $\\int _{H_x\\mathbb {S}^2} L_X^2 \\pi ^{\\ast }f \\alpha $ , we can use the parity properties (with respect to $\\psi $ ) of the functions $X_{\\theta }, \\; X_{\\phi }$ and $X_{\\psi }$ (which are respectively even, odd and even) to get rid of half of the above terms.",
"We obtain $\\pi \\Delta ^{F_{\\varepsilon }} f(p) = \\int _{H_p\\mathbb {S}^2} \\hspace{-4.2679pt}X_{\\theta }^2 \\; \\alpha \\; \\frac{\\partial ^2 f}{\\partial \\theta ^2} + \\int _{H_p\\mathbb {S}^2} \\hspace{-4.2679pt} X_{\\phi }^2 \\; \\alpha \\; \\frac{\\partial ^2 f}{\\partial \\phi ^2} \\\\+ \\int _{H_p\\mathbb {S}^2} \\left( X_{\\psi }\\frac{\\partial X_{\\phi }}{\\partial \\psi } + X_{\\phi }\\frac{\\partial X_{\\phi }}{\\partial \\phi } \\right) \\alpha \\; \\frac{\\partial f}{\\partial \\phi } \\, .$ Direct computation (with a little help from Maple) yields $\\Delta ^{F_{\\varepsilon }} = \\frac{2 \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{\\frac{3}{2}}}{\\sin ^2(\\phi )\\left(1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\right)} \\; \\frac{\\partial ^2 }{\\partial \\theta ^2} + 2 \\frac{1-\\varepsilon ^2\\sin ^2(\\phi )}{1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\; \\frac{\\partial ^2 }{\\partial \\phi ^2} \\\\+ \\frac{2 \\cos (\\phi )}{\\sin (\\phi )} \\left( 2 - \\frac{1}{1+ \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} -\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\right) \\frac{\\partial }{\\partial \\phi } \\, .$ This concludes the proof of Theorem REF ."
],
[
"Proof of Theorem ",
"We state the following property of spherical harmonics that will be useful in later computations : Proposition 3.2.11 Let $l\\in \\mathbb {N}$ , and $m\\in \\mathbb {Z}$ , such that $|m|\\le l$ , then the associated Legendre polynomial $P_l^m\\left(\\cos (\\phi )\\right)$ , denoted here by $\\tilde{P}_{l}^{m}$ , is a solution to the equation $\\frac{\\partial ^2 \\tilde{P}_{l}^{m}}{\\partial \\phi ^2} + \\frac{\\cos (\\phi )}{\\sin (\\phi )}\\frac{\\partial \\tilde{P}_{l}^{m}}{\\partial \\phi } +\\left(l(l+1) - \\frac{m^2}{\\sin ^2(\\phi )} \\right) \\tilde{P}_{l}^{m}= 0 ,$ They verify (see [1]) $\\left(2l-1\\right) \\cos (\\phi )\\tilde{P}^{m}_{l-1} &= (l-m)\\tilde{P}_{l}^{m}+ \\left(l+m-1\\right) \\tilde{P}^{m}_{l-2} \\, , \\\\\\sin (\\phi )\\frac{\\partial \\tilde{P}_{l}^{m}}{\\partial \\phi } & = l \\cos (\\phi )\\tilde{P}_{l}^{m}- (l+m) \\tilde{P}^{m}_{l-1} \\, , \\\\\\sin (\\phi )\\tilde{P}_{l}^{m}&= \\frac{1}{2l+1}\\left( \\tilde{P}_{l-1}^{m+1} - \\tilde{P}_{l+1}^{m+1}\\right) .$ The spherical harmonics form an orthogonal Hilbert basis of $L^2\\left(\\mathbb {S}^2\\right)$ and their norm is given by $||Y_{l}^{m}|| = \\sqrt{\\frac{4\\pi }{2l+1} \\frac{\\left(l+m\\right)!}{\\left(l-m\\right)!}}",
"\\, .$ We can now proceed with the proof.",
"Take $f$ an eigenfunction of $\\Delta ^{F_{\\varepsilon }}$ and $\\lambda $ the associated eigenvalue.",
"As the $Y_{l}^{m}$ form an Hilbert basis of $L^2\\left(\\mathbb {S}^2\\right)$ , there exist $a_l^m$ such that $f = \\sum _{l=0}^{+\\infty } \\sum _{|m|\\le l} a_l^m Y_{l}^{m},$ where the convergence is a priori in the $L^2$ -norm.",
"The elliptic regularity theorem implies that $f \\in C^{\\infty }\\left(\\mathbb {S}^2 \\right)$ .",
"Therefore the convergence above is uniform.",
"So $\\Delta ^{F_{\\varepsilon }} f = \\sum _{l=0}^{+\\infty } \\sum _{|m|\\le l} a_l^m \\Delta ^{F_{\\varepsilon }} Y_{l}^{m}$ .",
"Let $l,m$ be fixed.",
"The equation $\\langle \\Delta ^{F_{\\varepsilon }} f, Y_{l}^{m}\\rangle = \\lambda \\langle f, Y_{l}^{m}\\rangle $ yields $\\lambda a_l^m \\Vert Y_{l}^{m}\\Vert ^2 = \\sum _{k=0}^{+\\infty } \\sum _{|n|\\le k} a_k^n \\langle Y_{l}^{m}, \\Delta ^{F_{\\varepsilon }} Y_k^n \\rangle .$ Claim 3.2.12 For any $l,m$ we have $\\Delta ^{F_{\\varepsilon }} Y_{l}^{m}= -l(l+1) Y_{l}^{m}\\\\+ \\frac{\\varepsilon ^2}{\\left(1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right)^2} \\Biggl [ \\left(1+2\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right)l\\left( l-1\\right) \\sin ^2(\\phi )Y_{l}^{m}\\\\+ \\left(2 m^2 \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)\\right)Y_{l}^{m}+ 2 \\frac{l^2 + m^2 +l}{2l+1} \\left(1+2\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right) Y_{l}^{m}\\\\- 2(l+m)(l+m-1)\\left(1+2\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right) Y_{l-2}^m \\Biggr ].$ By the formula (REF ) for $\\Delta ^{F_{\\varepsilon }}$ , we have $\\Delta ^{F_{\\varepsilon }} Y_{l}^{m}&= \\frac{2}{1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\left[ \\frac{1}{\\sin ^2(\\phi )} \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{\\frac{3}{2}} (-m^2) Y_{l}^{m}\\right.",
"\\\\& \\left.",
"+ e^{im\\theta } \\Biggl ( \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right) \\frac{\\partial ^2 }{\\partial \\phi ^2}\\tilde{P}_{l}^{m}\\right.",
"\\\\& \\left.",
"+\\frac{\\cos (\\phi )}{\\sin (\\phi )} \\left(\\varepsilon ^2\\sin ^2(\\phi )+ \\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )} \\right)\\frac{\\partial }{\\partial \\phi }\\tilde{P}_{l}^{m}\\Biggr ) \\right],\\\\$ Applying first Equation (REF ) and then Equation (), we get $\\Delta ^{F_{\\varepsilon }} Y_{l}^{m}&= \\frac{2}{1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\left[ -m^2 \\frac{1}{\\sin ^2(\\phi )} \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{\\frac{3}{2}} Y_{l}^{m}\\right.",
"\\\\& \\left.",
"+ e^{im\\theta } \\Biggl ( \\frac{\\cos (\\phi )}{\\sin (\\phi )} \\left(-1+2\\varepsilon ^2\\sin ^2(\\phi )+ \\sqrt{1- \\varepsilon ^2\\sin ^2(\\phi )} \\right) \\frac{\\partial }{\\partial \\phi }\\tilde{P}_{l}^{m}\\right.\\\\& \\left.",
"-\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)\\left( l(l+1) - \\frac{m^2}{\\sin ^2(\\phi )} \\right) \\tilde{P}_{l}^{m}\\Biggr ) \\right] \\\\&= \\frac{2}{1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}} \\Biggl [ \\left( 1- \\varepsilon ^2\\sin ^2(\\phi )\\right)\\left(-l(l+1) + \\frac{m^2}{ \\sin ^2(\\phi )} \\left(1-\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right) \\right)Y_{l}^{m}\\\\& + \\left(-1+2\\varepsilon ^2\\sin ^2(\\phi )+ \\sqrt{1- \\varepsilon ^2\\sin ^2(\\phi )} \\right) l \\frac{\\cos ^2(\\phi )}{\\sin ^2(\\phi )} Y_{l}^{m}\\\\& -\\frac{l+m}{\\sin ^2(\\phi )} \\left( \\frac{l-m}{2l-1} Y_{l}^{m}+ \\frac{l+m-1}{2l-1} Y_{l-2}^m\\right) \\Biggr ].$ After a bit of rearranging, we get $\\Delta ^{F_{\\varepsilon }} Y_{l}^{m}&= -l(l+1) Y_{l}^{m}+ \\varepsilon ^2 \\Biggl [ \\frac{1+2\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}}{\\left(1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right)^2} l\\left( l-1\\right) \\sin ^2(\\phi )Y_{l}^{m}\\\\& + \\frac{2 m^2 \\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)}{\\left(1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right)^2} Y_{l}^{m}+ 2 \\frac{l^2 + m^2 +l}{2l+1} \\frac{1+2\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}}{\\left(1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right)^2} Y_{l}^{m}\\\\& -2(l+m)(l+m-1)\\frac{1+2\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}}{\\left(1+\\sqrt{1-\\varepsilon ^2\\sin ^2(\\phi )}\\right)^2} Y_{l-2}^m \\Biggr ],$ which gives our claim after some more simplifications.",
"Using the claim, Equation (REF ) becomes $\\lambda a_l^m \\Vert Y_{l}^{m}\\Vert ^2 = \\sum _{k=0}^{+\\infty } a_k^m \\langle Y_{l}^{m}, \\Delta ^{F_{\\varepsilon }} Y_k^m \\rangle .$ Now, we can use an expansion of $\\Delta ^{F_{\\varepsilon }} Y_k^m $ in powers of $\\varepsilon $ .",
"Claim 3.2.13 For any $l,m$ , we have $\\Delta ^{F_{\\varepsilon }} Y_{l}^{m}= -l(l+1) Y_{l}^{m}+ \\varepsilon ^2 \\Biggl [\\frac{3 l (l+1)}{4} \\sin ^2(\\phi )Y_{l}^{m}\\\\+ \\left( \\frac{m^2}{2} + \\frac{3\\left(l(l+1) +m^2\\right)}{ 2l+1} \\right) Y_{l}^{m}+ \\frac{3}{2} (l+m)(l+m-1) Y_{l-2}^m \\Biggr ] + O\\left(\\varepsilon ^4\\right).$ The claim follows once again from a straightforward computation.",
"Using this second claim and the orthogonality of the spherical harmonics, Equation (REF ) now reads $\\lambda a_l^m \\Vert Y_{l}^{m}\\Vert ^2 = -l(l+1) a_l^m \\Vert Y_{l}^{m}\\Vert ^2 + a_l^m \\varepsilon ^2 \\Biggl [\\frac{3 l (l+1)}{4} \\langle \\sin ^2(\\phi )Y_{l}^{m}, Y_{l}^{m}\\rangle \\\\+ \\left( \\frac{m^2}{2} + \\frac{3\\left(l(l+1) +m^2\\right)}{ 2l+1} \\right) \\Vert Y_{l}^{m}\\Vert ^2 \\Biggr ] \\\\+ \\sum _{k \\ne l} a_k^m \\varepsilon ^2 \\Biggl [\\frac{3 k (k+1)}{4} \\langle \\sin ^2(\\phi )Y_k^m, Y_{l}^{m}\\rangle + \\frac{3}{2} (k+m)(k+m-1) \\langle Y_{k-2}^m, Y_{l}^{m}\\rangle \\Biggr ] + O\\left(\\varepsilon ^4\\right).$ Claim 3.2.14 There is at most one $l$ such that $\\frac{1}{a_l^m}$ is bounded independently of $\\varepsilon $ .",
"Equation (REF ) shows that, if $\\frac{1}{a_l^m}$ is bounded as $\\varepsilon $ tends to 0, then $\\lambda $ tends to $-l(l+1)$ .",
"Therefore we can have only one such $l$ .",
"Let $l$ be given by the previous claim, (REF ) reduces to $\\lambda = -l(l+1) + \\frac{\\varepsilon ^2}{\\Vert Y_{l}^{m}\\Vert ^2 } \\Biggl [\\frac{3 l (l+1)}{4} \\langle \\sin ^2(\\phi )Y_{l}^{m}, Y_{l}^{m}\\rangle \\\\+ \\left( \\frac{m^2}{2} + \\frac{3\\left(l(l+1) +m^2\\right)}{ 2l+1} \\right) \\Vert Y_{l}^{m}\\Vert ^2 \\Biggr ] + o\\left(\\varepsilon ^2\\right).$ Some more computations (using Equations (), (REF ) and the orthogonality of the spherical harmonics) give $\\frac{\\langle \\sin ^2(\\phi )Y_{l}^{m}, Y_{l}^{m}\\rangle }{\\Vert Y_{l}^{m}\\Vert ^2 } = 2\\frac{l^2+l-1+ m^2}{ \\left(2l+3 \\right)\\left(2l-1 \\right)} \\, ,$ so that $ \\lambda = -l(l+1) + \\varepsilon ^2 \\Biggl [\\frac{3 l (l+1)}{2} \\frac{l^2+l-1+ m^2}{ \\left(2l+3 \\right)\\left(2l-1 \\right)} \\\\+ \\left( \\frac{m^2}{2} + \\frac{3\\left(l(l+1) +m^2\\right)}{ 2l+1} \\right) \\Biggr ] + o\\left(\\varepsilon ^2\\right).$ From this equation, we deduce Claim 3.2.15 There can only be one $m$ such that $\\frac{1}{a_l^m}$ or $\\frac{1}{a_l^{-m}}$ is bounded independently of $\\varepsilon $ .",
"Otherwise, we would find two different coefficients in $\\varepsilon ^2$ for $\\lambda $ .",
"We sum up what we proved, namely that there exist unique $l,m\\in \\mathbb {N}$ , $a,b \\in and $ g S2 such that $f = a Y_{l}^{m}+ b Y_l^{-m} + g \\,.$ Furthermore, for any $p\\in \\mathbb {S}^2$ , $|g(p)|$ tends to 0 with $\\varepsilon $ and the associated eigenvalue verifies Equation (REF ).",
"That is, we proved Theorem REF ."
],
[
"First eigenvalue and volume",
"We finish by proving Corollary REF .",
"Recall : Corollary REF The smallest non-zero eigenvalue of $-\\Delta ^{F_{\\varepsilon }}$ is $\\lambda _1 = 2 - 2 \\varepsilon ^2 = \\frac{8\\pi }{\\text{vol}_{\\Omega }\\left(\\mathbb {S}^2\\right) }.$ It is of multiplicity two and the eigenspace is generated by $Y_1^1$ and $Y_1^{-1}$ .",
"Computation using either (REF ) or directly Theorem REF gives $\\Delta ^{F_{\\varepsilon }} Y_1^1 = (-2 +2 \\varepsilon ^2) Y_1^1$ and $\\Delta ^{F_{\\varepsilon }} Y_1^{-1} = (-2 +2 \\varepsilon ^2) Y_1^{-1}$ .",
"It also yields $\\Delta ^{F_{\\varepsilon }} Y_1^0 = -2 Y_1^0 $ , now Theorem REF shows that the eigenfunctions for the first (non-zero) eigenvalue must live in the vicinity of the space generated by $Y_1^1, \\; Y_1^0$ and $Y_1^{-1}$ , therefore $\\lambda _1 = 2 - 2 \\varepsilon ^2$ .",
"Now using Equation (REF ) we get that the Finsler volume form for $(\\mathbb {S}^2,F_{\\varepsilon })$ is $\\Omega ^{F_{\\varepsilon }} = \\frac{\\sin (\\phi )}{\\left(1-\\varepsilon ^2\\sin ^2(\\phi )\\right)^{3/2}} d\\theta \\wedge d\\phi \\, .$ So $\\text{vol}_{\\Omega } \\left(\\mathbb {S}^2\\right) = \\frac{4\\pi }{1-\\varepsilon ^2} \\, ,$ and hence, $\\lambda _1 = \\frac{8\\pi }{\\text{vol}_{\\Omega }\\left(\\mathbb {S}^2\\right) } \\, .$"
],
[
"Spectrum and geometry at infinity",
"Our focus here will be the study of the links between the Finsler–Laplace operator and the dynamics or geometry for Finsler metrics with negative curvature (in the sense of equation (REF )).",
"In Riemannian manifolds of negative curvature, there are (at least) three natural classes of measures on the boundary at infinity : the Liouville (or visual) measure class, which is obtained by pushing the Lebesgue measure on unit spheres to the boundary via the geodesic flow ; the Patterson-Sullivan measure class, which can be obtained from the Bowen-Margulis measure via the Kaimanovich correspondence (see [75]) ; and the Harmonic measure class which is linked to the Laplace–Beltrami operator in a way that we will explicit later.",
"In the case of surfaces we have a famous rigidity phenomenon : when two of those classes are equivalent, it forces the Riemannian metric to be of constant curvature (this is due to Katok [79], [78] and Ledrappier [84]).",
"In higher dimensions, Ledrappier [84] showed that equality between the Harmonic and Patterson-Sullivan classes is equivalent to $\\lambda _1 = \\frac{h^2}{4}$ , where $\\lambda _1$ is the bottom of the spectrum of $\\Delta $ and $h$ is the topological entropy.",
"In [24], G. Besson, G. Courtois and S. Gallot proved that $\\lambda _1 = \\frac{h^2}{4}$ implies that the manifold is a symmetric space.",
"When we started studying the Finsler–Laplace operator in negative curvature, our goal was to generalize some of (or get counter-examples to) the above results.",
"Unfortunately, this is still out of reach.",
"The first difficulty we stumbled upon was the existence of harmonic measures associated with our Finsler–Laplace operator.",
"Many papers prove their existence in the Riemannian case, or even for weighted Laplace operators (see the remark after Lemma REF ) when the symbol is of negative curvature, but none, to my knowledge, was made for our more general case.",
"However, Ancona gives in [7] a very general theorem that implies existence of harmonic measures.",
"Sections REF and REF are devoted to stating Ancona's theorem and proof that it applies to our case.",
"But beforehand, we start by recalling some geometrical and dynamical properties of negatively curved Finsler manifolds and use them to give an upper bound for the first eigenvalue of the Finsler–Laplace operator in terms of topological entropy.",
"If not stated otherwise, in this chapter, $M$ is a closed manifold of dimension $n$ endowed with a Finsler metric of negative curvature $F$ and $\\widetilde{M}$ is a fixed universal cover of $M$ endowed with the lifted Finsler metric $\\widetilde{F}$ ."
],
[
"Negatively curved Finsler manifolds",
"Manifolds of negative Finsler curvature enjoy many of the same dynamical and geometrical features of Riemannian ones.",
"We will recall here two of those."
],
[
"Gromov-hyperbolicity",
"Egloff, in his Ph.D Thesis [48] (the reader can also refer to [50] as it is available on-line when the dissertation is not), studied the Finsler equivalent of Cartan Hadamard manifolds that he called uniform Finsler Hadamard manifolds.",
"Note that in Egloff's definition, uniform refers to a control of the quadratic forms $\\left( \\frac{\\partial ^2 F^2}{\\partial v_i v_j} \\right)$ , not to a control of the curvature.",
"We do not enter into more details as, for us, uniform Finsler Hadamard manifolds will just be the universal cover of a closed manifold of non-positive curvature.",
"Such manifolds are in particular homeomorphic to $\\mathbb {R}^n$ and Egloff studied the property of the Finsler distance and the existence of a visual boundary.",
"He proved : Theorem 4.1.1 (Egloff [50]) Let $\\widetilde{M}$ be a uniform Finsler Hadamard manifold of strictly negative curvature.",
"Then $\\widetilde{M}$ is Gromov-hyperbolic.",
"Remark 4.1.2 Note that Egloff only studied reversible metrics, as is normally the case in metric geometry.",
"However Fang and Foulon [51] proved that the same theorem holds for non-reversible metrics (with an appropriate definition of Gromov-hyperbolicity).",
"We very briefly recall some facts about Gromov-hyperbolic spaces.",
"Proofs, better explanations and much more can be found in [44] or [68].",
"Let $(V,d)$ be a complete, locally compact, geodesic (i.e., there exists at least one distance-minimizing curve between two points), simply connected metric space.",
"Let $x,a, b\\in V$ .",
"Then the Gromov product at $x$ of $a$ and $b$ is defined as $\\lbrace a, b\\rbrace _x = \\frac{1}{2}\\left( d(a,x) +d(x,b) -d(a,b) \\right),$ The metric space $V$ is called Gromov-hyperbolic if there exists $\\delta >0$ such that, for any $x, a, b, c\\in V$ , $\\min \\left( \\lbrace a, b\\rbrace _x , \\lbrace a, c\\rbrace _x \\right) \\le \\delta \\, .$ If we want to make explicit the constant $\\delta $ , we say that $V$ is $\\delta $ -hyperbolic.",
"The Gromov product is very useful, but unfortunately it is hard (at least for me) to get an insight of what being Gromov-hyperbolic represents using the above definition.",
"An equivalent definition uses geodesic triangles : The space $V$ is Gromov-hyperbolic if there exists $\\delta >0$ such that for any geodesic triangle $(a,b,c) \\subset V$ , any side is contained in the $\\delta $ -neighborhood of the union of the remaining sides.",
"A Gromov-hyperbolic space admits a boundary at infinity $V(\\infty )$ .",
"One way to define it is to take equivalence classes of geodesic rays ; if $O\\in V$ is a base point, two geodesic rays $\\gamma _1, \\gamma _2 \\colon \\mathbb {R}^+ \\rightarrow V$ issuing from $O$ are equivalent if $d(\\gamma _1(t),\\gamma _2(t) )$ stays bounded for any $t\\in \\mathbb {R}^+$ .",
"Consider the elements in $V$ as endpoints of geodesic rays starting at $O$ and endow the set of all rays with the uniform convergence on compact topology.",
"Then $\\overline{V}:= V \\cup V(\\infty )$ with the quotient topology is compact, $V$ is a dense open set in $\\overline{V}$ and $\\partial \\overline{V} = V(\\infty )$ .",
"This boundary is traditionally called the visual boundary of $V$ and is independent of the base point $O$ .",
"Using only the Gromov product, we can also define a boundary, which turns out to be homeomorphic to the visual boundary.",
"The advantage of this presentation is that it comes naturally equipped with a metric.",
"Fix a base point $O\\in V$ .",
"A sequence $(x_n)$ in $V$ is a Gromov-sequence if $\\lbrace x_i, x_j \\rbrace _O \\rightarrow +\\infty $ when $i,j \\rightarrow +\\infty $ .",
"Two Gromov-sequences $(x_n)$ and $(y_n)$ are equivalent if $\\lbrace x_i, y_i \\rbrace _O \\rightarrow +\\infty $ when $i \\rightarrow +\\infty $ .",
"Then the set of equivalence classes of Gromov-sequences is the Gromov boundary of $V$ .",
"For $\\xi , \\eta \\in V(\\infty )$ , we define the Gromov product of $\\xi $ and $\\eta $ by $\\lbrace \\xi , \\eta \\rbrace _O := \\inf \\liminf _{n \\rightarrow +\\infty } \\lbrace a_n , b_n \\rbrace _O \\, ,$ where the infimum is taken over all sequences $a_n$ converging to $\\xi $ and $b_n$ converging to $\\eta $ .",
"We can now describe the metric on the boundary : let $\\epsilon >0$ and set, for any $\\xi , \\eta \\in V(\\infty )$ , $\\rho _{\\epsilon }(\\xi , \\eta ):=e^{-\\epsilon \\lbrace \\xi , \\eta \\rbrace _O}.$ Unfortunately, $\\rho _{\\epsilon }$ does not yet verify the triangle inequality, but can be slightly altered in order to do so.",
"A chain between $\\xi , \\eta \\in V(\\infty )$ is a finite sequence $\\xi = \\xi _0, \\xi _1,\\dots , \\xi _n = \\eta $ in $V(\\infty )$ and we write $\\mathcal {C}_{\\xi ,\\eta }$ for the set of chains between $\\xi $ and $\\eta $ .",
"Let $c= (\\xi _0, \\xi _1, \\dots , \\xi _n) \\in \\mathcal {C}_{\\xi ,\\eta }$ , define $\\rho _{\\epsilon }(c) &:= \\sum _{i=0}^{n-1} \\rho _{\\epsilon }(\\xi _i, \\xi _{i+1}), \\\\d_{G,\\epsilon }(\\xi ,\\eta ) &:= \\inf \\lbrace \\rho _{\\epsilon }(c) \\mid c \\in \\mathcal {C}_{\\xi ,\\eta } \\rbrace .$ Proposition 4.1.3 If $\\epsilon >0$ is chosen such that $ e^{\\epsilon \\delta } < \\sqrt{2}$ , then $d_{G,\\epsilon }$ is a distance on $V(\\infty )$ , compatible with the above topology.",
"Furthermore, $(3 - 2 e^{\\epsilon \\delta }) \\rho _{\\epsilon }(\\xi ,\\eta ) \\le d_{G,\\epsilon }(\\xi ,\\eta ) \\le \\rho _{\\epsilon }(\\xi ,\\eta ).$ We call $d_{G,\\epsilon }$ a Gromov metric on $V(\\infty )$ .",
"The proof of the above proposition is given in Chapter 7 of [68].",
"Remark that the boundary of a Gromov-hyperbolic space admits a Hölder structure (see [44])."
],
[
"Geodesic flow and entropy",
"As is the case in Riemannian geometry, negatively curved Finsler metrics gives hyperbolic dynamics : Theorem 4.1.4 (Foulon [60]) The geodesic flow of a negatively curved Finsler manifold is a (contact) Anosov flow.",
"We give the precise definition of an Anosov flow and some of its basic properties in the second part of this dissertation (see Section REF ).",
"For the moment, we just recall that it is uniformly hyperbolic, i.e., in the tangent space there exist one direction of (uniform) exponential expansion and one direction of (uniform) exponential shrinking.",
"Let $h$ denote the topological entropy associated with the geodesic flow of $F$ (see, for instance [80] for equivalent definitions).",
"Manning [85] proved that the topological entropy for the geodesic flow of a Riemannian metric of non-positive curvature is the same as the volume entropy, i.e., the exponential growth of the volume of balls.",
"It turns out that this is still true for Finsler metrics Theorem 4.1.5 (Egloff [49]) Let $(M,F)$ be a compact Finsler manifold and $h$ the topological entropy of the geodesic flow.",
"Let $x\\in \\widetilde{M}$ and $B(R)$ be the ball of radius $R$ centered at $x$ , and set $h_{\\text{vol}} := \\lim _{R \\rightarrow +\\infty }\\frac{1}{R}\\log \\left( \\int _{B(R)} \\Omega ^{\\widetilde{F}} \\right).$ Then $h \\ge h_{\\text{vol}}$ .",
"Furthermore, if $F$ is of non-positive curvature, then $h = h_{\\text{vol}} \\, .$ Note that Egloff proved the above result with another volume form, however as $M$ is compact, there exists a constant controlling the ratio of two different volume forms, and this constant disappears when we consider exponential growth.",
"Note that Egloff [49] also showed, using the Anosov property, that the visual boundary of $\\widetilde{M}$ admits a Hölder structure with a constant depending on the Lyapunov exponents."
],
[
"Bounds for the first eigenvalue",
"We denote by $\\lambda _1$ the infimum of the essential spectrum of $-\\Delta ^{\\widetilde{F}}$ .",
"Recall that it is given by the infimum of the Rayleigh quotients (see Proposition REF )."
],
[
"A dynamical upper bound",
"As in the Riemannian case, we have an upper bound for $\\lambda _1$ depending only on the dimension of the manifold and the topological entropy of the geodesic flow.",
"Proposition 4.2.1 If $M$ is of dimension $n$ and $h$ is the topological entropy, then $\\lambda _1 \\le n \\frac{h^2}{4} \\,.$ Remark 4.2.2 This bound is far less sharp than in the Riemannian case, where we have $\\lambda _1^{\\textrm {Riem}} \\le h^2/4$ .",
"The additional $n$ appears in the proof because we don't know how to control locally the Finsler metric.",
"It would be interesting to decide whether we could improve this bound to the Riemannian one or if there exist Finsler metrics with $h^2/4 < \\lambda _1 \\le nh^2/4$ .",
"The proof follows the Riemannian one and is based on the following : Claim 4.2.3 Let $x_0 \\in \\widetilde{M}$ and $\\rho (x):= d(x_0,x)$ , then $\\exp (-s \\rho (x))$ is in $L^2(\\widetilde{M})$ for any $s> \\frac{h}{2}$ .",
"By Theorem REF , we have $h = \\lim _{R \\rightarrow +\\infty }\\frac{1}{R}\\log \\left( \\int _{B(R)} \\Omega ^{\\widetilde{F}} \\right).$ Therefore, if $s > \\frac{h}{2}$ , $exp(-s \\rho (x))$ is in $L^2(\\widetilde{M})$ .",
"[Proof of Proposition REF ] Our goal is to find an upper bound for the Rayleigh quotient of $e^{-s\\rho (x)}$ .",
"We have $L_X \\pi ^{\\ast } \\exp (-s \\rho ) (x,\\xi ) = -s \\left(L_X\\pi ^{\\ast } \\rho \\right)(x,\\xi ) \\exp (-s \\rho (x)).$ So, using Fubini theorem, $\\int _{H\\widetilde{M}} \\left(L_X \\pi ^{\\ast } \\exp (-s \\rho ) \\right)^2 A \\wedge dA^{n-1}= \\int _{x\\in \\widetilde{M}} s^2 \\left(\\int _{\\xi \\in H_x\\widetilde{M}} \\left(L_X\\pi ^{\\ast } \\rho (x,\\xi )\\right)^2 \\alpha \\right) \\exp (- 2 s \\rho (x)) \\Omega .$ To deduce the proposition, we just have to bound $\\int _{\\xi \\in H_x\\widetilde{M}} \\left(L_X\\pi ^{\\ast } \\rho (x,\\xi )\\right)^2 \\alpha $ because $\\lambda _1$ is the infimum of the Rayleigh quotient.",
"Since $\\left|L_X\\pi ^{\\ast } \\rho (x,\\xi ) \\right|\\le 1$ , we have $\\lambda _1 &\\le \\frac{n}{\\operatorname{\\mathrm {vol}_{\\mathrm {Eucl}}}\\left(\\mathbb {S}^{n-1}\\right)} \\frac{ s^2 \\int _{\\widetilde{M}} \\exp (- 2 s \\rho ) \\int _{H_{x}\\widetilde{M}} \\alpha \\, \\Omega }{\\Vert \\exp (-s \\rho ) \\Vert ^2} \\\\&\\le n s^2 \\, .$"
],
[
"A topological lower bound",
"In this section we do not need the negative curvature assumption.",
"Bounds on $\\lambda _1$ can be obtained through bounds for the Laplace–Beltrami operator associated with the symbol of $\\Delta ^{\\widetilde{F}}$ : Proposition 4.2.4 Let $g_{\\sigma }$ be the Riemannian metric on $\\widetilde{M}$ given by the dual of the symbol of $\\Delta ^{\\widetilde{F}}$ , $\\Delta ^{\\sigma }$ its associated Laplace–Beltrami operator and $\\lambda _1(\\sigma )$ the infimum of the spectrum of $-\\Delta ^{\\sigma }$ .",
"Then, if we let $a\\in C^{\\infty }(\\widetilde{M})$ such that $\\Omega ^{\\widetilde{F}}= a \\Omega ^{g_{\\sigma }}$ , we have $\\lambda _1(\\sigma ) \\frac{\\sup _{x\\in \\widetilde{M}} a(x)}{\\inf _{x\\in \\widetilde{M}} a(x)} \\ge \\lambda _1 \\ge \\lambda _1(\\sigma ) \\frac{\\inf _{x\\in \\widetilde{M}} a(x)}{\\sup _{x\\in \\widetilde{M}} a(x)} \\, .$ Remark 4.2.5 As $a$ is a $\\pi _1(M)$ -invariant function and $M$ is compact, $a$ is bounded, so the above makes sense.",
"Recall once again that $\\lambda _1 = \\inf \\frac{\\int _{H\\widetilde{M}} \\left( L_X \\pi ^{\\ast } f \\right)^2 A \\wedge dA^{n-1}}{\\int _{\\widetilde{M}} f^2 \\Omega ^F} \\,,$ where the infimum is taken over all functions in $H^1(\\widetilde{M})$ .",
"We will prove the lower bound.",
"The proof for the upper bound follows along the same lines.",
"$\\int _{H\\widetilde{M}} \\left( L_X \\pi ^{\\ast } f \\right)^2 A \\wedge dA^{n-1}&= \\int _{x\\in \\widetilde{M}} \\left(\\int _{H_x \\widetilde{M}} \\left( L_X \\pi ^{\\ast } f \\right)^2 \\alpha ^F\\right) \\Omega ^F \\\\&= \\int _{\\widetilde{M}} \\Vert \\nabla f \\Vert _{\\sigma }^2 \\Omega ^F \\\\&\\ge \\int _{\\widetilde{M}} \\Vert \\nabla f \\Vert _{\\sigma }^2 \\Omega ^{g_{\\sigma }} \\inf _{x\\in \\widetilde{M}} a(x) \\, .$ Therefore, for any $f\\in H^1(\\widetilde{M})$ , $\\frac{\\int _{H\\widetilde{M}} \\left( L_X \\pi ^{\\ast } f \\right)^2 A \\wedge dA^{n-1}}{\\int _{\\widetilde{M}} f^2 \\Omega ^F} \\ge \\frac{\\int _{\\widetilde{M}} \\Vert \\nabla f \\Vert _{\\sigma }^2 \\Omega ^{g_{\\sigma }} }{\\int _{\\widetilde{M}} f^2 \\Omega ^{g_{\\sigma }} } \\frac{\\inf _{x\\in \\widetilde{M}} a(x)}{\\sup _{x\\in \\widetilde{M}} a(x)} \\, .$ Hence the result.",
"We do not know much about lower bounds for $\\lambda _1(\\sigma )$ because we do not know anything on the metric $g_{\\sigma }$ , but Brooks [26] gave a purely topological condition for it to be strictly positive.",
"Using the above proposition leads to a generalization of Brooks theorem to the Finsler–Laplace operator : Theorem 4.2.6 One has $\\lambda _1 = 0 $ if and only if $\\pi _1(M)$ is amenable.",
"This is a direct application of the proposition together with the main result of [26]."
],
[
"Harmonic measures and the Martin Boundary",
"We are going to leave Finsler geometry for a bit, to give some basics about potential theory.",
"As an introduction, we recall how harmonic measures are obtained in Riemannian geometry.",
"Given a Riemannian metric $g$ on $\\widetilde{M}$ , one way to construct the harmonic measure associated with a point $x\\in \\widetilde{M}$ is by defining the measure of a Borel set $U \\subset \\widetilde{M}(\\infty )$ as the probability for a Brownian motion for $g$ leaving $x$ to end in $U$ .",
"In Riemannian geometry, Brownian motion can be thought of in two ways : one is as the limit of a random walk ; the other as the diffusion associated with the Laplacian.",
"We don't know how to generalize the first approach to our Finsler setting, without losing all the Finslerian information, but the second approach comes in fact from the more general theory of elliptic equations and hence will apply in our context.",
"A related way of viewing harmonic measures is given via solutions to the Dirichlet problem at infinity.",
"If $\\widetilde{M}$ is a Cartan–Hadamard manifold of bounded, strictly negative scalar curvature, we have the following result : Theorem 4.3.1 (Anderson [8], Sullivan [106]) Let $f \\in C^0(\\widetilde{M}(\\infty ))$ .",
"There exists a unique function $u_f \\in C^{\\infty }(\\widetilde{M})$ such that $\\left\\lbrace \\begin{aligned}\\Delta u_f &= 0 \\quad \\textrm {on } \\widetilde{M}\\\\u_f(x) &\\rightarrow f(\\xi ) \\quad \\textrm {when } x\\rightarrow \\xi , \\; \\xi \\in \\widetilde{M}(\\infty ) \\,.\\end{aligned}\\right.$ Now, to define the harmonic measures, take $x$ in $\\widetilde{M}$ .",
"There exists a positive linear functional on $C^0(\\widetilde{M}(\\infty ))$ given by $f \\mapsto u_f(x)$ .",
"This defines a probability measure $\\mu _x$ on $\\widetilde{M}(\\infty )$ which is the harmonic measure at $x$ .",
"Furthermore, the solution to the Dirichlet problem at infinity is given by $u_f(x) = \\int _{\\xi \\in \\widetilde{M}(\\infty )} f(\\xi ) \\,d\\mu _x (\\xi ) \\, .$ We wish to extend this construction to our Finsler–Laplace operator.",
"One way to prove Theorem REF (see [8], [9], [6]) is to study the Martin Boundary associated with $\\Delta $ and show that the harmonic functions are given by Equation (REF ) where $\\mu _x$ is a measure on the Martin boundary.",
"Ancona [7], [6] showed that this method works for a very general class of elliptic operators, requiring no Riemannian setting, but just assumptions on the operator and Gromov-hyperbolicity.",
"The redaction of the next section has been a (possibly failed) challenge for me.",
"My main goal was to present Ancona's theorem (Theorem REF below), but, in order to make it somewhat understandable, we need a good deal of background which is, if my personal example is significant, not in the usual toolbox of the average student in Finsler geometry or dynamical systems.",
"Therefore, I tried to include some of the basics I learned on potential theory but not too much for space and time issues.",
"I also tried to give the proofs that seem to me to be important for the understanding of this theory, while using only the tools I introduce.",
"So, all in all, the resulting redaction is probably disappointing for many reasons but I hope that it gives a reasonable idea of the theory used to prove Ancona's result and to obtain harmonic measures."
],
[
"Some potential theory and the Martin Boundary",
"In this section, we will recall the construction of the Martin compactification of $\\widetilde{M}$ .",
"Our main reference is [7], but if you want to get back to the roots, see [86]."
],
[
"The Green function",
"We start out by recalling some definitions.",
"In the following, $L$ will be a second-order uniformly elliptic differential operator.",
"We will furthermore always assume that $L$ is self-adjoint with respect to some volume form $\\Omega $ on $\\widetilde{M}$ , even if this is not necessary for the general theory.",
"Recall that an operator is elliptic if, for any local coordinates, the symbol of $L$ is a positive-definite matrix.",
"It is uniformly elliptic if the positivity of the symbol is uniform, i.e., there exists a constant $c$ depending only on $L$ , such that, if $(\\sigma _{ij})$ is the symbol of $L$ in a local chart, then $\\sum \\sigma _{ij}x_i x_j \\ge c \\sum x_i^2$ .",
"Definition 4.3.2 Let $U \\subset \\widetilde{M}$ .",
"A $L$ -harmonic function on $U$ is a $C^2$ function such that $Lu=0$ on $U$ .",
"A relatively compact open set $V$ is called (Dirichlet-)regular if for any $f\\in C(\\partial V)$ there exists a unique function $u\\in C(\\overline{V})$ harmonic on $V$ such that $u=f$ on $\\partial V$ .",
"If $V$ is a regular open set and $x\\in V$ , we denote by $\\mu _{x}^{V}$ the harmonic measure at $x$ relatively to $V$ .",
"That is, the only Borel measure such that, for any $f \\in C(\\partial V)$ , $H_f^{V}(x) = \\int _{\\partial V} f(\\xi ) d\\mu _x^{V}(\\xi )$ is harmonic on $V$ .",
"A function $u: U \\rightarrow ]-\\infty , +\\infty ]$ is $L$ -superharmonic on $U$ if $u$ is lower semi-continuous and if, for any regular set $V \\subset \\subset U$ , $u \\ge H_{u}^V$ on $V$ .",
"A superharmonic function $s$ is non-degenerate on $U$ if for any $V \\subset \\subset U$ regular, $s_{|V}$ is harmonic.",
"A function $u$ is a $L$ -potential on $U$ if $u \\ge 0$ , superharmonic on $U$ and such that any $L$ -harmonic function on $U$ smaller than $u$ is non-positive.",
"Remark 4.3.3 A $C^2$ function is superharmonic if and only if $L u \\le 0$ .",
"Harmonic functions enjoy the following fundamental property : Proposition 4.3.4 (Harnack's principle) Let $U$ be a domain of $\\widetilde{M}$ , then for any $p\\in U$ , the set of functions $\\lbrace u \\mid u \\; L-\\textrm {harmonic, positive on U}, \\; u(p)=1 \\rbrace $ is compact for the topology of uniform convergence on compact subsets of $U$ .",
"Proposition 4.3.5 (Harnack's inequality) Let $u$ be a harmonic function on a bounded domain $U$ of $\\widetilde{M}$ .",
"Then there exists a constant $c$ such that $\\sup _{x\\in U} u(x) \\le c \\inf _{x\\in U} u(x)\\, .$ Another fundamental piece of the theory is the maximum (or minimum) principle : Proposition 4.3.6 (Minimum Principle) Let $U$ be a bounded domain in $\\widetilde{M}$ and $u\\in C^2(U)\\cap C^0(\\overline{U})$ such that $Lu\\le 0$ .",
"If $u(y) \\ge 0$ for every $y\\in \\partial U$ , then $u(x) \\ge 0$ on $U$ .",
"Note that there exists also a global version : Proposition 4.3.7 Suppose that $u$ is a potential on $\\widetilde{M}$ , harmonic outside a closed set $F$ and continuous on $\\partial F$ .",
"If $s$ is a non-negative superharmonic function on $\\widetilde{M}$ such that $s \\ge u$ on $\\partial F$ , then $s \\ge u$ on $\\widetilde{M}\\setminus F$ .",
"To get on with potential theory, we need to assume that the operator $P = L -\\frac{\\partial }{\\partial t}$ admits a fundamental solution, the heat kernel of $L$ .",
"Definition 4.3.8 The heat kernel of $L$ is a positive function $p(x,y,t)$ defined on $M\\times M \\times \\mathbb {R}$ , vanishing identically when $t\\le 0$ , continuous for $t>0$ and $x \\ne y$ , $C^2$ with respect to $y$ , $C^1$ in $t$ and such that : For any fixed $x\\in M $ , $P p(x, \\cdot , \\cdot ) = 0 \\quad \\text{on} \\; M\\times \\mathbb {R}\\setminus \\lbrace (x,0)\\rbrace ;$ For any bounded continuous function $f$ on $M$ , $\\lim _{t\\rightarrow 0} \\int _{y\\in M} p(x,y,t) f(y) \\, \\Omega _y = f(x)\\,.$ Such a function always exists for a uniformly elliptic operator with uniform Hölder continuous coefficients.",
"See, for instance, [65] for the construction using the parametrix method in $\\mathbb {R}^n$ , or [12] for a diffusion approach on manifolds.",
"For an approach more specifically adapted to the case we will be interested in in the next sections, the reader can consult [70], [71].",
"Definition 4.3.9 The Green function of $L$ is defined, for $(x,y) \\in M\\times M$ , by $G(x,y) = \\int _0 ^{+\\infty } p(x,y,t) dt\\, .$ Remark 4.3.10 When $G$ is not identically infinite, then, for any fixed $x$ , $G(x,\\cdot )$ is a $L$ -potential.",
"Note also that, if $L$ is self-adjoint, then $G(x,y) = G(y,x)$ .",
"The following result gives us a characterization of when the Green function is not identically infinite : Theorem 4.3.11 ([7], Théorèmes 1 and 13) The following propositions are equivalent : There exist $x_0,y_0$ in $M$ such that $G(x_0,y_0) <\\infty $ ; The function $G$ is finite and continuous on $M\\times M \\setminus \\lbrace x=y\\rbrace $ and, for $x \\in M$ , $G(x, \\cdot )$ is a $C^2$ $L$ -harmonic function on $M\\setminus \\lbrace x\\rbrace $ ; There exists a strictly positive $L$ -potential on $M$ ; There exists a superharmonic, non-degenerate, non-negative function on $M$ which is not $L$ -harmonic.",
"In the following, $L$ will be assumed to satisfy one of the above equivalent properties."
],
[
"Martin Compactification",
"Let $O\\in \\widetilde{M}$ be fixed, for $x\\in \\widetilde{M}$ , let $K_x(y):= \\frac{G(x,y)}{G(x,O)} \\, .$ We say that a sequence $(x_n)$ converging to infinity in $M$ converges to a Martin point if $(K_{x_n})$ is pointwise convergent.",
"By the Harnack principle (see Proposition REF ) any sequence converging to infinity admits a subsequence converging to a Martin point and, if we denote by $K_{\\xi }$ the function associated to a Martin point $\\xi $ , then $K_{\\xi }$ is a non-negative harmonic function such that $K_{\\xi }(O)=1$ .",
"By definition, we say that two sequences define the same Martin point if and only if the limit functions are the same.",
"We will therefore often think of $\\xi $ and $K_{\\xi }$ as the same thing.",
"We write $\\mathcal {M}\\widetilde{M}$ for the set of Martin points and $\\hat{M} = \\widetilde{M}\\cup \\mathcal {M}\\widetilde{M}$ .",
"We define on $\\hat{M}$ the following metric : for $x,x^{\\prime }\\in \\hat{M}$ $\\rho (x,x^{\\prime }) = \\sup _{y\\in B(O,1)} |K_{x}(y) - K_{x^{\\prime }}(y)|,$ where $B(O,1)\\subset M$ is the ball of center $O$ and radius 1.",
"Proposition 4.3.12 (Martin [86]) The space $\\hat{M}$ equipped with the metric $\\rho $ is a complete, compact space for which $\\mathcal {M}\\widetilde{M}$ is the boundary and $M$ the interior.",
"Furthermore, the topology induced on $M$ coincides with its natural topology.",
"The space $\\hat{M}$ is called the (L-)Martin compactification of $M$ .",
"The fact that $\\rho $ is a metric is straightforward from the definition.",
"Both topologies coincide on $M$ : If $x\\in M$ and $x_n$ is a sequence that converges to $x$ (for the distance on $M$ ), then, by continuity of the Green function, $\\rho (x,x_n)$ also tends to zero.",
"$\\mathcal {M}\\widetilde{M}$ is compact : Let $\\left( \\xi _n \\right) $ be a sequence of points in $\\mathcal {M}\\widetilde{M}$ and $\\left(x_n^i\\right)$ a sequence of points converging to $\\xi _n$ .",
"Choose $\\left( K_n \\right)$ a sequence of compact sets such that $K_{n+1} \\supset K_n$ and $M = \\cup K_n$ .",
"Write $y_n$ for a point in the sequence $x_n^i$ outside of $K_n$ and such that $\\rho (y_n, \\xi _n) \\le \\frac{1}{n}$ .",
"Then $\\left(y_n\\right)$ is a sequence that leaves all compact sets of $M$ , therefore admits a subsequence converging to a Martin point $\\xi $ and by construction, $\\rho (\\xi _n, \\xi )$ tends to zero.",
"$\\hat{M}$ is compact : If $\\left(x_n\\right)$ is a sequence in $\\hat{M}$ , then either there is a subsequence that stays in a compact set of $M$ , or there is a subsequence that leaves every compact set of $M$ and there is a subsequence converging to a Martin point, or, finally, there is a subsequence staying in $\\mathcal {M}\\widetilde{M}$ , and we apply the previous fact.",
"$M$ is open inside $\\hat{M}$ with boundary $\\mathcal {M}\\widetilde{M}$ : this is obvious.",
"Let $\\mathcal {H}_+$ be the convex cone of positive $L$ -harmonic functions and write ${\\mathcal {K}:= \\lbrace u \\in \\mathcal {H}_+ \\mid u(O)=1 \\rbrace }$ for the subset of normalized $L$ -harmonic functions.",
"The space $\\mathcal {K}$ is a base of $\\mathcal {H}_+$ , it is a convex set and we denote by $E$ its extremal points.",
"A harmonic function $u$ such that $u/u(O)$ is in $E$ is called minimal.",
"A function $u$ is minimal iff $u$ does not dominate any other harmonic function apart from multiples of itself (see [86]).",
"Theorem 4.3.13 (Martin [86]) For any $u\\in E$ , there exists a Martin point $\\xi $ such that $K_{\\xi } = u$ .",
"The proof can be found, for instance, p. 31 of [7].",
"The Martin boundary does not in general coincide with $E$ , but the following result shows the importance of the case when $\\mathcal {M}\\widetilde{M}$ is reduced to its minimal part.",
"Proposition 4.3.14 For any $u \\in \\mathcal {H}_+$ , there exists a unique positive and finite Borel measure $\\mu _u$ on $E$ such that, for $x \\in M$ , $u(x) = \\int _{\\xi \\in E} K_{\\xi }(x) d\\mu _u(\\xi ) \\, .$ Remark 4.3.15 If the Martin boundary is reduced to $E$ , then any harmonic function is obtained as the integral of the $K_{\\xi }$ over $\\mathcal {M}\\widetilde{M}$ .",
"It is not quite Equation (REF ) that we are ultimately looking for, but it is getting closer.",
"The existence is given by Choquet's theorem and the uniqueness follows from the fact that $\\mathcal {K}$ is a base of a cone $\\mathcal {H}_+$ which is a lattice (see [92])."
],
[
"Ancona's Theorem",
"In [6], Ancona shows that, for an operator of the form $L= \\Delta + \\langle B, \\nabla \\rangle $ , where $\\Delta $ and $\\nabla $ come from a Riemannian metric of bounded negative curvature with $B$ satisfying certain conditions, then the Martin boundary is homeomorphic to the visual one.",
"In our case, we do not know whether the Riemannian metric we obtain from the symbol is of negative curvature, and hence cannot apply directly this result.",
"However, the proof Ancona gives remains true for a generic Gromov-hyperbolic space with a suitable elliptic operator and he presents it in the general setting in [7].",
"Theorem 4.3.16 (Ancona) Let $\\widetilde{M}$ be a Gromov-hyperbolic space, $L$ a self-adjoint elliptic operator, and denote by $d(\\cdot ,\\cdot )$ the distance on $\\widetilde{M}$ .",
"Suppose that $L$ verifies : There exist strictly positive constants $r_0, \\, \\tau , \\, c_1$ , and $c_2$ such that For all $ m \\in \\widetilde{M}$ , there exists a function $\\theta : B(m,r_0) \\rightarrow \\mathbb {R}^n $ , such that for all $ x,y \\in B(m,r_0) $ , we have $c_2^{-1} d(x,y) \\le \\Vert \\theta (x) -\\theta (y) \\Vert \\le c_2 d(x,y)\\,;$ For all $ x_0 \\in \\widetilde{M}$ and $\\forall 0\\le t \\le \\tau $ the Green function $g_t$ relative to $B(x_0,r_0)$ of the operator $L + t \\operatorname{Id}$ satisfies : for all $x,y \\in B(x_0,r_0/2)$ $\\left\\lbrace \\begin{aligned}c_1 &\\le g_t(x,y) \\\\g_t(x,y) &\\le c_2, \\quad \\text{if } d(x,y) \\ge \\frac{r_0}{4} \\,;\\end{aligned}\\right.$ There exists $\\varepsilon $ , $0<\\varepsilon <\\frac{1}{2}$ , such that for all $ t < 2\\varepsilon $ , $L + t\\operatorname{Id}$ admits a Green function $G_t$ on $\\widetilde{M}$ .",
"Then the Martin compactification of $\\widetilde{M}$ is homeomorphic to $\\widetilde{M}\\cup \\widetilde{M}(\\infty )$ and the Martin boundary $\\mathcal {M}\\widetilde{M}$ is reduced to its minimal part.",
"Furthermore, if we have chosen a base point $O$ and denote by $K_{\\xi } \\in \\mathcal {M}\\widetilde{M}$ the harmonic function corresponding to $\\xi \\in \\widetilde{M}(\\infty )$ , then the application $(\\xi ,x) \\mapsto K_{\\xi }(x)$ is continuous on $\\widetilde{M}(\\infty ) \\times \\widetilde{M}$ .",
"Remark 4.3.17 Ancona proves also that, given the above conditions on $L$ , the Green function tends to zero on the boundary : there exist constants $c>0$ and $\\alpha >0$ , depending on $L$ , such that, for any $x,y \\in \\widetilde{M}$ , if $d(x,y) \\ge 2$ , $G(x,y) \\le c e^{-\\alpha d(x,y)}.$ The main point in the proof is to give an estimate for the Green function of $L$ : Theorem 4.3.18 (Ancona, Theorem 6.1 of [7]) If $\\widetilde{M}$ is a Gromov-hyperbolic manifold and $L$ verifies Conditions (H1), (H2) and (H3), then, for any geodesic segment $[x,z]$ in $\\widetilde{M}$ and any point $y$ on it satisfying ${\\min \\left( d(x,y) , d(y,z) \\right) \\ge 1}$ , there exists a constant $c>0$ depending only on $L$ and the $\\delta $ given by the Gromov-hyperbolicity such that the Green function $G$ of $L$ verifies $c^{-1} G(y,x)G(z,y) \\le G(z,x) \\le c G(y,x) G(z,y)\\, .$ Note that this theorem contains in fact two results : one, hard and general, giving estimates of the Green function along what Ancona calls $\\phi $ -chain (Theorem 5.2 of [7]) and one, easier, showing that, when the space is Gromov-hyperbolic, you obtain $\\phi $ -chain by following geodesics.",
"Assuming this result, we can start the : [Sketch of proof of Theorem REF ] Ancona splits his proof into three steps : Choose a base point $O \\in \\widetilde{M}$ , let $\\xi \\in \\widetilde{M}(\\infty )$ and $\\gamma \\colon \\mathbb {R}^+ \\rightarrow \\widetilde{M}$ be a geodesic ray issuing from $O$ and ending in $\\xi $ .",
"Set $x_j := \\gamma \\left((j+1) \\delta \\right)$ and $V_j := \\lbrace x\\in \\widetilde{M}\\mid \\lbrace x,x_j \\rbrace _O \\ge j\\delta \\rbrace $ where $\\delta $ is a constant coming from the Gromov-hyperbolicity assumption.",
"For $x\\in \\widetilde{M}$ , recall that $K_x(y) := G(x,y)/G(x,O)$ .",
"Then, using Theorem REF , we can deduce that there exists a constant $c$ such that, for any $x\\in V_{j+1}$ and $y\\in \\widetilde{M}\\setminus V_j$ , $c^{-2} K_{x_j}(y) \\le K_{x}(y) \\le c^2 K_{x_j}(y)\\, .",
"$ Now take a sequence $(z_n)$ in $\\widetilde{M}$ converging to $\\xi \\in \\widetilde{M}(\\infty )$ and such that $K_{z_n}$ converges to a harmonic function $h$ , i.e., the sequence $(z_n)$ is a Martin point.",
"By Gromov-hyperbolicity, $V_j$ is a base of neighborhood of $\\xi $ (we can deduce that from the characterization of the visual boundary via the Gromov-compactification, see Section REF ).",
"So, we can deduce that for $ j \\ge 2$ and $y\\in \\widetilde{M}\\setminus V_j$ , $c^{-2} K_{x_j}(y) \\le h(y) \\le c^2 K_{x_j}(y)\\, .$ Note that, as a conclusion to that first step, we proved the following : If $h$ is a harmonic function corresponding to a Martin point $(z_n)$ which, as a sequence in $\\widetilde{M}$ , converges to $\\xi \\in \\widetilde{M}(\\infty )$ , then, for any $\\xi ^{\\prime } \\ne \\xi \\in \\widetilde{M}(\\infty )$ , there exists a neighborhood $V$ of $\\xi ^{\\prime }$ such that $h$ is bounded above by a multiple of $K_{x_j}$ on $\\widetilde{M}\\cap V$ .",
"Using the Harnack principle, we can deduce that $h$ is bounded by a multiple of $G(\\cdot , O)$ , with a constant depending on the point $\\xi ^{\\prime }$ .",
"Now define $\\mathcal {H}_{\\xi }$ the cone of non-negative $L$ -harmonic functions on $\\widetilde{M}$ that are bounded above by a multiple of $G(\\cdot , O)$ in a neighborhood of any $\\xi ^{\\prime } \\in \\widetilde{M}(\\infty )$ different from $\\xi $ .",
"Ancona proves that, for any $h \\in \\mathcal {H}_{\\xi }$ , the following holds : for any $j\\ge 2$ and $y\\in \\widetilde{M}\\setminus V_j $ , $c^{-2} h(O) K_{x_j}(y) \\le h(y) \\le c^2 h(O) K_{x_j}(y)\\, .$ From this we deduce that, for any $h,h^{\\prime } \\in \\mathcal {H}_{\\xi }$ and any $y\\in \\widetilde{M}$ , we have $c^{-4} h(O) h^{\\prime }(y) \\le h^{\\prime }(0)h(y) \\le c^2 h(O) h^{\\prime }(y)\\, ,$ that is, any element of $\\mathcal {H}_{\\xi }$ is bounded by a multiple of any other (non-zero) elements.",
"Recall that, in the first step, we showed that $\\mathcal {H}_{\\xi }$ is not reduced to zero.",
"Now the second step induces that $\\mathcal {H}_{\\xi }$ is one-dimensional.",
"Indeed, take two strictly positive elements $h,h^{\\prime } \\in \\mathcal {H}_{\\xi }$ and set $\\lambda := \\sup \\left\\lbrace \\frac{h(y)}{h^{\\prime }(y)} : y\\in \\widetilde{M}\\right\\rbrace $ .",
"By definition, $\\lambda h^{\\prime } - h \\in \\mathcal {H}_{\\xi }$ , but, by our choice of $\\lambda $ , $\\lambda h^{\\prime } - h$ cannot be bounded below by a multiple of $h^{\\prime }$ .",
"We set $K_{\\xi }$ the unique element of $\\mathcal {H}_{\\xi }$ such that $K_{\\xi }(O)= 1$ .",
"Given a point $\\xi \\in \\widetilde{M}(\\infty )$ and any sequence $(z_n)$ in $\\widetilde{M}$ converging to $\\xi $ , by Equation (REF ) and the Harnack principle, we know that $(K_{z_n})$ must converge simply and the third step shows that the limit must be $K_{\\xi }$ .",
"Moreover $K_{\\xi }$ is minimal because if $u$ is a positive harmonic function bounded above by $K_{\\xi }$ , then $u$ is in $\\mathcal {H}_{\\xi }$ , so if $u(O)=1$ then $u= K_{\\xi }$ .",
"So we constructed an application from $\\widetilde{M}(\\infty )$ to the minimal part of $\\mathcal {M}\\widetilde{M}$ .",
"Let $\\xi ,\\xi ^{\\prime }$ be two distinct points in $\\widetilde{M}(\\infty )$ , then $K_{\\xi } \\ne K_{\\xi ^{\\prime }}$ .",
"Indeed, otherwise we would have shown that $K_{\\xi }$ is bounded above by a multiple of $G(\\cdot , O)$ on all of $\\widetilde{M}(\\infty )$ and therefore on all of $\\widetilde{M}$ .",
"Hence $G(\\cdot , O)$ would be bounded below by a positive harmonic function, which is impossible because $y \\mapsto G(y,O)$ is a potential.",
"So our application $\\widetilde{M}(\\infty ) \\rightarrow \\mathcal {M}\\widetilde{M}$ is injective, and we have everything to see that it is also surjective : take a Martin sequence $(z_n)$ such that $\\left(K_{z_n}\\right)$ converges to $h$ , there exists a subsequence that converges to an element $\\xi \\in \\widetilde{M}(\\infty )$ so $h = K_{\\xi }$ and therefore $(z_n)$ must converge to $\\xi $ .",
"Remark 4.3.19 Note that the proof gives the following characterization of $K_{\\xi }$ : it is the only harmonic function such that $K_{\\xi }(O)=1$ and that is bounded by a multiple of $ G(\\cdot , O) $ in a neighborhood of every point in $\\widetilde{M}(\\infty ) \\setminus \\lbrace \\xi \\rbrace $ .",
"The second condition really gives a meaning to “$K_{\\xi }$ is zero on $\\widetilde{M}(\\infty ) \\setminus \\lbrace \\xi \\rbrace $ ”.",
"Some corollary of this result is that the Dirichlet problem at infinity for an operator satisfying (H1) to (H3) has a unique solution.",
"The proof is once again copied from [7].",
"Note that it is also a direct transcription of the classical proof of the Dirichlet Problem for the Euclidean disc via the Poisson integral formula (see for instance [69]).",
"Theorem 4.3.20 (Dirichlet problem at infinity) Let $(\\widetilde{M},L)$ be as above and $L({1}) =0$ .",
"Then, for any $f\\in C(\\widetilde{M}(\\infty ))$ , there exists a unique $u \\in C(\\widetilde{M}\\cup \\widetilde{M}(\\infty ))$ such that $u = f$ on $\\widetilde{M}(\\infty )$ and $Lu= 0$ on $\\widetilde{M}$ .",
"Furthermore, if we choose a base point $O$ on $\\widetilde{M}$ , we can write $u(x):= \\int _{\\xi \\in \\widetilde{M}(\\infty )} K_{\\xi }(x) f(\\xi ) d\\mu (\\xi )\\,.$ Note that uniqueness is a direct consequence of the maximum principle.",
"With Ancona's theorem, applying Proposition REF to 1 shows that we have a measure $\\mu $ on $\\widetilde{M}(\\infty )$ such that ${1}(x) = \\int _{\\xi \\in \\widetilde{M}(\\infty )} K_{\\xi }(x) d\\mu (\\xi ) \\, ,$ where we chose a base point $O \\in \\widetilde{M}$ for the normalization of $K_{\\xi }$ .",
"Now let $u(x):= \\int _{\\xi \\in \\widetilde{M}(\\infty )} K_{\\xi }(x) f(\\xi ) d\\mu (\\xi )\\, ,$ $u$ is $L$ -harmonic by definition, so we just need to prove that $u(x)$ tends to $f(\\xi )$ when $x \\rightarrow \\xi $ .",
"Let $\\xi _0 \\in \\widetilde{M}(\\infty )$ and $V$ a neighborhood of $\\xi _0$ , we write $u(x)- f(\\xi _0):= \\int _{\\widetilde{M}(\\infty ) \\setminus V} K_{\\xi }(x) (f(\\xi )-f(\\xi _0) ) d\\mu (\\xi ) + \\int _{V} K_{\\xi }(x) (f(\\xi )-f(\\xi _0) ) d\\mu (\\xi )\\, .$ Now, for $x$ in a smaller neighborhood of $\\xi _0$ , $K_{\\xi }(x)$ is bounded above by a multiple of $G(x, O)$ (by the first step in the proof of Ancona's Theorem), so the first part in the RHS of the above equation is small.",
"The second part is easily seen to be small because $ \\int _{\\widetilde{M}(\\infty )} K_{\\xi }(x) d\\mu (\\xi ) = 1$ , which proves the theorem.",
"In fact, we can say more about the regularity of the identification between the Martin and the visual boundary.",
"Recall (Proposition REF ) that there is a natural metric on the boundary of a Gromov-hyperbolic space $d_{G, \\epsilon }$ , where $\\epsilon $ is a small real depending only on $\\delta $ , and that we have, for $\\xi , \\eta \\in \\widetilde{M}(\\infty )$ , $ (3 - 2 e^{\\epsilon \\delta }) e^{-\\epsilon \\lbrace \\xi , \\eta \\rbrace _O}\\le d_{G,\\epsilon }(\\xi ,\\eta ) \\le e^{-\\epsilon \\lbrace \\xi , \\eta \\rbrace _O}.$ Moreover, the boundary admits a Hölder structure (see [44]).",
"Theorem 4.3.21 Let $(\\widetilde{M}, L)$ satisfying the conditions of Theorem REF .",
"There exists a constant $\\alpha >0$ depending on $\\widetilde{M}$ and $L$ such that the identification $\\mathcal {M}\\widetilde{M}\\rightarrow \\widetilde{M}(\\infty )$ is $\\alpha $ -Hölder.",
"Moreover, if we denote by $K(O,x,\\xi )$ the Poisson kernel normalized at $O$ , for any compact $K \\subset \\widetilde{M}$ , the application $(O,x,\\xi ) \\in K\\times K \\times \\widetilde{M}(\\infty ) \\mapsto K(O,x,\\xi )$ is Hölder-continuous.",
"The above result is a direct consequence of the following, more technical result : Theorem 4.3.22 Let $(\\widetilde{M}, L)$ satisfying the conditions of Theorem REF and $U$ an unbounded domain in $\\widetilde{M}$ .",
"Suppose that $u$ and $v$ are two harmonic functions on $U$ , continuous on $\\overline{U}$ , and such that $u_{|\\overline{U}\\cap \\widetilde{M}(\\infty )}= v_{|\\overline{U}\\cap \\widetilde{M}(\\infty )}=0$ .",
"Then the quotient $u/v$ has a $C^{\\alpha }$ extension to $\\overline{U}\\cap \\widetilde{M}(\\infty )$ , where $\\alpha $ depends on $M$ and $L$ .",
"As far as I know there is no published proof of that result in this generality.",
"However, the hard part is due to Ancona [6] and Anderson and Schoen [9], the only personal contribution being the following lemma.",
"Note also that there are some closely related results of this generality : in [74], M. Izumi, S. Neshveyev and R. Okayasu prove that the Martin kernel for a random walk on a hyperbolic group is Hölder continuous.",
"Lemma 4.3.23 Let $\\gamma $ be a geodesic ray from $O$ to a point $\\xi _0 \\in \\widetilde{M}(\\infty )$ .",
"We define $A_i := \\gamma (4i\\delta )$ and $A_i^{\\prime } :=\\gamma ((4i+2)\\delta )$ .",
"Let $V_i := \\lbrace x\\in M \\mid \\lbrace x, A^{\\prime }_i \\rbrace _O > 4 i \\delta \\rbrace $ .",
"Let $\\xi $ and $\\eta $ be the two points in $\\partial \\left(\\widetilde{M}(\\infty ) \\cap \\overline{V_i} \\right)$ , then $ \\lbrace \\xi , \\eta \\rbrace _O \\le \\left( 4(i + 1) + 2\\right) \\delta + 17 \\delta \\, .$ Therefore, for $i \\ge 12 + \\dfrac{-\\log (3-2\\exp (\\epsilon \\delta ))}{2\\epsilon \\delta }$ , $\\sup _{a,b \\in \\overline{V_i}} d_{G,\\epsilon }(a,b) \\ge e^{-8 i \\epsilon \\delta } \\,.$ First note that $\\sup _{a,b \\in \\overline{V_i}} d_{G,\\epsilon }(a,b) \\ge (3 - 2 e^{\\epsilon \\delta })e^{-\\epsilon \\lbrace \\xi ,\\eta \\rbrace _O}$ (see Equation (REF )), so proving Equation (REF ) will indeed give us the result.",
"Now remark that, for any point $p_0$ on the geodesic $(\\xi ,\\eta )$ , $\\lbrace \\xi ,\\eta \\rbrace _O \\le d(O, p_0)$ .",
"Indeed, if we take two sequences $(a_n)$ and $(b_n)$ on $(\\xi ,\\eta )$ converging respectively to $\\xi $ and $\\eta $ , we have $\\lbrace \\xi ,\\eta \\rbrace _O &= \\lim _{n \\rightarrow +\\infty } \\frac{1}{2}\\left( d(a_n, O) + d(b_n, O) - d(a_n, b_n) \\right) \\\\&\\le \\lim _{n \\rightarrow +\\infty } \\frac{1}{2}\\left( d(a_n,p_0 ) + d(p_0, O) + d(b_n, p_0) + d(p_0, O) - d(a_n, b_n) \\right) \\\\&\\le d(p_0,O)\\, .$ Let $p:= \\gamma \\cap (\\xi ,\\eta )$ (we take $i$ big enough so that this intersection exists).",
"Depending on the position of $p$ , we have two possibilities : Either $d(O,p) \\le d(O, A_i^{\\prime }) = (4i +2) \\delta $ which implies Equation (REF ), Or $d(O,p) \\ge d(O, A_i^{\\prime })$ ; then, $p$ is in $V_i$ (because the part of $\\gamma $ after $A_i$ is in $V_i$ ), so we again have two possible cases ; Either $p \\notin V_{i+1}$ and therefore $d(O,p) \\le d(O, A_{i+1}) = 4(i+1) \\delta $ which again implies Equation (REF ) ; Or $p \\in V_{i+1}$ ; in that case, we know (using Scholie 3.1 of [7]) that the geodesic ray $[p,\\xi )$ must pass no further than $17\\delta $ from $A_{i+1}^{\\prime }$ ; if we denote by $p^{\\prime }$ the point realizing that distance, we have $d(O,p^{\\prime }) \\le 17\\delta + d(O, A_{i+1}^{\\prime }) = \\left( 4(i + 1) + 2\\right) \\delta + 17 \\delta $ , which ends the proof.",
"[Proof of Theorem REF ] Let $\\xi _0 $ be an interior point of $ \\overline{U} \\cap \\widetilde{M}(\\infty )$ , $O \\in U$ a base point and $\\gamma $ a geodesic ray from $O$ to $\\xi _0$ .",
"Set $A_i := \\gamma (4 i \\delta )$ and $A^{\\prime }_i := \\gamma (4(i+2)\\delta ) $ , where $\\delta >0$ is given by the Gromov-hyperbolicity of $\\widetilde{M}$ .",
"Now we define $V_i:= \\lbrace x\\in \\widetilde{M}\\mid \\lbrace x, A^{\\prime }_i \\rbrace _O >4i\\delta \\rbrace $ , the $V_i$ 's form a basis of neighborhoods of $\\xi _0$ and are such that $\\overline{V_{i+1}} \\subset V_i$ .",
"So for $i$ big enough, $\\overline{V_i} \\subset U $ .",
"Replacing their $C_i$ 's by $V_i$ , we can copy verbatim the proof of Theorem 6.2 of [9] and obtain that $u/v$ admits a radial extension $\\varphi $ to $\\overline{U}$ and that there exists a constant $c_1>0$ , depending on $\\delta $ and $L$ , such that $ \\sup _{x\\in \\overline{V_i}} \\varphi (x) - \\inf _{x\\in \\overline{V_i}} \\varphi (x) \\le c_1^i \\varphi (O)\\, .$ The only change that needs to be done is to use Ancona's Harnack inequality at infinity given by Theorem 5' of [6], instead of the Harnack inequality Anderson and Schoen use.",
"Theorem 5' applies because the $(A_i, V_i)$ form a $\\phi $ -chain (see the proof of Theorem 6.1 in [7]).",
"Now, by Lemma REF above, we have, for $i \\ge c_2$ , where $c_2$ depends only on $\\delta $ (as $\\epsilon $ depends only on $\\delta $ ), $ \\sup _{a,b \\in \\overline{V_i}} e^{-\\lbrace a,b\\rbrace _O} \\ge e^{-8 i \\epsilon \\delta }.$ So putting together Equation (REF ) and Equation (REF ), we obtain that, for $y$ and $y^{\\prime }$ sufficiently far from $O$ , setting $\\alpha := (1/8 \\epsilon \\delta ) \\log \\left(1/c_1 \\right) >0$ $\\left| \\varphi (y) - \\varphi (y^{\\prime }) \\right| \\le \\left[e^{-\\lbrace y,y^{\\prime }\\rbrace _O} \\right]^{\\alpha } \\varphi (O) \\, .$ This proves that the extension $\\varphi $ is in fact $C^{\\alpha }$ .",
"Finally we can deduce Theorem REF from Theorem REF (see [9]) : [Proof of Theorem REF ] Recall that, for $\\xi \\in \\widetilde{M}(\\infty )$ , the Poisson Kernel normalized at $O$ , $K(O, \\cdot , \\xi )$ is such that $K(O, x, \\xi ) = \\lim _{n \\rightarrow \\infty } \\frac{G(x,y_n)}{G(O,y_n)}\\,,$ where $(y_n)$ is a sequence converging to $\\xi $ .",
"Applying Theorem REF to the $G(x,y_n)$ gives that there exists a constant $C$ such that, for any $x\\in B(O,1)$ and any distinct $\\xi , \\xi ^{\\prime } \\in \\widetilde{M}(\\infty )$ $\\left| K(O, x, \\xi ) - K(O, x, \\xi ^{\\prime } ) \\right| \\le C \\left[ d_G (\\xi , \\xi ^{\\prime } ) \\right]^{\\alpha },$ which proves the theorem."
],
[
"Existence of Finsler–Laplace harmonic measures",
"We finally get back to Finsler geometry.",
"Recall that $\\widetilde{M}$ is the universal cover of a closed manifold $M$ and $\\widetilde{F}$ is a Finsler metric of negative flag curvature lifted from $M$ .",
"From now on, we assume that $\\widetilde{F}$ is reversible.",
"Indeed Ancona's Theorem is proved only for symmetric distance.",
"Note however, that Fang and Foulon [51] showed that, for irreversible Finsler metric of negative curvature, there exist two boundaries at infinity : one is given by following the geodesics into the future and the other into the past.",
"It seems very probable that if we consider a non-symmetric distance and redo the steps of Ancona's proof, we should obtain identifications of the past and forwards boundaries with the Martin boundary by taking the Poisson kernels $K_{\\xi }$ along forward, respectively backwards, geodesics.",
"But, if this is true, proving it will remain a project for later (or for any interested reader).",
"Theorem 4.4.1 If $(M,F)$ is a closed reversible Finsler manifold of strictly negative curvature and $(\\widetilde{M},\\widetilde{F})$ its universal cover, then $\\Delta ^{\\widetilde{F}}$ verifies Conditions (H1) to (H3) of Theorem REF .",
"Before starting the proof, let us state the main corollary : Corollary 4.4.2 There is a $C^{\\alpha }$ identification between the Martin and the visual boundaries of $(\\widetilde{M},\\widetilde{F})$ and the Dirichlet problem at infinity for $\\Delta ^{\\widetilde{F}}$ admits a unique solution.",
"That is, for any $f\\in C(\\widetilde{M}(\\infty ))$ , there exists a unique $u \\in C(\\widetilde{M}\\cup \\widetilde{M}(\\infty ))$ such that $u = f$ on $\\widetilde{M}(\\infty )$ and $\\Delta ^{\\widetilde{F}}u= 0$ on $\\widetilde{M}$ .",
"Furthermore, for any $x\\in \\widetilde{M}$ , there exists a measure $\\mu _x$ , called the harmonic measure for $\\Delta ^{\\widetilde{F}}$ such that $u(x):= \\int _{\\xi \\in \\widetilde{M}(\\infty )} f(\\xi ) d\\mu _x(\\xi )\\, .$ The first part of the corollary is Theorem REF and the second part is Theorem REF with $d\\mu _x := K_{\\xi }(x) d\\mu $ .",
"Condition (H1) just means that $\\widetilde{M}$ is of “bounded geometry” and it would stay true for any uniform Finsler Hadamard manifold in Egloff's sense.",
"Conditions (H2) and (H3) come from the following : Proposition 4.4.3 The operator $\\Delta ^{\\widetilde{F}}$ is coercive, i.e., there exists $c>0$ such that, for ${f \\in C^{\\infty }_0}$ , $\\int _{H\\widetilde{M}} \\left( L_X \\pi ^{\\ast } f \\right)^2 A \\wedge dA^{n-1}\\ge c \\int _{\\widetilde{M}} f^2 \\Omega \\, .$ By Theorem REF , we know that $\\lambda _1$ is strictly positive because $\\pi _1(M)$ is Gromov-hyperbolic, hence not amenable, and the characterization of $\\lambda _1$ as the infimum of the Rayleigh quotient (see Proposition REF ) shows that we can take $c$ to be $\\lambda _1$ .",
"We can deduce (H3) from there (see [6], we recall the proof below) : Corollary 4.4.4 (Weak coercivity) There exists $\\varepsilon $ , $0<\\varepsilon <\\frac{1}{2}$ , such that for all $ t < 2\\varepsilon $ , $\\Delta ^{\\widetilde{F}} + t\\operatorname{Id}$ admits a Green function $G_t$ on $\\widetilde{M}$ .",
"Remark 4.4.5 Recall that, as a weighted Laplace operator, there always exists a heat kernel for $\\Delta ^{\\widetilde{F}}$ (see for instance [71]).",
"Therefore we can apply Theorem REF to decide whether a Green function exists.",
"Take $\\varepsilon $ smaller than $c/2$ .",
"Then, for any $t <2 \\varepsilon $ , $L:= \\Delta ^{\\widetilde{F}} + t\\operatorname{Id}$ is still coercive and therefore we have a coercive bilinear form $q_L$ associated with $L$ and continuous on $H^1_0(\\widetilde{M})$ .",
"Indeed, just set $q_L(u,v):= - \\int _{\\widetilde{M}} u\\left( \\Delta ^{\\widetilde{F}} v + tv\\right) \\; \\Omega $ .",
"We will construct a $L$ -superharmonic positive function $s$ and use Theorem REF to conclude.",
"Take a positive test function $f \\in C^{\\infty }_0(\\widetilde{M})$ .",
"There exists (by Lax–Milgram Theorem) an $s \\in H_0^1(\\widetilde{M})$ such that, for any $\\varphi \\in C_0^{\\infty }(\\widetilde{M})$ , $q_L(s, \\varphi ) = \\int _{\\widetilde{M}} f \\varphi \\, \\Omega \\, .$ Now $s$ is positive, because if we let $s^-:= \\max \\lbrace 0, - s \\rbrace $ , then $q_L (s^-, s^-) = - q_L (s, s^-) \\le 0$ (because of the above equation, using a suitable $C^{\\infty }$ approximation of $s^-$ ).",
"As $s$ is a weak solution of $Ls = - f$ , $s$ is superharmonic and we can choose $f$ so that $s$ is strictly positive and Theorem REF proves the claim.",
"Lemma 4.4.6 There exist strictly positive constants $r_0, \\, \\tau _0, \\, c_1$ , and $c_2$ such that For all $ m \\in \\widetilde{M}$ , there exists a function $\\theta : B(m,r_0) \\rightarrow \\mathbb {R}^n $ , such that, for all $ x,y \\in B(m,r_0) $ we have $c_2^{-1} d(x,y) \\le || \\theta (x) -\\theta (y) || \\le c_2 d(x,y) \\,;$ For all $x_0 \\in \\widetilde{M}$ and $\\forall 0\\le t \\le \\tau _0 $ the Green function $g_t$ relative to $B(x_0,r_0)$ of the operator $\\Delta ^{\\widetilde{F}} + t \\operatorname{Id}$ satisfies, for all $ x,y \\in B(x_0,r_0/2)$ $\\left\\lbrace \\begin{aligned}c_1 &\\le g_t(x,y) \\\\g_t(x,y) &\\le c_2, \\quad \\text{if } d(x,y) \\ge \\frac{r_0}{4} \\,.\\end{aligned}\\right.$ Remark 4.4.7 These two conditions follow from the fact that $\\widetilde{M}$ and $\\Delta ^{\\widetilde{F}}$ are well-adapted (in the terminology of Ancona [7]), i.e., that $\\widetilde{M}$ verifies (H1) and that the push-forwards of $\\Delta ^{\\widetilde{F}}$ by $\\theta $ have coefficients with Hölder norms bounded by uniform constants.",
"Condition (H2) just means that the Green function of $\\Delta ^{\\widetilde{F}}$ should behave like a Green function of a uniformly elliptic operator on $\\mathbb {R}^n$ .",
"For (H1), first remark that, for any $m \\in \\widetilde{M}$ , the exponential map at $m$ is a diffeomorphism from $T_m\\widetilde{M}$ to $\\widetilde{M}$ , and we can take $\\theta = \\exp _m^{-1}$ .",
"For any real number $r_0$ , there exists a constant $c= c(r_0) $ depending only on $r_0$ such that (H1) is satisfied.",
"The constant $c$ exists by compactness of $B(m,r_0)$ and does not depend on $m$ by compactness of $M$ .",
"The constant $c_2$ will be determined by Condition (H2), as long as we take $c_2 \\ge c(r_0)$ .",
"Let $\\tau _0 < \\lambda _1$ and fix $0 \\le t_0 \\le \\tau _0$ , let $L:= \\theta _{\\ast }\\left(\\Delta ^{\\widetilde{F}} + t_0\\operatorname{Id}\\right)$ .",
"The operator $L$ is a uniformly elliptic operator on $U= \\theta (B(x_0,r_0))$ with smooth bounded coefficients (by compactness of $M$ ).",
"Therefore (see, for instance [105] or Chapter 1 of [65], Equations (6.12) and (6.13) in particular), the operator $L$ admits a Green function $g^L$ and there exist two constants $c_1$ and $c_2$ (depending on the Hölder norm of the coefficients of $L$ ) such that, for $u,v$ a bounded distance apart and a bounded distance from $\\partial U$ , we have $g^L(u,v) \\ge c_1$ and $g^L(u,v) \\le c_2$ .",
"As $g^t$ is the pullback of $g^L$ by $\\theta $ and again using the compactness of $M$ , we have a uniform control on all those bounds and hence have proven our lemma."
],
[
"Ergodic property of harmonic measures",
"In this section, we will adapt ergodic results on the harmonic measures to our case.",
"The result and proof are based on Ledrappier's work on harmonic measures for negatively curved Riemannian manifolds in [83].",
"A measure class on a space $V$ is a set $\\lbrace \\mu _x \\rbrace _{x\\in V}$ indexed by $V$ such that, for any $x\\in V$ , $\\mu _x$ is a measure on $V$ and, for $y\\in V$ , $\\mu _x$ and $\\mu _y$ are equivalent.",
"All the measures we consider are Radon measures.",
"If a group $\\Gamma $ acts on $V$ , we say that a measure class is invariant by $\\Gamma $ if, for $x\\in V$ and $\\gamma \\in \\Gamma $ , $\\mu _{\\gamma \\cdot x} = \\gamma _{\\ast } \\mu _x\\, ,$ where $\\gamma _{\\ast }\\mu _x$ is defined by, for $U \\subset V$ measurable, $\\gamma _{\\ast }\\mu _x(U) := \\mu _{x} \\left(\\gamma ^{-1} \\cdot U \\right)$ .",
"Remark that, if $\\lbrace \\mu _x\\rbrace $ is an invariant measure class, then the measures $\\mu _x$ are quasi-invariant, that is, for any $\\gamma \\in \\Gamma $ , $\\mu _{x}$ and $\\gamma _{\\ast } \\mu _x$ are equivalent.",
"For an invariant measure class (or a quasi-invariant measure), we can use the traditional definition of ergodicity : an invariant measure class is ergodic if, for any measurable set $U$ invariant under $\\Gamma $ , then $U$ is either of measure (for any $\\mu _x$ ) full or null.",
"In this section, $(M,F)$ is still a closed reversible Finsler manifold of negative curvature, $(\\widetilde{M},\\widetilde{F})$ its universal cover with the lifted metric.",
"For any $x \\in \\widetilde{M}$ , we denote by $\\widetilde{\\mu } _x$ the harmonic measure on $\\widetilde{M}(\\infty )$ associated to $\\Delta ^{\\widetilde{F}}$ .",
"Lemma 4.5.1 The harmonic measures $\\lbrace \\widetilde{\\mu } _x \\rbrace $ form a measure class invariant by the action of $\\pi _1(M)$ on $\\widetilde{M}(\\infty )$ .",
"Moreover, for $x,y\\in \\widetilde{M}$ , the Radon-Nykodim derivative of $\\widetilde{\\mu } _x$ and $\\widetilde{\\mu } _y$ is $\\frac{d\\widetilde{\\mu } _x}{d\\widetilde{\\mu } _y}(\\xi ) = K(y,x,\\xi )\\, ,$ where $x\\mapsto K(y,x,\\xi )$ is the Poisson Kernel normalized at $y$ .",
"We can obtain the harmonic measure $\\mu _x$ in the following way : let $O$ be a point in $\\widetilde{M}$ and $K(O,x,\\xi )$ the Poisson kernel normalized at $O$ .",
"If $\\widetilde{\\mu } _O$ is the measure on $\\widetilde{M}(\\infty )$ such that $\\int _{\\xi \\in \\widetilde{M}(\\infty )} K(O,x,\\xi ) d\\widetilde{\\mu } _O(\\xi ) =1\\, ,$ then $\\widetilde{\\mu } _x$ is such that, for any Borel set $U \\subset \\widetilde{M}(\\infty )$ , $ \\widetilde{\\mu } _x(U) = \\int _U K(O,x,\\xi ) d\\widetilde{\\mu } (\\xi )$ .",
"From this we see that all the measures are equivalent and that their Radon-Nykodim derivative is given by $\\frac{d\\widetilde{\\mu } _x}{d\\widetilde{\\mu } _y}(\\xi ) = \\frac{K(O,x,\\xi )}{K(O,y,\\xi )} \\, .$ Now recalling the characterization of $K(O,\\cdot ,\\xi )$ given by the proof of Theorem REF (see Remark REF ), we see that $\\frac{K(O,x,\\xi )}{K(O,y,\\xi )} = K(y,x,\\xi )$ .",
"Indeed, the map ${ x \\mapsto K(O,x,\\xi )/K(O,y,\\xi )}$ is harmonic, normalized at $y$ and such that it tends to zero when $x$ tends to $\\xi ^{\\prime } \\ne \\xi $ .",
"So we have $\\frac{d\\widetilde{\\mu } _x}{d\\widetilde{\\mu } _y}(\\xi ) = K(y,x,\\xi ) \\, .$ Using the fact that the $\\mu _x$ solves the Dirichlet problem at infinity, we get that, for any $x\\in \\widetilde{M}$ and $f \\in C^0(\\widetilde{M}(\\infty ))$ , $\\int _{\\xi \\in \\widetilde{M}(\\infty )} f(\\xi ) d\\mu _{\\gamma \\cdot x} = \\int _{\\xi \\in \\widetilde{M}(\\infty )} f\\circ \\gamma (\\xi ) d\\mu _{ x} = \\int _{\\eta \\in \\widetilde{M}(\\infty )} f(\\eta ) d\\left(\\gamma _{\\ast } \\mu _{x} \\right) (\\eta )\\, ,$ so we do have $\\mu _{\\gamma \\cdot x} = \\gamma _{\\ast } \\mu _{x}$ .",
"Let $\\tau : H\\widetilde{M}\\rightarrow \\widetilde{M}(\\infty )$ be the application sending an element $(x,v) \\in H\\widetilde{M}$ to the point at infinity obtained as the limit of the geodesic ray leaving $x$ in the direction $v$ .",
"For any fixed $x\\in \\widetilde{M}$ , $\\tau _x$ is a (Hölder)-homeomorphism.",
"As the harmonic measures are $\\pi _1(M)$ -invariant, we can define the spherical harmonic measures $\\lbrace \\bar{\\mu }_y, y\\in M \\rbrace $ as measures on $H_yM$ by setting $\\bar{\\mu }_y := \\mu _{\\widetilde{y}} \\circ \\tau _{\\widetilde{y}}\\, ,$ where $\\widetilde{y}\\in \\widetilde{M}$ is any lift of $y\\in M$ .",
"Theorem 4.5.2 Let $(M,F)$ and $\\widetilde{\\mu } _x$ be as before.",
"We have the following properties : (i) The harmonic measure class $\\lbrace \\widetilde{\\mu } _x \\rbrace $ is ergodic for the action of $\\pi _1(M)$ on $\\widetilde{M}(\\infty )$ ; (ii) For any $x\\in \\widetilde{M}$ , the product measure $\\widetilde{\\mu } _x \\otimes \\widetilde{\\mu } _x$ is ergodic for the action of $\\pi _1(M)$ on $\\partial ^2\\widetilde{M}:= \\widetilde{M}(\\infty ) \\times \\widetilde{M}(\\infty ) \\setminus \\text{diagonal}$ ; (iii) There exists a unique $ \\phi ^{t} $ -invariant measure $\\mu $ on $HM$ such that the family of spherical harmonics $\\bar{\\mu }_x$ is a family of transverse measures.",
"Moreover, $(HM, \\phi ^{t} , \\mu )$ is ergodic.",
"Note that, by the following result of Kaimanovich, which still holds in this context, proving (iii) gives the theorem.",
"Theorem 4.5.3 (Kaimanovich [75]) There exists a convex isomorphism between the cone of Radon measures on $\\partial ^2 \\widetilde{M}$ and the cone of Radon measures on $H\\widetilde{M}$ invariant by $ \\tilde{ \\phi }^t $ .",
"Similarly, there exists a convex isomorphism between the cone of Radon measures on $\\partial ^2 \\widetilde{M}$ invariant by $\\pi _1(M)$ and the cone of Radon measures on $HM$ invariant by $ \\phi ^{t} $ .",
"So if we construct $\\mu $ , the Kaimanovich correspondence shows that there exists a weight $f \\colon \\partial ^2 \\widetilde{M}\\rightarrow \\mathbb {R}$ such that the measure $f \\widetilde{\\mu } _x \\otimes \\widetilde{\\mu } _x$ is invariant by $\\pi _1(M)$ (see [75]).",
"And proving that $\\mu $ is ergodic for the flow proves that $f \\widetilde{\\mu } _x \\otimes \\widetilde{\\mu } _x$ is ergodic.",
"To prove (iii) of Theorem REF , it suffice to copy verbatim the proof of Proposition 3 in [83] (using that our Poisson kernel is Hölder continuous by Theorem REF ).",
"We get : Proposition 4.5.4 There exists a Hölder continuous function $F_0$ on $M$ such that the spherical harmonic measures $\\bar{\\mu }_x$ can be chosen as a family of transverse measures for the equilibrium state $\\mu $ of $F_0$ .",
"But we know (see [25]) that an equilibrium state is ergodic, and so we have (iii)."
],
[
"Basics on Anosov flows",
"In all this part, we will be interested in Anosov flows on closed 3-manifolds, but we can state the definition in any dimension : Definition 5.1.1 Let $M$ be a compact manifold and $ \\phi ^{t} \\colon M \\rightarrow M$ a $C^1$ flow on $M$ .",
"The flow $ \\phi ^{t} $ is called Anosov if there exists a splitting of the tangent bundle ${TM = \\mathbb {R}\\cdot X \\oplus E^{ss} \\oplus E^{uu}}$ preserved by $D \\phi ^{t} $ and two constants $a,b >0$ such that : $X$ is the generating vector field of $ \\phi ^{t} $ ; For any $v\\in E^{ss}$ and $t>0$ , $\\Vert D \\phi ^{t} (v)\\Vert \\le be^{-at}\\Vert v \\Vert \\, ;$ For any $v\\in E^{uu}$ and $t>0$ , $\\Vert D\\phi ^{-t}(v)\\Vert \\le be^{-at}\\Vert v \\Vert \\, .$ In the above, $\\Vert \\cdot \\Vert $ is any Riemannian or Finsler metric on $M$ .",
"The subbundle $E^{ss}$ (resp.",
"$E^{uu}$ ) is called the strong stable distribution (resp.",
"strong unstable distribution).",
"It is a classical result ([10]) that $E^{ss}$ , $E^{uu}$ , $\\mathbb {R}\\cdot X \\oplus E^{ss}$ and $\\mathbb {R}\\cdot X \\oplus E^{uu}$ are integrable.",
"We denote by $\\mathcal {F}^{ss}$ , $\\mathcal {F}^{uu}$ , $\\mathcal {F}^{s} $ and $\\mathcal {F}^{u} $ the respective foliations and we call them the strong stable, strong unstable, stable and unstable foliations.",
"In all the following, if $x \\in M$ , then $\\mathcal {F}^{s} (x)$ (resp.",
"$\\mathcal {F}^{u} (x)$ ) is the leaf of the foliation $\\mathcal {F}^{s} $ (resp.",
"$\\mathcal {F}^{u} $ ) containing $x$ .",
"Another kind of flow that will appear is a pseudo-Anosov flow.",
"This type of flows is the generalization of suspensions of pseudo-Anosov diffeomorphisms.",
"They should be thought of as Anosov flows everywhere apart from a finite number of periodic orbits where the stable and unstable foliations are singular.",
"For foundational works on pseudo-Anosov flows, see [87], [88], [89].",
"Definition 5.1.2 A flow $\\psi ^t$ on a closed 3-manifold $M$ is called pseudo-Anosov if it satisfies the following conditions : For each $x\\in M$ , the flow line $t\\mapsto \\psi ^t(x)$ is $C^1$ , not a single point, and the tangent vector field is $C^0$ ; There is a finite number of periodic orbits, called singular orbits, such that the flow is smooth off of the singular orbits ; The flow lines of $\\psi ^t$ are contained in two possibly singular 2-dimensional foliations $\\Lambda ^s$ and $\\Lambda ^u$ satisfying : outside of the singular orbits, the foliations are not singular, are transverse to each other and their leaves intersect exactly along the flow lines of $\\psi ^t$ .",
"A leaf containing a singularity is homeomorphic to $P\\times [0,1] / f$ where $P$ is a $p$ -prong in the plane and $f$ is a homeomorphism from $P\\times \\lbrace 1\\rbrace $ to $P\\times \\lbrace 0\\rbrace $ .",
"We will always assume that $p\\ge 3$ ; In a stable leaf, all orbits are forward asymptotic ; in an unstable leaf, they are all backward asymptotic.",
"In the definition of Anosov flow, we asked for $ \\phi ^{t} $ to be at least $C^1$ but we will only care about smooth (i.e., $C^{\\infty }$ ) flows.",
"Note that the foliations however might not be very regular.",
"We further assume that $ \\phi ^{t} $ is transversally oriented, i.e., there exists an orientation on $M$ given by an orientation on each leaf of $\\mathcal {F}^{s} $ together with an orientation on each leaf of $\\mathcal {F}^{uu}$ .",
"Note that this hypothesis will be essential for the description we give of skewed Anosov flows, for instance, in order to have an orientation on the leaf spaces (to be defined below).",
"However, it can be achieved by taking the lift of the flow to a two-fold cover (four-fold if the manifold is not orientable).",
"Both Sergio Fenley and Thierry Barbot — at the same time and independently — started studying Anosov flow via their transversal geometry, that is via the study of the space of orbits.",
"We will follow their lead and use their works throughout this part.",
"So some of the main objects of study here will be the orbit and the leaf spaces that we define as follow.",
"Let $\\widetilde{M}$ be the universal cover of $M$ and $\\pi \\colon \\widetilde{M}\\rightarrow M$ the canonical projection.",
"The flow $ \\phi ^{t} $ and all the foliations lift to $\\widetilde{M}$ and we denote them respectively by $ \\tilde{ \\phi }^t $ , $\\widetilde{\\mathcal {F}}^{ss} $ , $\\widetilde{\\mathcal {F}}^{s} $ , $\\widetilde{\\mathcal {F}}^{uu}$ and $\\widetilde{\\mathcal {F}}^{u} $ .",
"Now we can define The orbit space of $ \\phi ^{t} $ as $\\widetilde{M}$ quotiented out by the relation “being on the same orbit of $ \\tilde{ \\phi }^t $ ”.",
"We denote it by $ \\mathcal {O} $ .",
"The stable (resp.",
"unstable) leaf space of $ \\phi ^{t} $ as $\\widetilde{M}$ quotiented out by the relation “being on the same leaf of $\\widetilde{\\mathcal {F}}^{s} $ (resp.",
"$\\widetilde{\\mathcal {F}}^{u} $ )”.",
"We denote them by $ \\mathcal {L} ^{s} $ and $ \\mathcal {L} ^{u} $ respectively.",
"Note that the foliations $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ obviously project to two transverse foliations of $ \\mathcal {O} $ .",
"We will keep the same notations for the projected foliations, hoping that it will not lead to any confusion.",
"For (pseudo)-Anosov flows in 3-manifolds, the orbit space is always homeomorphic to $\\mathbb {R}^2$ (see [17] and [55] for the Anosov case and [54] for the pseudo-Anosov case).",
"The leaf spaces $ \\mathcal {L} ^{s} $ and $ \\mathcal {L} ^{u} $ however are in general non-Hausdorff 1-manifolds, but still connected and simply-connected.",
"In this work, we are specially interested in one particular case : Definition 5.1.3 An Anosov flow is called $\\mathbb {R}$ -covered if $ \\mathcal {L} ^{s} $ and $ \\mathcal {L} ^{u} $ are homeomorphic to $\\mathbb {R}$ .",
"Remark that to prove that a flow is $\\mathbb {R}$ -covered, we just need to show that one of the leaf spaces is homeomorphic to $\\mathbb {R}$ : Theorem 5.1.4 (Barbot [16], Fenley [55]) If $ \\mathcal {L} ^{s} $ is Hausdorff, then $ \\mathcal {L} ^{u} $ is Hausdorff and vice versa.",
"Let us also recall the following result of A. Verjovsky which is fundamental for the following (and already used in the proof of the above mentioned results) : Proposition 5.1.5 (Verjovsky [109]) Let $ \\phi ^{t} $ be an Anosov flow on a 3-manifold $M$ .",
"Then : Periodic orbits of $ \\phi ^{t} $ are not null-homotopic ; Leaves of $\\widetilde{\\mathcal {F}}^{s} $ (resp.",
"$\\widetilde{\\mathcal {F}}^{u} $ ) are homeomorphic to $\\mathbb {R}^{2}$ ; $\\widetilde{M}$ is homeomorphic to $\\mathbb {R}^3$ .",
"Verjovsky's result is in fact more general ; the above stays true (with obvious modifications) for codimension one Anosov flows, i.e., Anosov flows such that (say) the strong unstable foliation $\\mathcal {F}^{uu}$ is one-dimensional.",
"In the same article, Verjovsky also proved the following result : Proposition 5.1.6 If $ \\phi ^{t} $ is a codimension 1 Anosov flow, then any leaf of $\\widetilde{\\mathcal {F}}^{uu}$ intersects at most once a leaf of $\\widetilde{\\mathcal {F}}^{s} $ .",
"Note that this result does not tell you that there always is an intersection.",
"Indeed, we say that a $\\mathbb {R}$ -covered flow is skewed if, for every leaf $L^{u} \\in \\widetilde{\\mathcal {F}}^{u} $ , there exists a leaf $L^{s}$ of $\\widetilde{\\mathcal {F}}^{s} $ such that $L^u \\cap L^s = \\emptyset $ and vice-versa.",
"We have the following : Theorem 5.1.7 (Barbot [17]) If $ \\phi ^{t} $ is a $\\mathbb {R}$ -covered flow on a 3-manifold $M$ , then either $ \\phi ^{t} $ is skewed, or it is a topologically conjugated to a suspension of an Anosov diffeomorphism.",
"An $\\mathbb {R}$ -covered Anosov flow is always transitive (see [17]), i.e., admits a dense orbit.",
"If it is skewed, it is even more : Proposition 5.1.8 (Barbot) A skewed $\\mathbb {R}$ -covered Anosov flow is topologically mixing.",
"The proof is given in Remark 2.2 of [19].",
"Topologically mixing means that given two open sets, there exists a time after which the image by the flow of one set always intersects the other (see [80]).",
"In all the rest, we will be considering skewed $\\mathbb {R}$ -covered Anosov flows.",
"Note that a geodesic flow of a negatively curved surface is a skewed $\\mathbb {R}$ -covered Anosov flow.",
"More generally, any contact Anosov flow is skewed $\\mathbb {R}$ -covered ([19])"
],
[
"Orbit space and fundamental group",
"It is easy to see that the action of the fundamental group of $M$ on $\\widetilde{M}$ projects to the orbit and leaf spaces.",
"We can even say a bit more about this action : Proposition 5.1.9 Let $ \\phi ^{t} $ be an Anosov flow on $M$ .",
"The stabilizer by $\\pi _1(M)$ of a point in $ \\mathcal {O} $ , $ \\mathcal {L} ^{s} $ or $ \\mathcal {L} ^{u} $ is either trivial or cyclic.",
"If $\\gamma \\in \\pi _1(M) $ fixes a point $O \\in \\mathcal {O} $ , then $O$ is a hyperbolic fixed point of $\\gamma $ .",
"If $\\gamma \\in \\pi _1(M) $ fixes a point $l \\in \\mathcal {L} ^{s} $ (or $ \\mathcal {L} ^{u} $ ), then $l$ is either an attractor or a repeller for the action of $\\gamma $ .",
"The proof can be found in [16] and holds once again for codimension 1 Anosov flows.",
"One fundamental remark of Fenley in [55] is the following : Proposition 5.1.10 (Fenley) Let $ \\phi ^{t} $ be a skewed, $\\mathbb {R}$ -covered Anosov flow in a 3-manifold $M$ .",
"Then, there exist two functions $\\eta ^s \\colon \\mathcal {L} ^{s} \\rightarrow \\mathcal {L} ^{u} $ and $\\eta ^u \\colon \\mathcal {L} ^{u} \\rightarrow \\mathcal {L} ^{s} $ that are monotonous, $\\pi _1(M)$ -equivariant and $C^{\\alpha }$ .",
"Furthermore, $\\eta ^u \\circ \\eta ^s$ and $\\eta ^s \\circ \\eta ^u$ are strictly increasing and we can define $\\eta \\colon \\mathcal {O} \\rightarrow \\mathcal {O} $ by $\\eta (o):= \\eta ^u \\left( \\widetilde{\\mathcal {F}}^{u} (o)\\right) \\cap \\eta ^s\\left(\\widetilde{\\mathcal {F}}^{s} (o) \\right) .$ Let $L^s \\in \\mathcal {L} ^{s} $ .",
"Define $I : = \\lbrace L^u \\in \\mathcal {L} ^{u} \\mid L^u \\cap L^s \\ne \\emptyset \\rbrace $ .",
"The set $I$ is an open, connected subset in $ \\mathcal {L} ^{s} \\simeq \\mathbb {R}$ .",
"Hence $\\partial I$ consists of 2 elements and, as $ \\phi ^{t} $ is transversally oriented, $ \\mathcal {L} ^{s} $ as a natural orientation.",
"So we can set $\\eta ^s(L^s)$ to be the largest element of $\\partial I$ .",
"The function $\\eta ^u$ can be defined in exactly the same fashion.",
"Monotonicity is trivial to check using the definition, as is the equivariance under the fundamental group.",
"Hölder continuity is done in [19].",
"Using this result, we can get a better picture of the space of orbits : Let $\\Gamma (\\eta ^s) := \\lbrace \\left( \\lambda ^s , \\eta ^s(\\lambda ^s) \\right) , \\; \\lambda ^s \\in \\mathcal {L} ^{s} \\rbrace \\subset \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u} $ and ${\\Gamma (\\eta ^u) := \\lbrace \\left( \\eta ^u(\\lambda ^u) , \\lambda ^u \\right) , \\; \\lambda ^u \\in \\mathcal {L} ^{u} \\rbrace } \\subset \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u} $ be the graphs of $\\eta ^s$ and $\\eta ^u$ respectively.",
"Then $ \\mathcal {O} $ is the subset of $ \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u} $ in between $\\Gamma (\\eta ^s)$ and $\\Gamma (\\eta ^u)$ , and the foliations $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ in $ \\mathcal {O} $ are just given by vertical and horizontal lines (see Figure REF ).",
"Figure: The space 𝒪 \\mathcal {O} seen in ℒ s ×ℒ u \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u}"
],
[
"Free homotopy class of periodic orbits",
"In [55], Fenley constructed examples of skewed $\\mathbb {R}$ -covered Anosov flows on atoroidal, not Seifert-fibered spaces.",
"So in particular, these flows are not geodesic flows.",
"It turns out that, if you consider the free homotopy class of periodic orbits, Fenley's examples behave in a very different way : Theorem 5.1.11 (Fenley [55]) If $M$ is atoroidal and not Seifert-fibered, then the free homotopy class of a periodic orbit of $ \\phi ^{t} $ contains infinitely many distinct periodic orbits.",
"For the geodesic flow of a negatively curved (Riemannian or Finsler) manifold, it is a classical result ([82]) that there is at most one periodic orbit in a free homotopy class (and exactly one geodesic for each element in the fundamental group of the manifold).",
"We give a sketch of proof of this result as it is useful for the understanding of such flows.",
"[Sketch of proof] Let $\\alpha $ be a periodic orbit of $M$ , and $\\widetilde{\\alpha }$ a lift to the universal cover.",
"There exists an element $\\gamma \\in \\pi _1(M)$ such that $\\gamma $ leaves $\\widetilde{\\alpha }$ invariant.",
"For any $i \\in \\mathbb {Z}$ , $\\eta ^i(\\widetilde{\\alpha })$ is also left invariant by $\\gamma $ (by the previous proposition) and hence its projection on $M$ is a periodic orbit.",
"Just by looking at the action of $\\gamma $ on $ \\mathcal {O} $ we can deduce that $\\eta ^i(\\widetilde{\\alpha })$ and $\\eta ^{i+1}(\\widetilde{\\alpha })$ have reverse directions (see Figure REF ).",
"Then, there is some work to show that, if $\\gamma $ was the generator of the stabilizer of $\\widetilde{\\alpha }$ for the action of $\\pi _1(M)$ , then it is the generator for any $\\eta ^i(\\widetilde{\\alpha })$ .",
"This proves that $\\eta ^{2i}(\\widetilde{\\alpha })$ are all freely homotopic.",
"Finally, using the topological assumptions, Fenley shows that the projections of $\\eta ^{2i}(\\widetilde{\\alpha })$ to $M$ are all distinct (otherwise, there would be a $\\mathbb {Z}^2$ in $\\pi _1(M)$ ).",
"In the following, we will often abuse terminology and say that an orbit $\\widetilde{\\alpha }$ of $ \\tilde{ \\phi }^t $ is periodic if its projection to $M$ is periodic, or equivalently, if $\\widetilde{\\alpha }$ is stabilized by an element of the fundamental group."
],
[
"Lozenges",
"In [55], Fenley introduced the notion of lozenges, which is a kind of basic block in the orbit space and is fundamental to the study of (not only $\\mathbb {R}$ -covered) (pseudo)-Anosov flow.",
"Figure: A lozenge with corners α\\alpha , β\\beta and sides A,B,C,DA,B,C,DDefinition 5.1.12 A lozenge $L$ in $ \\mathcal {O} $ is a subset of $ \\mathcal {O} $ such that (see Figure REF ) : There exist two points $\\alpha ,\\beta \\in L$ and four half leaves $A \\subset \\widetilde{\\mathcal {F}}^{s} (\\alpha )$ , $B \\subset \\widetilde{\\mathcal {F}}^{u} (\\alpha )$ , $C \\subset \\widetilde{\\mathcal {F}}^{s} (\\beta )$ and $D \\subset \\widetilde{\\mathcal {F}}^{u} (\\beta )$ verifying : For any $\\lambda ^s \\in \\mathcal {L} ^{s} $ , $\\lambda ^s \\cap B \\ne \\emptyset $ if and only if $\\lambda ^s \\cap D\\ne \\emptyset $ , For any $\\lambda ^u \\in \\mathcal {L} ^{u} $ , $\\lambda ^u \\cap A \\ne \\emptyset $ if and only if $\\lambda ^u \\cap C \\ne \\emptyset $ , The half-leaf $A$ does not intersect $D$ and $B$ does not intersect $C$ .",
"Then, $L := \\lbrace \\alpha ,\\beta \\rbrace \\cup \\lbrace p \\in \\mathcal {O} \\mid \\widetilde{\\mathcal {F}}^{s} (p) \\cap B \\ne \\emptyset , \\; \\widetilde{\\mathcal {F}}^{u} (p) \\cap A \\ne \\emptyset \\rbrace .$ The points $\\alpha $ and $\\beta $ are called the corners of $L$ and $A,B,C$ and $D$ are called the sides.",
"Note that in our definition, we do not count the sides as part of a lozenge, but we do include the two corners.",
"Definition 5.1.13 A chain of lozenges is a union (finite or infinite) of lozenges $L_i$ such that two consecutive lozenges $L_i$ and $L_{i+1}$ always share a corner.",
"There are basically two configurations for consecutive lozenges in a chain : either they share a side, or they don't.",
"The first case is characterized by the fact that there exists a leaf intersecting the interior of both lozenges, while it cannot happen in the second case (see Figure REF ) Figure: The two types of consecutive lozenges in a chainIn the case at hand, lozenges and chain of lozenges are pretty nice : Proposition 5.1.14 (Fenley [55]) Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow, and $C = \\bigcup L_i$ a chain of lozenges.",
"Let $p_{i-1}$ and $p_{i}$ be the two corners of $L_i$ , then $p_{i} = \\eta (p_{i-1})$ .",
"Furthermore, if $p_{i}$ is the shared corner with $L_{i+1}$ , then the union of the sides through $p_i$ of $L_i$ and $L_{i+1}$ is $\\widetilde{\\mathcal {F}}^{s} (p_i) \\cup \\widetilde{\\mathcal {F}}^{u} (p_i)$ .",
"In other words, consecutive lozenges never share a side.",
"In particular, an (un)stable leaf cannot intersect the interior of more than one lozenge in $C$ .",
"Corollary 5.1.15 If $L$ is a lozenge such that one of its corners is fixed by an element $\\gamma $ of $\\pi _1(M)$ , then $\\gamma $ stabilizes the whole lozenge and fix the other corner.",
"Now, if we assume furthermore that $M$ is atoroidal and not a Seifert-fibered space, then we have : If $C$ is a chain of lozenges with corners $p_i$ and if one corner is a periodic orbit, then every corner is a periodic orbit, the loops $\\lbrace \\pi (p_{2i})\\rbrace $ (resp.",
"$\\lbrace \\pi (p_{2i+1})\\rbrace $ ) are in the free homotopy class of $\\pi (p_0)$ (resp.",
"$\\pi (p_1)$ ) and $C$ is stabilized by the deck transformation fixing the $p_i$ .",
"Conversely, if $\\alpha $ is a periodic orbit of $ \\phi ^{t} $ and $\\widetilde{\\alpha }$ a lift to $\\widetilde{M}$ , then $\\widetilde{\\alpha }$ is a corner of an infinite maximal chain of lozenges $C$ .",
"The stabilizer of $C$ in $\\pi _1(M)$ is generated by one element.",
"We call the projection to $M$ of all the corners of $C$ the double free homotopy class of $\\alpha $ .",
"Remark 5.1.16 The difference between the free homotopy class of a periodic orbit and the double free homotopy class, is that in the latter, we forget about the orientation of the curves.",
"Figure: The action of an element γ∈π 1 (M)\\gamma \\in \\pi _1(M) stabilizing a chain of lozengesThe first assertion is easy : if $\\gamma $ stabilizes some corner, then it stabilizes the other corner because it is an image of the first one by a power of $\\eta $ and $\\eta $ commutes with deck transformations.",
"Now, this implies that $\\gamma $ stabilizes every side of $L$ , hence stabilizes $L$ .",
"The second assertion is a trivial application of the first one and the fact that free homotopy classes are obtained by powers of $\\eta $ (see [55] or the sketch of proof of Theorem REF ).",
"Finally, the only hard part of the last assertion is that the stabilizer of $C$ is some cyclic group.",
"It amounts to the same thing as we already admitted in the proof of Theorem REF , i.e., that the projections to $M$ of the corners are distinct.",
"Indeed, the image by a deck transformation of a lozenge is a lozenge and in particular it sends corner to corner.",
"Note that Fenley obviously first studied these lozenges and obtained the above results and then deduced Theorem REF .",
"I hope the reader will forgive the liberty I took with the order in which I present the results.",
"My goal was not to give complete proofs but merely an idea of what these flows look like.",
"Remark 5.1.17 Looking at the orientation of the sides of a lozenge, we can see that they come in two different types.",
"Recall that the transverse orientability of $ \\phi ^{t} $ gives an orientation on each leaf of $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ seen in $ \\mathcal {O} $ .",
"So any orbit defines two half stable leafs (positive and negative) and two half unstable leafs.",
"Now, let $p$ be a corner of a lozenge $L$ .",
"The sides of $L$ going through $p$ — call them $A$ for the stable and $B$ for the unstable — could either be both positive, both negative, or of different signs.",
"It is quite easy to see that the stable (resp.",
"unstable) side of the other corner needs to have switched sign from $B$ (resp.",
"from $A$ ).",
"So each lozenge could be of two types, either $(+,+,-,-)$ or $(+,-,-,+)$ , but evidently, all the lozenges of the same (transversally orientable) flow are of the same type ([55]).",
"In the sequel, we will consider flows such that lozenges are of the type $(+,+,-,-)$ (see Figure REF ).",
"Figure: The two possible orientations of lozenges"
],
[
"Foliations and slitherings",
"We leave for a bit Anosov flows to digress about ($\\mathbb {R}$ -covered) foliations.",
"Thurston in [107] introduced the following notion : Definition 5.1.18 The manifold $M$ slithers around the circle if there exists a fibration $s \\colon \\widetilde{M} \\rightarrow S^1$ such that $\\pi _1(M)$ acts by bundle automorphisms of $s$ , i.e., an element $\\gamma $ of $\\pi _1(M)$ sends a fiber $s^{-1}(x)$ to a possibly different fiber.",
"Such a map $s$ is called a slithering.",
"A slithering $s$ defines a foliation $\\mathcal {F}(s)$ on $M$ , just by taking the leaves to be the projections on $M$ of the connected components of $s^{-1}(x)$ .",
"Reciprocally, we say that a foliation comes from a slithering if it is obtained in that way.",
"It is immediate that foliations coming from slitherings are $\\mathbb {R}$ -covered.",
"Here, $\\mathbb {R}$ -covered means that the leaf space of the foliation is $\\mathbb {R}$ .",
"Recall that a taut foliation is a foliation that admits a closed transversal.",
"Note that the foliations we are interested in are always taut.",
"A skewed $\\mathbb {R}$ -covered Anosov flow is transitive, so each strong leaf is dense (see [93]).",
"Hence, in order to get a closed path transverse to the weak stable foliation, we can just follow a strong unstable leaf until we are close to where we began and close it up in a transverse way.",
"Candel managed to apply the classical uniformization theorem to taut foliations : Theorem 5.1.19 (Candel's Uniformization Theorem [34]) Let $\\mathcal {F}$ be a taut foliation on an atoroidal $M$ .",
"Then there exists a Riemannian metric such that its restriction to every leaf is hyperbolic.",
"Using the metric given by Candel's uniformization theorem, we can define a boundary at infinity for any leaf $\\lambda $ of $\\widetilde{\\mathcal {F}}$ that we denote by $S^1_{\\infty }(\\lambda )$ .",
"Thurston explained how to stitch those boundaries together to obtain a “universal circle” that takes into account the topological and geometrical information of the foliation.",
"Let us write $\\mathcal {L}$ for the leaf space of the foliation $\\mathcal {F}$ .",
"Definition 5.1.20 (Universal circle) Let $\\mathcal {F}$ be a $\\mathbb {R}$ -covered taut foliation on an atoroidal 3-manifold $M$ .",
"A universal circle for $\\mathcal {F}$ is a circle $S^1_{\\text{univ}}$ together with the following data : There is a faithful representation $\\rho _{\\text{univ}} \\colon \\pi _1(M) \\rightarrow \\text{Homeo}^+\\left(S^1_{\\text{univ}}\\right)\\,;$ For every leaf $\\lambda $ of $\\widetilde{\\mathcal {F}}$ , there is a monotone map $\\Psi _{\\lambda } \\colon S^1_{\\text{univ}}\\rightarrow S^1_{\\infty }(\\lambda ).$ Moreover, the map $\\Psi \\colon S^1_{\\text{univ}}\\times \\mathcal {L} \\rightarrow E_{\\infty }$ defined by $\\Psi (\\cdot , \\lambda ) := \\Psi _{\\lambda }(\\cdot )$ is continuous.",
"Here, $E_{\\infty }:= \\bigcup _{\\lambda \\in \\mathcal {L}} S^1_{\\infty }(\\lambda )$ is given the largest topology such that the endpoint map $e\\colon T^1 \\widetilde{\\mathcal {F}} \\rightarrow E_{\\infty }$ , which associates the endpoint in $ S^1_{\\infty }(\\lambda )$ of a geodesic ray defined by an element of the unit tangent bundle of the leaf $\\lambda $ , is continuous.",
"For every leaf $\\lambda $ of $\\widetilde{\\mathcal {F}}$ and any $\\gamma \\in \\pi _1(M)$ the following diagram commutes : ${S^1_{\\text{univ}}[r]^{\\rho _{\\text{univ}}(\\gamma )} [d]_{\\Psi _{\\lambda }} & S^1_{\\text{univ}}[d]^{\\Psi _{\\gamma \\cdot \\lambda }} \\\\S^1_{\\infty }(\\lambda ) [r]_{\\gamma } & S^1_{\\infty }(\\gamma \\cdot \\lambda )}$ This definition is taken from [32].",
"Note however that Calegari defines universal circles for any kind of taut foliations and hence needs a last condition that is empty for $\\mathbb {R}$ -covered foliations.",
"Theorem 5.1.21 (Thurston [107], Fenley [56], Calegari [31], Calegari and Dunfield [33]) A foliation coming from a slithering defines a universal circle.",
"The different sources for this result are in fact different generalizations of the original result due to Thurston.",
"Definition 5.1.22 (Regulating flow) A flow $\\psi ^t$ on $M$ is said to be regulating for $\\mathcal {F}$ if $\\psi ^t$ is transverse to $\\mathcal {F}$ and, when lifted to the universal cover, any orbit of $\\widetilde{\\psi }^t$ intersects every leaf of $\\widetilde{\\mathcal {F}}$ .",
"In other words, we have a homeomorphism between any orbit of $\\widetilde{\\psi }^t$ and the leaf space of $\\mathcal {F}$ .",
"The main result concerning slitherings is probably the following, but note that Fenley and Calegari obtained that result for a larger class of foliations : Theorem 5.1.23 (Thurston, Fenley, Calegari [31]) If $\\mathcal {F}$ is a foliation coming from a slithering on an atoroidal, aspherical closed 3-manifold $M$ , then it admits a pseudo-Anosov regulating flow $\\psi ^t \\colon M \\rightarrow M$ .",
"Fenley proves even more about these pseudo-Anosov regulating flows : Theorem 5.1.24 (Fenley [52], [53]) Let $\\mathcal {F}$ be a foliation coming from a slithering on an atoroidal, aspherical closed 3-manifold $M$ .",
"Then, up to topological conjugacy, there is only one regulating pseudo-Anosov flow $\\psi ^t$ .",
"Furthermore, the orbit space of $\\psi ^t$ is a disc that admits $S^1_{\\text{univ}}$ as a natural boundary.",
"In order to have a better picture of the above results, let us describe very roughly how these regulating pseudo-Anosov flows are obtained.",
"The first step is to construct a lamination of $S^1_{\\text{univ}}$ .",
"Let us recall the definition, Definition 5.1.25 Let $(a,b)$ and $(c,d)$ be two pairs of points in $S^1$ .",
"We say that they intersect if $(c,d)$ is contained in different components of $S^1\\setminus \\lbrace a,b\\rbrace $ .",
"Calegari [32] says that the two pairs are linked.",
"We will justify later (Remark REF ) why we use this name.",
"Definition 5.1.26 A lamination of $S^1$ is a closed subset of the set of unordered pairs of distinct points in $S^1$ with the property that no two elements of the lamination intersect.",
"Now, the first (big) step towards Theorem REF is : Theorem 5.1.27 (Thurston, Fenley, Calegari) If $\\mathcal {F}$ is a foliation coming from a slithering on an atoroidal, aspherical closed 3-manifold $M$ , then the associated universal circle $S^1_{\\text{univ}}$ admits two laminations $\\Lambda ^{\\pm }_{\\text{univ}}$ which are preserved under the natural action of $\\pi _1(M)$ on $S^1_{\\text{univ}}$ .",
"While proving that result, they also construct two laminations $\\widetilde{\\Lambda }^{\\pm }$ of $\\widetilde{M}$ such that they are transverse, $\\pi _1(M)$ -invariant and the intersection of $\\widetilde{\\Lambda }^{+}$ (or $\\widetilde{\\Lambda }^{+}$ ) with a leaf of $\\mathcal {F}$ is a geodesic (for the leaf-wise hyperbolic metric).",
"The regulating pseudo-Anosov flow is then obtained from $\\widetilde{\\Lambda }^{\\pm }$ by “collapsing” the complementary regions, thus obtaining two transverse singular foliations, and taking the flow to be the line field generated by the intersection.",
"Note also that, in our case, any lamination in $\\widetilde{M}$ , $\\pi _1(M)$ -invariant and obtained from the laminations $\\Lambda ^{\\pm }_{\\text{univ}}$ will give rise to a regulating pseudo-Anosov flow (see [32])."
],
[
"Skewed $\\mathbb {R}$ -covered Anosov Flows",
"One of the motivations of Thurston in [107] to study foliations coming from slithering was Fenley's examples.",
"Indeed : Proposition 5.2.1 (Thurston) Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow.",
"Then the foliations $\\mathcal {F}^{s} $ and $\\mathcal {F}^{u} $ come from slitherings.",
"Let $C:= \\mathcal {L} ^{s} / \\eta ^u \\circ \\eta ^s$ and $\\pi _C \\colon \\mathcal {L} ^{s} \\rightarrow C$ the projection.",
"As $ \\mathcal {L} ^{s} $ is homeomorphic to $\\mathbb {R}$ and $\\eta ^u \\circ \\eta ^s$ is a strictly increasing, continuous, $\\pi _1(M)$ -equivariant function (by Proposition REF ), we have that $C$ is homeomorphic to $S^1$ and that the action of $\\pi _1(M)$ on $ \\mathcal {L} ^{s} $ descends to an action by bundle automorphisms on $C$ .",
"Hence we can define $s^s \\colon \\widetilde{M}\\rightarrow C$ as the canonical projection from $\\widetilde{M}$ to $ \\mathcal {L} ^{s} $ , then to $C$ and it is clear that $\\mathcal {F}^{s} $ comes from the slithering $s^s$ .",
"The same thing applies for $\\mathcal {F}^{u} $ .",
"So a skewed $\\mathbb {R}$ -covered Anosov flow comes with two slitherings and we can apply the theory developed by Thurston.",
"Note that those foliations are linked : Proposition 5.2.2 A regulating flow for $\\mathcal {F}^{s} $ transverse to $\\mathcal {F}^{u} $ is also regulating for $\\mathcal {F}^{u} $ and vice-versa.",
"Any lift of a(n) (un)stable leaf separates $\\widetilde{M}$ into two connected components, hence the result.",
"Note that, when the skewed $\\mathbb {R}$ -covered Anosov flow we consider is just a geodesic flow on a negatively curved surface $\\Sigma $ , we have an obvious regulating flow coming to mind : here, $M$ corresponds to $H\\Sigma $ , so consider the flow that just push vectors along the fibers, without moving the base point (in other words, the flow generating rotations).",
"It is clear that this flow is regulating.",
"Now $\\widetilde{H\\Sigma }$ is (not surprisingly) homeomorphic to $\\mathbb {R}\\times \\widetilde{\\Sigma } $ , where the first coordinate is given by how much a vector is turned with respect to a fixed direction (taken as a point on the visual boundary of $\\widetilde{\\Sigma }$ ).",
"It turns out that we can always have this kind of identification whenever we consider a skewed $\\mathbb {R}$ -covered Anosov flow (see also Figure REF ) : [50][h] (-2,-1)(2,7) viewpoint=30 30 30,Decran=60 [algebraic]helice(t)cos((2*Pi/3)*t)sin((2*Pi/3)*t)t [object=courbe,r=0.001,range=0 6,linecolor=blue,linewidth=0.02,resolution=360,function=helice] [algebraic]geodesic(t)(1-t)*(-0.70710) + t*cos((2*Pi/3)*3)(1-t)*(-0.70710) + t*sin((2*Pi/3)*3)3 [algebraic]unstableleaf(s,t)(1-t)*(-sqrt(2)/2) + t*cos((2*Pi/3)*s)(1-t)*(-sqrt(2)/2) + t*sin((2*Pi/3)*s)s [object=surfaceparametree, opacity=0.7, linecolor=[cmyk]1,0,1,0.5, base= 1.875 4.875 0 1, action = draw, function=unstableleaf, linewidth=0.5,ngrid=30 1] [algebraic]stableleaf(s,t)(1-t)*cos((2*Pi/3)*(1+s)) + t*cos((2*Pi/3)*1)(1-t)*sin((2*Pi/3)*(1+s)) + t*sin((2*Pi/3)*1)1 [object=surfaceparametree, linecolor=red, base= 1.875 4.875 0 1, action = draw, function=stableleaf, linewidth=0.5,ngrid=20 1] [object=cylindrecreux,h=6,r=1,action=draw,linewidth=0.01,ngrid=3 25,opacity = 0.1,fillcolor=red,incolor=white](0,0,0) Using Proposition REF , we can represent $ \\tilde{ \\phi }^t $ in the following way : $\\widetilde{M}$ is identified with a solid cylinder where each horizontal slice is a stable leaf.",
"On a stable leaf, the orbits of $ \\tilde{ \\phi }^t $ are lines all pointing towards the same point on the boundary at infinity of the leaf.",
"We represented a stable leaf, with some orbits on it, in red.",
"The blue curve represents the point at infinity where orbits ends.",
"It is a way of seeing $ \\mathcal {L} ^{s} $ “slithers”.",
"An unstable leaf now, represented in green, is given by fixing the $(x,y)$ -coordinates (i.e., the points that project to the same point on the universal circle) and taking the lines pointing towards the blue curve.",
"Finally, the orbits of the regulating pseudo-Anosov flow $\\widetilde{\\psi }^t$ are vertical curves inside the cylinder and stabilize the foliation by vertical straight lines on the boundary.",
"Proposition 5.2.3 Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow and $\\psi ^t$ a regulating flow for both $\\mathcal {F}^{s} $ and $\\mathcal {F}^{u} $ .",
"Then, we can construct two continuous identifications of the universal cover : $I^s \\colon &\\widetilde{M} \\rightarrow \\mathcal {L} ^{s} \\times \\mathcal {O}(\\psi )\\,, & I^u \\colon &\\widetilde{M} \\rightarrow \\mathcal {L} ^{u} \\times \\mathcal {O}(\\psi ) \\,, \\\\& x \\mapsto \\left( \\widetilde{\\mathcal {F}}^{s} (x), \\widetilde{\\psi }^t (x) \\right) & & x \\mapsto \\left( \\widetilde{\\mathcal {F}}^{u} (x),\\widetilde{\\psi }^t (x) \\right)$ where $ \\mathcal {O} (\\psi )$ is the orbit space of $\\psi ^t$ .",
"Injectivity and surjectivity are given by the definition of a regulating flow.",
"Continuity follows from the fact that $\\widetilde{\\mathcal {F}}^{s} $ is a foliation of $\\widetilde{M}$ , $\\widetilde{\\psi }^t$ is transverse to it and $ \\mathcal {O} (\\psi )$ is homeomorphic to $\\mathbb {R}^2$ .",
"Note that, for any orbit of $ \\tilde{ \\phi }^t $ , using the slithering given by $\\mathcal {F}^{s} $ , it is possible to “project” this orbit onto the universal circle.",
"Indeed, an orbit determines two leaves $L^s \\in \\mathcal {L} ^{s} $ and $L^u \\in \\mathcal {L} ^{u} $ .",
"Then, using $\\eta ^u \\colon \\mathcal {L} ^{u} \\rightarrow \\mathcal {L} ^{s} $ (defined in Proposition REF ), we get a pair of points $(L^s, \\eta ^u(L^u))$ in $ \\mathcal {L} ^{s} $ and this in turn determines two distinct points in the universal circle.",
"When we consider the reciprocal image of this application, we obtain : Lemma 5.2.4 Two distinct points on the universal circle define a (countable) collection of orbits of $ \\phi ^{t} $ .",
"Instead of giving a rigorous proof of that lemma, I would like to refer the reader to Figure REF : two points on $S^1_{\\text{univ}}$ determine two vertical lines on the outside of the cylinder, and every time one of these lines intersects $ \\mathcal {L} ^{s} $ , we obtain one orbit."
],
[
"Isotopy and co-cylindrical class",
"The question that started our study of these kinds of Anosov flows was the following : Suppose you are given a skewed $\\mathbb {R}$ -covered Anosov flow in a hyperbolic 3-manifold $M$ .",
"Any periodic orbit is freely homotopic to infinitely many other orbits.",
"In other words, we have a family of knots in $M$ .",
"Then are these knots different ?",
"Here, we understand “different” in the sense of traditional knot theory, i.e., two knots are equivalent if there exists an isotopy between them.",
"And if some of these knots are indeed different, can we say more about them ?",
"That is, can we develop a kind of knot theory adapted to Anosov flows ?",
"It turns out that there is no knot theory in that case.",
"Indeed we will show below that any freely homotopic orbits are isotopic.",
"We will also study a related question : among a free homotopy class, can we say when two orbits are boundaries of an embedded cylinder ?",
"Indeed, an isotopy between two orbits gives an immersed cylinder.",
"So it seems natural to wonder whether this can be made into an embedding.",
"Furthermore, Barbot [18] (and later together with Fenley [20]) studied embeddings of tori in manifolds equipped with an Anosov flow.",
"This is in some sense the atoroidal equivalent.",
"The results in the following sections are joint work with Sergio Fenley and will be published with full details later."
],
[
"Isotopy class of periodic orbits",
"Let us start by giving the definition of isotopy we will use here : Definition 5.3.1 Two curves $c_1$ and $c_2$ in $M$ are isotopic if there exists a continuous application $H \\colon S^1 \\times [0,1] \\rightarrow M$ such that $H(S^1,0)= c_1$ , $H(S^1,0)= c_2$ and, for any $t\\in [0,1]$ , $H(S^1,t)$ is an embedding of $S^1$ in $M$ .",
"Among isotopic orbits, we define : Definition 5.3.2 Two curves $c_1$ and $c_2$ in $M$ are co-cylindrical if there exists an embedded annulus $A$ in $M$ such that $\\partial A = c_1 \\cup c_2$ .",
"Note that this is not an equivalence relation as it is clearly non-transitive.",
"However, as we will see, its study is quite interesting.",
"Let us start by considering geodesic flows for a minute.",
"In that case, the question of isotopy is trivial (because there is, at most, one periodic orbit in a free homotopy class).",
"What is not trivial however is answering the following question : given a periodic orbit $\\alpha $ , is there an embedded torus in $H\\Sigma $ containing $\\alpha $ ?",
"If you suppose that $\\alpha $ is simple, then the answer is clearly yes.",
"Indeed, just take $\\lbrace (x,v) \\in H\\Sigma \\mid x \\in \\pi (\\alpha ) \\rbrace $ .",
"If the orbit is non-simple however, it turns out that there is no such embedded torus.",
"To my everlasting surprise, this kind of condition will remain true for any skewed $\\mathbb {R}$ -covered Anosov flow.",
"But before studying co-cylindrical classes, we can use the work of Thurston, Fenley and Calegari to answer our first question and deduce that the isotopy classes are the same as the double free-homotopy classes : Theorem 5.3.3 (Barthelmé, Fenley) Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow on a closed atoroidal, not Seifert fibered, 3-manifold.",
"If $\\alpha _i$ is a double free homotopy class of periodic orbits of $ \\phi ^{t} $ , then all the $\\alpha _i$ s are isotopic.",
"[Sketch of proof] We are going to construct an isotopy between $\\alpha _0$ and $\\alpha _1$ .",
"As isotopy is an equivalence relation, it will show that all free homotopic orbits are isotopic.",
"Let $\\psi ^t$ be a regulating flow for $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ , $\\widetilde{\\alpha _{0} } $ a lift of $\\alpha _0$ to $\\widetilde{M}$ and ${\\widetilde{\\alpha _{1} } = \\eta (\\widetilde{\\alpha _{0} } )}$ .",
"For any $x \\in \\widetilde{\\alpha _{0} } $ there exists a time $T(x)$ such that $\\widetilde{\\psi }^{T(x)}(x) \\in \\widetilde{\\mathcal {F}}^{s} (\\widetilde{\\alpha _{1} } )$ .",
"Let $C:=\\lbrace \\psi ^t(\\pi (x)) \\mid x\\in \\widetilde{\\alpha _{0} } , 0\\le t \\le T(x) \\rbrace $ .",
"It is an immersed cylinder with one boundary $\\alpha _0$ and the other one a closed curve on $\\mathcal {F}^{s} (\\alpha _1)$ .",
"Let us call the second boundary component $\\alpha _1^{\\prime }$ .",
"Up to a $C^1$ modification of $\\psi ^t$ , we can show that there is only a finite number of (transverse) intersections of $\\alpha _0$ with $C$ .",
"We can therefore find a continuous time change $\\Psi ^t$ of $\\psi ^t$ such that, for some $t_1\\in \\mathbb {R}$ , $\\alpha _{1}^{\\prime } = \\Psi ^{t_1}(\\alpha _{0})$ .",
"As $\\Psi ^t$ is a flow, for any $t\\in [0,t_1]$ , $\\Psi ^t(\\alpha _0)$ is an embedded $S^1$ in $M$ .",
"We produced an isotopy from $\\alpha _0$ to $\\Psi ^{t_1}(\\alpha _0)$ .",
"Now, $\\Psi ^{t_1}(\\alpha _0)$ is freely homotopic to $\\alpha _1$ on the surface $\\mathcal {F}^{s} (\\alpha _1)$ , hence is isotopic."
],
[
"Co-cylindrical class",
"We will now show the link between having two co-cylindrical periodic orbits and simple chain of lozenges.",
"This is essentially based on Barbot's work [18].",
"In [18] (see also [20]) Barbot studied embedded tori in (toroidal) 3-manifolds supporting skewed $\\mathbb {R}$ -covered Anosov flows, showing that they could be put in a quasi-transverse position (i.e., transverse to the flow, apart from along some periodic orbits).",
"We will use his work to obtain properties of embedded annuli : Theorem 5.3.4 Let $\\alpha $ and $\\beta $ be two orbits in the same free homotopy class, choose coherent lifts $\\widetilde{\\alpha }$ and $\\widetilde{\\beta }$ , and denote by $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ the chain of lozenges between $\\widetilde{\\alpha }$ and $\\widetilde{\\beta }$ .",
"If $\\alpha $ and $\\beta $ are co-cylindrical, then $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is simple, i.e., if we denote by $(\\widetilde{\\alpha _{i} } )_{i=0 \\dots n}$ the corners of the lozenges in $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ , with $\\widetilde{\\alpha _{0} } = \\widetilde{\\alpha }$ and $\\widetilde{\\alpha _{n} } = \\widetilde{\\beta }$ , then $\\left( \\pi _1(M) \\cdot \\widetilde{\\alpha _{i} } \\right) \\cap B(\\widetilde{\\alpha }, \\widetilde{\\beta }) = \\lbrace \\widetilde{\\alpha _{i} } \\rbrace .$ Conversely, if $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is simple, then there exists an embedded annulus, called a Birkhoff annulus, with boundary $\\alpha \\cup \\beta $ .",
"Construction of an embedded Birkhoff annulus from a simple chain of lozenges is done in [18], hence proving the converse part.",
"To prove that, if $\\alpha $ and $\\beta $ are co-cylindrical, then $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is simple, we have to re-prove Lemma 7.6 of [18] (or equivalently step 1 of the proof of Theorem 6.10 of [20]) when, instead of having an embedded torus, we just have an embedded cylinder.",
"Let $C$ be an embedded cylinder such that $\\partial C = \\lbrace \\alpha , \\beta \\rbrace $ and $\\widetilde{C}$ the lift of $C$ in $\\widetilde{M}$ such that its boundary is on $\\widetilde{\\alpha }$ and $\\widetilde{\\beta }$ .",
"Let us also denote the generator of the stabilizer of $\\widetilde{\\alpha }$ by $\\gamma \\in \\pi _1(M)$ .",
"Following [18], we can construct a embedded plane $\\widetilde{C}_0$ in $\\widetilde{M}$ such that $\\widetilde{C}_0$ is $\\gamma $ -invariant, $\\widetilde{C}_0$ contains all the $\\widetilde{\\alpha _{i} } $ , $\\widetilde{C}_0$ is transverse to $ \\tilde{ \\phi }^t $ except along the $\\widetilde{\\alpha _{i} } $ , the projection of $\\widetilde{C}_0$ to $ \\mathcal {O} $ is $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ .",
"Barbot's trick to obtain such a plan is, for every lozenge in $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ , to take a simple curve $\\bar{c}$ from one corner of the lozenge to the other (for instance $\\widetilde{\\alpha _{i} } $ and $\\widetilde{\\alpha _{i+1} } $ ).",
"Then, lift $\\bar{c}$ to $\\widetilde{c}\\subset \\widetilde{M}$ such that $\\widetilde{c}$ is transverse to $ \\tilde{ \\phi }^t $ .",
"Now choose an embedded rectangle $R_i$ in $\\widetilde{M}$ such that $R_i$ is bounded by $\\widetilde{c}$ , $\\gamma \\cdot \\widetilde{c}$ , and the two pieces of $\\widetilde{\\alpha _{i} } $ and $\\widetilde{\\alpha _{i+1} } $ between the endpoints of $\\widetilde{c}$ and $\\gamma \\cdot \\widetilde{c}$ .",
"Then define $\\widetilde{C}_0$ as the orbit under $\\gamma $ of the unions of the rectangles $R_i$ .",
"From now on, we copy the proof of [20].",
"Suppose that $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is not simple.",
"Then there exist $\\widetilde{\\alpha _{i} } $ and $h \\in \\pi _1(M)$ such that $\\theta := h \\cdot \\widetilde{\\alpha _{i} } $ intersects the interior of $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ .",
"Then $\\theta $ intersects $\\widetilde{C}_0$ in a single point $p$ .",
"Let $\\theta ^+$ and $\\theta ^-$ be the two rays in $\\theta $ defined by $p$ .",
"If we denote by $\\widetilde{V}$ the subset of $\\widetilde{M}$ delimited by $\\widetilde{\\mathcal {F}}^{s} (\\widetilde{\\alpha })$ and $\\widetilde{\\mathcal {F}}^{s} (\\widetilde{\\beta })$ and containing $\\widetilde{C}_0$ , then $\\widetilde{C}_0$ separates $\\widetilde{V}$ in two components.",
"Claim 5.3.5 Either $\\theta ^+$ or $\\theta ^-$ stays a bounded distance away from $\\widetilde{C}_0$ .",
"Assume they don't : for any $R>0$ , there exist points $q_R^-,q_R^+$ on $\\theta ^-, \\theta ^+$ such that $d(q_R^{\\pm }, \\widetilde{C}_0) >R$ .",
"As $\\pi (\\widetilde{C}_0)$ and $C$ are freely homotopic, there exists $R_0$ such that $\\widetilde{C}$ is contained in the $R_0$ -neighborhood of $\\widetilde{C}_0$ .",
"Then, for any $R > 2 R_0$ , any path in $\\widetilde{V}$ joining two points $q^-$ and $q^+$ such that $d(q^{\\pm }, q_R^{\\pm })<R$ must intersect $\\widetilde{C}$ .",
"Now, $\\pi (\\theta )$ is freely homotopic to a curve in $C$ , and, as $C$ is embedded in an oriented manifold, it must be two-sided.",
"So $\\pi (\\theta )$ is homotopic to a curve disjoint from $C$ .",
"(Note that this is the only point where we use the fact that $C$ is embedded).",
"Lifting it gives a homotopy from $\\theta $ to a curve $\\theta _1$ disjoint from $\\widetilde{C}$ .",
"But homotopies move points a bounded distance away : there exists $r>0$ such that, for any $R>0$ , there are two points $m_R^{\\pm }$ on $\\theta _1$ such that $d(m_R^{\\pm }, q_R^{\\pm }) <r$ .",
"Choose $R > \\max \\lbrace 2 R_0, r \\rbrace $ , according to the above, the segment in $\\theta _1$ from $m_R^-$ to $m_R^+$ must intersect $\\widetilde{C}$ hence a contradiction.",
"We assume that $\\theta ^+$ stays at a distance $\\le a_1$ from $\\widetilde{C_0}$ .",
"Let $g\\in \\pi _1(M)$ be the generator of the stabilizer of $\\theta $ .",
"Choose a sequence $\\left(p_i\\right)$ with $p_i := g^{n_i}\\cdot p \\in \\theta ^+$ .",
"Let $\\left(q_i\\right)$ be a sequence in $\\widetilde{C}_0$ such that $d(q_i,p_i) \\le a_1$ , up to a subsequence, we can assume that $\\pi (q_i)$ converges and as $\\pi (\\widetilde{C_0})$ is compact, we can even assume that $\\pi (q_i)$ is constant.",
"Now, up to another subsequence, we can assume that there are segments $u_i$ in $\\widetilde{V}$ from $p_i$ to $q_i$ such that $\\pi (u_i)$ converges in $M$ .",
"Adjusting once again, we can assume that $\\pi (u_i)$ is constant for big enough $i$ .",
"We consider the following closed curve in $\\widetilde{V}$ : start by a segment in $\\theta ^+$ from $p_i$ to $p_k$ , $k>i$ , then follow $u_k$ , then choose a segment in $\\widetilde{C}_0$ from $q_k$ to $q_i$ and close up along $u_i$ .",
"Since $\\pi (u_i) = \\pi (u_k)$ , this shows that there exists $n\\in \\mathbb {Z}$ such that $g^n(q_i)= q_k$ .",
"Hence, for some $n \\ne 0$ , $\\widetilde{C}_0$ is left invariant by $g^n$ , which implies that $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is also invariant by $g^n$ .",
"But $g^n \\cdot \\theta = \\theta $ , so $g^n$ leaves invariant a point in the interior of a lozenge as well as the whole lozenge, which is impossible.",
"Using the theorem, we can deduce the following property of co-cylindrical class : Proposition 5.3.6 If the co-cylindrical class of one orbit is finite, then all the co-cylindrical classes in the same double free homotopy class are finite.",
"Moreover, they all have the same cardinality.",
"This result just relies on the fact that the homeomorphism $\\eta $ of $ \\mathcal {O} $ , defined by applying $\\eta ^u$ and $\\eta ^s$ to respectively the unstable and stable leaf commutes with the action of $\\pi _1(M)$ .",
"Let $\\alpha _i$ be a double free homotopy class of periodic orbit.",
"Suppose that $\\alpha _0$ has $k$ elements in its co-cylindrical class.",
"Then it implies that, for any coherent lift $\\widetilde{\\alpha _{i} } $ of the $\\alpha _i$ , the chain of lozenges $B(\\widetilde{\\alpha _{0} } , \\widetilde{\\alpha _{k} } )$ is non-simple.",
"More precisely, we have an element $h\\in \\pi _1(M)$ such that $h \\cdot \\widetilde{\\alpha _{0} } $ is in $L(\\widetilde{\\alpha _{k-1} } , \\widetilde{\\alpha _{k} } )$ , the lozenge with corners $\\widetilde{\\alpha _{k-1} } $ and $\\widetilde{\\alpha _{k} } $ .",
"Indeed, recall that $\\widetilde{\\alpha _{i} } = \\eta ^i (\\widetilde{\\alpha _{0} } )$ , so, if $h \\cdot \\widetilde{\\alpha _{i} } \\in L(\\widetilde{\\alpha _{k-1} } , \\widetilde{\\alpha _{k} } )$ , then $h \\cdot \\widetilde{\\alpha _{0} } = h \\cdot \\eta ^{-i}(\\widetilde{\\alpha _{i} } ) = \\eta ^{-i}\\left(h \\cdot \\widetilde{\\alpha _{i} } \\right) \\in \\eta ^{-i}\\left(L(\\widetilde{\\alpha _{k-1} } , \\widetilde{\\alpha _{k} } ) \\right) = L(\\widetilde{\\alpha _{k-1-i} } , \\widetilde{\\alpha _{k-i} } ).$ So $\\alpha _0$ and $\\alpha _{k-i}$ would not be co-cylindrical.",
"Then, for any $i$ , $h \\cdot \\widetilde{\\alpha _{i} } \\in L(\\widetilde{\\alpha _{k-1 + i} } , \\widetilde{\\alpha _{k+i} } )$ , which proves that the number of orbits co-cylindrical to $\\alpha _i$ is at most $k$ , and again the same argument as above shows that it is also at least $k$ ."
],
[
"Action of the fundamental group on $S^1_{\\text{univ}}$ and co-cylindrical orbits",
"Action on $S^1_{\\text{univ}}$ and co-cylindrical orbits Thanks to Thurston's work in [107] we know that the fundamental group of a 3-manifold admitting a $\\mathbb {R}$ -covered foliation acts on the universal circle implying many results about the type of group it can be, as we can see for instance in [31], [33], [56].",
"There is a remarkable link between the existence of co-cylindrical orbits and the action of $\\pi _1(M)$ on pairs of points in $S^1_{\\text{univ}}$ .",
"Definition 5.4.1 Let $(\\alpha ^+, \\alpha ^-)$ and $(\\beta ^+, \\beta ^-)$ be two pairs of points in $S^1_{\\text{univ}}$ .",
"We say that $(\\alpha ^+, \\alpha ^-)$ and $(\\beta ^+, \\beta ^-)$ intersect if, for some order on $S^1_{\\text{univ}}$ , we have $\\alpha ^- < \\beta ^- <\\alpha ^+ < \\beta ^+.$ We will say that $(\\alpha ^+, \\alpha ^-)$ self-intersects if there exists $h\\in \\pi _1(M)$ such that $(\\alpha ^+, \\alpha ^-)$ and $(h \\cdot \\alpha ^+, h \\cdot \\alpha ^-)$ intersect.",
"Proposition 5.4.2 Let $\\alpha $ be a periodic orbit of $ \\phi ^{t} $ , $\\widetilde{\\alpha }$ a lift to $\\widetilde{M}$ and $(\\alpha ^+, \\alpha ^-)$ the projection of $\\widetilde{\\alpha }$ on $S^1_{\\text{univ}}$ .",
"The co-cylindrical class of $\\alpha $ is finite if and only if $(\\alpha ^+, \\alpha ^-)$ self-intersects.",
"If the co-cylindrical of $\\alpha $ is finite, then (by Theorem REF ) the chain of lozenges containing $\\widetilde{\\alpha }$ is non-simple.",
"So, there exists $h\\in \\pi _1(M)$ such that $h\\cdot \\widetilde{\\alpha } \\in L(\\widetilde{\\alpha _{i} } , \\widetilde{\\alpha _{i+1} } )$ .",
"Projecting that lozenge to $S^1_{\\text{univ}}$ shows that $(\\alpha ^+, \\alpha ^-)$ and $h \\cdot (\\alpha ^+, \\alpha ^-)$ intersect.",
"Reciprocally, if there exists $h \\in \\pi _1(M)$ such that $(\\alpha ^+, \\alpha ^-)$ and $h \\cdot (\\alpha ^+, \\alpha ^-)$ intersect, then $h\\cdot \\widetilde{\\alpha } \\in L(\\widetilde{\\alpha _{i} } , \\widetilde{\\alpha _{i+1} } )$ for some $i$ .",
"Hence, by Theorem REF , the co-cylindrical class of $\\alpha $ must be finite.",
"Let us announce the following result with Sergio Fenley, to be published later : Theorem 5.4.3 (Barthelmé, Fenley) Let $(\\alpha ^+, \\alpha ^-)$ be the projection on $S^1_{\\text{univ}}$ of a periodic orbit $\\widetilde{\\alpha }$ of $ \\tilde{ \\phi }^t $ .",
"Then $(\\alpha ^+, \\alpha ^-)$ self-intersects.",
"The theorem is proved by seeing $S^1_{\\text{univ}}$ as the boundary at infinity of the orbit space of a regulating pseudo-Anosov flow $\\psi ^t$ and using the transitivity of such flows (Mosher [87], proved that any pseudo-Anosov flow on an atoroidal manifold is transitive).",
"Remark 5.4.4 Suppose that $(\\alpha ^+, \\alpha ^-)$ self-intersects and denote by $(\\widetilde{\\alpha _{i} } )$ the orbits in $\\widetilde{M}$ projecting to $(\\alpha ^+, \\alpha ^-)$ and $\\alpha _i = \\pi (\\widetilde{\\alpha _{i} } )$ their projection to $M$ .",
"Then, for any $i$ , there exist a $j$ and a $t$ such that $\\psi ^t(\\alpha _i) \\cap \\alpha _j \\ne \\emptyset $ .",
"By flowing one orbit we get an actual intersection.",
"If we consider the geodesic flow case now, there is also a natural circle at infinity.",
"Just take the visual boundary $\\widetilde{\\Sigma }(\\infty )$ and the fundamental group $\\pi _1(H\\Sigma )$ naturally acts on it.",
"So, for a pair $(\\alpha ^+, \\alpha ^-)$ in $\\widetilde{\\Sigma }(\\infty )$ , to self-intersect in that case means that the only geodesic in $\\Sigma $ such that a lift of it has endpoints $(\\alpha ^+, \\alpha ^-)$ is non-simple.",
"Hence, in the geodesic flow case, there exist points on $\\widetilde{\\Sigma }(\\infty )$ representing a periodic orbit that does not self-intersect, in contrast with the atoroidal case we studied here.",
"As a corollary of Proposition REF and Theorem REF , we obtain : Theorem 5.4.5 (Barthelmé, Fenley) Every co-cylindrical class is finite.",
"Note that it is still an open question whether a co-cylindrical class can be non-trivial.",
"We only know that some are : Proposition 5.4.6 There exist periodic orbits of $ \\phi ^{t} $ with trivial co-cylindrical class.",
"Remark 5.4.7 For such an orbit, Proposition REF shows that every other orbit in the double free homotopy class must also have a trivial co-cylindrical class.",
"Let $V$ be a flow box of $ \\phi ^{t} $ , as $ \\phi ^{t} $ is transitive, we can pick a long segment of a dense orbit that $\\varepsilon $ -fills $V$ .",
"Then, by the Anosov Closing lemma (see [80]), we get a periodic orbit $\\alpha $ that $2\\varepsilon $ -fills $V$ .",
"Now, choose $x$ on one of the connected components of $\\alpha \\cap V$ .",
"If $\\varepsilon $ was chosen small enough, then there must exist $y$ on another connected component of $\\alpha \\cap V$ such that there is a close path $c$ staying in $V$, starting at $x$ going through the positive stable leaf of $x$ , then the negative unstable leaf of $y$ , then the negative stable leaf of $y$ and finally close up along the positive stable leaf of $x$ .",
"If we lift the path $c$ to the universal cover of $M$ and project it to the orbit space $ \\mathcal {O} $ , as $V$ has no topology, we see that the projection of the lift of $y$ must be inside the lozenge determined by the lift of $x$ (remember that we chose our flow so that the lozenges orientation is $(+,+,-,-)$ , otherwise, we would have to modify our path $c$ , see Figure REF ).",
"Hence the lozenge is non-simple and therefore the co-cylindrical class of $\\alpha $ is trivial."
],
[
"Some open questions",
"I wanted to end this dissertation with a list of questions I have about skewed $\\mathbb {R}$ -covered Anosov flow, because even if a lot of things are known, thanks mostly to T. Barbot and S. Fenley, the things that are unknown justify, at least in my view, a continuation of their study.",
"Let's start with the “topological” questions : P. Foulon and B. Hasselblatt [63] have constructed contact Anosov flows (i.e., Anosov flow preserving a contact form) on not Seifert-fibered spaces and it seems very likely that their construction often yields hyperbolic manifolds.",
"Now, contact Anosov flows are skewed and $\\mathbb {R}$ -covered (see [19]) and are the “nicest” flows from a regularity point of view (see [64]).",
"In [20], Barbot and Fenley showed that skewed $\\mathbb {R}$ -covered Anosov flows in Seifert-fibered spaces are (up to a finite cover) topologically conjugated to a geodesic flow on a closed surface.",
"A natural question is then : Is a skewed $\\mathbb {R}$ -covered Anosov flow on an atoroidal manifold always topologically conjugate to a contact Anosov flow ?",
"Indeed, it seems that the structure of $\\widetilde{M}$ given by the regulating pseudo-Anosov flow (see proposition REF ) is very rigid, so can we use that to show that $ \\phi ^{t} $ is topologically contact ?",
"(See [19] for a definition.)",
"And, from there, can we get an actual topological conjugacy ?",
"Given a continuous map $s \\colon \\mathbb {R}\\times \\mathbb {H}^2 \\rightarrow S^1$ such that, for all $t \\in \\mathbb {R}$ , $s(t, \\cdot )$ is constant and for any $x \\in \\mathbb {H}^2$ , $s(\\cdot , x)$ is strictly monotone, we can construct, using Figure REF , an Anosov flow on $\\mathbb {R}\\times \\mathbb {H}^2$ .",
"Now suppose that we are given a discreet group $\\Gamma $ acting in a “good” way on $\\mathbb {R}\\times \\mathbb {H}^2$ , then is the quotient flow a contact Anosov flow ?",
"And if that is true, then, can we get all contact Anosov flows on atoroidal 3-manifolds in this fashion ?",
"Finally, there are a lot of ergodic theoretical questions for these flows : A classical question (initiated by Bowen and Margulis) for Anosov flows is to count the number of closed orbits of length less than $R$ and find an asymptotic equivalent when $R$ gets big.",
"In [91], Parry and Policott prove that this number is asymptotic to $e^{hR}/hR$ where $h$ is the topological entropy.",
"Following them, Katsuda and Sunada [81] answered the question of counting closed orbits inside an homology class.",
"So it seems natural to ask, in the case of skewed $\\mathbb {R}$ -covered Anosov flows on atoroidal manifolds, whether we can give an equivalent to the number of closed orbits of length less than $R$ inside a free homotopy class.",
"A somewhat related question (asked by M. Crampon) is the following : let $\\alpha _i$ be the orbits in a free homotopy class of $ \\phi ^{t} $ , denote by $l_i$ the length of $\\alpha _i$ and $\\delta _{\\alpha _i}$ the Dirac measure on $\\alpha _i$ .",
"Let $\\mu _n := \\sum _{ |i| \\le n } \\frac{\\delta _{\\alpha _i}}{ l_i }\\,.$ The sequence $(\\mu _n)$ admits at least one weak limit $\\mu $ .",
"Can we show that this limit is unique and ergodic ?",
"If that is so, then what is the measure-entropy of $\\mu $ ?",
"Can we link that entropy to the previous counting question ?",
"french Je ne sais pas le reste."
],
[
"Basics on Anosov flows",
"In all this part, we will be interested in Anosov flows on closed 3-manifolds, but we can state the definition in any dimension : Definition 5.1.1 Let $M$ be a compact manifold and $ \\phi ^{t} \\colon M \\rightarrow M$ a $C^1$ flow on $M$ .",
"The flow $ \\phi ^{t} $ is called Anosov if there exists a splitting of the tangent bundle ${TM = \\mathbb {R}\\cdot X \\oplus E^{ss} \\oplus E^{uu}}$ preserved by $D \\phi ^{t} $ and two constants $a,b >0$ such that : $X$ is the generating vector field of $ \\phi ^{t} $ ; For any $v\\in E^{ss}$ and $t>0$ , $\\Vert D \\phi ^{t} (v)\\Vert \\le be^{-at}\\Vert v \\Vert \\, ;$ For any $v\\in E^{uu}$ and $t>0$ , $\\Vert D\\phi ^{-t}(v)\\Vert \\le be^{-at}\\Vert v \\Vert \\, .$ In the above, $\\Vert \\cdot \\Vert $ is any Riemannian or Finsler metric on $M$ .",
"The subbundle $E^{ss}$ (resp.",
"$E^{uu}$ ) is called the strong stable distribution (resp.",
"strong unstable distribution).",
"It is a classical result ([10]) that $E^{ss}$ , $E^{uu}$ , $\\mathbb {R}\\cdot X \\oplus E^{ss}$ and $\\mathbb {R}\\cdot X \\oplus E^{uu}$ are integrable.",
"We denote by $\\mathcal {F}^{ss}$ , $\\mathcal {F}^{uu}$ , $\\mathcal {F}^{s} $ and $\\mathcal {F}^{u} $ the respective foliations and we call them the strong stable, strong unstable, stable and unstable foliations.",
"In all the following, if $x \\in M$ , then $\\mathcal {F}^{s} (x)$ (resp.",
"$\\mathcal {F}^{u} (x)$ ) is the leaf of the foliation $\\mathcal {F}^{s} $ (resp.",
"$\\mathcal {F}^{u} $ ) containing $x$ .",
"Another kind of flow that will appear is a pseudo-Anosov flow.",
"This type of flows is the generalization of suspensions of pseudo-Anosov diffeomorphisms.",
"They should be thought of as Anosov flows everywhere apart from a finite number of periodic orbits where the stable and unstable foliations are singular.",
"For foundational works on pseudo-Anosov flows, see [87], [88], [89].",
"Definition 5.1.2 A flow $\\psi ^t$ on a closed 3-manifold $M$ is called pseudo-Anosov if it satisfies the following conditions : For each $x\\in M$ , the flow line $t\\mapsto \\psi ^t(x)$ is $C^1$ , not a single point, and the tangent vector field is $C^0$ ; There is a finite number of periodic orbits, called singular orbits, such that the flow is smooth off of the singular orbits ; The flow lines of $\\psi ^t$ are contained in two possibly singular 2-dimensional foliations $\\Lambda ^s$ and $\\Lambda ^u$ satisfying : outside of the singular orbits, the foliations are not singular, are transverse to each other and their leaves intersect exactly along the flow lines of $\\psi ^t$ .",
"A leaf containing a singularity is homeomorphic to $P\\times [0,1] / f$ where $P$ is a $p$ -prong in the plane and $f$ is a homeomorphism from $P\\times \\lbrace 1\\rbrace $ to $P\\times \\lbrace 0\\rbrace $ .",
"We will always assume that $p\\ge 3$ ; In a stable leaf, all orbits are forward asymptotic ; in an unstable leaf, they are all backward asymptotic.",
"In the definition of Anosov flow, we asked for $ \\phi ^{t} $ to be at least $C^1$ but we will only care about smooth (i.e., $C^{\\infty }$ ) flows.",
"Note that the foliations however might not be very regular.",
"We further assume that $ \\phi ^{t} $ is transversally oriented, i.e., there exists an orientation on $M$ given by an orientation on each leaf of $\\mathcal {F}^{s} $ together with an orientation on each leaf of $\\mathcal {F}^{uu}$ .",
"Note that this hypothesis will be essential for the description we give of skewed Anosov flows, for instance, in order to have an orientation on the leaf spaces (to be defined below).",
"However, it can be achieved by taking the lift of the flow to a two-fold cover (four-fold if the manifold is not orientable).",
"Both Sergio Fenley and Thierry Barbot — at the same time and independently — started studying Anosov flow via their transversal geometry, that is via the study of the space of orbits.",
"We will follow their lead and use their works throughout this part.",
"So some of the main objects of study here will be the orbit and the leaf spaces that we define as follow.",
"Let $\\widetilde{M}$ be the universal cover of $M$ and $\\pi \\colon \\widetilde{M}\\rightarrow M$ the canonical projection.",
"The flow $ \\phi ^{t} $ and all the foliations lift to $\\widetilde{M}$ and we denote them respectively by $ \\tilde{ \\phi }^t $ , $\\widetilde{\\mathcal {F}}^{ss} $ , $\\widetilde{\\mathcal {F}}^{s} $ , $\\widetilde{\\mathcal {F}}^{uu}$ and $\\widetilde{\\mathcal {F}}^{u} $ .",
"Now we can define The orbit space of $ \\phi ^{t} $ as $\\widetilde{M}$ quotiented out by the relation “being on the same orbit of $ \\tilde{ \\phi }^t $ ”.",
"We denote it by $ \\mathcal {O} $ .",
"The stable (resp.",
"unstable) leaf space of $ \\phi ^{t} $ as $\\widetilde{M}$ quotiented out by the relation “being on the same leaf of $\\widetilde{\\mathcal {F}}^{s} $ (resp.",
"$\\widetilde{\\mathcal {F}}^{u} $ )”.",
"We denote them by $ \\mathcal {L} ^{s} $ and $ \\mathcal {L} ^{u} $ respectively.",
"Note that the foliations $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ obviously project to two transverse foliations of $ \\mathcal {O} $ .",
"We will keep the same notations for the projected foliations, hoping that it will not lead to any confusion.",
"For (pseudo)-Anosov flows in 3-manifolds, the orbit space is always homeomorphic to $\\mathbb {R}^2$ (see [17] and [55] for the Anosov case and [54] for the pseudo-Anosov case).",
"The leaf spaces $ \\mathcal {L} ^{s} $ and $ \\mathcal {L} ^{u} $ however are in general non-Hausdorff 1-manifolds, but still connected and simply-connected.",
"In this work, we are specially interested in one particular case : Definition 5.1.3 An Anosov flow is called $\\mathbb {R}$ -covered if $ \\mathcal {L} ^{s} $ and $ \\mathcal {L} ^{u} $ are homeomorphic to $\\mathbb {R}$ .",
"Remark that to prove that a flow is $\\mathbb {R}$ -covered, we just need to show that one of the leaf spaces is homeomorphic to $\\mathbb {R}$ : Theorem 5.1.4 (Barbot [16], Fenley [55]) If $ \\mathcal {L} ^{s} $ is Hausdorff, then $ \\mathcal {L} ^{u} $ is Hausdorff and vice versa.",
"Let us also recall the following result of A. Verjovsky which is fundamental for the following (and already used in the proof of the above mentioned results) : Proposition 5.1.5 (Verjovsky [109]) Let $ \\phi ^{t} $ be an Anosov flow on a 3-manifold $M$ .",
"Then : Periodic orbits of $ \\phi ^{t} $ are not null-homotopic ; Leaves of $\\widetilde{\\mathcal {F}}^{s} $ (resp.",
"$\\widetilde{\\mathcal {F}}^{u} $ ) are homeomorphic to $\\mathbb {R}^{2}$ ; $\\widetilde{M}$ is homeomorphic to $\\mathbb {R}^3$ .",
"Verjovsky's result is in fact more general ; the above stays true (with obvious modifications) for codimension one Anosov flows, i.e., Anosov flows such that (say) the strong unstable foliation $\\mathcal {F}^{uu}$ is one-dimensional.",
"In the same article, Verjovsky also proved the following result : Proposition 5.1.6 If $ \\phi ^{t} $ is a codimension 1 Anosov flow, then any leaf of $\\widetilde{\\mathcal {F}}^{uu}$ intersects at most once a leaf of $\\widetilde{\\mathcal {F}}^{s} $ .",
"Note that this result does not tell you that there always is an intersection.",
"Indeed, we say that a $\\mathbb {R}$ -covered flow is skewed if, for every leaf $L^{u} \\in \\widetilde{\\mathcal {F}}^{u} $ , there exists a leaf $L^{s}$ of $\\widetilde{\\mathcal {F}}^{s} $ such that $L^u \\cap L^s = \\emptyset $ and vice-versa.",
"We have the following : Theorem 5.1.7 (Barbot [17]) If $ \\phi ^{t} $ is a $\\mathbb {R}$ -covered flow on a 3-manifold $M$ , then either $ \\phi ^{t} $ is skewed, or it is a topologically conjugated to a suspension of an Anosov diffeomorphism.",
"An $\\mathbb {R}$ -covered Anosov flow is always transitive (see [17]), i.e., admits a dense orbit.",
"If it is skewed, it is even more : Proposition 5.1.8 (Barbot) A skewed $\\mathbb {R}$ -covered Anosov flow is topologically mixing.",
"The proof is given in Remark 2.2 of [19].",
"Topologically mixing means that given two open sets, there exists a time after which the image by the flow of one set always intersects the other (see [80]).",
"In all the rest, we will be considering skewed $\\mathbb {R}$ -covered Anosov flows.",
"Note that a geodesic flow of a negatively curved surface is a skewed $\\mathbb {R}$ -covered Anosov flow.",
"More generally, any contact Anosov flow is skewed $\\mathbb {R}$ -covered ([19])"
],
[
"Orbit space and fundamental group",
"It is easy to see that the action of the fundamental group of $M$ on $\\widetilde{M}$ projects to the orbit and leaf spaces.",
"We can even say a bit more about this action : Proposition 5.1.9 Let $ \\phi ^{t} $ be an Anosov flow on $M$ .",
"The stabilizer by $\\pi _1(M)$ of a point in $ \\mathcal {O} $ , $ \\mathcal {L} ^{s} $ or $ \\mathcal {L} ^{u} $ is either trivial or cyclic.",
"If $\\gamma \\in \\pi _1(M) $ fixes a point $O \\in \\mathcal {O} $ , then $O$ is a hyperbolic fixed point of $\\gamma $ .",
"If $\\gamma \\in \\pi _1(M) $ fixes a point $l \\in \\mathcal {L} ^{s} $ (or $ \\mathcal {L} ^{u} $ ), then $l$ is either an attractor or a repeller for the action of $\\gamma $ .",
"The proof can be found in [16] and holds once again for codimension 1 Anosov flows.",
"One fundamental remark of Fenley in [55] is the following : Proposition 5.1.10 (Fenley) Let $ \\phi ^{t} $ be a skewed, $\\mathbb {R}$ -covered Anosov flow in a 3-manifold $M$ .",
"Then, there exist two functions $\\eta ^s \\colon \\mathcal {L} ^{s} \\rightarrow \\mathcal {L} ^{u} $ and $\\eta ^u \\colon \\mathcal {L} ^{u} \\rightarrow \\mathcal {L} ^{s} $ that are monotonous, $\\pi _1(M)$ -equivariant and $C^{\\alpha }$ .",
"Furthermore, $\\eta ^u \\circ \\eta ^s$ and $\\eta ^s \\circ \\eta ^u$ are strictly increasing and we can define $\\eta \\colon \\mathcal {O} \\rightarrow \\mathcal {O} $ by $\\eta (o):= \\eta ^u \\left( \\widetilde{\\mathcal {F}}^{u} (o)\\right) \\cap \\eta ^s\\left(\\widetilde{\\mathcal {F}}^{s} (o) \\right) .$ Let $L^s \\in \\mathcal {L} ^{s} $ .",
"Define $I : = \\lbrace L^u \\in \\mathcal {L} ^{u} \\mid L^u \\cap L^s \\ne \\emptyset \\rbrace $ .",
"The set $I$ is an open, connected subset in $ \\mathcal {L} ^{s} \\simeq \\mathbb {R}$ .",
"Hence $\\partial I$ consists of 2 elements and, as $ \\phi ^{t} $ is transversally oriented, $ \\mathcal {L} ^{s} $ as a natural orientation.",
"So we can set $\\eta ^s(L^s)$ to be the largest element of $\\partial I$ .",
"The function $\\eta ^u$ can be defined in exactly the same fashion.",
"Monotonicity is trivial to check using the definition, as is the equivariance under the fundamental group.",
"Hölder continuity is done in [19].",
"Using this result, we can get a better picture of the space of orbits : Let $\\Gamma (\\eta ^s) := \\lbrace \\left( \\lambda ^s , \\eta ^s(\\lambda ^s) \\right) , \\; \\lambda ^s \\in \\mathcal {L} ^{s} \\rbrace \\subset \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u} $ and ${\\Gamma (\\eta ^u) := \\lbrace \\left( \\eta ^u(\\lambda ^u) , \\lambda ^u \\right) , \\; \\lambda ^u \\in \\mathcal {L} ^{u} \\rbrace } \\subset \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u} $ be the graphs of $\\eta ^s$ and $\\eta ^u$ respectively.",
"Then $ \\mathcal {O} $ is the subset of $ \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u} $ in between $\\Gamma (\\eta ^s)$ and $\\Gamma (\\eta ^u)$ , and the foliations $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ in $ \\mathcal {O} $ are just given by vertical and horizontal lines (see Figure REF ).",
"Figure: The space 𝒪 \\mathcal {O} seen in ℒ s ×ℒ u \\mathcal {L} ^{s} \\times \\mathcal {L} ^{u}"
],
[
"Free homotopy class of periodic orbits",
"In [55], Fenley constructed examples of skewed $\\mathbb {R}$ -covered Anosov flows on atoroidal, not Seifert-fibered spaces.",
"So in particular, these flows are not geodesic flows.",
"It turns out that, if you consider the free homotopy class of periodic orbits, Fenley's examples behave in a very different way : Theorem 5.1.11 (Fenley [55]) If $M$ is atoroidal and not Seifert-fibered, then the free homotopy class of a periodic orbit of $ \\phi ^{t} $ contains infinitely many distinct periodic orbits.",
"For the geodesic flow of a negatively curved (Riemannian or Finsler) manifold, it is a classical result ([82]) that there is at most one periodic orbit in a free homotopy class (and exactly one geodesic for each element in the fundamental group of the manifold).",
"We give a sketch of proof of this result as it is useful for the understanding of such flows.",
"[Sketch of proof] Let $\\alpha $ be a periodic orbit of $M$ , and $\\widetilde{\\alpha }$ a lift to the universal cover.",
"There exists an element $\\gamma \\in \\pi _1(M)$ such that $\\gamma $ leaves $\\widetilde{\\alpha }$ invariant.",
"For any $i \\in \\mathbb {Z}$ , $\\eta ^i(\\widetilde{\\alpha })$ is also left invariant by $\\gamma $ (by the previous proposition) and hence its projection on $M$ is a periodic orbit.",
"Just by looking at the action of $\\gamma $ on $ \\mathcal {O} $ we can deduce that $\\eta ^i(\\widetilde{\\alpha })$ and $\\eta ^{i+1}(\\widetilde{\\alpha })$ have reverse directions (see Figure REF ).",
"Then, there is some work to show that, if $\\gamma $ was the generator of the stabilizer of $\\widetilde{\\alpha }$ for the action of $\\pi _1(M)$ , then it is the generator for any $\\eta ^i(\\widetilde{\\alpha })$ .",
"This proves that $\\eta ^{2i}(\\widetilde{\\alpha })$ are all freely homotopic.",
"Finally, using the topological assumptions, Fenley shows that the projections of $\\eta ^{2i}(\\widetilde{\\alpha })$ to $M$ are all distinct (otherwise, there would be a $\\mathbb {Z}^2$ in $\\pi _1(M)$ ).",
"In the following, we will often abuse terminology and say that an orbit $\\widetilde{\\alpha }$ of $ \\tilde{ \\phi }^t $ is periodic if its projection to $M$ is periodic, or equivalently, if $\\widetilde{\\alpha }$ is stabilized by an element of the fundamental group."
],
[
"Lozenges",
"In [55], Fenley introduced the notion of lozenges, which is a kind of basic block in the orbit space and is fundamental to the study of (not only $\\mathbb {R}$ -covered) (pseudo)-Anosov flow.",
"Figure: A lozenge with corners α\\alpha , β\\beta and sides A,B,C,DA,B,C,DDefinition 5.1.12 A lozenge $L$ in $ \\mathcal {O} $ is a subset of $ \\mathcal {O} $ such that (see Figure REF ) : There exist two points $\\alpha ,\\beta \\in L$ and four half leaves $A \\subset \\widetilde{\\mathcal {F}}^{s} (\\alpha )$ , $B \\subset \\widetilde{\\mathcal {F}}^{u} (\\alpha )$ , $C \\subset \\widetilde{\\mathcal {F}}^{s} (\\beta )$ and $D \\subset \\widetilde{\\mathcal {F}}^{u} (\\beta )$ verifying : For any $\\lambda ^s \\in \\mathcal {L} ^{s} $ , $\\lambda ^s \\cap B \\ne \\emptyset $ if and only if $\\lambda ^s \\cap D\\ne \\emptyset $ , For any $\\lambda ^u \\in \\mathcal {L} ^{u} $ , $\\lambda ^u \\cap A \\ne \\emptyset $ if and only if $\\lambda ^u \\cap C \\ne \\emptyset $ , The half-leaf $A$ does not intersect $D$ and $B$ does not intersect $C$ .",
"Then, $L := \\lbrace \\alpha ,\\beta \\rbrace \\cup \\lbrace p \\in \\mathcal {O} \\mid \\widetilde{\\mathcal {F}}^{s} (p) \\cap B \\ne \\emptyset , \\; \\widetilde{\\mathcal {F}}^{u} (p) \\cap A \\ne \\emptyset \\rbrace .$ The points $\\alpha $ and $\\beta $ are called the corners of $L$ and $A,B,C$ and $D$ are called the sides.",
"Note that in our definition, we do not count the sides as part of a lozenge, but we do include the two corners.",
"Definition 5.1.13 A chain of lozenges is a union (finite or infinite) of lozenges $L_i$ such that two consecutive lozenges $L_i$ and $L_{i+1}$ always share a corner.",
"There are basically two configurations for consecutive lozenges in a chain : either they share a side, or they don't.",
"The first case is characterized by the fact that there exists a leaf intersecting the interior of both lozenges, while it cannot happen in the second case (see Figure REF ) Figure: The two types of consecutive lozenges in a chainIn the case at hand, lozenges and chain of lozenges are pretty nice : Proposition 5.1.14 (Fenley [55]) Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow, and $C = \\bigcup L_i$ a chain of lozenges.",
"Let $p_{i-1}$ and $p_{i}$ be the two corners of $L_i$ , then $p_{i} = \\eta (p_{i-1})$ .",
"Furthermore, if $p_{i}$ is the shared corner with $L_{i+1}$ , then the union of the sides through $p_i$ of $L_i$ and $L_{i+1}$ is $\\widetilde{\\mathcal {F}}^{s} (p_i) \\cup \\widetilde{\\mathcal {F}}^{u} (p_i)$ .",
"In other words, consecutive lozenges never share a side.",
"In particular, an (un)stable leaf cannot intersect the interior of more than one lozenge in $C$ .",
"Corollary 5.1.15 If $L$ is a lozenge such that one of its corners is fixed by an element $\\gamma $ of $\\pi _1(M)$ , then $\\gamma $ stabilizes the whole lozenge and fix the other corner.",
"Now, if we assume furthermore that $M$ is atoroidal and not a Seifert-fibered space, then we have : If $C$ is a chain of lozenges with corners $p_i$ and if one corner is a periodic orbit, then every corner is a periodic orbit, the loops $\\lbrace \\pi (p_{2i})\\rbrace $ (resp.",
"$\\lbrace \\pi (p_{2i+1})\\rbrace $ ) are in the free homotopy class of $\\pi (p_0)$ (resp.",
"$\\pi (p_1)$ ) and $C$ is stabilized by the deck transformation fixing the $p_i$ .",
"Conversely, if $\\alpha $ is a periodic orbit of $ \\phi ^{t} $ and $\\widetilde{\\alpha }$ a lift to $\\widetilde{M}$ , then $\\widetilde{\\alpha }$ is a corner of an infinite maximal chain of lozenges $C$ .",
"The stabilizer of $C$ in $\\pi _1(M)$ is generated by one element.",
"We call the projection to $M$ of all the corners of $C$ the double free homotopy class of $\\alpha $ .",
"Remark 5.1.16 The difference between the free homotopy class of a periodic orbit and the double free homotopy class, is that in the latter, we forget about the orientation of the curves.",
"Figure: The action of an element γ∈π 1 (M)\\gamma \\in \\pi _1(M) stabilizing a chain of lozengesThe first assertion is easy : if $\\gamma $ stabilizes some corner, then it stabilizes the other corner because it is an image of the first one by a power of $\\eta $ and $\\eta $ commutes with deck transformations.",
"Now, this implies that $\\gamma $ stabilizes every side of $L$ , hence stabilizes $L$ .",
"The second assertion is a trivial application of the first one and the fact that free homotopy classes are obtained by powers of $\\eta $ (see [55] or the sketch of proof of Theorem REF ).",
"Finally, the only hard part of the last assertion is that the stabilizer of $C$ is some cyclic group.",
"It amounts to the same thing as we already admitted in the proof of Theorem REF , i.e., that the projections to $M$ of the corners are distinct.",
"Indeed, the image by a deck transformation of a lozenge is a lozenge and in particular it sends corner to corner.",
"Note that Fenley obviously first studied these lozenges and obtained the above results and then deduced Theorem REF .",
"I hope the reader will forgive the liberty I took with the order in which I present the results.",
"My goal was not to give complete proofs but merely an idea of what these flows look like.",
"Remark 5.1.17 Looking at the orientation of the sides of a lozenge, we can see that they come in two different types.",
"Recall that the transverse orientability of $ \\phi ^{t} $ gives an orientation on each leaf of $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ seen in $ \\mathcal {O} $ .",
"So any orbit defines two half stable leafs (positive and negative) and two half unstable leafs.",
"Now, let $p$ be a corner of a lozenge $L$ .",
"The sides of $L$ going through $p$ — call them $A$ for the stable and $B$ for the unstable — could either be both positive, both negative, or of different signs.",
"It is quite easy to see that the stable (resp.",
"unstable) side of the other corner needs to have switched sign from $B$ (resp.",
"from $A$ ).",
"So each lozenge could be of two types, either $(+,+,-,-)$ or $(+,-,-,+)$ , but evidently, all the lozenges of the same (transversally orientable) flow are of the same type ([55]).",
"In the sequel, we will consider flows such that lozenges are of the type $(+,+,-,-)$ (see Figure REF ).",
"Figure: The two possible orientations of lozenges"
],
[
"Foliations and slitherings",
"We leave for a bit Anosov flows to digress about ($\\mathbb {R}$ -covered) foliations.",
"Thurston in [107] introduced the following notion : Definition 5.1.18 The manifold $M$ slithers around the circle if there exists a fibration $s \\colon \\widetilde{M} \\rightarrow S^1$ such that $\\pi _1(M)$ acts by bundle automorphisms of $s$ , i.e., an element $\\gamma $ of $\\pi _1(M)$ sends a fiber $s^{-1}(x)$ to a possibly different fiber.",
"Such a map $s$ is called a slithering.",
"A slithering $s$ defines a foliation $\\mathcal {F}(s)$ on $M$ , just by taking the leaves to be the projections on $M$ of the connected components of $s^{-1}(x)$ .",
"Reciprocally, we say that a foliation comes from a slithering if it is obtained in that way.",
"It is immediate that foliations coming from slitherings are $\\mathbb {R}$ -covered.",
"Here, $\\mathbb {R}$ -covered means that the leaf space of the foliation is $\\mathbb {R}$ .",
"Recall that a taut foliation is a foliation that admits a closed transversal.",
"Note that the foliations we are interested in are always taut.",
"A skewed $\\mathbb {R}$ -covered Anosov flow is transitive, so each strong leaf is dense (see [93]).",
"Hence, in order to get a closed path transverse to the weak stable foliation, we can just follow a strong unstable leaf until we are close to where we began and close it up in a transverse way.",
"Candel managed to apply the classical uniformization theorem to taut foliations : Theorem 5.1.19 (Candel's Uniformization Theorem [34]) Let $\\mathcal {F}$ be a taut foliation on an atoroidal $M$ .",
"Then there exists a Riemannian metric such that its restriction to every leaf is hyperbolic.",
"Using the metric given by Candel's uniformization theorem, we can define a boundary at infinity for any leaf $\\lambda $ of $\\widetilde{\\mathcal {F}}$ that we denote by $S^1_{\\infty }(\\lambda )$ .",
"Thurston explained how to stitch those boundaries together to obtain a “universal circle” that takes into account the topological and geometrical information of the foliation.",
"Let us write $\\mathcal {L}$ for the leaf space of the foliation $\\mathcal {F}$ .",
"Definition 5.1.20 (Universal circle) Let $\\mathcal {F}$ be a $\\mathbb {R}$ -covered taut foliation on an atoroidal 3-manifold $M$ .",
"A universal circle for $\\mathcal {F}$ is a circle $S^1_{\\text{univ}}$ together with the following data : There is a faithful representation $\\rho _{\\text{univ}} \\colon \\pi _1(M) \\rightarrow \\text{Homeo}^+\\left(S^1_{\\text{univ}}\\right)\\,;$ For every leaf $\\lambda $ of $\\widetilde{\\mathcal {F}}$ , there is a monotone map $\\Psi _{\\lambda } \\colon S^1_{\\text{univ}}\\rightarrow S^1_{\\infty }(\\lambda ).$ Moreover, the map $\\Psi \\colon S^1_{\\text{univ}}\\times \\mathcal {L} \\rightarrow E_{\\infty }$ defined by $\\Psi (\\cdot , \\lambda ) := \\Psi _{\\lambda }(\\cdot )$ is continuous.",
"Here, $E_{\\infty }:= \\bigcup _{\\lambda \\in \\mathcal {L}} S^1_{\\infty }(\\lambda )$ is given the largest topology such that the endpoint map $e\\colon T^1 \\widetilde{\\mathcal {F}} \\rightarrow E_{\\infty }$ , which associates the endpoint in $ S^1_{\\infty }(\\lambda )$ of a geodesic ray defined by an element of the unit tangent bundle of the leaf $\\lambda $ , is continuous.",
"For every leaf $\\lambda $ of $\\widetilde{\\mathcal {F}}$ and any $\\gamma \\in \\pi _1(M)$ the following diagram commutes : ${S^1_{\\text{univ}}[r]^{\\rho _{\\text{univ}}(\\gamma )} [d]_{\\Psi _{\\lambda }} & S^1_{\\text{univ}}[d]^{\\Psi _{\\gamma \\cdot \\lambda }} \\\\S^1_{\\infty }(\\lambda ) [r]_{\\gamma } & S^1_{\\infty }(\\gamma \\cdot \\lambda )}$ This definition is taken from [32].",
"Note however that Calegari defines universal circles for any kind of taut foliations and hence needs a last condition that is empty for $\\mathbb {R}$ -covered foliations.",
"Theorem 5.1.21 (Thurston [107], Fenley [56], Calegari [31], Calegari and Dunfield [33]) A foliation coming from a slithering defines a universal circle.",
"The different sources for this result are in fact different generalizations of the original result due to Thurston.",
"Definition 5.1.22 (Regulating flow) A flow $\\psi ^t$ on $M$ is said to be regulating for $\\mathcal {F}$ if $\\psi ^t$ is transverse to $\\mathcal {F}$ and, when lifted to the universal cover, any orbit of $\\widetilde{\\psi }^t$ intersects every leaf of $\\widetilde{\\mathcal {F}}$ .",
"In other words, we have a homeomorphism between any orbit of $\\widetilde{\\psi }^t$ and the leaf space of $\\mathcal {F}$ .",
"The main result concerning slitherings is probably the following, but note that Fenley and Calegari obtained that result for a larger class of foliations : Theorem 5.1.23 (Thurston, Fenley, Calegari [31]) If $\\mathcal {F}$ is a foliation coming from a slithering on an atoroidal, aspherical closed 3-manifold $M$ , then it admits a pseudo-Anosov regulating flow $\\psi ^t \\colon M \\rightarrow M$ .",
"Fenley proves even more about these pseudo-Anosov regulating flows : Theorem 5.1.24 (Fenley [52], [53]) Let $\\mathcal {F}$ be a foliation coming from a slithering on an atoroidal, aspherical closed 3-manifold $M$ .",
"Then, up to topological conjugacy, there is only one regulating pseudo-Anosov flow $\\psi ^t$ .",
"Furthermore, the orbit space of $\\psi ^t$ is a disc that admits $S^1_{\\text{univ}}$ as a natural boundary.",
"In order to have a better picture of the above results, let us describe very roughly how these regulating pseudo-Anosov flows are obtained.",
"The first step is to construct a lamination of $S^1_{\\text{univ}}$ .",
"Let us recall the definition, Definition 5.1.25 Let $(a,b)$ and $(c,d)$ be two pairs of points in $S^1$ .",
"We say that they intersect if $(c,d)$ is contained in different components of $S^1\\setminus \\lbrace a,b\\rbrace $ .",
"Calegari [32] says that the two pairs are linked.",
"We will justify later (Remark REF ) why we use this name.",
"Definition 5.1.26 A lamination of $S^1$ is a closed subset of the set of unordered pairs of distinct points in $S^1$ with the property that no two elements of the lamination intersect.",
"Now, the first (big) step towards Theorem REF is : Theorem 5.1.27 (Thurston, Fenley, Calegari) If $\\mathcal {F}$ is a foliation coming from a slithering on an atoroidal, aspherical closed 3-manifold $M$ , then the associated universal circle $S^1_{\\text{univ}}$ admits two laminations $\\Lambda ^{\\pm }_{\\text{univ}}$ which are preserved under the natural action of $\\pi _1(M)$ on $S^1_{\\text{univ}}$ .",
"While proving that result, they also construct two laminations $\\widetilde{\\Lambda }^{\\pm }$ of $\\widetilde{M}$ such that they are transverse, $\\pi _1(M)$ -invariant and the intersection of $\\widetilde{\\Lambda }^{+}$ (or $\\widetilde{\\Lambda }^{+}$ ) with a leaf of $\\mathcal {F}$ is a geodesic (for the leaf-wise hyperbolic metric).",
"The regulating pseudo-Anosov flow is then obtained from $\\widetilde{\\Lambda }^{\\pm }$ by “collapsing” the complementary regions, thus obtaining two transverse singular foliations, and taking the flow to be the line field generated by the intersection.",
"Note also that, in our case, any lamination in $\\widetilde{M}$ , $\\pi _1(M)$ -invariant and obtained from the laminations $\\Lambda ^{\\pm }_{\\text{univ}}$ will give rise to a regulating pseudo-Anosov flow (see [32])."
],
[
"Skewed $\\mathbb {R}$ -covered Anosov Flows",
"One of the motivations of Thurston in [107] to study foliations coming from slithering was Fenley's examples.",
"Indeed : Proposition 5.2.1 (Thurston) Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow.",
"Then the foliations $\\mathcal {F}^{s} $ and $\\mathcal {F}^{u} $ come from slitherings.",
"Let $C:= \\mathcal {L} ^{s} / \\eta ^u \\circ \\eta ^s$ and $\\pi _C \\colon \\mathcal {L} ^{s} \\rightarrow C$ the projection.",
"As $ \\mathcal {L} ^{s} $ is homeomorphic to $\\mathbb {R}$ and $\\eta ^u \\circ \\eta ^s$ is a strictly increasing, continuous, $\\pi _1(M)$ -equivariant function (by Proposition REF ), we have that $C$ is homeomorphic to $S^1$ and that the action of $\\pi _1(M)$ on $ \\mathcal {L} ^{s} $ descends to an action by bundle automorphisms on $C$ .",
"Hence we can define $s^s \\colon \\widetilde{M}\\rightarrow C$ as the canonical projection from $\\widetilde{M}$ to $ \\mathcal {L} ^{s} $ , then to $C$ and it is clear that $\\mathcal {F}^{s} $ comes from the slithering $s^s$ .",
"The same thing applies for $\\mathcal {F}^{u} $ .",
"So a skewed $\\mathbb {R}$ -covered Anosov flow comes with two slitherings and we can apply the theory developed by Thurston.",
"Note that those foliations are linked : Proposition 5.2.2 A regulating flow for $\\mathcal {F}^{s} $ transverse to $\\mathcal {F}^{u} $ is also regulating for $\\mathcal {F}^{u} $ and vice-versa.",
"Any lift of a(n) (un)stable leaf separates $\\widetilde{M}$ into two connected components, hence the result.",
"Note that, when the skewed $\\mathbb {R}$ -covered Anosov flow we consider is just a geodesic flow on a negatively curved surface $\\Sigma $ , we have an obvious regulating flow coming to mind : here, $M$ corresponds to $H\\Sigma $ , so consider the flow that just push vectors along the fibers, without moving the base point (in other words, the flow generating rotations).",
"It is clear that this flow is regulating.",
"Now $\\widetilde{H\\Sigma }$ is (not surprisingly) homeomorphic to $\\mathbb {R}\\times \\widetilde{\\Sigma } $ , where the first coordinate is given by how much a vector is turned with respect to a fixed direction (taken as a point on the visual boundary of $\\widetilde{\\Sigma }$ ).",
"It turns out that we can always have this kind of identification whenever we consider a skewed $\\mathbb {R}$ -covered Anosov flow (see also Figure REF ) : [50][h] (-2,-1)(2,7) viewpoint=30 30 30,Decran=60 [algebraic]helice(t)cos((2*Pi/3)*t)sin((2*Pi/3)*t)t [object=courbe,r=0.001,range=0 6,linecolor=blue,linewidth=0.02,resolution=360,function=helice] [algebraic]geodesic(t)(1-t)*(-0.70710) + t*cos((2*Pi/3)*3)(1-t)*(-0.70710) + t*sin((2*Pi/3)*3)3 [algebraic]unstableleaf(s,t)(1-t)*(-sqrt(2)/2) + t*cos((2*Pi/3)*s)(1-t)*(-sqrt(2)/2) + t*sin((2*Pi/3)*s)s [object=surfaceparametree, opacity=0.7, linecolor=[cmyk]1,0,1,0.5, base= 1.875 4.875 0 1, action = draw, function=unstableleaf, linewidth=0.5,ngrid=30 1] [algebraic]stableleaf(s,t)(1-t)*cos((2*Pi/3)*(1+s)) + t*cos((2*Pi/3)*1)(1-t)*sin((2*Pi/3)*(1+s)) + t*sin((2*Pi/3)*1)1 [object=surfaceparametree, linecolor=red, base= 1.875 4.875 0 1, action = draw, function=stableleaf, linewidth=0.5,ngrid=20 1] [object=cylindrecreux,h=6,r=1,action=draw,linewidth=0.01,ngrid=3 25,opacity = 0.1,fillcolor=red,incolor=white](0,0,0) Using Proposition REF , we can represent $ \\tilde{ \\phi }^t $ in the following way : $\\widetilde{M}$ is identified with a solid cylinder where each horizontal slice is a stable leaf.",
"On a stable leaf, the orbits of $ \\tilde{ \\phi }^t $ are lines all pointing towards the same point on the boundary at infinity of the leaf.",
"We represented a stable leaf, with some orbits on it, in red.",
"The blue curve represents the point at infinity where orbits ends.",
"It is a way of seeing $ \\mathcal {L} ^{s} $ “slithers”.",
"An unstable leaf now, represented in green, is given by fixing the $(x,y)$ -coordinates (i.e., the points that project to the same point on the universal circle) and taking the lines pointing towards the blue curve.",
"Finally, the orbits of the regulating pseudo-Anosov flow $\\widetilde{\\psi }^t$ are vertical curves inside the cylinder and stabilize the foliation by vertical straight lines on the boundary.",
"Proposition 5.2.3 Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow and $\\psi ^t$ a regulating flow for both $\\mathcal {F}^{s} $ and $\\mathcal {F}^{u} $ .",
"Then, we can construct two continuous identifications of the universal cover : $I^s \\colon &\\widetilde{M} \\rightarrow \\mathcal {L} ^{s} \\times \\mathcal {O}(\\psi )\\,, & I^u \\colon &\\widetilde{M} \\rightarrow \\mathcal {L} ^{u} \\times \\mathcal {O}(\\psi ) \\,, \\\\& x \\mapsto \\left( \\widetilde{\\mathcal {F}}^{s} (x), \\widetilde{\\psi }^t (x) \\right) & & x \\mapsto \\left( \\widetilde{\\mathcal {F}}^{u} (x),\\widetilde{\\psi }^t (x) \\right)$ where $ \\mathcal {O} (\\psi )$ is the orbit space of $\\psi ^t$ .",
"Injectivity and surjectivity are given by the definition of a regulating flow.",
"Continuity follows from the fact that $\\widetilde{\\mathcal {F}}^{s} $ is a foliation of $\\widetilde{M}$ , $\\widetilde{\\psi }^t$ is transverse to it and $ \\mathcal {O} (\\psi )$ is homeomorphic to $\\mathbb {R}^2$ .",
"Note that, for any orbit of $ \\tilde{ \\phi }^t $ , using the slithering given by $\\mathcal {F}^{s} $ , it is possible to “project” this orbit onto the universal circle.",
"Indeed, an orbit determines two leaves $L^s \\in \\mathcal {L} ^{s} $ and $L^u \\in \\mathcal {L} ^{u} $ .",
"Then, using $\\eta ^u \\colon \\mathcal {L} ^{u} \\rightarrow \\mathcal {L} ^{s} $ (defined in Proposition REF ), we get a pair of points $(L^s, \\eta ^u(L^u))$ in $ \\mathcal {L} ^{s} $ and this in turn determines two distinct points in the universal circle.",
"When we consider the reciprocal image of this application, we obtain : Lemma 5.2.4 Two distinct points on the universal circle define a (countable) collection of orbits of $ \\phi ^{t} $ .",
"Instead of giving a rigorous proof of that lemma, I would like to refer the reader to Figure REF : two points on $S^1_{\\text{univ}}$ determine two vertical lines on the outside of the cylinder, and every time one of these lines intersects $ \\mathcal {L} ^{s} $ , we obtain one orbit."
],
[
"Isotopy and co-cylindrical class",
"The question that started our study of these kinds of Anosov flows was the following : Suppose you are given a skewed $\\mathbb {R}$ -covered Anosov flow in a hyperbolic 3-manifold $M$ .",
"Any periodic orbit is freely homotopic to infinitely many other orbits.",
"In other words, we have a family of knots in $M$ .",
"Then are these knots different ?",
"Here, we understand “different” in the sense of traditional knot theory, i.e., two knots are equivalent if there exists an isotopy between them.",
"And if some of these knots are indeed different, can we say more about them ?",
"That is, can we develop a kind of knot theory adapted to Anosov flows ?",
"It turns out that there is no knot theory in that case.",
"Indeed we will show below that any freely homotopic orbits are isotopic.",
"We will also study a related question : among a free homotopy class, can we say when two orbits are boundaries of an embedded cylinder ?",
"Indeed, an isotopy between two orbits gives an immersed cylinder.",
"So it seems natural to wonder whether this can be made into an embedding.",
"Furthermore, Barbot [18] (and later together with Fenley [20]) studied embeddings of tori in manifolds equipped with an Anosov flow.",
"This is in some sense the atoroidal equivalent.",
"The results in the following sections are joint work with Sergio Fenley and will be published with full details later."
],
[
"Isotopy class of periodic orbits",
"Let us start by giving the definition of isotopy we will use here : Definition 5.3.1 Two curves $c_1$ and $c_2$ in $M$ are isotopic if there exists a continuous application $H \\colon S^1 \\times [0,1] \\rightarrow M$ such that $H(S^1,0)= c_1$ , $H(S^1,0)= c_2$ and, for any $t\\in [0,1]$ , $H(S^1,t)$ is an embedding of $S^1$ in $M$ .",
"Among isotopic orbits, we define : Definition 5.3.2 Two curves $c_1$ and $c_2$ in $M$ are co-cylindrical if there exists an embedded annulus $A$ in $M$ such that $\\partial A = c_1 \\cup c_2$ .",
"Note that this is not an equivalence relation as it is clearly non-transitive.",
"However, as we will see, its study is quite interesting.",
"Let us start by considering geodesic flows for a minute.",
"In that case, the question of isotopy is trivial (because there is, at most, one periodic orbit in a free homotopy class).",
"What is not trivial however is answering the following question : given a periodic orbit $\\alpha $ , is there an embedded torus in $H\\Sigma $ containing $\\alpha $ ?",
"If you suppose that $\\alpha $ is simple, then the answer is clearly yes.",
"Indeed, just take $\\lbrace (x,v) \\in H\\Sigma \\mid x \\in \\pi (\\alpha ) \\rbrace $ .",
"If the orbit is non-simple however, it turns out that there is no such embedded torus.",
"To my everlasting surprise, this kind of condition will remain true for any skewed $\\mathbb {R}$ -covered Anosov flow.",
"But before studying co-cylindrical classes, we can use the work of Thurston, Fenley and Calegari to answer our first question and deduce that the isotopy classes are the same as the double free-homotopy classes : Theorem 5.3.3 (Barthelmé, Fenley) Let $ \\phi ^{t} $ be a skewed $\\mathbb {R}$ -covered Anosov flow on a closed atoroidal, not Seifert fibered, 3-manifold.",
"If $\\alpha _i$ is a double free homotopy class of periodic orbits of $ \\phi ^{t} $ , then all the $\\alpha _i$ s are isotopic.",
"[Sketch of proof] We are going to construct an isotopy between $\\alpha _0$ and $\\alpha _1$ .",
"As isotopy is an equivalence relation, it will show that all free homotopic orbits are isotopic.",
"Let $\\psi ^t$ be a regulating flow for $\\widetilde{\\mathcal {F}}^{s} $ and $\\widetilde{\\mathcal {F}}^{u} $ , $\\widetilde{\\alpha _{0} } $ a lift of $\\alpha _0$ to $\\widetilde{M}$ and ${\\widetilde{\\alpha _{1} } = \\eta (\\widetilde{\\alpha _{0} } )}$ .",
"For any $x \\in \\widetilde{\\alpha _{0} } $ there exists a time $T(x)$ such that $\\widetilde{\\psi }^{T(x)}(x) \\in \\widetilde{\\mathcal {F}}^{s} (\\widetilde{\\alpha _{1} } )$ .",
"Let $C:=\\lbrace \\psi ^t(\\pi (x)) \\mid x\\in \\widetilde{\\alpha _{0} } , 0\\le t \\le T(x) \\rbrace $ .",
"It is an immersed cylinder with one boundary $\\alpha _0$ and the other one a closed curve on $\\mathcal {F}^{s} (\\alpha _1)$ .",
"Let us call the second boundary component $\\alpha _1^{\\prime }$ .",
"Up to a $C^1$ modification of $\\psi ^t$ , we can show that there is only a finite number of (transverse) intersections of $\\alpha _0$ with $C$ .",
"We can therefore find a continuous time change $\\Psi ^t$ of $\\psi ^t$ such that, for some $t_1\\in \\mathbb {R}$ , $\\alpha _{1}^{\\prime } = \\Psi ^{t_1}(\\alpha _{0})$ .",
"As $\\Psi ^t$ is a flow, for any $t\\in [0,t_1]$ , $\\Psi ^t(\\alpha _0)$ is an embedded $S^1$ in $M$ .",
"We produced an isotopy from $\\alpha _0$ to $\\Psi ^{t_1}(\\alpha _0)$ .",
"Now, $\\Psi ^{t_1}(\\alpha _0)$ is freely homotopic to $\\alpha _1$ on the surface $\\mathcal {F}^{s} (\\alpha _1)$ , hence is isotopic."
],
[
"Co-cylindrical class",
"We will now show the link between having two co-cylindrical periodic orbits and simple chain of lozenges.",
"This is essentially based on Barbot's work [18].",
"In [18] (see also [20]) Barbot studied embedded tori in (toroidal) 3-manifolds supporting skewed $\\mathbb {R}$ -covered Anosov flows, showing that they could be put in a quasi-transverse position (i.e., transverse to the flow, apart from along some periodic orbits).",
"We will use his work to obtain properties of embedded annuli : Theorem 5.3.4 Let $\\alpha $ and $\\beta $ be two orbits in the same free homotopy class, choose coherent lifts $\\widetilde{\\alpha }$ and $\\widetilde{\\beta }$ , and denote by $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ the chain of lozenges between $\\widetilde{\\alpha }$ and $\\widetilde{\\beta }$ .",
"If $\\alpha $ and $\\beta $ are co-cylindrical, then $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is simple, i.e., if we denote by $(\\widetilde{\\alpha _{i} } )_{i=0 \\dots n}$ the corners of the lozenges in $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ , with $\\widetilde{\\alpha _{0} } = \\widetilde{\\alpha }$ and $\\widetilde{\\alpha _{n} } = \\widetilde{\\beta }$ , then $\\left( \\pi _1(M) \\cdot \\widetilde{\\alpha _{i} } \\right) \\cap B(\\widetilde{\\alpha }, \\widetilde{\\beta }) = \\lbrace \\widetilde{\\alpha _{i} } \\rbrace .$ Conversely, if $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is simple, then there exists an embedded annulus, called a Birkhoff annulus, with boundary $\\alpha \\cup \\beta $ .",
"Construction of an embedded Birkhoff annulus from a simple chain of lozenges is done in [18], hence proving the converse part.",
"To prove that, if $\\alpha $ and $\\beta $ are co-cylindrical, then $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is simple, we have to re-prove Lemma 7.6 of [18] (or equivalently step 1 of the proof of Theorem 6.10 of [20]) when, instead of having an embedded torus, we just have an embedded cylinder.",
"Let $C$ be an embedded cylinder such that $\\partial C = \\lbrace \\alpha , \\beta \\rbrace $ and $\\widetilde{C}$ the lift of $C$ in $\\widetilde{M}$ such that its boundary is on $\\widetilde{\\alpha }$ and $\\widetilde{\\beta }$ .",
"Let us also denote the generator of the stabilizer of $\\widetilde{\\alpha }$ by $\\gamma \\in \\pi _1(M)$ .",
"Following [18], we can construct a embedded plane $\\widetilde{C}_0$ in $\\widetilde{M}$ such that $\\widetilde{C}_0$ is $\\gamma $ -invariant, $\\widetilde{C}_0$ contains all the $\\widetilde{\\alpha _{i} } $ , $\\widetilde{C}_0$ is transverse to $ \\tilde{ \\phi }^t $ except along the $\\widetilde{\\alpha _{i} } $ , the projection of $\\widetilde{C}_0$ to $ \\mathcal {O} $ is $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ .",
"Barbot's trick to obtain such a plan is, for every lozenge in $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ , to take a simple curve $\\bar{c}$ from one corner of the lozenge to the other (for instance $\\widetilde{\\alpha _{i} } $ and $\\widetilde{\\alpha _{i+1} } $ ).",
"Then, lift $\\bar{c}$ to $\\widetilde{c}\\subset \\widetilde{M}$ such that $\\widetilde{c}$ is transverse to $ \\tilde{ \\phi }^t $ .",
"Now choose an embedded rectangle $R_i$ in $\\widetilde{M}$ such that $R_i$ is bounded by $\\widetilde{c}$ , $\\gamma \\cdot \\widetilde{c}$ , and the two pieces of $\\widetilde{\\alpha _{i} } $ and $\\widetilde{\\alpha _{i+1} } $ between the endpoints of $\\widetilde{c}$ and $\\gamma \\cdot \\widetilde{c}$ .",
"Then define $\\widetilde{C}_0$ as the orbit under $\\gamma $ of the unions of the rectangles $R_i$ .",
"From now on, we copy the proof of [20].",
"Suppose that $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is not simple.",
"Then there exist $\\widetilde{\\alpha _{i} } $ and $h \\in \\pi _1(M)$ such that $\\theta := h \\cdot \\widetilde{\\alpha _{i} } $ intersects the interior of $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ .",
"Then $\\theta $ intersects $\\widetilde{C}_0$ in a single point $p$ .",
"Let $\\theta ^+$ and $\\theta ^-$ be the two rays in $\\theta $ defined by $p$ .",
"If we denote by $\\widetilde{V}$ the subset of $\\widetilde{M}$ delimited by $\\widetilde{\\mathcal {F}}^{s} (\\widetilde{\\alpha })$ and $\\widetilde{\\mathcal {F}}^{s} (\\widetilde{\\beta })$ and containing $\\widetilde{C}_0$ , then $\\widetilde{C}_0$ separates $\\widetilde{V}$ in two components.",
"Claim 5.3.5 Either $\\theta ^+$ or $\\theta ^-$ stays a bounded distance away from $\\widetilde{C}_0$ .",
"Assume they don't : for any $R>0$ , there exist points $q_R^-,q_R^+$ on $\\theta ^-, \\theta ^+$ such that $d(q_R^{\\pm }, \\widetilde{C}_0) >R$ .",
"As $\\pi (\\widetilde{C}_0)$ and $C$ are freely homotopic, there exists $R_0$ such that $\\widetilde{C}$ is contained in the $R_0$ -neighborhood of $\\widetilde{C}_0$ .",
"Then, for any $R > 2 R_0$ , any path in $\\widetilde{V}$ joining two points $q^-$ and $q^+$ such that $d(q^{\\pm }, q_R^{\\pm })<R$ must intersect $\\widetilde{C}$ .",
"Now, $\\pi (\\theta )$ is freely homotopic to a curve in $C$ , and, as $C$ is embedded in an oriented manifold, it must be two-sided.",
"So $\\pi (\\theta )$ is homotopic to a curve disjoint from $C$ .",
"(Note that this is the only point where we use the fact that $C$ is embedded).",
"Lifting it gives a homotopy from $\\theta $ to a curve $\\theta _1$ disjoint from $\\widetilde{C}$ .",
"But homotopies move points a bounded distance away : there exists $r>0$ such that, for any $R>0$ , there are two points $m_R^{\\pm }$ on $\\theta _1$ such that $d(m_R^{\\pm }, q_R^{\\pm }) <r$ .",
"Choose $R > \\max \\lbrace 2 R_0, r \\rbrace $ , according to the above, the segment in $\\theta _1$ from $m_R^-$ to $m_R^+$ must intersect $\\widetilde{C}$ hence a contradiction.",
"We assume that $\\theta ^+$ stays at a distance $\\le a_1$ from $\\widetilde{C_0}$ .",
"Let $g\\in \\pi _1(M)$ be the generator of the stabilizer of $\\theta $ .",
"Choose a sequence $\\left(p_i\\right)$ with $p_i := g^{n_i}\\cdot p \\in \\theta ^+$ .",
"Let $\\left(q_i\\right)$ be a sequence in $\\widetilde{C}_0$ such that $d(q_i,p_i) \\le a_1$ , up to a subsequence, we can assume that $\\pi (q_i)$ converges and as $\\pi (\\widetilde{C_0})$ is compact, we can even assume that $\\pi (q_i)$ is constant.",
"Now, up to another subsequence, we can assume that there are segments $u_i$ in $\\widetilde{V}$ from $p_i$ to $q_i$ such that $\\pi (u_i)$ converges in $M$ .",
"Adjusting once again, we can assume that $\\pi (u_i)$ is constant for big enough $i$ .",
"We consider the following closed curve in $\\widetilde{V}$ : start by a segment in $\\theta ^+$ from $p_i$ to $p_k$ , $k>i$ , then follow $u_k$ , then choose a segment in $\\widetilde{C}_0$ from $q_k$ to $q_i$ and close up along $u_i$ .",
"Since $\\pi (u_i) = \\pi (u_k)$ , this shows that there exists $n\\in \\mathbb {Z}$ such that $g^n(q_i)= q_k$ .",
"Hence, for some $n \\ne 0$ , $\\widetilde{C}_0$ is left invariant by $g^n$ , which implies that $B(\\widetilde{\\alpha }, \\widetilde{\\beta })$ is also invariant by $g^n$ .",
"But $g^n \\cdot \\theta = \\theta $ , so $g^n$ leaves invariant a point in the interior of a lozenge as well as the whole lozenge, which is impossible.",
"Using the theorem, we can deduce the following property of co-cylindrical class : Proposition 5.3.6 If the co-cylindrical class of one orbit is finite, then all the co-cylindrical classes in the same double free homotopy class are finite.",
"Moreover, they all have the same cardinality.",
"This result just relies on the fact that the homeomorphism $\\eta $ of $ \\mathcal {O} $ , defined by applying $\\eta ^u$ and $\\eta ^s$ to respectively the unstable and stable leaf commutes with the action of $\\pi _1(M)$ .",
"Let $\\alpha _i$ be a double free homotopy class of periodic orbit.",
"Suppose that $\\alpha _0$ has $k$ elements in its co-cylindrical class.",
"Then it implies that, for any coherent lift $\\widetilde{\\alpha _{i} } $ of the $\\alpha _i$ , the chain of lozenges $B(\\widetilde{\\alpha _{0} } , \\widetilde{\\alpha _{k} } )$ is non-simple.",
"More precisely, we have an element $h\\in \\pi _1(M)$ such that $h \\cdot \\widetilde{\\alpha _{0} } $ is in $L(\\widetilde{\\alpha _{k-1} } , \\widetilde{\\alpha _{k} } )$ , the lozenge with corners $\\widetilde{\\alpha _{k-1} } $ and $\\widetilde{\\alpha _{k} } $ .",
"Indeed, recall that $\\widetilde{\\alpha _{i} } = \\eta ^i (\\widetilde{\\alpha _{0} } )$ , so, if $h \\cdot \\widetilde{\\alpha _{i} } \\in L(\\widetilde{\\alpha _{k-1} } , \\widetilde{\\alpha _{k} } )$ , then $h \\cdot \\widetilde{\\alpha _{0} } = h \\cdot \\eta ^{-i}(\\widetilde{\\alpha _{i} } ) = \\eta ^{-i}\\left(h \\cdot \\widetilde{\\alpha _{i} } \\right) \\in \\eta ^{-i}\\left(L(\\widetilde{\\alpha _{k-1} } , \\widetilde{\\alpha _{k} } ) \\right) = L(\\widetilde{\\alpha _{k-1-i} } , \\widetilde{\\alpha _{k-i} } ).$ So $\\alpha _0$ and $\\alpha _{k-i}$ would not be co-cylindrical.",
"Then, for any $i$ , $h \\cdot \\widetilde{\\alpha _{i} } \\in L(\\widetilde{\\alpha _{k-1 + i} } , \\widetilde{\\alpha _{k+i} } )$ , which proves that the number of orbits co-cylindrical to $\\alpha _i$ is at most $k$ , and again the same argument as above shows that it is also at least $k$ ."
],
[
"Action of the fundamental group on $S^1_{\\text{univ}}$ and co-cylindrical orbits",
"Action on $S^1_{\\text{univ}}$ and co-cylindrical orbits Thanks to Thurston's work in [107] we know that the fundamental group of a 3-manifold admitting a $\\mathbb {R}$ -covered foliation acts on the universal circle implying many results about the type of group it can be, as we can see for instance in [31], [33], [56].",
"There is a remarkable link between the existence of co-cylindrical orbits and the action of $\\pi _1(M)$ on pairs of points in $S^1_{\\text{univ}}$ .",
"Definition 5.4.1 Let $(\\alpha ^+, \\alpha ^-)$ and $(\\beta ^+, \\beta ^-)$ be two pairs of points in $S^1_{\\text{univ}}$ .",
"We say that $(\\alpha ^+, \\alpha ^-)$ and $(\\beta ^+, \\beta ^-)$ intersect if, for some order on $S^1_{\\text{univ}}$ , we have $\\alpha ^- < \\beta ^- <\\alpha ^+ < \\beta ^+.$ We will say that $(\\alpha ^+, \\alpha ^-)$ self-intersects if there exists $h\\in \\pi _1(M)$ such that $(\\alpha ^+, \\alpha ^-)$ and $(h \\cdot \\alpha ^+, h \\cdot \\alpha ^-)$ intersect.",
"Proposition 5.4.2 Let $\\alpha $ be a periodic orbit of $ \\phi ^{t} $ , $\\widetilde{\\alpha }$ a lift to $\\widetilde{M}$ and $(\\alpha ^+, \\alpha ^-)$ the projection of $\\widetilde{\\alpha }$ on $S^1_{\\text{univ}}$ .",
"The co-cylindrical class of $\\alpha $ is finite if and only if $(\\alpha ^+, \\alpha ^-)$ self-intersects.",
"If the co-cylindrical of $\\alpha $ is finite, then (by Theorem REF ) the chain of lozenges containing $\\widetilde{\\alpha }$ is non-simple.",
"So, there exists $h\\in \\pi _1(M)$ such that $h\\cdot \\widetilde{\\alpha } \\in L(\\widetilde{\\alpha _{i} } , \\widetilde{\\alpha _{i+1} } )$ .",
"Projecting that lozenge to $S^1_{\\text{univ}}$ shows that $(\\alpha ^+, \\alpha ^-)$ and $h \\cdot (\\alpha ^+, \\alpha ^-)$ intersect.",
"Reciprocally, if there exists $h \\in \\pi _1(M)$ such that $(\\alpha ^+, \\alpha ^-)$ and $h \\cdot (\\alpha ^+, \\alpha ^-)$ intersect, then $h\\cdot \\widetilde{\\alpha } \\in L(\\widetilde{\\alpha _{i} } , \\widetilde{\\alpha _{i+1} } )$ for some $i$ .",
"Hence, by Theorem REF , the co-cylindrical class of $\\alpha $ must be finite.",
"Let us announce the following result with Sergio Fenley, to be published later : Theorem 5.4.3 (Barthelmé, Fenley) Let $(\\alpha ^+, \\alpha ^-)$ be the projection on $S^1_{\\text{univ}}$ of a periodic orbit $\\widetilde{\\alpha }$ of $ \\tilde{ \\phi }^t $ .",
"Then $(\\alpha ^+, \\alpha ^-)$ self-intersects.",
"The theorem is proved by seeing $S^1_{\\text{univ}}$ as the boundary at infinity of the orbit space of a regulating pseudo-Anosov flow $\\psi ^t$ and using the transitivity of such flows (Mosher [87], proved that any pseudo-Anosov flow on an atoroidal manifold is transitive).",
"Remark 5.4.4 Suppose that $(\\alpha ^+, \\alpha ^-)$ self-intersects and denote by $(\\widetilde{\\alpha _{i} } )$ the orbits in $\\widetilde{M}$ projecting to $(\\alpha ^+, \\alpha ^-)$ and $\\alpha _i = \\pi (\\widetilde{\\alpha _{i} } )$ their projection to $M$ .",
"Then, for any $i$ , there exist a $j$ and a $t$ such that $\\psi ^t(\\alpha _i) \\cap \\alpha _j \\ne \\emptyset $ .",
"By flowing one orbit we get an actual intersection.",
"If we consider the geodesic flow case now, there is also a natural circle at infinity.",
"Just take the visual boundary $\\widetilde{\\Sigma }(\\infty )$ and the fundamental group $\\pi _1(H\\Sigma )$ naturally acts on it.",
"So, for a pair $(\\alpha ^+, \\alpha ^-)$ in $\\widetilde{\\Sigma }(\\infty )$ , to self-intersect in that case means that the only geodesic in $\\Sigma $ such that a lift of it has endpoints $(\\alpha ^+, \\alpha ^-)$ is non-simple.",
"Hence, in the geodesic flow case, there exist points on $\\widetilde{\\Sigma }(\\infty )$ representing a periodic orbit that does not self-intersect, in contrast with the atoroidal case we studied here.",
"As a corollary of Proposition REF and Theorem REF , we obtain : Theorem 5.4.5 (Barthelmé, Fenley) Every co-cylindrical class is finite.",
"Note that it is still an open question whether a co-cylindrical class can be non-trivial.",
"We only know that some are : Proposition 5.4.6 There exist periodic orbits of $ \\phi ^{t} $ with trivial co-cylindrical class.",
"Remark 5.4.7 For such an orbit, Proposition REF shows that every other orbit in the double free homotopy class must also have a trivial co-cylindrical class.",
"Let $V$ be a flow box of $ \\phi ^{t} $ , as $ \\phi ^{t} $ is transitive, we can pick a long segment of a dense orbit that $\\varepsilon $ -fills $V$ .",
"Then, by the Anosov Closing lemma (see [80]), we get a periodic orbit $\\alpha $ that $2\\varepsilon $ -fills $V$ .",
"Now, choose $x$ on one of the connected components of $\\alpha \\cap V$ .",
"If $\\varepsilon $ was chosen small enough, then there must exist $y$ on another connected component of $\\alpha \\cap V$ such that there is a close path $c$ staying in $V$, starting at $x$ going through the positive stable leaf of $x$ , then the negative unstable leaf of $y$ , then the negative stable leaf of $y$ and finally close up along the positive stable leaf of $x$ .",
"If we lift the path $c$ to the universal cover of $M$ and project it to the orbit space $ \\mathcal {O} $ , as $V$ has no topology, we see that the projection of the lift of $y$ must be inside the lozenge determined by the lift of $x$ (remember that we chose our flow so that the lozenges orientation is $(+,+,-,-)$ , otherwise, we would have to modify our path $c$ , see Figure REF ).",
"Hence the lozenge is non-simple and therefore the co-cylindrical class of $\\alpha $ is trivial."
],
[
"Some open questions",
"I wanted to end this dissertation with a list of questions I have about skewed $\\mathbb {R}$ -covered Anosov flow, because even if a lot of things are known, thanks mostly to T. Barbot and S. Fenley, the things that are unknown justify, at least in my view, a continuation of their study.",
"Let's start with the “topological” questions : P. Foulon and B. Hasselblatt [63] have constructed contact Anosov flows (i.e., Anosov flow preserving a contact form) on not Seifert-fibered spaces and it seems very likely that their construction often yields hyperbolic manifolds.",
"Now, contact Anosov flows are skewed and $\\mathbb {R}$ -covered (see [19]) and are the “nicest” flows from a regularity point of view (see [64]).",
"In [20], Barbot and Fenley showed that skewed $\\mathbb {R}$ -covered Anosov flows in Seifert-fibered spaces are (up to a finite cover) topologically conjugated to a geodesic flow on a closed surface.",
"A natural question is then : Is a skewed $\\mathbb {R}$ -covered Anosov flow on an atoroidal manifold always topologically conjugate to a contact Anosov flow ?",
"Indeed, it seems that the structure of $\\widetilde{M}$ given by the regulating pseudo-Anosov flow (see proposition REF ) is very rigid, so can we use that to show that $ \\phi ^{t} $ is topologically contact ?",
"(See [19] for a definition.)",
"And, from there, can we get an actual topological conjugacy ?",
"Given a continuous map $s \\colon \\mathbb {R}\\times \\mathbb {H}^2 \\rightarrow S^1$ such that, for all $t \\in \\mathbb {R}$ , $s(t, \\cdot )$ is constant and for any $x \\in \\mathbb {H}^2$ , $s(\\cdot , x)$ is strictly monotone, we can construct, using Figure REF , an Anosov flow on $\\mathbb {R}\\times \\mathbb {H}^2$ .",
"Now suppose that we are given a discreet group $\\Gamma $ acting in a “good” way on $\\mathbb {R}\\times \\mathbb {H}^2$ , then is the quotient flow a contact Anosov flow ?",
"And if that is true, then, can we get all contact Anosov flows on atoroidal 3-manifolds in this fashion ?",
"Finally, there are a lot of ergodic theoretical questions for these flows : A classical question (initiated by Bowen and Margulis) for Anosov flows is to count the number of closed orbits of length less than $R$ and find an asymptotic equivalent when $R$ gets big.",
"In [91], Parry and Policott prove that this number is asymptotic to $e^{hR}/hR$ where $h$ is the topological entropy.",
"Following them, Katsuda and Sunada [81] answered the question of counting closed orbits inside an homology class.",
"So it seems natural to ask, in the case of skewed $\\mathbb {R}$ -covered Anosov flows on atoroidal manifolds, whether we can give an equivalent to the number of closed orbits of length less than $R$ inside a free homotopy class.",
"A somewhat related question (asked by M. Crampon) is the following : let $\\alpha _i$ be the orbits in a free homotopy class of $ \\phi ^{t} $ , denote by $l_i$ the length of $\\alpha _i$ and $\\delta _{\\alpha _i}$ the Dirac measure on $\\alpha _i$ .",
"Let $\\mu _n := \\sum _{ |i| \\le n } \\frac{\\delta _{\\alpha _i}}{ l_i }\\,.$ The sequence $(\\mu _n)$ admits at least one weak limit $\\mu $ .",
"Can we show that this limit is unique and ergodic ?",
"If that is so, then what is the measure-entropy of $\\mu $ ?",
"Can we link that entropy to the previous counting question ?",
"french Je ne sais pas le reste."
]
] | 1204.0879 |
[
[
"The Power of Linear Programming for Valued CSPs"
],
[
"Abstract A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain.",
"An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum.",
"This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs).",
"Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation.",
"Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees."
],
[
"Introduction",
"The constraint satisfaction problem (CSP) provides a common framework for many theoretical and practical problems in computer science.",
"An instance can be vaguely described as a set of variables to be assigned values from the domains of the variables so that all constraints are satisfied [44].",
"The CSP is NP-complete in general and thus we are interested in restrictions which give rise to tractable classes of problems.",
"Following Feder & Vardi [21], we restrict the constraint language; that is, all constraint relations in a given instance must belong to a fixed, finite set of relations on the domain.",
"The most successful approach to classifying language-restricted CSPs is the so-called algebraic approach [30], [29], [7], which has led to several complexity classifications [6], [8], [4], [1] and algorithmic characterisations [2], [27] going beyond the seminal work of Schaefer [46].",
"Motivated by reasons both theoretical (optimisation problems are different from decision problems) and practical (many problems are over-constrained and hence have no solution, or under-constrained and hence have many solutions), we study valued constraint satisfaction problems (VCSPs) [5], [47].",
"A valued constraint language is a finite set of cost functions on the domain, and a VCSP instance is given by a weighted sum of cost functions from the language with the goal to minimise the sum.",
"(CSPs correspond to the case when the range of all cost functions is $\\lbrace 0,\\infty \\rbrace $ , and Max-CSPs correspond to the case when the range of all cost functions is $\\lbrace 0,1\\rbrace $ .With respect to exact solvability, Max-CSPs (“maximising the number of satisfied constraints”) are polynomial-time equivalent to Min-CSPs (“minimising the number of unsatisfied constraints”).",
"Therefore, with respect to exact solvability, Max-CSPs are polynomial-time equivalent to $\\lbrace 0,1\\rbrace $ -valued VCSPs.)",
"The VCSP framework is very robust and has also been studied under different names such as Min-Sum problems, Gibbs energy minimisation, Markov Random Fields (MRF), Conditional Random Fields (CRF) and others in several different contexts in computer science [40], [49], [15].",
"Given the generality of the VCSP, it is not surprising that only few complexity classifications are known.",
"In particular, only Boolean (on a 2-element domain) languages [11], [16] and conservative (containing all $\\lbrace 0,1\\rbrace $ -valued unary cost functions) languages [36] have been completely classified with respect to exact solvability.",
"On the algorithmic side, most known tractable languages are somewhat related to submodular functions on distributive lattices [11], [10], [32], [36].",
"An alternative approach for solving VCSPs is using linear programming (LP) and semidefinite programming (SDP); these have been used mostly for approximation [45], [39], [18], [3].",
"Contribution We study the power of the basic linear programming relaxation (BLP).",
"Our main result (Theorem REF ) is a precise characterisation of valued constraint languages for which BLP is a decision procedure.",
"In more detail, we characterise valued constraint languages over which VCSP instances can be solved exactly by a certain basic linear program.",
"Equivalently, we show precisely when a particular integer programming formulation of a VCSP has zero integrality gap.",
"The characterisation is algebraic in terms of fractional polymorphisms [9].",
"Our work is the first link between solving VCSPs exactly using LP and the algebraic machinery for VCSPs introduced by Cohen et al.",
"in [9], [12].",
"Part of the proof is inspired by the characterisation of width-1 CSPs [21], [19].",
"One of the main technical contributions is a construction of totally symmetric fractional polymorphisms of all arities (Theorem REF ).",
"This result allows us to demonstrate that several valued constraint languages are covered by our characterisation and thus are tractable; that is, VCSP instances over these languages can be solved exactly using BLP.",
"New tractable languages include: (1) submodular languages on arbitrary lattices; (2) bisubmodular (also known as $k$ -submodular) languages on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular languages on arbitrary trees.",
"The complexity of (subclasses of) these languages has been mentioned explicitly as open problems in [20], [37], [35], [26].",
"More generally, we show that any valued constraint language with a binary multimorphism in which at least one operation is a semi-lattice operation is tractable (cf.",
"Section ).",
"Our results cover all known tractable finite-valued constraint languages.",
"Related work Apart from identifying tractable classes of CSPs and VCSPs with respect to exact solvability, the approximability of Max-CSPs has attracted a lot of attention [17], [33], [31].",
"Under the assumption of the unique games conjecture [34], Raghavendra showed how to approximate all Max-CSPs and finite-valued VCSPs optimally [45].Note that Max-CSPs (=$\\lbrace 0,1\\rbrace $ -valued VCSPs) and finite-valued VCSPs, respectively, are called CSPs and Generalised CSPs (GCSPs), respectively, in [45].",
"For VCSPs that are tractable, Raghavendra's algorithms provide a PTAS, but it seems notoriously difficult to determine the approximation ratios of these algorithms.",
"Very recently, Max-CSPs that are robustly approximable have been characterised as those having bounded width [18], [39], [3].",
"Specifically, Kun et al.",
"studies the question of which Weighted Max-CSPsIn Weighted Max-CSPs, every constraint $f$ is $\\lbrace 0,c_f\\rbrace $ -valued, where $c_f$ is a positive constant.",
"Weighted Max-CPSs are a special case of VCSPs.",
"can be robustly approximated using BLP [39].",
"Their result is related but incomparable to ours as it applies to robust approximability and not to exact solvability, except for the special case of width-1 CSPs.",
"In particular, “solving” (“deciding”) for us means finding an optimum solution to a VCSP instance, which is an optimisation problem, whereas “solving” in [39] means (ignoring their results on robust approximability, which do not apply here) the basic LP formulation of a CSP instance finds a solution if one exists.Note that CSPs are defined as $\\lbrace 0,1\\rbrace $ -valued in [39] and not as $\\lbrace 0,\\infty \\rbrace $ -valued, as in this paper.",
"This is needed for the LP formulation and the measure of approximability.",
"After all, [39] deals with Max-CSPs.",
"We remark that our tractability results apply to the minimisation problem of VCSP instances (i.e., the objective function is given by a sum of “local” cost functions) but not to objective functions given by an oracle.",
"In particular, submodular functions given by an oracle can be minimised on distributive lattices [48], [28], diamonds [38], and several constructions on lattices preserving tractability have been identified [37], but it is widely open what happens on non-distributive lattices.",
"Similarly, bisubmodular functions given by an oracle can be minimised in polynomial-time on domains of size 3 [24], but the complexity is open on domains of larger size [26].",
"It is known that strongly tree-submodular functions given by an oracle can be minimised in polynomial time on binary trees [35], but the complexity is open on general (non-binary) trees.",
"Similarly, it is known that weakly tree-submodular functions given by an oracle can be minimised in polynomial time on chains and forks [35], but the complexity on (even binary) trees is open.",
"Extending the notion of (generalised) arc consistency for CSPs [41], [23] and several previously studied notions of arc consistencies for VCSPs [14], Cooper et al.",
"introduced optimal soft arc consistency (OSAC) [13], which is a linear program relaxation of a given VCSP instance.",
"Since OSAC is is a tighter relaxation than BLP (cf.",
"Appendix ), all tractable classes identified in this paper are solved by OSAC as well.",
"Similarly, since the basic SDP relaxation from [45] is tighter than BLP, all tractable cases identified in this paper are solved by it as well."
],
[
"Preliminaries",
"The set of non-negative rational numbers is denoted by $\\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"A signature $\\tau $ is a set of function symbols $f$ , each with an associated positive arity, $ar(f)$ .",
"A valued $\\tau $ -structure $A$ (also known as a valued constraint language, or just a language) consists of a domain $D = D(A)$ , together with a function $f^A : D^{ar(f)} \\rightarrow \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ , for each function symbol $f \\in \\tau $ .",
"(To be precise, these are finite-valued structures.",
"In Section , we will extend $\\mbox{$\\mathbb {Q}_{\\ge 0}$}$ with infinity.)",
"Let $A$ be a valued $\\tau $ -structure.",
"An instance of VCSP$(A)$ is given by a valued $\\tau $ -structure $I$ .",
"A solution to $I$ is a function $h : D(I) \\rightarrow D(A)$ , its measure given by $\\sum _{f \\in \\tau , {\\bar{x}} \\in D(I)^{ar(f)}} f^I({\\bar{x}}) f^A(h({\\bar{x}})).$ The goal is to find a solution of minimum measure.",
"This measure will be denoted by ${\\sf Opt}_A(I)$ .",
"For an $m$ -tuple $\\bar{t}$ , we denote by $\\lbrace \\bar{t}\\rbrace $ the set of elements in $\\bar{t}$ .",
"Furthermore, we denote by $[\\bar{t}]$ the multiset of elements in $\\bar{t}$ ."
],
[
"Fractional Homomorphisms",
"Let $A$ and $B$ be valued structures over the same signature $\\tau $ .",
"Let $B^A$ denote the set of all functions from $D(A)$ to $D(B)$ .",
"A fractional homomorphism from $A$ to $B$ is a function $\\omega : B^A \\rightarrow \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ , with $\\sum _{g \\in B^A} \\omega (g) = 1$ , such that for every function symbol $f \\in \\tau $ and tuple ${\\bar{a}} \\in D(A)^{ar(f)}$ , it holds that $\\sum _{g \\in B^A} \\omega (g) f^B(g({\\bar{a}})) \\le f^A({\\bar{a}}),$ where the functions $g$ are applied component-wise.",
"We write $A \\rightarrow _fB$ to indicate the existence of a fractional homomorphism.",
"Proposition 2.1 Assume that $A \\rightarrow _fB$ .",
"Then ${\\sf Opt}_A(I) \\ge {\\sf Opt}_B(I)$ , for every instance $I$ .",
"Let $\\omega $ be a fractional homomorphism from $A$ to $B$ , let $X = D(I)$ and let $h : X \\rightarrow A$ be an arbitrary solution.",
"Then, $\\sum _{f, \\bar{x}} f^I({\\bar{x}}) f^A(h({\\bar{x}})) \\ge \\sum _{f, \\bar{x}} f^I({\\bar{x}}) \\sum _{g \\in B^A} \\omega (g) f^B(g(h({\\bar{x}}))) = \\sum _{g \\in B^A} \\omega (g) \\sum _{f, \\bar{x}} f^I({\\bar{x}}) f^B(g(h({\\bar{x}}))),$ where the sums are over $f \\in \\tau $ and $\\bar{x} \\in X^{ar(f)}$ .",
"Hence, there exists a $g \\in B^A$ such that the measure of the solution $g\\circ h$ to $I$ as an instance of VCSP$(B)$ is no greater than the measure of the solution $h$ to $I$ as an instance of VCSP$(A)$ ."
],
[
"Fractional Polymorphisms",
"Let $A$ be a valued $\\tau $ -structure, and let $D = D(A)$ .",
"An $m$ -ary operation on $D$ is a function $g : D^m \\rightarrow D$ .",
"Let ${\\cal O}^{(m)}_{D}$ denote the set of all $m$ -ary operations on $D$ .",
"An $m$ -ary fractional operation is a function $\\omega : {\\cal O}^{(m)}_{D} \\rightarrow \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"Define $\\Vert \\omega \\Vert _1 := \\sum _{g} \\omega (g)$ .",
"An $m$ -ary fractional operation $\\omega $ is called an $m$ -ary fractional polymorphism [9] if $\\Vert \\omega \\Vert _1 = 1$ and for every function symbol $f \\in \\tau $ and tuples ${\\bar{a}_1}, \\dots , {\\bar{a}_m} \\in D^{ar(f)}$ , it holds that $ \\sum _{g \\in {\\cal O}^{(m)}_{D}} \\omega (g) f^A(g({\\bar{a}_1},\\dots ,{\\bar{a}_m})) \\le \\frac{1}{m} \\sum _{i=1}^m f^{A}({\\bar{a}_i}).",
"$ The set $\\lbrace g \\mid \\omega (g) > 0 \\rbrace $ of operations is called the support of $\\omega $ and is denoted by $\\mbox{\\rm supp}(\\omega )$ .",
"Let $S_m$ be the symmetric group on $\\lbrace 1,\\dots ,m\\rbrace $ .",
"An $m$ -ary operation $g$ is symmetricSymmetric operations are called totally symmetric in [39].",
"if for every permutation $\\pi \\in S_m$ , we have $ g(x_1,\\dots ,x_m) = g(x_{\\pi (1)},\\dots ,x_{\\pi (m)}).",
"$ Definition 1 A totally symmetric fractional polymorphism $\\omega $ is a fractional polymorphism such that if $g \\in \\mbox{\\rm supp}(\\omega )$ , then $g$ is symmetric.",
"The superposition of an $n$ -ary operation $h$ with $n$ $m$ -ary operations $g_1, \\dots , g_n$ is the $m$ -ary operation defined by $h[g_1,\\dots ,g_n](x_1,\\dots ,x_m) = h(g_1(x_1,\\dots ,x_m),\\dots ,g_n(x_1,\\dots ,x_m))$ .",
"A set of operations is called a clone if it contains all projections and is closed under superposition.",
"The smallest clone that contains a set of operations ${\\cal F}$ is called the clone generated by ${\\cal F}$ .",
"We say that an operation $f$ is generated by ${\\cal F}$ if it is contained in the clone generated by ${\\cal F}$ .",
"Definition 2 The superposition, $\\omega [g_1,\\dots ,g_n]$ , of an $n$ -ary fractional polymorphism $\\omega $ with $n$ $m$ -ary operations $g_1,\\dots ,g_n$ is the $m$ -ary fractional operation $\\omega ^{\\prime }$ , where $\\omega ^{\\prime }(h^{\\prime }) = \\sum _{h : h^{\\prime } = h[g_1,\\dots ,g_n]} \\omega (h).$ Note that in general $\\omega ^{\\prime }$ is not a fractional polymorphism, but it does satisfy the following inequality: $\\sum _{h^{\\prime } \\in {\\cal O}^{(m)}_{D}} \\omega ^{\\prime }(h^{\\prime }) f^A(h^{\\prime }({\\bar{a}}_1,\\dots ,{\\bar{a}}_m))& = & \\sum _{h \\in {\\cal O}^{(n)}_{D}} \\omega (h) f^A(h[g_1,\\dots ,g_n]({\\bar{a}}_1,\\dots ,{\\bar{a}}_m))\\\\& \\le &\\frac{1}{n} \\sum _{i=1}^n f^A(g_i({\\bar{a}}_1,\\dots ,{\\bar{a}}_m)),$ for every $f \\in \\tau $ and ${\\bar{a}}_1, \\dots {\\bar{a}}_m \\in D^{ar(f)}$ ."
],
[
"The Multiset-Structure $P^m(A)$",
"Let $A$ be a valued $\\tau $ -structure, $D = D(A)$ , and let $m \\ge 1$ .",
"We define the multiset-structureA similar structure for $\\lbrace 0,\\infty \\rbrace $ -valued languages was introduced in [39].",
"$P^m(A)$ as the valued structure with domain $\\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ , where $\\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ denotes the multisets of elements from $D$ of size $m$ , and for every $k$ -ary function symbol $f \\in \\tau $ , and $\\alpha _1,\\dots ,\\alpha _k \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ , $f^{P^m(A)}(\\alpha _1, \\dots , \\alpha _k) = \\frac{1}{m} \\min _{{\\bar{t}_i} \\in D^m : [{\\bar{t}_i}] = \\alpha _i} \\sum _{i=1}^m f^{A}({\\bar{t}_1}[i],\\dots ,{\\bar{t}_k}[i]).$ The following lemma follows from the definitions (the proof is in the appendix).",
"Lemma 2.2 Let $A$ be a valued structure and $m > 1$ .",
"Then $P^m(A) \\rightarrow _fA$ if and only if $A$ has an $m$ -ary totally symmetric fractional polymorphism."
],
[
"Basic Linear Programming Relaxation",
"Let $I$ and $A$ be valued structures over a common finite signature $\\tau $ .",
"Let $X = D(I)$ and $D = D(A)$ .",
"The basic LP relaxation (BLP) (sometimes also called the standard, or canonical LP relaxation) has variables $\\lambda _{f,{\\bar{x}},\\sigma }$ for $f \\in \\tau $ , ${\\bar{x}} \\in X^{ar(f)}$ , $\\sigma : \\lbrace {\\bar{x}}\\rbrace \\rightarrow D$ ; and variables $\\mu _{x}(a)$ for $x \\in X, a \\in D$ .",
"$\\begin{array}{lll}\\min & \\multicolumn{2}{l}{\\displaystyle \\sum _{f,{\\bar{x}}} \\sum _{\\sigma : \\lbrace {\\bar{x}}\\rbrace \\rightarrow D} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) \\lambda _{f,{\\bar{x}},\\sigma }} \\\\\\text{s.t.",
"}& \\displaystyle \\sum _{\\sigma : \\sigma (x) = a} \\lambda _{f,{\\bar{x}},\\sigma } = \\mu _{x}(a)& \\qquad \\text{ $\\forall f \\in \\tau , {\\bar{x}} \\in X^{ar(f)}, x \\in \\lbrace {\\bar{x}}\\rbrace , a \\in D$} \\\\& \\hspace*{8.00003pt} \\displaystyle \\sum _{a \\in D} \\mu _{x}(a) = 1& \\qquad \\text{ $\\forall x \\in X$} \\\\\\smallskip &\\quad 0 \\le \\lambda , \\mu \\le 1 & \\\\\\end{array}$ For any fixed $A$ , BLP is polynomial in the size of a given VCSP$(A)$ instance.",
"Let IP be the program obtained from (REF ) together with the constraints that all variables take values in the range $\\lbrace 0,1\\rbrace $ rather than $[0,1]$ .",
"This is an integer programming formulation of the original VCSP instance.",
"The interpretation of the variables in IP is as follows: $\\mu _{x}(a)=1$ iff variable $x$ is assigned value $a$ ; $\\lambda _{f,{\\bar{x}},\\sigma }=1$ iff constraint $f$ on scope ${\\bar{x}}$ is assigned tuple $\\sigma ({\\bar{x}})$ .",
"LP (REF ) is now a relaxation of IP and the question of whether (REF ) solves a given VCSP instance $I$ is the question of whether IP has a zero integrality gap."
],
[
"Characterisation",
"Definition 3 Let BLP$(I,A)$ denote the optimum of $(\\ref {eq:basiclp})$ .",
"We say that BLP solves VCSP$(A)$ if BLP$(I,A) = {\\sf Opt}_A(I)$ for every instance $I$ of VCSP$(A)$ .",
"Solving the BLP provides an optimum value of the VCSP.",
"To obtain an assignment achieving this value, we apply self-reduction: Successively try each possible value for a variable and solve the altered LP.",
"Once the new optimum matches the original one, proceed with the next variable.",
"Theorem 4.1 (Main) Let $A$ be a valued structure over a finite signature.",
"TFAE: BLP solves VCSP$(A)$ .",
"For every $m>1$ , $P^m(A) \\rightarrow _fA$ .",
"For every $m>1$ , $A$ has an $m$ -ary totally symmetric fractional polymorphism.",
"For every $n>1$ , $A$ has a fractional polymorphism $\\omega _n$ such that $\\mbox{\\rm supp}(\\omega _n)$ generates an $n$ -ary symmetric operation.",
"The rest of this section is devoted to proving Theorem REF .",
"We start with proving $(\\ref {main:2})\\Rightarrow (\\ref {main:1})$ .",
"Theorem 4.2 Assume that $P^m(A) \\rightarrow _fA$ for every $m > 1$ .",
"Then BLP solves VCSP$(A)$ .",
"Let $\\lambda ^*, \\mu ^*$ be an optimal solution to (REF ).",
"Let $M$ be a positive integer such that $M \\cdot \\lambda ^*$ and $M \\cdot \\mu ^*$ are both integral.",
"Let $\\nu : X \\rightarrow \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{M}\\big )\\hspace{-1.99997pt}\\big )$ be defined by mapping $x$ to the multiset in which the elements are distributed according to $\\mu ^*_x$ , i.e., the number of occurrences of $a$ in $\\nu (x)$ is equal to $M \\cdot \\mu ^*_{x}(a)$ for each $a \\in D$ .",
"Let $f$ be a $k$ -ary function symbol in $\\tau $ , and let $\\sum _{{\\bar{x}}, \\sigma : \\lbrace {\\bar{x}}\\rbrace \\rightarrow D} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) \\lambda ^*_{f,{\\bar{x}},\\sigma }= \\sum _{{\\bar{x}}} f^I({\\bar{x}})\\Big (\\sum _{\\sigma : \\lbrace \\bar{x}\\rbrace \\rightarrow D} \\lambda ^*_{f,{\\bar{x}},\\sigma } f^A(\\sigma ({\\bar{x}}))\\Big )$ be the sum of all terms of the objective function in which $f$ occurs; Now, write $M \\cdot \\sum _{\\sigma : \\lbrace \\bar{x}\\rbrace \\rightarrow D} \\lambda ^*_{f,{\\bar{x}},\\sigma } f^A(\\sigma ({\\bar{x}}))= f^A({\\bar{a}_1}) + \\dots + f^A({\\bar{a}_M}),$ where the ${\\bar{a}_i} \\in D^k$ are such that a $\\lambda ^*_{f,{\\bar{x}},\\sigma }$ -fraction are equal to $\\sigma ({\\bar{x}})$ .",
"Let ${\\bar{a}_i}^{\\prime } = ({\\bar{a}_1}[i],\\dots ,{\\bar{a}_M}[i])$ for $i = 1, \\dots , k$ .",
"$\\sum _{\\sigma : \\lbrace \\bar{x}\\rbrace \\rightarrow D} \\lambda ^*_{f,\\bar{x},\\sigma } f^A(\\sigma ({\\bar{x}}))& = & \\frac{1}{M} \\sum _{i=1}^{M} f^A({\\bar{a}_i})\\ =\\ \\frac{1}{M} \\sum _{i=1}^{M} f^A({\\bar{a}_1}^{\\prime }[i],\\dots ,{\\bar{a}_k}^{\\prime }[i]) \\\\& \\ge &\\frac{1}{M} \\min _{{\\bar{t}_i} \\in D^M : [{\\bar{t}_i}] = [{\\bar{a}_i}^{\\prime }]} \\sum _{i=1}^M f^A({\\bar{t}_1}[i],\\dots ,{\\bar{t}_k}[i])\\ =\\ f^{P^M(A)}(\\nu ({\\bar{x}})),$ where the last equality follows as the number of $a$ 's in $\\bar{a}^{\\prime }_i$ is $M \\cdot \\sum _{\\sigma : \\sigma (\\bar{x}[i]) = a} \\lambda ^*_{f,\\bar{x},\\sigma } = M \\cdot \\mu ^*_{\\bar{x}[i]}(a)$ .",
"We now have $BLP(I,A) & = &\\sum _{f, {\\bar{x}}} \\sum _{\\sigma : \\lbrace {\\bar{x}}\\rbrace \\rightarrow D} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) \\lambda ^*_{f,{\\bar{x}},\\sigma } \\\\& = &\\sum _{f \\in \\tau , {\\bar{x}}} f^I({\\bar{x}})\\Big (\\sum _{\\sigma : \\lbrace \\bar{x}\\rbrace \\rightarrow D} \\lambda ^*_{f,{\\bar{x}},\\sigma } f^A(\\sigma ({\\bar{x}}))\\Big ) \\\\& \\ge &\\sum _{f \\in \\tau , {\\bar{x}}} f^I({\\bar{x}}) f^{P^M(A)}(\\nu ({\\bar{x}})) \\\\& = & {\\sf Opt}_{P^M(A)}(I)$ It follows that ${\\sf Opt}_A(I) \\ge BLP(I,A) \\ge {\\sf Opt}_{P^M(A)}(I)$ .",
"Since $P^M(A) \\rightarrow _fA$ , the result then follows from Proposition REF .",
"To prove $(\\ref {main:1})\\Rightarrow (\\ref {main:2})$ , we express the existence of a fractional homomorphism $P^m(A) \\rightarrow _fA$ as a system of linear inequalities.",
"We then apply a variant of Farkas' Lemma to show that if for some $m > 1$ there is no such fractional homomorphism, then there exists an instance $I$ of VCSP$(A)$ with a strictly greater optimum than BLP$(I,A)$ (the proof is in the appendix).",
"Theorem 4.3 Let $A$ be a valued structure and assume that BLP solves VCSP$(A)$ .",
"Then $P^m(A) \\rightarrow _fA$ for every $m > 1$ .",
"Lemma REF proves $(\\ref {main:2})\\Leftrightarrow (\\ref {main:3})$ .",
"Since $(\\ref {main:3})\\Rightarrow (\\ref {main:4})$ follows trivially, it remains to show that $(\\ref {main:4})\\Rightarrow (\\ref {main:3})$ .",
"Theorem 4.4 Let $A$ be a valued structure and assume that for every $n>1$ , $A$ has a fractional polymorphism $\\omega _n$ that generates an $n$ -ary symmetric operation.",
"Then, for every $m > 1$ , $A$ has an $m$ -ary totally symmetric fractional polymorphism.",
"For an $m$ -ary operation $g$ , let $\\tilde{g}$ denote the equivalence class of $g$ under the relation: $g \\sim g^{\\prime } \\Leftrightarrow g(x_1,\\dots ,x_m) = g^{\\prime }(x_{\\pi (1)},\\dots ,x_{\\pi (m)})\\text{ for some $\\pi \\in S_m$.",
"}$ Note that we have $|\\tilde{g}| = 1$ if and only if $g$ is symmetric.",
"We say that a fractional operation $\\omega $ is weight-symmetric if $\\omega (g) = \\omega (g^{\\prime }) \\text{ whenever $g \\sim g^{\\prime }$}.$ We construct an $m$ -ary totally symmetric fractional polymorphism by building a rooted tree in a number of stages.",
"At each stage of the construction, every node $u$ of the tree contains an $m$ -ary weight-symmetric fractional operation with support on a single equivalence class of $\\sim $ .",
"For a node $u$ , we will also denote this fractional operation by $u$ .",
"Since $u$ is weight-symmetric, it follows that $u(g) = u(g^{\\prime })$ for all $g, g^{\\prime } \\in \\mbox{\\rm supp}(u)$ .",
"This common weight for the operations in the support of $u$ will be denoted by $w(u)$ .",
"A node $u$ with $|\\mbox{\\rm supp}(u)| = 1$ will be called final.",
"The following invariants are maintained throughout the construction.",
"Every non-leaf node has at least one final child.",
"For every node $u$ , we have $w(u) > 0$ .",
"For every non-leaf node $v$ , $\\sum _{\\text{$u$ is a child of $v$}} \\Vert u\\Vert _1 = \\Vert v\\Vert _1.$ For every non-leaf node $v$ , every $f \\in \\tau $ , and all tuples ${\\bar{a}_1}, \\dots , {\\bar{a}_m} \\in D^{ar(f)}$ , $\\sum _{g} v(g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m}))\\ge \\sum _{\\text{$u$ is a child of $v$}} \\sum _{g} u(g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m})).$ We say that a leaf $u$ is covered (by $v$ ), and that $v$ is a covering node (of $u$ ) if $\\mbox{\\rm supp}(u) = \\mbox{\\rm supp}(v)$ and $u$ is a (proper) descendant of $v$ .",
"We say that $v$ is a minimal covering node of $u$ if no descendant of $v$ is a covering node of $u$ .",
"At the beginning of the construction, the tree consists of a single root $r$ with $\\mbox{\\rm supp}(r)$ being the set of $m$ -ary projections and $w(r) = \\frac{1}{m}$ .",
"We then apply the following two steps: Expansion: A leaf $u$ that is not final and not covered is chosen to be expanded.",
"This amounts to adding a finite non-empty set of children to $u$ while maintaining the invariants.",
"The expansion step is repeated until no longer applicable.",
"Pruning: A leaf that is not final and covered is removed together with a number of internal nodes while maintaining the invariants.",
"The pruning step is repeated until no longer applicable.",
"Since there is a finite number of $m$ -ary operations, and hence a finite number of equivalence classes of $\\sim $ , it follows that, eventually, every leaf in the tree that is not final must be covered.",
"Hence, the expansion step is only applicable a finite number of times.",
"Each round of pruning shrinks the tree by at least one node, but no final leaf is ever removed.",
"Therefore we eventually obtain a tree containing only final leaves, at which time the pruning step is no longer applicable.",
"Let $\\mathcal {L}$ be the set of leaves in the final tree.",
"By repeated application of invariant (REF ), starting from the root, $\\sum _{u \\in \\mathcal {L}} u$ is then an $m$ -ary totally symmetric fractional polymorphism.",
"Expansion.",
"We expand a leaf $u$ with $|\\mbox{\\rm supp}(u)| = n$ as follows: Let $\\omega $ be a $k$ -ary fractional polymorphism of $A$ such that $\\mbox{\\rm supp}(\\omega )$ generates an $n$ -ary symmetric operation $t$ .",
"We will define a sequence of $m$ -ary weight-symmetric fractional operations $\\nu _i$ , each with $\\Vert \\nu _i\\Vert _1 = \\Vert u\\Vert _1$ .",
"Let $\\nu _0 = u$ .",
"Assume that $\\nu _{i-1}$ has been defined for some $i \\ge 1$ .",
"Let $l_{i-1} = \\min \\lbrace \\nu _{i-1}(g) \\mid g \\in \\mbox{\\rm supp}(\\nu _{i-1}) \\rbrace $ be the minimum weight of an operation in the support of $\\nu _{i-1}$ .",
"The fractional operation $\\nu _i$ is obtained by subtracting from $\\nu _{i-1}$ an equal amount of weight from each operation in $\\mbox{\\rm supp}(\\nu _{i-1})$ and adding this weight as superpositions of $\\omega $ by all possible choices of operations in $\\nu _{i-1}$ .",
"The amount subtracted from each operation is $\\frac{1}{2} l_{i-1}$ so that every operation in $\\mbox{\\rm supp}(\\nu _{i-1})$ is also in $\\mbox{\\rm supp}(\\nu _i)$ .",
"Formally $\\nu _i$ is defined as follows: $ \\nu _i =\\nu _{i-1} -\\frac{1}{2} l_{i-1} \\chi _{i-1} +\\sum _{(g_1,\\dots ,g_k) \\in {\\rm supp}(\\nu _{i-1})^k} \\frac{1}{2} l_{i-1} \\frac{1}{K} \\omega [g_1,\\dots ,g_k],$ where $K = |\\mbox{\\rm supp}(\\nu _{i-1})|^k$ and $\\chi _{i-1}$ is the indicator function of $\\mbox{\\rm supp}(\\nu _{i-1})$ .",
"By definition $\\Vert \\nu _i\\Vert _1 = \\Vert \\nu _{i-1}\\Vert _1 = \\Vert u\\Vert _1$ .",
"To verify that $\\nu _i$ is weight-symmetric, it suffices to verify that the sum $ \\sum _{(g_1,\\dots ,g_k) \\in {\\rm supp}(\\nu _{i-1})^k} \\omega [g_1,\\dots ,g_k]$ is weight-symmetric.",
"Let $g \\sim g^{\\prime }$ , let $\\pi \\in S_m$ be such that $g(x_1,\\dots ,x_m) =g^{\\prime }(x_{\\pi (1)},\\dots ,x_{\\pi (m)})$ and let $g^{\\prime }_j(x_1,\\dots ,x_m) =g_j(x_{\\pi (1)},\\dots ,x_{\\pi (m)})$ for $1 \\le j \\le m$ .",
"Since $\\nu _{i-1}$ is weight-symmetric, it follows that $g_i \\in {\\rm supp}(\\nu _{i-1})$ if and only if $g^{\\prime }_i \\in {\\rm supp}(\\nu _{i-1})$ .",
"Therefore the terms $\\omega (h) h[g_1,\\dots ,g_k]$ in (REF ) such that $g = h[g_1,\\dots ,g_k]$ are in bijection with the terms $\\omega (h) h[g^{\\prime }_1,\\dots ,g^{\\prime }_k]$ such that $g^{\\prime } = h[g^{\\prime }_1,\\dots ,g^{\\prime }_k]$ .",
"So the fractional operation in (REF ) assigns the same weight to $g$ and $g^{\\prime }$ .",
"Let $e$ be an expression for $t$ consisting of superpositions of projections and operations from $\\mbox{\\rm supp}(\\omega )$ .",
"We recursively define the nested depth, $d = d(e)$ , of $e$ as follows: $d(p) = 0$ for every projection $p$ ; and $d(h[g_1,\\dots ,g_k]) = 1+\\max _{1\\le i \\le k} d(g_i)$ .",
"Let $\\mbox{\\rm supp}(u) = \\lbrace g_1,\\dots ,g_n\\rbrace $ .",
"Using $\\mbox{\\rm supp}(\\nu _0) = \\mbox{\\rm supp}(u)$ and the fact that $\\mbox{\\rm supp}(\\nu _i)$ contains all superpositions of operations in $\\mbox{\\rm supp}(\\nu _{i-1})$ , it follows that $t[g_1,\\dots ,g_n] \\in \\mbox{\\rm supp}(\\nu _d)$ .",
"Now, we add a child $v$ to $u$ for every equivalence class in the set $\\lbrace \\tilde{g} \\mid g \\in \\mbox{\\rm supp}(\\nu _d) \\rbrace $ .",
"For an added child $v$ with $\\mbox{\\rm supp}(v) = \\tilde{g}$ , we let $w(v) = \\nu _d(g)$ .",
"Invariant (REF ) holds as $t[g_1,\\dots ,g_n] \\in \\mbox{\\rm supp}(\\nu _d)$ is symmetric: for all $\\pi \\in S_m$ there is a $\\pi ^{\\prime } \\in S_n$ such that $t[g_1,\\dots ,g_n](x_{\\pi (1)},\\dots ,x_{\\pi (m)}) = t[g_{\\pi ^{\\prime }(1)},\\dots ,g_{\\pi ^{\\prime }(n)}](x_1,\\dots ,x_m) = t[g_1,\\dots ,g_n](x_1,\\dots ,x_m)$ .",
"Invariants (REF ) and (REF ) hold by construction.",
"For each $i \\ge 1$ , we have $\\sum _{g} \\nu _{i-1}(g) f^A(g({\\bar{a}}_1,\\dots ,{\\bar{a}}_m)) \\ge \\sum _{g} \\nu _{i}(g) f^A(g({\\bar{a}}_1,\\dots ,{\\bar{a}}_m))$ for all $f \\in \\tau $ and ${\\bar{a}}_1, \\dots {\\bar{a}}_m \\in D^{ar(f)}$ .",
"Therefore invariant (REF ) also holds after expanding $u$ .",
"Pruning.",
"The pruning step maintains an additional invariant, namely that every leaf that is not final is covered.",
"Pruning is accomplished as follows.",
"Pick a minimal covering node $v$ .",
"Let $\\nu = \\nu _v + \\nu _\\bot $ be the fractional operation induced by the leaves in the subtree rooted at $v$ , where $\\nu _v$ is the part of $\\nu $ with the same support as $v$ and $\\nu _\\bot $ is the part of $\\nu $ with support disjoint from $v$ .",
"Inductively, by invariant (REF ), $\\sum _{g} v(g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m})) \\ge \\sum _{g} \\nu _v(g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m})) +\\sum _{g} \\nu _\\bot (g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m})),$ for all $f \\in \\tau $ and ${\\bar{a}}_1, \\dots {\\bar{a}}_m \\in D^{ar(f)}$ .",
"We simplify this inequality as follows.",
"$\\sum _{g} v(g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m})) \\ge \\sum _{g} \\frac{1}{1-\\kappa } \\nu _\\bot (g) f^A(g({\\bar{a}_1}, \\dots , {\\bar{a}_m})),$ where $\\kappa = \\Vert \\nu _v\\Vert _1/\\Vert v\\Vert _1$ .",
"Remove all nodes below $v$ and add a new child $u$ to $v$ for every equivalence class in the set $\\lbrace \\tilde{g} \\mid g \\in \\mbox{\\rm supp}(\\nu _\\bot ) \\rbrace $ .",
"For an added child $u$ with $\\mbox{\\rm supp}(u) = \\tilde{g}$ , we let $w(u) = \\frac{1}{1-\\kappa } \\nu _\\bot (g)$ .",
"By invariant (REF ) the node $v$ is guaranteed to have at least one final child (leaf).",
"Hence, by invariant (REF ), $\\nu _\\bot $ is not identically 0.",
"By induction on (REF ), it follows that $\\Vert v\\Vert _1 > \\Vert \\nu _v\\Vert _1$ , so $\\kappa < 1$ and the new weights are defined and positive.",
"So invariant (REF ) holds.",
"Furthermore, $\\sum _{\\text{$u$ is a child of $v$}} \\Vert u\\Vert _1 =\\frac{1}{1-\\kappa } \\Vert \\nu _\\bot \\Vert _1 =\\frac{1}{1-\\kappa } (\\Vert \\nu \\Vert _1 - \\Vert \\nu _v\\Vert _1) =\\frac{1}{1-\\kappa } (\\Vert v\\Vert _1 - \\Vert \\nu _v\\Vert _1) =\\Vert v\\Vert _1,$ so invariant (REF ) holds.",
"Invariant (REF ) holds by construction.",
"Finally, invariant (REF ) also holds since for any final child $u$ of $v$ , there is still a child of $v$ with support on the same equivalence class as $u$ .",
"Only nodes in the subtree rooted at $v$ have been removed and every child added to $v$ has the same support as a previous leaf of this subtree.",
"Every such leaf $u$ that was not final was covered by a node above $v$ in the tree.",
"Hence, all leaves in the tree that are not final are still covered.",
"General-valued Structures The valued structures we have dealt with so far were in fact finite-valued; i.e., for each function symbol $f\\in \\tau $ , the range of $f^A$ was $\\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"We now discuss how our result can be extended to the general-valued case, in which for each function symbol $f\\in \\tau $ , the range of $f^A$ is $\\mbox{$\\overline{\\mathbb {Q}}_{\\ge 0}$}=\\mbox{$\\mathbb {Q}_{\\ge 0}$}\\cup \\lbrace \\infty \\rbrace $ .",
"(We define $c+\\infty =\\infty +c=\\infty $ for all $c\\in \\mbox{$\\overline{\\mathbb {Q}}_{\\ge 0}$}$ , and $0\\infty =\\infty 0=0$ .)",
"Inspired by the OSAC algorithm [13], the algorithm for general-valued structures, denoted by BLP$_g$ , works in two stages.",
"Firstly, the instance is made arc consistent using a standard arc consistency algorithm [23].The algorithm from [23] is sometimes called generalised arc consistency algorithm to emphasise the fact that it works for CSPs of arbitrary arities, not only for binary CSPs [41].",
"Secondly, BLP (i.e., linear program (1) from Section ) is solved.",
"Definition 4 Let $A$ be a general-valued $\\tau $ -structure, and let $D=D(A)$ .",
"An $m$ -ary operation $g:D^m\\rightarrow D$ is a polymorphism of $A$ if for every function symbol $f\\in \\tau $ and tuples ${\\bar{a}_1},\\ldots ,{\\bar{a}_m}\\in D^{ar(f)}$ , it holds that ${\\sf Feas}(f^A(g({\\bar{a}_1},\\dots ,{\\bar{a}_m}))) \\le \\sum _{i=1}^m {\\sf Feas}(f^{A}({\\bar{a}_i})),$ where ${\\sf Feas}(\\infty )=\\infty $ and ${\\sf Feas}(c)=0$ for any $c\\in \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"From Definitions REF and REF , if $\\omega $ is a fractional polymorphism of a general-valued structure $A$ , then every $g\\in \\mbox{\\rm supp}(\\omega )$ is a polymorphism of $A$ .",
"In fact, any operation $g$ that is generated by $\\mbox{\\rm supp}(\\omega )$ (i.e., $g$ belongs to the clone generated by $\\mbox{\\rm supp}(\\omega )$ ) is a polymorphism of $A$ .",
"(For finite-valued structures, trivially, every operation $g$ is a polymorphism.)",
"Arc consistency (also known as $(1,k)$ -consistency) [23] is a decision procedure for precisely those $\\lbrace 0,\\infty \\rbrace $ -valued structures $A$ that are closed under a set function $g : 2^{D(A)} \\setminus \\lbrace \\emptyset \\rbrace \\rightarrow D(A)$ [19], [21].",
"This condition has recently been shown to be equivalent to the requirement that $A$ should have symmetric polymorphisms of all arities [39].",
"Our main theorem (Theorem REF ) also holds for general-valued structures: Theorem 5.1 Let $A$ be a general-valued structure over a finite signature.",
"TFAE: BLP$_g$ solves VCSP$(A)$ .",
"For every $m>1$ , $P^m(A) \\rightarrow _fA$ .",
"For every $m>1$ , $A$ has an $m$ -ary totally symmetric fractional polymorphism.",
"For every $n>1$ , $A$ has a fractional polymorphism $\\omega _n$ such that $\\mbox{\\rm supp}(\\omega _n)$ generates an $n$ -ary symmetric operation.",
"Note that $(\\ref {gmain:2})$ implies that $A$ has symmetric polymorphisms of all arities.",
"This follows from Lemma REF , which holds for general-valued structures, and the above-mentioned fact that any $g\\in \\mbox{\\rm supp}(\\omega )$ , where $\\omega $ is a fractional polymorphism of $A$ , is a polymorphism of $A$ .",
"The same argument guarantees symmetric polymorphisms of all arities in $(\\ref {gmain:3})$ and $(\\ref {gmain:4})$ ; in $(\\ref {gmain:4})$ , we use that fact than any operation generated by $\\mbox{\\rm supp}(\\omega _n)$ is a polymorphism of $A$ .",
"Theorem REF proves $(\\ref {gmain:2})\\Rightarrow (\\ref {gmain:1})$ since the assumption of having symmetric polymorphisms of all arities guarantees a feasible solution to (REF ).",
"From the discussion above on arc consistency, if BLP$_g$ solves VCSP$(A)$ , then $A$ has symmetric polymorphisms of all arities since arc consistency decides the existence of a finite-valued solution.",
"Furthermore, the proof of Theorem REF shows that having symmetric polymorphisms of all arities but not having a fractional homomorphism $P^m(A) \\rightarrow _fA$ implies that BLP$_g$ does not solve VCSP$(A)$ .",
"This gives $(\\ref {gmain:1})\\Rightarrow (\\ref {gmain:2})$ .",
"$(\\ref {gmain:2})\\Leftrightarrow (\\ref {gmain:3})$ and $(\\ref {gmain:3})\\Leftrightarrow (\\ref {gmain:4})$ are proved the same way as in Theorem REF , by Lemma REF and Theorem REF , respectively.",
"Remark 1 BLP$_g$ for general-valued structures uses the arc consistency algorithm [23] and BLP.",
"Since Kun et al [39] have shown that BLP solves CSPs (i.e., $\\lbrace 0,\\infty \\rbrace $ -valued VCSPs), if represented by $\\lbrace 0,1\\rbrace $ -valued structures, of width 1 – thus providing an alternative to the standard arc consistency algorithm [23] – a different approach is to combine Theorem REF from this paper with [39] and solve general-valued structures using only BLP with an amended objective function which takes care of infinite costs using a large (but polynomial), instance-dependent constant.",
"Tractable Valued Constraint Languages As before, we denote $D=D(A)$ .",
"A binary multimorphism [11] of a valued structure $A$ is a pair $\\langle g_1, g_2 \\rangle $ of binary functions $g_1,g_2:D^2\\rightarrow D$ such that for every function symbol $f \\in \\tau $ and tuples $\\bar{a}_1,\\bar{a}_2 \\in D^{ar(f)}$ , it holds that $f^A(g_1(\\bar{a}_1,\\bar{a}_2))+f^A(g_2(\\bar{a}_1,\\bar{a}_2)) \\le f^A(\\bar{a}_1)+f^A(\\bar{a}_2).$ (Multimorphisms are a special case of fractional polymorphisms.)",
"Since any semi-lattice operationA semi-lattice operation is associative, commutative, and idempotent.",
"generates symmetric operations of all arities, we get: Corollary 6.1 (of Theorem REF ) Let $A$ be a valued structure with a binary multimorphism $\\langle g_1, g_2 \\rangle $ where either $g_1$ or $g_2$ is a semi-lattice operation.",
"Then $A$ is tractable.",
"We now give examples of valued structures (i.e., valued constraint languages) defined by such binary multimorphisms.",
"Example 1 Let $(D; \\wedge , \\vee )$ be an arbitrary lattice on $D$ .",
"Assume that a valued structure $A$ has the multimorphism $\\langle \\wedge , \\vee \\rangle $ .",
"Then VCSP$(A)$ is tractable.",
"The tractability of $A$ was previously known only for distributive lattices [48], [28] and (finite-valued) diamonds [38], see also [37].",
"Example 2 A pair of operations $\\langle g_1, g_2 \\rangle $ is called a symmetric tournament pair (STP) if both $g_1$ and $g_2$ are commutative, conservative ($g_1(x,y)\\in \\lbrace x,y\\rbrace $ and $g_2(x,y)\\in \\lbrace x,y\\rbrace $ for all $x,y\\in D$ ), and $g_1(x,y)\\ne g_2(x,y)$ for all $x,y\\in D$ .",
"Let $A$ be a finite-valued structure with an STP multimorphism $\\langle g_1, g_2 \\rangle $ .",
"It is known that if a finite-valued structure admits an STP multimorphism, it also admits a submodularity multimorphism.",
"This result is implicitly contained in [10].Namely, the STP might contain cycles, but [10] tells us that on cycles we have, in the finite-valued case, only unary cost functions.",
"It follows that the cost functions admitting the STP must be submodular with respect to some total order.",
"Therefore, BLP solves any instance from VCSP$(A)$ .",
"Example 3 Assume that a valued structure $A$ is bisubmodular [24].",
"This means that $D=\\lbrace 0,1,2\\rbrace $ and $A$ has a multimorphism $\\langle \\min _0, \\max _0 \\rangle $ [11], where $\\min _0(x,x)=x$ for all $x\\in D$ and $\\min _0(x,y)=0$ for all $x,y\\in D,x\\ne y$ ; $\\max _0(x,y)=0$ if $0\\ne x\\ne y\\ne 0$ and $\\max _0(x,y)=\\max (x,y)$ otherwise, where $\\max $ returns the larger of its two arguments with respect to the normal order of integers.",
"Since $\\min _0$ is a semi-lattice operation, $A$ is tractable.",
"The tractability of (finite-valued) $A$ was previously known only using a general algorithm for bisubmodular functions given by an oracle [24], [43].",
"Example 4 Assume that a valued structure $A$ is weakly tree-submodular on an arbitrary tree [35].",
"The meet (which is defined as the highest common ancestor) is again a semi-lattice operation.",
"The same holds for strongly tree-submodular structures since strong tree-submodularity implies weak tree-submodularity [35].",
"The tractability of weakly tree-submodular valued structures was previously known only for chains and forks [35].",
"The tractability of strongly tree-submodular valued structures was previously known only for binary trees [35].",
"Example 5 Note that the previous example applies to all trees, not just binary ones.",
"In particular, it applies to the tree consisting of one root with $k$ children.",
"This is equivalent to structures with $D=\\lbrace 0,1,\\dots ,k\\rbrace $ and the multimorphism $\\langle \\min _0, \\max _0 \\rangle $ from Example REF .",
"This is a natural generalisation of submodular ($k=1$ ) and bisubmodular ($k=2$ ) functions, known as $k$ -submodular functions [26].",
"The tractability of $k$ -submodular valued structures for $k>2$ was previously open.",
"Example 6 Let $b$ and $c$ be two distinct elements of $D$ and let $(D;<)$ be a partial order which relates all pairs of elements except for $b$ and $c$ .",
"A pair $\\langle g_1,g_2\\rangle $ , where $g_1,g_2:D^2\\rightarrow D$ are two binary operations, is a 1-defect multimorphism if $g_1$ and $g_2$ are both commutative and satisfy the following conditions: If $\\lbrace x,y\\rbrace \\ne \\lbrace b,c\\rbrace $ , then $g_1(x,y)=x\\wedge y$ and $g_2(x,y)=x\\vee y$ .",
"If $\\lbrace x,y\\rbrace =\\lbrace b,c\\rbrace $ , then $\\lbrace g_1(x,y),g_2(x,y)\\rbrace \\cap \\lbrace x,y\\rbrace =\\emptyset $ , and $g_1(x,y)<g_2(x,y)$ .",
"The tractability of valued structures that have a 1-defect multimorphism has recently been shown in [32].",
"We now show that valued structures with a 1-defect multimorphism are solvable by BLP$_g$ .",
"Without loss of generality, we assume that $g_1(b,c)<b,c$ and write $g=g_1$ .",
"(Otherwise, $g_2(b,c)>b,c$ , and $g_2$ is used instead.)",
"Using $g$ , we construct a symmetric $m$ -ary operation $f(x_1,\\dots ,x_m)$ .",
"Let $f_1,\\ldots ,f_M$ be the $M={m\\atopwithdelims ()2}$ terms $g(x_i,x_j)$ .",
"Let $f=g(f_1,g(f_2,\\dots ,g(f_{M-1},f_M)\\dots ))$ .",
"There are three possible cases: $\\lbrace b,c\\rbrace \\lnot \\subseteq {x_1,...,x_m}$ .",
"Then $g$ acts as $\\wedge $ , which is a semi-lattice operation, hence so does $f$ .",
"$\\lbrace b,c\\rbrace \\subseteq \\lbrace x_1,...,x_m\\rbrace $ and $g(b,c)\\le x_1,\\dots ,x_m$ .",
"Then $f_i=g(b,c)$ for some $1\\le i\\le M$ , and $g(f_i,f_j)=g(b,c)$ for all $1\\le j\\le M$ , so $f(x_1,\\dots ,x_m)=g(b,c)$ .",
"$\\lbrace b,c\\rbrace \\subseteq \\lbrace x_1,\\dots ,x_m\\rbrace $ and there is a variable $x_p$ for some $1\\le p\\le m$ such that $x_p\\le g(b,c)$ and $x_p\\le x_1,\\dots ,x_m$ .",
"Then $g(x_p,x_q)=x_p$ for all $1\\le q\\le m$ so $f_i=x_p$ for some $1\\le i\\le M$ and $g(f_i,f_j)=x_p$ for all $1\\le j\\le M$ , so $f(x_1,\\dots ,x_m)=x_p$ .",
"Conclusions We have characterised precisely for which valued structures the basic linear programming relaxation (BLP) is a decision procedure.",
"This implies tractability of several previously open classes of VCSPs including several generalisations of submodularity.",
"In fact, BLP solves all known tractable finite-valued structures.",
"The main result does not give a decidability criterion for testing whether a valued structure is solvable by BLP.",
"Interestingly, all known tractable finite-valued structures have a binary multimorphism.",
"It is possible that every (finite-)valued structure solvable by BLP admits a fixed-arity multimorphism, which would give a polynomial-time checkable condition.",
"An intriguing open question is whether our tractability results hold in the oracle-value model; that is, for objective functions which are not given explicitly as a sum of cost functions, but only by an oracle.",
"For instance, the maximisation problem for submodular functions on distributive lattices, known to be NP-complete, allows for good approximation algorithms in both models [22].",
"appendix Proof of Lemma REF Lemma 1.1 Let $A$ be a valued structure and $m > 1$ .",
"Then $P^m(A) \\rightarrow _fA$ if and only if $A$ has an $m$ -ary totally symmetric fractional polymorphism.",
"Let $D = D(A)$ .",
"$(\\Rightarrow )$ Let $\\omega $ be a fractional homomorphism from $P^m(A)$ to $A$ .",
"Let $h : D^m \\rightarrow \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ be the function that sends a tuple $(a_1,\\dots ,a_m) \\in D^m$ to its multiset $[a_1,\\dots ,a_m]$ .",
"Then the composition $\\omega \\circ h$ is the desired fractional polymorphism.",
"$(\\Leftarrow )$ Let $\\omega $ be an $m$ -ary totally symmetric fractional polymorphism.",
"Every operation $g \\in \\mbox{\\rm supp}(\\omega )$ induces a function $g^{\\prime } : \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big ) \\rightarrow D$ .",
"Define $\\omega ^{\\prime }$ as the fractional homomorphism with $\\omega ^{\\prime }(g^{\\prime }) = \\omega (g)$ .",
"Let $f$ be a $k$ -ary function symbol in the signature of $A$ , let $\\alpha _1, \\dots , \\alpha _k \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ be arbitrary, and pick ${\\bar{a}_i}$ with $[\\bar{a}_i] = \\alpha _i$ that minimises $\\sum _{i=1}^m f^A({\\bar{a}_1}[i],\\dots ,{\\bar{a}_k}[i])$ , for $1 \\le i \\le k$ .",
"$\\sum _{g^{\\prime }} \\omega ^{\\prime }(g^{\\prime }) f^{A}(g^{\\prime }(\\alpha _1),\\dots ,g^{\\prime }(\\alpha _k))& = & \\sum _{g} \\omega (g) f^{A}(g({\\bar{a}_1}),\\dots ,g({\\bar{a}_k}))\\\\& \\le & \\frac{1}{m} \\sum _{i=1}^m f^{A}(a_1[i],\\dots ,a_k[i])\\\\& = & \\frac{1}{m} \\min _{{\\bar{t}_i} \\in D^m : [{\\bar{t}_i}] = \\alpha _i} \\sum _{i=1}^m f^A({\\bar{t}_1}[i],\\dots ,{\\bar{t}_k}[i])\\\\& = & \\frac{1}{m} \\sum _{i=1}^m f^{P^m(A)}(\\alpha _1,\\dots ,\\alpha _k).$ Hence, $P^m(A) \\rightarrow _fA$ .",
"Proof of Theorem REF The following variant of Farkas' Lemma is due to Gale [25] (cf.",
"Mangasarian [42]).",
"Lemma 2.1 Let $A \\in \\mbox{$\\mathbb {R}$}^{m \\times n}$ and ${\\bar{b}} \\in \\mbox{$\\mathbb {R}$}^m$ .",
"Then exactly one of the two holds: $A{\\bar{x}} \\le {\\bar{b}}$ for some ${\\bar{x}} \\in \\mbox{$\\mathbb {R}$}^n$ ; or $A^T{\\bar{y}} = 0$ , ${\\bar{b}}^T{\\bar{y}} = -1$ for some ${\\bar{y}} \\in \\mbox{$\\mathbb {R}$}_{\\ge 0}$ .",
"Theorem 2.2 Let $A$ be a valued structure and assume that BLP solves VCSP$(A)$ .",
"Then $P^m(A) \\rightarrow _fA$ for every $m > 1$ .",
"Let $\\tau $ be the signature of $A$ and let $D = D(A)$ .",
"We prove the contrapositive.",
"Assume that there is an integer $m > 1$ such that $P^m(A)$ does not have a fractional homomorphism to $A$ .",
"Let $\\Omega $ denote the set of functions from $\\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ to $D$ .",
"We are assuming that the following system of inequalities does not have a solution $\\omega : \\Omega \\rightarrow \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"$\\sum _{g \\in \\Omega } \\omega (g) f^A(g(\\bar{\\alpha })) & \\le &f^{P^m(A)}(\\bar{\\alpha })\\quad \\text{ $\\forall f \\in \\tau , \\bar{\\alpha } \\in \\textstyle \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$}\\\\ \\sum _{g \\in \\Omega } \\omega (g) & = & 1 \\\\\\omega (g) & \\ge & 0 \\quad \\text{ $\\forall g \\in \\Omega $.",
"}$ In order to apply Lemma REF , we rewrite the equality $\\sum _{g} \\omega (g) = 1$ into two inequalities $\\sum _{g} \\omega (g) \\le 1$ and $-\\sum _{g} \\omega (g) \\le -1$ .",
"The last set of inequalities are rewritten to the form $-\\omega (g) \\le 0$ for each $g \\in \\Omega $ .",
"We have one variable for each inequality, i.e., $y(f,\\bar{\\alpha })$ for $f \\in \\tau $ , and $\\bar{\\alpha } \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$ .",
"Additionally, we have two variables $z_+, z_-$ for the two inequalities involving the constant 1 and one variable $w(g)$ for each $g \\in \\Omega $ .",
"$\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^A(g(\\bar{\\alpha })) + z_+ - z_- - w(g) & = & 0\\\\\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }) + z_+ - z_- & = & -1 \\\\y, z_+, z_-, w & \\ge & 0$ We can isolate $z_++z_-$ in the last equality, $z_++z_- = -1 - \\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }),$ which substituted into the first set of equalities implies that there is a solution $y(f,\\bar{\\alpha }), w(g)$ such that, for each $g \\in \\Omega $ , $\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^A(g(\\bar{\\alpha })) = w(g) + 1 + \\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }).$ We therefore find that there is a solution to the following system: $\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^A(g(\\bar{\\alpha })) & > & \\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha })\\quad \\text{ $\\forall g \\in \\Omega $} \\\\y(f,\\bar{\\alpha }) & \\ge & 0 \\quad \\text{ $\\forall f, \\bar{\\alpha }$.}",
"$ Let $I$ be the instance on variables $\\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ .",
"For each $k$ -ary function symbol $f \\in \\tau $ , and $\\bar{\\alpha } \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$ , define $f^I(\\bar{\\alpha }) = y(f,\\bar{\\alpha }).$ We now give a solution $\\lambda , \\mu $ to the basic LP (REF ) with an objective value equal to the right-hand side of (REF ).",
"Each variable $\\mu _{\\alpha }(a)$ is assigned the value of the multiplicity of $a$ in $\\alpha $ divided by $m$ .",
"Given $f, \\bar{\\alpha }$ , let ${\\bar{t}_1},\\dots ,{\\bar{t}_k} \\in D^m$ be such that $f^{P^m(A)}(\\bar{\\alpha }) = \\frac{1}{m} \\sum _{i=1}^m f^A(\\bar{t}_1[i],\\dots ,\\bar{t}_k[i])$ , and assign values to the $\\lambda $ -variables as follows: $\\lambda _{f,\\bar{\\alpha },\\sigma } =\\frac{1}{m} |\\lbrace i \\mid \\sigma (\\bar{\\alpha }[j]) = \\bar{t}_j[i] \\text{ for all $j$} \\rbrace |$ Note that $\\sum _{\\sigma : \\sigma (\\bar{\\alpha }[j]) = a} \\lambda _{f,\\bar{\\alpha },\\sigma } = \\mu _{\\bar{\\alpha }[j]}(a)$ for all $1 \\le j \\le k$ and $a \\in D$ .",
"Furthermore, $\\lambda $ is defined so that $f^{P^{m}(A)}(\\bar{\\alpha }) = \\sum _{\\sigma : \\lbrace \\bar{\\alpha }\\rbrace \\rightarrow D} f^A(\\sigma (\\bar{\\alpha })) \\lambda _{f,\\bar{\\alpha },\\sigma }$ .",
"Hence, the variables $\\lambda , \\mu $ satisfy the basic LP (REF ), and we have $BLP(I,A) \\le \\sum _{f,\\bar{\\alpha }} f^I(\\bar{\\alpha }) \\sum _{\\sigma : \\lbrace \\bar{\\alpha }\\rbrace \\rightarrow D} f^A(\\sigma (\\bar{\\alpha })) \\lambda _{f,\\bar{\\alpha },\\sigma }=\\sum _{f,\\bar{\\alpha }} f^I(\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }),$ where the sums are over $f \\in \\tau $ and $\\bar{\\alpha } \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$ .",
"It now follows from (REF ) and (REF ) that the measure of any solution $g : \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big ) \\rightarrow D$ to $I$ is strictly greater than BLP$(I,A)$ .",
"Consequently, BLP does not solve VCSP$(A)$ .",
"Optimal Soft Arc Consistency In this section we define optimal soft arc consistency, which is closely related to BLP given by (REF ) in Section .",
"Let $I$ and $A$ be valued structures over a common finite signature.",
"Let $X = D(I)$ and $D = D(A)$ .",
"We will group the terms of an instance with respect to their scope.",
"Let $S \\subseteq X$ .",
"The terms of this scope are those of the form $f^I({\\bar{x}})f^A(\\sigma ({\\bar{x}}))$ , where $\\lbrace {\\bar{x}}\\rbrace = S$ , and $\\text{ar}(f) = |{\\bar{x}}|$ .",
"For each scope $S$ , $x \\in S$ , and $\\sigma : S \\rightarrow D$ , we have a variable $y_{S,x}(\\sigma (x))$ .",
"For each $x \\in X$ , we have a variable $z_x$ .",
"Establishing optimal soft arc consistency (OSAC) amounts to solving the following linear program [13]: $\\begin{array}{lll}\\max & \\displaystyle \\sum _{x} z_x & \\\\\\text{s.t.",
"}&\\displaystyle \\sum _{\\lbrace {\\bar{x}}\\rbrace = S, f} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) - \\displaystyle \\sum _{x \\in S} y_{S,x}(\\sigma (x)) \\ge 0&\\qquad \\text{ $\\forall S \\subseteq X, \\sigma : S \\rightarrow D$} \\\\&\\hspace*{3.99994pt}\\displaystyle \\sum _{u} u^I(x) u^A(\\sigma (x)) - z_x + \\displaystyle \\sum _{S : x \\in S} y_{S,x}(\\sigma (x)) \\ge 0&\\qquad \\text{ $\\forall x \\in X, \\sigma : \\lbrace x\\rbrace \\rightarrow D$} \\\\\\end{array}$ We refer the reader to [13] for more details, but the idea behind (REF ) is that it gives the maximum lower bound on ${\\sf Opt}_A(I)$ among all arc-consistency closures of the given instance $I$ , where the closure is obtained by repeated calls of three basic operations called Extend, Project, and UnaryProject.",
"We will be interested in the dual of (REF ).",
"The dual has variables $\\lambda _{S,\\sigma }$ for $S \\subseteq X$ and $\\sigma : S \\rightarrow D$ , and variables $\\mu _{x}(a)$ for $x \\in X, a \\in D$ .",
"$\\begin{array}{lll}\\min & \\multicolumn{2}{l}{\\displaystyle \\sum _{S \\subseteq X,\\sigma } \\Big ( \\sum _{\\lbrace {\\bar{x}}\\rbrace = S, f} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) \\Big ) \\lambda _{S,\\sigma } +\\displaystyle \\sum _{x \\in X, \\sigma } \\Big ( \\sum _{u} u^I(x) u^A(\\sigma (x))\\Big ) \\mu _{x}(\\sigma (x))} \\\\\\text{s.t.",
"}& \\displaystyle \\sum _{\\sigma : \\sigma (x) = a} \\lambda _{S,\\sigma } = \\mu _{x}(a)& \\qquad \\text{ $\\forall S \\subseteq X, x \\in S, a \\in D$} \\\\& \\hspace*{8.00003pt} \\displaystyle \\sum _{a \\in D} \\mu _{x}(a) = 1& \\qquad \\text{ $\\forall x \\in X$} \\\\\\smallskip &\\quad \\lambda , \\mu \\ge 0 & \\\\\\end{array}$ Note that (REF ) is a tighter relaxation than (REF ) as it has only one variable $\\lambda $ for all constraints with the same scope (seen as a set) of variables.",
"In (REF ), different constraints have different variables $\\lambda $ even if the scopes (seen as sets) are the same.",
"Consequently, OSAC solves all problems solved by BLP.",
"Moreover, since the basic SDP relaxation of a VCSP$(A)$ instance is tighter than BLP [45], the basic SDP relaxation also solves all tractable cases identified in this paper."
],
[
"General-valued Structures",
"The valued structures we have dealt with so far were in fact finite-valued; i.e., for each function symbol $f\\in \\tau $ , the range of $f^A$ was $\\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"We now discuss how our result can be extended to the general-valued case, in which for each function symbol $f\\in \\tau $ , the range of $f^A$ is $\\mbox{$\\overline{\\mathbb {Q}}_{\\ge 0}$}=\\mbox{$\\mathbb {Q}_{\\ge 0}$}\\cup \\lbrace \\infty \\rbrace $ .",
"(We define $c+\\infty =\\infty +c=\\infty $ for all $c\\in \\mbox{$\\overline{\\mathbb {Q}}_{\\ge 0}$}$ , and $0\\infty =\\infty 0=0$ .)",
"Inspired by the OSAC algorithm [13], the algorithm for general-valued structures, denoted by BLP$_g$ , works in two stages.",
"Firstly, the instance is made arc consistent using a standard arc consistency algorithm [23].The algorithm from [23] is sometimes called generalised arc consistency algorithm to emphasise the fact that it works for CSPs of arbitrary arities, not only for binary CSPs [41].",
"Secondly, BLP (i.e., linear program (1) from Section ) is solved.",
"Definition 4 Let $A$ be a general-valued $\\tau $ -structure, and let $D=D(A)$ .",
"An $m$ -ary operation $g:D^m\\rightarrow D$ is a polymorphism of $A$ if for every function symbol $f\\in \\tau $ and tuples ${\\bar{a}_1},\\ldots ,{\\bar{a}_m}\\in D^{ar(f)}$ , it holds that ${\\sf Feas}(f^A(g({\\bar{a}_1},\\dots ,{\\bar{a}_m}))) \\le \\sum _{i=1}^m {\\sf Feas}(f^{A}({\\bar{a}_i})),$ where ${\\sf Feas}(\\infty )=\\infty $ and ${\\sf Feas}(c)=0$ for any $c\\in \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"From Definitions REF and REF , if $\\omega $ is a fractional polymorphism of a general-valued structure $A$ , then every $g\\in \\mbox{\\rm supp}(\\omega )$ is a polymorphism of $A$ .",
"In fact, any operation $g$ that is generated by $\\mbox{\\rm supp}(\\omega )$ (i.e., $g$ belongs to the clone generated by $\\mbox{\\rm supp}(\\omega )$ ) is a polymorphism of $A$ .",
"(For finite-valued structures, trivially, every operation $g$ is a polymorphism.)",
"Arc consistency (also known as $(1,k)$ -consistency) [23] is a decision procedure for precisely those $\\lbrace 0,\\infty \\rbrace $ -valued structures $A$ that are closed under a set function $g : 2^{D(A)} \\setminus \\lbrace \\emptyset \\rbrace \\rightarrow D(A)$ [19], [21].",
"This condition has recently been shown to be equivalent to the requirement that $A$ should have symmetric polymorphisms of all arities [39].",
"Our main theorem (Theorem REF ) also holds for general-valued structures: Theorem 5.1 Let $A$ be a general-valued structure over a finite signature.",
"TFAE: BLP$_g$ solves VCSP$(A)$ .",
"For every $m>1$ , $P^m(A) \\rightarrow _fA$ .",
"For every $m>1$ , $A$ has an $m$ -ary totally symmetric fractional polymorphism.",
"For every $n>1$ , $A$ has a fractional polymorphism $\\omega _n$ such that $\\mbox{\\rm supp}(\\omega _n)$ generates an $n$ -ary symmetric operation.",
"Note that $(\\ref {gmain:2})$ implies that $A$ has symmetric polymorphisms of all arities.",
"This follows from Lemma REF , which holds for general-valued structures, and the above-mentioned fact that any $g\\in \\mbox{\\rm supp}(\\omega )$ , where $\\omega $ is a fractional polymorphism of $A$ , is a polymorphism of $A$ .",
"The same argument guarantees symmetric polymorphisms of all arities in $(\\ref {gmain:3})$ and $(\\ref {gmain:4})$ ; in $(\\ref {gmain:4})$ , we use that fact than any operation generated by $\\mbox{\\rm supp}(\\omega _n)$ is a polymorphism of $A$ .",
"Theorem REF proves $(\\ref {gmain:2})\\Rightarrow (\\ref {gmain:1})$ since the assumption of having symmetric polymorphisms of all arities guarantees a feasible solution to (REF ).",
"From the discussion above on arc consistency, if BLP$_g$ solves VCSP$(A)$ , then $A$ has symmetric polymorphisms of all arities since arc consistency decides the existence of a finite-valued solution.",
"Furthermore, the proof of Theorem REF shows that having symmetric polymorphisms of all arities but not having a fractional homomorphism $P^m(A) \\rightarrow _fA$ implies that BLP$_g$ does not solve VCSP$(A)$ .",
"This gives $(\\ref {gmain:1})\\Rightarrow (\\ref {gmain:2})$ .",
"$(\\ref {gmain:2})\\Leftrightarrow (\\ref {gmain:3})$ and $(\\ref {gmain:3})\\Leftrightarrow (\\ref {gmain:4})$ are proved the same way as in Theorem REF , by Lemma REF and Theorem REF , respectively.",
"Remark 1 BLP$_g$ for general-valued structures uses the arc consistency algorithm [23] and BLP.",
"Since Kun et al [39] have shown that BLP solves CSPs (i.e., $\\lbrace 0,\\infty \\rbrace $ -valued VCSPs), if represented by $\\lbrace 0,1\\rbrace $ -valued structures, of width 1 – thus providing an alternative to the standard arc consistency algorithm [23] – a different approach is to combine Theorem REF from this paper with [39] and solve general-valued structures using only BLP with an amended objective function which takes care of infinite costs using a large (but polynomial), instance-dependent constant."
],
[
"Tractable Valued Constraint Languages",
"As before, we denote $D=D(A)$ .",
"A binary multimorphism [11] of a valued structure $A$ is a pair $\\langle g_1, g_2 \\rangle $ of binary functions $g_1,g_2:D^2\\rightarrow D$ such that for every function symbol $f \\in \\tau $ and tuples $\\bar{a}_1,\\bar{a}_2 \\in D^{ar(f)}$ , it holds that $f^A(g_1(\\bar{a}_1,\\bar{a}_2))+f^A(g_2(\\bar{a}_1,\\bar{a}_2)) \\le f^A(\\bar{a}_1)+f^A(\\bar{a}_2).$ (Multimorphisms are a special case of fractional polymorphisms.)",
"Since any semi-lattice operationA semi-lattice operation is associative, commutative, and idempotent.",
"generates symmetric operations of all arities, we get: Corollary 6.1 (of Theorem REF ) Let $A$ be a valued structure with a binary multimorphism $\\langle g_1, g_2 \\rangle $ where either $g_1$ or $g_2$ is a semi-lattice operation.",
"Then $A$ is tractable.",
"We now give examples of valued structures (i.e., valued constraint languages) defined by such binary multimorphisms.",
"Example 1 Let $(D; \\wedge , \\vee )$ be an arbitrary lattice on $D$ .",
"Assume that a valued structure $A$ has the multimorphism $\\langle \\wedge , \\vee \\rangle $ .",
"Then VCSP$(A)$ is tractable.",
"The tractability of $A$ was previously known only for distributive lattices [48], [28] and (finite-valued) diamonds [38], see also [37].",
"Example 2 A pair of operations $\\langle g_1, g_2 \\rangle $ is called a symmetric tournament pair (STP) if both $g_1$ and $g_2$ are commutative, conservative ($g_1(x,y)\\in \\lbrace x,y\\rbrace $ and $g_2(x,y)\\in \\lbrace x,y\\rbrace $ for all $x,y\\in D$ ), and $g_1(x,y)\\ne g_2(x,y)$ for all $x,y\\in D$ .",
"Let $A$ be a finite-valued structure with an STP multimorphism $\\langle g_1, g_2 \\rangle $ .",
"It is known that if a finite-valued structure admits an STP multimorphism, it also admits a submodularity multimorphism.",
"This result is implicitly contained in [10].Namely, the STP might contain cycles, but [10] tells us that on cycles we have, in the finite-valued case, only unary cost functions.",
"It follows that the cost functions admitting the STP must be submodular with respect to some total order.",
"Therefore, BLP solves any instance from VCSP$(A)$ .",
"Example 3 Assume that a valued structure $A$ is bisubmodular [24].",
"This means that $D=\\lbrace 0,1,2\\rbrace $ and $A$ has a multimorphism $\\langle \\min _0, \\max _0 \\rangle $ [11], where $\\min _0(x,x)=x$ for all $x\\in D$ and $\\min _0(x,y)=0$ for all $x,y\\in D,x\\ne y$ ; $\\max _0(x,y)=0$ if $0\\ne x\\ne y\\ne 0$ and $\\max _0(x,y)=\\max (x,y)$ otherwise, where $\\max $ returns the larger of its two arguments with respect to the normal order of integers.",
"Since $\\min _0$ is a semi-lattice operation, $A$ is tractable.",
"The tractability of (finite-valued) $A$ was previously known only using a general algorithm for bisubmodular functions given by an oracle [24], [43].",
"Example 4 Assume that a valued structure $A$ is weakly tree-submodular on an arbitrary tree [35].",
"The meet (which is defined as the highest common ancestor) is again a semi-lattice operation.",
"The same holds for strongly tree-submodular structures since strong tree-submodularity implies weak tree-submodularity [35].",
"The tractability of weakly tree-submodular valued structures was previously known only for chains and forks [35].",
"The tractability of strongly tree-submodular valued structures was previously known only for binary trees [35].",
"Example 5 Note that the previous example applies to all trees, not just binary ones.",
"In particular, it applies to the tree consisting of one root with $k$ children.",
"This is equivalent to structures with $D=\\lbrace 0,1,\\dots ,k\\rbrace $ and the multimorphism $\\langle \\min _0, \\max _0 \\rangle $ from Example REF .",
"This is a natural generalisation of submodular ($k=1$ ) and bisubmodular ($k=2$ ) functions, known as $k$ -submodular functions [26].",
"The tractability of $k$ -submodular valued structures for $k>2$ was previously open.",
"Example 6 Let $b$ and $c$ be two distinct elements of $D$ and let $(D;<)$ be a partial order which relates all pairs of elements except for $b$ and $c$ .",
"A pair $\\langle g_1,g_2\\rangle $ , where $g_1,g_2:D^2\\rightarrow D$ are two binary operations, is a 1-defect multimorphism if $g_1$ and $g_2$ are both commutative and satisfy the following conditions: If $\\lbrace x,y\\rbrace \\ne \\lbrace b,c\\rbrace $ , then $g_1(x,y)=x\\wedge y$ and $g_2(x,y)=x\\vee y$ .",
"If $\\lbrace x,y\\rbrace =\\lbrace b,c\\rbrace $ , then $\\lbrace g_1(x,y),g_2(x,y)\\rbrace \\cap \\lbrace x,y\\rbrace =\\emptyset $ , and $g_1(x,y)<g_2(x,y)$ .",
"The tractability of valued structures that have a 1-defect multimorphism has recently been shown in [32].",
"We now show that valued structures with a 1-defect multimorphism are solvable by BLP$_g$ .",
"Without loss of generality, we assume that $g_1(b,c)<b,c$ and write $g=g_1$ .",
"(Otherwise, $g_2(b,c)>b,c$ , and $g_2$ is used instead.)",
"Using $g$ , we construct a symmetric $m$ -ary operation $f(x_1,\\dots ,x_m)$ .",
"Let $f_1,\\ldots ,f_M$ be the $M={m\\atopwithdelims ()2}$ terms $g(x_i,x_j)$ .",
"Let $f=g(f_1,g(f_2,\\dots ,g(f_{M-1},f_M)\\dots ))$ .",
"There are three possible cases: $\\lbrace b,c\\rbrace \\lnot \\subseteq {x_1,...,x_m}$ .",
"Then $g$ acts as $\\wedge $ , which is a semi-lattice operation, hence so does $f$ .",
"$\\lbrace b,c\\rbrace \\subseteq \\lbrace x_1,...,x_m\\rbrace $ and $g(b,c)\\le x_1,\\dots ,x_m$ .",
"Then $f_i=g(b,c)$ for some $1\\le i\\le M$ , and $g(f_i,f_j)=g(b,c)$ for all $1\\le j\\le M$ , so $f(x_1,\\dots ,x_m)=g(b,c)$ .",
"$\\lbrace b,c\\rbrace \\subseteq \\lbrace x_1,\\dots ,x_m\\rbrace $ and there is a variable $x_p$ for some $1\\le p\\le m$ such that $x_p\\le g(b,c)$ and $x_p\\le x_1,\\dots ,x_m$ .",
"Then $g(x_p,x_q)=x_p$ for all $1\\le q\\le m$ so $f_i=x_p$ for some $1\\le i\\le M$ and $g(f_i,f_j)=x_p$ for all $1\\le j\\le M$ , so $f(x_1,\\dots ,x_m)=x_p$ ."
],
[
"Conclusions",
"We have characterised precisely for which valued structures the basic linear programming relaxation (BLP) is a decision procedure.",
"This implies tractability of several previously open classes of VCSPs including several generalisations of submodularity.",
"In fact, BLP solves all known tractable finite-valued structures.",
"The main result does not give a decidability criterion for testing whether a valued structure is solvable by BLP.",
"Interestingly, all known tractable finite-valued structures have a binary multimorphism.",
"It is possible that every (finite-)valued structure solvable by BLP admits a fixed-arity multimorphism, which would give a polynomial-time checkable condition.",
"An intriguing open question is whether our tractability results hold in the oracle-value model; that is, for objective functions which are not given explicitly as a sum of cost functions, but only by an oracle.",
"For instance, the maximisation problem for submodular functions on distributive lattices, known to be NP-complete, allows for good approximation algorithms in both models [22]."
],
[
"Proof of Lemma ",
"Lemma 1.1 Let $A$ be a valued structure and $m > 1$ .",
"Then $P^m(A) \\rightarrow _fA$ if and only if $A$ has an $m$ -ary totally symmetric fractional polymorphism.",
"Let $D = D(A)$ .",
"$(\\Rightarrow )$ Let $\\omega $ be a fractional homomorphism from $P^m(A)$ to $A$ .",
"Let $h : D^m \\rightarrow \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ be the function that sends a tuple $(a_1,\\dots ,a_m) \\in D^m$ to its multiset $[a_1,\\dots ,a_m]$ .",
"Then the composition $\\omega \\circ h$ is the desired fractional polymorphism.",
"$(\\Leftarrow )$ Let $\\omega $ be an $m$ -ary totally symmetric fractional polymorphism.",
"Every operation $g \\in \\mbox{\\rm supp}(\\omega )$ induces a function $g^{\\prime } : \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big ) \\rightarrow D$ .",
"Define $\\omega ^{\\prime }$ as the fractional homomorphism with $\\omega ^{\\prime }(g^{\\prime }) = \\omega (g)$ .",
"Let $f$ be a $k$ -ary function symbol in the signature of $A$ , let $\\alpha _1, \\dots , \\alpha _k \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ be arbitrary, and pick ${\\bar{a}_i}$ with $[\\bar{a}_i] = \\alpha _i$ that minimises $\\sum _{i=1}^m f^A({\\bar{a}_1}[i],\\dots ,{\\bar{a}_k}[i])$ , for $1 \\le i \\le k$ .",
"$\\sum _{g^{\\prime }} \\omega ^{\\prime }(g^{\\prime }) f^{A}(g^{\\prime }(\\alpha _1),\\dots ,g^{\\prime }(\\alpha _k))& = & \\sum _{g} \\omega (g) f^{A}(g({\\bar{a}_1}),\\dots ,g({\\bar{a}_k}))\\\\& \\le & \\frac{1}{m} \\sum _{i=1}^m f^{A}(a_1[i],\\dots ,a_k[i])\\\\& = & \\frac{1}{m} \\min _{{\\bar{t}_i} \\in D^m : [{\\bar{t}_i}] = \\alpha _i} \\sum _{i=1}^m f^A({\\bar{t}_1}[i],\\dots ,{\\bar{t}_k}[i])\\\\& = & \\frac{1}{m} \\sum _{i=1}^m f^{P^m(A)}(\\alpha _1,\\dots ,\\alpha _k).$ Hence, $P^m(A) \\rightarrow _fA$ ."
],
[
"Proof of Theorem ",
"The following variant of Farkas' Lemma is due to Gale [25] (cf.",
"Mangasarian [42]).",
"Lemma 2.1 Let $A \\in \\mbox{$\\mathbb {R}$}^{m \\times n}$ and ${\\bar{b}} \\in \\mbox{$\\mathbb {R}$}^m$ .",
"Then exactly one of the two holds: $A{\\bar{x}} \\le {\\bar{b}}$ for some ${\\bar{x}} \\in \\mbox{$\\mathbb {R}$}^n$ ; or $A^T{\\bar{y}} = 0$ , ${\\bar{b}}^T{\\bar{y}} = -1$ for some ${\\bar{y}} \\in \\mbox{$\\mathbb {R}$}_{\\ge 0}$ .",
"Theorem 2.2 Let $A$ be a valued structure and assume that BLP solves VCSP$(A)$ .",
"Then $P^m(A) \\rightarrow _fA$ for every $m > 1$ .",
"Let $\\tau $ be the signature of $A$ and let $D = D(A)$ .",
"We prove the contrapositive.",
"Assume that there is an integer $m > 1$ such that $P^m(A)$ does not have a fractional homomorphism to $A$ .",
"Let $\\Omega $ denote the set of functions from $\\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ to $D$ .",
"We are assuming that the following system of inequalities does not have a solution $\\omega : \\Omega \\rightarrow \\mbox{$\\mathbb {Q}_{\\ge 0}$}$ .",
"$\\sum _{g \\in \\Omega } \\omega (g) f^A(g(\\bar{\\alpha })) & \\le &f^{P^m(A)}(\\bar{\\alpha })\\quad \\text{ $\\forall f \\in \\tau , \\bar{\\alpha } \\in \\textstyle \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$}\\\\ \\sum _{g \\in \\Omega } \\omega (g) & = & 1 \\\\\\omega (g) & \\ge & 0 \\quad \\text{ $\\forall g \\in \\Omega $.",
"}$ In order to apply Lemma REF , we rewrite the equality $\\sum _{g} \\omega (g) = 1$ into two inequalities $\\sum _{g} \\omega (g) \\le 1$ and $-\\sum _{g} \\omega (g) \\le -1$ .",
"The last set of inequalities are rewritten to the form $-\\omega (g) \\le 0$ for each $g \\in \\Omega $ .",
"We have one variable for each inequality, i.e., $y(f,\\bar{\\alpha })$ for $f \\in \\tau $ , and $\\bar{\\alpha } \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$ .",
"Additionally, we have two variables $z_+, z_-$ for the two inequalities involving the constant 1 and one variable $w(g)$ for each $g \\in \\Omega $ .",
"$\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^A(g(\\bar{\\alpha })) + z_+ - z_- - w(g) & = & 0\\\\\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }) + z_+ - z_- & = & -1 \\\\y, z_+, z_-, w & \\ge & 0$ We can isolate $z_++z_-$ in the last equality, $z_++z_- = -1 - \\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }),$ which substituted into the first set of equalities implies that there is a solution $y(f,\\bar{\\alpha }), w(g)$ such that, for each $g \\in \\Omega $ , $\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^A(g(\\bar{\\alpha })) = w(g) + 1 + \\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }).$ We therefore find that there is a solution to the following system: $\\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^A(g(\\bar{\\alpha })) & > & \\sum _{f,\\bar{\\alpha }} y(f,\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha })\\quad \\text{ $\\forall g \\in \\Omega $} \\\\y(f,\\bar{\\alpha }) & \\ge & 0 \\quad \\text{ $\\forall f, \\bar{\\alpha }$.}",
"$ Let $I$ be the instance on variables $\\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )$ .",
"For each $k$ -ary function symbol $f \\in \\tau $ , and $\\bar{\\alpha } \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$ , define $f^I(\\bar{\\alpha }) = y(f,\\bar{\\alpha }).$ We now give a solution $\\lambda , \\mu $ to the basic LP (REF ) with an objective value equal to the right-hand side of (REF ).",
"Each variable $\\mu _{\\alpha }(a)$ is assigned the value of the multiplicity of $a$ in $\\alpha $ divided by $m$ .",
"Given $f, \\bar{\\alpha }$ , let ${\\bar{t}_1},\\dots ,{\\bar{t}_k} \\in D^m$ be such that $f^{P^m(A)}(\\bar{\\alpha }) = \\frac{1}{m} \\sum _{i=1}^m f^A(\\bar{t}_1[i],\\dots ,\\bar{t}_k[i])$ , and assign values to the $\\lambda $ -variables as follows: $\\lambda _{f,\\bar{\\alpha },\\sigma } =\\frac{1}{m} |\\lbrace i \\mid \\sigma (\\bar{\\alpha }[j]) = \\bar{t}_j[i] \\text{ for all $j$} \\rbrace |$ Note that $\\sum _{\\sigma : \\sigma (\\bar{\\alpha }[j]) = a} \\lambda _{f,\\bar{\\alpha },\\sigma } = \\mu _{\\bar{\\alpha }[j]}(a)$ for all $1 \\le j \\le k$ and $a \\in D$ .",
"Furthermore, $\\lambda $ is defined so that $f^{P^{m}(A)}(\\bar{\\alpha }) = \\sum _{\\sigma : \\lbrace \\bar{\\alpha }\\rbrace \\rightarrow D} f^A(\\sigma (\\bar{\\alpha })) \\lambda _{f,\\bar{\\alpha },\\sigma }$ .",
"Hence, the variables $\\lambda , \\mu $ satisfy the basic LP (REF ), and we have $BLP(I,A) \\le \\sum _{f,\\bar{\\alpha }} f^I(\\bar{\\alpha }) \\sum _{\\sigma : \\lbrace \\bar{\\alpha }\\rbrace \\rightarrow D} f^A(\\sigma (\\bar{\\alpha })) \\lambda _{f,\\bar{\\alpha },\\sigma }=\\sum _{f,\\bar{\\alpha }} f^I(\\bar{\\alpha }) f^{P^m(A)}(\\bar{\\alpha }),$ where the sums are over $f \\in \\tau $ and $\\bar{\\alpha } \\in \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big )^{ar(f)}$ .",
"It now follows from (REF ) and (REF ) that the measure of any solution $g : \\big (\\hspace{-1.99997pt}\\big (\\genfrac{}{}{0.0pt}{}{D}{m}\\big )\\hspace{-1.99997pt}\\big ) \\rightarrow D$ to $I$ is strictly greater than BLP$(I,A)$ .",
"Consequently, BLP does not solve VCSP$(A)$ ."
],
[
"Optimal Soft Arc Consistency",
"In this section we define optimal soft arc consistency, which is closely related to BLP given by (REF ) in Section .",
"Let $I$ and $A$ be valued structures over a common finite signature.",
"Let $X = D(I)$ and $D = D(A)$ .",
"We will group the terms of an instance with respect to their scope.",
"Let $S \\subseteq X$ .",
"The terms of this scope are those of the form $f^I({\\bar{x}})f^A(\\sigma ({\\bar{x}}))$ , where $\\lbrace {\\bar{x}}\\rbrace = S$ , and $\\text{ar}(f) = |{\\bar{x}}|$ .",
"For each scope $S$ , $x \\in S$ , and $\\sigma : S \\rightarrow D$ , we have a variable $y_{S,x}(\\sigma (x))$ .",
"For each $x \\in X$ , we have a variable $z_x$ .",
"Establishing optimal soft arc consistency (OSAC) amounts to solving the following linear program [13]: $\\begin{array}{lll}\\max & \\displaystyle \\sum _{x} z_x & \\\\\\text{s.t.",
"}&\\displaystyle \\sum _{\\lbrace {\\bar{x}}\\rbrace = S, f} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) - \\displaystyle \\sum _{x \\in S} y_{S,x}(\\sigma (x)) \\ge 0&\\qquad \\text{ $\\forall S \\subseteq X, \\sigma : S \\rightarrow D$} \\\\&\\hspace*{3.99994pt}\\displaystyle \\sum _{u} u^I(x) u^A(\\sigma (x)) - z_x + \\displaystyle \\sum _{S : x \\in S} y_{S,x}(\\sigma (x)) \\ge 0&\\qquad \\text{ $\\forall x \\in X, \\sigma : \\lbrace x\\rbrace \\rightarrow D$} \\\\\\end{array}$ We refer the reader to [13] for more details, but the idea behind (REF ) is that it gives the maximum lower bound on ${\\sf Opt}_A(I)$ among all arc-consistency closures of the given instance $I$ , where the closure is obtained by repeated calls of three basic operations called Extend, Project, and UnaryProject.",
"We will be interested in the dual of (REF ).",
"The dual has variables $\\lambda _{S,\\sigma }$ for $S \\subseteq X$ and $\\sigma : S \\rightarrow D$ , and variables $\\mu _{x}(a)$ for $x \\in X, a \\in D$ .",
"$\\begin{array}{lll}\\min & \\multicolumn{2}{l}{\\displaystyle \\sum _{S \\subseteq X,\\sigma } \\Big ( \\sum _{\\lbrace {\\bar{x}}\\rbrace = S, f} f^I({\\bar{x}}) f^A(\\sigma ({\\bar{x}})) \\Big ) \\lambda _{S,\\sigma } +\\displaystyle \\sum _{x \\in X, \\sigma } \\Big ( \\sum _{u} u^I(x) u^A(\\sigma (x))\\Big ) \\mu _{x}(\\sigma (x))} \\\\\\text{s.t.",
"}& \\displaystyle \\sum _{\\sigma : \\sigma (x) = a} \\lambda _{S,\\sigma } = \\mu _{x}(a)& \\qquad \\text{ $\\forall S \\subseteq X, x \\in S, a \\in D$} \\\\& \\hspace*{8.00003pt} \\displaystyle \\sum _{a \\in D} \\mu _{x}(a) = 1& \\qquad \\text{ $\\forall x \\in X$} \\\\\\smallskip &\\quad \\lambda , \\mu \\ge 0 & \\\\\\end{array}$ Note that (REF ) is a tighter relaxation than (REF ) as it has only one variable $\\lambda $ for all constraints with the same scope (seen as a set) of variables.",
"In (REF ), different constraints have different variables $\\lambda $ even if the scopes (seen as sets) are the same.",
"Consequently, OSAC solves all problems solved by BLP.",
"Moreover, since the basic SDP relaxation of a VCSP$(A)$ instance is tighter than BLP [45], the basic SDP relaxation also solves all tractable cases identified in this paper."
]
] | 1204.1079 |
[
[
"Compatible quadratic Poisson brackets related to a family of elliptic\n curves"
],
[
"Abstract We construct nine pairwise compatible quadratic Poisson structures such that a generic linear combination of them is associated with an elliptic algebra in n generators.",
"Explicit formulas for Casimir elements of this elliptic Poisson structure are obtained."
],
[
"Introduction",
"Two Poisson brackets $\\lbrace \\cdot ,\\cdot \\rbrace _0$ and $\\lbrace \\cdot ,\\cdot \\rbrace _1$ defined on the same finite dimensional vector space are said to be compatible if $ \\lbrace \\cdot ,\\cdot \\rbrace _{u}=\\lbrace \\cdot ,\\cdot \\rbrace _0+u \\lbrace \\cdot ,\\cdot \\rbrace _1$ is a Poisson bracket for any constant $u$ .",
"Note that if $\\lbrace \\cdot ,\\cdot \\rbrace _{u_1,...,u_k}=\\lbrace \\cdot ,\\cdot \\rbrace _0+u_1\\lbrace \\cdot ,\\cdot \\rbrace _1+...+u_k \\lbrace \\cdot ,\\cdot \\rbrace _k$ is a Poisson bracket for arbitrary $u_1,...,u_k$ , then all brackets $\\lbrace \\cdot ,\\cdot \\rbrace _0,...,\\lbrace \\cdot ,\\cdot \\rbrace _k$ are Poisson and pairwise compatible.",
"Compatible Poisson structures play an important role in the theory of integrable systems [1], [2] and in differential geometry [3], [4].",
"A lot of examples of compatible Poisson structures are known [2].",
"Most of these are linear in certain coordinates.",
"However, quadratic Poisson structures are also interesting.",
"While the theory of linear Poisson structures is well-understood and possesses a classification theoryThe theory of linear Poisson structures coincides with the theory of Lie algebras., the theory of quadratic Poisson algebras is more complicated.",
"If the dimension of a linear space is larger than four, then no classification results for quadratic Poisson structures on this space are available.",
"All known examples can be divided into two classes: rational and elliptic.",
"In the elliptic case structure constants of a Poisson bracket are modular functions of a parameter $\\tau $ , a modular parameter of an elliptic curve.",
"This elliptic curve appears naturally as a symplectic leaf of this elliptic Poisson structure [5].",
"Let $Q_n(\\tau ,\\eta )$ be an associative algebra defined by $n$ generators $\\lbrace x_i;i\\in \\mathbb {Z}/n\\mathbb {Z}\\rbrace $ and quadratic relations [5] $\\sum _{r\\in \\mathbb {Z}/n\\mathbb {Z}}{1\\over \\theta _{j-i-r}(-\\eta ,\\tau )\\theta _r(\\eta ,\\tau )}x_{j-r}x_{i+r}=0,$ for all $i\\ne j\\in \\mathbb {Z}/n\\mathbb {Z}$ .",
"Here $\\theta _i(z,\\tau )$ are $\\theta $ -functions with characteristics (see Appendix).",
"It is known that for generic $\\eta $ the algebra $Q_n(\\tau ,\\eta )$ has the same size of graded components as the polynomial ring $\\mathbb {C}[x_1,...,x_n]$ .",
"Moreover, if $\\eta =0$ , then $Q_n(\\tau ,\\eta )$ is isomorphic to $\\mathbb {C}[x_1,...,x_n]$ .",
"Therefore, for any fixed $\\tau $ we have a flat deformation of a polynomial ring.",
"Let $q_n(\\tau )$ be the corresponding Poisson algebra.",
"Symplectic leaves of this Poisson structure are known [5].",
"In particular, the center of this Poisson algebra is generated by one homogeneous polynomial of degree $n$ if $n$ is odd and by two homogeneous polynomials of degree ${n\\over 2}$ if $n$ is even.",
"One can pose the following problems: Do there exist Poisson structures compatible with the one in $q_n(\\tau )$ ?",
"Construct a maximal number of Poisson structures pairwise compatible and compatible with the one in $q_n(\\tau )$ .",
"It is easy to study these questions in the cases $n=3,~4$ .",
"Let $n=3$ .",
"The Poisson bracket in $q_3(\\tau )$ can be written as $\\lbrace x_{\\sigma _1},x_{\\sigma _2}\\rbrace ={\\partial P\\over \\partial x_{\\sigma _3}}$ where $P$ is a certain homogeneous cubic polynomial in $x_1,x_2,x_3$ and $\\sigma $ is an arbitrary even permutation.",
"Moreover, this formula defines a Poisson bracket for an arbitrary polynomial $P$ and all these brackets are pairwise compatible.",
"In particular, there exist 10 linearly independent quadratic Poisson brackets because there are 10 linearly independent homogeneous cubic polynomials in 3 variables.",
"Let $n=4$ .",
"The Poisson bracket in $q_4(\\tau )$ can be written as $\\lbrace x_{\\sigma _1},x_{\\sigma _2}\\rbrace =\\det \\left(\\begin{array}{cc}{\\partial P\\over \\partial x_{\\sigma _3}}&{\\partial P\\over \\partial x_{\\sigma _4}} \\vspace{6.0pt}\\\\{\\partial R\\over \\partial x_{\\sigma _3}}&{\\partial R\\over \\partial x_{\\sigma _4}}\\end{array}\\right)$ where $P,~R$ are certain homogeneous quadratic polynomials in $x_1,...,x_4$ and $\\sigma $ is an arbitrary even permutation.",
"Moreover, this formula defines a Poisson bracket for arbitrary polynomials $P$ and $R$ .",
"If we fix $R$ and vary $P$ , we obtain an infinite family of pairwise compatible Poisson brackets.",
"In particular, there exist 9 pairwise compatible quadratic Poisson brackets.",
"Indeed, there are 10 quadratic polynomials in 4 variables and $P$ should not be proportional to $R$ .",
"If $n>4$ , then the situation is more complicated because the similar construction for $q_n(\\tau )$ , $n>4$ does not exists.",
"In this paper we construct nine pairwise compatible quadratic Poisson brackets Three of these where constructed in [6].",
"for arbitrary $n$ .",
"A generic linear combination of these Poisson brackets is isomorphic to $q_n(\\tau )$ where $\\tau $ depends on coefficients in this linear combination.",
"Moreover, we think that this family of Poisson brackets is maximal.",
"We have checked, that for $n=5,6,...,40$ there are no quadratic Poisson brackets that are compatible with all our nine Poisson brackets and are linearly independent of them.",
"For these values of $n$ there are no Poisson brackets compatible with all ours that are constant, linear, cubic and quartic.",
"Let us describe the contents of the paper.",
"In section we construct nine compatible quadratic Poisson structures on a certain $n$ -dimensional linear space ${\\cal F}_n$ .",
"This construction is slightly different for even and odd $n$ .",
"It is summarized in Remarks 1, 1$^{\\prime }$ and 2, 2$^{\\prime }$ as an algorithm for the computation of Poisson brackets between $x_i$ and $x_j$ for $i,j=0,2,3,...,n$ .",
"In section we explain the functional version of the same construction.",
"In sections we describe symplectic leaves and Casimir elements of our Poisson algebras (see also [5], [6]).",
"In the Conclusion we outline several open problems.",
"In the Appendix we collect some notations and standard facts about elliptic and $\\theta $ -functions (see [8], [5] for details)."
],
[
"Notations",
"Our construction of Poisson brackets is slightly different for even and odd $n$ .",
"We will use index $ev$ (resp.",
"$od$ ) for objects related to even (resp.",
"odd) $n$ .",
"Let $P_{ev}(t)=a_0+a_1t+a_2t^2+a_3t^3+a_4t^4,~~P_{od}(t)=a_0+a_1t+a_2t^2+a_3t^3,~~Q(t)=b_0+b_1t+b_2t^2$ be arbitrary polynomials of degree not larger then four, three and two correspondingly.",
"Let ${\\cal F}_{ev}$ be a commutative associative algebra defined by generators $f,~g$ and the relation $g^2=P_{ev}(f)+Q(f)g.$ Let ${\\cal F}_{od}$ be a commutative associative algebra defined by generators $f,~g$ and the relation $(f+c)g^2=P_{od}(f)+Q(f)g.$ Here $a_0,...,a_4,b_0,b_1,b_2,c$ are constants.",
"Let $D$ be a derivation of ${\\cal F}_{ev}$ and ${\\cal F}_{od}$ defined on its generators by $D(f)=2g-Q(f),~~~D(g)=P_{ev}^{\\prime }(f)+Q^{\\prime }(f)g$ for ${\\cal F}_{ev}$ and by $D(f)=2(f+c)g-Q(f),~~~D(g)=P_{od}^{\\prime }(f)+Q^{\\prime }(f)g-g^2$ for ${\\cal F}_{od}$ .",
"Let $\\cal F$ be either ${\\cal F}_{ev}$ or ${\\cal F}_{od}$ .",
"It is clear that $\\cal F\\otimes \\cal F$ is generated by $f_1=f\\otimes 1,~f_2=1\\otimes f,~g_1=g\\otimes 1,~g_2=1\\otimes g$ as an associative algebra.",
"For an arbitrary element $h\\in \\cal F$ we will use the notations $h_1=h\\otimes 1,~h_2=1\\otimes h$ for the corresponding elements in $\\cal F\\otimes \\cal F$ .",
"Let $\\lambda _{ev}\\in Frac({\\cal F}_{ev}\\otimes {\\cal F}_{ev})$ be an element of a field of fractions of ${\\cal F}_{ev}\\otimes {\\cal F}_{ev}$ defined by $(f_1-f_2)\\lambda _{ev}=g_1+g_2-{1\\over 2}Q(f_1)-{1\\over 2}Q(f_2),$ or by $(g_1-g_2)\\lambda _{ev}={P_{ev}(f_1)-P_{ev}(f_2)\\over f_1-f_2}+{Q(f_1)-Q(f_2)\\over 2(f_1-f_2)}(g_1+g_2).$ These definitions are equivalent by virtue of (REF ).",
"Let $\\lambda _{od}\\in Frac({\\cal }F_{od}\\otimes {\\cal F}_{od})$ be an element of a field of fractions of ${\\cal F}_{od}\\otimes {\\cal F}_{od}$ defined by $(f_1-f_2)\\lambda _{od}=(f_1+c)g_1+(f_2+c)g_2-{1\\over 2}Q(f_1)-{1\\over 2}Q(f_2),$ or by $(g_1-g_2)\\lambda _{od}={P_{od}(f_1)-P_{od}(f_2)\\over f_1-f_2}+{Q(f_1)-Q(f_2)\\over 2(f_1-f_2)}(g_1+g_2)-g_1g_2.$ These definitions are equivalent by virtue of (REF ).",
"Note that ${H(f_1)-H(f_2)\\over f_1-f_2}\\in S^2\\cal F\\subset \\cal F\\otimes \\cal F$ for an arbitrary polynomial $H$ .",
"Indeed, ${f_1^m-f_2^m\\over f_1-f_2}=f_1^{m-1}+f_1^{m-2}f_2+...+f_2^{m-1}\\in S^2\\cal F$ .",
"We define elements $x_0,x_2,x_3,x_4,...\\in \\cal F$ by $x_{2i}=f^i,~x_{2i+3}=f^ig,~i=0,1,2,...$ Let ${\\cal F}_n\\subset \\cal F$ be an $n$ -dimensional linear space with a basis $\\lbrace x_0,x_2,x_3,...,x_n\\rbrace =\\lbrace x_0,x_i;~2\\le i\\le n\\rbrace $ .",
"Note that ${\\cal F}_n\\subset \\cal F$ is not a subalgebra of $\\cal F$ .",
"We assume ${\\cal F}_n\\subset {\\cal } F_{ev}$ if $n$ is even and ${\\cal F}_n\\subset {\\cal } F_{od}$ if $n$ is odd.",
"We will identify $S^*{\\cal F}_n$ with a polynomial algebra $\\mathbb {C}[x_0,x_2,..,x_n]$ in $n$ variables.",
"In particular: $f_1^if_2^j+f_1^jf_2^i=x_{2i}x_{2j},~f_1^if_2^jg_2+f_1^jf_2^ig_1=x_{2i}x_{2j+3},~(f_1^if_2^j+f_1^jf_2^i)g_1g_2=x_{2i+3}x_{2j+3}.$"
],
[
"Construction in the case of even $n$",
"Proposition 1.",
"The following formula $\\lbrace \\phi ,\\psi \\rbrace =n\\lambda _{ev}(\\phi _1\\psi _2-\\psi _1\\phi _2)+\\phi _1D(\\psi _2)+\\phi _2D(\\psi _1)-\\psi _1D(\\phi _2)-\\psi _2D(\\phi _1)$ defines a quadratic Poisson bracket on the polynomial ring $S^*{\\cal F}_n=\\mathbb {C}[x_0,x_2,..,x_n]$ where $n$ is even.",
"Here $\\phi ,\\psi \\in {\\cal F}_n$ and $\\lbrace \\phi ,\\psi \\rbrace \\in S^2{\\cal F}_n$ .",
"This Poisson bracket is linear with respect to coefficients $a_0,...,a_4,b_0,...,b_2$ of polynomials $P_{ev},~Q$ and, therefore, can be written in the form $\\lbrace \\cdot ,\\cdot \\rbrace =\\lbrace \\cdot ,\\cdot \\rbrace _0+\\sum _{i=0}^4a_i\\lbrace \\cdot ,\\cdot \\rbrace _{i,1}+\\sum _{j=0}^2b_j\\lbrace \\cdot ,\\cdot \\rbrace _{j,2}$ where $\\lbrace \\cdot ,\\cdot \\rbrace _0,~\\lbrace \\cdot ,\\cdot \\rbrace _{i,1},~\\lbrace \\cdot ,\\cdot \\rbrace _{j,2}$ are pairwise compatible.",
"Therefore, for each even $n$ we have constructed nine compatible quadratic Poisson brackets in $n$ variables.",
"Proof.",
"The Jacobi identity is a consequence of a functional construction described in the next section.",
"Let us check linearity with respect to coefficients of $P_{ev},~Q$ .",
"Each of $\\phi ,\\psi \\in {\\cal F}_n\\subset {\\cal F}_{ev}$ can be of the form $R(f)$ or $R(f)g$ where $R$ is a polynomial.",
"Therefore, we have three cases: Case 1.",
"Let $\\phi =R(f),~\\psi =T(f)$ .",
"We have $\\lbrace \\phi ,\\psi \\rbrace &=&\\lbrace R(f),T(f)\\rbrace \\\\&=&n\\lambda _{ev}(R(f_1)T(f_2)-R(f_2)T(f_1))+R(f_1)D(T(f_2))+R(f_2)D(T(f_1))\\\\& &-T(f_1)D(R(f_2))-T(f_2)D(R(f_1)) \\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}\\left(g_1+g_2-{1\\over 2}Q(f_1)-{1\\over 2}Q(f_2)\\right)+ \\\\& &(R(f_2)T^{\\prime }(f_1)-T(f_2)R^{\\prime }(f_1))(2g_1-Q(f_1))+(R(f_1)T^{\\prime }(f_2)-T(f_1)R^{\\prime }(f_2))(2g_2-Q(f_2))$ Case 2.",
"Let $\\phi =R(f),~\\psi =T(f)g$ .",
"We have $\\lbrace \\phi ,\\psi \\rbrace &=&\\lbrace R(f),T(f)\\rbrace \\\\&=&n\\lambda _{ev}(R(f_1)T(f_2)g_2-R(f_2)T(f_1)g_1)+R(f_1)D(T(f_2)g_2)+R(f_2)D(T(f_1)g_1)\\\\& &-T(f_1)g_1D(R(f_2))-T(f_2)g_2D(R(f_1))\\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}g_1g_2-{n\\over 2}{Q(f_1)-Q(f_2)\\over f_1-f_2}\\left(R(f_1)T(f_2)g_2+R(f_2)T(f_1)g_1\\right)+\\\\& &n{R(f_1)T(f_2)P_{ev}(f_2)-R(f_2)T(f_1)P_{ev}(f_1)\\over f_1-f_2}\\\\& &+R(f_1)T(f_2)(P_{ev}^{\\prime }(f_2)+Q^{\\prime }(f_2)g_2)+R(f_2)T(f_1)(P_{ev}^{\\prime }(f_1)+Q^{\\prime }(f_1)g_1)\\\\& &+R^{\\prime }(f_1)T(f_2)(Q(f_1)-2g_1)g_2+R^{\\prime }(f_2)T(f_1)(Q(f_2)-2g_2)g_1\\\\& &+R(f_1)T^{\\prime }(f_2)(2P_{ev}(f_2)+Q(f_2)g_2)+R(f_2)T^{\\prime }(f_1)(2P_{ev}(f_1)+Q(f_1)g_1).$ Here we used $\\lambda _{ev}(R(f_1)T(f_2)g_2-R(f_2)T(f_1)g_1)=$ $(g_1+g_2){R(f_1)T(f_2)-R(f_2)T(f_1)\\over 2(f_1-f_2)}\\lambda _{ev}(f_1-f_2)-{1\\over 2}(R(f_1)T(f_2)+R(f_2)T(f_1))\\lambda _{ev}(g_1-g_2).$ Case 3.",
"Let $\\phi =R(f)g,~\\psi =T(f)g$ .",
"We have $\\lbrace \\phi ,\\psi \\rbrace &=&\\lbrace R(f),T(f)\\rbrace \\\\&=&n\\lambda _{ev}((R(f_1)T(f_2)-R(f_2)T(f_1))g_1g_2)+R(f_1)g_1D(T(f_2)g_2)+R(f_2)g_2D(T(f_1)g_1)\\\\& &-T(f_1)g_1D(R(f_2)g_2)-T(f_2)g_2D(R(f_1)g_1) \\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}(g_1^2g_2+g_1g_2^2-{1\\over 2}Q(f_1)g_1g_2-{1\\over 2}Q(f_2)g_1g_2)\\\\& &+(R(f_2)T^{\\prime }(f_1)-T(f_2)R^{\\prime }(f_1))(2g_1-Q(f_1))g_1g_2\\\\& &+(R(f_1)T^{\\prime }(f_2)-T(f_1)R^{\\prime }(f_2))(2g_2-Q(f_2))g_1g_2\\\\& &+(R(f_1)T(f_2)-R(f_2)T(f_1))(g_1D(g_2)-D(g_1)g_2) \\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}(P_{ev}(f_1)g_2+g_1P_{ev}(f_2)+{1\\over 2}Q(f_1)g_1g_2+{1\\over 2}Q(f_2)g_1g_2)\\\\& &+(R(f_2)T^{\\prime }(f_1)-T(f_2)R^{\\prime }(f_1))(2P_{ev}(f_1)+Q(f_1)g_1)g_2\\\\& &+(R(f_1)T^{\\prime }(f_2)-T(f_1)R^{\\prime }(f_2))(2P_{ev}(f_2)+Q(f_2)g_2)g_1\\\\& &+(R(f_1)T(f_2)-R(f_2)T(f_1))(P_{ev}^{\\prime }(f_2)g_1+Q^{\\prime }(f_2)g_1g_2-P_{ev}^{\\prime }(f_1)g_2-Q^{\\prime }(f_1)g_1g_2)$ In these computations we use formulas (REF ) and (REF ) where $f,g$ are replaced by $f_1,g_1$ or $f_2,g_2$ .",
"Note that in each case we obtain an expression for $\\lbrace \\phi ,\\psi \\rbrace $ that is linear non-homogeneous in $P_{ev},~Q$ and bi-linear non-homogeneous in $g_1,~g_2$ .",
"Using identifications (REF ) we can write each of these expressions as a quadratic homogeneous polynomial in $x_0,x_2,x_3,...,x_n$ with coefficients linear in $a_0,...,a_4,b_0,...,b_2$ .",
"Remark 1.",
"Let us describe an algorithm for the computation of $\\lbrace x_i,x_j\\rbrace $ as a quadratic polynomial in $x_0,x_2,x_3,...,x_n$ .",
"One uses the formulas for $\\lbrace \\phi ,\\psi \\rbrace $ from the proof of the proposition 1 where $R(f)=f^i,~T(f)=f^j$ and $P_{ev},~Q$ are given by (REF ).",
"For the computation of $\\lbrace x_{2i},x_{2j}\\rbrace $ the formula of case 1 is used, for the computation of $\\lbrace x_{2i},x_{2j+3}\\rbrace $ the formula of case 2 is used, and for the computation of $\\lbrace x_{2i+3},x_{2j+3}\\rbrace $ the formula of case 3 is used.",
"These formulas give a polynomial in $f_1,~f_2,~g_1,~g_2$ linear in $g_1,~g_2$ .",
"This polynomial is symmetric under transformations $f_1\\leftrightarrow f_2,~g_1\\leftrightarrow g_2$ .",
"By using identifications (REF ) the expressions can be written as a polynomial quadratic in $x_0,x_2,x_3,...,x_n$ .",
"Remark 2.",
"In all cases of the above remark $\\lbrace x_i,x_j\\rbrace $ is linear non-homogeneous with respect to the eight coefficients $a_0,...,b_2$ of polynomials $P_{ev},~Q$ .",
"Therefore, nine compatible Poisson brackets can be obtained in the following way: $\\lbrace x_i,x_j\\rbrace _0=\\lbrace x_i,x_j\\rbrace |_{a_0=...=b_2=0},~\\lbrace x_i,x_j\\rbrace _{k,1}={\\partial \\lbrace x_i,x_j\\rbrace \\over \\partial a_k},~\\lbrace x_i,x_j\\rbrace _{k,2}={\\partial \\lbrace x_i,x_j\\rbrace \\over \\partial b_k}.$"
],
[
"Construction in the case of odd $n$",
"Proposition 1$^{\\prime }$ .",
"The following formula $\\lbrace \\phi ,\\psi \\rbrace =(n\\lambda _{od}+g_2-g_1+{1\\over 2}b_2(f_1-f_2))(\\phi _1\\psi _2-\\psi _1\\phi _2)+\\phi _1D(\\psi _2)+\\phi _2D(\\psi _1)-\\psi _1D(\\phi _2)-\\psi _2D(\\phi _1)$ defines a quadratic Poisson bracket on the polynomial ring $S^*{\\cal F}_n=\\mathbb {C}[x_0,x_2,..,x_n]$ where $n$ is odd.",
"Here $\\phi ,\\psi \\in {\\cal F}_n$ and $\\lbrace \\phi ,\\psi \\rbrace \\in S^2{\\cal F}_n$ .",
"This Poisson bracket is linear with respect to $c$ and coefficients $a_0,...,a_3,b_0,...,b_2$ of polynomials $P_{od},~Q$ and, therefore, can be written in the form $\\lbrace \\cdot ,\\cdot \\rbrace =\\lbrace \\cdot ,\\cdot \\rbrace _0+\\sum _{i=0}^3a_i\\lbrace \\cdot ,\\cdot \\rbrace _{i,1}+\\sum _{j=0}^2b_j\\lbrace \\cdot ,\\cdot \\rbrace _{j,2}+c\\lbrace \\cdot ,\\cdot \\rbrace _{3}$ where $\\lbrace \\cdot ,\\cdot \\rbrace _0,~\\lbrace \\cdot ,\\cdot \\rbrace _{i,1},~\\lbrace \\cdot ,\\cdot \\rbrace _{j,2},~\\lbrace \\cdot ,\\cdot \\rbrace _{3}$ are pairwise compatible.",
"Therefore, for each odd $n$ we have constructed nine compatible quadratic Poisson brackets in $n$ variables.",
"Proof.",
"The Jacobi identity is a consequence of a functional construction described in the next section.",
"Let us check linearity with respect to $c$ and coefficients of $P_{od},~Q$ .",
"Each of $\\phi ,\\psi \\in {\\cal F}_n\\subset {\\cal F}_{ev}$ can be of the form $R(f)$ or $R(f)g$ where $R$ is a polynomial.",
"Therefore, we have three cases: Case 1.",
"Let $\\phi =R(f),~\\psi =T(f)$ .",
"We have $\\lbrace \\phi ,\\psi \\rbrace &=&\\lbrace R(f),T(f)\\rbrace \\\\&=&(n\\lambda _{od}+g_2-g_1+{1\\over 2}b_2(f_1-f_2))(R(f_1)T(f_2)-R(f_2)T(f_1))+\\\\& &R(f_1)D(T(f_2))+R(f_2)D(T(f_1))\\\\& &-T(f_1)D(R(f_2))-T(f_2)D(R(f_1)) \\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}\\left((f_1+c)g_1+(f_2+c)g_2-{1\\over 2}Q(f_1)-{1\\over 2}Q(f_2)\\right)+\\\\& &(g_2-g_1+{1\\over 2}b_2(f_1-f_2))(R(f_1)T(f_2)-R(f_2)T(f_1))+ \\\\& &(R(f_2)T^{\\prime }(f_1)-T(f_2)R^{\\prime }(f_1))(2(f_1+c)g_1-Q(f_1))+\\\\& &(R(f_1)T^{\\prime }(f_2)-T(f_1)R^{\\prime }(f_2))(2(f_2+c)g_2-Q(f_2))$ Case 2.",
"Let $\\phi =R(f),~\\psi =T(f)g$ .",
"We have $\\lbrace \\phi ,\\psi \\rbrace &=&\\lbrace R(f),T(f)\\rbrace \\\\&=&(n\\lambda _{od}+g_2-g_1+{1\\over 2}b_2(f_1-f_2))(R(f_1)T(f_2)g_2-R(f_2)T(f_1)g_1)+\\\\& &R(f_1)D(T(f_2)g_2)+R(f_2)D(T(f_1)g_1)\\\\& &-T(f_1)g_1D(R(f_2))-T(f_2)g_2D(R(f_1))\\\\&=&{n\\over 2}{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}(f_1+f_2+2c)g_1g_2-\\\\& &{n\\over 2}{Q(f_1)-Q(f_2)\\over f_1-f_2}\\left(R(f_1)T(f_2)g_2+R(f_2)T(f_1)g_1\\right)+\\\\& &{n-2\\over 2}(R(f_1)T(f_2)+R(f_2)T(f_1))g_1g_2+\\\\& &n{R(f_1)T(f_2)P_{od}(f_2)-R(f_2)T(f_1)P_{od}(f_1)\\over f_1-f_2}+\\\\& &{1\\over 2}b_2(f_1-f_2)(R(f_1)T(f_2)g_2-R(f_2)T(f_1)g_1)\\\\& &+R(f_1)T(f_2)(P_{od}^{\\prime }(f_2)+Q^{\\prime }(f_2)g_2)+R(f_2)T(f_1)(P_{od}^{\\prime }(f_1)+Q^{\\prime }(f_1)g_1)\\\\& &+R^{\\prime }(f_1)T(f_2)(Q(f_1)-2(f_1+c)g_1)g_2+R^{\\prime }(f_2)T(f_1)(Q(f_2)-2(f_2+c)g_2)g_1\\\\& &+R(f_1)T^{\\prime }(f_2)(2P_{od}(f_2)+Q(f_2)g_2)+R(f_2)T^{\\prime }(f_1)(2P_{od}(f_1)+Q(f_1)g_1).$ Here we used $\\lambda _{od}(R(f_1)T(f_2)g_2-R(f_2)T(f_1)g_1)=$ $(g_1+g_2){R(f_1)T(f_2)-R(f_2)T(f_1)\\over 2(f_1-f_2)}\\lambda _{od}(f_1-f_2)-{1\\over 2}(R(f_1)T(f_2)+R(f_2)T(f_1))\\lambda _{od}(g_1-g_2).$ Case 3.",
"Let $\\phi =R(f)g,~\\psi =T(f)g$ .",
"We have $\\lbrace \\phi ,\\psi \\rbrace &=&\\lbrace R(f),T(f)\\rbrace \\\\&=&(n\\lambda _{od}+g_2-g_1+{1\\over 2}b_2(f_1-f_2))((R(f_1)T(f_2)-R(f_2)T(f_1))g_1g_2)+\\\\& &R(f_1)g_1D(T(f_2)g_2)+R(f_2)g_2D(T(f_1)g_1)\\\\& &-T(f_1)g_1D(R(f_2)g_2)-T(f_2)g_2D(R(f_1)g_1) \\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}((f_1+c)g_1^2g_2+(f_2+c)g_1g_2^2-{1\\over 2}Q(f_1)g_1g_2-{1\\over 2}Q(f_2)g_1g_2)+\\\\& &(R(f_1)T(f_2)-R(f_2)T(f_1))({1\\over 2}b_2(f_1-f_2)g_1g_1+g_1g_2^2-g_1^2g_2)\\\\& &+(R(f_2)T^{\\prime }(f_1)-T(f_2)R^{\\prime }(f_1))(2(f_1+c)g_1-Q(f_1))g_1g_2\\\\& &+(R(f_1)T^{\\prime }(f_2)-T(f_1)R^{\\prime }(f_2))(2(f_2+c)g_2-Q(f_2))g_1g_2\\\\& &+(R(f_1)T(f_2)-R(f_2)T(f_1))(g_1D(g_2)-D(g_1)g_2) \\\\&=&n{R(f_1)T(f_2)-R(f_2)T(f_1)\\over f_1-f_2}(P_{od}(f_1)g_2+g_1P_{od}(f_2)+{1\\over 2}Q(f_1)g_1g_2+{1\\over 2}Q(f_2)g_1g_2)\\\\& &+{b_2\\over 2}(R(f_1)T(f_2)-R(f_2)T(f_1))(f_1-f_2)g_1g_2\\\\& &+(R(f_2)T^{\\prime }(f_1)-T(f_2)R^{\\prime }(f_1))(2P_{od}(f_1)+Q(f_1)g_1)g_2\\\\& &+(R(f_1)T^{\\prime }(f_2)-T(f_1)R^{\\prime }(f_2))(2P_{od}(f_2)+Q(f_2)g_2)g_1\\\\& &+(R(f_1)T(f_2)-R(f_2)T(f_1))(P_{od}^{\\prime }(f_2)g_1+Q^{\\prime }(f_2)g_1g_2-P_{od}^{\\prime }(f_1)g_2-Q^{\\prime }(f_1)g_1g_2)$ In these computations we use formulas (REF ) and (REF ) where $f,g$ are replaced by $f_1,g_1$ or $f_2,g_2$ .",
"Note that in each case we obtain an expression for $\\lbrace \\phi ,\\psi \\rbrace $ that is linear non-homogeneous in $P_{od},~Q$ and bi-linear non-homogeneous in $g_1,~g_2$ .",
"Using identifications (REF ) we can write each of these expressions as a quadratic homogeneous polynomial in $x_0,x_2,x_3,...,x_n$ with coefficients linear in $a_0,...,a_3,b_0,...,b_2,c$ .",
"Remark 1$^\\prime $ .",
"Let us describe an algorithm for the computation of $\\lbrace x_i,x_j\\rbrace $ as a quadratic polynomial in $x_0,x_2,x_3,...,x_n$ .",
"One uses the formulas for $\\lbrace \\phi ,\\psi \\rbrace $ from the proof of the proposition 1$^\\prime $ where $R(f)=f^i,~T(f)=f^j$ and $P_{od},~Q$ are given by (REF ).",
"For the computation of $\\lbrace x_{2i},x_{2j}\\rbrace $ the formula of case 1 is used, for the computation of $\\lbrace x_{2i},x_{2j+3}\\rbrace $ the formula of case 2 is used, and for the computation of $\\lbrace x_{2i+3},x_{2j+3}\\rbrace $ the formula of case 3 is used.",
"These formulas give a polynomial in $f_1,~f_2,~g_1,~g_2$ linear in $g_1,~g_2$ .",
"This polynomial is symmetric under transformations $f_1\\leftrightarrow f_2,~g_1\\leftrightarrow g_2$ .",
"By using identifications (REF ) the expressions can be written as a polynomial quadratic in $x_0,x_2,x_3,...,x_n$ .",
"Remark 2$^\\prime $ .",
"In all cases of the above remark $\\lbrace x_i,x_j\\rbrace $ is linear non-homogeneous with respect to $c$ and the seven coefficients $a_0,...,b_2$ of polynomials $P_{od},~Q$ .",
"Therefore, nine compatible Poisson brackets can be obtained in the following way: $\\lbrace x_i,x_j\\rbrace _0=\\lbrace x_i,x_j\\rbrace |_{a_0=...=b_2=c=0},~\\lbrace x_i,x_j\\rbrace _{k,1}={\\partial \\lbrace x_i,x_j\\rbrace \\over \\partial a_k},~\\lbrace x_i,x_j\\rbrace _{k,2}={\\partial \\lbrace x_i,x_j\\rbrace \\over \\partial b_k},~\\lbrace x_i,x_j\\rbrace _{3}={\\partial \\lbrace x_i,x_j\\rbrace \\over \\partial c}.$ Remark 3.",
"If we replace $n$ in the formulas (REF ), (REF ) by an arbitrary constant $\\alpha $ , then these formulas still define Poisson brackets on the polynomial algebra $\\mathbb {C}[x_0,x_2,x_3,...]$ .",
"However, $\\mathbb {C}[x_0,x_2,x_3,...,x_n]$ $\\subset $ $\\mathbb {C}[x_0,x_2,x_3,...]$ is closed with respect to these brackets only if $\\alpha =n$ ."
],
[
"General constructions",
"Recall a general construction of associative algebras and Poisson structures [5].",
"Let $\\lambda (x,y)$ be a meromorphic function in two variables.",
"We construct an associative algebra $A_{\\lambda }$ through: $A_{\\lambda }=\\mathbb {C}\\oplus F_1\\oplus F_2\\oplus F_3\\oplus ...$ where $F_m$ is the space of symmetric meromorphic functions in $m$ variables and a product $f\\star g\\in F_{a+b}$ of $f\\in F_a,~g\\in F_b$ is defined by: $f\\star g(z_1,...,z_{a+b})={1\\over a!b!",
"}\\sum _{\\sigma \\in S_{a+b}}f(z_{\\sigma _1},...,z_{\\sigma _a})g(z_{\\sigma _{a+1}},...,z_{\\sigma _{a+b}})\\prod _{1\\le p\\le a,~ a+1\\le q\\le a+b}\\lambda (z_{\\sigma _p},z_{\\sigma _q}).$ Note that this formula defines an associative product for an arbitrary function $\\lambda $ and this product is non-commutative if $\\lambda $ is not symmetric.",
"In particular, if $\\lambda (x,y)=1+{1\\over 2}\\epsilon \\mu (x,y)+o(\\epsilon )$ , then we obtain a Poisson algebra.",
"Assume $\\mu (x,y)=-\\mu (y,x)$ .",
"For $f,~g\\in F_1$ we get the following formulas for the associative commutative product $fg\\in F_2$ and the Poisson bracket: $fg(x,y)=f(x)g(y)+g(x)f(y),~\\lbrace f,g\\rbrace =(f(x)g(y)-g(x)f(y))\\mu (x,y).$ If $\\lambda =\\lambda (x-y)$ , then the formula for the product can be deformed in the following way: $f\\star g(z_1,...,z_{a+b})={1\\over a!b!",
"}\\sum _{\\sigma \\in S_{a+b}}f(z_{\\sigma _1},...,z_{\\sigma _a})g(z_{\\sigma _{a+1}}+ap,...,z_{\\sigma _{a+b}}+ap)\\prod _{1\\le p\\le a,~ a+1\\le q\\le a+b}\\lambda (z_{\\sigma _p}-z_{\\sigma _q})$ where $p$ is an arbitrary constant.",
"The corresponding formula for the Poisson brackets (if we set $p=\\epsilon \\alpha $ ) is: $\\lbrace f,g\\rbrace =(f(x)g(y)-g(x)f(y))\\mu (x-y)+\\alpha (f(x)g^{\\prime }(y)+f(y)g^{\\prime }(x)-g(x)f^{\\prime }(y)-g(y)f^{\\prime }(x)).$ We will need the following generalization of the last formula: $\\lbrace f,g\\rbrace &=&(f(x)g(y)-g(x)f(y))(\\mu (x-y)+\\nu (x)-\\nu (y)) \\nonumber \\\\& &+\\alpha (f(x)g^{\\prime }(y)+f(y)g^{\\prime }(x)-g(x)f^{\\prime }(y)-g(y)f^{\\prime }(x)).",
"$ Here $\\nu $ is an arbitrary function.",
"This formula is obtained from the previous one by transformation $f\\rightarrow \\kappa f,~g\\rightarrow \\kappa g,~\\lbrace f,g\\rbrace \\rightarrow \\kappa ^2\\lbrace f,g\\rbrace $ where $\\alpha \\kappa ^{\\prime }=-\\nu $ .",
"Note that the algebra $A_{\\lambda }$ is very large.",
"One can construct associative algebras (and the corresponding Poisson algebras) of a reasonable size by a suitable choice of spaces $F_{\\alpha }$ and function $\\lambda $ .",
"For example, the algebra $Q_n(\\tau ,\\eta )$ and the Poisson algebra $q_n(\\tau )$ can be constructed in this way [5].",
"See [7] for the functional construction of a wider class of Poisson algebras."
],
[
"The case of even $n$",
"It is clear that equation (REF ) defines an elliptic curve in $\\mathbb {C}^2$ with coordinates $f,~g$ .",
"Therefore, one can find elliptic functions $f=f(z),~g=g(z)$ such that $g(z)^2=P_{ev}(f(z))+Q(f(z))g(z).$ Moreover, one can assume (see (REF )) $f^{\\prime }(z)=2g(z)-Q(f(z)),~~~g^{\\prime }(z)=P_{ev}^{\\prime }(f(z))+Q^{\\prime }(f(z))g(z).$ Note that elliptic functions $f(z),~g(z)$ have a form: $ f(z)=c_1+c_2\\zeta (z,\\tau )+c_3\\zeta (z-u,\\tau ),~~~g(z)=c_4+c_5\\zeta (z,\\tau )+c_6\\zeta (z-u,\\tau )+c_7\\zeta ^{\\prime }(z,\\tau )+c_8\\zeta ^{\\prime }(z-u,\\tau ).$ Here $\\zeta (z,\\tau )$ is the Weierstrass elliptic function, $\\tau $ is a modular parameter and constants $c_1,...,c_8, u, \\tau $ are determined by $a_0,...,b_2$ .",
"There exists an elliptic function in two variables $\\mu _{ev}(z_1,z_2)$ such that $(f(z_1)-f(z_2))\\mu _{ev}(z_1,z_2)=g(z_1)+g(z_2)-{1\\over 2}Q(f(z_1))-{1\\over 2}Q(f(z_2)),$ $(g(z_1)-g(z_2))\\mu _{ev}(z_1,z_2)={P_{ev}(f(z_1))-P_{ev}(f(z_2))\\over f(z_1)-f(z_2)}+{Q(f(z_1))-Q(f(z_2))\\over 2(f(z_1)-f(z_2))}(g(z_1)+g(z_2)).$ This function has the form $\\mu _{ev}(z_1,z_2)=\\zeta (z_1-z_2,\\tau )+k_1\\zeta (z_1,\\tau )+k_2\\zeta (z_1-u,\\tau )-k_1\\zeta (z_2,\\tau )-k_2\\zeta (z_2-u,\\tau )$ for some constants $k_1,~k_2$ such that $k_1+k_2=1$ .",
"Let ${\\cal F}_n$ be the space of elliptic functions in one variable with periods 1 and $\\tau $ , holomorphic outside $z=0,~u$ modulo periods and having poles of order not larger than $n$ at $z=0,~u$ .",
"It is clear that $\\lbrace e_0(z),e_2(z),e_3(z),e_4(z),...,e_n(z)\\rbrace $ is a basis of the linear space ${\\cal F}_n$ where we define $e_{2i}(z)=f(z)^i,~e_{2i+3}(z)=f(z)^ig(z),~i=0,1,2,... .$ We will identify $S^m{\\cal F}_n$ with the space of symmetric elliptic functions in $m$ variables $\\lbrace h(z_1,...,z_m)\\rbrace $ holomorphic if $z_k\\ne 0,u$ modulo periods and having poles of order not larger than $n$ at $z_k=0,~u$ .",
"We construct a bilinear operator $\\lbrace ,\\rbrace : \\Lambda ^2{\\cal F}_n\\rightarrow S^2{\\cal F}_n$ as follows: for $\\phi ,\\psi \\in {\\cal F}_n$ we set $\\lbrace \\phi ,\\psi \\rbrace (z_1,z_2)&=&n\\mu _{ev}(z_1,z_2)(\\phi (z_1)\\psi (z_2)-\\psi (z_1)\\phi (z_2))\\nonumber \\\\& &+\\phi (z_1)\\psi ^{\\prime }(z_2)+\\phi (z_2)\\psi ^{\\prime }(z_1)-\\psi (z_1)\\phi ^{\\prime }(z_2)-\\psi (z_2)\\phi ^{\\prime }(z_1).",
"$ Proposition 2.",
"The formula (REF ) defines a Poisson structure on the polynomial algebra $S^{*}{\\cal F}_n$ .",
"This Poisson bracket is linear with respect to coefficients $a_0,...,a_4,b_0,...,b_2$ of polynomials $P_{ev},~Q$ and, therefore, can be written in the form $\\lbrace \\cdot ,\\cdot \\rbrace = \\lbrace \\cdot ,\\cdot \\rbrace _0+\\sum _{i=0}^4a_i\\lbrace \\cdot ,\\cdot \\rbrace _{i,1}+\\sum _{j=0}^2b_j\\lbrace \\cdot ,\\cdot \\rbrace _{j,2}$ where $\\lbrace \\cdot ,\\cdot \\rbrace _0,~\\lbrace \\cdot ,\\cdot \\rbrace _{i,1},~\\lbrace \\cdot ,\\cdot \\rbrace _{j,2}$ are pairwise compatible.",
"Therefore, for each even $n$ we have constructed nine compatible quadratic Poisson brackets in $n$ variables.",
"Proof.",
"This is just a reformulation of the Proposition 1.",
"Formula (REF ) is a special case of (REF ) and therefore the Jacobi identity for (REF ) is satisfied.",
"One can check straightforwardly that if $\\phi ,\\psi \\in {\\cal F}_n$ , then $\\lbrace \\phi ,\\psi \\rbrace (z_1,z_2)$ given by (REF ) is a symmetric elliptic function in two variables $z_1,z_2$ having poles of order not larger than $n$ at $z_1,z_2=0,u$ and therefore $\\lbrace \\phi ,\\psi \\rbrace (z_1,z_2)\\in S^2{\\cal F}_n$ ."
],
[
"The case of odd $n$",
"It is clear that equation (REF ) defines an elliptic curve in $\\mathbb {C}^2$ with coordinates $f,~g$ .",
"Therefore, one can find elliptic functions $f=f(z),~g=g(z)$ such that $(f(z)+c)g(z)^2=P_{od}(f(z))+Q(f(z))g(z).$ Moreover, one can assume (see (REF )) $f^{\\prime }(z)=2(f(z)+c)g(z)-Q(f(z)),~~~g^{\\prime }(z)=P_{od}^{\\prime }(f(z))+Q^{\\prime }(f(z))g(z)-g(z)^2.$ Note that elliptic functions $f(z),~g(z)$ have a form: $ f(z)=c_1+c_2\\zeta (z,\\tau )+c_3\\zeta (z-u,\\tau ),~~~g(z)=c_4+c_5\\zeta (z,\\tau )+c_6\\zeta (z-u,\\tau )+c_7\\zeta (z-v,\\tau ).$ Here $\\zeta (z,\\tau )$ is the Weierstrass elliptic function, $\\tau $ is a modular parameter and constants $c_1,...,c_7, u, v, \\tau $ are determined by $a_0,...,b_2,c$ .",
"There exists an elliptic function in two variables $\\mu _{od}(z_1,z_2)$ such that $(f(z_1)-f(z_2))\\mu _{od}(z_1,z_2)=(f(z_1)+c)g(z_1)+(f(z_2)+c)g(z_2)-{1\\over 2}Q(f(z_1))-{1\\over 2}Q(f(z_2)),$ $(g(z_1)-g(z_2))\\mu _{od}(z_1,z_2)={P(f(z_1))-P(f(z_2))\\over f(z_1)-f(z_2)}+{Q(f(z_1))-Q(f(z_2))\\over 2(f(z_1)-f(z_2))}(g(z_1)+g(z_2))-g(z_1)g(z_2).$ This function has the form $\\mu _{od}(z_1,z_2)=\\zeta (z_1-z_2,\\tau )+k_1\\zeta (z_1,\\tau )+k_2\\zeta (z_1-u,\\tau )-k_1\\zeta (z_2,\\tau )-k_2\\zeta (z_2-u,\\tau )$ for some constants $k_1,~k_2$ such that $k_1+k_2=1$ .",
"Let ${\\cal F}_n$ be the space of elliptic functions in one variable with periods 1 and $\\tau $ , holomorphic outside $z=0,~u,~v$ modulo periods and having poles of order not larger than $n$ at $z=0,~u$ and not larger than one at $z=v$ .",
"It is clear that $\\lbrace e_0(z),e_2(z),e_3(z),e_4(z),...,e_n(z)\\rbrace $ is a basis of the linear space ${\\cal F}_n$ where we define $e_{2i}(z)=f(z)^i,~e_{2i+3}(z)=f(z)^ig(z),~i=0,1,2,... .$ We will identify $S^m{\\cal F}_n$ with the space of symmetric elliptic functions in $m$ variables $\\lbrace h(z_1,...,z_m)\\rbrace $ holomorphic if $z_k\\ne 0,u,v$ modulo periods and having poles of order not larger than $n$ at $z_k=0,~u$ and not larger than one at $z_k=v$ .",
"We construct a bilinear operator $\\lbrace ,\\rbrace : \\Lambda ^2{\\cal F}_n\\rightarrow S^2{\\cal F}_n$ as follows: for $\\phi ,\\psi \\in {\\cal F}_n$ we set $\\lbrace \\phi ,\\psi \\rbrace (z_1,z_2)&=&(n\\mu _{od}(z_1,z_2)+g(z_2)-g(z_1)+{1\\over 2}b_2(f(z_1)-f(z_2)))(\\phi (z_1)\\psi (z_2)-\\psi (z_1)\\phi (z_2))\\nonumber \\\\& &+\\phi (z_1)\\psi ^{\\prime }(z_2)+\\phi (z_2)\\psi ^{\\prime }(z_1)-\\psi (z_1)\\phi ^{\\prime }(z_2)-\\psi (z_2)\\phi ^{\\prime }(z_1).",
"$ Proposition 2$^{\\prime }$ .",
"The formula (REF ) defines a Poisson structure on the polynomial algebra $S^{*}{\\cal F}_n$ .",
"This Poisson bracket is linear with respect to $c$ and coefficients $a_0,...,a_3,b_0,...,b_2$ of polynomials $P_{od},~Q$ and, therefore, can be written in the form $\\lbrace \\cdot ,\\cdot \\rbrace = \\lbrace \\cdot ,\\cdot \\rbrace _0+\\sum _{i=0}^3a_i\\lbrace \\cdot ,\\cdot \\rbrace _{i,1}+\\sum _{j=0}^2b_j\\lbrace \\cdot ,\\cdot \\rbrace _{j,2}+c\\lbrace \\cdot ,\\cdot \\rbrace _3$ where $\\lbrace \\cdot ,\\cdot \\rbrace _0,~\\lbrace \\cdot ,\\cdot \\rbrace _{i,1},~\\lbrace \\cdot ,\\cdot \\rbrace _{j,2},~\\lbrace \\cdot ,\\cdot \\rbrace _3$ are pairwise compatible.",
"Therefore, for each odd $n$ we have constructed nine compatible quadratic Poisson brackets in $n$ variables.",
"Proof.",
"This is just a reformulation of the Proposition 1$^{\\prime }$ .",
"Formula (REF ) is a special case of (REF ) and therefore the Jacobi identity for (REF ) is satisfied.",
"One can check straightforwardly that if $\\phi ,\\psi \\in {\\cal F}_n$ , then $\\lbrace \\phi ,\\psi \\rbrace (z_1,z_2)$ given by (REF ) is a symmetric elliptic function in two variables $z_1,z_2$ having poles of order not larger than $n$ at $z_1,z_2=0,u$ and not larger than one at $z_1,z_2=v$ and therefore $\\lbrace \\phi ,\\psi \\rbrace (z_1,z_2)\\in S^2{\\cal F}_n$ ."
],
[
"Symplectic leaves and Casimir elements",
"For $p,n\\in \\mathbb {N}$ we denote by $b_{p,n}$ the Poisson algebra spanned by the elements $\\lbrace h(f_1,g_1,...,f_p,g_p)e_1^{\\alpha _1}...e_p^{\\alpha _p};\\alpha _1,...,\\alpha _p\\in \\mathbb {Z}_{\\ge 0}\\rbrace $ as a linear space, where $h$ is a rational function, $f_i,g_i$ for each $i=1,...,p$ are subject to relation (REF ) if $n$ is even and (REF ) if $n$ is odd where $f,g$ are replaced by $f_i,g_i$ .",
"In other wordsRecall that ${\\cal F}={\\cal F}_{ev}$ if $n$ is even and ${\\cal F}={\\cal F}_{od}$ if $n$ is odd., $h\\in Frac(\\otimes ^p\\cal F)$ .",
"A Poisson bracket on $b_{p,n}$ is defined as follows: $\\lbrace f_i,e_j\\rbrace =-D(f_i)e_j,~\\lbrace g_i,e_j\\rbrace =-D(g_i)e_j,~\\lbrace f_i,e_i\\rbrace ={n-2\\over 2}D(f_i)e_j,~\\lbrace g_i,e_i\\rbrace ={n-2\\over 2}D(g_i)e_i$ where $i\\ne j$ , $D$ is defined by (REF ) for even $n$ and (REF ) for odd $n$ .",
"We also assume $\\lbrace e_i,e_j\\rbrace =n\\lambda _{i,j,ev}e_ie_j$ if $n$ is even and $\\lbrace e_i,e_j\\rbrace =\\left(n\\lambda _{i,j,od}+{1\\over 2}b_2(f_i-f_j)+g_j-g_i\\right)e_ie_j$ if $n$ is odd.",
"Here $\\lambda _{i,j,ev}$ (resp.",
"$\\lambda _{i,j,od}$ ) is given by (REF ) (resp.",
"(REF )) where $f_1,g_1,f_2,g_2$ are replaced by $f_i,g_i,f_j,g_j$ correspondingly.",
"All brackets between $f_i,g_j$ are zero.",
"Let us define a linear map $\\phi _p:{\\cal F}_n\\rightarrow b_{p,n}$ by the formula $\\phi _p(x_{2j})=\\sum _{i=1}^pf^j_ie_i,~~~\\phi _p(x_{2j+3})=\\sum _{i=1}^pf^j_ig_ie_i.$ There is a unique extension of this map to the homomorphism of commutative algebras $S^{*}({\\cal F}_n)\\rightarrow b_{p,n}$ which we also denote by $\\phi _p$ .",
"Proposition 3.",
"The map $\\phi _p:S^{*}({\\cal F}_n)\\rightarrow b_{p,n}$ is a homomorphism of Poisson algebras.",
"Proof.",
"One can check straightforwardly that $\\phi _p(\\lbrace r,s\\rbrace )=\\sum _{i,j=1}^p\\lbrace r,s\\rbrace _{i,j}e_ie_j$ where $r,s$ are arbitrary elements from ${\\cal F}_n$ and $\\lbrace r,s\\rbrace _{i,j}$ is obtained from $\\lbrace r,s\\rbrace \\in {\\cal F}_n\\otimes {\\cal F}_n$ replacing $f_1,g_1,f_2,g_2$ by $f_i,g_i,f_j,g_j$ correspondingly.",
"This implies the proposition.",
"It is known [5], [6] that if $2p<n$ , then the map $\\phi _p$ defines a $2p$ dimensional symplectic leaf of the Poisson algebra $S^{*}({\\cal F}_n)$ .",
"Moreover, central elements of the Poisson algebra $S^{*}({\\cal F}_n)$ belong to $\\ker \\phi _p$ for $2p<n$ .",
"One can check that for $p={n\\over 2}-1$ for even $n$ (resp.",
"$p={n-1\\over 2}$ for odd $n$ ) the ideal $\\ker \\phi _p$ is generated by two elements of degree ${n\\over 2}$ (resp.",
"by one element of degree $n$ ).",
"We denote these elements by $C_0^{{n\\over 2}},C_1^{{n\\over 2}}$ if $n$ is even (resp.",
"by $C^n$ if $n$ is odd).",
"The center of the Poisson algebra $S^{*}({\\cal F}_n)$ is a polynomial algebra generated by $C_0^{{n\\over 2}},C_1^{{n\\over 2}}$ if $n$ is even (resp.",
"by $C^{n}$ if $n$ is odd).",
"Let us describe these elements explicitly (see also [6]).",
"Let $n$ be even.",
"We define elements $x_i,~i\\in \\mathbb {Z}$ by the formula (REF ).",
"It is clear that $x_i\\in Frac({\\cal F}_{ev})$ if $i=1,-1,-2,...$ and $x_0,x_2,...,x_n\\in {\\cal F}_n$ .",
"If ${n\\over 2}$ is even we setThese formulas work for $n>4$ .",
"If $n=4$ we set $C_0^{2}&\\!\\!=\\!\\!&\\det \\left((x_0,x_2)^t(x_0,x_2)\\right),\\\\C_1^{2}&\\!\\!=\\!\\!&\\det \\left((x_0,x_1)^t(x_2,x_3)\\right)-\\det \\left((x_0,x_2)^t(x_2,a_4x_{4}+b_2x_{3})\\right)-\\det \\left((a_0x_{-2}+b_0x_1,x_0)^t(x_0,x_2)\\right)$ $C_0^{{n\\over 2}}&\\!\\!=\\!\\!&\\det \\left((x_0,x_2,..,x_{{n\\over 2}})^t(x_0,x_2,..,x_{{n\\over 2}})\\right),\\\\C_1^{{n\\over 2}}&\\!\\!=\\!\\!&\\ \\ \\ \\det \\left((x_0,x_1,x_2,..,x_{{n\\over 2}-1})^t(x_2,..,x_{{n\\over 2}+1})\\right)\\\\&&-\\det \\left((x_0,x_1,x_2,..,x_{{n\\over 2}-2},x_{{n\\over 2}})^t(x_2,..,x_{{n\\over 2}},a_4x_{{n\\over 2}+2}+b_2x_{{n\\over 2}+1})\\right)\\\\&&-\\det \\left((a_0x_{-2}+b_0x_1,x_0,x_2,..,x_{{n\\over 2}-1})^t(x_0,x_2,x_4,..,x_{{n\\over 2}+1})\\right) \\\\&&+\\det \\left( (a_0x_{-2}+b_0x_1,x_0,x_2,..,x_{{n\\over 2}-2},x_{{n\\over 2}})^t (x_0,x_2,x_4,..,x_{{n\\over 2}},a_4x_{{n\\over 2}+2}+b_2x_{{n\\over 2}+1}) \\right)$ and if ${n\\over 2}$ is odd $C_0^{{n\\over 2}}&\\!\\!\\!=\\!\\!\\!&\\det \\left((x_0,x_2,..,x_{{n\\over 2}})^t(x_0,x_2,..,x_{{n\\over 2}})\\right)-\\det \\left((x_0,x_2,..,x_{{n\\over 2}-1},x_{{n\\over 2}+1})^t(x_0,x_2,..,x_{{n\\over 2}-1},b_2x_{{n\\over 2}}+a_4x_{{n\\over 2}+1})\\right),\\\\C_1^{{n\\over 2}}&\\!\\!\\!=\\!\\!\\!&\\det \\left((x_2,..,x_{{n\\over 2}+1})^t(x_0,x_1,x_2,..,x_{{n\\over 2}-1})\\right)-\\det \\left((x_{-2},x_0,x_2,..,x_{{n\\over 2}-1})^t(a_0x_0+b_0x_3,x_2,x_4,..,x_{{n\\over 2}+1})\\right) .$ In these formulas we use the product in the algebra $Frac({\\cal F}_{ev})$ for computing products of vector components.",
"Therefore entries of our matrices of the form $v^tw$ are linear combinations of $x_i\\in Frac({\\cal F}_{ev})$ , $i\\in \\mathbb {Z}$ .",
"On the other hand, for computing determinants we use the product in $S^{*}(Frac({\\cal F}_{ev}))$ .",
"So these determinants are polynomials of degree ${n\\over 2}$ in $x_i,~i\\in \\mathbb {Z}$ .For example, $\\det \\left((x_0,x_2)^t(x_0,x_2)\\right)=\\det \\left((1,f)^t(1,f)\\right)=\\det \\left(\\begin{array}{cc}1 & f \\\\f & f^2 \\end{array}\\right)=\\det \\left(\\begin{array}{cc}x_0 & x_2 \\\\x_2 & x_4 \\end{array}\\right)=x_0x_4-x_2^2.$ Moreover, it turns out that in our linear combinations of determinants all terms with $x_i\\notin \\lbrace x_0,x_2,...,x_n\\rbrace $ cancel out and $C_0^{{n\\over 2}},C_1^{{n\\over 2}}$ are polynomials in $x_0,x_2,...,x_n$ of degree ${n\\over 2}$ .",
"Let $n$ be odd.",
"We define elements $y_i\\in Frac({\\cal F}_{od})$ for $i\\in \\mathbb {Z}$ by $y_{2i}=(f+c)^i,~y_{2i+3}=(f+c)^ig.$ It is clear that $y_i\\in Frac({\\cal F}_{od})$ if $i=1,-1,-2,...$ and $y_0,y_2,...,y_n\\in {\\cal F}_n$ .",
"Moreover, $y_0,y_2,...,y_n$ can be written as linear combinations of $x_0,x_2,...,x_n\\in {\\cal F}_n$ .",
"If ${n+1\\over 2}$ is even we setThese formulas work for $n>3$ .",
"If $n=3$ we set $\\tilde{C}_0^{2}&\\!\\!=\\!\\!&\\det \\left((y_0,y_2)^t(y_{-2},y_0)\\right),\\\\\\tilde{C}_1^{2}&\\!\\!=\\!\\!&\\det \\left((y_0,y_3)^t(y_0,y_3)\\right)-\\det \\left((y_0,y_2)^t(y_0,a_3y_{2}+b_2y_{3})\\right)+\\det \\left((Q(-c)y_0,y_3)^t(y_{-2},y_0)\\right)$ $\\tilde{C}_0^{{n+1\\over 2}}&\\!\\!=\\!\\!&\\det \\left((y_0,y_2,y_4,..,y_{{n+3\\over 2}})^t(y_{-2},y_0,y_2,..,y_{{n-1\\over 2}})\\right),\\\\\\tilde{C}_1^{{n+1\\over 2}}&\\!\\!=\\!\\!&\\ \\ \\ \\det \\left((y_0,y_2,..,y_{{n-1\\over 2}},y_{{n+3\\over 2}})^t(y_0,y_2,..,y_{{n-1\\over 2}},y_{{n+3\\over 2}})\\right)\\\\&&-\\det \\left((y_0,y_2,..y_{{n+1\\over 2}})^t(y_0,y_2,..,y_{{n-1\\over 2}},a_3y_{{n+1\\over 2}}+b_2y_{{n+3\\over 2}})\\right)\\\\&&-\\det \\left((Q(-c)y_0,y_2,..,y_{{n-1\\over 2}},y_{{n+3\\over 2}})^t(y_{-2},y_0,y_2,y_4,..,y_{{n-1\\over 2}},y_{{n+3\\over 2}})\\right) \\\\&&+\\det \\left( (Q(-c)y_0,y_2,..,y_{{n+1\\over 2}})^t (y_{-2},y_0,y_2,y_4,..,y_{{n-1\\over 2}},a_3y_{{n+1\\over 2}}+b_2y_{{n+3\\over 2}}) \\right)$ and if ${n+1\\over 2}$ is odd $\\tilde{C}_0^{{n+1\\over 2}}&\\!\\!\\!=\\!\\!\\!&\\det \\left((y_0,y_2,y_4,..,y_{{n+1\\over 2}},y_{{n+5\\over 2}})^t(y_{-2},y_0,y_2,..,y_{{n-3\\over 2}},y_{{n+1\\over 2}})\\right)\\\\ &&-\\det \\left((y_0,y_2,y_4,..,y_{{n+3\\over 2}})^t(y_{-2},y_0,y_2,..,y_{{n-3\\over 2}},b_2y_{{n+1\\over 2}}+a_3y_{{n-1\\over 2}})\\right),\\\\\\tilde{C}_1^{{n+1\\over 2}}&\\!\\!\\!=\\!\\!\\!&\\det \\left((y_0,y_2,..,y_{{n+1\\over 2}})^t(y_0,y_2,..,y_{{n+1\\over 2}})\\right)-\\det \\left((y_{-2},y_0,y_2,y_4,..,y_{{n+1\\over 2}})^t(Q(-c)y_0,y_2,..,y_{{n+1\\over 2}})\\right) .$ In these formulas we use the product in the algebra $Frac({\\cal F}_{od})$ for computing products of vector components.",
"Therefore entries of our matrices of the form $v^tw$ are linear combinations of $y_i\\in Frac({\\cal F}_{od})$ , $i\\in \\mathbb {Z}$ .",
"On the other hand, for computing determinants we use the product in $S^{*}(Frac({\\cal F}_{od}))$ .",
"So these determinants are polynomials of degree ${n+1\\over 2}$ in $y_i,~i\\in \\mathbb {Z}$ .",
"Moreover, it turns out that in our linear combinations of determinants all terms with $y_i\\notin \\lbrace y_{-2},y_0,y_2,...,y_n\\rbrace $ cancel out and $\\tilde{C}_0^{{n+1\\over 2}},\\tilde{C}_1^{{n+1\\over 2}}$ are polynomials in $y_{-2},y_0,y_2,...,y_n$ of degree ${n+1\\over 2}$ .",
"These polynomials are linear in $y_{-2}$ and therefore can be written as $\\tilde{C}_i^{{n+1\\over 2}}=A_i+B_iy_{-2}$ , $i=0,1$ where $A_i,B_i$ are polynomials in $y_0,y_2,...,y_{n}$ .",
"We set $C^n=A_0B_1-A_1B_0$ ."
],
[
"Conclusion",
"In this paper we have constructed nine pairwise compatible quadratic Poisson structures on a linear space of arbitrary dimension.",
"It seems that this family of Poisson structures is maximal if the dimension of linear space is larger than four.",
"We think that the following problems deserve further investigation: Study the differential and algebraic geometry of these compatible Poisson structures to explain geometrically why these structures exist and why the number of them is exactly nine.",
"Do there exist other Poisson structures compatible with the one in $q_n(\\tau )$ where $n\\ge 5$ ?",
"There exist other elliptic Poisson algebras, for example $q_{n,k}(\\tau )$ where $1\\le k<n$ and $n,~k$ are coprime [5].",
"Functional constructions of these Poisson algebras can be found in [7].",
"Do there exist Poisson structures compatible with the one in $q_{n,k}(\\tau )$ ?",
"Note that $q_{n,1}(\\tau )=q_n(\\tau ),~q_{n,n-1}(\\tau )$ is trivial so the first nontrivial example other than $q_n(\\tau )$ is $q_{5,2}(\\tau )$ .",
"We plan to address these problems elsewhere."
],
[
"Appendix: Elliptic and $\\theta $ -functions",
"tocsectionAppendix: Elliptic and $\\theta $ -functions Fix $\\tau \\in \\mathbb {C}$ such that $Im~\\tau >0$ .",
"Let $\\Gamma =\\lbrace k+l\\tau ;~k,l\\in \\mathbb {Z}\\rbrace \\subset \\mathbb {C}$ be an integral lattice generated by 1 and $\\tau $ .",
"The Weierstrass zeta function is defined as follows: $\\zeta (z)={1\\over z}+\\sum _{\\omega \\in \\Gamma \\setminus \\lbrace 0\\rbrace }\\left({1\\over z-\\omega }+{1\\over \\omega }+{z\\over \\omega ^2}\\right)$ The function $\\zeta (z)$ is not elliptic but one has $\\zeta (z+\\omega )=\\zeta (z)+\\eta (\\omega )$ where $\\eta :\\Gamma \\rightarrow \\mathbb {C}$ is a $\\mathbb {Z}$ -linear function.",
"The functions $\\zeta (z_1-z_2)-\\zeta (z_1)+\\zeta (z_2)$ and $\\zeta ^{\\prime }(z)$ are elliptic.",
"Moreover, a function $c_1\\zeta (z-u_1)+...+c_m\\zeta (z-u_m)$ is elliptic in $z$ if $c_1+...+c_m=0$ .",
"Let $n\\in \\mathbb {N}$ .",
"We denote by $\\Theta _n(\\tau )$ the space of the entire functions of one variable satisfying the following relations: $f(z+1)=f(z),~~~f(z+\\tau )=(-1)^ne^{-2\\pi inz}f(z)$ It is known [8] that $\\dim \\Theta _n(\\tau )=n$ , every function $f\\in \\Theta _n(\\tau )$ has exactly $n$ zeros modulo $\\Gamma $ (counted according to their multiplicities), and the sum of these zeros modulo $\\Gamma $ is equal to zero.",
"Let $\\theta (z)=\\sum _{\\alpha \\in \\mathbb {Z}}(-1)^{\\alpha }e^{2\\pi i(\\alpha z+{\\alpha (\\alpha -1)\\over 2}\\tau )}$ .",
"It is clear that $\\theta (z)\\in \\Theta _n(\\tau )$ .",
"We have $\\theta (0)=0$ and this is the only zero modulo $\\Gamma $ .",
"Moreover, there exist functions $\\lbrace \\theta _{\\alpha }(z); \\alpha \\in \\mathbb {Z}/n\\mathbb {Z}\\rbrace \\subset \\Theta _n(\\tau )$ .",
"These functions are uniquely defined (up to multiplication by a common constant) by the following identities: $ \\theta _{\\alpha }\\left(z+{1\\over n} \\right)= e^{ 2\\pi i{\\alpha \\over n}}\\theta _{\\alpha }(z),~~~\\theta _{\\alpha }\\left(z+{1\\over n}\\tau \\right)=-e^{-2\\pi i(z+{1\\over n}-{n-1\\over 2n}\\tau )}\\theta _{\\alpha }(z)$ and form a basis of the linear space $\\Theta _n(\\tau )$ .",
"Note that $\\zeta (z)$ as well as $\\theta (z),~\\theta _{\\alpha }(z)$ are functions in two variables: $z\\in \\mathbb {C}$ and modular parameter $\\tau $ .",
"Therefore, they can be written as $\\zeta (z,\\tau )$ and $\\theta (z,\\tau ),~\\theta _{\\alpha }(z,\\tau )$ .",
"tocsectionReferences"
]
] | 1204.1299 |
[
[
"The regular and profinite representations of residually finite groups"
],
[
"Abstract Let $G$ be a residually finite group.",
"To any decreasing sequence $\\mathcal S = (H_n)_n $ of finite index subgroups of $G$ is associated a unitary representation $\\rho_{\\mathcal S}$ of $G$ in the Hilbert space $\\bigoplus_{n=0}^{+\\infty} \\ell^2 (G/H_n) $.",
"This paper investigates the following question: when does the representation $\\rho_{\\mathcal S} $ weakly contain the regular representation $\\lambda$ of $G$?"
],
[
"Introduction",
"Let $G$ be a countable residually finite group and $ H$ be a finite index, not necessarily normal, subgroup.",
"The group $G$ acts on $G/H $ and we let $\\lambda _{G/H}$ denote the representation that $G$ admits on $\\ell ^2 \\left(G/H \\right)$ .",
"That is, we let $\\lambda _{G/H}(g)(\\xi )(x)=\\xi (g^{-1}\\cdot x)$ with $g\\in G, \\ \\xi \\in \\ell ^2 \\left(G/H \\right) $ and $x \\in G/H$ .",
"In this paper, we will mainly be interested in representations of the form $\\rho _{\\mathcal {S}} := \\bigoplus _{n=0}^{+\\infty } \\lambda _{G/H_n}$ where $\\mathcal {S} = \\left( H_n\\right)_n $ is a decreasing sequence of finite index, not necessarily normal, subgroups of $G$ .",
"Such representations are called profinite representations.",
"We will investigate the link between such representations and the regular representation $\\lambda $ of $G$ .",
"Recall that $\\lambda $ is the representation that $G$ admits on $\\ell ^2 \\left( G\\right)$ and defined for all $g\\in G, \\ \\xi \\in \\ell ^2 \\left(G \\right) $ and $h \\in G$ by $\\lambda (g)(\\xi )(h)=\\xi (g^{-1}h).$ As $\\rho _{\\mathcal {S}}$ is a sum of finite representations (i.e.",
"factorizing through a finite index subgroup of $G$ ) and $\\lambda $ is a $C^0$-representation (i.e.",
"the coefficients $\\left\\langle \\lambda (g)(\\xi )\\ | \\ \\psi \\right\\rangle $ tend to 0 when $g$ tends to infinity), as soon as $G$ is infinite, none of these two representations can be a subrepresentation of the other [6].",
"However, one can expect a weaker link between them, namely weak containment.",
"The goal of this work can be stated as follows: when does the representation $\\rho _{\\mathcal {S} } $ weakly contain the regular representation $\\lambda $ of $G$ ?",
"Let us recall the definition of weak containment.",
"Definition 0.1 If $G$ is a countable group and $\\left(\\pi _1, \\mathcal {H} _1\\right)$ , $\\left(\\pi _2,\\mathcal {H} _2 \\right)$ are two unitary representations of $G$ , we say that $\\pi _2$ weakly contains $\\pi _1$ (and write $\\pi _1 \\prec \\pi _2$ ) if and only if for all $\\epsilon >0$ , $\\xi \\in \\mathcal {H} _1$ and $ K \\subset G$ finite, there are vectors $\\nu _1,\\nu _2,\\dots ,\\nu _n$ in $\\mathcal {H} _2$ such that $\\forall g \\in K, \\ \\left| \\left\\langle \\pi _1(g)\\xi | \\xi \\right\\rangle - \\sum _{i=1 }^n \\left\\langle \\pi _2 (g)\\nu _i | \\nu _i \\right\\rangle \\right| < \\epsilon .$ There is an equivalent formulation of this property in the framework of $C^*$ -algebras [6], [8]: for every unitary representation $\\left(\\pi ,\\mathcal {H} \\right) $ of $G$ , we extend $\\pi $ linearly to get a $*$ -homomorphism $\\ell ^1 \\left(G\\right) \\xrightarrow{} \\mathcal {B} \\left( \\mathcal {H} \\right)$ where $\\mathcal {B} \\left( \\mathcal {H} \\right) $ denotes the algebra of bounded operators on $\\mathcal {H} $ .",
"One then defines $C^{*}_{\\pi }\\left( G \\right):= \\overline{\\pi \\left( \\ell ^1 \\left(G\\right) \\right)}$ where the closure is taken with respect to the $C^*$ -operator norm $\\Vert \\cdot \\Vert _{_{\\mathcal {B} \\left( \\mathcal {H} \\right) }}$ on $\\mathcal {B} \\left( \\mathcal {H} \\right) $ .",
"Then, $\\pi _1 \\prec \\pi _2 & \\Longleftrightarrow & C^{*}_{\\pi _2}\\left( G \\right)\\twoheadrightarrow C^{*}_{\\pi _1}\\left( G \\right) \\\\& \\Longleftrightarrow & \\forall f \\in \\ell ^1\\left( G \\right),\\ \\Vert \\pi _1 (f) \\Vert _{_{\\mathcal {B} \\left(\\mathcal {H} _1 \\right)}} \\le \\Vert \\pi _2 (f) \\Vert _{_{\\mathcal {B} \\left( \\mathcal {H} _2 \\right)}} \\\\& \\Longleftrightarrow & \\forall f \\in \\ell ^1\\left( G \\right),\\ \\text{sp}\\left(\\pi _1 (f) \\right) \\subset \\text{sp}\\left(\\pi _2 (f) \\right)$ where $\\text{sp}\\left(M \\right) $ denotes the spectrum of a operator $M$ .",
"In particular, one remarks that $\\pi _1$ and $\\pi _2$ are weakly isomorphic (written $\\pi _1 \\simeq \\pi _2$ and defined to be $\\pi _1 \\prec \\pi _2$ and $\\pi _2 \\prec \\pi _1$ ) if and only if the $C^*$ -algebras $C^{*}_{\\pi _1}\\left( G \\right)$ and $ C^{*}_{\\pi _2}\\left( G \\right) $ are isomorphic.",
"The representation $\\rho _{\\mathcal {S}}$ comes from an action of $G$ on a countable set.",
"In the preliminary Section , we introduce the notion of locally somewhere free action of a countable group $G$ on a measure space $\\left(X,\\nu \\right)$ : Definition (REF ) We say that the action of $G$ on $\\left(X,\\nu \\right)$ is locally somewhere free (l.s.f.",
"for short) if for every finite subset $K \\subset G$ , there is a Borel subset $F \\subset X$ of positive measure such that the non-trivial elements of $K$ fixe almost no point in $F$ : $\\forall \\gamma \\ne 1 \\in K,\\ x\\in F \\Rightarrow \\gamma \\cdot x \\ne x \\text{ a.e.",
"}$ If $X$ is metrizable, locally compact and separable, and $\\nu $ is a Radon measure quasi-preserved by $G$ , then one can prove the following: Proposition (REF ) If the action of $G$ on $X$ is l.s.f.",
"then the regular representation $\\lambda $ of $\\Gamma $ is weakly contained in $\\rho _X$ .",
"Here, $\\rho _X$ denotes the canonical unitary representation that $G$ admits on $L^2\\left(X\\right)$ .",
"In order to use this result, we remark in Section that there is a correspondence between profinite representations and actions on rooted trees.",
"More precisely, to any decreasing sequence $\\mathcal {S} = \\left( H_n\\right)_n $ of finite index subgroups of $G$ , is associated a rooted tree $\\mathcal {T}_{\\mathcal {S}}$ together with a spherically transitive action of $G$ on it.",
"This correspondence is one-to-one and the representation $\\rho _{\\mathcal {S}}$ is isomorphic to the permutational representation $\\rho _{\\mathcal {T}_{\\mathcal {S}}}$ coming from the action of $G$ on $\\mathcal {T}_{\\mathcal {S}}$ .",
"In the sequel, we mostly use this interpretation to adress our problem.",
"We thus briefly recall in Section some well-known facts about the automorphism group $\\text{Aut}\\left(\\mathcal {T} \\right)$ of a rooted tree.",
"It turns out that the property $\\lambda \\prec \\rho _{\\mathcal {T}}$ is linked to the size of the stabilizers $\\text{Stab}_G (v\\mathcal {T}) $ in $G$ of the subtrees $v\\mathcal {T}$ of $\\mathcal {T}$ .",
"In Section , we use Proposition REF of Section in order to prove the following: Theorem (REF ) Let $G$ be a countable group acting faithfully on a rooted tree $\\mathcal {T}$ .",
"If for every vertex $v\\in \\mathcal {T}^0$ , the stabilizer $\\text{Stab}_G (v\\mathcal {T})$ is trivial, then $\\lambda \\prec \\rho _{\\mathcal {T}}$ .",
"More generally, if the set $\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_G (v\\mathcal {T})$ is finite and has cardinality $n$ , then $\\lambda \\prec \\rho _{\\mathcal {T}}^{\\otimes n}$ .",
"One always has $\\lambda \\prec \\bigoplus _{n=1}^{+\\infty }\\rho _{\\mathcal {T}}^{\\otimes n} $ .",
"The proof of the first part amounts to show that the triviality of the stabilizers $\\text{Stab}_G (v\\mathcal {T})$ is equivalent to the action of $G$ on $\\mathcal {T}$ to be l.s.f., so that Proposition REF can be applied.",
"In Section , we study the inverse implication of this part of the Theorem.",
"Theorem (REF ) Let $G$ be a countable group in which the normalizer $N_G \\left(H \\right)$ of any non-central finite group $H$ has infinite index in $G$ .",
"Suppose that $G$ acts spherically transitively on a rooted tree $\\mathcal {T}$ .",
"If there exists a subtree $v\\mathcal {T}$ whose stabilizer $\\text{Stab}_G \\left( v\\mathcal {T}\\right)$ in $G$ is not trivial, then the $*$ -homomorphism $\\rho _\\mathcal {T}$ defined on $G̏$ is not faithful.",
"In particular, $\\lambda \\nprec \\rho _{\\mathcal {S}}$ .",
"Here, we have to make an algebraic assumption on $G$ .",
"Indeed, the sufficient condition in Theorem REF .REF is not necessary in general (see Example REF ).",
"In the last section, we illustrate the results of the previous two.",
"In particular, we show that for the following classes of group, any faithful and spherically transitive action on a rooted tree is l.s.f.",
": torsion free Gromov hyperbolic groups, uniform lattices in a connected simple real Lie group $G$ with finite center and $\\mathbb {R}$ -rank 1, irreducible lattices in a connected semisimple real Lie group with finite centre, no compact factor and $\\mathbb {R}$ -rank $\\ge 2$ .",
"Thus, if $G$ belongs to one of these classes, then any faithful profinite representation of $G$ weakly contains the regular.",
"Here is an application: Corollary (REF ) Let $\\Gamma $ be non-elementary, torsion free, residually finite hyperbolic group.",
"Let $\\mathcal {S} = \\left( H_n\\right)_n $ be a decreasing sequence of finite index subgroups of $\\Gamma $ such that the representation $\\rho _{\\mathcal {S}}$ is faithful.",
"Let $M_F := \\frac{1}{2|F|}\\sum _{g\\in F} g+g^{-1}$ be the Markov operator associated to a finite generating set $F$ of $\\Gamma $ which does not contain 1.",
"Then $\\left[ -\\frac{1}{|F|}, \\frac{\\sqrt{2|F|-1}}{|F|}\\right] \\subset \\overline{\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right)} .$ Finally, we study the case of weakly branched subgroups of the automorphism group $\\text{Aut}\\left(\\mathcal {T} \\right)$ of a regular rooted tree $\\mathcal {T}$ .",
"They form an interesting and wide class of residually finite groups which provides important examples (infinite, periodic and finitely generated groups [1], finitely generated groups with intermediate growth [11], amenable groups not belonging to the class SG [14], [4], finitely generated groups with non-uniform exponential growth [23], [22], [2]).",
"A direct consequence of their definition is that their action on $\\mathcal {T}$ is not l.s.f.",
"Thus, the results of §REF imply that a weakly branched group cannot belong to one the classes (REF ),(REF ) or (REF ) above.",
"It seems to the author that this was not known.",
"Finally, we prove the following proposition, which in particular indicates that Theorem REF .REF is optimal: Proposition (REF ) Let $G$ be a weakly branched subgroup of $\\text{Aut}\\left(\\mathcal {T} \\right)$ .",
"For every $n>0$ , the $*$ -homomorphism $\\rho _{\\mathcal {T}}^{\\otimes n}$ defined on $G̏$ is not faithful.",
"In particular, the representation $\\rho _{\\mathcal {T}}^{\\otimes n}$ does not weakly contain the regular $\\lambda $ ."
],
[
"Locally somewhere free actions and weak containment of the regular representation",
"Let $G$ be a countable group.",
"Let $\\left(X,\\nu \\right)$ be a metrizable, locally compact and separable space endowed with a Radon measure $\\nu $ , on which $G$ acts.",
"If the action is measurable and quasi-preserves $\\nu $ , then it yields the following unitary representation: $\\begin{array}{ccc}G & \\overset{\\rho _X}{\\longrightarrow } & \\mathcal {U} \\left( L^2 \\left(X\\right)\\right) \\\\g & \\longrightarrow & f \\rightarrow \\rho _X\\left(g\\right)\\left(f\\right)\\end{array}$ where $\\rho _X\\left(g\\right)\\left(f\\right) $ is defined by $\\rho _X\\left(g\\right)\\left(f\\right)\\left(x\\right)= \\left(\\frac{dg^{-1}\\nu }{d\\nu }\\left(x\\right)\\right)^{1/2} f\\left(g^{-1}\\cdot x \\right) $ with $\\frac{dg^{-1}\\nu }{d\\nu } $ the Radon-Nikodym derivative.",
"Definition 1.1 We say that the action of $G$ on $\\left(X,\\nu \\right)$ is locally somewhere free (l.s.f.",
"for short) if for every finite subset $K \\subset G$ , there is a Borel subset $F \\subset X$ of positive measure such that the non-trivial elements of $K$ fixe almost no point in $F$ : $\\forall \\gamma \\ne 1 \\in K,\\ x\\in F \\Rightarrow \\gamma \\cdot x \\ne x \\text{ a.e.",
"}$ Proposition 1.2 If the action of $G$ on $X$ is l.s.f.",
"then the regular representation $\\lambda $ of $\\Gamma $ is weakly contained in $\\rho _X$ .",
"Let $K$ be a finite subset of $X$ and $F$ a subset of $X$ of positive measure such that the non-trivial elements of $K$ fixe almost no point in $F$ .",
"Let us consider a distance $d$ on $X$ compatible with its topology and for each positive integer $n$ , the following measurable set: $E_n := \\left\\lbrace x \\in F \\ | \\ \\forall \\gamma \\ne 1 \\in K, \\ d(\\gamma \\cdot x, x) > 1/n\\right\\rbrace .$ By definition, $\\nu \\left(F \\setminus \\cup _n E_n \\right)= 0$ and therefore, there is a positive integer $k$ such that $\\nu \\left( E_k\\right)>0$ .",
"Since $X$ is separable, there is ball $B\\left(x,l\\right)$ of radius $l<\\frac{1}{4k}$ such that $U_K:=B\\left(x,l\\right) \\cap E_k$ has non-zero measure.",
"It satisfies $\\forall \\gamma \\ne 1 \\text{ in } K, \\ \\gamma \\left( U_K\\right) \\cap U_K = \\emptyset .$ Let us now prove that $\\lambda \\prec \\rho _X$ .",
"We denote by $\\delta _{1}$ the Dirac function over the identity element of $G$ .",
"This is a cyclic vector for the regular representation $\\lambda $ , i.e.",
"the family $\\left\\lbrace \\lambda \\left(g\\right)\\left( \\delta _{1}\\right)= \\delta _{g} \\ | \\ g\\in G \\right\\rbrace $ is total in $\\ell ^2 \\left(\\Gamma \\right)$ .",
"Therefore, by a result of Fell [9], it is enough to check (REF ) in Definition REF for $\\xi =\\delta _{1}$ .",
"As such, we consider $K$ a finite subset of $\\Gamma $ and $U:=U_K$ the measurable subset associated to $K$ defined previously.",
"If we let $\\chi _{U} \\in L^2\\left(X\\right) $ be the characteristic function of $U$ , then (REF ) implies that for $g\\in K$ $\\left< \\lambda \\left(g\\right) \\left(\\delta _{1}\\right) \\ | \\ \\delta _1\\right> = \\delta _{1,g} =\\left< \\rho _X\\left( g\\right)\\left(\\frac{\\chi _{U} }{ \\nu \\left(U\\right)} \\right)\\ | \\ \\frac{\\chi _{U} }{ \\nu \\left(U\\right)} \\right>$ where $\\delta _{1,g}=1$ if $1=g$ and 0 otherwise.",
"Example 1.3 The action of a non-elementary, torsion free Gromov hyperbolic group on its boundary $\\left(\\partial G,\\nu \\right)$ endowed with a Patterson-Sullivan measure $\\nu $ fullfills all the conditions of the previous proposition (see [5]).",
"It is l.s.f.",
"since each element in $G$ admits exactly two fixed points in $\\partial G$ , and the measure $\\nu $ has no atom.",
"We stress that a faithful action is not necessarily l.s.f.",
"; the last chapter will give such examples."
],
[
"Rooted trees",
"Let $\\bar{d}=d_0,d_1,\\dots ,d_n,\\dots $ be a sequence of integers with $d_i \\ge 2$ for all $i$ .",
"We define the rooted tree $\\mathcal {T}_{\\bar{d}}$ as follows: $\\mathcal {T}_{\\bar{d}}$ is an infinite, locally finite tree endowed with the usual metric $dist$ carried by any graph, and for which: there is a particular vertex $\\varnothing $ called the root, the degree $\\textrm {deg}(v)$ of any vertex $v$ depends only on its distance to the root and is more precisely given by $\\textrm {deg}(\\varnothing )=d_0, \\text{ and for } n=dist(\\varnothing ,v)\\ge 1, \\ \\textrm {deg}(v)=d_n +1.",
"$ Let $\\mathcal {T}_{\\bar{d}}^0$ denotes the set vertices of $\\mathcal {T}_{\\bar{d}}$ .",
"If $n$ is a non-negative integer, the set of vertices whose distance to the root equals $n$ is called the $n$ -th level of $\\mathcal {T}_{\\bar{d}}$ and is denoted by $L_n$ .",
"One can give $\\mathcal {T}_{\\bar{d}}$ a planar graph structure as follows: for every $n$ , one labels each vertex of $L_n$ by a finite sequence $i_1 i_2 \\dots i_n$ with $i_j \\in \\left\\lbrace 1,2,\\dots ,d_j \\right\\rbrace $ such that 2 vertices $i_1 i_2 \\dots i_n \\in L_n$ and $i^{\\prime }_1 i^{\\prime }_2 \\dots i^{\\prime }_{n+1} \\in L_{n+1}$ are connected by an edge if and only if $i_1 i_2 \\dots i_n = i^{\\prime }_1 i^{\\prime }_2 \\dots i^{\\prime }_n $ .",
"Every level $L_n$ is then endowed with the lexicographic order.",
"Given any vertex $v$ , one defines the subtree $v\\mathcal {T}_{\\bar{d}}$ to be the connected subgraph of $\\mathcal {T}_{\\bar{d}}$ whose vertices $w$ are descendants of $v$ (i.e.",
"$v \\in \\left[\\varnothing ,w \\right] $ where $\\left[\\varnothing ,w \\right]$ denotes the unique geodesic path linking $\\varnothing $ and $w$ ).",
"We refer to Fig.",
"REF for a less rigourous but more visual presentation of these definitions.",
"Figure: Rooted tree 𝒯\\mathcal {T}A countable group $G$ is said to act on a rooted tree $\\mathcal {T}_{\\bar{d}}$ if it acts on the underlying graph while fixing the root.",
"Such an action induces a unitary representation $\\rho _{\\mathcal {T}_{\\bar{d}}}$ of $G$ on the Hilbert space $\\ell ^2 \\left( \\mathcal {T}_{\\bar{d}}^0\\right)$ defined by: $\\forall g \\in G, \\ \\forall \\xi \\in \\ell ^2 \\left( \\mathcal {T}_{\\bar{d}}^0\\right), \\ \\forall v \\in \\mathcal {T}_{\\bar{d}}^0, \\ \\rho _{\\mathcal {T}_{\\bar{d}}}(g)(\\xi )(v)=\\xi (g^{-1}\\cdot v).$ Let us also define, for every vertex $v \\in \\mathcal {T}_{\\bar{d}}^0$ and every non-negative integer $n$ the following stabilizers: $\\text{Stab}_G (v):=\\left\\lbrace g \\in G \\ | \\ g(v)=v\\right\\rbrace , \\quad \\text{Stab}_G (L_n):=\\left\\lbrace g \\in G \\ | \\ g(v)=v, \\ \\forall v \\in L_n \\right\\rbrace $ which both have finite index in $G$ , since $G$ preserves the finite sets $L_n$ .",
"The representation $\\rho _{\\mathcal {T}_{\\bar{d}}} $ will be faithful if and only if the action of $G$ on $\\mathcal {T}_{\\bar{d}} $ is, that is if and only if $\\bigcap _{n=0}^{\\infty }\\text{Stab}_G (L_n)$ is trivial.",
"The action of $G$ on $\\mathcal {T}_{\\bar{d}}$ is said spherically transitive if $G$ acts transitively on each level.",
"In this case, one has $\\forall n, \\ \\forall v \\in L_n, \\ \\text{Stab}_G (L_n)=\\bigcap _{g\\in G}g\\text{Stab}_G (v)g^{-1} .$ One notices that there is a correspondence between spherically transitive actions of $G$ on a rooted tree $\\mathcal {T}_{\\bar{d}}$ and decreasing sequences $\\mathcal {S}=\\left( H_n\\right)_n $ of finite index subgroups of $G$ .",
"Moreover, this correspondence respects the representations $\\rho _{\\mathcal {T}_{\\bar{d}}} $ and $\\rho _{\\mathcal {S}} $ .",
"Indeed, suppose $G$ acts spherically transitively on $\\mathcal {T}_{\\bar{d}}$ and let $\\left( v_n\\right)_n$ be the sequence with $v_n$ the left most vertex $11\\dots 1$ of $L_n$ .",
"Then, $\\mathcal {S}\\left(\\mathcal {T}_{\\bar{d}} \\right):=\\left(\\text{Stab}_G (v_n)\\right)_n$ is a decreasing sequence of finite index subgroups of $G$ fullfilling $\\forall n,\\ \\left|\\text{Stab}_G (v_{n})/\\text{Stab}_G (v_{n+1}) \\right|=d_n .$ Moreover, since the actions of $G$ on $L_n$ and on $G/\\text{Stab}_G(v_n)$ are isomorphic, the representations $\\rho _{\\mathcal {T}_{\\bar{d}}} $ and $\\rho _{\\mathcal {S} (\\mathcal {T}_{\\bar{d}})} $ are isomorphic.",
"Conversely, let $\\mathcal {S}=\\left( H_n\\right)_n $ be a decreasing sequence of finite index subgroups of $G$ .",
"Without lost of generality, one can assume that $H_0=G$ and $\\mathcal {S}$ is strictly decreasing.",
"Then, we can construct a rooted tree $\\mathcal {T}\\left(\\mathcal {S} \\right)$ as follows: - its vertices are the cosets $g_{n,k}H_n$ , - two vertices $g_{n,k}H_n $ and $g_{n^{\\prime },k^{\\prime }}H_{n^{\\prime }}$ are connected by an edge if and only if $n^{\\prime }=n+1$ and $g_{n^{\\prime },k^{\\prime }}H_{n^{\\prime }} \\subset g_{n,k}H_n$ .",
"By construction, the root of $\\mathcal {T}\\left(\\mathcal {S} \\right)$ is represented by $G$ and, more generally, the sets $L_n $ and $ G/H_n$ are equal: $\\mathcal {T}\\left(\\mathcal {S} \\right)$ is actually the rooted tree $\\mathcal {T}_{\\bar{d}}$ where $\\bar{d}=d_0,d_1,\\dots $ is caracterised by $d_n=\\left| H_n/H_{n+1} \\right|$ .",
"Moreover, it follows that $G$ acts on $L_n$ for all $n$ , and thus on $\\mathcal {T}\\left( \\mathcal {S} \\right)$ .",
"The representations $\\rho _{\\mathcal {S}}$ and $\\rho _{\\mathcal {T}\\left( \\mathcal {S} \\right)}$ are clearly isomorphic.",
"This correspondence implies for instance that a countable group $G$ acts faithfully on a rooted tree if ond only if it is residually finite.",
"We will soon see that this picture is relevant for our purpose.",
"More precisely, the weak containment of the regular representation $\\lambda $ in $\\rho _{\\mathcal {S}}$ , equivalently in $\\rho _{\\mathcal {T}}$ , is linked to the size of the stabilizers $\\text{Stab}_G \\left( v\\mathcal {T}\\right)$ of subtrees defined by $\\forall v \\in \\mathcal {T}^0, \\ \\text{Stab}_G \\left( v\\mathcal {T}\\right) := \\left\\lbrace g\\in G \\ | \\ g(x)=x, \\ \\forall x\\in v\\mathcal {T}\\right\\rbrace .$ Before starting the study of our problem, we recall in next section some convinient facts about the group of the automorphisms of a rooted tree."
],
[
"The automorphism group of a rooted tree",
"By $\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) $ we denote the group of the automorphisms of $\\mathcal {T}_{\\bar{d}}$ that fixe the root $\\varnothing $ .",
"This short section aims to recall some convenient tools to describe such automorphisms.",
"We refer to [25], [19], [20], [21] for more details.",
"For all $n$ , the group $\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) $ preserves the level $L_n$ as well as the finite set $\\left\\lbrace v\\mathcal {T}_{\\bar{d}} \\ | \\ v \\in L_n\\right\\rbrace $ of subtrees rooted at its vertices.",
"Thanks to the self-similar structure of $\\mathcal {T}_{\\bar{d}} $ , the group $\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) $ admits a natural decomposition in terms of the automorphisms group of other rooted trees.",
"More precisely, we have just noticed that every $g$ in $\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right)$ induces a permutation $g_1$ on the set $L_1$ , as well as an isomorphism $\\varphi _{g(v)}(g)$ from $v\\mathcal {T}_{\\bar{d}}$ onto $g(v)\\mathcal {T}_{\\bar{d}}$ , for every vertex $v$ in $L_1$ .",
"These two subtrees are canonically isomorphic to $\\mathcal {T}_{\\sigma (\\bar{d})}$ where $\\sigma $ denotes the shift on $\\mathbb {R}^{\\mathbb {N}}$ (i.e.",
"$\\sigma (\\bar{d})=d_1,d_{2},\\dots $ ) and $\\varphi _{g(v)}(g)$ can be seen therefore as an element of $\\text{Aut}\\left(\\mathcal {T}_{\\sigma (\\bar{d})} \\right) $ .",
"It is easy to see that this data completely determines the action of $g$ on $\\mathcal {T}_{\\bar{d}} $ .",
"In fact, we have the following decomposition: $\\begin{array}{ccc}\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) & \\overset{\\Phi }{\\longrightarrow } & \\left( \\text{Aut}\\left(\\mathcal {T}_{\\sigma (\\bar{d})} \\right) \\times \\dots \\times \\text{Aut}\\left(\\mathcal {T}_{\\sigma (\\bar{d})} \\right) \\right) \\rtimes \\mathfrak {S}_{d_0} \\\\g & \\longrightarrow & ( \\varphi _{1}(g), \\dots , \\varphi _{d_0}(g)) .",
"g_1\\end{array}$ where $\\mathfrak {S}_{d_0}$ denotes the symetric group on the set of $d_0$ elements $L_1$ .",
"Its action on $\\left( \\text{Aut}\\left(\\mathcal {T}_{\\sigma (\\bar{d})} \\right) \\times \\dots \\times \\text{Aut}\\left(\\mathcal {T}_{\\sigma (\\bar{d})} \\right) \\right) $ is the permutation of the coordinates.",
"The isomorphism $\\Phi $ is called the recursion isomorphism.",
"Generalizing further, we denote by $\\Phi ^{(n)}$ the decomposition of $\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) $ with respect to the level $L_n$ : $\\begin{array}{ccc}\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) & \\overset{\\Phi ^{(n)}}{\\longrightarrow } & \\left(\\prod _{w \\in L_n} \\text{Aut}\\left(\\mathcal {T}_{\\sigma ^n(\\bar{d})} \\right) \\right) \\rtimes \\text{Aut}\\left(\\rm T_{\\bar{d},n}\\right) \\\\g & \\longrightarrow & \\left( \\varphi _w \\right)_{w\\in L_n} .\\ g_n\\end{array}$ where $\\rm T_{\\bar{d},n} $ is the restriction of $\\mathcal {T}_{\\bar{d}}$ to its $n$ -th first levels, and where $\\text{Aut}(\\rm T_{\\bar{d},n} )$ is the restriction of $\\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) $ to this stable subgraph $\\rm T_{\\bar{d},n} $ .",
"We remark that an element $g\\in \\text{Aut}\\left(\\mathcal {T}_{\\bar{d}} \\right) $ fixes the restriction $\\rm T_{\\bar{d},n} $ if and only if $g_n$ equals 1, which is also equivalent to $g$ fixing the $n$ -th level $L_n$ ."
],
[
"A sufficient condition",
"Let $G$ be a countable group and $\\mathcal {S} = \\left( H_k\\right)_k $ a strictly decreasing sequence of finite index subgroups.",
"To simplify the notations, let $\\mathcal {T}$ denote the associated rooted tree $\\mathcal {T}\\left(\\mathcal {S} \\right)$ .",
"The goal of this section is to find a condition which ensures that $\\rho _{\\mathcal {S}}$ , or equivalently $\\rho _{\\mathcal {T}}$ , weakly contains the regular representation $\\lambda $ of $G$ on $\\ell ^2 \\left( G \\right)$ .",
"This is of course only possible if the representation $\\rho _{\\mathcal {S}}$ is faithful, i.e.",
"if the action on $\\mathcal {T}$ is faithful.",
"What follows deals also with the representations $\\rho _{\\mathcal {S}}^{\\otimes n} \\simeq \\rho _{\\mathcal {T}}^{\\otimes n}$ that $G$ admits on $\\left(\\bigoplus _{k=0}^{+\\infty } \\ell ^2 \\left( G/H_k\\right) \\right)^{\\otimes n}=\\left(\\ell ^2 \\left( \\mathcal {T}^0 \\right) \\right)^{\\otimes n} = \\ell ^2 \\left( \\mathcal {T}^0 \\times \\dots \\times \\mathcal {T}^0\\right)$ coming from the diagonal action of $G$ on $\\mathcal {T}^n$ .",
"Theorem 4.1 Let $G$ be a countable group acting faithfully on a rooted tree $\\mathcal {T}$ .",
"If for every vertex $v\\in \\mathcal {T}^0$ , the stabilizer $\\text{Stab}_G (v\\mathcal {T})$ is trivial, then $\\lambda \\prec \\rho _{\\mathcal {T}}$ .",
"More generally, if the set $\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_G (v\\mathcal {T})$ is finite and has cardinality $n$ , then $\\lambda \\prec \\rho _{\\mathcal {T}}^{\\otimes n}$ .",
"One always has $\\lambda \\prec \\bigoplus _{n=1}^{+\\infty }\\rho _{\\mathcal {T}}^{\\otimes n} $ .",
"i.",
"Let us show that the action of $G$ on the countable set $ \\mathcal {T}^0$ is l.s.f.",
"so that Proposition REF applies.",
"If this is not the case, then there exists a finite subset $F$ of $G$ not containing 1 and such that $F\\cap \\text{Stab}_G \\left( v \\right) $ is non-empty, for all $v \\in \\mathcal {T}^0$ .",
"Let $\\Upsilon $ be the map that associates to a vertex $v\\in \\mathcal {T}^0$ this non-empty finite set $F\\cap \\text{Stab}_G \\left( v \\right) $ , and let $v_0 \\in \\mathcal {T}^0$ be such that $\\left| \\Upsilon \\left(v_0\\right) \\right| $ is minimal.",
"It is clear that if a vertex $y$ is a descendant of a vertex $x$ (i.e.",
"$y\\in x\\mathcal {T}$ ), then $\\Upsilon \\left( y \\right) \\subset \\Upsilon \\left( x \\right)$ .",
"By minimality of the cardinality of $\\Upsilon \\left( v_0 \\right)$ and the previous remark, we have $\\forall v\\in v_0 \\mathcal {T}, \\ \\Upsilon \\left( v \\right)= \\Upsilon \\left( v_0 \\right)$ and therefore any element in $\\Upsilon \\left( v_0 \\right)$ , by construction necessarily non-trivial, fixes the subtree $v_0 \\mathcal {T}$ .",
"This conclusion contradicts the hypothesis of REF .REF .",
"ii.",
"Once again, we want to use Proposition REF so suppose that $F$ is a finite subset of $G$ not containing 1.",
"Let $F_{free}$ be the subset of $F$ consisting of its elements which do not fix any subtree: $F_{free} = F \\setminus \\left( F \\cap \\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_G (v\\mathcal {T}) \\right).$ We set $\\left\\lbrace h_1,h_2,\\dots ,h_k \\right\\rbrace = F \\setminus F_{free}.$ For each $h_i$ , there is an element $v_i \\in \\mathcal {T}^0$ such that $\\text{Stab}_G (v_i) \\bigcap \\left(F_{free} \\cup \\left\\lbrace h_i\\right\\rbrace \\right)=\\emptyset .$ Indeed, let $w\\in \\mathcal {T}^0$ such that $h_i(w)\\ne w$ : by the definitions of $w$ and $F_{free}$ , no subtree of $w\\mathcal {T}^0$ can be fixed by an element of $F_{free} \\cup \\left\\lbrace h_i\\right\\rbrace $ .",
"We thus can apply the method of REF .",
"to prove the existence of such a $v_i$ in the subtree $w\\mathcal {T}^0$ .",
"Now, we have an element $\\left( v_1,\\dots , v_k \\right) \\in \\left( \\mathcal {T}^0\\right)^k$ such that $F \\cap \\text{Stab}_G \\left( \\left\\lbrace \\left( v_1,\\dots , v_k \\right) \\right\\rbrace \\right)=F \\cap \\bigcap _{i=1}^{k} \\text{Stab}_G \\left( v_i\\right)=\\emptyset $ where $k= \\left|F \\setminus F_{free} \\right| =\\left| F \\cap \\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_G (v\\mathcal {T})\\right| \\le \\left|\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_G (v\\mathcal {T})\\right|=n.$ Completing the sequence $ \\left( v_1,\\dots , v_k \\right) $ by any vertices $v_{k+1}, \\dots , v_n$ , we get an element in $\\left( \\mathcal {T}^0\\right)^n $ whose stabilizer in $G$ does not intersect $F$ .",
"Therefore, the action on $\\left(\\mathcal {T}^0\\right)^n$ is l.s.f.",
"and Proposition REF concludes.",
"iii.",
"It is clear that for an action $G$ on $X$ , the action of $G$ on $\\bigsqcup _{n\\in \\mathbb {N}} X^n$ is l.s.f.",
"as soon as the one on $X$ is faithful.",
"But $\\bigoplus _{n=1}^{+\\infty }\\rho _{\\mathcal {T}}^{\\otimes n} = \\rho _{\\bigsqcup _{n\\in \\mathbb {N}} \\left(\\mathcal {T}^0\\right)^n}$ so that Proposition REF applies again.",
"Remark 4.2 (on the l.s.f.",
"condition for actions on rooted trees) The first part of the proof shows that the triviality of all the stabilizers $\\text{Stab}_G \\left(v\\mathcal {T}\\right)$ implies that the action is l.s.f.",
"If the action is spherically transitive, then the converse holds.",
"Besides, it is easy to see that if the stabilizer of an infinite geodesic path in $\\mathcal {T}$ is trivial, then the action is l.s.f.",
"In particular, if $\\mathcal {S} = \\left( H_n\\right)_n $ is a decreasing sequence of finite index subgroups of $G$ with trivial intersection, then the action on the associated rooted tree $\\mathcal {T}_{\\mathcal {S}}$ is l.s.f.",
"and the representation $\\rho _{\\mathcal {S}}$ weakly contains the regular.",
"However, this condition is not necessary in general for the action to be l.s.f.",
"(see for instance the realization of the lamplighter group $\\left(\\oplus _{n\\in \\mathbb {Z}} \\mathbb {Z}/2\\mathbb {Z}\\right)\\rtimes \\mathbb {Z}$ as an automaton group in [13]).",
"Remark 4.3 Notice that, for every positive integer $n$ , $\\rho _{\\mathcal {S}}^{\\otimes n}$ is a subrepresentation of $\\rho _{\\mathcal {S}}^{\\otimes n+1}$ .",
"Indeed, the root of $\\mathcal {T}$ is fixed by $G$ , therefore there are invariant vectors under $G$ in $\\ell ^2 \\left( \\mathcal {T}^0 \\right) $ and so the trivial representation $\\epsilon $ is a subrepresentation of $\\rho _{\\mathcal {S}}$ .",
"It is thus not a surprise that the sufficient condition in REF .REF is weaker than the one in REF .REF .",
"In [21] we give, for every positive integer $n$ , some concrete examples of pair $\\left(G,\\mathcal {S}\\right)$ for which $\\rho _{\\mathcal {S}}^{\\otimes 2^n}$ weakly contains $\\lambda $ , but $\\rho _{\\mathcal {S}}^{\\otimes 2^n -1}$ does not.",
"Remark 4.4 The statement REF .REF can be compared with the well known fact that if $G$ is a finite group and $\\rho $ is a faithful representation of $G$ , then there is a positive integer $N$ such that every irreducible representation of $G$ appears in $\\rho ^{\\otimes N}$ .",
"Here we only get that the regular representation which weakly contains every unitary representation of $G$ if and only if $G$ is amenable.",
"is weakly contained in the sum of all the $\\rho _{\\mathcal {S}}^{\\otimes n}$ .",
"But as we will see in the last section, there are examples of pair $\\left(G,\\mathcal {S}\\right)$ for which $\\lambda $ is not weakly contained in $\\rho _{\\mathcal {S}}^{\\otimes n}$ for all $n$ , and so the statement of REF .REF is optimal."
],
[
"A necessary condition",
"In this section, we study the inverse implication of Theorem REF .REF .",
"It turns out to be true only under an algebraic assumption on $G$ .",
"After the proof of Theorem REF , we will give an example (see REF ) of an action of a group on a rooted tree which shows that the sufficient condition of Theorem REF .REF is not necessary in general and explains the additional assumption in Theorem REF below.",
"Theorem 5.1 Let $G$ be a countable group in which the normalizer $N_G \\left(H \\right)$ of any non-central finite group $H$ has infinite index in $G$ .",
"Suppose that $G$ acts spherically transitively on a rooted tree $\\mathcal {T}$ .",
"If there exists a subtree $v\\mathcal {T}$ whose stabilizer $\\text{Stab}_G \\left( v\\mathcal {T}\\right)$ in $G$ is not trivial, then the $*$ -homomorphism $\\rho _\\mathcal {T}$ defined on $G̏$ is not faithful.",
"Remark 5.2 The conclusion of REF clearly implies that the representation $\\rho _\\mathcal {T}$ does not weakly contain the regular representation $\\lambda $ of $G$ ; indeed $\\lambda $ always defines a faithful representation of $G̏$ .",
"However, these two conclusions are in general not equivalent.",
"Indeed, the regular representation of $ \\mathbb {Z}$ extended linearly to $$ is, via Fourier transform, given by multiplication on $L^2 \\left( S^1 \\right) $ by functions $S^1 \\ni z \\rightarrow \\sum _{i=-l}^l \\alpha _i z^i$ with $\\alpha _i \\in .",
"If one restricts this representation to $ C ( A ) $ where $ A$ is a infinite closed strict subset of $ S1$, one gets a representation which induces a faithful $ *$-homomorphism of $$ (because a non-zero function $ z i=-ll i zi$ has finitely many zeros) and which cannot weakly contain the regular (because there are some functions $ f=z i=-ll i zi$ such that $ f$ is onlyreached in $ S1 A$).$ Let $v$ be a vertex of the $n$ -th level $L_n$ of $\\mathcal {T}$ such that $\\text{Stab}_G \\left( v\\mathcal {T}\\right) $ is not trivial.",
"First note that such a group is not normal, thus non-central in $G$ .",
"Indeed, the $G$ -action on the $n$ -th level $L_n$ is transitive, hence $\\bigcap _{g\\in G} g^{-1}\\text{Stab}_G \\left( v\\mathcal {T}\\right)g=\\bigcap _{g\\in G} \\text{Stab}_G \\left( g\\left(v\\right) \\mathcal {T}\\right) =\\left\\lbrace 1\\right\\rbrace .$ Moreover, this subgroup is normalized by the stabilizer $\\text{Stab}_G (L_n)$ of the $n$ -th level.",
"The later having finite index in $G$ , our additional assumption on $G$ implies that $\\text{Stab}_G \\left( v\\mathcal {T}\\right) $ is an infinite group.",
"Therefore, it has a non-trivial intersection with the finite index subgroup $\\text{Stab}_G (L_n)$ .",
"Now, let us define for every subset $A$ of $L_n $ , the subgroup $\\text{Stab}_G \\left( L_n ; A \\mathcal {T}\\right) $ of $G$ of the elements which fixe $L_n$ as well as all the subtrees rooted at a vertex in $A$ : $\\text{Stab}_G \\left( L_n ; A \\mathcal {T}\\right) := \\text{Stab}_G \\left( L_n \\right) \\cap \\bigcap _{a \\in A} \\text{Stab}_G \\left( a \\mathcal {T}\\right) .$ The first paragraph of the proof concludes that $\\text{Stab}_G \\left( L_n ; \\left\\lbrace v\\right\\rbrace \\mathcal {T}\\right) $ is not trivial.",
"So let $A_0$ be a subset of $L_n$ such that $\\text{Stab}_G \\left( L_n ; A_0 \\mathcal {T}\\right) $ is non-trivial, and such that among the subsets of $L_n$ with this property, the cardinality of $A_0$ is maximal.",
"Denote by $\\left( A_0,A_1,\\dots ,A_N \\right) $ the orbit of $A_0$ under the $G$ -action.",
"One has: Property 5.3 $\\bigcup _{i=0}^{N} A_i = L_n$ because $G$ acts transitively on $L_n$ .",
"$\\forall i=1\\dots N,\\ \\exists \\alpha _i \\in G $ $ \\text{Stab}_G (L_n,A_{i} \\mathcal {T})=\\text{Stab}_G (L_n,\\alpha _i (A_{0}) \\mathcal {T})=\\alpha _i \\text{Stab}_G (L_n,A_0 \\mathcal {T}) \\alpha _{i}^{-1} \\ne \\lbrace 1\\rbrace .",
"$ For every $i \\ne j$ , $g_i \\in \\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right)$ and $g_j \\in \\text{Stab}_G \\left( L_n ; A_j \\mathcal {T}\\right)$ , one has $\\left[ g_i,g_j\\right]:=g_ig_jg_i^{-1}g_j^{-1} \\in \\text{Stab}_G \\left( L_n ; (A_i \\cup A_j) \\mathcal {T}\\right) =\\left\\lbrace 1 \\right\\rbrace .$ Therefore every element in $\\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right) $ commutes with every element in $\\text{Stab}_G \\left( L_n ; A_j \\mathcal {T}\\right) $ .",
"The properties (REF ) and (REF ) are clear.",
"Let us check (REF ).",
"If we write down the decomposition of $g_i$ and $g_j$ with respect to the level $L_n$ (see Section ), we get $\\Phi ^{(n)} \\left(g_i \\right)&=& \\left( *,\\dots ,*,1,*,\\dots ,1,\\dots \\right) \\\\\\Phi ^{(n)} \\left(g_j \\right)&=& \\left( *,\\dots ,1,1,*,\\dots ,*,\\dots \\right)$ where the 1s appear respectively in the positions corresponding to $ x\\in A_i$ and $ y\\in A_j$ .",
"Therefore $\\Phi ^{(n)} \\left( \\left[g_i,g_j \\right] \\right)&=& \\left( *,\\dots ,1,1,*,\\dots ,1,\\dots \\right)$ where the 1s appear in the positions corresponding to $ z\\in A_i \\cup A_j$ .",
"Thus, we conclude that $\\left[g_i,g_j \\right] $ is an element of the group $ \\text{Stab}_G \\left( L_n ; \\left( A_i \\cup A_j\\right) \\mathcal {T}\\right) $ which is by construction trivial since $|A_i \\cup A_j|>|A_0| $ .",
"Now, for all sequences $g_0,g_1,\\dots ,g_N$ with $g_i \\in \\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right) $ , we define $M(g_0,g_1,\\dots ,g_N)=\\prod _{i=0}^{N} \\left( 1-g_i \\right) \\in G̏ .$ The following two lemmas clearly imply Theorem REF .",
"Lemma 5.4 For every sequence $g_0,g_1,\\dots ,g_N$ with $g_i \\in \\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right) $ , $\\rho _{\\mathcal {T}} \\left(M(g_0,g_1,\\dots ,g_N) \\right)=0 .$ As we have already seen, the decomposition of the element $g_i \\in \\text{Aut} \\left( \\mathcal {T}\\right)$ with respect to the $n$ -th level $ L_n$ is given by $\\Phi ^{(n)} (g_i)= \\left( *,\\dots ,*,1,*,\\dots ,1,\\dots \\right)$ where the 1s appear in the positions corresponding to $w\\in E_i$ .",
"Thus, the operator $\\rho _{\\mathcal {T}}\\left( 1-g_i \\right)$ restricts to 0 on the invariant subspace $\\left\\lbrace f \\in \\ell ^2 \\left( \\mathcal {T}^0\\right) \\ | \\ \\text{supp}(f) \\subset \\bigcup _{k=0}^n L_k \\cup \\bigcup _{w\\in E_i} w \\mathcal {T}\\right\\rbrace .",
"$ Because $\\bigcup _{i=0}^{N} E_i = L_n $ (Property REF ), $\\rho _{\\mathcal {T}} \\left(M(g_0,g_1,\\dots ,g_N) \\right)=0 $ .",
"Lemma 5.5 There exists a sequence $g_0,g_1,\\dots ,g_N$ with $g_i \\in \\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right) $ such that $M(g_0,g_1,\\dots ,g_N) \\ne 0 .$ The group $\\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right) $ is non-central in $G$ and is normalized by the finite index subgroup $\\text{Stab}_G \\left( L_n \\right) $ .",
"Hence, it is infinite by assumption.",
"Two cases arise: 1st case: there exists an element $g_0 \\in \\text{Stab}_G \\left( L_n ; A_0 \\mathcal {T}\\right)$ whose order is infinite.",
"The groups $\\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right) $ being conjugated (Property REF ), we can choose $g_i \\in \\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right)$ of infinite order.",
"These elements commute (Property REF ) and thus generate a group of the form $\\mathbb {Z}^l \\oplus K $ where $K$ is a finite abelian group.",
"If $m$ is the cardinality of $K$ , we have $H \\overset{\\textrm {def}}{:=} \\left\\langle g_0 ^m, \\dots ,g_N ^m \\right\\rangle = \\mathbb {Z}^{l^{\\prime }}$ with $l^{\\prime } \\ne 0$ because, for instance, $g_0 ^m$ has infinite order.",
"It is well known that the algebra $H̏$ is a domain, thus $M(g_0 ^m,g_1 ^m,\\dots ,g_N ^m) =\\prod _{i=0}^{N} \\left( 1-g_i ^m \\right) \\ne 0 $ .",
"2nd case: the groups $\\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right)$ are periodic (and infinite).",
"Let us construct inductively a sequence $g_0,g_1,\\dots ,g_N $ such that: - for all $i$ , $g_i$ is a non trivial element of $\\text{Stab}_G \\left( L_n ; A_i \\mathcal {T}\\right)$ , - for all $i$ , $g_i \\notin \\left\\langle g_0,\\dots , g_{i-1} \\right\\rangle .$ The initial step of the induction is clear.",
"Suppose $g_0,\\dots g_i$ are already constructed, with $i<N$ .",
"The group $K=\\left\\langle g_0 , \\dots ,g_i \\right\\rangle $ is generated by torsion elements which commute (Property REF ): $K$ is a finite group and thus we can choose $g_{i+1}$ in the infinite group $\\text{Stab}_G \\left( L_n ; A_{i+1} \\mathcal {T}\\right) $ which is not in $K$ .",
"Now $M(g_0 , \\dots , g_N )=\\prod _{i=0}^{N} \\left( 1-g_i \\right) \\ne 0 $ .",
"Indeed, expanding this product, one sees that its nullity implies a relation of the form $1=g_{i_1}g_{i_2} \\dots g_{i_l} \\textrm { with } i_1<i_2< \\dots <i_l \\textrm { and } l \\textrm { odd}.$ In particular it would follow that $g_{i_l} \\in \\left\\langle g_{i_1},\\dots , g_{i_{l-1}} \\right\\rangle $ which is by construction impossible.",
"In the next example, we construct a group $G$ together with a spherically transitive action on a rooted tree $\\mathcal {T}$ with non-trivial stabilizers of subtrees.",
"These stabilizers are finite, non-central and normalized by a finite index subgroup of $G$ .",
"Thus, $G$ is excluded from the framework of Theorem REF and in fact, we will prove that the profinite representation of $G$ on $\\ell ^2 \\left( \\mathcal {T}^0 \\right) $ weakly contains the regular $\\lambda $ .",
"Example 5.6 On the finite set $X=\\left\\lbrace 1,2,3,4,5,6\\right\\rbrace $ (which will play the role of the first level $L_1$ of $\\mathcal {T}$ ), consider the permutations $\\alpha = \\left(1,3,5 \\right)\\left(2,4,6\\right), \\quad \\beta _r= \\left(1,2 \\right)\\left(3,4\\right).$ Also let $H$ be the group generated by $\\alpha $ and $\\beta _r$ ; we denote by $\\rho _1$ the permutational representation that $H$ admits on $\\ell ^2 \\left( X \\right) \\simeq 6$ via its action on $X$ .",
"Property 5.7 The group $K$ generated by $\\beta _r$ , $\\beta _l:=\\alpha \\beta \\alpha ^{-1}$ and $\\beta _m:=\\alpha ^2 \\beta \\alpha ^{-2}$ is a non-central normal subgroup in $H$ and $H = K \\rtimes \\left\\langle \\alpha \\right\\rangle = \\left(\\mathbb {Z}/2\\mathbb {Z}\\oplus \\mathbb {Z}/2\\mathbb {Z}\\right)\\rtimes \\mathbb {Z}/3\\mathbb {Z}.$ The representation $\\rho _1$ extended linearly to $H̏$ defines a faithful $*$ -homomorphism into $\\mathcal {B} \\left( \\ell ^2 \\left( X \\right) \\right) \\simeq \\text{End} \\left( 6 \\right)$ .",
"(A) One has $\\beta _l=\\alpha \\beta \\alpha ^{-1}= \\left( 3,4\\right) \\left( 5,6\\right) \\text{ and }\\beta _m=\\alpha ^2 \\beta \\alpha ^{-2}=\\left( 1,2\\right) \\left( 5,6\\right)$ so that $\\beta \\alpha \\beta \\alpha ^{-1}=\\alpha ^2 \\beta \\alpha ^{-2}$ .",
"Therefore, the group $\\left\\langle \\beta ,\\alpha \\beta \\alpha ^{-1},\\alpha ^2 \\beta \\alpha ^{-2} \\right\\rangle $ is $\\mathbb {Z}/2\\mathbb {Z}\\oplus \\mathbb {Z}/2\\mathbb {Z}$ and is normal in $H$ .",
"Thus, $H=\\left\\langle \\beta ,\\alpha \\beta \\alpha ^{-1},\\alpha ^2 \\beta \\alpha ^{-2} \\right\\rangle \\rtimes \\left\\langle \\alpha \\right\\rangle = \\left(\\mathbb {Z}/2\\mathbb {Z}\\oplus \\mathbb {Z}/2\\mathbb {Z}\\right)\\rtimes \\mathbb {Z}/3\\mathbb {Z}.$ (B) We want to prove that every irreducible representation of $H$ appears in $\\rho _1$ .",
"Let us first compute the character $\\tau $ of the representation $\\rho _1$ .",
"Here, the map $\\tau $ sends a permutation in $H$ to the cardinality of its fixed points set: $\\tau \\left( h\\right) = \\left\\lbrace \\begin{array}{ll}6 & \\text{if $h=1$}\\\\2 & \\text{if $h=\\beta _l,\\beta _m$ or $\\beta _r$}\\\\0 & \\text{otherwise.",
"}\\end{array} \\right.$ Let $\\pi $ be an irreducible representation of $H$ whose character is denoted by $\\psi $ .",
"We want to prove that $\\left\\langle \\tau ,\\psi \\right\\rangle $ is non zero, i.e.",
"$\\frac{1}{|H|} \\sum _{h \\in H} \\tau \\left( h \\right)\\psi \\left( h \\right) \\ne 0.$ Thanks to the above computation, this is equivalent to $6 \\psi \\left( 1\\right) + 2\\left( \\psi \\left( \\beta _r\\right)+\\psi \\left( \\beta _m\\right)+\\psi \\left( \\beta _l\\right) \\right) \\ne 0.$ Since the $\\beta _*$ are conjugate in $G$ , their image under $\\pi $ have the same trace.",
"Thus, we want to show (whatever $*$ is) that one of the following three equivalent statements is true: $6 \\left( \\psi \\left( 1\\right) +\\psi \\left( \\beta _*\\right) \\right)\\ne 0 &\\Longleftrightarrow & \\dim \\pi +\\psi \\left( \\beta _*\\right)\\ne 0 \\\\&\\Longleftrightarrow & \\pi \\left( \\beta _*\\right) \\ne -1.$ Now the last assertion is clear because $\\beta _r \\beta _l=\\beta _m$ and it is impossible for the $\\pi \\left(\\beta _* \\right)$ 's to all equal $-1$ .",
"Next consider the action of the group $\\mathbb {Z}$ on $\\mathcal {T}_{\\tilde{2}}:=\\mathcal {T}_{2,2,\\dots ,2,\\dots }$ given by the sequence of subgroups $\\mathcal {S} = \\left(2\\mathbb {Z},4\\mathbb {Z},\\dots ,2^n\\mathbb {Z},\\dots \\right)$ .",
"This action is generated by a single element; let $s \\in \\text{Aut}\\left(\\mathcal {T}_{\\tilde{2}} \\right)$ be this generator.",
"Before continuing our construction, we remark that Theorem REF .REF and the remark REF implies that $\\rho _{\\mathcal {S}}$ weakly contains the regular representation of $\\mathbb {Z}$ .",
"The group $\\mathbb {Z}$ is amenable, hence these two representations are weakly isomorphic i.e.",
"$C^{*}_{\\rho _{\\mathcal {S}}}\\left( \\mathbb {Z}\\right)=C^{*}_{\\rho _{\\mathcal {T}_{\\tilde{2}}}} \\left( \\mathbb {Z}\\right)\\simeq C^{*}_{\\lambda } \\left( \\mathbb {Z}\\right).$ We now consider the rooted tree $\\mathcal {T}_{6,2,2,\\dots ,2,\\dots }$ whose first level $L_1$ is the set $X$ on which $H$ acts, and whose subtrees rooted at $L_1$ are all $\\mathcal {T}_{\\tilde{2}} $ on which $\\mathbb {Z}$ acts.",
"Definition 5.8 The group $G$ is the subgroup of $\\text{Aut} \\left(\\mathcal {T}_{6,2,2,\\dots ,2,\\dots } \\right)$ generated by the elements $\\bar{\\alpha }$ , $\\bar{\\beta _r} $ and $ \\bar{s} $ defined via the recursion map (see Section 2): $\\Phi (\\bar{\\alpha })&=&\\left(1,1,1,1,1,1 \\right) \\alpha , \\\\\\Phi (\\bar{\\beta _r})&=&\\left(1,1,1,1,1,1 \\right) \\beta _r, \\\\\\Phi (\\bar{s})&=&\\left(s,s,s,s,s,s \\right).$ The subgroup $\\left\\langle \\bar{\\alpha },\\bar{\\beta _r} \\right\\rangle $ is isomorphic to $H$ and the element $\\bar{s} $ generates a copy of $\\mathbb {Z}$ in $G$ which clearly commutes with $H$ .",
"Thus, $G=H \\oplus \\mathbb {Z}.$ The group $G$ acts spherically transitively on $\\mathcal {T}_{6,2,2,\\dots }$ because $H$ acts transitively on its first level $L_1$ , and by construction $\\mathbb {Z}$ acts spherically transitively on all the subtrees $\\mathcal {T}_{\\tilde{2}}$ rooted at it.",
"Moreover, the stabilizer of such a subtree is not trivial; for instance, the two right-most one (issued from the vertices labelled by 5 and 6) are fixed under the action of $\\bar{\\beta _r} $ .",
"The next proposition shows that the sufficient condition in Theorem REF .REF is not necessary and explains the additional algebraic assumption in REF .",
"Proposition 5.9 The representation $\\rho _{\\mathcal {T}_{6,2,2,\\dots }}$ of $G$ on $\\ell ^2 \\left(\\mathcal {T}_{6,2,2,\\dots }^0 \\right)$ is weakly isomorphic to its regular $\\lambda $ .",
"To simplify notation, let us write $\\rho $ for $\\rho _{\\mathcal {T}_{6,2,2,\\dots }}$ .",
"Maybe the easiest way to prove REF is to use the language of $C^*$ -algebras.",
"We want to prove that $C^{*}_{\\rho }\\left( G \\right) = C^{*}_{\\lambda } \\left( G \\right).$ We have $C^{*}_{\\lambda }\\left( G \\right)&=&C^{*}_{\\lambda }\\left(\\mathbb {Z}\\right)\\otimes C^{*}_{\\lambda }\\left(H\\right)\\quad \\text{by (\\ref {Gdecomp})}\\\\&=&C^{*}_{\\lambda }\\left(\\mathbb {Z}\\right)\\otimes H̏ \\quad \\text{because $H$ is finite.",
"}$ Thus Proposition REF is a consequence of (REF ) and the following lemma.",
"Lemma 5.10 $C^{*}_{\\rho }\\left( G \\right) = C^{*}_{\\rho _{\\mathcal {T}_{\\bar{2}}}} \\left( \\mathbb {Z}\\right) \\otimes H̏$ Consider the restriction $\\rho ^{\\prime } $ of $\\rho $ to the invariant subspace $\\mathcal {H} $ of $\\ell ^2 \\left(\\mathcal {T}_{6,2,2,\\dots }^0 \\right)$ consisting of functions null at the root of $\\mathcal {T}_{6,2,2,\\dots } $ : $\\mathcal {H} &=& \\left\\lbrace f \\in \\ell ^2 \\left(\\mathcal {T}_{6,2,2,\\dots }^0 \\right) \\ |\\ \\text{supp} (f) \\subset \\bigcup _{v \\in L_1} v\\mathcal {T}_{6,2,2,\\dots } \\right\\rbrace \\\\&=& \\bigoplus _{v \\in L_1} \\ell ^2 \\left(\\mathcal {T}_{\\tilde{2}}^0 \\right).$ The above restriction amounts to the removal of a copy of $\\epsilon $ (the trivial representation) from $\\rho $ .",
"But $\\rho ^{\\prime }$ still contains $\\epsilon $ since the constant functions on a level $L_n$ belong to $\\ell ^2 \\left( L_n \\right) \\subset \\mathcal {H}$ .",
"Therefore $C^{*}_{\\rho }\\left( G \\right) = C^{*}_{\\rho ^{\\prime }}\\left( G \\right).$ Now, the decomposition (REF ) of $\\mathcal {H} $ implies that $C^{*}_{\\rho ^{\\prime }}\\left( G \\right) $ is a subalgebra of $\\mathcal {A} \\otimes \\text{End} \\left(6 \\right)$ where $\\mathcal {A}$ is the $C^*$ -algebra generated by the restrictions $\\varphi _v \\left( g\\right) \\in \\mathcal {U} \\left( \\ell ^2 \\left(\\mathcal {T}_{\\tilde{2}}^0 \\right)\\right)$ with $v\\in L_1$ and $g\\in G$ .",
"By Definition REF of the group $G$ , the algebra $\\mathcal {A}$ is generated by $ s$ and hence is $ C^{*}_{\\rho _{\\mathcal {T}_{\\tilde{2}}}} \\left( \\mathbb {Z}\\right) $ .",
"Moreover, the only elements in $G$ acting non-trivially on $L_1$ (i.e.",
"inducing a non-trivial permutation of the factors $\\ell ^2 \\left(\\mathcal {T}_{\\tilde{2}}^0 \\right) $ in $\\mathcal {H} $ ) are the elements of the subgroup $H$ .",
"Summarizing, we have $C^{*}_{\\rho ^{\\prime }}\\left( G \\right)= C^{*}_{\\rho _{\\mathcal {T}_{\\tilde{2}}}} \\left( \\mathbb {Z}\\right) \\otimes \\rho _1 \\left( H̏ \\right)$ where $\\rho _1$ is the permutational representation of $H$ on $\\ell ^2 \\left( X \\right)$ defined at the beginning of this paragraph, i.e.",
"the restriction of $\\rho $ to $\\ell ^2 \\left( L_1 \\right)$ .",
"Lemma REF is now a consequence of Property REF .REF ."
],
[
"Examples and applications",
"In this last section, we want to illustrate the results of the previous two.",
"In particular, we will see that the sufficient condition of REF .REF is automatically fullfilled for many lattices in semisimple real Lie groups, independently on the choice of a sequence $\\mathcal {S}$ defining a faithful representation $\\rho _{\\mathcal {S}}$ .",
"We will also see that for weakly branched groups, the situation is diametrically different."
],
[
"Higher $\\mathbb {R}$ -rank case.",
"The following proposition is a direct consequence of the Margulis Normal Subgroup Theorem [16], [17], [24].",
"Proposition 6.1 Let $G$ be a connected semisimple real Lie group with finite centre, no compact factor and $\\mathbb {R}$ -rank $\\ge 2$ .",
"Let $\\Gamma $ be an irreducible lattice in $G$ acting faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ .",
"Then for all subtrees $v\\mathcal {T}$ of $\\mathcal {T}$ , the stabilizer $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)$ is trivial.",
"Let $v$ be a vertex of $\\mathcal {T}$ .",
"The group $\\text{Stab}_{\\Gamma } \\left(v \\right)$ has finite index in $\\Gamma $ , hence is also an irreducible lattice in $G$ .",
"The stabilizer $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ of the subtree rooted at $v$ is a normal subgroup of $\\text{Stab}_{\\Gamma } \\left( v \\right)$ .",
"Moreover, $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ has infinite index in $\\Gamma $ (because $\\Gamma $ acts spherically transitively on the subtree $v\\mathcal {T}$ ).",
"Therefore it has infinite index in $\\text{Stab}_{\\Gamma } \\left( v \\right)$ .",
"By the Margulis Normal Subgroup Theorem, $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ is central in $G$ and thus in $\\Gamma $ .",
"From this follows that $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ is trivial, because $\\Gamma $ acts transitively on $L_n$ so that $\\bigcap _{\\gamma \\in \\Gamma } \\gamma ^{-1}\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)\\gamma =\\left\\lbrace 1\\right\\rbrace .$"
],
[
"Hyperbolic groups and $\\mathbb {R}$ -rank=1 case.",
"The first result of this paragraph deals with Gromov hyperbolic groups.",
"We refer to [12], [10] for details and proofs of the general facts which we use in the proof below.",
"Proposition 6.2 Let $\\Gamma $ be a Gromov hyperbolic group.",
"Assume that $\\Gamma $ is non-elementary and acts faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ .",
"Then $\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_{\\Gamma } (v\\mathcal {T})$ is finite.",
"Remark 6.3 This result is optimal.",
"Indeed, in Example REF , replace the 6 subtrees rooted at the first level on which $\\mathbb {Z}$ acts diagonally by 6 rooted trees $\\mathcal {T}_{\\bar{d}}$ on which a residually finite hyperbolic group $\\Gamma $ acts faihfully and spherically transitively.",
"The subgroup of $\\text{Aut}\\left( \\mathcal {T} _{6,\\bar{d}}\\right)$ that we get is then $H \\times \\Gamma $ and is hyperbolic because $H$ is finite.",
"Here again, by construction, its action on $\\mathcal {T} _{6,\\bar{d}} $ is faithful and spherically transitive, but the subgroup $H$ fixes the subtrees rooted at the two right-most vertices of the first level.",
"First, we prove that for every vertex $v\\in \\mathcal {T}$ , the stabilizer $\\text{Stab}_{\\Gamma } (v\\mathcal {T})$ is finite.",
"If this is not the case, being a subgroup of a hyperbolic group, $\\text{Stab}_{\\Gamma } (v\\mathcal {T})$ contains an element of infinite order $\\gamma $ .",
"Let $n$ be the level of $v$ , i.e.",
"$v \\in L_n$ ; replacing $\\gamma $ by the non trivial element $\\gamma ^{N !",
"}$ where $N=\\left| L_n \\right|$ , we can assume also that $\\gamma $ fixes the $n$ -th level $L_n$ .",
"Once again, we consider for every subset $\\mathcal {A}$ of $L_n$ $\\text{Stab}_{\\Gamma } \\left( L_n ; A \\mathcal {T}\\right) := \\text{Stab}_{\\Gamma } \\left( L_n \\right) \\cap \\bigcap _{a \\in A} \\text{Stab}_{\\Gamma } \\left( a \\mathcal {T}\\right) .$ Above, we showed that $\\text{Stab}_{\\Gamma } \\left( L_n ; \\left\\lbrace v \\right\\rbrace \\mathcal {T}\\right)$ is not trivial, and even contains a infinite order element.",
"So consider $\\mathcal {A}_{max}$ a subset of maximal cardinality for which $\\text{Stab}_{\\Gamma } \\left( L_n ; \\mathcal {A}_{max} \\mathcal {T}\\right)$ contains an infinite order element, say $\\gamma $ .",
"Of course, $\\mathcal {A}_{max}$ is a strict subset of $L_n$ .",
"Hence, let us choose $v \\in \\mathcal {A}_{max} $ , $w \\in L_n \\setminus \\mathcal {A}_{max}$ , and $\\sigma \\in \\Gamma $ that such $\\sigma (v)=w $ (this is possible because the action of $\\Gamma $ on $L_n$ is assumed transitive).",
"Lemma 6.4 There exists $h\\in \\text{Stab}_{\\Gamma } \\left( L_n \\right)$ and integer $k$ such that $\\left[\\gamma ^k, h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\right]:= \\gamma ^k h \\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\gamma ^{-k} h \\sigma \\gamma ^{-k} \\sigma ^{-1} h^{-1}$ has infinite order.",
"It is sufficient to prove that there is an $h \\in \\text{Stab}_{\\Gamma } \\left( L_n \\right)$ such that $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right)\\cap \\text{Fix}_{\\partial \\Gamma } \\left(h\\sigma \\gamma \\sigma ^{-1} h^{-1}\\right)=\\emptyset $ where $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right)=\\left\\lbrace \\gamma ^{+\\infty },\\gamma ^{-\\infty } \\right\\rbrace $ and $\\text{Fix}_{\\partial \\Gamma } \\left( h\\sigma \\gamma \\sigma ^{-1} h^{-1} \\right)=\\left\\lbrace h\\sigma (\\gamma ^{+\\infty }),h\\sigma (\\gamma ^{-\\infty }) \\right\\rbrace $ are the pairs of fixed points of $\\gamma $ and $ h\\sigma \\gamma \\sigma ^{-1} h^{-1} $ in the boundary $\\partial \\Gamma $ of $\\Gamma $ .",
"Indeed, it is known that in that case, the group generated by $\\gamma ^k$ and $ \\left(h\\sigma \\gamma \\sigma ^{-1} h^{-1}\\right)^k =h\\sigma \\gamma ^k\\sigma ^{-1} h^{-1} $ is the free group $\\mathbb {F} _2$ , as soon as $k$ is big enough.",
"If $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right)\\cap \\text{Fix}_{\\partial \\Gamma } \\left(\\sigma \\gamma \\sigma ^{-1}\\right)=\\emptyset $ , we take $h=1$ .",
"If this is not the case, we want to prove that $\\text{Stab}_{\\Gamma } \\left( L_n \\right)$ is not included in the subgroup $P$ of $G$ consisted of the elements preserving $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right) $ .",
"This is easy because this last group contains $\\left\\langle \\gamma \\right\\rangle $ as a finite index subgroup, whereas $\\text{Stab}_{\\Gamma } \\left( L_n \\right)$ has finite index in $\\Gamma $ : as $\\Gamma $ is non-elementary, $\\text{Stab}_{\\Gamma } \\left( L_n \\right)$ is not amenable and thus cannot be a subgroup of $P$ (which is quasi-isometric to $\\mathbb {Z}$ ).",
"Fix $h$ and $k$ like in the statement of the previous lemma.",
"Since $\\gamma ^k $ belongs to $ \\text{Stab}_{\\Gamma } \\left( L_n ; \\mathcal {A}_{max} \\mathcal {T}\\right)$ and $h$ is in $\\text{Stab}_{\\Gamma } ( L_n)$ , the element $h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1}$ belongs to the group $ \\text{Stab}_{\\Gamma } \\left( L_n ; \\sigma (\\mathcal {A}_{max}) \\mathcal {T}\\right)$ .",
"If we write down the decomposition of these two elements with respect to the level $L_n$ , we get $\\Phi ^{(n)} \\left(\\gamma ^k \\right)&=& \\left( *,\\dots ,*,1,*,\\dots ,1,\\dots \\right) \\\\\\Phi ^{(n)} \\left( h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1}\\right)&=& \\left( *,\\dots ,1,1,*,\\dots ,*,\\dots \\right)$ where the 1s appear respectively in the positions corresponding to $ x\\in \\mathcal {A}_{max}$ and $ y\\in \\sigma \\left(\\mathcal {A}_{max}\\right)$ .",
"Therefore $\\Phi ^{(n)} \\left( \\left[\\gamma ^k, h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\right] \\right)&=& \\left( *,\\dots ,1,1,*,\\dots ,1,\\dots \\right)$ where the 1s appear in the positions corresponding to $ z\\in \\mathcal {A}_{max} \\cup \\sigma (\\mathcal {A}_{max})$ .",
"Lemma REF then implies that $\\left[\\gamma ^k, h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\right] $ is an infinite order element of the group $ \\text{Stab}_{\\Gamma } \\left( L_n ;\\mathcal {B} \\mathcal {T}\\right) $ with $\\mathcal {B} =\\mathcal {A}_{max} \\cup \\sigma (\\mathcal {A}_{max})$ .",
"This contradicts the maximality of $\\left|\\mathcal {A}_{max} \\right| $ because $w \\in \\sigma (\\mathcal {A}_{max})\\setminus \\mathcal {A}_{max}$ and so $\\left|\\mathcal {A}_{max} \\right|<\\left|\\mathcal {B} \\right| $ .",
"We just proved that all the stabilizers $\\text{Stab}_G\\left(v\\mathcal {T}\\right)$ are finite.",
"To complete the proof of REF , recall that in a hyperbolic group, there are only finitely many conjugacy classes of finite group.",
"Hence, there exists a non-negative integer $N$ such that $&\\forall k \\in \\mathbb {N}, \\ \\forall v \\in L_{N+k},\\ \\exists g\\in G,\\ i\\le N \\text{ and } w \\in L_i \\text{ such that } & \\\\&\\text{Stab}_{\\Gamma } (v\\mathcal {T}) = g \\text{Stab}_{\\Gamma } (w\\mathcal {T}) g^{-1}.&$ As $ g \\text{Stab}_{\\Gamma } (w\\mathcal {T}) g^{-1} = \\text{Stab}_{\\Gamma } \\left( g(w) \\mathcal {T}\\right) $ , we conclude that $\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_{\\Gamma } (v\\mathcal {T}) = \\bigcup _{i=1}^{N} \\bigcup _{v \\in L_i} \\text{Stab}_{\\Gamma } (v\\mathcal {T})$ is finite.",
"As a direct consequence of this proposition and Theorem REF .REF , we obtain: Corollary 6.5 Let $\\Gamma $ be non-elementary, residually finite hyperbolic group.",
"Let $\\mathcal {S}$ be a decreasing sequence of finite index subgroups of $\\Gamma $ such that the representation $\\rho _{\\mathcal {S}}$ is faithful.",
"Then, there exists a positive integer $n$ such that the representation $\\rho _{\\mathcal {S}}^{\\otimes n}$ of $\\Gamma $ weakly contains the regular $\\lambda $ .",
"The next application shows how to use Proposition REF to get information on the spectrum of Schreier graphs.",
"Corollary 6.6 Let $\\Gamma $ be non-elementary, torsion free, residually finite hyperbolic group.",
"Let $\\mathcal {S} = \\left( H_n\\right)_n $ be a decreasing sequence of finite index subgroups of $\\Gamma $ such that the representation $\\rho _{\\mathcal {S}}$ is faithful.",
"Let $M_F := \\frac{1}{2|F|}\\sum _{g\\in F} g+g^{-1}$ be the Markov operator associated to a finite generating set $F$ of $\\Gamma $ which does not contain 1.",
"Then $\\left[ -\\frac{1}{|F|}, \\frac{\\sqrt{2|F|-1}}{|F|}\\right] \\subset \\overline{\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right)} .$ As $M$ is self-adjoint, the closure of $\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right) $ equals the spectrum of $\\rho _{\\mathcal {S}}\\left( M \\right)$ and is a compact subset of $\\mathbb {R}$ .",
"Now, the assumption that $\\Gamma $ is torsion free, Proposition REF and Theorem REF .REF together imply that $\\rho _{\\mathcal {S}}$ weakly contains the regular representation $\\lambda $ of $\\Gamma $ .",
"In particular, $\\text{sp}\\left( \\lambda \\left( M_F\\right)\\right) \\subset \\text{sp} \\left( \\rho _{\\mathcal {S}}\\left(M_F \\right) \\right)=\\overline{\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right)}.$ Thanks to Proposition 5 in [7], we know that $\\text{sp}\\left( \\lambda \\left( M\\right)\\right)$ contains a pair $\\left\\lbrace m, M \\right\\rbrace $ with $m\\le -\\frac{1}{|F|} $ and $M \\ge \\frac{\\sqrt{2|F|-1}}{|F|}$ .",
"Since the Baum–Connes conjecture is true for hyperbolic groups [15], [18] and $\\Gamma $ is torsion free, it fullfills the Kadison-Kaplansky conjecture.",
"Therefore, the spectrum $\\text{sp}\\left( \\lambda \\left( M\\right)\\right) $ is connected.",
"This proves Corollary REF .",
"We conclude with an analogous result to Proposition REF , in the case of uniform lattices in $\\mathbb {R}$ -rank 1 Lie groups.",
"Corollary 6.7 Let $\\Gamma $ be a uniform lattice in a connected simple real Lie group $G$ with finite center and $\\mathbb {R}$ -rank 1.",
"Assume that $\\Gamma $ acts faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ .",
"Then for all subtrees $v\\mathcal {T}$ of $\\mathcal {T}$ , the stabilizer $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)$ is trivial and thus the representation $\\rho _{\\mathcal {T}}$ of $\\Gamma $ weakly contains the regular $\\lambda $ .",
"Such a uniform lattice is hyperbolic.",
"Let $v$ be a vertex of $\\mathcal {T}$ .",
"Proposition REF implies that the stabilizer $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) $ is a finite group.",
"Therefore, its normalizer $N_G \\left(\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) \\right)$ in $G$ is Zariski closed.",
"Moreover, $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)$ is a normal subgroup of $\\text{Stab}_{\\Gamma } \\left( v \\right)$ ; the later, having finite index in $\\Gamma $ , is also a lattice in $G$ .",
"By the Borel Density Theorem, the Zariski closure of $\\text{Stab}_{\\Gamma } \\left( v \\right)$ is $G$ and therefore $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) $ is a finite normal subgroup of $G$ .",
"Hence, $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) $ has to be central in $G$ , thus in $\\Gamma $ .",
"Finally, $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) = \\bigcap _{\\gamma \\in \\Gamma } \\gamma \\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)\\gamma ^{-1}=\\left\\lbrace 1 \\right\\rbrace ,$ the last equality coming from the transitivity of the $\\Gamma $ -action on $L_n$ .",
"Theorem REF .REF applies."
],
[
"Weakly branched groups",
"This last section deals with natural examples of couple $\\left(G,\\mathcal {T}\\right) $ where $G$ is a finitely generated group acting faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ for which the conditions in Theorem REF .REF and REF .REF are ”far” from being true.",
"These examples come from the class of weakly branched groups.",
"Definition 6.8 Let $\\mathcal {T}$ be a regular rooted tree, which means that $\\mathcal {T}= \\mathcal {T}_{\\tilde{d}}$ where $\\tilde{d}$ is the constant sequence $\\tilde{d}=d,d,\\dots ,d,\\dots $ with $d>1$ .",
"A finitely generated subgroup $G$ of $\\text{Aut}\\left(\\mathcal {T} \\right)$ is said weakly branched if its action on $\\mathcal {T}$ is spherically transitive and $\\forall v\\in \\mathcal {T}^0, \\ \\text{Rist}_G \\left(v\\right) := \\bigcap _{w\\in L_n \\setminus \\left\\lbrace v\\right\\rbrace } \\text{Stab}_G \\left( w\\mathcal {T}\\right)$ is non-trivial.",
"The subgroup $\\text{Rist}_G \\left(v\\right) $ consists of elements which only act on the subtree $v\\mathcal {T}$ .",
"It is called the rigid stabilizer of $v$ .",
"It is easy to see that if $G$ is weakly branched, these rigid stabilizers are infinite.",
"So a fortiori are the stabilizers $\\text{Stab}_G \\left(v\\mathcal {T}\\right)$ of subtrees.",
"We refer to [25], [3] for a survey on weakly branched groups.",
"Concerning the link between $\\rho _{\\mathcal {T}}$ and $\\lambda $ for such a weakly branched group, one has: Proposition 6.9 Let $G$ be a weakly branched subgroup of $\\text{Aut}\\left(\\mathcal {T} \\right)$ .",
"For every $n>0$ , the $*$ -homomorphism $\\rho _{\\mathcal {T}}^{\\otimes n}$ defined on $G̏$ is not faithful.",
"In particular, the representation $\\rho _{\\mathcal {T}}^{\\otimes n}$ does not weakly contain the regular $\\lambda $ .",
"The method is the same that the one we used for the proof of Theorem REF .",
"First, we see that remark REF implies that we need only to prove the proposition for integers of the form $d^n -1$ .",
"For every vertex $v$ in the $n$ -th level $L_n$ of $\\mathcal {T}$ , choose $g_v$ a non-trivial element in the infinite group $\\text{Rist}_G \\left(w\\right)$ .",
"We have $\\Phi ^{(n)}\\left(g_v \\right) = \\left(1,1,\\dots ,1,\\varphi _v \\left(g_v\\right),1,\\dots \\right)$ where the non-trivial element $\\varphi _v \\left( g_v \\right)$ appears in the position corresponding to $v$ .",
"It is clear that the $g_v$ 's commute.",
"Let $M=\\prod _{v\\in L_n} \\left( 1- g_v \\right) \\in G̏.$ Then, $M \\ne 0$ .",
"Indeed, the nullity of $M$ would imply the existence of a subset $\\mathcal {A}$ of $L_n$ (of odd cardinality) such that $1=\\prod _{v\\in \\mathcal {A}} g_v,$ and this is impossible because, if $w$ is any vertex in $\\mathcal {A}$ , (REF ) implies that $\\varphi _w \\left(\\prod _{v\\in \\mathcal {A}} g_v \\right)=\\varphi _w \\left(g_w \\right)\\ne 1.$ Let us show that $\\rho _{\\mathcal {T}}^{\\otimes d^n -1} \\left( M \\right)$ is 0.",
"This is equivalent to proving that for every $n$ -tuple $\\left(z_1,\\dots ,z_{d^n -1} \\right)$ consisting of elements in $\\mathcal {T}^0$ , $\\rho _{\\mathcal {T}}^{\\otimes d^n -1} \\left( M \\right) \\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right)=0 $ There is necessarily a vertex $v_0$ among the $d^n$ in $L_n$ such that the subtree $v_0\\mathcal {T}$ does not contain any $z_i$ .",
"By construction, $\\rho _{\\mathcal {T}}^{\\otimes d^n -1} \\left(1-g_{v_0} \\right)\\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right)=0$ As the $g_v$ 's commute, $M=\\left( \\prod _{v \\in L_n \\setminus \\left\\lbrace v_0\\right\\rbrace } \\left( 1-g_v\\right) \\right) (1-g_{v_0})$ and this implies $\\rho _{\\mathcal {T}}^{\\otimes d^n -1}\\left( M \\right) \\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right) =0 $ ."
],
[
"Examples and applications",
"In this last section, we want to illustrate the results of the previous two.",
"In particular, we will see that the sufficient condition of REF .REF is automatically fullfilled for many lattices in semisimple real Lie groups, independently on the choice of a sequence $\\mathcal {S}$ defining a faithful representation $\\rho _{\\mathcal {S}}$ .",
"We will also see that for weakly branched groups, the situation is diametrically different."
],
[
"Higher $\\mathbb {R}$ -rank case.",
"The following proposition is a direct consequence of the Margulis Normal Subgroup Theorem [16], [17], [24].",
"Proposition 6.1 Let $G$ be a connected semisimple real Lie group with finite centre, no compact factor and $\\mathbb {R}$ -rank $\\ge 2$ .",
"Let $\\Gamma $ be an irreducible lattice in $G$ acting faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ .",
"Then for all subtrees $v\\mathcal {T}$ of $\\mathcal {T}$ , the stabilizer $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)$ is trivial.",
"Let $v$ be a vertex of $\\mathcal {T}$ .",
"The group $\\text{Stab}_{\\Gamma } \\left(v \\right)$ has finite index in $\\Gamma $ , hence is also an irreducible lattice in $G$ .",
"The stabilizer $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ of the subtree rooted at $v$ is a normal subgroup of $\\text{Stab}_{\\Gamma } \\left( v \\right)$ .",
"Moreover, $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ has infinite index in $\\Gamma $ (because $\\Gamma $ acts spherically transitively on the subtree $v\\mathcal {T}$ ).",
"Therefore it has infinite index in $\\text{Stab}_{\\Gamma } \\left( v \\right)$ .",
"By the Margulis Normal Subgroup Theorem, $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ is central in $G$ and thus in $\\Gamma $ .",
"From this follows that $\\text{Stab}_{\\Gamma }\\left( v\\mathcal {T}\\right)$ is trivial, because $\\Gamma $ acts transitively on $L_n$ so that $\\bigcap _{\\gamma \\in \\Gamma } \\gamma ^{-1}\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)\\gamma =\\left\\lbrace 1\\right\\rbrace .$"
],
[
"Hyperbolic groups and $\\mathbb {R}$ -rank=1 case.",
"The first result of this paragraph deals with Gromov hyperbolic groups.",
"We refer to [12], [10] for details and proofs of the general facts which we use in the proof below.",
"Proposition 6.2 Let $\\Gamma $ be a Gromov hyperbolic group.",
"Assume that $\\Gamma $ is non-elementary and acts faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ .",
"Then $\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_{\\Gamma } (v\\mathcal {T})$ is finite.",
"Remark 6.3 This result is optimal.",
"Indeed, in Example REF , replace the 6 subtrees rooted at the first level on which $\\mathbb {Z}$ acts diagonally by 6 rooted trees $\\mathcal {T}_{\\bar{d}}$ on which a residually finite hyperbolic group $\\Gamma $ acts faihfully and spherically transitively.",
"The subgroup of $\\text{Aut}\\left( \\mathcal {T} _{6,\\bar{d}}\\right)$ that we get is then $H \\times \\Gamma $ and is hyperbolic because $H$ is finite.",
"Here again, by construction, its action on $\\mathcal {T} _{6,\\bar{d}} $ is faithful and spherically transitive, but the subgroup $H$ fixes the subtrees rooted at the two right-most vertices of the first level.",
"First, we prove that for every vertex $v\\in \\mathcal {T}$ , the stabilizer $\\text{Stab}_{\\Gamma } (v\\mathcal {T})$ is finite.",
"If this is not the case, being a subgroup of a hyperbolic group, $\\text{Stab}_{\\Gamma } (v\\mathcal {T})$ contains an element of infinite order $\\gamma $ .",
"Let $n$ be the level of $v$ , i.e.",
"$v \\in L_n$ ; replacing $\\gamma $ by the non trivial element $\\gamma ^{N !",
"}$ where $N=\\left| L_n \\right|$ , we can assume also that $\\gamma $ fixes the $n$ -th level $L_n$ .",
"Once again, we consider for every subset $\\mathcal {A}$ of $L_n$ $\\text{Stab}_{\\Gamma } \\left( L_n ; A \\mathcal {T}\\right) := \\text{Stab}_{\\Gamma } \\left( L_n \\right) \\cap \\bigcap _{a \\in A} \\text{Stab}_{\\Gamma } \\left( a \\mathcal {T}\\right) .$ Above, we showed that $\\text{Stab}_{\\Gamma } \\left( L_n ; \\left\\lbrace v \\right\\rbrace \\mathcal {T}\\right)$ is not trivial, and even contains a infinite order element.",
"So consider $\\mathcal {A}_{max}$ a subset of maximal cardinality for which $\\text{Stab}_{\\Gamma } \\left( L_n ; \\mathcal {A}_{max} \\mathcal {T}\\right)$ contains an infinite order element, say $\\gamma $ .",
"Of course, $\\mathcal {A}_{max}$ is a strict subset of $L_n$ .",
"Hence, let us choose $v \\in \\mathcal {A}_{max} $ , $w \\in L_n \\setminus \\mathcal {A}_{max}$ , and $\\sigma \\in \\Gamma $ that such $\\sigma (v)=w $ (this is possible because the action of $\\Gamma $ on $L_n$ is assumed transitive).",
"Lemma 6.4 There exists $h\\in \\text{Stab}_{\\Gamma } \\left( L_n \\right)$ and integer $k$ such that $\\left[\\gamma ^k, h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\right]:= \\gamma ^k h \\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\gamma ^{-k} h \\sigma \\gamma ^{-k} \\sigma ^{-1} h^{-1}$ has infinite order.",
"It is sufficient to prove that there is an $h \\in \\text{Stab}_{\\Gamma } \\left( L_n \\right)$ such that $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right)\\cap \\text{Fix}_{\\partial \\Gamma } \\left(h\\sigma \\gamma \\sigma ^{-1} h^{-1}\\right)=\\emptyset $ where $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right)=\\left\\lbrace \\gamma ^{+\\infty },\\gamma ^{-\\infty } \\right\\rbrace $ and $\\text{Fix}_{\\partial \\Gamma } \\left( h\\sigma \\gamma \\sigma ^{-1} h^{-1} \\right)=\\left\\lbrace h\\sigma (\\gamma ^{+\\infty }),h\\sigma (\\gamma ^{-\\infty }) \\right\\rbrace $ are the pairs of fixed points of $\\gamma $ and $ h\\sigma \\gamma \\sigma ^{-1} h^{-1} $ in the boundary $\\partial \\Gamma $ of $\\Gamma $ .",
"Indeed, it is known that in that case, the group generated by $\\gamma ^k$ and $ \\left(h\\sigma \\gamma \\sigma ^{-1} h^{-1}\\right)^k =h\\sigma \\gamma ^k\\sigma ^{-1} h^{-1} $ is the free group $\\mathbb {F} _2$ , as soon as $k$ is big enough.",
"If $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right)\\cap \\text{Fix}_{\\partial \\Gamma } \\left(\\sigma \\gamma \\sigma ^{-1}\\right)=\\emptyset $ , we take $h=1$ .",
"If this is not the case, we want to prove that $\\text{Stab}_{\\Gamma } \\left( L_n \\right)$ is not included in the subgroup $P$ of $G$ consisted of the elements preserving $\\text{Fix}_{\\partial \\Gamma } \\left(\\gamma \\right) $ .",
"This is easy because this last group contains $\\left\\langle \\gamma \\right\\rangle $ as a finite index subgroup, whereas $\\text{Stab}_{\\Gamma } \\left( L_n \\right)$ has finite index in $\\Gamma $ : as $\\Gamma $ is non-elementary, $\\text{Stab}_{\\Gamma } \\left( L_n \\right)$ is not amenable and thus cannot be a subgroup of $P$ (which is quasi-isometric to $\\mathbb {Z}$ ).",
"Fix $h$ and $k$ like in the statement of the previous lemma.",
"Since $\\gamma ^k $ belongs to $ \\text{Stab}_{\\Gamma } \\left( L_n ; \\mathcal {A}_{max} \\mathcal {T}\\right)$ and $h$ is in $\\text{Stab}_{\\Gamma } ( L_n)$ , the element $h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1}$ belongs to the group $ \\text{Stab}_{\\Gamma } \\left( L_n ; \\sigma (\\mathcal {A}_{max}) \\mathcal {T}\\right)$ .",
"If we write down the decomposition of these two elements with respect to the level $L_n$ , we get $\\Phi ^{(n)} \\left(\\gamma ^k \\right)&=& \\left( *,\\dots ,*,1,*,\\dots ,1,\\dots \\right) \\\\\\Phi ^{(n)} \\left( h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1}\\right)&=& \\left( *,\\dots ,1,1,*,\\dots ,*,\\dots \\right)$ where the 1s appear respectively in the positions corresponding to $ x\\in \\mathcal {A}_{max}$ and $ y\\in \\sigma \\left(\\mathcal {A}_{max}\\right)$ .",
"Therefore $\\Phi ^{(n)} \\left( \\left[\\gamma ^k, h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\right] \\right)&=& \\left( *,\\dots ,1,1,*,\\dots ,1,\\dots \\right)$ where the 1s appear in the positions corresponding to $ z\\in \\mathcal {A}_{max} \\cup \\sigma (\\mathcal {A}_{max})$ .",
"Lemma REF then implies that $\\left[\\gamma ^k, h\\sigma \\gamma ^k \\sigma ^{-1} h^{-1} \\right] $ is an infinite order element of the group $ \\text{Stab}_{\\Gamma } \\left( L_n ;\\mathcal {B} \\mathcal {T}\\right) $ with $\\mathcal {B} =\\mathcal {A}_{max} \\cup \\sigma (\\mathcal {A}_{max})$ .",
"This contradicts the maximality of $\\left|\\mathcal {A}_{max} \\right| $ because $w \\in \\sigma (\\mathcal {A}_{max})\\setminus \\mathcal {A}_{max}$ and so $\\left|\\mathcal {A}_{max} \\right|<\\left|\\mathcal {B} \\right| $ .",
"We just proved that all the stabilizers $\\text{Stab}_G\\left(v\\mathcal {T}\\right)$ are finite.",
"To complete the proof of REF , recall that in a hyperbolic group, there are only finitely many conjugacy classes of finite group.",
"Hence, there exists a non-negative integer $N$ such that $&\\forall k \\in \\mathbb {N}, \\ \\forall v \\in L_{N+k},\\ \\exists g\\in G,\\ i\\le N \\text{ and } w \\in L_i \\text{ such that } & \\\\&\\text{Stab}_{\\Gamma } (v\\mathcal {T}) = g \\text{Stab}_{\\Gamma } (w\\mathcal {T}) g^{-1}.&$ As $ g \\text{Stab}_{\\Gamma } (w\\mathcal {T}) g^{-1} = \\text{Stab}_{\\Gamma } \\left( g(w) \\mathcal {T}\\right) $ , we conclude that $\\bigcup _{v\\in \\mathcal {T}^0} \\text{Stab}_{\\Gamma } (v\\mathcal {T}) = \\bigcup _{i=1}^{N} \\bigcup _{v \\in L_i} \\text{Stab}_{\\Gamma } (v\\mathcal {T})$ is finite.",
"As a direct consequence of this proposition and Theorem REF .REF , we obtain: Corollary 6.5 Let $\\Gamma $ be non-elementary, residually finite hyperbolic group.",
"Let $\\mathcal {S}$ be a decreasing sequence of finite index subgroups of $\\Gamma $ such that the representation $\\rho _{\\mathcal {S}}$ is faithful.",
"Then, there exists a positive integer $n$ such that the representation $\\rho _{\\mathcal {S}}^{\\otimes n}$ of $\\Gamma $ weakly contains the regular $\\lambda $ .",
"The next application shows how to use Proposition REF to get information on the spectrum of Schreier graphs.",
"Corollary 6.6 Let $\\Gamma $ be non-elementary, torsion free, residually finite hyperbolic group.",
"Let $\\mathcal {S} = \\left( H_n\\right)_n $ be a decreasing sequence of finite index subgroups of $\\Gamma $ such that the representation $\\rho _{\\mathcal {S}}$ is faithful.",
"Let $M_F := \\frac{1}{2|F|}\\sum _{g\\in F} g+g^{-1}$ be the Markov operator associated to a finite generating set $F$ of $\\Gamma $ which does not contain 1.",
"Then $\\left[ -\\frac{1}{|F|}, \\frac{\\sqrt{2|F|-1}}{|F|}\\right] \\subset \\overline{\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right)} .$ As $M$ is self-adjoint, the closure of $\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right) $ equals the spectrum of $\\rho _{\\mathcal {S}}\\left( M \\right)$ and is a compact subset of $\\mathbb {R}$ .",
"Now, the assumption that $\\Gamma $ is torsion free, Proposition REF and Theorem REF .REF together imply that $\\rho _{\\mathcal {S}}$ weakly contains the regular representation $\\lambda $ of $\\Gamma $ .",
"In particular, $\\text{sp}\\left( \\lambda \\left( M_F\\right)\\right) \\subset \\text{sp} \\left( \\rho _{\\mathcal {S}}\\left(M_F \\right) \\right)=\\overline{\\bigcup _{n} \\text{sp} \\left( \\lambda _{\\Gamma /H_n}\\left(M_F \\right) \\right)}.$ Thanks to Proposition 5 in [7], we know that $\\text{sp}\\left( \\lambda \\left( M\\right)\\right)$ contains a pair $\\left\\lbrace m, M \\right\\rbrace $ with $m\\le -\\frac{1}{|F|} $ and $M \\ge \\frac{\\sqrt{2|F|-1}}{|F|}$ .",
"Since the Baum–Connes conjecture is true for hyperbolic groups [15], [18] and $\\Gamma $ is torsion free, it fullfills the Kadison-Kaplansky conjecture.",
"Therefore, the spectrum $\\text{sp}\\left( \\lambda \\left( M\\right)\\right) $ is connected.",
"This proves Corollary REF .",
"We conclude with an analogous result to Proposition REF , in the case of uniform lattices in $\\mathbb {R}$ -rank 1 Lie groups.",
"Corollary 6.7 Let $\\Gamma $ be a uniform lattice in a connected simple real Lie group $G$ with finite center and $\\mathbb {R}$ -rank 1.",
"Assume that $\\Gamma $ acts faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ .",
"Then for all subtrees $v\\mathcal {T}$ of $\\mathcal {T}$ , the stabilizer $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)$ is trivial and thus the representation $\\rho _{\\mathcal {T}}$ of $\\Gamma $ weakly contains the regular $\\lambda $ .",
"Such a uniform lattice is hyperbolic.",
"Let $v$ be a vertex of $\\mathcal {T}$ .",
"Proposition REF implies that the stabilizer $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) $ is a finite group.",
"Therefore, its normalizer $N_G \\left(\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) \\right)$ in $G$ is Zariski closed.",
"Moreover, $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)$ is a normal subgroup of $\\text{Stab}_{\\Gamma } \\left( v \\right)$ ; the later, having finite index in $\\Gamma $ , is also a lattice in $G$ .",
"By the Borel Density Theorem, the Zariski closure of $\\text{Stab}_{\\Gamma } \\left( v \\right)$ is $G$ and therefore $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) $ is a finite normal subgroup of $G$ .",
"Hence, $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) $ has to be central in $G$ , thus in $\\Gamma $ .",
"Finally, $\\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right) = \\bigcap _{\\gamma \\in \\Gamma } \\gamma \\text{Stab}_{\\Gamma } \\left( v\\mathcal {T}\\right)\\gamma ^{-1}=\\left\\lbrace 1 \\right\\rbrace ,$ the last equality coming from the transitivity of the $\\Gamma $ -action on $L_n$ .",
"Theorem REF .REF applies."
],
[
"Weakly branched groups",
"This last section deals with natural examples of couple $\\left(G,\\mathcal {T}\\right) $ where $G$ is a finitely generated group acting faithfully and spherically transitively on a rooted tree $\\mathcal {T}$ for which the conditions in Theorem REF .REF and REF .REF are ”far” from being true.",
"These examples come from the class of weakly branched groups.",
"Definition 6.8 Let $\\mathcal {T}$ be a regular rooted tree, which means that $\\mathcal {T}= \\mathcal {T}_{\\tilde{d}}$ where $\\tilde{d}$ is the constant sequence $\\tilde{d}=d,d,\\dots ,d,\\dots $ with $d>1$ .",
"A finitely generated subgroup $G$ of $\\text{Aut}\\left(\\mathcal {T} \\right)$ is said weakly branched if its action on $\\mathcal {T}$ is spherically transitive and $\\forall v\\in \\mathcal {T}^0, \\ \\text{Rist}_G \\left(v\\right) := \\bigcap _{w\\in L_n \\setminus \\left\\lbrace v\\right\\rbrace } \\text{Stab}_G \\left( w\\mathcal {T}\\right)$ is non-trivial.",
"The subgroup $\\text{Rist}_G \\left(v\\right) $ consists of elements which only act on the subtree $v\\mathcal {T}$ .",
"It is called the rigid stabilizer of $v$ .",
"It is easy to see that if $G$ is weakly branched, these rigid stabilizers are infinite.",
"So a fortiori are the stabilizers $\\text{Stab}_G \\left(v\\mathcal {T}\\right)$ of subtrees.",
"We refer to [25], [3] for a survey on weakly branched groups.",
"Concerning the link between $\\rho _{\\mathcal {T}}$ and $\\lambda $ for such a weakly branched group, one has: Proposition 6.9 Let $G$ be a weakly branched subgroup of $\\text{Aut}\\left(\\mathcal {T} \\right)$ .",
"For every $n>0$ , the $*$ -homomorphism $\\rho _{\\mathcal {T}}^{\\otimes n}$ defined on $G̏$ is not faithful.",
"In particular, the representation $\\rho _{\\mathcal {T}}^{\\otimes n}$ does not weakly contain the regular $\\lambda $ .",
"The method is the same that the one we used for the proof of Theorem REF .",
"First, we see that remark REF implies that we need only to prove the proposition for integers of the form $d^n -1$ .",
"For every vertex $v$ in the $n$ -th level $L_n$ of $\\mathcal {T}$ , choose $g_v$ a non-trivial element in the infinite group $\\text{Rist}_G \\left(w\\right)$ .",
"We have $\\Phi ^{(n)}\\left(g_v \\right) = \\left(1,1,\\dots ,1,\\varphi _v \\left(g_v\\right),1,\\dots \\right)$ where the non-trivial element $\\varphi _v \\left( g_v \\right)$ appears in the position corresponding to $v$ .",
"It is clear that the $g_v$ 's commute.",
"Let $M=\\prod _{v\\in L_n} \\left( 1- g_v \\right) \\in G̏.$ Then, $M \\ne 0$ .",
"Indeed, the nullity of $M$ would imply the existence of a subset $\\mathcal {A}$ of $L_n$ (of odd cardinality) such that $1=\\prod _{v\\in \\mathcal {A}} g_v,$ and this is impossible because, if $w$ is any vertex in $\\mathcal {A}$ , (REF ) implies that $\\varphi _w \\left(\\prod _{v\\in \\mathcal {A}} g_v \\right)=\\varphi _w \\left(g_w \\right)\\ne 1.$ Let us show that $\\rho _{\\mathcal {T}}^{\\otimes d^n -1} \\left( M \\right)$ is 0.",
"This is equivalent to proving that for every $n$ -tuple $\\left(z_1,\\dots ,z_{d^n -1} \\right)$ consisting of elements in $\\mathcal {T}^0$ , $\\rho _{\\mathcal {T}}^{\\otimes d^n -1} \\left( M \\right) \\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right)=0 $ There is necessarily a vertex $v_0$ among the $d^n$ in $L_n$ such that the subtree $v_0\\mathcal {T}$ does not contain any $z_i$ .",
"By construction, $\\rho _{\\mathcal {T}}^{\\otimes d^n -1} \\left(1-g_{v_0} \\right)\\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right)=0$ As the $g_v$ 's commute, $M=\\left( \\prod _{v \\in L_n \\setminus \\left\\lbrace v_0\\right\\rbrace } \\left( 1-g_v\\right) \\right) (1-g_{v_0})$ and this implies $\\rho _{\\mathcal {T}}^{\\otimes d^n -1}\\left( M \\right) \\left(\\delta _{z_1}\\otimes \\dots \\otimes \\delta _{z_{d^n -1}} \\right) =0 $ ."
]
] | 1204.1514 |
[
[
"Geometric Heat Flux for Classical Thermal Transport in Interacting Open\n Systems"
],
[
"Abstract We study classical heat conduction in a dissipative open system composed of interacting oscillators.",
"By exactly solving a twisted Fokker-Planck equation which describes the full counting statistics of heat flux flowing through the system, we identify the geometric-phase-like effect and examine its impact on the classical heat transport.",
"Particularly, we find that the nonlinear interaction as well as the closely related temperature-dependence of system-parameters are crucial in manifesting the geometric-phase contribution of heat flux.",
"Finally, we propose an electronic experiment based on RC circuits to verify our theoretical predictions."
],
[
"Supplementary Material for “Geometric Heat Flux of Classical Thermal Transport in Interacting Open Systems”",
"In this supplement, we are going to (1) analytically solve the eigen-problem of the twisted Fokker-Planck equation, (2) expose the condition of so-called “adiabatic”, (3) detail the derivations of geometric-phase effect in generating functions, which finally leads to the geometric heat flux.",
"Transport behaviors in the long time limit are of our central interest.",
"They are governed by the ground state of the twisted Fokker-Planck operator $L_{\\chi }$ .",
"We thus need first exactly solve the eigen-problem of the twisted Fokker-Planck equation, with time-independent $L_{\\chi }$ : $\\partial _t z=L_{\\chi }z,$ with $L_{\\chi }=k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)\\frac{\\partial ^2}{\\partial y^2}+k\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{2k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)y\\frac{\\partial }{\\partial y}+k^2\\left(\\frac{k_BT_1}{\\gamma _1}\\left(\\mathbf {i}\\chi \\right)^2-\\frac{1}{\\gamma _1}\\mathbf {i}\\chi \\right)y^2+k\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right).$ Note that in the main text, we set $k_B=1$ for the sake of clarity.",
"We make an ansatz of the solution: $z(y,\\chi ,t)=\\sum _{n=0}^{\\infty }C_ne^{\\lambda _nt}e^{-\\frac{y^2}{4T}}f_n(y),$ where $C_n$ is the coefficient depending on initial conditions, $\\lambda _n$ denotes the $n$ -th eigenvalue of $L_{\\chi }$ , $e^{-\\frac{y^2}{4T}}f_n(y)$ is the corresponding eigenfunction, and $T$ is a parameter to be determined in follows.",
"Put this ansatz into the twisted Fokker-Planck equation, and utilize the following relations: $\\frac{\\partial }{\\partial y}e^{\\frac{-y^2}{4T}}f_n(y)&=&-\\frac{y}{2T}e^{\\frac{-y^2}{4T}}f_n(y)+e^{\\frac{-y^2}{4T}}\\frac{\\partial f_n(y)}{\\partial y},\\\\\\frac{\\partial ^2}{\\partial y^2}e^{\\frac{-y^2}{4T}}f_n(y)&=&-\\frac{1}{2T}e^{\\frac{-y^2}{4T}}f_n(y)+\\frac{y^2}{4T^2}e^{\\frac{-y^2}{4T}}f_n(y)-\\frac{y}{2T}e^{\\frac{-y^2}{4T}}\\frac{\\partial f_n(y)}{\\partial y}-\\frac{y}{2T}e^{\\frac{-y^2}{4T}}\\frac{\\partial f_n(y)}{\\partial y}+e^{\\frac{-y^2}{4T}}\\frac{\\partial ^2f_n(y)}{\\partial y^2},$ we then have $\\bigg \\lbrace k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)\\frac{\\partial ^2}{\\partial y^2}+\\left[k\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{2k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)-k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)\\frac{1}{T}\\right]y\\frac{\\partial }{\\partial y}\\\\+\\left[\\frac{k_B}{4T^2}\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)-\\frac{k}{2T}\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{2k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)+k^2\\left(\\frac{k_BT_1}{\\gamma _1}\\left(\\mathbf {i}\\chi \\right)^2-\\frac{1}{\\gamma _1}\\mathbf {i}\\chi \\right)\\right]y^2+\\\\k\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)-\\frac{k_B}{2T}\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)-\\lambda _n\\bigg \\rbrace f_n(y)=0.$ To eliminate the term related to $y\\partial _y$ , we found that one needs to set $T=\\frac{k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)}{k\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{2k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)}.$ Thus, the above equation reduces to: $\\left\\lbrace k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)\\frac{\\partial ^2}{\\partial y^2}+\\bigg [-\\frac{k^2\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{2k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)^2}{4k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)}+k^2\\left(\\frac{k_BT_1}{\\gamma _1}\\left(\\mathbf {i}\\chi \\right)^2-\\frac{1}{\\gamma _1}\\mathbf {i}\\chi \\right)\\bigg ]y^2+\\frac{k}{2}\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}\\right)-\\lambda _n\\right\\rbrace f_n(y)=0.$ Let us set $r_{1,2}=k/\\gamma _{1,2}$ , $D_{1,2}=k_BT_{1,2}/\\gamma _{1,2}$ , then we have: $\\left\\lbrace \\left(D_1+D_2\\right)\\frac{\\partial ^2}{\\partial y^2}+\\left[-\\frac{\\left(r_1+r_2-2\\mathbf {i}\\chi kD_1\\right)^2}{4\\left(D_1+D_2\\right)}+\\left(\\mathbf {i}\\chi \\right)^2k^2D_1-\\mathbf {i}\\chi kr_1\\right]y^2+\\frac{r_1+r_2}{2}-\\lambda _n\\right\\rbrace f_n(y)=0,$ $\\left\\lbrace \\left(D_1+D_2\\right)\\frac{\\partial ^2}{\\partial y^2}-\\frac{\\left(r_1+r_2\\right)^2}{4\\left(D_1+D_2\\right)}\\left[1+\\frac{4k\\mathbf {i}\\chi }{\\left(r_1+r_2\\right)^2}\\left(D_2r_1-D_1r_2-\\mathbf {i}\\chi kD_1D_2\\right)\\right]y^2+\\frac{r_1+r_2}{2}-\\lambda _n\\right\\rbrace f_n(y)=0.$ Then set $r=r_1+r_2$ , $D=D_1+D_2$ , $\\theta =\\sqrt{1+\\frac{4k^2D_1D_2\\chi }{r^2}(\\mathbf {i}(1/T_2-1/T_2)-\\chi )}$ , we can simplify the above equation as $\\left\\lbrace D\\frac{\\partial ^2}{\\partial y^2}-\\frac{r^2}{4D}\\theta ^2y^2+\\frac{r}{2}-\\lambda _n\\right\\rbrace f_n(y)=0$ We further set $Y=\\sqrt{\\frac{r\\theta }{2D}} y$ and $F_n(Y)=f_n(y)$ , which finally leads to the eigen-problem of quantum harmonic oscillator's Schrödinger's equation: $\\bigg \\lbrace \\frac{\\partial ^2}{\\partial Y^2}-Y^2+\\frac{1}{\\theta }(1-\\frac{2\\lambda _n}{r})\\bigg \\rbrace F_n(Y)=0,$ This equation requires $\\frac{1}{\\theta }\\left(1-\\frac{2\\lambda _n}{r}\\right)=1+2n$ , with $n=0,1,2,\\ldots $ .",
"From any textbook of quantum mechanics which solves the Schrödinger's equation of quantum harmonic oscillator, we know the eigenvalue: $\\lambda _n=\\frac{r}{2}\\left[1-\\left(1+2n\\right)\\theta \\right],$ and the eigenfunction $f_n(y)=e^{-\\frac{r\\theta }{4D}y^2}H_n(\\sqrt{\\frac{r\\theta }{2D}}y),$ where $H_n$ denotes the $n$ -th order Hermite polynomials.",
"Since we already have $T=D/(r-2\\mathbf {i}\\chi kD_1)$ , as a consequence, the ansatz of the solution now reads: $z(y,\\chi ,t)=\\sum _{n=0}^{\\infty }C_ne^{\\lambda _nt}e^{-\\frac{r(1+\\theta )-2\\mathbf {i}\\chi kD_1}{4D}y^2}H_n(\\sqrt{\\frac{r\\theta }{2D}}y)$ Substitute it back to the twisted Fokker-Planck equation $\\partial _t z=L_{\\chi }z$ , we have $L_{\\chi }e^{-\\frac{r(1+\\theta )-2\\mathbf {i}\\chi kD_1}{4D}y^2}H_n(\\sqrt{\\frac{r\\theta }{2D}}y)=\\lambda _ne^{-\\frac{r(1+\\theta )-2i\\chi kD_1}{4D}y^2}H_n(\\sqrt{\\frac{r\\theta }{2D}}y),$ such that $\\lambda _n$ is indeed the eigenvalue of the operator $L_{\\chi }$ , the right eigenfunction for the operator $L_{\\chi }$ is, $\\psi _n(y,\\chi )=e^{-\\frac{r(1+\\theta )-2\\mathbf {i}\\chi kD_1}{4D}y^2}H_n(\\sqrt{\\frac{r\\theta }{2D}}y).$ Straightforwardly, the corresponding left eigenfunction, which is bi-orthonormal with the right one, reads, $\\varphi _n(y,\\chi )=\\frac{1}{2^nn!",
"}\\sqrt{\\frac{r\\theta }{2\\pi D}}e^{\\frac{r(1-\\theta )-2\\mathbf {i}\\chi kD_1}{4D}y^2}H_n(\\sqrt{\\frac{r\\theta }{2D}}y),$ This left eigenfunction of $L_{\\chi }$ , is just the right eigenfunction of the adjoint operator $L_{\\chi }^+$ , which has the property [38]: $L_{\\chi }^+\\varphi _n(y,\\chi )=\\lambda _n\\varphi _n(y,\\chi ),$ corresponding to $L_{\\chi }\\psi _n(y,\\chi )=\\lambda _n\\psi _n(y,\\chi ),$ so that we have the scalar product $\\int ^{+\\infty }_{-\\infty }dy\\varphi _m(y,\\chi )L_{\\chi }\\psi _n(y,\\chi )=\\int ^{+\\infty }_{-\\infty }dy L_{\\chi }^+\\varphi _m(y,\\chi )\\psi _n(y,\\chi )=\\int ^{+\\infty }_{-\\infty }dy \\psi _n(y,\\chi )L_{\\chi }^+\\varphi _m(y,\\chi ).$ We also may thus normalize the functions according to $\\int ^{+\\infty }_{-\\infty }dy\\varphi _m(y,\\chi )\\psi _n(y,\\chi )=\\delta _{mn}$ Under the adjoint operation “+” We note here that, by the definition in this work, the adjoint operation “+” is similar to the transpose operation in matrix, but not the conjugate transpose., one can easily find that $y\\rightarrow y, \\frac{\\partial }{\\partial y}\\rightarrow -\\frac{\\partial }{\\partial y},y\\frac{\\partial }{\\partial y}\\rightarrow -\\frac{\\partial }{\\partial y}y=-\\textbf {1}-y\\frac{\\partial }{\\partial y}.$ Therefore, from Eq.",
"(19), we have $L_{\\chi }^+=k_B\\left(\\frac{T_1}{\\gamma _1}+\\frac{T_2}{\\gamma _2}\\right)\\frac{\\partial ^2}{\\partial y^2}-k\\left(\\frac{1}{\\gamma _1}+\\frac{1}{\\gamma _2}-\\frac{2k_BT_1}{\\gamma _1}\\mathbf {i}\\chi \\right)y\\frac{\\partial }{\\partial y}+k^2\\left(\\frac{k_BT_1}{\\gamma _1}\\left(\\mathbf {i}\\chi \\right)^2-\\frac{1}{\\gamma _1}\\mathbf {i}\\chi \\right)y^2+k\\frac{k_BT_1}{\\gamma _1}\\mathbf {i}\\chi .$ Then, following the similar procedure, one is able to arrive at the left eigenfunction Eq.",
"(35).",
"Behaviors in the long time limit are of our central interest.",
"They are governed by the ground state of the twisted Fokker-Planck operator $L_{\\chi }$ , of which the eigenvalue, $\\lambda _0(\\chi )$ , possesses the least negative real part.",
"In other words, for time-independent $L_{\\chi }$ , $\\lim _{t\\rightarrow \\infty }Z(\\chi ,t)\\sim e^{\\lambda _0(\\chi )t}$ and in turn $\\lim _{t\\rightarrow \\infty }\\langle \\langle Q^n\\rangle \\rangle /t=\\partial ^n_{\\mathbf {i}\\chi } \\lambda _0(\\chi )|_{\\chi =0}$ .",
"If the system is under time-dependent modulation such that the Fokker-Planck operator is time-dependent $L_{\\chi }(t)$ , then as discussed in the main text, the full counting statistics rely on the instantaneous right and left ground-state $\\psi _0(y,\\chi ,t)$ and $\\varphi _0(y,\\chi ,t)$ .",
"Therefore, we only needs the information about the ground state.",
"From Eqs.",
"(30, 34, 35), we finally have the instantaneous ground state: $&\\lambda _0(\\chi ,t)=\\frac{r}{2}\\left(1-\\theta \\right)=\\frac{1}{2}\\left(\\frac{k}{\\gamma _1}+\\frac{k}{\\gamma _2}-\\sqrt{(\\frac{k}{\\gamma _1}+\\frac{k}{\\gamma _2})^2+\\mathbf {i}\\chi \\frac{4k^2}{\\gamma _1\\gamma _2}\\big (k_BT_2-k_BT_1-\\mathbf {i}\\chi k_B^2T_1T_2\\big )}\\right), \\\\&\\psi _0(y,\\chi ,t)=\\exp \\left({-\\frac{r+r\\theta -2\\mathbf {i}\\chi kD_1}{4D}y^2}\\right), \\quad \\varphi _0(y,\\chi ,t)=\\sqrt{\\frac{r\\theta }{2\\pi D}}\\exp \\left({\\frac{r-r\\theta -2\\mathbf {i}\\chi kD_1}{4D}y^2}\\right).$ Similar solutions in a Fokker-Planck equation without the counting parameter $\\chi $ were discussed in [38], [39].",
"Similar results about the eigenvalue and the right eigenfunction, but with the left eigenfunction absent, were given by [40]."
],
[
"Condition for “adiabatic”",
"From Eq.",
"(3), we write down the first three term for the sake of clarity: $z(y,\\chi ,t)&=&\\widetilde{C}_0e^{\\lambda _0t}+\\widetilde{C}_1e^{\\lambda _1t}+\\widetilde{C}_2e^{\\lambda _2t}+... \\nonumber \\\\&=&\\widetilde{C}_0e^{\\lambda _0t}\\left(1+\\frac{\\widetilde{C}_1}{\\widetilde{C}_0}e^{-(\\lambda _0-\\lambda _1)t}+\\frac{\\widetilde{C}_2}{\\widetilde{C}_0}e^{-(\\lambda _0-\\lambda _2)t}+...\\right),$ where $\\widetilde{C}_n$ are some unimportant coefficients here.",
"Clearly, $1/(\\lambda _0(\\chi )-\\lambda _n(\\chi ))$ depicts the characteristic relaxation time of the $n$ -th mode to the ground state.",
"Let us set the counting parameter $\\chi =0$ to look into spectrum of the operator $L_{\\chi =0}$ in the real physical space.",
"From Eq.",
"(30), we can see that the spectrum is ordered by $n$ and the eigenvalue of the ground state $\\lambda _0(\\chi =0)=0$ .",
"This is apparent, since it corresponds to the long time steady state of the evolution.",
"So, the rate of relaxation to the steady state is determined by the inverse energy gap $1/(\\lambda _0(\\chi =0)-\\lambda _1(\\chi =0))$ .",
"As a consequence, the system's characteristic time scale of relaxation is given as $T_c=\\left.\\frac{1}{\\lambda _0(\\chi )-\\lambda _1(\\chi )}\\right|_{\\chi =0}=\\frac{\\gamma _1\\gamma _2}{k(\\gamma _1+\\gamma _2)},$ where $\\lambda _1(\\chi =0)=k/\\gamma _1+k/\\gamma _2$ has the physics meaning of the effective damping rate.",
"Therefore, given the system, which is already in its steady state after a long time evolution, as long as we drive it slowly such that the driving period $T_p\\gg T_c$ , the system can always dwell in its steady state (ground state).",
"In other words, $T_p\\gg T_c$ is right the so-called “adiabatic” condition."
],
[
"Geometric phase contribution in cumulant generating functions",
"We assume the system is already at its steady state, or say ground state.",
"And then, time-dependent modulations are imposed on the system adiabatically.",
"Thus, we can make the ansatz $z(y,\\chi ,t)=C_0(t)e^{\\int _0^t\\lambda _0(\\chi ,t^{\\prime })dt^{\\prime }}\\psi _0(y,\\chi ,t)$ , and substitute it into the time-dependent twisted Fokker-Planck equation $\\partial _t z(y,\\chi ,t)=L_{\\chi }(t)z(y,\\chi ,t)$ .",
"We then obtain $\\dot{C}_0(t)e^{\\int _0^t\\lambda _0(\\chi ,t^{\\prime })dt^{\\prime }}\\psi _0(y,\\chi ,t)+C_0(t)e^{\\int _0^t\\lambda _0(\\chi ,t^{\\prime })dt^{\\prime }}\\dot{\\psi }_0(y,\\chi ,t)=0.$ Utilizing the bi-orthonormal condition $\\int ^{\\infty }_{-\\infty } dy\\varphi _m(y,\\chi ,t)\\psi _n(y,\\chi ,t)=\\delta _{mn}$ , we left multiply $\\varphi _0$ , and do integral over $y$ , such that: $\\dot{C}_0(t)=-C_0(t)\\int ^{\\infty }_{-\\infty }dy\\varphi _0(y,\\chi ,t)\\dot{\\psi }_0(y,\\chi ,t).$ Therefore, in the adiabatic limit, we obtain $C_0(t)=C_0(0)e^{-\\int _0^tdt^{\\prime }\\int ^{\\infty }_{-\\infty }dy\\varphi _0(y,\\chi ,t^{\\prime })\\dot{\\psi }_0(y,\\chi ,t^{\\prime })}.$ Consequently, the characteristic function finally reads $Z(\\chi ,t)= \\left[\\int ^{\\infty }_{-\\infty } dy C_0(0)\\psi _0(y,\\chi ,t)\\right]\\exp \\left[{\\int _0^t\\lambda _0(\\chi ,t^{\\prime })dt^{\\prime }}\\right]\\exp \\left[{-\\int _0^tdt^{\\prime }\\int ^{\\infty }_{-\\infty } dy\\varphi _0(y,\\chi ,t^{\\prime })\\dot{\\psi }_0(y,\\chi ,t^{\\prime })}\\right].$ The first exponent, the time integral of the instantaneous ground-state eigenvalue, is an analog of the dynamic phase.",
"While the second additional exponent resulting from the time-evolving of ground eignstates is an analog of the geometric phase.",
"Different from the conventional phase of the wave function in quantum mechanics, here the “phase” refers to the cumulant generating function in the exponent of the characteristic function, which will contribute to the full counting statistics of the quantities of interest.",
"Similar geometric phase contribution to the generating function so as to the full counting statistics is firstly discovered by Sinitsyn and Nemenman in discrete chemical kinetics [18].",
"Successively, the cumulant generating function of adiabatically driven systems can be separated into two parts – the dynamic phase contribution and the geometric phase contribution: $G(\\chi )=\\lim _{t\\rightarrow \\infty }\\frac{1}{t}\\ln Z(\\chi ,t)=G_{\\mathrm {dyn}}+G_{\\mathrm {geom}}$ , with $G_{\\mathrm {dyn}}&=\\frac{1}{T_p}\\int ^{T_p}_0dt\\lambda _0(\\chi ,t), \\\\G_{\\mathrm {geom}}&=-\\frac{1}{T_p}\\int ^{T_p}_0dt\\int ^{\\infty }_{-\\infty }dy\\varphi _0(y,\\chi ,t)\\dot{\\psi }_0(y,\\chi ,t).$ Note that here the contribution of initial conditions $t^{-1}\\ln \\big [\\int ^{\\infty }_{-\\infty } dyC_0(0)\\psi _0(y,\\chi ,t)\\big ]$ is assumed negligible in the long time limit.",
"The dynamic phase contribution survives whenever system parameters are static, or experience single or multiple modulations.",
"While the existence of the geometric phase contribution requires at least two parameter modulations.",
"For the case of periodically driving $(u_1(t), u_2(t))$ , which could be chosen from $k, \\gamma _j, T_j$ , using Stokes theorem, we have $G_{\\mathrm {geom}}=-\\frac{1}{T_p}\\iint _{u_1u_2}du_1du_2\\mathcal {F}_{u_1u_2}(\\chi ),$ where the subscript $u_1u_2$ denotes the integral area enclosed by the modulating contour of $(u_1(t), u_2(t))$ , and $\\mathcal {F}_{u_1u_2}(\\chi )\\equiv \\int ^{\\infty }_{-\\infty } dy \\left[\\frac{\\partial \\varphi _0}{\\partial u_1}\\frac{\\partial \\psi _0}{\\partial u_2}-\\frac{\\partial \\varphi _0}{\\partial u_2}\\frac{\\partial \\psi _0}{\\partial u_1}\\right]$ is a classical analog of the quantum mechanical Berry curvature.",
"Different from the curvature usually defined for discrete Hilbert space of matrix-like Hamiltonian, the curvature here is in the continuous function space, indicated by the integral over infinity.",
"Clearly, the curvature $\\mathcal {F}_{u_1u_2}(\\chi )$ is also of pure geometric origin, since it is independent of the modulation speed."
]
] | 1204.0859 |
[
[
"Continuous Markov Random Fields for Robust Stereo Estimation"
],
[
"Abstract In this paper we present a novel slanted-plane MRF model which reasons jointly about occlusion boundaries as well as depth.",
"We formulate the problem as the one of inference in a hybrid MRF composed of both continuous (i.e., slanted 3D planes) and discrete (i.e., occlusion boundaries) random variables.",
"This allows us to define potentials encoding the ownership of the pixels that compose the boundary between segments, as well as potentials encoding which junctions are physically possible.",
"Our approach outperforms the state-of-the-art on Middlebury high resolution imagery as well as in the more challenging KITTI dataset, while being more efficient than existing slanted plane MRF-based methods, taking on average 2 minutes to perform inference on high resolution imagery."
],
[
"Introduction",
"Over the past few decades we have witnessed a great improvement in performance of stereo algorithms.",
"Most modern approaches frame the problem as inference on a Markov random field (MRF) and utilize global optimization techniques such as graph cuts or message passing [3] to reason jointly about the depth of each pixel in the image.",
"A leading approach to stereo vision uses slanted-plane MRF models which were introduced a decade ago in [4].",
"Most methods [5], [6], [7], [8], [9], [10] assume a fixed set of superpixels on a reference image, say the left image of the stereo pair, and model the surface under each superpixel as a slanted plane.",
"The MRF typically has a robust data term scoring the assigned plane in terms of a matching score induced by the plane on the pixels contained in the superpixel.",
"This data term often incorporates an explicit treatment of occlusion — pixels in one image that have no corresponding pixel in the other image [11], [12], [8], [13].",
"Slanted-plane models also typically include a robust smoothness term expressing the belief that the planes assigned to adjacent superpixels should be similar.",
"A major issue with slanted-plane stereo models is their computational complexity.",
"For example, [13] reports an average of approximately one hour of computation for each low-resolution Middebury stereo pair.",
"This makes these approaches impractical for applications such as robotics or autonomous driving.",
"A main source of difficulty is that each plane is defined by three continuous parameters and inference for continuous MRFs with non-convex energies is computationally challenging.",
"This paper contains two contributions.",
"First, we introduce the use of junction potentials, described below, into this class of models.",
"Second, we show that particle methods can achieve strong performance with reasonable inference times on the high-resolution, in-the-wild KITTI dataset [2].",
"Junction potentials originate in early line labeling algorithms [14], [15].",
"These algorithms assign labels to the lines of a line drawing where the label indicate whether the line represents a discontinuity due to changes in depth (an occlusion), surface orientation (a corner), lighting (a shadow) or albedo (paint).",
"A junction is a place where three lines meet.",
"Only certain combinations of labels are physically realizable at junctions.",
"The constraints on label combinations at junctions often force the labeling of the entire line drawing [14].",
"Here, as in recent work on monocular image interpretation [16], [17], [18], we label the boundaries between image segments rather than the lines of a line drawing with labels –“left occlusion”, “right occlusion”, “hinge” or “coplanar”.",
"In our model the occlusion labels play a role in the data term where they are interpreted as expressing ownership of the pixels that compose the boundary between segments — an occlusion boundary is “owned” by the foreground object.",
"Our second contribution is to show that particle methods can be used to implement high performance inference in high resolution imagery with reasonable running time.",
"Particle methods avoid premature commitment to any fixed quantization of continuous variables and hence allow a precise exploration of the continuous space.",
"Our particle inference method is based on the recently developed particle convex belief propagation (PCBP) [19].",
"We learn the contribution of each potential via the primal-dual optimization framework of [20].",
"In the remainder of the paper we first review related work.",
"We then introduce our continuous MRF model for stereo and show how to do learning and inference in this model.",
"Finally, we demonstrate the effectiveness of our approach in estimating depth from stereo pairs and show that it outperforms the state-of-the-art in the high resolution Middlebury imagery [1] as well as in the more challenging KITTI dataset [2]."
],
[
"Related Work",
"In the past few years much progress has been made towards solving the stereo problem, as evidenced by Scharstein et al.",
"overview [21].",
"Local methods typically aggregate image statistics in a small window, thus imposing smoothness implicitly.",
"Optimization is usually performed using a winner-takes-all strategy, which selects for each pixel the disparity with the smallest value under some distance metric [21].",
"Traditional local methods [22] often suffer from border bleeding effects or struggle with correspondence ambiguities.",
"Approaches based on adaptive support windows [23], [24] adjust their computations locally to improve performance, especially close to border discontinuities.",
"This results in better performance at the price of more computation.",
"Hirschmüller proposed semi-global matching [25], an approach which extends polynomial time 1D scan-line methods to propagate information along 16 orientations.",
"This reduces streaking artifacts and improves accuracy compared to traditional methods.",
"In this paper we employ this technique to compute a disparity map from which we build our potentials.",
"In [26], [27] disparities are `grown' from a small set of initial correspondence seeds.",
"Though these methods produce accurate results and can be faster than global approaches, they do not provide dense matching and struggle with textureless and distorted image areas.",
"Approaches to reduce the search space have been investigated for global stereo methods [28], [29] as well as local methods [30].",
"Dense and accurate matching can be obtained by global methods, which enforce smoothness explicitly by minimizing an MRF-based energy function.",
"These MRFs can be formulated at the pixel level [31], however, the smoothness is then defined very locally.",
"Slatend-plane MRF models for stereo vision were introduced in [4] and have been since very widely used [5], [6], [7], [9], [10], [13].",
"In the context of this literature, our work has several distinctive features.",
"First, we use a novel model involving “boundary labels”, “junction potentials”, and “edge ownership”.",
"Second, for inference we employ the convex form of the particle norm-product belief propagation [32], which we refer to as particle convex belief propagation (PCBP) [19].",
"In contrast, some previous works used particle belief propagation (PBP) [33], [34], [10] which correspond to non-convex norm-product with the Bethe entropy approximation.",
"The efficiency and convexity of PCBP makes it possible to evaluate our approach on hundreds of high-resolution images [2], whereas previous empirical evaluations of slanted-plane models have largely been restricted to the low-resolution versions of the small number of highly controlled Middlebury images.",
"Third, we use a training algorithm based on primal-dual approximate inference [35] which allow us to effectively learn the importance of each potential.",
"Figure: Impossible cases of 3-way junctions.",
"(a) 3 cyclic occlusions, (b) hinge and 2 occlusion with opposite directions, (c) coplanar and 2 occlusion with opposite directions, (d) 2 hinge and occlusion, (e) 2 coplanar and occlusion, (f) 2 coplanar and hinge, (g) hinge, coplanar, and occlusion (superpixel with coplanar boundary is in front)."
],
[
"Continuous MRFs for stereo",
"In this section we describe our approach to joint reasoning of boundary labels and depth.",
"We reason at the segment level, employing a richer representation than a discrete disparity label.",
"In particular, we formulate the problem as inference in a hybrid conditional random field, which contains continuous and discrete random variables.",
"The continuous random variables represent, for each segment, the disparities of all pixels contained in that segment in the form of a 3D slanted plane.",
"The discrete random variables indicate for each pair of neighboring segments, whether they are co-planar, they form a hinge or there is a depth discontinuity (indicating which plane is in front of which).",
"Figure: Valid 4-way junctions.",
"(a) 4 coplanar boundaries, (b)-(d) 2 coplanar vertical boundaries and 2 occlusion/hinge horizontal boundaries, (e)-(g) 2 coplanar horizontal boundaries and 2 vertical occlusion/hinge boundariese.",
"A 4-way junction only appears in a region of uniform color.More formally, let $y_i= (\\alpha _i, \\beta _i, \\gamma _i) \\in \\Re ^3$ be a random variable representing the $i-$ th slanted 3D plane.",
"We can compute the disparities of each pixel belonging to the $i-$ th segment as follows $\\hat{d_i}(\\mathbf {p}, \\mathbf {y}_i) = \\alpha _i(u - c_{ix}) + \\beta _i(v - c_{iy}) + \\gamma _i$ with $\\mathbf {p}= (u,v)$ , and $\\mathbf {c}_i = (c_{ix}, c_{iy})$ the center of the $i$ -th segment.",
"We have defined $\\gamma _i$ to be the disparity in the segment center as it improves the efficiency of PCBP inference.",
"Let $o_{i,j} \\in \\lbrace co, hi, lo, ro \\rbrace $ be a discrete random variable representing whether two neighboring planes are coplanar, form a hinge or an occlusion boundary.",
"Here, $lo$ implies that plane $i$ occludes plane $j$ , and $ro$ represents that plane $j$ occludes plane $i$ .",
"We define our hybrid conditional random field as follows $p(\\mathbf {y},\\mathbf {o}) = \\frac{1}{Z} \\prod _i \\psi _i(\\mathbf {y}_i)\\prod _{\\alpha }\\psi _{\\alpha }(\\mathbf {y}_{\\alpha })\\prod _\\beta \\psi _{\\beta }(\\mathbf {o}_{\\beta })\\prod _{\\gamma } \\psi _\\gamma (\\mathbf {y}_\\gamma , \\mathbf {o}_\\gamma )\\vspace{-5.69046pt}$ where $\\mathbf {y}$ represents the set of all 3D slanted planes, and $\\mathbf {o}$ the set of all discrete random variables.",
"The unitary potentials are represented as $\\psi _i$ , while $\\psi _\\alpha , \\psi _\\beta , \\psi _\\gamma $ encode potential functions over sets of continuos, discrete or mixture of both types of variables.",
"Note that $\\mathbf {y}$ contains three random variables for every segments in the image, and there is a random variable $o_{i,j}$ for each pair of neighboring segments.",
"In the following, we describe the different potentials we employed for our joint occlusion boundary and depth reasoning.",
"For clarity, we describe the potentials in the log domain, i.e., $\\mathbf {w}^T\\phi _i = \\log (\\psi _i)$ (similarly for potentials over cliques).",
"The weights $\\mathbf {w}$ will be learned using structure prediction methods."
],
[
"Occlusion Boundary and Segmentation Potentials",
"Our approach takes as input a disparity image computed by any matching algorithm.",
"In particular, in this paper we employ semi-global block matching [25].",
"Most matching methods return estimated disparity values on a subset of pixels.",
"Let ${\\cal F}$ be the set of all pixels whose initial disparity has been estimated, and let ${\\cal D}(\\mathbf {p})$ be the disparity of pixel $\\mathbf {p}\\in {\\cal F}$ .",
"Our model jointly reasons about segmentation in the form of occlusion boundaries as well as depth.",
"We define potentials for each of these tasks individually as well as potentials which link both tasks.",
"We start by defining truncated quadratic potentials, which we will employ in the definition of some of our potentials, i.e., $\\phi ^{TP}_i(\\mathbf {p}, \\mathbf {y}_i, K) = \\min \\left(\\left|\\mathcal {D}(\\mathbf {p}) - \\hat{d_i}(\\mathbf {p}, \\mathbf {y}_i)\\right|, K\\right)^2$ with $K$ a constant threshold, and $\\hat{d_i}(\\mathbf {p}, \\mathbf {y}_i)$ the disparity of pixel $\\mathbf {p}$ estimated as in Eq.",
"REF .",
"Note that we have made the quadratic potential robust via the $\\min $ function.",
"We now describe each of the potentials employed in more details."
],
[
"We define truncated quadratic unitary potentials for each segment expressing the fact that the plane should agree with the results of the matching algorithm, $\\phi _i^{\\mathrm {seg}}(\\mathbf {y}_i) =\\sum _{\\mathbf {p}\\in S_i \\cap {\\cal F}} \\phi ^{TP}_i(\\mathbf {p},\\mathbf {y}_i,K)$ where $S_i$ is the set of pixels in segment $i$ ."
],
[
"We employ 3-way potentials linking our discrete and continuous variables.",
"In particular, these potentials express the fact that when two neighboring planes are hinge or coplanar they should agree on the boundary, and when a segment occludes another segment, the boundary should be explained by the occluder.",
"We thus define $\\phi _{ij}^{\\mathrm {bdy1}}(o_{ij}, \\mathbf {y}_i, \\mathbf {y}_j) = {\\left\\lbrace \\begin{array}{ll}\\sum _{\\mathbf {p}\\in B_{ij} \\cap {\\cal F}} \\phi _i^{TP}(\\mathbf {p}, \\mathbf {y}_i,K) & \\text{if $o_{ij}$ = $lo$ }\\\\\\sum _{\\mathbf {p}\\in B_{ij} \\cap {\\cal F}} \\phi ^{TP}_j(\\mathbf {p}, \\mathbf {y}_j,K) & \\text{if $o_{ij}$ = $ro$ }\\\\\\frac{1}{2}\\sum _{\\mathbf {p}\\in B_{ij} \\cap \\mathcal {F}}\\phi ^{TP}_i(\\mathbf {p}, \\mathbf {y}_i,K) + \\phi ^{TP}_j(\\mathbf {p}, \\mathbf {y}_j,K)& \\text{if $o_{ij}=hi \\vee co$}\\end{array}\\right.",
"}$ where $B_{ij}$ is the set of pixels around the boundary (within 2 pixels of the boundary) between segments $i$ and $j$ .",
"Table: Comparison with the state of the art KITTI datasetFigure: Examples from the KITTI.",
"(Left) Original images.",
"(Right) Disparity errors."
],
[
"We introduce an additional potential which ensures that the discrete occlusion labels match well the disparity observations.",
"We do so by defining $\\phi ^{\\mathrm {occ}}_{ij}(\\mathbf {y}_{\\mathrm {front}},\\mathbf {y}_{\\mathrm {back}})$ to be a penalty term which penalizes occlusion boundaries that are not supported by the data $\\phi ^{\\mathrm {occ}}_{ij}(\\mathbf {y}_{\\mathrm {front}}, \\mathbf {y}_{\\mathrm {back}}) = {\\left\\lbrace \\begin{array}{ll}\\lambda _{\\mathrm {imp}} &\\text{if $\\exists {\\mathbf {p}\\in B_{ij}}: \\hspace{5.69046pt}\\hat{d_i}(\\mathbf {p},\\mathbf {y}_{\\mathrm {front}}) < \\hat{d_j}(\\mathbf {p}, \\mathbf {y}_{\\mathrm {back}})$ } \\\\0 & \\text{otherwise}\\end{array}\\right.",
"}$ We also define $\\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_i)$ to be a function which penalizes negative disparities $\\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_i) = {\\left\\lbrace \\begin{array}{ll}\\lambda _{imp} & \\text{if $\\min _{\\mathbf {p}\\in B_{ij}} \\hat{d_i}(\\mathbf {p}, \\mathbf {y}_i) < 0$} \\\\0 & \\text{otherwise}\\end{array}\\right.",
"}$ We impose a regularization on the type of occlusion boundary, where we prefer simpler explanations (i.e., coplanar is preferable than hinge which is more desirable than occlusion).",
"We encode this preference by defining $\\lambda _{\\mathrm {occ}} >\\lambda _{\\mathrm {hinge}}> 0$ .",
"We thus define our computability potential $\\begin{small}\\phi _{ij}^{\\mathrm {bdy2}}(o_{ij}, \\mathbf {y}_i, \\mathbf {y}_j) = {\\left\\lbrace \\begin{array}{ll}\\lambda _{\\mathrm {occ}} + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_i) + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_j) + \\phi ^{\\mathrm {occ}}_{ij}(\\mathbf {y}_i, \\mathbf {y}_j) & \\text{if $o_{ij}=lo$} \\\\\\lambda _{\\mathrm {occ}} + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_i) + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_j) + \\phi ^{\\mathrm {occ}}_{ij}(\\mathbf {y}_j, \\mathbf {y}_i)& \\text{if $o_{ij}=ro$} \\\\\\lambda _{\\mathrm {hinge}} + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_i) + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_j) + \\frac{1}{|B_{ij}|}\\sum _{\\mathbf {p}\\in B_{ij}} \\Delta d_{ij} & \\text{if $o_{ij}= hi$} \\\\\\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_i) + \\phi ^{\\mathrm {neg}}_{ij}(\\mathbf {y}_j) + \\frac{1}{|S_i \\cup S_j|}\\sum _{\\mathbf {p}\\in S_i \\cup S_j}\\Delta d_{i,j} & \\text{if $o_{ij}=co$}\\end{array}\\right.",
"}\\end{small}$ with $\\Delta d_{i,j} =(\\hat{d_i}(\\mathbf {p}, \\mathbf {y}_i) - \\hat{d_j}(\\mathbf {p}, \\mathbf {y}_j))^2$ .",
"Table: Comparison with the state-of-the-art on Middlebury high-resolution imagery.",
"The baselines are provided by the author of Table: Difference in performance when employing different segmentation methods to compute superpixels on the KITTI dataset.",
"Note that employing an intersection of SLIC+UCM superpixels works best for the same amount of superpixels."
],
[
"Following work on occlusion boundary reasoning [15], [16], we utilize higher order potentials to encode whether a junction of three planes is possible.",
"We refer the reader to Fig.",
"REF for an illustration of these cases.",
"We thus define the compatibility of a junction $\\lbrace i,j,k\\rbrace $ to be $\\phi _{ijk}^{\\mathrm {jct}}(o_{ij}, o_{jk}, o_{ik}) = {\\left\\lbrace \\begin{array}{ll}\\lambda _{\\mathrm {imp}} & \\mathrm {if \\ impossible \\ case} \\\\0 & \\mathrm {otherwise}\\end{array}\\right.",
"}$ We also defined a potential encoding the feasibility of a junction of four planes (see Fig.",
"REF ) as follows $\\phi _{pqrs}^{crs}(o_{pq}, o_{qr}, o_{rs}, o_{ps}) = {\\left\\lbrace \\begin{array}{ll}\\lambda _{\\mathrm {imp}} & \\mathrm {if \\ impossible \\ case} \\\\0 & \\mathrm {otherwise}\\end{array}\\right.",
"}$ Note that, although these potentials are high order, they only involve variables with small number of states, i.e., 4 states."
],
[
"Finally, we employ a simple color potential to reason about segmentation, which is defined in terms of the $\\chi $ -squared distance between color histograms of neighboring segments.",
"This potential encodes the fact that we expect segments which are coplanar to have similar color statistics (i.e., histograms), while the entropy of this distribution is higher when the planes form an occlusion boundary or a hinge.",
"Note that this trend is shown in Fig.",
"REF (left) for the KITTI [2] dataset.",
"The statistics are less meaningful in the case of the Middelbury high resolution imagery [1], as this dataset is captured in a control environment.",
"We thus reflect these statistics in the following potential $\\phi _{ij}^{\\mathrm {col}}(o_{ij}) = {\\left\\lbrace \\begin{array}{ll}\\min \\left(\\kappa \\cdot \\chi ^2(h_i, h_j), \\lambda _{\\mathrm {col}}\\right) & \\text{if $o_{ij}=co$}\\\\\\lambda _{\\mathrm {col}} & \\text{otherwise}\\end{array}\\right.",
"}$ with $\\kappa $ a scalar and $\\chi ^2(h_i, h_j)$ the $\\chi $ -squared distance between the color histograms of segments $i$ and $j$ ."
],
[
"Inference in Continuous MRFs",
"Now that we have defined the model, we can turn our attention to inference, which is defined as computing the MAP estimate as follows $(\\text{inference})\\hspace{8.5359pt} \\, \\,\\text{arg} \\max _{\\mathbf {y},\\mathbf {o}} \\frac{1}{Z} \\prod _i \\psi _i(\\mathbf {y}_i)\\prod _{\\alpha }\\psi _{\\alpha }(\\mathbf {y}_{\\alpha })\\prod _\\beta \\psi _{\\beta }(\\mathbf {o}_{\\beta })\\prod _{\\gamma } \\psi _\\gamma (\\mathbf {y}_\\gamma , \\mathbf {o}_\\gamma )\\vspace{-5.69046pt}$ Inference in this model is in general NP hard.",
"Our inference is also particularly challenging since, unlike traditional MRF stereo formulations, we have defined a hybrid MRF, which reasons about continuous as well as discrete variables.",
"While there is a vast literature on discrete MRF inference, only a few attempts have focussed on solving the continuous case.",
"The exact MAP solution can only be recovered in very restrictive cases.",
"For example when the potentials are quadratic and diagonally dominated, an algorithm called Gaussian Belief propagation [41] returns the optimal solution.",
"For general potentials, one can approximate the messages using mixture models, or via particles.",
"In this paper we make use of particle convex belief propagation (PCBP) [19], a technique that is guarantee to converge and gradually approach the optimum.",
"This works very well in practice, yielding state-of-the-art results.",
"PCBP is an iterative algorithm that works as follows: For each random variable, particles are sampled around the current solution.",
"These samples act as labels in a discretized MRF which is solved to convergence using convex belief propagation [32].",
"The current solution is then updated with the MAP estimate obtained on the discretized MRF.",
"This process is repeated for a fixed number of iterations.",
"In our implementation, we use the distributed message passing algorithm of [42] to solve the discretized MRF at each iteration.",
"Algorithm REF depicts PCBP for our formulation.",
"At each iteration, to balance the trade off between exploration and exploitation, we decrease the values of the standard deviations $\\sigma _\\alpha , \\sigma _\\beta $ and $\\sigma _\\gamma $ of the normal distributions from which the plane random variables are drawn.",
"[h] PCBP for stereo estimation and occlusion boundary reasoning Set $N$ Initialize slanted planes $\\mathbf {y}_i^0 = (\\alpha _i^0, \\beta _i^0,\\gamma _i^0)$ via local fitting $\\forall i$ Initialize $\\sigma _\\alpha , \\sigma _\\beta $ and $\\sigma _\\gamma $ $t = 1$ to $\\#$ iters Sample $N$ times $\\forall i$ from $\\alpha _i \\sim {\\cal N}(\\alpha _i^{t-1},\\sigma _\\alpha )$ , $\\beta _i \\sim {\\cal N}(\\beta _i^{t-1},\\sigma _\\beta )$ , $\\gamma _i \\sim {\\cal N}(\\gamma _i^{t-1},\\sigma _\\gamma )$ $(\\mathbf {o}^{t},\\mathbf {y}^t)$ $\\leftarrow $ Solve the discretized MRF using convex BP Update $\\sigma ^c_{\\alpha } = \\sigma ^c_{\\beta } = 0.5\\times \\exp (-c/10)$ and $\\sigma ^c_{\\gamma } = 5.0\\times \\exp (-c/10)$ Return $\\mathbf {o}^t$ , $\\mathbf {y}^t$"
],
[
"Learning in Continuous MRFs",
"We employ the algorithm of [20] for learning.",
"Given a set of training images and corresponding depth labels, the goal of learning is to estimate the weights which minimize the surrogate partition loss.",
"However, our learning problem, as opposed to the one defined in [20], contains a mixture of continuous and discrete variables.",
"Therefore the surrogate partition loss in our setting requires to integrate over the continuous variables.",
"We note that our continuous variables have robust quadratic potentials, thus integrating over them can be efficiently estimated by discretizing the continuous variables.",
"In practice, summing over 30 particles gives a good approximation for the integral.",
"Table: Performance changes when employing different segmentation methods to compute superpixels on the Middlebury high-resolution dataset.",
"Employing an intersection of SLIC+UCM superpixels works best for the same amount of superpixels.Figure: Importance of the number of superpixels: KITTI results as a function of the number of superpixels.",
"Even with a small number our approach still outperforms the baselines.",
"(Right) The inference time scales linearly with the number of superpixels"
],
[
"Experimental Evaluation",
"We perform exhaustive experiments on two publicly available datasets: Middebury high resolution images [1] as well as the more challenging KITTI dataset [2].",
"For all experiments, we employ the same parameters which have been validated on the training set.",
"We use a disparity difference threshold $K = 5.0$ pixels, and set $\\lambda _{\\mathrm {occ}} = 15$ , $\\lambda _{\\mathrm {hinge}} = 3$ , $\\lambda _{\\mathrm {imp}} = 30$ and $\\lambda _{\\mathrm {col}} = 30$ .",
"For the color potential, we use a color histogram with 64 bins and set $\\kappa = 60$ .",
"Unless otherwise stated, we employ 10 particles and 5 iterations of re-sampling for PCBP [19], and run each iteration of convex BP to convergence.",
"For learning, we use a value of $C$ equal to the number of examples and unless otherwise stated use a CRF, i.e., $\\epsilon =1$ .",
"We learned the importance of each potential, thus 6 parameters.",
"We employ two different metrics.",
"The first one measures the average number of non-occluded pixels which error is bigger than a fixed threshold.",
"To test the extrapolation capabilities of the different approaches, the second metric computes the average number of pixels (including the occluding ones) which error is bigger than a fixed threshold.",
"Figure: Importance of the number of re-sampling iterations: on KITTI.Table: Training set size: Estimation errors as a function of the training set size.",
"Note that very few images are needed to learn good parameters.We now describe the characteristics of the databases we evaluate our approach on.",
"Our first dataset consists on high resolution images from the Middebury dataset [1], which have an average resolution of $1239.2\\times 1038.0$ pixels.",
"We employ 5 images for training (i.e., Books, Laundry, Moebius, Reindeer, Bowling2) and 9 images for testing (i.e., Cones, Teddy, Art, Aloe, Dolls, Baby3, Cloth3, Lampshade2, Rocks2).",
"We also evaluate our approach on the KITTI dataset [2], which is the only real-world stereo dataset with accurate ground truth.",
"It is composed of 194 training and 195 test high-resolution images ($1237.1 \\times 374.1$ pixels) captured from an autonomous driving platform driving around in a urban environment.",
"The ground truth is generated by means of a Velodyne sensor which is calibrated with the stereo pair.",
"This results in semi-dense ground truth covering approximately 30 % of the pixels.",
"We employ 10 images for training, and utilize the remaining 184 images for validation purposes."
],
[
"We begin our experimentation by comparing our approach with the state-of-the-art.",
"Table REF depicts results of our approach and the baselines in terms of the two metrics for the KITTI dataset.",
"Note that our approach significantly outperforms all the baselines in all settings (i.e., thresholds bigger than 2, 3, 4 and 5 pixels).",
"Table REF depicts similar comparisons for high resolution Middlebury.",
"Once more, our approach outperforms significantly the baselines in all settings.",
"Fig.",
"REF depicts an illustrative set of example results for the KITTI dataset.",
"Note that despite the challenges that the images pose, our approach does a good job at estimating disparities."
],
[
"We next investigate how the segmentation strategy affects the stereo estimation.",
"Towards this goal we evaluate the results of our approach when employing UCM segments [43], SLIC superpixels [44] or the intersection of both as input.",
"Table REF depicts results on the KITTI dataset.",
"UCM performs very poorly as the number of superpixels on average is very small, and some of the superpixels are very large.",
"Therefore, a single 3D plane is a poor representation for the disparities in those large segments.",
"SLIC performs quite well, but the intersection of SLIC and UCM superpixels outperforms the other strategies.",
"This is also expected, as UCM respects the boundaries much better than SLIC.",
"Note that as shown in Table REF similar results are observed for the Middlebury dataset.",
"Table: Oracle performance: Oracle, our approach and initial fit on KITTI.Table: Oracle performance: Oracle, our approach and initial fit on Middlebury."
],
[
"We next investigate how well our approach scales with the number of superpixels in terms of computatinal complexity as well as accuracy.",
"Fig.",
"REF shows results for the KITTI dataset when varying the number of superpixels.",
"Our approach reduces performance gracefully when reducing the amount of superpixels.",
"Note that inference scales linearly with the number of superpixels, taking on average $5.5$ minutes per high resolution image when employing 1200 superpixels and $2.5$ minutes when using 300."
],
[
"We evaluate the effects of varying the number of resampling iterations on the performance of our approach.",
"As shown in Fig.",
"REF , our approach converges to a good local optima after only 2 resampling iterations.",
"This reduces the inference cost from 5.5 minutes per high-resolution image for 5 iterations to 2.2 minutes for 2 iterations."
],
[
"We evaluate the effect of increasing the training set size in Table REF .",
"Even when training with a single image we outperform all baselines."
],
[
"We next evaluate the best performance that our model can achieve, by fitting the model to the ground truth disparities.",
"This shows an upper-bound on the performance that our method could ever achieve if we were able to learn an energy that has its MAP at the ground truth, and if we were able to solve the NP-hard inference problem.",
"Tables REF and REF depict the oracle performance in terms of both the occluded an non-occluded pixels for both datasets.",
"Note that as KITTI does not release the test ground truth, we compute this values using 10 images for training and the rest of the training set for testing.",
"We also report performance of our initialization which is computed by fitting a local plane to the results of semi-global block matching [25].",
"Note that the oracle can achieve great performance, showing that the errors due to the 3D slanted plane discretization are negligible."
],
[
"We investigate the robustness of our approach to noise by building a synthetic dataset, which is composed of 10 images for training and 90 images for test.",
"Each Image has a resolution of $320\\times 240$ pixels and contains several planes.",
"The average number of superpixels is 108.0.",
"We create ${\\cal D}(\\mathbf {p})$ by sampling 3 to 5 points at random on the boundaries and generating disparities by corrupting the ground truth with Gaussian noise of varying standard deviation.",
"Table REF shows RMS errors for disparity as well as percentage of boundary variables wrongly estimated.",
"Table: Robustness to noise: RMS as well as boundary error as a function of noise."
],
[
"We evaluate the performance of our learning algorithm as a function of its parameter $\\epsilon $ which ranges from CRFs for $\\epsilon =1$ to structural SVMs for $\\epsilon =0$ .",
"Fig.",
"REF depicts performance on the KITTI dataset.",
"Our approach results in state-of-the-art performance for all settings.",
"Table: Family of structure prediction problems: for the KITTI dataset.",
"Note that ϵ=0\\epsilon =0 is structural SVM and ϵ=1\\epsilon =1 is CRF."
],
[
"We evaluate the importance of incorporating different tasks loss in the learning framework of [20].",
"In particular, we employ RMS as well as the same loss that we employ for evaluation.",
"Note that the loss has little effect.",
"Table: Loss functions: Performance changes when employing different loss functions for learning in the KITTI dataset"
],
[
"Conclusion",
"We have presented a novel stereo slanted-plane MRF model that reasons jointly about occlusion boundaries as well as depth.",
"We have formulated the problem as inference in a hybrid MRF composed of both continuous (i.e., slanted 3D planes) and discrete (i.e., occlusion boundaries) random variables, which we have tackled using particle convex belief propagation.",
"We have demonstrated the effectiveness of our approach on high resolution imagery from Middlebury as well as the more challenging KITTI dataset.",
"In the future we plan to investigate alternative inference algorithms as well as other segmentation potentials."
]
] | 1204.1393 |
[
[
"Characteristics of shape and knotting in ideal rings"
],
[
"Abstract We present two descriptions of the the local scaling and shape of ideal rings, primarily featuring subsegments.",
"Our focus will be the squared radius of gyration of subsegments and the squared internal end to end distance, defined to be the average squared distance between vertices $k$ edges apart.",
"We calculate the exact averages of these values over the space of all such ideal rings, not just a calculation of the order of these averages, and compare these to the equivalent values in open chains.",
"This comparison will show that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths.",
"These results will be corroborated by numerical experiments.",
"They will be used to analyze the convergence of our generation method and the effect of knotting on these characteristics of shape."
],
[
"Introduction",
"Long strings of connected molecules, called polymers, are central structures in life and physical sciences, as well as engineering.",
"Prominent examples are DNA, proteins, polystyrene, and silicone.",
"With regard to DNA, Fiers and Sinsheimer first showed that the DNA of a specific virus is a single-stranded ring [8].",
"Because of this closure condition, DNA, like other closed polymers, can be knotted.",
"In 1976, Liu et al.",
"discovered examples of knotted DNA, which was followed by the discovery of topoisomerases, enzymes which knot and unknot DNA [2], [13].",
"These discoveries suggest that knotting plays an important role in the behavior and shape of DNA and other polymers.",
"We will examine ideal rings, embedded equilateral polygons in $\\mathbb {R}^3$ , which provide a model for polymers under the $\\theta $ -condition, where excluded volume can be ignored [3], [9], [22].",
"We will compare ideal rings with ideal chains, random walks in $\\mathbb {R}^3$ , to determine the effect of the closure constraint as well as the effect of knotting.",
"These models will be used to analyze the relationship between shape, scale, knotting and two physical characteristics specific to their local structure.",
"We define the average squared radius of gyration of subsegments, calculated by averaging the standard squared radius of gyration of each subsegment of length $k$ , and the squared internal end to end distance, the average distance between vertices $k$ edges apart.",
"Rather than finding approximations of these averages, we will determine the exact theoretical averages for these descriptions of shape, as well as discuss numerical simulations and show how these characteristics are affected by knotting.",
"In numerical studies examining squared end to end distance, squared end to end distance is refered to internal end to end distance or two point correlation [21], [24], [26].",
"These characteristics are simplified in open chains because subsegments of length $k$ in chains of length 100 are identical to subsegments of length $k$ in chains of length $100,000$ : the ambient length has no effect on the behavior of subsegments.",
"However, in the case of ideal rings, subsegments of length $k$ in a ring of length $n$ do not behave the same as a subsegment of length $k$ in a ring of length $m \\ne n$ .",
"Thus, subsegment behavior relies on both $k$ and $n$ .",
"It will be shown that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths, which may correspond with the difference in $\\theta $ -temperature between open chain polymers and ring polymers [22].",
"Understanding in the case of closed chains may also allow us to use these charactereistics to identify and describe the knotted portions of open chains [11], [21]."
],
[
"An Introduction to Ideal Rings and Their Notation",
"Definition 1 An ideal ring, $P$ , is an $n$ -edged equilateral polygon embedded in $\\mathbb {R}^3$ , with one vertex at the origin.",
"Let $e_1,...,e_n$ be unit vectors such that the $k^{\\textrm {th}}$ vertex of $P$ is $v_k = \\displaystyle \\sum _{i=1}^k e_i$ .",
"Let each $e_i$ be called an edge vector of $P$ .",
"The fact that $P$ is a polygon is equivalent to requiring closure, that is, that $v_n = \\displaystyle \\sum _{i=1}^n e_i = 0.$ Let $\\mathcal {P}_n$ denote the space of all such polygons.",
"The careful reader may note that these ideal rings are more specific than usual.",
"We require these polygons to be based at the origin, and be oriented.",
"This definition makes notation concise, and does not affect the averages we will calculate, as they are independent of the base point and orientation.",
"For each $P \\in \\mathcal {P}_n$ , the edge vectors $e_i$ are all identically distributed.",
"Therefore, when $i \\ne j$ , $(e_i \\cdot e_j)$ is a random variable that does not depend on $i$ and $j$ .",
"We will compare these with ideal chains, a similar population but without the closure constraint.",
"Definition 2 An open chain, $W$ , also sometimes called an ideal chain or ideal open chain, is an $n$ -edged random walk in $\\mathbb {R}^3$ , where each edge has unit length.",
"Let $e_1,...,e_n$ be unit vectors such that the $k^{\\textrm {th}}$ vertex is given by $v_k = \\displaystyle \\sum _{i=1}^k e_i$ .",
"As with ideal rings, let $e_i$ be called an edge vector.",
"Let $\\mathcal {W}_n$ denote the space of all such open chains.",
"Again, the careful reader will see that these are also based at the origin for ease of notation."
],
[
"Squared End to End Distance in Ideal Chains",
"For comparison, we will first calculate the squared end to end distance in ideal chains, as in [6].",
"Lemma 1 In $\\mathcal {W}_n$ , the average squared end to end distance is $n$ .",
"Let $W \\in \\mathcal {W}_n$ .",
"Calculating the end to end distance of $W$ : $\\left|\\left| \\displaystyle \\sum _{i=1}^n e_i \\right|\\right|^2 & = & (\\sum _{i=1}^n e_i \\cdot \\sum _{i=1}^n e_i) \\\\& = & \\sum _{i = 1}^n (e_i \\cdot e_i) + 2 \\left( \\sum _{i = 1}^{n-1} \\sum _{j = 1}^n (e_i \\cdot e_j) \\right) \\\\$ For random walks $W \\in \\mathcal {W}_n$ , the direction of each edge vector is completely uncorrelated with the direction of the previous edge vector, and each edge vector $e_i$ is a uniformly distributed, unit length, random vector.",
"Thus, for $i = j$ , $(e_i \\cdot e_j) = 1$ , and for $i \\ne j$ the average value of $(e_i \\cdot e_j)$ , $\\left< e_i \\cdot e_j \\right>$ , is 0.",
"Thus we have that the average end to end distance, taken over all $W \\in \\mathcal {W}_n$ , is $\\left<\\left|\\left| \\displaystyle \\sum _{i=1}^n e_i \\right|\\right|^2\\right> = n.$"
],
[
"Average Edge Product in Rings",
"Now we will consider ideal rings.",
"As noted before, $(e_i \\cdot e_j)$ is independent of $i$ and $j$ .",
"Here, we find an average value for $(e_i \\cdot e_j)$ , taken over all $i$ , $j$ for all $P \\in \\mathcal {P}_n$ .",
"Definition 3 Consider the space of ideal rings, $\\mathcal {P}_n$ , for some $n$ .",
"Let $r_n$ denote the average of the set $\\mathcal {R}_n$ , $\\displaystyle \\mathcal {R}_n = \\lbrace (e_i\\cdot e_j) : e_i,e_j \\textrm { are edge vectors of some } P \\in \\mathcal {P}_n, i \\ne j\\rbrace .$ We will call $r_n$ the average edge product.",
"The following Lemma is generally known, and is foundational to the following proofs [10].",
"Lemma 2 For all $n\\in \\mathbb {N}$ , the average edge product, over all ideal rings of length $n$ , is given by $\\displaystyle r_n = \\frac{-1}{n-1}$ .",
"For all $P \\in \\mathcal {P}_n$ , $\\displaystyle \\sum _{i=1}^n e_i = 0$ .",
"Then squaring both sides we have $(0 \\cdot 0) = 0& = & (\\displaystyle \\sum _{i=1}^n e_i) \\cdot (\\displaystyle \\sum _{i=1}^n e_i ) \\\\& = & \\sum _{i=1}^n(e_i\\cdot e_i)+2\\sum _{1\\le i < j \\le n} (e_i \\cdot e_j).$ Taking the average over all $P \\in \\mathcal {P}$ , we replace $(e_i \\cdot e_j)$ with $\\left<e_i \\cdot e_j\\right> = r_n$ : $0 = n+ 2 \\frac{n(n-1)}{2} r_n.$ Solving for $r_n$ we have $r_n = \\frac{-n}{n(n-1)} = \\frac{-1}{n-1}.$"
],
[
"Squared End to End Distance",
"In open chains, the average squared end to end distance of a subsegments of length $k$ in a chain of length $n$ is identical to the average squared end to end distance of chains of length $k$ .",
"That is, the ambient chain has no effect on the shape of the subsegment.",
"The same is not true in ideal rings, where the closure constraint plays a pivotal role in local scale, as we will see.",
"Definition 4 For any $P \\in \\mathcal {P}_n$ , define $d_{P}(k,j) = \\displaystyle \\left|\\left|\\sum _{i=j}^{j+k} e_i \\right|\\right|^2 = ( \\sum _{i=j}^{j+k} e_i)\\cdot ( \\sum _{i=j}^{j+k} e_i),$ the squared distance between the $j^{\\textrm {th}}$ and $j+k^{\\textrm {th}}$ vertex.",
"We call $d_{P}(k,j)$ the squared end-to-end distance of length $k$ at $j$ of $P$ .",
"Define $A(k,n)$ to be the average value of $\\lbrace d_{P}(k,j): 1\\le j \\le n, P \\in \\mathcal {P}_n\\rbrace ,$ that is, let $A(k,n)$ denote the average squared end to end distance of a subsegment of length $k$ in an ideal ring of length $n$ .",
"We will call $A(k,n)$ the average end to end distance of length $k$.",
"Figure: In the above image, we have end to end distances marked for k=2k=2, k=3k = 3 and k=4k = 4.",
"We would take the average of their squares to find the average squared end to end distance for k=2,3k=2,3 and 4.Theorem 3 $\\displaystyle A(k,n)=\\frac{k(n-k)}{n-1}$ First we have that $d_{P}(k,j) = \\left|\\left| \\displaystyle \\sum _{i=j}^{j+k} e_i \\right|\\right|^2 &=& (\\displaystyle \\sum _{i=j}^{j+k} e_i) \\cdot (\\displaystyle \\sum _{i=j}^{j+k} e_i )\\\\&=& \\sum _{i=j}^{j+k} (e_i\\cdot e_i)+2\\sum _{j\\le i < m \\le j+k} (e_i \\cdot e_m) \\\\&=& k+2\\sum _{j\\le i < m \\le j+k} (e_i \\cdot e_m)$ We have shown that independent of $P$ , $k$ , $i$ and $j$ , the average value of $\\displaystyle (e_i \\cdot e_m) = \\frac{-1}{n-1}$ .",
"Therefore the average value of $d_{P}(k,j)$ is, independent of $P$ , $i$ and $j$ , $\\displaystyle k+2\\sum _{j\\le i < m \\le j+k}r_n$ .",
"This average is precisely $A(k,n)$ .",
"We conclude that $A(k,n) & = & k+2\\sum _{j\\le i < m \\le j+k}r_n \\\\& = & k + 2 \\frac{(k-1)(k)}{2} \\frac{-1}{n-1} \\\\&=& \\frac{k(n - k)}{n-1} \\\\$ Therefore the average value of $\\parallel v_k \\parallel ^2$ is $\\displaystyle A(k,n) = \\frac{k(n-k)}{n-1}$ .",
"We can verify some key features of this identity immediately.",
"At $k = 1$ and $k = n-1$ , we have that the average squared end to end distance is 1.",
"Vertices one edge apart must be distance 1 apart, as we would expect, so the theoretical average agrees with the physical reality.",
"Also, the function is symmetric about $\\frac{n}{2}$ , as a segment of length $k$ and its complement, a segment of length $n-k$ , should have the same end to end distance.",
"Lastly, the function achieves its maximum at $\\frac{n}{2}$ , as after that point, on average, the end to end distance must decrease due to the closure condition.",
"Witz showed through a different argument that squared end to end distance scales like $\\frac{k(n-k)}{n}$ , which is close to our value [25].",
"This is an approximation though, as for $k = 1$ , we have $\\frac{n-1}{n} \\ne 1$ , though it is close for large values of $k$ and $n$ .",
"As in [6] and Lemma REF , for open chains, the average squared end to end distance for ideal chains is $n$ (or $nb^2$ where $b$ is the length of the segments.)",
"The squared end to end distance of subsegments of length $k$ within an open chain of length $n$ will be identical to the squared end to end distance of an open chain of length $k$ .",
"Therefore, in the open case, the average squared end to end distance of a subsegment of length $k$ within a chain of length $n$ is $k$ .",
"As we can see in Figure REF , the average squared end to end distance of subsegments of an open chain and the average squared end to end distance subsegments of an ideal ring look very different for subsegments of very short lengths when $n =50$ .",
"We observe that for the average squared end to end distance of segments of length $k$ in open chains to be within $\\frac{1}{100}^{\\textrm {th}}$ of the average squared end to end distance of segments of length $k$ in ideal rings, we would need $ k \\approx \\frac{n}{100}$ or smaller.",
"For length 50, as in the figure below, $k$ must be length 1, which is a very strong restraint.",
"Figure: Here we compare the theoretical average squared end to end distance of subsegments of length kk for open chains (blue) and ideal rings (red and dashed), both with total length n=50n = 50."
],
[
"Average Center of Mass",
"For the following definition and Lemma, it is helpful to recall that these ideal rings are based at the origin, $\\displaystyle \\sum _{i = 1}^n e_i = 0$ , as we stipulated originally.",
"Definition 5 Let $n \\in \\mathbb {N}$ .",
"Let $P\\in \\mathcal {P}_n$ .",
"Define $c_P$ , the center of mass of $P$, to be $\\displaystyle c_P = \\frac{1}{n} \\sum _{k=1}^n v_k $ , the average of the vertices.",
"Lemma 4 For any $P \\in \\mathcal {P}_n$ , $\\displaystyle \\parallel c_P \\parallel ^2 = \\frac{1}{n^2} (\\sum _{k=1}^n (n-k+1)^2 + 2 \\sum _{1 \\le i < j \\le n} (n-j+1)(n-i+1)(e_i \\cdot e_j)).",
"$ $\\displaystyle \\parallel c_P \\parallel ^2 &=& (\\frac{1}{n} \\sum _{k=1}^n v_k \\cdot \\frac{1}{n} \\sum _{k=1}^n v_k) \\\\&=& \\frac{1}{n^2} ( \\sum _{k=1}^n \\sum _{i=1}^k e_i \\cdot \\sum _{k=1}^n \\sum _{j=1}^k e_j)) \\\\&=& \\frac{1}{n^2} (\\sum _{k=1}^n (n-k+1)^2 + 2 \\sum _{1 \\le i < j \\le n} (n-j+1)(n-i+1)(e_i \\cdot e_j))$ Definition 6 As above, let us denote the average of the set $\\lbrace \\parallel c_P \\parallel ^2 : P \\in \\mathcal {P}_n\\rbrace $ as $\\parallel c_n \\parallel ^2$ , the average center of mass of $\\mathcal {P}_n$ .",
"Lemma 5 $\\displaystyle \\parallel c_n \\parallel ^2 = \\frac{n+1}{12} .$ From Lemma REF , we have that for any $P \\in \\mathcal {P}_n$ , $\\displaystyle \\parallel c_P \\parallel ^2 = \\frac{1}{n^2} (\\sum _{k=1}^n (n-k+1)^2 + 2 \\sum _{1 \\le i < j \\le n} (n-j+1)(n-i+1)(e_i \\cdot e_j)).$ We will replace $(e_i \\cdot e_j)$ with $r_n$ to find $\\parallel c_n \\parallel ^2$ : $\\parallel c_n \\parallel ^2 &=& \\frac{1}{n^2} (\\sum _{k=1}^n (n-k+1)^2 + 2 \\sum _{1 \\le i < j \\le n} (n-j+1)(n-i+1)r_n).\\\\& = & \\frac{1}{n^2} (\\sum _{k=1}^n (n-k+1)^2) + \\frac{2}{n^2} \\sum _{j = 2}^n \\sum _{i = 1}^{j-1}\\left( (n-j+1)(n-i+1)\\left( \\frac{-1}{n-1}\\right)\\right).$ Simplifying the first term, we have $\\displaystyle \\frac{1}{n^2} (\\sum _{k=1}^n (n-k+1)^2) = \\frac{2n^2 + 3n +1}{6n}.$ Likewise the second term, $\\displaystyle \\frac{2}{n^2} \\sum _{j = 2}^n \\sum _{i = 1}^{j-1}\\left( (n-j+1)(n-i+1)\\left( \\frac{-1}{n-1}\\right)\\right)$ is simplified to $\\frac{-1}{n^2(n-1)} \\sum _{j = 2}^n \\left(j^3 - (3n+4)j^2 + (2n^2 + 7n +5)j - (2n^2+4n+2) \\right).$ The fact that we are summing from $j = 2$ rather than $j = 1$ complicates this calculation.",
"To fix this, we change the sum so that it starts at $j = 1$ , and at the end, we will subtract off the $j = 1$ term: $1 - 3n-4 + 2n^2 + 7n +5 - 2n^2-4n-2 = 0.$ Because the $j = 1$ term is 0, we may change the summation to start at $j = 1$ .",
"This allows further simplification, and the second term is finally $ \\displaystyle -\\left(\\frac{3n^4 + 2n^3 -3n^2 - 2n}{12n^2(n-1)}\\right).$ Combining these two terms, we have $\\parallel c_P \\parallel ^2 & = & \\frac{2n^2 + 3n +1}{6n} - \\frac{3n^4 + 2n^3 -3n^2 - 2n}{12n^2(n-1)} \\\\&=& \\frac{(n+1)}{12}\\\\$"
],
[
"Squared Radius of Gyration",
"With the average center of mass defined, we may ask for the average squared difference between the center of mass and the vertices.",
"This average difference is precisely the average squared radius of gyration.",
"Definition 7 Let $P \\in \\mathcal {P}_n$ with vertices $v_1,...,v_n$ .",
"With $c_P$ as above, define the squared radius of gyration of $P$ to be $\\displaystyle R_G^2(P) = \\frac{1}{n} \\sum _{k=1}^n \\parallel v_k - c_P \\parallel ^2.$ Definition 8 We will define the average of the set $\\lbrace R_G^2(P) : P\\in \\mathcal {P}_n\\rbrace $ to be $R_{G,n}^2$ , the average squared radius of gyration of $\\mathcal {P}_n$ .",
"Theorem 6 The average squared radius of gyration for all $P \\in \\mathcal {P}_n$ is $R_{G,n}^2 = \\frac{n+1}{12}$ .",
"For any $P \\in \\mathcal {P}_n$ , we have that $\\displaystyle R_G^2(P) &=& \\frac{1}{n} \\sum _{k=1}^n \\parallel v_k - c_P \\parallel ^2\\\\&=& \\frac{1}{n} \\sum _{k=1}^n (\\parallel v_k \\parallel ^2 - 2 (v_k \\cdot c_P) + \\parallel c_P \\parallel ^2) \\\\&=& \\frac{1}{n}\\sum _{k=1}^n \\parallel v_k \\parallel ^2 - 2 ( \\sum _{k=1}^n\\frac{v_k}{n} \\cdot c_P) + \\frac{1}{n}(n\\parallel c_P \\parallel ^2) \\\\&=& \\frac{1}{n}\\sum _{k=1}^n \\parallel v_k \\parallel ^2 - 2 ( c_P \\cdot c_P)+ \\parallel c_P \\parallel ^2\\\\&=& \\frac{1}{n}\\sum _{k=1}^n \\parallel v_k \\parallel ^2 - \\parallel c_P \\parallel ^2$ From Theorem REF we have that the average value of $ \\parallel v_k \\parallel ^2 $ is $\\frac{k(n-k)}{n-1} $ .",
"From Lemma REF we have that the average value of $\\parallel c_P \\parallel ^2$ is $ \\frac{n+1}{12}$ .",
"Replacing these, we can find the average squared radius of gyration over all $P \\in \\mathcal {P}_n$ , that is, $R_{G,n}^2$ (rather than $R_G^2(P) $ as above.)",
"$\\displaystyle R_{G,n}^2&=& \\left(\\frac{1}{n}\\sum _{k=1}^n \\frac{k(n-k)}{n-1}\\right) - \\frac{n+1}{12} \\\\& = & \\frac{n^2 - 1 }{12(n-1)} = \\frac{n+1}{12} = \\frac{n}{12} + \\frac{1}{12}.$ Therefore the average squared radius of gyration scales like $\\frac{n}{12}$ , agreeing with Zimm and Stockmayer's estimate [27]."
],
[
"Average Squared Radius of Gyration of Subsegments of Length $k$",
"The scaling of the average squared radius of gyration of ideal rings and chains is influenced by the presence of knotting [16].",
"As a consequence, one expects that the average squared radius of gyration of subsegments will also be affected by the presence or absence of knotting.",
"Determining a correlation between knotting and shape characteristics will be affected by the nature of knotting.",
"If the knotted sections of the rings are small, involving relatively few edges, we may not see the influence of those edges in the squared radius of gyration, which can be obscured by the behavior of the unknotted complement.",
"In order to detect local behavior, we study characteristics of shape that are local.",
"In open chains, the average squared radius of gyration of subsegments of length $k$ inside of larger chains of length $n$ is identical to the average squared radius of gyration of segments of length $k$ .",
"That is, the ambient chain has no effect of the shape of the subsegment.",
"This is not true for ideal rings, where we will see that the closure constraint plays a pivotal role in local shape.",
"However, these characteristics may play a pivotal role in identifying and characterizing the knotted portions of open chains.",
"Definition 9 Let $P \\in \\mathcal {P}_n$ .",
"Define $P_{i,k}$ to be the translated subsegment of $P$ of length $k$ beginning with the $i^{\\textrm {th}}$ edge vector, $e_i$ .",
"That is, $P_{i,k}$ is a segment starting at the origin, where the $j^{\\textrm {th}}$ vertex is given by $\\displaystyle \\sum _{m = i}^{i+j} e_m$ .",
"For ease of notation, we will relabel the position and edge vectors so that the $j^{\\textrm {th}}$ vertex is given by $v_j^{\\prime } = \\displaystyle \\sum _{m = 1}^{j} e_m^{\\prime }$ .",
"So $P_{i,k}$ is isomorphic to a subsegment of $P$ , though we've done some relabeling.",
"Now we would like to find the center of mass and the squared radius of gyration of $P_{i,k}$ .",
"Definition 10 Let $\\displaystyle c_{P_{i,k}} = \\frac{1}{k} \\sum _{j = 1}^{k} v_j^{\\prime } $ , the translated center of mass of $P_{i,k}$.",
"Then we define the squared radius of gyration of $P_{i,k}$ to be $R_{G}^2(P_{i,k}) = \\frac{1}{k} \\displaystyle \\sum _{j=1}^{k} \\parallel v_j^{\\prime } - c_{P_{i,k}}\\parallel ^2.$ For $P \\in \\mathcal {P}$ and $k < n$ , define the average squared radius of gyration of subsegments of length $k$ as $R_{G,k}^2(P) = \\frac{1}{n} \\displaystyle \\sum _{j = 1}^{n} R_{G}^2(P_{i,k}) .$ Now, we have the analogous definitions when we take the averages of $c_{P_{i,k}}$ and $R_{G,k}^2(P)$ over all $P \\in \\mathcal {P}_n$ .",
"Definition 11 Let $c_{k,n}$ be average center of mass of a translated subsegment of length $k$, the average of $\\lbrace c_{P_{i,k}}:i \\le n\\textrm { and }P\\in \\mathcal {P}_n\\rbrace $ .",
"For some $n$ , let $R_{G,n,k}^2$ be the average squared radius of gyration of subsegment of length $k$ to be the average of $\\lbrace R_{G,k}^2(P): P \\in \\mathcal {P}_n\\rbrace $ .",
"As in the last section, in order to compute $R_{G,n,k}^2$ , we must first find $c_{k,n}$ .",
"Lemma 7 $\\displaystyle \\parallel c_{k,n} \\parallel ^2 = \\frac{(2k^2 + 3k + 1)2n -3k(k+1)^2}{12k(n-1)}.$ Let us begin with some $\\displaystyle \\parallel c_{P_{m,k}} \\parallel ^2.",
"$ $\\displaystyle \\parallel c_{P_{m,k}} \\parallel ^2 &=& \\frac{1}{k^2} (\\sum _{j=1}^{k} v_j^{\\prime } \\cdot \\sum _{j=1}^{k} v_j^{\\prime })\\\\&=& \\frac{1}{k^2} (\\sum _{j=1}^{k} \\sum _{i = 1}^j e_i^{\\prime } \\cdot \\sum _{j=1}^{k} \\sum _{i = 1}^j e_i^{\\prime })\\\\&=& \\frac{1}{k^2} \\left(\\sum _{i=1}^{k} (k - i + 1)^2 + 2 \\sum _{j=2}^{k}\\sum _{i = 1}^{j-1} (k - j + 1)(k - i + 1)(e_i^{\\prime } \\cdot e_j^{\\prime })\\right)\\\\$ Because $(e_i^{\\prime } \\cdot e_j^{\\prime }) = (e_{i+m}^{\\prime } \\cdot e_{j+m}^{\\prime })$ , the product is still independent of $i$ and $j$ , and we can replace $(e_i^{\\prime } \\cdot e_j^{\\prime })$ with $\\displaystyle r_n = \\frac{-1}{n-1}$ .",
"This will let us take the average over all such $P_{m,k}$ to obtain $\\parallel c_{k,n} \\parallel ^2$ : $\\displaystyle \\parallel c_{k,n} \\parallel ^2 = \\frac{1}{k^2} \\left(\\sum _{i=1}^{k} (k - i + 1)^2 + 2 \\sum _{j=2}^{k}\\sum _{i = 1}^{j-1} (k - j + 1)(k - i + 1)\\left(\\frac{-1}{n-1}\\right)\\right).",
"$ There are two sums to evaluate.",
"The first is straightforward: $\\displaystyle \\sum _{i=1}^{k} (k - i + 1)^2= \\frac{k(k+1)(2k+1)}{6}$ The second sum is $S = \\displaystyle 2 \\sum _{j=2}^{k}\\sum _{i = 1}^{j-1} (k - j + 1)(k - i + 1)\\left(\\frac{-1}{n-1}\\right).$ $S & = & \\frac{-2}{n-1} \\sum _{j=2}^{k}\\sum _{i = 1}^{j-1} (k - j + 1)(k - i + 1)\\\\& = & \\frac{-1}{n-1} \\sum _{j=2}^{k}\\left(j^3 - (3k +4)j^2 + (2k^2 +7k +5) j - 2(k+1)^2 \\right)\\\\$ Evaluating $j^3 - (3k +4)j^2 + (2k^2 +7k +5) j - 2(k+1)^2$ at $j = 1$ we have $1 - 3k +4 + 2k^2 +7k +5 - 2k^2 - 4k -2 = 0$ .",
"Thus we can replace the lower bound of our sum, $j = 2$ with $j = 1$ with no penalty.",
"$S & = & \\frac{-1}{n-1} \\sum _{j=1}^{k}\\left(j^3 - (3k +4)j^2 + (2k^2 +7k +5) j - 2(k+1)^2 \\right)\\\\& = & \\frac{ 2k + 3k^2 - 2k^3 - 3k^4}{12(n-1)}$ Combining these terms, we have $\\parallel c_{k,n} \\parallel ^2 &=& \\frac{1}{k^2}(\\frac{k(k+1)(2k+1)}{6} + \\frac{ 2k + 3k^2 - 2k^3 - 3k^4}{12(n-1)})\\\\& = & \\frac{(2k^2 + 3k + 1)2n - 3k(k+1)^2}{12k(n-1)}$ We note that when $k = n$ , we have $\\displaystyle \\parallel c_{n,n} \\parallel ^2 & = & \\frac{(2n^2 + 3n + 1)2n - 3n(n+1)^2}{12n(n-1)}\\\\& = & \\frac{n^3 - n}{12n(n-1)} = \\frac{n+1}{12}\\\\$ which agrees with Lemma REF .",
"Theorem 8 The average squared radius of gyration of a subsegment of length $k$ , taken over all such subsegments in all $P \\in \\mathcal {P}_n$ is $R_{G,n,k}^2 = \\frac{(k^2 - 1)(2n - k)}{12k(n-1)}$ .",
"For some $P_{i,k}$ we have $R_{G}^2(P_{i,k}) &=& \\frac{1}{k} \\sum _{j=1}^{k} \\parallel v_j^{\\prime } - c_{P_{i,k}}\\parallel ^2\\\\&=& \\frac{1}{k}\\sum _{j=1}^{k}\\parallel v_j^{\\prime }\\parallel - 2 (\\frac{1}{k}\\sum _{j=1}^{k} v_j^{\\prime } \\cdot c_{P_{i,k}}) + \\frac{1}{k}\\sum _{j=1}^{k}\\parallel c_{P_{i,k}}\\parallel ^2\\\\&=& \\frac{1}{k}\\sum _{j=1}^{k}\\parallel v_j^{\\prime }\\parallel - 2 ( c_{P_{i,k}}\\cdot c_{P_{i,k}}) + \\frac{1}{k}k\\parallel c_{P_{i,k}}\\parallel ^2\\\\&=& \\frac{1}{k}\\sum _{j=1}^{k}\\parallel v_j^{\\prime }\\parallel - \\parallel c_{P_{i,k}}\\parallel ^2\\\\$ From Lemma REF , $\\parallel c_{P_{i,k}}\\parallel ^2$ is, on average, $\\displaystyle \\parallel c_{k,n} \\parallel ^2 =\\frac{(2k^2 + 3k + 1)2n - 3k(k+1)^2}{12k(n-1)}$ .",
"Likewise, average value of $\\parallel v_j^{\\prime }\\parallel ^2$ , the end to end distance of the segment $P_{i,k}$ , is given in Theorem REF .",
"By replacing $\\parallel c_{P_{i,k}}\\parallel ^2$ and $\\parallel v_j^{\\prime }\\parallel ^2$ with the averages for these values, we can find $R_{G,n,k}^2$ : $R_{G,n,k}^2 &=& \\frac{1}{k}\\sum _{j=1}^{k} \\frac{j(n-j)}{n-1} - \\frac{(2k^2 + 3k + 1)2n - 3k(k+1)^2}{12k(n-1)}\\\\&=&\\frac{(k^2 - 1)(2n - k)}{12k(n-1)}\\\\$ We note that when $k = n$ , we have $\\displaystyle R_{G,n,n}^2 &=& \\frac{(n^2 - 1)(n)}{12n(n-1)}\\\\& = & \\frac{n+1}{12}\\\\$ which agrees with Theorem REF .",
"Now for a fixed $n$ , we have a function that returns the average squared radius gyration of a subsegment of length $k$ ."
],
[
"Comparison Between Ideal Rings and Open Ideal Chains",
"For comparison, consider ideal chains.",
"From Lemma REF , which also holds for ideal chains, we know that for any $W \\in \\mathcal {W}_n$ , the squared center of mass is given by $\\parallel c_W \\parallel ^2 = \\frac{1}{n^2} \\left(\\sum _{k=1}^n (n-k+1)^2 + 2 \\sum _{1 \\le i < j \\le n} (n-j+1)(n-i+1)(e_i \\cdot e_j)\\right).$ When the average is taken over all ideal chains $W \\in \\mathcal {W}_n$ , define the average ${c_{\\mathcal {W}_n}}$ .",
"Likewise, let the squared radius of gyration of the ideal chain $W$ be $R_G^2(W)$ , and let the average over all such ideal chains be ${R_{G,\\mathcal {W}_n}^2}$ .",
"Lemma 9 $R_{G,\\mathcal {W}_n}^2 = \\frac{n^2 - 1}{6n}.$ First, we find $\\parallel c_{\\mathcal {W}_n} \\parallel ^2$ , which is simplified by that fact that in $\\mathcal {W}_n$ , the average value of $(e_i \\cdot e_j)$ is 0 for $i \\ne j$ .",
"$\\parallel c_{_n} \\parallel ^2 & = & \\frac{1}{n^2} \\left(\\sum _{k=1}^n (n-k+1)^2 + 2 \\sum _{1 \\le i < j \\le n} (n-j+1)(n-i+1)(e_i \\cdot e_j)\\right)\\\\& = & \\frac{1}{n^2} \\left(\\sum _{k=1}^n (n-k+1)^2\\right)\\\\& =& \\frac{2n^2 + 3n +1}{6n}.$ As in the first part of Theorem REF , the squared radius of gyration of some open chain, $R_G^2(W)$ , is $R_G^2(W) = \\frac{1}{n}\\sum _{k=1}^n \\parallel v_k \\parallel ^2 - \\parallel c_W \\parallel ^2.$ Replacing $\\parallel c_W \\parallel ^2$ with the average $\\parallel c_{\\mathcal {W}_n} \\parallel ^2$ and $\\parallel v_k \\parallel ^2$ with its average value, $k$ , from Lemma REF , we can find the average radius of gyration for open chains, $R_{G,{\\mathcal {W}_n}}^2$ : $R_{G,{\\mathcal {W}_n}}^2 & = & \\frac{1}{n}\\sum _{k=1}^n k - \\parallel c_{\\mathcal {W}_n} \\parallel ^2 \\\\& = & \\frac{1}{n} \\left(\\frac{n(n+1)}{2}\\right) - \\frac{2n^2 + 3n +1}{6n} \\\\& = & \\frac{n^2 - 1}{6n}.$ For open chains, the squared radius of gyration of a subsegment of length $k$ is the same as for a chain of length $k$ .",
"So we may compare $R_{G,n,k}^2 =\\displaystyle \\frac{(k^2 - 1)(2n - k)}{12k(n-1)}$ and $\\displaystyle R_{G,\\mathcal {W}_n}^2 = \\frac{k^2 - 1}{6k}$ .",
"Figure: Here we compare the average squared radius of gyration of subsegments of length kk for open chains (blue) and ideal rings (red and dashed).As with squared end to end distance, we can see that the average squared radius of gyration of subsegments in an ideal ring is radically different from the average squared radius of gyration of subsegments in an open ideal chain.",
"For ideal rings and chains of length $n$ , in order to have the averages squared radius of gyration of a subsegment of length $k$ for ideal rings to be within $\\frac{1}{100}$ of the squared radius of gyration of a subsegment of length $k$ in an ideal chain, we must have each other, we must have that the length of the subsegment considered, $k$ , is less than $\\frac{n}{50}$ .",
"So for $n=1000$ , $k$ must be 20 or less.",
"Recall, with squared end to end distance, for the averages to be within $1\\%$ of each other, the segments had to be less than $\\frac{1}{100}^{\\textrm {th}}$ of the length, so $k$ had to be length 10 or shorter if $n = 1000$ .",
"This suggests that the squared end to end distance characterization is more sensitive than squared radius of gyration of subsegments, as it is more affected by the closure condition.",
"Once again, we observe that the local scale of an ideal ring differs significantly from that of a ideal chain at all but the smallest length scales."
],
[
"Experimental Methods",
"Generating a random sample from $\\mathcal {P}_n$ is much trickier than generating a random walk.",
"With the latter, we can uniformly sample points edge vectors on the unit sphere.",
"The complication in $\\mathcal {P}_n$ is the closure constraint: we cannot just generate random walks and hope that they close.",
"In order to randomly sample $\\mathcal {P}_n$ we have two step process called the hedgehog method [1].",
"First, we find a starting point, then we use crankshaft rotations to sample the space.",
"After some number of moves, our sample is independent of the starting point, and is a random element of $\\mathcal {P}_n$ ."
],
[
"Hedgehog Method",
"The hedgehog method begins with selecting $n$ points uniformly on the unit sphere, and label them $e_1, ... , e_n$ .",
"Then, we add to that list each $e_i$ 's negative, $-e_i$ .",
"Thus we have $2n$ edge vectors, $e_1, -e_1, e_2, -e_2, ...,e_n, -e_n$ .",
"Figure: The edge vectors e 1 ,-e 1 ,e 2 ,-e 2 ,...,e n ,-e n e_1, -e_1, e_2, -e_2, ...,e_n, -e_n are plotted above, hence this is called the hedgehog method.We randomly permute these, getting a list of edge vectors $e_1^{\\prime }, e_2^{\\prime }, ... e_{2n}^{\\prime }$ .",
"Adding in each edge vector's negative ensures the all important closure condition is met."
],
[
"Crankshaft Rotations",
"This is a good starting point, but for each edge vector, its exact opposite is also an edge vector, which is undesirable.",
"We finish the hedgehog method by performing crankshaft rotations.",
"We begin a crankshaft rotation by randomly selecting two non-parallel edge vectors $e_j$ and $e_k$ .",
"These are rotated about the axis determined by $e_j + e_k$ , by a random angle $\\theta $ .",
"The form of these rotations is given by the following: $e_j & \\longmapsto & \\frac{e_j + e_k}{2} + \\frac{e_j - e_k}{2}\\cos (\\theta ) + \\frac{e_j \\times e_k}{\\parallel e_j + e_k \\parallel }\\sin (\\theta )\\\\e_j & \\longmapsto & \\frac{e_k + e_j}{2} + \\frac{e_k - e_j}{2}\\cos (\\theta ) + \\frac{e_k \\times e_j}{\\parallel e_k + e_j \\parallel }\\sin (\\theta )$ Because the sum $e_j+e_k$ is conserved, the modified sequence of edge vectors and vertices still satisfies the closure condition.",
"Because any equilateral polygon can be deformed by a finite sequence of crankshaft rotations to the regular polygon, a finite series of crankshaft rotations will take us from one polygon to another random polygon [1].",
"We performed $6n$ crankshaft rotations on each sample polygon."
],
[
"Experimental Results",
"We use the above methods to randomly generate ideal rings.",
"Numerically, we will have two primary foci: how quickly does the average for an ensemble converge to these theoretical values, and how will knotting affect these characteristics?"
],
[
"Convergence",
"For various population sizes, $10^1$ , $10^2$ , $10^3$ , $10^4$ and $10^5$ , we generated that many 50 edged polygons with 150 crankshaft rotations.",
"For each polygon, we found the squared radius of gyration, and took the average over the population.",
"For each population size, we did this 10 times.",
"We then compared these averages to the theoretical average.",
"We define $E(n)$ to be the difference between the experimental average and the theoretical average with $n$ samples.",
"Figure REF shows the convergence result.",
"Figure: For population sizes 10 1 10^1, 10 2 10^2, 10 3 10^3, 10 4 10^4 and 10 5 10^5 we computed the average radius of gyration, and plotted it against the theoretical value for the same length.",
"Note that this is a log scale plot of the data.For population sizes of 10000 and larger, we have excellent convergence to the theoretical average.",
"Using linear regression, we can estimate that $E(n)\\sim Cn^{-1.559}$ , providing for excellent convergence for large $n$ ."
],
[
"Knotting",
"Knotted ideal rings, specifically trefoils, were generated by first constructing ideal rings of length 50, then calculating their knot type, and saving those of the given knot type.",
"By the length of a knot in a ring we mean the length of the shortest subchain that contains the knot.",
"The knot length is determined by examining subsegments of progressively longer length, starting at all possible locations.",
"Each open segment is situated inside of a large sphere.",
"A random collection of points on the surface of the sphere is selected.",
"The ends of the segment are closed to each point, and the knot type of each of these closures is then calculated.",
"This yields a spectrum of knot types, as in [14], [15], [18], [19], [23].",
"A segment is considered to be a trefoil if the closure is a trefoil with some tolerance (greater than 50%).",
"The knotted portion is identified as the shortest segment which is a trefoil, for which the complement is unknotted.",
"Figure: Here we compare the average squared end to end distance for a phantom population of ideal rings (blue) and a population of randomly sampled trefoils (red and dashed).Above we compare the average squared end to end distance for a phantom population of ideal rings and a population of randomly sampled trefoils.",
"We can see that for length 50, the average for the trefoils is smaller than the average for the whole space.",
"This suggests that for length 50, knotting compresses the polygon, making vertices closer together.",
"Looking at the maximum squared end to end distance, we may ask what length curve has the same maximum, $10.2291$ .",
"Solving, we have that a curve of $39.89$ , approximately 40, has the given maximum.",
"That suggests that the average shortening caused by knotting is about 10 edges of length, and that on average a trefoil of length 50 has an effective length of 40.",
"The average length of the knotted portion of these trefoils is $16.4$ .",
"We predict that the difference between the length 10 reduction we saw above and the average trefoil length, $16.4$ , can be accounted for by examining the end to end distance of the knotted portion.",
"As in Figure REF , the difference could be explained by an average knotted portion of 16 to 17 edges with an average end to end distance around 6.",
"Figure: A hypothetical average knotted section with knot length 16 edges and end to end distance, (red and dashed), of 6.Figure: Here we compare the average squared radius of gyration of subsegments of a phantom population of ideal rings (blue) and of a population of randomly sampled trefoils (red and dashed).As with squared end to end distance, the average for the trefoils is smaller than the average for the whole space.",
"Again, this suggests that for length 50, knotting compresses the polygon, making vertices closer together.",
"We know that ideal trefoils, and indeed any collection of some fixed knot type, will for some small number of edges have mean squared radius of gyration less than the the mean for the total population, and for larger number of edges, they will have a greater average squared radius of gyration [5].",
"For example, the average squared radius of gyration of trefoils is smaller than the average squared radius of gyration of the whole population for lengths less than 175, and the opposite is true for lengths greater than 200 [5].",
"We expect similar behavior for squared end to end distance.",
"We can set the average squared radius of gyration of the population of trefoils, $3.5768$ equal to our function for squared radius of gyration, $\\frac{n+1}{12}$ .",
"Solving for $n$ we have $n = 41.9216$ , suggesting that the average shortening caused by knotting is about 8 edges of length.",
"We can compare this with the average shortening prediction from squared end to end distance, 10.",
"These differ by $4\\%$ of the total length, 50."
],
[
"Conclusion",
"These theoretical averages have many potential applications.",
"Primarily, they can be used as a criterion to determine effectiveness of sampling methods.",
"By comparing the the squared radius of gyration of subsegments or the squared internal end to end distance of polygons generated a given sampling method and the theoretical average, we can determine the effectiveness of the generation scheme and corroborate numerical simulations.",
"Further, in the above generation method, we can use this convergence to determine how many crankshaft rotations are needed to sample the space of polygons uniformly, as in [1].",
"As the previous section highlights, squared end to end distance and squared radius of gyration of subsegments may be used to predict knot length, which is computationally expensive to calculate.",
"This will allow us to examine the growth of the knot length as $n \\rightarrow \\infty $ , to determine if average knot length is bounded, or grows proportionally with $n^{1/2}$ or $n$ , allowing us to ascertain if knotting is strongly local, local or global on average [4], [7], [12], [14], [15], [17], [20]."
],
[
"Acknowledgments",
"The authors would like to thank Eric Rawdon, both for his wonderful data and feedback on the paper.",
"Eric was responsible for finding the trefoils and their knot lengths, a truly impressive computational task."
]
] | 1204.1366 |
[
[
"Elliptic curves with p-Selmer growth for all p"
],
[
"Abstract It is known, that for every elliptic curve over Q there exists a quadratic extension in which the rank does not go up.",
"For a large class of elliptic curves, the same is known with the rank replaced by the 2-Selmer group.",
"We show, however, that there exists a large supply of semistable elliptic curves E/Q whose 2-Selmer group goes up in every bi-quadratic extension and for any odd prime p, the p-Selmer group goes up in every D_{2p}-extension and every elementary abelian p-extension of rank at least 2.",
"We provide a simple criterion for an elliptic curve over an arbitrary number field to exhibit this behaviour.",
"We also discuss generalisations to other Galois groups."
],
[
"Introduction",
"In [6] it is shown that there exist elliptic curves over number fields for which in every quadratic extension of the base field either the rank goes up or the Tate-Shafarevich group becomes infinite.",
"Equivalently, every quadratic twist of such a curve has either positive rank or an infinite Tate-Shafarevich group (the latter is of course conjectured to never happen).",
"Over $$ , such curves do not exist by the combined work of Kolyvagin [10] and Bump–Friedberg–Hoffstein–Murty–Murty–Waldspurger [3], [12], [16].",
"In fact, it is conjectured that half of all quadratic twists of an elliptic curve over $$ have rank 0.",
"Moreover, if $E/$ has no cyclic 4-isogeny, then there exists a quadratic extension $F/$ such that the size of the 2-Selmer group of $E$ over $F$ is the same as over $$ [15], [11], [9].",
"As we shall show, however, if we allow only slightly bigger extensions of $$ than quadratic, then there are lots of elliptic curves over $$ whose Selmer groups grow in size in all such extensions.",
"Below, $S^p(E/F)$ will denote the $p$ -Selmer group of $E$ over a number field $F$ , which we recall to be an elementary abelian $p$ -group defined as the the kernel of the map $H^1(F,E[p])\\longrightarrow \\prod _{} H^1(F_{},E)$ .",
"Here, the product runs over all places of $F$ , and each $(\\bar{F}_/F_)$ is identified with a subgroup of $(\\bar{F}/F)$ .",
"Let $E/K$ be a semistable elliptic curve over a number field with positive rank.",
"Let $p$ and $G$ be one of the following combinations of a prime number and a finite group: $p=2$ , $G\\cong C_2\\times C_2$ ; $p$ is odd, $G\\cong D_{2p}$ , the dihedral group of order $2p$ ; $p$ is odd, and either $G\\cong C_p\\times C_p$ or $G\\cong C_p\\rtimes C_q$ , where $q$ is an odd prime, and $C_q$ acts faithfully on $C_p$ .",
"Suppose further that if $p$ and $G$ are as in (REF ), then the rank of $E/K$ is greater than the number of primes of $K$ at which $E$ has non-split multiplicative reduction, while if $p$ and $G$ are as in (REF ), then the rank of $E/K$ is greater than the number of primes $v$ of non-split multiplicative reduction for which $_v(\\Delta (E))$ is even.",
"Then, the following conclusions hold.",
"If $(E/K)[p^\\infty ]$ is finite, then we have that for every Galois extension $F/K$ with Galois group $G$ , either the order of the $p$ -primary part of the Tate-Shafarevich group changes at some step in $F/K$ , or $E(F)\\supsetneq E(K)$ .",
"If $(E/K)[p]=0$ , then for every Galois extension $F/K$ with Galois group $G$ , either $\\#(E/F)[p^\\infty ] >0$ , or $E(F)\\supsetneq E(K)$ .",
"If $(E/K)[p]=0$ and $E(K)[p]=0$ , then for every Galois extension $F/K$ with Galois group $G$ , we have $\\#S^p(E/F) > \\#S^p(E/K)$ .",
"The first few curves over $$ in Cremona's database that satisfy the hypotheses of Theorem (REF ) and (REF ) for all the combinations of $p$ and $G$ listed there are 91b1, 91b2, 91b3, 123a1, 123a2, 141a1, 142a1, 155a1, all of rank 1 and with no primes of non-split multiplicative reduction, and with trivial Tate-Shafarevich groups [8].",
"Out of these, 91b3, 123a2, 141a1, 142a1 have trivial torsion subgroup over $$ and thus also satisfy the stronger hypothesis of Theorem (REF ) for all the pairs $p$ , $G$ .",
"Also, the huge majority of rank 2 curves over $$ are expected to have trivial Tate-Shafarevich groups [4].",
"For example the curve with Cremona label 389a1 almost certainly satisfies all the hypotheses of Theorem () for all $p$ , $G$ .",
"Unfortunately, we do not even know that the Tate-Shafarevich group is finite for a single rank 2 elliptic curve.",
"In principle, triviality of the $p$ -part for any fixed prime $p$ can be checked algorithmically using descent (see [14] and the references therein).",
"Such checks have been performed for thousands of higher rank curves, including 389a1, and thousands of primes in [13] (using methods quite different from descent).",
"The following example illustrates that the above results are not a parity phenomenon.",
"Let $E$ be the curve with Cremona label 65a1.",
"It has rank 1 over $$ and square-free discriminant, and $(E/)$ is trivial, so the conclusion of Theorem (REF ) holds for all combinations of $p$ and $G$ listed in the theorem, and in particular for $p=2$ , $G\\cong C_2\\times C_2$ .",
"Let $F$ be the bi-quadratic field $(\\sqrt{3},\\sqrt{5})$ .",
"Then we have $E() = E(F)$ .",
"However, $(E/F)[2]\\cong C_2\\times C_2$ , as predicted by Theorem .",
"This cannot be detected by root numbers.",
"We will collect the necessary ingredients of the proof in great generality, with no assumption on the Galois group of the extension $F/K$ , although we will simplify the exposition by assuming early on that $E$ is semistable.",
"Then we will perform the necessary calculations in the case of dihedral and bi-quadratic extensions, thereby proving Theorem for (REF ) and (REF ).",
"In the last section, we will discuss generalisations to other Galois groups, such as those of Theorem (REF ).",
"We will also explain why our approach cannot be pushed any further than that.",
"This will rely on a certain representation theoretic classification [2].",
"I would like to thank Vladimir Dokchitser for several very helpful remarks.",
"I would also like to thank the anonymous referee for carefully reading the manuscript and for suggestions that helped greatly improve the exposition."
],
[
"Tamagawa numbers and regulators",
"We begin with a simplifying reduction to a seemingly special case: Let $E/K$ be a semistable elliptic curve over a number field.",
"Let the prime $p$ and the finite group $G$ be one of the combinations of Theorem .",
"Theorem holds for all extensions $F/K$ with Galois group $G$ if and only if it holds for all $F/K$ with Galois group $G$ that satisfy the following additional conditions: $(E/F)[p^\\infty ]$ is finite (and consequently, by the inflation-restriction exact sequence, so are the $p$ -primary parts of the Tate-Shafarevich groups over all subfields); $E(K)\\otimes _p = E(F)\\otimes _p$ under the natural inclusion map.",
"If $(E/K)[p^\\infty ]$ is infinite, then Theorem is empty.",
"Also, if $(E/K)[p^\\infty ]$ is finite, then clearly the conclusions of Theorem automatically hold for all $F/K$ with Galois group $G$ for which $(E/F)[p^\\infty ]$ is infinite.",
"This proves (REF ).",
"Finally, if $E(K)\\otimes _p \\ne E(F)\\otimes _p$ , then either $(E/F)>(E/K)$ , or a point of infinite order on $E(K)$ becomes $p$ -divisible over $F$ , or $\\#E(F)[p^\\infty ] > \\#E(K)[p^\\infty ]$ .",
"In all three cases, the conclusions (REF ) and (REF ) of Theorem obviously hold.",
"As for (REF ), certainly if $(E/F)>(E/K)$ or if $\\#E(F)[p^\\infty ] > \\#E(K)[p^\\infty ]$ , then the assumption $(E/K)[p]=0$ forces $\\#S^p(E/F) > \\#S^p(E/K)$ .",
"If on the other hand a point of infinite order on $E(K)$ becomes $p$ -divisible, then by Kummer theory, $E(F)[p]$ must be non-trivial, so the assumption $E(K)[p]=0$ again forces $\\#S^p(E/F) > \\#S^p(E/K)$ .",
"We will therefore henceforth restrict our attention to semistable elliptic curves $E/K$ and to Galois extensions $F/K$ such that $E/F$ satisfies the additional conditions of Lemma .",
"Let $G$ be a finite group.",
"A formal $$ -linear combination of representatives of conjugacy classes of subgroups $\\Theta =\\sum _H n_HH$ is called a Brauer relation if the virtual permutation representation $\\bigoplus G/H]^{\\oplus n_H}$ is zero.",
"For a detailed discussion of the concept of Brauer relations, see the introduction to [2].",
"Let $E/K$ be an elliptic curve over a number field, $F/K$ a Galois extension with Galois group $G$ , and $\\Theta =\\sum _H n_HH$ a Brauer relation.",
"There is a corresponding relation between $L$ -functions of $E$ over the intermediate fields: $\\prod _H L(E/F^H,s)^{n_H}=1.$ If $E$ is semistable and has finite Tate-Shafarevich groups over all intermediate extensions of $F/K$ , then a combination of various compatibility results on the Birch and Swinnerton-Dyer conjecture yields a relationsee [7] and the remarks at the beginning of [7], as well as [1] between arithmetic invariants of $E$ over the intermediate fields.",
"We shall recall the necessary notation shortly: $\\prod _H \\left(\\frac{c(E/F^H)\\#(E/F^H)(E/F^H)}{|E(F^H)_{}|^2}\\right)^{n_H} = 1.$ Moreover, if only the $p$ -primary parts of the Tate-Shafarevich groups are assumed to be finite for some prime $p$ , then the $p$ -part of equation (REF ) holds.",
"Here, $c$ denotes the product of Tamagawa numbersIt is here that we use the assumption that $E$ is semistable.",
"Otherwise, the Tamagawa numbers have to be re-normalised.",
"See [7] for the general case.",
"of $E$ over the finite places of the respective field.",
"Recall that the regulator of an elliptic curve is the determinant of the Néron-Tate height pairing evaluated on any basis of the free part of the Mordell-Weil group.",
"Note that since each of the regulators is some real number, in general transcendental, it does not make any sense to speak of its $p$ -part.",
"However, since the quotient $\\prod _H (E/F^H)^{n_H}$ is a rational number (this is an immediate consequence of [5]), it does make sense to speak of the $p$ -parts of the regulator quotient and of the remaining terms.",
"The precise normalisation of the Néron-Tate height pairing that enters the Birch and Swinnerton-Dyer conjecture will be crucial for us.",
"If $M/K$ is a finite extension of fields, and if $\\langle \\cdot ,\\cdot \\rangle _K$ , respectively $\\langle \\cdot ,\\cdot \\rangle _M$ denotes the Néron-Tate height pairing on $E(K)$ , respectively on $E(M)$ , then for any $P,Q\\in E(K)$ , $\\langle P,Q\\rangle _M = [M:K]\\langle P,Q\\rangle _K$ .",
"In particular, if $E/K$ does not acquire any new points of infinite order over $F$ , then the regulator quotient in (REF ) does not vanish in general, but rather $\\prod _H (E/F^H)^{n_H} = \\prod _H \\frac{1}{|H|^{n_HE(K)}}.$ More generally, if $E(K)\\otimes _p = E(F)\\otimes _p$ , then the $p$ -part of equation (REF ) holds.",
"In summary, if $E/K$ is a semistable elliptic curve with $(E/F)[p^\\infty ]$ finite and if $E(K)\\otimes _p= E(F)\\otimes _p$ , then $\\prod _H \\#(E/F^H)^{n_H}c(E/F^H)^{n_H} =_{p^{\\prime }} \\prod |H|^{n_HE(K)}.$ Here, $=_{p^{\\prime }}$ means that both sides have the same $p$ -adic valuation."
],
[
"Dihedral extensions",
"Suppose that $G\\cong D_{2p}$ , where $p$ is an odd prime.",
"There is a Brauer relation in $G$ of the form $\\Theta = 1 - 2C_2 - C_p + 2G$ .",
"For this relation, we have $\\prod |H|^{n_H} = \\frac{(2p)^2}{4p} = p,$ so that the right hand side of equation (REF ) is $p^{E(K)}$ .",
"If $v$ is a place of $K$ , write $c_v(E/K)$ for the Tamagawa number at $v$ , and $c_v(E/F^H)$ for the product of Tamagawa numbers at places of $F^H$ above $v$ .",
"Write $c_v(E/\\Theta )$ for the quotient $\\frac{c_v(E/K)^2c_v(E/F)}{c_v(E/F^{C_2})^2c_v(E/F^{C_p})}$ .",
"Similarly, write $\\#(E/\\Theta )[p^\\infty ]$ for the $p$ -primary part of the corresponding quotient of sizes of Tate-Shafarevich groups.",
"Finally, let $M$ denote the intermediate quadratic extension $M=F^{C_p}$ .",
"The following table gives the possible values of the $p$ -part of $c_v(E/\\Theta )$ , depending on the reduction type of $E$ at $v$ (horizontal axis) and on the splitting behaviour of $v$ in $F/K$ (vertical axis): Table: NO_CAPTIONIt follows immediately from this table and from equation (REF ) that if $E/K$ is semistable, if $(E/F)[p^\\infty ]$ is finite, if $E(K)\\otimes _p= E(F)\\otimes _p$ , and if the rank of $E/K$ is greater than the number of primes of non-split multiplicative reduction, then $\\#(E/\\Theta )[p^\\infty ]>1$ , and thus at least one of $\\#(E/K)[p^\\infty ]$ , $\\#(E/F)[p^\\infty ]$ is strictly larger than at least one of $\\#(E/M)[p^\\infty ]$ , $\\#(E/F^{C_2})[p^\\infty ]$ .",
"This, together with Lemma concludes the proof of Theorem for $G\\cong D_{2p}$ ."
],
[
"Bi-quadratic extensions",
"We now apply the same reasoning to $G\\cong C_2\\times C_2$ , with $p=2$ .",
"Denote the three distinct subgroups of order 2 by $C_2^a$ , $C_2^b$ , $C_2^c$ .",
"The space of Brauer relations in $G$ is generated by the relation $\\Theta = 1 - C_2^a - C_2^b - C_2^c +2G$ , for which we have $\\prod |H|^{n_H} = \\frac{16}{8} = 2.$ Again writing $c_v(E/\\Theta )$ for the corresponding quotient of Tamagawa numbers of $E$ at places above $v$ over the corresponding intermediate fields of $F/K$ , the following table gives the possible values of $c_v(E/\\Theta )$ : Table: NO_CAPTIONAs above, this table together with equation (REF ) and with Lemma proves Theorem for $G\\cong C_2\\times C_2$ ."
],
[
"Generalisation to other Galois groups",
"If $G$ is a subgroup of a group $\\tilde{G}$ , then by transitivity of induction, a Brauer relation $\\Theta $ in $G$ automatically gives a Brauer relation $_{\\tilde{G}/G}\\Theta $ in $\\tilde{G}$ .",
"Also, if $G$ is a quotient of a group $\\Gamma $ , $G=\\Gamma /N$ , then a Brauer relation $\\Theta =\\sum _H n_HH$ in $G$ gives rise to a Brauer relation $_{\\Gamma /N}\\Theta =\\sum _Hn_HNH$ in $\\Gamma $ .",
"In general, in order to prove by the same technique as above that the size of the $p$ -Selmer group of some elliptic curve grows in all Galois extensions with Galois group $G$ , we need to have a Brauer relation $\\Theta =\\sum _H n_H H$ in $G$ such that $_p(\\prod |H|^{n_H})\\ne 0$ .",
"This quantity is clearly invariant under inductions and lifts of Brauer relations.",
"[[2], Corollary 9.2] Let $p$ be a prime number.",
"A finite group $\\tilde{G}$ has a Brauer relation $\\Theta =\\sum _H n_H H$ with $_p(_\\Theta ())\\ne 0$ if and only if $\\tilde{G}$ has a subquotient $G$ isomorphic either to $C_p\\times C_p$ or to $C_p\\rtimes C_q$ with $C_q$ cyclic of prime order acting faithfully on $C_p$ .",
"Moreover, in the former case $\\Theta $ can be taken to be induced and/or lifted from the relation $1 - \\sum _{U\\le _p G} U + pG$ ; while in the latter case $\\Theta $ can be taken to be induced and/or lifted from the relation $1 - q C_q - C_p + qG$ .",
"Suppose that $\\tilde{G}$ is a finite group that has a quotient isomorphic to $G$ .",
"If some inequalities between the rank of an elliptic curve and the number of places of certain reduction types ensure that sizes of Tate-Shafarevich groups or Mordell-Weil groups grow in all $G$ -extensions of number fields, as in Theorem , then clearly the same inequalities imply the same conclusions for $\\tilde{G}$ -extensions.",
"On the other hand, suppose that $G$ is a subgroup of a finite group $\\tilde{G}$ .",
"Suppose that some condition on the rank of $E$ and on the number of places of given reduction types implies the conclusions of Theorem (REF ), say, for all Galois extensions with Galois group $G$ .",
"Some care is then needed to deduce the same result for $\\tilde{G}$ -extensions.",
"Indeed, if $F/K$ is Galois with Galois group $\\tilde{G}$ , then one would like to deduce the conclusion of Theorem (REF ) for $F/K$ by applying the theorem to the extension $F/F^G$ .",
"But in order to satisfy the required inequalities for the number of places of $F^G$ , one might need to impose stronger conditions on $E/K$ , since the number of places of given reduction type might grow in $F^G/K$ (by at most a factor of $[\\tilde{G}:G]$ ).",
"This is of course a straightforward modification.",
"Thus, we may restrict attention to the groups $C_p\\times C_p$ and $C_p\\rtimes C_q$ as in Proposition .",
"When $G\\cong C_p\\times C_p$ and $\\Theta =1 - \\sum _{U\\le _p G} U + pG$ , we have $\\prod |H|^{n_H} = \\frac{(p)^{2p}}{p^{p+1}} = p^{p-1}.$ When $G\\cong C_p\\rtimes C_q$ and $\\Theta =1 - q C_q - C_p + qG$ , we have $\\prod |H|^{n_H} = \\frac{(pq)^q}{pq^q} = p^{q-1}.$ Having already dealt with such groups of even order, we may now restrict our attention to groups of odd order, so that only the primes of $K$ at which $E$ has split multiplicative reduction contribute to the $p$ -part of the corresponding Tamagawa number quotients.",
"Here are the possible values of $c_v(E/\\Theta )$ when $E/K$ has split multiplicative reduction at $v$ , first for $G\\cong C_p\\times C_p$ and $\\Theta =1 - \\sum _{U\\le _p G} U + pG$ , and then for $G\\cong C_p\\rtimes C_q$ and $\\Theta =1 - q C_q - C_p + qG$ , where $p$ and $q$ are odd primes: Table: NO_CAPTION, Table: NO_CAPTION.",
"These tables together with equation (REF ) finish the proof of Theorem for $G\\cong C_p\\times C_p$ and $G\\cong C_p\\rtimes C_q$ ."
]
] | 1204.1166 |
[
[
"Microcanonical Jet-fragmentation in proton-proton collisions at LHC\n Energy"
],
[
"Abstract In this paper, we show that the distribution of the longitudinal momentum fraction of charged hadrons $dN/dz$ inside jets stemming from proton-proton collisions at $\\sqrt{s}$ = 7 TeV center of mass energy can be described by a statistical jet-fragmentation model.",
"This model combines microcanonical statistics and super-statistics induced by multiplicity fluctuations.",
"The resulting scale dependence of the parameters of the model turns out to be similar to what was observed in electron-positron annihilations in Urmossy, Barnaf\\\"oldi, and Bir\\'o."
],
[
"Introduction",
"When calculating the spectrum of hadrons produced in high-energy proton-proton ($pp$ ) collisions using perturbative quantum chromodynamics (pQCD) improved parton model calculations [1], hadron production is described by fragmentation functions [2], [3].",
"Though the evolution of these fragmentation functions with the scale $Q^2$ can be understood within the framework of pQCD [4], [5], [6], their actual form at a given scale $Q^2 = s_0$ can not be deduced from QCD.",
"In Refs.",
"[7], [8], [9], it has been shown that fragmentation functions of charged hadrons, $\\pi $ -s, $K$ -s and $\\Lambda $ -s produced in electron-positron ($e^+e^-$ ) collisions can be described by a simple statistical physical model.",
"This model treats hadrons formed in a jet as a microcanonical ensemble and obtains fragmentation functions via smearing the single hadron distribution in a jet over the distribution of charged hadron multiplicity measured in $e^+e^-$ annihilations.",
"The hadron multiplicity distribution is an imput in this model and its derivation is out of the scope of this paper.",
"Such calculations in the microcanonical ensemble have been performed by other groups to describe hadron multiplicity distributions from $e^+e^-$ to nucleus-nucleus ($AA$ ) collisions [10] - [12].",
"The hadron multiplicity distribution used in Ref.",
"[7], namely the Euler-Gamma distribution, belongs to the family of the Koba – Nielsen – Olesen (KNO) type distributions [19], [20] that describe measurements in $e^+e^-$ collisions well [21], [22], [23].",
"If hadrons created in a single event are distributed according to the Boltzmann – Gibbs distribution, and the multiplicity or the temperature parameter of the distribution fluctuates according to the Euler-Gamma function, then the average hadron spectrum will be the Tsallis – Pareto (or q-canonical) distribution [24] - [29].",
"Similarly, if in a single event hadrons have microcanonical distribution, and multiplicity fluctuates according to the Euler-Gamma distribution, then the resulting average hadron spectrum will be a microcanonical generalisation of the Tsallis – Pareto (or q-microcanonical) distribution [7], [8].",
"It is interesting, that the q-microcanonical distribution can be obtained within the framework of non-additive thermodynamics too [30].",
"In this paper, we point out that the fragmentation functions measured in $pp$ collisions at $\\sqrt{s}$ = 7 TeV collision energy [32], [33] can be described by the q-microcanonical distribution too.",
"Furthermore, the parameters of the distribution show similar scale dependence as seen in Refs.",
"[34], [35].",
"The structure of this paper is as follows: in Section , we provide a description of the microcanonical fragmentation model [7].",
"Section contains the fits of the model to fragmentation functions and multiplicity distributions measured in $pp$ collision at $\\sqrt{s}$ = 7 TeV center of mass energy at the LHC [32], [33].",
"Finally, the conclusion is presented in Section ."
],
[
"Microcanonical ",
"If the process of the creation of hadrons $h_1,\\,\\dots \\,,\\,h_N$ by the leading parton $p_{L}$ of a jet with multiplicity $N$ is such that the corresponding cross-section is simply proportional to the phase space available for the hadrons, restricted only by the energy conservation, $d\\sigma ^{h_1,\\,\\dots ,\\,h_N} &=& |\\mathcal {M}|^2\\,\\delta ^{(4)}\\left(\\sum \\limits _j p^\\mu _{h_j} - P^\\mu _{p_L} \\right)\\,d\\Omega \\nonumber \\\\&\\propto & \\delta \\left(\\sum \\limits _j \\epsilon _{h_j} - E_{jet} \\right)\\,d\\Omega \\;, $ then the hadrons created in the fragmentation process form a microcanonical ensemble.",
"In Eq.",
"(REF ), $\\Omega $ is the phase space of the created hadrons, $p^\\mu _{h_j}$ is the four-momentum, $\\epsilon _{h_j}$ is the energy of the hadron $h_j$ , $P^\\mu _{p_L}$ is the four-momentum of the leading parton $p_{L}$ , $E_{jet} = P^0_{p_L}$ is the energy of the jet, and $\\mathcal {M}$ is the matrix amplitude describing the process.",
"This way, neglecting hadron masses, the energy distribution of a hadron inside a jet with multiplicity $N$ equals to (see Ref.",
"[7]) $f_N(\\epsilon ) = A_{mc}\\; (1-x)^{D(N-1)-1}\\;, $ where $x = \\epsilon /E_{jet}$ , $\\epsilon $ is the energy of the hadron, $D$ is the effective dimensionality of the jet and $A_{mc} = {D\\,N - 1 \\atopwithdelims ()D}\\, DN /(\\, k_D\\, E_{jet}^D\\,)$ follows from the normalisation condition $N = \\int d\\Omega _p \\int dp\\,p^{D-1} f_N(\\epsilon )\\;, $ with $k_D = \\int d\\Omega _p$ being the angular part of the momentum space integral.",
"Eq.",
"(REF ) follows from the microcanonical momentum space volume at fixed energy and multiplicity, $\\Omega _{N}(E) &=& \\frac{1}{N!",
"}\\int \\prod _{i=1}^{N} d^Dp_i \\:\\delta \\left( E-\\sum _{j=1}^N\\epsilon _j\\right)\\nonumber \\\\&=& \\frac{\\bigl [k_D\\,\\Gamma (D)\\bigr ]^N}{N!\\,\\Gamma (DN)} E^{N\\,D\\,-\\,1}\\;, $ and the one-particle distribution is obtained as $f_N(\\epsilon ) = \\frac{\\Omega _{N-1}(E-\\epsilon )}{\\Omega _N(E)}\\;.",
"$ As particles in a jet form a microcanonical ensemble, an entropy $S_{jet} = \\ln \\Omega _{N}(E_{jet})$ and so a thermodynamical temperature $\\frac{1}{T_{jet}} = \\frac{\\partial S_{jet}}{\\partial E_{jet}} = \\frac{DN-1}{ E_{jet}} $ based on the zeroth law of thermodynamics [30], [31] can be associated to them.",
"Microcanonical treatment of hadron production has also been proposed in Refs.",
"[7] - [18] for $e^+e^-$ , $pp$ and $AA$ reactions.",
"The main difference between our approach and the ones discussed in [13] - [18] is that in order to analyse the distribution of charge-averaged hadrons inside jets of very high energy and small jetcone, we do not deal with charge conservation and neglect masses and transverse momenta of hadrons (transverse with respect to the jet direction).",
"Thus, jet masses are neglected in our calculations: $M^2_{jet} = \\left(\\sum p^\\mu _i\\right)^2 = \\left(\\sum \\epsilon _i\\right)^2 - \\left(\\sum \\mathbf {p}_i\\right)^2 \\approx 0$ .",
"Consequently, the conservation of four-momentum is equivalent to energy conservation inside a one-dimensional directed jet.",
"This way, instead of the jet mass $M_{jet}$ the jet energy $E_{jet}$ would control the distribution of hadron multiplicity.",
"This multiplicity distribution we do not derive here, we rely on empirical fits to measurements instead.",
"In Refs.",
"[24] - [29], it has been shown that special event-by-event fluctuation patterns of the temperature or of the particle multiplicity can result in power-law tailed average particle spectra.",
"This applies even if in each event, particles are distributed according to the Boltzmann – Gibbs distribution.",
"In Refs.",
"[19] - [23], it has been argued that an approximate Koba – Nielsen – Olesen (KNO) scaling of the multiplicity distribution of charged hadrons holds for electron-positron collisions (though the scaling is weakly violated by the scale evolution of the strong fine structure constant $\\alpha _s(Q^2)$ ).",
"If we consider multiplicity fluctuations of the form $p(N) = \\frac{\\beta ^{\\,\\alpha }}{\\Gamma (\\alpha )} (N-N_0)^{\\alpha -1} e^{-\\beta \\, (N-N_0)}\\;, $ and microcanonical single hadron distribution inside each jet (cf.",
"Eq.",
"(REF )) the multiplicity averaged hadron spectrum becomes $\\frac{1}{\\sigma }\\frac{d\\sigma }{d^Dp} &=& \\sum _{N=N_0}^{\\infty } p(N)\\, f_N(\\epsilon )\\nonumber \\\\&\\approx & \\dfrac{A\\, ( 1-x)^{D(N_0-1)-1} }{\\left( 1 - \\dfrac{q-1}{T/E_{jet}}\\ln (1-x)\\right)^{1/(q-1)}} \\;.",
"$ This result can be obtained by replacing the discrete sum by an integral, and using Stirling's formula $n!\\approx \\sqrt{2\\pi n}\\,(n/e)^n$ .",
"In terms of the integration variable $\\xi = N-N_0$ , only the highest power is taken into account.",
"In Eq.",
"(REF ), $N_0$ is the minimal number of hadrons that must be produced in the fragmentation process.",
"The newly introduced parameters are: $q = 1 + 1/(\\alpha +D + 1)$ and $T = E_{jet}\\,\\beta \\,/\\,[D (\\alpha +D+1)]$ .",
"The parameter $q$ measures the deviation of Eq.",
"(REF ) from the microcanonical distribution Eq.",
"(REF ).",
"For $e^+e^-$ data, $q>1$ holds, however, in the limit of $q\\rightarrow 1$ , the hadron distribution $\\frac{1}{\\sigma }\\frac{d\\sigma }{dx} \\rightarrow A\\, x^{\\,D-1} ( 1-x)^{D(\\overline{N}-1)-1} \\;, $ is recovered with $\\overline{N} = \\alpha /\\beta + N_0$ being the mean multiplicity.",
"Since the multiplicity fluctuates from jet to jet, so does the thermodynamical temperature introduced in Eq.",
"(REF ).",
"The distribution of $T_{jet}$ can be obtained from Eqs.",
"(REF ) and (REF ): $p(T_{jet}) = \\frac{\\beta ^{\\,\\alpha }}{\\Gamma (\\alpha )} \\frac{D}{E_{jet} }\\,\\theta ^2 \\,(\\theta -\\theta _0)^{\\alpha -1} \\,e^{-\\beta \\, (\\theta -\\theta _0)} $ with $\\theta = E_{jet}/(D\\,T_{jet})$ , $\\theta _0 = E_{jet}/(D\\,T_{jet\\,0})$ and $T_{jet\\,0} = E_{jet}/[D\\,N_0 - 1]$ .",
"The mean value of the thermodynamical temperature is $\\overline{T_{jet}} = \\frac{E_{jet}}{D\\,\\left(\\overline{N} - N_0 \\right)} \\, \\frac{\\alpha }{\\alpha -1} \\;+\\; {\\cal O}\\left(1\\,/\\,\\overline{N}^{\\,2}\\right)\\;.",
"$ The $T$ parameter appearing in the multiplicity averaged hadron spectrum Eq.",
"(REF ) on the other hand may be referred to as “mean equipartition temperature”.",
"It is proportional to the average energy per particle in a jet $\\left\\langle \\frac{\\epsilon }{N-N_0} \\right\\rangle _{N,\\vec{p}}&=& \\sum _N p(N) \\int d^Dp\\,f_{N}(\\epsilon ) \\left(\\frac{\\epsilon }{N-N_0} \\right) \\nonumber \\\\&=& \\frac{E_{jet}}{\\overline{N}-N_0} \\frac{\\alpha }{\\alpha -1} \\nonumber \\\\&=& \\frac{D\\,T}{1-(D+2)(q-1)} \\;, $ In the limit $q\\rightarrow 1$ the mean energy per particle tends to the familiar result: $\\left\\langle \\frac{\\epsilon }{N-N_0} \\right\\rangle _{N,\\vec{p}}\\quad \\rightarrow \\quad D\\,T\\;, \\qquad \\left(\\text{if}\\; q\\rightarrow 1\\right) \\;.",
"$ It is also worth noting that if $\\overline{N}\\gg N_0$ the usual equipartition formula holds for $\\overline{T_{jet}}$ : $\\left\\langle \\frac{\\epsilon }{N-N_0} \\right\\rangle _{N,\\vec{p}}\\quad \\rightarrow \\quad D\\,\\overline{T_{jet}}\\;, \\qquad \\left(\\text{if}\\; \\overline{N}/ N_0 \\rightarrow 0 \\right) \\;.",
"$ Finally, from Eqs.",
"(REF ) and (REF ), one can conclude that $T = T_{jet}\\, [1 - (D+2)(q-1)] \\; +\\; {\\cal O}\\left( 1\\,/\\,\\overline{N}^{\\,2} \\right) \\;.",
"$"
],
[
"Analysis of Fragmentation Functions Measured in $\\sqrt{s}$ = 7 TeV Proton-Proton Collisions",
"In the jet-analysis reported in Refs.",
"[32], [33], very narrow jet-cones of $R = \\sqrt{\\Delta \\eta ^2 + \\Delta \\phi ^2} = 0.6$ were used where $\\Delta \\phi $ and $\\Delta \\eta $ are the azimuthal angle and pseudorapidity of the hadrons relative to that of the jet.",
"($\\eta = -\\ln \\tan \\theta $ , with $\\theta $ being the polar angle.)",
"For such a jetcone, it is reasonable to make the approximation $z = \\dfrac{\\mathbf {p}_{_h}\\, \\mathbf {P}_{jet}}{|\\mathbf {P}_{jet}|^2} = x\\,\\cos \\Delta \\theta \\approx x \\;.",
"$ Furthermore, jets may be considered to be one-dimensional bunches of ultra-relativistic particles.",
"This way, the four-momentum of the jet can be approximated as $P^{\\mu }_{ jet} &=& (M_T \\cosh y, M_T \\sinh y, \\mathbf {P}_{T} )\\nonumber \\\\&\\approx & (P_T \\cosh \\eta , P_T \\sinh \\eta , \\mathbf {P}_{T} ) \\;, $ where $M_T$ and $\\mathbf {P}_{T}$ are the transverse energy and momentum of the jet, and $y = 0.5 \\ln [(E_{jet}+P_{jet\\,z})/(E_{jet}-P_{jet\\,z})]$ .",
"In the following, we will analyse jets mainly transverse to the beam direction ($\\eta = 0$ ), thus, we may use $E_{jet}\\approx P_{T}$ .",
"Finally, the $z$ distribution of charged hadrons takes the form $\\frac{1}{N_{jet}}\\frac{dN}{dz} \\approx \\dfrac{A\\, z^{\\,D-1} ( 1-z)^{D(N_0-1)-1} }{\\left( 1 - \\dfrac{q-1}{T^{\\ast }}\\ln (1-z)\\right)^{1/(q-1)}} \\;, $ with $T^{\\ast }=T/P_{Tjet}$ .",
"In the canonical limit, $z\\ll 1$ Eq.",
"(REF ) tends to the Boltzmann – Gibbs distribution, and Eq.",
"(REF ) approaches the q-canonical distribution $\\frac{1}{N_{jet}}\\frac{dN}{dz}\\rightarrow A \\, \\Bigg [ 1+\\frac{q-1}{T^{\\ast }}z \\Bigg ]^{-1/(q-1)} \\;.",
"$ Fits of Eq.",
"(REF ), (REF ) and (REF ) to data on fragmentation functions and multiplicity distributions measured in $pp$ collisions at $\\sqrt{s}$ = 7 TeV [32], [33] are shown in Figs.",
"REF - REF .",
"Figs.",
"REF and REF show that the q-microcanonical (or microcanonical Tsallis – Pareto) distribution, Eq.",
"(REF ), describes data on $dN/dz$ well.",
"The low $z$ deviation of the model from the data is assumably due to the low $p_T$ cut used in the jet analysis.",
"Particles with transverse momentum less than $p_{T0}=0.5$ GeV/c were not taken into account in the jet analysis, and the downward curl of the measured data at low $z$ from Eq.",
"(REF ) starts around $z_{0}=p_{T0}\\,/P_{Tjet}$ .",
"Similar conclusions may be drawn from the data-over-theory plots shown in Fig.",
"REF for the q-canonical distribution, Eq.",
"(REF ), except that this distribution describes data for $z\\lessapprox 0.2$ only.",
"The evolution of the fitted $q$ and $T$ parameters of Eqs.",
"(REF ) and (REF ) with the transverse momentum of the jet are shown in Figs.",
"REF - REF .",
"The “mean equipartition temperature” parameter scaled by the transverse momentum of the jet $T^{\\ast } = T/P_{Tjet}$ shows power-law dependence on $P_{Tjet}$ , $T^{\\ast } = \\left(P_{Tjet}/Q_{0} \\right)^{\\mu }\\;, $ while for the parameter $q$ , both a power-law, $q = \\left(P_{Tjet}/Q_{0} \\right)^{\\mu }\\;, $ and a double-logarithmic ansatz, $q = 1 + \\mu \\ln \\ln \\left(P_{Tjet}/Q_{0} \\right)\\;, $ fit.",
"The $Q_0$ and $\\mu $ parameters of the q-canonical and the q-microcanonical distributions approximately coincide within errors.",
"For $q$ , Eq.",
"(REF ) was successfully used in Ref.",
"[35] to adjust the $Q^2$ evolution of a fragmentation function of the form of Eq.",
"(REF ) to that of an AKK type one [2].",
"In $pp$ collisions, $Q=P_{Tjet}$ seems to be a good choice.",
"The $q$ and $T$ parameters show similar scale dependence for fragmentation functions of protons, $K^{0}$ -s, $\\pi ^{0}$ -s, $\\Lambda $ -s and charge-averaged hadrons produced in $e^+e^-$ annihilations as well as for the transverse momentum spectra of charged hadrons stemming from $pp$ collisions [7], [8], [9], [34].",
"Fig.",
"REF shows that Eq.",
"(REF ) describes data on multiplicity distributions well, except for $N<3$ , where measured data are higher than what the Euler-Gamma distribution predicts.",
"This effect is perhaps due to the small conesize.",
"In a jet with one or two very energetic particles, the others have very small energies, and thus may fly out of the jetcone.",
"This way, the number of jets with only a few particles increases in this type of jet analysis.",
"As a consequence, the $\\alpha $ parameter of Eq.",
"(REF ), which is greatly influenced by the number of low multiplicity jets, can not be determined reliably.",
"Thus, it is not so disconcerting that the $q$ parameter predicted from multiplicity fits takes lower values (of around $q=1.1$ ) than that obtained from fits to $dN/dz$ data.",
"It is important to note that in Ref.",
"[33], jets were reconstructed from charged particles only, while in Ref.",
"[32], calorimetric measurements were used in the jet reconstruction, so both neutral and charged particles were included in the analysis.",
"For this reason, we fitted Eqs.",
"(REF ) - (REF ) to $q$ and $T$ values obtained for jets with high transverse momenta only (25 GeV/c $\\le P_{Tjet} \\le $ 500 GeV/c from [32]).",
"Nevertheless, $q$ and $T$ values for jets with low transverse momenta (4 GeV/c $\\le P_{Tjet} \\le $ 40 GeV/c obtained from [33]) show a tendency similar to that of the high $P_{Tjet}$ jets.",
"The $T^{\\ast }$ parameters of the low $P_{Tjet}$ dataset are approximately 10% higher, than what the fit of Eq.",
"(REF ) to the high $P_{Tjet}$ dataset predicts (see Fig.",
"REF ).",
"From Eq.",
"(REF ), it can be seen that $T^{\\ast }$ is proportional to the inverse of the multiplicity.",
"This way, the ratio of $T^{\\ast }$ -s obtained in the two different analysis' is proportional to the ratio of the total multiplicity to that of charged particles.",
"If as an estimate, we used the ratio of charged to neutral pions, we would get a factor of 3/2 for the ratio of $T^{\\ast }$ -s obtained from the two different analysis'.",
"This value is somewhat higher than what can be seen in Fig.",
"REF .",
"Eq.",
"(REF ) describes both datasets.",
"For the high $P_{Tjet}$ dataset, the power of the $1-z$ factor in the numerator takes the value $D(N_0-1)-1 = +1$ , while this quantity decreases from 1 to -1 for the low $P_{Tjet}$ dataset as $P_{Tjet}$ decreases from 40 GeV/c to 4 GeV/c.",
"Figure: Measured distributions of the longitudinal momentum fraction zz of hadrons inside jets with various transverse momenta (data of jets with P Tjet =4-6,⋯,24-40P_{Tjet}= \\left[4-6\\right],\\dots ,\\left[24-40\\right] GeV/c and with P Tjet =25-40,⋯,400-500P_{Tjet}= \\left[25-40\\right],\\dots ,\\left[400-500\\right] GeV/c are published in Ref.",
"and in Ref.",
"respectively) and fitted 1 dimensional q-microcanonical distributions (Eq.",
"() with D=1D=1, and N 0 =3N_0=3 for the high P Tjet P_{Tjet} dataset and N 0 =1,1,1,2,3N_0=1,1,1,2,3 (from bottom to top) for the low P Tjet P_{Tjet} dataset).Figure: Ratios of measured dN/dzdN/dz distributions and fitted 1 dimensional q-canonical distributions (Eq.",
"() with D=1D=1) for jets with various transverse momenta (data of graphs are published in Refs.",
", ).Figure: Ratios of measured dN/dzdN/dz distributions and fitted 1 dimensional q-microcanonical distributions (Eq.",
"() with D=1D=1.",
"For the values of N 0 N_{0}, see the caption of Fig. )",
"for jets with various transverse momenta (data of graphs are published in Refs.",
", ).Figure: Measured multiplicity distributions of charged hadrons inside jets with various transverse momenta and rapidity and fitted Euler-Gamma distributions (Eq. ()).",
"Data of graphs are published in Refs.",
", .Figure: Fitted values of the qq parameter in Eq.",
"() with D=1D=1 to measured dN/dzdN/dz distributions shown in Fig.",
".Figure: Fitted values of the T * T^{\\ast } parameter in Eq.",
"() with D=1D=1 to measured dN/dzdN/dz distributions shown in Fig.",
".Figure: Fitted values of the qq parameter in Eq.",
"() with D=1D=1 (for the values of N 0 N_{0}, see the caption of Fig. )",
"to measured dN/dzdN/dz distributions shown in Fig.",
".Figure: Fitted values of the T * T^{\\ast } parameter in Eq.",
"() with D=1D=1 (for the values of N 0 N_{0}, see the caption of Fig. )",
"to measured dN/dzdN/dz distributions shown in Fig.",
"."
],
[
"Conclusions",
"This paper shows that the statistical jet-fragmentation model [7], [8] describes the $dN/dz$ distribution of hadrons in jets created in proton-proton reactions at $\\sqrt{s}$ = 7 TeV center of mass energy [32], [33].",
"This model combines microcanonical statistics (which has also been used in the description of different hadronic observables in high-energy phenomena in Refs.",
"[10]-[17]) with super-statistics [24]-[29] stemming from multiplicity fluctuations emerging in proton-proton as well as in electron-positron and nucleus-nucleus collisions [19]-[23].",
"It turns out that the parameters of the $dN/dz$ distribution of charged hadrons in jets in proton-proton collisions (Sect. )",
"show similar scale dependence as the parameters of fragmentation functions in electron-positron annihilations [7], [8], [9] and of transverse momentum spectra of charged hadrons stemming from proton-proton collisions [34].",
"These scale evolutions are consistent with the DGLAP equations [35].",
"Finally, it is pointed out that the $dN/dz$ distributions obtained from two different jet analysis' and different kinematical ranges [32], [33] both can be described by the microcanonical jet-fragmentation model."
],
[
"Acknowledgement",
"The authors thank Brian Cole and Eric Feng for the discussions.",
"This work was supported by the Hungarian OTKA grants K68108, K104260, NK77816 and by the bilateral Hungarian–South-African project NIH TET 10-1 2011-0061, ZA-15/2009.",
"One of the authors (GGB) thanks the János Bolyai Research Scolarship from the Hungarian Academy of Sciences."
]
] | 1204.1508 |
[
[
"Fast ALS-based tensor factorization for context-aware recommendation\n from implicit feedback"
],
[
"Abstract Albeit, the implicit feedback based recommendation problem - when only the user history is available but there are no ratings - is the most typical setting in real-world applications, it is much less researched than the explicit feedback case.",
"State-of-the-art algorithms that are efficient on the explicit case cannot be straightforwardly transformed to the implicit case if scalability should be maintained.",
"There are few if any implicit feedback benchmark datasets, therefore new ideas are usually experimented on explicit benchmarks.",
"In this paper, we propose a generic context-aware implicit feedback recommender algorithm, coined iTALS.",
"iTALS apply a fast, ALS-based tensor factorization learning method that scales linearly with the number of non-zero elements in the tensor.",
"The method also allows us to incorporate diverse context information into the model while maintaining its computational efficiency.",
"In particular, we present two such context-aware implementation variants of iTALS.",
"The first incorporates seasonality and enables to distinguish user behavior in different time intervals.",
"The other views the user history as sequential information and has the ability to recognize usage pattern typical to certain group of items, e.g.",
"to automatically tell apart product types or categories that are typically purchased repetitively (collectibles, grocery goods) or once (household appliances).",
"Experiments performed on three implicit datasets (two proprietary ones and an implicit variant of the Netflix dataset) show that by integrating context-aware information with our factorization framework into the state-of-the-art implicit recommender algorithm the recommendation quality improves significantly."
],
[
"Introduction",
"Recommender systems are information filtering algorithms that help users in information overload to find interesting items (products, content, etc).",
"Users get personalized recommendations that contain typically a few items deemed to be of user's interest.",
"The relevance of an item with respect to a user is predicted by recommender algorithms; items with the highest prediction scores are displayed to the user.",
"Recommender algorithms are usually sorted into two main approaches: the content based filtering (CBF) and the collaborative filtering (CF).",
"CBF algorithms use user metadata (e.g.",
"demographic data) and item metadata (e.g.",
"author, genre, etc.)",
"and predict user preference using these attributes.",
"In contrast, CF methods do not use metadata, but only data of user–item interactions.",
"Depending on the nature of the interactions, CF algorithms can be further classified into explicit and implicit feedback based methods.",
"In the former case, users provide explicit information on their item preferences, typically in form of user ratings.",
"In the latter case, however, users express their item preferences only implicitly, as they regularly use an online system; typical implicit feedback types are viewing and purchasing.",
"Obviously, implicit feedback data is less reliable as we will detail later.",
"CF algorithms proved to be more accurate than CBF methods, if sufficient preference data is available [1].",
"CF algorithms can be classified into memory-based and model-based ones.",
"Until recently, memory-based solutions were concerned as the state-of-the-art.",
"These are neighbor methods that make use of item or user rating vectors to define similarity, and they calculate recommendations as a weighted average of similar item or user rating vectors.",
"In the last few years, model-based methods gained enhanced popularity, because they were found to be much more accurate in the Netflix Prize, a community contest launched in late 2006 that provided the largest explicit benchmark dataset (100M ratings) [2] for a long time.",
"Model-based methods build generalized models that intend to capture user preference.",
"The most successful approaches are the latent factor algorithms.",
"These represent each user and item as a feature vector and the rating of user $u$ for item $i$ is predicted as the scalar product of these vectors.",
"Different matrix factorization (MF) methods are applied to compute these vectors, which approximate the partially known rating matrix using alternating least squares (ALS) [3], gradient [4] and coordinate descent method [5], conjugate gradient method [6], singular value decomposition [7], or a probabilistic framework [8].",
"Explicit feedback based methods are able to provide accurate recommendations if enough ratings are available.",
"In certain application areas, such as movie rental, travel applications, video streaming, users have motivation to provide ratings to get better service, better recommendations, or award or punish a certain vendor.",
"However, in general, users of an arbitrary online service do not tend to provide ratings on items even if such an option is available, because (1) when purchasing they have no information on their satisfaction (2) they are not motivated to return later to the system to rate.",
"In such cases, user preferences can only be inferred by interpreting user actions (also called events).",
"For instance, a recommender system may consider the navigation to a particular product page as an implicit sign of preference for the item shown on that page [9].",
"The user history specific to items are thus considered as implicit feedback on user taste.",
"Note that the interpretation of implicit feedback data may not necessarily reflect user satisfaction which makes the implicit feedback based preference modeling a difficult task.",
"For instance, a purchased item could be disappointing for the user, so it might not mean a positive feedback.",
"We can neither interpret missing navigational or purchase information as negative feedback, that is, such information is not available.",
"Despite its practical importance, this harder but more realistic task has been less studied.",
"The proposed solutions for the implicit task are often the algorithms for the explicit problems that had been modified in a way that they can handle the implicit task.",
"The classical MF methods only consider user-item interaction (ratings or events) when building the model.",
"However, we may have additional information related to items, users or events, which are together termed contextual information, or briefly context.",
"Context can be, for instance, the time or location of recommendation, social networks of users, or user/item metadata [10].",
"Integrating context can help to improve recommender models.",
"Tensor factorization have been suggested as a generalization of MF for considering contextual information [11].",
"However, the existing methods only work for the explicit problem.",
"In this work, we developed a tensor factorization algorithm that can efficiently handle the implicit recommendation task.",
"The novelty of our work is threefold: (1) we developed a fast tensor factorization method—coined iTALS—that can efficiently factorize huge tensors; (2) we adapted this general tensor factorization to the implicit recommendation task; (3) we present two specific implementations of this general implicit tensor factorization that consider different contextual information.",
"The first variant uses seasonality which was also used in [11] for the explicit problem.",
"The second algorithm applies sequentiality of user actions and is able to learn association rule like usage patterns.",
"By using these patterns we can tell apart items or item categories having been purchased with different repetitiveness, which improves the accuracy of recommendations.",
"To our best knowledge, iTALS is the first factorization algorithm that uses this type of information.",
"This paper is organized as follows.",
"Section briefly reviews related work on context-aware recommendation algorithms and tensor factorization.",
"In Section we introduce our tensor factorization method and its application to the implicit recommendation task.",
"Section shows two application examples of our factorization method: (1) we show how seasonality can be included in recommendations and (2) we discuss how a recommendation algorithm can learn repetitiveness patterns from the dataset.",
"Section presents the results of our experiments, and Section sums up our work and derive the conclusions.",
"We will use the following notation in the rest of this paper: [noitemsep,topsep=0pt,parsep=0pt,partopsep=0pt] $A\\circ B\\circ \\ldots \\rightarrow $ The Hadamard (elementwise) product of $A$ , $B$ , ...The operands are of equal size, and the result's size is also the same.",
"The element of the result at $(i,j,k,\\ldots )$ is the product of the element of $A$ , $B$ , ...at $(i,j,k,\\ldots )$ .",
"This operator has higher precedence than matrix multiplication in our discussion.",
"$A_{\\bullet ,i} / A_{i,\\bullet } \\rightarrow $ The $i^{\\rm th}$ column/row of matrix $A$ .",
"$A_{i_1,i_2,\\ldots } \\rightarrow $ The $(i_1,i_2,\\ldots )$ element of tensor/matrix $A$ .",
"$K \\rightarrow $ The number of features, the main parameter of factorization.",
"$D \\rightarrow $ The number of dimensions of the tensor.",
"$T \\rightarrow $ A $D$ dimensional tensor that contains only zeroes and ones (preference tensor).",
"$W \\rightarrow $ A tensor with the exact same size as $T$ (weight tensor).",
"$S_i \\rightarrow $ The size of $T$ in the $i^{\\rm th}$ dimension ($i=1,\\ldots , D$ ).",
"$N^+ \\rightarrow $ The number of non-zero elements in tensor $T$ .",
"$M^{(i)} \\rightarrow $ A $K\\times S_i$ sized matrix.",
"Its columns are the feature vectors for the entities in the $i^{\\rm th}$ dimension."
],
[
"Related work",
"Context-aware recommender systems [12] emerged as an important research topic in the last years and entire workshops are devoted to this topic on major conferences (CARS series started in 2009 [13], CAMRA in 2010 [14]).",
"The application fields of context-aware recommenders include among other movie [15] and music recommendation [16], point-of-interest recommendation (POI) [17], citation recommendation [18].",
"Context-aware recommender approaches can be classified into three main groups: pre-filtering, post-filtering and contextual modeling [10].",
"Baltrunas and Amatriain [16] proposed a pre-filtering approach by partitioned user profiles into micro-profiles based on the time split of user event falls, and experimented with different time partitioning.",
"Post-filtering ignores the contextual data at recommendation generation, but disregards irrelevant items (in a given context) or adjust recommendation score (according to the context) when the recommendation list is prepared; see a comparison in [19].",
"The tensor factorization based solutions, including our proposed approach, falls into the contextual modeling category.",
"Tensor factorization incorporates contextual information into the recommendation model.",
"Let us have a set of items, users and ratings (or events) and assume that additional context of the ratings is available (e.g.",
"time of the rating).",
"Having $C$ different contexts, the rating data can be cast into a $C+2$ dimensional tensor.",
"The first dimension corresponds to users, the second to items and the subsequent $C$ dimensions $[3,\\ldots ,C+2]$ are devoted to contexts.",
"We want to decompose this tensor into lower rank matrices and/or tensors in a way that the reconstruction the original tensor from its decomposition approximates well the original tensor.",
"Approximation accuracy is calculated at the known positions of the tensor using RMSE as error measure.",
"In [11], a sparse HOSVD [20] method is presented that decomposes a $D$ dimensional sparse tensor into $D$ matrices and a $D$ dimensional tensor.",
"If the size of the original tensor is $S_1\\times S_2\\times \\cdots \\times S_{D}$ and the number of features is $K$ then the size of the matrices are $S_1\\times K$ , $S_2\\times K$ , ..., $S_{D}\\times K$ and the size of the tensor is $K\\times K\\times \\cdots \\times K$ .",
"The authors use gradient descent on the known ratings to find the decomposition, and by doing so, the complexity of one iteration of their algorithm scales linearly with the number of non-missing values in the original tensor (number of rating) and cubically with the number of features ($K$ ).",
"This is much less than the cost of the dense HOSVD, which is $O(K\\cdot (S_1+\\cdots +S_{D})^{D})$ .",
"A further improvement was proposed by Rendle et al [21], where the computational complexity was reduced so that their method scales linearly both with the number of explicit ratings and with the number of features.",
"However, if the original tensor is large and dense like for the implicit recommendation task then neither method scales well."
],
[
"ALS based fast tensor factorization",
"In this section we present iTALS, a general ALS-based tensor factorization algorithm that scales linearly with the non-zero element of a dense tensor (when appropriate weighting is used) and cubically with the number of features.",
"This property makes our algorithm suitable to handle the context-aware implicit recommendation problem.",
"Let $T$ be a tensor of zeroes and ones and let $W$ contain weights to each element of $T$ .",
"$T_{u,i,c_1,\\cdots ,c_C}$ is 1 if user $u$ has (at least one) event on item $i$ while the context-state of $j^{\\rm {th}}$ context dimension was $c_j$ , thus the proportion of ones in the tensor is very low.",
"An element of $W$ is 1 if the corresponding element in $T$ is 0 and greater than 1 otherwise.",
"Instead of using the form of the common HOSVD decomposition ($D$ matrices and a $D$ dimensional tensor) we decompose the original $T$ tensor into $D$ matrices.",
"The size of the matrices are $K\\times S_1, K\\times S_2, \\ldots , K\\times S_{D}$ .",
"The prediction for a given cell in $T$ is the elementwise product of columns from $M^{(i)}$ low rank matrices.",
"Equation REF describes the model.",
"$\\hat{T}_{i_1,i_2,\\ldots ,i_{D}}=1^TM^{(1)}_{\\bullet ,i_1}\\circ M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}$ We want to minimize the loss function of equation REF : $L(M^{(1)},\\ldots ,M^{(D)})=\\sum _{i_1=1,\\ldots ,i_{D}=1}^{S_1,\\ldots ,S_{D}}W_{i_1,\\ldots ,i_{D}}\\left(T_{i_1,\\ldots ,i_{D}}-\\hat{T}_{i_1,\\ldots ,i_{D}}\\right)^2$ If all but one $M^{(i)}$ is fixed, $L$ is convex in the non-fixed variables.",
"We use this method to minimize the loss function.",
"$L$ reaches its minimum (in $M^{(i)}$ ) where its derivate with respect to $M^{(i)}$ is zero.",
"Since the derivate of $L$ is linear in $M^{(i)}$ the columns of the matrix can be computed separately.",
"For the $(i_1)^{\\rm {th}}$ column of $M^{(1)}$ : ${\\begin{array}{c}0=\\frac{\\partial L}{\\partial M^{(1)}_{\\bullet ,i_1}} =-2\\underbrace{\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots ,S_{D}}W_{i_2,\\ldots ,i_{D}}T_{i_1,\\ldots ,i_{D}}\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)}_{\\mathcal {O}} +\\\\2\\underbrace{\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots ,S_{D}}W_{i_2,\\ldots ,i_{D}}\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)^TM^{(1)}_{\\bullet ,i_1}}_{\\mathcal {I}}\\end{array}}$ It takes $O(DKN^+_{i_1})$ time to compute $\\mathcal {O}$ in equation REF , because only $N^+_{i_1}$ cells of $T$ for $i_1$ in the first dimension contain ones, the others are zeroes.",
"For every column it yields a complexity of $O(DKN^+)$ .",
"The naive computation of $\\mathcal {I}$ however is very expensive computationally: $O(K\\prod _{i=2}^D{S_i})$ .",
"Therefore we transform $\\mathcal {I}$ by using $W_{i_2,\\ldots ,i_{D}}=W^{\\prime }_{i_2,\\ldots ,i_{D}}+1$ and get: ${\\begin{array}{c}\\mathcal {I}=\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots ,S_{D}}W^{\\prime }_{i_2,\\ldots ,i_{D}}\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)^TM^{(1)}_{\\bullet ,i_1}+\\\\+\\underbrace{\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots S_{D}}\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)^T}_{\\mathcal {J}}M^{(1)}_{\\bullet ,i_1}\\end{array}}$ The first part in equation REF can be calculated in $O(K^2N^+_{i_1})$ as $W^{\\prime }_{i_2,\\ldots ,i_{D}}=(W_{i_2,\\ldots ,i_{D}}-1)$ and the weights for the zero elements of $T$ are ones.",
"This step is the generalization of the Hu et.",
"al's adaptation of ALS to the implicit problem [22].",
"The total complexity of calculating all columns of the matrix is $O(K^2N^+)$ .",
"$\\mathcal {J}$ is the same for all columns of $M^{(1)}$ (independent of $i_1$ ) and thus can be precomputed.",
"However the cost of directly computing $\\mathcal {J}$ remains $O(K\\prod _{i=2}^D{S_i})$ .",
"Observe the following: $\\begin{aligned}\\mathcal {J}_{j,k}=&\\left(\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots ,S_{D}}\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)^T\\right)_{j,k}=\\\\=&\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots ,S_{D}}\\left(M^{(2)}_{j,i_2}\\cdot \\ldots \\cdot M^{(D)}_{j,i_{D}}\\right)\\left(M^{(2)}_{k,i_2}\\cdot \\ldots \\cdot M^{(D)}_{k,i_{D}}\\right)=\\\\=&\\left(\\sum _{i_2=1}^{S_2}{M^{(2)}_{j,i_2}M^{(2)}_{k,i_2}}\\right)\\cdot \\ldots \\cdot \\left(\\sum _{i_D=1}^{S_D}{M^{(D)}_{j,i_D}M^{(D)}_{k,i_D}}\\right)\\end{aligned}$ Using equation REF we can transform the second part from equation REF into the following form: ${\\begin{array}{c}\\mathcal {J}=\\sum _{i_2=1,\\ldots ,i_{D}=1}^{S_2,\\ldots ,S_{D}}\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)\\left(M^{(2)}_{\\bullet ,i_2}\\circ \\cdots \\circ M^{(D)}_{\\bullet ,i_{D}}\\right)^T=\\\\=\\underbrace{\\left(\\sum _{i_2=1}^{S_2}{M^{(2)}_{\\bullet ,i_2}\\left(M^{(2)}_{\\bullet ,i_2}\\right)^T}\\right)}_{\\mathcal {M}^{(2)}}\\circ \\cdots \\circ \\underbrace{\\left(\\sum _{i_D=1}^{S_D}{M^{(D)}_{\\bullet ,i_D}\\left(M^{(D)}_{\\bullet ,i_D}\\right)^T}\\right)}_{\\mathcal {M}^{(D)}}\\end{array}}$ The members of equation REF can be computed in $O(S_iK^2)$ time.",
"From the $\\mathcal {M}^{(i)}$ matrices the expression can be calculated in $O(K^2D)$ time.",
"Note that the $\\mathcal {M}^{(i)}$ is needed for computing all but the $i^{\\rm {th}}$ matrix but only changes if $M^{(i)}$ changed.",
"Therefore we count the cost of computing $\\mathcal {M}^{(i)}$ to the cost of recomputing $M^{(i)}$ .",
"To get the desired column of the matrix we need to invert a $K\\times K$ sized matrix per column (see equation REF ).",
"That requires $O(K^3S_1)$ time for all columns of $M^{(1)}$ .",
"The columns of the other matrices can be calculated similarly.",
"[!h] Fast ALS-based tensor factorization for implicit feedback recommendations Input: $T$ : a $D$ dimensional $S_1 \\times \\cdots \\times S_{D}$ sized tensor of zeroes and ones; $W$ : a $D$ dimensional $S_1 \\times \\cdots \\times S_{D}$ sized tensor containing the weights; $K$ : number of features; $E$ : number of epochs Output: $\\lbrace M^{(i)}\\rbrace _{i=1,\\ldots , D}$ $K\\times S_i$ sized low rank matrices procedure iTALS$T$ , $W$ , $K$ , $E$ [1] $i=1,\\ldots ,D$ $M^{(i)} \\leftarrow $ Random $K\\times S_i$ sized matrix $\\mathcal {M}^{(i)} \\leftarrow M^{(i)}(M^{(i)})^T$ $e=1,\\ldots ,E$ $i=1,\\ldots ,D$ $C^{(i)} \\leftarrow \\mathcal {M}^{(\\ell _1)}\\circ \\cdots \\circ \\mathcal {M}^{(\\ell _{D-1})}, (i\\notin \\lbrace \\ell _1,\\ldots ,\\ell _{D-1}\\rbrace )$ $T^{(i)} \\leftarrow $ UnfoldTensor$T$ ,$i$ $j_i=1,\\ldots ,S_i$ $C^{(i)}_{j_i} \\leftarrow C^{(i)}$ $O^{(i)}_{j_i} \\leftarrow 0$ $t: \\lbrace t\\in T^{(i)}_{j_i}, t\\ne 0\\rbrace $ $\\lbrace j_\\ell |\\ell \\ne i\\rbrace \\leftarrow $ Indices of $t$ in $T$ $W_t \\leftarrow $ GetWeight$W$ ,$t$ $v \\leftarrow M^{(\\ell _1)}\\circ \\cdots \\circ M^{(\\ell _{D-1})}, (i\\notin \\lbrace \\ell _1,\\ldots ,\\ell _{D-1}\\rbrace )$ $C^{(i)}_{j_i} \\leftarrow C^{(i)}_{j_i} + vW_tv^T$ and $O^{(i)}_{j_i} \\leftarrow O^{(i)}_{j_i} + W_tv$ $M^{(i)}_{\\bullet ,j_i} \\leftarrow (C^{(i)}_{j_i}+\\lambda I)^{-1}O^{(i)}_{j_i}$ $\\mathcal {M}^{(i)} \\leftarrow M^{(i)}(M^{(i)})^T$ return $\\lbrace M^{(i)}\\rbrace _{i=1\\dots D}$ end procedure The total cost of computing $M^{(i)}$ is $O(K^3S_i+K^2N^++KDN^+)$ that can be simplified to $O(K^3S_i+K^2N^+)$ using that usually $D\\ll K$ .",
"Therefore the cost of computing each matrix once is $O\\left(K^3\\sum _{i=1}^{D}{S_i}+K^2N^+\\right)$ .",
"Thus the cost of an epoch is linear in the number of the non-zero examples and cubical in the number of features.",
"The cost is also linear in the number of dimensions of the tensor and the sum of the length of the tensors in each dimension.",
"We will also show in Section REF that the $O(K^2)$ part is dominant when dealing with practical problems.",
"The complexity of the gradient descent method for implicit feedback is $O(K\\prod _{i=1}^{D}{S_i})$ that is linear in the number of features but the $\\prod _{i=1}^{D}{S_i}$ part makes impossible to run it on real life datasets.",
"Sampling can be applied to reduce that cost but it is not trivial how to sample in the implicit feedback case.",
"The pseudocode of the suggested iTALS (Tensor factorization using ALS for implicit recommendation problem) is given in Algorithm .",
"There we use two simple functions.",
"${UnfoldTensor}{T,i}$ unfolds tensor $T$ by its $i^{\\rm th}$ dimension.",
"This step is used for the sake of clarity, but with proper indexing we would not need to actually unfold the tensor.",
"${GetWeight}{W,t}$ gets the weight from the weight tensor $W$ for the $t$ element of tensor $T$ and creates a diagonal matrix from it.",
"The size of $W_t$ is $K\\times K$ and it contains the weight for $t$ in its main diagonal and 0 elsewhere.",
"The pseudocode follows the deduction above.",
"In line we precompute $\\mathcal {M}^{(i)}$ .",
"We create the column independent part from equation REF in line .",
"We add the column dependent parts to each side of equation REF in lines – and compute the desired column in line .",
"In this step we use regularization to avoid numerical instability and overfitting of the model.",
"After each column of $M^{(i)}$ is computed $\\mathcal {M}^{(i)}$ is recomputed in line ."
],
[
"Context-aware iTALS algorithm",
"In this section we derive two specific algorithms from the generic iTALS method presented in Section .",
"The first method uses seasonality as context, the second considers the user history as sequential data, and learns meta-rules about sequentiality and repetitiveness."
],
[
"Seasonality",
"Many application areas of recommender systems exhibit the seasonality effect, therefore seasonal data is an obvious choice as context [23].",
"Strong periodicity can be observed in most of the human activities: as people have regular daily routines, they also follow similar patterns in TV watching at different time of a day, they do their summer/winter vacation around the same time in each year.",
"Taking the TV watching example, it is probable that horror movies are typically watched at night and animation is watched in the afternoon or weekend mornings.",
"Seasonality can be also observed in grocery shopping or in hotel reservation data.",
"In order to consider seasonality, first we have to define the length of season.",
"During a season we do not expect repetitions in the aggregated behavior of users, but we expect that at the same time offset in different seasons, the aggregated behavior of the users will be similar.",
"The length of the season depends on the data.",
"For example it is reasonable to set the season length to be 1 day for VoD consumption, however, this is not an appropriate choice for shopping data, where 1 week or 1 month is more justifiable.",
"Having the length of the season determined, we need to create time bands (bins) in the seasons.",
"These time bands are the possible context-states.",
"Time bands specify the time resolution of a season, which is also data dependent.",
"We can create time bands with equal or different length.",
"For example, every day of a week are time bands of equal length, but 'morning', 'around noon', 'afternoon', 'evening', 'late evening', 'night' could be time bands of a day with different length.",
"Obviously, these two steps require some a-priori knowledge about the data or the recommendation problem, but iTALS is not too sensitive to minor deviations related to the length and the resolution of the season.",
"In the next step, events are assigned to time bands according to their time stamp.",
"Thus, we can create the (user, item, time band) tensor.",
"We factorize this tensor using the iTALS algorithm and we get feature vectors for each user, for each item and for each time band.",
"When a recommendation is requested for user $u$ at time $t$ , first the time band of $t$ is determined and then the preference value for each item using the feature vector of user $u$ and the feature vector of time band $tb_t$ is calculated."
],
[
"Sequentiality",
"Recommendation algorithms often recommend items from categories that the user likes.",
"For example if the user often watches horror movies then the algorithm will recommend her horror movies.",
"This phenomenon is even stronger if time decay is applied and so recent events have greater weights.",
"Pushing newer events can increase accuracy, because similar items will be recommended.",
"This functioning can be beneficial in some application fields, like VoD recommendation, but will fail in such cases where repetitiveness in user behavior with respect to items can not be observed.",
"A typical example for that is related to household appliance products: if a user buys a TV set and then she gets further TV sets recommended, she will not probably purchase another one.",
"In such a case, complementary or related goods are more appropriate to recommend, DVD players or external TV-tuners for example.",
"On the other hand, the purchase of a DVD movie does not exclude at all the purchase of another one.",
"Whether recommendation of similar items is reasonable, depends on the nature of the item and behavior of the user.",
"Next, we propose an approach to integrate the repetitiveness of purchase patterns into the latent factor model.",
"Using association rules is a possible approach to specify item purchase patterns.",
"Association rules [24] are often used to determine which products are bought frequently together and it was reported that in certain cases association rule based recommendations yield the best performance [25].",
"In our setting, we can extract purchase patterns from the data using association rule mining on the subsequent user events within a given time window.",
"There are two possibilities: we can generate category–category rules, or category–item rule, thus having usage patterns: if a user bought an item from category $A$ then she will buy an item from category $B$ next time, or if a user bought an item from category $A$ then she will buy an item $X$ next time.",
"We face, however, with the following problems, when attempting to use such patterns in recommendations: (1) the parameter selection (minimum support, minimum confidence and minimum lift) influences largely the performance, their optimization may be slow; (2) rules with negated consequents (e.g.",
"bought from $A$ will not buy from $B$ ) are not found at all; (3) with category–category rules one should devise further weighting/filtering to promote/demote the items in the pushed category; (4) the category–item rules are too specific therefore either one gets too many rules or the rules will overfit.",
"We show how repetitiveness related usage patterns can be efficiently integrated into recommendation model using the the iTALS algorithm.",
"Let us now consider the category of last purchased item as the context for the next recommendation.",
"The tensor has again three dimensions: users, items and item categories.",
"The $(i,u,c)$ element of the tensor means that user $u$ bought item $i$ and the user's latest purchase (before buying $i$ ) was an item from category $c$ .",
"Using the examples above: the user bought a given DVD player after the purchase of a TV set.",
"After factorizing this tensor we get feature vectors for the item categories as well.",
"These vectors act as weights in the feature space that reweight the user–item relations.",
"For example, assuming that the first item feature means “having large screen” then the first feature of the TV category would be low as such items are demoted.",
"If the second item feature means “item can play discs” then the second feature of the TV category would be high as these items are promoted.",
"The advantage of this method is that it learns the usage patterns from the data globally by producing feature vectors that reweight the user–item relations.",
"One gets simple but general usage patterns using the proposed solution that integrates seamlessly into the common factorization framework: no post-processing is required to define promotional/demotional weights/filters.",
"We can generalize the concept described above to take into account several recent purchases.",
"We could create a $C+2$ dimensional tensor, where the $[3,\\ldots ,C+2]$ dimensions would represent the item categories of the last $C$ purchases, but the resulting tensor would be very sparse as we increase $C$ .",
"Instead we remain at a three dimensional tensor but we set simultaneously $C$ item categories to 1 for each user–item pair.",
"We may also decrease the weights in $W$ for those additional $C-1$ cells as they belong to older purchases.",
"Thus we may control the effect of previous purchases based on their recency.",
"When recommending, we have to compute the (weighted) average of the feature vectors of the corresponding categories and use that vector as the context feature vector."
],
[
"Experiments",
"We used five databases to validate our algorithms.",
"Three of them contain genuine implicit feedback data (LastFM 1K and 2 proprietary), while the other two are implicit variants of explicit feedback data.",
"The LastFM 1K [26] dataset contains listening habits of $\\sim $ 1 000 users on songs of $\\sim $ 170 000 artists (artists are considered items).",
"The training set contains all events until 28/04/2009.",
"The test set contains the events of the next day following the training period.",
"In VoD consumption dataset, with 8 weeks of training data we tested on the data of the next day.",
"Thus, all test events occurred after the last train event.",
"The training set contains 22.5 million events and 17 000 items.",
"The online grocery dataset contains only purchase events.",
"We used a few years' data for training and one month for testing.",
"The training set contains 6.24 million events and 14 000 items.",
"The two explicit feedback datasets are the Netflix [2] and the MovieLens 10M [27].",
"We kept the five star ratings for the former and ratings of 4.5 and above for the latter and used them as positive implicit feedback.",
"For train-test splits we used the splitting dates 15/12/2005 and 01/12/2008, respectively.",
"Table: Recall@20 for all datasets and algorithms using factorization with 20 and 40 features; in each row, the best and second best results are highlighted by bold and slanted typesetting, respectivelyWe determined the seasonality for each dataset, that is, the periodicity patterns observed in the data.",
"As for the VoD data, we defined a day as the season and defined custom time intervals as time bands ('morning', 'around noon', 'afternoon', 'evening', 'late evening', 'night' and 'dawn'), because people watch and channels broadcast different programs at different time of the day.",
"For LastFM 1K and MovieLens we also used a day as the season and time bands of 30 minutes.",
"For the Grocery data we defined a week as the season and the days of the week as the time bands.",
"The argument here is that people tend to follow different shopping behavior on weekdays and weekends.",
"For the Netflix data only the day of the rating is available, so we decided to define a week as the season and the days of the week as time bands.",
"In our next experiment, we used item category with Grocery and Netflix datasets, genre with VoD and MovieLens and artists for LastFM as the category of the item for the meta-rule learning algorithm.",
"We experimented with using the last 1, 2, 5 events prior to the current event of the users.",
"We compared the two iTALS variants to the basic iALS as well as to a context-aware baseline for implicit feedback data.",
"This method, referred as implicit CA (iCA) baseline, is the composite of several iALS models.",
"For each context state we train a model using only the events with the appropriate context, e.g., with the VoD we train 7 models for the 7 time bands.",
"The context of the recommendation request (e.g.",
"time of day) selects the model for the prediction.",
"This baseline treats context-states independently.",
"Due to its long running time we used iCA only with seasonality, as #(time bands) $\\ll $ #(preceding item categories).",
"Every algorithm has three common parameters: the number of features, the number of epochs and the regularization parameter.",
"We set the number of features to 20 and 40 commonly used in literature [1], [7].",
"The number of epochs was set to 10 as the ranked list of items hardly changes after 10 epochs.",
"The regularization was proportional to the support of the given item/user/context.",
"We did not use any other heuristics like time decay to focus on the pure performance of the algorithms.",
"The weights in $W$ were proportional to the number of events belonging to the given cell of the tensor.",
"We measured recall and precision on the $N=1,\\ldots ,50$ interval.",
"We consider items relevant to a user if the user has at least one event for that item in the test set.",
"Recall@$N$ is the ratio of relevant items on the ranked topN recommendations for the user relative to the number of the user's events in the test set.",
"Precision@$N$ is the ratio of the number of returned relevant items (for each user) and the number of total returned items.",
"Greater values mean better performance.",
"Figure: Running times of iTALS compared to iALS on the Grocery and LastFM 1K datasets.Table REF contains recall@20 values for every experiment.",
"Recall@20 is important in practical application as the user usually sees maximum the top 20 items.",
"Using context, the performance is increased overall.",
"The selection of the appropriate context is crucial.",
"In our experiments seasonality improved performance on all datasets.",
"The sequentiality patterns caused large improvements on the Grocery and LastFM 1K datasets (significantly surpassed the results with the seasonality) but did not increased performance on the movie databases (VoD, Netflix, MovieLens).",
"By including seasonality the performance is increased by an average of $30\\%$ for the VoD data.",
"This agrees with our assumption that the VoD consumption has a very strong daily repetitiveness and the behavior in different time bands can be well segmented.",
"The results increased by an additional $35\\%$ when we used iTALS instead of the context-aware baseline.",
"The genre of the previously watched movies can also improve performance, however its extent is only around $10\\%$ .",
"On the other two movie datasets iCA did not improve the performance significantly.",
"We assume that this is due to the explicit–implicit transformation because the transformed implicit feedback is more reliable and also results a sparser tensor.",
"The iTALS using seasonality however could achieve $30\\%$ and $80\\%$ improvement on Netflix and MovieLens respectively.",
"Inclusion of the sequentiality patterns increased the performance on Grocery and LastFM 1K datasets by more than $90\\%$ and $300\\%$ (compared to iALS, recall that no sequential iCA baseline is calculated).",
"Interestingly, the model using the last category is the best with 20 features, but with 40 features the model using last two categories becomes better.",
"We conjecture that this is connected to the greater expressive power of the model with more features.",
"With seasonality the performance also improved by more than $50\\%$ and $75\\%$ , respectively, on these datasets.",
"We expected that the usage pattern learning will perform better on Grocery and LastFM 1K datasets than on the movie datasets as sequentiality is rather important in shopping and music listening than seasonality.",
"Figure REF shows the precision–recall curves.",
"The order of the performance of the algorithms is the same as with the recall@20.",
"Observe that the distance between the curves of the iTALS variants and the curve of the iALS is larger when we use 40 features.",
"Recall that the feature vectors of the context works as a reweighting of the user–item relation.",
"If the resolution of this relation is finer, the reweighting can be more efficient and each factor describes a more specific item property, so the behavior in different context can be described more specifically.",
"Thus, increasing the number of features results in larger performance increase for the context-aware iTALS variants than for iALS."
],
[
"Running times",
"We compared the running times of the iTALS and iALS algorithms in terms of $K$ (see Figure REF ).",
"The experiments were run on a laptop with an Intel Core i5 2410M 2.3GHz processor using only one core.",
"We depict only curves for 2 datasets, since others are similar.",
"We made several runs for each $K$ ; the median of the epoch running times are shown (dashed lines).",
"The solid lines show the computation time for one feature matrix.",
"Observe that iTALS scales quadratically with $K$ as iALS; the (re)computation time of one feature matrix is basically the same.",
"Since iTALS recomputes more feature matrices its running time per epoch is larger.",
"Importantly, even if the number of context-states is large (as with the sequential iTALS on LastFM 1K) the $O(K^2)$ part of the complexity remains dominant.",
"This is because the number of non-zero elements in $T$ is much larger than the number of different items/users/contex-states in every case where the usage of context-aware approaches is justified."
],
[
"Conclusion",
"In this paper we presented an efficient ALS-based tensor factorization method for the context-aware implicit feedback recommendation problem.",
"Our method, coined iTALS, scales linearly with the number of non-zeroes in the tensor, thus it works well on implicit data.",
"We presented two specific examples for context-aware implicit scenario with iTALS.",
"When using the seasonality as context, we efficiently segmented periodical user behavior in different time bands.",
"When exploiting sequentiality in the data, the model was able to tell apart items having different repetitiveness in usage pattern.",
"These variants of iTALS allow us to analyze user behavior by integrating arbitrary contextual information within the well-known factorization framework.",
"Experiments performed on five large datasets show that proposed algorithms can greatly improve the performance.",
"Compared to iALS and iCA, our algorithm attained an increase in recall@20 up to $300\\%$ and $35\\%$ .",
"One should, however, avoid creating a high dimensional tensors because the number of non-zero elements remains the same no matter how many context types are integrated; so tensors with more dimensions become sparser and thus the results may be poorer than with only a few context dimensions used.",
"Our work opens up a new path for context-aware recommendations in the most common implicit feedback task when only the user history but no rating is available.",
"Future work will include the characterization of the relation between reweighting, context features and the number of features ($K$ ) as well as the design of further context-aware iTALS-based recommendation algorithms."
],
[
"Acknowledgment",
"Authors thank Gábor Takács for his very useful comments on the paper."
]
] | 1204.1259 |
[
[
"Numerical Invariants through Convex Relaxation and Max-Strategy\n Iteration"
],
[
"Abstract In this article we develop a max-strategy improvement algorithm for computing least fixpoints of operators on on the reals that are point-wise maxima of finitely many monotone and order-concave operators.",
"Computing the uniquely determined least fixpoint of such operators is a problem that occurs frequently in the context of numerical program/systems verification/analysis.",
"As an example for an application we discuss how our algorithm can be applied to compute numerical invariants of programs by abstract interpretation based on quadratic templates."
],
[
"Motivation",
"Finding tight invariants for a given program or system is crucial for many applications related to program respectively system verification.",
"Examples include linear recursive filters and numerical integration schemes.",
"Abstract Interpretation as introduced by [4] reduces the problem of finding tight invariants to the problem of finding the uniquely determined least fixpoint of a monotone operator.",
"In this article, we consider the problem of inferring numerical invariants using abstract domains that are based on templates.",
"That is, in addition to the program or system we want to analyze, a set of templates is given.",
"These templates are arithmetic expressions in the program/system variables.",
"The goal then is to compute small safe upper bounds on these templates.",
"We may, for instance, be interested in computing a safe upper bound on the difference $_1 - _2$ of two program/system variables $_1$ , $_2$ (at some specified control point of the program).",
"Examples for template-based numerical invariants include intervals (upper and lower bounds on the values of the numerical program variables) [3], zones (intervals and additionally upper and lower bounds on the differences of program variables) [10], [18], [11], octagons (zones and additionally upper und lower bounds on the sum of program variables) [12], and, more generally, linear templates (also called template polyhedra, upper bounds on arbitrary linear functions in the program variables, where the functions a given a priori) [15].",
"In this article, we focus on quadratic templates as considered by [1].",
"That is, a priori, a set of linear and quadratic functions in the program variables (the templates) is given and we are interested in computing small upper bounds on the values of these functions.",
"An example for a quadratic template is represented by the quadratic polynomial $2 x_1^2 + 3 x_2^2 + 2 x_1 x_2$ , where $x_1$ and $x_2$ are program variables.",
"When using such a template-based numerical abstract domain, the problem of finding the minimal inductive invariant, that can be expressed in the abstract domain specified by the templates, can be recast as a purely mathematical optimization problem, where the goal is to minimize a vector $(_1,\\ldots ,_n)$ subject to a set of inequalities of the form $_i \\ge f(_1,\\ldots ,_n).$ Here, $f$ is a monotone operator.",
"The variables $_1,\\ldots ,_n$ take values in $\\cup \\lbrace \\pm \\infty \\rbrace $ .",
"The variables are representing upper bounds on the values of the templates.",
"Accordingly, the vector $(_1,\\ldots ,_n)$ is to be minimized w.r.t.",
"the usual component-wise ordering.",
"Because of the monotonicity of the operators $f$ occurring in the right-hand sides of the inequalities and the completeness of the linearly ordered set $\\cup \\lbrace \\pm \\infty \\rbrace $ , the fixpoint theorem of Knaster/Tarski ensures the existence of a uniquely determined least solution.",
"Computing the least solution of such a constraint system is a difficult task.",
"Even if we restrict our consideration to the special case of intervals as an abstract domain, which is, if the program variables are denoted by $x_1,\\ldots ,x_n$ , specified by the templates $-x_1,x_1,\\ldots ,-x_n,x_n$ , the static analysis problem is at least as hard as solving mean payoff games.",
"The latter problem is a long outstanding problem which is in $\\mathsf {NP}$ and in $\\mathsf {coNP}$ , but not known to be in $\\mathsf {P}$ .",
"A generic way of solving systems of constraints of the form (REF ) with right-hand sides that are monotone and variables that range over a complete lattice is given through the abstract interpretation framework of [4].",
"Solving constraint systems in this framework is based on Kleene fixpoint iteration.",
"However, in our case the lattice has infinite ascending chains.",
"In this case, termination of the fixpoint iteration is ensured through an appropriate widening (see [4]).",
"Widening, however, buys termination for precision.",
"Although the lost precision can be partially recovered through a subsequently performed narrowing iteration, there is no guarantee that the computed result is minimal."
],
[
"Main Contribution",
"In this article, we study the case where the operators $f$ in the right-hand side of the systems of equations of the form (REF ) are not only monotone, but additionally order-concave or even concave (concavity implies order-concavity, but not vice-versa).",
"In the static program analysis application we consider in this article, the end up in this comfortable situation by considering a semi-definite relaxation of the abstract semantics.",
"The concavity of the mappings $f$ , however, does not imply that the problem can be formulated as a convex optimization problem.",
"The feasible space of the resulting mathematical optimization problem is normally neither order-convex nor order-concave and thus neither convex nor concave.",
"In consequence, convex optimization methods cannot be directly applied.",
"For the linear case (obtained when using used linear templates), we solved a long outstanding problem — namely the problem of solving mean payoff games in polynomial time — if we would be able to formalize the problem through a linear programming problem that can be constructed in polynomial time.",
"In this article, we exploit the fact that the operators $f$ that occur in the right-hand sides of the system of inequalities of the form (REF ) we have to solve are not only order-concave, but also monotone.",
"In other words: we do not require convexity of the feasible space, but we do require monotonicity in addition to the order-concavity.",
"The main contribution of this article is an algorithm for computing least solutions of such systems of inequalities.",
"The algorithm is based on strategy iteration.",
"That is, we consider the process of solving the system of inequality as a game between a maximizer and a minimizer.",
"The maximizer aims at minimizing the solution, whereas the minimizer aims at minimizing it.",
"The algorithm iteratively constructs a winning strategy for the maximizer — a so-called max-strategy.",
"It uses convex optimization techniques as sub-routines to evaluate parts of the constructed max-strategy.",
"The concrete convex optimization technique that is used for the evaluation depends on the right-hand sides.",
"In some cases linear programming is sufficient (see [5], [8]), In other cases more sophisticated convex optimization techniques are required.",
"The application we study in this article will require semi-definite programming.",
"P An important example for monotone and order-concave operators are the operators that are monotone and affine.",
"The class of monotone and order-concave operators is closed under the point-wise infimum operator.",
"The point-wise infimum of a set of monotone and affine functions, for instance, is monotone and order-concave.",
"Another example is the $\\sqrt{\\phantom{x}}$ -operator, which is defined by $\\sqrt{x} = \\sup \\; \\lbrace y \\in \\mid y^2 \\le x \\rbrace $ for all $x \\in $ .",
"P An example for a system of inequalities of the class we are considering in this article is the following system of inequalities: $_1 &\\ge \\frac{1}{2} &_1 &\\ge \\sqrt{_}2 &_2 &\\ge _1 &_2 &\\ge 1 + \\sqrt{ _2 - 1 }$ The uniquely determined least solution of the system (REF ) of inequalities is $_1 = _2 = 1$ .",
"We remind the reader again that the important property here is that the right-hand sides of (REF ) are monotone and order-concave.",
"P The least solution of the system (REF ) is also the uniquely determined optimal solution of the following convex optimization problem: $\\max \\;\\;& _1 + _2 & \\text{subject to} &&_1 &\\le \\sqrt{_}2 &_2 &\\le _1$ P Observe that the above convex optimization problem is in some sense a “subsystem” of the system (REF ).",
"Such a “subsystem”, which we will call a max-strategy later on, is obtained from the system (REF ) by selecting exactly one inequality of the form $_i \\ge e_i$ from (REF ) for each variable $_i$ and replacing the relation $\\le $ by the relation $\\ge $ .",
"Note that there are exponentially many max-strategies.",
"The algorithm we present in this article starts with a max-strategy and assigns a value to it.",
"It then iteratively improves the current max-strategy and assigns a new value to it until the least solution is found.",
"We utilize the monotonicity and the order-concavity of the right-hand sides to prove that our algorithm always terminates with the least solution after at most exponentially many improvement steps.",
"Each improvement step can be executed by solving linearly many convex optimization problems, each of which can be constructed in linear time.",
"P As a second contribution of this article, we show how any algorithm for solving such systems of inequalities, e.g., our max-strategy improvement algorithm, can be applied to infer numerical invariants based on quadratic templates.",
"The method is based on the relaxed abstract semantics introduced by [1]."
],
[
"Related Work",
"P The most closely related work is the work of [1].",
"They apply the min-strategy improvement approach of [2] to the problem of inferring quadratic invariants of programs.",
"In order to do so, they introduced the relaxed abstract semantics we are going to use in this article.",
"[1] in fact use the dual version of the relaxed abstract semantics we use in this article.",
"However, this minor difference does not have any practical consequences.",
"Their method, however, has several drawbacks compared to the method we present in this article.",
"The first drawback is that it does not necessarily terminate after finitely many steps.",
"In addition, even if it terminates, the computed solution is not guaranteed to be minimal.",
"On the other hand, their approach also has substantial advantages that are especially important in practice.",
"Firstly, it can be stopped at any time with a safe over-approximation to the least solution.",
"Secondly, the computational steps that have to be performed are quite cheap compared the the ones we have to perform for the method we propose in this article.",
"This is caused by the fact that the semi-definite programming problems (or in more general cases: convex programming problems) that have to be solved in each iteration are reasonable small.",
"We refer to [9] for a detailed comparison between the max- and the min-strategy approach."
],
[
"Previous Publications",
"P Parts of this work were previously published in the proceedings of the Seventeenth International Static Analysis Symposium (SAS 2010).",
"In contrast to the latter version, this article contains the full proofs and the precise treatment of infinities.",
"In order to simplify some argumentations and to deal with infinities, we modified some definitions quite substentially.",
"In addition to these improvements, we provide a much more detailed study of different classes of order-concave functions and the consequences for our max-strategy improvement algorithm.",
"We do not report on experimental results in this article.",
"Such reports can be found in the article in the proceedings of the Seventeenth International Static Analysis Symposium (SAS 2010)."
],
[
"Structure",
"P This article is structured as follows: Section is dedicated to preliminaries.",
"We study the class of monotone and order-concave operators in Section .",
"The results we obtain in Section are important to prove the correctness of our max-strategy improvement algorithm.",
"The method and its correctness proof is presented in Section .",
"In Section , we discuss the important special cases where the right-hand sides of the system of inequalities are parametrized convex optimization problems.",
"This can be used to evaluate strategies more efficiently.",
"These special cases are important, since they are present especially in the program analysis applications we mainly have in mind.",
"In Section , we finally explain how our methods can be applied to a numerical static program analysis based on quadratic templates.",
"We conclude with Section ."
],
[
"Preliminaries",
"P"
],
[
"Vectors and Matrices",
"We denote the $i$ -th row (resp.",
"$j$ -th column) of a matrix $A$ by $A_{i\\cdot }$ (resp.",
"$A_{\\cdot j}$ ).",
"Accordingly, $A_{i \\cdot j}$ denotes the component in the $i$ -th row and the $j$ -th column.",
"We also use these notations for vectors and vector valued functions $f : X\\rightarrow Y^k$ , i.e., $f_{i\\cdot } (x) = (f(x))_{i\\cdot }$ for all $i \\in \\lbrace 1,\\ldots ,k\\rbrace $ and all $x \\in X$ .",
"P"
],
[
"Sets, Functions, and Partial Functions",
"We write $A \\operatorname{{\\dot{\\cup }}}B$ for the disjoint union of the two sets $A$ and $B$ , i.e., $A \\operatorname{{\\dot{\\cup }}}B$ stands for $A \\cup B$ , where we assume that $A \\cap B = \\emptyset $ .",
"For sets $X$ and $Y$ , $X \\rightarrow Y$ denotes the set of all functions from $X$ to $Y$ , and $X \\rightsquigarrow Y$ denotes the set of all partial functions from $X$ to $Y$ .",
"Note that $X \\rightarrow Y \\subseteq X \\rightsquigarrow Y \\subseteq X \\times Y$ .",
"Accordingly, we apply the set operators $\\cup $ , $\\cap $ , and $\\setminus $ also to partial functions.",
"For $X^{\\prime } \\subseteq X$ , the restriction $f|_{X^{\\prime }} : X^{\\prime } \\rightarrow Y$ of a function $f : X \\rightarrow Y$ to $X^{\\prime }$ is defined by $f|_{X^{\\prime }} := f \\cap X^{\\prime } \\times Y$ .",
"The domain and the codomain of a partial function $f$ are denoted by $\\operatorname{\\mathsf {dom}}(f)$ and $\\operatorname{\\mathsf {codom}}(f)$ , respectively.",
"For $f : X \\rightarrow Y$ and $g : X \\rightsquigarrow Y$ , we define $f \\oplus g : X \\rightarrow Y$ by $f \\oplus g:=f|_{X\\setminus \\operatorname{\\mathsf {dom}}(g)} \\cup g$ .",
"P"
],
[
"Partially Ordered Sets",
"Let $ be a partially ordered set (partially ordered by the binary relation $$).Two elements $ x, y are called comparable if and only if $x \\le y$ or $y \\le x$ .",
"For all $x \\in ,we set $ x := { y y x }$ and $ x := { y y x }$.We denote the \\emph {least upper bound} andthe \\emph {greatest lower bound} ofa set $ X by $\\bigvee X$ and $\\bigwedge X$ , respectively, provided that it exists.",
"The least element $\\bigvee \\emptyset = \\bigwedge (resp.\\ the greatest element $ = ) is denoted by $\\bot $ (resp.",
"$\\top $ ), provided that it exists.",
"A subset $C \\subseteq is called a \\emph {chain}if and only if $ C$ is linearly ordered by $$,i.e., it holds $ x y$ or $ y x$ for all $ x,y C$.For every subset $ X of a set $ that is partially ordered by $$,we set $ X := { y x X .",
"x y }$.The set $ X is called upward closed w.r.t.",
"$$ if and only if $X{^ = X.We omit the reference to , if is clear from the context.",
"}$ P"
],
[
"Monotonicity",
"Let $1, 2$ be partially ordered sets (partially ordered by $\\le $ ).",
"A mapping $f : 1 \\rightarrow 2$ is called monotone if and only if $f(x) \\le f(y)$ for all $x,y \\in 1$ with $x \\le y$ .",
"A monotone function $f$ is called upward-chain-continuous (resp.",
"downward-chain-continuous) if and only if $f(\\bigvee C) = \\bigvee f(C)$ (resp.",
"$f(\\bigwedge C) = \\bigwedge f(C)$ ) for every non-empty chain $C$ with $\\bigvee C \\in \\operatorname{\\mathsf {dom}}(f)$ (resp.",
"$\\bigwedge C \\in \\operatorname{\\mathsf {dom}}(f)$ ).",
"It is called chain-continuous if and only if it is upward-chain-continuous and downward-chain-continuous.",
"P"
],
[
"Complete Lattices",
"A partially ordered set $is called a \\emph {complete lattice}if and only if$ X$ and $ X$ exist for all $ X .",
"If $ is a complete lattice and $ x , then the sublattices ${\\ge x}$ and ${\\le x}$ are also complete lattices.",
"On a complete lattice $,we define the binary operators $$ and $$ by{\\begin{@align}{1}{-1}x \\vee y := \\bigvee \\lbrace x, y \\rbrace \\text{ and }x \\wedge y := \\bigwedge \\lbrace x, y \\rbrace &&\\text{for all } x, y \\in \\end{@align}}$P respectively.",
"If the complete lattice $ is a\\emph {complete linearly ordered set}(for instance $ = { }$),then $$ is the binary \\emph {maximum} operatorand $$ the binary \\emph {minimum} operator.For all binary operators $ { , }$,we also consider $ x1 xk$as the application of a $ k$-ary operator.This will cause no problems,since the binary operators $$ and $$are associative and commutative.$ P"
],
[
"Fixpoints",
"Assume that the set $ is partially ordered by $$ and $ f: is a unary operator on $.An element $ x is called fixpoint (resp.",
"pre-fixpoint, resp.",
"post-fixpoint) of $f$ if and only if $x = f(x)$ (resp.",
"$x \\le f(x)$ , resp.",
"$x \\ge f(x)$ ).",
"The set of all fixpoints (resp.",
"pre-fixpoints, resp.",
"post-fixpoints) of $f$ is denoted by $(f)$ (resp.",
"$(f)$ , resp.",
"$(f)$ ).",
"We denote the least (resp.",
"greatest) fixpoint of $f$ — provided that it exists — by $\\mu f$ (resp.",
"$\\nu f$ ).",
"If the partially ordered set $ is a complete lattice and $ f$ is monotone,then the fixpoint theorem of Knaster/Tarski \\cite {Tarski55} ensures the existence of $ f$ and $ f$.Moreover,we have$ f = (f)$and dually$ f = (f)$.$ P We write $\\mu _{\\ge x} f$ (resp.",
"$\\nu _{\\le x} f$ ) for the least element in the set $(f) \\cap {\\ge x}$ (resp.",
"$(f) \\cap {\\le x}$ ).",
"The existence of $\\mu _{\\ge x} f$ (resp.",
"$\\nu _{\\le x} f$ ) is ensured if ${\\ge x}$ is a complete lattice and $f|_{{\\ge x}}$ (resp.",
"$f|_{{\\le x}}$ ) is a monotone operator on ${\\ge x}$ (resp.",
"${\\le x}$ ), i.e., if ${\\ge x}$ (resp.",
"${\\le x}$ ) is closed under the operator $f$ .",
"The latter condition is, for instance, fulfilled if $ is a complete lattice, $ f$ is a monotone operator on $ , and $x$ is a pre-fixpoint (resp.",
"post-fixpoint) of $f$ .",
"P"
],
[
"The Complete Lattice $^n$",
"The set of real numbers is denoted by $$ , and the complete linearly ordered set $\\cup \\lbrace \\pm \\infty \\rbrace $ is denoted by $$ .",
"Therefore, the set $^n$ is a complete lattice that is partially ordered by $\\le $ , where we write $x \\le y$ if and only if $x_{i\\cdot } \\le y_{i\\cdot }$ for all $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"As usual, we write $x < y$ if and only if $x \\le y$ and $x \\ne y$ .",
"We write $x \\ll y$ if and only if $x_{i\\cdot } < y_{i\\cdot }$ for all $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"For $f : ^n \\rightsquigarrow ^m$ , we set $\\operatorname{\\mathsf {fdom}}(f) := \\lbrace x \\in \\operatorname{\\mathsf {dom}}(f) \\cap ^n \\mid f(x) \\in ^m \\rbrace .$ P"
],
[
"The Vector Space $^n$",
"The standard base vectors of the Euclidian vector space $^n$ are denoted by $e_1,\\ldots ,e_n$ .",
"We denote the maximum norm on $^n$ by ${\\cdot }$ , i.e., $x = \\max \\; \\lbrace {x_{i\\cdot }} \\mid i \\in \\lbrace 1,\\ldots ,n \\rbrace \\rbrace $ for all $x \\in ^n$ .",
"A vector $x \\in ^n$ with $x = 1$ is called a unit vector."
],
[
"Morcave Operators",
"P In this section, we introduce a notion of order-concavity for functions from the set $^n \\rightarrow ^m$ .",
"We then study the properties of functions that are monotone and order-concave.",
"The results obtained in this section are used in Section to prove the correctness of our max-strategy improvement algorithm."
],
[
"Monotone Operators on $^n$",
"P In this subsection, we collect important properties about monotone operators on $^n$ .",
"We start with the following auxiliary lemma: P Let $d, d^{\\prime } \\in ^n$ with $d \\gg 0$ and $d^{\\prime } \\ge 0$ .",
"There exist $j \\in \\lbrace 1,\\ldots ,n\\rbrace $ and $\\lambda , \\lambda _1,\\ldots ,\\lambda _n \\ge 0$ such that $\\lambda _j = 0$ and $\\lambda d = d^{\\prime } + \\sum _{i = 1}^n \\lambda _i e_i$ .",
"P Since $d \\gg 0$ , there exist a $j \\in \\lbrace 1,\\ldots ,n\\rbrace $ and a $\\lambda \\ge 0$ such that $\\lambda d - d^{\\prime } \\ge 0$ and $(\\lambda d - d^{\\prime })_{j \\cdot } = 0$ .",
"Thus, there exist $\\lambda _1,\\ldots ,\\lambda _n$ with $\\lambda _j = 0$ such that $\\lambda d - d^{\\prime } = \\sum _{i = 1}^n \\lambda _i e_i$ .",
"$\\Box $ P We now provide a sufficient criterium for a fixpoint $x$ of a monotone partial operator $f$ on $^n$ for being the greatest pre-fixpoint of $f$ .P Note that, since $^n$ is not a complete lattice, the greatest pre-fixpoint of $f$ is not necessarily the greatest fixpoint of $f$ .",
"The greatest fixpoint of the monotone operators $f_1, f_2$ defined by $f_1(x) = \\frac{1}{2}x$ and $f_2(x) = 2 x$ for all $x \\in $ , for instance, is 0.",
"This is also the greatest pre-fixpoint of $f_1$ , but not the greatest pre-fixpoint of $f_2$ , since $f_2$ has no greatest pre-fixpoint.",
"Such sufficient criteria are crucial to prove the correctness of our max-strategy improvement algorithm.",
"P Let $f : ^n \\rightsquigarrow ^n$ be monotone with $\\operatorname{\\mathsf {dom}}(f)$ upward closed, $f(x) = x$ , and $k \\in $ .",
"Assume that, for every $\\epsilon > 0$ , there exists a unit vector $d_\\epsilon \\gg 0$ such that $f^k(x + \\lambda d_\\epsilon ) \\ll x + \\lambda d_\\epsilon $ for all $\\lambda \\ge \\epsilon $ .",
"Then, $y \\le x$ for all $y$ with $y \\le f(y)$ , i.e., $x$ is the greatest pre-fixpoint of $f$ .",
"P We show $y \\lnot \\le x \\Rightarrow y \\lnot \\le f(y)$ .",
"For that, we first show the following statement: $y > x \\Rightarrow y \\lnot \\le f(y)$ P For that, let $y > x$ .",
"Let $\\epsilon := {y - x}$ .",
"By Lemma REF , there exist $\\lambda ,\\lambda _1,\\ldots ,\\lambda _n \\ge 0$ with $\\lambda _j = 0$ for some $j \\in \\lbrace 1,\\ldots ,n\\rbrace $ such that $\\overline{y} := x + \\lambda d_\\epsilon = y + \\sum _{i = 1}^n \\lambda _i e_i$ holds.",
"We necessarily have $\\lambda \\ge \\epsilon $ .",
"Using the monotonicity of $f$ and the fact that $f^k(\\overline{y}) \\ll \\overline{y}$ holds by assumption, we get $f^k_{j\\cdot }(y) \\le f^k_{j\\cdot }(\\overline{y}) < \\overline{y}_{j\\cdot } = y_{j\\cdot }$ .",
"Therefore, $y \\lnot \\le f(y)$ .",
"Thus, we have shown (REF ).",
"Now, let $y \\lnot \\le x$ .",
"Thus, $y^{\\prime } := x \\vee y > x$ .",
"Using (REF ) we get $y ^{\\prime } \\lnot \\le f(y^{\\prime })$ .",
"For the sake of contradiction assume that $y \\le f(y)$ holds.",
"Then we get $f(y^{\\prime }) = f(x \\vee y) \\ge f(x) \\vee f(y) \\ge x \\vee y = y^{\\prime }$ — contradiction.",
"$\\Box $ P In the remainder of this article, we only use the following corollary of Lemma REF : P Let $f : ^n \\rightsquigarrow ^n$ be monotone with $\\operatorname{\\mathsf {dom}}(f)$ upward closed, $f(x) = x$ , and $k \\in $ .",
"Assume that there exists a unit vector $d \\gg 0$ such that $f^k(x + \\lambda d) \\ll x + \\lambda d$ for all $\\lambda > 0$ .",
"Then, $y \\le x$ for all $y$ with $y \\le f(y)$ , i.e., $x$ is the greatest pre-fixpoint of $f$ .",
"$\\Box $"
],
[
"Monotone and Order-Concave Operators on $^n$",
"P A set $X \\subseteq ^n$ is called order-convex if and only if $ \\lambda x + (1-\\lambda ) y \\in X$ for all comparable $x,y \\in X$ and all $\\lambda \\in [0,1]$ .",
"It is called convex if and only if this condition holds for all $x,y \\in X$ .",
"Every convex set is order-convex, but not vice-versa.",
"If $n = 1$ , then every order-convex set is convex.",
"Every upward closed set is order-convex, but not necessarily convex.",
"P A partial function $f : ^n \\rightsquigarrow ^m$ is called order-convex (resp.",
"order-concave) if and only if $\\operatorname{\\mathsf {dom}}(f)$ is order-convex and $f(\\lambda x + (1-\\lambda )y)\\le \\text{(resp.\\ $\\ge $)}\\;\\lambda f(x) + (1-\\lambda ) f(y)$ P for all comparable $x,y \\in \\operatorname{\\mathsf {dom}}(f)$ and all $\\lambda \\in [0,1]$ (cf.",
"[14]).",
"A partial function $f : ^n \\rightsquigarrow ^m$ is called convex (resp.",
"concave) if and only if $\\operatorname{\\mathsf {dom}}(f)$ is convex and $f(\\lambda x + (1-\\lambda )y)\\le \\text{(resp.\\ $\\ge $)}\\;\\lambda f(x) + (1-\\lambda ) f(y)$ P for all $x,y \\in \\operatorname{\\mathsf {dom}}(f)$ and all $\\lambda \\in [0,1]$ (cf.",
"[14]).",
"Every convex (resp.",
"concave) partial function is order-convex (resp.",
"order-concave), but not vice-versa.",
"Note that $f$ is (order-)concave if and only if $-f$ is (order-)convex.",
"Note also that $f$ is (order-)convex (resp.",
"(order-)concave) if and only if $f_{i\\cdot }$ is (order-)convex (resp.",
"(order-)concave) for all $i = 1,\\ldots ,m$ .",
"If $n = 1$ , then every order-convex (resp.",
"order-concave) partial function is convex (resp.",
"concave).",
"Every order-convex/order-concave partial function is chain-continuous.",
"Every convex/concave partial function is continuous.",
"P The set of (order-)convex (resp.",
"(order-)concave) partial functions is not closed under composition.",
"The functions $f, g$ defined by $f(x) = (x-2)^2$ and $g(x) = \\frac{1}{x}$ for all $x \\in $ , for instance, are both convex and thus also order-convex.",
"However, $f \\circ g$ with $(f \\circ g)(x) = (\\frac{1}{x} - 2)^2$ for all $x \\in $ is neither convex nor order-convex.",
"P In contrast to the set of all order-concave partial functions, the set of all partial functions that are monotone and order-concave is closed under composition: P Let $f : ^m \\rightsquigarrow ^n$ and $g : ^l \\rightsquigarrow ^m$ be monotone and order-convex (resp.",
"order-concave).",
"Assume that $\\operatorname{\\mathsf {codom}}(g) \\subseteq \\operatorname{\\mathsf {dom}}(f)$ .",
"Then $f \\circ g$ is monotone and order-convex (resp.",
"order-concave).",
"P We assume that $f$ and $g$ are order-convex.",
"The other case can be proven dually.",
"Let $x,x^{\\prime } \\in \\operatorname{\\mathsf {dom}}(g)$ with $x \\le x^{\\prime }$ , $y = g(x)$ , $y^{\\prime } = g(x^{\\prime })$ .",
"Since $g$ is monotone, we get $y \\le y^{\\prime }$ .",
"Since $f$ is monotone, we get $(f \\circ g)(x)=f(g(x))=f(y)\\le f(y^{\\prime })=f(g(x^{\\prime }))=(f \\circ g)(x^{\\prime })$ .",
"Hence, $f\\circ g$ is monotone.",
"Let $\\lambda \\in [0,1]$ .",
"Then $(f\\circ g)(\\lambda x + (1-\\lambda ) x^{\\prime })=f(g(\\lambda x + (1-\\lambda ) x^{\\prime })\\le f(\\lambda g(x) + (1-\\lambda ) g(x^{\\prime }))=f(\\lambda y + (1-\\lambda ) y^{\\prime })\\le \\lambda f(y) + (1-\\lambda ) f ( y^{\\prime } )=\\lambda f(g(x)) + (1-\\lambda ) f ( g(x^{\\prime }) )=\\lambda (f\\circ g)(x) + (1-\\lambda ) (f\\circ g)(x^{\\prime })$ , because $f$ is monotone, and $f$ and $g$ are order-convex.",
"Hence, $f\\circ g$ is order-convex.",
"$\\Box $"
],
[
"Fixpoints of Monotone and Order-concave Operators on $^n$",
"P We now study the fixpoints of monotone and order-concave partial operators on $^n$ .",
"We are in particular interested in developing a simple sufficient criterium for a fixpoint of a monotone and order-concave partial operator on $^n$ for being the greatest pre-fixpoint of this partial operator.",
"To prepare this, we first show the following lemma: P Let $f : ^n \\rightsquigarrow ^n$ be order-convex (resp.",
"order-concave).",
"Let $x, x^* \\in \\operatorname{\\mathsf {dom}}(f)$ with $x^* = f(x^*)$ , $x \\gg \\text{(resp.\\ $\\ll $)} \\, f(x)$ , $d := x^* - x \\gg 0$ .",
"Then, $x^* + \\lambda d \\ll \\text{(resp.\\ $\\gg $)} \\, f(x^* + \\lambda d)$ for all $\\lambda > 0$ with $x^* + \\lambda d \\in \\operatorname{\\mathsf {dom}}(f)$ .",
"P We only consider the case that $f$ is order-convex.",
"The proof for the case that $f$ is order-concave can be carried out dually.",
"Let $\\lambda > 0$ .",
"Assume for the sake of contradiction that there exists some $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ such that $(x^* + \\lambda d)_{i\\cdot } \\ge f_{i\\cdot }(x^* + \\lambda d)$ .",
"Since $f_{i\\cdot }$ is order-convex and $x_{i\\cdot } > f_{i \\cdot }(x)$ holds, it follows $x^*_{i\\cdot } > f_{i\\cdot }(x^*)$ — contradiction.",
"$\\Box $ Let $F \\subseteq ^m\\rightarrow ^n$ be a set of (order-)concave functions.",
"Then $g$ defined by $g(x) := \\bigwedge \\lbrace f(x) \\mid f \\in F \\rbrace $ for all $x \\in ^m$ is (order-)concave.",
"?",
"??",
"TODO ?",
"??",
"Let $f, g : ^m\\rightarrow ^n$ be order-concave, and $a \\in _{\\ge 0}$ .",
"Then $f + g$ and $af$ are order-concave.",
"?",
"??",
"TODO ?",
"??",
"$\\Box $ ?",
"??",
"Das folgende könnte man auf den endlichen Bereich erstmal beschränken, oder ?",
"??",
"P We now use the results obtained so far to prove the following sufficient criterium for a fixpoint of a monotone and order-concave partial operator for being the greatest pre-fixpoint.",
"P Let $f : ^n \\rightsquigarrow ^n$ be monotone and order-concave with $\\operatorname{\\mathsf {dom}}(f)$ upward closed.",
"Let $x^*$ be a fixpoint of $f$ , $x$ be a pre-fixpoint of $f$ with $x \\ll x^*$ , and $\\mu _{\\ge x} f = x^*$ .",
"Then, $x^*$ is the greatest pre-fixpoint of $f$ .",
"P Since $f$ is chain-continuous and $x \\ll x^*$ is a pre-fixpoint of $f$ , there exists some $k \\in $ such that $x \\ll f^k(x)$ .",
"Let $x^{\\prime }$ be a pre-fixpoint of $f$ .",
"Let $d := x^* - x$ .",
"Note that $d \\gg 0$ .",
"Since $f^k|_{^n_{\\ge x}} = (f|_{^n_{\\ge x}})^k$ is monotone and order-concave by Lemma REF , and $x^*$ is a fixpoint of $f^k$ and thus of $f^k|_{^n_{\\ge x}}$ , we get $f^k(x^* + \\lambda d)=f^k|_{^n_{\\ge x}}(x^* + \\lambda d)\\ll x^* + \\lambda d$ for all $\\lambda \\in $ by Lemma REF .",
"Thus, Lemma REF gives us $x^{\\prime } \\le x^*$ .",
"$\\Box $ P Let us consider the monotone and concave partial operator $\\sqrt{\\cdot }: \\rightsquigarrow $ .",
"The points 0 and 1 are fixpoints of $\\sqrt{\\cdot }$ , since $0 = \\sqrt{0}$ , and $1 = \\sqrt{1}$ .",
"Since $\\frac{1}{2}$ is a pre-fixpoint of $\\sqrt{\\cdot }$ , $\\frac{1}{2} < 1$ , and $\\mu _{\\ge \\frac{1}{2}} \\sqrt{\\cdot }= 1$ , Lemma REF implies that 1 is the greatest pre-fixpoint of $\\sqrt{\\cdot }$ .",
"Observe that for the fixpoint 0, there is no pre-fixpoint $x \\in $ of $\\sqrt{\\cdot }$ with $x < 0$ .",
"Therefore, Lemma REF cannot be applied.",
"$\\Box $ P The following example shows that the criterium of Lemma REF is sufficient, but not necessary: P Let $f : \\rightarrow $ be defined by $f(x) = 0 \\wedge x$ for all $x \\in $ .",
"Recall that $\\wedge $ denotes the minimum operator.",
"Then, 0 is the greatest pre-fixpoint of $f$ .",
"However, there does not exist a $x \\in $ with $x < 0$ such that $\\mu _{\\ge x} f = 0$ , since $\\mu _{\\ge x} f = x$ for all $x \\le 0$ .",
"Therefore, Lemma REF cannot be applied to show that 0 is the greatest pre-fixpoint of $f$ .",
"$\\Box $ P The set $^n$ can be identified with the set $\\lbrace 1,\\ldots ,n\\rbrace \\rightarrow $ which can be identified with the set $\\rightarrow $ , whenever $= n$ .",
"In the remainder of this article, we therefore identify the set $(\\rightarrow ) \\rightsquigarrow (\\rightarrow )$ with the set $^n \\rightsquigarrow ^n$ — provided that $= n$ .",
"Usually, we use $= \\lbrace _1,\\ldots ,_n \\rbrace $ .",
"We use one or the other representation depending on which representation is more convenient in the given context.",
"P Our next goal is to weaken the preconditions of Lemma REF , i.e., we aim at providing a weaker sufficient criterium for a fixpoint of a monotone and order-concave partial operator for being the greatest pre-fixpoint than the one provided by Lemma REF .",
"The weaker sufficient criterium we are going to develop can, for instance, be applied to the following example: P Let us consider the monotone and order-concave partial operator $f : ^2 \\rightsquigarrow ^2$ defined by $f(x_1,x_2):=(x_2 + 1 \\wedge 0, \\; \\sqrt{x}_1 )$ for all $x_1,x_2 \\in $ .",
"Then, $x^* = (x^*_1,x^*_2) = (0, 0)$ is the greatest pre-fixpoint of $f$ .",
"In order to prove this, assume that $y = (y_1,y_2) > x^*$ is a pre-fixpoint of $f$ , i.e., $y_1 \\le y_2 + 1$ , $y_1 \\le 0$ , and $y_2 \\le \\sqrt{y_1}$ .",
"It follows immediately that $y_1 \\le 0$ and thus $y_2 \\le \\sqrt{y_1} \\le \\sqrt{0} = 0$ .",
"P Lemma REF is not applicable to prove that $x^*$ is the greatest pre-fixpoint of $f$ , because there is no pre-fixpoint $x$ of $f$ with $x \\ll x^*$ .",
"The situation is even worse: there is no $x \\in \\operatorname{\\mathsf {dom}}(f)$ with $x \\ll x^*$ .",
"P We observe that, locally at $x^* = (0,0)$ , the first component $f_{1\\cdot }$ of $f$ does not depend on the second argument in the following sense: For every $y = (y_1,y_2) \\in ^2$ with $y_1 = x^*_1 = 0$ and $y_2 > x^*_2 = 0$ , we have $f_{1\\cdot }(y) = 0 = f_{1\\cdot }(x^*)$ .",
"The weaker sufficient criterium we develop in the following takes this into account.",
"That is, we will assume that the set of variables can be partitioned according to their dependencies.",
"The sufficient criterium of Lemma REF should then hold for each partition.",
"In this example this means: there exists some $x_1 < x^*_1$ with $x_1 \\le f_{1\\cdot }(x_1,x^*_2) = f_{1\\cdot }(x_1,0)$ and $\\mu _{\\ge x_1} f_{1\\cdot }(\\cdot ,0) = x^*_1 = 0$ , and there exists some $x_2 < x^*_2$ with $x_2 \\le f_{2\\cdot }(x^*_1,x_2) = f_{2\\cdot }(0,x_2)$ and $\\mu _{\\ge x_2} f_{2\\cdot }(0,\\cdot ) = x^*_2 = 0$ .",
"We could choose $x_1 = x_2 = -1$ , for instance.",
"$\\Box $ P In order to derive a sufficient criterium that is weaker than the sufficient criterium of Lemma REF , we should, as suggested in Example REF , partition the variables according to their dependencies.",
"In order to define a suitable notion of dependencies, let $$ be a set of variables, $f : (\\rightarrow )\\rightsquigarrow (\\rightarrow )$ be a monotone partial operator, and $\\rho : \\rightarrow $ .",
"For $_1 \\operatorname{{\\dot{\\cup }}}_2 = $ , we write $_1 \\stackrel{f,\\rho }{\\rightarrow }_2$ if and only if $_1 = \\emptyset $ , $_2 = \\emptyset $ , or there exists an $\\rho ^{\\prime } : _2 \\rightarrow $ with $\\rho \\oplus \\rho ^{\\prime } \\in \\operatorname{\\mathsf {dom}}(f)$ and $\\rho ^{\\prime } \\ll \\rho |_{_2}$ such that $f (\\rho \\oplus \\rho ^{\\prime }) |_{_1} = f(\\rho )|_{_1}$ .",
"P Informally spoken, $_1 \\stackrel{f,\\rho }{\\rightarrow }_2$ states that — locally at $\\rho $ — the values of the variables from the set $_1$ do not depend on the values of the variables from the set $_2$ .",
"Dependencies are only admitted in the opposite direction — from $_1$ to $_2$ .",
"P Let us again consider the monotone and order-concave partial operator $f : ^2 \\rightsquigarrow ^2$ from Example REF defined by $f(x_1,x_2):=(x_2 + 1 \\wedge 0, \\; \\sqrt{x}_1 )$ for all $x_1,x_2 \\in $ .",
"Note that $f$ is not a total operator, since $\\sqrt{x_1}$ and thus $f(x_1,x_2)$ is undefined for all $x_1 < 0$ .",
"Moreover, let $x := (0, 0 )$ .",
"Recall that we identify the set $^2$ with the set $\\lbrace _1, _2\\rbrace \\rightarrow $ .",
"Especially, we identify $x$ with the function $\\lbrace _1 \\mapsto 0,\\; _2 \\mapsto 0 \\rbrace $ .",
"Then, we have $\\lbrace _1 \\rbrace \\stackrel{f, x}{\\rightarrow }\\lbrace _2\\rbrace $ .",
"That is, locally at $x$ , the first component $f_{1\\cdot }$ of $f$ does not depend on the second argument.",
"In other words: locally at $x$ , one can strictly decrease the value of the second argument without changing the value of the first component $f_{1\\cdot }$ of $f$ .",
"However, the second component $f_{2\\cdot }$ of $f$ may, locally at $x$ , depend on the first argument.",
"In this example, this is actually the case: Locally at $x$ , we cannot decrease the value of the first argument without changing the value of the second component $f_{2\\cdot }$ of $f$ .",
"$\\Box $ P If the partial operator $f$ is monotone and order-concave, then the statement $_1 \\stackrel{f,\\rho }{\\rightarrow }_2$ also implies that, locally at $\\rho $ , the values of the $_1$ -components of $f$ do not increase if the values of the variables from $_2$ increase: P Assume that $f : (\\rightarrow ) \\rightsquigarrow (\\rightarrow )$ is monotone and order-concave.",
"If $_1 \\stackrel{f,\\rho }{\\rightarrow } _2$ , then $(f(\\rho \\oplus \\rho ^{\\prime }))|_{_1} = (f(\\rho ))|_{_1}$ for all $\\rho ^{\\prime } : _2 \\rightarrow $ with $\\rho ^{\\prime } \\ge \\rho |_{_2}$ and $\\rho \\oplus \\rho ^{\\prime } \\in \\operatorname{\\mathsf {dom}}(f)$ .",
"$\\Box $ P For $_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k = $ , we write $_1 \\stackrel{f,\\rho }{\\rightarrow }\\cdots \\stackrel{f,\\rho }{\\rightarrow }_k$ if and only if $k = 1$ or $_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j\\stackrel{f,\\rho }{\\rightarrow }_{j+1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k$ for all $j \\in \\lbrace 1,\\ldots ,k-1\\rbrace $ .",
"P Let $$ and $ be sets,$ f : ( ($,and$ 1 2 = $.For $ 2 : 2 , we define $f \\leftarrow \\rho _2 : (_1\\rightarrow \\rightsquigarrow (_1\\rightarrow $ by $(f \\leftarrow \\rho _2) (\\rho _1)&:=(f ( \\rho _1 \\cup \\rho _2 )) |_{_1}&&\\text{for all }\\rho _1 : _1 \\rightarrow $ P Informally spoken, $f \\leftarrow \\rho _2$ is the function that is obtained from $f$ by fixing the values of the variables from the set $_2$ according to variable assignment $\\rho _2$ and afterwards removing all variables from the set $_2$ .",
"P Let us again consider the monotone and order-concave partial operator $f : ^2 \\rightsquigarrow ^2$ from Examples REF and REF that is defined by $f(x_1,x_2):=(x_2 + 1 \\wedge 0, \\; \\sqrt{x}_1 )$ for all $x_1,x_2 \\in $ .",
"Let again $x := (0, 0)$ be identified with $x = \\lbrace _1 \\mapsto 0, \\; _2 \\mapsto 0 \\rbrace $ .",
"Then $(f \\leftarrow x|_{\\lbrace _2\\rbrace })(\\rho _1)=\\lbrace _1 \\mapsto 0 \\rbrace $ for all $\\rho _1 : \\lbrace _1\\rbrace \\rightarrow $ , and $(f \\leftarrow x|_{\\lbrace _1\\rbrace })(\\rho _2)=\\lbrace _2 \\rightarrow 0 \\rbrace $ for all $\\rho _2 : \\lbrace _2\\rbrace \\rightarrow $ .",
"$\\Box $ P The weaker sufficient criterium for a fixpoint of a monotone and order-concave partial operator for being the greatest pre-fixpoint of this partial operator can now be formalized as follows: [Feasibility] P Let $f : (\\rightarrow ) \\rightsquigarrow (\\rightarrow )$ be monotone and order-concave.",
"A fixpoint $\\rho ^*$ of $f$ is called feasible if and only if there exist $_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k = $ with $_1 \\stackrel{f,\\rho ^*}{\\rightarrow }\\cdots \\stackrel{f,\\rho ^*}{\\rightarrow }_k$ such that, for each $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ , there exists some pre-fixpoint $\\rho : _j\\rightarrow $ of $f \\leftarrow \\rho ^*|_{\\setminus _j}$ with $\\rho \\ll \\rho ^*|_{_j}$ such that $\\mu _{\\ge \\rho } (f \\leftarrow \\rho ^*|_{\\setminus _j}) = \\rho ^*|_{_j}$ .",
"$\\Box $ P Let us again consider the monotone and order-concave partial operator $f : ^2 \\rightsquigarrow ^2$ from the Examples REF , REF , and REF that is defined by $f(x_1,x_2):=(x_2 + 1 \\wedge 0, \\; \\sqrt{x}_1 )$ for all $x_1,x_2 \\in $ .",
"We show that $x := (0, 0 )$ is a feasible fixpoint of $f$ .",
"From Example REF , we know that Lemma REF is not applicable to prove that $x$ is the greatest pre-fixpoint.",
"Recall that we can identify the set $^2$ with the set $\\lbrace _1, _2\\rbrace \\rightarrow $ , and hence $x$ with $\\lbrace _1\\mapsto 0 ,\\; _2\\mapsto 0 \\rbrace $ .",
"We have $\\lbrace _1 \\rbrace \\stackrel{f, x}{\\rightarrow }\\lbrace _2 \\rbrace $ .",
"Moreover, $\\lbrace _1 \\mapsto -1 \\rbrace \\ll x|_{\\lbrace _1\\rbrace }$ is a pre-fixpoint of $f \\leftarrow x|_{\\lbrace _2\\rbrace }$ with $ \\mu _{\\ge \\lbrace _1 \\mapsto -1 \\rbrace } (f \\leftarrow x|_{\\lbrace _2\\rbrace })=x|_{\\lbrace _1\\rbrace },$ and $\\lbrace _2 \\mapsto -1 \\rbrace \\ll x|_{\\lbrace _2\\rbrace }$ is a pre-fixpoint of $f \\leftarrow x|_{\\lbrace _1\\rbrace }$ with $ \\mu _{\\ge \\lbrace _2 \\mapsto -1 \\rbrace } (f \\leftarrow x|_{\\lbrace _1\\rbrace })=x|_{\\lbrace _2\\rbrace }$ .",
"Thus, $x$ is a feasible fixpoint of $f$ .",
"$\\Box $ P We now show that feasibility is indeed sufficient for a fixpoint to be the greatest pre-fixpoint.",
"Since any fixpoint that fulfills the criterium given by Lemma REF is feasible, but, as the Examples REF and REF show, not vice-versa, the following lemma is a strict generalization of Lemma REF .",
"P Let $f : (\\rightarrow ) \\rightsquigarrow (\\rightarrow )$ be monotone and order-concave with $\\operatorname{\\mathsf {dom}}(f)$ upward closed, and $\\rho ^*$ be a feasible fixpoint of $f$ .",
"Then, $\\rho ^*$ is the greatest pre-fixpoint of $f$ .",
"P Since $\\rho ^*$ is a feasible fixpoint of $f$ , there exists $_1\\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k = $ with $_1 \\stackrel{f,\\rho ^*}{\\rightarrow }\\cdots \\stackrel{f,\\rho ^*}{\\rightarrow }_k$ such that, for each $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ , there exists some pre-fixpoint $\\rho _j$ of $f \\leftarrow \\rho ^*|_{\\setminus _j}$ with $\\rho _j \\ll \\rho ^*|_{_j}$ and $\\mu _{\\ge \\rho _j} (f \\leftarrow \\rho ^*|_{\\setminus _j}) = \\rho ^*|_{_j}$ .",
"Let $\\rho ^{\\prime }$ be a pre-fixpoint of $f$ with $\\rho ^{\\prime } \\ge \\rho ^*$ (it is sufficient to consider this case, since the statement that $\\rho ^{\\prime \\prime }$ is a pre-fixpoint of $f$ implies that $\\rho ^{\\prime } := \\rho ^* \\vee \\rho ^{\\prime \\prime } \\ge \\rho ^*$ is also a pre-fixpoint of $f$ ).",
"We show by induction on $j$ that $\\rho ^{\\prime }|_{_{1}\\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j}}=\\rho ^*|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j}}$ for all $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ .",
"P Firstly, assume that $j = 1$ .",
"Since $_1 \\stackrel{f,\\rho ^*}{\\rightarrow }_2 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k$ , Lemma REF gives us $\\rho ^*|_{_1}=(f(\\rho ^*))|_{_1}=(f \\leftarrow \\rho ^*|_{\\setminus _1})(\\rho ^*|_{_1})=(f \\leftarrow \\rho ^{\\prime }|_{\\setminus _1})(\\rho ^*|_{_1})$ .",
"Using the monotonicity we thus get $\\mu _{\\ge \\rho _1} (f \\leftarrow \\rho ^{\\prime }|_{\\setminus _1}) = \\rho ^*|_{_1}$ .",
"Hence, Lemma REF gives us that $\\rho ^*|_{_1}$ is the greatest pre-fixpoint of $f \\leftarrow \\rho ^{\\prime }|_{\\setminus _1}$ .",
"Thus, $\\rho ^{\\prime }|_{_1} = \\rho ^*|_{_1}$ .",
"P Now, assume that $j \\in \\lbrace 2,\\ldots ,k\\rbrace $ and $\\rho ^{\\prime }|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j-1}}=\\rho ^*|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j-1}}$ .",
"It remains to show that $\\rho ^{\\prime }|_{_{j}}=\\rho ^*|_{_{j}}$ .",
"Since $_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j}\\stackrel{f,\\rho ^*}{\\rightarrow }_{j+1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k$ and $\\rho ^{\\prime }|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j-1}}=\\rho ^*|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j-1}}$ , Lemma REF gives us that $\\rho ^*|_{_j}=(f(\\rho ^*))|_{_j}=(f \\leftarrow \\rho ^*|_{\\setminus _j})(\\rho ^*|_{_j})=(f \\leftarrow \\rho ^{\\prime }|_{\\setminus _j})(\\rho ^*|_{_j})$ .",
"By monotonicity, we thus get $\\mu _{\\ge \\rho _j} (f \\leftarrow \\rho ^{\\prime }|_{\\setminus _j}) = \\rho ^*|_{_j}$ .",
"Hence, Lemma REF gives us that $\\rho ^*|_{_j}$ is the greatest pre-fixpoint of $(f \\leftarrow \\rho ^{\\prime }|_{\\setminus _j})$ .",
"Hence $\\rho ^{\\prime }|_{_j} = \\rho ^*|_{_j}$ .",
"Thus, we get $\\rho ^{\\prime }|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j}}=\\rho ^*|_{_{1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{j}}$ .",
"$\\Box $"
],
[
"Morcave Operators on $^n$",
"P We now study total operators on $$ that are monotone and order-concave.",
"For that, we firstly extend the notion of order-concavity that is defined for partial operators on $$ to total operators on $$ .",
"Before doing so, we start with the following observation: P Let $f : ^n \\rightarrow ^m$ be monotone.",
"Then, $\\operatorname{\\mathsf {fdom}}(f)$ is order-convex.",
"P Let $x, y \\in \\operatorname{\\mathsf {fdom}}(f)$ with $x \\le y$ and $\\lambda \\in [0,1]$ .",
"Because of the monotonicity of $f$ , we get $< f(x) \\le f(\\lambda x + (1-\\lambda ) y) \\le f(y) < \\infty $ .",
"Hence, $\\lambda x + (1-\\lambda ) y \\in \\operatorname{\\mathsf {fdom}}(f)$ .",
"This proves the statement.",
"$\\Box $ P We extend the notion of (order-)convexity/(order-)concavity from $^n \\rightsquigarrow $ to $^n \\rightarrow $ as follows: let $f : ^n \\rightarrow $ , and $I : \\lbrace 1,\\ldots ,n \\rbrace \\rightarrow \\lbrace , \\mathsf {id}, \\infty \\rbrace $ be a mapping.",
"Here, $$ denotes the function that assigns $$ to every argument, $\\mathsf {id}$ denotes the identity function, and $\\infty $ denotes the function that assigns $\\infty $ to every argument.",
"We define the mapping $f^{(I)} : ^{n} \\rightarrow $ by $f^{(I)} (x)&:= f(I(1)(x_{1\\cdot }),\\ldots ,I(n)(x_{n\\cdot }))&&\\text{for all } x \\in ^n.$ P A function $f : ^n \\rightarrow $ is called (order-)concave if and only if the following conditions are fulfilled for all mappings $I : \\lbrace 1,\\ldots ,n \\rbrace \\rightarrow \\lbrace , \\mathsf {id}, \\infty \\rbrace $ : $\\operatorname{\\mathsf {fdom}}(f^{(I)})$ is (order-)convex.",
"$f^{(I)}|_{\\operatorname{\\mathsf {fdom}}(f^{(I)})}$ is (order-)concave.",
"If $\\operatorname{\\mathsf {fdom}}(f^{(I)}) \\ne \\emptyset $ , then $f^{(I)}(x) < \\infty $ for all $x \\in ^n$ .",
"P Note that, by Lemma REF , condition 1 is fulfilled for every monotone function $f : ^n\\rightarrow $ and every mapping $I : \\lbrace 1,\\ldots ,n \\rbrace \\rightarrow \\lbrace , \\mathsf {id}, \\infty \\rbrace $ .",
"A monotone operator is order-concave if and only if the following conditions are fulfilled for all mappings $I : \\lbrace 1,\\ldots ,n\\rbrace \\rightarrow \\lbrace ,\\mathsf {id},\\infty \\rbrace $ : $\\operatorname{\\mathsf {fdom}}(f^{(I)})$ is upward closed w.r.t.",
"$^n$ .",
"$f^{(I)}|_{\\operatorname{\\mathsf {fdom}}(f^{(I)})}$ is order-concave.",
"P In order to get more familiar with the above definition, we consider a few examples of order-concave operators on $$ : P We consider the operators $f : ^2\\rightarrow $ and $g : ^2\\rightarrow $ that are defined by $f (x_1, x_2)&:=\\sqrt{x_1},&g (x_1, x_2)&:={\\left\\lbrace \\begin{array}{ll}\\sqrt{x_1} & \\text{if } x_2 < \\infty \\\\x_1^2 & \\text{if } x_2 = \\infty \\end{array}\\right.",
"}&&\\text{for all } x_1,x_2 \\in .$ P Then, $f|_{^2} = g|_{^2} = \\lbrace (x_1,x_2) \\mapsto \\sqrt{x_1} \\mid x_1, x_2 \\in \\rbrace $ is a monotone and concave operator on the convex set $\\operatorname{\\mathsf {fdom}}(f) = \\operatorname{\\mathsf {fdom}}(g) = \\times $ .",
"Nevertheless, $f$ is monotone and order-concave whereas $g$ is neither monotone nor order-concave.",
"In order to show that $g$ is not order-concave, let $I : \\lbrace 1,2\\rbrace \\rightarrow \\lbrace ,\\mathsf {id},\\infty \\rbrace $ be defined by $I(1) = \\mathsf {id}$ and $I(2) = \\infty $ .",
"Then, $g^{(I)}(x_1,x_2) = x_1^2$ for all $x_1,x_2 \\in $ .",
"Hence, $\\operatorname{\\mathsf {fdom}}(g^{(I)}) = ^2$ .",
"Obviously, $g^{(I)}|_{^2}$ is not order-concave.",
"Therefore, $g$ is not order-concave.",
"P Another example for a monotone and order-concave operator is the function $h : ^2\\rightarrow $ defined by $h (x_1, x_2)&={\\left\\lbrace \\begin{array}{ll}\\sqrt{x_1} & \\text{if } x_2 < \\infty \\\\\\sqrt{x_1} + 1 & \\text{if } x_2 = \\infty \\end{array}\\right.",
"}&&\\text{for all } x_1,x_2 \\in .$ P Although $h$ is an order-concave operator on $$ , it is not upward-chain-continuous, since, for $C = \\lbrace (0,i) \\mid i \\in \\rbrace $ , we have $h(\\bigvee C) = h(0,\\infty ) = 1 > 0 = \\bigvee \\lbrace 0 \\rbrace = \\bigvee h(C)$ .",
"We study different classes of monotone and order-concave functions in the remainder of this article.",
"$\\Box $ P A mapping $f : ^n \\rightarrow ^m$ is called (order-)concave if and only if $f_{i\\cdot }$ is (order-)concave for all $i \\in \\lbrace 1,\\ldots ,m\\rbrace $ .",
"A mapping $f : ^n \\rightarrow ^m$ is called (order-)convex if and only if $-f$ is (order-)concave.",
"P One property we expect from the set of all order-concave functions from $^n$ in $^m$ is that it is closed under the point-wise infimum operation.",
"This is indeed the case: P Let $\\mathcal {F}$ be a set of (order-)concave functions from $^n$ in $^m$ .",
"The function $g : ^n\\rightarrow ^m$ defined by $g(x) := \\bigwedge \\lbrace f(x) \\mid f \\in \\mathcal {F} \\rbrace $ for all $x \\in ^n$ is (order-)concave.",
"P The statement can be proven straightforwardly.",
"Note that $g(x) = (\\infty ,\\ldots , \\infty )$ for all $x \\in ^n$ if $\\mathcal {F} = \\emptyset $ .",
"In this case, $g$ is concave.",
"$\\Box $ P Monotone and order-concave functions play a central role in the remainder of this article.",
"For the sake of simplicity, we give names to important classes of monotone and order-concave functions: [Morcave, Mcave, Cmorcave, and Cmcave Functions] P A mapping $f : ^n\\rightarrow ^m$ is called morcaveif and only if it is monotone and order-concave.",
"It is called mcaveif and only if it is monotone and concave.",
"It is called cmorcave(resp.",
"cmcave ) if and only if it is morcave (resp.",
"mcave) and $f^{(I)}_{i\\cdot }$ is upward-chain-continuous on $\\lbrace x \\in ^n \\mid f^{(I)}_{i\\cdot }(x) > \\rbrace $ for all $I : \\lbrace 1,\\ldots ,n\\rbrace \\rightarrow \\lbrace ,\\mathsf {id},\\infty \\rbrace $ and all $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"$\\Box $ P Figure REF shows the graph of a morcave function $f : ^2 \\rightarrow $ .",
"$\\Box $ Figure: Graph of a morcave operator f: 2 →f : ^2 \\rightarrow .P An important cmcave operator for our applications is the operator $\\wedge $ on $^n$ : P The operator $\\vee $ on $^n$ is monotone and convex, but not order-concave.",
"The operator $\\wedge $ on $^n$ is cmcave, but not order-convex.",
"$\\Box $ P Next, we extend the definition of affine functions from $^n\\rightarrow ^m$ to a definition of affine functions from $^n\\rightarrow ^m$ .",
"[Affine Functions] P A function $f : ^n\\rightarrow ^m$ is called affine if and only if there exist some $A \\in ^{m \\times n}$ and some $b \\in ^m$ such that $f(x) = Ax + b$ for all $x \\in ^n$ .",
"A function $f : ^n \\rightarrow ^m$ is called affine if and only if there exist some $A \\in ^{m \\times n}$ and some $b \\in ^m$ such that $f(x) = Ax + b$ for all $x \\in ^n$ .",
"$\\Box $ P In the above definition and throughout this article, we use the convention that $+ \\infty = $ .",
"Observe that an affine function $f$ with $f(x) = Ax+b$ is monotone, whenever all entries of the matrix $A$ are non-negative.",
"P Every affine function $f : ^n \\rightarrow ^m$ is concave and convex.",
"Every monotone and affine function $f : ^n \\rightarrow ^m$ is cmcave.",
"$\\Box $ P In contrast to the class of monotone and order-concave operators on $$ , the class of morcave operators on $$ is not closed under functional composition, as the following example shows: P We consider the functions $f : \\rightarrow $ and $g : \\rightarrow $ defined by $f(x)&:={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } x = \\\\1 & \\text{if } x > \\end{array}\\right.",
"}&g(x)&:={\\left\\lbrace \\begin{array}{ll}& \\text{if } x < 0 \\\\0 & \\text{if } x \\ge 0\\end{array}\\right.",
"}&&\\text{for all } x \\in .$ P The functions $f$ and $g$ are both morcave — even cmcave.",
"However, observe that $(f\\circ g)(x) = f(g(x))&:={\\left\\lbrace \\begin{array}{ll}0 & \\text{if } x < 0 \\\\1 & \\text{if } x \\ge 0\\end{array}\\right.",
"}&&\\text{for all } x \\in .$ P Then, $f \\circ g$ is monotone, but not order-concave.",
"$\\Box $ P As we will see, the composition $f \\circ g$ of two morcave operators $f$ and $g$ is again morcave, if $f$ is additionally strict in the following sense: a function $f : ^n\\rightarrow $ is called strict if and only if $f(x) = $ for all $x \\in ^n$ with $x_{k\\cdot } = $ for some $k \\in \\lbrace 1,\\ldots ,n\\rbrace $ .",
"P Let $f : ^m \\rightarrow $ and $g : ^n \\rightarrow ^m$ be morcave.",
"Assume additionally that $f$ is strict.",
"Then $f \\circ g$ is morcave.",
"P Since $f$ and $g$ are monotone, $f \\circ g$ is also monotone.",
"In order to show that $f \\circ g$ is order-concave, let $I : \\lbrace 1,\\ldots ,n \\rbrace \\rightarrow \\lbrace , \\mathsf {id}, \\infty \\rbrace $ and $h := (f\\circ g)^{(I)}$ .",
"P The set $\\operatorname{\\mathsf {fdom}}(h)$ is order-convex by Lemma REF , since $h$ is monotone.",
"P Let $x, y \\in \\operatorname{\\mathsf {fdom}}(h)$ with $x \\le y$ , $\\lambda \\in [0,1]$ , and $z := \\lambda x + (1-\\lambda )y$ .",
"Moreover, let $x^{\\prime } := g^{(I)}(x)$ , $y^{\\prime } := g^{(I)}(y)$ , and $z^{\\prime } := g^{(I)}(z)$ .",
"The strictness of $f$ implies that $z^{\\prime } \\gg (,\\ldots ,)$ .",
"Since $g^{(I)}$ is monotone, we get $x^{\\prime } \\le y^{\\prime }$ .",
"We define $I^{\\prime } : \\lbrace 1,\\ldots ,m\\rbrace \\rightarrow \\lbrace ,\\mathsf {id},\\infty \\rbrace $ by $I^{\\prime }(k)&={\\left\\lbrace \\begin{array}{ll}\\mathsf {id} & \\text{if } z^{\\prime }_{k\\cdot } \\in \\\\\\mathsf {\\infty } & \\text{if } z^{\\prime }_{k\\cdot } = \\infty \\end{array}\\right.",
"}&&\\text{for all } k \\in \\lbrace 1,\\ldots ,m\\rbrace .$ P We get: $h(z)&=f(g^{(I)}(z))\\\\&=f^{(I^{\\prime })}(g^{(I)}(z))\\\\&\\ge f^{(I^{\\prime })}(\\lambda g^{(I)}(x) + (1-\\lambda ) g^{(I)}(y))\\\\&\\qquad \\qquad \\qquad \\text{(Monotonicity, Order-Concavity)}\\\\&=f^{(I^{\\prime })}(\\lambda x^{\\prime } + (1-\\lambda ) y^{\\prime })\\\\&\\ge \\lambda f^{(I^{\\prime })}( x^{\\prime } ) + (1-\\lambda ) f^{(I^{\\prime })}(y^{\\prime })& \\text{(Order-Concavity)}\\\\&=\\lambda f^{(I^{\\prime })}( g^{(I)}(x) ) + (1-\\lambda ) f^{(I^{\\prime })}( g^{(I)}(y) )\\\\&\\ge \\lambda f( g^{(I)}(x) ) + (1-\\lambda ) f( g^{(I)}(y) )& \\text{($f \\le f^{(I^{\\prime })}$)}\\\\&=\\lambda h(x) + (1-\\lambda ) h ( y )$ P Hence, $h|_{\\operatorname{\\mathsf {fdom}}(h)}$ is order-concave.",
"P Now, assume that $\\operatorname{\\mathsf {fdom}}(h) \\ne \\emptyset $ .",
"That is, there exists some $y \\in ^n$ with $h(y) = f(g^{(I)}(y)) \\in $ .",
"Since $f$ is strict, we get $y^{\\prime } := g^{(I)}(y) \\gg (,\\ldots ,)$ .",
"Let $I^{\\prime } : \\lbrace 1,\\ldots ,m \\rbrace \\rightarrow \\lbrace , \\mathsf {id}, \\infty \\rbrace $ be defined by $I^{\\prime }(k)&={\\left\\lbrace \\begin{array}{ll}\\mathsf {id} & \\text{if } y^{\\prime }_{k\\cdot } \\in \\\\\\mathsf {\\infty } & \\text{if } y^{\\prime }_{k\\cdot } = \\infty \\end{array}\\right.",
"}&&\\text{for all } k \\in \\lbrace 1,\\ldots ,m\\rbrace .$ P Since $g$ is order-concave, we get $g^{(I)}_{k\\cdot }(x) < \\infty $ for all $x \\in ^n$ and all $k \\in \\lbrace 1,\\ldots ,m\\rbrace $ with $y^{\\prime }_{k\\cdot } \\in $ .",
"Since $f$ is order-concave, we get $f^{(I^{\\prime })}(x) < \\infty $ for all $x \\in ^n$ .",
"Thus, by monotonicity, we get $f^{(I^{\\prime })} \\circ g^{(I)}(x) = f^{(I^{\\prime })}(g^{(I)}(x)) < \\infty $ for all $x \\in ^n$ .",
"Since we have $h = (f \\circ g)^{(I)} \\le f^{(I^{\\prime })} \\circ g^{(I)}$ by construction, we get $h(x) < \\infty $ for all $x \\in ^n$ .",
"$\\Box $ P In this section, we present our $\\vee $ -strategy improvement algorithm for computing least solutions of systems of $\\vee $ -morcave equations and prove its correctness."
],
[
"Systems of $\\vee $ -morcave Equations",
"P Assume that a fixed finite set $$ of variables and a complete linearly ordered set $ is given.Assume that $ is partially ordered by $\\le $ .",
"We consider equations of the form $= e$ over $,where $$ is a variableand $ e$ is an expression over $ .",
"A system $$ of (fixpoint-)equations over $ is a finiteset$ { 1 = e1,...,n = en } $ of equations,where$ 1,...,n$ are pairwise distinct variables.We denote the set $ {1, ..., n}$ of variablesoccurring in $$ by $$.We drop the subscript, whenever it is clear from the context.$ P For a variable assignment $\\rho : \\rightarrow ,an expression $ e$ is mapped to a value$ e$by setting$ := ()$, and$ f(e1,...,ek):= f(e1,...,ek)$,where $$, $ f$ is a $ k$-ary operator ($ k=0$ is possible; then $ f$ is a constant),for instance $ +$, and$ e1,...,ek$ are expressions.For every system $$ of equations,we define the unary operator$$on$ by setting $({}\\rho )():= {e}\\rho $ for all equations $= e$ from $$ and all $\\rho : \\rightarrow .A \\emph {solution} is a fixpoint of $$,i.e.,it is a variable assignment $$such that $ = $.We denote the set of all solutions of $$ by $ ()$.$P The set $\\rightarrow of all \\emph {variable assignments}is a complete lattice.For $ , ' : , we write $\\rho \\ll \\rho ^{\\prime }$ (resp.",
"$\\rho \\gg \\rho ^{\\prime }$ ) if and only if $\\rho () < \\rho ^{\\prime }()$ (resp.",
"$\\rho () > \\rho ^{\\prime }()$ ) for all $\\in $ .",
"For $d \\in ,$d$ denotes the variable assignment$ { d }$.A variable assignment $$ with $$is called \\emph {finite}.A pre-solution (resp.\\ post-solution) is a variable assignment $$such that$$ (resp.\\ $$) holds.The set of pre-solutions(resp.\\ the set of post-solutions)is denotedby $ ()$ (resp.\\ $ ()$).The least solution (resp.\\ the greatest solution)of a system $$ of equations is denoted by$$ (resp.\\ $$),provided that it exists.For a pre-solution $$ (resp.\\ for a post-solution $$),$$ (resp.\\ $$)denotes the least solution that is greater than or equal to $$(resp.\\ the greatest solution that is less than or equal to $$).$P An expression $e$ (resp.",
"an (fixpoint-)equation $= e$ is called monotone if and only if $e$ is monotone.",
"In our setting, the fixpoint theorem of Knaster/Tarski can be stated as follows: every system $$ of monotone fixpoint equations over a complete lattice has a least solution $\\mu $ and a greatest solution $\\nu $ .",
"Furthermore, we have $\\mu = \\bigwedge ()$ and $\\nu = \\bigvee ()$ .",
"[$\\vee $ -morcave Equations] P An expression $e$ (resp.",
"fixpoint equation $= e$ ) over $$ is called morcave (resp.",
"cmorcave, resp.",
"mcave, resp.",
"cmcave) if and only if ${e}$ is morcave (resp.",
"cmorcave, resp.",
"mcave, resp.",
"cmcave).",
"An expression $e$ (resp.",
"fixpoint equation $= e$ ) over $$ is called $\\vee $ -morcave (resp.",
"$\\vee $ -cmorcave, resp.",
"mcave, resp.",
"cmcave) if and only if $e = e_1 \\vee \\cdots \\vee e_k$ , where $e_1,\\ldots ,e_k$ are morcave (resp.",
"cmorcave, resp.",
"mcave, resp.",
"cmcave).",
"$\\Box $ P The square root operator $\\sqrt{\\cdot }: \\rightarrow $ (defined by $\\sqrt{x} := \\sup \\; \\lbrace y \\in \\mid y^2 \\le x \\rbrace $ for all $x \\in $ ) is cmcave.",
"The least solution of the system $= \\lbrace = \\frac{1}{2} \\vee \\sqrt{\\rbrace }$ of $\\vee $ -cmcave equations is $\\mu = 1$ .",
"$\\Box $ [$\\vee $ -strategies] P A $\\vee $ -strategy $\\sigma $ for a system $$ of equations is a function that maps every expression $e_1 \\vee \\cdots \\vee e_k$ occurring in $$ to one of the immediate sub-expressions $e_j$ , $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ .",
"We denote the set of all $\\vee $ -strategies for $$ by $_$ .",
"We drop the subscript, whenever it is clear from the context.",
"The application $(\\sigma )$ of $\\sigma $ to $$ is defined by $(\\sigma ) := \\lbrace = \\sigma (e) \\mid = e \\in \\rbrace $ .",
"P The two $\\vee $ -strategies $\\sigma _1,\\sigma _2$ for the system $$ of $\\vee $ -cmcave equations defined in Example REF lead to the systems $(\\sigma _1) = \\lbrace = \\frac{1}{2} \\rbrace $ and $(\\sigma _2) = \\lbrace = \\sqrt{\\rbrace }$ of cmcave equations.",
"$\\Box $"
],
[
"The Strategy Improvement Algorithm",
"P We now present the $\\vee $ -strategy improvement algorithm in a general setting.",
"That is, we consider arbitrary systems of monotone equations over arbitrary complete linearly ordered sets $$ .",
"The algorithm iterates over $\\vee $ -strategies.",
"It maintains a current $\\vee $ -strategy $\\sigma $ and a current approximate $\\rho $ to the least solution.",
"A so-called $\\vee $ -strategy improvement operator is used to determine a next, improved $\\vee $ -strategy $\\sigma ^{\\prime }$ .",
"Whether or not a $\\vee $ -strategy $\\sigma ^{\\prime }$ is an improvement of the current $\\vee $ -strategy $\\sigma $ may depend on the current approximate $\\rho $ : [Improvements] P Let $$ be a system of monotone equations over a complete linearly ordered set.",
"Let $\\sigma , \\sigma ^{\\prime } \\in $ be $\\vee $ -strategies for $$ and $\\rho $ be a pre-solution of $(\\sigma )$ .",
"The $\\vee $ -strategy $\\sigma ^{\\prime }$ is called an improvement of $\\sigma $ w.r.t.",
"$\\rho $ if and only if the following conditions are fulfilled: If $\\rho \\notin ()$ , then ${(\\sigma ^{\\prime })}\\rho > \\rho $ .",
"For all expressions $e = e_1 \\vee \\cdots \\vee e_k$ of $$ the following holds: If $\\sigma ^{\\prime }(e) \\ne \\sigma (e)$ , then ${\\sigma ^{\\prime }(e)} \\rho > {\\sigma (e)} \\rho $ .",
"A function $$ that assigns an improvement of $\\sigma $ w.r.t.",
"$\\rho $ to every pair $(\\sigma ,\\rho )$ , where $\\sigma $ is a $\\vee $ -strategy and $\\rho $ is a pre-solution of $(\\sigma )$ , is called a $\\vee $ -strategy improvement operator.",
"If it is impossible to improve $\\sigma $ w.r.t.",
"$\\rho $ , then we necessarily have $(\\sigma ,\\rho ) = \\sigma $ .",
"$\\Box $ P Consider the system $ = \\lbrace _1 = _2 + 1 \\wedge 0, _2 = -1 \\vee \\sqrt{_}1 \\rbrace $ of $\\vee $ -cmcave equations.",
"Let $\\sigma _1$ and $\\sigma _2$ be the $\\vee $ -strategies for $$ such that $(\\sigma _1) &= \\lbrace _1 = _2 + 1 \\wedge 0, _2 = -1 \\rbrace ,\\text{ and}\\\\(\\sigma _2) &= \\lbrace _1 = _2 + 1 \\wedge 0, _2 = \\sqrt{_}1 \\rbrace .$ The variable assignment $\\rho := \\lbrace _1 \\mapsto 0, _2 \\mapsto -1 \\rbrace $ is a solution and thus also a pre-solution of $(\\sigma _1)$ .",
"The $\\vee $ -strategy $\\sigma _2$ is an improvement of the $\\vee $ -strategy $\\sigma _1$ w.r.t.",
"$\\rho $ .",
"$\\Box $ P We can now formulate the $\\vee $ -strategy improvement algorithm for computing least solutions of systems of monotone equations over complete linearly ordered sets.",
"This algorithm is parameterized with a $\\vee $ -strategy improvement operator $$ .",
"The input is a system $$ of monotone equations over a complete linearly ordered set, a $\\vee $ -strategy $\\sigma _{\\mathrm {init}}$ for $$ , and a pre-solution $\\rho _{\\mathrm {init}}$ of $(\\sigma _{\\mathrm {init}})$ .",
"In order to compute the least and not just some solution, we additionally require that $\\rho _{\\mathrm {init}} \\le \\mu $ holds: P [H] $\\\\[0mm]\\begin{array}{@{}l@{\\text{:}\\,}l@{}}\\text{Parameter}&\\text{A $\\vee $-strategy improvement operator $$}\\\\[1mm]\\text{Input}&\\left\\lbrace \\begin{array}{@{}l@{}}\\text{\\!\\!-A system $$ of monotone equationsover a complete linearly ordered set } \\\\\\text{\\!\\!-A $\\vee $-strategy $\\sigma _{\\mathrm {init}}$ for $$} \\\\\\text{\\!\\!-A pre-solution $\\rho _{\\mathrm {init}}$of $(\\sigma _{\\mathrm {init}})$with $\\rho _{\\mathrm {init}} \\le \\mu $} \\\\\\end{array}\\right.\\\\[5mm]\\text{Output}&\\text{The least solution $\\mu $ of $$}\\end{array} \\\\[2mm]\\sigma \\sigma _{\\mathrm {init}} ; \\\\\\rho \\rho _{\\mathrm {init}} ; \\\\(\\rho \\notin ()) \\;\\lbrace \\\\\\hspace*{14.22636pt} \\sigma (\\sigma ,\\rho ) ; \\\\ \\hspace*{14.22636pt}\\rho \\mu _{\\ge \\rho } {(\\sigma )} ; \\\\\\rbrace \\\\\\rho ;\\\\[-3mm]$ The $\\vee $ -Strategy Improvement Algorithm We consider the system $\\textstyle =\\left\\lbrace =\\vee \\frac{1}{2} \\vee \\sqrt{\\vee }\\frac{7}{8} + \\sqrt{- \\frac{47}{64}}\\right\\rbrace $ of $\\vee $ -cmorcave equations.",
"We start with the $\\vee $ -strategy $\\sigma _0$ that leads to the system $(\\sigma _0) = \\lbrace = \\rbrace $ of cmorcave equations.",
"Then $\\rho _0 := $ is a feasible solution of $(\\sigma _0)$ .",
"Since $\\rho _0 \\notin ()$ , we improve $\\sigma _0$ w.r.t.",
"$\\rho _0$ to the $\\vee $ -strategy $\\sigma _1$ that gives us $(\\sigma _1) = \\left\\lbrace = \\frac{1}{2} \\right\\rbrace .$ Then, $\\rho _1 := \\mu _{\\ge \\rho _0} {\\sigma _1} = \\lbrace \\mapsto \\frac{1}{2} \\rbrace $ .",
"Since $\\sqrt{\\frac{1}{2}} > \\frac{1}{2}$ and $\\frac{7}{8} + \\sqrt{\\frac{1}{2} - \\frac{47}{64}} < \\frac{1}{2}$ hold, we improve the strategy $\\sigma _1$ w.r.t.",
"$\\rho _1$ to the $\\vee $ -strategy $\\sigma _2$ with $(\\sigma _2) = \\lbrace = \\sqrt{\\rbrace }.$ We get $\\rho _2 := \\mu _{\\ge \\rho _1} { \\sigma _2 } = \\lbrace \\mapsto 1\\rbrace $ .",
"Since $\\frac{7}{8} + \\sqrt{1 - \\frac{47}{64}}>\\frac{7}{8} + \\sqrt{1 - \\frac{60}{64}}=\\frac{9}{8}> 1$ , we get $\\sigma _3 = \\lbrace = \\frac{7}{8} + \\sqrt{- \\frac{47}{64}} \\rbrace $ .",
"Finally we get $\\rho _3:=\\mu _{\\ge \\rho _2} {\\sigma _3}=\\lbrace \\mapsto 2 \\rbrace $ .",
"The algorithm terminates, because $\\rho _3$ solves $$ .",
"Therefore, $\\rho _3 = \\mu $ .",
"$\\Box $ P In the following lemma, we collect basic properties that can be proven by induction straightforwardly: P Let $$ be a system of monotone equations over a complete linearly ordered set.",
"For all $i \\in $ , let $\\rho _i$ be the value of the program variable $\\rho $ and $\\sigma _i$ be the value of the program variable $\\sigma $ in the $\\vee $ -strategy improvement algorithm (Algorithm REF ) after the $i$ -th evaluation of the loop-body.",
"The following statements hold for all $i \\in $ : $\\rho _i \\le \\mu $ .",
"$\\rho _i \\in ((\\sigma _{i+1}))$ .",
"If $\\rho _i < \\mu $ , then $\\rho _{i+1} > \\rho _i$ .",
"If $\\rho _i = \\mu $ , then $\\rho _{i+1} = \\rho _i$ .",
"If the execution of the $\\vee $ -strategy improvement algorithm terminates, then the least solution $\\mu $ of $$ is computed.",
"$\\Box $ P In the following, we apply our algorithm to solve systems of $\\vee $ -morcave equations.",
"In the next subsection, we show that our algorithm terminates in this case.",
"More precisely, it returns the least solution at the latest after considering every $\\vee $ -strategy at most $$ times.",
"We additionally provide an important characterization of $\\mu _{\\ge \\rho }{(\\sigma )}$ which allows us to compute it using convex optimization techniques.",
"Here, $\\sigma $ are the $\\vee $ -strategies and $\\rho $ are the pre-solutions $\\rho $ of $(\\sigma )$ that can be encountered during the execution of the algorithm."
],
[
"Feasibility",
"P In this subsection, we extend the notion of feasibility as defined in Definition REF .",
"We then show that feasibility is preserved during the execution of the $\\vee $ -strategy improvement algorithm.",
"In the next subsection, we finally make use of the feasibility.",
"We denote by $[x_1/_1 , \\ldots , x_n/_n]$ the equation system that is obtained from the equation system $$ by simultaneously replacing, for all $i \\in \\lbrace 1,\\ldots ,n\\rbrace $ , every occurrence of a variable from the set $_i$ in the right-hand sides of $$ by the value $x_i$ .",
"[Feasibility] P Let $$ be a system of morcave equations.",
"A finite solution $\\rho $ of $$ is called ($$ -)feasible if and only if $\\rho $ is a feasible fixpoint of $$ .",
"A pre-solution $\\rho $ of $$ with $\\rho \\gg $ is called ($$ -)feasible if and only if $\\rho ^{\\prime }|_{^{\\prime }}$ is a feasible finite solution of $^{\\prime } := \\lbrace = e \\in \\mid \\in ^{\\prime } \\rbrace [\\infty / (\\setminus ^{\\prime })] $ , where $\\rho ^{\\prime } := \\mu _{\\ge \\rho } $ and $^{\\prime } := \\lbrace \\in \\mid \\rho ^{\\prime }() < \\infty \\rbrace $ .",
"A pre-solution $\\rho $ of $$ is called feasible if and only if $e = $ for all $= e \\in $ with $e \\rho = $ , and $\\rho |_{^{\\prime }}$ is a feasible pre-solution of $^{\\prime } := \\lbrace = e \\in \\mid \\in ^{\\prime } \\rbrace [/ (\\setminus ^{\\prime })]$ , where $^{\\prime } := \\lbrace \\mid = e \\in , e \\rho > \\rbrace $ .",
"$\\Box $ P We consider the system $= \\lbrace = \\sqrt{\\rbrace }$ of mcave equations.",
"For all $x \\in $ , let $\\underline{x} := \\lbrace \\mapsto x \\rbrace $ .",
"From Example REF , we know that the solution $\\underline{0}$ is not feasible, whereas the solution $\\underline{1}$ is feasible.",
"Thus, $\\underline{x}$ is a feasible pre-solution for all $x \\in (0,1]$ .",
"Note that $\\underline{1}$ is the only feasible finite solution of $$ and thus, by Lemma REF , the greatest finite pre-solution of $$ .",
"$\\Box $ P Let us consider the system $= \\lbrace _1 = _2 + 1 \\wedge 0, _2 = \\sqrt{_}1 \\rbrace $ of mcave equations.",
"From Example REF it follows that $\\rho := \\lbrace _1 \\mapsto 0, _2 \\mapsto 0 \\rbrace $ is a feasible finite fixpoint of $$ .",
"Thus, $\\lbrace _1 \\mapsto 0, _2 \\mapsto x \\rbrace $ is a feasible pre-solution for all $x \\in [-1,0]$ .",
"The solution $\\lbrace _1 \\mapsto , _2 \\mapsto \\rbrace $ is not feasible, since the right-hand sides evaluate to $$ , although they are not $$ .",
"$\\Box $ P The following two lemma imply that our $\\vee $ -strategy improvement algorithm stays in the feasible area, whenever it is started in the feasible area.",
"P Let $$ be a system of morcave equations and $\\rho $ be a feasible pre-solution of $$ .",
"Every pre-solution $\\rho ^{\\prime }$ of $$ with $\\rho \\le \\rho ^{\\prime } \\le \\mu _{\\ge \\rho }$ is feasible.",
"The statement is an immediate consequence of the definition.",
"$\\Box $ P Let $$ be a system of $\\vee $ -morcave equations, $\\sigma $ be a $\\vee $ -strategy for $$ , $\\rho $ be a feasible solution of $(\\sigma )$ , and $\\sigma ^{\\prime }$ be an improvement of $\\sigma $ w.r.t.",
"$\\rho $ .",
"Then $\\rho $ is a feasible pre-solution of $(\\sigma ^{\\prime })$ .",
"P Let $\\rho ^* := \\mu _{\\ge \\rho }{(\\sigma ^{\\prime })}$ .",
"We w.l.o.g.",
"assume that $\\ll \\rho ^* \\ll $ .",
"Hence, $\\rho \\ll $ .",
"Let $^\\mathsf {old}&:= \\lbrace \\in \\mid \\rho () > \\rbrace , \\text{and } \\\\^{\\mathsf {old}}&:= \\lbrace = e \\in (\\sigma ) \\mid \\in ^\\mathsf {old}\\rbrace [/ (\\setminus ^\\mathsf {old})].$ P Hence, $\\rho |_{^\\mathsf {old}}$ is a feasible finite solution of $^{\\mathsf {old}}$ , i.e., a feasible finite fixpoint of $^{\\mathsf {old}}$ .",
"Therefore, there exist $_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_k = ^\\mathsf {old}$ with $_1\\stackrel{^{\\mathsf {old}},\\rho |_{^\\mathsf {old}}}{\\rightarrow }\\cdots \\stackrel{^{\\mathsf {old}},\\rho |_{^\\mathsf {old}}}{\\rightarrow }_k$ P such that, for each $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ , there exists some pre-fixpoint $\\rho ^{\\prime }$ of ${^{\\mathsf {old}}} \\leftarrow \\rho |_{^\\mathsf {old}\\setminus _j}$ with $\\rho ^{\\prime } \\ll \\rho |_{_j}$ such that $\\mu _{\\ge \\rho ^{\\prime }}({^{\\mathsf {old}}} \\leftarrow \\rho |_{^\\mathsf {old}\\setminus _j})=\\rho |_{_j}$ .",
"P Let $^{\\mathsf {imp}}:= \\lbrace \\in \\mid \\rho ^*() > \\rho () \\rbrace $ , $_j^{\\prime } := _j \\setminus ^{\\mathsf {imp}}$ for all $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ , and $_{k+1}^{\\prime } := ^{\\mathsf {imp}}$ .",
"Obviously, we have $_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k+1}^{\\prime } = $ .",
"It remains to show that the following properties are fulfilled: P $_1^{\\prime }\\stackrel{{(\\sigma ^{\\prime })},\\rho ^*}{\\rightarrow }\\cdots \\stackrel{{(\\sigma ^{\\prime })},\\rho ^*}{\\rightarrow }_{k+1}^{\\prime }$ P For each $j \\in \\lbrace 1,\\ldots ,k+1\\rbrace $ , there exists some pre-fixpoint $\\rho ^{\\prime }$ with $\\rho ^{\\prime } \\ll \\rho ^* |_{_j^{\\prime }}$ such that $\\mu _{\\ge \\rho ^{\\prime }}({(\\sigma ^{\\prime })} \\leftarrow \\rho ^*|_{\\setminus _j^{\\prime }})=\\rho ^*|_{_j^{\\prime }}$ .",
"P In order to prove statement 1, let $j \\in \\lbrace 1,\\ldots ,k\\rbrace $ .",
"We have to show that $_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }\\stackrel{{(\\sigma ^{\\prime })},\\rho ^*}{\\rightarrow }_{j+1}^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k+1}^{\\prime }.$ P Since $_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j\\stackrel{{^{\\mathsf {old}}},\\rho |_{^\\mathsf {old}}}{\\rightarrow }_{j+1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k}$ , there exists some variable assignment $\\rho ^{\\prime } : _{j+1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k} \\rightarrow $ with $\\rho ^{\\prime } \\ll \\rho |_{_{j+1} \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k}}$ such that $({^{\\mathsf {old}}} (\\rho |_{^\\mathsf {old}} \\oplus \\rho ^{\\prime }))|_{_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j}=({^{\\mathsf {old}}} (\\rho |_{^\\mathsf {old}}))|_{_1 \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j}.$ P We define $\\rho ^{\\prime \\prime } : _{j+1}^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k+1}^{\\prime } \\rightarrow $ by $\\rho ^{\\prime \\prime } (){=}{\\left\\lbrace \\begin{array}{ll}\\rho ^{\\prime }() &\\text{if } \\in _{j+1}^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k}^{\\prime } \\\\\\rho () &\\text{if } \\in _{k+1}^{\\prime } \\text{ and } \\in ^\\mathsf {old}\\\\\\rho ^*() - 1 &\\!\\text{if } \\in _{k+1}^{\\prime } \\text{ and } \\notin ^\\mathsf {old}\\\\\\end{array}\\right.",
"}&&\\text{for all } \\in _{j+1}^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k+1}^{\\prime }.$ O By construction, we have $\\rho ^{\\prime \\prime }\\ll \\rho ^*|_{_{j+1}^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_{k+1}^{\\prime }}$ .",
"Hence, we get $({(\\sigma ^{\\prime })} (\\rho ^*))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}&\\ge ({(\\sigma ^{\\prime })} (\\rho ^* \\oplus \\rho ^{\\prime \\prime }))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}& \\text{($\\rho ^* \\ge \\rho ^* \\oplus \\rho ^{\\prime \\prime }$)}\\\\&\\ge ({^{\\mathsf {old}}} (\\rho |_{^\\mathsf {old}} \\oplus \\rho ^{\\prime }))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}\\\\&=({^{\\mathsf {old}}} (\\rho |_{^\\mathsf {old}}))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}& \\text{(because of (\\ref {eq:l:impr:1}))}\\\\&=({^{\\mathsf {old}}} (\\rho ^*|_{^\\mathsf {old}}))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}& \\!\\!\\!\\!\\text{(because of Lemma \\ref {l:partition:nach:oben:unabhae})}\\\\&=({(\\sigma ^{\\prime })} (\\rho ^*))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}.$ Thus, $({(\\sigma ^{\\prime })} (\\rho ^* \\oplus \\rho ^{\\prime \\prime }))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}=({(\\sigma ^{\\prime })} (\\rho ^*))|_{_1^{\\prime } \\operatorname{{\\dot{\\cup }}}\\cdots \\operatorname{{\\dot{\\cup }}}_j^{\\prime }}$ .",
"This proves statement 1.",
"P In order to prove statement 2, let $j \\in \\lbrace 1,\\ldots ,k+1\\rbrace $ .",
"We distinguish 2 cases.",
"Firstly, assume that $j \\le k$ .",
"Since $\\rho |_{^\\mathsf {old}}$ is a feasible finite fixpoint of ${^{\\mathsf {old}}}$ , there exists some pre-fixpoint $\\rho ^{\\prime }$ with $\\rho ^{\\prime } \\ll \\rho |_{_j} = \\rho ^* |_{_j}$ such that $\\mu _{\\ge \\rho ^{\\prime }} ({^{\\mathsf {old}}} \\leftarrow \\rho |_{^\\mathsf {old}\\setminus _j})=\\rho |_{_j}=\\rho ^* |_{_j}$ .",
"Using monotonicity, we get $\\mu _{\\ge \\rho ^{\\prime }} ({^{\\mathsf {old}}} \\leftarrow \\rho ^*|_{^\\mathsf {old}\\setminus _j})=\\rho |_{_j}=\\rho ^* |_{_j}$ .",
"Hence, $\\rho ^{\\prime }|_{_j^{\\prime }} : _j^{\\prime } \\rightarrow $ , $\\rho ^{\\prime }|_{_j^{\\prime }} \\ll \\rho |_{_j^{\\prime }} = \\rho ^* |_{_j^{\\prime }}$ , and $\\mu _{\\ge \\rho ^{\\prime }|_{_j^{\\prime }}} ({^{\\mathsf {old}}} \\leftarrow \\rho ^*|_{^\\mathsf {old}\\setminus _j^{\\prime }})=\\mu _{\\ge \\rho ^{\\prime }|_{_j^{\\prime }}}({(\\sigma ^{\\prime })} \\leftarrow \\rho ^*|_{\\setminus _j^{\\prime }})=\\rho ^*|_{_j^{\\prime }}$ .",
"This proves statement 2 for $j \\le k$ .",
"Now, assume that $j = k+1$ .",
"By definition of $_{k+1}^{\\prime }$ , $\\rho |_{_{k+1}^{\\prime }} \\ll \\rho ^*|_{_{k+1}^{\\prime }}$ .",
"Moreover, we get immediately that $\\rho |_{_{k+1}^{\\prime }}$ is a pre-fixpoint of ${(\\sigma ^{\\prime })} \\leftarrow \\rho ^*|_{\\setminus _{k+1}^{\\prime }}$ and $\\mu _{\\ge \\rho |_{_{k+1}^{\\prime }}}({(\\sigma ^{\\prime })} \\leftarrow \\rho ^*|_{\\setminus _{k+1}^{\\prime }})=\\rho ^*|_{_{k+1}^{\\prime }}$ .",
"This proves statement 2.",
"$\\Box $ P We continue Example REF .",
"Obviously, $\\rho = \\lbrace _1 \\mapsto 0, _2 \\mapsto -1 \\rbrace $ is a feasible solution of $(\\sigma _1) = \\lbrace _1 = _2 + 1 \\wedge 0, _2 = -1 \\rbrace $ .",
"The $\\vee $ -strategy $\\sigma _2$ is an improvement of the $\\vee $ -strategy $\\sigma _1$ w.r.t.",
"$\\rho $ .",
"By lemma REF , $\\rho $ is also a feasible pre-solution of $(\\sigma _2) = \\lbrace _1 = _2 + 1 \\wedge 0, _2 = \\sqrt{_}1 \\rbrace $ .",
"The fact that $\\rho $ is a feasible pre-solution of $(\\sigma _2)$ is also shown in Example REF .",
"$\\Box $ P The above two lemmas ensure that our $\\vee $ -strategy improvement algorithm stays in the feasible area, whenever it is started in the feasible area.",
"In order to start in the feasible area, we in the following simply assume w.l.o.g.",
"that each equation of $$ is of the form $= \\vee e$ .",
"We say that such a system of fixpoint equations is in standard form.",
"Then, we start our $\\vee $ -strategy improvement algorithm with a $\\vee $ -strategy $\\sigma _\\mathrm {init}$ such that $(\\sigma _\\mathrm {init}) = \\lbrace = \\mid \\in \\rbrace $ .",
"In consequence, $$ is a feasible solution of $(\\sigma _\\mathrm {init})$ .",
"We get: P Let $$ be a system of $\\vee $ -morcave equations.",
"For all $i \\in $ , let $\\rho _i$ be the value of the program variable $\\rho $ and $\\sigma _i$ be the value of the program variable $\\sigma $ in the $\\vee $ -strategy improvement algorithm (Algorithm REF ) after the $i$ -th evaluation of the loop-body.",
"Then, $\\rho _i$ is a feasible pre-solution of $(\\sigma _{i+1})$ for all $i \\in $ .",
"$\\Box $ P We again consider the system $=\\lbrace _1 = \\vee _2 + 1 \\wedge 0, \\;_2 = \\vee -1 \\vee \\sqrt{_}1\\rbrace $ of $\\vee $ -morcave equations introduced in Example REF .",
"A run of our $\\vee $ -strategy improvement algorithm gives us $(\\sigma _0) &= \\lbrace _1 = ,\\; _2 = \\rbrace &\\rho _0 &= \\lbrace _1 \\mapsto ,\\; _2 \\mapsto \\rbrace \\\\(\\sigma _1) &= \\lbrace _1 = ,\\; _2 = -1 \\rbrace &\\rho _1 &= \\lbrace _1 \\mapsto ,\\; _2 \\mapsto -1 \\rbrace \\\\(\\sigma _2) &= \\lbrace _1 = _2 + 1 \\wedge 0 ,\\; _2 = -1 \\rbrace &\\rho _2 &= \\lbrace _1 \\mapsto 0 ,\\; _2 \\mapsto -1 \\rbrace \\\\(\\sigma _3) &= \\lbrace _1 = _2 + 1 \\wedge 0 ,\\; _2 = \\sqrt{_1} \\rbrace &\\rho _3 &= \\lbrace _1 \\mapsto 0 ,\\; _2 \\mapsto 0 \\rbrace $ By Lemma REF , $\\rho _i$ is a feasible pre-solution of $(\\sigma _{i+1})$ for all $i = \\lbrace 0,1,2\\rbrace $ .",
"$\\Box $"
],
[
"Evaluating $\\vee $ -Strategies / Solving Systems of Morcave Equations",
"P It remains to develop a method for computing $\\mu _{\\ge \\rho }{}$ under the assumption that $\\rho $ is a feasible pre-solution of the system $$ of morcave equations.",
"This is an important step in our $\\vee $ -strategy improvement algorithm (Algorithm REF ).",
"Before doing this, we introduce the following notation for the sake of simplicity: P Let $$ be a system of morcave equations and $\\rho $ a pre-solution of $$ .",
"Let $^_\\rho &:= \\lbrace \\mid = e \\in ,\\; {e} \\rho = \\rbrace \\\\^\\infty _\\rho &:= \\lbrace \\mid = e \\in ,\\; {e} \\rho = \\infty \\rbrace \\\\^{\\prime }_\\rho &:= \\setminus (^_\\rho \\cup ^\\infty _\\rho )= \\lbrace \\mid = e \\in ,\\; {e} \\rho \\in \\rbrace \\\\^{\\prime }_\\rho &= \\lbrace = e \\in \\mid \\in ^{\\prime }_\\rho \\rbrace [/^_\\rho , \\infty /^\\infty _\\rho ]$ P The pre-solution $\\mathsf {suppresol}_\\rho $ of $$ is defined by $\\mathsf {suppresol}_\\rho ()&:={\\left\\lbrace \\begin{array}{ll}& \\text{if } \\in ^_\\rho \\\\\\sup \\; \\lbrace \\hat{\\rho }() \\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; \\hat{\\rho }\\le {^{\\prime }}\\hat{\\rho }\\rbrace & \\text{if }\\in ^{\\prime }_\\rho \\\\\\infty & \\text{if }\\in ^\\infty _\\rho \\end{array}\\right.",
"}$ for all $\\in $ .",
"$\\Box $ P The variables assignment $\\mathsf {suppresol}_\\rho $ is by construction a pre-solution of $$ , but, as we will see in Example REF , not necessarily a solution of $$ .",
"$\\Box $ O Under some constraints, we can compute $\\mathsf {suppresol}_\\rho $ by solving $$ convex optimisation problems of linear size.",
"This can be done by general convex optimization methods.",
"For further information on convex optimization, we refer, for instance, to [13].",
"Let $$ be a system of mcave equations and $\\rho $ a pre-solution of $$ .",
"Then, the pre-solution $\\mathsf {suppresol}_\\rho $ of $$ can be computed by solving at most $$ convex optimization problems.",
"Let $^_\\rho $ , $^\\infty _\\rho $ , $^{\\prime }_\\rho $ , and $^{\\prime }_\\rho $ be defined as in Definition REF .",
"We have to compute $\\mathsf {suppresol}_\\rho ()=\\sup \\; \\lbrace \\hat{\\rho }() \\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; \\hat{\\rho }\\le {^{\\prime }}\\hat{\\rho }\\rbrace =\\sup \\; \\lbrace \\hat{\\rho }() \\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; (\\mathsf {id} - {^{\\prime }}) \\hat{\\rho }\\le 0 \\rbrace $ for all $\\in ^{\\prime }_\\rho $ .",
"Here, $\\mathsf {id}$ denotes the identity function.",
"Therefore, since $\\mathsf {id}$ is affine, ${^{\\prime }}$ is concave (considered as a function that maps values from $^{\\prime }_\\rho \\rightarrow $ to values from $^{\\prime }_\\rho \\rightarrow (\\cup \\lbrace \\rbrace $ ), and thus $- {^{\\prime }}\\hat{\\rho }$ is convex (considered as a function that maps values from $^{\\prime }_\\rho \\rightarrow $ to values from $^{\\prime }_\\rho \\rightarrow (\\cup \\lbrace \\infty \\rbrace $ ), the mathematical optimization problem $\\sup \\; \\lbrace \\hat{\\rho }() \\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; (\\mathsf {id} - {^{\\prime }})\\hat{\\rho }\\le 0 \\rbrace $ is a convex optimization problem.",
"$\\Box $ P We will use $\\mathsf {suppresol}_\\rho $ iteratively to compute $\\mu _{\\ge \\rho }$ under the assumption that $\\rho $ is a feasible pre-solution of the system $$ of morcave equations.",
"As a first step in this direction, we prove the following lemma, which gives us at least a method for computing $\\mu _{\\ge \\rho }$ under the assumption that $$ is a system of cmorcaveequations.",
"P Let $$ be a system of morcave equations and $\\rho $ a feasible pre-solution of $$ .",
"Let $^_\\rho $ , $^\\infty _\\rho $ , and $^{\\prime }_\\rho $ be defined as in Definition REF ((REF ) - ()).",
"Then: $\\mu _{\\ge \\rho }()&= \\mathsf {suppresol}_\\rho () = && \\text{for all } \\in ^_\\rho \\\\\\mu _{\\ge \\rho }()&\\ge \\mathsf {suppresol}_\\rho ()&& \\text{for all }\\in ^{\\prime }_\\rho \\\\\\mu _{\\ge \\rho }()&= \\mathsf {suppresol}_\\rho () = \\infty && \\text{for all }\\in ^\\infty _\\rho $ P If $$ is a system of cmorcave equations, then the inequality in () is in fact an equality, i.e., we have $\\mu _{\\ge \\rho }&=\\mathsf {suppresol}_\\rho .$ P Let $^{\\prime }_\\rho $ be defined as in Definition REF ().",
"We first prove (REF ) - ().",
"Let $x \\in $ .",
"If $\\in ^_\\rho \\cup ^\\infty _\\rho $ , then the statement is obviously fulfilled, because $\\rho $ is feasible and thus $e=$ for all equations $= e$ from $$ with $e \\rho = $ .",
"This gives us (REF ) and ().",
"Assume now that $\\in ^{\\prime }_\\rho $ .",
"Let $\\rho ^{\\prime } := \\rho |_{^{\\prime }_\\rho }$ and $\\rho ^* := \\mu _{\\ge \\rho ^{\\prime }}{^{\\prime }_\\rho }$ .",
"We have to show that $\\rho ^*()\\ge \\sup \\;\\lbrace \\hat{\\rho }() \\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; \\hat{\\rho }\\le {^{\\prime }_\\rho }\\hat{\\rho }\\rbrace .$ P If $\\rho ^*() = \\infty $ , there is nothing to prove.",
"Therefore, assume that $\\rho ^*() < \\infty $ .",
"Then $\\rho ^*() \\in $ .",
"Let $^{\\prime \\prime }_\\rho := \\lbrace ^{\\prime \\prime } \\in ^{\\prime }_\\rho \\mid \\rho ^*(^{\\prime \\prime }) < \\infty \\rbrace $ .",
"Then, $^{\\prime \\prime }_\\rho = \\lbrace ^{\\prime \\prime } \\in ^{\\prime }_\\rho \\mid \\rho ^*(^{\\prime \\prime }) \\in \\rbrace $ .",
"Let $^{\\prime \\prime }_\\rho := \\lbrace ^{\\prime \\prime } = e \\in ^{\\prime }_\\rho \\mid ^{\\prime \\prime } \\in ^{\\prime \\prime }_\\rho \\rbrace [\\infty /(^{\\prime }_\\rho \\setminus ^{\\prime \\prime }_\\rho )]$ , and $\\rho ^{\\prime \\prime } := \\rho |_{^{\\prime \\prime }_\\rho }$ .",
"The pre-solution $\\rho ^{\\prime \\prime }$ of $^{\\prime \\prime }_\\rho $ is feasible.",
"Hence, $\\rho ^*|_{^{\\prime \\prime }_\\rho }$ is a feasible finite pre-solution of $^{\\prime \\prime }_\\rho $ , i.e., a feasible finite fixpoint of ${^{\\prime \\prime }_\\rho }$ .",
"Therefore, we finally get () using Lemma REF .",
"P Before we actually prove (REF ), we start with an easy observation.",
"The sequence $({^{\\prime }_\\rho }^k \\rho ^{\\prime })_{k\\in }$ is increasing, because $\\rho ^{\\prime }$ is a pre-solution of $^{\\prime }_\\rho $ .",
"Further ${^{\\prime }_\\rho }^k \\rho ^{\\prime } : ^{\\prime }_\\rho \\rightarrow $ and ${^{\\prime }_\\rho }^k \\rho ^{\\prime } \\le {^{\\prime }_\\rho } ( {^{\\prime }_\\rho }^k \\rho ^{\\prime } )$ for all $k \\in $ .",
"Hence, we get $\\sup \\;\\lbrace \\hat{\\rho }()\\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; \\hat{\\rho }\\le {^{\\prime }_\\rho }\\hat{\\rho }\\rbrace \\ge \\sup \\;\\lbrace ({^{\\prime }_\\rho }^k \\rho ^{\\prime })() \\mid k \\in \\rbrace $ P Now, assume that $$ is a system of cmorcave equations.",
"In order to prove (REF ), it remains to show that $ \\rho ^*()\\le \\sup \\;\\lbrace \\hat{\\rho }() \\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; \\hat{\\rho }\\le {^{\\prime }_\\rho }\\hat{\\rho }\\rbrace $ .",
"Since ${^{\\prime }_\\rho }$ is monotone and upward-chain-continuous on $\\lbrace \\hat{\\rho }: ^{\\prime }_\\rho \\rightarrow \\mid \\hat{\\rho }\\ge \\rho ^{\\prime } \\rbrace $ , we have $\\rho ^*=\\bigvee \\lbrace {^{\\prime }_\\rho }^k \\rho ^{\\prime } \\mid k \\in \\rbrace $ .",
"Using (REF ), this gives us $ \\rho ^*()\\le \\sup \\;\\lbrace \\hat{\\rho }()\\mid \\hat{\\rho }:^{\\prime }_\\rho \\rightarrow ,\\; \\hat{\\rho }\\le {^{\\prime }_\\rho }\\hat{\\rho }\\rbrace $ , as desired.",
"$\\Box $ P If the equations are morcave but not cmorcave, then the inequality in () can indeed be strict as the following example shows.",
"P Let us consider the following system $$ of morcave equations: $_1 &= 1 &_2 &= _1 + _2 &_3 &={\\left\\lbrace \\begin{array}{ll}0 &\\text{if } _2 < \\infty \\\\1 &\\text{if } _2 = \\infty \\end{array}\\right.",
"}$ P Observe that the third equation is not cmorcave, since, for the ascending chain $C = \\lbrace \\lbrace _2 \\mapsto k \\rbrace \\mid k \\in \\rbrace $ , we have $\\bigvee \\lbrace {e} \\rho \\mid \\rho \\in C \\rbrace = 0 < 1 = {e} (\\bigvee C)$ , where $e$ denotes the right-hand side of the third equation.",
"The variable assignment $\\rho := \\lbrace _1 \\mapsto 0 ,\\; _2 \\mapsto 0 ,\\; _3 \\mapsto 0 \\rbrace $ is a feasible pre-solution, since $\\rho ^*:=\\mu _{\\ge \\rho } {}=\\lbrace _1 \\mapsto 1 ,\\; _2 \\mapsto \\infty ,\\; _3 \\mapsto 1 \\rbrace $ P is a feasible solution of $$ .",
"Now, let the variable assignment $\\rho _1$ be defined by $\\rho _1 &:= \\mathsf {suppresol}_\\rho .$ P Lemma REF gives us $\\rho _1 \\le \\rho ^*$ , but not $\\rho _1 = \\rho ^*$ .",
"Indeed, we have $\\rho _1=\\lbrace _1 \\mapsto 1 ,\\; _2 \\mapsto \\infty ,\\; _3 \\mapsto 0 \\rbrace <\\rho ^*.$ P We emphasize that $\\rho _1(_3) = 0$ , because ${e}\\hat{\\rho }= 0$ for all $\\hat{\\rho }: \\rightarrow $ , where $e$ denotes the right-hand side of the third equation of REF .",
"P How we can actually compute $\\rho ^*$ , remains an open question.",
"The discontinuity at $_2 = \\infty $ is the reason for the strict inequality in (REF ).",
"However, since upward discontinuities can only be present at $\\infty $ , there are at most $n$ upward discontinuities, where $n$ is the number of variables of the equation system.",
"Hence, we could think of using (REF ) to get over at least one discontinuity.",
"P Let us perform a second iteration for the example.",
"We know that $\\rho _1 \\le \\rho ^*$ .",
"Moreover, by definition, $\\rho _1$ is also a feasible pre-solution of $$ .",
"For the variable assignment $\\rho _2$ that is defined by $\\rho _2 := \\mathsf {suppresol}_{\\rho _1}$ we obviously have $\\rho ^* = \\rho _2$ .",
"We will see that this method can always be applied.",
"More precisely, we can always compute $\\rho ^*$ after performing at most $n$ such iterations.",
"$\\Box $ P In order to deal not only with systems of cmorcave equations, but also with systems of morcave equations, we use Lemma REF iteratively until we reach a solution.",
"That is, we generalize the statement of Lemma REF as follows: P Let $$ be a system of morcave equations and $\\rho $ a feasible pre-solution of $$ .",
"For all $i \\in $ , let $\\mathsf {suppresol}^i_\\rho $ be defined by $\\mathsf {suppresol}^0_\\rho &:= \\rho \\\\\\mathsf {suppresol}^{i+1}_\\rho &:= \\mathsf {suppresol}_{\\mathsf {suppresol}^i_\\rho }&& \\text{for all } i \\in .$ Then, the following statements hold: $(\\mathsf {suppresol}^i_\\rho )_{i\\in }$ is an increasing sequence of feasible pre-solutions of $$ .",
"$\\mathsf {suppresol}^i_\\rho \\le \\mu _{\\ge \\rho }$ for all $i \\in $ .",
"$\\mathsf {suppresol}^{X}_\\rho = \\mu _{\\ge \\rho }$ .",
"$\\mathsf {suppresol}_\\rho = \\mu _{\\ge \\rho }$ , whenever $$ is a system of $cmorcave $ equations.",
"P The first two statements can be proven by induction on $i$ using Lemma REF .",
"The third statement follows from the fact that, for any feasible pre-solution $\\rho $ of a system $$ of morcave equations, $\\mathsf {suppresol}_\\rho < \\mu _{\\ge \\rho }$ implies that there exists some variable $\\in $ such that $\\rho () < \\infty $ and $\\mathsf {suppresol}_\\rho () = \\infty $ .",
"The fourth statement is the second statement of Lemma REF .",
"$\\Box $ P For the situation in Example REF , we have $ \\mu _{\\ge \\rho } {} = \\mathsf {suppresol}^3_\\rho = \\mathsf {suppresol}^2_\\rho > \\mathsf {suppresol}^1_\\rho > \\rho .$ $\\Box $ P Because of the definition of $\\mathsf {suppresol}_\\rho $ (see Definition REF ), Lemma REF implies the following corollary: P Let $$ be a system of morcave equations and $\\rho $ a feasible pre-solution of $$ .",
"Then, the value $\\mu _{\\ge \\rho }$ only depends on $$ and $^\\infty _\\rho := \\lbrace \\mid = e \\in ,\\; {e} \\rho = \\infty \\rbrace $ .",
"$\\Box $"
],
[
"Termination",
"P It remains to show that our $\\vee $ -strategy improvement algorithm (Algorithm REF ) terminates.",
"That is, we have to come up with an upper bound on the number of iterations of the loop.",
"In each iteration, we have to compute $\\mu _{\\ge \\rho }{(\\sigma )}$ , where $\\rho $ is a feasible pre-solution of $(\\sigma )$ .",
"This has to be done until we have found a solution.",
"By Corollary REF , $\\mu _{\\ge \\rho }{(\\sigma )}$ only depends on the $\\vee $ -strategy $\\sigma $ and the set $^\\infty _\\rho := \\lbrace \\mid = e \\in (\\sigma ) ,\\; {e} \\rho = \\infty \\rbrace $ .",
"During the run of our $\\vee $ -strategy improvement algorithm, the set $^\\infty _\\rho $ monotonically increases.",
"This implies that we have to consider each $\\vee $ -strategy $\\sigma $ at most $$ times.",
"That is, the number of iterations of the loop is bounded from above by $\\cdot $ .",
"Summarizing, we have shown our main theorem: P Let $$ be a system of $\\vee $ -morcave equations in standard form.",
"Our $\\vee $ -strategy improvement algorithm computes $\\mu $ and performs at most $|| \\cdot ||$ $\\vee $ -strategy improvement steps.",
"$\\Box $ P In our experiments, we did not observe the exponential worst-case behavior.",
"All examples we know of require linearly many $\\vee $ -strategy improvement steps.",
"We are also not aware of a class of examples, where we would be able to observe the exponential worst-case behavior.",
"Therefore, our conjecture is that for practical examples our algorithm terminates after linearly many iterations.",
"P In the static program analysis application that we discuss in Section , the right-hand sides of the fixpoint equation systems we have to solve are maxima of finitely many parametrized optimization problems.",
"In this special situation, we can evaluate $\\vee $ -strategies more efficiently than by solving general convex optimization problems as described in Section (see Lemma REF , REF , and REF ).",
"We provide a in-depth study of this special situation in this section."
],
[
"Parametrized Optimization Problems",
"P We now consider the case that a system $$ of fixpoint equations is given, where the right-hand sides are parametrized optimization problems.",
"In this article, we call an operator $g : ^n \\rightarrow $ a parametrized optimization problem if and only if $g(x) &= \\sup \\,\\lbrace f(y) \\mid y \\in Y(_1,\\ldots ,_n) \\rbrace && \\text{for all } x \\in ^n,$ P where $f : ^{k}\\rightarrow $ is an objective function, and $Y : ^n \\rightarrow 2^{^{k}}$ is a mapping that assigns a set $Y(x) \\subseteq ^k$ of states to any vector of bounds $x \\in ^n$ .",
"The parametrized optimization problem $g$ is monotone on $^n$ , whenever $Y$ is monotone on $^n$ .",
"It is monotone on $^n$ and upward chain continuous on $g^{-1}(\\setminus \\lbrace \\rbrace )$A monotone function $g : ^n\\rightarrow $ is upward chain continuous on an upward closed set $X \\subseteq ^n$ if and only if $g(\\bigvee C) = \\bigvee g(C)$ for all non-empty chains $C \\subseteq X$ .",
"whenever $f$ is continuous on $^k$ and $Y$ is monotone on $^n$ and upward chain continuous on $Y^{-1}(2^{^k} \\setminus \\lbrace \\emptyset \\rbrace )$A monotone function $Y : ^n \\rightarrow 2^{^k}$ is upward chain continuous on an upward closed set $X \\subseteq ^n$ if and only if $Y(\\bigvee C) = \\bigcup Y(C)$ for all chains $C \\subseteq X$ ..",
"In the following, we are concerned with the latter situation.",
"A parametrized optimization problem $g$ that is monotone on $^n$ and upward chain continuous on $g^{-1}(\\setminus \\lbrace \\rbrace )$ is called upward chain continuous parametrized optimization problem.",
"P Assume that $Y$ and $f$ are given by $Y(x) &:= \\lbrace y \\in ^k \\mid Ay \\le x \\rbrace && \\text{for all } x \\in ^n, \\text{ and} \\\\f(y) &:= b + c^\\top y && \\text{for all } y \\in ^k,$ P where $A \\in ^{n\\times k}$ , $b \\in $ , and $c \\in ^k$ .",
"Then, $g$ is defined through Equation (REF ) is an upward chain continuous parametrized optimization problems.",
"To be more precise, it is a parametrized linear programming problem (to be defined).",
"Although this is also an interesting case (cf.",
"[8]), in the following, we mainly focus on the more general case where the right-hand sides are parametrized semi-definite programming problems (to be defined).",
"In this example, the right-hand side is not only upward chain continuous, it is even cmcave.",
"To be more precise, on the set of points where it returns a value greater than $$ it is a point-wise minimum of finitely many monotone and affine operators.",
"$\\Box $"
],
[
"Fixpoint Equations with Parametrized Optimization Problems",
"Assume now that we have a system of fixpoint equations, where the right-hand sides are point-wise maxima of finitely many upward chain continuous parametrized optimization problems.",
"If we use our $\\vee $ -strategy improvement algorithm to compute the least solution, then, for each $\\vee $ -strategy improvement step, we have to compute $\\mu _{\\ge \\rho _0} {}$ for a system $$ of fixpoint equations whose right-hand sides are upward chain continuous parametrized optimization problems, and $\\rho _0$ is a pre-solution of $$ .",
"We study this case in the following: P Assume that $$ is a system of fixpoint equations, where the right-hand sides are upward chain continuous parametrized optimization problems.",
"For simplicity and without loss of generality, we additionally assume that a variable assignment $\\rho _0 : \\rightarrow $ is given such that $\\ll \\rho _0 \\le \\rho _0 \\ll .$ P We are interested in computing the pre-solution $\\mathsf {suppresol}_{\\rho _0}$ of $$ .",
"In the case at hand, this means that we need to compute $\\rho ^* : \\rightarrow $ that is defined by $\\rho ^*() &:= \\sup \\, \\lbrace \\rho () \\mid \\rho : \\rightarrow \\text{ and } \\rho \\le \\rho \\rbrace && \\text{ for all } \\in .$ P"
],
[
"Algorithm ",
"As a start, we firstly consider the case where all right-hand sides are upward chain continuous parametrized optimisation problems of the form $\\sup \\,\\lbrace f(y) \\mid y \\in Y(_1,\\ldots ,_n) \\rbrace $ , where $ \\sup \\,\\lbrace f(y) \\mid y \\in Y(x_1,\\ldots ,x_n) \\rbrace = \\max \\,\\lbrace f(y) \\mid y \\in Y(x_1,\\ldots ,x_n) \\rbrace $ P for all $x_1,\\ldots ,x_n \\in $ with $< \\sup \\,\\lbrace f(y) \\mid y \\in Y(x_1,\\ldots ,x_n) \\rbrace < \\infty $ .",
"We say that such a parametrized optimization problem attains its optimal value for all parameter values.",
"Parametrized linear programming problems, for instance, are parametrized optimization problems that attain their optimal values for all parameter values.",
"In the case at hand, the variable assignment $\\rho ^*$ can be characterized as follows: $\\rho ^*() &:= \\sup \\,\\lbrace \\rho () \\mid \\rho : _{)} \\rightarrow \\text{ and } \\rho \\le {)}\\rho \\rbrace &&\\text{for all } \\in ,$ P where the constraint system $)$ is obtained from $$ by replacing every equation $= \\sup \\;\\lbrace f(y) \\mid y \\in Y(_1,\\ldots ,_n) \\rbrace $ P with the constraints $&\\le f(_1,\\ldots ,_{k}) &(_1,\\ldots ,_{k}) &\\in Y(_1,\\ldots ,_n),$ P where $_1,\\ldots ,_{k}$ are fresh variables.",
"P As we will see in the remainder of this section, the above characterization enable us to compute $\\rho ^*$ using specialized convex optimization techniques.",
"If, for instance, the right-hand sides are parametrized linear programming problems (to be defined), then we can compute $\\rho ^*$ through linear programming.",
"Likewise, if the right-hand sides are parametrized semi-definite programming problems (to be defined), then we can compute $\\rho ^*$ through semi-definite programming.",
"P Let us consider the system $$ of equations that consist of the following equations: $_1 &= \\sup \\; \\lbrace x_1^{\\prime } \\in \\mid x_1^{\\prime } \\in ,\\; x_1^{\\prime } \\le 0 \\rbrace \\\\_2 &=\\sup \\;\\lbrace x_2^{\\prime \\prime } \\in \\mid x_2^{\\prime },x_2^{\\prime \\prime } \\in ,\\; 0 \\le x_2^{\\prime } \\le _1 ,\\; x_2^{\\prime \\prime } \\le 1 \\rbrace &$ P We aim at computing the variable assignment $\\rho ^* : \\rightarrow $ defined by $\\rho ^*() &:=\\sup \\, \\lbrace \\rho () \\mid \\rho : \\rightarrow \\text{ and } \\rho \\le \\rho \\rbrace && \\text{ for all } \\in .$ P All right-hand sides of the equations are upward continuous parametrized optimization problems that attain their optimal value for all parameter values.",
"Hence, we can apply the above described method to compute $\\rho ^*$ .",
"If we do so, the system $)$ of inequalities consist of the following inequalities: $_1 &\\le _1^{\\prime } &_1^{\\prime } &\\le 0 &_2 &\\le _2^{\\prime \\prime } &0 &\\le _2^{\\prime } \\le _1 &_2^{\\prime \\prime } \\le 1.$ O According to Equation (REF ), for all $i \\in \\lbrace 1,2\\rbrace $ , we thus have $\\rho ^*(_i) &= \\sup \\; \\lbrace _i\\mid _1, _1^{\\prime },_2,_2^{\\prime },_2^{\\prime \\prime } \\in ,\\\\&\\qquad _1 \\le _1^{\\prime } ,_1^{\\prime } \\le 0 ,_2 \\le _2^{\\prime \\prime } ,0 \\le _2^{\\prime } \\le _1 ,_2^{\\prime \\prime } \\le 1\\rbrace $ P Observe that these optimization problems are actually linear programming problems.",
"Solving these linear programming problems gives us, as desired, $\\rho ^* = \\lbrace _1\\mapsto 0 ,\\, _2\\mapsto 1 \\rbrace $ .",
"$\\Box $ P"
],
[
"Algorithm ",
"If we are not in the nice situation that all parametrized optimization problems attain their optimal values for all parameter values, then we have to apply a more sophisticated method to compute $\\rho ^*$ .",
"The following example, that is obtained from Example REF , illustrates the need for more sophisticated methods.",
"P We now slightly modify the fixpoint equation system $$ from Example REF by replacing Equation (REF ) by the equation $_1 = \\sup \\; \\lbrace x_1^{\\prime } \\in \\mid x_1^{\\prime } \\in ,\\; x_1^{\\prime } < 0 \\rbrace $ .",
"That is, we are now concerned with strict inequality instead of non-strict inequality.",
"In consequence, the parametrized optimization problem does not attain its optimal value for any parameter value.",
"The fixpoint equation system $$ now consists of the following equations: $_1 &= \\sup \\; \\lbrace x_1^{\\prime } \\in \\mid x_1^{\\prime } \\in ,\\; x_1^{\\prime } < 0 \\rbrace \\\\_2 &=\\sup \\;\\lbrace x_2^{\\prime \\prime } \\in \\mid x_2^{\\prime },x_2^{\\prime \\prime } \\in ,\\; 0 \\le x_2^{\\prime } \\le _1 ,\\; x_2^{\\prime \\prime } \\le 1 \\rbrace &$ This modification does not change the value of $\\rho ^*$ (defined by Equation (REF )), since the right-hand side of the first equation still evaluates to 0.",
"However, the system $)$ of inequalities is now given by $_1 &< _1^{\\prime } &_1^{\\prime } &\\le 0 &_2 &\\le _2^{\\prime \\prime } &0 &\\le _2^{\\prime } \\le _1 &_2^{\\prime \\prime } \\le 1.$ P Since the above inequalities imply $0 \\le _2^{\\prime } \\le _1 < _1^{\\prime } \\le 0$ and thus $0 < 0$ , there is no solution to the above inequalities.",
"Therefore, we cannot apply the methods we applied in Example REF to compute $\\rho ^*$ .",
"$\\Box $ P We now describe a more sophisticated method to compute $\\rho ^*$ .",
"For all variable assignments $\\rho _0$ and $\\rho $ , we define the system $_{\\rho _0,\\rho }$ of equations as follows: $\\nonumber _{\\rho _0,\\rho }&:=\\left\\lbrace = \\rho _0() \\mid = e \\in \\text{ and } \\rho _0() \\ge e \\rho \\right\\rbrace \\\\&\\qquad \\quad \\cup \\left\\lbrace = e \\mid = e \\in \\text{ and } \\rho _0() < e \\rho \\right\\rbrace $ O That is, $_{\\rho _0,\\rho }$ contains all equations $=e$ of $$ whose right-hand sides $e$ evaluate under $\\rho $ to a value greater than $\\rho _0()$ .",
"The other equations of $$ are replaced by $=\\rho _0()$ .",
"We again assume that $\\rho _0$ is a variable assignment with $\\ll \\rho _0 \\le {}\\rho _0 \\ll $ .",
"For all $k \\in _{>0}$ , we then define the variable assignment $\\rho _k$ inductively by $\\rho _k() &:= \\sup \\,\\lbrace \\rho () \\mid \\rho : \\rightarrow \\text{ and } \\rho \\le {_{\\rho _0,\\rho _{k-1}}}\\rho \\rbrace && \\text{ for all } \\in .$ Now, $\\rho ^*$ is the limit of the sequence $(\\rho _k)_{k\\in }$ and the sequence reaches its limits after at most $$ steps: P The sequence $(\\rho _{k})_{k\\in }$ of variables assignments is increasing, $\\rho _k \\le \\rho ^*$ for all $k \\in $ , $\\rho _{k+1} > \\rho _k$ if $\\rho _k < \\rho ^*$ , and $\\rho _{} = \\rho ^*$ .",
"Moreover, $\\rho _k() = \\sup \\,\\lbrace \\rho () \\mid \\rho : _{_{\\rho _0,\\rho _{k-1}})}\\rightarrow \\text{ and } \\rho \\le {_{\\rho _0,\\rho _{k-1}})}\\rho \\rbrace $ for all $k$ and all $\\in $ .",
"$\\Box $ P Let us again consider the fixpoint equation system $$ from Example REF .",
"We again aim at computing the variable assignment $\\rho ^* : \\rightarrow $ that is defined by $\\rho ^*() :=\\sup \\, \\lbrace \\rho () \\mid \\rho : \\rightarrow \\text{ and } \\rho \\le \\rho \\rbrace $ for all $\\in $ .",
"Since $\\ll \\rho _0 \\le \\rho _0 \\ll $ for $\\rho _0 := \\lbrace _1\\mapsto 0 ,\\; _2 \\mapsto 0 \\rbrace $ , we can apply the method we just developed.",
"The system $_{\\rho _0,\\rho _0}$ is given by $_1 &= 0 &_2 &=\\sup \\;\\lbrace x_2^{\\prime \\prime } \\in \\mid x_2^{\\prime },x_2^{\\prime \\prime } \\in ,\\; 0 \\le x_2^{\\prime } \\le _1 ,\\; x_2^{\\prime \\prime } \\le 1 \\rbrace $ P Therefore, the constraint system $_{\\rho _0,\\rho _0})$ is given by $_1 &\\le 0 &_2 &\\le _2^{\\prime \\prime } &0 &\\le _2^{\\prime } \\le _1 &_2^{\\prime \\prime } \\le 1$ P Solving the optimization problems that aims at maximizing $_1$ and $_2$ , respectively, we get $\\rho _1 = \\lbrace _1\\mapsto 0 ,\\; _2\\mapsto 1 \\rbrace $ .",
"We then construct the fixpoint equation system $_{\\rho _0,\\rho _1}$ .",
"The system $_{\\rho _0,\\rho _1}$ is equal to the system $_{\\rho _0,\\rho _0}$ , and thus $_{\\rho _0,\\rho _1})$ is equal to the system $_{\\rho _0,\\rho _0})$ .",
"Therefore, we get $\\rho ^* = \\rho _1$ by Lemma REF .",
"$\\Box $ P"
],
[
"Algoritm ",
"In our static program analysis application we discuss in the next section, we have the comfortable situation that our right-hand sides are not only upward continuous parametrized optimization problems, but they are additionally cmcave.",
"We can utilize this in order to simplify the above developed procedure EvalForGen.",
"The following lemma is the key ingredient for this optimization: P Let $\\rho $ be a feasible pre-solution of a system $$ of cmorcave equations.",
"For all $\\in $ , we have $\\mu _{\\ge \\rho }() > \\rho ()$ if and only if $({}^{{}} \\rho ) () > \\rho ()$ .",
"[Sketch] P Since $\\rho $ is a feasible pre-solution of $$ , we can w.l.o.g.",
"assume that ${e}\\rho > $ for all equations $= e$ of $$ .",
"Therefore, $$ is upward chain continuous on $(\\rightarrow )_{\\ge \\rho }$ .",
"The statement finally follows from the fact that $$ is additionally monotone and order-concave.",
"$\\Box $ Assume now that we want to use our $\\vee $ -strategy improvement algorithm to compute the least solution of a system of $\\vee $ -cmorcave equations.",
"In each $\\vee $ -strategy improvement step, we are then in the situation that we have to compute $\\rho ^* := \\mu _{\\ge \\rho }$ , where $\\rho $ is a feasible pre-solution of a system $$ of morcave equations (cf.",
"Lemma REF ).",
"By Lemma REF , we can compute the set $^{\\prime } := \\lbrace \\in \\mid \\rho ^*() > \\rho () \\rbrace $ by performing $$ Kleene iteration steps.",
"We then construct the equation system $^{\\prime } := \\lbrace = e \\in \\mid \\in ^{\\prime } \\rbrace \\cup \\lbrace = \\rho () \\mid \\in \\setminus ^{\\prime }\\rbrace $ By construction, we get: $\\rho ^*() = \\sup \\,\\lbrace \\rho () \\mid \\rho : _{^{\\prime })} \\rightarrow \\text{ and } \\rho \\le {^{\\prime })}\\rho \\rbrace $ for all $\\in $ .",
"$\\Box $ P In consequence, we can compute $\\rho ^*$ by performing $$ Kleene iteration steps followed by solving $$ optimization problems.",
"O Let us again consider the fixpoint equation system $$ from Example REF and REF .",
"That is, $$ consists of the following equations: $_1 &= \\sup \\; \\lbrace x_1^{\\prime } \\in \\mid x_1^{\\prime } \\in ,\\; x_1^{\\prime } < 0 \\rbrace \\\\_2 &=\\sup \\;\\lbrace x_2^{\\prime \\prime } \\in \\mid x_2^{\\prime },x_2^{\\prime \\prime } \\in ,\\; 0 \\le x_2^{\\prime } \\le _1 ,\\; x_2^{\\prime \\prime } \\le 1 \\rbrace &$ The fixpoint equation system $$ is a system of cmorcave equations.",
"The pre-solution $\\rho := \\lbrace _1\\mapsto 0, _2 \\mapsto 0\\rbrace $ of $$ is feasible.",
"Moreover, we have $\\ll \\rho \\le \\rho \\ll $ .",
"We aim at computing $\\rho ^* := \\mu _{\\ge \\rho } $ .",
"We have ${}^{} \\rho = {}^{2} \\rho = \\lbrace _1\\mapsto 0, _2 \\mapsto 1\\rbrace $ .",
"By Lemma REF , we thus get $^{\\prime } := \\lbrace \\in \\mid \\rho ^*() > \\rho () \\rbrace = \\lbrace _2 \\rbrace $ (cf.",
"(REF )).",
"Lemma REF finally gives us $\\rho ^*(_i)=\\sup \\;\\lbrace _i \\mid _1,_1^{\\prime },_2,_2^{\\prime },_2^{\\prime \\prime } \\in ,_1 \\le 0 ,_2 \\le _2^{\\prime \\prime } ,0 \\le _2^{\\prime } \\le _1 ,_2^{\\prime \\prime } \\le 1\\rbrace .$ for all $i \\in \\lbrace 1,2\\rbrace $ (cf.",
"Example REF ).",
"This is the desired result.",
"We performed two Kleene iteration steps and solved two mathematical optimization problems.",
"$\\Box $ P We now introduce parameterized linear programming problems.",
"We do this as follows.",
"For all $A \\in ^{k\\times m}$ and all $c \\in ^m$ , we define the operator $\\mathbf {LP}_{A, c}: ^k \\rightarrow $ which solves a parametrized linear programming problem by $\\mathbf {LP}_{A, c}(b)&:=\\sup \\,\\lbrace c^\\top x\\mid x \\in ^m\\text{ and }A x \\le b\\rbrace && \\text{for all }b \\in ^k.$ We use the LP-operators in the right-hand sides of fixpoint equation systems: (LP-equations, $\\vee $ -LP-equations) A fixpoint equation $=e$ is called LP-equation if and only if $e$ is a parametrized linear programming problem.",
"It is called $\\vee $ -LP-equation if and only if $e$ is a point-wise maximum of finitely many semi-definite programming problems.",
"$\\Box $ P LP-operators have the following important properties: P The following statements hold for all $A \\in ^{k\\times m}$ and all $c \\in ^m$ : The operator $\\mathbf {LP}_{A, c}$ is cmcave.",
"$\\mathbf {LP}_{A, c}(b) = \\max \\, \\lbrace c^\\top x \\mid x \\in ^m \\text{ and } A x \\le b\\rbrace $ for all $b \\in ^k$ with $< \\mathbf {LP}_{A, c}(b) < \\infty $ .",
"That is, the parametrized optimization problem $\\mathbf {LP}_{A, c}$ attains its optimal value for all parameter values.",
"P We do not prove the first statement, since, as we will see, it is just a special case of Lemma REF (see below).",
"This second statement is a direct consequence of the fact that the optimal value of a feasible and bounded linear programming problem is attained at the edges of the feasible space.",
"$\\Box $ O If we apply our $\\vee $ -strategy improvement algorithm for solving a system of $\\vee $ -LP-equations, then, because of Lemma REF , we have the convenient situation that we can apply Algorithm EvalForMaxAtt instead of its more general variant EvalForGen for evaluating a single $\\vee $ -strategy that is encountered during the $\\vee $ -strategy iteration (see Section REF ).",
"We thus obtain the following result: P If $$ is a system of $\\vee $ -LP-equations, then the evaluation of a $\\vee $ -strategy that is encountered during the $\\vee $ -strategy iteration can be performed by solving ${}$ linear programming problems, each of which can be constructed in polynomial time.",
"In consequence, a $\\vee $ -strategy improvement step can be performed in polynomial time.",
"$\\Box $ Theorem REF implies that our $\\vee $ -strategy improvement algorithm terminates after at most $\\cdot \\Sigma $ $\\vee $ -strategy improvement steps, whenever it runs on a system $$ of $\\vee $ -LP-equations.",
"A consequence of the fact that we can evaluate $\\vee $ -strategies in polynomial time is the following decision problem is in $\\mathsf {NP}$ : Decide whether or not, for a given system $$ of $\\vee $ -LP-equations, a given variable $\\in $ , and a given value $b \\in $ , the statement $\\mu () \\le b$ holds.",
"This decision problem is at least as hard as the problem of computing the winning regions in mean payoff games.",
"However, whether or not it is $\\mathsf {NP}$ -hard is an open question."
],
[
"Parameterized Semi-Definite Programming Problems",
"P As a strict generalization of parameterized linear programming problems, we now introduce parameterized semi-definite programming problems.",
"Before we can do so, we have to briefly introduce semi-definite programming.",
"P"
],
[
"Semi-definite Programming",
"${n}$ (resp.",
"$S^{n\\times n}_+$ ) denotes the set of symmetric matrices (resp.",
"the set of positive semidefinite matrices).",
"$\\preceq $ denotes the Löwner ordering of symmetric matrices, i.e., $A \\preceq B$ if and only if $B - A \\in S^{n\\times n}_+$ .",
"$\\mathrm {Tr}(A)$ denotes the trace of a square matrix $A \\in ^{n \\times n}$ , i.e., $\\mathrm {Tr}(A) = \\sum _{i = 1}^n A_{i \\cdot i}$ .",
"The inner product of two matrices $A$ and $B$ is denoted by $A \\bullet B$ , i.e., $A \\bullet B = \\mathrm {Tr}(A^\\top B)$ .",
"For $\\mathcal {A}= (A_1,\\ldots ,A_m)$ with $A_i \\in ^{n \\times n}$ for all $i =1,\\ldots ,m$ , we denote the vector $(A_1 \\bullet X , \\ldots , A_m \\bullet X)^\\top $ by $\\mathcal {A}(X)$ .",
"For all $x \\in ^n$ , the dyadic matrix $X(x)$ is defined by $X(x):=\\begin{pmatrix}1 \\\\ x \\end{pmatrix} (1, x^\\top ).$ O We consider semidefinite programming problems (SDP problems for short) of the form $z^*=\\sup \\;\\lbrace C \\bullet X \\mid X \\in S^{n\\times n}_+,\\mathcal {A}(X) = a,(X) \\le b\\rbrace ,$ where $\\mathcal {A}= (A_1,\\ldots ,A_m)$ , $a \\in ^m$ , $A_1,\\ldots ,A_m \\in n$ , $= (B_1,\\ldots ,B_k)$ , $B_1,\\ldots ,B_k \\in n$ , $b \\in ^k$ , and $C \\in n$ .",
"The set $\\lbrace X \\in S^{n\\times n}_+\\mid \\mathcal {A}(X) = a,(X) \\le b\\rbrace $ is called the feasible space.",
"The problem is called feasible if and only if the feasible space is non-empty.",
"It is called infeasible otherwise.",
"An element of the feasible space is called feasible solution.",
"The value $z^*$ is called optimal value.",
"The problem is called bounded iff $z^* < \\infty $ .",
"It is called unbounded, otherwise.",
"A feasible solution $X^*$ is called an optimal solution if and only if $z^* = C \\bullet X^*$ .",
"In contrast to the situation for linear programming, there exist feasible and bounded semi-definite programming problem that have no optimal solution.",
"For semi-definite programming problems, fast algorithms exist.",
"Semi-definite programming is polynomial time solvable if an a priori bound on the size of the solutions is known and provided as an input.",
"For more detailed information on semi-definite programming, or, more generally, on convex optimization, we refer, for instance, to [17], [13].",
"P"
],
[
"Parametrized SDP Problems",
"For $\\mathcal {A}= (A_1,\\ldots ,A_m)$ , $A_1,\\ldots ,A_m \\in n$ , $a \\in ^m$ , $= (B_1,\\ldots ,B_k)$ , $B_1,\\ldots ,B_k \\in n$ , and $C \\in n$ , we define the operator $\\mathbf {SDP}_{\\mathcal {A}, a, , C}: ^k \\rightarrow $ which solves a parametrized SDP problem by $\\mathbf {SDP}_{\\mathcal {A}, a, , C}(b)&:=\\sup \\,\\lbrace C {\\bullet } X \\mid X {\\in } S^{n\\times n}_+,\\mathcal {A}(X) = a,(X) \\le b\\rbrace &&\\!\\!\\!\\!\\text{for all }b \\in ^k.$ P The SDP-operators generalizes the LP-operators in the same way as semi-definite programming generalizes linear programming.",
"That is, for every LP-operator we can construct an equivalent SDP-operator.",
"(SDP-equations, $\\vee $ -SDP-equations) A fixpoint equation $=e$ is called SDP-equation if and only if $e$ is a parametrized semi-definite programming problem.",
"It is called $\\vee $ -SDP-equation if and only if $e$ is a point-wise maximum of finitely many semi-definite programming problems.",
"$\\Box $ For this article, the following properties of SDP-operators are important: P The operator $\\mathbf {SDP}_{\\mathcal {A}, a, , C}$ is cmcave.",
"O Let $f := \\mathbf {SDP}_{\\mathcal {A}, a, , C}$ .",
"For all $b \\in ^k$ , let $M(b) := \\lbrace X \\in S^{n\\times n}_+ \\mid \\mathcal {A}(X) = a, (X) \\le b \\rbrace $ .",
"Therefore, $f(b) = \\sup \\;\\lbrace C \\bullet X \\mid X \\in M(b) \\rbrace $ for all $b \\in ^k$ .",
"We do not need to consider all $I : \\lbrace 1,\\ldots ,k\\rbrace \\rightarrow \\lbrace ,\\mathsf {id},\\infty \\rbrace $ , because, for all $I : \\lbrace 1,\\ldots ,k\\rbrace \\rightarrow \\lbrace ,\\mathsf {id},\\infty \\rbrace $ , $f^{(I)}$ can be obtained by choosing appropriate $\\mathcal {A}, a, \\mathcal {B}, C$ .",
"The fact that $f$ is monotone is obvious.",
"Firstly, we show that $f(b) < \\infty $ holds for all $b \\in ^k$ , whenever $\\operatorname{\\mathsf {fdom}}(f) \\ne \\emptyset $ .",
"For the sake of contradiction assume that there exist $b_1,b_2 \\in ^k$ such that $f(b_1) \\in $ and $f(b_2) = \\infty $ hold.",
"Note that $M(b_i)$ are convex sets for all $i \\in \\lbrace 1,2\\rbrace $ .",
"Thus, there exists some $D \\in S^{n\\times n}_+$ such that $C \\bullet D > 0$ and $M(b_2) + \\lbrace \\lambda D \\mid \\lambda \\in \\rbrace \\subseteq M(b_2)$ hold.",
"Therefore, $\\mathcal {A}(D) = 0$ and $(D) \\le 0$ .",
"Let $X_1 \\in S^{n\\times n}_+$ with $\\mathcal {A}(X_1) = a$ and $(X_1) \\le b_1$ .",
"Then $\\mathcal {A}(X_1 + \\lambda D)=\\mathcal {A}(X_1) + \\lambda \\mathcal {A}(D)=a$ and $(X_1 + \\lambda D)=(X_1) + \\lambda (D)\\le b_1$ hold for all $\\lambda > 0$ .",
"Thus, $f(b_1) = \\infty $ — contradiction.",
"Thus, $f(b) < \\infty $ holds for all $b \\in ^k$ , whenever $\\operatorname{\\mathsf {fdom}}(f) \\ne \\emptyset $ .",
"O Next, we show that $\\operatorname{\\mathsf {fdom}}(f)$ is convex and $f|_{\\operatorname{\\mathsf {fdom}}(f)}$ is concave.",
"Assume that $\\operatorname{\\mathsf {fdom}}(f) \\ne \\emptyset $ .",
"Thus, $f(b) < \\infty $ for all $b \\in ^k$ .",
"Let $b_1, b_2 \\in \\operatorname{\\mathsf {fdom}}(f)$ , $\\lambda \\in [0,1]$ , and $b := \\lambda b_1 + (1-\\lambda ) b_2$ .",
"In order to show that $\\lambda M(b_1) + (1 - \\lambda ) M(b_2)\\subseteq M(b)$ holds, let $X_i \\in M(b_i)$ , $i =1,2$ , and $X = \\lambda X_1 + (1 - \\lambda ) X_2$ .",
"Since $X_i \\in S^{n\\times n}_+$ , $\\mathcal {A}(X_i) = a$ , and $(X_i) \\le b_i$ for all $i=1,2$ , we have $X \\in S^{n\\times n}_+$ , $\\mathcal {A}(X) = \\lambda \\mathcal {A}(X_1) + (1-\\lambda )\\mathcal {A}(X_2)= a$ , $(X)=\\lambda (X_1) + (1 - \\lambda ) (X_1)\\le \\lambda b_1 + (1-\\lambda ) b_2=b$ .",
"Therefore, $X \\in M(b)$ .",
"Using (REF ), we finally get: $f( b )&=\\sup \\lbrace C \\bullet X\\mid X \\in M(b)\\rbrace \\\\&\\ge \\lambda \\sup \\lbrace C \\bullet X_1 \\mid X_1 \\in M(b_1) \\rbrace +(1-\\lambda )\\sup \\lbrace C \\bullet X_2 \\mid X_2 \\in M(b_2) \\rbrace \\\\&=\\lambda f(b_1)+(1-\\lambda ) f(b_2)>$ O Therefore, $\\operatorname{\\mathsf {fdom}}(f)$ is convex and $f|_{\\operatorname{\\mathsf {fdom}}(f)}$ is concave.",
"O It remains to show that $f$ is upward chain continuous on $f^{-1}(\\setminus \\lbrace \\rbrace )$ .",
"For that, let $B \\subseteq f^{-1}(\\setminus \\lbrace \\rbrace )$ be a chain.",
"We have $\\textstyle f(\\bigvee B)&=\\textstyle \\sup \\; \\lbrace C \\bullet X \\mid X \\in M(\\bigvee B) \\rbrace \\\\&=\\textstyle \\sup \\; \\lbrace C \\bullet X \\mid X \\in \\bigcup \\lbrace M(b) \\mid b \\in B \\rbrace \\rbrace && \\text{($M$ is continuous)}\\\\&=\\textstyle \\sup \\; \\lbrace \\sup \\; \\lbrace C \\bullet X \\mid X \\in M(b) \\rbrace \\mid b \\in B \\rbrace \\\\&=\\textstyle \\sup \\; \\lbrace f(b) \\mid b \\in B \\rbrace $ O This proves that $f$ is upward chain continuous on $f^{-1}(\\setminus \\lbrace \\rbrace )$ .",
"$\\Box $ P The next example shows that the square root operator can be expressed through a SDP-operator: P The square root operator $\\sqrt{\\cdot }: \\rightarrow $ is defined by $\\sqrt{b} := \\sup \\lbrace x \\in \\mid x^2 \\le b \\rbrace $ for all $b \\in $ .",
"Note that $\\sqrt{b} = $ for all $b < 0$ , and $\\sqrt{\\infty }= \\infty $ .",
"Let $\\textstyle \\mathcal {A}:= \\begin{pmatrix}\\begin{pmatrix} 1 & 0 \\\\ 0 & 0 \\end{pmatrix}\\end{pmatrix},\\quad a := 1,\\quad := \\begin{pmatrix}\\begin{pmatrix} 0 & 0 \\\\ 0 & 1 \\end{pmatrix}\\end{pmatrix},\\quad C := \\begin{pmatrix} 0 & \\frac{1}{2} \\\\ \\frac{1}{2} & 0 \\end{pmatrix}.$ P For $x,b \\in $ , the statement $x^2 \\le b$ is equivalent to the statement $\\exists b^{\\prime } .",
"x^2 \\le b^{\\prime } \\le b$ .",
"By the Schur complement theorem (c.f.",
"Section 3, Example 5 of [17], for instance), this is equivalent to $\\exists b^{\\prime } .\\begin{pmatrix}1 & x \\\\x & b^{\\prime }\\end{pmatrix}\\succeq 0\\wedge b^{\\prime } \\le b.$ P This is equivalent to $ \\textstyle \\exists X \\in S^{2\\times 2}_+ .x = X_{1 \\cdot 2} = X_{2 \\cdot 1}\\wedge \\mathcal {A}(X) = a\\wedge (X) \\le b$ .",
"Thus, $ \\sqrt{b} =\\mathbf {SDP}_{\\mathcal {A}, a, , C}(b)$ for all $b \\in $ .",
"$\\Box $ O If $$ is a system of $\\vee $ -SDP-equations, then, because of Lemma REF , we have the convenient situation that we can apply Algorithm EvalForCmorcave instead of its more general variant EvalForGen (see Section REF ) to evaluate the $\\vee $ -strategies that are encountered during the $\\vee $ -strategy iteration.",
"This case is in particular interesting for the static program analysis application we will describe in Section .",
"O If $$ is a system of $\\vee $ -SDP-equations, then the evaluation of a $\\vee $ -strategy that is encountered during the $\\vee $ -strategy iteration can be performed by performing $$ Kleene iteration steps and subsequently solving ${}$ semi-definite programming problems, each of which can be constructed in polynomial time.",
"$\\Box $ Theorem REF implies that our $\\vee $ -strategy improvement algorithm terminates after at most $\\cdot \\Sigma $ $\\vee $ -strategy improvement steps, whenever it runs on a system $$ of $\\vee $ -LP-equations."
],
[
"Quadratic Zones and Relaxed Abstract Semantics",
"O In this section, we apply our $\\vee $ -strategy improvement algorithm to a static program analysis problem.",
"For that, we first introduce our programming model as well as its collecting and its abstract semantics.",
"We then relax the abstract semantics along the same lines as [1] using Shor's semidefinite relaxation schema.",
"Finally, we show how we can use our finding to compute the relaxation of the abstract semantics."
],
[
"Collecting Semantics",
"In our programming model, we consider statements of the following two forms: $x := Ax + b$ , where $A \\in ^{n \\times n}$ , and $b \\in ^n$ (affine assignments) $x^\\top Ax + 2 b^\\top x \\le c$ , where $A \\in n$ , $b \\in ^n$ , and $c \\in $ (quadratic guards) Here, $x \\in ^n$ denotes the vector of program variables.",
"We denote the set of statements by $$ .",
"The collecting semantics ${s} : 2^{^n} \\rightarrow 2^{^n}$ of a statement $s \\in $ is defined by: ${x := Ax + b}X&:= \\lbrace Ax + b \\mid x \\in X \\rbrace && \\text{for all } X \\subseteq ^n\\\\{x^\\top A x + 2 b^\\top x \\le c}X&:= \\lbrace x \\in X \\mid x^\\top A x + 2 b^\\top x \\le c \\rbrace && \\text{for all } X \\subseteq ^n$ O A program $G$ is a triple $(N,E,, I)$ , where $N$ is a finite set of control-points, $E \\subseteq N \\times \\times N$ is a finite set of control-flow edges, $\\in N$ is the start control-point, and $I \\subseteq ^n$ is a set of initial values.",
"The collecting semantics $$ of a program $G = (N,E,, I)$ is then the least solution of the following constraint system: $[]&\\supseteq I&[v]&\\supseteq {s} ([u])&& \\text{for all } (u,s,v) \\in E$ Here, the variables $[v]$ , $v \\in N$ take values in $2^{^n}$ .",
"The components of the collecting semantics $$ are denoted by $[v]$ for all $v \\in N$ ."
],
[
"Quadratic Zones and Abstract Semantics",
"Along the lines of [1], we define quadratic zones as follows: A set $P$ of templates $p : ^n \\rightarrow $ is a quadratic zone if and only if every template $p \\in P$ can be written as $p(x) &= x^\\top A_p x + 2 b_p^\\top x&& \\text{for all } x \\in ^n,$ where $A_p \\in n$ and $b_p \\in ^n$ for all $p \\in P$ .",
"In the remainder of this article, we assume that $P = \\lbrace p_1,\\ldots ,p_m \\rbrace $ is a finite quadratic zone.",
"Moreover, we assume w.l.o.g.",
"that $p_i \\ne 0$ for all $i = 1,\\ldots ,m$ .",
"The abstraction $\\alpha : 2^{^n} \\rightarrow P \\rightarrow $ and the concretization $\\gamma : (P \\rightarrow ) \\rightarrow 2^{^n}$ are defined as follows: $\\gamma (v)&\\textstyle :=\\lbrace x \\in ^n \\mid \\forall p \\in P .",
"p(x) \\le v(p) \\rbrace && \\text{for all } v : P \\rightarrow \\\\\\alpha (X)&\\textstyle :=\\bigwedge \\lbrace v : P \\rightarrow \\mid \\gamma (v) \\supseteq X \\rbrace && \\text{for all } X \\subseteq ^n$ As shown by [1], $\\alpha $ and $\\gamma $ form a Galois-connection.",
"The elements from $\\gamma (P \\rightarrow )$ and the elements from $\\alpha (2^{^n})$ are called closed.",
"$\\alpha (\\gamma (v))$ is called the closure of $v : P \\rightarrow $.",
"Accordingly, $\\gamma (\\alpha (X))$ is called the closure of $X \\subseteq ^n$.",
"As usual, the abstract semantics ${s}^\\sharp : (P \\rightarrow ) \\rightarrow P \\rightarrow $ of a statement $s$ is defined by ${s}^\\sharp := \\alpha \\circ {s} \\circ \\gamma $ .",
"The abstract semantics $^\\sharp $ of a program $G = (N,E,,I)$ is then the least solution of the following constraint system: $^\\sharp []&\\ge \\alpha (I)&^\\sharp [v]&\\ge {s}^\\sharp (^\\sharp [u])&& \\text{for all } (u,s,v) \\in E$ Here, the variables $^\\sharp [v]$ , $v \\in N$ take values in $P \\rightarrow $ .",
"The components of the abstract semantics $^\\sharp $ are denoted by $^\\sharp [v]$ for all $v \\in N$ ."
],
[
"Relaxed Abstract Semantics",
"O The problem of deciding, whether or not, for a given quadratic zone $P$ , a given $v : P \\rightarrow $ , a given $p \\in P$ , and a given $q \\in $ , $\\alpha (\\gamma (v))(p) \\le q$ holds, is NP-hard (cf.",
"[1]) and thus intractable.",
"Therefore, we use the relaxed abstract semantics $^\\mathcal {R}$ introduced by [1].",
"It is based on Shor's semidefinite relaxation schema.",
"In order to fit it into our framework, we have to switch to the semi-definite dual.",
"This is not a disadvantage.",
"It is actually an advantage, since we gain additional precision through this step.",
"[${x := Ax + b}^\\mathcal {R}$ ] We define the relaxed abstract semantics ${x := Ax + b}^\\mathcal {R}: (P \\rightarrow ) \\rightarrow P \\rightarrow $ of an affine assignment $x := Ax + b$ by $&{x := Ax + b}^\\mathcal {R}v \\, (p)\\\\&\\qquad :=\\textstyle \\sup \\lbrace \\overline{A} (p) {\\bullet } X\\mid \\forall p^{\\prime } \\in P .",
"\\overline{A}_{p^{\\prime }} {\\bullet } X \\le v(p^{\\prime }),X \\succeq 0, X_{1\\cdot 1} = 1 \\rbrace $ for all $v : P \\rightarrow $ and all $p \\in P$ , where, for all $p^{\\prime } \\in P$ , $A(p) := A^\\top A_p A ,\\quad b(p) := A^\\top A_p b + A^\\top b_p,\\quad c(p) := b^\\top A_p b + 2 b_p^\\top b\\\\\\overline{A} (p):=\\begin{pmatrix}c(p) & b^\\top (p) \\\\b(p) & A(p)\\end{pmatrix},\\quad \\overline{A}_{p^{\\prime }}:=\\begin{pmatrix}0 & b_{p^{\\prime }}^\\top \\\\b_{p^{\\prime }} & A_{p^{\\prime }}\\end{pmatrix}.\\text{$\\Box $}$ [${x^\\top A x + 2 b^\\top x \\le c}^\\mathcal {R}$ ] We define the relaxed abstract semantics ${x^\\top A x + 2 b^\\top x \\le c}^\\mathcal {R}: (P \\rightarrow ) \\rightarrow P \\rightarrow $ of a quadratic guard $x^\\top A x + 2 b^\\top x \\le c$ by $&\\;{x^\\top A x + 2 b^\\top x \\le c}^\\mathcal {R}v \\, (p)\\\\&\\qquad :=\\;\\textstyle \\sup \\lbrace \\overline{A}_p {\\bullet } X\\mid \\forall p^{\\prime } \\in P .",
"\\overline{A}_{p^{\\prime }} {\\bullet } X \\le v(p^{\\prime }),\\widetilde{A} {\\bullet } X \\le 0,X \\succeq 0,X_{1\\cdot 1} = 1 \\rbrace $ for all $v : P \\rightarrow $ and all $p \\in P$ , where, for all $p^{\\prime } \\in P$ , $\\widetilde{A}:=\\begin{pmatrix}-c & b^\\top \\\\b & A\\end{pmatrix},\\quad \\overline{A}_{p^{\\prime }}:=\\begin{pmatrix}0 & b_{p^{\\prime }}^\\top \\\\b_{p^{\\prime }} & A_{p^{\\prime }}\\end{pmatrix}.\\text{$\\Box $}$ The relaxed abstract semantics ${\\cdot }^\\mathcal {R}$ is the semidefinite dual of the one used by [1].",
"By weak-duality, it is at least as precise as the one used by [1].",
"Next, we show that the relaxed abstract semantics is indeed a relaxation of the abstract semantics, and that the relaxed abstract semantics of a statement is expressible through a SDP-operator.",
"P The following statements hold for every statement $s \\in $ : ${s}^\\sharp \\le {s}^\\mathcal {R}$ For every $i \\in \\lbrace 1,\\ldots ,m\\rbrace $ , there exist $\\mathcal {A},a,,C$ such that ${s}^\\mathcal {R}v \\, (p_i) = \\mathbf {SDP}_{\\mathcal {A}, a, , C}(v(p_1),\\ldots ,v(p_m))$ for all $v : P \\rightarrow $ .",
"From $s$ , the values $\\mathcal {A}$ , $a$ , $$ , and $ can be computed in polynomial time.\\Box $ Since the second statement is obvious, we only prove the first one.",
"We only consider the case that $s$ is an affine assignment $x := Ax + b$ .",
"The case that $s$ is a quadratic guard can be treated along the same lines.",
"Let $v : P \\rightarrow $ , $p \\in P$ , and $v^{\\prime } := {x := Ax + b}^\\sharp v$ .",
"Then, $v^{\\prime }(p)&=\\sup \\lbrace p(Ax + b) \\mid x \\in ^n , \\forall p^{\\prime } \\in P .",
"p^{\\prime }(x) \\le v(p^{\\prime })\\rbrace \\\\&=\\sup \\lbrace x^\\top A(p) x + 2 b^\\top (p) x + c(p)\\mid \\\\&\\qquad \\qquad \\qquad \\qquad x \\in ^n,\\forall p^{\\prime } \\in P .",
"x^\\top A_{p^{\\prime }} x + 2 b_{p^{\\prime }}^\\top x \\le v(p^{\\prime })\\rbrace \\\\&=\\textstyle \\sup \\left\\lbrace (1,x^\\top )\\overline{A}(p)(1,x^\\top )^\\top \\mid \\forall p^{\\prime } \\in P.(1,x^\\top )\\overline{A}_{p^{\\prime }}(1,x^\\top )^\\top \\le v(p^{\\prime })\\right\\rbrace \\\\&=\\textstyle \\sup \\left\\lbrace \\overline{A}(p) {\\bullet } X(x)\\mid \\forall p^{\\prime } \\in P.\\overline{A}_{p^{\\prime }}{\\bullet }X(x)\\le v(p^{\\prime })\\right\\rbrace \\\\&\\le \\textstyle \\sup \\left\\lbrace \\overline{A}(p) {\\bullet } X\\mid \\forall p^{\\prime } \\in P.\\overline{A}_{p^{\\prime }}{\\bullet }X\\le v(p^{\\prime }),X \\succeq 0,X_{1 \\cdot 1} = 1\\right\\rbrace .$ The last inequality holds, because $X(x) \\succeq 0$ and $X(x)_{1 \\cdot 1} = 1$ for all $x \\in ^n$ .",
"This completes the proof of statement 1.",
"$\\Box $ A relaxation of the closure operator $\\alpha \\circ \\gamma $ is given by ${x := x}^\\mathcal {R}$ .",
"That is, $\\alpha \\circ \\gamma \\le {x := x}^\\mathcal {R}$ .",
"The relaxed abstract semantics $^\\mathcal {R}$ of a program $G = (N,E,,I)$ is finally defined as the least solution of the following constraint system: $^\\mathcal {R}[]&\\ge \\alpha (I)&^\\mathcal {R}[v]&\\ge {s}^\\mathcal {R}(^\\mathcal {R}[u])&& \\text{for all } (u,s,v) \\in E$ Here, the variables $^\\mathcal {R}[v]$ , $v \\in N$ take values in $P \\rightarrow $ .",
"The components of the relaxed abstract semantics $^\\mathcal {R}$ are denoted by $^\\mathcal {R}[v]$ for all $v \\in N$ .",
"P Because of Lemma REF , the relaxed abstract semantics of a program is a safe over-approximation of its abstract semantics.",
"If all templates and all guards are linear, then the relaxed abstract semantics is precise (cf.",
"[1]): We have $^\\sharp \\le ^\\mathcal {R}$ .",
"Moreover, if all templates and all guards are linear, then $^\\sharp = ^\\mathcal {R}$ .",
"$\\Box $ Computing Relaxed Abstract Semantics We now use our $\\vee $ -strategy improvement algorithm to compute the relaxed abstract semantics $^\\mathcal {R}$ of a program $G = (N,E,,I)$ w.r.t.",
"a given finite quadratic zone $P = \\lbrace p_1,\\ldots ,p_m \\rbrace $ .",
"For that, we define $ to be the constraint system{\\begin{@align}{1}{-1}_{,p}&\\ge \\alpha (I)(p)&& \\text{for all } p \\in P\\\\_{v,p}&\\ge ({s}^\\mathcal {R}(_{u,p_1},\\ldots ,_{u,p_m})^\\top )(p)&& \\text{for all } (u, s, v) \\in E\\text{, and all } p \\in P\\end{@align}}$ which uses the variables $= \\lbrace _{v,p} \\mid v \\in N ,\\; p \\in P \\rbrace $ .",
"The value of the variable $_{v,p}$ is the bound on the template $p$ at control-point $v$ .",
"Because of Lemma REF , from $ we can construct a system $$ of SDP-equationswith $ = in polynomial time.",
"Finally, we have: $^\\mathcal {R}[v](p) = \\mu {}(_{v,p})$ for all $v \\in N$ and all $p \\in P$ .",
"$\\Box $ Since $$ is a system of $\\vee $ -SDP-equations, by Theorem REF and Theorem REF , we can compute the least solution $\\mu $ of $$ using our $\\vee $ -strategy improvement algorithm.",
"Thus, we have finally shown the following main result for the static program analysis application: We can compute the relaxed abstract semantics $^\\mathcal {R}$ of a program $G = (N,E, ,I)$ using our $\\vee $ -strategy improvement algorithm.",
"Each $\\vee $ -strategy improvement step can by performed by performing $N \\cdot P$ Kleene iteration steps and solving $N \\cdot P$ SDP problems, each of which can be constructed in polynomial time.",
"The number of strategy improvement steps is exponentially bounded by the product of the number of merge points in the program and the number of program variables.",
"$\\Box $ O In order to give a complete picture of our method, we now discuss the harmonic oscillator example of [1] in detail.",
"The program consists only of the simple loop $()\\;x := Ax,$ where $x = (x_1,x_2)^\\top \\in ^2$ is the vector of program variables and $A=\\begin{pmatrix}1 & 0.01 \\\\-0.01 & 0.99\\end{pmatrix}.$ We assume that the two-dimensional interval $I = [0,1] \\times [0,1]$ is the set of initial states.",
"The set of control-points just consists of $$ , i.e.",
"$N = \\lbrace \\rbrace $ .",
"The set $P = \\lbrace p_1,\\ldots ,p_5 \\rbrace $ of templates is given by $p_1(x_1,x_2) &= -x_1 &p_2(x_1,x_2) &= x_1 &p_3(x_1,x_2) &= -x_2 \\\\p_4(x_1,x_2) &= x_2 &p_5(x_1,x_2) &= 2 x_1^2 + 3 x_2^2 + 2 x_1 x_2$ The abstract semantics is thus given by the least solution of the following system of $\\vee $ -SDP-equations: $_{,p_1}&=\\vee 0 \\vee \\mathbf {SDP}_{\\mathcal {A},a,,C_1} (_{,p_1},_{,p_2},_{,p_3},_{,p_4},_{,p_5})\\\\_{,p_2}&=\\vee 1 \\vee \\mathbf {SDP}_{\\mathcal {A},a,,C_2} (_{,p_1},_{,p_2},_{,p_3},_{,p_4},_{,p_5})\\\\_{,p_3}&=\\vee 0 \\vee \\mathbf {SDP}_{\\mathcal {A},a,,C_3} (_{,p_1},_{,p_2},_{,p_3},_{,p_4},_{,p_5})\\\\_{,p_4}&=\\vee 1 \\vee \\mathbf {SDP}_{\\mathcal {A},a,,C_4} (_{,p_1},_{,p_2},_{,p_3},_{,p_4},_{,p_5})\\\\_{,p_5}&=\\vee 7 \\vee \\mathbf {SDP}_{\\mathcal {A},a,,C_5} (_{,p_1},_{,p_2},_{,p_3},_{,p_4},_{,p_5})$ Here $\\mathcal {A}= \\left( \\begin{pmatrix}1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0& 0 & 0 \\end{pmatrix}\\right)\\qquad a = (1)$ $=\\left(\\begin{pmatrix} 0 & -0.5 & 0 \\\\ -0.5 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix},\\begin{pmatrix} 0 & 0.5 & 0 \\\\ 0.5 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix},\\begin{pmatrix} 0 & 0 & -0.5 \\\\ 0 & 0 & 0 \\\\ -0.5 & 0 & 0 \\end{pmatrix},\\begin{pmatrix} 0 & 0 & 0.5 \\\\ 0 & 0 & 0 \\\\ 0.5 & 0 & 0 \\end{pmatrix},\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 2 & 1 \\\\ 0 & 1 & 3 \\end{pmatrix}\\right)$ $C_1 &= \\begin{pmatrix} 0 & -0.5 & -0.005 \\\\ -0.5 & 0 & 0 \\\\ -0.005 & 0 & 0 \\end{pmatrix}&C_2 &= \\begin{pmatrix} 0 & 0.5 & 0.005 \\\\ 0.5 & 0 & 0 \\\\ 0.005 & 0 & 0 \\end{pmatrix}\\\\C_3 &= \\begin{pmatrix} 0 & 0.005 & -0.495 \\\\ 0.005 & 0 & 0 \\\\ -0.495 & 0 & 0 \\end{pmatrix}&C_4 &= \\begin{pmatrix} 0 & -0.005 & 0.495 \\\\ -0.005 & 0 & 0 \\\\ 0.495 & 0 & 0 \\end{pmatrix}\\\\C_5 &= \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 1.9803 & 0.9802 \\\\ 0 & 0.9802 & 2.9603 \\end{pmatrix}$ In this example we have $3^5 = 243$ different $\\vee $ -strategies.",
"Assuming that the algorithm always chooses the best local improvement, in the first step it switch to the $\\vee $ -strategy that is given by the finite constants.",
"At each equation, it then can switch to the $\\mathbf {SDP}$ -expression, but then, because it constructs a strictly increasing sequence, it can never return to the constant.",
"Summarizing, because of the simple structure, it is clear that our $\\vee $ -strategy improvement algorithm will perform at most 6 $\\vee $ -strategy improvement steps.",
"In fact our prototypical implementation performs 4 $\\vee $ -strategy improvement steps on this example.",
"$\\Box $ Conclusion O We introduced and studied systems of $\\vee $ -morcave equations — a natural and strict generalization of systems of rational equations that were previously studied by [5], [8].",
"We showed how the $\\vee $ -strategy improvement approach from [6], [5] can be generalized to solve these fixpoint equation systems.",
"We provided full proves and a in-depth discussion on the different cases.",
"On the practical side, we showed that our algorithm can be applied to perform static program analysis w.r.t.",
"quadratic templates using the relaxed abstract semantics of [1] (based on Shor's semi-definite relaxation schema).",
"This analysis can, for instance, be used to verify linear recursive filters and numerical integration schemes.",
"In the conference article that appears in the proceedings of the Seventeenth International Static Analysis Symposium (SAS 2010) we report on experimental results that were obtained through our proof-of-concept implementation [7].",
"For future work, we are interested in studying the use of other convex relaxation schemes to deal with more sophisticated cases, a problem already posed by [1].",
"This would partially abolish the restriction to affine assignments and quadratic guards.",
"Currently, we apply our $\\vee $ -strategy improvement algorithm only to numerical static analysis of programs.",
"It remains to investigate in how far the $\\vee $ -strategy improvement algorithm we developed can be applied to other applications — maybe in other fields of computer science.",
"Since our methods are solving quite general fixpoint problems, we have some hope that this is the case.",
"Natural candidates could perhaps be found in the context of two-players zero-sum games."
],
[
"Conclusion",
"O We introduced and studied systems of $\\vee $ -morcave equations — a natural and strict generalization of systems of rational equations that were previously studied by [5], [8].",
"We showed how the $\\vee $ -strategy improvement approach from [6], [5] can be generalized to solve these fixpoint equation systems.",
"We provided full proves and a in-depth discussion on the different cases.",
"On the practical side, we showed that our algorithm can be applied to perform static program analysis w.r.t.",
"quadratic templates using the relaxed abstract semantics of [1] (based on Shor's semi-definite relaxation schema).",
"This analysis can, for instance, be used to verify linear recursive filters and numerical integration schemes.",
"In the conference article that appears in the proceedings of the Seventeenth International Static Analysis Symposium (SAS 2010) we report on experimental results that were obtained through our proof-of-concept implementation [7].",
"For future work, we are interested in studying the use of other convex relaxation schemes to deal with more sophisticated cases, a problem already posed by [1].",
"This would partially abolish the restriction to affine assignments and quadratic guards.",
"Currently, we apply our $\\vee $ -strategy improvement algorithm only to numerical static analysis of programs.",
"It remains to investigate in how far the $\\vee $ -strategy improvement algorithm we developed can be applied to other applications — maybe in other fields of computer science.",
"Since our methods are solving quite general fixpoint problems, we have some hope that this is the case.",
"Natural candidates could perhaps be found in the context of two-players zero-sum games."
]
] | 1204.1147 |
[
[
"Evidence for a Peierls phase-transition in a three-dimensional multiple\n charge-density waves solid"
],
[
"Abstract The effect of dimensionality on materials properties has become strikingly evident with the recent discovery of graphene.",
"Charge ordering phenomena can be induced in one dimension by periodic distortions of a material's crystal structure, termed Peierls ordering transition.",
"Charge-density waves can also be induced in solids by strong Coulomb repulsion between carriers, and at the extreme limit, Wigner predicted that crystallization itself can be induced in an electrons gas in free space close to the absolute zero of temperature.",
"Similar phenomena are observed also in higher dimensions, but the microscopic description of the corresponding phase transition is often controversial, and remains an open field of research for fundamental physics.",
"Here, we photoinduce the melting of the charge ordering in a complex three-dimensional solid and monitor the consequent charge redistribution by probing the optical response over a broad spectral range with ultrashort laser pulses.",
"Although the photoinduced electronic temperature far exceeds the critical value, the charge-density wave is preserved until the lattice is sufficiently distorted to induce the phase transition.",
"Combining this result with it ab initio} electronic structure calculations, we identified the Peierls origin of multiple charge-density waves in a three-dimensional system for the first time."
],
[
"Acknowledgements",
"The authors acknowledge useful discussions with D. van der Marel, J. Demsar, C. Giannetti and A. Pasquarello.",
"The Lu$_5$ Ir$_4$ Si$_{10}$ single crystal have been provided by J.",
"A. Mydosh.",
"We thank U. Roethlisberger (EPFL, Lausanne) and ACRC (Bristol) for computer time.",
"This work was supported by the ERC grant N$^{\\circ }$ 258697 USED and by the Swiss NSF via contract $20020-127231/1$ .",
"Supporting Information: Evidence for a Peierls phase-transition in a three-dimensional multiple charge-density waves solid.",
"B. Mansart Laboratory for Ultrafast Electrons, ICMP, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH Laboratory of Ultrafast Spectroscopy, ISIC, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH [][email protected] M. Cottet Laboratory for Ultrafast Electrons, ICMP, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH T. J. Penfold Laboratory of Ultrafast Spectroscopy, ISIC, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH Laboratory of Computational Chemistry and Biochemistry, ISIC, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH SwissFEL, PSI, CH-5232 Villigen, CH S. B. Dugdale H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK R. Tediosi Département de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, CH M. Chergui Laboratory of Ultrafast Spectroscopy, ISIC, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH F. Carbone Laboratory for Ultrafast Electrons, ICMP, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, CH"
],
[
"Experimental setup",
"We performed an experimental study of the transient optical properties of Lu$_5$ Ir$_4$ Si$_{10}$ using pump-probe broadband reflectivity.",
"Our experimental setup, located at the Laboratory of Ultrafast Spectroscopy, EPFL, Switzerland, is presented in Fig.",
"S1.",
"An amplified Ti:sapphire laser system provides 1.55 eV pulses of 40 fs duration at a repetition rate of 1kHz.",
"This laser beam is separated into two paths by a beamsplitter.",
"The pump beam, whose energy is fixed at 1.55 eV, and the probe beam which is converted into a supercontinuum by a CaF$_2$ nonlinear crystal.",
"The probe spectrum covers a range between 1.5 and 3.2 eV.",
"The fluence of the white light probe is several orders of magnitude lower than the pump beam and has no effect on the measurements.",
"The pump pulse is used to excite the system at the time $t=0$ , and the consequent variations of reflectivity are recorded with a delay $t$ by changing the optical path difference between pump and probe (which can be achieved by setting a variable delay in the probe arm).",
"Small reflectivity changes (down to the order of 10$^{-3}$ ) are detectable owing to the signal-to-noise ratio of 10$^4$ , obtained by the use of an optical chopper on the pump beam which reduces its repetition rate by a factor of two.",
"This allows the recording, for each time delay, of a single-shot spectrum with and without the pumping pulse.",
"Furthermore, the variations of the white light intensity are taken into account using a reference signal, measured just before the sample position.",
"Two identical spectrometers are used to detect the transient reflectivity signal and the reference one.",
"By dispersing the incident light with optical gratings and sending it to an array detector, we obtain the complete spectrum for each single pulse with an energy resolution better than 1 meV.",
"Subsequently, the group velocity dispersion introduced by the chromatic aberration of the optics is corrected before the transient reflectivity analysis.",
"The pump-beam diameter on the sample surface was 250 $\\mu $ m, while that of the probe was set to 100 $\\mu $ m in order to probe a uniformly excited area.",
"Both spot sizes and spatial overlap were rigorously verified using a beam profiler and a charge-coupled device camera.",
"During this experiment, the pump fluence was varied between 0.8 and 3.1 mJ/cm$^2$ .",
"Prior to these experiments, the sample was oriented using X-ray diffraction and polished to obtain good optical quality.",
"Its surface contained both (001) and (100) directions.",
"Therefore, pump and probe polarizations were changed independently with quarter-waveplates and were oriented either parallel or orthogonal to the $c$ -axis (containing the Lu chains).",
"Finally, in order to cool down to 10 K and precisely monitor the sample temperature, the latter was placed in a liquid He cryostat.",
"Such kind of experimental setup has been successfully employed to investigate photoexcited states in strongly correlated systems, such as the charge-transfer insulator CuGeO$_3$ [S1], as well as in bulk graphite and graphene paper [S2]."
],
[
"Spectral Weight analysis",
"The definition of the optical spectral weight ($SW$ ) in a solid, given in Eq.",
"(1) of the main article, makes clear that the evaluation of $N_{eff}$ requires the knowledge of the real part of the optical conductivity down to zero frequency.",
"Since this is not experimentally feasible, common procedures (including extrapolations or modeling based on the Drude-Lorentz formalism) are invoked to obtain the whole $\\sigma _1(\\omega )$ spectrum.",
"Recently, it has been shown mathematically that according to the theorems on analytical continuation of holomorphic functions, and taking advantage of the Kramers-Kronig relation between the real and the imaginary part of the complex optical conductivity ($\\sigma (\\omega ) = \\sigma _1(\\omega )+i\\sigma _2(\\omega )$ ), the partial $SW$ integral can be re-written as a function of a limited spectral interval in the form of Carleman-Goluzin-Krylov equations [S3]: $SW(\\Omega _c) = Re~\\lbrace \\lim _{n\\rightarrow \\infty }\\int ^{\\omega _{max}}_{\\omega _{min}} u_n(\\omega ^\\prime ,\\Omega _c)\\sigma (\\omega ^\\prime )d\\omega ^\\prime \\rbrace $ where $u_n(\\omega ^{\\prime },\\Omega _c) = \\int _{0}^{\\Omega _c}Q_n(\\omega ^{\\prime },\\omega )d\\omega $ are properly chosen kernel functions.",
"In the case of real experimental data (with finite energy resolution and statistical noise), Eq.",
"(REF ) cannot be strictly applied, and a numerical procedure based on the idea of the analytical continuation has to be invoked in order to obtain the $SW$ integral in a model-independent fashion [S4].",
"A code called “Devin” has been developed which allows the $SW$ integral to be extracted from the complex dielectric function in a limited data range in a model-independent fashion [S4].",
"We performed a Drude-Lorentz analysis of our experimental data, and verified the results via the model-independent routine “Devin”.",
"The approach is described below in more detail.",
"To obtain the complex dielectric function, we fitted our transient reflectivity data to a highly flexible Drude-Lorentz function [S5]; the underlying Drude-Lorentz model of the optical data is based on the knowledge of the static optical spectra between a few meV and 4.6 eV, data taken from Ref. [S6].",
"For the fitting of the transient data, the Lorentz oscillators in our experimental range are allowed to change in order to reproduce the dynamical reflectivity.",
"This procedure yields the transient $\\epsilon _1$ and $\\epsilon _2$ , displayed together with the static spectra in Fig.",
"S2.",
"In our experimental frequency range, the real and imaginary part of the dielectric constant have rather similar absolute values (Fig.",
"S2A and E).",
"For this reason, the optical reflectivity, defined as $R = \\mid \\frac{1-\\surd \\epsilon }{1+\\surd \\epsilon }\\mid ^2$ , is equally sensitive to the reactive and the absorptive components of the dielectric function.",
"As a consequence, the complex $\\epsilon $ obtained through the Drude-Lorentz fitting of the optical reflectivity matches very well the one obtained via spectroscopic ellipsometry (see Fig.",
"S2A and E and Ref. [S6]).",
"The transient dielectric function is displayed in Fig.",
"S2(B-D, F-H) for light polarized along the $a$ -axis and the $c$ -axis at 10 K, and for light polarized along the $c$ -axis at room temperature.",
"Such a Drude-Lorentz modeling of the data also yields the spectral weight; the overall static $N_{eff}$ spectrum for both light polarizations is displayed in Fig.",
"S3.",
"In the same plot, we show the derivative of $N_{eff}$ as a function of the number of carriers which exhibits an inflection point between 1 and 1.5 eV for light polarized along the $c$ -axis; this inflection point is found to be between 10 and 15 carriers.",
"Since Lu has a $5d^1$ electronic configuration, and we found a strong hybridization with at least one or two Ir $5d$ bands, we expect between 10 and 20 electrons to form the material conduction band and therefore the low energy $c$ -axis conductivity should be mainly sensitive to these orbitals.",
"By integrating the transient conductivity, we calculated $N_{eff}$ at each time step and obtained the time evolution of $N_{eff}$ via Drude-Lorentz model, presented as 3D color maps in Fig.",
"2(A-C) of the main article.",
"In order to verify that the transient changes of the $SW$ we obtained by extrapolating the Drude-Lorentz model down to low frequency are robust and do not depend on the latter extrapolation, we fed into the “Devin” routine the real and imaginary parts of the dielectric function as an input in order to generate the $SW$ up to a given frequency cut-off.",
"Through this procedure it is also possible to obtain an estimate of the uncertainty related to the knowledge of the spectrum being in a limited range with a limited resolution and with a given noise level.",
"Because we are interested in the relative changes of the $SW$ induced by pump pulses, we use as an error our estimated accuracy in the determination of the transient dielectric function, which is of the order of $10^{-4}$ , while the resolution of our spectrometer is approximately 1 meV.",
"The transient dielectric function is fed into “Devin” and the model-independent $SW$ changes are shown in Fig.",
"2(A-B) of the main article as white lines with errorbars.",
"The fact that both these analyses give consistent results validate our description of the transient optical reflectivity.",
"The reason why we can successfully obtain information on the CDW-induced spectral weight transfer from a limited spectrum in the visible light region is that, in this sample, the optical anisotropy, related to the Lu one-dimensional chains, strongly manifests itself around 2 eV.",
"This is visible in the static reflectivity spectrum (Fig.",
"1 of the main article), and also in the temperature-dependent dielectric constants obtained via ellipsometry and reported in Ref.",
"[S6], which in fact show a particularly large CDW-induced change between 1.5 and 2.5 eV.",
"Also, in strongly correlated solids, a phase transition can induce spectral weight transfers across a very broad energy range, extending to the visible light region.",
"In these cases, it has been shown that the spectral region in the proximity of the screened plasma frequency of the material is very sensitive to these shifts [S7, S8].",
"Here, the screened plasma frequency along the $c$ -axis (direction of the charge-density-wave) lies within our experimental range (zero crossing of $\\epsilon _1$ in Fig.",
"S3A)."
],
[
"Three-temperature model",
"The two-temperature model was first introduced by Kaganov et al.",
"[S9] in order to describe the energy transfer between the electron and lattice subsystems, in the case where the electrons are driven to a higher temperature than the phonons.",
"Its validity relies on the electronic temperature being of the order of the Debye temperature (366 K in Lu$_5$ Ir$_4$ Si$_{10}$ [S10]) and on the isotropy of the electron-phonon (e-ph) coupling function.",
"For anisotropic materials like cuprates [S11] or iron-pnictides [S12], a selective coupling between electrons and a subset of the total phonon modes may be taken into account using the three-temperature model (3TM), governed by the following equations: $\\begin{array}{l}2 C_e \\frac{\\partial T_e}{\\partial t}= \\frac{2(1-R)}{l_s}I(t)+\\frac{\\partial }{\\partial z}(\\kappa _e \\frac{\\partial T_e}{\\partial z})-g(T_e-T_h)\\\\~\\\\\\alpha C_L \\frac{\\partial T_h}{\\partial t} = g(T_e-T_h)-g_c(T_h-T_c)\\\\~\\\\(1-\\alpha )C_{L} \\frac{\\partial T_c}{\\partial t}= g_c(T_h-T_c)\\\\~\\\\\\end{array}$ where $T_e$ , $T_h$ and $T_c$ are the temperatures of electrons, efficiently coupled phonons (“hot phonons\") and remaining modes (“cold phonons\"), respectively.",
"$C_e$ =$\\gamma T_e$ is the electronic specific heat, $C_L$ is that of the lattice.",
"$\\alpha $ is the fraction of efficiently coupled modes, $R$ is the static reflectivity and $l_s$ is the penetration depth (both at the pump energy).",
"$I(t)$ the pumping intensity, and $\\kappa _e$ is the electronic thermal conductivity (which is ignored in our case).",
"Indeed, the electronic diffusion term $\\frac{\\partial }{\\partial z}(\\kappa _e \\frac{\\partial T_e}{\\partial z})$ changes neither the relaxation rates nor the e-ph thermalization time.",
"The constant $g$ governs the energy transfer rate from electrons to hot phonons, and is related to the second moment of the Eliashberg function $\\lambda \\left\\langle \\omega ^2 \\right\\rangle $ through $g=\\frac{6\\hbar \\gamma }{\\pi k_B} \\lambda \\left\\langle \\omega ^2 \\right\\rangle $ [S13], $\\lambda $ being the dimensionless e-ph coupling constant.",
"$g_c$ governs the energy relaxation from hot phonons to the remaining lattice.",
"To perform the 3TM simulations, we used an iteration procedure, calculating at each time step the depth-dependent temperature profiles.",
"At each depth and time step, we iterate the electronic and lattice part of the specific heat.",
"The electronic part is taken as $C_e=\\gamma T_e$ , with $\\gamma $ =23.42 $mJ\\cdot mol^{-1}\\cdot K^{-2}$ [S10], and the lattice part is taken from Ref. [S14].",
"The e-ph coupling constant $\\lambda $ was obtained via these simulations, given a logarithmic average of the phonon energies $\\left\\langle \\omega ^2 \\right\\rangle $ of 30 meV, obtained by static optical spectroscopy in Ref. [S6].",
"The e-ph coupling constant should in any case depend on the probing energy; as a check, we simulated the experimental data for several probe wavelengths in order to show the result constancy in the experimental energy range.",
"The constant obtained through these simulations is given in Fig.",
"S4C, from which we may notice its energy-independence in the range 1.6 –2 eV.",
"In fact, as this energy range is far from the electronic inter-band transitions (except very close to 2 eV, see the Lorentz oscillator positions in Fig.",
"1B of the article), the probe is mainly sensitive to intra-band transitions.",
"Considering $N_{eff}$ , both experimental and theoretical determinations show that the carriers involved in the optical conductivity at 1.5 eV are mainly the Lu 5$d$ and some Ir 5$d$ , which mainly belong to the conduction band of the material; in fact, the plasma edge of the solid is close to 1.7 eV along the $c$ -axis.",
"Therefore, even though the numerous possible inter-band transitions should give a non-negligible contribution to the optical absorption, screening effects prevent the existence of long lived excited states at those energies (1.5 eV) in such a metallic system, and the typical electron-hole lifetime should not exceed 10-20 fs.",
"Therefore, on the time-scales of our experiments, $>$ 50 fs, most if not all of the light excitation creates electron-hole pairs which relax instantaneously back in the conduction band with an excess energy that can be described by an effective temperature.",
"Consequently, applying the 3TM in this energy range allows the measurement of the coupling between electrons at the Fermi level and lattice.",
"The linear dependence of the fast relaxation time as a function of the maximum electronic temperature reached (see Fig.",
"S4A), as expected considering this relaxation as due to e-ph coupling [S13], reflects also the applicability of the 3TM in our case.",
"We may notice the excellent agreement between the parameters obtained from this effective temperature model and the calculated partial e-ph coupling constants presented in the following section.",
"Indeed, both the fraction of efficiently coupled phonons (Fig.",
"S4D) and the sum of their partial e-ph coupling (Fig.",
"S4C) may be retrieved by 3TM simulations, enhancing our confidence in the reliability of this model in describing our transient data."
],
[
"Electronic structure calculations",
"Two separate sets of calculations were performed, the first using the highly accurate all-electron full-potential linearized augmented plane-wave method (often referred to as FP-LAPW), as implemented in the ELK code [S15], and the second using pseudopotentials, as implemented within the QUANTUM ESPRESSO code [S16], for the calculation of the phonons and e-ph coupling.",
"The lattice constants used were the experimental ones ($a=12.4936$ Å, $c=4.1852$ Å), as were the atomic positions [S17], giving a unit cell of two formula units (38 atoms).",
"The calculations made using the ELK code [S15] had a cutoff (for planewaves in the interstitial region) determined by $k_{\\rm max}=7.0/R_{\\rm min}$ , where $R_{\\rm min}$ is the smallest muffin-tin radius.",
"These muffin-tin radii were Lu (2.3328 a.u.",
"), Ir (2.3654 a.u.)",
"and Si (1.8369 a.u.).",
"The self-consistent cycle was carried out on a mesh of 64 $k$ -points in the irreducible (one sixteenth) of the Brillouin zone, and the Fermi surface evaluated from the converged potential on a denser mesh of 1936 $k$ -points within one eighth of the Brillouin zone.",
"The calculations presented are scalar-relativistic and omit spin-orbit coupling (after the effect of its inclusion was investigated and found to be negligibly small at the Fermi level.)",
"The exchange-correlation functional was that of Perdew-Wang/Ceperley-Alder, Ref. [S18].",
"The calculated band structure in the energy range [-8 eV, +2 eV] is given in Fig.",
"S5A.",
"In this range, the latter is composed of strongly hybridized Lu $5d$ and $4f$ , Ir $5d$ and Si $2p$ , the Lu $4f$ giving localized density of states around -4.5 eV, while electrons at the Fermi surface come mainly from Lu $5d$ , Ir $5d$ and Si $2p$ as shown in Fig.",
"S5B and Fig.",
"5 of the main article.",
"A large amount of “interstitial\" electronic density is also found at the Fermi level, indicating their delocalized nature as well as the strong hybridization degree in this system.",
"The e-ph coupling resulting from the mass enhancement may be calculated from the density of states (DOS) at the Fermi level, following the formula $N(E_F)=N_b(E_F) \\frac{m}{m^*}=N_b(E_F) (1+\\lambda )$ , where $N(E_F)$ is the renormalized DOS and $N_b(E_F)$ the bare one, both at the Fermi level $E_F$ ; $m$ is the free electron mass and $m^*$ the effective one.",
"The renormalized DOS obtained by specific heat measurement being 0.26/eV/atom/spin, (i.e.",
"19.76/eV), and the bare DOS calculated here of 11.02/eV if one considers that 30$\\%$ of the Fermi surface is gapped, one obtains $(1+\\lambda )=1.79$ giving an e-ph coupling constant $\\lambda =0.79$ .",
"We also present in Fig.",
"S6 the real and imaginary parts of the calculated susceptibility for two bands, corresponding of the Fermi surfaces indicated by arrows in Fig.",
"5A of the main article.",
"One may notice the peak at two different wavevector, in both real and imaginary parts.",
"These peaks appear at nesting wavevectors which are different between the two presented bands, indicating the formation of several charge-density waves.",
"For the phonon and e-ph coupling calculations, linear response within the pseudopotential approach provided by the QUANTUM ESPRESSO code was used [S16].",
"The Lu pseudopotential had the $f$ electrons as core states, and the resulting electronic structure was verified against that produced by the all-electron ELK calculation.",
"Both ultrasoft and norm-conserving pseudopotentials were used, the latter needed for the calculation of the dielectric function with the EPSILON package within QUANTUM ESPRESSO.",
"The cut-offs for the energy and charge density were 40 Ry and 400 Ry, respectively.",
"Phonons at the $\\Gamma $ point were calculated from self-consistent calculations performed on a (6,6,16) $k$ -point mesh and e-ph parameters were evaluated on a denser mesh of (12,12,32).",
"For the calculations of dielectric function, the routine EPSILON within the QUANTUM ESPRESSO package was used.",
"Here, Brillouin zone integrations were carried out on a mesh of over 6000 $k$ -points with a Gaussian smearing of 0.25 eV applied to the evaluation of the inter-band contribution.",
"We present the partial e-ph coupling constants for each of the 114 modes of the system as a function of the mode energy at $q=0$ in Fig.",
"S7B.",
"Two strongly coupled modes are found having respectively a partial e-ph coupling constant of 0.166 and 0.131, involving mainly Si atom motion since the latter are the lighest elements of the crystal structure.",
"One of these two modes is represented with yellow arrows in Fig.",
"5C of the main paper.",
"The sum over all modes gives $\\lambda =0.97$ , so in good agreement with the value obtained from the DOS renormalization at the Fermi level (when one considers that the electron-phonon calculation has only been made at the $\\Gamma $ point).",
"The histogram of partial e-ph coupling constants is shown in Fig.",
"S7A.",
"From such a representation, one is able to extract the partial e-ph coupling of some percentage of the most efficiently coupled modes, as represented with a red-dashed line in Fig.",
"S7A.",
"This area contains 21 modes, (or 18 $\\%$ of the phonons), and summing up their $\\lambda $ constant one obtain 0.62.",
"It indicates that the 18 $\\%$ of the phonon modes having the largest e-ph coupling have, all together, an e-ph coupling constant of 0.62.",
"This observation is in excellent agreement with the results of 3TM simulations, also indicating the reliability of these electronic structure and phonon calculations.",
"This value of $\\lambda $ gives a superconducting transition temperature of $\\approx $ 9 K using the McMillan formula [S19], as experimentally observed in these system above the critical pressure, and suggests that conventional e-ph mediated superconductivity takes place in Lu$_{5}$ Ir$_{4}$ Si$_{10}$ , obeying Bardeen-Cooper-Shrieffer theory.",
"Indeed, since upon photoexcitation the CDW is molten, and therefore does not compete anymore with superconductivity (as in the system under pressure), obtaining a similar $\\lambda $ value may reflect effectively the order parameter giving rise to superconductivity and not the one related to Peierls ordering.",
"Figure: Experimental Setup.Figure: (A) and (E) Static dielectric function for both light polarizations.",
"Color maps: transient dielectric function at different temperatures and along the different orientations.Figure: Effective number of carriers N eff N_{eff} as a function of energy, and its derivative dN eff dω\\frac{d N_{eff}}{d \\omega } plotted as a function of N eff N_{eff}; (A) measured data, and (B) obtained through electronic structure calculations described in the text.Figure: Three temperature model analysis.",
"(A) Fluence dependence of the signal obtained at 10K, with pump and probe parallel to cc-axis, at an energy of 1.7 eV; in the inset the fast decay time, associated with the electron-phonon coupling time, is plotted as a function of the maximum electronic temperature reached in average in the skin-depth together with its linear fit.",
"(B) Electron, hot phonon and cold phonon temperatures as a function of time delay obtained by simulating the transient reflectivity shown in open circles in the inset, with the simulation as the solid line; this curve corresponds to a geometry where both pump and probe were parallel to the cc-axis, at a pump fluence of 3.1 mJ/cm 2 ^2, and a probing energy of 1.7 eV.",
"(C) electron-phonon coupling constant obtained from the simulations, for probe polarization parallel to cc-axis or within the (a,ba,b) plane, simulated at a pump fluence of 3.1 mJ/cm 2 ^2 as a function of the probing energy; (D) corresponding fraction of coupled modes.Figure: Electronic band structure.",
"(A) Calculated electronic band structure using the ELK code.",
"(B) Density of states (note that the contribution of the localised Lu 4ff states is truncated).Figure: Calculated real and imaginary part of the susceptibility, for two of the bands close to the Fermi level (labeled 278 and 280).Figure: Calculated electron-phonon coupling.",
"(A) Histogram of the electron-phonon coupling constants λ\\lambda ; the red area represents the 18%\\% more coupled modes, whose total lambda gives 0.62.",
"(B) Energy distribution of the λ\\lambda ."
]
] | 1204.0883 |
[
[
"Decoherence on a two-dimensional quantum walk using four- and two-state\n particle"
],
[
"Abstract We study the decoherence effects originating from state flipping and depolarization for two-dimensional discrete-time quantum walks using four-state and two-state particles.",
"By quantifying the quantum correlations between the particle and position degree of freedom and between the two spatial ($x-y$) degrees of freedom using measurement induced disturbance (MID), we show that the two schemes using a two-state particle are more robust against decoherence than the Grover walk, which uses a four-state particle.",
"We also show that the symmetries which hold for two-state quantum walks breakdown for the Grover walk, adding to the various other advantages of using two-state particles over four-state particles."
],
[
"Introduction",
"Quantum walks are a close quantum analog of classical random walks, in which the evolution of a particle is given by a series of superpositions in position space [1], [2], [3], [4], [5].",
"Recently they have emerged as an efficient tool to carry out quantum algorithms [6], [7] and have been suggested as an explanation for wavelike energy transfer within photosynthetic systems [8], [9].",
"They have applications in the coherent control of atoms and Bose-Einstein condensates in optical lattices [10], [11], the creation of topological phases [12], and the generation of entanglement [13].",
"Quantum walks therefore have the potential to serve as a framework to simulate, control and understand the dynamics of a variety of physical and biological systems.",
"Experimental implementations of quantum walks in last few years have included NMR [14], [15], [16], cold ions [17], [18], photons [19], [22], [20], [21], [23], [24], and ultracold atoms [25], which has drawn further interest of the wider scientific community to their study.",
"The two most commonly studied forms of quantum walks are the continuous-time [26] and the discrete-time evolutions [5], [27], [28], [29], [30], [31], [32].",
"In this work we will focus on the discrete-time quantum walk and just call it quantum walk for simplicity.",
"If we consider a one-dimensional (1D) example of a two-state particle initially in the state $|\\Psi _{\\rm in}\\rangle = \\left( \\cos (\\delta /2)| 0 \\rangle +e^{i\\eta }\\sin (\\delta /2)| 1 \\rangle \\right)\\otimes |\\psi _{0}\\rangle ,$ then the operators to implement the walk are defined on the coin (particle) Hilbert space $\\mathcal {H}_c$ and the position Hilbert space $\\mathcal {H}_p$ [$\\mathcal {H} =\\mathcal {H}_c \\otimes \\mathcal {H}_p$ ].",
"A full step is given by first using the the unitary quantum coin $\\hat{B} (\\theta ) \\equiv \\begin{bmatrix}\\cos (\\theta ) & ~~\\sin (\\theta ) \\\\\\sin (\\theta ) & -\\cos (\\theta )\\end{bmatrix},$ and then following it by a conditional shift operation $\\hat{S}_x \\equiv \\sum _x \\left[ |0 \\rangle \\langle 0|\\otimes |\\psi _{x-1}\\rangle \\langle \\psi _x| + |1 \\rangle \\langle 1|\\otimes |\\psi _{x+1}\\rangle \\langle \\psi _x| \\right].$ The state after $t$ steps of evolution is therefore given by $|\\Psi _t\\rangle = [\\hat{S}_x [\\hat{B}(\\theta ) \\otimes \\hat{{\\mathbb {1}}}]]^t|\\Psi _{\\rm in}\\rangle \\;.$ All experimental implementations of quantum walks reported by today have used effectively 1D dynamics.",
"A natural extension of 1D quantum walks to higher dimension is to enlarge the Hilbert space of the particle with one basis state for each possible direction of evolution at the vertices.",
"Therefore, the evolution has to be defined using an enlarged coin operation followed by an enlarged conditioned shift operation.",
"For a two-dimensional (2D) rectangular lattice the dimension of the Hilbert space of the particle will be four and a four dimensional coin operation has to be used.",
"Two examples of this are given by using either the degree four discrete Fourier operator (DFO) [Fourier walk] or the Grover diffusion operator (GDO) [Grover walk] as coin operations [33], [34], [35].",
"An alternative extension to two and higher $(d)$ dimensions is to use $d$ coupled qubits as internal states to evolve the walk [36], [37].",
"Both these methods are experimentally demanding and beyond the capability of current experimental set ups.",
"Surprisingly however, two alternative schemes to implement quantum walks on a 2D lattice were recently proposed which use only two-state particles.",
"In one of these a single two-state particle is evolved in one dimension followed by the evolution in other dimension using a Hadamard coin operation [38], [39].",
"In the other, a two-state particle is evolved in one dimension followed by the evolution in the other using basis states of different Pauli operators as translational states [40], [41].",
"In this work we expand the understanding of 2D quantum walks by studying the effects decoherence has on the four-state Grover walk and the two two-state walks mentioned above.",
"The environmental effects are modeled using a state-flip and a depolarizing channel and we quantify the quantum correlations using a measure based on the disturbance induced by local measurements [42].",
"While in the absence of noise the probability distributions for all three schemes are identical, the quantum correlations built up during the evolutions differ significantly.",
"However, due to the difference in the size of the particles Hilbert space for the Grover walk and the two-state walks, quantum correlations generated between the particle and the position space cannot be compared.",
"The quantum correlations between the two spatial dimensions ($x-y$ ), obtained after tracing out the particle state, on the other hand, can be compared and we will show that they are larger for the walks using the two-state particles.",
"When taking the environmental effects into account, we find that all three schemes lead to different probability distribution and decoherence is strongest for the Grover walk, therefore making the two-state walks more robust for maintaining quantum correlations.",
"Interestingly, we also find that certain symmetries which hold for the two-state quantum walk in the presence of noise do not hold for a Grover walk.",
"Together with the specific initial state and the coin operation required for the evolution of the Grover walk, this reduces the chances to identify an equivalence class of operations on a four-state particle to help experimentally implement the quantum walk in any physical system that allows to manipulate the four internal states of the coin.",
"This article is organized as follows : In Section we define the three schemes for the 2D quantum walk used to study the decoherence and in Section we define the measure we use to quantify the quantum correlations.",
"In Section , the effect of decoherence in the presence of a state-flip noise channel and a depolarizing channel are presented and we compare the quantum correlations between the $x$ and $y$ directions for the three schemes.",
"We finally show in Section that the state-flip and phase-flip symmetries, which hold for the two-state quantum walk, breakdown for the four-state walk and conclude in Section .",
"For a Grover walk of degree four the coin operation is given by [33], [34] $\\hat{G} = \\frac{1}{2}\\begin{bmatrix}-1 & ~~1 & ~~1 & ~~1 \\\\~~1 & -1 & ~~1 & ~~1 \\\\~~1 & ~~1 & -1 & ~~1 \\\\~~1 & ~~1 & ~~1 & -1\\end{bmatrix},$ and the shift operator is S(x, y) x, y [|0 0|| x-1, y-1x, y| + |1 1|| x-1, y+1x, y| + |2 2|| x+1, y-1x, y| + |3 3|| x+1, y+1x, y| ], where $|\\psi _{x,y} \\rangle =|\\psi _x \\rangle \\otimes |\\psi _y \\rangle $ .",
"It is well known that the operation $[\\hat{S}_{(x,y)} [\\hat{G} \\otimes \\hat{{\\mathbb {1}}}]]^t$ results in maximal spread of the probability distribution only for the very specific initial state $|\\Psi _{\\rm in}^4\\rangle = \\frac{1}{2}\\left(|0\\rangle -|1\\rangle -|2\\rangle +|3\\rangle \\right)\\otimes |\\psi _{0, 0}\\rangle ,$ whereas the walk is localized at the origin for any other case [43], [44].",
"Choosing $|\\Psi _{\\rm in}^4\\rangle $ and evolving it for $t$ steps one finds $|\\Psi ^4(t)\\rangle &=&\\sum _{x=-t}^t \\sum _{y=-t}^t[ \\mathcal {A}_{(x, y),t}|0 \\rangle +\\mathcal {B}_{(x, y),t}|1 \\rangle \\nonumber \\\\&&\\;\\;\\;+ \\mathcal {C}_{(x, y),t}|2 \\rangle +\\mathcal {D}_{(x, y),t}|3 \\rangle ]\\otimes |\\psi _{(x, y)}\\rangle $ where $\\mathcal {A}_{(x, y),t}$ , $\\mathcal {B}_{(x, y),t}$ , $\\mathcal {C}_{(x, y),t}$ and $\\mathcal {D}_{(x, y),t}$ are given by the iterative relations $\\mathcal {A}_{(x, y),t} &=& \\frac{1}{2} [-\\mathcal {A}_{(x+1, y+1),t-1} + \\mathcal {B}_{(x+1, y+1),t-1} \\nonumber \\\\&& + \\mathcal {C}_{(x+1, y+1),t-1} + \\mathcal {D}_{(x+1, y+1),t-1} ] \\\\\\mathcal {B}_{(x, y),t} &=& \\frac{1}{2} [\\mathcal {A}_{(x+1, y-1),t-1} - \\mathcal {B}_{(x+1, y-1),t-1} \\nonumber \\\\&&+ \\mathcal {C}_{(x-1, y-1),t-1} + \\mathcal {D}_{(x+1, y-1),t-1} ] \\\\\\mathcal {C}_{(x, y),t} &=& \\frac{1}{2} [\\mathcal {A}_{(x-1, y+1),t-1} + \\mathcal {B}_{(x-1, y+1),t-1} \\nonumber \\\\&& - \\mathcal {C}_{(x-1, y+1),t-1} + \\mathcal {D}_{(x-1, y+1),t-1} ]\\\\\\mathcal {D}_{(x, y),t} &=& \\frac{1}{2} [\\mathcal {A}_{(x-1, y-1),t-1} + \\mathcal {B}_{(x-1, y-1),t-1} \\nonumber \\\\&& + \\mathcal {C}_{(x-1, y-1),t-1} - \\mathcal {D}_{(x-1, y-1),t-1} ].$ This results in the probability distribution P4s = x =-tt y =-tt [|A(x, y),t|2 +|B(x, y),t|2 + |C(x, y),t|2+|D(x, y),t|2] which is shown in Fig.",
"REF for $t=25$ .",
"Figure: (Color online) Probability distribution for the Grover after 25 steps.An identical distribution using the two-state particle with the |0〉+i|1〉 2\\frac{|0\\rangle + i |1\\rangle }{\\sqrt{2}} can be obtained using the alternate walk with the coin operation B(π/4)B(\\pi /4) and the Pauli walk after 25 steps."
],
[
"Alternate walk",
"Very recently a 2D quantum walk was suggested which used only a two-state particle which walks first only along the $x$ -axis followed by a step along the $y$ -axis [38].",
"This walk can results in the same probability distribution as the Grover walk and its evolution is given by $|\\Psi _1\\rangle = (\\hat{S}_{(0,y)}[\\hat{B}(\\theta ) \\otimes \\hat{{\\mathbb {1}}}])(\\hat{S}_{(x, 0)}[\\hat{B}(\\theta ) \\otimes \\hat{{\\mathbb {1}}}])|\\Psi _{\\rm in} \\rangle ,$ where $\\hat{S}_{(x, 0)} &\\equiv & \\sum _{x, y} \\big [ |0 \\rangle \\langle 0|\\otimes |\\psi _{x-1, y}\\rangle \\langle \\psi _{x, y}| \\nonumber \\\\&&\\;\\;\\;\\; + |1 \\rangle \\langle 1|\\otimes |\\psi _{x+1, y}\\rangle \\langle \\psi _{x, y}| \\big ] \\\\\\hat{S}_{(0, y)} &\\equiv & \\sum _{x, y} \\big [ |0 \\rangle \\langle 0|\\otimes |\\psi _{x, y-1}\\rangle \\langle \\psi _{x, y}| \\nonumber \\\\&&\\;\\;\\;\\; + |1 \\rangle \\langle 1|\\otimes |\\psi _{x, y+1}\\rangle \\langle \\psi _{x, y}| \\big ].$ Using a coin operation with $\\theta = \\pi /4$ , the state of the walk after $t$ steps can then be calculated as $|\\Psi (t)\\rangle &=& \\hat{W}(\\pi /4)^t |\\Psi _{\\rm in}\\rangle \\nonumber \\\\&=& \\sum _{x=-t}^t \\sum _{y=-t}^t \\left[ \\mathcal {A}_{(x, y),t}|0 \\rangle +\\mathcal {B}_{(x, y),t}|1 \\rangle \\right]\\otimes |\\psi _{(x,y)}\\rangle \\nonumber ,\\\\$ where $\\hat{W}(\\pi /4) = (\\hat{S}_{(0,y)} [\\hat{B}(\\pi /4) \\otimes \\hat{{\\mathbb {1}}}])(\\hat{S}_{(x, 0)} [\\hat{B}(\\pi /4) \\otimes \\hat{{\\mathbb {1}}}])$ , and $\\mathcal {A}_{(x, y),t}$ and $\\mathcal {B}_{(x, y),t}$ are given by the coupled iterative relations A(x, y),t = 12 [ A(x+1, y+1),t-1 + A(x-1, y+1),t-1 + B(x+1, y+1),t-1 - B(x-1, y+1),t-1 ] B(x, y),t = 12 [ A(x+1, y-1),t-1 - A(x-1, y-1),t-1 + B(x+1, y-1),t-1 + B(x-1, y-1),t-1 ].",
"The resulting probability distribution is then $P_{2s} = \\sum _{x =-t}^t \\sum _{y =-t}^t \\left[|\\mathcal {A}_{(x,y),t}|^2 + |\\mathcal {B}_{(x, y),t}|^2 \\right],$ which for the initial state $|\\Psi _{\\rm in}\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle + i |1\\rangle )\\otimes |\\psi _{0,0}\\rangle $ gives the same probability distribution as the four-state Grover walk (see Fig.",
"REF )."
],
[
"Pauli walk",
"A further scheme to implement a 2D quantum walk using only a two-state particle can be constructed using different Pauli basis states as translational states for the two axis.",
"For convenience we can choose the eigenstates of the Pauli operator $\\hat{\\sigma }_3= \\begin{bmatrix} 1 & ~~0\\\\ 0 & -1 \\end{bmatrix}$ , $|0\\rangle $ and $|1\\rangle $ as basis states for $x-$ axis and eigenstates of $\\hat{\\sigma }_1 = \\begin{bmatrix} 0 &1\\\\ 1 & 0 \\end{bmatrix}$ , $|+\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle +|1\\rangle )$ and $|-\\rangle = \\frac{1}{\\sqrt{2}}(|0\\rangle - |1\\rangle )$ as basis states for $y-$ axis [40], which also implies that $|0\\rangle = \\frac{1}{\\sqrt{2}}(|+\\rangle + |-\\rangle )$ and $|1\\rangle = \\frac{1}{\\sqrt{2}}(|+\\rangle - |-\\rangle )$ .",
"In this scheme a coin operation is not necessary and each step of the walk can be implemented by the operation $\\hat{S}_{\\sigma _3} \\equiv \\hat{S}_{(x, 0)}$ followed by the operation S1 x, y [ |+ +||x, y-1x, y| + |- -||x, y+1x, y|].",
"The state after $t$ steps of quantum walk is then given by $|\\Psi _t\\rangle &=&[\\hat{S}_{\\sigma _1}\\hat{S}_{\\sigma _3}]^t |\\Psi _{\\rm in}\\rangle \\nonumber \\\\&=& \\sum _{x =-t}^t \\sum _{y =-t}^t \\left[\\mathcal {A}_{(x, y),t}|0\\rangle +\\mathcal {B}_{(x, y),t}|1\\rangle \\right]\\otimes |\\psi _{x,y}\\rangle ,\\nonumber \\\\$ where $\\mathcal {A}_{(x, y),t}$ and $\\mathcal {B}_{(x,y),t}$ are given by the coupled iterative relations A(x, y),t = 12 [A(x+1, y+1),t-1 + B(x-1, y+1),t-1 + A(x+1, y-1),t-1 - B(x-1, y-1),t-1 ] B(x, y),t =12 [B(x-1, y+1),t-1 + A(x+1, y+1),t-1 + B(x-1, y-1),t-1 - A(x+1, y-1),t-1 ].",
"The probability distribution $P_{2s\\sigma } = \\sum _{x =-t}^t \\sum _{y =-t}^t\\left[|\\mathcal {A}_{(x, y),t}|^2 + |\\mathcal {B}_{(x, y),t}|^2 \\right]$ is again equivalent to the distribution obtained using the Grover walk and therefore also to the alternative walk for the initial state $|\\Psi _{\\rm in}\\rangle =\\frac{1}{\\sqrt{2}}(|0\\rangle + i |1\\rangle )\\otimes |\\psi _{0,0}\\rangle $ (see Fig.",
"REF ).",
"While the shift operator for the Grover walk is defined by a single operation, experimentally it has to be implemented as a two shift operations.",
"For example, to shift the state $|0\\rangle $ from $(x, y)$ to $(x-1, y-1)$ , it has first to be shifted along one axis followed by the other, very similarly to the way it is done in the two-state quantum walk schemes.",
"Therefore, a two-state quantum walk in 2D has many advantages over a four-state quantum walk.",
"The two-state walk using different Pauli basis states for the different axes has the further advantage of not requiring a coin operation at all, making the experimental task even simpler in physical systems where access to different Pauli basis states as translational states is available.",
"One can, of course, also consider including a coin operation in the Pauli walk, which would result in a different probability distributions [41].",
"For general initial states, quantum walks in 2D with a coin operation $\\in U(2)$ can result in a many non-localized probability distribution in position space.",
"This is a further difference to the Grover walk which is very specific with respect to the initial state of the four-state particle and the coin operation.",
"Quantifying non-classical correlations inherent in a certain state is currently one of the most actively studied topics in physics (see for example [45]).",
"While many of the suggested methods involve optimization, making them computationally hard, Luo [42] recently proposed a computable measure that avoids this complication: if one considers a bipartite state $\\rho $ living in the Hilbert space ${\\cal H}_A \\otimes {\\cal H}_B$ , one can define a reasonable measure of the total correlations between the systems $A$ and $B$ using the mutual information $I(\\rho ) = S(\\rho _A) + S(\\rho _B) - S(\\rho ),$ where $S(\\cdot )$ denotes von Neumann entropy and $\\rho _A$ and $\\rho _B$ are the respective reduced density matrices.",
"If $\\rho _A = \\sum _j p_A^j\\Pi _A^j$ and $\\rho _B = \\sum _j p_B^j\\Pi _B^j$ , then the measurement induced by the spectral components of the reduced states is $\\Pi (\\rho ) \\equiv \\sum _{j,k} \\Pi _A^j \\otimes \\Pi _B^k \\rho \\Pi _A^j \\otimes \\Pi _B^k.$ Given that $I[\\Pi (\\rho )]$ is a good measure of classical correlations in $\\rho $ , one may consider a measure for quantum correlations defined by the so-called Measurement Induced Disturbance (MID) [42] $Q(\\rho ) = I(\\rho ) - I[\\Pi (\\rho )].$ MID does not involve any optimization over local measurements and can be seen as a loose upper bound on quantum discord [46].",
"At the same time it is known to capture most of the detailed trends in the behaviour of quantum correlations during quantum walks [47].",
"Therefore, we will use MID ($Q(\\rho )$ ) in the following to quantify quantum correlations for the different 2D quantum walk evolutions.",
"Figure: (Color online) Quantum correlations Q(ρ pp )Q(\\rho _{pp}) and Q(ρ xy )Q(\\rho _{xy}) for the differentschemes in the absence of noise.Despite having the same probability distributions in the absence of noise, the MIDs for the four-state walk and the two-state walks differ.",
"In Fig.",
"REF (a) we show the MID between the particle and the position degree of freedom, $Q(\\rho _{pp})$ for all three walks and find that it is significantly higher for the Grover walk.",
"Where $\\rho _{pp}$ is $\\tilde{\\rho }_{4s}(t)$ for Grover walk, $\\tilde{\\rho }_{2s}(t)$ for alternate walk and $\\tilde{\\rho }_{2s\\sigma }(t)$ for the Pauli walk.",
"However, due to the difference in the size of the particles degree of freedom for the Grover and the two-state walks, a direct comparison of the quantum correlations $Q(\\rho _{pp})$ does not make sense.",
"Among the two-state schemes on the other hand, we see that the Pauli walk has a larger $Q(\\rho _{pp})$ in comparison to the alternate walk.",
"A fair comparison between all systems can be made by looking at the quantum correlations generated between the two spatial dimensions $x$ and $y$ , $Q(\\rho _{xy})$ (see Fig.",
"REF (b)).",
"$\\rho _{xy}$ is obtained by tracing out the particle degree of freedom from from complete density matrix [$\\tilde{\\rho }_{4s}(t)$ , $\\tilde{\\rho }_{2s}(t)$ and $\\tilde{\\rho }_{2s\\sigma }(t)$ ] comprising of the particle and the position space.",
"We find that $Q(\\rho _{xy})$ is identical for both two-state schemes and exceeding the Grover walk result.",
"This behaviour is similar to the one described in Refs.",
"[38], [39], where the entanglement created during the Grover walk was compared with the alternate walk using the negativity of the partial transpose, in its generalization for higher-dimensional systems [48], [49]."
],
[
"Decoherence",
"The effects of noise on 1D quantum walks has been widely studied [51], [52], [53], [50], [47] but the implications in 2D settings are less well known [37], [54], [55].",
"In particular no study has been done on either of the two-state schemes presented in the previous section and we therefore now compare their decoherence properties to the Grover walk, using a state-flip and a depolarizing channel as noise models.",
"We show that this leads to differing probability distributions and has an effect on the amounts of quantum correlations as well."
],
[
"Grover walk",
"For a two-state particle, state-flip noise simply induces a bit flip [$\\sigma _1 = \\begin{bmatrix} 0 & 1\\\\ 1 & 0 \\end{bmatrix}$ ] but for the Grover walk, the state-flip noise on the four basis states can change one state to 23 other possible permutations.",
"Therefore, the density matrix after $t$ steps in the presence of a state-flip noise channel can be written as $\\hat{\\rho }_{4s}(t) &=& \\frac{p}{k}\\left[\\sum _{i =1}^k\\hat{f}_i\\hat{S}_4\\hat{\\rho }_{4s}(t-1)\\hat{S}_4{^\\dagger } \\hat{f}_i{^\\dagger } \\right]\\nonumber \\\\& &+(1-p)[\\hat{S}_4\\hat{\\rho }_{4s}(t-1)\\hat{S}_4{^\\dagger }],$ where $p$ is the noise level, $S_4= \\hat{S}_{(x,y)} [\\hat{G} \\otimes \\hat{{\\mathbb {1}}}]$ , and the $\\hat{f}_{i}$ are the state-flip operations.",
"Figure: (Color online) Probability distribution ofthe Grover walk when subjected to a different state-flip noise level withk=23k=23 after 15 steps.",
"(a) and (b) are for noise levels, p=0.1p=0.1 and p=0.9p=0.9, respectively.Figure: (Color online) Quantum correlations created by the Grover walk in the presence of a noise channel including all possible state flips (k=23k=23).For a noisy channel with all the 23 possible flips one has $k=23$ and in Figs.",
"REF and REF we show the probability distribution of the Grover for weak ($p=0.1$ ) and strong ($p=0.9$ ) noise levels after 15 step.",
"Compared to the distribution in the absence of noise (see Fig.",
"REF ) a progressive reduction in the quantum spread is clearly visible.",
"Note that for $p=1$ the walk corresponds to a fully classical evolution.",
"In Fig.",
"REF (a) we show the quantum correlations between the particle state with the position space, $Q(\\rho _{pp})$ , as a function of number of steps $t$ .",
"With increasing noise level, a decrease in $Q(\\rho _{pp})$ is seen, whereas for the quantum correlations between the $x$ and $y$ spatial dimensions, $Q(\\rho _{xy})$ , the same amount of noise mainly leads to a decrease in the positive slope (see Fig.",
"REF (b))."
],
[
"Two-state walks",
"The evolution of each step of the two-state quantum walk comprises of a move along one axis followed a move along the other.",
"Therefore, the walk can be subjected to a noise channel after evolution along each axis or after each full step of the walk.",
"In the first case, the noise level $p^{\\prime }=\\frac{p}{2}$ is applied two times during each step in oder to be equivalent to the application of a noise of strength $p$ in the second case.",
"For the alternate walk the evolution of the density matrix with a bit-flip noise channel applied after evolution along each axis is then given by 2s(t) = p2 [ 1 Sx 2s(t-1) Sx 1 ] + (1-p2)[Sx 2s(t-1) Sx ] 2s(t) = p2 [ 1Sy 2s(t) Sy1 ] + (1-p2 )[Sy 2s(t) Sy ], where $\\hat{\\sigma }_1 = \\begin{bmatrix}0 & 1 \\\\1 & 0\\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}}$ , $\\hat{S}_y = \\hat{S}_{(0,y)} [\\hat{B}(\\theta ) \\otimes \\hat{{\\mathbb {1}}}]$ , and $\\hat{S}_x=\\hat{S}_{(x, 0)} [\\hat{B}(\\theta ) \\otimes \\hat{{\\mathbb {1}}}]$ .",
"Similarly, the density matrix with a bit-flip noise applied after the evolution along each axis for the Pauli walk is given by 2s(t) = p2 [1 S3 2s(t-1) S3 1 ] + (1-p2)[S3 2s(t-1) S3 ] 2s(t) = p2 [ 1 S1 2s(t) S11 ] + (1-p2 )[S1 2s(t) S1 ].",
"Figure: Pauli walkFigure: Pauli WalkFigure: Pauli walkIn Figs.",
"REF and REF the probability distributions for the alternate walk after 25 steps of noisy evolution with $p=0.1$ and $p=0.9$ are shown and Figs.",
"REF and REF show the same for the Pauli walk.",
"It can be seen that the bit-flip noise channel acts symmetrically on both axes for the alternate walk, but asymmetrically on the Pauli walk.",
"This is due to the fact that the bit-flip noise applied along the axis in which the $\\sigma _1$ Pauli basis is used leaves the state unchanged.",
"A completely classical evolution is recovered for $p=1$ ($p^{\\prime }=0.5$ ) and an evolution with $p=2$ is equivalent to one with $p=0$ .",
"Evolving the density matrix and calculating the MID for a noiseless evolution ($p=0$ ), one can see from Fig.",
"REF that the initial difference in $Q(\\rho _{pp})$ between the Pauli walk and the alternate walk decreases during the evolution and eventually both values settle around 1.5.",
"For a noisy evolution, however, the initial difference in $Q(\\rho _{pp})$ does not decrease over time and we find a higher value for the Pauli walk compared to the alternate walk.",
"Similarly, careful examination of Fig.",
"REF shows that the $Q(\\rho _{xy})$ for the alternate walk and the Pauli walk are identical in the absence of noise, but differ for noisy evolution, with the alternate walk being affected stronger than the Pauli walk.",
"The density matrix for the second case, that is, with a noisy channel applied only once after one full step of walk evolution, for both two-state walks is given by 2s(t) = p [ 1Q(t-1)1 ] + (1-p) Q(t-1) 2s(t) = p [ 1 Q(t-1) 1 ] + (1-p) Q(t-1) where Q(t-1) = Sy Sx 2s(t-1) Sx Sy Q(t-1) = S1 S3 2s(t-1) S3 S1.",
"In this case maximum decoherence and a completely classical evolution is obtained for $p=0.5$ and the evolution with $p=1$ is equivalent to the one with $p=0$ .",
"The obtained probability distributions are almost identical for both walks and differ only slightly from the ones obtained for the alternate walk with noise applied after evolution along each axis (see Fig.",
"REF and REF ).",
"The correlation functions $Q(\\rho _{pp})$ and $Q(\\rho _{xy})$ behave very similar for both walks (see Figs.",
"REF and REF ) and we can conclude that the presence of bit-flip noise on both two-state walks when applied after a full step leads to equally strong decoherence.",
"Figure: Pauli walkFigure: Pauli walkTo describe depolarizing noise we use the standard model in which the density matrix of our two-state system is replaced by a linear combination of a completely mixed and an unchanged state, $\\hat{\\rho } = \\frac{p}{3}\\left(\\hat{\\sigma }_{1}\\hat{\\rho } \\hat{\\sigma }_{1} +\\hat{\\sigma }_{2}\\hat{\\rho } \\hat{\\sigma }_{2}+\\hat{\\sigma }_{3}\\hat{\\rho } \\hat{\\sigma }_{3} \\right) +(1-p)\\hat{\\rho },$ where $\\hat{\\sigma }_1$ , $\\hat{\\sigma }_2$ and $\\hat{\\sigma }_3$ are the standard Pauli operators.",
"To be able to compare the effects of the depolarizing channel on the Grover walk and the two-state walks we will apply the noise only once after each full step."
],
[
"Grover walk",
"For the four-state particle the depolarizing noise channel comprises of all possible state flips, phase flips and their combinations.",
"State-flip noise alone leads to 23 possible changes in the four-state system and adding the phase-flip noise and all combinations of these two is unfortunately a task beyond current computational ability.",
"Therefore let us first briefly investigate the possibility of approximating the state flip noise by restricting ourselves to only a subset of flips.",
"One example would be a noisy channel with only 6 possible flips ($k=6$ ) between two of the four basis states $\\hat{f}_1 = \\begin{bmatrix} 0 & 1 & 0& 0 \\\\ 1 & 0 & 0 & 0\\\\ 0 & 0 & 1 & 0\\\\ 0 & 0 & 0 & 1 \\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}} ~~;~~\\hat{f}_2 = \\begin{bmatrix} 0 & 0 & 1 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 1 & 0 & 0 & 0\\\\ 0 & 0 & 0 & 1 \\end{bmatrix} \\otimes \\hat{\\mathbb {1}}\\nonumber \\\\\\hat{f}_3 = \\begin{bmatrix} 0 & 0 & 0 & 1 \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 1& 0\\\\ 1 & 0 & 0 & 0 \\end{bmatrix} \\otimes \\hat{\\mathbb {1}} ~~;~~\\hat{f}_4 = \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 1 & 0\\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 0 & 1 \\end{bmatrix} \\otimes \\hat{\\mathbb {1}} \\nonumber \\\\\\hat{f}_5 = \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 0 & 0 & 1\\\\ 0 & 0 & 1& 0\\\\ 0 & 1 & 0 & 0 \\end{bmatrix} \\otimes \\hat{\\mathbb {1}} ~~;~~\\hat{f}_6 = \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0\\\\ 0 & 0 & 0& 1\\\\ 0 & 0 & 1 & 0 \\end{bmatrix} \\otimes \\hat{\\mathbb {1}}$ and another a channel where only cyclic flips ($k=3$ ) of all the basis states can appear $\\hat{f}_1 = \\hat{f} \\otimes \\hat{{\\mathbb {1}}} ~~;~~ \\hat{f}_2 = \\hat{f}^2\\otimes \\hat{{\\mathbb {1}}} ~~;~~\\hat{f}_3 = \\hat{f}^3\\otimes \\hat{{\\mathbb {1}}}$ with $\\hat{f}= \\begin{bmatrix} 0 & 1 & 0 & 0 \\\\ 0 & 0 & 1 & 0\\\\ 0 & 0& 0 & 1\\\\ 1 & 0 & 0 & 0 \\end{bmatrix}$ .",
"The probability distributions for these two approximations are visually very similar to the situation where all possible state-flips are taken into account ($k=23$ ) and in Fig.",
"REF we compare the results obtained for the the $x-y$ spatial quantum correlations for a noise level of $p=0.1$ .",
"Figure: (Color online) Quantum correlations between the xx and yy spatial dimensions for the Grover walk in the presence of a state-flip noise channel with p=0.1p=0.1.",
"The noise is modelled as state-flips including all possible flips (k=23k=23), flips between only two of the basis states (k=6k=6) and cyclic flips of all the four basis state (k=3k=3)One can see that the spatial quantum correlations are affected stronger by the $k=3$ than by the full $k=23$ flip noise.",
"This implies that the two- and three state flips included in $k=23$ acts as reversals of cyclic flips, thereby reducing the effect of noise.",
"Since the trends for the decrease of the quantum correlation are functionally similar for $k=3, 6$ and 23, we will use the model with $k=3$ cyclic flips as the state-flip noise channel for the Grover walk in this section.",
"Similarly, taking into account the computational limitations and following the model adopted for the state-flip, we will use a cyclic phase-flips to model the phase-flip noise, $\\hat{r}_1 = \\hat{r} \\otimes \\hat{\\mathbb {1}}~~;~~ \\hat{r}_2 =\\hat{r}^2\\otimes \\hat{\\mathbb {1}} ~~;~~ \\hat{r}_3 = \\hat{r}^{3}\\otimes \\hat{\\mathbb {1}},$ where $\\hat{r} = \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & \\omega & 0 & 0\\\\ 0 & 0 & \\omega ^2 & 0 \\\\ 0 & 0 & 0 & \\omega ^3 \\end{bmatrix}$ with $\\omega = e^{\\frac{2\\pi i}{4}}$ .",
"Figure: (Color online) Probability distribution of (a) the Groverwalk and (b) the two-state walks when subjected to a depolarizing channel.",
"For the Grover walk the channel is given by Eq.",
"() and for all walks the noise level is p=0.1p=0.1.",
"The distribution is shown after 15 steps of evolution.Both these approximations for state-flip and phase-flip noise will makes the complex depolarization noise manageable for numerically treatment.",
"The density matrix of the Grover walk can then be written as $\\hat{\\rho }_{4s}(t) &=& \\frac{p}{15} \\left[ \\sum _{i =1}^{3} \\hat{f}_i \\hat{S}_4 \\hat{\\rho }_{4s}(t-1) \\hat{S}_4{^\\dagger } \\hat{f}_i{^\\dagger } \\right] \\nonumber \\\\&+& \\frac{p}{15} \\left[ \\sum _{j=1}^{3} \\hat{r}_j \\hat{S}_4 \\hat{\\rho }_{4s}(t-1) \\hat{S}_4{^\\dagger } \\hat{r}_j{^\\dagger } \\right]\\nonumber \\\\&+& \\frac{p}{15} \\left[ \\sum _{i=1}^{3} \\sum _{j=1}^{3} \\hat{r}_j \\hat{f}_i \\hat{S}_4 \\hat{\\rho }_{4s}(t-1) \\hat{S}_4{^\\dagger } \\hat{f}_i^{\\dagger } \\hat{r}_j{^\\dagger } \\right]\\nonumber \\\\& &+ (1-p)[\\hat{S}_4 \\hat{\\rho }_{4s}(t-1) \\hat{S}_4{^\\dagger }]$ and the probability distribution for this walk is shown in Fig.",
"REF for $p=0.1$ .",
"The quantum correlations $Q(\\rho _{pp})$ and $Q(\\rho _{xy})$ are shown in Fig.",
"REF .",
"With increasing noise level, a decrease in $Q(\\rho _{pp})$ is seen, whereas for the quantum correlations between the $x$ and $y$ spatial dimensions, $Q(\\rho _{xy})$ , the same amount of noise mainly leads to a decrease in the positive slope (see Fig.",
"REF ).",
"From this we can conclude that the general trend in the quantum correlaltion due to state-flip noise (Fig.",
"REF ) and depolarizing noise is the same but the effect is slightly stronger when including the depolarizing channel.",
"Figure: Grover walk"
],
[
"Two-state walks",
"The depolarizing channels for the alternate walk and Pauli walk can be written as 2s(t) = p3 [i=13 i Q(t-1) i ] + (1-p) Q(t-1) 2s(t) = p3 [i=13 i Q(t-1) i ] + (1-p) Q(t-1), where $\\hat{\\sigma }_1 = \\begin{bmatrix}0 & 1 \\\\1 & 0\\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}}$ , $\\hat{\\sigma }_2 = \\begin{bmatrix}0 & -i \\\\i & ~~0\\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}}$ , $\\hat{\\sigma }_3 = \\begin{bmatrix}1 & ~~0 \\\\0 & -1\\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}}$ and $\\hat{Q}(t-1)$ and $\\hat{Q}_{\\sigma }(t-1)$ are given by Eqs.",
"(REF ) and (REF ).",
"Similarly to the situation where we considered only state-flip noise after one complete step (see Figs.",
"REF and REF ) we find again that the quantum correlations for both walks behave nearly identical and only differ slightly in strength compared to the case of state-flip noise alone (see Figs.",
"REF and REF ).",
"Figure: Pauli walkFigure: Pauli walk"
],
[
"Robustness of two-state walk",
"From the preceding sections we note that the $x-y$ spatial correlations, $Q(\\rho _{xy})$ , have a larger absolute value for the two-state walks compared to the Grover walk and that the presence of noise affects all schemes in a similar manner.",
"To quantify and better illustrate the effect the noise has we therefore calculate $R(\\rho _{xy}) = \\frac{ Q(\\rho _{xy})\\quad \\text{for noisy walk}}{Q(\\rho _{xy}) \\quad \\text{for noiseless walk}},$ as a function of number of steps, which gives the rate of decrease in the quantum correlations.",
"In Figs.",
"REF and REF we show this quantity in the presence of state-flip or depolarizing noise, respectively, for a noise level of $p=0.2$ .",
"One can clearly see that in both cases the two-state walks are more robust against the noise at any point during the evolution.",
"Note that the Grover walk only produces correlations from step 2 on, which is the reason for its graph starting later.",
"Figure: (Color online) Relative decay of the quantum correlations Q(ρ xy )Q(\\rho _{xy}) in the presence of a state-flip noise channel with p=0.2p=0.2.Figure: (Color online) Relative decay of the quantum correlations Q(ρ xy )Q(\\rho _{xy}) in the presence of a depolarizing noise channel with p=0.2p=0.2."
],
[
"Breakdown of state-flip and phase-flip symmetries for\nfour-state walks",
"The quantum walk of a two-state particle in 1D is known to remain unaltered in the presence of unitary operations which equally effect each step of the evolution.",
"This is due to the existence of symmetries [52], [53], which can help to identify different variants of the same quantum walk protocol and which can be useful in designing experimental implementation.",
"For example, in a recent scheme used to implement a one-dimensional quantum walk using atoms in an optical lattice [56], the conditional shift operator also flipped the state of the atom with every shift in position space.",
"However, the existence of a bit-flip symmetry in the system allowed to implement the walk without the need for compensation of these bit-flips.",
"In this section we look at possible symmetries in the walks discussed above and show that the bit-flip and phase-flip symmetries, which are present in the evolution of the two-state particle are absent in the evolution of the four-state particle.",
"The density matrix for a two-state quantum walk in the presence of a noisy channel will evolve through a linear combination of noisy operations on the state and the unaffected state itself.",
"As an example we illustrate the symmetry due to bit-flip operations in the alternate walk with bit-flip noise after evolution of one complete step.",
"The density matrix in this case is given by $\\hat{\\rho }_{2s}(t) &=& p \\Big [ \\hat{\\sigma }_1 \\hat{S}_y \\hat{S}_x \\hat{\\rho }_{2s}(t-1) \\hat{S}_x^{\\dagger } \\hat{S}_y^{\\dagger } \\hat{\\sigma }^{\\dagger }_1 \\Big ] \\nonumber \\\\& & +(1-p) \\Big [ \\hat{S}_y \\hat{S}_x \\hat{\\rho }_{2s} (t-1) \\hat{S}^{\\dagger }_{x}\\hat{S}^{\\dagger }_{y} \\Big ],$ where $\\hat{\\sigma _1} = \\begin{bmatrix}0 & 1 \\\\1 & 0\\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}}$ .",
"When the noise level is $p=1$ this expression reduces to $\\hat{\\rho }_{2s}(t) &=& \\Big [ \\hat{\\sigma }_1 \\hat{S}_y \\hat{S}_x \\hat{\\rho }_{2s}(t-1) \\hat{S}_x^{\\dagger } \\hat{S}_y^{\\dagger } \\hat{\\sigma }^{\\dagger }_1 \\Big ] \\nonumber \\\\&=&\\hat{S}^{\\prime }_y \\hat{S}_x \\hat{\\rho }_{2s}(t-1) \\hat{S}_x^{\\dagger } (\\hat{S}_y^{\\prime })^{\\dagger }.$ where in the second line the bit-flip operation has been absorbed into the evolution operator $\\hat{S}^{\\prime }_{y}$ .",
"This replaces $|0\\rangle \\langle 0|$ and $|1\\rangle \\langle 1|$ in $\\hat{S}_{y}$ by $|1\\rangle \\langle 0|$ and $|0\\rangle \\langle 1|$ , respectively.",
"Similarly, for a phase-flip, $\\hat{\\sigma _1}$ in Eq.",
"(REF ) is replaced by $\\hat{\\sigma _3} = \\begin{bmatrix}1 & 0 \\\\0 & -1\\end{bmatrix}\\otimes \\hat{{\\mathbb {1}}}$ leading to $|1\\rangle \\langle 1|$ in $\\hat{S}_{y}$ being replaced by $-|1\\rangle \\langle 1|$ to construct $\\hat{S}^{\\prime }_{y}$ .",
"An alternative way to look at this is to absorb the bit-flip or phase-flip operation into the coin operation.",
"For a two-state walk using the Hadamard coin operation, the bit-flip operation after each steps corresponds to the coin operation taking the form, $\\hat{H}^{\\prime } =\\frac{1}{\\sqrt{2}}\\begin{bmatrix}1 & &-1 \\\\1 && ~~1\\end{bmatrix}$ and the phase-flip after each steps corresponds to the coin operation taking the form, $\\hat{H}^{\\prime \\prime } = \\frac{1}{\\sqrt{2}}\\begin{bmatrix}~~1 & & 1 \\\\-1 && 1\\end{bmatrix}.$ For a bit-flip or phase-flip of noise level $p=1$ the Hadamard coin operation $\\hat{H}$ can therefore be recast into a noiseless ($p=0$ ) quantum walk evolution using $\\hat{H}^{\\prime }$ and $\\hat{H}^{\\prime \\prime }$ as coin operation.",
"A noise level of $p=1$ then returns a probability distribution equivalent to the noiseless evolution and consequently the maximum bit-flip and phase-flip noise level for a two state walk corresponds to $p=0.5$ .",
"This is a symmetry within the alternate walk, which also holds for the Pauli walk.",
"A state-flip noise channel for the four-state walk, on the other hand, evolves the state into a linear combination of all possible flips between the four basis states and an unchanged state for all values of $p$ except for $p=0$ (see Eq.",
"REF ).",
"That is, only when $p=0$ , Eq.",
"(REF ) reduces to $\\hat{\\rho }_{4s}(t) =\\hat{S}_4\\hat{\\rho }_{4s}(t-1)\\hat{S}_4{^\\dagger },$ whereas for any non-zero $p$ including $p=1$ , Eq.",
"(REF ) takes the form $\\hat{\\rho }_{4s}(t) &=& \\frac{1}{k}\\left[\\sum _{i =1}^k\\hat{f}_i\\hat{S}_4\\hat{\\rho }_{4s}(t-1)\\hat{S}_4{^\\dagger } \\hat{f}_i{^\\dagger } \\right],$ Any attempt to absorb the noise operations $\\hat{f}_i$ into the shift operator or the coin operation leads to $k$ different results, which have to be applied with probability $\\frac{1}{k}$ .",
"Therefore, in contrast to the two-state evolution, a state flip noise level of $p=1$ does not result in a pure evolution equivalent to the situation for $p=0$ .",
"However, if the state-flip noise is restricted to one possible operation ($\\hat{f}_i$ ), the density matrix is no longer a linear combination of noisy operations and the unchanged state.",
"Thus, for $p=1$ in Eq.",
"(REF ) a single noise operation can be absorbed into the GDO by changing the form of $G$ (see Eq.",
"(REF )).",
"This absence of a useful symmetry for the four-state quantum walk reduces the chances to find an equivalent class of four-state quantum walk evolutions.",
"Furthermore, since the four-state quantum walk requires a specific form of coin operation to implement the walk, any possible absorption will not result in a quantum walk in 2D, which is a significant difference to the two-state walks."
],
[
"Conclusion",
"In this work we have studied the decoherence properties of three different schemes that realize a quantum walk in two-dimensions, namely the Grover (four-state) walk, the alternate walk and the Pauli walk.",
"The noise for two-state particle evolution was modeled using a bit-flip channel and depolarizing channel.",
"For the four-state evolution, different possible state-flip channels were explored and we have shown a channel with 3 cyclic flips between all the four states can be used as a very good approximation to the full situation.",
"Similarly, we presented a possible model for the depolarizing channel of the four-state quantum walk.",
"Using MID as a measure for the quantum correlations within the state, our studies have shown that the two-state quantum walk evolution is in general more robust against decoherence from state-flip and depolarizing noise channels.",
"Following earlier studies on bit- and phase-flip symmetries in two-state quantum walks in 1D, we have shown that they also hold for two-state quantum walks in 2D, but break down for four-state 2D quantum walks.",
"With the larger robustness against decoherence, the existence of symmetries which allow freedom of choice with respect to the initial state and the coin operation and the much easier experimental control, we conclude that two-state particles can be conveniently used to implement quantum walks in 2D compared to schemes using higher dimensional coins.",
"An other important point to be noted is the straightforward extendability of the both two-state schemes to higher dimensions by successively carrying out the evolution in each dimension.",
"Acknowledgement: We acknowledge support from Science Foundation Ireland under Grant No.",
"10/IN.1/I2979.",
"We would like to thank Carlo Di Franco and Gianluca Giorgi for helpful discussions."
]
] | 1204.1287 |
[
[
"On a supercongruence conjecture of Rodriguez-Villegas"
],
[
"Abstract In examining the relationship between the number of points over $\\mathbb{F}_p$ on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, Rodriguez-Villegas identified numerically 22 possible supercongruences.",
"We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the $p$-th Fourier coefficient of a modular form."
],
[
"Introduction and Statement of Results",
"This work was supported by the UCD Ad Astra Research Scholarship program.",
"Let $\\mathbb {F}_{p}$ denote the finite field with $p$ , a prime, elements.",
"In [21] Rodriguez-Villegas examined the relationship between the number of points over $\\mathbb {F}_p$ on certain Calabi-Yau manifolds and truncated generalized hypergeometric series which correspond to a particular period of the manifold.",
"In doing so, he identified numerically 22 possible supercongruences which can be categorized by the dimension, $D$ , of the manifold as outlined below.",
"We first define the truncated generalized hypergeometric series.",
"For a complex number $a$ and a non-negative integer $n$ let ${\\left({a}\\right)}_{n}$ denote the rising factorial defined by ${\\left({a}\\right)}_{0}:=1 \\quad \\textup {and} \\quad {\\left({a}\\right)}_{n} := a(a+1)(a+2)\\cdots (a+n-1) \\textup { for } n>0.$ Then, for complex numbers $a_i$ , $b_j$ and $z$ , with none of the $b_j$ being negative integers or zero, we define the truncated generalized hypergeometric series ${{_rF_s} \\left[ \\begin{array}{ccccc} a_1, & a_2, & a_3, & \\cdots , & a_r \\vspace{3.61371pt}\\\\\\phantom{a_1} & b_1, & b_2, & \\cdots , & b_s \\end{array}\\Big | \\; z \\right]}_{m}:=\\sum ^{m}_{n=0}\\frac{{\\left({a_1}\\right)}_{n} {\\left({a_2}\\right)}_{n} {\\left({a_3}\\right)}_{n} \\cdots {\\left({a_r}\\right)}_{n}}{{\\left({b_1}\\right)}_{n} {\\left({b_2}\\right)}_{n} \\cdots {\\left({b_s}\\right)}_{n}}\\; \\frac{z^n}{{n!",
"}}.$ We also let $\\phi (\\cdot )$ denote Euler's totient function and $\\left(\\frac{\\cdot }{p}\\right)$ the Legendre symbol modulo $p$ .",
"For $D=1$ , associated to certain elliptic curves, 4 supercongruences were identified.",
"They were all of the form ${_{2}F_1} \\Biggl [ \\begin{array}{cc} \\frac{1}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv \\left(\\frac{-t}{p}\\right)\\pmod {p^2},$ where $\\phi (d) \\le 2$ , $1\\le t \\le 3$ and $p$ is a prime not dividing $d$ .",
"These cases have been proven by Mortenson [19], [20].",
"For $D=2$ another 4 supercongruences were identified which relate to certain modular K3 surfaces.",
"These were all of the form ${_{3}F_2} \\Biggl [ \\begin{array}{ccc} \\frac{1}{2}, & \\frac{1}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d}} & 1, &1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv a(p)\\pmod {p^2},$ where $\\phi (d) \\le 2$ , $p$ is a prime not dividing $d$ and $a(p)$ is the $p$ -th Fourier coefficient of a weight three modular form on a congruence subgroup of $SL(2,\\mathbb {Z})$ .",
"The case when $d=2$ was originally conjectured by Beukers and Stienstra [5] and was first proven by Van Hamme [23].",
"Subsequently, proofs were also provided by Ishikawa [10] and Ahlgren [1].",
"The other $D=2$ cases are dealt with by Mortenson [18] where they have been proven for $p\\equiv 1 \\pmod {d}$ and up to sign otherwise.",
"The remaining 14 supercongruence conjectures relate to Calabi-Yau threefolds (i.e.",
"$D=3$ ).",
"The threefolds in question are complete intersections of hypersurfaces, of which 13 are discussed by Batyrev and van Straten in [3].",
"The supercongruences can be expressed as either ${_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d_1}, & 1-\\frac{1}{d_1}, & \\frac{1}{d_2}, & 1-\\frac{1}{d_2}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv b(p)\\pmod {p^3},$ where $\\phi (d_i) \\le 2$ and $p$ is a prime not dividing $d_i$ , or ${_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d}, & \\frac{r}{d}, & 1-\\frac{r}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv b(p)\\pmod {p^3},$ where $\\phi (d) =4$ , $1<r<d-1$ with $\\gcd (r,d)=1$ , $p$ is a prime not dividing $d$ and $b(p)$ is the $p$ -th Fourier coefficient of a weight four modular form on a congruence subgroup of $SL(2,\\mathbb {Z})$ .",
"Only one of these cases has been proven (Kilbourn [11]).",
"It is of the first type with $d_1=d_2=2$ and is an extension of the Apéry number supercongruence [2].",
"Let $f(z):= f_1(z)+5f_2(z)+20f_3(z)+25f_4(z)+25f_5(z)=\\sum _{n=1}^{\\infty } c(n) q^n$ where $f_i(z):=\\eta ^{5-i}(z) \\hspace{2.0pt} \\eta ^4(5z) \\hspace{2.0pt} \\eta ^{i-1}(25z)$ , $\\eta (z):=q^{\\frac{1}{24}} \\prod _{n=1}^{\\infty }(1-q^n)$ is the Dedekind eta function and $q:=e^{2 \\pi i z}$ .",
"Then $f$ is a cusp form of weight four on the congruence subgroup $\\Gamma _0(25)$ .",
"We now list one of the outstanding conjectures of type (REF ).",
"Conjecture 1.1 (Rodriguez-Villegas [21]) If $p \\ne 5$ is prime and $c(p)$ is as defined in (REF ), then ${_4F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{5}, & \\frac{2}{5}, & \\frac{3}{5}, & \\frac{4}{5} \\\\\\phantom{\\frac{1}{5},} & 1, & 1, & 1\\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\equiv c(p) \\pmod {p^3}.$ The main result of this paper is the following theorem.",
"Theorem 1.2 Conjecture REF is true.",
"The main approach for proving these types of supercongruences has been to use congruence relations between truncated generalized hypergeometric series and Greene's hypergeometric functions over finite fields [1], [2], [11], [16], [18], [19], [20].",
"However, as noted in [15], many results using this approach are restricted to primes in certain congruence classes (e.g.",
"$p\\equiv 1 \\pmod {d}$ in some of the $D=2$ cases described above).",
"In [15], this author extends Greene's hypergeometric functions to the $p$ -adic setting and establishes congruences between this new function and certain truncated generalized hypergeometric series.",
"These congruences cover all 22 hypergeometric series outlined above and are valid for all primes required in each of these cases, thus providing a framework for proving all 22 cases.",
"The proof of Theorem REF relies on one of these congruences along with counting the number of rational points on a modular Calabi-Yau threefold over $\\mathbb {F}_p$ .",
"Specifically, let $p$ be an odd prime and let $n \\in \\mathbb {Z}^{+}$ .",
"For $1 \\le i \\le n+1$ , let $\\frac{m_i}{d_i} \\in \\mathbb {Q} \\cap \\mathbb {Z}_p$ such that $0<\\frac{m_i}{d_i}<1$ .",
"Let $\\Gamma _p{\\left({\\cdot }\\right)}$ denote Morita's $p$ -adic gamma function, $\\left\\lfloor x \\right\\rfloor $ denote the greatest integer less than or equal to $x$ and $\\langle x \\rangle $ denote the fractional part of $x$ , i.e.",
"$x- \\left\\lfloor x \\right\\rfloor $ .",
"Then define ${_{n+1}G} \\left( \\tfrac{m_1}{d_1}, \\tfrac{m_2}{d_2}, \\cdots , \\tfrac{m_{n+1}}{d_{n+1}} \\right)_p\\\\:= \\frac{-1}{p-1} \\sum _{j=0}^{p-2}{\\left((-1)^j\\Gamma _p{\\bigl ({\\tfrac{j}{p-1}}\\bigr )}\\right)}^{n+1}\\prod _{i=1}^{n+1} \\frac{\\Gamma _p{\\bigl ({\\langle \\frac{m_i}{d_i}-\\frac{j}{p-1}\\rangle }\\bigr )}}{\\Gamma _p{\\bigl ({\\frac{m_i}{d_i}}\\bigr )}}(-p)^{-\\lfloor {\\frac{m_i}{d_i}-\\frac{j}{p-1}\\rfloor }}.$ An example of one the supercongruence results from [15] is the following theorem.",
"Theorem 1.3 ([15] Thm.",
"2.7) Let $r, d \\in \\mathbb {Z}$ such that $2 \\le r \\le d-2$ and $\\gcd (r,d)=1$ .",
"Let $p$ be an odd prime such that $p\\equiv \\pm 1 \\pmod {d}$ or $p\\equiv \\pm r \\pmod {d}$ with $r^2 \\equiv \\pm 1 \\pmod {d}$ .",
"If $s(p) := \\Gamma _p{\\left({\\tfrac{1}{d}}\\right)} \\Gamma _p{\\left({\\tfrac{r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-r}{d}}\\right)}\\Gamma _p{\\left({\\tfrac{d-1}{d}}\\right)}$ , then ${_{4}G} \\Bigl (\\tfrac{1}{d} , \\tfrac{r}{d}, 1-\\tfrac{r}{d} , 1-\\tfrac{1}{d}\\Bigr )_p\\\\\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{d}, & \\frac{r}{d}, & 1-\\frac{r}{d}, & 1-\\frac{1}{d}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}+s(p)\\hspace{1.0pt} p\\pmod {p^3}.$ Taking $d=5$ in Theorem REF yields ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p-s(p)\\hspace{1.0pt} p\\equiv {_{4}F_3} \\Biggl [ \\begin{array}{cccc} \\frac{1}{5}, & \\frac{2}{5}, & \\frac{3}{5}, & \\frac{4}{5}\\vspace{3.61371pt}\\\\\\phantom{\\frac{1}{d_1}} & 1, & 1, & 1 \\end{array}\\bigg | \\; 1 \\Biggr ]_{p-1}\\pmod {p^3}.$ Therefore Theorem REF will be established (after checking the case when $p=2$ ) on proving the following theorem.",
"Theorem 1.4 If $p \\ne 5$ is an odd prime, $s(p):= \\Gamma _p{\\left({\\frac{1}{5}}\\right)} \\Gamma _p{\\left({\\frac{2}{5}}\\right)} \\Gamma _p{\\left({\\frac{3}{5}}\\right)} \\Gamma _p{\\left({\\frac{4}{5}}\\right)}$ and $c(p)$ is as defined in (REF ), then ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p-s(p)\\hspace{1.0pt} p&=c(p).$ As mentioned above, ${_{n+1}G}$ extends Greene's hypergeometric function over finite fields, which was introduced in [7].",
"Let $\\widehat{\\mathbb {F}_p^{*}}$ denote the group of multiplicative characters of $\\mathbb {F}_p^{*}$ .",
"We extend the domain of $\\chi \\in \\widehat{\\mathbb {F}_p^{*}}$ to $\\mathbb {F}_{p}$ by defining $\\chi (0):=0$ (including the trivial character $\\varepsilon $ ) and denote $\\overline{B}$ as the inverse of $B$ .",
"For $A$ , $B \\in \\widehat{\\mathbb {F}_p^{*}}$ , define $\\binom{A}{B} :=\\frac{B(-1)}{p} \\sum _{x \\in \\mathbb {F}_{p}} A(x) \\overline{B}(1-x).$ Then for $A_0,A_1,\\cdots , A_n, B_1, B_2, \\cdots , B_n \\in \\widehat{\\mathbb {F}_p^{*}}$ and $x \\in \\mathbb {F}_{p}$ , define the Gaussian hypergeometric series by ${_{n+1}F_n} {\\Biggl ( \\begin{array}{cccc} A_0, & A_1, & \\cdots , & A_n \\\\\\phantom{A_0} & B_1, & \\cdots , & B_n \\end{array}\\bigg | \\; x \\Biggr )}_{p}:= \\frac{p}{p-1} \\sum _{\\chi } \\binom{A_0 \\chi }{\\chi } \\prod _{i=1}^{n} \\binom{A_i \\chi }{B_i \\chi }\\, \\chi (x).$ We recover the Gaussian hypergeometric series from $_{n+1}G$ via the following result.",
"Proposition 1.5 ([15] Prop.",
"2.2) Let $n \\in \\mathbb {Z}^{+}$ and, for $1 \\le i \\le n+1$ , let $\\frac{m_i}{d_i} \\in \\mathbb {Q}$ such that $0<\\frac{m_i}{d_i}<1$ .",
"Let $p \\equiv 1 \\pmod {d_i}$ be prime and let $\\rho _i$ be the character of order $d_i$ of $\\mathbb {F}_p^*$ given by $\\overline{\\omega }^{\\frac{p-1}{d_i}}$ , where $\\omega $ is the Teichmüller character.",
"Then ${_{n+1}G} \\left( \\tfrac{m_1}{d_1}, \\tfrac{m_2}{d_2}, \\cdots , \\tfrac{m_{n+1}}{d_{n+1}} \\right)_p=(-p)^n \\, {_{n+1}F_n} {\\Bigg ( \\begin{array}{cccc} \\rho _1^{m_1}, & \\rho _2^{m_2}, & \\cdots , & \\rho _{n+1}^{m_{n+1}} \\\\\\phantom{\\rho _1^{m_1}} & \\varepsilon , & \\cdots , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )}_{p}.$ Using Proposition REF it is easy to see the following corollary to Theorem REF .",
"Corollary 1.6 If $p \\equiv 1 \\pmod {5}$ is prime and $c(p)$ is as defined in (REF ), then $-p^3 \\; _4F_3 \\Biggl ( \\begin{array}{cccc} \\chi _5, & \\chi _5^2, & \\chi _5^3, & \\chi _5^4 \\\\\\phantom{\\chi _5} & \\varepsilon , & \\varepsilon , & \\varepsilon \\end{array}\\bigg | \\; 1 \\Biggr )_p- p= c(p).$ The remainder of the paper is organized as follows.",
"Section 2 recalls some properties of Gauss and Jacobi sums, and the $p$ -adic gamma function.",
"The proof of Theorems REF and REF appear in Section 3."
],
[
"Preliminaries",
"We briefly recall some properties of Gauss and Jacobi sums and the $p$ -adic gamma function, and also develop some preliminary results which we will use in Section 3.",
"Throughout we let $\\mathbb {Z}_p$ , $\\mathbb {Q}_p$ and $\\mathbb {C}_p$ denote the ring of $p$ -adic integers, the field of $p$ -adic numbers and the $p$ -adic completion of the algebraic closure of $\\mathbb {Q}_p$ , respectively."
],
[
"Gauss and Jacobi Sums",
"We first recall some properties of multiplicative characters.",
"In particular, we note the following orthogonal relations.",
"For $\\chi \\in \\widehat{\\mathbb {F}_p^{*}}$ we have $\\sum _{x \\in \\mathbb {F}_p} \\chi (x)={\\left\\lbrace \\begin{array}{ll}p-1 & \\text{if $\\chi = \\varepsilon $} ,\\\\0 & \\text{if $\\chi \\ne \\varepsilon $} ,\\end{array}\\right.",
"}$ and, for $x \\in \\mathbb {F}_p$ we have $\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (x)={\\left\\lbrace \\begin{array}{ll}p-1 & \\text{if $x=1$} ,\\\\0 & \\text{if $x \\ne 1$} .\\end{array}\\right.",
"}$ We now introduce some properties Gauss and Jacobi sums.",
"For further details see [4] and [9], noting that we have adjusted results to take into account $\\varepsilon (0)=0$ .",
"Let $\\zeta _p$ be a fixed primitive $p$ -th root of unity in $\\overline{\\mathbb {Q}_p}$ .",
"We define the additive character $\\theta : \\mathbb {F}_p \\rightarrow \\mathbb {Q}_p(\\zeta _p)$ by $\\theta (x):=\\zeta _p^{x}$ .",
"We note that $\\mathbb {Q}_p$ contains all $(p-1)$ -th roots of unity and in fact they are all in $\\mathbb {Z}^{*}_p$ .",
"Thus we can consider multiplicative characters of $\\mathbb {F}_p^{*}$ to be maps $\\chi : \\mathbb {F}_p^{*} \\rightarrow \\mathbb {Z}_{p}^{*}$ .",
"Recall then that for $\\chi \\in \\widehat{\\mathbb {F}_p^{*}}$ , the Gauss sum $g(\\chi )$ is defined by $g(\\chi ):= \\sum _{x \\in \\mathbb {F}_p} \\chi (x) \\theta (x).$ It easily follows from (REF ) that we can express the additive character as a sum of Gauss sums.",
"Specifically, for $x \\in \\mathbb {F}_p^*$ we have $\\theta (x)= \\frac{1}{p-1}\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} g(\\overline{\\chi }) \\, \\chi (x).$ The following important result gives a simple expression for the product of two Gauss sums.",
"For $\\chi \\in \\widehat{\\mathbb {F}_p^{*}}$ we have $g(\\chi )g(\\overline{\\chi })={\\left\\lbrace \\begin{array}{ll}\\chi (-1) p & \\text{if } \\chi \\ne \\varepsilon ,\\\\1 & \\text{if } \\chi = \\varepsilon .\\end{array}\\right.",
"}$ Another important product formula for Gauss sums is the Hasse-Davenport formula.",
"Theorem 2.1 (Hasse, Davenport [4] Thm 11.3.5) Let $\\chi $ be a character of order $m$ of $\\mathbb {F}_p^*$ for some positive integer $m$ .",
"For a character $\\psi $ of $\\mathbb {F}_p^*$ we have $\\prod _{i=0}^{m-1} g(\\chi ^i \\psi ) = g(\\psi ^m) \\psi ^{-m}(m)\\prod _{i=1}^{m-1} g(\\chi ^i).$ We now introduce generalized Jacobi sums.",
"Let $\\chi _1, \\chi _2, \\cdots , \\chi _k \\in \\widehat{\\mathbb {F}_p^{*}}$ .",
"Then the generalized Jacobi sum $J(\\chi _1, \\chi _2, \\cdots , \\chi _k)$ of order $k$ is defined by $J(\\chi _1, \\chi _2, \\cdots , \\chi _k):= \\sum _{\\begin{array}{c}t_1+t_2+ \\cdots + t_k=1\\\\ t_i \\in \\mathbb {F}_p\\end{array}} \\chi _1(t_1) \\chi _2(t_2) \\cdots \\chi _k(t_k).$ There is a general formula relating generalized Jacobi sums of order $k$ to ones of order $k-1$ .",
"One special case is the following.",
"If $\\chi _1\\chi _2\\cdots \\chi _k$ is trivial but at least one of $\\chi _1, \\chi _2, \\cdots , \\chi _k$ is non-trivial, then $J(\\chi _1, \\chi _2, \\cdots , \\chi _k)=-\\chi _k(-1) J(\\chi _1, \\chi _2, \\cdots , \\chi _{k-1}) \\; .$ We can relate generalized Jacobi sums to Gauss sums in the following way.",
"For $\\chi _1, \\chi _2, \\cdots , \\chi _k \\in \\widehat{\\mathbb {F}_p^{*}}$ not all trivial, $J(\\chi _1, \\chi _2, \\cdots , \\chi _k)={\\left\\lbrace \\begin{array}{ll}\\dfrac{g(\\chi _1)g(\\chi _2)\\cdots g(\\chi _k)}{g(\\chi _1 \\chi _2 \\cdots \\chi _k)}& \\qquad \\text{if } \\chi _1 \\chi _2 \\cdots \\chi _k \\ne \\varepsilon ,\\\\[18pt]-\\dfrac{g(\\chi _1)g(\\chi _2)\\cdots g(\\chi _k)}{p}&\\qquad \\text{if }\\chi _1 \\chi _2 \\cdots \\chi _k = \\varepsilon \\: .\\end{array}\\right.",
"}$ We now develop some new results which we will use in Section 3.",
"Lemma 2.2 Let $p\\equiv 1 \\pmod {5}$ be prime and let $\\psi $ be a character of order 5 of $\\mathbb {F}_p^*$ .",
"If $a,b,c \\in \\mathbb {Z}$ are such that $a+c$ , $b+c \\lnot \\equiv 0 \\pmod {5}$ , then $\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (-1) J(\\overline{\\chi } \\psi ^a,\\, \\overline{\\chi } \\psi ^b, \\, \\chi \\psi ^c)=-(p-1).$ $\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (-1) J(\\overline{\\chi } \\psi ^a,\\, \\overline{\\chi } \\psi ^b, \\, \\chi \\psi ^c)&=\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (-1) \\sum _{\\begin{array}{c}t_1+t_2+ t_3=1\\\\ t_i \\in \\mathbb {F}_p^*\\end{array}} \\overline{\\chi } \\psi ^a(t_1) \\, \\overline{\\chi } \\psi ^b(t_2) \\, \\chi \\psi ^c(t_3)\\\\&=\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (-1) \\sum _{\\begin{array}{c}t_1+t_2+ t_3=1\\\\ t_i \\in \\mathbb {F}_p^*\\end{array}}\\overline{\\chi }\\left(\\frac{t_1t_2}{t_3}\\right) \\psi (t_1^a \\; t_2^b \\; t_3^c)\\\\&= (p-1) \\sum _{\\begin{array}{c}t_1+t_2+ t_3=1\\\\ t_i \\in \\mathbb {F}_p^*\\\\-\\frac{t_1t_2}{t_3}=1\\end{array}} \\psi (t_1^a \\; t_2^b \\; t_3^c) \\qquad \\text{by (\\ref {for_TOrthCh}).",
"}$ All possible triples $(t_1,t_2,t_3)$ satisfying the conditions of the summation can be represented by $(t_1,1,-t_1)$ and $(1,t_2,-t_2)$ , not counting $(1,1,-1)$ twice.",
"Hence $\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (-1) & J(\\overline{\\chi } \\psi ^a,\\, \\overline{\\chi } \\psi ^b, \\, \\chi \\psi ^c)\\\\&=(p-1) \\left[\\sum _{t_1\\in \\mathbb {F}_p^*} \\psi (t_1^{a+c}\\: (-1)^c) + \\sum _{t_2\\in \\mathbb {F}_p^*} \\psi (t_2^{b+c} \\:(-1)^c) - \\psi ((-1)^c) \\right]\\\\&=(p-1)\\;\\psi ^c(-1) \\left[\\sum _{t_1\\in \\mathbb {F}_p^*} \\psi ^{a+c}(t_1) + \\sum _{t_2\\in \\mathbb {F}_p^*} \\psi ^{b+c}(t_2) - 1 \\right].$ Now $a+c$ , $b+c \\lnot \\equiv 0 \\pmod {5}$ so both $\\psi ^{a+c}$ and $\\psi ^{b+c}$ are non-trivial.",
"Thus, by (REF ), $\\displaystyle \\sum _{t_1\\in \\mathbb {F}_p^*} \\psi ^{a+c}(t_1) = \\sum _{t_2\\in \\mathbb {F}_p^*} \\psi ^{b+c}(t_2)=0.$ Also, $\\psi (-1)=\\psi ^5(-1) =1,$ which completes the proof.",
"Corollary 2.3 Let $p\\equiv 1 \\pmod {5}$ be prime and let $\\psi $ be a character of order 5 of $\\mathbb {F}_p^*$ .",
"If $a,b,c \\in \\mathbb {Z}$ are such that $a+c$ , $b+c \\lnot \\equiv 0 \\pmod {5}$ , then $\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} g(\\overline{\\chi } \\psi ^a) \\,g(\\overline{\\chi } \\psi ^b) \\, g(\\chi \\psi ^c) \\, g(\\chi \\overline{\\psi }{}^{a+b+c})=-p(p-1).$ Using (REF ) and (REF ) we see that $\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} g(\\overline{\\chi } \\psi ^a) \\,g(\\overline{\\chi } \\psi ^b) \\, & g(\\chi \\psi ^c) \\, g(\\chi \\overline{\\psi }{}^{a+b+c})\\\\&=-p\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} J\\left(\\overline{\\chi } \\psi ^a, \\, \\overline{\\chi } \\psi ^b, \\, \\chi \\psi ^c, \\, \\chi \\overline{\\psi }{}^{a+b+c}\\right) \\\\&=p\\sum _{\\chi \\in \\widehat{\\mathbb {F}_p^{*}}} \\chi (-1) \\; J\\left(\\overline{\\chi } \\psi ^a, \\, \\overline{\\chi } \\psi ^b, \\, \\chi \\psi ^c \\right).$ Applying Lemma REF then yields the result.",
"We now recall a formula for counting zeros of polynomials in affine space using the additive character.",
"If $f(x_1, x_2, \\ldots x_n) \\in \\mathbb {F}_p[x_1, x_2, \\ldots x_n]$ , then the number of points, $N_p^*$ , in $\\mathbb {A}^n(\\mathbb {F}_p)$ satisfying $f(x_1, x_2, \\ldots x_n) =0$ is given by $p N_p^* = p^n +\\sum _{y \\in \\mathbb {F}_p^*} \\sum _{x_1, x_2, \\ldots x_n \\in \\mathbb {F}_p}\\theta (y \\: f(x_1, x_2, \\ldots x_n)) \\; .$ In [13], Koblitz provides a formula for the number of points in $\\mathbb {P}^{n-1}(\\mathbb {F}_p)$ on the hypersurface $x_1^d + x_2^d + \\dots + x_n^d - d \\lambda x_1 x_2 \\dots x_n=0$ , for some $d, \\lambda \\in \\mathbb {F}_p$ , where $p \\equiv 1 \\pmod {d}$ .",
"Let $W:=\\lbrace w=(w_1, w_2, \\ldots , w_n) \\in \\mathbb {Z}^n : 0 \\le w_i < d, \\sum _{i=1}^n w_i \\equiv 0 \\pmod {d}\\rbrace .$ Let $T$ be a fixed generator for the group of characters of $\\mathbb {F}_p^*$ and set $t:=\\frac{p-1}{d}$ .",
"Define an equivalence relation $\\sim $ on $W$ by $w \\sim w \\prime \\textup { if } w- w\\prime \\textup { is a multiple modulo d of (1,1, \\ldots , 1)}.$ Define $N_p(0,w):={\\left\\lbrace \\begin{array}{ll}0 & \\text{if some but not all } w_i=0,\\\\[6pt]\\frac{p^{n-1}-1}{p-1} & \\text{if all } w_i=0,\\\\[6pt]\\frac{1}{p} \\prod _{i=1}^{n} g(T^{w_i t}) & \\text{if all } w_i \\ne 0.\\end{array}\\right.",
"}$ Then we have the following theorem.",
"Theorem 2.4 (Koblitz [13] Thm.",
"2) Let $N_p(\\lambda )$ be the number of points in $\\mathbb {P}^{n-1}(\\mathbb {F}_p)$ on $\\sum _{i=1}^n x_i^d - d \\lambda \\prod _{j=1}^{n} x_i=0$ , for some $d, \\lambda \\in \\mathbb {F}_p$ .",
"Let $W$ , $\\sim $ and $N_p(0,w)$ be defined by (REF ), (REF ) and (REF ) respectively.",
"Let $T$ be a fixed generator for the group of characters of $\\mathbb {F}_p^*$ , let $p \\equiv 1 \\pmod {d}$ and define $t:=\\frac{p-1}{d}$ .",
"Then $N_p(\\lambda ) = \\sum _{w \\in W} N_p(0,w) + \\frac{1}{p-1} \\sum _{[w] \\in W/ \\sim } \\; \\sum _{j=0}^{p-2}\\frac{\\prod _{i=1}^{n} g(T^{j+w_i t})}{g(T^{dj})}\\; T^{dj}(d \\lambda ).\\\\$"
],
[
"$p$ -adic preliminaries",
"We first define the Teichmüller character to be the primitive character $\\omega : \\mathbb {F}_p \\rightarrow \\mathbb {Z}^{*}_p$ satisfying $\\omega (x) \\equiv x \\pmod {p}$ for all $x \\in \\lbrace 0,1, \\ldots , p-1\\rbrace $ .",
"We now recall the $p$ -adic gamma function.",
"For further details, see [12].",
"Let $p$ be an odd prime.",
"For $n \\in \\mathbb {Z}^{+}$ we define the $p$ -adic gamma function as $\\Gamma _p{\\left({n}\\right)} &:= {(-1)}^n \\prod _{\\begin{array}{c}0<j<n\\\\p \\nmid j\\end{array}} j \\\\\\multicolumn{2}{l}{\\text{and extend to all $x \\in \\mathbb {Z}_p$ by setting $\\Gamma _p{\\left({0}\\right)}:=1$ and}}\\\\\\Gamma _p{\\left({x}\\right)} &:= \\lim _{n \\rightarrow x} \\Gamma _p{\\left({n}\\right)}$ for $x\\ne 0$ , where $n$ runs through any sequence of positive integers $p$ -adically approaching $x$ .",
"This limit exists, is independent of how $n$ approaches $x$ , and determines a continuous function on $\\mathbb {Z}_p$ with values in $\\mathbb {Z}^{*}_p$ .",
"We now state a product formula for the $p$ -adic gamma function.",
"If $m\\in \\mathbb {Z}^{+}$ , $p \\nmid m$ and $x=\\frac{r}{p-1}$ with $0\\le r \\le p-1$ then $\\prod _{h=0}^{m-1} \\Gamma _p{\\left({\\tfrac{x+h}{m}}\\right)}=\\omega \\left(m^{(1-x)(1-p)}\\right)\\Gamma _p{\\left({x}\\right)} \\prod _{h=1}^{m-1} \\Gamma _p{\\left({\\tfrac{h}{m}}\\right)}.$ The Gross-Koblitz formula [8] allows us to relate Gauss sums and the $p$ -adic gamma function.",
"Let $\\pi \\in \\mathbb {C}_p$ be the fixed root of $x^{p-1}+p=0$ which satisfies ${\\pi \\equiv \\zeta _p-1 \\pmod {{(\\zeta _p-1)}^2}}$ .",
"Then we have the following result.",
"Theorem 2.5 (Gross, Koblitz [8]) For $0 \\le j \\le p-2$ , $g(\\overline{\\omega }^j)=-\\pi ^j \\: \\Gamma _p{\\left({\\tfrac{j}{p-1}}\\right)}.$"
],
[
"Proofs",
"Let $N_p$ be the number of points in $\\mathbb {P}^4(\\mathbb {F}_p)$ on $x_1^5+x_2^5+x_3^5+x_4^5+ x_5^5-5 x_1 x_2 x_3 x_4 x_5=0$ .",
"Then from [17] (following the work of Schoen [22]), we have $c(p)={\\left\\lbrace \\begin{array}{ll}p^3 + 25 p^2 -100 p + 1 - N_p & \\text{if $p \\equiv 1\\phantom{,2} \\pmod {5}$},\\\\p^3 + \\phantom{25} p^2 \\: \\;\\phantom{-100 p} + 1 - N_p & \\text{if $p \\equiv 4\\phantom{,2} \\pmod {5}$}, \\\\p^3 + \\phantom{25} p^2 + \\phantom{10}2 p + 1 - N_p & \\text{if $p \\equiv 2,3 \\pmod {5}$}.\\end{array}\\right.",
"}$ Therefore, noting that $s(p)={\\left\\lbrace \\begin{array}{ll}+1 & \\text{if } p\\equiv 1, 4 \\pmod {5},\\\\-1 & \\text{if } p\\equiv 2,3 \\pmod {5},\\end{array}\\right.",
"}$ it suffices to prove $N_p = - {_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p +{\\left\\lbrace \\begin{array}{ll}p^3 + 25 p^2 -99 p + 1 & \\text{if $p \\equiv 1 \\pmod {5}$},\\\\p^3 + \\phantom{25} p^2 + \\phantom{99} p + 1 & \\text{if $p \\lnot \\equiv 1 \\pmod {5}$}.\\end{array}\\right.",
"}$ We will express both sides of (REF ) in terms of Gauss sums, starting with ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p$ .",
"Let $m_0:=-1$ , $m_{5}:=p-2$ , $d_0=d_{5}:=p-1$ and $m_i:=i$ , $d_i:=5$ for $1\\le i \\le 4$ .",
"Also, let $r_i=\\frac{p-1}{d_i}$ for $0 \\le i \\le 5$ .",
"Consequently, from definition (REF ) we get that ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p\\\\= \\frac{-1}{p-1} \\sum _{k=0}^{4} (-p)^k \\sum _{j=\\left\\lfloor m_k r_k \\right\\rfloor +1}^{\\left\\lfloor m_{k+1}r_{k+1}\\right\\rfloor }\\frac{{\\Gamma _p{\\bigl ({\\tfrac{j}{p-1}}\\bigr )}}^{4}}{\\Gamma _p{\\bigl ({1-\\tfrac{j}{p-1}}\\bigr )}}\\frac{\\prod _{h=0}^{4} \\Gamma _p{\\Bigl ({\\frac{k+1-\\frac{5j}{p-1}+h}{5}}\\Bigr )}}{\\prod _{h=1}^{4} \\Gamma _p{\\bigl ({\\frac{h}{5}}\\bigr )}}.$ For a given $k$ , we easily see that $k\\bigl (\\frac{p-1}{5}\\bigr ) \\le j \\le (k+1)\\bigl (\\frac{p-1}{5}\\bigr )$ , with equality on the left when $k=j=0$ .",
"Therefore $0 \\le (k+1)(p-1) -5j \\le p-1$ and we can apply (REF ) with $m=5$ and $x= k+1-\\frac{5j}{p-1}$ to get ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p\\\\= \\frac{-1}{p-1} \\sum _{k=0}^{4} (-p)^k \\sum _{j=\\left\\lfloor m_k r_k \\right\\rfloor +1}^{\\left\\lfloor m_{k+1}r_{k+1}\\right\\rfloor }\\frac{{\\Gamma _p{\\bigl ({\\tfrac{j}{p-1}}\\bigr )}}^{4}}{\\Gamma _p{\\bigl ({1-\\tfrac{j}{p-1}}\\bigr )}}\\Gamma _p{\\bigl ({k+1-\\tfrac{5j}{p-1}}\\bigr )}\\omega (5^{-5j}).$ We now use Theorem REF to convert this to an expression involving Gauss sums.",
"To satisfy the conditions of the theorem for all arguments of the $p$ -adic gamma functions above, we split off the term when $j=0$ .",
"For all other values of $j$ , we have $0 \\le (p-1)- j, (k+1)(p-1)-5j \\le p-2$ .",
"We then get ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p= \\frac{-1}{p-1} \\left[1+ \\sum _{j=1}^{p-2} \\frac{g(\\omega ^{-j})^4}{g(\\omega ^j)} \\hspace{2.0pt} g(\\omega ^{5j})\\hspace{2.0pt} \\omega (5^{-5j}) \\right].$ Now applying (REF ) yields ${_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p= -\\frac{1}{p-1} \\left[1+\\frac{1}{p} \\sum _{j=1}^{p-2}\\; g(\\omega ^{-j})^{5}\\, g(\\omega ^{5j}) \\; \\omega ^{-5j}(-5) \\right].$ We will now evaluate $N_p$ .",
"We first consider the case when $p \\equiv 1 \\pmod {5}$ .",
"By Theorem REF with $d=5, \\lambda =1$ we see that $N_p = \\sum _{w \\in W} N_p(0,w) + \\frac{1}{p-1} \\sum _{[w] \\in W/ \\sim } \\sum _{j=0}^{p-2}\\frac{\\prod _{i=1}^{5} g(T^{j+w_i t})}{g(T^{5j})}\\; T^{5j}(5),$ where $W$ , $\\sim $ and $N_p(0,w)$ are as defined in (REF ), (REF ) and (REF ) respectively, $T$ is a fixed generator for the group of characters of $\\mathbb {F}_p$ and $t=\\frac{p-1}{5}$ .",
"We now describe the elements of both $W$ and $W/ \\sim $ as they apply to our setting.",
"We first note that the contribution of any $(w_1, w_2, \\ldots , w_5)$ to the above formula is the same as any permutation of it.",
"We therefore list the elements of these sets up to permutation.",
"We will however indicate, using a superscript, the multiplicity with which it contributes to the formula, i.e., its total number of distinct permutations.",
"As $N_p(0,w)$ vanishes if some but not all of the $w_i$ are zero we will just list the elements of $W$ for which all $w_i$ are non-zero and call it $W^*$ .",
"Therefore $W^* = \\lbrace &(1,1,1,1,1)^1, (2,2,2,2,2)^1, (3,3,3,3,3)^1, (4,4,4,4,4)^1,\\\\& (1,1,1,3,4)^{20}, (1,2,2,2,3)^{20}, (2,3,3,3,4)^{20}, (1,2,4,4,4)^{20},\\\\& (1,1,2,2,4)^{30}, (2,2,3,3,4)^{30},(1,1,2,3,3)^{30}, (1,3,3,4,4)^{30} \\rbrace $ and $W/\\sim \\; = \\lbrace (0,0,0,0,0)^1, (0,1,2,3,4)^{24}, (0,0,0,1,4)^{20},\\\\ (0,0,0,2,3)^{20}, (0,0,1,1,3)^{30}, (0,0,1,2,2)^{30} \\rbrace .$ We now use $W^*$ to evaluate $N_p(0):=\\sum _{w \\in W} N_p(0,w)$ .",
"We note that many of the elements of $W^*$ are multiples modulo 5 of each other.",
"Thus $N_p(0) = \\frac{p^4 - 1}{p-1} + \\frac{20}{p} \\sum _{i=1}^{4} g(T^{it})^3 \\; g(T^{3it}) \\; g(T^{4it})\\\\ + \\frac{30}{p} \\sum _{i=1}^{4} g(T^{it})^2 \\; g(T^{2it})^2 \\; g(T^{4it}) + \\frac{1}{p} \\sum _{i=1}^{4} g(T^{it})^5.$ Applying (REF ) then yields $N_p(0)&= p^3 + p^2 + p + 1 + 50 \\sum _{i=1}^{4} g(T^{it})^2 \\; g(T^{3it}) + \\frac{1}{p} \\sum _{i=1}^{4} g(T^{it})^5.$ We now focus on the second sum on the right-hand side of (REF ) and evaluate it for each individual $[w] \\in W/\\sim $ (up to permutation).",
"We will denote each such minor sum as $S_{[w]}$ .",
"We start with $[w]= (0,0,0,0,0).$ So $S_{(0,0,0,0,0)} = \\frac{1}{p-1} \\sum _{j=0}^{p-2} \\frac{g(T^{j})^5}{g(T^{5j})} \\; T^{5j}(5).$ We isolate the terms where $j$ is a multiple of $t= \\frac{p-1}{5}$ and apply (REF ) to the other terms to get $S_{(0,0,0,0,0)}= \\frac{1}{p-1} \\left[1 - \\sum _{\\begin{array}{c}j=1\\\\t \\mid j\\end{array}}^{p-2} g(T^{j})^5 + \\frac{1}{p} \\sum _{\\begin{array}{c}j=1\\\\t \\nmid j\\end{array}}^{p-2} g(T^{j})^5 \\; g(T^{-5j})\\; T^{5j}(-5)\\right].$ Next considering $[w]=(0,1,2,3,4)$ yields $S_{(0,1,2,3,4)}=\\frac{24}{p-1} \\sum _{j=0}^{p-2} \\frac{g(T^j) \\; g(T^{j+t}) \\; g(T^{j+2t}) \\; g(T^{j+3t}) \\; g(T^{j+4t})}{g(T^{5j})} \\; T^{5j}(5).$ Combining Theorem REF , with $m=5$ and $\\psi =T^j$ , and (REF ) we see that $g(T^j) \\; g(T^{j+t})\\; g(T^{j+2t})\\; g(T^{j+3t})\\; g(T^{j+4t}) = g(T^{5j})\\; T^{-5j}(5)\\; p^2$ Therefore, we have that $S_{(0,1,2,3,4)}= 24 \\, p^2.$ The sums for the remaining elements, $(0,0,0,1,4)$ , $(0,0,0,2,3)$ , $(0,0,1,1,3)$ and $(0,0,1,2,2)$ , can all be evaluated in a similar manner to each other.",
"By definition $S_{(0,0,0,1,4)}=\\frac{20}{p-1} \\sum _{j=0}^{p-2} \\frac{g(T^j)^3 \\; g(T^{j+t}) \\; g(T^{j+4t})}{g(T^{5j})} \\; T^{5j}(5).$ Applying (REF ) gives us $S_{(0,0,0,1,4)}=\\frac{20 \\, p^2}{p-1} \\sum _{j=0}^{p-2} \\frac{g(T^j)^2}{g(T^{j+2t})\\; g(T^{j+3t})}.$ We now use (REF ) to clear denominators, being careful to deal with the cases $j=2t,3t$ separately.",
"Thus $S_{(0,0,0,1,4)}=\\frac{20}{p-1} \\left[\\; \\sum _{\\begin{array}{c}j=0\\\\j \\ne 2t, 3t\\end{array}}^{p-2} g(T^j)^2 \\; g(T^{-j+3t})\\; g(T^{-j+2t})\\right.",
"\\\\ \\left.\\phantom{\\sum _{\\begin{array}{c}j=0\\\\j \\ne 2t, 3t\\end{array}}^{p-2}} - p \\, g(T^{2t})^2 \\; g(T^t) - p \\, g(T^{3t})^2 \\; g(T^{4t}) \\right].$ We now apply Corollary REF with $a=3$ , $b=2$ and $c=0$ to get $S_{(0,0,0,1,4)}=-20p - 20 \\left[g(T^{2t})^2 \\; g(T^t) + g(T^{3t})^2 \\; g(T^{4t}) \\right].$ Similarly, we have $S_{(0,0,0,2,3)}=-20p - 20 \\left[g(T^{t})^2 \\; g(T^{3t}) + g(T^{4t})^2 \\; g(T^{2t}) \\right],$ $S_{(0,0,1,1,3)}=-30p - 30 \\left[g(T^{3t})^2 \\; g(T^{4t}) + g(T^{2t})^2 \\; g(T^{t}) \\right]$ and $S_{(0,0,1,2,2)}=-30p - 30 \\left[g(T^{t})^2 \\; g(T^{3t}) + g(T^{4t})^2 \\; g(T^{2t}) \\right].\\\\[9pt]$ Combining (REF ), (REF ), (REF ), (REF ), (REF ), (REF ) and (REF ) we get via (REF ) that $N_p = p^3 + 25 p^2 -99 p + 1 + \\frac{1}{p-1} \\left[1 + \\frac{1}{p} \\sum _{j=1}^{p-2} g(T^{j})^5 \\; g(T^{-5j})\\; T^{5j}(-5)\\right].$ Taking $T$ to be $\\omega ^{-j}$ gives us $N_p = p^3 + 25 p^2 -99 p + 1 - {_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p$ as required.",
"We now evaluate $N_p$ when $p \\lnot \\equiv 1 \\pmod {5}$ , using (REF ), and express our results in terms of Gauss sums using (REF ).",
"Let $\\overline{x}$ denote the tuple $(x_1, x_2, x_3, x_4, x_5)$ and define $f(\\overline{x}) := x_1^5+x_2^5+x_3^5+x_4^5+ x_5^5-5 x_1 x_2 x_3 x_4 x_5$ for brevity.",
"Also, let $N_p^A$ denote the number of points in $\\mathbb {A}^5(\\mathbb {F}_p)$ on $f(\\overline{x})=0.$ Then, as $f$ is homogeneous, $N_p = \\frac{N_p^A-1}{p-1}.$ From (REF ) we see that $p N_p^A= p^5 + \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{\\begin{array}{c}x_i \\in \\mathbb {F}_p\\\\some \\; x_i=0\\end{array}} \\theta (y \\: f(\\overline{x})) + \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{x_i \\in \\mathbb {F}_p^{*}} \\theta (y \\: f(\\overline{x})).$ We now consider the number of points $N_p^{\\prime }$ in $\\mathbb {A}^5(\\mathbb {F}_p)$ on $f^{\\prime }(\\overline{x}):=x_1^5+x_2^5+x_3^5+x_4^5+ x_5^5=0.$ As $x \\rightarrow x^5$ is an automorphism on $\\mathbb {F}_p$ when $p \\lnot \\equiv 1 \\pmod {5}$ , it is easy to see that $N_p^{\\prime }=p^4$ .",
"From (REF ) we also have that $p N_p^{\\prime }&= p^5 + \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{\\begin{array}{c}x_i \\in \\mathbb {F}_p\\\\some \\; x_i=0\\end{array}} \\theta (y \\: f^{\\prime }(\\overline{x})) + \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{x_i \\in \\mathbb {F}_p^{*}} \\theta (y \\: f^{\\prime }(\\overline{x})).$ Therefore $\\sum _{y \\in \\mathbb {F}_p^*} \\sum _{\\begin{array}{c}x_i \\in \\mathbb {F}_p\\\\some \\; x_i=0\\end{array}} \\theta (y \\: f^{\\prime }(\\overline{x})) =- \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{x_i \\in \\mathbb {F}_p^{*}} \\theta (y \\: f^{\\prime }(\\overline{x})).$ Noting that $\\sum _{y \\in \\mathbb {F}_p^*} \\sum _{\\begin{array}{c}x_i \\in \\mathbb {F}_p\\\\some \\; x_i=0\\end{array}} \\theta (y \\: f(\\overline{x})) =\\sum _{y \\in \\mathbb {F}_p^*} \\sum _{\\begin{array}{c}x_i \\in \\mathbb {F}_p\\\\some \\; x_i=0\\end{array}} \\theta (y \\: f^{\\prime }(\\overline{x}))$ yields $p N_p^A = p^5 + \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{x_i \\in \\mathbb {F}_p^{*}} \\theta (y \\: f(\\overline{x}))- \\sum _{y \\in \\mathbb {F}_p^*} \\sum _{x_i \\in \\mathbb {F}_p^{*}} \\theta (y \\: f^{\\prime }(\\overline{x})).$ We now convert the two sums on the right above, which we call $S_1$ and $S_2$ respectively, to expressions involving Gauss sums using (REF ).",
"Then, starting with $S_2$ , we have $S_2&= \\sum _{y, x_i \\in \\mathbb {F}_p^{*}} \\theta (y \\: x_1^5) \\theta (y \\: x_2^5)\\theta (y \\: x_3^5)\\theta (y \\: x_4^5)\\theta (y \\: x_5^5)\\\\&= \\frac{1}{(p-1)^5} \\sum _{a,b,c,d,e = 0}^{p-2} \\sum _{y, x_i \\in \\mathbb {F}_p^{*}} g(T^{-a}) \\; g(T^{-b}) \\; g(T^{-c}) \\; g(T^{-d}) \\; g(T^{-e}) \\; \\\\&\\qquad \\qquad \\qquad \\qquad \\; \\qquad \\qquad \\cdot T^a(y x_1^5)\\; T^b(y x_2^5)\\; T^c(y x_3^5)\\; T^d(y x_4^5)\\; T^e(y x_5^5)\\\\&=\\frac{1}{(p-1)^5} \\sum _{a,b,c,d,e = 0}^{p-2} \\sum _{x_i \\in \\mathbb {F}_p^{*}} g(T^{-a}) \\; g(T^{-b}) \\; g(T^{-c}) \\; g(T^{-d}) \\; g(T^{-e}) \\; \\\\& \\qquad \\qquad \\; \\qquad \\cdot T^a(x_1^5)\\; T^b(x_2^5)\\; T^c(x_3^5)\\; T^d(x_4^5)\\; T^e(x_5^5) \\sum _{y \\in \\mathbb {F}_p^{*}} T^{a+b+c+d+e}(y).$ We now apply (REF ) to the last summation on the right, which yields $(p-1)$ if $e=-a-b-c-d$ and zero otherwise.",
"So $S_2&=\\frac{1}{(p-1)^4} \\sum _{a,b,c,d = 0}^{p-2} \\sum _{x_i \\in \\mathbb {F}_p^{*}} g(T^{-a}) \\; g(T^{-b}) \\; g(T^{-c}) \\; g(T^{-d}) \\; g(T^{a+b+c+d}) \\; \\\\&\\qquad \\qquad \\qquad \\quad \\qquad \\qquad \\cdot T^a(x_1^5)\\; T^b(x_2^5)\\; T^c(x_3^5)\\; T^d(x_4^5)\\; T^{-a-b-c-d}(x_5^5)\\\\&=\\frac{1}{(p-1)^4} \\sum _{a,b,c,d = 0}^{p-2} \\sum _{x_2, x_3, x_4, x_5 \\in \\mathbb {F}_p^{*}} g(T^{-a}) \\, g(T^{-b}) \\, g(T^{-c}) \\, g(T^{-d}) \\, g(T^{a+b+c+d}) \\; \\\\&\\qquad \\qquad \\qquad \\qquad \\cdot T^b(x_2^5)\\; T^c(x_3^5)\\; T^d(x_4^5)\\; T^{-a-b-c-d}(x_5^5)\\sum _{x_1 \\in \\mathbb {F}_p^{*}} T^{5a}(x_1)\\; .$ We again apply (REF ) to the last summation on the right and continue in this manner isolating the sum for each $x_i$ in turn and applying (REF ).",
"This leads to $S_2=- \\sum _{x_5 \\in \\mathbb {F}_p^{*}} 1 = -(p-1).$ A similar evaluation of $S_1$ using (REF ) and (REF ) yields $S_1 = \\sum _{e=0}^{p-2} g(T^{-e})^5 \\; g(T^{5e}) T^{-5e}(-5).$ Accounting for (REF ), (REF ) in (REF ) we see that, via (REF ), $N_p&= \\frac{1}{p-1} \\left\\lbrace \\frac{1}{p} \\left[ p^5 + \\sum _{e=0}^{p-2} g(T^{-e})^5 \\; g(T^{5e}) T^{-5e}(-5) +p -1 \\right] - 1 \\right\\rbrace \\\\&= \\frac{p^4-1}{p-1} + \\frac{1}{p-1}\\left[1 + \\frac{1}{p} \\sum _{e=1}^{p-2} g(T^{-e})^5 \\; g(T^{5e}) T^{-5e}(-5) \\right].$ Taking $T$ to be $\\omega ^{j}$ gives us $N_p = p^3 + p^2 + p + 1 - {_{4}G} \\left(\\tfrac{1}{5} ,\\hspace{2.0pt} \\tfrac{2}{5},\\hspace{2.0pt} \\tfrac{3}{5} ,\\hspace{2.0pt} \\tfrac{4}{5}\\right)_p$ as required.",
"One easily checks the result for $p=2$ .",
"Combining Theorem REF with $d=5$ and Theorem REF yields the result."
],
[
"Remark",
"Using (REF ) to count points on certain algebraic varieties is by no means new.",
"This author first observed the technique in [6].",
"In the proof above, we have applied this technique in the case $p \\lnot \\equiv 1 \\pmod {5}$ .",
"We could have also applied this method to the case $p \\equiv 1 \\pmod {5}$ , choosing instead to use Theorem REF for reasons of brevity.",
"The two methods are essentially the same with Theorem REF encapsulating much of the work which must be done if (REF ) is used.",
"For a detailed account of how the case $p \\equiv 1 \\pmod {5}$ is established using (REF ), please see [14]."
]
] | 1204.1575 |
[
[
"Infrared studies of the Be star X Per"
],
[
"Abstract Photometric and spectroscopic results are presented for the Be star X Per/HD 24534 from near-infrared monitoring in 2010-2011.",
"The star is one of a sample of selected Be/X-ray binaries being monitored by us in the near-IR to study correlations between their X ray and near-IR behaviour.",
"Comparison of the star's present near-IR magnitudes with earlier records shows the star to be currently in a prominently bright state with mean J, H, K magnitudes of 5.49, 5.33 and 5.06 respectively.",
"The JHK spectra are dominated by emission lines of HeI and Paschen and Brackett lines of HI.",
"Lines of OI 1.1287 and 1.3165 micron are also present and their relative strength indicates, since OI 1.1287 is stronger among the two lines, that Lyman beta fluorescence plays an important role in their excitation.",
"Recombination analysis of the HI lines is done which shows that the Paschen and Brackett line strengths deviate considerably from case B predictions.",
"These deviations are attributed to the lines being optically thick and this supposition is verified by calculating the line center optical depths predicted by recombination theory.",
"Similar calculations indicate that the Pfund and Humphrey series lines should also be expected to be optically thick which is found to be consistent with observations reported in other studies.",
"The spectral energy distribution of the star is constructed and shown to have an infrared excess.",
"Based on the magnitude of the IR excess, which is modeled using a free-free contribution from the disc, the electron density in the disc is estimated and shown to be within the range of values expected in Be star discs."
],
[
"Introduction",
"The Be star X Per/HD 24534 is the optical/IR counterpart of the X-ray source 4U0352+30 and belongs to the class of Be/X-ray binaries.",
"The orbital parameters of the system have been estimated by [9] as follows: a period of 250 days, an eccentricity of 0.11 and an inclination angle between 26 and 33 degrees.",
"[37] found evidence of 13.9 min modulations of the X-ray flux using the data taken with Copernicus and Ariel 5 satellites.",
"[9] found similar modulation of 837 s in RXTE data, which possibly corresponds to the spin period of the neutron star.",
"The Be star was classified to be of O9.5 iiie type with a rotation velocity ($v~sini$ ) of 200 km s$^{-1}$ and lying at a distance of 1300 $\\pm $ 400 pc [28], [23].",
"[20] re-estimated the spectral type, $v~sini$ and distance using the data taken during a low-luminosity disc-less phase (1989-91) to be B0Ve, 215 $\\pm $ 10 km s$^{-1}$ and 700 $\\pm $ 300 pc, respectively.",
"From optical and infrared photometric data spanning a decade (1987-95), [26] estimated the spectral type and distance as B0V and 900 $\\pm $ 300 pc, respectively, during the disc-less phase.",
"From optical spectroscopy and infrared photometry during the period 1988-90, [23] identified the loss of the circumstellar disc in X Per.",
"This was based on the change of the H$\\alpha $ profile from emission to absorption, an associated decrease in the infrared flux and the flattening of the infrared spectrum.",
"[10] used this dataset to study the astrophysical parameters of X Per since the loss of disc revealed the normal B-type star.",
"They estimated the spectral type of the star to be O9.5 iiie and set a lower age limit of 6 Myr for X Per system.",
"From high resolution optical spectroscopy and $V$ band photometry, [8] identified an episode of complete disc loss during 1988 May – 1989 June, characterised by reduction in flux of 0.6 mag in $V$ band and the presence of absorption profiles of H$\\alpha $ and Hei 6678 Å lines.",
"[25] identified an extended low state during 1974-77 which may be associated with a disc loss event from the analysis of optical, infrared and X-ray observations of X Per over a period of 25 years.",
"[34] reported the interesting formation of a double circumstellar disc in X Per, inferred from the quadruple emission peak structure in Hei 6678 Å line.",
"X Per was monitored as part of a program to observe Be/X-ray binaries in the near-IR using the 1.2m Mt.",
"Abu telescope.",
"The first photometric observations indicated that X Per was brighter by $\\sim $ 0.7 mag in $J, H, K$ compared to its listed 2MASS values providing the initial motivation to continue observing the object.",
"At present, photometry and spectroscopy spanning 8 nights spaced over a period of 3 months are reported.",
"Since much of this work is related to near-IR spectroscopy, it is worthwhile to summarize the major near-IR spectroscopic studies of Be stars that are relevant to this work.",
"A large sample of 57 and 66 stars in the $H$ and $K$ bands respectively of spectral types O9 – B9 and luminosity classes III, IV and V were studied by [31] and [7] in two studies separately devoted to the $H$ and $K$ bands respectively.",
"The major emphasis of these studies was on characterization of the stars based on the lines of different species seen in their spectra.",
"Both studies serve as good templates for comparing or contrasting newly obtained $H$ and $K$ spectroscopic data of other Be stars.",
"A significant extension into understanding the $L$ band spectra of Be stars was recently made by [13].",
"These authors used simultaneous $K$ and $L$ band spectroscopy to understand the circumstellar envelope properties from the Brackett, Pfund and Humphrey lines of hydrogen seen in the spectra.",
"From the ISO spectra, [18] used the line flux ratio of Hu(14)/Br$\\alpha $ and Hu(14)/Pf$\\gamma $ as a diagnostic tool to constrain the geometry of the ionized circumstellar material.",
"In the $J$ band, there appears to be a paucity of spectroscopic results - either of isolated stars or of larger samples - although this band contains certain diagnostic lines of considerable physical interest as discussed in section 3.1."
],
[
"Observations",
"The photometric and spectroscopic observations of X Per were carried out from the 1.2m Mt.",
"Abu telescope, operated by the Physical Research Laboratory.",
"The log of the photometric observations along with derived $JHK$ magnitudes is given in Table 1.",
"The log of the spectroscopic observations is given in Table 2.",
"The near-IR $JHK$ spectra presented here were obtained at similar dispersions of $\\sim $ 9.5 Å /pixel in each of the $J, H, K$ bands using the Near-Infrared Imager/Spectrometer with a 256 $\\times $ 256 HgCdTe NICMOS3 array.",
"The 1 arc second wide slit images to 2 pixels on the detector thereby yielding a resolving power of 800–1000 in the near-IR bands.",
"A set of two spectra were taken with the object dithered to two positions along the slit.",
"The spectra were extracted using IRAF and wavelength calibration was done using a combination of OH sky lines and telluric lines that register with the stellar spectra.",
"Following the standard procedure, the object spectra were then ratioed with the spectra of a comparison star (SAO 56762; A5V, $T_{eff}$ = 8200 K [27]) observed at similar airmass as the object.",
"Prior to the ratioing process the hydrogen Paschen and Brackett absorption lines in the comparison stars spectrum are removed using a Gaussian fit using IRAF.",
"The ratioed spectra were then multiplied by a blackbody curve at the effective temperature of the comparison star to yield the final spectra.",
"Photometry in the $JHK$ bands was done in photometric sky conditions using the imaging mode of the NICMOS3 array.",
"Several frames, in five dithered positions offset typically by 20 arcsec, were obtained of both the program object and a selected standard star (SAO 56762; A5V) in each of the $J, H, K$ filters.",
"Near-IR $JHK$ magnitudes were then derived using IRAF tasks and following the regular procedure followed by us for photometric reduction (e.g.",
"[3])."
],
[
"Photometric results and general characteristics of the spectra",
"The light curve of X Per taken over 6 epochs is shown in Figure 1, indicating no significant variations over this period.",
"The mean near-IR brightness is high when compared with the compilation of $JHK$ magnitudes collected over 25 years by [35].",
"For example, two of the sets of data with lowest $J, H, K$ magnitudes (highest flux values) recorded by these authors are 5.82, 5.21 and 5.15 on 1987 August 30 and 5.44, 5.41 and 5.29 respectively between 1994 September 16 – 20.",
"This enhanced near-IR brightness could be an indication of the accumulation of more material in the disc, resulting from episodes of stellar mass loss events.",
"Regarding the spectra, the recombination emission lines of hydrogen and helium are seen to dominate the $JHK$ spectra.",
"The prominent lines seen are Paschen $\\beta $ 1.2818 $\\mu $ m, Paschen $\\gamma $ 1.0938 $\\mu $ m and Hei 1.0830 $\\mu $ m in the $J$ band (Figure 2); Brackett 10 to 18 and Hei 1.7002 $\\mu $ m in the $H$ band (Figure 3) and Brackett $\\gamma $ 2.1656 $\\mu $ m, Hei 2.058, 2.1120, 2.1132 $\\mu $ m in the $K$ band (Figure 4).",
"The $H$ and $K$ band spectra of X Per are similar to those of classical Be stars observed by [31] and [7].",
"Based on their $K$ band spectra, [7] classified Be stars into five groups based on the strength and presence of the Br$\\gamma $ , Hei and Mgii 2.138, 2.144 $\\mu $ m features which are seen in the $K$ band.",
"Group i candidates are those stars which show Br$\\gamma $ in emission along with Hei line features which can either be in emission or in absorption.",
"It was also seen that all Group i candidates belonged to spectral class B3 or earlier.",
"The observed presence of both Br$\\gamma $ and Hei in emission in X Per would indicate that it belongs to Group i and its spectral type is hence expected to be earlier than B3 — this is consistent with its present spectral classification of O9.5 iiie.",
"An interesting aspect of the spectroscopy is the presence of the Oi 1.1287 $\\mu $ m and 1.3165 $\\mu $ m lines in the $J$ band spectra.",
"The relative strengths of these lines can help discriminate whether the Ly$\\beta $ fluorescence mechanism is operational or not in the star.",
"The Ly$\\beta $ fluorescence mechanism was proposed by [4] wherein due to the near coincidence of wavelengths, hydrogen Ly$\\beta $ photons at 1025.72 Å can pump the Oi ground state resonance line at 1025.77 Å thereby populating the Oi 3d$^3$ D$^0$ level.",
"The subsequent downward cascade produces the 11287, 8446 and 1304 Å lines in emission thereby enhancing the strengths of these lines.",
"It is expected that $W$ (1.3165)/$W$ (1.1287) $\\ge $ 1 if continuum fluorescence is the significant excitation mechanism for these lines ( [33], [14]; $W$ is the equivalent width).",
"On the other hand, if excitation by the Ly$\\beta $ fluorescence process is significant, the 1.1287 $\\mu $ m line should become stronger of the two lines.",
"Since we measure a mean value of $W$ (1.3165)/$W$ (1.1287) = 0.43 for for all epochs of our observations, it is implied that the Ly$\\beta $ fluorescence process is operative and has a significant role in the excitation of the Oi lines.",
"The equivalent widths of the prominent lines, measured in Å are given in Table 3 – the typical error in the measurement of the equivalent width values is 10 %.",
"Figure: Near-IR light-curve of X Per during the period 2010 December to 2011 February.Table: Journal of the photometric observations.The errors on the JHKJHK magnitudes are shown in brackets.Table: Journal of the spectroscopic observations.Figure: Flux calibrated JJ band spectra of X Per are displayed at differentepochs with an offset between adjacent spectra for clarity.",
"The amount of offset inunits of 10 -15 ^{-15} Wcm -2 ^{-2}μ\\mu m -1 ^{-1} is shown in brackets after thedate of observation.Figure: Flux calibrated HH band spectra of X Per are displayed at differentepochs with an offset between adjacent spectra for clarity.",
"The amount of offset inunits of 10 -15 ^{-15} Wcm -2 ^{-2}μ\\mu m -1 ^{-1} is shown in brackets after thedate of observation.Figure: Flux calibrated KK band spectra of X Per are displayed at differentepochs with an offset between adjacent spectra for clarity.",
"The amount of offset inunits of 10 -15 ^{-15} Wcm -2 ^{-2}μ\\mu m -1 ^{-1} is shown in brackets after thedate of observation."
],
[
"Analysis of the continuum",
"We construct and analyse the spectral energy distribution of the object in this section based on the near-IR magnitudes.",
"Since the $JHK$ photometric estimates during the period of observation do not change much (Table 1), it is adequate to model the SED by considering the data for one representative epoch - we have taken the 2010 December 10 data for this purpose.",
"Figure 5 shows the SED where the $JHK$ spectra have been dereddened using the corresponding $JHK$ photometric values and $E(B-V)$ = 0.39 [10] using the task DEREDDEN in IRAF.",
"We have shown a blackbody curve at T = 31400 K corresponding to the effective temperature determined for X Per [10].",
"In spite of a thorough search of available databases, we are unable to locate any $V$ band measurement contemporaneous with our near-IR observations.",
"We have thus used the long-term compilation of photometric magnitudes of [35] and assume that the colors of the star (for e.g.",
"$(V-J)$ , $(V-H)$ etc.)",
"should remain fairly the same at similar brightness levels.",
"That is, similar $JHK$ magnitudes in Telting's and the present study should be accompanied by similar $V$ magnitudes.",
"As mentioned earlier, our $JHK$ values are similar to that obtained on 1994 September 16 – 20 when X Per was in a high brightness state.",
"Hence we have taken the $V$ magnitude as 6.24, corresponding to 1994 September 25, as given in [35].",
"The blackbody curve in Figure 5 has hence been anchored to this $V$ band magnitude which is also deredenned using $E(B-V)$ = 0.39.",
"From Figure 5 it is evident that a blackbody curve poorly fits the SED of X Per and an infrared excess is seen which we attribute to free-free (f-f) emission from the disc.",
"The observed free-free excess can be modeled to obtain an average value of the electron density in the disc.",
"Given a distance $D$ to the object, a volume $v$ for the emitting ionized gas of the disc, the observed flux $F$ (in units of W cm$^{-2}$ $\\mu $ m$^{-1}$ ) due to f-f contribution will be given by $F = j_{\\lambda ff} \\times v / 4 \\pi D^2$ where the free-free volume emission coefficient, $j_{\\lambda ff}$ (in units of W cm$^{-3}$ $\\mu $ m$^{-1}$ ) can be calculated from $j_{\\lambda ff} = 2.05\\times 10^{-30} \\lambda ^{-2} z^2 g T_s^{-1/2} n_e n_i exp(-c2 / \\lambda T_s)$ In the above $\\lambda $ is the wavelength of emission in $\\mu $ m, $z$ is the charge, $g$ is the Gaunt factor, $T_s$ is the disc temperature, $n_e$ and $n_i$ are the electron and ion densities respectively and $c2$ = 1.438 cm K [2].",
"In the case of a circumstellar disc, the volume of the emission region ($v$ ) is reasonably estimated as $\\pi $$R_s$$^2$$H_s$ , where $R_s$ and $H_s$ are the disc radius and thickness respectively.",
"In the case of Be stars, the disc thickness can be approximated to be one-fifth of stellar radius [11].",
"We assume $g$ and $z$ to be unity, $n_e$ = $n_i$ for a pure hydrogen shell, and adopt a distance to the object of 1300 pc from [10].",
"The calculated values of free-free electron flux as a function of wavelength is shown as dotted line in Figure 5 for a temperature of 10,000 K and $n_e$ = 4$\\times $ 10$^{11}$ cm$^{-3}$ .",
"The free-free contribution, computed for a choice of these parameters, when added to the blackbody curve is found to reproduce the observed SED much better than a blackbody alone.",
"This value of electron density is comparable with that expected from observational modeling (e.g.",
"[29]; [5]; [12]).",
"A realistic model should take into account the optical depth effects while calculating the continuum emission.",
"Figure: The spectral energy distribution of X Per for the epoch 2010 December 10is shown in the figure.",
"The flux calibrated JHKJHK spectra are dereddenedusing E(B-V)E(B-V) = 0.39.",
"The blackbody corresponding to the central stars temperature ofT = 31400 K is shown by adashed line, the free-free contribution from the disc by a dotted lineand their co-added sum by a solid line.",
"Also shown is the VV band fluxcorresponding to high brightness state (filled circle).",
"For further details see section3.2.Figure: Recombination case B analysis of the Brackett lines of hydrogen are shown for X Perwhere the Br line fluxes are all normalized with respect to the line flux of Pa β\\beta .The lines shown are Br10 – 18 (1.7367 – 1.5346 μ\\mu m) and Brγ\\gamma .A large deviation from case B predictions is seen especially in the behavior of the higher Br lines.Filled triangles indicate the observations taken on 2010 December 10 while the filledcircles correspond to that on 2011 January 26.Solid lines represent case B values for T e T_e = 10 4 ^4 K,n e n_e = 10 10 ^{10} (shown as continuous line), 10 12 ^{12} (shown as dashed line)and 10 14 ^{14} cm -3 ^{-3} (shown as dot-dashed line).Table: Measured equivalent widths of the emission lines in Å."
],
[
" Recombination analysis of the hydrogen lines",
"Of the Hi lines, only two of the Paschen series lines could be covered in the spectra presented here viz.",
"Pa$\\beta $ at 1.2818 $\\mu $ m and Pa$\\gamma $ at 1.0938 $\\mu $ m. Whenever recorded, Pa$\\gamma $ is found to be stronger than Pa$\\beta $ contrary to what is expected (see Figure 2).",
"This indicates that these Paschen lines are optically thick since the expected ratio in recombination case B conditions is $I$ (Pa$\\beta $ )/$I$ (Pa$\\gamma $ ) $\\sim $ 1.57 – 2.01 for typical densities and temperatures prevailing in Be star discs (i.e.",
"$T_e$ = 10$^4$ K, $n_e$ in the range 10$^{10}$ to 10$^{14}$ cm$^{-3}$ ; here $I$ is the line intensity in units of erg cm$^{-2}$ s$^{-1}$ whose values are taken from [32]).",
"In essence, Pa$\\beta $ is always expected to be stronger than Pa$\\gamma $ under optically thin case B conditions.",
"Optical depth effects are more clearly seen in the Brackett series lines viz.",
"Br$\\gamma $ in the $K$ band and Br10 to 18 in the $H$ band.",
"In Figure 6, we present plots of the observed strength of Br lines versus their predicted intensities under recombination case B conditions.",
"The line fluxes used in this figure were derived from the spectra which were flux calibrated by the broad-band $JHK$ magnitudes of Table 1 and dereddened using a value of $A_V$ = 1.19 [36].",
"We have presented the data only for 2010 December 10 and 2011 January 26 in Figure 6 since there is considerable cluttering and loss of clarity if the observed data of all 5 days are presented.",
"But we found that a similar trend for the line strengths, as presented in Figure 6, is seen too for the data of the other days.",
"The case B line intensities are from [32] for a temperature $T_e$ = 10$^4$ K and for three representative values of the electron density $n_e$ = 10$^{10}$ , 10$^{12}$ and 10$^{14}$ cm$^{-3}$ , respectively.",
"As can be seen from Figure 6, the observed line intensities deviate significantly from the optically thin case B values.",
"This indicates that Brackett lines are optically thick during all the epochs of observation of X Per.",
"This observed behaviour of the Br line strengths is consistent with that seen in Be stars in general.",
"[31], from their $H$ band spectroscopy of 57 Be stars, showed that the strengths of Br 11 to Br 18, relative to each other, do not in general fit case B theory particularly well.",
"Being a paper devoted to the $H$ band, they did not include Br$\\gamma $ in their analysis, but its inclusion here in Figure 6 brings out the deviation from case B values even more clearly.",
"From recombination theory it is qualitatively expected, under optically thin conditions, that when strengths of lines of the same series are compared, a lower line of the series should be stronger than a higher line.",
"For example, it is expected that Br$\\gamma $ (corresponding to a transition between levels 7–4) is expected to be stronger than any higher line of the series like Br10 or Br11 (transitions between 10—4 and 11–4 respectively).",
"But the reverse is actually being observed here.",
"It is interesting to note that lines of even the higher Pfund (Pf) and Humphrey (Hu) series could in general be optically thick in Be stars.",
"For e.g., of the 8 Be stars for which the $L$ band spectra have been presented by [13], the spectra of EW Lac and BK Cam cover several of the Pf and Hu lines viz.",
"Pf 8, 9, 10 and 17 – 27 & Hu 14 – 25.",
"Analysis of their strengths, and comparison with case B predictions, shows a strong departure from the optically thin case (Figures 4 and 5 in [13]).",
"We analyzed the case B model values to check whether opacity effects are expected to affect the strengths of the Br line shown in Figure 6 under the density conditions prevailing in Be discs.",
"It is verified from [16] and [32] that line center optical depth values can be significant when the densities become large as in Be star discs.",
"The above studies tabulate the value of the opacity factor $\\Omega $$_{n,n^{\\prime }}$ for different Hi lines for transitions between levels $(n,n^{\\prime })$ at different temperatures and densities (equation 29 of [16]).",
"From this value of $\\Omega $$_{n,n^{\\prime }}$ , the optical depth at line-center $\\tau $$_{n,n^{\\prime }}$ is to be calculated using $\\tau $ = $n_e$$n_i$$\\Omega $$L$ , where $L$ is the path length in cm.",
"As a representative example we consider a Br line photon, originating in the disc, trying to escape across the thickness of the disc.",
"Then $L$ may reasonably be approximated as the thickness of the circumstellar disc which is typically assumed to be 1/5 times the stellar radius.",
"For X Per, the stellar radius is estimated as 13$\\times $$R_\\odot $ [10] for which $L$ comes out to be 1.82$\\times $ 10$^{11}$ cm.",
"The corresponding value of the optical depth $\\tau $ , from [32], is then found to be $\\sim $ 13500 for Br$\\gamma $ and this then decreases monotonically down the series to a value of $\\tau $ $\\sim $ 300 for Br 18 (these $\\tau $ values are computed at representative values of $T_e$ = 10$^4$ K and $n_e$ = 10$^{13}$ cm$^{-3}$ ).",
"At a lower density of $n_e$ = 10$^{11}$ cm$^{-3}$ , the $\\tau $ values decrease by approximately a factor of $\\sim $ 10000 (since $\\tau $ $\\propto $ $n_en_i$ ) to $\\tau $ $\\sim $ 1 so lines like Br$\\gamma $ still remain significantly thick.",
"Thus recombination theory does predict that the Br and Pa lines should in general be optically thick.",
"Going beyond the Pa and Br lines, it may also be easily verified from the [32] data that the optical depth in the Humphrey and Pfund lines are also large - of similar magnitude as the $\\tau $ values for the Br lines - at similar densities considered here.",
"This is likely to explain the observed deviations of the strength of these lines from case B predictions as seen in the data of [13].",
"Unrelated to Be stars but worth mentioning in this connection, is the analogous behaviour of the Pa and Br line strengths arising from the ionized ejecta of novae.",
"Here too, similar optical depth effects are sometimes seen as for example in the spectra of nova Oph 1998 (V2487 Oph) and nova Sgr 2001 (V4643 Sgr) studied by [19] and [1], respectively.",
"The large optical depths seen in the Br and Pa lines in such cases are well explained by a model developed by [19] who show that the relatively larger intensities of the higher members of the Paschen and Brackett series arise because of emission from high-density or optically thick emission line gas.",
"In effect, it is only at high densities (around 10$^{10}$ – 10$^{12}$ ) which occur in novae ejecta just after outburst before expansion dilutes the ejecta, that optical depth effects become pronounced."
],
[
"Discussion",
"In Be stars the situation regarding optical depth effects could be complex since the optical depth in a line will also depend on the region from where the line emanates.",
"Interferometric results indicate that different lines originate from different regions in the circumstellar disc, i.e., regions of different electron density and hence different $\\tau $ values.",
"The presence of density enhancements due to spiral waves in the disk could complicate matters further ([38], [15]http://utmost.cl.utoledo.edu).",
"Disc size estimates that are now available show considerable variation in sizes from measurements around the H$\\alpha $ emission line (e.g.",
"[24], [30]; several papers by Tycner and collaborators); in the infrared 10 $\\mu $ m $N$ band continuum [6]; in the near-IR $K$ band (e.g.",
"[12]); in the $H$ band continuum and also in $K$ band Br$\\gamma $ and Hei 2.058 $\\mu $ m emission lines [21].",
"A summary of the interferometric results available upto fairly recent times may be found in [12] and Monnier (2003; and references therein).",
"Variations in disc sizes, though not measured directly, are also suggested by the modeling of [17] for the Oi and Paschen lines."
],
[
"Acknowledgments",
"We thank the referee, Prof. Douglas R. Gies, for his suggestions and comments which helped in improving the manuscript.",
"The research work at Physical Research Laboratory is funded by the Department of Space, Government of India."
]
] | 1204.1496 |
[
[
"Creation and characterization of vortex clusters in atomic Bose-Einstein\n condensates"
],
[
"Abstract We show that a moving obstacle, in the form of an elongated paddle, can create vortices that are dispersed, or induce clusters of like-signed vortices in 2D Bose-Einstein condensates.",
"We propose new statistical measures of clustering based on Ripley's K-function which are suitable to the small size and small number of vortices in atomic condensates, which lack the huge number of length scales excited in larger classical and quantum turbulent fluid systems.",
"The evolution and decay of clustering is analyzed using these measures.",
"Experimentally it should prove possible to create such an obstacle by a laser beam and a moving optical mask.",
"The theoretical techniques we present are accessible to experimentalists and extend the current methods available to induce 2D quantum turbulence in Bose-Einstein condensates."
],
[
"Introduction",
"Turbulent superfluid helium has been shown to exhibit features typical of ordinary, classical turbulence, on scales larger than the average inter-vortex spacing, such as the same Kolmogorov energy spectrum in three dimensional (3D) turbulence [1], [2], [3], [4], [5], [6], [7], [8] as well as differences, such as non-Gaussian velocity statistics [9], [10], [11], [7].",
"Two dimensional (2D) turbulence is very different from 3D turbulence.",
"In 3D turbulence, on scales larger than the average inter-vortex spacing, there is a flow of energy from large scales to small scales through the Richardson cascade, whereby large eddies are broken up into smaller and smaller eddies giving a Kolmogorov energy spectrum with $k^{-5/3}$ scaling.",
"In 2D turbulence there is an inverse energy cascade, where energy flows from the scale of energy injection (small scales), to larger scales as like-charged vortices cluster [12].",
"This phenomenon is established in classical fluids and is thought to be the mechanism responsible for Jupiter's great red spot [13], [14].",
"The concept of an inverse energy cascade was introduced by Onsager [15] for a point-vortex gas consisting of many many vortices, where it was found clusters of like-signed point vortices have a negative temperature (further detail can be found in subsequent analyses [16], [17]).",
"Beyond the seminal work of Onsager, there is no fundamental theory that exists for small systems of vortices.",
"In Bose-Einstein condensates, while there have been a number of theoretical investigations [18], [19], [20], [21] and some experimental work [22], the question of whether an inverse cascade is a feature of a system of turbulent vortices in quasi-two dimensional systems remains open; this is firstly because these condensates are relatively new and secondly (and more importantly) because, due to their relatively small size, they lack the large range of length scales typical of other 2D turbulent flows (such as planetary atmospheres).",
"On the other hand, ultra-cold Bose condensed atomic gases are ideal to investigate the dynamics of few-vortex systems as well as turbulent dynamics of many vortices.",
"The dimensionality of such condensates can be easily controlled, allowing direct study of vortices in two and three dimensions.",
"Advanced experimental methods for the imaging and detection of vortices in Bose–Einstein condensates (BECs) have been developed [23] and new theoretical proposals have shown that vortices can be easily manipulated in BECs [24].",
"There are many techniques that can be applied to nucleate vortices in BECs, such as: engineering the condensate phase profile [25], [26]; stirring the condensate with a blue or red detuned laser (for experiments see [27], [28] and theory [29], [30], [24]); mixing and merging condensates of well defined phases [31], [32], [33]; moving a condensate past a defect [28]; rotating the trapping potential or thermal cloud [34], [35], [36], [37], [38], [39]; and cooling the condensate with a rapid quench through the phase transition (Kibble-Zurek mechanism) [40], [41], [42].",
"Finally, vortices can be nucleated from dynamical instabilities, such as through the decay of the snake instability of a soliton [43], [44], the bending wave instability of a vortex ring [45], [46] or surface mode excitations of the condensate [47], [48].",
"In the first experimental demonstration of 3D quantum turbulence in a BEC, tangled vortices were created by shaking the condensate with an oscillatory trapping potential [47], [48], [49], [50].",
"In this paper we add to the numerous existing techniques applied to induce vortices in BECs, and apply a moving object with an elliptical paddle shape to create vortices.",
"We demonstrate that the trajectory of the optical paddle through the condensate can be controlled to create vortices that are both well distributed or clustered into groups with like-winding.",
"To determine if vortices are indeed clustered, and if this clustering increases or decreases significantly with time, we develop tools which are more suitable to the small size and the relative small number of vortices which can be generated in ultracold atom BECs.",
"Drawing on the wealth of available statistical pattern recognition methods, we analyze our data using Besag's function [51], a modification of Ripley's K-function [52], [53], which has been extensively applied across a variety of scientific fields to measure clustering and clumping of discrete objects e.g.",
"[54], [55], [56], [57], [58], [59], [60].",
"Motivated by Besag's function, we develop some new measures of independent clustering when the system is comprised of two unique types of discrete objects.",
"In our case, these are vortices with `$+$ ' or `$-$ ' winding in a BEC.",
"These techniques can distinguish between the cases of mixed clusters of vortices and independent clusters of vortices of like-winding in the condensate.",
"After reviewing our theoretical model, we analyze vortex structures obtained by different forms of stirring by applying both Besag's function and nearest neighbor techniques.",
"Finally, we summarize our main conclusions, that the resulting vortex clustering is strongly dependent on the trajectory of the moving paddle.",
"We find no compelling evidence of an inverse cascade in these small systems, in the sense that we don't see an increase in clustering over time."
],
[
"Numerical Model",
"We simulate the dynamics of a trapped two-dimensional Bose-Einstein condensate stirred by an optical paddle by integrating the following 2D dimensionless time-dependent Gross-Pitaevskii equation (GPE): $i \\frac{\\partial \\psi }{\\partial t} = -\\frac{1}{2}\\nabla ^{2}\\psi + \\frac{{r}^{2}}{2}\\psi +V_{P}\\psi + \\kappa _{2d} |\\psi |^{2}\\psi \\,,$ where the interaction strength, $\\kappa _{2d}= 2\\sqrt{2\\pi }aN / a_{z} $ , is written in terms of the scattering length, $a$ , and the total number of condensate atoms, N. $\\psi $ is scaled as $\\psi =a_{r}\\Psi /\\sqrt{N}$ , where $\\Psi $ is the condensate wavefunction and is normalized to unity i.e.",
"$\\int \\text{d\\bf {x}}|\\psi |^{2}=1$ .",
"Lengths and time are scaled as $\\tilde{r}/a_{r}=r$ and $\\tilde{t}\\omega _{r}=t$ , where $\\tilde{r}$ and $\\tilde{t}$ are dimensional with units of meters and seconds respectively.",
"$a_{z}= \\sqrt{\\hbar /(m\\omega _{z})}$ and $a_{r}= \\sqrt{\\hbar /(m\\omega _{r})}$ are the axial and radial harmonic oscillator lengths, determined by the axial and radial trapping frequencies, $\\omega _{z}$ and $\\omega _{r}$ respectively.",
"The potential, $V_{P}$ , describes a far-off-resonance blue-detuned laser beam shaped into a paddle that follows a rotating and circular stirring trajectory given by $\\nonumber V_{P} = V_{0}\\exp \\left[-{\\eta }^{2}\\frac{\\left(\\tilde{x}\\cos (\\omega t)-\\tilde{y}\\sin (\\omega t)\\right)^{2}}{d^{2}} \\right.",
"\\\\ \\left.",
"- \\frac{\\left(\\tilde{y}\\cos (\\omega t)+\\tilde{x}\\sin (\\omega t)\\right)^{2}}{d^{2}}\\right]\\,,$ where $\\tilde{x} = x-v\\sin (t)$ and $\\tilde{y}=y-v\\cos (t)$ .",
"$V_{0}$ gives the peak strength of the potential and is selected to be $V_{0}\\sim 2.6 \\mu $ , where $\\mu $ is the chemical potential of the BEC.",
"$\\eta $ determines the paddle elongation and $d$ the width.",
"Experimentally, the paddle can be shaped by shining a far-off-resonance blue detuned laser through a mask as in [61].",
"For a paddle rotating with a frequency, $\\omega $ , at the center of the condensate, $\\tilde{x} = x$ and $\\tilde{y} = y$ .",
"For a paddle moving at a constant radius from the center of the condensate without rotating, we take $\\omega =1$ .",
"In all simulations the paddle is initially linearly ramped up to its maximum stirring frequency, $\\omega $ , after which the condensate is stirred at constant $\\omega $ until $t_{S}=20$ .",
"The paddle is then ramped off linearly over $t=5$ by making the replacement $V_{P} \\rightarrow \\left(1-\\frac{(t-t_{S})}{5}\\right)V_{P},$ in Eq.",
"(REF ).",
"For paddle sizes and stirring frequencies which result in vortex formation, the condensate dynamics are evolved for a further time of $t=45$ .",
"Numerically, Eq.",
"(REF ) is solved pseudo-spectrally with periodic boundary conditions and integrated in time by applying an adaptive 4th-5th order Runge-Kutta method with the help of xmds [62].",
"The initial state for our simulations is obtained by a short propagation of the 2D GPE in imaginary time (by making the replacement $\\tau = -it$ in Eq.",
"(REF )), and applying a stationary paddle potential.",
"The condensate is parameterized by the nonlinear interaction strength $\\kappa _{2d} = 10399$ .",
"For example, by choosing experimentally relevant axial and radial trapping frequencies of $\\omega _{z}=2\\pi \\times 50$ Hz and $\\omega _{r}=2 \\pi \\times 5$ Hz, this describes a condensate of $2.2\\times 10^{6}$ $^{23}$ Na ($6\\times 10^{5}$ $^{87}$ Rb) atoms with scattering length $a = 2.75$ nm ($a=5.29$ nm).",
"Our simulations are run on a grid of spatial extent $-20$ to 20 with gridsize $N_{g}=512$ (see appendix)."
],
[
"Vortex Generation",
"This paper looks at three stirring motions of the paddle, $V_{\\text{I}}$ , $V_{\\text{II}}$ and $V_{\\text{III}}$ , each of which are specific realizations of Eq.",
"(REF ), corresponding to a rotating paddle, a paddle moving on a deferent from the condensate center and a paddle moving along a deferent while rotating, respectively (see figure 1, top row).",
"The stirring motions of the paddle studied generate contrasting vortex configurations, as discussed below.",
"For each case, we track the total number of vortices nucleated, as shown in figure 3a.",
"In all cases, at later times, vortices are lost to the edge of the condensate.",
"The condensate edge is selected by identifying where the density falls to less than $30\\%$ of the maximum condensate density at that time.",
"The resulting profile is then smoothed to give the condensate edge.",
"When two vortices are closer than a critical separation distance and are of opposite winding, vortex pair annihilation occurs, a mechanism which also reduces the total number of vortices in the condensate.",
"For all simulations the total angular momentum of the condensate is also tracked, given by $\\langle L_{z} \\rangle = -i\\int \\text{d\\bf {x}}\\psi ^{*}\\left(x\\frac{\\partial }{\\partial y}-y\\frac{\\partial }{\\partial x}\\right)\\psi \\,.$"
],
[
"Case I: Paddle rotating at the condensate center",
"Firstly we look at a paddle rotating about its center with frequency $\\omega $ at the center of the condensate.",
"This is modeled by evolving Eq.",
"(REF ) with $V_{\\text{I}}=V_{P}(\\tilde{x}\\rightarrow x, \\tilde{y}\\rightarrow y)$ .",
"The smallest paddle size we consider is $d=0.5$ , rotating at a frequency $\\omega = 6$ .",
"The rotating motion of the paddle produces circular spiral sound waves, which at late times interfere with each other giving a wave interference pattern (see the first column of figure 1).",
"Paddles with a larger width ($d=1$ ), rotating at frequencies of $\\omega = 4, 6$ and 8, nucleate vortices in addition to creating spiral sound waves.",
"The density profile for a paddle rotating at frequency $\\omega = 4$ is show in the second column of figure 1 (see supplementary movie 1 for a comparison of paddles rotating at frequencies $\\omega = 4$ and $\\omega = 8$ ).",
"A greater rate of rotation increases the number of vortices initially nucleated, as expected.",
"In all cases, vortices are initially nucleated from the ends of the paddle with winding opposite to the direction of rotation of the paddle, as depicted in figure 2.",
"At subsequent times, when the local superfluid velocity surpasses the critical velocity for vortex nucleation [63], vortices of both signs are nucleated from both the center and ends of the paddle.",
"A centered rotating paddle imparts a small amount of angular momentum to the condensate.",
"The angular momentum imparted is proportional to the frequency of the rotating paddle (see figure 3b).",
"Note that after paddles of frequency $\\omega = 6$ and 8 have been ramped off, the condensate angular momentum saturates to approximately the same value.",
"Figure: (Color online)Schematic diagram of vortices nucleated from a paddle rotating at the condensate center, V I V_{\\text{I}} as the paddle is ramped up to ω=8\\omega =8.",
"Positive vortices are identified by (pink) magenta+\\color {magenta}{+} signs and negative vortices by (blue) cyan∘\\color {cyan}{\\circ } signs, the paddle profile is shown in purple.Figure: Angular MomentumThe second stirring motion of the laser paddle we simulate is a paddle stirring the condensate at a constant radius from the condensate center, or traveling along a deferent, $V_{\\text{II}}=V_{P}(\\omega =1)$ (see figure 1, column 3, top row, for a schematic of the paddle motion).",
"This paddle trajectory creates vortices which tend to group initially in like-signed clusters as shown by the progressive time samples of the condensate density profiles in figure 1, column 3 (see also supplementary movie 2).",
"The condensate gains a large amount of angular momentum (refer to figure 3b), and consequently at final times there is a significant imbalance in the total number of vortices with positive and negative winding.",
"The final stirring trajectory of the laser paddle we simulate, $V_{\\text{III}}=V_{P}$ , is a combination of the two previous motions, with the paddle moved at a constant radius from the condensate center while rotating at a small frequency.",
"This motion can also be described as a paddle rotating and moving along the deferent (see figure 1, column 4, top row, for a schematic of the paddle motion).",
"The effect of adding the rotational motion of the paddle to its trajectory mixes the clusters produced, resulting in smaller groups consisting of 3 and 4 vortices (compare the vortex distributions in figure 1 columns 3 and 4 and supplementary movie 2).",
"While initially (for times $t<12$ ) the angular momentum transfered to the condensate is reduced in comparison to that of paddle $V_{\\text{II}}$ (see figure 3b), this stirring trajectory results in the greatest transfer of angular momentum to the condensate at later times."
],
[
"Ripley's K-function",
"We analyze the clustering of vortices formed by stirring a 2D condensate with a paddle by applying Ripley's K-function, a statistical pattern analysis method used as a measure of spatial clustering.",
"In the context of clustering of like-signed vortices, Ripley's K-function is dependent on the total number of liked signed vortices, $N$ , within the total condensate area, $A$ , and can be expressed as $K(r) = \\frac{A}{N^{2}}\\sum _{i=1}^{N}\\sum _{j=1}^{N}f_{ij}(r)\\,,$ where $f_{ij}(r) = 1$ for a vortex, $j$ , within a distance $r$ of the reference vortex, $i$ , with like-winding.",
"Otherwise, $f_{ij}(r) = 0$ if $i=j$ , or if the distance between vortex $i$ and $j$ is greater than $r$ .",
"That is $f_{ij} = \\left[\\begin{array}{ll}1 & \\forall \\, r_{ij}<r, \\, i \\ne j \\\\0 & \\forall \\, r_{ij}>r \\,\\,\\text{or}\\,\\, i = j\\end{array}\\right.$ here $r_{ij}$ is the distance from a reference vortex $i$ to the comparison vortex $j$ with like-winding.",
"This is depicted in figure 4.",
"Figure: The function f ij (r)f_{ij}(r) in Ripley's K-function.Figure: Besag's function L(r/r c )L(r/r_{c}) (see Eq.",
"9) for vortices of negative winding at varying times (see legend) for a condensate with vortices nucleated by a paddle with trajectory V II V_{\\text{II}}.",
"Parameters: d=1d=1, ω=1\\omega =1 and v=4v=4.Ripley's K-function looks at the number of like-signed vortices within a radius, $r$ , from the position of an arbitrarily chosen vortex, $i$ , at its center (see figure 4 and 5).",
"If the number of vortices with like-winding per unit area within this radius, $r$ , is greater than the overall number of like-signed vortices per unit area for the whole condensate, then the vortices are said to be clustered.",
"Clustering results in $K(r)$ increasing faster than if vortices of either sign are distributed in a spatially random manner, that is, if they follow a Poisson distribution.",
"Ripley's-K function for a poisson-distributed data set takes the form $K(r)=\\pi r^{2}$ .",
"For a linear scaling of Poisson-distributed data, it is useful to normalize Ripley's K-function to $H(r)=\\sqrt{K(r)/\\pi }$ .",
"Ripley's L-function, also commonly known as Besag's function, is obtained from further normalization of Ripley's K-function: $L(r) = \\sqrt{K(r)/\\pi }-r.$ As the condensate area, $A$ , does not necessarily remain constant over all times, we scale $r$ by the characteristic condensate radius $r_{c}=\\sqrt{A/\\pi }$ for that time and in our subsequent analysis evaluate $L(r/r_{c}) =\\sqrt{\\frac{A}{\\pi (Nr_{c})^{2}}\\sum _{i=1}^{N}\\sum _{j=1}^{N}f_{ij}(r/r_{c})}-\\frac{r}{r_{c}},$ which simplifies to: $L(r/r_{c}) =\\sqrt{\\frac{1}{N^{2}}\\sum _{i=1}^{N}\\sum _{j=1}^{N}f_{ij}(r/r_{c})}-\\frac{r}{r_{c}}.$ Besag's function is zero for like-signed vortices which are randomly distributed, takes positive values for vortices clustered over that spatial scale, and is negative if the vortex distribution is dispersed.",
"That is: $L(r/r_{c}) = \\left[\\begin{array}{rl}1 & \\text{Clustered} \\\\0 & \\text{Random} \\\\-1 & \\text{Dispersed}\\end{array}\\right.$ The radius around a centered vortex containing, on average, the most like-signed vortices per area, is called the radius of maximal aggregation, and is given by the value of $r$ which maximizes $L(r)$ [59].",
"Figure: (Color Online) Schematic diagrams of independent (A) and co-clustering (B) in systems with two distinct types of objects, represented by pink pluses and blue circles respectively.For a paddle rotating at a constant radius from the condensate center, evaluating $L(r/r_c)$ (see equation 9) for positive vortices, as seen in figure 5, shows that the clustering of vortices decreases with time.",
"Although the vortices are clustered, the amount of clustering is not constant or increasing in time, giving no evidence of an inverse cascade for this system.",
"Figure: (Color online) Comparison of evolution second nearest neighbor data, C 2 (t)C_{2}(t), (top figure) and fourth nearest neighbor data, C 4 (t)C_{4}(t), (bottom figure) for paddle trajectories V II V_{\\text{II}}, V III V_{\\text{III}} and V I V_{\\text{I}} with v=8v=8 represented by green triangles, blue squares and pink diamonds respectively.",
"Time is measured from the beginning of stirring the condensate.Figure: (Color online) Comparison of evolution second nearest neighbor data, C 2 (t)C_{2}(t) (top figure), and fourth nearest neighbor data, C 4 (t)C_{4}(t) (bottom figure), for vortices created by rotating a paddle at the condensate center (V I V_{\\text{I}}) with v=4v=4 (orange ++), v=6v=6 (purple crosses) and v=8v=8 (pink diamonds).",
"Time is measured from the beginning of stirring the condensate."
],
[
"Measures of independent clustering of like-signed vortices",
"While Besag's function gives a measure of the clustering of vortices with the same winding, it does not discriminate between cases where like and opposite signed vortices are clustered in the same spatial region and cases where like-signed vortices are clustered in spatially independent regions (refer to figure 6 for a schematic illustration).",
"It is necessary to make this distinction when looking for a measure of an inverse cascade process, as the clustering of like signed vortices is not expected to occur in the same spatial region as clustered vortices of the opposite sign.",
"To address this issue we define a new measure of clustering which uses the sign of nearest neighboring vortices to determine if clustering occurs in regions that are spatially independent.",
"We express this measure of clustering based on looking at the sign of all $j$ neighboring vortices up to the $B$ th nearest neighbor of an arbitrary reference vortex $i$ as $C_{B}(t) = \\frac{1}{N}\\sum _{i=1}^{N}\\sum _{j=1}^{B}\\frac{c_{ij}(t)}{B}\\,.$ Here $c_{ij} = 1$ if the vortex $i$ and its $j^{th}$ nearest neighbor are of the same sign and $c_{ij} =0$ if vortex $i$ and its $j^{th}$ nearest neighbor are of opposite sign.",
"If vortex $i$ is separated by distance greater than $R_c=r_{c}/3$ to its $j^{th}$ nearest neighbor, then $c_{ij} = 0$ .",
"$B$ is the maximum nearest neighbor to the reference vortex $i$ .",
"It is necessary that the value of $R_{c}$ chosen is greater than the average inter-vortex separation distance and on order of the largest cluster size.",
"Vortices closer than $R_{c}$ to the condensate edge will bias the calculation of $C_{B}(t)$ as they have an area less than $\\pi R_{c}^2$ surrounding them, which their nearest neighboring vortices could inhabit.",
"To correct for these edge effects we omit these vortices which are less than $R_{c}$ from the condensate edge from the set of reference vortices, but still include them in the set of comparison vortices for vortices a distance greater than $R_{c}$ from the condensate edge.",
"Systems for which $C_{B}=0.5$ are randomly distributed and when $C_{B}$ takes values greater than $0.5$ the objects are clustered.",
"We note that these measures can be applied generically to investigate cases of co and independent clustering of two discrete objects and could be simply extended to look at co and independent clustering of many discrete objects.",
"A comparison of the evolution of $C_{B}(t)$ for the simulation runs described previously is shown in figures 7 and 8.",
"From the nearest neighbor analysis we learn: From figure 7, we can see that a paddle moving at a constant radius from the condensate center ($V_{\\text{II}}$ ) creates vortices that are initially very clustered, indicated by $C_{2}$ and $C_{4}$ taking values very near one.",
"After the paddle is turned off the vortices remain clustered, with $C_{2}$ and $C_{4}$ not decreasing below $0.5$ until $t\\approx 50$ .",
"At long times ($t > 50$ ) vortices become randomly distributed.",
"A paddle moving with trajectory $V_{\\text{III}}$ creates clusters that are initially smaller in size than purely moving the paddle at constant radius from the condensate center ($V_{\\text{II}}$ ).",
"After the paddle is turned off, the vortex distribution closely follows that of run $V_{\\text{II}}$ with vortices remaining clustered until long times ($t > 50$ ) when they become randomly distributed.",
"When the paddle is only rotated at the condensate center, $V_{\\text{I}}$ , vortices never become clustered (refer to figure 8).",
"For all cases, regardless of how the vortices are initially nucleated, evaluating $C_{B}(t)$ gives no evidence of a tendency of increasing clustering of like-signed vortices over a scale of $R_{c}$ in a turbulent 2D BEC after the laser paddle has been ramped off.",
"Our observations imply that clusters of like-signed vortices exist due to the way in which they were induced in the condensate but do not naturally tend to cluster."
],
[
"Conclusion",
"In this paper we have covered two main objectives: We have extended the available methods for creating vortices in 2D atomic Bose-Einstein condensates, demonstrating that a paddle can be used to stir a condensate in two quite different ways, creating long-lived vortex clusters or more randomly distributed vortices that are turbulent in two dimensions.",
"A new statistical measure of clustering based on analyzing nearest neighbor vortices was defined, motivated by a well known statistical spatial point pattern analysis technique, Besag's function.",
"These measures have been applied to analyze how vortices are distributed in 2D condensates.",
"We find that a paddle moved through the condensate at a constant radius from the center creates vortices of both positive and negative winding in clusters.",
"When the paddle is rotated at the condensate center, vortices created are initially clustered co-dependently in the same local spatial regions and later disperse throughout the condensate.",
"For a combination of both moving a paddle at a constant radius through the condensate while simultaneously rotating the paddle, the vortices induced are less clustered than if the paddle is only moved at a constant radius from the condensate center.",
"The later method can be applied to create long-lived vortex clusters in BECs.",
"The clusters are considered long lived in terms of two relevant sets of timescales; the constraining timescale determined by realistic experimental lifetimes of BECs and the timescales determined by the system size and properties.",
"The relevant timescales intrinsic to the system size and length-scales are determined by the average separation distance between vortices ($l_{sep}$ ) and is given by $\\tau _{sep} = l_{sep}^{2}/\\gamma $ where $\\gamma $ is the phase winding of a vortex and the largest turnover time $\\tau = \\pi r_{c}^{2}/2\\pi \\approx 35$ , is determined by the condensate size.",
"As the largest turnover time is smaller than the longevity of clusters, which persist for $\\sim 50$ (see $C(4)$ in Figure 7) we describe the clusters to be `long-lived'.",
"For our choice of experimental parameters, cluster lifetime is $\\sim 1$ s, which is also long-lived in comparison to the typical experimental lifetime of BECs, ($\\sim 10$ s).",
"The extent of clustering was quantitatively measured by evaluating two statistical measures of clustering; applying a modified Ripley's function and a technique based on comparing the sign of nearest neighboring vortices.",
"We did not observe an increase in clustering over time.",
"This was despite evolution times longer than the largest turnover time $\\tau = \\pi r_{c}^{2}/2\\pi \\approx 35$ , determined by the condensate size.",
"Our system contains too few vortices to determine if the relevant physical process for 2D turbulent systems in atomic Bose-gases is an inverse cascade of incompressible kinetic energy from small to large scales manifesting in a clustering of like-signed vortices.",
"In particular, it would be difficult to apply traditional methods used for large systems (planetary atmospheres, superfluid helium), based on Fourier transforming the velocity field and analyzing the spectra of energy and enstrophy over many decades in wavenumber space.",
"Our statistical analysis based on Ripley's K-function and on nearest neighbor methods provides a way to quantify an increase or decrease in the degree of vortex clustering.",
"As these methods are constructed from a knowledge of the position and winding of vortices in the system they are readily accessible experimentally.",
"Information on vortex location in condensates can be obtained experimentally through standard absorption imaging techniques, e.g.",
"[34], [35], [23], and winding of vortices is found by analysis techniques giving phase information, such as condensate interferometry, e.g.",
"[64], [27], [65].",
"A.",
"White thanks C. J.",
"Foster for the vortex detection algorithm based on the plaquette technique [66].",
"We thank A. Baggaley for useful discussions.",
"This work was supported by EPSRC grants EP/H027777/1 and BH101785."
],
[
"Error Checking",
"To check our results are independent of gridsize used, the simulation grid size was doubled from $N_{g}=512$ to $N_{g}=1024$ for a paddle rotating at the condensate center with $v=6$ and $d=1$ .",
"In figure 9 a comparison is made of the total vortex number and condensate angular momentum, when the grid size is doubled.",
"The angular momentum is calculated by evaluating Eq (REF ).",
"The reasonable agreement between rates of vortex production and elimination, as well as evolution of the condensate angular momentum in both runs establishes that the gridsize of $N_{g}=512$ applied in the simulations presented in the body of the paper is adequate.",
"The small variance in results from doubling the gridsize are attributed to the condensate edge selection routine used.",
"A further source of difference is the chaotic nature of vortex dynamics in turbulent systems.",
"A small amount of numerical noise would be enough to seed a difference in vortex trajectories."
]
] | 1204.1519 |
[
[
"Measurement of the underlying event in the Drell-Yan process in\n proton-proton collisions at sqrt(s) = 7 TeV"
],
[
"Abstract A measurement of the underlying event (UE) activity in proton-proton collisions at a center-of-mass energy of 7 TeV is performed using Drell--Yan events in a data sample corresponding to an integrated luminosity of 2.2 inverse femtobarns, collected by the CMS experiment at the LHC.",
"The activity measured in the muonic final state (q q-bar to opposite-sign muons) is corrected to the particle level and compared with the predictions of various Monte Carlo generators and hadronization models.",
"The dependence of the UE activity on the dimuon invariant mass is well described by PYTHIA and HERWIG++ tunes derived from the leading jet/track approach, illustrating the universality of the UE activity.",
"The UE activity is observed to be independent of the dimuon invariant mass in the region above 40 GeV, while a slow increase is observed with increasing transverse momentum of the dimuon system.",
"The dependence of the UE activity on the transverse momentum of the dimuon system is accurately described by MADGRAPH, which simulates multiple hard emissions."
],
[
"Introduction",
"In hadron-hadron scattering, the “underlying event” (UE) is defined as any hadronic activity that cannot be attributed to the particles originating from the hard scattering, which is characterized by a large momentum transfer, or to the hadronization of initial- and final-state radiation.",
"The UE activity is thus due to the hadronization of partonic constituents, not involved in the hard scattering, that have undergone multiple-parton interactions (MPIs) and to the hadronization of beam remnants that did not participate in other scatterings.",
"These semihard interactions cannot be completely described by perturbative quantum chromodynamics (QCD) and require a phenomenological description involving parameters that must be tuned with the help of data [1].",
"The experimental study of the UE probes various aspects of hadron production in high energy hadron-hadron collisions.",
"In particular it is sensitive to the interplay of perturbative methods describing the hard process and phenomenological models of the soft interactions that attempt to simultaneously describe MPIs, initial- and final-state radiation, the colour flow between final state partons, and the hadronisation process.",
"Understanding the UE in terms of particle and energy densities will lead to better modelling by Monte Carlo programs that are used in precise measurements of standard model processes and searches for new physics at high energies.",
"The UE affects the estimation of the efficiency of isolation criteria applied to photons and charged leptons, and the energy scale in jet identification.",
"It also affects the reconstruction efficiency for processes like H$\\rightarrow \\gamma \\gamma $ , where the primary vertex is partly determined from the charged particles originating from the UE.",
"Hard MPIs are an important background for new physics searches, e.g.",
"same-sign W production from MPIs [2] is a possible background to the same-sign double lepton SUSY searches [3].",
"The Compact Muon Solenoid (CMS) [4], ATLAS, and ALICE experiments have carried out UE measurements at centre-of-mass energies ($\\sqrt{s}$ ) of 0.9and 7using hadronic events (minimum-bias and single-jet triggered) containing a leading track-jet [5], [6] or a leading track [7], [8].",
"The analysis of the central charged particles and forward energy flow correlations in hard processes, e.g.",
"$\\rm pp \\rightarrow $ W()X $\\rightarrow \\ell \\nu (\\ell \\ell )$ X [9], provides supplementary insights into the nature of MPIs.",
"In this paper, we use the Drell–Yan (DY) process [10] with the muonic final state at a centre-of-mass energy of 7to perform a complementary UE measurement.",
"The DY process with muonic final state is experimentally clean and theoretically well understood, allowing the particles from the UE to be reliably identified.",
"The absence of QCD final-state radiation (FSR) permits a study of different kinematic regions with varying transverse momentum of $\\gamma ^{*}$ /due to harder or softer initial-state radiation (ISR).",
"The comparison of the UE measurement in DY events with QCD events having a leading track-jet is useful for probing the UE activity in different processes.",
"UE measurements using the DY process have been reported previously in proton-antiproton collisions at $\\sqrt{s}$ = 1.96 [11].",
"The UE activity at a given centre-of-mass energy is expected to increase with the momentum transfer of the interaction.",
"Events with a harder scale are expected to correspond, on average, to interactions with a smaller impact parameter and, in some models, to more MPIs [12], [13].",
"This increased activity is observed to reach a plateau for high energy scales corresponding to small impact parameter.",
"In this paper we investigate some aspects of the UE modelling in detail by measuring the invariant mass dependence of the UE activity for DY events with small transverse momentum of the DY system.",
"This measurement separates the scale dependence of the UE activity from the ISR effect.",
"The universality of the model parameters, denoted as tunes, implemented in the various MC programs is tested by comparing their predictions with our measurements.",
"The portability of the UE parameters across different event generators, combined in some cases with different parton distribution functions (PDFs), is investigated as well.",
"The modelling of the ISR is studied by measuring the UE activity as a function of the transverse momentum of the DY system.",
"Finally, the dependence of the UE activity on ISR and FSR is determined by comparing the measurements from DY events with previous results from hadronic events containing a leading jet where FSR also plays a role.",
"The outline of the paper is as follows.",
"Section 2 describes the various observables used in the present study.",
"Section 3 summarizes the different MC models used and corresponding UE parameters.",
"Section 4 presents experimental details: a brief detector description, data samples, event and track selection criteria, correction procedure, and systematic uncertainties.",
"Section 5 presents the results on UE activity measured in DY events and the comparison with the measurements based on a leading track-jet.",
"The main results are summarized in Section 6."
],
[
"Observables",
"The UE activity is measured in terms of particle and energy densities.",
"The particle density ($1/[\\Delta \\eta \\Delta (\\Delta \\phi )] \\langle N_{\\rm ch} \\rangle $ ) is computed as the average number of primary charged particles per unit pseudorapidity $\\eta $ and per unit azimuthal separation $\\Delta \\phi $ (in radians) between a track and the transverse momentum of the dimuon system.",
"The pseudorapidity is defined as $\\eta = -\\ln (\\tan (\\theta / 2))$ , where $\\theta $ is the polar angle measured with respect to the anticlockwise beam direction.",
"The azimuthal angle $\\phi $ is measured in the plane perpendicular to the beam axis.",
"The energy density ($1/[\\Delta \\eta \\Delta (\\Delta \\phi )] \\langle \\Sigma p_{T} \\rangle $ ) is expressed in terms of the average of the scalar sum of the transverse momenta of primary charged particles per unit pseudorapidity per unit azimuthal separation.",
"The ratio of the energy and particle densities, as well as the total charged-particle multiplicity $N_{\\rm ch}$ and the transverse momentum spectrum are also computed.",
"The charged-particle multiplicity and transverse momentum distributions are normalized to unit area and to the average number of charged particles per event, respectively.",
"Particles are considered as primary if they originate from the initial proton-proton interaction and are not the decay products of long-lived hadrons with a lifetime exceeding 10$^{-10}$ s. Apart from the muons from the DY process, all charged particles in the central region of the detector with pseudorapidity $|\\eta | < 2$ and with transverse momentum $p_{T} > 0.5$ are considered.",
"The spatial distribution of the tracks is categorized by the azimuthal separation $\\Delta \\phi $ .",
"Particle production in the away region ($|\\Delta \\phi | > 120^{\\circ }$ ) is expected to be dominated by the hardest ISR emissions, which balance the dimuon system.",
"The transverse region ($60^{\\circ } < |\\Delta \\phi | < 120^{\\circ }$ ) and towards region ($|\\Delta \\phi | < 60^{\\circ }$ ) are more sensitive to soft emissions and, in particular, those due to MPIs.",
"The relevant information about the hard and the soft processes is extracted from the tracking and the muon systems of the CMS detector and thus the derived observables are insensitive to the uncertainties of the calorimetric measurements.",
"The DY events with dimuon mass $M_{\\mu \\mu }$ around the resonance are the least contaminated by background processes (heavy-quark, , W+jets, and DY $\\rightarrow \\tau \\tau $ production) [14], [15] and best suited for the measurement of the UE activity.",
"The UE activity is studied as a function of the magnitude of the dimuon transverse momentum ($p_{T}^{\\mu \\mu }={ \\vec{p}^{\\mu }_{T,1}+\\vec{p}^{\\mu }_{T,2}}$ ) and as a function of $M_{\\mu \\mu }$ .",
"The dependence of the UE activity on $p_{T}^{\\mu \\mu }$ for high-mass dimuon pairs effectively probes the ISR spectrum.",
"In order to minimize the background contamination, the $p_{T}^{\\mu \\mu }$ dependence is studied only in the narrow mass window $81 < M_{\\mu \\mu } < 101$ .",
"In contrast to the study of the UE activity in hadronic events using a leading track-jet [5], [6], this energy scale is sufficiently large to saturate the MPI contributions.",
"This observation is verified by studying the UE activity as a function of the dimuon mass in a wider mass range, where the total transverse momentum of the dimuon system is kept to a minimum by requiring $p_{T}^{\\mu \\mu } < 5$ ."
],
[
"Monte Carlo models",
"The UE dynamics are studied through the comparison of the observables in data with various tunes of 6 [16] and its successor 8 [17], [18].",
"(version 5) [19], [20], which simulates up to six final-state fermions (including the muons), and [21], which includes next-to-leading-order corrections on the hardest emission, are also compared to our measurements.",
"For these two generators, softer emissions are simulated by $p_{T}$ -ordered parton showers using 6 tunes and matched with the hard process produced by the generators.",
"Hadronization in 6 and 8 is based on the Lund string fragmentation model [22].",
"The measurements are also compared to predictions of the ++ [23] angular-ordered parton shower and cluster hadronization model [24], [25].",
"The UE contributions from MPIs rely on modelling and tuning of the parameters in the MC generators.",
"The MPI model of PYTHIA relies on two fundamental assumptions [12]: The ratio of the 2$\\rightarrow $ 2 partonic cross section, integrated above a transverse momentum cutoff scale, and the total of the hadronic cross section is a measure of the amount of MPIs.",
"The cutoff scale $p_{0T}$ is introduced to regularize an otherwise diverging partonic cross section, $\\sigma (p_{T}) = \\sigma (p_{0T})\\frac{p_{T}^{4}}{(p_{T}^{2} + p_{0T}^{2})^{2}}\\ ,$ with $p_{0T} (\\sqrt{s}) = p_{0T} (\\sqrt{s_{0}})\\left( \\frac{\\sqrt{s}}{\\sqrt{s_{0}}} \\right)^{\\epsilon } \\ .$ Here $\\sqrt{s_{0}} = 1.8$ and $\\epsilon $ is a parameter characterizing the energy dependence of the cutoff scale.",
"The number of MPIs in an event has a Poisson distribution with a mean that depends on the overlap of the matter distribution of the hadrons in impact-parameter space.",
"The MPI model used here [26] includes showering of the MPI process, which is interleaved with the ISR.",
"The tunes of the models vary mainly in the MPI regularization parameters, $p_{0T}$ and $\\epsilon $ , in the amount of colour reconnection, and in the PDF used.",
"The Z1 tune [27] of 6 adopts the results of a global tuning performed by the ATLAS Collaboration [28] and uses the fragmentation and colour reconnection parameters of the ATLAS AMBT1 tune [29].",
"The parameters of the Z1 tune related to the MPI regularization cutoff and its energy dependence are adjusted to describe previous CMS measurements of the UE activity in hadronic events [6] and uses the CTEQ5L PDF.",
"The Z2 tune of 6 is an update of the Z1 tune using CTEQ6L1 [30], the default used in most CMS generators; the regularization cutoff value at the nominal energy of $\\sqrt{s_{0}}$ = 1.8is optimized to 1.832.",
"The value of the energy evolution parameter for the Z2 tune is 0.275, as for the Z1 tune.",
"The 4C [31] tune of 8 follows a similar procedure as the ATLAS AMBT1 tune, but includes ALICE multiplicity data as well.",
"The values of the $p_{0T}(\\sqrt{s_0})$ and $\\epsilon $ parameters for the 4C tune are 2.085and 0.19, respectively.",
"The effective value of $p_{0T}$ at $\\sqrt{s} = 7$ is about 2.7for both the Z2 and 4C tunes.",
"The LHC-UE7-2 tune of ++ is based on ATLAS measurements of the UE activity in hadronic events [7].",
"The regularization cutoff parameter $p_{0T}$ for the LHC-UE7-2 tune is 3.36at $\\sqrt{s}$ = 7.",
"The CTEQ6L1 PDF is used in conjunction with 6 Z2, 8 4C, Z2, and ++ LHC-UE7-2, while CT10 [32] is used for , and CTEQ5L for the 6 Z1 simulations.",
"A comparison of these models with the measurements is presented in Section 5."
],
[
"Experimental methods",
"The present analysis is performed with a sample of proton-proton collisions corresponding to an integrated luminosity of 2.2, collected in March–August 2011 using the CMS detector [4].",
"Muons are measured in the pseudorapidity range $|\\eta | < 2.4$ with a detection system consisting of three subsystems: Drift Tubes, Cathode Strip Chambers, and Resistive Plate Chambers.",
"Matching track segments from the muon detector to the tracks measured in the inner tracker results in a transverse momentum resolution between 1% and 5% for $p_T$ values up to 1.",
"The tracker subsystem consists of 1440 silicon-pixel and 15 148 silicon-strip detector modules, and it measures charged particle trajectories within the nominal pseudorapidity range $|\\eta | < 2.5$ .",
"The tracker is designed to provide a transverse impact parameter resolution of about 100and a transverse momentum resolution of about 0.7% for 1charged particles at normal incidence ($\\eta $ = 0).",
"The detector response is simulated in detail using the GEANT4 package [33].",
"The simulated signal and background events, including heavy-quark, , W+jets, and DY $\\rightarrow \\tau \\tau $ production, are processed and reconstructed in the same manner as collision data."
],
[
"Event and track selection",
"The trigger requires the presence of at least two muon candidates.",
"In periods of lower instantaneous luminosity both muons were required to have $p_{T} > 7$ , while in other periods the transverse momentum requirements were 13and 8for the leading and subleading muons, respectively.",
"The trigger efficiency is above 95% for the offline selected DY events with the requirement of $81 < M_{\\mu \\mu } < 101$ .",
"The offline selection requires exactly two muons reconstructed in the muon detector and the silicon tracker.",
"Muon candidates are required to satisfy identification criteria based on the number of hits in the muon stations and tracker, transverse impact parameter with respect to the beam axis, and normalized $\\chi ^{2}$ of the global fit [15].",
"The backgrounds from jets misidentified as muons and from semileptonic decays of heavy quarks are suppressed by applying an isolation condition on the muon candidates.",
"The isolation variable $I$ for muons is defined as $I = \\left\\lbrace \\Sigma \\left[p_{T} (\\text{tracks}) + E_{T}(\\mathrm {EM}) + E_{T}(\\mathrm {HAD})\\right] - \\pi (\\Delta R)^{2}\\rho \\right\\rbrace /p_{T}^{\\mu },$ where the sum is defined in a cone of radius $\\Delta R=\\sqrt{(\\Delta \\phi )^{2} + (\\Delta \\eta )^{2}}=0.3$ around the muon direction; $\\Delta \\eta $ and $\\Delta \\phi $ are the pseudorapidity and azimuthal separation between the muon and tracks or calorimetric towers.",
"Here $p_{T}$ (tracks) is the transverse momentum of tracks, excluding muons, with $p_{T}>$ 1, $E_{T} ({\\rm EM})$ is the transverse energy deposited in the electromagnetic calorimeter, $E_{T}({\\rm HAD})$ is the transverse energy deposited in the hadronic calorimeter, and $\\rho $ is the average energy density [34] in the calorimeter and tracker originating from additional inelastic pp interactions (pile-up) in the same bunch crossing as the DY interaction.The calculation of $\\rho $ takes into account the number of reconstructed primary vertices in the event; the average value of $\\rho $ is 5.6.",
"A muon is considered to be isolated if $I < 0.15$ .",
"Because of the energy density correction, the isolation efficiency is independent of the number of pile-up interactions.",
"The selected muons are required to have opposite charges, transverse momenta larger than 20, and pseudorapidity $|\\eta | < 2.4$ .",
"Both muons are required to be associated with the same vertex, which is designated as the signal vertex.",
"The selected signal vertex is required to be within $\\pm $ 18of the nominal interaction point as measured along the $z$ direction.",
"At least five tracks are required to be associated with the signal vertex, and the transverse displacement of the signal vertex from the beam axis is required to be less than 2.",
"These criteria select a pure sample of DY events with a total background contribution of less than 0.5% as estimated from simulated events.",
"Tracks, excluding the selected muons, are considered for the UE measurement if they are well reconstructed in the silicon-pixel and the silicon-strip tracker, have $p_{T} > 0.5$ and $|\\eta | < 2$ , and originate from the signal vertex.",
"To reduce the number of improperly reconstructed tracks, a high purity reconstruction algorithm [35] is used.",
"The high purity algorithm requires stringent cuts on the number of hits, the normalized $\\chi ^{2}$ of the track fit, and the consistency of the track originating from a pixel vertex.",
"To reduce the contamination of secondary tracks from decays of long-lived particles and photon conversions, the distances of closest approach between the track and the signal vertex in the transverse plane and in the longitudinal direction are required to be less than 3 times the respective uncertainties.",
"Tracks with poorly measured momenta are removed by requiring $\\sigma (p_{T})/p_{T} < 5\\%$ , where $\\sigma (p_{T})$ is the uncertainty on the $p_{T}$ measurement.",
"These selection criteria reject about 10% of primary tracks and 95% of misreconstructed and secondary tracks.",
"The selected tracks have a contribution of about 2% from misreconstructed and secondary tracks."
],
[
"Corrections and systematic uncertainties",
"The UE observables, discussed in Section 2, are corrected for detector effects and selection efficiencies.",
"The measured observables are corrected to reflect the activity from all primary charged particles with transverse momentum $p_{T} > 0.5$ and pseudorapidity $|\\eta | < 2$ .",
"The particle and energy densities are corrected using a bin-by-bin technique.",
"In the bin-by-bin technique, the correction factor is calculated by taking the bin-by-bin ratio of the particle level and detector level distributions for simulated events and then the measured quantity is multiplied by this correction factor.",
"There is a small growth in the particle and energy densities with increasing $p_{T}^{\\mu \\mu }$ and $M_{\\mu \\mu }$ in the towards and transverse regions.",
"Because of this slow growth of densities the bin migration in $p_{T}^{\\mu \\mu }$ and $M_{\\mu \\mu }$ has a small effect on the measurements, therefore a bin-by-bin method is considered to be sufficiently precise.",
"There is a fast rise in the energy and particle densities in the away region with the increase of $p_{T}^{\\mu \\mu }$ , but corrected results using a bin-by-bin method are consistent with correction obtained from a Bayesian [36] technique.",
"The transverse momenta of the charged particles have very good resolution and are corrected using a bin-by-bin method.",
"In this analysis the average of the calculated correction factors from 6 Z2, 6 D6T, and Z2 is used to correct the experimental distributions.",
"The maximum deviation from the average correction factor is taken as the model-dependent systematic uncertainty, estimated to be 0.7–1.4% for the particle and energy densities.",
"In the case of charged-particle multiplicity, there is substantial bin migration and the corrected results using the Bayesian [36] and bin-by-bin techniques differ by 10–15%.",
"Therefore the charged-particle multiplicity is corrected using a Bayesian unfolding technique with a response matrix obtained using the 6 Z2 tune.",
"The systematic uncertainty related to the correction procedure is calculated by unfolding the data with response matrices obtained using different tunes.",
"In the analyzed data, there are on average 6–7 collisions in each bunch crossing.",
"Tracks originating from these pile-up interactions cause the UE activity to be overestimated, so the measurements are corrected for the presence of pile-up interactions.",
"The correction factor is calculated as the ratio of the UE activity for simulated events with and without pile-up.",
"The uncertainty in the modelling of the pile-up events is estimated by varying the mean of the expected number of pile-up events by $\\pm $ 1.",
"This uncertainty in pile-up modelling affects the particle and energy densities by 0.3–1.0%.",
"The effect due to pile-up events is small because only the tracks associated with the same vertex as the muon pair are used.",
"The results are also cross-checked with low pile-up 7data collected during 2010 and the differences are found to be negligible.",
"Table: Summary of the systematic uncertainties on the particle and energy densities (in percent).",
"The first three rows show the systematic uncertainties for the particle density in the towards, transverse, and away regions.",
"The last three rows report the systematic uncertainties for the energy density.",
"The numbers outside the parentheses refer to the case where the densities are measured as a function of M μμ M_{\\mu \\mu } and those in the parentheses correspond to the measurements as a function of p T μμ p_{T}^{\\mu \\mu }.We also consider possible systematic effects related to trigger requirements, different beam-axis positions in data and simulation, various track selection criteria, muon isolation, and misidentification of tracks.",
"The combined systematic uncertainty related to trigger conditions, the varying beam-axis position, and track selection is less than 0.5%.",
"The systematic uncertainty due to isolation is calculated by removing the isolation condition in the simulated events used for the correction and is found to be 0.8–2.5% for the particle and energy densities.",
"The yield of secondary tracks originating from the decay of long-lived particles is not correctly predicted by the simulation [37].",
"To estimate the effect of secondary tracks, a subset of simulated events is created by rejecting tracks that do not have a matching primary charged particle at the generator level.",
"The uncertainty is evaluated by correcting the measurements with this subset of the simulated events, containing fewer secondary tracks, and is found to be 0.7–1.0% for the particle and energy densities.",
"Though the total contribution of background processes is very small, it affects the measurement at higher $p_{T}^{\\mu \\mu }$ (50–100) and small $M_{\\mu \\mu }$ (40–60) where the contamination from and DY$\\rightarrow \\tau \\tau $ background processes is 1% and 5%, respectively.",
"The particle and energy densities differ between DY$\\rightarrow \\tau \\tau $ and DY$\\rightarrow \\mu \\mu $ (the signal process) by 20%.",
"The particle (energy) density for the background is two times (four times) that for the signal process.",
"Combination of the differences in the densities for background processes and relative background contributions gives a systematic uncertainty of 0.2–0.9%.",
"Table REF summarizes the dominant systematic uncertainties on the particle and energy densities.",
"The total systematic uncertainty on the particle and energy densities is in the range 1.5–3.0%, whereas the uncertainties on the track multiplicity and $p_{T}$ spectra reach 10% in the tail (not reported in Table REF ).",
"In all figures, inner error bars represent the statistical uncertainty only, while outer error bars account for the quadratic sum of statistical and systematic uncertainties."
],
[
"Results",
"The UE activity in DY events, for charged particles with $p_{T} > 0.5$ and $|\\eta | < 2.0$ , is presented as a function of $M_{\\mu \\mu }$ and $p_{T}^{\\mu \\mu }$ .",
"The multiplicity and the transverse momentum distributions are also presented for two different sets of events, $p_T^{\\mu \\mu } < 5$ and $81 < M_{\\mu \\mu } < 101$ .",
"Finally, the UE activity in the transverse region is compared with that measured in hadronic events using a leading track-jet."
],
[
"Underlying event in the Drell–Yan process",
"The energy-scale dependence of the MPI activity is studied by limiting the ISR.",
"To accomplish this we require the muons to be back-to-back in the transverse plane with $p_T^{\\mu \\mu } < 5$ and measure the dependence of the UE activity on the dimuon mass, $M_{\\mu \\mu }$ .",
"The resulting particle and energy densities are shown in Fig.",
"REF .",
"Because the activity is almost identical in the towards and transverse regions, they are combined as $|\\Delta \\phi | < 120^{\\circ }$ .",
"The contribution of ISR to the UE activity is small after requiring $p_{T}^{\\mu \\mu } < 5$ , as shown by the prediction of ++ without MPIs.",
"This figure also illustrates the dominant role of MPIs in our current models as they generate more than 80% of the UE activity in these ISR-reduced events.",
"The lack of dependence of the UE activity on $M_{\\mu \\mu }$ within the range under study (40–140) indicates that the activity due to MPIs is constant at energy scales down to 40.",
"The quantitative description by model tunes based on the minimum-bias and UE observables in hadronic events is illustrated by the MC/Data ratios in Fig.",
"REF .",
"In general, 6 Z2, 8 4C, and ++ LHC-UE7-2 describe the densities well, whereas the Z2 tune used together with the generator underestimates both densities by 5–15%.",
"Both and ++ model tunes derived from the UE measurement in hadronic events using the leading jet/track approach describe the UE activity in the Drell–Yan events equally well and hence illustrate a certain universality of the underlying event across QCD and electroweak processes in hadronic collisions.",
"Figure: Top: The UE activity as a function of the dimuon invariant mass (M μμ M_{\\mu \\mu }) for events with p T μμ <5p_{T}^{\\mu \\mu } < 5for charged particles having Δφ<120 ∘ \\Delta \\phi < 120^{\\circ }: (left) particle density; (centre) energy density; (right) ratio of the energy and particle densities.",
"The predictions of 6 Z2, Z2, 8 4C, and ++ LHC-UE7-2 (with and without MPIs) are also displayed.",
"In the top right plot, the structure around 60–80for ++ without MPIs reflects the influence of photon radiation by final-state muons, which is enhanced below the resonance.",
"Bottom: Ratios of the predictions of various MC models and the measurement.",
"The inner band shows the statistical uncertainity of data whereas the outer band represents the total uncertainty.Dependence of the UE activity on the transverse momentum of the dimuon system is shown in Fig.",
"REF in the towards, transverse, and away regions (top to bottom) for events having $M_{\\mu \\mu }$ between 81and 101.",
"At this high energy scale, the $p_{T}^{\\mu \\mu }$ dependence of the UE activity is sensitive to the ISR.",
"The slope in the $p_{T}^{\\mu \\mu }$ dependence of the UE activity is identical for a model with and without MPIs and is therefore mainly due to ISR.",
"The predictions of ++ without MPIs underestimate the measurements in the away region as well because the MPIs produce particles uniformly in all directions.",
"The UE activity does not fall to zero when $p_{T}^{\\mu \\mu }\\rightarrow 0$ because of the presence of the hard scale set by $M_{\\mu \\mu }$ .",
"The particle and energy densities in the away region rise sharply with $p_{T}^{\\mu \\mu }$ and, because of momentum conservation mainly sensitive to the spectrum of the hardest emission, are equally well described by all tunes and generators considered.",
"In the towards and transverse regions there is a slow growth in the particle and energy densities with increasing $p_{T}^{\\mu \\mu }$ .",
"The energy density increases more than the particle density, implying a continuous increase in the average transverse momentum of the charged particles with $p_{T}^{\\mu \\mu }$ .",
"This effect is also reflected in the ratio of the energy density to the particle density.",
"The activity in the towards region is qualitatively similar to that in the transverse region.",
"Quantitatively, the activity is higher in the transverse region than the towards region, an effect caused by the spill-over contributions from the recoil activity in the away region, which balances the dimuon system.",
"This observation is visible in Fig.",
"REF at small $p_{T}^{\\mu \\mu }$ , where the radiation contribution is small and the activity in the transverse region is the same as that in the towards region.",
"Figure: The UE activity in the towards (upper row), transverse (centre row), and away (bottom row) regions as functions of p T μμ p_{T}^{\\mu \\mu } for events satisfying 81<M μμ <10181 < M_{\\mu \\mu } < 101: (left) particle density;(centre) energy density; (right) the ratio of the energy density and the particle densities.",
"Predictions of Z2, Z2, 8 4C, and ++ LHC-UE7-2 (with and without MPIs) are superimposed.Figure REF presents the ratios of the predictions of various MC models to the measurements for the observables shown in Fig.",
"REF .",
"Statistical fluctuations in the data induce correlated fluctuations for the various MC/data ratios.",
"in conjunction with 6 tune Z2 describes the $p_{T}^{\\mu \\mu }$ dependence of the UE activity very well, both qualitatively and quantitatively.",
"8 4C and ++ describe the $p_{T}^{\\mu \\mu }$ dependence of the particle density within 10–15%, but fail to describe the energy density.",
"8 4C and ++ agree better with data as $p_{T}^{\\mu \\mu }$ approaches zero.",
"The combination of the Z2 tune with fails to describe the energy density in the towards and transverse regions, but gives a reasonable description of the particle density.",
"This observation, combined with the information in Fig.",
"REF , indicates that the discrepancies are not necessarily due to a flaw in the UE tune, but to an inadequate description of the multiple hard emissions and the different sets of PDFs used with .",
"At small $p_{T}^{\\mu \\mu }$ the comparisons with 6 Z2 and Z2 are similar to those in Ref.",
"[38], where 6 gives a good description of the $p_{T}^{\\mu \\mu }$ spectrum while underestimates the $p_{T}^{\\mu \\mu }$ .",
"Figure: Ratios, as functions of p T μμ p_{T}^{\\mu \\mu }, of the predictions of various MC models to the measurements in the towards (upper row), transverse (centre row), and away (bottom row) regions for events satisfying 81<M μμ <10181 < M_{\\mu \\mu } < 101: (left) particle density; (centre) energy density; (right) the ratio of the energy density and particle densities.",
"The inner band shows the statistical uncertainty on the data whereas the outer band represents the total uncertainty.Figure REF shows the distributions of charged particle multiplicity (top row) and transverse momentum (bottom row).",
"Figure REF (left) shows a comparison of the normalized distributions in the away, transverse, and towards regions for events satisfying $81 < M_{\\mu \\mu } < 101$ .",
"As expected, the transverse and towards regions have fewer charged particles with a softer $p_{T}$ spectrum than the away region.",
"Figure REF (centre) shows the comparison of the normalized distributions in the transverse region for two different subsets of the selected events, one with $81 < M_{\\mu \\mu } < 101$ and one with $p_{T}^{\\mu \\mu } < 5$ .",
"The charged particle multiplicity is decreased and the $p_{T}$ spectrum is softer when $p_{T}^{\\mu \\mu } < 5$ is required, because of the reduced contribution of ISR.",
"Figure REF (right) shows the comparison of the normalized distributions with the predictions of various simulations in the transverse region for events satisfying $81 < M_{\\mu \\mu } < 101$ .",
"The charge multiplicity distribution is described well, within 10–15%, by Z2 and 8 4C.",
"The $p_{T}$ spectrum is described within 10–15% by Z2, whereas 8 4C, Z2, and ++ LHC-UE7-2 have softer $p_{T}$ spectra.",
"The various MC programs achieve a similar level of agreement with data in the towards region as in the transverse region.",
"Figure: Distributions of the charged particle multiplicity (upper row) and transverse momentum (bottom row) of the selected tracks.",
"The left plots show the comparisons of the normalized distributions in the away, transverse, and towards regions for events satisfying 81<M μμ <10181 < M_{\\mu \\mu } < 101.",
"Comparisons of the normalized distributions in the transverse region are shown in the centre plots, requiring 81<M μμ <10181 < M_{\\mu \\mu } < 101or p T μμ <5p_{T}^{\\mu \\mu } < 5 .",
"The right plots show the comparisons of the normalized distributions in the transverse region with the predictions of various simulations for events satisfying 81<M μμ <10181 < M_{\\mu \\mu } < 101."
],
[
"Comparison with the UE activity in hadronic events ",
"The UE activity was previously measured as a function of leading jet $p_{T}$ in hadronic events for charged particles with pseudorapidity $|\\eta | < 2$ and with transverse momentum $p_{T} > 0.5$ [6].",
"Figure REF shows the comparison of the UE activity measured in the hadronic and the DY events (around the Z peak) in the transverse region as a function of $p_{T}^{\\rm leading~jet}$ and $p_{T}^{\\mu \\mu }$ , respectively.",
"For the hadronic events two components are visible: a fast rise for $p_{T}^{\\rm leading~jet} \\lesssim 10$ due to an increase in the MPI activity, followed by an almost constant particle density and a slow increase in the energy density with $p_{T}^{\\rm leading~jet}$ .",
"The increase in the particle and energy densities for $p_{T}^{\\rm leading~jet} \\gtrsim 10$ is mainly due to the increase of ISR and FSR.",
"Owing to the presence of a hard energy scale ($81 < M_{\\mu \\mu } < 101$ ), densities in the DY events do not show a sharply rising part, but only a slow growth with $p_{T}^{\\mu \\mu }$ due to the ISR contribution.",
"For $p_{T}^{\\mu \\mu }$ and $p_{T}^{\\rm leading~jet} > 10$ , DY events have a smaller particle density with a harder $p_{T}$ spectrum compared to the hadronic events, as can be seen in Fig.",
"REF .",
"This distinction is due to the different nature of radiation in the hadronic and DY events.",
"Drell–Yan events have only initial-state QCD radiation initiated by quarks, which fragment into a smaller number of hadrons carrying a larger fraction of the parent parton energy, whereas the hadronic events have both initial- and final-state QCD radiation predominantly initiated by gluons with a softer fragmentation into hadrons.",
"Similar behavior is observed for the track-jet measurement where the UE activity is higher by 10–20% for gluon-dominated processes, as estimated from simulation.",
"Figure: Comparison of the UE activity measured in hadronic and Drell–Yan events (around the resonance peak) as a function of p T leading jetp_{T}^\\text{leading~jet} and p T μμ p_{T}^{\\mu \\mu }, respectively: (left) particle density, (centre) energy density, and (right) ratio of energy and particle densities in the transverse region."
],
[
"Summary",
"We have used Drell–Yan events to measure the UE activity in proton-proton collisions at $\\sqrt{s} = 7$ , which were recorded with the CMS detector at the LHC.",
"The DY process provides a UE measurement where a clean separation of the hard interaction from the soft component is possible.",
"After excluding the muons from the DY process, the towards ($|\\Delta \\phi |<60^{\\circ }$ ) and the transverse ($60^{\\circ }<|\\Delta \\phi |<120^{\\circ }$ ) regions are both sensitive to initial-state radiation and multiple parton interactions.",
"The DY process provides an effective way to study the dependence of the UE activity on the hard interaction scale, which is related to the invariant mass of the dimuon pair.",
"The influence of the ISR is probed by the dependence on the transverse momentum of the muon pair.",
"The UE activity is observed to be independent of the dimuon mass above 40, after limiting the recoil activity, which confirms the MPI saturation at this scale.",
"The UE activity in the DY events with no hard ISR is well described by 6 and with the Z2 tune and the CTEQ6L PDF.",
"The Z2 tune does not agree with the data if used with PDFs other than CTEQ6L, as in the case of the simulation.",
"The 8 4C and ++ LHC-UE7-2 tunes provide good descriptions of the energy-scale dependence of the UE activity.",
"Thus the dependence of the UE activity on the energy scale is well described by tunes derived from hadronic events, illustrating the universality of MPIs in different processes.",
"This universality is also indicated by the similarity between the UE activity in DY and hadronic events, although these events have different types of radiation.",
"In addition, there is some ambiguity in the definition of the hard scale for both types of events.",
"The UE activity in the towards and transverse regions shows a slow growth with the transverse momentum of the muon pair and provides an important probe of the ISR.",
"The leading-order matrix element generator provides a good description of the UE dependence on dimuon transverse momentum.",
"However, , , and ++, which do not simulate the multiple hard emissions with sufficient accuracy, underestimate the energy density, but describe the particle density reasonably well.",
"These measurements provide important input for further tuning or improvements of the Monte Carlo models and also for the understanding of the dynamics of QCD."
],
[
"Acknowledgements",
"We wish to congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine.",
"We thank the technical and administrative staff at CERN and other CMS institutes.",
"This work was supported by the Austrian Federal Ministry of Science and Research; the Belgium Fonds de la Recherche Scientifique, and Fonds voor Wetenschappelijk Onderzoek; the Brazilian Funding Agencies (CNPq, CAPES, FAPERJ, and FAPESP); the Bulgarian Ministry of Education and Science; CERN; the Chinese Academy of Sciences, Ministry of Science and Technology, and National Natural Science Foundation of China; the Colombian Funding Agency (COLCIENCIAS); the Croatian Ministry of Science, Education and Sport; the Research Promotion Foundation, Cyprus; the Estonian Academy of Sciences and NICPB; the Academy of Finland, Finnish Ministry of Education and Culture, and Helsinki Institute of Physics; the Institut National de Physique Nucléaire et de Physique des Particules / CNRS, and Commissariat à l'Énergie Atomique et aux Énergies Alternatives / CEA, France; the Bundesministerium für Bildung und Forschung, Deutsche Forschungsgemeinschaft, and Helmholtz-Gemeinschaft Deutscher Forschungszentren, Germany; the General Secretariat for Research and Technology, Greece; the National Scientific Research Foundation, and National Office for Research and Technology, Hungary; the Department of Atomic Energy and the Department of Science and Technology, India; the Institute for Studies in Theoretical Physics and Mathematics, Iran; the Science Foundation, Ireland; the Istituto Nazionale di Fisica Nucleare, Italy; the Korean Ministry of Education, Science and Technology and the World Class University program of NRF, Korea; the Lithuanian Academy of Sciences; the Mexican Funding Agencies (CINVESTAV, CONACYT, SEP, and UASLP-FAI); the Ministry of Science and Innovation, New Zealand; the Pakistan Atomic Energy Commission; the State Commission for Scientific Research, Poland; the Fundação para a Ciência e a Tecnologia, Portugal; JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); the Ministry of Science and Technologies of the Russian Federation, the Russian Ministry of Atomic Energy and the Russian Foundation for Basic Research; the Ministry of Science and Technological Development of Serbia; the Ministerio de Ciencia e Innovación, and Programa Consolider-Ingenio 2010, Spain; the Swiss Funding Agencies (ETH Board, ETH Zurich, PSI, SNF, UniZH, Canton Zurich, and SER); the National Science Council, Taipei; the Scientific and Technical Research Council of Turkey, and Turkish Atomic Energy Authority; the Science and Technology Facilities Council, UK; the US Department of Energy, and the US National Science Foundation.",
"Individuals have received support from the Marie-Curie programme and the European Research Council (European Union); the Leventis Foundation; the A. P. Sloan Foundation; the Alexander von Humboldt Foundation; the Belgian Federal Science Policy Office; the Fonds pour la Formation à la Recherche dans l'Industrie et dans l'Agriculture (FRIA-Belgium); the Agentschap voor Innovatie door Wetenschap en Technologie (IWT-Belgium); and the Council of Science and Industrial Research, India."
],
[
"The CMS Collaboration ",
"Yerevan Physics Institute, Yerevan, Armenia S. Chatrchyan, V. Khachatryan, A.M. Sirunyan, A. Tumasyan Institut für Hochenergiephysik der OeAW, Wien, Austria W. Adam, T. Bergauer, M. Dragicevic, J. Erö, C. Fabjan, M. Friedl, R. Frühwirth, V.M.",
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"Van Spilbeeck Vrije Universiteit Brussel, Brussel, Belgium F. Blekman, S. Blyweert, J.",
"D'Hondt, R. Gonzalez Suarez, A. Kalogeropoulos, M. Maes, A. Olbrechts, W. Van Doninck, P. Van Mulders, G.P.",
"Van Onsem, I. Villella Université Libre de Bruxelles, Bruxelles, Belgium O. Charaf, B. Clerbaux, G. De Lentdecker, V. Dero, A.P.R.",
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"Pol, M.H.G.",
"Souza Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil W.L.",
"Aldá Júnior, W. Carvalho, A. Custódio, E.M. Da Costa, C. De Oliveira Martins, S. Fonseca De Souza, D. Matos Figueiredo, L. Mundim, H. Nogima, V. Oguri, W.L.",
"Prado Da Silva, A. Santoro, S.M.",
"Silva Do Amaral, L. Soares Jorge, A. Sznajder Instituto de Fisica Teorica, Universidade Estadual Paulista, Sao Paulo, Brazil T.S.",
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"Fernandez Perez Tomei, E. M. Gregores3, C. Lagana, F. Marinho, P.G.",
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"Novaes, Sandra S. Padula Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria V. Genchev1, P. Iaydjiev1, S. Piperov, M. Rodozov, S. Stoykova, G. Sultanov, V. Tcholakov, R. Trayanov, M. Vutova University of Sofia, Sofia, Bulgaria A. Dimitrov, R. Hadjiiska, A. Karadzhinova, V. Kozhuharov, L. Litov, B. Pavlov, P. Petkov Institute of High Energy Physics, Beijing, China J.G.",
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"Jiang, D. Liang, S. Liang, X. Meng, J. Tao, J. Wang, J. Wang, X. Wang, Z. Wang, H. Xiao, M. Xu, J. Zang, Z. Zhang State Key Lab.",
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"Osorio Oliveros, J.C. Sanabria Technical University of Split, Split, Croatia N. Godinovic, D. Lelas, R. Plestina5, D. Polic, I. Puljak1 University of Split, Split, Croatia Z. Antunovic, M. Dzelalija, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, S. Duric, K. Kadija, J. Luetic, S. Morovic University of Cyprus, Nicosia, Cyprus A. Attikis, M. Galanti, J. Mousa, C. Nicolaou, F. Ptochos, P.A.",
"Razis Charles University, Prague, Czech Republic M. Finger, M. Finger Jr. Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt Y. Assran6, A. Ellithi Kamel7, S. Khalil8, M.A.",
"Mahmoud9, A. Radi8$^{, }$ 10 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia A. Hektor, M. Kadastik, M. Müntel, M. Raidal, L. Rebane, A. Tiko Department of Physics, University of Helsinki, Helsinki, Finland V. Azzolini, P. Eerola, G. Fedi, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland S. Czellar, J. Härkönen, A. Heikkinen, V. Karimäki, R. Kinnunen, M.J. Kortelainen, T. Lampén, K. Lassila-Perini, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, T. Peltola, E. Tuominen, J. Tuominiemi, E. Tuovinen, D. Ungaro, L. Wendland Lappeenranta University of Technology, Lappeenranta, Finland K. Banzuzi, A. Korpela, T. Tuuva Laboratoire d'Annecy-le-Vieux de Physique des Particules, IN2P3-CNRS, Annecy-le-Vieux, France D. Sillou DSM/IRFU, CEA/Saclay, Gif-sur-Yvette, France M. Besancon, S. Choudhury, M. Dejardin, D. Denegri, B. Fabbro, J.L.",
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"Physikalisches Institut A, Aachen, Germany M. Ata, J. Caudron, E. Dietz-Laursonn, M. Erdmann, A. Güth, T. Hebbeker, C. Heidemann, K. Hoepfner, T. Klimkovich, D. Klingebiel, P. Kreuzer, D. Lanske$^{\\textrm {\\dag }}$ , J. Lingemann, C. Magass, M. Merschmeyer, A. Meyer, M. Olschewski, P. Papacz, H. Pieta, H. Reithler, S.A. Schmitz, L. Sonnenschein, J. Steggemann, D. Teyssier, M. Weber RWTH Aachen University, III.",
"Physikalisches Institut B, Aachen, Germany M. Bontenackels, V. Cherepanov, M. Davids, G. Flügge, H. Geenen, M. Geisler, W. Haj Ahmad, F. Hoehle, B. Kargoll, T. Kress, Y. Kuessel, A. Linn, A. Nowack, L. Perchalla, O. Pooth, J. Rennefeld, P. Sauerland, A. Stahl, M.H.",
"Zoeller Deutsches Elektronen-Synchrotron, Hamburg, Germany M. Aldaya Martin, W. Behrenhoff, U. Behrens, M. Bergholz14, A. Bethani, K. Borras, A. Cakir, A. Campbell, E. Castro, D. Dammann, G. Eckerlin, D. Eckstein, A. Flossdorf, G. Flucke, A. Geiser, J. Hauk, H. Jung1, M. Kasemann, P. Katsas, C. Kleinwort, H. Kluge, A. Knutsson, M. Krämer, D. Krücker, E. Kuznetsova, W. Lange, W. Lohmann14, B. Lutz, R. Mankel, I. Marfin, M. Marienfeld, I.-A.",
"Melzer-Pellmann, A.B.",
"Meyer, J. Mnich, A. Mussgiller, S. Naumann-Emme, J. Olzem, A. Petrukhin, D. Pitzl, A. Raspereza, P.M. Ribeiro Cipriano, M. Rosin, J. Salfeld-Nebgen, R. Schmidt14, T. Schoerner-Sadenius, N. Sen, A. Spiridonov, M. Stein, J. Tomaszewska, R. Walsh, C. Wissing University of Hamburg, Hamburg, Germany C. Autermann, V. Blobel, S. Bobrovskyi, J. Draeger, H. Enderle, J. Erfle, U. Gebbert, M. Görner, T. Hermanns, K. Kaschube, G. Kaussen, H. Kirschenmann, R. Klanner, J. Lange, B. Mura, F. Nowak, N. Pietsch, C. Sander, H. Schettler, P. Schleper, E. Schlieckau, M. Schröder, T. Schum, H. Stadie, G. Steinbrück, J. Thomsen Institut für Experimentelle Kernphysik, Karlsruhe, Germany C. Barth, J. Berger, T. Chwalek, W. De Boer, A. Dierlamm, G. Dirkes, M. Feindt, J. Gruschke, M. Guthoff1, C. Hackstein, F. Hartmann, M. Heinrich, H. Held, K.H.",
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"Simonis, F.M.",
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"Ziebarth Institute of Nuclear Physics \"Demokritos\", Aghia Paraskevi, Greece G. Daskalakis, T. Geralis, S. Kesisoglou, A. Kyriakis, D. Loukas, I. Manolakos, A. Markou, C. Markou, C. Mavrommatis, E. Ntomari University of Athens, Athens, Greece L. Gouskos, T.J. Mertzimekis, A. Panagiotou, N. Saoulidou, E. Stiliaris University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas1, P. Kokkas, N. Manthos, I. Papadopoulos, V. Patras, F.A.",
"Triantis KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary A. Aranyi, G. Bencze, L. Boldizsar, C. Hajdu1, P. Hidas, D. Horvath15, A. Kapusi, K. Krajczar16, F. Sikler1, G. Vesztergombi16 Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, J. Molnar, J. Palinkas, Z. Szillasi, V. Veszpremi University of Debrecen, Debrecen, Hungary J. Karancsi, P. Raics, Z.L.",
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"Mehta, N. Nishu, L.K.",
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"Singh, J. Singh, S.P.",
"Singh University of Delhi, Delhi, India S. Ahuja, B.C.",
"Choudhary, A. Kumar, A. Kumar, S. Malhotra, M. Naimuddin, K. Ranjan, V. Sharma, R.K. Shivpuri Saha Institute of Nuclear Physics, Kolkata, India S. Banerjee, S. Bhattacharya, S. Dutta, B. Gomber, Sa.",
"Jain, Sh.",
"Jain, R. Khurana, S. Sarkar Bhabha Atomic Research Centre, Mumbai, India R.K. Choudhury, D. Dutta, S. Kailas, V. Kumar, A.K.",
"Mohanty1, L.M.",
"Pant, P. Shukla Tata Institute of Fundamental Research - EHEP, Mumbai, India T. Aziz, S. Ganguly, M. Guchait17, A. Gurtu18, M. Maity19, G. Majumder, K. Mazumdar, G.B.",
"Mohanty, B. Parida, A. Saha, K. Sudhakar, N. Wickramage Tata Institute of Fundamental Research - HECR, Mumbai, India S. Banerjee, S. Dugad, N.K.",
"Mondal Institute for Research in Fundamental Sciences (IPM), Tehran, Iran H. Arfaei, H. Bakhshiansohi20, S.M.",
"Etesami21, A. Fahim20, M. Hashemi, H. Hesari, A. Jafari20, M. Khakzad, A. Mohammadi22, M. Mohammadi Najafabadi, S. Paktinat Mehdiabadi, B. Safarzadeh23, M. Zeinali21 INFN Sezione di Bari $^{a}$ , Università di Bari $^{b}$ , Politecnico di Bari $^{c}$ , Bari, Italy M. Abbrescia$^{a}$$^{, }$$^{b}$ , L. Barbone$^{a}$$^{, }$$^{b}$ , C. Calabria$^{a}$$^{, }$$^{b}$ , S.S. Chhibra$^{a}$$^{, }$$^{b}$ , A. Colaleo$^{a}$ , D. Creanza$^{a}$$^{, }$$^{c}$ , N. De Filippis$^{a}$$^{, }$$^{c}$$^{, }$ 1, M. De Palma$^{a}$$^{, }$$^{b}$ , L. Fiore$^{a}$ , G. Iaselli$^{a}$$^{, }$$^{c}$ , L. Lusito$^{a}$$^{, }$$^{b}$ , G. Maggi$^{a}$$^{, }$$^{c}$ , M. Maggi$^{a}$ , N. Manna$^{a}$$^{, }$$^{b}$ , B. Marangelli$^{a}$$^{, }$$^{b}$ , S. My$^{a}$$^{, }$$^{c}$ , S. Nuzzo$^{a}$$^{, }$$^{b}$ , N. Pacifico$^{a}$$^{, }$$^{b}$ , A. Pompili$^{a}$$^{, }$$^{b}$ , G. Pugliese$^{a}$$^{, }$$^{c}$ , F. Romano$^{a}$$^{, }$$^{c}$ , G. Selvaggi$^{a}$$^{, }$$^{b}$ , L. Silvestris$^{a}$ , G. Singh$^{a}$$^{, }$$^{b}$ , S. Tupputi$^{a}$$^{, }$$^{b}$ , G. Zito$^{a}$ INFN Sezione di Bologna $^{a}$ , Università di Bologna $^{b}$ , Bologna, Italy G. Abbiendi$^{a}$ , A.C. Benvenuti$^{a}$ , D. Bonacorsi$^{a}$ , S. Braibant-Giacomelli$^{a}$$^{, }$$^{b}$ , L. Brigliadori$^{a}$ , P. Capiluppi$^{a}$$^{, }$$^{b}$ , A. Castro$^{a}$$^{, }$$^{b}$ , F.R.",
"Cavallo$^{a}$ , M. Cuffiani$^{a}$$^{, }$$^{b}$ , G.M.",
"Dallavalle$^{a}$ , F. Fabbri$^{a}$ , A. Fanfani$^{a}$$^{, }$$^{b}$ , D. Fasanella$^{a}$$^{, }$ 1, P. Giacomelli$^{a}$ , C. Grandi$^{a}$ , S. Marcellini$^{a}$ , G. Masetti$^{a}$ , M. Meneghelli$^{a}$$^{, }$$^{b}$ , A. Montanari$^{a}$ , F.L.",
"Navarria$^{a}$$^{, }$$^{b}$ , F. Odorici$^{a}$ , A. Perrotta$^{a}$ , F. Primavera$^{a}$ , A.M. Rossi$^{a}$$^{, }$$^{b}$ , T. Rovelli$^{a}$$^{, }$$^{b}$ , G. Siroli$^{a}$$^{, }$$^{b}$ , R. Travaglini$^{a}$$^{, }$$^{b}$ INFN Sezione di Catania $^{a}$ , Università di Catania $^{b}$ , Catania, Italy S. Albergo$^{a}$$^{, }$$^{b}$ , G. Cappello$^{a}$$^{, }$$^{b}$ , M. Chiorboli$^{a}$$^{, }$$^{b}$ , S. Costa$^{a}$$^{, }$$^{b}$ , R. Potenza$^{a}$$^{, }$$^{b}$ , A. Tricomi$^{a}$$^{, }$$^{b}$ , C. Tuve$^{a}$$^{, }$$^{b}$ INFN Sezione di Firenze $^{a}$ , Università di Firenze $^{b}$ , Firenze, Italy G. Barbagli$^{a}$ , V. Ciulli$^{a}$$^{, }$$^{b}$ , C. Civinini$^{a}$ , R. D'Alessandro$^{a}$$^{, }$$^{b}$ , E. Focardi$^{a}$$^{, }$$^{b}$ , S. Frosali$^{a}$$^{, }$$^{b}$ , E. Gallo$^{a}$ , S. Gonzi$^{a}$$^{, }$$^{b}$ , M. Meschini$^{a}$ , S. Paoletti$^{a}$ , G. Sguazzoni$^{a}$ , A. Tropiano$^{a}$$^{, }$ 1 INFN Laboratori Nazionali di Frascati, Frascati, Italy L. Benussi, S. Bianco, S. Colafranceschi24, F. Fabbri, D. Piccolo INFN Sezione di Genova, Genova, Italy P. Fabbricatore, R. Musenich INFN Sezione di Milano-Bicocca $^{a}$ , Università di Milano-Bicocca $^{b}$ , Milano, Italy A. Benaglia$^{a}$$^{, }$$^{b}$$^{, }$ 1, F. De Guio$^{a}$$^{, }$$^{b}$ , L. Di Matteo$^{a}$$^{, }$$^{b}$ , S. Fiorendi$^{a}$$^{, }$$^{b}$ , S. Gennai$^{a}$$^{, }$ 1, A. Ghezzi$^{a}$$^{, }$$^{b}$ , S. Malvezzi$^{a}$ , R.A. Manzoni$^{a}$$^{, }$$^{b}$ , A. Martelli$^{a}$$^{, }$$^{b}$ , A. Massironi$^{a}$$^{, }$$^{b}$$^{, }$ 1, D. Menasce$^{a}$ , L. Moroni$^{a}$ , M. Paganoni$^{a}$$^{, }$$^{b}$ , D. Pedrini$^{a}$ , S. Ragazzi$^{a}$$^{, }$$^{b}$ , N. Redaelli$^{a}$ , S. Sala$^{a}$ , T. Tabarelli de Fatis$^{a}$$^{, }$$^{b}$ INFN Sezione di Napoli $^{a}$ , Università di Napoli \"Federico II\" $^{b}$ , Napoli, Italy S. Buontempo$^{a}$ , C.A.",
"Carrillo Montoya$^{a}$$^{, }$ 1, N. Cavallo$^{a}$$^{, }$ 25, A.",
"De Cosa$^{a}$$^{, }$$^{b}$ , O. Dogangun$^{a}$$^{, }$$^{b}$ , F. Fabozzi$^{a}$$^{, }$ 25, A.O.M.",
"Iorio$^{a}$$^{, }$ 1, L. Lista$^{a}$ , M. Merola$^{a}$$^{, }$$^{b}$ , P. Paolucci$^{a}$ INFN Sezione di Padova $^{a}$ , Università di Padova $^{b}$ , Università di Trento (Trento) $^{c}$ , Padova, Italy P. Azzi$^{a}$ , N. Bacchetta$^{a}$$^{, }$ 1, P. Bellan$^{a}$$^{, }$$^{b}$ , D. Bisello$^{a}$$^{, }$$^{b}$ , A. Branca$^{a}$ , R. Carlin$^{a}$$^{, }$$^{b}$ , P. Checchia$^{a}$ , T. Dorigo$^{a}$ , U. Dosselli$^{a}$ , F. Gasparini$^{a}$$^{, }$$^{b}$ , U. Gasparini$^{a}$$^{, }$$^{b}$ , A. Gozzelino$^{a}$ , K. Kanishchev$^{a}$$^{, }$$^{c}$ , S. Lacaprara$^{a}$$^{, }$ 26, I. Lazzizzera$^{a}$$^{, }$$^{c}$ , M. Margoni$^{a}$$^{, }$$^{b}$ , M. Mazzucato$^{a}$ , A.T. Meneguzzo$^{a}$$^{, }$$^{b}$ , M. Nespolo$^{a}$$^{, }$ 1, L. Perrozzi$^{a}$ , N. Pozzobon$^{a}$$^{, }$$^{b}$ , P. Ronchese$^{a}$$^{, }$$^{b}$ , F. Simonetto$^{a}$$^{, }$$^{b}$ , E. Torassa$^{a}$ , M. Tosi$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Triossi$^{a}$ , S. Vanini$^{a}$$^{, }$$^{b}$ , P. Zotto$^{a}$$^{, }$$^{b}$ , G. Zumerle$^{a}$$^{, }$$^{b}$ INFN Sezione di Pavia $^{a}$ , Università di Pavia $^{b}$ , Pavia, Italy P. Baesso$^{a}$$^{, }$$^{b}$ , U. Berzano$^{a}$ , S.P.",
"Ratti$^{a}$$^{, }$$^{b}$ , C. Riccardi$^{a}$$^{, }$$^{b}$ , P. Torre$^{a}$$^{, }$$^{b}$ , P. Vitulo$^{a}$$^{, }$$^{b}$ , C. Viviani$^{a}$$^{, }$$^{b}$ INFN Sezione di Perugia $^{a}$ , Università di Perugia $^{b}$ , Perugia, Italy M. Biasini$^{a}$$^{, }$$^{b}$ , G.M.",
"Bilei$^{a}$ , B. Caponeri$^{a}$$^{, }$$^{b}$ , L. Fanò$^{a}$$^{, }$$^{b}$ , P. Lariccia$^{a}$$^{, }$$^{b}$ , A. Lucaroni$^{a}$$^{, }$$^{b}$$^{, }$ 1, G. Mantovani$^{a}$$^{, }$$^{b}$ , M. Menichelli$^{a}$ , A. Nappi$^{a}$$^{, }$$^{b}$ , F. Romeo$^{a}$$^{, }$$^{b}$ , A. Santocchia$^{a}$$^{, }$$^{b}$ , S. Taroni$^{a}$$^{, }$$^{b}$$^{, }$ 1, M. Valdata$^{a}$$^{, }$$^{b}$ INFN Sezione di Pisa $^{a}$ , Università di Pisa $^{b}$ , Scuola Normale Superiore di Pisa $^{c}$ , Pisa, Italy P. Azzurri$^{a}$$^{, }$$^{c}$ , G. Bagliesi$^{a}$ , T. Boccali$^{a}$ , G. Broccolo$^{a}$$^{, }$$^{c}$ , R. Castaldi$^{a}$ , R.T. D'Agnolo$^{a}$$^{, }$$^{c}$ , R. Dell'Orso$^{a}$ , F. Fiori$^{a}$$^{, }$$^{b}$ , L. Foà$^{a}$$^{, }$$^{c}$ , A. Giassi$^{a}$ , A. Kraan$^{a}$ , F. Ligabue$^{a}$$^{, }$$^{c}$ , T. Lomtadze$^{a}$ , L. Martini$^{a}$$^{, }$ 27, A. Messineo$^{a}$$^{, }$$^{b}$ , F. Palla$^{a}$ , F. Palmonari$^{a}$ , A. Rizzi$^{a}$$^{, }$$^{b}$ , A.T. Serban$^{a}$ , P. Spagnolo$^{a}$ , R. Tenchini$^{a}$ , G. Tonelli$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Venturi$^{a}$$^{, }$ 1, P.G.",
"Verdini$^{a}$ INFN Sezione di Roma $^{a}$ , Università di Roma \"La Sapienza\" $^{b}$ , Roma, Italy L. Barone$^{a}$$^{, }$$^{b}$ , F. Cavallari$^{a}$ , D. Del Re$^{a}$$^{, }$$^{b}$$^{, }$ 1, M. Diemoz$^{a}$ , C. Fanelli$^{a}$$^{, }$$^{b}$ , D. Franci$^{a}$$^{, }$$^{b}$ , M. Grassi$^{a}$$^{, }$ 1, E. Longo$^{a}$$^{, }$$^{b}$ , P. Meridiani$^{a}$ , F. Micheli$^{a}$$^{, }$$^{b}$ , S. Nourbakhsh$^{a}$ , G. Organtini$^{a}$$^{, }$$^{b}$ , F. Pandolfi$^{a}$$^{, }$$^{b}$ , R. Paramatti$^{a}$ , S. Rahatlou$^{a}$$^{, }$$^{b}$ , M. Sigamani$^{a}$ , L. Soffi$^{a}$$^{, }$$^{b}$ INFN Sezione di Torino $^{a}$ , Università di Torino $^{b}$ , Università del Piemonte Orientale (Novara) $^{c}$ , Torino, Italy N. Amapane$^{a}$$^{, }$$^{b}$ , R. Arcidiacono$^{a}$$^{, }$$^{c}$ , S. Argiro$^{a}$$^{, }$$^{b}$ , M. Arneodo$^{a}$$^{, }$$^{c}$ , C. Biino$^{a}$ , C. Botta$^{a}$$^{, }$$^{b}$ , N. Cartiglia$^{a}$ , R. Castello$^{a}$$^{, }$$^{b}$ , M. Costa$^{a}$$^{, }$$^{b}$ , N. Demaria$^{a}$ , A. Graziano$^{a}$$^{, }$$^{b}$ , C. Mariotti$^{a}$$^{, }$ 1, S. Maselli$^{a}$ , E. Migliore$^{a}$$^{, }$$^{b}$ , V. Monaco$^{a}$$^{, }$$^{b}$ , M. Musich$^{a}$ , M.M.",
"Obertino$^{a}$$^{, }$$^{c}$ , N. Pastrone$^{a}$ , M. Pelliccioni$^{a}$ , A. Potenza$^{a}$$^{, }$$^{b}$ , A. Romero$^{a}$$^{, }$$^{b}$ , M. Ruspa$^{a}$$^{, }$$^{c}$ , R. Sacchi$^{a}$$^{, }$$^{b}$ , A. Solano$^{a}$$^{, }$$^{b}$ , A. Staiano$^{a}$ , P.P.",
"Trapani$^{a}$$^{, }$$^{b}$ , A. Vilela Pereira$^{a}$ INFN Sezione di Trieste $^{a}$ , Università di Trieste $^{b}$ , Trieste, Italy S. Belforte$^{a}$ , F. Cossutti$^{a}$ , G. Della Ricca$^{a}$$^{, }$$^{b}$ , B. Gobbo$^{a}$ , M. Marone$^{a}$$^{, }$$^{b}$ , D. Montanino$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Penzo$^{a}$ Kangwon National University, Chunchon, Korea S.G. Heo, S.K.",
"Nam Kyungpook National University, Daegu, Korea S. Chang, J. Chung, D.H. Kim, G.N.",
"Kim, J.E.",
"Kim, D.J.",
"Kong, H. Park, S.R.",
"Ro, D.C.",
"Son Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea J.Y.",
"Kim, Zero J. Kim, S. Song Konkuk University, Seoul, Korea H.Y.",
"Jo Korea University, Seoul, Korea S. Choi, D. Gyun, B. Hong, M. Jo, H. Kim, T.J. Kim, K.S.",
"Lee, D.H.",
"Moon, S.K.",
"Park, E. Seo, K.S.",
"Sim University of Seoul, Seoul, Korea M. Choi, S. Kang, H. Kim, J.H.",
"Kim, C. Park, I.C.",
"Park, S. Park, G. Ryu Sungkyunkwan University, Suwon, Korea Y. Cho, Y. Choi, Y.K.",
"Choi, J. Goh, M.S.",
"Kim, B. Lee, J. Lee, S. Lee, H. Seo, I. Yu Vilnius University, Vilnius, Lithuania M.J. Bilinskas, I. Grigelionis, M. Janulis Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-de La Cruz, R. Lopez-Fernandez, R. Magaña Villalba, J. Martínez-Ortega, A. Sánchez-Hernández, L.M.",
"Villasenor-Cendejas Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico H.A.",
"Salazar Ibarguen Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico E. Casimiro Linares, A. Morelos Pineda, M.A.",
"Reyes-Santos University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand A.J.",
"Bell, P.H.",
"Butler, R. Doesburg, S. Reucroft, H. Silverwood National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan M. Ahmad, M.I.",
"Asghar, H.R.",
"Hoorani, S. Khalid, W.A.",
"Khan, T. Khurshid, S. Qazi, M.A.",
"Shah, M. Shoaib Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland G. Brona, M. Cwiok, W. Dominik, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski Soltan Institute for Nuclear Studies, Warsaw, Poland H. Bialkowska, B. Boimska, T. Frueboes, R. Gokieli, M. Górski, M. Kazana, K. Nawrocki, K. Romanowska-Rybinska, M. Szleper, G. Wrochna, P. Zalewski Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal N. Almeida, P. Bargassa, A. David, P. Faccioli, P.G.",
"Ferreira Parracho, M. Gallinaro, P. Musella, A. Nayak, J. Pela1, P.Q.",
"Ribeiro, J. Seixas, J. Varela, P. Vischia Joint Institute for Nuclear Research, Dubna, Russia S. Afanasiev, I. Belotelov, P. Bunin, I. Golutvin, I. Gorbunov, A. Kamenev, V. Karjavin, V. Konoplyanikov, G. Kozlov, A. Lanev, P. Moisenz, V. Palichik, V. Perelygin, S. Shmatov, V. Smirnov, A. Volodko, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St Petersburg), Russia S. Evstyukhin, V. Golovtsov, Y. Ivanov, V. Kim, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, V. Sulimov, L. Uvarov, S. Vavilov, A. Vorobyev, An.",
"Vorobyev Institute for Nuclear Research, Moscow, Russia Yu.",
"Andreev, A. Dermenev, S. Gninenko, N. Golubev, M. Kirsanov, N. Krasnikov, V. Matveev, A. Pashenkov, A. Toropin, S. Troitsky Institute for Theoretical and Experimental Physics, Moscow, Russia V. Epshteyn, M. Erofeeva, V. Gavrilov, M. Kossov1, A. Krokhotin, N. Lychkovskaya, V. Popov, G. Safronov, S. Semenov, V. Stolin, E. Vlasov, A. Zhokin Moscow State University, Moscow, Russia A. Belyaev, E. Boos, M. Dubinin4, L. Dudko, A. Ershov, A. Gribushin, O. Kodolova, I. Lokhtin, A. Markina, S. Obraztsov, M. Perfilov, S. Petrushanko, L. Sarycheva$^{\\textrm {\\dag }}$ , V. Savrin, A. Snigirev P.N.",
"Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin, I. Dremin, M. Kirakosyan, A. Leonidov, G. Mesyats, S.V.",
"Rusakov, A. Vinogradov State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, V. Grishin1, V. Kachanov, D. Konstantinov, A. Korablev, V. Krychkine, V. Petrov, R. Ryutin, A. Sobol, L. Tourtchanovitch, S. Troshin, N. Tyurin, A. Uzunian, A. Volkov University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia P. Adzic28, M. Djordjevic, M. Ekmedzic, D. Krpic28, J. Milosevic Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain M. Aguilar-Benitez, J. Alcaraz Maestre, P. Arce, C. Battilana, E. Calvo, M. Cerrada, M. Chamizo Llatas, N. Colino, B.",
"De La Cruz, A. Delgado Peris, C. Diez Pardos, D. Domínguez Vázquez, C. Fernandez Bedoya, J.P. Fernández Ramos, A. Ferrando, J. Flix, M.C.",
"Fouz, P. Garcia-Abia, O. Gonzalez Lopez, S. Goy Lopez, J.M.",
"Hernandez, M.I.",
"Josa, G. Merino, J. Puerta Pelayo, I. Redondo, L. Romero, J. Santaolalla, M.S.",
"Soares, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain C. Albajar, G. Codispoti, J.F.",
"de Trocóniz Universidad de Oviedo, Oviedo, Spain J. Cuevas, J. Fernandez Menendez, S. Folgueras, I. Gonzalez Caballero, L. Lloret Iglesias, J.M.",
"Vizan Garcia Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain J.A.",
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"Cabrillo, A. Calderon, S.H.",
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"Rodríguez-Marrero, A. Ruiz-Jimeno, L. Scodellaro, M. Sobron Sanudo, I. Vila, R. Vilar Cortabitarte CERN, European Organization for Nuclear Research, Geneva, Switzerland D. Abbaneo, E. Auffray, G. Auzinger, P. Baillon, A.H. Ball, D. Barney, C. Bernet5, W. Bialas, G. Bianchi, P. Bloch, A. Bocci, H. Breuker, K. Bunkowski, T. Camporesi, G. Cerminara, T. Christiansen, J.A.",
"Coarasa Perez, B. Curé, D. D'Enterria, A.",
"De Roeck, S. Di Guida, M. Dobson, N. Dupont-Sagorin, A. Elliott-Peisert, B. Frisch, W. Funk, A. Gaddi, G. Georgiou, H. Gerwig, M. Giffels, D. Gigi, K. Gill, D. Giordano, M. Giunta, F. Glege, R. Gomez-Reino Garrido, P. Govoni, S. Gowdy, R. Guida, L. Guiducci, M. Hansen, P. Harris, C. Hartl, J. Harvey, B. Hegner, A. Hinzmann, H.F. Hoffmann, V. Innocente, P. Janot, K. Kaadze, E. Karavakis, K. Kousouris, P. Lecoq, P. Lenzi, C. Lourenço, T. Mäki, M. Malberti, L. Malgeri, M. Mannelli, L. Masetti, G. Mavromanolakis, F. Meijers, S. Mersi, E. Meschi, R. Moser, M.U.",
"Mozer, M. Mulders, E. Nesvold, M. Nguyen, T. Orimoto, L. Orsini, E. Palencia Cortezon, E. Perez, A. Petrilli, A. Pfeiffer, M. Pierini, M. Pimiä, D. Piparo, G. Polese, L. Quertenmont, A. Racz, W. Reece, J. Rodrigues Antunes, G. Rolandi31, T. Rommerskirchen, C. Rovelli32, M. Rovere, H. Sakulin, F. Santanastasio, C. Schäfer, C. Schwick, I. Segoni, A. Sharma, P. Siegrist, P. Silva, M. Simon, P. Sphicas33, D. Spiga, M. Spiropulu4, M. Stoye, A. Tsirou, G.I.",
"Veres16, P. Vichoudis, H.K.",
"Wöhri, S.D.",
"Worm34, W.D.",
"Zeuner Paul Scherrer Institut, Villigen, Switzerland W. Bertl, K. Deiters, W. Erdmann, K. Gabathuler, R. Horisberger, Q. Ingram, H.C. Kaestli, S. König, D. Kotlinski, U. Langenegger, F. Meier, D. Renker, T. Rohe, J. Sibille35 Institute for Particle Physics, ETH Zurich, Zurich, Switzerland L. Bäni, P. Bortignon, M.A.",
"Buchmann, B. Casal, N. Chanon, Z. Chen, A. Deisher, G. Dissertori, M. Dittmar, M. Dünser, J. Eugster, K. Freudenreich, C. Grab, P. Lecomte, W. Lustermann, P. Martinez Ruiz del Arbol, N. Mohr, F. Moortgat, C. Nägeli36, P. Nef, F. Nessi-Tedaldi, L. Pape, F. Pauss, M. Peruzzi, F.J. Ronga, M. Rossini, L. Sala, A.K.",
"Sanchez, M.-C. Sawley, A. Starodumov37, B. Stieger, M. Takahashi, L. Tauscher$^{\\textrm {\\dag }}$ , A. Thea, K. Theofilatos, D. Treille, C. Urscheler, R. Wallny, H.A.",
"Weber, L. Wehrli, J. Weng Universität Zürich, Zurich, Switzerland E. Aguilo, C. Amsler, V. Chiochia, S. De Visscher, C. Favaro, M. Ivova Rikova, B. Millan Mejias, P. Otiougova, P. Robmann, A. Schmidt, H. Snoek, M. Verzetti National Central University, Chung-Li, Taiwan Y.H.",
"Chang, K.H.",
"Chen, C.M.",
"Kuo, S.W.",
"Li, W. Lin, Z.K.",
"Liu, Y.J.",
"Lu, D. Mekterovic, R. Volpe, S.S. Yu National Taiwan University (NTU), Taipei, Taiwan P. Bartalini, P. Chang, Y.H.",
"Chang, Y.W.",
"Chang, Y. Chao, K.F.",
"Chen, C. Dietz, U. Grundler, W.-S. Hou, Y. Hsiung, K.Y.",
"Kao, Y.J.",
"Lei, R.-S. Lu, D. Majumder, E. Petrakou, X. Shi, J.G.",
"Shiu, Y.M.",
"Tzeng, X. Wan, M. Wang Cukurova University, Adana, Turkey A. Adiguzel, M.N.",
"Bakirci38, S. Cerci39, C. Dozen, I. Dumanoglu, E. Eskut, S. Girgis, G. Gokbulut, I. Hos, E.E.",
"Kangal, G. Karapinar, A. Kayis Topaksu, G. Onengut, K. Ozdemir, S. Ozturk40, A. Polatoz, K. Sogut41, D. Sunar Cerci39, B. Tali39, H. Topakli38, D. Uzun, L.N.",
"Vergili, M. Vergili Middle East Technical University, Physics Department, Ankara, Turkey I.V.",
"Akin, T. Aliev, B. Bilin, S. Bilmis, M. Deniz, H. Gamsizkan, A.M. Guler, K. Ocalan, A. Ozpineci, M. Serin, R. Sever, U.E.",
"Surat, M. Yalvac, E. Yildirim, M. Zeyrek Bogazici University, Istanbul, Turkey M. Deliomeroglu, E. Gülmez, B. Isildak, M. Kaya42, O. Kaya42, S. Ozkorucuklu43, N. Sonmez44 National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine L. Levchuk University of Bristol, Bristol, United Kingdom F. Bostock, J.J. Brooke, E. Clement, D. Cussans, H. Flacher, R. Frazier, J. Goldstein, M. Grimes, G.P.",
"Heath, H.F. Heath, L. Kreczko, S. Metson, D.M.",
"Newbold34, K. Nirunpong, A.",
"Poll, S. Senkin, V.J.",
"Smith, T. Williams Rutherford Appleton Laboratory, Didcot, United Kingdom L. Basso45, K.W.",
"Bell, A. Belyaev45, C. Brew, R.M.",
"Brown, D.J.A.",
"Cockerill, J.A.",
"Coughlan, K. Harder, S. Harper, J. Jackson, B.W.",
"Kennedy, E. Olaiya, D. Petyt, B.C.",
"Radburn-Smith, C.H.",
"Shepherd-Themistocleous, I.R.",
"Tomalin, W.J.",
"Womersley Imperial College, London, United Kingdom R. Bainbridge, G. Ball, R. Beuselinck, O. Buchmuller, D. Colling, N. Cripps, M. Cutajar, P. Dauncey, G. Davies, M. Della Negra, W. Ferguson, J. Fulcher, D. Futyan, A. Gilbert, A. Guneratne Bryer, G. Hall, Z. Hatherell, J. Hays, G. Iles, M. Jarvis, G. Karapostoli, L. Lyons, A.-M. Magnan, J. Marrouche, B. Mathias, R. Nandi, J. Nash, A. Nikitenko37, A. Papageorgiou, M. Pesaresi, K. Petridis, M. Pioppi46, D.M.",
"Raymond, S. Rogerson, N. Rompotis, A.",
"Rose, M.J. Ryan, C. Seez, P. Sharp, A. Sparrow, A. Tapper, S. Tourneur, M. Vazquez Acosta, T. Virdee, S. Wakefield, N. Wardle, D. Wardrope, T. Whyntie Brunel University, Uxbridge, United Kingdom M. Barrett, M. Chadwick, J.E.",
"Cole, P.R.",
"Hobson, A. Khan, P. Kyberd, D. Leslie, W. Martin, I.D.",
"Reid, P. Symonds, L. Teodorescu, M. Turner Baylor University, Waco, USA K. Hatakeyama, H. Liu, T. Scarborough The University of Alabama, Tuscaloosa, USA C. Henderson Boston University, Boston, USA A. Avetisyan, T. Bose, E. Carrera Jarrin, C. Fantasia, A. Heister, J. St. John, P. Lawson, D. Lazic, J. Rohlf, D. Sperka, L. Sulak Brown University, Providence, USA S. Bhattacharya, D. Cutts, A. Ferapontov, U. Heintz, S. Jabeen, G. Kukartsev, G. Landsberg, M. Luk, M. Narain, D. Nguyen, M. Segala, T. Sinthuprasith, T. Speer, K.V.",
"Tsang University of California, Davis, Davis, USA R. Breedon, G. Breto, M. Calderon De La Barca Sanchez, M. Caulfield, S. Chauhan, M. Chertok, J. Conway, R. Conway, P.T.",
"Cox, J. Dolen, R. Erbacher, M. Gardner, R. Houtz, W. Ko, A. Kopecky, R. Lander, O.",
"Mall, T. Miceli, R. Nelson, D. Pellett, J. Robles, B. Rutherford, M. Searle, J. Smith, M. Squires, M. Tripathi, R. Vasquez Sierra University of California, Los Angeles, Los Angeles, USA V. Andreev, K. Arisaka, D. Cline, R. Cousins, J. Duris, S. Erhan, P. Everaerts, C. Farrell, J. Hauser, M. Ignatenko, C. Jarvis, C. Plager, G. Rakness, P. Schlein$^{\\textrm {\\dag }}$ , J. Tucker, V. Valuev, M. Weber University of California, Riverside, Riverside, USA J. Babb, R. Clare, J. Ellison, J.W.",
"Gary, F. Giordano, G. Hanson, G.Y.",
"Jeng47, H. Liu, O.R.",
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"Smith, J. Swanson †: Deceased 1: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 2: Also at National Institute of Chemical Physics and Biophysics, Tallinn, Estonia 3: Also at Universidade Federal do ABC, Santo Andre, Brazil 4: Also at California Institute of Technology, Pasadena, USA 5: Also at Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France 6: Also at Suez Canal University, Suez, Egypt 7: Also at Cairo University, Cairo, Egypt 8: Also at British University, Cairo, Egypt 9: Also at Fayoum University, El-Fayoum, Egypt 10: Now at Ain Shams University, Cairo, Egypt 11: Also at Soltan Institute for Nuclear Studies, Warsaw, Poland 12: Also at Université de Haute-Alsace, Mulhouse, France 13: Also at Moscow State University, Moscow, Russia 14: Also at Brandenburg University of Technology, Cottbus, Germany 15: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 16: Also at Eötvös Loránd University, Budapest, Hungary 17: Also at Tata Institute of Fundamental Research - HECR, Mumbai, India 18: Now at King Abdulaziz University, Jeddah, Saudi Arabia 19: Also at University of Visva-Bharati, Santiniketan, India 20: Also at Sharif University of Technology, Tehran, Iran 21: Also at Isfahan University of Technology, Isfahan, Iran 22: Also at Shiraz University, Shiraz, Iran 23: Also at Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Teheran, Iran 24: Also at Facoltà Ingegneria Università di Roma, Roma, Italy 25: Also at Università della Basilicata, Potenza, Italy 26: Also at Laboratori Nazionali di Legnaro dell' INFN, Legnaro, Italy 27: Also at Università degli studi di Siena, Siena, Italy 28: Also at Faculty of Physics of University of Belgrade, Belgrade, Serbia 29: Also at University of California, Los Angeles, Los Angeles, USA 30: Also at University of Florida, Gainesville, USA 31: Also at Scuola Normale e Sezione dell' INFN, Pisa, Italy 32: Also at INFN Sezione di Roma; Università di Roma \"La Sapienza\", Roma, Italy 33: Also at University of Athens, Athens, Greece 34: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 35: Also at The University of Kansas, Lawrence, USA 36: Also at Paul Scherrer Institut, Villigen, Switzerland 37: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia 38: Also at Gaziosmanpasa University, Tokat, Turkey 39: Also at Adiyaman University, Adiyaman, Turkey 40: Also at The University of Iowa, Iowa City, USA 41: Also at Mersin University, Mersin, Turkey 42: Also at Kafkas University, Kars, Turkey 43: Also at Suleyman Demirel University, Isparta, Turkey 44: Also at Ege University, Izmir, Turkey 45: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 46: Also at INFN Sezione di Perugia; Università di Perugia, Perugia, Italy 47: Also at University of Sydney, Sydney, Australia 48: Also at Utah Valley University, Orem, USA 49: Also at Institute for Nuclear Research, Moscow, Russia 50: Also at University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia 51: Also at Los Alamos National Laboratory, Los Alamos, USA 52: Also at Argonne National Laboratory, Argonne, USA 53: Also at Erzincan University, Erzincan, Turkey 54: Also at Kyungpook National University, Daegu, Korea"
]
] | 1204.1411 |
[
[
"On the computation of Manin products for operads"
],
[
"Abstract In the theory of binary quadratic operads, the white and black products of operads (called Manin products) play an important role.",
"Given two such operads, the computation of either of their Manin products is a routine task.",
"We present and describe a computer program that helps to compute white and black Manin products of binary quadratic operads.",
"The same utility allows to find the Koszul-dual operads.",
"In particular, we compute the white product of the operads Lie and As (governing the varieties of Lie and associative algebras, respectively).",
"It turns out that the resulting operad is magmatic, i.e., defines the variety of all algebras with one bilinear operation."
],
[
"Introduction",
"In a series recent works related to Leibniz algebras, Rota—Baxter operators, and dendriform algebras, the important role of the white and black products of operads (called Manin products [1], they are denoted by $\\circ $ and $\\bullet $ , respectively) has been established.",
"As a general reference in the operad theory one may apply to [2].",
"However, most of examples that can be found in the literature deal with white products $\\mathcal {P}\\circ \\mathrm {Perm}$ , $\\mathcal {P}\\circ \\mathrm {ComTrias}$ , and black products $\\mathcal {P}\\bullet \\mathrm {PreLie}$ , $\\mathcal {P}\\bullet \\mathrm {PostLie}$ .",
"Here $\\mathcal {P}$ is a binary quadratic operad, the definitions of $\\mathrm {Perm}$ , $\\mathrm {ComTrias}$ , $\\mathrm {PreLie}$ , and $\\mathrm {PostLie}$ can be found in the reference preprint [3].",
"Another class of operads has been considered in [5], where white products also coincide with Hadamard products.",
"These examples of products are “degenerate” in the sense that for both $\\mathrm {Perm}$ and $\\mathrm {ComTrias}$ the white products $\\mathcal {P}\\circ \\mathrm {Perm}$ , $\\mathcal {P}\\circ \\mathrm {ComTrias}$ coincide with the Hadamard products $\\mathcal {P}\\otimes \\mathrm {Perm}$ , $\\mathcal {P}\\otimes \\mathrm {ComTrias}$ [4].",
"Since the white and black Manin products of binary quadratic operads are dual with respect to Koszul duality of operads, i.e., $(\\mathcal {P} \\bullet \\mathcal {Q})^!",
"= \\mathcal {P}^!",
"\\circ \\mathcal {Q}^!",
"$ , and $\\mathrm {PreLie}=\\mathrm {Perm}^!",
"$ , $\\mathrm {PostLie}= \\mathrm {ComTrias}^!",
"$ , we have $\\mathcal {P}\\bullet \\mathrm {PreLie}= (\\mathcal {P}^!\\otimes \\mathrm {Perm})^!$ , $\\mathcal {P}\\bullet \\mathrm {PostLie}= (\\mathcal {P}^!\\otimes \\mathrm {ComTrias})^!$ .",
"The computation of a Manin product may require a lot of computations even in the simplest cases.",
"In a recent paper [6], the following question has been stated: What is the black product of Com and As?",
"(Equivalently, this is the dual to the white product of Lie and As.)",
"In this note, we describe in elementary terms the process of computation of Manin products for binary quadratic operads with no nontrivial unary operators.",
"We also describe an utility called manin designed to perform the corresponding computations over the field of rational numbers.",
"The source code of the utility is available at http://math.nsc.ru/LBRT/a1/pavelsk/manin2.zip.",
"It is designed by GNU Pascal version 20070904 based on gcc-4.1.3 20080704 for Ubuntu 2.1-4.1.2-29ubuntu2 (Free Software Foundation, Inc.).",
"The executable file for Windows XP is compiled by Borland Delphi 5.0."
],
[
"Computing Manin products",
"Let $\\mathbb {k}$ be a base field.",
"If $\\mathcal {P}$ is a variety of binary algebras over $\\mathbb {k}$ defined by multilinear identities then $\\mathcal {P}$ is governed by an operad which is also denoted by $\\mathcal {P}$ .",
"A binary quadratic operad corresponds to a variety whose defining identities have degrees 2 or 3.",
"To define such an operad, we need a space of (binary) operations $E$ and the space of relations $R$ , see [1] for details.",
"The space of multilinear terms of degree 3 is identified with $E(3) = \\mathbb {k}S_3\\otimes _{\\mathbb {k}S_2}(E\\otimes E)$ where the action of $(12)\\in S_2 $ on $E\\otimes E$ is given by $\\mathrm {id}\\,\\otimes (12)$ .",
"If $\\mu $ is an element of an $S_2$ -module $E$ , representing binary operation $(x_1,x_2)\\mapsto x_1 * x_2$ then $\\mu ^{(12)}$ corresponds to $(x_1,x_2)\\mapsto x_2* x_1$ .",
"If $\\mu _1 $ and $\\mu _2 $ represent two binary operations $(x_1,x_2)\\mapsto x_1 *_k x_2$ ($k=1,2$ ) then $\\mu _1\\otimes \\mu _2 \\in E\\otimes E$ corresponds to the following function: $(x_1,x_2,x_3)\\mapsto (x_1 *_2 x_2)*_1 x_3 $ .",
"It is convenient to identify the basic elements of $E(3)$ with appropriate monomials in formal variables $x_1,x_2,x_3$ as stated in the table below.",
"Table: NO_CAPTIONSuppose $\\mathcal {P}_1$ and $\\mathcal {P}_2$ are two binary quadratic operads, and $\\mathcal {P}_i(1)= 1$ , $i=1,2$ .",
"Then $\\mathcal {P}_i=\\mathcal {P}(E_i, R_i)$ , where $E_i$ are the spaces of binary operations considered as $S_2$ -modules, $R_i$ are the spaces of quadratic relations.",
"Assume $\\dim E_i<\\infty $ .",
"Recall that the Hadamard product $\\mathcal {P}=\\mathcal {P}_1\\otimes \\mathcal {P}_2$ is given by the rule $\\mathcal {P}(n)= \\mathcal {P}_1(n)\\otimes \\mathcal {P}_2(n)$ , $n\\ge 1$ , and the composition maps are expanded on $\\mathcal {P}(n)$ in the componentwise way.",
"It the same way, the structure of an $S_n$ -module is defined on $\\mathcal {P}(n)$ : A permutation $\\sigma \\in S_n$ acts on $\\mathcal {P}_1(n)\\otimes \\mathcal {P}_2(n)$ as $\\sigma \\otimes \\sigma $ .",
"By definition [1], the white product $\\mathcal {P}_1\\circ \\mathcal {P}_2$ is the sub-operad of $\\mathcal {P}_1\\otimes \\mathcal {P}_2$ generated by the space of operations $E= E_1\\otimes E_2$ .",
"It is known to be a binary quadratic operad.",
"To compute the space of relations, consider the map $\\tau : E(3) \\rightarrow E_1(3)\\otimes E_2(3)$ defined by $\\tau : \\sigma \\otimes _{\\mathbb {k}S_2} ((a_1\\otimes b_1)\\otimes (a_2\\otimes b_2))\\mapsto (\\sigma \\otimes _{\\mathbb {k}S_2}(a_1\\otimes a_2) )\\otimes (\\sigma \\otimes _{\\mathbb {k}S_2}(b_1\\otimes b_2) ),$ $a_i\\in E_1$ , $b_i\\in E_2$ , $\\sigma \\in S_3$ .",
"This is a well-defined $S_3$ -linear map.",
"Obviously, $\\tau $ is injective.",
"Denote by $D(E_1,E_2)$ the image of $\\tau $ .",
"Since $\\mathcal {P}_i(3)=E_i(3)/R_i$ for $i=1,2$ , the desired space of relations (a subspace in $E(3)$ ) is exactly the kernel of $E(3) \\overset{\\tau }{\\rightarrow } E_1(3)\\otimes E_2(3) \\overset{\\tau _1\\otimes \\tau _2}{\\rightarrow } \\mathcal {P}_1(3)\\otimes \\mathcal {P}_2(3),$ where $\\tau _i: E_i(3)\\rightarrow \\mathcal {P}_i(3)$ are the natural epimorphisms.",
"The kernel of $\\tau _1\\otimes \\tau _2$ is equal to $R_1\\otimes E_2(3) + E_1(3)\\otimes R_2$ .",
"It remains to find the intersection of $\\mathrm {Ker}\\,(\\tau _1\\otimes \\tau _2)$ with $D(E_1,E_2)$ and apply $\\tau ^{-1}$ to get $R$ —the space of relations, defining $\\mathcal {P}_1\\circ \\mathcal {P}_2$ .",
"For every finite-dimensional $S_n$ -module $M$ , let $M^\\vee $ stand for the dual space of $M$ considered as an $S_n$ -module with respect to sgn-twisted action: $\\langle f^{\\sigma }, e\\rangle = -\\langle f,e^{\\sigma }\\rangle $ , $f\\in M^\\vee $ , $e\\in E$ , $\\sigma \\in S_n$ .",
"Recall that if $\\mathcal {P} = \\mathcal {P}(E,R)$ , $R\\subseteq E(3)$ , then the Koszul dual operad $\\mathcal {P}^!$ is defined as $\\mathcal {P}(E^\\vee , R^\\perp )$ , where $E^\\vee $ is the dual space to $E$ endowed with sgn-twisted $S_2$ -action and $R^\\perp $ is the subspace of $E^\\vee (3)\\simeq E(3)^\\vee $ orthogonal to $R$ .",
"To get the black product of two binary quadratic operads $\\mathcal {P}_1$ and $\\mathcal {P}_2$ , it is enough to compute $\\mathcal {P}_1\\bullet \\mathcal {P}_2 = \\big (\\mathcal {P}_1^!\\circ \\mathcal {P}_2^!\\big )^!.$"
],
[
"Presenting initial data",
"Each input operad $\\mathcal {P}(E,R)$ is described in a separate file (e.g., the operad of Leibniz algebras—in the file leib).",
"The description consists of several parts.",
"The first line of the file contains the number $n=\\dim E$ .",
"The next $n$ lines present the action of $(12)\\in S_2$ on $E$ : Each $i$ th line $x_{i1}\\ x_{i2}\\ \\dots \\ x_{in}$ consists of coordinates of $e_i^{(12)} = x_{i1}e_1 + x_{i2}e_2 + \\dots + x_{in}e_n$ .",
"Below, the number of defining relation should be stated in a separate line (followed by a comment, e.g., what an operad is defined by this file).",
"After that, the list of relations $R$ comes.",
"They are presented as integer coordinate rows in the following standard basis of $E(3)$ : ${\\begin{array}{c}a_1 = \\mathrm {id}\\,\\otimes (e_1\\otimes e_1), \\ \\dots ,\\ a_n = \\mathrm {id}\\,\\otimes (e_1\\otimes e_n), \\\\a_{n+1}=\\mathrm {id}\\,\\otimes (e_2\\otimes e_1), \\ \\dots ,\\ a_{2n} = \\mathrm {id}\\,\\otimes (e_2\\otimes e_n), \\\\\\dotfill \\\\a_{n(n-1)+1}=\\mathrm {id}\\,\\otimes (e_n\\otimes e_1), \\dots , \\ a_{n^2} = \\mathrm {id}\\,\\otimes (e_n\\otimes e_n),\\\\a_{n^2+1}=(13)\\otimes (e_1\\otimes e_1), \\ \\dots ,\\ a_{2n^2}= (13)\\otimes (e_n\\otimes e_n), \\\\a_{2n^2+1}=(23)\\otimes (e_1\\otimes e_1), \\ \\dots ,\\ a_{3n^2}= (23)\\otimes (e_n\\otimes e_n).",
"\\\\\\end{array}}$ Here we identify the space $E(3) = \\mathbb {k}S_3\\otimes _{S_2} (E\\otimes E)$ with a sum of three copies of $E\\otimes E$ , i.e., $E(3)\\simeq V_3\\otimes E\\otimes E$ , where $V_3$ is formally spanned by $\\mathrm {id}\\,,(13),(23)\\in S_3$ .",
"The rest of the file can be used as a notebook, the program does not read these data.",
"For example, let us state the content of the file as describing the operad governing associative algebras.",
"2 % e_1 = x_1 x_2, e_2 = x_2 x_1 0 1 1 0 6 % ident.",
"of associative algebra 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 0 -1 0 0 0 0 0 0 0 0 1 0 a_1=e \\o e, a_2=e \\o e^{12}, a_3=e^{(12)}\\o e, a_4=e^{12}\\o e^{12} a_5=(13)a_1, a_6=(13)a_2, a_7=(13)a_3, a_8=(13)a_4 a_9=(23)a_1, a_{10}=(23)a_2, a_{11}=(23)a_3, a_{12}=(23)a_4"
],
[
"Usage of the utility",
"To compute the white or black product of operads described in files file1 and file2, type manin w file1 file2 or manin b file1 file2 respectively.",
"To compute the Koszul-dual operad $\\mathcal {P}^!$ to an operad $\\mathcal {P}$ described in file1, type manin d file1 The output is written into two files: result and result.amx.",
"The first one contains a description of the resulting operad $\\mathcal {P}(E,R)$ in the same form as the input files do, i.e., after a minor editing (and, possibly, commenting) it can be used as an input file.",
"The second one contains the description in a “human-readable” AMS-TeX format.",
"To write down the identities one should assign binary operations to basic vectors of $E(3)$ and rewrite the relations in terms of these operations (as in the table stated above).",
"Note that the final form of identities highly depends on this assignment."
],
[
"Overview of the units",
"The main program manin.pas uses three units: lspace, dynarr, and shmidt.",
"The unit lspace contains the definitions of main types of data: Vector and Space.",
"Vectors are presented as lists of integers, spaces—as lists of vectors.",
"Also, in the unit lspace the main arithmetic operations with vectors and spaces are defined (sum, tensor product, intersection), as well as input-output routines.",
"For intersection, the Hauss reduction method is implemented: If $V_1$ is a $\\mathbb {Q}$ -linear span of vectors $a_1,\\dots , a_m\\in \\mathbb {Z}^n$ , $V_2$ —of $b_1,\\dots , b_l\\in \\mathbb {Z}^n$ then the basis of $V_1\\cap V_2$ can be found as follows.",
"Consider a matrix of size $(k+l)\\times (2n)$ given by $\\begin{pmatrix}\\hbox{--- $a_1$ ---} & \\hbox{--- $a_1$ ---} \\\\\\hdots & \\hdots \\\\\\hbox{--- $a_m$ ---} & \\hbox{--- $a_m$ ---} \\\\\\hbox{--- $b_1$ ---} & \\hbox{--- $0$ ---} \\\\\\hdots & \\hdots \\\\\\hbox{--- $b_l$ ---} & \\hbox{--- $0$ ---}\\end{pmatrix}$ and apply elementary transformations of rows to make it a trapezoid.",
"All vectors remaining in the right half of the table, opposite to zero vectors in the left half, span the intersection $V_1\\cap V_2$ .",
"The unit dynarr is just a description of two-dimensional dynamic arrays and procedures allowing to convert a list of vectors to an array and converse.",
"In the unit shmidt, the orthogonalization procedure (Gram—Schmidt process) is implemented.",
"We use this process to compute the orthogonal complements.",
"Namely, in order to find the orthogonal component of a vector $v\\in \\mathbb {Z}^n$ relative to a subspace $V$ spanned by $a_1,\\dots , a_m \\in \\mathbb {Z}^n$ , we first make the vectors $a_1,\\dots , a_m $ pairwise mutually orthogonal and then compute $v_1 = \\langle a_1, a_1\\rangle v - \\langle a_1, v\\rangle a_1$ , $v_2 = \\langle a_2, a_2\\rangle v_1 - \\langle a_2, v_1\\rangle a_2$ , and so on (cancellation of coefficients is applied on each step).",
"Thus, to find an orthogonal complement $V^\\perp $ for $V\\subset \\mathbb {Q}^n$ , we find orthogonal components of all standard basic vectors $e_1,\\dots , e_n$ relative to $V$ and then apply Hauss reduction process.",
"The Hauss reduction method as well as the Gram—Schmidt process are encoded in such a way that both input and output lists of vectors have integer coordinates."
],
[
"White product of Lie and Perm",
"The white product of the operads governing Lie and Perm algebras is known to be the operad of Leibniz algebras [7].",
"Entering manin w lie perm we obtain the following relations (written in result.amx; here we have just replaced the AMS-TeX commands \\pmatrix and \\endpmatrix with the corresponding environment): Space of operations $E$ : $a_{1}, \\dots , a_{2}$ $S_2$ acts by: $\\begin{pmatrix}0 & -1 \\\\-1 & 0 \\\\\\end{pmatrix} $ Relations: $+1(23)\\otimes _{S_2}(a_{1}\\otimes a_{1}) -1(23)\\otimes _{S_2}(a_{1}\\otimes a_{2}) $ $+1(13)\\otimes _{S_2}(a_{1}\\otimes a_{1}) -1(13)\\otimes _{S_2}(a_{1}\\otimes a_{2}) $ $-1(id)\\otimes _{S_2}(a_{1}\\otimes a_{1}) +1(13)\\otimes _{S_2}(a_{2}\\otimes a_{2}) +1(23)\\otimes _{S_2}(a_{2}\\otimes a_{1}) $ $+1(id)\\otimes _{S_2}(a_{2}\\otimes a_{1}) -1(13)\\otimes _{S_2}(a_{2}\\otimes a_{1}) -1(23)\\otimes _{S_2}(a_{1}\\otimes a_{2}) $ $+1(id)\\otimes _{S_2}(a_{1}\\otimes a_{2}) -1(13)\\otimes _{S_2}(a_{2}\\otimes a_{2}) -1(23)\\otimes _{S_2}(a_{2}\\otimes a_{1}) $ $+1(id)\\otimes _{S_2}(a_{2}\\otimes a_{2}) -1(13)\\otimes _{S_2}(a_{1}\\otimes a_{2}) -1(23)\\otimes _{S_2}(a_{2}\\otimes a_{2}) $ Now, let us interpret $a_1$ as the operation $[x_1x_2]$ .",
"Then, according to the obtained $S_2$ -action on $E$ , $a_2=-a_1^{(12)}$ , i.e., $a_2$ should be interpreted as $-[x_2x_1]$ .",
"Thus the six relations above turn into ${\\begin{array}{c}[[x_1x_3]x_2]+[[x_3x_1]x_2], \\\\[[x_3x_2]x_1]+[[x_2x_3]x_1], \\\\-[[x_1x_2]x_3] + [x_1[x_2x_3]] - [x_2[x_1x_3]], \\\\-[x_3[x_1x_2]] + [x_1[x_3x_2]] + [[x_3 x_1]x_2], \\\\-[[x_2x_1]x_3] - [x_1[x_2x_3]] + [x_2[x_1x_3]], \\\\[x_3[x_2x_1]] + [[x_2x_3]x_1] - [x_2[ x_3x_1]].\\end{array}}$ These are corollaries of the only identity $[x[yz]]-[y[xz]] -[[xy]z]$ , the left Leibniz identity."
],
[
"The black product of PreLie and As",
"The operad of dendriform algebras [8] is known to be the black product of operads governing the varieties of pre-Lie and associative algebras (see also [9]).",
"The command manin b prelie as generates the following output: Space of operations $E$ : $a_{1}, \\dots , a_{4}$ $S_2$ acts by: $\\begin{pmatrix}0 & 0 & 0 & -1 \\\\0 & 0 & -1 & 0 \\\\0 & -1 & 0 & 0 \\\\-1 & 0 & 0 & 0 \\\\\\end{pmatrix} $ Relations: $-1(id)\\otimes _{S_2}(a_{4}\\otimes a_{4}) +1(13)\\otimes _{S_2}(a_{1}\\otimes a_{1}) -1(13)\\otimes _{S_2}(a_{1}\\otimes a_{3}) $ $+1(id)\\otimes _{S_2}(a_{3}\\otimes a_{3}) -1(13)\\otimes _{S_2}(a_{2}\\otimes a_{2}) +1(13)\\otimes _{S_2}(a_{2}\\otimes a_{4}) $ $-1(id)\\otimes _{S_2}(a_{4}\\otimes a_{2}) +1(13)\\otimes _{S_2}(a_{3}\\otimes a_{1}) $ $+1(id)\\otimes _{S_2}(a_{2}\\otimes a_{2}) -1(id)\\otimes _{S_2}(a_{2}\\otimes a_{4}) -1(13)\\otimes _{S_2}(a_{3}\\otimes a_{3}) $ $+1(id)\\otimes _{S_2}(a_{3}\\otimes a_{1}) -1(13)\\otimes _{S_2}(a_{4}\\otimes a_{2}) $ $+1(id)\\otimes _{S_2}(a_{1}\\otimes a_{1}) -1(id)\\otimes _{S_2}(a_{1}\\otimes a_{3}) -1(13)\\otimes _{S_2}(a_{4}\\otimes a_{4}) $ $+1(13)\\otimes _{S_2}(a_{4}\\otimes a_{1}) +1(23)\\otimes _{S_2}(a_{1}\\otimes a_{1}) -1(23)\\otimes _{S_2}(a_{1}\\otimes a_{3}) $ $-1(id)\\otimes _{S_2}(a_{4}\\otimes a_{1}) -1(23)\\otimes _{S_2}(a_{1}\\otimes a_{2}) +1(23)\\otimes _{S_2}(a_{1}\\otimes a_{4}) $ $+1(id)\\otimes _{S_2}(a_{3}\\otimes a_{2}) +1(23)\\otimes _{S_2}(a_{2}\\otimes a_{1}) -1(23)\\otimes _{S_2}(a_{2}\\otimes a_{3}) $ $+1(13)\\otimes _{S_2}(a_{3}\\otimes a_{2}) +1(23)\\otimes _{S_2}(a_{2}\\otimes a_{2}) -1(23)\\otimes _{S_2}(a_{2}\\otimes a_{4}) $ $+1(13)\\otimes _{S_2}(a_{4}\\otimes a_{3}) +1(23)\\otimes _{S_2}(a_{3}\\otimes a_{1}) $ $-1(id)\\otimes _{S_2}(a_{2}\\otimes a_{1}) +1(id)\\otimes _{S_2}(a_{2}\\otimes a_{3}) -1(23)\\otimes _{S_2}(a_{3}\\otimes a_{2}) $ $-1(13)\\otimes _{S_2}(a_{2}\\otimes a_{1}) +1(13)\\otimes _{S_2}(a_{2}\\otimes a_{3}) +1(23)\\otimes _{S_2}(a_{3}\\otimes a_{3}) $ $-1(id)\\otimes _{S_2}(a_{4}\\otimes a_{3}) +1(23)\\otimes _{S_2}(a_{3}\\otimes a_{4}) $ $-1(id)\\otimes _{S_2}(a_{1}\\otimes a_{2}) +1(id)\\otimes _{S_2}(a_{1}\\otimes a_{4}) -1(23)\\otimes _{S_2}(a_{4}\\otimes a_{1}) $ $+1(13)\\otimes _{S_2}(a_{3}\\otimes a_{4}) +1(23)\\otimes _{S_2}(a_{4}\\otimes a_{2}) $ $+1(id)\\otimes _{S_2}(a_{3}\\otimes a_{4}) -1(23)\\otimes _{S_2}(a_{4}\\otimes a_{3}) $ $+1(13)\\otimes _{S_2}(a_{1}\\otimes a_{2}) -1(13)\\otimes _{S_2}(a_{1}\\otimes a_{4}) -1(23)\\otimes _{S_2}(a_{4}\\otimes a_{4}) $ The 18 relations above split into three orbits with respect to the action of $S_3$ .",
"The representatives of these orbits are: $-\\mathrm {id}\\,\\otimes _{S_2}(a_{4}\\otimes a_{2}) +(13)\\otimes _{S_2}(a_{3}\\otimes a_{1}) $ , $-\\mathrm {id}\\,\\otimes _{S_2}(a_{4}\\otimes a_{4}) +(13)\\otimes _{S_2}(a_{1}\\otimes a_{1}) - (13)\\otimes _{S_2}(a_{1}\\otimes a_{3}) $ , $\\mathrm {id}\\,\\otimes _{S_2}(a_{3}\\otimes a_{3}) - (13)\\otimes _{S_2}(a_{2}\\otimes a_{2}) + (13)\\otimes _{S_2}(a_{2}\\otimes a_{4}) $ .",
"Let us interpret $x_1\\succ x_2$ as $a_1$ and $x_2\\prec x_1$ —as $a_2$ .",
"Then $a_3$ corresponds to $-x_1\\prec x_2$ and $a_4$ —to $-x_2\\succ x_1$ .",
"Hence, the defining identities of $\\mathrm {PreLie}\\bullet \\mathrm {As}$ are: ${\\begin{array}{c}x_3\\succ (x_2\\prec x_1) - (x_3\\succ x_2)\\prec x_1, \\\\(x_3\\succ x_2)\\succ x_1 +(x_3\\prec x_2)\\succ x_1 - x_3\\succ (x_2\\succ x_1), \\\\x_1\\prec (x_2\\prec x_3) + x_1\\prec (x_2\\succ x_3) - (x_1\\prec x_2)\\prec x_3.\\end{array}}$"
],
[
"Black product of Com and As",
"The command manin b as comm generates the following output: Space of operations $E$ : $a_{1}, \\dots , a_{2}$ $S_2$ acts by: $\\begin{pmatrix}0 & -1 \\\\-1 & 0 \\\\\\end{pmatrix} $ Relations: $+1(id)\\otimes _{S_2}(a_{1}\\otimes a_{1}) $ $+1(id)\\otimes _{S_2}(a_{1}\\otimes a_{2}) $ $+1(id)\\otimes _{S_2}(a_{2}\\otimes a_{1}) $ $+1(id)\\otimes _{S_2}(a_{2}\\otimes a_{2}) $ $-1(13)\\otimes _{S_2}(a_{1}\\otimes a_{1}) $ $-1(13)\\otimes _{S_2}(a_{1}\\otimes a_{2}) $ $-1(13)\\otimes _{S_2}(a_{2}\\otimes a_{1}) $ $-1(13)\\otimes _{S_2}(a_{2}\\otimes a_{2}) $ $-1(23)\\otimes _{S_2}(a_{1}\\otimes a_{1}) $ $-1(23)\\otimes _{S_2}(a_{1}\\otimes a_{2}) $ $-1(23)\\otimes _{S_2}(a_{2}\\otimes a_{1}) $ $-1(23)\\otimes _{S_2}(a_{2}\\otimes a_{2}) $ This is clear that such an operad corresponds to the variety of 3-nilpotent algebras.",
"As a corollary (which is also easy to check by means of manin w lie as), the white product $\\mathrm {Lie}\\circ \\mathrm {As}$ is the magmatic algebra."
]
] | 1204.0894 |
[
[
"Finding room for antilinear terms in the Hamiltonian"
],
[
"Abstract Although the Hamiltonian in quantum physics has to be a linear operator, it is possible to make quantum systems behave as if their Hamiltonians contained antilinear (i.e., semilinear or conjugate-linear) terms.",
"For any given quantum system, another system can be constructed that is physically equivalent to the original one.",
"It can be designed, despite the Wightman reconstruction theorem, so that antilinear operators in the original system become linear operators in the new system.",
"Under certain conditions, these operators can then be added to the new Hamiltonian.",
"The new quantum system has some unconventional features, a hidden degeneracy of the vacuum and a subtle distinction between the Hamiltonian and the observable of energy, but the physical equivalence guarantees that its states evolve like those in the original system and that corresponding measurements produce the same results.",
"The same construction can be used to make time-reversal linear."
],
[
"Introduction",
"Fundamental principles of quantum physics require the Hamiltonian to be a linear operator.",
"Wigner's unitary-antiunitary theorem [19], in particular, tells us that symmetries can be described, up to a phase, by unitary or antiunitary operators.",
"This implies that generators of continuous symmetries should be linear operators (e.g., [18]).",
"As the Hamiltonian is the generator of temporal displacements, it should also be linear.",
"For certain applications, which we will discuss in section , it would, however, help if some terms in the Hamiltonian could be antilinear.",
"Such antilinear operators, which are also called conjugate-linear or semilinear, are known from time-reversal, commute with real numbers, but anticommute with the imaginary unit $i$ (e.g., [13]).",
"Here we show a general procedure to make quantum systems behave as if their Hamiltonian contained antilinear terms.",
"For any given quantum system, we can construct another physically equivalent system where such antilinear terms become linear and where they can be added to the Hamiltonian.",
"This result may surprise, because the Wightman reconstruction theorem tells us that any quantum field theory can be reconstructed, up to a unitary transformation, from its correlation functions [14], and because a unitary transformation cannot turn antilinear terms into linear ones.",
"However, the Wightman theorem applies only to systems where the vacuum state is unique (up to a complex phase).",
"We will design the new quantum system so that its vacuum states are degenerate (encompassing at least two orthogonal vectors of the Hilbert space).",
"This degeneracy is hidden, because of the physical equivalence to the original quantum system, but it provides room for the new system to differ from the original one by more than just a unitary transformation.",
"To construct the new quantum system, given the original one, we proceed in two steps.",
"Given any quantum system, called system A, we first construct a system B, with degeneracy, and then a system C where antilinear operators turn linear.",
"The first step, in section , is almost trivial.",
"By taking the direct sum of the Hilbert space with itself, and by lifting the observables and the Hamiltonian to the resulting space, we introduce a twofold degeneracy of the vacuum and of other states.",
"This new system B has more possible states than the original one, but its observables fulfill a certain constraint which makes some states indistinguishable.",
"To prove physical equivalence, we will identify corresponding states in both systems, show that they evolve in parallel, and show that they produce the same results during quantum measurements.",
"The second step, in section , is less obvious.",
"In system B we introduce an operator $j$ that toggles between degenerate states.",
"It somewhat resembles the imaginary unit $i$ because: $ j^2 = -1 \\, ; \\;\\; j^\\dag = -j$ (see eq. ()).",
"It was motivated by a similar operator $\\breve{J}$ known from quantum physics on real Hilbert spaces [15], [16], but, unlike $\\breve{J}$ , it exists alongside $i$ .",
"We then construct system C by replacing some occurrences of $i$ with $j$ .",
"The Schrödinger and the von Neumann equation retain their usual form, with $i$ instead of $j$ , and vectors differing by a complex phase $\\exp (i a)$ will still belong to the same physical state, but in observables and in the Hamiltonian $j$ may replace $i$ .",
"Physical equivalence can again be shown, so that the degeneracy remains hidden, but antilinear operators, which anticommute with $i$ , may turn into linear operators that anticommute with $j$ .",
"Section shows how this construction makes room for antilinear terms in the Hamiltonian.",
"Antilinear operators $H^A_2$ , which cannot be directly added to the original Hamiltonian $H^A$ , become linear operators $H^C_2$ , in system C, and may be added to the new Hamiltonian $H^C$ .",
"States then behave as if they would evolve under $H^A + H^A_2$ , which is not linear, even though they actually evolve under the linear Hamiltonian $H^C + H^C_2$ .",
"Part of this construction is known, especially for the Dirac equation, from the field of quantum simulations.",
"It has been shown before that such a system C can simulate any state of system A and that antilinear operators in system A become linear operators in system C [3].",
"Here it is shown that the converse also holds, with system A being able to simulate any state of system C, so that both systems can be regarded as physically equivalent.",
"That is, each state in one system corresponds to (at least) one state in the other system, corresponding states evolve in parallel, and they produce the same results, with the same probabilities, when corresponding observables are measured.",
"While mathematically different, such systems cannot be told apart by physical observations.",
"We will also take a closer look at mixed states, compare the measurement process in both systems, and address a feature of system C which is unconventional: By construction, the observable of energy of system C will differ from its Hamiltonian (section REF ).",
"Both operators will, however, obey the usual rules, such as being self-adjoint, and their difference will be rather subtle.",
"Due to the physical equivalence, of systems A and C, this feature seems permissible.",
"As linear Hamiltonians have been very successful at describing physics, the importance of adding antilinear terms is not self-evident, so section also takes a few steps towards possible applications.",
"It indicates how these results may be used to turn an antilinear time-reversal operator $T^A$ into a linear time-reversal operator $T^C$ , to replace right-handed Weyl spinors by left-handed ones, or to make room for continuous symmetries with antilinear generators.",
"The appendix summarizes useful properties of sums of linear and antilinear operators and their adjoints.",
"For simplicity, we will not attempt to prove all mathematical results in full generality.",
"Some relations are proven only for finite Hilbert spaces, even though they may hold on infinite Hilbert spaces as well."
],
[
"From system A to B",
"Let us consider any quantum system.",
"We call it system A, its Hamiltonian $H^A$ , and its Hilbert space $\\mathcal {H}^A$ .",
"The Hamiltonian $H^A$ is linear, and, for simplicity, we assume that it does not explicitly depend on time.",
"The vacuum $\\Theta ^A$ , the state of lowest energy, is assumed to be unique (although the same procedure would work if the vacuum was already degenerate).",
"We also assume that any positive semidefinite, self-adjoint operator $\\rho $ of trace 1 constitutes a possible density operator of this system (although the results may be generalized to systems with superselection rules).",
"We do not assume that every self-adjoint operator constitutes an observable, but let system A be characterized by a given set of observables $O^A$ .",
"The goal of this section is to construct another system B, with Hamiltonian $H^B$ and Hilbert space $\\mathcal {H}^B$ , which is physically equivalent to the original one but has a hidden, twofold degeneracy of the vacuum.",
"This will demonstrate how two equivalent systems may differ by more than just a unitary transformation.",
"The construction itself will be trivial, and even showing the equivalence will not be hard, but we have to go through this in some detail as we could not find it published elsewhere."
],
[
"Doubling the dimension",
"The new space $\\mathcal {H}^B$ is simply constructed as the direct sum: $ \\mathcal {H}^B \\;=\\; \\mathcal {H}^A \\oplus \\mathcal {H}^A$ For finite-dimensional vector spaces, this step would double the dimension.",
"Any vector of the new space $\\mathcal {H}^B$ can be written as a pair $(\\Psi , $$ \\Phi )$ with the vectors $\\Psi $ and $\\Phi $ taken from $\\mathcal {H}^A$ .",
"The inner product of two such pairs $(\\Psi , $$ \\Phi )$ and $(\\Psi ^{\\prime }, $$ \\Phi ^{\\prime })$ is defined as the sum of two inner products from $\\mathcal {H}^A$ : $ \\Psi ^\\dag \\Psi ^{\\prime } + \\Phi ^\\dag \\Phi ^{\\prime }$ With this inner product, $\\mathcal {H}^B$ also becomes a Hilbert space [4].",
"Any operator $M^A$ on the original space $\\mathcal {H}^A$ can be lifted to an analogous operator $M^B$ on $\\mathcal {H}^B$ via: $ M^B (\\Psi , \\Phi ) \\;=\\; (M^A \\Psi , M^A \\Phi )$ for any $\\Psi $ and $\\Phi $ in $\\mathcal {H}^A$ .",
"Clearly, this preserves all the algebraic relationships between operators.",
"For example: $ \\nonumber R^A = M^A + N^A &\\Rightarrow & R^B = M^B + N^B\\\\ R^A = M^A N^A &\\Rightarrow & R^B = M^B N^B$ for any operators $M^A$ and $N^A$ , linear or otherwise, on $\\mathcal {H}^A$ .",
"When $M^A$ is linear, $M^B$ is also linear, and their traces and adjoints can be lifted along the same lines.",
"On finite Hilbert spaces, this is rather trivial, as traces and adjoints are then defined for any linear operator with full domain.",
"When lifted to $\\mathcal {H}^B$ , the trace doubles: $ \\mathrm {Tr}(M^B) \\;=\\; 2 \\mathrm {Tr}(M^A)$ since it is calculated by summing over basis vectors and since eq.",
"(REF ) doubles the size of the basis.",
"It is also obvious that the adjoint can be lifted via: $ R^A = (M^A)^\\dag &\\Rightarrow & R^B = (M^B)^\\dag $ since the inner product on $\\mathcal {H}^B$ , from eq.",
"(REF ), was based in a natural way on the inner product of $\\mathcal {H}^A$ .",
"In particular, a self-adjoint operator stays self-adjoint, as it is lifted to $\\mathcal {H}^B$ , and a unitary operator stays unitary.",
"On infinite Hilbert spaces, these relations will naturally hold only for operators whose trace or adjoint is defined at all.",
"To construct system B, the original observables $O^A$ and the Hamiltonian $H^A$ are lifted, via eq.",
"(REF ), to $\\mathcal {H}^B$ and become the new observables $O^B$ and Hamiltonian $H^B$ : $ O^B (\\Psi , \\Phi ) &=& (O^A \\Psi , O^A \\Phi )\\\\ H^B (\\Psi , \\Phi ) &=& (H^A \\Psi , H^A \\Phi )$ The original vacuum $\\Theta ^A$ gives rise to two states of lowest energy $(\\Theta ^A, 0)$ and $(0, \\Theta ^A)$ in system B, so that the new vacuum becomes twofold degenerate.",
"In fact, every energy eigenstate becomes at least twofold degenerate.",
"Lifting density operators to $\\mathcal {H}^B$ takes more care because of their normalization.",
"Usually, density operators are required to have a trace of 1, but the lifting via (REF ) would double the trace.",
"We thus define, for any density operator $\\rho ^A$ in system A, a corresponding density operator $\\rho ^B$ in system B as: $ \\rho ^B (\\Psi , \\Phi ) \\;=\\; \\frac{1}{2} (\\rho ^A \\Psi , \\rho ^A \\Phi )$ The normalization factor $1/2$ ensures that $\\rho ^B$ has the same trace as $\\rho ^A$ .",
"We will see below that such normalization factors cancel so that the laws of quantum physics keep their usual shape in system B.",
"Because $\\rho ^A$ is linear, self-adjoint, and positive semidefinite, it follows, from eq.",
"(REF ) and eq.",
"(REF ), that $\\rho ^B$ is also linear, self-adjoint, and positive semidefinite.",
"To lift a pure state, described by a vector $\\Psi ^A$ , from $\\mathcal {H}^A$ to $\\mathcal {H}^B$ , we can first turn it into a density operator $\\rho ^A = \\Psi ^A (\\Psi ^A)^\\dag $ and then use eq.",
"(REF ).",
"This turns it into a mixed state $\\rho ^B$ of rank two.",
"Alternatively, we could also identify the pure state in system A with another pure state in system B: $ \\Psi ^B = (\\Psi ^A, 0)$ Section REF will show that these states $\\rho ^B$ and $\\Psi ^B$ are physically indistinguishable, in system B, so both can be identified with $\\Psi ^A$ ."
],
[
"Physical equivalence",
"Since most relationships between operators are preserved, as we pass from system A to system B, the physical equivalence is not hard to show.",
"Let us first consider the temporal evolution, the measurement process, and then, in section REF , the only non-trivial issue, the number of states.",
"The evolution of any state $\\rho ^A$ in system A, between quantum measurements, is given by the von Neumann equation: $ \\frac{d}{dt} \\rho ^A(t) \\;=\\; -i [H^A, \\rho ^A(t)]$ (with $\\hbar $ set to 1).",
"Lifting $H^A$ and $\\rho ^A$ , via eq.",
"() and eq.",
"(REF ), and using the simple relations (REF ) gives: $ \\frac{d}{dt} \\rho ^B(t) \\;=\\; -i [H^B, \\rho ^B(t)]$ as the normalization factor $1/2$ , from eq.",
"(REF ), occurs on both sides and cancels.",
"Corresponding states $\\rho ^A$ and $\\rho ^B$ thus evolve in parallel.",
"The results of measurements are also the same in systems A and B.",
"For any observable $O^A$ of system A, the possible results are the eigenvalues $\\lambda _n$ in the spectral expansion: $O^A \\;= \\; \\sum _n \\lambda _n E^A_n$ (where no two $\\lambda _n$ are equal).",
"Here the $E^A_n$ are orthogonal projections onto eigenspaces with: $ \\nonumber E^A_n E^A_m = \\delta _{nm} E^A_n \\; ; \\;\\; (E^A_n)^\\dag = E^A_n$ for any $n$ and $m$ .",
"For simplicity, we have assumed that $O^A$ has discrete spectrum, as it would on a finite Hilbert space, but the generalization of the following to continuous spectra is straightforward.",
"All the operators in this expansion can be lifted to the Hilbert space $\\mathcal {H}^B$ via eq.",
"(REF ).",
"Because of eq.",
"(REF ) to (REF ), these lifted operators $O^B$ and $E^B_n$ will satisfy the same relations as the original ones: $ O^B \\;=\\; \\sum _n \\lambda _n E^B_n$ $ \\nonumber E^B_n E^B_m = \\delta _{nm} E^B_n \\; ; \\;\\; (E^B_n)^\\dag = E^B_n$ $O^B$ thus has the same eigenvalues as $O^A$ .",
"Even though $E^B_n$ has twice the rank of $E^A_n$ , the possible results $\\lambda _n$ of measurements are the same in systems A and B.",
"Both systems also agree in the probability of any particular result.",
"In system A, the probability of measuring the result $\\lambda _n$ , in state $\\rho ^A$ , for the observable $O^A$ is given by $\\mathrm {Tr}(\\rho ^A E^A_n)$ (with $\\mathrm {Tr}(\\rho ^A) = 1$ ).",
"The analogous probability, in system B, is given by $\\mathrm {Tr}(\\rho ^B E^B_n)$ .",
"Combining eq.",
"(REF ), (REF ), and (REF ) shows that both values match: $ \\mathrm {Tr}(\\rho ^A E^A_n) \\;=\\; \\mathrm {Tr}(\\rho ^B E^B_n)$ since the factor 2 from eq.",
"(REF ) cancels the factor $1/2$ from eq.",
"(REF ).",
"The same trace appears at the collapse of a wave function, where it keeps the trace of the density operators at 1: $ \\rho ^A \\rightarrow \\frac{E^A_n \\rho ^A E^A_n}{\\mathrm {Tr}(\\rho ^A E^A_n)} \\; ; \\;\\;\\rho ^B \\rightarrow \\frac{E^B_n \\rho ^B E^B_n}{\\mathrm {Tr}(\\rho ^B E^B_n)}$ According to eq.",
"(REF ), both traces are the same, and it follows that the relation (REF ) between $\\rho ^A$ and $\\rho ^B$ continues to hold after a collapse of the wave function.",
"These results almost suffice to show the physical equivalence of the systems A and B.",
"They guarantee that corresponding states $\\rho ^A$ and $\\rho ^B$ , in both systems, evolve analogously and produce the same results during measurements.",
"The only remaining issue is system B having more states than system A.",
"We will resolve this issue, in section REF , by showing that certain states in system B are physically indistinguishable."
],
[
"The operators $V$ and {{formula:90f8f197-fb67-439d-aeb7-3dcb385e2f7b}}",
"Before showing this, let us first introduce two linear operators $V$ and $j$ on $\\mathcal {H}^B$ .",
"The former is defined as: $ V (\\Psi ,\\Phi ) \\;=\\; (\\Phi , 0)$ for any $\\Psi $ and $\\Phi $ from $\\mathcal {H}^A$ .",
"Its adjoint satisfies: $ V^\\dag (\\Psi ,\\Phi ) \\;=\\; (0, \\Psi )$ By using $V$ and $V^\\dag $ , we can switch between degenerate states.",
"By definition, these operators have similar properties as fermionic field operators: $ V^2 \\;=\\; 0 \\;=\\; (V^\\dag )^2$ $ V V^\\dag + V^\\dag V \\;=\\; 1$ and it follows that: $ V V^\\dag V \\;=\\; V (V^\\dag V + V V^\\dag ) \\;=\\; V$ $ V^\\dag V V^\\dag \\;=\\; V^\\dag (V V^\\dag + V^\\dag V) \\;=\\; V^\\dag $ The linear operator $j$ , on $\\mathcal {H}^B$ , is defined as: $ j \\;=\\; V^\\dag - V$ As already mentioned in eq.",
"(REF ), $j$ satisfies: $ j^2 &=& V^\\dag (- V) + (- V) V^\\dag \\;=\\; -1\\\\ j^\\dag &=& V - V^\\dag \\;=\\; -j$ and is therefore unitary.",
"It somewhat resembles an operator $\\breve{J}$ used, instead of the imaginary unit $i$ , on real Hilbert spaces [15], [16].",
"Unlike $\\breve{J}$ , our $j$ exists alongside $i$ and toggles between degenerate states.",
"We can use $V$ to identify corresponding states in systems A and B.",
"Let us first consider any linear operator $N^B$ that was constructed by lifting an operator $N^A$ from $\\mathcal {H}^A$ via eq.",
"(REF ).",
"According to eq.",
"(REF ), any such an operator $N^B$ commutes with $V$ : $ \\nonumber V N^B (\\Psi , \\Phi ) \\;=\\; (N^A \\Phi , 0) \\;=\\; N^B V (\\Psi , \\Phi )$ Similarly, it commutes with $V^\\dag $ and thus with $j$ : $ [V, N^B] = 0; \\; [V^\\dag , N^B] = 0; \\; [j, N^B] = 0$ On the other hand, any operator $M^B$ , on $\\mathcal {H}^B$ , that fulfills the constraint (REF ) can be constructed by lifting an operator $M^A$ from $\\mathcal {H}^A$ .",
"To show this, we choose $M^A$ so that for any $\\Psi $ in $\\mathcal {H}^A$ : $ (M^A \\Psi , 0) \\;=\\; V V^\\dag M^B (\\Psi , 0)$ Here the term $V V^\\dag $ , together with eq.",
"(REF ) and (REF ), guarantees that the second component of $(M^A \\Psi , 0)$ is indeed zero.",
"For any $\\Phi $ in $\\mathcal {H}^A$ , we then get: $ \\nonumber (M^A \\Psi , 0) &=& V V^\\dag M^B V V^\\dag (\\Psi , \\Phi )\\\\ \\nonumber (0, M^A \\Phi ) &=& V^\\dag V V^\\dag M^B V (\\Psi , \\Phi )$ Because $M^B$ fulfills the constraint (REF ), and because of eq.",
"(REF ) and (REF ), this becomes: $ \\nonumber (M^A \\Psi , 0) &=& V V^\\dag M^B (\\Psi , \\Phi )\\\\ \\nonumber (0, M^A \\Phi ) &=& V^\\dag V M^B (\\Psi , \\Phi )$ Summing both relations gives: $ (M^A \\Psi , M^A \\Phi ) \\;=\\; M^B (\\Psi , \\Phi )$ which proves that $M^B$ can be constructed by lifting $M^A$ from $\\mathcal {H}^A$ via eq.",
"(REF )."
],
[
"Indistinguishable states",
"Using $V$ and $j$ , we can now resolve the issue of system B having more possible states than system A.",
"For any density operator $\\rho ^B_1$ on $\\mathcal {H}^B$ , we find another density operator $\\rho ^B_2$ on $\\mathcal {H}^B$ that can be generated, via eq.",
"(REF ), by lifting a density operator $\\rho ^A$ from $\\mathcal {H}^A$ .",
"We then show that all three density operators are physically equivalent, as they evolve in parallel and lead to the same results in quantum measurements, and conclude that the larger number of states, in system B, remains hidden.",
"For any given density operator $\\rho ^B_1$ , on $\\mathcal {H}^B$ , the new $\\rho ^B_2$ is chosen as: $ \\rho ^B_2 \\;=\\; \\frac{1}{4} \\sum _{a,b = 0}^1 (V^\\dag + V)^a j^b \\rho ^B_1j^{-b} (V^\\dag + V)^{-a}$ This choice is motivated by $j$ and $(V^\\dag + V)$ being unitary, due to eq.",
"(REF ), (), and: $ \\nonumber (V^\\dag + V)(V^\\dag + V)^\\dag \\;=\\; V^\\dag V + V V^\\dag \\;=\\; 1$ With $\\rho ^B_1$ being self-adjoint and positive semidefinite, it follows that $\\rho ^B_2 $ is also self-adjoint and positive semidefinite.",
"Due to the the cyclic property of the trace, we get: $ \\;\\;\\; \\mathrm {Tr}((V^\\dag + V)^a j^b \\rho ^B_1j^{-b} (V^\\dag + V)^{-a}) \\;=\\; \\mathrm {Tr}(\\rho ^B_1)$ so that $\\rho ^B_2$ has the same trace as $\\rho ^B_1$ and qualifies as density operator.",
"Let us first show that $\\rho ^B_2$ commutes with $V$ and $V^\\dag $ .",
"For any operator $R$ on $\\mathcal {H}^B$ , the sum: $ \\nonumber \\sum _{a = 0}^1 (V^\\dag + V)^a R (V^\\dag + V)^{-a}$ commutes with $(V^\\dag + V)$ since $(V^\\dag + V)^2 = 1$ .",
"Because eq.",
"(REF ) is based on such a sum, $\\rho ^B_2 $ commutes with $(V^\\dag + V)$ .",
"Similarly, we can use: $ \\nonumber (V^\\dag + V) j \\;=\\; -V^\\dag V + V V^\\dag \\;=\\; - j (V^\\dag + V)$ to move $j$ and $j^{-1}$ past $(V^\\dag + V)$ in eq.",
"(REF ).",
"From $j^2 = -1$ , it then follows that $\\rho ^B_2 $ commutes with $j$ .",
"Taken together, this proves that $\\rho ^B_2 $ commutes with $2V = V^\\dag + V - j$ and with $2V^\\dag = V^\\dag + V + j$ : $ \\nonumber [\\rho ^B_2, V] \\;=\\; 0 \\;=\\; [\\rho ^B_2, V^\\dag ]$ and fulfills the constraint (REF ).",
"We can therefore find, via eq.",
"(REF ), an operator on $\\mathcal {H}^A$ that becomes $\\rho ^B_2$ when lifted to $\\mathcal {H}^B$ .",
"Calling this operator $\\rho ^A/2$ , we get: $ \\rho ^B_2 (\\Psi , \\Phi ) \\;=\\; \\frac{1}{2} (\\rho ^A \\Psi , \\rho ^A \\Phi )$ (for any $\\Psi $ and $\\Phi $ in $\\mathcal {H}^A$ ).",
"We already know, from section REF , that these states $\\rho ^A$ and $\\rho ^B_2$ are physically equivalent.",
"All that remains to be shown is that $\\rho ^B_2$ is equivalent to $\\rho ^B_1$ .",
"To show this, we recall that all the observables $O^B$ of system B were lifted from $\\mathcal {H}^A$ .",
"Because of eq.",
"(REF ), they commute with $V$ , $V^\\dag $ , and $j$ .",
"This also holds for the Hamiltonian $H^B$ which thus commutes with all the factors surrounding $\\rho ^B_1$ in the definition (REF ) of $\\rho ^B_2$ .",
"This implies that the relation (REF ) remains valid as $\\rho ^B_1$ and $\\rho ^B_2$ evolve under the Hamiltonian $H^B$ in the von Neumann equation.",
"We also know, from eq.",
"(REF ), that the $E^B_n$ , which project onto eigenspaces of observables, were lifted from $\\mathcal {H}^A$ and also commute with $V$ , $V^\\dag $ , and $j$ .",
"As the collapse of the wave function, in eq.",
"(REF ), is described by $E^B_n$ , it follows that the relation (REF ), between $\\rho ^B_1$ and $\\rho ^B_2$ , remains valid during this collapse.",
"Finally, the probabilities of measuring results $\\lambda _n$ , for any observable $O^B$ , are also the same in state $\\rho ^B_1$ and state $\\rho ^B_2$ .",
"This follows from the analogue of eq.",
"(REF ) $ \\nonumber \\mathrm {Tr}((V^\\dag + V)^a j^b \\rho ^B_1 E^B_nj^{-b} (V^\\dag + V)^{-a}) \\;=\\; \\mathrm {Tr}(\\rho ^B_1 E^B_n)$ Because $E^B_n$ commutes with $(V^\\dag + V)$ and $j$ , we conclude from eq.",
"(REF ) that: $\\mathrm {Tr}(\\rho ^B_2 E^B_n) \\;=\\; \\mathrm {Tr}( \\rho ^B_1 E^B_n )$ Taken together, this shows that the three density operators $\\rho ^B_1$ , $\\rho ^B_2$ , and $\\rho ^A$ evolve in parallel and produce the same results during measurements.",
"Any density operator $\\rho ^B_1$ in system B thus corresponds to a physically equivalent operator $\\rho ^A$ in system A, and vice versa, via a many-to-one relationship.",
"Both systems are physically equivalent, and since $\\rho ^B_1$ and $\\rho ^B_2$ cannot be distinguished by observations, the larger number of states in system B remains hidden.",
"In particular, the vacuum degeneracy of system B stays hidden.",
"We can also conclude that it does not matter whether a pure state $\\Psi ^A$ is lifted to $\\mathcal {H}^B$ via eq.",
"(REF ) or (REF ).",
"The former choice results in $ \\rho ^B_2 \\;=\\; \\frac{1}{2} \\Psi ^B_1 (\\Psi ^B_1)^\\dag + \\frac{1}{2} \\Psi ^B_2 (\\Psi ^B_2)^\\dag $ with $\\Psi ^B_1 = (\\Psi ^A, 0)$ and $\\Psi ^B_2 = (0, \\Psi ^A)$ .",
"By contrast, eq.",
"(REF ) results in: $\\rho ^B_1 \\;=\\; \\Psi ^B_1 (\\Psi ^B_1)^\\dag $ Inserting this $\\rho ^B_1$ into eq.",
"(REF ), and using $V^\\dag \\Psi ^B_1 = \\Psi ^B_2$ , reproduces the $\\rho ^B_2$ from eq.",
"(REF ).",
"The two states are thus physically indistinguishable and can both be identified with $\\Psi ^A$ ."
],
[
"Physical arguments",
"As this proof of physical equivalence was rather formal, let us briefly discuss it.",
"Firstly, it should be noted that the degeneracy of states depends on the convention that every state is described by a density operator.",
"Such states are sometimes called quantum “microstates\" in contrast to the quantum “macrostates\" (not thermodynamic macrostates) that can actually be distinguished by observables [7].",
"The two microstates $\\rho ^B_1$ and $\\rho ^B_2$ , which cannot be distinguished, would belong to the same macrostate.",
"One might thus argue, for example, that all vacuum states belong to the same macrostate and are not really degenerate, but this seems to be mostly a matter of terminology.",
"No matter how states are defined, the two systems A and B will differ by more than a unitary transformation, but still be physically equivalent.",
"Secondly, one might worry that doubling the number of microstates somehow violates the Pauli exclusion principle.",
"It would be violated if we doubled, in atomic physics, the number of electrons in each orbital.",
"However, this is not what we have done here.",
"In a system with $n$ electrons, this doubling of orbitals could increase the number of possible states by as much as $2^n$ .",
"By contrast, in our system B, we have only doubled the number of possible states, no matter how many electrons the system contains.",
"Finally, one might worry that doubling the number of states affects the sum over states in thermodynamics.",
"The Gibbs formula for the entropy increases by $k_B \\ln (2)$ when we double the number of states (assuming that corresponding states are assigned equal probability).",
"This would be a problem if we could measure the absolute value of the Gibbs entropy, and not only the relative changes of entropy in the second law of thermodynamics.",
"For our systems A and B, this problem cannot arise.",
"It is widely accepted that all of physics can be based on quantum physics, so that any measurement can, at least in principle, be regarded as a quantum measurement (e.g., [7]).",
"When we measure, for example, the pressure, volume, and temperature in a Carnot cycle, all these measurements could be described, in principle, as observing the positions of the dials of certain instruments, and such a measurement of position $Q$ can be put into the usual quantum-mechanical forms $\\mathrm {Tr}(\\rho Q)$ or $\\Psi ^\\dag Q \\Psi $ .",
"As long as these quantum measurements are the same, in systems A and B, we will arrive at the same physical conclusions, find the same thermodynamic laws, and observe the same entropy.",
"In system A, or other systems with unique vacuum, this entropy will, as usually, agree with the Gibbs entropy formula.",
"In system B, it may not agree, simply because the Gibbs entropy formula was not designed for systems with a hidden vacuum degeneracy.",
"We can generalize this entropy formula to such systems as well, by subtracting a term $k_B \\ln (2)$ for any twofold, hidden degeneracy of the vacuum, and will thereby get the same value as in system A.",
"Subtracting this term may seem unconventional, but it is similar to dividing sums over states by $N!$ in systems with $N$ identical particles.",
"In the end, the proof of sections REF to REF always guarantees that the twofold degeneracy is hidden.",
"Other kinds of vacuum degeneracy are also known from other contexts, especially spontaneously broken symmetries (e.g., [9]), and it is accepted that such a degeneracy may remain hidden."
],
[
"From system B to C",
"Introducing vacuum degeneracy was the first step towards our goal of making room for antilinear terms in the Hamiltonian.",
"In the second, less obvious, step, we will replace some occurrences of the imaginary unit $i$ , in observables, by the linear operator $j$ from eq.",
"(REF ).",
"We will not replace all occurrences of $i$ , in the laws of quantum physics, which would be analogous to doing quantum physics on a real Hilbert space [16].",
"A physical state is commonly identified with a unit ray in Hilbert space, that is, with a set of vectors differing only by a complex phase $\\exp (i a)$ .",
"This interpretation will work here as well, before and after the replacement of $i$ , so that this particular $\\exp (i a)$ is not replaced by $\\exp (j a)$ .",
"Furthermore, we will take care that the abstract Schrödinger equation and the von Neumann equation maintain their usual form with the factor $i$ , not $j$ .",
"We will not insist that all commutation relations between observables, such as $[Q, P] = i$ for position $Q$ and momentum $P$ , maintain their usual form but let $i$ replace $j$ in such relations.",
"Nevertheless, the resulting quantum system will still be physically equivalent to the original one.",
"This will happen partly because $j$ shares some properties with $i$ , and partly because the expectation values $\\mathrm {Tr}(\\rho O)$ and $\\Psi ^\\dag O \\Psi $ in quantum measurements are always real so that $i$ cannot be measured directly."
],
[
"The operators $K$ , {{formula:116db53c-d261-49c8-b40b-c1b847f79a9d}} , and {{formula:667d3988-94cb-4729-87fb-63f2b91eb49c}}",
"To properly replace $i$ with $j$ , we first need to consider three other operators $K$ , $L$ , and $U$ .",
"The first operator $K$ is well-known and simply takes the complex conjugate of the vector to its right $K \\Psi \\;=\\; \\Psi ^* $ .",
"It is thus antilinear.",
"On finite-dimensional, complex vector spaces, one can find $K \\Psi $ simply by taking the complex conjugate of each component of $\\Psi $ .",
"On infinite-dimensional Hilbert spaces, $K$ can be defined by specifying a basis that stays invariant under $K$ , but we do not need to review these details here, as we will only need a few basic properties of $K$ .",
"Like other operators, $K$ can be lifted from $\\mathcal {H}^A$ to $\\mathcal {H}^B$ .",
"We will use it mostly on $\\mathcal {H}^B$ and, for simplicity, write it as $K$ , not $K^B$ .",
"Like any operator lifted from $\\mathcal {H}^A$ , this $K$ obeys eq.",
"(REF ): $[V, K] = 0; \\; [V^\\dag , K] = 0; \\; [j, K] = 0$ Two other basic properties of $K$ are $K^2 = 1$ and: $ \\left\\langle K \\Psi , K \\Phi \\right\\rangle &=& \\left\\langle \\Psi , \\Phi \\right\\rangle ^*$ where $\\left\\langle \\Psi , \\Phi \\right\\rangle $ is the inner product of $\\Psi $ and $\\Phi $ (which we identify with $\\Psi ^\\dag \\Phi $ so that it is linear in the second argument).",
"That is, taking the conjugate of two vectors also takes the conjugate of their inner product.",
"Consequently, $K$ maps any orthonormal basis of the Hilbert space, consisting of vectors $\\Gamma _n$ , into another orthonormal basis consisting of vectors $K \\Gamma _n$ : $ \\left\\langle K \\Gamma _m , K \\Gamma _n \\right\\rangle &=& \\delta _{mn}$ since the Kronecker delta $\\delta _{mn}$ is a real number.",
"The complex conjugation thus acts somewhat like a unitary transformation, yet being antilinear, it is not exactly unitary (but antiunitary).",
"In particular, it does not keep the trace of a linear operator $B$ invariant (with $\\mathrm {Tr}(KBK) = \\mathrm {Tr}(B)^*$ due to eq.",
"(REF )).",
"All this does not fix $K$ completely, as we might, for example, still choose whether $K$ commutes with position $Q$ or with momentum $P$ in the canonical commutation relation $[Q, P] = i$ , but we do not need to specify this here, as any definition of $K$ with the above properties suffices for our purposes.",
"The second operator $L$ can be regarded as a counterpart to $K$ in the same sense that our $j = V^\\dag - V$ , from eq.",
"(REF ), is a counterpart to $i$ .",
"It is defined on the degenerate Hilbert space $\\mathcal {H}^B$ as: $ L \\;=\\; V V^\\dag - V^\\dag V$ with the $V$ from eq.",
"(REF ).",
"It is self-adjoint, and, being linear, it commutes with $i$ .",
"It anticommutes with $j$ : $L j \\;=\\; -V^\\dag - V \\;=\\; - j L$ due to eq.",
"(REF ) to (REF ).",
"By contrast, $K$ anticommutes with $i$ and commutes with $j$ .",
"Like $K$ , this $L$ squares to: $ L^2 \\;=\\; V V^\\dag V V^\\dag + V^\\dag V V^\\dag V \\;=\\; 1$ and is thus unitary ($L^\\dag = L = L^{-1}$ ).",
"Since $K$ commutes with $V$ and $V^\\dag $ , it also commutes with $L$ .",
"All in all, the factors $i$ , $j$ , $K$ , and $L$ all commute with each other, except for $iK = -Ki$ and $jL = -Lj$ .",
"They square either to $i^2 = -1 = j^2$ or to $K^2 = 1 = L^2$ .",
"The third operator $U$ is defined as: $ U \\;=\\; \\frac{1}{2}\\left( 1 - ij + KL + ijKL \\right)$ in terms of the $j$ , $K$ , and $L$ from eq.",
"(REF ), (REF ), and (REF ).",
"Since $i$ , $j$ , and $L$ are linear, but $K$ antilinear, this $U$ is neither linear nor antilinear.",
"Recall that the product of a linear and an antilinear operator is antilinear, so $KL$ is antilinear, and $U$ is a sum of linear and antilinear terms.",
"In general, the sum $M = B + A$ of a linear operator $B$ and an antilinear operator $A$ is called “real-linear\" [6], so $U$ is real-linear.",
"Note that any real-linear operator commuting with $i$ is, by definition, linear.",
"The properties of real-linear operators are summarized in the appendix, but, for now, we can deal with $U$ simply by using the explicit expression (REF ) in terms of linear and antilinear operators.",
"To interpret $U$ , we use that $\\frac{1}{2}( 1 - ij)$ is an orthogonal projection because of $(ij)^\\dag = ij$ , from eq.",
"(), and because of: $\\left( \\frac{1}{2}( 1 - ij) \\right)^2 &=& \\frac{1}{4}\\left( 1 - 2ij + i^2j^2 \\right) \\nonumber \\\\ \\nonumber &=& \\frac{1}{2}( 1 - ij)$ Since $[ij,KL]=0$ , this projection commutes with $U$ , so $U$ acts separately on two orthogonal complements of the Hilbert space, the range and the kernel of this projection, consisting of vectors $\\Psi $ with $ij \\Psi = -\\Psi $ and $ij \\Psi = +\\Psi $ , respectively.",
"The first two terms of eq.",
"(REF ): $ \\nonumber \\frac{1}{2}\\left( 1 - ij \\right)$ act on the range of this projection and keep it invariant.",
"The other two terms of eq.",
"(REF ): $ \\nonumber \\frac{1}{2}\\left( 1 + ij \\right) KL$ act on the kernel of this projection and multiply it by $KL$ .",
"From eq.",
"(REF ), we know that $L$ is unitary, and from eq.",
"(REF ) we know that $K$ maps any orthonormal basis into another orthonormal basis.",
"It follows that the operators $KL$ and $U$ , despite being real-linear, also map any orthonormal basis into another orthonormal basis.",
"Writing the basis vectors as $\\Gamma _n$ , we thus get: $ \\left\\langle U\\Gamma _m, U\\Gamma _n \\right\\rangle \\;=\\; \\delta _{mn}$ Like the complex conjugation $K$ from eq.",
"(REF ), this $U$ therefore resembles a unitary transformation, except for not being linear.",
"In particular, it is bounded, and we will even find $U^\\dag = U^{-1}$ in eq.",
"(REF ).",
"The same interpretation of $U$ , together with $(KL)^2 = 1$ , also tells us that $U$ squares to 1: $U^2 &=& \\frac{1}{4} \\left( 1 - ij \\right)^2 + \\frac{1}{4} \\left( 1 + ij \\right)^2 \\left(KL \\right)^2\\nonumber \\\\ &=& \\frac{1}{2} ( 1 - ij ) + \\frac{1}{2} ( 1 + ij ) \\;=\\; 1$ It is thus invertible with $U^{-1} = U$ ."
],
[
"Interchanging $i$ and {{formula:f2c8ac7e-7db9-4c84-a851-333993b6bc43}}",
"Using the real-linear $U$ , from eq.",
"(REF ), we can now introduce the desired transformation that interchanges $i$ and $j$ and turns system B into system C. We simply transform any vector $\\Psi $ and any operator $M$ of the Hilbert space $\\mathcal {H}^B$ as: $ \\Psi \\;\\rightarrow \\; U \\Psi \\; ; \\;\\;M \\;\\rightarrow \\; U M U^{-1}$ Since $U$ is bounded, any bounded $M$ turns into a bounded $U M U^{-1}$ .",
"Since $U^2 = 1$ , applying the transformation twice leads back to the original $M$ .",
"For the most part, we will apply this transformation only to linear operators that commute with $j$ , for example, to the observables $O^B$ of system B.",
"For any such operator and, more generally, for any operator $M$ that commutes with $ij$ , the transformation (REF ) becomes: $U M U^{-1} &=& \\frac{1}{4} (1 - ij)^2 M + \\frac{1}{4} (1 + ij)^2 KL M KL\\nonumber \\\\ &=&\\frac{1-ij}{2} M + \\frac{1+ij}{2} KL M KL$ where we have used $(1-ij)(1+ij) = 0$ in the first step.",
"The product of two antilinear operators, like $K$ and $L M KL$ , is linear, so any linear $M$ that commutes with $j$ transforms into a linear $U M U^{-1}$ .",
"In particular, $i$ transforms into $j$ and vice versa: $U i U^{-1} &=& \\frac{1-ij}{2} i + \\frac{1+ij}{2} (-i) \\;=\\; j\\\\ U j U^{-1} &=& \\frac{1-ij}{2} j + \\frac{1+ij}{2} (-j) \\;=\\; i$ while their product $ij$ commutes with $U$ .",
"Occasionally, we will apply this transformation to antilinear operators that commute with $j$ .",
"For any such operator and, more generally, for any operator $N$ that anticommutes with $ij$ , the transformation (REF ) becomes: $U N U^{-1} &=& \\frac{1}{4} (1 - ij)^2 N KL + \\frac{1}{4} (1 + ij)^2 KL N\\nonumber \\\\ &=&\\frac{1-ij}{2} N KL + \\frac{1+ij}{2} KL N$ where we have again used $(1-ij)(1+ij) = 0$ .",
"In particular, $K$ transforms into $L$ and vice versa: $U K U^{-1} &=& \\frac{1-ij}{2} L + \\frac{1+ij}{2} L \\;=\\; L\\\\ U L U^{-1} &=& \\frac{1-ij}{2} K + \\frac{1+ij}{2} K \\;=\\; K$ while their product $KL$ commutes with $U$ .",
"As before, any real-linear $N$ that commutes with $j$ maps into a $U N U^{-1}$ that commutes with $U j U^{-1} = i$ and is linear.",
"This already indicates how we are going to achieve our goal of finding room for antilinear terms $N$ in the Hamiltonian.",
"We will turn them into linear terms $U N U^{-1}$ that can be added to the new Hamiltonian."
],
[
"The adjoint $U^\\dag $",
"A slight complication, when using a real-linear $U$ on the complex Hilbert space, comes from finding the adjoint $U^\\dag $ of a real-linear operator.",
"Recall that the usual definition for the adjoint $B^\\dag $ of a linear operator $B$ : $ \\left\\langle B^\\dag \\Psi , \\Phi \\right\\rangle &=& \\left\\langle \\Psi , B \\Phi \\right\\rangle $ cannot be used to define the adjoint of an antilinear operator since then the left-hand side would be linear in $\\Phi $ and the right-hand side antilinear.",
"Instead, the adjoint $A^\\dag $ of an antilinear operator $A$ , which is familiar from time-reversal, is the unique operator satisfying (e.g., [13]): $ \\left\\langle A^\\dag \\Psi , \\Phi \\right\\rangle &=& \\left\\langle \\Psi , A \\Phi \\right\\rangle ^*$ for any vectors $\\Psi $ and $\\Phi $ in the complex Hilbert space.",
"Due to the complex conjugation on the right-hand side, both sides are linear in $\\Phi $ and $\\Psi $ , with the adjoint $A^\\dag $ being an antilinear operator.",
"Since eq.",
"(REF ) and (REF ) differ, neither of them will, in general, hold for a sum of linear and antilinear operators, that is, for real-linear operators.",
"However, both linear and antilinear operators $M$ satisfy the relation: $ \\Re \\left\\langle M^\\dag \\Psi , \\Phi \\right\\rangle &=& \\Re \\left\\langle \\Psi , M \\Phi \\right\\rangle $ which results from taking the real part of eq.",
"(REF ) or (REF ).",
"It is thus plausible to define the adjoint $M^\\dag $ of real-linear operators $M$ so that it also fulfills this relation (see appendix).",
"Unfortunately, this definition of the adjoint of real-linear operators is only sometimes [6], but not always [11], used in the mathematical literature.",
"We will use it here because it has convenient properties for our purpose.",
"It agrees with how the adjoint of linear and antilinear operators is defined in eq.",
"(REF ) and (REF ), and obeys similar rules as the adjoint of linear operators.",
"The appendix shows: $ (M^\\dag )^\\dag &=& M \\\\ (M + N)^\\dag &=& M^\\dag + N^\\dag \\\\ (M N)^\\dag &=& N^\\dag M^\\dag $ for any real-linear operators $M$ and $N$ where these adjoints exist.",
"It should be noted that the last rule, the product rule, may no longer hold when $N^\\dag $ is replaced by the adjoint $\\Psi ^\\dag $ of a vector $\\Psi $ .",
"This happens because $\\Psi ^\\dag $ does not map the Hilbert space into itself, but into the field $\\mathbb {C}$ of complex numbers.",
"The same issue already occurs for antilinear operators.",
"When we write the inner product as $\\left\\langle \\Psi , \\Phi \\right\\rangle =$$\\Psi ^\\dag \\Phi $ , then eq.",
"(REF ) tells us that $(M \\Psi )^\\dag \\Phi $ is, in general, not the same as $\\Psi ^\\dag M^\\dag \\Phi $ .",
"To deal with such expressions, we should use eq.",
"(REF ) instead.",
"Apart from this, we can use the adjoints of real-linear operators about as easily as the adjoints of linear operators.",
"With these relations, we can find the adjoint $U^\\dag $ of the real-linear operator $U$ from eq.",
"(REF ).",
"First, comparing eq.",
"(REF ) and (REF ), we find the familiar relation $K^\\dag = K$ .",
"From eq.",
"() and (REF ), we also know $j^\\dag = -j$ and $L^\\dag = L$ .",
"Taking the adjoint of eq.",
"(REF ) gives: $U^\\dag \\;=\\; \\frac{1}{2}\\left( 1 - ji + LK + LKji \\right) \\;=\\; U $ With $U^2=1$ , from eq.",
"(REF ), this yields: $ U^\\dag \\;=\\; U^{-1}$ The transformation $U$ thus keeps the norm of any vector $\\Psi $ invariant: $|| U \\Psi ||^2 &=& \\Re \\left\\langle U \\Psi , U \\Psi \\right\\rangle \\nonumber \\\\&=& \\Re \\left\\langle U^\\dag U \\Psi , \\Psi \\right\\rangle \\nonumber \\\\ &=& || \\Psi ||^2$ where eq.",
"(REF ) was used.",
"Due to eq.",
"(), any real-linear operator $M$ also obeys: $ \\left( U M U^{-1} \\right)^\\dag \\;=\\; U M^\\dag U^{-1}$ In most of the following applications, a linear $M$ will be turned into a $U M U^{-1}$ that is also linear.",
"In such cases, eq.",
"(REF ) implies that a self-adjoint $M$ is turned into a self-adjoint $U M U^{-1}$ , and a unitary $M$ is turned into a unitary $U M U^{-1}$ .",
"Incidentally, as $U$ was defined, in eq.",
"(REF ), in terms of linear and antilinear operators, we could prove these relations by using only the familiar adjoints of linear and antilinear operators, but it is useful to know that they hold for the adjoint of real-linear operators as well."
],
[
"The new observables $O^C$",
"Using this transformation $ U $ , we now construct our system C where antilinear terms become linear and find a place in the Hamiltonian.",
"Recall that, in section REF , we started with system A, with Hamiltonian $H^A$ , observables $O^A$ , and Hilbert space $\\mathcal {H}^A$ , and constructed an equivalent system B with Hamiltonian $H^B$ , observables $O^B$ , and Hilbert space $\\mathcal {H}^B$ .",
"Any density operator $\\rho ^A$ in system A corresponded to a density operator $\\rho ^B$ in system B.",
"We now take this one step further and apply the transformation $U$ , from eq.",
"(REF ), to any observable $O^B$ .",
"The resulting observable, for system C, is: $ O^C \\;=\\; U O^B U^{-1}$ Since all the observables $O^B$ of the earlier system were lifted from $\\mathcal {H}^A$ , they commute with $V$ and $V^\\dag $ (eq.",
"(REF )) and therefore with the operators $j$ and $L$ from eq.",
"(REF ) and (REF ).",
"Being linear, $O^B$ also commutes with $i$ .",
"Consequently, $O^C$ commutes with $U j U^{-1} = i$ and with $U i U^{-1} = j$ .",
"Any real-linear operator commuting with $i$ is linear, so $O^C$ is linear and, due to eq.",
"(REF ), also self-adjoint.",
"Instead of eq.",
"(REF ), we can also use eq.",
"(REF ) to express $O^C$ .",
"It simplifies to: $ O^C \\;=\\; \\frac{1-ij}{2} O^B + \\frac{1+ij}{2} K O^B K$ because of $[O^B,L]=0$ , $[K,L]=0$ , and $L^2=1$ .",
"Many relations between the observables of system B remain valid in system C. For example, it follows directly from eq.",
"(REF ) and (REF ) that adding or multiplying two observables, or multiplying them with a real number, gives analogous results in both systems B and C. Multiplying an observable with the imaginary unit $i$ would not give analogous results, since $U$ does not commute with $i$ , but this is of no concern here as it would not give a self-adjoint observable either.",
"Incidentally, we could also split an observable $O^B$ into a real and an imaginary part: $ \\nonumber \\Re O^B = \\frac{1}{2} (O^B + K O^B K) \\; ; \\;\\; \\Im O^B = \\frac{1}{2i} (O^B - K O^B K)$ with $O^B = \\Re O^B + i \\Im O^B$ .",
"The new observable from eq.",
"(REF ) would turn out to be: $O^C \\;=\\; \\Re O^B + j \\Im O^B$ which illustrates how $i$ is replaced by $j$ .",
"The same transformation $U$ relates states $\\Psi ^B$ in system B to corresponding states $\\Psi ^C$ in system C: $ \\Psi ^C \\;=\\; U \\Psi ^B$ For mixed states, described by density operators, the rules of correspondence are more complex and will be derived in section REF .",
"As $U$ is so similar to a unitary transformation, it is straightforward to see that the new observables $O^C$ produce the same results as the previous observables $O^B$ .",
"To simplify notation, let us consider observables with discrete spectrum, although the generalization to continuous spectrum is straightforward.",
"In eq.",
"(REF ), we have already used the spectral expansion $O^B = \\sum _n \\lambda _n E^B_n$ of an observable in system B.",
"The eigenvalues $\\lambda _n$ are real and the $E^B_n$ are orthogonal projections onto eigenspaces.",
"Since $O^B$ commutes with $j$ , these $E^B_n$ also commute with $j$ .",
"Applying $U$ , we find the analogous expansion: $ O^C \\;=\\; \\sum _n \\lambda _n E^C_n$ where the eigenvalues $\\lambda _n$ remain the same and the operators $E^C_n = U E^B_n U^{-1}$ retain the properties from eq.",
"(REF ): $ E^C_n E^C_m = \\delta _{nm} E^C_n \\; ; \\;\\; (E^C_n)^\\dag = E^C_n$ due to eq.",
"(REF ).",
"As $E^B_n$ commutes with $j$ , $E^C_n$ commutes with $U j U^{-1} = i$ and is not only real-linear, but linear.",
"It is thus an orthogonal projection, and eq.",
"(REF ) describes the spectrum of $O^C$ .",
"When $O^B$ is observed in experiments, the possible results are the values $\\lambda _n$ .",
"According to eq.",
"(REF ), the same results are observed when measuring $O^C$ .",
"The probability of seeing each result, in state $\\Psi ^B$ or state $\\Psi ^C$ , is also the same: $(\\Psi ^C)^\\dag E^C_n \\Psi ^C &=&\\Re \\left\\langle \\Psi ^C, E^C_n \\Psi ^C \\right\\rangle \\nonumber \\\\ &=&\\Re \\left\\langle U \\Psi ^B, U E^B_n \\Psi ^B \\right\\rangle \\nonumber \\\\ &=&\\Re \\left\\langle U^\\dag U \\Psi ^B, E^B_n \\Psi ^B \\right\\rangle \\nonumber \\\\ &=& (\\Psi ^B)^\\dag E^B_n \\Psi ^B$ where we have used eq.",
"(REF ), (REF ), and the fact that the expectation value of a self-adjoint operator is real.",
"In particular, the mean observed value of the observable is the same in both systems: $(\\Psi ^C)^\\dag O^C \\Psi ^C &=& \\sum _n \\lambda _n (\\Psi ^C)^\\dag E^C_n \\Psi ^C\\nonumber \\\\ &=&(\\Psi ^B)^\\dag O^B \\Psi ^B $ This shows that the transformation $U$ , despite not being linear, does not affect the results of observations.",
"Both systems also reach corresponding states after a collapse of the wavefunction.",
"Apart from a normalization factor, the state $\\Psi ^B$ becomes $E^B_n \\Psi ^B$ and the state $\\Psi ^C$ becomes $E^C_n \\Psi ^C$ .",
"The relationship $\\Psi ^C = U \\Psi ^B$ thus continues to hold after a collapse of the wave function: $ E^C_n \\Psi ^C \\;=\\; U E^B_n U^{-1} U \\Psi ^B \\;=\\; U E^B_n \\Psi ^B$ and the norms of these states also stay equal due to eq.",
"(REF ).",
"Section REF will show that this also holds for mixed states, so that corresponding observations in both quantum systems give the same results."
],
[
"The new Hamiltonian $H^C$",
"To show the physical equivalence of both systems, we still have to show that corresponding states evolve in parallel.",
"When $\\Psi ^C(0) = U \\Psi ^B(0)$ holds at time 0, then it should continue to hold at any later time: $ \\Psi ^C(t) = U \\Psi ^B(t)$ The main issue here is the form of the abstract Schrödinger equation.",
"In system B it reads, as usually: $ \\frac{d}{dt} \\Psi ^B(t) \\;=\\; -i H^B \\Psi ^B(t)$ After the transformation $U$ , this becomes: $ \\frac{d}{dt} \\Psi ^C(t) \\;=\\; -j U H^B U^{-1} \\Psi ^C(t)$ where $U H^B U^{-1}$ is the observable of energy, in system C, and $j$ replaces $i$ due to $U i U^{-1} = j$ .",
"By construction, eq.",
"(REF ) guarantees that relation (REF ) continues to hold as $\\Psi ^C(t)$ evolves.",
"However, the substitution of $j$ , at such a central place of quantum physics, seems awkward.",
"To avoid it, and retain the usual form of the Schrödinger equation, we take another step that seems somewhat less awkward.",
"We let the Hamiltonian of system C, which we write as $H^C$ , differ slightly from the observable of energy which we continue to write as $U H^B U^{-1}$ .",
"Specifically, we set the Hamiltonian to: $ H^C \\;=\\; -ij \\, U H^B U^{-1}$ so that eq.",
"(REF ) becomes: $ \\frac{d}{dt} \\Psi ^C(t) \\;=\\; -i H^C \\Psi ^C(t)$ with the usual factor $i$ instead of $j$ .",
"Like any observable $O^C$ from eq.",
"(REF ), the observable $U H^B U^{-1}$ is linear, self-adjoint, and commutes with $j$ .",
"With $j^\\dag = -j$ , it follows that the Hamiltonian $H^C$ is also linear, commutes with $j$ , and is self-adjoint: $\\left( H^C \\right)^\\dag &=& -U H^B U^{-1} (-j) (-i) \\;=\\; H^C$ as it should be to guarantee the condition of unitarity.",
"Moreover, it follows that $U H^B U^{-1}$ commutes with $H^C$ so that the energy $(\\Psi ^C)^\\dag U H^B U^{-1} \\Psi ^C$ is conserved.",
"Section REF will show that the same Hamiltonian $H^C$ also appears in the von Neumann equation of system C. Making this distinction, between the Hamiltonian $H^C$ and the observable $U H^B U^{-1}$ of energy, is unconventional.",
"Even when non-Hermitian Hamiltonians are used in quantum mechanics [2], it is commonly assumed that the Hamiltonian should be equal to the observable of energy and thus have only real eigenvalues since energies are real-valued.",
"In our case, both $H^C$ and $U H^B U^{-1}$ have real eigenvalues, since they are self-adjoint, but only those of $U H^B U^{-1}$ denote energy values.",
"Their two spectra are, however, closely related.",
"The factor $-ij$ , from eq.",
"(REF ), has the properties $ \\nonumber (-ij)^2 = 1 \\; ; \\;\\; (-ij)^\\dag (-ij) = 1$ which makes it drop out of many calculations.",
"It is self-adjoint and commutes with the self-adjoint operator $U H^B U^{-1}$ , so both operators can be diagonalized simultaneously.",
"Since $-ij$ squares to 1, it constitutes a “grading\" operator whose eigenvalues are either $+1$ or $-1$ , and the eigenvalues of the Hamiltonian $H^C$ can differ from those of the observable $U H^B U^{-1}$ by at most a sign.",
"In fact, making a distinction between the Hamiltonian and the observable of energy is not without precedent.",
"As a trivial example, consider an experiment where we measure all the energy within a box except for the energy of neutrinos passing, without interaction, through this box.",
"The observable being measured in this experiment will contain no contribution from neutrinos, but the Hamiltonian will still contain the terms describing neutrino propagation.",
"More generally, such a distinction between Hamiltonian and observable of energy is likely to crop up whenever the Hamiltonian describes some process, involving neutrinos, dark matter, or perhaps degenerate vacuum states, that we cannot observe directly.",
"As another example, consider the Hamiltonian in gauge theories.",
"The observable of momentum $m \\dot{Q}$ will, in general, differ from the generator $P$ of spatial displacements, by more than just a factor $\\hbar $ , because the observable $m \\dot{Q}$ (or at least its expectation value) stays invariant under gauge transformations while the generator $P$ does not [7].",
"Presumably the same also applies, for certain choices of the gauge, to the temporal dimension, where displacements are generated by the Hamiltonian.",
"If we chose a gauge that varies in time, and derived the Hamiltonian, it would presumably also vary in time, even if the observable of energy did not.",
"Our distinction between the Hamiltonian and the observable of energy is not exactly the same as in these simple examples, but it also seems permissible as long as it produces the correct physical predictions.",
"The distinction between $H^C$ and $U H^B U^{-1}$ is arguably the most unconventional feature of system C, but there are other ones.",
"In many relations between operators, where $i$ appears explicitly, it will be replaced by $j$ .",
"The canonical commutation relation $[Q^B,P^B] = i$ between position $Q^B$ and momentum $P^B$ of a particle in one dimension, for example, will become: $ [Q^C, P^C] = U [Q^B, P^B] U^{-1} = U i U^{-1} = j$ It has been argued that such commutation relations always need a term like $i$ on the right-hand side so that the Heisenberg uncertainty relation holds [16], but our $j$ is similar enough to $i$ to meet this requirement.",
"Physical equivalence guarantees that the standard deviations of $Q^C$ and $P^C$ , which can be observed, keep their usual values and satisfy the uncertainty relation."
],
[
"Mixed states",
"So far, the physical equivalence of the two systems B and C has been proven only for pure states.",
"Section REF has shown that two corresponding states $\\Psi ^B(t)$ and $\\Psi ^C(t)$ evolve in parallel, between measurements, and section REF has shown that they produce the same results when corresponding observables $O^B$ and $O^C$ are measured.",
"To finish this proof, let us now show the same results for mixed states.",
"As in section REF , the density operators $\\rho ^B$ or $\\rho ^C$ are constrained only by the usual requirements.",
"They have to be linear, self-adjoint, and positive semidefinite with trace 1.",
"Unlike the observables $O^B$ or $O^C$ , they do not have to commute with $j$ .",
"This prevents us from simply applying the transformation $U$ to find corresponding density operators.",
"When $\\rho ^B$ does not commute with $j$ , then $U \\rho ^B U^{-1}$ does not commute with $U j U^{-1}=i$ and is not linear.",
"To find the correct relation between $\\rho ^B$ and $\\rho ^C$ , let us first consider a density operator $\\rho ^B$ of finite rank.",
"With the spectral theorem of self-adjoint operators, it can be written as: $ \\rho ^B \\;=\\; \\sum _n p_n \\Psi ^B_n (\\Psi ^B_n)^\\dag $ where the non-negative coefficients $p_n$ add up to 1 and the vectors $\\Psi ^B_n$ are orthogonal to each other and normalized to $||\\Psi ^B_n|| = 1$ .",
"As usually, we can interpret this as a statistical mixture of pure states $\\Psi ^B_n$ occurring with probability $p_n$ .",
"From eq.",
"(REF ), we know, for each of these states $\\Psi ^B_n$ in system B, the corresponding state $\\Psi ^C_n$ in system C: $ \\Psi ^C_n \\;=\\; U \\Psi ^B_n$ This implies that the corresponding density operator $\\rho ^C$ , in system C, is given by: $ \\rho ^C \\;=\\; \\sum _n p_n (U \\Psi ^B_n) (U \\Psi ^B_n)^\\dag $ We have seen, in section REF , that $(U \\Psi ^B_n)^\\dag $ is not necessarily the same as $(\\Psi ^B_n)^\\dag U^\\dag $ , since $U$ is not linear, so the expression (REF ) is not the same as $U \\rho ^B U^\\dag $ .",
"Unlike $U \\rho ^B U^\\dag $ , this $\\rho ^C$ is always linear since $(U \\Psi ^B_n)$ is just another vector without any antilinear or real-linear properties.",
"The vectors $U \\Psi ^B_n$ are still normalized to 1 and orthogonal to each other, as eq.",
"(REF ) tells us that $U$ maps any orthonormal basis into another orthonormal basis.",
"Consequently, $\\rho ^C$ can be interpreted, like $\\rho ^B$ , as a statistical mixture of pure states with probability $p_n$ .",
"Like $\\rho ^B$ , the $\\rho ^C$ from eq.",
"(REF ) is self-adjoint and positive semidefinite with trace $\\sum _n p_n = 1$ .",
"The equivalence of $\\rho ^B$ and $\\rho ^C$ , within their respective quantum systems, follows from the equivalence of the pure states $\\Psi ^B_n$ and $\\Psi ^C_n = U \\Psi ^B_n$ .",
"From section REF , we know that measuring an observable $O^B$ , in state $\\Psi ^B_n$ , and the corresponding observable $O^C$ , in state $\\Psi ^C_n$ , produces the same results.",
"This easily generalizes to density operators.",
"In particular, it follows from: $ \\nonumber \\mathrm {Tr}(\\rho ^B O^B) &=& \\sum _n p_n (\\Psi ^B_n)^\\dag O^B \\Psi ^B_n\\\\ \\nonumber \\mathrm {Tr}(\\rho ^C O^C) &=& \\sum _n p_n (\\Psi ^C_n)^\\dag O^C \\Psi ^C_n$ and the earlier result (REF ) that the expectation values of the observables are the same: $\\mathrm {Tr}(\\rho ^B O^B) \\;=\\; \\mathrm {Tr}(\\rho ^C O^C)$ The same equality holds, due to eq.",
"(REF ), for the probabilities $\\mathrm {Tr}(\\rho ^B E^B_n)$ or $\\mathrm {Tr}(\\rho ^C E^C_n)$ of observing any particular eigenvalue of $O^B$ or $O^C$ .",
"Similarly, the results of section REF can be used to show that the density matrices $\\rho ^B$ and $\\rho ^C$ evolve in parallel.",
"We find their evolution by applying the abstract Schrödinger equation, from eq.",
"(REF ) and (REF ), to the vectors $\\Psi ^B_n$ and $\\Psi ^C_n$ from eq.",
"(REF ) to (REF ).",
"This gives the von Neumann equation: $ \\frac{d}{dt} \\rho ^B(t) &=& -i [H^B, \\rho ^B(t)] \\\\\\frac{d}{dt} \\rho ^C(t) &=& -i [H^C, \\rho ^C(t)]$ where the Hamiltonian $H^C$ , in system C, is again given by the $ -ij \\, U H^B U^{-1}$ from eq.",
"(REF ).",
"Here we have used: $ \\nonumber (H^C \\Psi ^C_n)^\\dag \\;=\\; (\\Psi ^C_n)^\\dag H^C$ which holds trivially since $H^C$ is linear and self-adjoint.",
"Note that the von Neumann equation keeps its usual form with a factor $i$ , not $j$ .",
"In fact, if we had not already included the factor $-ij$ in the Hamiltonian $H^C$ , in eq.",
"(REF ), we would have to be careful where to put it now since $j$ does not necessarily commute with $\\rho ^C$ .",
"It is known that other placements of $j$ , within the context of quantum physics on real Hilbert spaces, may lead to difficulties [8].",
"Both density matrices also continue to evolve in parallel after a collapse of the wave function.",
"Again, we can show this by applying the corresponding result for pure states, from eq.",
"(REF ), to the density matrices in eq.",
"(REF ) and (REF ).",
"Apart from a trivial prefactor, which keeps the trace at 1, this gives: $\\rho ^B &\\rightarrow & E^B_m \\rho ^B E^B_m \\\\\\rho ^C &\\rightarrow & E^C_m \\rho ^C E^C_m$ where we have used that $E^C_m$ , from eq.",
"(REF ), is linear and self-adjoint with $(E^C_m \\Psi ^C_n)^\\dag =$$(\\Psi ^C_n)^\\dag E^C_m$ .",
"The collapse thus takes the same familiar form in systems B and C. These results can be generalized to density matrices that are not of finite rank.",
"They clearly still hold when the sum over eigenstates in eq.",
"(REF ) is infinite, and it is straightforward to generalize them to a continuous spectrum as well.",
"In fact, there is another, more general way to show the same results.",
"We could rewrite the linear density operator $\\rho ^B$ , from eq.",
"(REF ), in terms of a real-linear operator $\\rho ^B_R$ with: $\\rho ^B &=& \\rho ^B_R - i \\rho ^B_R i \\\\\\rho ^B_R \\Phi &=& \\sum _n p_n \\Psi ^B_n \\Re (\\Psi ^B_n)^\\dag \\Phi $ for any vector $\\Phi $ in $\\mathcal {H}^B$ .",
"Here $\\Re $ acts on the whole subsequent product $(\\Psi ^B_n)^\\dag \\Phi $ and not just the first factor.",
"This decomposition is analogous to the more familiar eq.",
"(REF ) in the appendix.",
"After putting $\\rho ^B$ into this form, we could then use eq.",
"(REF ) to write $\\rho ^C$ from eq.",
"(REF ) as: $\\rho ^C \\;=\\; U \\rho ^B_R U^\\dag - i U \\rho ^B_R U^\\dag i$ and conclude that $\\rho ^B_R$ simply becomes $U \\rho ^B_R U^\\dag $ in system C, even though $\\rho ^B$ itself does not transform in such a simple way.",
"Even without using the physical equivalence of pure states, it would then be straightforward to prove the physical equivalence of these density operators (not shown)."
],
[
"Finding room for antilinear terms in the Hamiltonian",
"This physical equivalence of the quantum systems A, B, and C, shown in the previous sections, may be useful for several applications involving antilinear operators.",
"Our main goal, from the introduction, was to find room for antilinear terms in the Hamiltonian.",
"That is, we would like to take a quantum system, with the usual, linear Hamiltonian $H^A$ , and add an antilinear term $H^A_2$ so that states evolve as: $ \\frac{d}{dt} \\Psi ^A(t) \\;=\\; -i (H^A + H^A_2) \\Psi ^A(t)$ While this makes sense as a differential equation, it does not make sense as a Schrödinger equation because the Hamiltonian would not be linear.",
"However, we can pass from system A to the system C, replace $H^A$ by the Hamiltonian $H^C$ from eq.",
"(REF ), replace $H^A_2$ by an analogous term: $ H^C_2 \\;=\\; -ij U H^B_2 U^{-1}$ and replace $\\Psi ^A$ by $\\Psi ^C = U (\\Psi ^A, 0)$ according to eq.",
"(REF ) and (REF ).",
"Lifting eq.",
"(REF ) to the new Hilbert space $\\mathcal {H}^B$ , and applying the transformation $U$ , gives: $\\frac{d}{dt} \\Psi ^C(t) &=& -U i (H^B + H^B_2) U^{-1} \\Psi ^C(t)\\nonumber \\\\&=& -i (H^C + H^C_2) \\Psi ^C(t)$ as in eq.",
"(REF ) to (REF ).",
"In this equivalent form, the differential equation can be interpreted as a Schrödinger equation.",
"Like other operators lifted to $\\mathcal {H}^B$ , the $H^B_2$ obeys eq.",
"(REF ) and commutes with $j$ .",
"Consequently, $U H^B_2 U^{-1}$ commutes with $U j U^{-1} = i$ and is linear, so $H^C_2$ is also linear.",
"In fact, we could start with any real-linear $H^A_2$ , not just antilinear ones, and $H^C_2$ would still be linear.",
"Though $H^A_2$ cannot be added directly to the Hamiltonian of system A, we can thus construct an equivalent system C where the corresponding term $H^C_2$ can be added.",
"Not every antilinear term $H^A_2$ can be added in this way.",
"The main restriction is that the resulting Hamiltonian $H^C + H^C_2$ should still be self-adjoint, as required by the condition of unitarity.",
"Because $H^C$ is self-adjoint, $H^C_2$ has to be self-adjoint.",
"This condition is satisfied by any $H^A_2$ with: $ (i H^A_2)^\\dag \\;=\\; -i H^A_2$ as we then get $-i H^B_2 = (i H^B_2)^\\dag $ .",
"Applying $U$ yields: $ \\nonumber \\left(j U H^B_2 U^{-1}\\right)^\\dag \\;=\\; -j U H^B_2 U^{-1}$ due to eq.",
"(REF ) and $U i U^{-1} = j$ .",
"Since $U H^B_2 U^{-1}$ is linear, we can conclude from eq.",
"(REF ) that: $H^C_2 \\;=\\; (H^C_2)^\\dag $ By reversing this argument, we can also show that condition (REF ) is necessary for $H^C_2$ to be self-adjoint.",
"Incidentally, if we let vectors evolve directly under eq.",
"(REF ), their norm stays constant, due to: $ \\nonumber \\exp (-i H^A t - i H^A_2 t)^\\dag \\;=\\; \\exp (i H^A t + i H^A_2 t)$ and eq.",
"(REF ).",
"By contrast, the inner product of two distinct vectors does not necessarily stay constant since it is, in general, not real.",
"This illustrates the underlying reason why an antilinear $H^A_2$ cannot be added directly in system A.",
"It might be possible to find a way around this issue, and add $H^A_2$ directly to $H^A$ , but this would probably require that we change the laws of quantum physics substantially.",
"We may have to treat two vectors $\\Psi $ and $i \\Psi $ , differing only by a phase $i$ , as distinct, yet indistinguishable, states instead of the same physical state.",
"By passing from system A to system C, we avoid this tricky issue.",
"As the Hamiltonian $H^C + H^C_2$ is linear and self-adjoint, two vectors $\\Psi $ and $i \\Psi $ can, as usually, be regarded as belonging to the same physical state, and the inner product of any two vectors will stay constant while they evolve.",
"It should be acknowledged that adding the new term to the Hamiltonian in eq.",
"(REF ) can change the physical properties of system C substantially so that the vacuum degeneracy may no longer be hidden and observables may take other forms.",
"In particular, the subtle distinction between the observable of energy and the Hamiltonian, from eq.",
"(REF ), might vanish.",
"The precise form of observables depends, however, on the details of the quantum system and cannot be explored here."
],
[
"Linear time-reversal",
"As another application, consider the case where system A has time-reversal symmetry $T^A$ .",
"Usually, this $T^A$ is an antilinear operator and there are good reasons for this [19].",
"For example, when describing a particle with position $Q^A$ and momentum $P^A$ , we would like $P^A$ to reverse under $T^A$ and $Q^A$ to stay invariant.",
"The canonical commutation relation $[Q^A, P^A] = i$ then requires that $T^A$ anticommutes with $i$ .",
"More generally, $T^A$ should commute with the observable of energy: $ [T^A, H^A] \\;=\\; 0$ so that it keeps energies invariant.",
"It should also anticommute with the term $iH^A$ in the abstract Schrödinger equation so that it can reverse time.",
"Again, this forces $T^A$ to be antilinear, and due to Wigner's theorem, even antiunitary [19]: $ (T^A)^\\dag \\;=\\; (T^A)^{-1}$ Interestingly, neither of these arguments holds in system C. As it is physically equivalent to system A, it should also have a time-reversal operator.",
"We can find this $T^C$ in analogy to the observables $O^C$ from section REF .",
"We lift $T^A$ to $\\mathcal {H}^B$ , where it becomes $T^B$ , and then set: $T^C \\;=\\; U T^B U^{-1}$ Just like $O^C$ , this $T^C$ turns out to be linear.",
"Since $T^A$ anticommutes with $i$ , its lifted version $T^B$ also anticommutes with $i$ but commutes, like other lifted operators, with $j$ .",
"This implies that $T^C$ anticommutes with $U i U^{-1} = j$ but commutes with $U j U^{-1} = i$ and is therefore linear.",
"From eq.",
"(REF ), we get $(T^B)^\\dag = (T^B)^{-1}$ and, with eq.",
"(REF ): $(T^C)^\\dag \\;=\\; (T^C)^{-1}$ so that $T^C$ is not only linear but unitary.",
"Furthermore, it follows from eq.",
"(REF ) and the other properties of $T^A$ that $T^C$ commutes with the observable $U H^B U^{-1}$ of energy and the observable $Q^C = U Q^B U^{-1}$ of position, but anticommutes with the observable $P^C = U P^B U^{-1}$ of momentum, just as a time-reversal operator should.",
"$T^C$ thus anticommutes with the product $Q^C P^C$ , but this does not prevent it from being linear, since we know from eq.",
"(REF ) that the canonical commutation relation $[Q^C, P^C] = j$ now contains $j$ , not $i$ , on the right-hand side.",
"Despite being linear, $T^C$ can thus anticommute with both sides of this rule.",
"Similarly, $T^C$ can reverse time in the Schrödinger equation despite being linear and commuting with the observable $U H^B U^{-1}$ of energy.",
"This is only possible because of the unconventional distinction between the observable $U H^B U^{-1}$ and the Hamiltonian $H^C$ from eq.",
"(REF ).",
"As $T^C$ commutes with $i$ , anticommutes with $j$ , and commutes with $U H^B U^{-1}$ , it anticommutes with the Hamiltonian: $\\lbrace T^C, H^C \\rbrace \\;=\\;\\lbrace T^C, -ij \\, U H^B U^{-1} \\rbrace \\;=\\; 0$ and thus anticommutes with the term $i H^C$ in the Schrödinger equation (REF ).",
"If $\\Psi ^C(t)$ is a solution of that equation, then $T^C \\Psi ^C(t)$ solves the time-reversed equation: $\\frac{d}{dt} T^C \\Psi ^C(t) &=& +i H^C T^C \\Psi ^C(t)$ While system C is mathematically more complicated than system A in some respects, the hidden degeneracy and the substitution of $j$ for $i$ , it thus has a linear time-reversal operator $T^C$ and is simpler in this respect.",
"It would be interesting to explore whether such a $T^C$ , or its generalization to CPT, can be embedded in a continuous set of linear symmetries."
],
[
"Continuous symmetries",
"A similar argument also holds for generators of continuous symmetries.",
"Usually, such generators $G$ have to be linear, so that the symmetry $1 + i \\epsilon G$ , for infinitesimal $\\epsilon $ , is linear and abides by Wigner's unitary-antiunitary theorem [19].",
"However, a real-linear operator in system A corresponds to a linear operator in system C, and it thus makes sense to consider real-linear generators $G^A$ corresponding to linear $G^C$ .",
"To keep the norm constant, such a continuous symmetry would have to obey: $(1 + i \\epsilon G^A)^\\dag \\;=\\; (1 + i \\epsilon G^A)^{-1}$ Its generator would therefore be constrained by $(i G^A)^\\dag = -i G^A$ , like the $H^A_2$ from eq.",
"(REF ), but it would not necessarily have to be linear or self-adjoint."
],
[
"Fermionic mass terms",
"It is well known that any Dirac spinor, describing a fermion, can be split into a left- and a right-handed Weyl spinor, and that a left-handed Weyl spinor $\\psi _L$ can be turned into a right-handed spinor $\\psi _R$ via [9]: $ \\psi _R \\;=\\; i \\sigma ^2 \\psi _L^*$ (where $\\sigma ^2$ is a Pauli spin matrix and $\\psi _L$ and $\\psi _R$ denote classical fields).",
"This transformation is antilinear as it involves complex conjugation.",
"We could use it, in principle, to replace any right-handed Weyl spinors in classical field theories by left-handed ones.",
"Fermionic mass terms, which normally couple a right-handed spinor $\\psi _R$ to a left-handed spinor $\\phi _L$ , will then involve the complex conjugation $K$ : $m \\phi _L^\\dag \\psi _R \\;=\\; m \\phi _L^\\dag i \\sigma ^2 K \\psi _L$ Such an application, concerning the Majorana equation, has been explored in the field of quantum simulations [3].",
"It may also be interesting for grand unified theories, especially the one based on $SO(10)$ , where all the 16 left-handed Weyl spinors, from one generation of particles, are combined in a 16-dimensional representation, and the 16 right-handed Weyl spinors are combined similarly (see [1] for a recent introduction).",
"After replacing the right-handed spinors by left-handed ones, it may be possible to combine these representations further, for example, to the 32-dimensional representation of $SO(12)$ (not shown)."
],
[
"Larger degeneracy",
"As a final application, let us discuss briefly how the procedure could be used to introduce a vacuum degeneracy that is more than just twofold.",
"We could, for example, iterate the step from section REF .",
"After introducing another twofold degeneracy, the Hilbert space would become: $\\mathcal {H}^B \\oplus \\mathcal {H}^B \\;=\\; \\mathcal {H}^A \\oplus \\mathcal {H}^A \\oplus \\mathcal {H}^A \\oplus \\mathcal {H}^A$ and the degeneracy would be fourfold.",
"Two linear operators: $ \\nonumber V_1 (\\Psi ^A_1,\\Psi ^A_2,\\Psi ^A_3,\\Psi ^A_4) &=& (\\Psi ^A_2, 0 ,\\Psi ^A_4,0) \\\\ \\nonumber V_2 (\\Psi ^A_1,\\Psi ^A_2,\\Psi ^A_3,\\Psi ^A_4) &=& (\\Psi ^A_3,\\Psi ^A_4,0,0)$ analogous to the $V$ from eq.",
"(REF ), could then be used to switch between degenerate states.",
"The operator $j$ could still be defined, for example, as: $ \\nonumber j = V_1^\\dag - V_1$ and substituted for $i$ , as before.",
"This does not affect the other operator $V_2$ , which might then be used for other purposes.",
"When the observables and the Hamiltonian are treated as before, for the twofold degeneracy, all the resulting quantum systems will still be physically equivalent.",
"It may even be possible to adapt this framework so that not all the states acquire the same degeneracy.",
"So far, we have associated each state $\\Phi $ with a “twin\" state $j \\Phi $ and thereby doubled the number of states.",
"Alternatively, it may be possible to introduce a twin creation operator $b_n^\\dag $ for each known creation operator $a_n^\\dag $ and, more generally, a twin field operator for each known field operator.",
"Let us briefly sketch the basic idea behind this in a simple example.",
"Consider a quantum system that was constructed, via the usual Fock-space procedure, from a unique vacuum and a finite number of fermionic creation operators $a_n^\\dag $ , on a lattice, with the usual properties: $ \\nonumber \\lbrace a_n, a_m^\\dag \\rbrace = \\delta _{mn} \\; ; \\;\\; \\lbrace a_n, a_m \\rbrace = 0$ where the index $n$ subsumes all their quantum numbers including position.",
"We also presume that the (normal-ordered) Hamiltonian $H^A$ contains only products $a_n^\\dag a_m$ of two such operators.",
"As before, the goal is to replace any explicit occurrence of the imaginary unit $i$ , in observables or in $H^A$ , with another term.",
"For this, we introduce twin operators $b_n^\\dag $ that are fermionic creation operators with exactly the same properties as the original $a_n^\\dag $ (and with $\\lbrace b_n^\\dag ,a_n\\rbrace = 0$ and $\\lbrace b_n,a_n\\rbrace = 0$ ).",
"Using both $a_n^\\dag $ and $b_n^\\dag $ in the construction of the Fock space produces much more states $\\Psi $ than using only $a_n^\\dag $ , so the number of states needs to be restricted.",
"A suitable constraint could be that any physical state $\\Psi $ satisfies: $ (a_n + i b_n) \\Psi \\;=\\; 0$ for any index $n$ .",
"One can check that such a constraint compensates for the larger number of creation operators (not shown).",
"To ensure that this constraint continues to hold, as $\\Psi $ evolves in time, the Hamiltonian $H^A$ has to be modified accordingly.",
"A suitable choice may be to replace, in $iH^A$ , any term $r a_n^\\dag a_m$ with real prefactor $r$ by: $r a_n^\\dag a_m \\;\\rightarrow \\; r \\left( a_n^\\dag a_m + b_n^\\dag b_m \\right)$ and to replace any term $ir a_n^\\dag a_m$ with imaginary prefactor $ir$ by: $i r a_n^\\dag a_m \\;\\rightarrow \\; r \\left( b_n^\\dag a_m - a_n^\\dag b_m \\right)$ These substitutions, like our earlier substitution of $j$ for $i$ in eq.",
"(REF ), remove any explicit occurrence of $i$ .",
"Like eq.",
"(REF ), they maintain most of the algebraic relations of the original terms.",
"For example, taking the adjoint of $ir a_n^\\dag a_m$ interchanges the indices $n$ and $m$ and adds a minus sign, and an analogous relation holds for the substituted term: $ \\nonumber r \\left( b_n^\\dag a_m - a_n^\\dag b_m \\right)^\\dag \\;=\\; -r \\left( b_m^\\dag a_n - a_m^\\dag b_n \\right)$ Furthermore, these substitutions agree with the constraint (REF ).",
"From: $[ a_n + ib_n, a_n^\\dag a_m + b_n^\\dag b_m ] \\!\\!\\!\\!&=& \\!\\!\\!\\!",
"\\lbrace a_n, a_n^\\dag \\rbrace a_m + i \\lbrace b_n, b_n^\\dag \\rbrace b_m\\nonumber \\\\ \\nonumber &=& a_m + i b_m$ it follows that the constraint (REF ) will hold for $(a_n^\\dag a_m + b_n^\\dag b_m)\\Psi $ if it holds for $\\Psi $ .",
"Similarly, from: $ \\nonumber [ a_n + ib_n, b_n^\\dag a_m - a_n^\\dag b_m ]&=&i (a_m + i b_m)$ it follows that the constraint (REF ) will hold for $(b_n^\\dag a_m - a_n^\\dag b_m) \\Psi $ if it holds for $\\Psi $ (not shown).",
"The constraint will thus continue to hold as $\\Psi $ evolves in time, and it seems possible that this quantum system, with twin field operators instead of twin vacuum states, is also physically equivalent to the original one.",
"Other examples may be constructed along similar lines.",
"Perhaps one can even construct a vacuum that contains a Dirac sea built from such twin field operators, so that the vacuum degeneracy becomes extremely large, yet remains hidden."
],
[
"Discussion",
"While the last remarks about twin field operators remain speculative, the main results, based on twin vacuum states, are rigorous.",
"A few steps, involving the trace or the adjoint of operators, were taken, for simplicity, only on finite Hilbert spaces, but we have indicated how to generalize them to infinite Hilbert spaces as well.",
"Essentially, we have shown two results.",
"Firstly, for any quantum system A with unique vacuum, another, physically equivalent system B, can be constructed where the vacuum and other states are degenerate but the degeneracy is hidden.",
"Secondly, this system B has room for an operator $j$ , which somewhat resembles the imaginary unit $i$ , and we can construct another system C, still physically equivalent to systems A and B, by substituting $j$ for $i$ at certain places in observables and the Hamiltonian.",
"Antilinear operators in system $A$ then correspond to linear operators in system $C$ .",
"The mathematics behind the first result was rather trivial, since it involved little more than taking the direct sum of the Hilbert space with itself, so only the physical arguments from section REF might be contentious.",
"There we presumed that any measurement in physics can, at least in principle, be described by the expectation value $\\Psi ^\\dag O \\Psi $ or $\\mathrm {Tr}(\\rho O)$ of an observable $O$ .",
"It is commonly assumed, in quantum physics, that all measurements can be described in this way (e.g., [7]), but there does not seem to be any extensive discussion of this issue.",
"If other measurements were possible, they might perhaps reveal the degeneracy and invalidate our first result.",
"The second result, concerning the substitution of $j$ for $i$ , was mathematically less trivial as it involved the transformation $ U $ , from eq.",
"(REF ), which is just real-linear, not linear.",
"The appendix indicates how this result may simplify if quantum physics was formulated on a real Hilbert space, instead of a complex one, along the lines investigated elsewhere [15], [16], [17].",
"On such a real vector space, $ U $ corresponds simply to an orthogonal transformation, and the physical equivalence would become more obvious.",
"However, it is not hard to deal with this transformation $ U $ directly on the complex Hilbert space, by using the convenient properties of real-linear operators, and their adjoints, summarized in the appendix.",
"The resulting quantum system C, after the substitution of $j$ for $i$ , has some unconventional features, especially the subtle distinction between the observable of energy and the Hamiltonian from eq.",
"(REF ).",
"However, due to the physical equivalence, it makes the same experimental predictions as the original system A, so there does not seem to be any physical reason why its unconventional features should be prohibited.",
"For physical applications, this substitution of $j$ for $i$ may be interesting because it can turn antilinear operators into linear ones.",
"When system A has a time-reversal symmetry $T^A$ , which is antilinear, then system C after the substitution of $j$ for $i$ , will have a corresponding operator $T^C$ that is linear but can still be used to reverse time (section REF ).",
"It would be interesting to explore in more detail what this approach may tell us about the CPT-theorem, about the generators of continuous symmetries from section REF , or about the fermionic mass terms from section REF .",
"Perhaps the most interesting application of these results is that they allow quantum systems to behave as if an antilinear term $H^A_2$ had been added to the Hamiltonian $H^A$ .",
"We cannot directly add it in system A, without loosing the linearity of the Hamiltonian, but we can add the corresponding term $H^C_2$ in the physically equivalent system C where it becomes linear.",
"To guarantee the condition of unitarity, any such term $H^A_2$ has to satisfy $(i H^A_2)^\\dag = -i H^A_2$ from eq.",
"(REF ), but this still allows a wide range of antilinear terms for the Hamiltonian.",
"Even though linear Hamiltonians have been very successful in physics, it would be interesting to study, for example, gauge symmetries with antilinear generators, since we can now find room for them in the Hamiltonian."
],
[
"Acknowledgment",
"I would like to express my gratitude to Prof. Herbert Spohn for his comments on this manuscript."
],
[
"Appendix",
"This appendix reviews and derives some properties of real-linear operators [6], additive operators [10], [11], and their adjoints.",
"The real-linear operator $U$ , from eq.",
"(REF ), was introduced as a sum of linear operators and antilinear (conjugate-linear or semilinear) operators.",
"Antilinear operators are those that satisfy: $M (\\alpha \\Psi + \\beta \\Phi ) \\;=\\; \\alpha ^* M \\Psi + \\beta ^* M \\Phi $ for any complex numbers $\\alpha $ and $\\beta $ and vectors $\\Psi $ and $\\Phi $ in the complex Hilbert space.",
"By contrast, real-linear operators [6] are those that satisfy: $ M (a \\Psi + b \\Phi ) \\;=\\; a M \\Psi + b M \\Phi $ for any real numbers $a$ and $b$ and vectors $\\Psi $ and $\\Phi $ .",
"Here we consider only operators on a complex Hilbert space, that is, maps of the Hilbert space into itself.",
"Clearly, any antilinear operator is real-linear, but linear operators and sums of linear and antilinear operators are also real-linear.",
"Real-linear operators form an algebra, that is, the sum or product of two real-linear operators is again real-linear since it again obeys eq.",
"(REF ).",
"If a real-linear $M$ commutes with the imaginary unit $i$ , then it is linear.",
"If it anticommutes with $i$ , it is antilinear.",
"Conversely, any operator satisfying eq.",
"(REF ) can be written as sum $M = B+A$ of a linear operator $B$ and an antilinear $A$ : $ B = \\frac{1}{2} \\left( M - i M i \\right)\\; ; \\;\\; A = \\frac{1}{2} \\left( M + i M i \\right)$ since $B$ commutes with $i$ and $A$ anticommutes with $i$ .",
"This decomposition is unique (since another such decomposition $M = B^{\\prime } + A^{\\prime }$ would imply that $B^{\\prime } - B = A - A^{\\prime }$ is both linear and antilinear, commutes and anticommutes with $i$ , and thus vanishes).",
"We have avoided using the trace of real-linear or antilinear operators because: $ \\nonumber \\left\\langle i\\Gamma _1, A i\\Gamma _1 \\right\\rangle \\;=\\; - \\left\\langle \\Gamma _1, A \\Gamma _1 \\right\\rangle $ for any antilinear $A$ .",
"When computing the trace, we should sum over such terms, but the result would depend on whether we sum over $\\Gamma _1$ or $i \\Gamma _1$ , and thus depend on the choice of basis.",
"It may be useful to know, for some applications, that the trace of a real-linear $M$ can still be defined, independent of the basis, as the trace of the linear part of $M$ : $ \\mathrm {Tr}(M) \\;=\\; \\frac{1}{2} \\mathrm {Tr}( M - i M i )$ This $\\mathrm {Tr}(M)$ obeys the rules $\\mathrm {Tr}(M + N) = \\mathrm {Tr}(M) + \\mathrm {Tr}(N)$ and, on finite vector spaces, $\\Re \\mathrm {Tr}(MN) = \\Re \\mathrm {Tr}(NM)$ (not shown).",
"Real-linear operators have also been studied in the context of “additive\" operators.",
"By definition, an operator is additive if it satisfies: $M (\\Psi + \\Phi ) \\;=\\; M \\Psi + M \\Phi $ for any vectors $\\Psi $ and $\\Phi $ in the complex Hilbert space.",
"Clearly, any real-linear operator is also additive.",
"On the other hand, any additive operator that is continuous is also real-linear [10].",
"The properties of additive operators have been studied in detail [11], [12], [13].",
"Unfortunately, the study of additive operators differs from the study of real-linear operators in how the adjoint is defined [6], [11].",
"To avoid confusion, let us compare the two approaches.",
"Since any real-linear operator $M$ can be decomposed uniquely, via eq.",
"(REF ), into a linear part $B$ and an antilinear part $A$ , the adjoint of $M = B + A$ can be defined as [6]: $ M^\\dag \\;=\\; B^\\dag + A^\\dag $ where the adjoints $B^\\dag $ and $A^\\dag $ are, as usually, given by eq.",
"(REF ) and (REF ).",
"This is the approach that we have used here.",
"It defines the adjoint for any real-linear operator $M$ whose linear part $B$ and antilinear part $A$ have well-defined adjoints.",
"We have already seen, in eq.",
"(REF ), that it implies: $ \\Re \\left\\langle M^\\dag \\Psi , \\Phi \\right\\rangle \\;=\\; \\Re \\left\\langle \\Psi , M \\Phi \\right\\rangle $ for any vectors $\\Psi $ and $\\Phi $ .",
"Since $B^\\dag $ is linear and $A^\\dag $ antilinear, $M^\\dag $ is real-linear.",
"Eq.",
"(REF ) also implies: $(M^\\dag )^\\dag \\;=\\; M$ because this relation holds for both linear and antilinear operators.",
"The real part $\\Re \\left\\langle \\ldots \\right\\rangle $ of the complex-valued inner product, in eq.",
"(REF ), acts similarly to the real-valued inner product on a real vector space.",
"The full, complex-valued, inner product can be reconstructed from such real parts: $ \\Re \\left\\langle \\Psi , \\Phi \\right\\rangle - i \\, \\Re \\left\\langle \\Psi , i\\Phi \\right\\rangle \\;=\\; \\left\\langle \\Psi , \\Phi \\right\\rangle $ Using this, eq.",
"(REF ) can be rewritten as: $ \\left\\langle M^\\dag \\Psi , \\Phi \\right\\rangle \\;=\\; \\Re \\left\\langle \\Psi , M \\Phi \\right\\rangle - i \\, \\Re \\left\\langle \\Psi , M i\\Phi \\right\\rangle $ Inserting either a linear or an antilinear operator for $M$ reproduces the usual definitions (REF ) and (REF ) of their adjoints, which thus follow from eq.",
"(REF ).",
"Consequently, eq.",
"(REF ) fixes the adjoint $M^\\dag $ of any real-linear operator uniquely and could be used, instead of eq.",
"(REF ), (REF ), and (REF ), as definition of $M^\\dag $ .",
"By employing the rules (REF ), (REF ), and (REF ), one can work with real-linear operators on a complex Hilbert space almost as easily as with linear operators on a real Hilbert space where analogous rules hold.",
"For any real-linear operators $M$ and $N$ on the complex Hilbert space, eq.",
"(REF ) gives: $(M + N)^\\dag \\;=\\; M^\\dag + N^\\dag $ Furthermore, the relation (REF ): $R^A = (M^A)^\\dag &\\Rightarrow & R^B = (M^B)^\\dag $ for operators lifted from $\\mathcal {H}^A$ to $\\mathcal {H}^B$ holds even when $M^A$ is not linear but real-linear.",
"It holds because the definition (REF ) of the adjoint involves only the inner product and because the inner product on $\\mathcal {H}^B$ was derived, in section REF , from the inner product on $\\mathcal {H}^A$ .",
"Finally, eq.",
"(REF ) gives: $ \\Re \\left\\langle N^\\dag M^\\dag \\Psi , \\Phi \\right\\rangle \\;=\\; \\Re \\left\\langle \\Psi , M N \\Phi \\right\\rangle $ Since eq.",
"(REF ) fixes the adjoint uniquely, this implies: $(M N)^\\dag \\;=\\; N^\\dag M^\\dag $ for any real-linear $M$ and $N$ whose adjoints $M^\\dag $ , $N^\\dag $ , and $(M N)^\\dag $ exist.",
"Setting $N$ to a complex number $\\alpha $ gives $(M \\alpha )^\\dag = \\alpha ^* M^\\dag $ .",
"When $M$ is invertible, then setting $N = M^{-1}$ gives: $\\left( M^{-1} \\right)^\\dag \\;=\\; \\left( M^\\dag \\right)^{-1}$ As mentioned in section REF , the similar relation $(N \\Psi )^\\dag = \\Psi ^\\dag N^\\dag $ does not, in general, hold when $\\Psi $ is a vector and $N$ an antilinear or real-linear operator on the Hilbert space.",
"To handle such an expression properly, we have to use eq.",
"(REF ).",
"It can be rewritten, with $N = M^\\dag $ , as: $ (N \\Psi )^\\dag \\Phi \\;=\\; \\Re \\Psi ^\\dag N^\\dag \\Phi - i \\Re \\Psi ^\\dag N^\\dag i \\Phi $ (where $\\Re $ acts on the whole product to its right, not just the first factor).",
"Apart from this complication, the adjoint defined by eq.",
"(REF ) can be used almost as easily as the adjoint of linear operators.",
"In the study of additive operators $M$ , by Sharma and colleagues, the adjoint was defined in another way [11].",
"Let us write this adjoint as $M^*$ to distinguish it from the above $M^\\dag $ .",
"It can be characterized by the diagram in fig.",
"1a which is familiar from more formal definitions of the adjoint of linear operators (e.g., [12]).",
"While $M$ acts on the complex vector space $\\mathcal {H}$ , its adjoint $M^*$ acts on another space, the dual $\\widetilde{\\mathcal {H}}$ .",
"In the usual case, with linear $M$ , this dual consists of all bounded linear functionals $f\\!:\\!",
"\\mathcal {H} \\rightarrow \\mathbb {C}$ .",
"That is, $M^*$ maps any functional $f$ into a functional $M^* f$ .",
"The adjoint $M^*$ is defined as the unique operator that makes the diagram 1a commute (where “id\" is the identity map on $\\mathbb {C}$ ).",
"Using the Riesz representation theorem, this $M^*$ on the dual $\\widetilde{\\mathcal {H}}$ can then be turned into the more familiar adjoint acting on $\\mathcal {H}$ .",
"Figure: Alternative definitions for the adjoint M * M^* or M † M^\\dag of a real-linear operator MM on the complex vector space ℋ\\mathcal {H}.To generalize this familiar definition to antilinear or real-linear operators $M$ , one has to change some aspect of diagram 1a.",
"Otherwise, the composition of $M^* f$ and id would be linear, but the composition of $M$ and $f$ would be antilinear or real-linear.",
"Sharma and colleagues proposed to use bounded, additive functionals $f$ instead of just linear ones.",
"This resulted in $M^* $ always being linear, even when $M$ was antilinear [11].",
"Essentially, such an $M^*$ maps a linear functional $\\alpha f$ into an antilinear functional $\\alpha M^* f$ , thereby commutes with complex numbers $\\alpha $ , and becomes linear.",
"Sharma carefully distinguished it from the usual definition (REF ) for the adjoint of antilinear $A$ , where $A^\\dag $ is an antilinear operator on $\\mathcal {H}$ , not a linear operator on the dual $\\widetilde{\\mathcal {H}}$ .",
"We can avoid such complications, and reproduce our adjoint from eq.",
"(REF ), by changing diagram 1a in another way.",
"Instead of letting the functionals $f\\!:\\!",
"\\mathcal {H} \\rightarrow \\mathbb {C}$ become additive, we replace them by functionals $f$ mapping $\\mathcal {H}$ to real numbers, not complex ones (diagram 1b).",
"We also require them to be real-linear ($f(a\\Psi + b\\Phi ) = a f(\\Psi ) + b f(\\Phi )$ for any $a,b$ in $\\mathbb {R}$ and $\\Psi , \\Phi $ in $\\mathcal {H}$ ).",
"Both branches in diagram 1b are then real-linear, which avoids the above problem of only one branch being linear.",
"It is then straightforward to define $M^\\dag $ in the usual way, as the unique operator that makes diagram 1b commute, and to turn it, via the Riesz representation theorem for real spaces, into an operator on $\\mathcal {H}$ (not shown).",
"The upshot of all this is that $M^\\dag $ becomes the unique operator on $\\mathcal {H}$ satisfying: $\\Re \\left\\langle M^\\dag \\Phi , \\Psi \\right\\rangle \\;=\\; \\Re \\left\\langle \\Phi , M \\Psi \\right\\rangle $ which is precisely how we defined $M^\\dag $ in eq.",
"(REF ) above.",
"Even from an abstract point of view, this definition of the adjoint $M^\\dag $ is thus a reasonable alternative to the definition of $M^*$ by Sharma.",
"Incidentally, all these mathematical concepts would simplify if we formulated quantum physics not on a complex Hilbert space but on a real one.",
"It is known that such a step is possible and leads to a physically equivalent description as long as the real Hilbert space is constructed properly with twice the dimension of the complex one [12], [16], [17].",
"Any real-linear operator on the complex space corresponds to a linear operator on the real space, and vice versa.",
"In particular, the complex conjugation $K$ and the imaginary unit $i$ , treated as operator on the complex space, correspond to linear operators on the real space which can be written in a block-diagonal form.",
"For $i$ , each block is commonly written as $\\left( \\begin{array}{cc} 0 & -1 \\\\ 1 & 0 \\end{array} \\right)$, and, for $K$ , it is written as $\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right)$.",
"This can be used to clarify the relation between $i$ and $K$ , on one side, and our operators $j$ and $L$ , on the other side, since $j$ and $L$ would take the same form if $V$ , in eq.",
"(REF ) or (REF ), was identified with $\\left( \\begin{array}{cc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right)$.",
"Moreover, the adjoint $A^\\dag $ of real-linear operators from eq.",
"(REF ), defined on the complex space, would become the standard adjoint of linear operators on the real space.",
"Our transformation $ U $ , from eq.",
"(REF ) and (REF ), would become simply an orthogonal transformation, and other steps would simplify as well.",
"Using a real Hilbert space would also have other advantages [5], and we avoided it here only because it would make the laws of quantum physics look unfamiliar."
]
] | 1204.1309 |
[
[
"Controlling phase separation of a two-component Bose-Einstein condensate\n by confinement"
],
[
"Abstract We point out that the widely accepted condition g11g22<g122 for phase separation of a two-component Bose-Einstein condensate is insufficient if kinetic energy is taken into account, which competes against the intercomponent interaction and favors phase mixing.",
"Here g11, g22, and g12 are the intra- and intercomponent interaction strengths, respectively.",
"Taking a d-dimensional infinitely deep square well potential of width L as an example, a simple scaling analysis shows that if d=1 (d=3), phase separation will be suppressed as L\\rightarrow0 (L\\rightarrow\\infty) whether the condition g11g22<g122 is satisfied or not.",
"In the intermediate case of d=2, the width L is irrelevant but again phase separation can be partially or even completely suppressed even if g11g22<g122.",
"Moreover, the miscibility-immiscibility transition is turned from a first-order one into a second-order one by the kinetic energy.",
"All these results carry over to d-dimensional harmonic potentials, where the harmonic oscillator length {\\xi}ho plays the role of L. Our finding provides a scenario of controlling the miscibility-immiscibility transition of a two-component condensate by changing the confinement, instead of the conventional approach of changing the values of the g's."
],
[
"Introduction",
"Phase separation is a ubiquitous phenomenon in nature [1], [2].",
"A most prominent example familiar to everyone is that oil and water do not mix.",
"Besides that, the phenomenon of water in coexistence with its vapor can also be understood as a type of phase separation [3].",
"In general, two phases mix or not depending on which configuration minimizes the energy or free energy of the whole system.",
"With the realization of Bose-Einstein condensation in ultracold atomic gases, another example of phase separation is offered by two-component Bose-Einstein condensates (BECs) [4], [5], [6], [7], [8].",
"In such a system, phase mixing or separation means the two condensates overlap or not spatially, which correspond to different interaction energies.",
"A widely accepted condition for phase separation, which is based on the consideration of minimizing the interaction energy [9], [10], is given by $g_{12}>\\sqrt{g_{11}g_{22}}.$ Here $g_{11}$ and $g_{22}$ are the intra-component interaction strengths of components 1 and 2, respectively, while $g_{12}$ is the interaction strength between them [11].",
"This condition is intuitively reasonable since if the inter-component interaction is too strong, the two components would like to get separated from each other.",
"Experimentally, controlled miscibility-immiscibility transition of a two-component BEC based on the idea of adjusting the values of the $g$ 's using Feshbach resonance and so as to get (REF ) satisfied or not has been demonstrated recently [12], [13].",
"Now the point is that though the condition above is very appealing in its simplicity and usefulness, it has great limitations.",
"In its derivation, the condensates are assumed to be uniform and the kinetic energy associated with the boundary/interface layers is neglected.",
"The problem is then reduced to minimizing the total interaction energy, or more specifically, to weighing the inter-component interaction against the intra-component interaction.",
"This approximation is legitimate if the widths of the boundary/interface layers are much smaller than the extension of the condensates, or in other words, if the boundary/interface layers are well defined.",
"However, this condition is not necessarily satisfied in all circumstances.",
"Actually, some simple scaling analysis may tell us when it will fail.",
"Consider a condensate trapped in a $d$ -dimensional container of size $L$ .",
"The characteristic (average) density of the condensate is on the order of $L^{-d}$ .",
"According to the mean-field (Gross-Pitaevskii) theory, the healing length of the condensate, which determines the widths of the boundary/interface layers, will be on the order of $L^{d/2}$ [9], [10].",
"Thus we see that in one and three dimensional cases, it makes sense to say boundary/interface layers only in the limits of $L\\rightarrow \\infty $ and $L\\rightarrow 0$ , respectively.",
"In the opposite limits, the “boundary/interface” layers overtake the condensates themselves in size, which signals that the kinetic energy will dominate the interaction energy and should no longer be neglected.",
"The two dimensional case is more subtle in that the widths of the boundary/interface layers scale in the same way with the sizes of the condensates, which at least means that the kinetic energy should not be neglected a priori.",
"The analysis above indicates that the kinetic energy is likely to play a vital role in determining the configuration of a two-component BEC.",
"Moreover, we note that the kinetic energy acts against the inter-component interaction.",
"The latter is responsible for phase separation while the former tries to expand the condensates and thus favors phase mixing.",
"Therefore, it is expected that phase separation can be suppressed by the kinetic energy in some circumstances even if the condition (REF ) is satisfied [15].",
"Notably, according to the argument above, the significance of the kinetic energy can be controlled by changing the size of the container.",
"That is, the phase mixing-demixing transition can be controlled by a geometrical method, instead of the mechanical method of changing the values of the $g$ 's, which is based on (REF ) and is demonstrated in Refs.",
"[12], [13]."
],
[
"A two-component BEC in an infinitely deep square well potential",
"The considerations above have led us to investigate the scenario of suppressing phase separation in a two-component BEC by kinetic energy.",
"We will start from the simplest and most generic case of a two-component BEC in a $d$ -dimensional infinitely deep square well potential (of width $L$ ).",
"The Dirichlet boundary condition implies that the condensate wave functions must be non-uniform and the kinetic energy is at least on the order of $L^{-2}$ .",
"On the contrary, inside the well, the potential energy is zero.",
"Therefore, we have a pure competition between the kinetic energy and the inter-component interaction energy, if the intra-component interactions are set zero [note that in this case, condition (REF ) is satisfied].",
"In this simplest model, in all dimensions ($d=1$ , 2, 3), we do observe that phase separation can be completely suppressed by the kinetic energy in some regime.",
"Of course, different dimensions have different features.",
"But all these effects and features carry over to the more realistic case of $d$ -dimensional harmonic potentials.",
"In the mean-field theory and at zero-temperature, the energy functional of a two-component BEC in a $d$ -dimensional infinitely deep square well potential $\\Omega = [-L/2,+L/2]^d$ is of the form $E[\\psi _1,\\psi _2] &=& \\int _{\\Omega } d \\vec{r} \\bigg \\lbrace \\sum _{\\alpha =1,2} \\frac{N_\\alpha \\hbar ^2}{2m_\\alpha }|\\nabla \\psi _\\alpha |^2 \\quad \\nonumber \\\\&& \\quad + \\frac{1}{2} \\sum _{\\alpha ,\\beta =1,2} g_{\\alpha \\beta }N_\\alpha N_\\beta |\\psi _\\alpha |^2|\\psi _\\beta |^2\\bigg \\rbrace .$ Here the two condensate wave functions are normalized to unity $\\int _\\Omega d \\vec{r} |\\psi _{1,2}|^2=1$ , and $\\psi _{1,2}=0$ on the boundary.",
"Note that throughout this paper we are only concerned with the ground configuration of the system, therefore all the wave functions can be taken to be real and positive.",
"The parameters $g_{11}$ , $g_{22}$ , and $g_{12}=g_{21}$ are the effective intra- and inter-component interaction strengths.",
"Finally, $N_{1,2}$ and $m_{1,2}$ are the atom numbers and atom masses of the two species, respectively.",
"Now we should note that for an arbitrary set of parameters, in the ground configuration, almost definitely, the two wave functions do overlap but do not coincide with each other (this can be easily understood in terms of the Gross-Pitaevskii equations for $\\psi _{1,2}$ ).",
"In this case, it is far from trivial to distinguish phase separation and phase mixing.",
"A method proposed in [14] is to consider the centers of mass of the two condensates: $\\vec{r}_{m\\alpha }=\\int _\\Omega d \\vec{r}|\\psi _\\alpha |^2 \\vec{r}, \\quad \\alpha =1,2.$ This idea is motivated by the observation that in some regime, both the two condensates are symmetric with respect to the origin while in other regime, both of them are asymmetric with respect to the origin, and more importantly, they are shifted in opposite directions [14].",
"Apparently, the former case is with $\\vec{r}_{m1 }=\\vec{r}_{m2 }=0$ and it is appropriate to call it phase-mixed while the latter case is with $\\vec{r}_{m1 }\\ne 0 \\ne \\vec{r}_{m2 }$ and it is appropriate to call it phase-separated.",
"Therefore, the offset between the two centers of mass $\\vec{r}_{m1}-\\vec{r}_{m2 }$ can serve as an order parameter for the miscibility-immiscibility transition of the system.",
"Though this order parameter works well for a general case, we will not use it much in this paper.",
"Actually, instead of studying a general case, we shall focus on the symmetric energy functional case, i.e., the case when $m_1=m_2=m$ , $N_1=N_2=N$ , and $g_{11} =g_{22}$ .",
"The reason is that this special case not only captures all the essential physics, but also has an extra merit.",
"That is, now it is possible to have $\\psi _1=\\psi _2$ , which corresponds to a completely mixed configuration.",
"Therefore, in this special case, an appropriate order parameter is the overlap between the two condensate wave functions (or more precisely, $1-\\eta $ , if phase separation is concerned): $\\eta =\\int _\\Omega d \\vec{r} \\psi _1 \\psi _2,$ which takes values between 0 and 1.",
"If $\\eta \\ll 1$ , it would be fair to say the system shows phase separation.",
"Otherwise, if $\\eta $ is close to 1, or more precisely if $1-\\eta \\ll 1$ , it would be fair to say the system shows phase mixing.",
"In the intermediate case, the system is partially phase-separated and partially phase-mixed.",
"Now make the transform $\\psi _{1,2}(\\vec{r})= L^{-d/2}\\phi _{1,2}(\\vec{x})$ with $\\vec{r}= L \\vec{x}$ .",
"Then $\\int _{\\Omega _0} d \\vec{x} |\\phi _{1,2}|^2=1$ and $\\phi _{1,2}=0$ on the boundary of $\\Omega _0$ , where $\\Omega _0=[-1/2,+1/2]^d$ .",
"In terms of the rescaled wave functions $\\phi _{1,2}$ , $\\eta =\\int _{\\Omega _0} d \\vec{x}\\phi _1 \\phi _2 $ , and the energy functional (REF ), under the assumption above, can be rewritten as $E[\\phi _1,\\phi _2]=\\frac{N\\hbar ^2}{m L^2} \\int _{\\Omega _0} d\\vec{x} \\bigg \\lbrace \\frac{1}{2}|\\nabla \\phi _1|^2+ \\frac{1}{2}|\\nabla \\phi _2|^2 \\ \\quad \\quad \\quad \\nonumber \\\\+ \\frac{1}{2}\\left(\\beta _{11} |\\phi _1|^4 +\\beta _{22} |\\phi _2|^4 +2\\beta _{12} |\\phi _1|^2|\\phi _2|^2\\right)\\bigg \\rbrace ,$ with the reduced dimensionless parameters $\\beta _{ij}$ defined as $\\beta _{ij} = \\frac{Nmg_{ij}}{ \\hbar ^2 L^{d-2}},\\quad i,j=1,2.$ These parameters are measures of the importance of the interactions.",
"In the curl bracket, the coefficients of the kinetic terms are constant, yet the coefficients of the interaction terms (the $\\beta $ 's) scale with $L$ as $L^{2-d}$ .",
"This fact has some important consequences.",
"If $d=1$ , there are two different limits.",
"In the limit of $L\\rightarrow \\infty $ (loose confinement), the kinetic terms are dominated by the interaction terms and thus the ground state can be determined by simply minimizing the interaction energy.",
"In this limit, the textbook analysis is valid and we have phase separation if condition (REF ) is satisfied or phase mixing otherwise.",
"In the opposite limit of $L\\rightarrow 0$ (tight confinement), the kinetic terms will dominate and the two rescaled wave functions can be well approximated by the ground state of the square well potential, i.e., $\\phi _{1,2}(x)\\simeq \\sqrt{2}\\cos (\\pi x)$ .",
"In this limit, phase separation will be suppressed whatever the values of the $g$ 's are, even if (REF ) is fulfilled.",
"The three dimensional case is the inverse of the one dimensional case.",
"In the limit of $L\\rightarrow 0$ , the kinetic terms are negligible and the criterion of phase separation (REF ) is valid.",
"In the other limit of $L\\rightarrow \\infty $ , the kinetic terms dominate and phase separation is suppressed regardless of the condition (REF ).",
"The two dimensional case is another story.",
"The parameter $L$ simply drops out in the curl bracket.",
"It is no use to adjust the width of the well to enhance the importance of the kinetic energy or the interaction energy relatively.",
"The kinetic and interaction energies should be treated on an equal footing, which means the analysis leading to criterion (REF ) may be invalid.",
"Figure: (Color online) (a) The overlap factor η\\eta as afunction of the reduced parameter β 12 \\beta _{12} [seeEq.",
"()] in different dimensions (infinitely deepsquare well potential case, g 11 =g 22 =0g_{11}=g_{22}=0).",
"Note thatfor all values of dd, there exists a critical valueβ 12 c ≠0\\beta _{12}^c\\ne 0, below which η\\eta attains itsmaximal possible value 1.",
"(b) a schematic plot of η\\eta versus the width of the square well in differentdimensions.",
"Note the counter-intuitive fact that in thethree dimensional case (d=3d=3), the stronger we squeeze thesystem (the smaller LL is), the stronger phase separationis (the smaller η\\eta is).We have checked all these predictions numerically.",
"Note that on the problem of phase separation, the intra-component interactions are on the same side as the kinetic energy—they both try to delocalize the condensates.",
"Therefore, to highlight the effect of kinetic energy, we shall set $g_{11}= g_{22}=0$ ($\\beta _{11}=\\beta _{22}=0$ ), so that the kinetic energy is the only element acting against phase separation.",
"As we shall see below, this special case also admits a simple analytical analysis.",
"We have solved the ground state of the system in all dimensions for a given value of $\\beta _{12}$ [16].",
"The overlap factor $\\eta $ is plotted versus $\\beta _{12}$ in Fig.",
"REF a.",
"We observe that in all dimensions, there exists a critical value of $\\beta _{12}$ (denoted as $\\beta _{12}^c$ ), below which the two condensates wave functions are equal ($\\eta =1 $ ).",
"That is, for $\\beta _{12}\\le \\beta _{12}^c$ , phase separation is completely suppressed.",
"Above the critical value, phase separation develops ($\\eta <1$ ) as $\\beta _{12}$ increases, but is still greatly suppressed for a wide range of value of $\\beta _{12}$ .",
"It should be stressed that though in Fig.",
"REF a the curves of $\\eta -\\beta _{12}$ are qualitatively similar to each another for all values of $d$ (the plateau of $\\eta =1$ is always located in the direction of $\\beta _{12} \\rightarrow 0$ ), the curves of $\\eta -L$ will be quite different.",
"The reason is that $\\beta _{12} \\propto L^{2-d}$ .",
"Figure REF b is a schematic plot of $\\eta $ versus $L$ in all the three cases.",
"It shows that $\\eta $ as a function of $L$ is monotonically decreasing, constant, and monotonically increasing in one, two, and three dimensions, respectively.",
"This means that to suppress phase separation, in one dimension we should tighten the confinement, in three dimensions we should loosen the confinement, while in two dimensions it is useless to change the confinement.",
"Overall, Fig.",
"REF confirms the initial conjecture that kinetic energy can suppress phase separation.",
"As a hindsight, we can actually understand why phase separation can be suppressed in the limits of $L\\rightarrow 0$ in one dimension and $L\\rightarrow \\infty $ in three dimensions.",
"Consider two different configurations.",
"The first one is a phase-separated one—the two condensates occupy the left and right halves of the container separately.",
"The second one is a phase-mixed one—the two condensates both occupy the whole space available and thus overlap significantly.",
"Compared with the first configuration, the second one costs more inter-component interaction energy which is on the order of $L^{-d}$ , but saves more kinetic energy which is on the order of $L^{-2}$ .",
"The second configuration (phase-mixed) is more economical in energy in the limit of $L\\rightarrow 0$ and $L\\rightarrow \\infty $ , in the cases of $d=1$ and $d=3$ , respectively.",
"The case of $d=2$ is more subtle and which configuration wins depends on parameters other than $L$ .",
"A remarkable fact revealed in Fig.",
"REF but not so obvious in our arguments is that in the symmetric case with $\\beta _{11}=\\beta _{22}=0$ , $\\eta =1$ for $\\beta _{12}\\le \\beta _{12}^c$ , which is on the order of unity.",
"This is a stronger fact than $\\eta \\rightarrow 1$ as $\\beta _{12}\\rightarrow 0$ as we argued.",
"Actually, the general observation is that for $\\beta _{11}=\\beta _{22}>0$ , $\\eta =1$ for $\\beta _{12}$ smaller than its critical value $\\beta _{12}^c$ , which is larger than $\\beta _{11}$ .",
"This fact has rich meanings.",
"On the one hand, it demonstrates that the kinetic energy is very effective—phase separation can be completely suppressed by it even if $\\beta _{12} >\\beta _{11}=\\beta _{22}$ , i.e., when (REF ) is satisfied.",
"On the other hand, it strongly indicates that as $\\beta _{12}$ crosses the critical value, the system undergoes a second order phase transition which can fit in the Landau formalism.",
"The picture is that the exchange symmetry $\\phi _1 \\leftrightarrow \\phi _2 $ of the energy functional (REF ) is preserved for $\\beta _{12} <\\beta _{12}^c$ , but is spontaneously broken as $\\beta _{12}$ surpasses $\\beta _{12}^c$ .",
"We have been able to prove the first point rigorously on the mathematical level (see Appendix ).",
"However, it is also desirable to develop a physical understanding of the two points.",
"This can be achieved by studying a two-component BEC in a double-well potential (see Appendix ) or using a variational approach [17].",
"We note that in the limit of $\\beta _{12} \\rightarrow 0$ , $\\phi _{1,2}$ both converge to the (non-degenerate) ground state of a single particle in the $[-1/2,+1/2]^d$ infinitely deep square well.",
"As $\\beta _{12}$ is turned on, the two wave functions are deformed and excited states mix in.",
"Because the energies of the excited states grow up quadratically, we cutoff at the first excited level and take the following ansatz for the two condensate wave functions $\\phi _1= c_0 \\varphi _0 + c_1 \\varphi _1, \\quad \\phi _2= c_0\\varphi _0 - c_1 \\varphi _1.$ Here $\\varphi _0 $ is the ground state, while $\\varphi _1$ is one of the possibly degenerate first excited states.",
"The coefficients $c_{0,1}$ are real and satisfy the normalization condition $c_0^2 + c_1^2=1$ .",
"Obviously, complete phase mixing would correspond to $c_1=0$ while partial phase separation to $c_1\\ne 0$ .",
"Our numerical simulations indicate that (this is also supported by the variational approach itself, see Appendix ) in the two dimensional case, when phase separation occurs, the two condensates are shifted either along $x$ or $y$ direction; in the three dimensional case, when phase separation occurs, the two condensates are shifted either along $x$ or $y$ or $z$ direction.",
"This fact motivates us to choose $\\varphi _1$ in the following form $&&d=1: \\varphi _1= w_1(x); \\\\&&d=2: \\varphi _1= w_0(x) w_1(y) \\ \\text{ or } \\ w_1(x) w_0(y) ; \\\\&&d=3: \\varphi _1= w_0(x) w_0(y)w_1(z) \\ \\text{ or } \\ w_0(x) w_1(y)w_0(z)\\quad \\nonumber \\\\&&\\quad \\quad \\quad \\quad \\quad \\quad \\text{ or } w_1(x) w_0(y)w_0(z),\\quad \\quad $ where $w_0(x)=\\sqrt{2} \\cos (\\pi x)$ and $w_1(x)=\\sqrt{2}\\sin (2 \\pi x)$ are the ground and first excited states of a single particle in the one dimensional $[-1/2,+1/2]$ infinitely deep square well potential.",
"Substituting Eqs.",
"(REF ) and () into (REF ), we get the reduced energy functional $\\widetilde{E}=E/(N\\hbar ^2/mL^2)$ as $&&d=1: \\widetilde{E}(c_1) = (3\\pi ^2 -5 \\beta _{12})c_1^2 + 5 \\beta _{12}c_1^4 +\\text{const}; \\nonumber \\\\&&d=2: \\widetilde{E}(c_1) = \\left(3\\pi ^2 -\\frac{15}{2} \\beta _{12} \\right)c_1^2 + \\frac{15}{2} \\beta _{12}c_1^4 +\\text{const}; \\nonumber \\\\&&d=3:\\widetilde{E} (c_1) = \\left(3\\pi ^2 -\\frac{45}{4}\\beta _{12} \\right)c_1^2 + \\frac{45}{4}\\beta _{12}c_1^4 +\\text{const}.\\nonumber $ These are nothing but the Landau's expression of the free energy in a second-order phase transition, with $c_1$ playing the role of the order parameter here.",
"We immediately determine the critical values of $\\beta _{12}$ by putting the coefficients of $c_1^2$ to zero.",
"Specifically, $\\beta _{12}^c=\\frac{3\\pi ^2}{5}$ , $\\frac{2\\pi ^2}{5}$ , and $\\frac{4\\pi ^2}{15}$ for $d=1$ , $d=2$ , and $d=3$ , respectively.",
"These values agree with those extracted from Fig.",
"REF very well.",
"The relative errors are within 1%, 9%, and 19%, respectively.",
"The deviation increases with $d$ because in higher dimensions, the degeneracy of the excited states increases and the two-mode approximation in (REF ) becomes less accurate.",
"In the expressions of $\\widetilde{E}$ , we can actually see how the kinetic energy suppresses phase separation.",
"The term $3\\pi ^2 c_1^2$ comes from the kinetic energy difference of the two modes $\\varphi _{1,2}$ .",
"Without this term, the critical value $\\beta _{12}^c$ would be zero instead of being finite.",
"For a general case without the exchange symmetry $\\phi _1\\leftrightarrow \\phi _2$ , the appropriate order parameter is no longer $\\eta $ but $\\vec{r}_{m1 }-\\vec{r}_{m2 }$ .",
"However, the second order transition picture still holds.",
"Specifically, $\\vec{r}_{m1 }=0=\\vec{r}_{m2 }$ for $\\beta _{12}$ smaller than some critical value $\\beta _{12}^c$ which is larger than $\\sqrt{\\beta _{11}\\beta _{22}}$ .",
"Overall, this asymmetric case is more involved than the symmetric case above because there are more parameters.",
"Hopefully, a systematic study will be presented in a follow-up work."
],
[
"A two-component BEC in a harmonic potential",
"So far, we have focused on the ideal case of infinitely deep square wells.",
"Experimentally, it is harmonic potentials that are most readily realized.",
"Therefore, it is necessary to see whether analogous results hold for harmonic potentials.",
"One concern is that the extra potential energy may blur the picture.",
"However, after some similar rescaling, we shall see that all the results persist.",
"The energy functional of a two-component BEC in a $d$ -dimensional isotropic harmonic potential is $\\frac{E}{N}= \\int _{R^d} d \\vec{r} \\bigg \\lbrace \\frac{\\hbar ^2}{2m}\\sum _{\\alpha =1,2}|\\nabla \\psi _\\alpha |^2+\\frac{1}{2}m\\omega _d^2 |\\vec{r}|^2 \\sum _{\\alpha =1,2} |\\psi _\\alpha |^2 \\nonumber \\\\+ \\frac{N}{2}\\left(g_{11} |\\psi _1|^4 + g_{22} |\\psi _2|^4 +2g_{12} |\\psi _1|^2|\\psi _2|^2 \\right)\\bigg \\rbrace .\\ \\ $ Here again we have assumed equal mass and equal number for the two species.",
"The two condensate wave functions are normalized to unity, i.e., $ \\int d \\vec{r}|\\psi _{1,2}|^2=1 $ .",
"Now make the transform $\\psi _{1,2}(\\vec{r})= \\xi _{ho}^{-d/2} \\phi _{1,2}(\\vec{x})$ with $\\vec{r}= \\xi _{ho} \\vec{x}$ , where $\\xi _{ho}=\\sqrt{\\hbar /m\\omega _d}$ is the characteristic length of the harmonic potential.",
"We have then $\\int d\\vec{x} |\\phi _{1,2}|^2=1$ .",
"In terms of $\\phi _{1,2}$ , the energy functional can be rewritten as $\\frac{E}{N \\hbar \\omega _d}= \\int _{R^d} d \\vec{x} \\bigg \\lbrace \\frac{1}{2}\\sum _{\\alpha =1,2}|\\nabla \\phi _\\alpha |^2+\\frac{1}{2}|\\vec{x}|^2 \\sum _{\\alpha =1,2}|\\phi _\\alpha |^2\\quad \\nonumber \\\\+ \\frac{1}{2}\\left(\\beta _{11} |\\phi _1|^4 + \\beta _{22} |\\phi _2|^4 +2\\beta _{12} |\\phi _1|^2|\\phi _2|^2 \\right)\\bigg \\rbrace .\\quad $ Here the reduced interaction strengths are defined as $\\beta _{ij}= \\frac{N m g_{ij} \\xi ^{2-d}_{ho}}{\\hbar ^2}\\propto \\omega _d^{(d-2)/2},\\ i,j=1,2.$ We now have a similar situation as before.",
"The importance of the interactions can be changed by changing the value of $\\xi _{ho}$ , which plays the role of $L$ in our previous example.",
"The interactions will be negligible if $d=1$ and $\\xi _{ho} \\rightarrow 0$ or if $d=3$ and $\\xi _{ho}\\rightarrow \\infty $ .",
"In this case, the rescaled wave functions $\\phi _{1,2}$ will be close to the ground state of the harmonic oscillator, i.e., $\\phi _{1,2} \\simeq \\pi ^{-d/2} \\exp (-\\vec{x}^2/2)$ , and phase separation is suppressed regardless of the values of the $g$ 's.",
"The interactions will become significant if $d=1$ and $\\xi _{ho}\\rightarrow \\infty $ or $d=3$ and $\\xi _{ho}\\rightarrow 0$ .",
"In this case, the kinetic energy can be neglected and we enter the Thomas-Fermi regime.",
"In this regime, the criterion (REF ) will be a faithful one for phase separation.",
"We have verified these predictions numerically.",
"In Fig.",
"REF , we have shown the overlap factor $\\eta \\equiv \\int d \\vec{x} \\phi _1 \\phi _2$ versus the reduced inter-component interaction strength $\\beta _{12}$ in all dimensions (with $g_{11}=g_{22}=g_{12}/1.05$ ).",
"Again, we see that phase separation is completely suppressed for $\\beta _{12}$ below some critical value $\\beta _{12}^c$ .",
"Figure: (Color online) (a)-(c) The overlap factor η\\eta as a function of the reduced parameter β 12 \\beta _{12} [seeEq.",
"()] in different dimensions (isotropicharmonic potential case, g 11 =g 22 =g 12 /1.05g_{11}=g_{22}=g_{12}/1.05).",
"Notethat for all value of dd, there exists a critical valueβ 12 c ≠0\\beta _{12}^c\\ne 0, below which η\\eta attains itsmaximal possible value 1.",
"(d) a schematic plot of η\\eta versus the characteristic frequency ω d \\omega _d of theharmonic potential in different dimensions.Let us now consider the possibility of experimentally observing the immiscibility-miscibility transition by adjusting the confinement, e.g., the frequency $\\omega _d$ .",
"In cold atom experiments, the harmonic potential is often of the form $V(\\vec{r})= \\frac{1}{2}m [\\omega _\\perp ^2 (x^2+y^2) + \\omega _z^2 z^2]$ .",
"To get a three dimensional isotropic potential, we set $\\omega _\\perp = \\omega _z$ .",
"An effectively one (two) dimensional potential can be obtained in the limit of $\\omega _\\perp \\gg \\omega _z$ ($\\omega _\\perp \\ll \\omega _z$ ).",
"For these three different geometries of the potential, the interaction strengths (the $g$ 's) relate to the $s$ -wave scattering lengths (the $a$ 's) as $g_{ij} \\!&=& \\frac{4\\pi \\hbar a_{ij}}{m}, \\ \\omega _d=\\omega _z=\\omega _\\perp , \\ d=3; \\\\g_{ij} \\!&=& \\frac{2 \\sqrt{2\\pi } \\hbar ^{3/2} \\omega _z^{1/2} a_{ij}}{m^{1/2}},\\ \\omega _d= \\omega _\\perp \\!",
"\\ll \\omega _z, \\ d=\\!2;\\quad \\\\g_{ij} \\!&=& 2\\hbar a_{ij} \\omega _\\perp , \\ \\omega _d= \\omega _z \\ll \\omega _\\perp , \\ d=1.$ Using Eqs.",
"(REF ) and (), we can study the possibility of tuning $\\beta _{12}$ across the critical value $\\beta _{12}^c$ .",
"We study each case individually (the mass $m$ is taken to be that of $^{23}$ Na): (i) $d=3$ .",
"Suppose $N=10^4$ , $a_{12}=40 $ $a_{\\text{B}}$ .",
"The critical value of $ \\omega _d$ is $2\\pi \\times 560$ Hz, which can be covered in current experiments.",
"(ii) $d=2$ .",
"Suppose $N=10^{4}$ , $a_{12}=40$ $a_{\\text{B}}$ , and the transverse frequency $\\omega _{\\perp }=2\\pi \\times 2.6$ Hz.",
"The critical value of the longitudinal frequency $\\omega _{z}$ is $2\\pi \\times 140$ Hz, which is realizable in current experiments [12].",
"(iii) $d=1$ .",
"Suppose $N=2\\times 10^{3}$ , $a_{12}=40$ $a_{\\text{B}}$ , and the transverse frequency $\\omega _{\\perp }=2\\pi \\times 130$ Hz.",
"The critical value of the longitudinal frequency $\\omega _{z}$ is $2\\pi \\times 19$ Hz, which is realizable in current experiments.",
"Here the number of atoms is one or two orders smaller than its typical value in experiments.",
"This explains why the criterion (REF ) is a reliable one in the experiments in [12], [13].",
"They work in a regime where the kinetic energy is indeed negligible.",
"However, with the advance of imaging techniques, hopefully future experiments can work with a relatively small number of atoms and observe the miscibility-immiscibility transition by changing the confinement."
],
[
"Conclusions",
"To conclude, we have demonstrated that kinetic energy can play a vital role in determining the configuration of a two-component BEC.",
"It renders the empirical condition of phase separation $g_{11} g_{22} < g_{12}^2$ insufficient and it also modifies the picture of phase separation.",
"To be specific, phase separation can be completely suppressed even if this condition is fulfilled.",
"Moreover, the phase mixing to phase separation transition is now known to be a second-order, continuous transition instead of a first-order, discontinuous one as in the usual view.",
"From the experimental point of view, our results may provide a new scenario of controlling the transition of phase mixing-demixing of a two-component BEC.",
"Instead of adjusting the interaction strengths, one can just change the confinement, the characteristic size of the container."
],
[
"Acknowledgments",
"We are grateful to Weizhu Bao, L. You, and C. H. Lee for helpful discussions.",
"This work is supported by NSF of China under Grant No.",
"11091240226, the Ministry of Science and Technology of China 973 program (2012CV821400) and NSFC-1190024.",
"Y. Cai acknowledges support from the Academic Research Fund of Ministry of Education of Singapore grant R-146-000-120-112."
],
[
"Rigorous justification",
"Here, we consider the energy functional as $E[\\phi _1,\\phi _2]= \\int _{\\Omega _0} d\\vec{x} \\bigg \\lbrace \\frac{1}{2}|\\nabla \\phi _1|^2+ \\frac{1}{2}|\\nabla \\phi _2|^2\\qquad \\qquad \\qquad \\\\+ \\frac{1}{2}\\left(\\beta _{11} |\\phi _1|^4 +\\beta _{22} |\\phi _2|^4 +2\\beta _{12} |\\phi _1|^2|\\phi _2|^2\\right)\\nonumber \\bigg \\rbrace ,$ where $\\Omega _0=[-\\frac{1}{2},\\frac{1}{2}]^d$ ($d=1,2,3$ ), $\\beta _{11}=\\beta _{22}=\\beta $ .",
"Let $\\phi _g$ be the unique positive ground state of the energy functional $E_s[\\phi ]\\equiv E[\\phi ,\\phi ]$ , and $\\mu _g$ be the corresponding chemical potential.",
"The functions $\\phi _{1,2}$ are normalized to unity by the usual $L^2$ -norm.",
"Let $(\\phi ^g_1,\\phi _2^g)$ be the positive ground state of (REF ).",
"For $\\beta _{12}\\le \\beta $ , $E[\\sqrt{\\rho _1},\\sqrt{\\rho _2}]$ ($\\rho _1\\equiv |\\phi _1|^2$ , $\\rho _2\\equiv |\\phi _2|^2$ ) is strictly convex in $(\\rho _1,\\rho _2)$ [18], [19], and the positive ground state is unique, i.e., $\\phi _1^g=\\phi _2^g=\\phi _g$ , $\\eta =1$ .",
"We are going to prove that there exists a critical value $\\beta _{12}^c>\\beta $ such that when $\\beta _{12}<\\beta _{12}^c$ , there holds $\\phi _1^g=\\phi _2^g=\\phi _g$ , i.e., $\\eta =1$ .",
"From now on, we concentrate on the case of $\\beta _{12}\\ge \\beta $ and assume that $\\beta _{12}=\\beta +\\beta ^\\prime $ , $0\\le \\beta ^\\prime \\le 1$ .",
"Simple calculation shows that $& E[\\phi _1^g,\\phi _2^g]-E[\\phi _g,\\phi _g] = \\int _{\\Omega _0} d\\vec{x} \\sum \\limits _{\\alpha =1,2}\\bigg \\lbrace \\frac{1}{2}|\\nabla (\\phi _\\alpha ^g-\\phi _g)|^2\\\\&+(\\beta +\\beta _{12}) |\\phi _g|^2|\\phi _\\alpha ^g-\\phi _g|^2+\\frac{\\beta -\\beta _{12}}{2} \\left(|\\phi _\\alpha ^g|^2-|\\phi _g|^2\\right)^2\\\\&+\\nabla (\\phi _\\alpha ^g-\\phi _g)\\cdot \\nabla \\phi _g+2(\\beta +\\beta _{12})|\\phi _g|^2\\phi _g(\\phi _\\alpha ^g-\\phi _g)\\bigg \\rbrace \\\\&+ \\frac{\\beta _{12}}{2}\\left(|\\phi _1^g|^2+|\\phi _2^g|^2-2|\\phi _g|^2\\right)^2.$ Making use of the Euler-Lagrange equation of $\\phi _g$ , $\\mu _g\\phi _g=-\\frac{1}{2}\\nabla ^2\\phi _g+(\\beta +\\beta _{12})|\\phi _g|^2\\phi _g,$ denoting $e_\\alpha =\\phi _\\alpha ^g-\\phi _g$ ($\\alpha =1,2$ ), and noticing $\\int _{\\Omega _0}e_\\alpha \\phi _g=-\\frac{1}{2}\\Vert e_\\alpha \\Vert _2^2$ , we obtain $& E[\\phi _1^g,\\phi _2^g]-E[\\phi _g,\\phi _g] = \\int _{\\Omega _0} d\\vec{x} \\sum \\limits _{\\alpha =1,2}\\bigg \\lbrace \\frac{1}{2}|\\nabla e_\\alpha |^2\\\\&+(\\beta +\\beta _{12}) |\\phi _g|^2|e_\\alpha |^2-\\frac{\\beta ^\\prime }{2} \\left(|\\phi _\\alpha ^g|^2-|\\phi _g|^2\\right)^2\\\\&+2\\mu _g \\phi _g e_\\alpha \\bigg \\rbrace + \\frac{\\beta _{12}}{2}\\left(|\\phi _1^g|^2+|\\phi _2^g|^2-2|\\phi _g|^2\\right)^2.$ Now, the operator $L_g=-\\frac{1}{2}\\nabla ^2+(\\beta +\\beta _{12})|\\phi _g|^2$ admits eigenvalues as $\\mu _g<\\mu _1\\le \\mu _2\\le \\cdots $ , and the eigenfunction $\\phi _g$ corresponds to $\\mu _g$ , $w_k\\in H_0^1$ with $\\Vert w_k\\Vert _2=1$ corresponds to $\\mu _k$ ($k\\ge 1$ ).",
"The reason $\\phi _g$ is the ground state comes from the positivity of $\\phi _g$ and the uniqueness of the positive ground state of $L_g$ .",
"Expand $e_\\alpha $ as $e_\\alpha =c_g^\\alpha \\phi _g+\\sum _{k=1}^\\infty c^\\alpha _kw_k$ , then $(c_g^\\alpha )^2+\\sum _{k=1}^\\infty |c^\\alpha _k|^2=\\Vert e_\\alpha \\Vert _2^2$ , $c_g^\\alpha =\\int _{\\Omega _0}e_\\alpha \\phi _g=-\\frac{1}{2}\\Vert e_\\alpha \\Vert _2^2$ and we can derive that $&\\int _{\\Omega _0} d\\vec{x} \\bigg \\lbrace \\frac{1}{2}|\\nabla e_\\alpha |^2+(\\beta +\\beta _{12}) |\\phi _g|^2|e_\\alpha |^2+2\\mu _g \\phi _g e_\\alpha \\bigg \\rbrace \\\\&= \\mu _g(c_g^\\alpha )^2+\\sum \\limits _{k=1}^\\infty \\mu _k|c^\\alpha _k|^2-\\mu _g\\Vert e_\\alpha \\Vert _2^2\\\\&\\ge (\\mu _1-\\mu _g)(\\Vert e_\\alpha \\Vert _2^2-(c_g^\\alpha )^2)\\\\&=(\\mu _1-\\mu _g)\\Vert e_\\alpha \\Vert _2^2(1-\\Vert e_g^\\alpha \\Vert _2^2/4).$ Now, firstly, we need a lower bound for $\\mu _1-\\mu _g$ , the so-called fundamental gap, which has been solved recently by A. Ben and C. Julie [20].",
"Using equation (REF ), applying elliptic theory with convex domain $\\Omega _0$ , it is easy to verify that $\\phi _g\\in H^2(\\Omega _0)$ and hence belongs to $C^{0,\\gamma }(\\overline{\\Omega _0})$ ($0<\\gamma <\\frac{1}{2}$ ) by Sobolev embedding.",
"Approximating $\\Omega _0$ by convex domain $\\Omega _\\varepsilon $ (with smooth boundary) and applying Schauder estimates, we shall have $\\phi _g\\in C^{2,\\gamma }(\\overline{\\Omega _\\varepsilon })$ and there exists some $c>0$ such that $|\\phi _g|^2+c|\\vec{x}|^2$ is convex (as Hessian matrix of $|\\phi _g|^2$ is bounded by Schauder estimates).",
"Hence, we can apply the results in Ref.",
"[20] to get ($D_\\varepsilon $ is the diameter of $\\Omega _\\varepsilon $ ) $\\mu _1^\\varepsilon -\\mu _g^\\varepsilon \\ge \\frac{3\\pi ^2}{D_\\varepsilon ^2},$ where $\\mu _g^\\varepsilon $ and $\\mu _1^\\varepsilon $ are the first and second eigenvalues, respectively, of $L_g$ in $H_0^1(\\Omega _\\varepsilon )$ .",
"By Min-max principles, letting $\\varepsilon \\rightarrow 0$ , we have $\\mu _g^\\varepsilon \\rightarrow \\mu _g$ and $\\mu _1^\\varepsilon \\rightarrow \\mu _1$ .",
"Hence we find $\\mu _1-\\mu _g\\ge \\frac{3\\pi ^2}{D^2},$ where $D$ is the diameter of $\\Omega _0$ [or if we assume $\\Omega _0$ is a convex domain with smooth boundaries, (REF ) follows directly].",
"Secondly, we have $\\Vert e_\\alpha \\Vert _2^2\\le \\int _{\\Omega _0}d\\vec{x}(|\\phi _\\alpha ^g|^2+|\\phi _g|^2)=2$ .",
"Thirdly, we would like to derive $L^\\infty $ bounds of $\\phi _g$ and $\\phi _\\alpha ^g$ .",
"The Euler-Lagrange equation for $\\phi _\\alpha ^g$ reads as $\\mu _\\alpha ^g\\phi _\\alpha ^g=-\\frac{1}{2}\\nabla ^2\\phi _\\alpha ^g+\\beta |\\phi _\\alpha ^g|^2\\phi _\\alpha ^g+\\beta _{12}|\\phi _{\\alpha ^\\prime }^g|^2\\phi _\\alpha ^g,$ with $\\alpha ^\\prime \\ne \\alpha $ .",
"For the nonlinear eigenvalues, we have the estimates $\\mu _\\alpha ^g\\le 2E[\\phi _1^g,\\phi _2^g]\\le 2E_s[\\phi _g]$ , $\\mu _g\\le E_s[\\phi _g]$ and $E_s[\\phi _g]$ can be bounded by choosing any test function (like the ground state of $-\\Delta $ ), which gives $E_s[\\phi _g]\\le \\widetilde{C}(1+\\beta )$ ($\\widetilde{C}$ depends on $\\Omega _0$ ).",
"If $\\beta \\ge 1$ , considering the point $x_0\\in \\Omega _0$ where $\\phi _g$ attains its maximum, then $\\Delta \\phi _g(x_0)\\le 0$ and from (REF ), we have $\\mu _g\\phi _g(x_0)\\ge (\\beta +\\beta _{12})|\\phi _g(x_0)|^2\\phi _g(x_0),$ which gives $\\Vert \\phi _g\\Vert _{\\infty }^2\\le \\frac{\\mu _g}{\\beta +\\beta _{12}}\\le 2\\widetilde{C}$ .",
"Similarly, we can obtain the $L^\\infty $ bound for $\\phi _\\alpha ^g$ using the Euler-Lagrange equation and $\\Vert \\phi _\\alpha ^g\\Vert _{\\infty }^2\\le \\frac{\\mu _\\alpha ^g}{\\beta }\\le 4\\widetilde{C}$ .",
"Thus, $\\Vert \\phi _g+\\phi _\\alpha ^g\\Vert _{\\infty }^2\\le 12\\widetilde{C}$ .",
"Combining the three observations above, we get $E[\\phi _1^g,\\phi _2^g]-E[\\phi _g,\\phi _g]\\ge \\sum \\limits _{\\alpha =1,2}\\bigg \\lbrace \\frac{3\\pi ^2}{2D^2}-\\frac{12\\beta ^\\prime \\widetilde{C}}{2} \\bigg \\rbrace \\Vert e_\\alpha \\Vert _2^2,$ which implies that for $0\\le \\beta ^\\prime \\le \\min \\lbrace \\frac{\\pi ^2}{4D^2\\widetilde{C}},1\\rbrace $ , there must hold $e_\\alpha =0$ , i.e., $\\eta =1$ .",
"For $\\beta \\in [0,1]$ , the approach above is not good.",
"In this case, we see that $\\mu _\\alpha ^g\\le 4\\widetilde{C}$ and $\\mu _g\\le 2\\widetilde{C}$ .",
"Using Sobolev inequality, in one dimension ($d=1$ ), we can find that $\\Vert \\phi _\\alpha ^g\\Vert _{\\infty }^2\\le \\Vert \\nabla \\phi _\\alpha ^g\\Vert _{2}\\Vert \\phi _\\alpha ^g\\Vert _{2}\\le \\sqrt{\\mu _\\alpha ^g}\\le 2\\sqrt{\\widetilde{C}}.$ Similarly, $\\Vert \\phi _g\\Vert _{\\infty }^2\\le \\sqrt{2\\widetilde{C}}$ .",
"For two and three dimensions ($d=2,3$ ), recalling (REF ) and (REF ), we can obtain from elliptic theory and Sobolev inequalities that there exist constants $C_1, C_2>0$ only depending on $\\Omega _0$ such that $\\Vert \\phi _\\alpha ^g\\Vert _{\\infty }\\le C_1\\Vert \\phi _\\alpha ^g\\Vert _{H^2}\\le C_2\\cdot \\Vert \\mu _\\alpha ^g\\phi _\\alpha ^g-\\beta |\\phi _\\alpha ^g|^2\\phi _\\alpha ^g-\\beta _{12}|\\phi _{\\alpha ^\\prime }^g|^2\\phi _\\alpha ^g\\Vert _{2}$ , and $\\Vert \\phi _g\\Vert _{\\infty }\\le C_2\\Vert \\mu _g\\phi _g-(\\beta +\\beta _{12})|\\phi _g|^2\\phi _g\\Vert _{2}$ .",
"In two and three dimensions, using Sobolev inequality, we have $\\Vert \\phi _\\alpha ^g\\Vert _{6}\\le C_3\\Vert \\nabla \\phi _\\alpha ^g\\Vert _{2}\\le C_3\\sqrt{\\mu _\\alpha ^g}$ ($C_3$ depends on $\\Omega _0$ ).",
"Cauchy inequality leads to $\\Vert \\phi _\\alpha ^g\\Vert _{\\infty }\\le C_2(\\mu _\\alpha ^g+\\beta \\Vert \\phi _\\alpha ^g\\Vert _{6}^{3}+\\beta _{12}\\Vert \\phi _\\alpha ^g\\Vert _{6}\\Vert \\phi _{\\alpha ^\\prime }^g\\Vert _{6}^2),$ and thus $\\Vert \\phi _\\alpha ^g\\Vert _{\\infty }^2\\le C_4$ ($C_4$ depends on $\\Omega _0$ ).",
"Similarly, $\\Vert \\phi _g\\Vert _{\\infty }^2\\le C_5$ ($C_5$ depends on $\\Omega _0$ ).",
"Eventually, we have in all dimensions ($d=1,2,3$ ), there exists a constant $C_{\\Omega _0}$ depending only on $\\Omega _0$ such that $\\Vert \\phi _\\alpha ^g+\\phi _g\\Vert _\\infty ^2\\le C_{\\Omega _0}$ .",
"Similar to the case with $\\beta \\ge 1$ , we have $& E[\\phi _1^g,\\phi _2^g]-E[\\phi _g,\\phi _g] \\ge \\sum \\limits _{\\alpha =1,2}\\left\\lbrace \\frac{3\\pi ^2}{2D^2}-\\frac{\\beta ^\\prime C_{\\Omega _0}}{2} \\right\\rbrace \\Vert e_\\alpha \\Vert _2^2,$ which leads to the conclusion that when $\\beta ^\\prime <\\min \\lbrace 1,\\frac{3\\pi ^2}{D^2C_{\\Omega _0}}\\rbrace $ , $\\phi _\\alpha ^g=\\phi _g$ , i.e., $\\eta =1$ .",
"In summary, for all $\\beta \\ge 0$ , if we choose $\\beta _{12}^c=\\beta +\\min \\lbrace 1,\\frac{3\\pi ^2}{D^2C_{\\Omega _0}},\\frac{\\pi ^2}{4D^2\\widetilde{C}}\\rbrace >\\beta $ , then for all $0\\le \\beta _{12}<\\beta _{12}^c$ , we shall have $\\eta =1$ ."
],
[
"Phase separation as a spontaneous symmetry\nbreaking",
"Consider a two-component BEC in a symmetric double-well potential.",
"Under the two-mode approximation, the mean-field energy functional is $E &=& -J_a(\\psi _{a1}^* \\psi _{a2}+ \\psi _{a2}^* \\psi _{a1})-J_b(\\psi _{b1}^* \\psi _{b2}+ \\psi _{b2}^* \\psi _{b1}) \\nonumber \\\\&& +\\frac{1}{2 }U_a (|\\psi _{a1}|^4 + |\\psi _{a2}|^4) + \\frac{1}{2 }U_b(|\\psi _{b1}|^4 + |\\psi _{b2}|^4) \\nonumber \\\\&& +V(|\\psi _{a1}|^2|\\psi _{b1}|^2 + |\\psi _{a2}|^2|\\psi _{b2}|^2).$ Here $J_a$ and $J_b$ are the hopping amplitudes of the two types of atoms, and $U_a$ and $U_b$ are the intra-component onsite interaction strengths, while $V$ is the inter-component one.",
"The complex numbers $\\psi _{a1}$ and $\\psi _{b1}$ ($\\psi _{a2}$ and $\\psi _{b2}$ ) are the amplitudes of the two condensate wave functions on the left (right) trap.",
"They are constrained by the total atom numbers, i.e., $|\\psi _{a1}|^2 + |\\psi _{a2}|^2 =N_a$ and $|\\psi _{b1}|^2 + |\\psi _{b2}|^2 =N_b$ .",
"For the sake of simplicity, in the following we shall assume $J_a=J_b=J\\ge 0$ , $U_a=U_b=U \\ge 0$ , and $N_a=N_b=N$ .",
"As far as the ground state is concerned, it is legitimate to assume the $\\psi $ 's real and positive.",
"Therefore, we can write $\\psi _{a1}=\\sqrt{N_{a1}}$ , $\\psi _{b1}=\\sqrt{N_{b1}}$ and similarly for other $\\psi $ 's.",
"First assume tunneling is turned off, i.e.",
"$J=0$ .",
"Let $N_{a1}= \\frac{1}{2} N + \\delta _a$ and $N_{b1}= \\frac{1}{2}N - \\delta _b$ .",
"The energy (REF ) is $E(\\delta _a, \\delta _b) = U(\\delta _a^2 +\\delta _b^2)- 2V\\delta _a \\delta _b +\\text{const}.$ It is readily determined that if $U>V$ , the ground state is of $\\delta _a=\\delta _b=0$ .",
"The two condensates are both distributed evenly between the two wells, which is a completely mixed configuration.",
"If $U<V$ [the counterpart of (REF ) in the present context], the ground state is of $(\\delta _a,\\delta _b)=\\pm (N/2,N/2)$ , which corresponds to complete phase separation—the two condensates occupy the two wells separately.",
"Therefore, without tunneling, the miscibility-immiscibility transition is a first-order phase transition with the critical point being $V^c =U$ .",
"Now turn on the tunneling.",
"For the sake of simplicity, suppose $\\delta _a=\\delta _b=\\delta $ .",
"The energy as a function of the order parameter $\\delta $ is $E &=&-4J \\sqrt{ \\left(\\frac{N}{2} \\right)^2-\\delta ^2} + 2(U-V)\\delta ^2 +\\text{const} \\nonumber \\\\&=& \\left[\\frac{4J}{N}+ 2(U-V) \\right]\\delta ^2 + \\frac{4J}{N^3}\\delta ^4+ o(\\delta ^4)+ \\text{const} .\\quad \\quad $ Here we have the familiar Landau formalism for second order phase transitions.",
"The coefficient of the quartic term is positive but the sign of the quadratic term changes from positive to negative as $V$ surpasses the critical value $V^c= U + 2J/N$ .",
"Corresponding, $\\delta =0 $ is turned from a minimum to a maximum point and phase separation develops.",
"Here we note that the tunneling, the kinetic term in the present context, has two consequences.",
"First, the first-order transition is turned into a second-order one.",
"Second, the transition point is up shifted from $U$ to $U+2J/N$ .",
"This is reasonable since phase separation costs kinetic energy.",
"What presented in Figs.",
"REF and REF are parallel to these results but in continuum (multi-mode) cases."
],
[
"Justification of the form of $\\varphi _1$ in Eq. (",
"In this Appendix, we show why among all the (degenerate) first excited states, the one in Eq.",
"() is selected.",
"For $d=2$ , the ansatz more general than Eq.",
"(REF ) is $\\phi _1 &=& c_0 \\varphi _0 + c_x \\varphi _x+ c_y\\varphi _y, \\\\\\phi _2 &=& c_0\\varphi _0 - c_x \\varphi _x- c_y \\varphi _y,$ with $\\varphi _x= w_1(x)w_0(y)$ , $\\varphi _y= w_0(x)w_1(y)$ , and $c_0$ , $c_{x}$ , $c_{y}$ being some real variables under the constraint $c_0^2 +c_x^2 +c_y^2 =1$ .",
"Substituting Eq.",
"() into Eq.",
"(REF ), we get the reduced energy functional $\\widetilde{E}=E/(N\\hbar ^2/mL^2)$ as a function of $c_{x,y}$ as $\\tilde{E}[c_x,c_y] &=& 2\\pi ^2 +\\frac{9}{4}\\beta _{12}+\\left(3\\pi ^2 -\\frac{15}{2}\\beta _{12} \\right)(c_x^2+c_y^2) \\nonumber \\\\& & +\\frac{15}{2}\\beta _{12}\\left(c_x^2 +c_y^2 \\right)^2+ \\frac{3}{2}\\beta _{12}c_x^2 c_y^2.$ We see that for $\\beta \\le \\beta _{12}^c =\\frac{2}{5}\\pi ^2$ , the minimum is at $c_x=c_y=0$ .",
"For $\\beta _{12} >\\beta _{12}^c$ , the minimum is no longer at the origin.",
"However, for a fixed value of $c_x^2+c_y^2$ , $\\tilde{E}$ is minimized when the last term in Eq.",
"(REF ) vanishes or when $c_x=0$ or $c_y=0$ .",
"That is why the particular ansatz in Eqs.",
"(REF ) and () is appropriate and enough.",
"We note that due to the symmetry of the trap, the reduced energy functional is invariant under the transform $(c_x,c_y) \\rightarrow (\\pm c_x,\\pm c_y)$ and $(c_x,c_y) \\rightarrow ( c_y, c_x)$ .",
"This symmetry is broken when phase separation occurs.",
"Similar analysis applies for $d=3$ .",
"In this case, the ansatz more general than Eq.",
"(REF ) is $\\phi _1&=& c_0 \\varphi _0 + c_x \\varphi _x+ c_y\\varphi _y+c_z \\varphi _z, \\\\\\phi _2 &= & c_0\\varphi _0 - c_x \\varphi _x- c_y \\varphi _y -c_z \\varphi _z,$ with $\\varphi _x= w_1(x)w_0(y)w_0(z)$ , $\\varphi _y= w_0(x)w_1(y)w_0(z)$ , $\\varphi _z= w_0(x)w_0(y)w_1(z)$ , and $c_0$ , $c_{x}$ , $c_{y}$ , $c_z$ being some real variables under the constraint $c_0^2 +c_x^2 +c_y^2+c_z^2 =1$ .",
"Substituting Eq.",
"() into Eq.",
"(REF ), we get the reduced energy functional $\\tilde{E} $ as a function of $c_{x,y,z}$ as $\\tilde{E} &=& 3\\pi ^2 +\\frac{27}{8}\\beta _{12}+\\left(3\\pi ^2 -\\frac{45}{4} \\beta _{12}\\right)(c_x^2+c_y^2+c_z^2)\\quad \\quad \\ \\nonumber \\\\&&\\quad \\quad +\\frac{45 }{4} \\beta _{12} \\left(c_x^2 +c_y^2 +c_z^2\\right)^2 \\nonumber \\\\&& \\quad \\quad + \\frac{9}{4} \\beta _{12} \\left(c_x^2 c_y^2 +c_y^2 c_z^2+c_z^2 c_x^2\\right).$ We see that for $\\beta \\le \\beta _{12}^c =\\frac{4}{15}\\pi ^2$ , the minimum is at $c_x=c_y=c_z=0$ .",
"For $\\beta _{12} >\\beta _{12}^c$ , the minimum is no longer at the origin.",
"However, for a fixed value of $c_x^2+c_y^2+c_z^2$ , $\\tilde{E}$ is minimized when the last term in Eq.",
"(REF ) vanishes or when two of the three $c$ 's are zero.",
"Again, we see that the particular ansatz in Eqs.",
"(REF ) and () is appropriate and enough."
]
] | 1204.1256 |
[
[
"Geometrical frustration of an extended Hubbard diamond chain in the\n quasi-atomic limit"
],
[
"Abstract We study the geometrical frustration of extended Hubbard model on diamond chain, where vertical lines correspond to the hopping and repulsive Coulomb interaction terms between sites, while the rest of them represent only the Coulomb repulsion one.",
"The phase diagrams at zero temperature show quite curious phases: five types of frustrated states and four types of non-frustrated ones, ordered antiferromagnetically.",
"Although decoration transformation was derived to spin coupling systems, this approach can be applied to electron coupling ones.",
"Thus the extended Hubbard model can be mapped onto another effective extended Hubbard model in atomic limit with additional three and four-body couplings.",
"This effective model is solved exactly through transfer matrix method.",
"In addition, using the exact solution of this model we discuss several thermodynamic properties away from half filled band, such as chemical potential behavior, electronic density, entropy, where we study the geometrical frustration, consequently we investigate the specific heat as well."
],
[
"Introduction",
"The Hubbard model is one of the simplest model for strongly interacting electron systems.",
"In general, rigorous analysis of the model is a difficult task, only in particular case a number of rigorous exact results have been obtained .",
"On the other hand, geometrical frustration in strongly correlated electron systems have attracted a great deal of interest over the past decades , .",
"For instance, due to the competition between the nearest and the next-nearest exchange coupling, the inorganic spin-Peierls compound exhibits a transition from a gapless phase to a gaped dimerized ground state .",
"The quantum phase transition point from the gapless phase to the gapped dimerized phase of this model was first determined by Okamoto and Nomura.",
"The interplay between geometrical frustration and strong electron correlation results in a complicated phase diagram, containing many interesting phases: spin-gaped metallic phase, disorder magnetic insulation phase, Heisenberg insulator .",
"The Hubbard model on the triangular lattice shows a transition from a paramagnetic metal to an antiferromagnetic insulator as the Hubbard on-site repulsion gradually grows .",
"Using the numerical diagonalization of finite size system Hida predicted the appearance of the magnetization plateau in one of the pioneering theoretical papers .",
"In the framework of the transfer-matrix and dynamical recursive approaches the frustrated magnetization plateaus were obtained for ferromagnetic-ferromagnetic-antiferromagnetic, kagome chains as well as zigzag ladder with multi-spin exchanges , , .",
"For small nanoclusters by the exact numerical diagonalization the average electron density, magnetization plateaus via chemical potential or magnetic field have been studied in Hubbard model , .",
"Recently geometrical frustration of Hubbard model was widely studied, particularly the diamond chain structure such as considered by Derzhko et al.",
", , where they discuss the frustration for a special class of lattice.",
"Montenegro-Filho and Coutinho-Filho also considered the doped $AB_{2}$ Hubbard chain, both in the weak coupling and the infinite-$U$ limit (atomic limit).",
"They studied a quite interesting phases as a function of hole doping away from half-filled band.",
"Mancini , has presented the exact solution of extended one-dimensional Hubbard model in the atomic limit, obtaining the chemical potential plateaus of the particle density, of the on-site potential at zero temperature and studied the thermodynamic charge susceptibility, compressibility at finite temperature, as well as other physical quantities.",
"Vidal et al.",
"also discussed two interacting particles evolving in a diamond chain structure embedded in a magnetic field.",
"As the particles interact, it leads to the strong localization induced by the magnetic field for the particular value of a flux.",
"Analogous model was also studied by Rossler and Mainemer .",
"The Hubbard model in other quasi-one-dimensional triangular structure also was studied by Wang .",
"The latter indicated that for small hopping term, the system exhibits short range antiferromagnetic correlation, whereas, when the hopping terms become greater than critical point, there is a transition from an antiferromagnetic state to a frustrated one.",
"Moreover, the insulator-metal transition takes place at hopping interaction even greater than another critical point.",
"Further Gulacsi et al.",
", also discussed the diamond Hubbard chain in a magnetic field and a wide range of properties such as flat-band ferromagnetism and correlation-induced metallic, half-metallic process.",
"The spinless versions of the Hubbard model on diamond chain was also recently investigated through exact analytical solution, as well as Lopes and Dias performed a detailed investigation using the exact diagonalization approach.",
"In the last decade several diamond chain structures were discussed.",
"Some of them were motivated by real materials such as $\\mathrm {Cu_{3}(CO_{3})_{2}(OH)_{2}}$ known as azurite, which is an interesting quantum antiferromagnetic model, described by Heisenberg model on generalized diamond chain.",
"Honecker et al.",
"studied the dynamic and thermodynamic properties for this model.",
"Besides, Pereira et al.",
"investigate the magnetization plateau of delocalized interstitial spins on diamond chain, as well as they detect magnetocaloric effect in kinetically frustrated diamond chain .",
"Quite recently, Lisnii studied a distorted diamond Ising-Hubbard chain and that model exhibits geometrical frustration also.",
"A further investigation regarding the exact evidence for the spontaneous antiferromagnetic long-range order in the two-dimensional hybrid model of localized Ising spins and itinerant electrons was discussed by Strečka et al.",
", .",
"Moreover, the thermodynamics of the Ising-Heisenberg model on diamond-like chain was widely discussed in references , , , , and also the thermal entanglement was considered by Ananikian et al.",
".",
"On the other hand, the analytical exact solution is rather amazing, since the exact result is always useful to manipulate against the numerical results.",
"Therefore, the exact solutions of the models are of great importance, so our main goal is to obtain an exact solution for the extended Hubbard model on diamond chain in quasi-atomic limit.",
"The research of the extended Hubbard diamond chain model without the hopping of electrons between the nodal sites is based on the three main reasons.",
"First of all 1/3 magnetization plateau, the double peaks both in the magnetic susceptibility and specific heat was observed in the experimental measurements, according to the experiments of the natural mineral azurite.",
"Theoretical calculations of the one-dimensional Hubbard model, as well as the experimental result of the exchange dimer (interstitial sites) parameter and their descriptions of the various theoretical models are presented.",
"It should be noted that the dimers (interstitial sites) exchange much more strongly than those nodal sites.",
"There were proposed various types of theoretical Heisenberg approximate methods: the renormalization of the density matrix renormalization group of the transfer matrix, the density functional theory, the high temperature expansion, Lanczos diagonalization on a diamond chain are to explain the experimental measurements (magnetization plateau and the double top) in natural mineral azurite.",
"All of these theoretical studies are approximate.",
"There is another possibility.",
"Since dimer interaction is much higher than the rest, it can be represented as an exactly solvable Ising-Heisenberg model.",
"In addition, experimental data on the magnetization plateau coincide with the approximation of Ising-Heisenberg model , , , and the extended Hubbard model without the hopping of electrons between the nodal sites on a diamond chain.",
"This is the first reason.",
"The second one is the quantum block-block entanglement, carried out by exact diagonalization technique in one-dimensional extended Hubbard model, calculated for finite size (L=10).",
"When the absolute value of nearest-neighbor Coulomb interactions (our case) becomes smaller, the effect of the hoping term and the on-site interaction cannot be neglected.",
"Finally, we would like to point out that although the results, obtained in this paper are for the 2-site (dimer) system, their qualitative features are the same as for big size system ones.",
"And, the third reason is the experimental observation of the double peaks both in the magnetic susceptibility and specific heat , , may be described exactly by the extended Hubbard diamond chain model without the hopping of electrons or holes between the nodal sites.",
"This phenomenon is particularly important in the quantum case.",
"This paper is organized as follows: in Sec.",
"we present the extended Hubbard model on diamond chain.",
"We have studied the phase diagram at zero temperature in Sec. .",
"Further, in Sec.",
", we present the exact solution of the model.",
"In Sec.",
", we have discussed the thermodynamic properties of the model, such as electronic density, internal energy, compressibility, entropy and specific heat away from the half-filled band.",
"Finally, Sec.",
"contains the concluding remarks."
],
[
"The Extended Hubbard model",
"The extended Hubbard model on diamond chain will be considered in this paper, as represented schematically in FIG.",
"REF .",
"In the present model we consider the hopping term $t$ between sites $a$ and $b$ , besides, there are onsite Coulomb repulsion interaction term denoted by $U$ and the nearest neighbor repulsion interaction term, denoted by $V$ .",
"We assume also that this model has an arbitrary particle density, so, we will include a chemical potential term noted by $\\mu $ , therefore the Hamiltonian of this model can be expressed by: Figure: (Color online) Schematic representation of extendedHubbard model on diamond chain.$H=\\sum _{i=1}^{N}H_{i,i+1},$ with $N$ being the number of cell (sites $a$ , $b$ and $c$ ), whereas $H_{i,i+1} $ is given by $\\begin{array}{l}H_{i,i+1}= -t\\sum \\limits _{\\sigma =\\downarrow ,\\uparrow }\\left(a_{i,\\sigma }^{\\dagger }b_{i,\\sigma }+a_{i,\\sigma }b_{i,\\sigma }^{\\dagger }\\right)-\\\\-\\ \\mu \\left(n_{i}^{a}+n_{i}^{b}+\\tfrac{1}{2}(n_{i}^{c}+n_{i+1}^{c})\\right)+\\\\+\\ U\\left(n_{i,\\uparrow }^{a}n_{i,\\downarrow }^{a}+n_{i,\\uparrow }^{b}n_{i,\\downarrow }^{b}+\\tfrac{1}{2}(n_{i,\\uparrow }^{c}n_{i,\\downarrow }^{c}+n_{i+1,\\uparrow }^{c}n_{i+1,\\downarrow }^{c})\\right)+\\\\+\\ V\\left(n_{i}^{a}n_{i}^{b}+\\tfrac{1}{2}(n_{i}^{a}+n_{i}^{b})(n_{i}^{c}+n_{i+1}^{c})\\right),\\end{array}$ with $a_{i,\\sigma }$ , and $b_{i,\\sigma }$ ($a_{i,\\sigma }^{\\dagger }$ and $b_{i,\\sigma }^{\\dagger }$ ) being the Fermi annihilation (creation) operator for the Hubbard model, while by $\\sigma $ we represent the electron spin, and by $n_{i,\\sigma }^{\\alpha }$ we mean the number operator, with $\\alpha =\\lbrace a,b,c\\rbrace $ , using this number operator we define also the following operators $n_{i}^{\\alpha }=n_{i,\\uparrow }^{\\alpha }+n_{i,\\downarrow }^{\\alpha }$ .",
"In order to contract the Hamiltonian (REF ), we can define the following operators $\\begin{array}{rcl}p_{i,i+1} & =&\\tfrac{1}{2}(n_{i}^{c}+n_{i+1}^{c}),\\\\q_{i,i+1} & =&\\tfrac{1}{2}(n_{i,\\uparrow }^{c}n_{i,\\downarrow }^{c}+n_{i+1,\\uparrow }^{c}n_{i+1,\\downarrow }^{c}),\\end{array}$ using these operators, we can rewrite the Hamiltonian (REF ), as follows: $\\begin{array}{ccl}H_{i,i+1}&= & -t\\sum \\limits _{\\sigma =\\downarrow ,\\uparrow }\\left(a_{i,\\sigma }^{\\dagger }b_{i,\\sigma }+a_{i,\\sigma }b_{i,\\sigma }^{\\dagger }\\right)-\\\\&-&(\\mu -Vp_{i,i+1})\\left(n_{i}^{a}+n_{i}^{b}\\right)+ \\\\& +&U\\left(n_{i,\\uparrow }^{a}n_{i,\\downarrow }^{a}+n_{i,\\uparrow }^{b}n_{i,\\downarrow }^{b}\\right)+\\\\&+&Vn_{i}^{a}n_{i}^{b}-\\mu p_{i,i+1}+Uq_{i,i+1}.\\end{array}$ The symmetry particle-hole can be analyzed in a similar way, as discussed in reference .",
"The phase diagram in the half-filled band, can be obtained using the following restriction for the chemical potential $\\mu =U/2+2V$ .",
"This relation could be obtained by using the symmetry particle-hole in a similar way, as discussed in reference."
],
[
"The phase diagram",
"In order to study the phase diagram at zero temperature, first we need to diagonalize the Hamiltonian (REF ) at sites $a$ and $b$ .",
"Where by $m=\\lbrace 0,..,4\\rbrace $ we will denote the total number of mobile electrons per unit cell at sites $a$ and $b$ , while by $n_{c}$ we mean the number of electrons per unit cell at site-c.",
"The Hamiltonian at sites $a$ and $b$ can be written as $16\\times 16$ matrix, but this matrix can be constructed as a block matrix, where in the largest block matrix we have $4\\times 4$ , therefore the eigenvalues and eigenvectors of this matrix are expressed below as: (a) State with $m=0$ particle $\\lambda _{0}=D_{0},\\quad |S_{0}\\rangle =|0,0\\rangle ,$ where $D_{m}=-(\\mu -Vp_{i,i+1})m-\\mu p_{i,i+1}+Uq_{i,i+1}$ , While the state $\\quad |S_{0}\\rangle =|0,0\\rangle $ means there are no particle at site $a$ and $b,$ respectively.",
"(b) State with $m=1$ particle $\\lambda _{\\sigma }^{(\\pm )}=D_{1}\\pm t,\\quad |S_{\\sigma }^{(\\pm )}\\rangle =\\tfrac{1}{\\sqrt{2}}\\left(|0,\\sigma \\rangle \\mp |\\sigma ,0\\rangle \\right),$ with $|S_{\\sigma }^{(\\pm )}\\rangle $ we mean there is one particle at site $a$ or $b$ with either spin up or down.",
"(c) State with $m=2$ particles $\\lambda _{2\\sigma } & =D_{2}+V,\\quad |S_{2\\sigma }\\rangle =|\\sigma ,\\sigma \\rangle ,\\\\\\lambda _{\\downarrow \\hspace{0.0pt}\\uparrow }^{(1)} & =D_{2}+U,\\quad |S_{\\downarrow \\hspace{0.0pt}\\uparrow }^{(1)}\\rangle =\\tfrac{1}{\\sqrt{2}}\\left(|\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow ,0\\rangle -|0,\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow \\rangle \\right),\\\\\\lambda _{\\downarrow \\hspace{0.0pt}\\uparrow }^{(\\pm )} & =D_{2}+V+\\frac{\\theta _{\\pm }t}{2},\\\\|S_{\\downarrow \\hspace{0.0pt}\\uparrow }^{(\\pm )}\\rangle & =\\tfrac{1}{\\zeta _{\\mp }}\\left(|\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow ,0\\rangle +|0,\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow \\rangle +\\theta _{\\mp }\\left(|\\uparrow ,\\downarrow \\rangle +|\\downarrow ,\\uparrow \\rangle \\right)\\right)\\\\\\lambda _{\\downarrow \\hspace{0.0pt}\\uparrow }^{(2)} & =D_{2}+V,\\quad |S_{\\downarrow \\hspace{0.0pt}\\uparrow }^{(2)}\\rangle =\\tfrac{1}{\\sqrt{2}}\\left(|\\uparrow ,\\downarrow \\rangle -|\\downarrow ,\\uparrow \\rangle \\right),$ where $\\zeta _{\\pm }= & \\sqrt{2+\\tfrac{\\theta _{\\pm }^{2}}{8}},\\\\\\theta _{\\pm }= & \\frac{U-V\\pm \\sqrt{(U-V)^{2}+16t^{2}}}{t},$ the states $|S_{2\\sigma }\\rangle $ and $|S_{\\downarrow \\hspace{0.0pt}\\uparrow }^{(\\pm )}\\rangle $ are defined in a similar way as defined in the previous case, two particles both with spin up or down and two particles with opposite spins.",
"(d) State with $m=3$ particles $\\lambda _{\\downarrow \\hspace{0.0pt}\\uparrow \\sigma }^{(\\pm )}=D_{3}+U+2V\\pm t,\\quad |S_{\\downarrow \\hspace{0.0pt}\\uparrow \\sigma }^{(\\pm )}\\rangle =\\tfrac{1}{\\sqrt{2}}\\left(|\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow ,\\sigma \\rangle \\mp |\\sigma ,\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow \\rangle \\right),$ the states $|S_{\\downarrow \\hspace{0.0pt}\\uparrow \\sigma }^{(\\pm )}\\rangle $ correspond to two particles with opposite spin and one spin (e) State with $m=4$ particles $\\lambda _{\\downarrow \\hspace{0.0pt}\\uparrow \\downarrow \\hspace{0.0pt}\\uparrow }=D_{4}+2U+4V,\\quad |S_{\\downarrow \\hspace{0.0pt}\\uparrow \\downarrow \\hspace{0.0pt}\\uparrow }\\rangle =|\\hspace{0.0pt}\\downarrow \\hspace{0.0pt}\\uparrow ,\\downarrow \\hspace{0.0pt}\\uparrow \\rangle .$ the state $|S_{\\downarrow \\hspace{0.0pt}\\uparrow \\downarrow \\hspace{0.0pt}\\uparrow }\\rangle $ means there are two spins at each site with opposite spins.",
"It is worth to note that, the Hamiltonian (REF ), has 256 eigenvalues per diamond plaquette.",
"From here after, we will consider only repulsive onsite Coulomb interaction ($U>0$ ) as well as repulsive Coulomb interaction between nearest neighbor ($V>0$ ).",
"Besides $V<U$ , in order to make more realistic our discussion.",
"Afterwards we will discuss the phase diagram at zero temperature, the present model exhibits 9 states which are tabulated in table REF , for all possible numbers of particles available in the chain.",
"For the empty particle and fully filled particles, we have one state $|\\mathsf {S_{0}\\rangle =|Vac\\rangle }$ and $|\\mathsf {S_{6}\\rangle =|Full\\rangle ,}$ respectively.",
"While for one particle per unit cell and five particles per unit cell, we also have one corresponding frustrated state $|\\mathsf {S_{1}\\rangle =|FRU_{1}\\rangle }$ and $|\\mathsf {S_{5}\\rangle =|FRU_{5}\\rangle ,}$ respectively.",
"However, for 2 particles per unit cell, we have two states, one configuration is an antiferromagnetic state ($\\mathsf {|S_{2}\\rangle =|AFM_{2}\\rangle }$ ), whereas other configuration is a frustrated state ($\\mathsf {|\\bar{S}_{2}\\rangle =|FRU_{2}\\rangle }$ ).",
"Due to particle-hole symmetry the analyze for the case of four particles (or two holes) becomes analogous for the case of two particles, hence, we have one antiferromagnetically ordered state ($\\mathsf {|S_{4}\\rangle =|AFM_{4}\\rangle }$ ) and another frustrated state ($\\mathsf {|\\bar{S}_{4}\\rangle =|FRU_{4}\\rangle }$ ).",
"Furthermore, for the special case of half-filled particles (with 3 particles or holes per unit cell) we also could have two possible states, one frustrated state $\\mathsf {|S_{3}\\rangle =|FRU_{3}\\rangle }$ and another antiferromagnetically ordered state ($|\\bar{\\mathsf {S}}_{3}\\rangle =|\\mathsf {AFM_{3}\\rangle }$ ), however the case $|\\mathsf {AFM_{3}}\\rangle $ occurs only when $V>U,$ this case we ignore, since, it is too artificial one.",
"Table: The first column n t n_t represents the number of particlesper unit cell, in the second and third column is tabulated the groundstate energy per unit cell and its corresponding ground state respectively.In FIG.",
"REF we illustrate the phase transition at zero temperature of these states.",
"In FIG.",
"REF (a) we illustrate the phase diagram of $V/U$ versus $\\mu /U$ , for fixed value of $t/U=1$ , where we display seven states $\\mathsf {S}_{i}$ ($i=0..6$ ).",
"While in FIG.",
"REF (b) we display the phase diagram of $t/U$ versus $\\mu /U$ , for fixed value of $V/U=0.25$ .",
"This phase diagram illustrates 9 states, the seven states are already shown in FIG.",
"REF (a) and two additional ones $|\\mathsf {FRU}_{2}\\rangle $ and $|\\mathsf {FRU}_{4}\\rangle $ .",
"The fully transition boundary functions between states are tabulated in table REF .",
"The first and third columns correspond to the boundary states, while the second and fourth columns correspond to the boundary functions.",
"In FIG.",
"REF , by dashed line we represent the half-filled band curve of the extended Hubbard model on diamond chain.",
"Figure: Phase diagrams at zero temperature: (a) displaysthe phase diagram of V/UV/U versus μ/U\\mu /U for fixed value of t/U=1t/U=1.Whereas in (b) displays the phase diagram of t/Ut/U versus μ/U\\mu /U for fixedvalue of V/U=0.25V/U=0.25.Table: The first and third columns represent the boundarybetween two states tabulated in table , while the secondand fourth columns mean the function of boundaries.",
"For simplicity we use the following notation V ' =V/UV^{\\prime }=V/U and t ' =t/Ut^{\\prime }=t/UOther variants of the phase diagram at zero temperature could also be discussed, although the main properties already had been discussed in FIG.",
"REF ."
],
[
"Exact solution",
"The method to be used will be the decoration transformation early proposed by Syozi and Fisher , more later this approach was the subject of study by Rojas et al.",
", for the case of multi-spins.",
"As well as by Strečka for the hybrid system (e.g.",
"Ising-Heisenberg).",
"Another interesting variant of this approach is also discussed by us, where we propose a direct transformation instead of several step by step one , an illustrative successful application of the last approach was performed in reference .",
"The decoration transformation approach is widely used to solve spin models, however the decoration transformation approach can also be applied to electron coupling system, such have been used by the authors for the case of spinless fermion on diamond structure.",
"Therefore, in order to use the decoration transformation approach we can write the Boltzmann factor for extended Hubbard model on diamond chain as follows: $\\begin{array}{l}w_{n_{i}^{c},n_{i+1}^{c}}= \\displaystyle \\mathrm {e}^{-\\beta D_{0}}+4\\left(\\mathrm {e}^{-\\beta D_{1}}+\\mathrm {e}^{-\\beta (D_{3}+U+2V)}\\right)\\cosh (\\beta t)\\\\\\qquad +\\displaystyle \\mathrm {e}^{-\\beta D_{2}}\\left(\\mathrm {e}^{-\\beta U}+3\\mathrm {e}^{-\\beta V}\\right) +\\mathrm {e}^{-\\beta (D_{4}+2U+4V)}+\\\\\\qquad +\\displaystyle \\mathrm {e}^{-\\beta (D_{2}+V)}\\left(\\mathrm {e}^{-\\beta \\theta _{+}t/2}+\\mathrm {e}^{-\\beta \\theta _{-}t/2}\\right),\\end{array}$ where the $\\theta _{\\pm }$ already were defined in equation (REF ).",
"The extended Hubbard model on diamond chain can be mapped onto an effective extended Hubbard-like model in atomic limit, where it involves additional three-body and four-body interaction.",
"Therefore, the effective extended Hubbard model becomes like: $\\begin{array}{ccl}\\tilde{H}_{i,i+1}&= & -\\tilde{\\mu }n_{i}^{c}+\\tilde{U}n_{i,\\uparrow }^{c}n_{i,\\downarrow }^{c} \\\\& +&\\tilde{V}\\left(n_{i,\\uparrow }^{c}+n_{i,\\downarrow }^{c}\\right)\\left(n_{i+1,\\uparrow }^{c}+n_{i+1,\\downarrow }^{c}\\right) \\\\& +&\\tilde{W}_{3}n_{i,\\uparrow }^{c}n_{i,\\downarrow }^{c}\\left(n_{i+1,\\uparrow }^{c}+n_{i+1,\\downarrow }^{c}\\right)\\\\& +&\\tilde{W}_{4}n_{i,\\uparrow }^{c}n_{i,\\downarrow }^{c}n_{i+1,\\uparrow }^{c}n_{i+1,\\downarrow }^{c},\\end{array}$ where $\\tilde{\\mu }$ could be interpreted as effective chemical potential, in a similar way, $\\tilde{U}$ labels the onsite Coulomb repulsion coupling, meanwhile $\\tilde{V}$ corresponds to the nearest neighbor repulsion coupling, the next terms, $\\tilde{W}_{3}$ will be interpreted as three body interaction term, whereas $W_{4}$ as four body coupling.",
"All the above parameters could be obtained as a function of Hamiltonian (REF ) parameters, when decoration transformation is performed The Boltzmann factor for effective Hubbard model in atomic limit becomes like: $\\tilde{w}_{n_{i}^{c},n_{i+1}^{c}}=A\\,\\mathrm {e}^{-\\beta \\tilde{H}_{i,i+1}},$ the factor $A$ is an additional variable to be determined in terms of the original Hamiltonian (REF ).",
"By the use of decoration transformation, , we will impose the following condition $w_{n_{i}^{c},n_{i+1}^{c}}=\\tilde{w}_{n_{i}^{c},n_{i+1}^{c}},$ using Eq.",
"(REF ) we can find all parameters of the effective Hamiltonian (REF ) and the factor $A$ .",
"To solve the effective Hubbard model with up to four-body coupling, we can use the transfer matrix method, similarly to that used in reference , .",
"Therefore we symmetrize the Hamiltonian by exchanging $i\\rightarrow i+1$ and $i+1\\rightarrow i$ , thus, the transfer matrix becomes symmetric, for our case, this transfer matrix could be expressed by: ${\\bf T}=\\left[\\begin{array}{cccc}w_{0,0} & w_{0,1} & w_{0,1} & w_{0,2}\\\\w_{0,1} & w_{1,1} & w_{1,1} & w_{1,2}\\\\w_{0,1} & w_{1,1} & w_{1,1} & w_{1,2}\\\\w_{0,2} & w_{1,2} & w_{1,2} & w_{2,2}\\end{array}\\right],$ where the Boltzmann factor is given by Eq.",
"(REF ), and using a convenient notation, such as $\\begin{array}{ccl}w_{0,0}(x)&= &\\displaystyle 1+2x\\left(1+\\frac{x^{2}}{z^{4}y^{2}}\\right)\\left(\\frac{1}{\\gamma ^{2}}+\\gamma ^{2}\\right)+ \\\\& +&\\displaystyle x^{2}\\left(\\frac{3}{z^{2}}+\\frac{1}{y^{2}}+\\frac{1}{yz\\varsigma }+\\frac{\\varsigma }{yz}\\right)+\\frac{x^{4}}{y^{4}z^{8}},\\end{array}$ with $ x=\\mathrm {e}^{2\\beta \\mu }$ , $y=\\mathrm {e}^{\\frac{1}{2}\\beta U}$ , $z=\\mathrm {e}^{\\frac{1}{2}\\beta V}$ , $\\gamma =\\mathrm {e}^{\\frac{1}{2}\\beta t}$ and $\\varsigma =\\mathrm {e}^{\\frac{1}{2}\\beta \\sqrt{(U-V)^{2}+16t^{2}}}$ .",
"Whereas the remaining Boltzmann factors could be expressed easily through the function $w_{0,0}(x)$ defined in Eq.",
"(REF ) as follows: $w_{n_{1},n_{2}}(x)= \\frac{x^{(n_{1}+n_{2})/2}}{y^{[n_{1}/2]+[n_{2}/2]}}w_{0,0}\\left(\\frac{x}{z^{n_{1}+n_{2}}}\\right),$ by $[\\ldots ]$ inside braces we mean the least integer of any real number.",
"Thereafter, all elements of matrix (REF ) are well expressed just in terms of $w_{0,0}(x)$ .",
"The Boltzmann factors for the effective Hubbard model (REF ) with three and four-body terms are given by: $\\begin{array}{cl}\\tilde{w}_{0,0}= & A,\\\\\\tilde{w}_{0,1}= & Ar;\\quad r=\\mathrm {e}^{\\beta \\tilde{\\mu }/2},\\\\\\tilde{w}_{0,2}= & Ars;\\quad s=\\mathrm {e}^{-\\beta \\tilde{U}/2},\\\\\\tilde{w}_{1,1}= & Ar^{2}t;\\quad t=\\mathrm {e}^{-\\beta \\tilde{V}},\\\\\\tilde{w}_{1,2}= & Ar^{3}st^{2}u;\\quad u=\\mathrm {e}^{-\\beta \\tilde{W}_{3}/2},\\\\\\tilde{w}_{2,2}= & Ar^{4}s^{2}t^{4}u^{4}v.\\end{array}$ The determinant of transfer matrix becomes a quartic equation of the form $\\mathrm {det}({\\bf T}-\\Lambda )=\\Lambda ^{4}+a_{3}\\Lambda ^{3}+a_{2}\\Lambda ^{2}+a_{1}\\Lambda $ , where the coefficients become: $\\begin{array}{rcl}a_{1}&=&-2w_{0,0}w_{1,1}w_{2,2}+2w_{0,2}^{2}w_{1,1}+2w_{0,1}^{2}w_{2,2}+\\\\&&+2w_{0,0}w_{1,2}^{2}-4w_{0,2}w_{0,1}w_{1,2},\\\\a_{2}&=&-2w_{0,1}^{2}+2w_{0,0}w_{1,1}+w_{0,0}w_{2,2}+\\\\&&+2w_{1,1}w_{2,2}-w_{0,2}^{2}-2w_{1,2}^{2},\\\\a_{3}&=&-w_{0,0}-w_{2,2}-2w_{1,1},\\end{array}$ or alternatively the coefficients also can be expressed, using the effective Hamiltonian parameters, thus, the coefficients of the quartic equation are given by: $\\begin{array}{cl}a_{1}= & A\\left(-2r^{6}t^{5}s^{2}u^{4}v+2r^{6}s^{2}t+2r^{6}s^{2}t^{4}u^{4}v+\\right.\\\\& \\left.+2r^{6}s^{2}t^{4}u^{2}-4r^{6}s^{2}t^{2}u\\right),\\\\a_{2}= & A\\left(-2r^{2}+2r^{2}t+r^{4}s^{2}t^{4}u^{4}v+2r^{6}t^{5}s^{2}u^{4}v-\\right.\\\\& \\left.-r^{4}s^{2}-2r^{6}s^{2}t^{4}u^{2}\\right),\\\\a_{3}= & -A\\left(1+r^{4}s^{2}t^{4}u^{4}v+2r^{2}t\\right).\\end{array}$ Therefore the roots of algebraic quartic equation may be reduced to a cubic one, solutions of which are given in the following way: $\\Lambda _{j}=2\\sqrt{Q}\\cos \\left(\\tfrac{\\phi +2\\pi j}{3}\\right)-\\frac{1}{3}a_{3}\\quad j=0,1,2,$ with $\\begin{array}{ccl}\\phi &=& \\arccos \\left(\\tfrac{R}{\\sqrt{Q^{3}}}\\right),\\\\Q&= & \\displaystyle \\frac{a_{3}^{2}-3a_{2}}{9},\\\\R&= & \\displaystyle \\frac{9a_{2}a_{3}-27a_{1}-2a_{3}^{3}}{54}.\\end{array}$ Furthermore, we also have additional trivial solution $\\Lambda _{3}=0$ , of the algebraic quartic equation.",
"Hence, the largest eigenvalue of the transfer matrix will be $\\Lambda _{0}$ , which is expressed by Eq.",
"(REF ).",
"Once known the largest eigenvalue of transfer matrix, we are able to obtain the partition function straightforwardly, and thereafter the free energy is given by $f=-\\frac{1}{\\beta }\\ln (\\Lambda _{0})$ .",
"Using the free energy per unit cell, we may obtain the thermodynamic properties and how the model behaves when the number particles are changing away from the half filled band."
],
[
"Thermodynamic properties",
"In order to study the thermodynamic properties we will use the exact free energy $\\displaystyle f=-\\frac{1}{\\beta }\\ln (\\Lambda _{0})$ as a starting point.",
"Therefore, we will proceed our discussion of thermodynamic properties as a function of temperature, chemical potential and electronic density.",
"Assuming that we are considering only repulsive onsite Coulomb interaction ($U>0$ ) as well as repulsive Coulomb interaction between nearest neighbor ($V>0$ ).",
"Besides $V<U$ , in order to make more realistic our discussion."
],
[
"The electronic density",
"We will explore the electronic density $\\displaystyle \\rho =-\\frac{\\partial f}{\\partial \\mu }$ , as a function of chemical potential as well as the hopping term.",
"In FIG.",
", we plot the chemical potential $\\mu /U$ versus $t/U$ , for fixed value of temperature $T/U=0.01$ and nearest neighbor coupling $V/U=0.1$ .",
"With gray scale, we mean electronic density between $0<\\rho <2$ , the darkest region corresponds to the lowest electronic density, while the brightest region corresponds to fully filled electrons.",
"Figure: Electronic density per site for T/U=0.01T/U=0.01 and V/U=0.1V/U=0.1, asa function of the t/Ut/U and μ/U\\mu /U, where the black region correspondsto empty particles, while the white region corresponds to fully filledelectrons or empty holes, by different levels of gray regions we representthe intermediate electronic density.In FIG.",
"(a), we plot the electronic density as a function of chemical potential at low temperature for fixed value of hopping term $t/U=0.5$ and $V/U=0.1$ , showing six plateaus at $\\rho =0$ , $1/3$ , $2/3$ , 1, $4/3$ , $5/3$ and 2, this phenomenon vanishes as soon as the temperature increases.",
"Whereas in FIG.",
"(b) we plot the same quantity but for large hopping term $t/U=2$ and $V/U=0.1$ .",
"In this case the plateaus corresponding to electronic densities $\\rho =1/3$ , $2/3$ , $4/3$ and $5/3$ , which turn away from the half filled band $\\rho =1$ .",
"Moreover, the plateaus for densities $2/3$ and $4/3$ become larger, as we can show in FIG.",
", which is also in agreement with FIG.",
"REF .",
"Figure: Acknowledgment"
]
] | 1204.0858 |
[
[
"Connectivity of Large Wireless Networks under A Generic Connection Model"
],
[
"Abstract This paper provides a necessary and sufficient condition for a random network with nodes Poissonly distributed on a unit square and a pair of nodes directly connected following a generic random connection model to be asymptotically almost surely connected.",
"The results established in this paper expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log-normal model to the more generic and more practical random connection model."
],
[
"Introduction",
"Connectivity is one of the most fundamental properties of wireless multi-hop networks [2], [3], [4].",
"A network is said to be connected if there is a path between any pair of nodes.",
"Extensive research has been done on connectivity problems using the well-known random geometric graph and the unit disk connection model, which is usually obtained by randomly and uniformly distributing $n$ vertices in a given area and connecting any two vertices iff (if and only if) their Euclidean distance is smaller than or equal to a given threshold $r(n)$ [5], [3].",
"Significant outcomes have been obtained [2], [6], [3].",
"Particularly, Penrose [7], [8] and Gupta and Kumar [2] proved using different techniques that if the transmission range is set to $r\\left(n\\right)=\\sqrt{(\\log n+c\\left(n\\right))/(\\pi n)}$ , a random network formed by uniformly placing $n$ nodes on a unit-area disk in $\\Re ^{2}$ is asymptotically almost surely (a.a.s.)",
"connected as $n\\rightarrow \\infty $ iff $c\\left(n\\right)\\rightarrow \\infty $ .",
"[An event $\\xi $ is said to occur almost surely if its probability equals to one; an event $\\xi _{n}$ depending on $n$ is said to occur a.a.s.",
"if its probability tends to one as $n\\rightarrow \\infty $ ].",
"Specifically, Penrose's result is based on the fact that in the above random network as $n\\rightarrow \\infty $ the longest edge of the minimum spanning tree converges in probability to the minimum transmission range required for the above network to have no isolated nodes [7], [8], [3].",
"Gupta and Kumar's result is based on a key finding in continuum percolation theory [9]: consider an infinite network with nodes distributed on $\\Re ^{2}$ following a Poisson distribution with density $\\rho $ ; and suppose that a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g\\left(x\\right)$ , independent of the event that another distinct pair of nodes are directly connected.",
"Here, $g:\\Re ^{+}\\rightarrow \\left[0,1\\right]$ satisfies the conditions of rotational invariance, non-increasing monotonicity and integral boundedness [9].",
"Denote the above network by $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g,\\Re ^{2}\\right)$ .",
"As $\\rho \\rightarrow \\infty $ , a.a.s.",
"$\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g,\\Re ^{2}\\right)$ has only a unique infinite component and isolated nodes.",
"The work of Gupta and Kumar is however incomplete to the extent that the above result obtained in continuum percolation theory for an infinite network cannot, counter to intuition, be directly applied to a finite (or asymptotically infinite) network on a finite (or asymptotically infinite) area in $\\Re ^{2}$ [10].",
"In addition to the above work based on the unit disk connection model, there is also limited work [11], [12] dealing with the necessary condition for a random network to be connected under the log-normal shadowing connection model.",
"Under the log-normal shadowing connection model, two nodes are directly connected if the received power at one node from the other node, whose attenuation follows the log-normal model, is greater than a given threshold.",
"The results in [11], [12] however rely on the assumption that the node isolation events are independent.",
"This assumption has only been justified using simulations.",
"Some work also exists on the analysis of the asymptotic distribution of the number of isolated nodes [13], [14], [15], [3] under the assumption of a unit disk model.",
"In [13], Yi et al.",
"considered a total of $n$ nodes distributed independently and uniformly on a unit-area disk and each node may be active independently with some probability $p$ .",
"A node is considered to be isolated if it is not directly connected to any of the active nodes.",
"Using some complicated geometric analysis, they showed that if all nodes have a maximum transmission range $r(n)=\\sqrt{\\left(\\log n+\\xi \\right)/(\\pi pn)}$ for some constant $\\xi $ , the total number of isolated nodes is asymptotically Poissonly distributed with mean $e^{-\\xi }$ .",
"In [14], [15], Franceschetti et al.",
"derived essentially the same result using the Chen-Stein technique.",
"A similar result can also be found in the earlier work of Penrose [3] in a continuum percolation setting.",
"In this paper, we consider a network where all nodes are distributed on a unit square $A\\triangleq [-\\frac{1}{2},\\frac{1}{2}]^{2}$ following a Poisson distribution with known density $\\rho $ and a pair of nodes are directly connected following a generic random connection model $g_{r_{\\rho }}$ , to be rigorously defined in Section .",
"Denote the above network by $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ , where $\\mathcal {X}_{\\rho }$ denotes the set of nodes in the network.",
"We give the sufficient and necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected as $\\rho \\rightarrow \\infty $ .",
"The results in this paper expand the above results on network connectivity to a more generic random connection model, with the unit disk model and the log-normal model being two special cases, thus providing an important link that allows the expansion of other associated results on connectivity to the random connection model.",
"The main contributions of this paper are: Using the Chen-Stein technique [16], [17], we show that the distribution of the number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ asymptotically converges to a Poisson distribution as $\\rho \\rightarrow \\infty $ .",
"This result readily leads to a necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected as $\\rho \\rightarrow \\infty $ ; We show that as $\\rho \\rightarrow \\infty $ , the number of components in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ of unbounded order converges to one.",
"This result, together with the result in [10] that the number of components of finite order $k>1$ in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ asymptotically vanishes as $\\rho \\rightarrow \\infty $ , allows us to conclude that as $\\rho \\rightarrow \\infty $ , a.a.s.",
"there are only a unique unbounded component and isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"The above results allow us to establish that the sufficient and necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected is that there is no isolated node in the network.",
"On that basis, we obtain the asymptotic probability that $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ forms a connected network as $\\rho \\rightarrow \\infty $ and the sufficient and necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected.",
"The rest of this paper is organized as follows: Section introduces the network model and problem setting; Section establishes a necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected; Section first establishes a sufficient condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected and on that basis, together with the results in Section , then establishes the sufficient and necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected; finally Section concludes the paper."
],
[
"Network Model and Problem Setting",
"We consider a network where all nodes are distributed on a unit square $A\\triangleq [-\\frac{1}{2},\\frac{1}{2}]^{2}$ following a Poisson distribution with known density $\\rho $ and a pair of nodes are directly connected following a random connection model, viz.",
"a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g_{r_{\\rho }}(x)\\triangleq g(x/r_{\\rho })$ , where $g:\\left[0,\\infty \\right)\\rightarrow \\left[0,1\\right]$ , independent of the event that another pair of nodes are directly connected.",
"Here $r_{\\rho }=\\sqrt{(\\log \\rho +b)/(C\\rho )}$ and $b$ is a constant.",
"The reason for choosing this particular form of $r_{\\rho }$ is that the analysis becomes nontrivial when $b$ is a constant.",
"Other forms of $r_{\\rho }$ can be accommodated by dropping the assumption that $b$ is constant, i.e.",
"$b$ becomes a function of $\\rho $ , and allowing $b\\rightarrow \\infty $ or $b\\rightarrow -\\infty $ as $\\rho \\rightarrow \\infty $ .",
"The results are rapidly attainable, and we discuss these situations separately in Sections and .",
"The function $g$ is usually required to satisfy the following properties of monotonicity, integral boundedness and rotational invariance [15], [9]Throughout this paper, we use the non-bold symbol, e.g.",
"$x$ , to denote a scalar and the bold symbol, e.g.",
"$x$ , to denote a vector.",
": $& g\\left(x\\right)\\le g\\left(y\\right) & {\\textstyle whenever}\\;\\; x\\ge y\\\\& 0<C\\triangleq \\int _{\\Re ^{2}}g(\\left\\Vert x\\right\\Vert )dx<\\infty $ where $\\left\\Vert \\right\\Vert $ represents the Euclidean norm.",
"We refer readers to [15], [9] for detailed discussions on the random connection model.",
"Equations (REF ) and () allow us to conclude that [10]The following notations and definitions are used throughout the paper: $f\\left(z\\right)=o_{z}(h\\left(z\\right))$ iff $\\lim _{z\\rightarrow \\infty }\\frac{f\\left(z\\right)}{h\\left(z\\right)}=0$ ; $f\\left(z\\right)=\\omega _{z}(h\\left(z\\right))$ iff $h\\left(z\\right)=o_{z}\\left(f\\left(z\\right)\\right)$ ; $f\\left(z\\right)=\\Theta _{z}(h\\left(z\\right))$ iff there exist a sufficiently large $z_{0}$ and two positive constants $c_{1}$ and $c_{2}$ such that for any $z>z_{0}$ , $c_{1}h\\left(z\\right)\\ge f\\left(z\\right)\\ge c_{2}h\\left(z\\right)$ ; $f\\left(z\\right)\\sim _{z}h\\left(z\\right)$ iff $\\lim _{z\\rightarrow \\infty }\\frac{f\\left(z\\right)}{h\\left(z\\right)}=1$ ; $g\\left(x\\right)=o_{x}(1/x^{2})$ However, we require $g$ to satisfy the more restrictive requirement that $g\\left(x\\right)=o_{x}(1/(x^{2}\\log ^{2}x))$ The condition (REF ) is only slightly more restrictive than (REF ) in that for an arbitrarily small positive constant $\\varepsilon $ , $1/x^{2+\\varepsilon }=o_{x}(1/(x^{2}\\log ^{2}x))$ .",
"The more restrictive requirement is needed to ensure that the impact of the truncation effect on connectivity is asymptotically vanishingly small as $\\rho \\rightarrow \\infty $ [10].",
"For convenience we also assume that $g$ has infinite support when necessary.",
"Our results however apply to the situation when $g$ has bounded support, which forms a special case and actually makes the analysis easier.",
"Denote the above network by $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"It is obvious that under a unit disk model where $g(x)=1$ for $x\\le 1$ and $g(x)=0$ for $x>1$ , $r_{\\rho }$ corresponds to the critical transmission range for connectivity [2].",
"Thus the above model incorporates the unit disk model as a special case.",
"A similar conclusion can also be drawn for the log-normal connection model."
],
[
"Necessary Condition for ",
"In this section, as an intermediate step to obtaining the main result, we first and temporarily consider a network with the same node distribution and connection model as $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ however with nodes deployed on a unit torus $A^{T}\\triangleq [-\\frac{1}{2},\\frac{1}{2}]^{2}$ .",
"Denote the network on the torus by $\\mathcal {G}^{T}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"We show that as $\\rho \\rightarrow \\infty $ , the distribution of the number of isolated nodes in $\\mathcal {G}^{T}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ , denoted by $W^{T}$ , asymptotically converges to a Poisson distribution with mean $e^{-b}$ .",
"We then extend the above result to $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"On that basis, we obtain a necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected as $\\rho \\rightarrow \\infty $ ."
],
[
"Distribution of the number of isolated nodes on a torus",
"In this subsection, we analyze the distribution of the number of isolated nodes in $\\mathcal {G}^{T}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"The use of a toroidal rather than planar region as a tool in analyzing network properties is well known [3].",
"The unit torus $A^{T}=[-\\frac{1}{2},\\frac{1}{2}]^{2}$ that is commonly used in random geometric graph theory is essentially the same as a unit square $A=[-\\frac{1}{2},\\frac{1}{2}]^{2}$ except that the distance between two points on a torus is defined by their toroidal distance, instead of Euclidean distance.",
"Thus a pair of nodes in $\\mathcal {G}^{T}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ , located at $x_{1}$ and $x_{2}$ respectively, are directly connected with probability $g_{r_{\\rho }}(\\left\\Vert x_{1}-x_{2}\\right\\Vert ^{T})$ where $\\left\\Vert x_{1}-x_{2}\\right\\Vert ^{T}$ denotes the toroidal distance between the two nodes.",
"For a unit torus $A^{T}=[-\\frac{1}{2},\\frac{1}{2}]^{2}$ , the toroidal distance is given by [3]: $\\left\\Vert x_{1}-x_{2}\\right\\Vert ^{T}\\triangleq \\min \\lbrace \\left\\Vert x_{1}+z-x_{2}\\right\\Vert :z\\in \\mathbb {Z}^{2}\\rbrace $ In this section, whenever the difference between a torus and a square affects the parameter being discussed, we use superscript $^{T}$ to mark the parameter in a torus while the unmarked parameter is associated with a square.",
"We note the following relation between toroidal distance and Euclidean distance on a square area centered at the origin: $\\left\\Vert x_{1}-x_{2}\\right\\Vert ^{T}\\le \\left\\Vert x_{1}-x_{2}\\right\\Vert \\;\\;\\textrm {and}\\;\\;\\left\\Vert x\\right\\Vert ^{T}=\\left\\Vert x\\right\\Vert $ which will be used in the later analysis.",
"The main result of this subsection is given in Theorem REF .",
"Theorem 1 The distribution of the number of isolated nodes in $\\mathcal {G}^{T}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ converges to a Poisson distribution with mean $e^{-b}$ as $\\rho \\rightarrow \\infty $ .",
"See Appendix I."
],
[
"Distribution of the number of isolated nodes on a square",
"We now consider the asymptotic distribution of the number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"Let $W$ be the number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ and $W^{E}$ be the number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ due to the boundary effect.",
"Using the coupling technique, it can be readily shown that $W=W^{E}+W^{T}$ [10].",
"Using the above equation, Theorem REF , Lemma 2 in [10]Let $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ be a network with nodes Poissonly distributed on a square $A_{\\frac{1}{r_{\\rho }}}=[-\\frac{1}{2r_{\\rho }},\\frac{1}{2r_{\\rho }}]^{2}$ with density $\\lambda =(\\log \\rho +b)/C$ and a pair of nodes separated by an Euclidean distance $x$ are directly connected with probability $g(x)$ , independent of other connections.",
"Results in [10] are derived for $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ .",
"By proper scaling, it is straightforward to extend the results for $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ to $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"Therefore we ignore the difference., which showed that $\\lim _{\\rho \\rightarrow \\infty }\\Pr (W^{E}=0)=1$ , and Slutsky's theorem [18], the following result on the asymptotic distribution of $W$ can be readily obtained.",
"Theorem 2 The distribution of the number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ converges to a Poisson distribution with mean $e^{-b}$ as $\\rho \\rightarrow \\infty $ .",
"Corollary REF follows immediately from Theorem REF .",
"Corollary 3 As $\\rho \\rightarrow \\infty $ , the probability that there is no isolated node in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ converges to $e^{-e^{-b}}$ .",
"Now we relax requirement that $b$ is a constant to obtain a necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected.",
"Specifically, consider the situation when $b\\rightarrow -\\infty $ or $b\\rightarrow \\infty $ as $\\rho \\rightarrow \\infty $ .",
"Note that the property that the network $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ has no isolated node is an increasing property (For an arbitrary network, a particular property is termed increasing if the property is preserved when more connections (edges) are added into the network.).",
"Using a coupling technique similar to that used in [15] and with a few simple steps (omitted), the following theorem and corollary can be obtained, which form a major contribution of this paper: Theorem 4 In $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ , if $b\\rightarrow \\infty $ as $\\rho \\rightarrow \\infty $ , a.a.s.",
"there is no isolated node in the network; if $b\\rightarrow -\\infty $ as $\\rho \\rightarrow \\infty $ , a.a.s.",
"the network has at least one isolated node.",
"Corollary 5 $b\\rightarrow \\infty $ is a necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected as $\\rho \\rightarrow \\infty $ ."
],
[
"Sufficient Condition for ",
"In this section, we continue to investigate the sufficient condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected.",
"In [10] we showed that vanishing of components of finite order $k>1$ in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g,\\Re ^{2}\\right)$ as $\\rho \\rightarrow \\infty $ (as shown in [9]) does not necessarily carry the conclusion that components of finite order $k>1$ in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ also vanish as $\\rho \\rightarrow \\infty $ , contrary perhaps to intuition.",
"Then, we presented a result for the vanishing of components of finite order $k>1$ in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ as $\\rho \\rightarrow \\infty $ to fill this theoretical gap [10].",
"On the basis of the above results, we shall further demonstrate in this section that a.a.s.",
"the number of unbounded components in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ is one as $\\rho \\rightarrow \\infty $ .",
"A sufficient condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected readily follows.",
"In [9], it was shown that there can be at most one unbounded component in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g,\\Re ^{2}\\right)$ .",
"However due to the truncation effect [10], it appears difficult to establish such a conclusion using [9].",
"Indeed differently from $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g,\\Re ^{2}\\right)$ in which an unbounded component may exist for a finite $\\rho $ , it can be easily shown that for any finite $\\rho $ , $\\Pr \\left(\\left|\\mathcal {X}_{\\rho }\\right|<\\infty \\right)=1$ , i.e.",
"the total number of nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ is almost surely finite.",
"It then follows that for any finite $\\rho $ almost surely there is no unbounded component in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"In this paper, we solve the above conceptual difficulty involving use of the term “unbounded component” by considering the number of components in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ of order greater than $M$ , denoted by $\\xi _{>M}$ , where $M$ is an arbitrarily large positive integer.",
"We then show that $\\lim _{M\\rightarrow \\infty }\\lim _{\\rho \\rightarrow \\infty }\\Pr (\\xi _{>M}=1)=1$ .",
"The analytical result is summarized in the following theorem, which forms a further major contribution of this paper: Theorem 6 As $\\rho \\rightarrow \\infty $ , a.a.s.",
"the number of unbounded components in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ is one.",
"See Appendix II Remark 7 Proof of the type of results in Theorem REF usually requires some complicated geometric analysis.",
"Particularly the proof of Lemma REF in Appendix II, which forms a foundation of the proof of Theorem REF , needs sophisticated geometric analysis.",
"In this paper, we omitted the proof of Lemma REF because the proof is exactly the same as the proof of Theorem REF , which in turn relies on some results established in [10].",
"We refer interested readers to the proof of Theorem 1 in [10] for techniques on handling geometric obstacles involved in analyzing the boundary effect and to the proof of Theorem 4 in [10] for techniques on handling geometric obstacles involved in analyzing the number of components in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"An implication of Theorem REF is that for an arbitrarily small positive constant $\\varepsilon $ , there exists large positive constants $M_{0}$ and $\\rho _{0}$ such that for all $M>M_{0}$ and $\\rho >\\rho _{0}$ , $\\Pr (\\xi _{>M}=1)>1-\\varepsilon $ .",
"From (REF ) in Appendix II, it can further be concluded that for a particular positive integer $M$ and an arbitrarily small positive constant $\\varepsilon $ , there exists $\\rho _{0}$ such that for all $\\rho >\\rho _{0}$ , $\\Pr (\\xi _{>M}=1)>1-\\frac{e^{-\\left(M+1\\right)b}}{\\left(M+1\\right)!",
"}-\\varepsilon $ The following corollary can be obtained from [10] and Theorem REF : Corollary 8 As $\\rho \\rightarrow \\infty $ , a.a.s.",
"$\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ forms a connected network iff there is no isolated node in it.",
"Let $\\xi $ be the total number of components in $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"It is clear that $\\xi =\\xi _{1}+\\sum _{k=2}^{M}\\xi _{k}+\\xi _{>M}$ , where $\\xi _{k}$ is the number of components of order $k$ .",
"Noting that $\\xi =1$ iff $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ forms a connected network, it suffices to show that $\\lim _{\\rho \\rightarrow \\infty }\\Pr \\left(\\xi =1|\\xi _{1}=0\\right)=1$ .$ $ We observe that $& \\Pr \\left(\\xi =1,\\xi _{1}=0\\right)\\nonumber \\\\\\ge & \\Pr (\\xi _{1}=0,\\sum _{k=2}^{M}\\xi _{k}=0,\\xi _{>M}=1)\\nonumber \\\\= & \\Pr (\\xi _{1}=0)-(\\Pr (\\overline{\\sum _{k=2}^{M}\\xi _{k}=0})+\\Pr (\\overline{\\xi _{>M}=1}))$ where in (REF ) $\\overline{\\xi _{>M}=1}$ represents the complement of the event $\\xi _{>M}=1$ and (REF ) results as a consequence of the union bound.",
"Further note that (REF ) is valid for any value of $M$ and that $\\Pr \\left(\\xi _{1}=0\\right)$ converges to a non-zero constant $e^{-e^{-b}}$ as $\\rho \\rightarrow \\infty $ (Theorem REF ).",
"Using the above results, [10] which showed that $\\lim _{\\rho \\rightarrow \\infty }\\Pr (\\sum _{k=2}^{M}\\xi _{k}=0)=1$ , and (REF ), and following a few simple steps (omitted), it can be shown that for an arbitrarily small positive constant $\\varepsilon $ , by choosing $M$ to be sufficiently large, there exists $\\rho _{0}$ such that for all $\\rho >\\rho _{0}$ , $\\Pr \\left(\\xi =1|\\xi _{1}=0\\right)>1-\\varepsilon $ .",
"As an easy consequence of Theorem REF and Corollary REF , the following theorem can be established: Theorem 9 As $\\rho \\rightarrow \\infty $ , the probability that $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ forms a connected network converges to $e^{-e^{-b}}$ .",
"Using the above theorem and a similar analysis as that leading to Theorem REF and Corollary REF , the following theorem on the sufficient and necessary condition for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ to be a.a.s.",
"connected can be obtained: Theorem 10 As $\\rho \\rightarrow \\infty $ , $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ is a.a.s.",
"connected iff $b\\rightarrow \\infty $ ; $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ is a.a.s.",
"disconnected iff $b\\rightarrow -\\infty $ ."
],
[
"Conclusion and Further Work",
"Following the seminal work of Penrose [5], [3] and Gupta and Kumar [2] on the asymptotic connectivity of large-scale random networks with Poisson node distribution and under the unit disk model, there is general expectation that there is a range of connection functions for which the above results [5], [3], [2] obtained assuming the unit disk model can carry over.",
"However, for quite a long time, both the asymptotic laws that the network should follow and the conditions on the connection function required for the network to be a.a.s.",
"connected under a more generic setting have been unknown.",
"In this paper, we filled in the gaps by providing the sufficient and necessary condition for a network with nodes Poissonly distributed on a unit square and following a generic random connection model to be a.a.s.",
"connected as $\\rho \\rightarrow \\infty $ .",
"The conditions on the connection function required in order for the above network to be a.a.s.",
"connected were also provided.",
"Therefore, the results in the paper constitute a significant advance of the earlier work by Penrose [5], [3] and Gupta and Kumar [2] from the unit disk model to the more generic random connection model and bring models addressed by theoretical research closer to reality.",
"However, there remain significant challenges ahead.",
"The results in this paper rely on three main assumptions: a) the connection function $g$ is isotropic, b) the random events underpinning generation of a connection are independent, c) nodes are Poissonly distributed.",
"We conjecture that assumption a) is not a critical assumption, i.e.",
"under some mild conditions, e.g.",
"nodes are independently and randomly oriented, assumption a) can be removed while our results are still valid.",
"It is part of our future work plan to validate the conjecture.",
"Our results however critically rely on assumption b), which is not necessarily valid in some real networks due to channel correlation and interference, where the latter effect makes the connection between a pair of nodes dependent on the locations and activities of other nearby nodes.",
"In [19] we have done some preliminary work on network connectivity considering the impact of interference.",
"The work essentially uses a de-coupling approach to solve the challenges of connection correlation caused by interference and suggests that when some realistic constraints are considered, i.e.",
"carrier-sensing, the connectivity results will be very close to those obtained under a unit disk model.",
"This conclusion is in contrast with that [20] obtained under an ALOHA multiple-access protocol.",
"A more thorough investigation is yet to be done.",
"The major obstacle in dealing with the impact of channel correlation is that there is no widely accepted model in the wireless communication community capturing the impact of channel correlation on connections.",
"Finally, it is a logical move after our work to consider connectivity of networks with nodes following a generic distribution other than Poisson.",
"It is part of our future work plan to tackle the problem."
],
[
"Appendix I: Proof of Theorem ",
"Our proof relies on the use of the Chen-Stein bound [16], [17].",
"We first establish some preliminary results that allow us to use the Chen-Stein bound for the analysis of number of isolated nodes in $\\mathcal {G}^{T}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"Divide the unit torus into $m^{2}$ non-overlapping squares each with size $\\frac{1}{m^{2}}$ .",
"Denote the $i_{m}^{th}$ square by $A_{i_{m}}$ .",
"Define two sets of indicator random variables $J_{i_{m}}^{T}$ and $I_{i_{m}}^{T}$ with $i_{m}\\in \\Gamma _{m}\\triangleq \\lbrace 1,\\ldots m^{2}\\rbrace $ , where $J_{i_{m}}^{T}=1$ iff there exists exactly one node in $A_{i_{m}}$ , otherwise $J_{i_{m}}^{T}=0$ ; $I_{i_{m}}^{T}=1$ iff there is exactly one node in $A_{i_{m}}$ and that node is isolated, $I_{i_{m}}^{T}=0$ otherwise.",
"Obviously $J_{i_{m}}^{T}$ is independent of $J_{j_{m}}^{T},j_{m}\\in \\Gamma _{m}\\backslash \\left\\lbrace i_{m}\\right\\rbrace $ .",
"Denote the center of $A_{i_{m}}^{T}$ by $x_{i_{m}}$ and without loss of generality we assume that when $J_{i_{m}}^{T}=1$ , the associated node in $A_{i_{m}}$ is at $x_{i_{m}}$In this paper we are mainly concerned with the case that $m\\rightarrow \\infty $ , i.e.",
"the size of the square is vanishingly small.",
"Therefore the actual position of the node in the square is not important..",
"Observe that for any fixed $m$ , the values of $\\Pr \\left(I_{i_{m}}^{T}=1\\right)$ and $\\Pr \\left(J_{i_{m}}^{T}=1\\right)$ do not depend on the particular index $i_{m}$ on a torus.",
"However both the set of indices $\\Gamma _{m}$ and a particular index $i_{m}$ depend on $m$ .",
"As $m$ changes, the square associated with $I_{i_{m}}^{T}$ and $J_{i_{m}}^{T}$ also changes.",
"Remark 11 In this paper, we are only interested in the limiting values of various parameters associated with a sub-square as $m\\rightarrow \\infty $ .",
"Also because of the consideration of a torus, the value of a particular index $i_{m}$ does not affect the discussion of the associated parameters, i.e.",
"these parameters $I_{i_{m}}^{T}$ and $J_{i_{m}}^{T}$ do not depend on $i_{m}$ .",
"Therefore in the following, we omit some straightforward discussions on the convergence of various parameters, e.g.",
"$i_{m}$ , $x_{i_{m}}$ , $I_{i_{m}}^{T}$ and $J_{i_{m}}^{T}$ , as $m\\rightarrow \\infty $ .",
"Without causing ambiguity, we drop the explicit dependence on $m$ in our notations for convenience.",
"As an easy consequence of the Poisson node distribution, $\\Pr (J_{i}^{T}=1)\\sim _{m}\\rho /m^{2}$ .",
"Using [9], $\\Pr (I_{i}^{T}=1)=\\Pr (I_{i}^{T}=1|J_{i}^{T}=1)\\Pr (J_{i}^{T}=1)$ and the property of a torus (see also [10]), it can be shown that $\\Pr (I_{i}^{T}=1) & \\sim _{m} & \\frac{\\rho }{m^{2}}e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx}\\nonumber \\\\& = & \\frac{\\rho }{m^{2}}e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x\\right\\Vert ^{T}}{r_{\\rho }})dx}$ Now consider the event $I_{i}^{T}I_{j}^{T}=1,i\\ne j$ , conditioned on the event that $J_{i}^{T}J_{j}^{T}=1$ , meaning that both nodes having been placed inside $A_{i}$ and $A_{j}$ respectively are isolated.",
"Following the same steps leading to (REF ), it can be shown that $& \\lim _{m\\rightarrow \\infty }\\Pr (I_{i}^{T}I_{j}^{T}=1|J_{i}^{T}J_{j}^{T}=1)\\nonumber \\\\= & (1-g(\\frac{\\left\\Vert x_{i}-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))\\exp [-\\int _{A}\\rho (g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})\\nonumber \\\\+ & g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }})-g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))dx]$ where the term $(1-g(\\frac{\\left\\Vert x_{i}-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))$ is due to the requirement that the two nodes located inside $A_{i}$ and $A_{j}$ cannot be directly connected given that they are both isolated nodes.",
"Observe also that $\\Pr (I_{i}^{T}I_{j}^{T}=1)=\\Pr (J_{i}^{T}J_{j}^{T}=1)\\Pr (I_{j}^{T}I_{j}^{T}=1|J_{i}^{T}J_{j}^{T}=1)$ .",
"Now using the above equation, (REF ) and (REF ), it can be established that $& \\frac{\\Pr (I_{i}^{T}I_{j}^{T}=1)}{\\Pr (I_{i}^{T}=1)\\Pr (I_{j}^{T}=1)}\\nonumber \\\\\\sim _{m} & (1-g(\\frac{\\left\\Vert x_{i}-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))e^{\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }})dx}$ Now we are ready to use the Chen-Stein bound to prove Theorem REF .",
"Particularly, we will show using the Chen-Stein bound that $W^{T}=\\lim _{m\\rightarrow \\infty }\\sum _{i\\in \\Gamma _{m}}I_{i}^{T}$ asymptotically converges to a Poisson distribution with mean $e^{-b}$ as $\\rho \\rightarrow \\infty $ .",
"The following theorem gives a formal statement of the Chen-Stein bound: Theorem 12 [17] For a set of indicator random variables $I_{i},\\; i\\in \\Gamma $ , define $W\\triangleq \\sum _{i\\in \\Gamma }I_{i}$ , $p_{i}\\triangleq E\\left(I_{i}\\right)$ and $\\eta \\triangleq E\\left(W\\right)$ .",
"For any choice of the index set $\\Gamma _{s,i}\\subset \\Gamma $ , $\\Gamma _{s,i}\\cap \\lbrace i\\rbrace =\\lbrace \\textrm {ￜ}\\rbrace $ , $& & d_{TV}(\\mathcal {L}\\left(W\\right),Po\\left(\\eta \\right))\\\\& \\le & \\sum _{i\\in \\Gamma }[(p_{i}^{2}+p_{i}E(\\sum _{j\\in \\Gamma _{s,i}}I_{j}))]\\min (1,\\frac{1}{\\eta })\\\\& + & \\sum _{i\\in \\Gamma }E(I_{i}\\sum _{j\\in \\Gamma _{s,i}}I_{j})\\min (1,\\frac{1}{\\eta })\\\\& + & \\sum _{i\\in \\Gamma }E|E\\lbrace I_{i}|(I_{j},j\\in \\Gamma _{w,i})\\rbrace -p_{i}|\\min (1,\\frac{1}{\\eta })$ where $\\mathcal {L}\\left(W\\right)$ denotes the distribution of $W$ , $Po\\left(\\eta \\right)$ denotes a Poisson distribution with mean $\\eta $ , $\\Gamma _{w,i}=\\Gamma \\backslash \\left\\lbrace \\Gamma _{s,i}\\cup \\lbrace i\\rbrace \\right\\rbrace $ and $d_{TV}$ denotes the total variation distance.",
"The total variation distance between two probability distributions $\\alpha $ and $\\beta $ on $\\mathbb {Z}^{+}$ is given by $d_{TV}\\left(\\alpha ,\\beta \\right)\\triangleq \\sup \\left\\lbrace \\left|\\alpha \\left(A\\right)-\\beta \\left(A\\right)\\right|:A\\subset \\mathbb {Z}^{+}\\right\\rbrace $ .",
"For convenience, we separate the bound in Theorem REF into three terms $b_{1}\\min (1,\\frac{1}{\\eta })$ , $b_{2}\\min (1,\\frac{1}{\\eta })$ and $b_{3}\\min (1,\\frac{1}{\\eta })$ where $b_{1}\\triangleq & \\sum _{i\\in \\Gamma }[(p_{i}^{2}+p_{i}E(\\sum _{j\\in \\Gamma _{s,i}}I_{j}))]\\\\b_{2}\\triangleq & \\sum _{i\\in \\Gamma }E(I_{i}\\sum _{j\\in \\Gamma _{s,i}}I_{j})\\\\b_{3}\\triangleq & \\sum _{i\\in \\Gamma }E|E\\lbrace I_{i}|(I_{j},j\\in \\Gamma _{w,i})\\rbrace -p_{i}|$ The set of indices $\\Gamma _{s,i}$ is often chosen to contain all those $j$ , other than $i$ , for which $I_{j}$ is “strongly” dependent on $I_{i}$ and the set $\\Gamma _{w,i}$ often contains all other indices apart from $i$ for which $I_{j}$ is at most “weakly” dependent on $I_{i}$ [16].",
"Remark 13 A main challenge in using the Chen-Stein bound to prove Theorem REF is that under the random connection model, the two events $I_{i}$ and $I_{j}$ may be correlated even when $x_{i}$ and $x_{j}$ are separated by a very large Euclidean distance.",
"Therefore the dependence structure is global, which significantly increases the complexity of the analysis.",
"In comparison, in applications where the dependence structure is local, by a suitable choice of $\\Gamma _{s,i}$ the $b_{3}$ term can be easily made to be 0 and the evaluation of the $b_{1}$ and $b_{2}$ terms involves the computation of the first two moments of $W$ only, which can often be achieved relatively easily.",
"An example is a random geometric network under the unit disk model.",
"If $\\Gamma _{s,i}$ is chosen to be a neighborhood of $i$ containing indices of all nodes whose distance to node $i$ is less than or equal to twice the transmission range, the $b_{3}$ term is easily shown to be 0.",
"It can then be readily shown that the $b_{1}$ and $b_{2}$ terms approach 0 as the neighborhood size of a node becomes vanishingly small compared to the overall network size as $\\rho \\rightarrow \\infty $ [14].",
"However this is certainly not the case for the random connection model.",
"Remark 14 The key idea involved using the Chen-Stein bound to prove Theorem REF is constructing a neighborhood of a node, i.e.",
"$\\Gamma _{s,i}$ in Theorem REF , such that a) the size of the neighborhood becomes vanishingly small compared with $A$ as $\\rho \\rightarrow \\infty $ .",
"This is required for the $b_{1}$ and $b_{2}$ terms to approach 0 as $\\rho \\rightarrow \\infty $ ; b) a.a.s.",
"the neighborhood contains all nodes that may have a direct connection with the node.",
"This is required for the $b_{3}$ term to approach 0 as $\\rho \\rightarrow \\infty $ .",
"Such a neighborhood is defined in the next paragraph.",
"Let $D^{T}\\left(x_{i},r\\right)\\triangleq \\lbrace x\\in A:\\left\\Vert x-x_{i}\\right\\Vert ^{T}\\le r\\rbrace $ and when $x_{i}$ is not within $r$ of the border of $A$ , $ $$D^{T}\\left(x_{i},r\\right)$ becomes the same as $D\\left(x_{i},r\\right)$ where $D\\left(x_{i},r\\right)\\triangleq \\lbrace x\\in A:\\left\\Vert x-x_{i}\\right\\Vert \\le r\\rbrace $ .",
"Further define the neighborhood of an index $i\\in \\Gamma $ as $\\Gamma _{s,i}\\triangleq \\lbrace j:x_{j}\\in D^{T}\\left(x_{i},2r_{\\rho }^{1-\\epsilon }\\right)\\rbrace \\backslash \\lbrace i\\rbrace $ and define the non-neighborhood of the index $i$ as $\\Gamma _{w,i}\\triangleq \\lbrace j:x_{j}\\notin D^{T}(x_{i},2r_{\\rho }^{1-\\epsilon })\\rbrace $ where $\\epsilon $ is a small positive constant and $\\epsilon \\in (0,\\frac{1}{2})$ .",
"It can be shown that $\\left|\\Gamma _{s,i}\\right|=m^{2}4\\pi r_{\\rho }^{2-2\\epsilon }+o_{m}(m^{2}4\\pi r_{\\rho }^{2-2\\epsilon })$ Note that in Theorem REF , $p_{i}=E(I_{i}^{T})$ and $E(I_{i}^{T})$ has been given in (REF ).",
"Further, as an easy consequence of (REF ) and [10] which showed that $\\lim _{\\rho \\rightarrow \\infty }E(W^{T})=\\lim _{\\rho \\rightarrow \\infty }\\rho e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x\\right\\Vert ^{T}}{r_{\\rho }})dx}=e^{-b}$ $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\eta =e^{-b}$ .",
"Using (REF ), $p_{i}=E(I_{i}^{T})$ and (REF ), it follows that $\\lim _{m\\rightarrow \\infty }m^{2}p_{i} & = & \\rho e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x\\right\\Vert ^{T}}{r_{\\rho }})dx}\\\\\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }m^{2}p_{i} & = & e^{-b}$ Next we shall evaluate the $b_{1}$ , $b_{2}$ and $b_{3}$ terms in the following three subsections separately and show that all three terms converge to 0 as $\\rho \\rightarrow \\infty $ ."
],
[
"An Evaluation of the $b_{1}$ Term",
"It can be shown that (following the equation, detailed explanations are given) $& \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\sum _{i\\in \\Gamma }(p_{i}^{2}+p_{i}E(\\sum _{j\\in \\Gamma _{s,i}}I_{j}^{T}))\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }m^{2}p_{i}E(\\sum _{j\\in \\Gamma _{s,i}\\cup \\left\\lbrace i\\right\\rbrace }I_{j}^{T})\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }(m^{2}p_{i})^{2}4\\pi r_{\\rho }^{2-2\\epsilon }\\\\= & \\lim _{\\rho \\rightarrow \\infty }4\\pi (\\rho e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx})^{2}(\\frac{\\log \\rho +b}{C\\rho })^{1-\\epsilon }\\\\= & 4\\pi e^{-2b}\\lim _{\\rho \\rightarrow \\infty }(\\frac{\\log \\rho +b}{C\\rho })^{1-\\epsilon }=0$ where (REF ) is used in obtaining (REF ); (REF ) and (REF ) are used in obtaining (); and (REF ) and () are used in obtaining ().",
"Therefore $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }b_{1}=0$ ."
],
[
"An Evaluation of the $b_{2}$ Term",
"For the $b_{2}$ term, assume that $\\rho $ is sufficiently large such that $\\frac{1}{2r_{\\rho }}>>2r_{\\rho }^{-\\epsilon }$ and let $A_{\\frac{1}{r_{\\rho }}}=[-\\frac{1}{2r_{\\rho }},\\frac{1}{2r_{\\rho }}]^{2}$ .",
"Using (REF ) in the first step; and first using some translation and scaling operations and then using (REF ) in the last step, equation (REF ) can be obtained.",
"Figure: NO_CAPTIONLetting $\\lambda \\triangleq \\frac{\\log \\rho +b}{C}$ for convenience, noting that (using (REF ) and ()) $\\lim _{\\rho \\rightarrow \\infty }\\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x\\right\\Vert ^{T})dx=\\lim _{\\rho \\rightarrow \\infty }\\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x-y\\right\\Vert ^{T})dx=C$ and that $1-g(\\left\\Vert y\\right\\Vert ^{T})\\le 1$ , it can further be shown following (REF ) that as $\\rho \\rightarrow \\infty $ , $& \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\sum _{i\\in \\Gamma }E(I_{i}^{T}\\sum _{j\\in \\Gamma _{s,i}}I_{j}^{T})\\nonumber \\\\\\le & e^{-2b}\\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D(0,2r_{\\rho }^{-\\epsilon })}e^{\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x\\right\\Vert ^{T})g(\\left\\Vert x-y\\right\\Vert ^{T})dx}dy$ In the following paragraphs, we will show that the right hand side of (REF ) converges to 0 as $\\rho \\rightarrow \\infty $ .",
"Using (REF ) and (), we assert that there exists a positive constant $r$ such that $g\\left(r^{-}\\right)(1-g\\left(r^{+}\\right))>0$ where $g\\left(r^{-}\\right)\\triangleq \\lim _{x\\rightarrow r^{-}}g\\left(x\\right)$ and $g\\left(r^{+}\\right)\\triangleq \\lim _{x\\rightarrow r^{+}}g\\left(x\\right)$ .",
"Indeed if $g$ is a continuous function, any positive constant $r$ with $g\\left(r\\right)>0$ satisfies the requirement; if $g$ is a discontinuous function, e.g.",
"a unit disk model, by choosing $r$ to be the transmission range, $g\\left(r^{-}\\right)\\left(1-g\\left(r^{+}\\right)\\right)=1$ .",
"In the following discussion we assume that $\\rho $ is sufficiently large such that $\\frac{1}{2r_{\\rho }}>>2r_{\\rho }^{-\\epsilon }>>r$ .",
"It can be shown using (REF ), () and (REF ) that for $y\\in D(0,2r_{\\rho }^{-\\epsilon })$ , $& \\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x\\right\\Vert ^{T})g(\\left\\Vert x-y\\right\\Vert ^{T})dx\\nonumber \\\\& \\le \\int _{\\Re ^{2}}g(\\left\\Vert x\\right\\Vert )g(\\left\\Vert x-y\\right\\Vert )dx\\nonumber \\\\& =C-\\int _{\\Re ^{2}}g(\\left\\Vert x\\right\\Vert )(1-g(\\left\\Vert x-y\\right\\Vert ))dx\\nonumber \\\\& \\le C-\\int _{D(0,r)\\backslash D(y,r)}g(\\left\\Vert x\\right\\Vert )(1-g(\\left\\Vert x-y\\right\\Vert ))dx\\nonumber \\\\& \\le C-g\\left(r^{-}\\right)(1-g\\left(r^{+}\\right))|D(0,r)\\backslash D(y,r)|$ Let $f\\left(x\\right)\\triangleq \\pi r^{2}-2r^{2}\\arcsin (\\sqrt{1-x^{2}/(4r^{2})})+rx\\sqrt{1-x^{2}/(4r^{2})}$ .",
"Using some simple geometric analysis, it can be shown that when $\\left\\Vert y\\right\\Vert >2r$ , $|D(0,r)\\backslash D(y,r)|=\\pi r^{2}$ ; and when $\\left\\Vert y\\right\\Vert \\le 2r$ , $|D(0,r)\\backslash D(y,r)|=f(\\left\\Vert y\\right\\Vert )$ .",
"Further, using the definition of $f\\left(x\\right)$ , it can be shown that when $\\left\\Vert y\\right\\Vert \\le r$ , $|D(0,r)\\backslash D(y,r)|\\ge \\sqrt{3}r\\left\\Vert y\\right\\Vert $ ; and when $\\left\\Vert y\\right\\Vert >r$ , $|D(0,r)\\backslash D(y,r)|\\ge (\\frac{\\pi }{3}+\\frac{\\sqrt{3}}{2})r^{2}$ .",
"For convenience, let $c_{1}\\triangleq g\\left(r^{-}\\right)(1-g\\left(r^{+}\\right))\\sqrt{3}r$ and $c_{2}\\triangleq g\\left(r^{-}\\right)(1-g\\left(r^{+}\\right))\\left(\\frac{\\pi }{3}+\\frac{\\sqrt{3}}{2}\\right)r^{2}$ .",
"Noting that $g\\left(r^{-}\\right)(1-g\\left(r^{+}\\right))>0$ , $c_{1}$ and $c_{2}$ are positive constants, independent of both $y$ and $\\rho $ .",
"As a result of (REF ) and the above inequalities on $|D(0,r)\\backslash D(y,r)|$ , it follows that $& \\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D(0,2r_{\\rho }^{-\\epsilon })}e^{\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x\\right\\Vert ^{T})g(\\left\\Vert x-y\\right\\Vert ^{T})dx}dy\\nonumber \\\\\\le & \\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D\\left(0,r\\right)}e^{\\lambda (C-c_{1}\\left\\Vert y\\right\\Vert )}dy\\nonumber \\\\+ & \\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D\\left(0,2r_{\\rho }^{-\\epsilon }\\right)\\backslash D\\left(0,r\\right)}e^{\\lambda \\left(C-c_{2}\\right)}dy$ For the first summand in the above equation, it can be shown that: $& \\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D\\left(0,r\\right)}e^{\\lambda (C-c_{1}\\left\\Vert y\\right\\Vert )}dy\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\frac{\\log \\rho +b}{C\\rho }\\int _{0}^{r}2\\pi ye^{\\frac{\\log \\rho +b}{C}(C-c_{1}y)}dy=0$ For the second summand in (REF ), by choosing $\\varepsilon <\\frac{c_{2}}{C}$ and using (REF ), it follows that $& \\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D(0,2r_{\\rho }^{-\\epsilon })\\backslash D\\left(0,r\\right)}e^{\\lambda (C-c_{2})}dy\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\frac{e^{b(1-\\frac{c_{2}}{C})}}{C}\\times \\frac{\\log \\rho +b}{\\rho ^{\\frac{c_{2}}{C}}}\\times \\pi (4r_{\\rho }^{-2\\epsilon }-r^{2})=0$ Combining (REF ), (REF ) and (REF ), it follows that $\\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda }{\\rho }\\int _{D(0,2r_{\\rho }^{-\\epsilon })}e^{\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x\\right\\Vert ^{T})g(\\left\\Vert x-y\\right\\Vert ^{T})dx}dy=0$ As a result of (REF ) and the above equation: $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }b_{2}=0$ ."
],
[
"An Evaluation of the $b_{3}$ Term",
"We first obtain an analytical expression of the term $E\\lbrace I_{i}|(I_{j},j\\in \\Gamma _{w,i})\\rbrace $ in $b_{3}$ .",
"Using the same procedure that results in (REF ), it can be obtained that (for convenience we use $g_{i}$ for $g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})$ and use $g_{ij}$ for $g(\\frac{\\left\\Vert x_{i}-x_{j}\\right\\Vert ^{T}}{r_{\\rho }})$ in the following equation): $& \\lim _{m\\rightarrow \\infty }\\frac{Pr(I_{i}^{T}=1,I_{j}^{T}=1,I_{k}^{T}=0)}{Pr(I_{i}^{T}=1)Pr(I_{j}^{T}=1,I_{k}^{T}=0)}\\nonumber \\\\= & \\lim _{m\\rightarrow \\infty }\\frac{Pr(I_{i}^{T}=1,I_{j}^{T}=1)-Pr(I_{i}^{T}=1,I_{j}^{T}=1,I_{k}^{T}=1)}{Pr(I_{i}^{T}=1)(Pr(I_{j}^{T}=1)-Pr(I_{j}^{T}=1,I_{k}^{T}=1))}\\nonumber \\\\\\sim _{m} & \\left(1-g_{ij}\\right)e^{\\int _{A}\\rho g_{i}g_{j}dx}\\nonumber \\\\\\times & \\frac{1-\\frac{\\rho }{m^{2}}(1-g_{ik})(1-g_{kj})e^{-\\int _{A}\\rho (g_{k}-g_{i}g_{k}-g_{k}g_{j}+g_{i}g_{j}g_{k})dx}}{1-\\frac{\\rho }{m^{2}}(1-g_{kj})e^{-\\int _{A}\\rho (g_{k}-g_{k}g_{j})dx}}$ Using (REF ), (), (REF ) and (REF ), it can be shown that when $j\\in \\Gamma _{w,i}$ (or equivalently $\\left\\Vert x_{i}-x_{j}\\right\\Vert ^{T}>2r_{\\rho }^{1-\\varepsilon }$ ), the integrals of some higher order terms inside the exponential function in (REF ) satisfy: $& \\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }})dx\\\\= & \\int _{D^{T}(x_{i},r_{\\rho }^{1-\\varepsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }})dx\\\\+ & \\int _{A\\backslash D^{T}(x_{i},r_{\\rho }^{1-\\varepsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }})dx\\\\\\le & 2C\\rho r_{\\rho }^{2}g(r_{\\rho }^{-\\varepsilon })\\sim _{\\rho }o_{\\rho }\\left(1\\right)$ Note also that $g_{ik}=g(\\frac{\\left\\Vert x_{i}-x_{k}\\right\\Vert ^{T}}{r_{\\rho }})=o_{\\rho }\\left(1\\right)$ for $k\\in \\Gamma _{w,i}$ .",
"Using the above equations and (REF ), it can be further shown following (REF ) that when $j,k\\in \\Gamma _{w,i}$ .",
"$& \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\frac{Pr(I_{i}^{T}=1,I_{j}^{T}=1,I_{k}^{T}=0)}{Pr(I_{i}^{T}=1)Pr(I_{j}^{T}=1,I_{k}^{T}=0)}\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\frac{Pr(I_{i}^{T}=1,I_{j}^{T}=1)}{Pr(I_{i}^{T}=1)Pr(I_{j}^{T}=1)}$ Equation (REF ) shows that the impact of those events, whose associated indicator random variables $I_{k}^{T}=0,k\\in \\Gamma _{w,i}$ , on the event $I_{i}^{T}=1$ is asymptotically vanishingly small, hence can be ignored.",
"Denote by $\\Gamma _{i}$ a random set of indices containing all indices $j$ where $j\\in \\Gamma _{w,i}$ and $I_{j}=1$ , i.e.",
"the node in question is also isolated, and denote by $\\gamma _{i}$ an instance of $\\Gamma _{i}$ .",
"Define $n\\triangleq \\left|\\gamma _{i}\\right|$ .",
"Following the same procedure that results in (REF ), it can be established that (with some verbose but straightforward discussions omitted) $& & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\frac{E\\lbrace I_{i}^{T}|(I_{j}^{T},j\\in \\Gamma _{w,i})\\rbrace }{\\frac{\\rho }{m^{2}}}\\nonumber \\\\& = & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\frac{E\\lbrace I_{i}^{T}|(I_{j}^{T}=1,j\\in \\gamma _{i})\\rbrace }{\\frac{\\rho }{m^{2}}}\\nonumber \\\\& = & \\lim _{\\rho \\rightarrow \\infty }E[e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})\\prod _{j\\in \\gamma _{i}}(1-g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))dx}\\nonumber \\\\& \\times & \\prod _{j\\in \\gamma _{i}}(1-g(\\frac{\\left\\Vert x_{i}-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))]$ Equation (REF ) gives an analytical expression of the term $E\\lbrace I_{i}^{T}|(I_{j}^{T},j\\in \\Gamma _{w,i})\\rbrace $ .",
"To solve the challenges associated with handling the absolute value term in $b_{3}$ , viz.",
"$|E\\lbrace I_{i}^{T}|(I_{j}^{T},j\\in \\Gamma _{w,i})\\rbrace -p_{i}|$ , we further obtain an upper and a lower bound of $I_{i}^{T}|(I_{j}^{T},j\\in \\Gamma _{w,i})$ , which allows us to remove the absolute value sign in the further analysis of $b_{3}$ .",
"Note that $x_{i}$ and $x_{j},j\\in \\Gamma _{w,i}$ is separated by a distance not smaller than $2r_{\\rho }^{-\\epsilon }$ .",
"Using (REF ), a lower bound on the value inside the expectation operator in (REF ) is given by $B_{L,i} & \\triangleq & (1-g(2r_{\\rho }^{-\\epsilon }))^{n}e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx}$ An upper bound on the value inside the expectation operator in (REF ) is given by $B_{U,i}\\triangleq e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})\\prod _{j\\in \\gamma _{i}}(1-g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))dx}$ Using $p_{i}=E(I_{i}^{T})$ and (REF ), it can be shown that $B_{U,i}\\ge \\lim _{m\\rightarrow \\infty }\\frac{m^{2}p_{i}}{\\rho }\\ge B_{L,i}$ Let us consider $E|E\\lbrace I_{i}|(I_{j},j\\in \\Gamma _{w,i})\\rbrace -p_{i}|$ now.",
"From (REF ), (REF ), (REF ) and (REF ), it is clear that $& & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\sum _{i\\in \\Gamma }E|E\\lbrace I_{i}^{T}|(I_{j}^{T},j\\in \\Gamma _{w,i})\\rbrace -p_{i}|\\nonumber \\\\& \\in & [0,\\max \\lbrace \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }m^{2}p_{i}-\\rho E(B_{L,i}),\\nonumber \\\\& & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\rho E(B_{U,i})-m^{2}p_{i}\\rbrace ]$ In the following we will show that both terms $\\lim _{m\\rightarrow \\infty }m^{2}p_{i}-\\rho E(B_{L,i})$ and $\\lim _{m\\rightarrow \\infty }\\rho E(B_{U,i})-m^{2}p_{i}$ in (REF ) approach 0 as $\\rho \\rightarrow \\infty $ .",
"First it can be shown following (REF ) that $& & \\lim _{m\\rightarrow \\infty }\\rho E(B_{L,i})\\nonumber \\\\& \\ge & \\lim _{m\\rightarrow \\infty }\\rho E[(1-ng(2r_{\\rho }^{-\\epsilon }))e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx}]\\nonumber \\\\& = & \\lim _{m\\rightarrow \\infty }\\rho (1-E\\left(n\\right)g(2r_{\\rho }^{-\\epsilon }))e^{-\\int _{A}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx}$ where $\\lim _{m\\rightarrow \\infty }E\\left(n\\right)$ is the expected number of isolated nodes in $A\\backslash D(x_{i},2r_{\\rho }^{1-\\epsilon })$ .",
"In the first step of the above equation, the inequality $\\left(1-x\\right)^{n}\\ge 1-nx$ for $0\\le x\\le 1$ and $n\\ge 0$ is used.",
"When $\\rho \\rightarrow \\infty $ , $r_{\\rho }^{1-\\epsilon }\\rightarrow 0$ and $r_{\\rho }^{-\\epsilon }\\rightarrow \\infty $ therefore $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }E\\left(n\\right)=\\lim _{\\rho \\rightarrow \\infty }E\\left(W^{T}\\right)=e^{-b}$ is a bounded value and $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }g\\left(2r_{\\rho }^{-\\epsilon }\\right)\\rightarrow 0$ , which is an immediate outcome of (REF ).",
"Using (REF ), it then follows that $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\frac{\\rho E\\left(B_{L,i}\\right)}{m^{2}p_{i}}\\ge \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }(1-E\\left(n\\right)g(2r_{\\rho }^{-\\epsilon }))=1$ Together with () and (REF ), we conclude that $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }m^{2}p_{i}-\\rho E\\left(B_{L,i}\\right)=0$ Now let us consider the second term $\\lim _{m\\rightarrow \\infty }\\rho E\\left(B_{U,i}\\right)-m^{2}p_{i}$ , it can be observed that $& \\lim _{m\\rightarrow \\infty }E\\left(B_{U,i}\\right)\\nonumber \\\\\\le & E[e^{-\\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})\\prod _{j\\in \\gamma _{i}}(1-g(\\frac{\\left\\Vert x-x_{j}\\right\\Vert ^{T}}{r_{\\rho }}))dx}]\\nonumber \\\\\\le & \\lim _{m\\rightarrow \\infty }E[e^{-\\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})\\prod _{j\\in \\gamma _{i}}(1-g(\\frac{r_{\\rho }^{1-\\epsilon }}{r_{\\rho }}))dx}]\\nonumber \\\\= & \\lim _{m\\rightarrow \\infty }E(e^{-(1-g(r_{\\rho }^{-\\epsilon }))^{n}\\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx})\\nonumber \\\\\\le & \\lim _{m\\rightarrow \\infty }E(e^{-(1-ng(r_{\\rho }^{-\\epsilon }))\\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx})$ where in the second step, the non-increasing property of $g$ , and the fact that $x_{j}$ is located in $A\\backslash D(x_{i},2r_{\\rho }^{1-\\epsilon })$ and $x$ is located in $D(x_{i},r_{\\rho }^{1-\\epsilon })$ , therefore $\\left\\Vert x-x_{j}\\right\\Vert ^{T}\\ge r_{\\rho }^{1-\\epsilon }$ is used.",
"It can be further demonstrated that the term $\\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert }{r_{\\rho }})dx$ in (REF ) have the following property: $\\eta \\left(\\varepsilon ,\\rho \\right) & \\triangleq & \\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx\\nonumber \\\\& = & \\rho r_{\\rho }^{2}\\int _{D(\\frac{x_{i}}{r_{\\rho }},r_{\\rho }^{-\\epsilon })}g(\\left\\Vert x-x_{i}/r_{\\rho }\\right\\Vert ^{T})dx\\nonumber \\\\& \\le & C\\rho r_{\\rho }^{2}=\\log \\rho +b$ For the other term $ng(r_{\\rho }^{-\\epsilon })$ in (REF ), choosing a positive constant $\\delta <2\\epsilon $ and using Markov's inequality, it can be shown that $Pr(n\\ge r_{\\rho }^{-\\delta })\\leqslant r_{\\rho }^{\\delta }E\\left(n\\right)$ .",
"Therefore $& & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }Pr(ng(r_{\\rho }^{-\\epsilon })\\eta (\\varepsilon ,\\rho )\\ge r_{\\rho }^{-\\delta }g(r_{\\rho }^{-\\epsilon })\\eta (\\varepsilon ,\\rho ))\\\\& \\le & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }r_{\\rho }^{\\delta }E\\left(n\\right)$ where $\\lim _{\\rho \\rightarrow \\infty }r_{\\rho }^{-\\delta }g(r_{\\rho }^{-\\epsilon })\\eta (\\varepsilon ,\\rho )=0$ due to (REF ), (REF ) and $\\delta <2\\epsilon $ , $\\lim _{\\rho \\rightarrow \\infty }r_{\\rho }^{B}=0$ for any positive constant $B$ , and $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }r_{\\rho }^{\\delta }E\\left(n\\right)=0$ due to that $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }E\\left(n\\right)=\\lim _{\\rho \\rightarrow \\infty }E(W^{T})=e^{-b}$ is a bounded value and that $\\lim _{\\rho \\rightarrow \\infty }r_{\\rho }^{\\delta }=0$ .",
"Therefore $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }Pr(ng(r_{\\rho }^{-\\epsilon })\\eta (\\varepsilon ,\\rho )=0)=1$ As a result of (REF ), (REF ), (REF ) and (REF ): $& & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\rho E(B_{U,i})\\nonumber \\\\& \\le & \\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\rho E(e^{-\\int _{D(x_{i},r_{\\rho }^{1-\\epsilon })}\\rho g(\\frac{\\left\\Vert x-x_{i}\\right\\Vert ^{T}}{r_{\\rho }})dx})\\nonumber \\\\& = & \\lim _{\\rho \\rightarrow \\infty }\\rho e^{-\\int _{D(0,r_{\\rho }^{1-\\varepsilon })}\\rho g(\\frac{\\left\\Vert x\\right\\Vert }{r_{\\rho }})dx}\\nonumber \\\\& = & \\lim _{\\rho \\rightarrow \\infty }\\rho e^{-\\rho r_{\\rho }^{2}(C-\\int _{\\Re ^{2}\\backslash D(0,r_{\\rho }^{-\\varepsilon })}g\\left(\\left\\Vert x\\right\\Vert \\right)dx)}\\nonumber \\\\& = & e^{-b}\\lim _{\\rho \\rightarrow \\infty }e^{\\rho r_{\\rho }^{2}\\int _{\\Re ^{2}\\backslash D(0,r_{\\rho }^{-\\varepsilon })}g\\left(\\left\\Vert x\\right\\Vert \\right)dx}=e^{-b}$ where the last step results because $& \\lim _{\\rho \\rightarrow \\infty }\\rho r_{\\rho }^{2}\\int _{\\Re ^{2}\\backslash D(0,r_{\\rho }^{-\\varepsilon })}g\\left(\\left\\Vert x\\right\\Vert \\right)dx\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\frac{\\int _{r_{\\rho }^{-\\varepsilon }}^{\\infty }2\\pi xg\\left(x\\right)dx}{\\frac{C}{\\log \\rho +b}}\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\frac{\\pi \\varepsilon r_{\\rho }^{-\\varepsilon }g(r_{\\rho }^{-\\varepsilon })r_{\\rho }^{-\\varepsilon -2}\\frac{\\log \\rho +b-1}{C\\rho ^{2}}}{\\frac{C}{\\rho (\\log \\rho +b)^{2}}}\\\\= & \\lim _{\\rho \\rightarrow \\infty }\\frac{\\pi \\varepsilon }{C}(\\log \\rho +b)^{2}r_{\\rho }^{-2\\varepsilon }o_{\\rho }(\\frac{1}{r_{\\rho }^{-2\\varepsilon }\\log ^{2}(r_{\\rho }^{-2\\varepsilon })})=0$ where L'Hï¿œpital's rule is used in reaching (REF ) and in the third step (REF ) is used.",
"Using (), (REF ) and (REF ), it can be shown that $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }\\rho E(B_{U,i})-m^{2}p_{i}=0$ As a result of (REF ), (REF ) and (REF ), $\\lim _{\\rho \\rightarrow \\infty }\\lim _{m\\rightarrow \\infty }b_{3}=0$ .",
"A combination of the analysis in subsections A, B and C completes this proof."
],
[
"Appendix II: Proof of Theorem ",
"For notational convenience, we prove the result for $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ and the result is equally valid for $\\mathcal {G}\\left(\\mathcal {X}_{\\rho },g_{r_{\\rho }},A\\right)$ .",
"The proof is based on analyzing the number of components in $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ of order greater than some integer $M$ as $\\rho \\rightarrow \\infty $ .",
"Specifically we will show that $\\lim _{M\\rightarrow \\infty }\\lim _{\\rho \\rightarrow \\infty }\\Pr (\\xi _{>M}=1)=1$ .",
"A direct analysis of $\\Pr (\\xi _{>M}=1)$ can be difficult.",
"In this paper, we first analyze $E(\\xi _{>M})$ and then use the result on $E(\\xi _{>M})$ to establish the desired asymptotic result on $\\Pr (\\xi _{>M}=1)$ .",
"Denote by $g_{1}\\left(x_{1},\\ldots ,x_{k}\\right)$ the probability that a set of $k$ nodes at non-random positions $x_{1}$ , $\\ldots $ , $x_{k}\\in A_{\\frac{1}{r_{\\rho }}}$ forms a connected component where nodes are connected randomly and independently following the connection function $g$ .",
"Denote by $g_{2}\\left(y;x_{1},x_{2},\\ldots ,x_{k}\\right)$ the probability that a node at non-random position $y$ is connected to at least one node in $\\left\\lbrace x_{1},x_{2},\\ldots ,x_{k}\\right\\rbrace $ .",
"As an easy consequence of [10], which showed that the expected number of components of order $k$ , denoted by $\\xi _{k}$ , in $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ is given by $E\\left(\\xi _{k}\\right)=\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}g_{1}(x_{1},\\ldots ,x_{k})e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots ,x_{k})dy}d(x_{1}\\cdots x_{k})$ , it follows that $& E\\left(\\xi _{>M}\\right)\\nonumber \\\\= & \\sum _{k=M+1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}(g_{1}(x_{1},\\ldots ,x_{k})\\nonumber \\\\& e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots ,x_{k})dy})d(x_{1}\\cdots x_{k})\\nonumber \\\\\\le & \\sum _{k=M+1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots ,x_{k})dy}d(x_{1}\\cdots x_{k})\\nonumber \\\\= & \\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots ,x_{k})dy}d(x_{1}\\cdots x_{k})\\nonumber \\\\- & \\sum _{k=1}^{M}\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots ,x_{k})dy}d(x_{1}\\cdots x_{k})$ In the following we show that as $\\rho \\rightarrow \\infty $ , the first term in (REF ) converges to $e^{e^{-b}}$ , and the second term in (REF ) after the “$-$ ” sign is lower-bounded by $\\sum _{k=1}^{M}\\frac{(e^{-b})^{k}}{k!",
"}$ .",
"The conclusion then follows that $E(\\xi _{>M})$ converge to 1 as $\\rho \\rightarrow \\infty $ and $M\\rightarrow \\infty $ .",
"Let us consider the first term in (REF ) now.",
"Let $\\Phi \\triangleq \\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}[1-g_{2}(y;x_{1},\\ldots x_{k})]dy$ for convenience.",
"It can be shown that $& \\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots x_{k})dy}d(x_{1},\\ldots x_{k})\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{\\Phi }d(x_{1},\\ldots x_{k})\\nonumber \\\\= & \\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\sum _{n=0}^{\\infty }\\frac{\\Phi ^{n}}{n!",
"}d(x_{1},\\ldots x_{k})\\nonumber \\\\= & \\sum _{n=0}^{\\infty }\\frac{1}{n!",
"}\\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n}d(x_{1},\\ldots x_{k})$ Next we shall show that in (REF ), $\\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n}d(x_{1},\\ldots x_{k})=(e^{-b})^{n}$ .",
"Given this result, conclusion readily follows from (REF ) that the first term in (REF ) converges to $e^{e^{-b}}$ .",
"A direct computation of the term $\\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n}d(x_{1},\\ldots x_{k})$ turns out to be very difficult.",
"To resolve the difficulty, we construct a random integer $X$ , depending on $\\rho $ , such that on the one hand, the pmf (probability mass function) of $X$ has an analytical form that can be easily related to the term $\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n}d(x_{1},\\ldots x_{k})$ ; and on the other hand using the Chen-Stein bound we are familiar with, the pmf can be shown to converge to a Poisson distribution as $\\rho \\rightarrow \\infty $ .",
"In this way, we are able to compute the above term using the intermediate random integer $X$ .",
"In the following, we give details of the analysis.",
"We first construct the random integer $X$ described in the last paragraph and demonstrate its properties related to our analysis.",
"Consider an additional independent Poisson point process $\\mathcal {X}^{\\prime }_{\\lambda }$ with nodes Poissonly distributed on $A_{\\frac{1}{r_{\\rho }}}$ and with density $\\lambda $ , being added to $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ .",
"Further, nodes in $\\mathcal {X}^{\\prime }_{\\lambda }$ are connected with nodes in $\\mathcal {X}_{\\lambda }$ following $g$ independently, i.e.",
"a node in $\\mathcal {X}^{\\prime }_{\\lambda }$ and a node in $\\mathcal {X}_{\\lambda }$ separated by an Euclidean distance $x$ are connected with probability $g\\left(x\\right)$ , independent of any other connection.",
"Let $X$ be the number of nodes in $\\mathcal {X}^{\\prime }_{\\lambda }$ that are not directly connected to any node in $\\mathcal {X}_{\\lambda }$ .",
"It is evident that, conditioned on $\\mathcal {X}_{\\lambda }=(x_{1},\\ldots x_{k})$ where $x_{1},\\ldots x_{k}\\in A_{\\frac{1}{r_{\\rho }}}$ and $|\\mathcal {X}_{\\lambda }|>0$ , a randomly chosen node in $\\mathcal {X}^{\\prime }_{\\lambda }$ at location $y$ is not directly connected to any node in $\\mathcal {X}_{\\lambda }$ with probability $1-g_{2}(y;x_{1},\\ldots ,x_{k})$ , which is determined by its location only.",
"It readily follows that the conditional distribution of $X$ , i.e.",
"$X|\\mathcal {X}_{\\lambda }=(x_{1},\\ldots x_{k})$ , is Poisson with mean $\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}[1-g_{2}(y;x_{1},\\ldots ,x_{k})]dy$ [9].",
"As a result of the above discussion: $\\Pr (X=m|\\mathcal {X}_{\\lambda }=(x_{1},\\ldots x_{k}))=\\frac{\\Phi ^{m}}{m!",
"}e^{-\\Phi }$ Obviously when $\\mathcal {X}_{\\lambda }=\\emptyset $ , $\\Pr (X=m|\\mathcal {X}_{\\lambda }=\\emptyset )=\\Pr (|\\mathcal {X}^{\\prime }_{\\lambda }|=m)$ .",
"Therefore the unconditional distribution of $X$ is given by: $& \\Pr \\left(X=m\\right)\\nonumber \\\\= & \\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\frac{\\Phi ^{m}}{m!",
"}e^{-\\Phi }d(x_{1},\\ldots x_{k})+\\frac{\\rho ^{m}}{m!",
"}e^{-2\\rho }$ Note that as $\\rho \\rightarrow \\infty $ , the term $\\frac{\\rho ^{m}}{m!",
"}e^{-2\\rho }$ in (REF ), which is associated with $\\mathcal {X}_{\\lambda }=\\emptyset $ , becomes vanishingly small.",
"Further note that $\\sum _{m=0}^{\\infty }\\frac{\\rho ^{m}}{m!",
"}e^{-2\\rho }=e^{-\\rho }\\rightarrow 0$ as $\\rho \\rightarrow \\infty $ , i.e.",
"as $\\rho \\rightarrow \\infty $ even the cumulative contribution to the cdf of $X$ is negligibly small.",
"If we define $g_{2}\\left(y;\\emptyset \\right)\\triangleq 0$ for completeness, we can also write (REF ) as $\\Pr \\left(X=m\\right)=\\sum _{k=0}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\frac{\\Phi ^{m}}{m!",
"}e^{-\\Phi }d(x_{1},\\ldots x_{k})$ Using (REF ), it can be readily shown that $& E\\left(X\\right)=\\sum _{m=0}^{\\infty }m\\Pr \\left(X=m\\right)\\nonumber \\\\= & \\sum _{k=0}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi d(x_{1},\\ldots x_{k})\\nonumber \\\\= & \\sum _{k=0}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\lbrace \\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}\\lbrace \\int _{A_{\\frac{1}{r_{\\rho }}}}[1-g(\\left\\Vert x-y\\right\\Vert )]dx\\rbrace ^{k}dy\\rbrace \\nonumber \\\\= & \\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x-y\\right\\Vert )dx}dy$ Comparing the above equation with [10], the conclusion readily follows that the above value is equal to the expected number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ , denoted by $W$ .",
"It then follows from [10], that $\\lim _{\\rho \\rightarrow \\infty }E\\left(X\\right)=e^{-b}$ .",
"In fact a stronger result that the distributions of $X$ and $W$ converge to the same Poisson distribution as $\\rho \\rightarrow \\infty $ can be established: Lemma 15 As $\\rho \\rightarrow \\infty $ , the distribution of $X$ converges to a Poisson distribution with mean $e^{-b}$ , i.e.",
"the total variation distance between the distribution of $X$ and a Poisson distribution with mean $e^{-b}$ reduces to 0 as $\\rho \\rightarrow \\infty $ .",
"Lemma REF can be proved using exactly the same steps as those used in proving Theorem REF .",
"Therefore the proof is omitted.",
"As a result of Lemma REF , for an arbitrary set of non-negative integers, denoted by $\\Gamma $ , $\\lim _{\\rho \\rightarrow \\infty }\\sum _{m\\in \\Gamma }\\Pr (X=m)=\\sum _{m\\in \\Gamma }\\frac{(e^{-b})^{m}}{m!",
"}e^{-e^{-b}}$ Now we are ready to continue our analysis on $\\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n}d(x_{1},\\ldots x_{k})$ .",
"Using (REF ) first and then using (REF ), it can be shown that for any positive integer $n$ : $& \\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n}d(x_{1},\\ldots x_{k})\\\\= & \\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\sum _{m=0}^{\\infty }\\Phi ^{n}\\frac{\\Phi ^{m}}{m!",
"}e^{-\\Phi }d(x_{1},\\ldots x_{k})\\\\= & \\sum _{m=0}^{\\infty }\\frac{1}{m!",
"}\\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}e^{-\\rho }\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}\\Phi ^{n+m}e^{-\\Phi }d(x_{1},\\ldots x_{k})\\\\= & \\sum _{m=0}^{\\infty }\\frac{1}{m!",
"}\\lim _{\\rho \\rightarrow \\infty }(\\Pr \\left(X=n+m\\right)-\\frac{\\rho ^{(n+m)}}{(n+m)!",
"}e^{-2\\rho })(n+m)!\\\\= & \\sum _{m=0}^{\\infty }\\frac{(e^{-b})^{n+m}}{m!",
"}e^{-e^{-b}}=(e^{-b})^{n}$ Using the above equation, it follows from (REF ) that $& \\lim _{\\rho \\rightarrow \\infty }\\sum _{k=1}^{\\infty }\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots x_{k})dy}d(x_{1},\\ldots x_{k})\\nonumber \\\\= & \\sum _{n=0}^{\\infty }\\frac{(e^{-b})^{n}}{n!",
"}=e^{e^{-b}}$ This deals with the first term on the right of (REF ).",
"Now we continue with the analysis of the second term in (REF ).",
"As an easy consequence of the union bound, $g_{2}\\left(y;x_{1},x_{2},\\ldots ,x_{k}\\right)\\le \\sum _{i=1}^{k}g\\left(\\left\\Vert y-x_{i}\\right\\Vert \\right)$ , it can then be shown that $& & \\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots x_{k})dy}d(x_{1}\\cdots x_{k})\\nonumber \\\\& \\ge & \\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}\\sum _{i=1}^{k}g(\\left\\Vert y-x_{i}\\right\\Vert )dy}d(x_{1}\\cdots x_{k})\\nonumber \\\\& = & \\frac{(\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g(\\left\\Vert x-y\\right\\Vert )dy}dx)^{k}}{k!",
"}$ and using [10], it can be further shown that $& \\lim _{\\rho \\rightarrow \\infty }\\frac{\\lambda ^{k}}{k!",
"}\\int _{(A_{\\frac{1}{r_{\\rho }}})^{k}}e^{-\\lambda \\int _{A_{\\frac{1}{r_{\\rho }}}}g_{2}(y;x_{1},\\ldots x_{k})dy}d(x_{1}\\cdots x_{k})\\nonumber \\\\\\ge & \\frac{(e^{-b})^{k}}{k!",
"}$ Note that (REF ) can also be obtained from Jensen's inequality.",
"Combining (REF ), (REF ) and (REF ), it follows that $\\lim _{\\rho \\rightarrow \\infty }E(\\xi _{>M})\\le e^{e^{-b}}-\\sum _{k=1}^{M}\\frac{(e^{-b})^{k}}{k!",
"}=1+\\frac{(\\eta _{M})^{M+1}}{(M+1)!",
"}$ where in the last step Taylor's theorem is used, $\\eta _{M}$ is a number depending on $M$ and $ $$0\\le \\eta _{M}\\le e^{-b}$ .",
"In Theorems REF and [10], we have established respectively that the asymptotic distribution of the number of isolated nodes in $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ is Poisson with mean $e^{-b}$ and the number of components in $\\mathcal {G}\\left(\\mathcal {X}_{\\lambda },g,A_{\\frac{1}{r_{\\rho }}}\\right)$ of order within $\\left[2,M\\right]$ vanishes as $\\rho \\rightarrow \\infty $ .",
"As a consequence of the above two results, $\\lim _{\\rho \\rightarrow \\infty }\\Pr (\\xi _{>M}\\ge 1)=1\\;\\;\\textrm {and}\\;\\;\\lim _{\\rho \\rightarrow \\infty }\\Pr (\\xi _{>M}=0)=0$ Further note that $& E(\\xi _{>M})=\\sum _{m=1}^{\\infty }m\\Pr (\\xi _{>M}=m)\\nonumber \\\\\\ge & \\Pr (\\xi _{>M}=1)+2\\sum _{m=2}^{\\infty }\\Pr (\\xi _{>M}=m)\\nonumber \\\\= & \\Pr (\\xi _{>M}=1)+2(1-\\Pr (\\xi _{>M}=1)-\\Pr (\\xi _{>M}=0))$ Combing the three equations (REF ), (REF ) and (REF ): $\\lim _{\\rho \\rightarrow \\infty }\\Pr (\\xi _{>M}=1)\\ge 1-\\frac{(\\eta _{M})^{M+1}}{\\left(M+1\\right)!",
"}$ As an easy consequence of the above equation: $\\lim _{M\\rightarrow \\infty }\\lim _{\\rho \\rightarrow \\infty }\\Pr \\left(\\xi _{>M}=1\\right)=1$ Guoqiang Mao (S'98–M'02–SM’08) received PhD in telecommunications engineering in 2002 from Edith Cowan University, Australia.",
"He joined the School of Electrical and Information Engineering, the University of Sydney in December 2002.",
"He has published over 100 papers in international journals and conferences.",
"His research interests include wireless multihop networks (e.g.",
"vehicular networks, mesh networks mobile networks, delay-tolerant networks, opportunistic networks), wireless sensor networks, wireless localization techniques, applied graph theory and network performance analysis.",
"He is a Senior Member of IEEE and an Associate Editor of IEEE Transactions on Vehicular Technology.",
"He has served as a program committee member in a large number of international conferences.",
"He was a symposium co-chair of IEEE PIMRC 2012, a publicity co-chair of 2007 SenSys and 2010 IEEE WCNC.",
"Brian D.O.",
"Anderson (S’62–M’66–SM’74–F’75– LF’07) was born in Sydney, Australia, and educated at Sydney University in mathematics and electrical engineering, with PhD in electrical engineering from Stanford University in 1966.",
"He is a Distinguished Professor at the Australian National University and Distinguished Researcher in National ICT Australia.",
"His awards include the IEEE Control Systems Award of 1997, the 2001 IEEE James H Mulligan, Jr Education Medal, and the Bode Prize of the IEEE Control System Society in 1992, as well as several IEEE and other best paper prizes.",
"He is a Fellow of the Australian Academy of Science, the Australian Academy of Technological Sciences and Engineering, the Royal Society, and a foreign associate of the US National Academy of Engineering.",
"He holds honorary doctorates from a number of universities, including Université Catholique de Louvain, Belgium, and ETH, Zürich.",
"He is a past president of the International Federation of Automatic Control and the Australian Academy of Science.",
"His current research interests are in distributed control, sensor networks and econometric modelling."
]
] | 1204.1400 |
[
[
"Role of three-body interactions in formation of bulk viscosity in liquid\n argon"
],
[
"Abstract With the aim of locating the origin of discrepancy between experimental and computer simulation results on bulk viscosity of liquid argon, a molecular dynamic simulation of argon interacting via ab initio pair potential and triple-dipole three-body potential has been undertaken.",
"Bulk viscosity, obtained using Green-Kubo formula, is different from the values obtained from modeling argon using Lennard-Jones potential, the former being closer to the experimental data.",
"The conclusion is made that many-body inter-atomic interaction plays a significant role in formation of bulk viscosity."
],
[
"Introduction",
"Argon above its melting temperature is a typical simple fluid.",
"Consisting of spherical atoms that interact via short-range repulsion and long-range attraction, and are heavy enough for the quantum effects to be small, fluid argon and heavier noble gases are an excellent choice of a real system to be used for testing various approaches in classical theory of fluids.",
"An inter-particle interaction in argon is commonly represented by a well known 12–6 Lennard-Jones pair potential [1], $v_{\\mathrm {LJ}}(r)=4\\epsilon _{\\mathrm {LJ}}\\left[\\left(\\frac{\\sigma _{\\mathrm {LJ}}}{r}\\right)^{12}-\\left(\\frac{\\sigma _{\\mathrm {LJ}}}{r}\\right)^6\\right].$ The two parameters, $\\sigma _{\\mathrm {LJ}}$ and $\\epsilon _{\\mathrm {LJ}}$ , are usually determined by fitting thermodynamic properties, derived from the potential (REF ) by theoretical or computational methods, to corresponding experimental data.",
"It is known that Lennard-Jones potential is only an approximation to real interaction in argon.",
"Several experimental results obtained for argon at large pressures are better explained if a larger steepness, compared to Lennard-Jones, of argon-argon interaction potential at small inter-atomic separation distances is taken into account [2], [3].",
"Accurate argon–argon interatomic potentials have been calculated by direct ab initio quantum chemical calculations [4], [5], [6] or obtained by inversion of experimental data [7].",
"Moreover, many-body dispersion, exchange and induced polarization contributions to inter-atomic interactions are not small and noticeably influence thermodynamic properties of argon [8], [9].",
"The most widely used of these contributions is triple-dipole dispersion interaction, derived by Axilrod and Teller [10], [11] and Muto [12], and account of this contribution in addition to ab initio pair potential is sufficient to describe thermodynamic properties of argon with good accuracy [13], [14], [15], [16], [17].",
"By virtue of Henderson theorem [18], [19], which states that, for fluids with only pairwise interactions, and under given conditions of temperature and density, the pair potential which gives rise to a given radial distribution function $g(r)$ is unique up to a constant, the thermodynamic properties of the system with many-body interactions can be described by a model system with an appropriate effective pair potential.",
"Generally, the effective potential depends on the thermodynamic state of the system and thermodynamic property to be described [20], [21], [22].",
"Van der Hoef and Madden [21] have demonstrated that the account of triple-dipole and dipole-dipole-quadrupole dispersion interactions moves the effective potential of argon towards Lennard-Jones form (REF ).",
"Moreover, the possibility of consistent description of many thermodynamic properties of argon, using Lennard-Jones potential in a wide domain of thermodynamic states [23], [24], [25], suggests that the state dependence of the effective potential is weak.",
"There is no analogous reason for kinetic properties of a system with many-body interactions to be equivalent to those of a system with a corresponding effective pair potential.",
"Nevertheless, experimental data on self-diffusion, shear viscosity and thermal conductivity coefficients of argon have been shown to be accurately described by Lennard-Jones model with the parameters obtained by fitting thermodynamic data [26], [27].",
"Bulk viscosity is a noticeable exception.",
"Bulk viscosity of argon has been measured experimentally [28], [29], [30], [31], [32], [33], [34], [35], and its behavior can be qualitatively described by the results of a molecular dynamics simulation of a Lennard-Jones system [36].",
"However, when results of simulations with Lennard-Jones potential are rescaled in an attempt to describe experimental data liquid argon, bulk viscosity, contrary to other kinetic properties, appears strongly underestimated (e.g.",
"up to 50% in Ref. [Fernandez:2004-157)].",
"In view of the above, I propose that the source of this discrepancy may lie in neglect of many-body interactions.",
"Previous molecular dynamics simulations of systems consisting of 108 particles interacting via ab initio pair potential and Axilrod-Teller-Muto (ATM) interaction indicated that a triple-dipole interaction does not affect the bulk viscosity of liquid xenon near its triple point [37] and dense gaseous krypton [38].",
"However, the error in the values of bulk viscosity obtained from molecular dynamics simulation of the systems with such a small number of particles can be quite large.",
"For example, the values of the reduced bulk viscosity of the Lennard-Jones systems consisting of 128 and 256 particles at the reduced temperature $T^*=0.722$ and the reduced density $\\rho ^*=0.8442$ , reported in Refs Heyes:1984-1363,Levesque:1987-143,Hoheisel:1987-7195,Meier:2005-014513, range from 0.89 to 1.47, with the ratio of the latter to the former of 1.65.",
"This paper presents the results of more accurate molecular dynamics simulations of a liquid consisting of 1372 argon atoms with ab initio+ATM interaction, which demonstrate that bulk viscosity, determined from Green-Kubo formulae, significantly changes with the account of three-body interaction, moving results towards experimental data."
],
[
"Interaction",
"Nasrabad et al [16] undertook a Monte Carlo simulation of argon using combination of ab initio pair interaction [4] and ATM triple-dipole dispersion interaction [10] to test their ability to predict vapor-liquid equilibrium.",
"Although more accurate ab initio pair potentials for argon have become available recently [5], [6], and other many-body contributions to inter-atom interaction can be calculated [8], we use the same interaction as Nasrabad et al because, being able to predict accurately the phase diagram of argon [16], it is computationally more efficient.",
"Specifically, the ab initio pair interaction potential used in the present work is described by a function [16] $u_2(r)=Ae^{-\\alpha r+\\beta r^2}+\\sum _{n=3}^5f_{2n}(r,b)\\frac{C_{2n}}{r^{2n}},$ where $f_{2n}(r,b)=1-e^{-br}\\sum _{k=0}^{2n}\\frac{(br)^k}{k!",
"},$ and numerical values of the parameters $A$ , $\\alpha $ , $\\beta $ , $b$ , and $C_{2n}$ are given in Ref. [Nasrabad:2004-6423].",
"The ATM triple-dipole interaction has form [10] $u_3(r_{12},r_{23},r_{31})=\\nu \\frac{1+3\\cos \\alpha \\cos \\beta \\cos \\gamma }{r_{12}^3r_{23}^3r_{31}^3},$ where the $r_{ik}$ are the lengths of the sides, $\\alpha $ , $\\beta $ , and $\\gamma $ are the angles of the triangle formed by three argon atoms, and $\\nu =7.32\\cdot 10^{-108}$ J$\\cdot $ m${}^9$ for argon [13], [14].",
"For simulations of argon using Lennard-Jones potential (REF ) the values $\\sigma _{\\mathrm {LJ}}=3.3952$ Å and $\\epsilon _{\\mathrm {LJ}}=116.79$ K are used [25]."
],
[
"Simulation",
"Meier et al [36] undertook a systematic study of the influence of the number of particles and the cutoff radius for pair interaction on the bulk viscosity of Lennard-Jones system.",
"In view of their results, simulations were performed in a cubic box containing $N=1372$ particles, and the cutoff radius for pair interactions was set to $5\\sigma _{\\mathrm {LJ}}$ .",
"Three-body interactions were cut off when the distance between any pair of the atoms in the triplet exceeded one quarter of the simulation box length (around $3\\sigma _{\\mathrm {LJ}}$ for the densities studied in this work).",
"Usual periodic boundary conditions and minimum image convention were applied.",
"The simulations were started with the particles in a face-centered-cubic lattice, with randomly assigned velocities.",
"Forces arising from three-body interactions were calculated using formulas given by Allen and Tildesley [42], and an expression for forces due to ab initio pair interaction was obtained by applying gradient operator to Eq.",
"(REF ).",
"Newton's equations of motion were solved using velocity-Verlet algorithm with the time step $\\Delta t\\cdot \\sqrt{\\epsilon _{\\mathrm {LJ}}/m}/\\sigma _{\\mathrm {LJ}}=0.003$ .",
"The runs were made at the experimental densities at various temperatures along the 40 atm isochore, taken from Ref. [Cowan:1972-1881].",
"Every simulation was initiated in the NVT ensemble and run for at least $2{\\cdot }10^5$ time steps to attain thermodynamic equilibrium.",
"After equilibration the thermostat was turned off and the NVE ensemble was invoked to calculate bulk and shear viscosities.",
"The length of the production period was $4{\\cdot }10^6$ time steps for the system interacting via Lennard-Jones potential, and between $10^6$ and $3{\\cdot }10^6$ time steps for the system with ab initio + ATM interaction, depending on the state point.",
"Bulk viscosity, $\\zeta $ , and shear viscosity, $\\eta $ , were calculated using Green-Kubo formulas [43]: $\\zeta =\\frac{V}{k_BT}\\int _0^\\infty \\left<\\delta p(t)\\delta p(t_0)\\right>dt,$ $\\eta =\\frac{V}{k_BT}\\int _0^\\infty \\left<\\sigma _{\\alpha \\beta }(t)\\sigma _{\\alpha \\beta }(t_0)\\right>dt,$ where $V$ is volume, $k_B$ is Boltzmann constant, $T$ is temperature, $t$ is time, $\\delta p=p-\\left<p\\right>$ is the deviation of the instantaneous pressure $p$ from its average value $\\left<p\\right>$ , $\\sigma _{\\alpha \\beta }$ is an off-diagonal element of the stress tensor, the angular brackets denote equilibrium ensemble averages over short trajectory sections of the phase-space trajectory of the system with multiple (every time step) time origins $t_0$ .",
"The stress tensor was calculated using formulae given by Lee and Cummings [44].",
"The integration in Eqs (REF ) and (REF ) was carried out up to $\\tau _L=L/c$ , where $L$ is simulation box length and $c$ is sound velocity taken from Ref. [Cowan:1972-1881].",
"Depending on the state point, the value of $\\tau _L$ was between 4.80 and 11.25 ps.",
"The statistical error in time correlation functions was estimated using formula given by Frenkel and Smit [45], $\\sigma \\left(\\left<X(t)X(0)\\right>\\right)\\approx \\sqrt{\\frac{2\\tau _X}{t_{\\mathrm {run}}}}\\left<X^2(0)\\right>,$ where $t_{\\mathrm {run}}$ is the length of the simulation, and the correlation time $\\tau _X$ was approximated as the time during which time correlation function decays $e\\approx 2.718$ times."
],
[
"Results",
"Fig.",
"REF and Table REF present simulation results for the bulk viscosity obtained using ab initio + ATM (Eqs (REF ) and (REF )) and Lennard-Jones (Eq.",
"(REF )) interaction, respectively.",
"Bulk viscosity, determined from Green-Kubo formulas, changes with the account of three-body interaction, moving towards experimental data.",
"However, this change is not sufficient to obtain numerical agreement with experiment, especially at lower densities.",
"Typical behavior of time correlation functions $C(t)=\\left<\\delta p(t)\\delta p(0)\\right>$ is shown in Fig.",
"REF .",
"Figure: Bulk viscosity of liquid argon at T=(90-140)T=(90{-}140) K. Error bars connected with solid and dashedlines correspond to the simulation results with ab initio + ATM and Lennard-Jonesinteraction, respectively.",
"Experimental points are taken from Refs [Cowan:1972-1881](circles, pressure 40 atm) and [Naugle:1966-4669] (square with error bar, pressure40 kg/cm 2 {}^2).Table: Bulk and shear viscosities of argon obrained from molecular dynamics simulations using Lennard-Jones(LJ) and ab initio pair + Axilrod-Teller-Muto three-body (AI+ATM) interaction, andcorresponding experimental data , .",
"Error in the simulation data iscalculated using Eq.",
"().Figure: Time-correlation functions C(t)C(t) used for calculation of bulk viscosity at density1.258 g/cm 3 {}^3.",
"Solid and dashed lines correspond to the simulation results withab initio + ATM and Lennard-Jones interaction, respectively.Fernandez et al [27] demonstrated that, contrary to bulk viscosity, the values of shear viscosity of argon obtained from molecular dynamics simulation of a Lennard-Jones system agree with experimental data.",
"Lee and Cummings [44] and Marcelli et al [46] found that the influence of triple-dipole interaction on shear viscosity of argon is small.",
"The results of the present simulation, shown in Fig.",
"REF and Table REF , agree with these findings.",
"Figure: Shear viscosity of liquid argon at T=(90-140)T=(90{-}140) K. Error bars connected with solid and dashedlines correspond to the simulation results with ab initio + ATM and Lennard-Jonesinteraction, respectively.",
"Dotted line corresponds to the interpolation data for pressure 40 atmtaken from Ref.",
"[NIST-old]."
],
[
"Conclusion",
"The message of this paper is that many-body interactions play a more substantial role in determining the value of the bulk viscosity than other transport coefficients.",
"The present results from the molecular dynamic simulation of liquid argon demonstrate that even account of a single many-body contribution, ATM triple-dipole interaction, shifts the values of the bulk viscosity of argon towards experimental data.",
"Larger sensitivity of the bulk viscosity to many-body interaction, compared to other transport coefficients, can be intuitively explained in the case of gaseous state.",
"Bulk viscosity of a non-relativistic monoatomic gas calculated from the Boltzmann equation, which takes into account only pair collisions of atoms, appears to be zero, in contrast to heat conductivity and shear viscosity which have non-zero values in the same approximation [47].",
"A non-zero value of bulk viscosity appears in the approximations corresponding to higher-order terms in the virial expansion [48], [49], which correspond to the explicit account of at least three-atom collisions which, in turn, are sensitive to three-body inter-atomic interaction.",
"I thank Prof. Jadran Vrabec, Dr Andrey Brukhno, and Dr Ian Halliday for stimulating discussions."
]
] | 1204.1235 |
[
[
"CMBR anisotropy in the framework of cosmological extrapolation of MOND"
],
[
"Abstract A modified gravity involving a critical acceleration, as empirically established at galactic scales and successfully tested by data on supernovae of type Ia, can fit the measured multipole spectrum of anisotropy in the cosmic microwave background radiation, so that a dark sector of Universe is constructively mimicked as caused by the dynamics beyond the general relativity.",
"Physical consequences, verifiable predictions and falsifiable issues are listed and discussed."
],
[
"Introduction",
"The general relativity as the theory of gravitation is triumphantly tested in “classical experiments” [1] on the deflection of light by the Sun, the perihelion precession of Mercury, the gravitational redshift of electromagnetic radiation, the time delay of signal from satellites due to the curved space-time, and the gyroscope precession during the orbital motion around the Earth as caused by the spatial curvature [2], [3].",
"In addition, an expanding Universe is the prediction being inherent for the general relativity.",
"Sure, the general relativity conceptually is the perfect theory of classic gravity.",
"In this respect, we usually expect that it is valid, indeed, until effects of quantum gravity would be essential at Planckian scales of energy that is unreachable in practice.",
"However, such the point of view, perhaps, is actually broken: The general relativity itself gives us a brilliant tool in order to search for indications, which signalize on breaking down its validity: while observing a motion by inertia, we get a curvature of space-time, which can be inserted into Einstein equations, that yields a tensor of energy-momentum for an appropriate substance, and if properties of the substance are mysterious and unpredictable, then we get a hint for suspecting of the incorrect description for the nature.",
"This is exactly the case of hypothetic dark matter (see, for instance, review in [4]): in the framework of general relativity, it should be inevitably introduced as a transparent pressureless substance dynamically isolated from the ordinary visible matter made of known, well studied particles, except the interaction via the gravity, so that properties of dark matter are artificially tuned.",
"This tuning has various aspects.",
"First, rotational curves in disc-like galaxies, i.e.",
"dependencies of rotation velocities of stars versus a distance to the galaxy center, if described by the law of Newtonian gravity, requires the introduction of dark matter with a tuned spatial distribution.",
"Unexpectedly, the dark matter halo is inessential in regions, wherein the gravitational acceleration caused by the visible matter, is greater than a critical value $\\tilde{g}_0$ , while the halo starts to dominate in regions, wherein the acceleration by the visible matter is less than $\\tilde{g}_0$ [4], [5], [6], [7].",
"That was M. Milgrom who first introduced the critical acceleration $\\tilde{g}_0$ in the description of rotational curves [8].",
"It is spectacular that the critical acceleration is universal: it does not depend on a genesis of disk-like galaxy, and it is the same for any studied disc-like galaxy.",
"Unbelievably, an amount and spatial distribution of dark matter is tuned to the amount and distribution of visible matter in order to form in cosmic collisions the dark halos in disc-like galaxies with the same universal critical acceleration.",
"In the framework of general relativity, there is no straightforward dynamical mechanism for a deduction of such the universal acceleration.",
"Anyway, the deduction looks to be very artificial, most probably, it certainly could be the fine tuning.",
"The critical acceleration subtly binds the dark matter to the visible matter.",
"If this relation is dynamical, then it is not due to Newtonian gravity, that does not include any critical acceleration.",
"Further, in deep regions of dark halo dominance, the rotation velocities tend to constant values $v_0$ , that empirically satisfy the baryonic Tully–Fisher law [9]: $v_0^4=GM\\tilde{g}_0,$ where $M$ stands for the mass of visible matter in disc-like galaxy, $G$ denotes the Newton constant.",
"Again, the dark matter halo is tuned, so that the star motion within the halo strictly correlates with the usual visible matter, while the constant $\\tilde{g}_0$ is universal [10].",
"Finally, in order to complete the first item of argumentation, features in distributions of visible matter, no doubt, are imprinted in rotational curves even in regions of dark matter dominance [4], hence, features of dark matter distributions are tuned to the visible matter, though we have no dynamical reasons for such the correlations in the framework of general relativity.",
"Moreover, the morphology of spatial distributions is absolutely different for the baryonic and dark matter in disc-like galaxies: an exponentially falling central bulge and thin disc of stars and gas in contrast to power-law decline of dark spherical halo.",
"Thus, the universal critical acceleration is the mysterious quantity for the general relativity, that cannot be predicted, while its notion emerged empirically.",
"The critical acceleration $\\tilde{g}_0$ reveals the fine tuning of hypothetic dark matter to the ordinary visible matter.",
"Second, in cosmology with the observed accelerated expansion of Universe by data on a dependence of brightness of type Ia supernovae versus the red shift [11], [12], [13], [14], [15], the general relativity was have to introduce the extended dark sector, which includes a homogeneous dark energy in addition to the nonhomogeneous dark matter.",
"In the simplest case, the dark energy can be represented by the cosmological constant, otherwise it should be described by a homogeneous fluid $X$ with the state parameter $w_X$ being the ratio of pressure $p_X$ to energy density $\\rho _X$ , $w_X=p_X/\\rho _X$ close to vacuum value of $-1$ , in contrast to the pressureless dark matter with $w_\\mathrm {DM}=0$ .",
"Evidently, the nature of dark matter and dark energy is very different.",
"But surprisingly, the energy density of dark energy, or the value of cosmological constant $\\Lambda $ , is finely tuned to the critical acceleration [4], so that $G\\rho _X\\sim \\Lambda \\sim \\tilde{g}_0^2.$ Therefore, the dark energy should be inherently connected to the unexpectedly correlated dynamics of dark matter and ordinary matter.",
"But the coincidence of (REF ) is mysterious for the general relativity.",
"Nevertheless, the general relativity applied to the cosmos still looks formally viable in the form of concordance model: the ordinary matter balanced with the cold dark matter (CDM) and cosmological constant $\\Lambda $ in the flat space, the $\\Lambda \\mbox{CDM}$ variant with a spatial curvature compatible with zero in limits of uncertainties.",
"Moreover, there are two important successes in the model: 1) a correct fitting for observed anisotropy of cosmic microwave background radiation (CMBR) [16] that becomes possible due to a tuned amount of non-baryonic dark matter, and 2) an appropriate baryon to photon ratio consistent with a current status of big bang nucleosynthesis (BBN) (see review by Fields B D, Sarkar S in [17]).",
"Indeed, the anisotropy of CMBR is suppressed by 5 orders of magnitude with respect to the CMBR temperature, and it is caused by a propagation of sound waves in a hot photon-electron-baryon medium up to a moment, when the electrons bind to nuclei to form neutral transparent gas.",
"The snapshot of Universe at the time of decoupling of photons evolves to us, and it represents acoustic peaks in the following multipole spectrum of temperature fluctuations $\\langle \\Delta T(n_1) \\Delta T(n_2)\\rangle =\\sum _l\\frac{2l+1}{4\\pi }\\,C_l\\,P_l(n_1\\cdot n_2),$ where $n_{1,2}$ denote directions in the celestial sphere, $P_l$ are Legendre polynomials of multipole number $l$ .",
"The spectrum, i.e.",
"$C_l$ , depends on the Universe evolution, a primary spectrum of inhomogeneity, and the propagation of inhomogeneity during the evolution.",
"Then, the Universe evolution is well described by $\\Lambda $ CMD [16] and it can be extrapolated to the age of Universe, when the snapshot of CMBR was done, i.e about 380 thousands years after the big bang.",
"The primary spectrum of inhomogeneity is suggested to be close to the Harison–Zeldovich distribution of so called “no-scale” limit at a spectral index $n_s(k)=1$ : $\\left\\langle \\frac{\\delta \\rho ^2(0)}{\\rho ^2}\\right\\rangle =A \\int \\left(\\frac{k}{k_0}\\right)^{n_s(k)-1}\\,\\mathrm {d\\ln k},$ where $\\delta \\rho (r)=\\rho (r)-\\rho $ denotes a contrast of energy density at comoving coordinate, $k$ is a wave vector conjugated to the comoving coordinate, $A$ stands for an amplitude at a reference value of $k_0$ .",
"The spectral index and amplitude are subjects to fit the observed spectrum of $C_l$ .",
"Finally, the propagation of inhomogeneity includes the sound and further smearing of waves by the gravitation.",
"The concordance model of cosmology in the framework of general relativity well fits $C_l$ with the flat space and tuned amount of dark components [16]: relative fractions of dark matter $\\Omega _\\mathrm {DM}\\approx 20$ % and dark energy in the approximation of cosmological constant $\\Omega _\\Lambda \\approx 76$ %.",
"The baryonic matter composes only $\\Omega _b\\approx 4$ %.",
"This value is dictated by heights of distinct initial three acoustic peaks in the multipole spectrum, $C_l$ .",
"That is the dark matter fraction, which regulates the relative heights and positions of peaks up to small variations due to the parameters of primary spectrum of inhomogeneity.",
"Next, the amount of baryonic matter and the temperature of CMBR fixes the baryon-to-photon ratio of densities $\\eta _b=n_b/n_\\gamma $ , that is the only free parameter in calculation of elements abundance during the big bang nucleosynthesis.",
"The current state of measurements of elements abundance extrapolated to the primary abundance is compatible with $\\eta _b$ extracted from the $\\Lambda $ CDM fit of CMBR anisotropy [17].",
"Thus, the success of concordance model stimulates direct searches for an appropriate heavy dark matter particle at colliders and underground big-volume experiments, sensitive to suppressed, but non-zero cross sections of dark matter interaction with the ordinary matter.",
"However, even a discovery of candidate for the dark matter particle would not withdraw the problem of ad hoc tuning of dark sector.",
"Moreover, it would sharpen the need to search for the dynamical reasons causing the adjustment of dark matter, i.e to look beyond the general relativity.",
"A model of gravity involving the critical acceleration should naturally include both empirical laws such as the Tully-Fisher relation at the galactic scales and correct descriptions of Universe evolution, observed features of CMBR, large scale structure and elements abundance, so that the model would give a successful approach being alternative to the general relativity in cosmology, of course.",
"At galactic scales, M. Milgrom invented the modified Newtonian dynamics (MOND) [8] stating the gravitational acceleration $g$ $g\\,\\zeta (g/\\tilde{g}_0)=-\\nabla \\phi _M,\\qquad \\zeta (y)=\\frac{y}{\\sqrt{1+y^2}},$ where the critical acceleration extracted from the modern analysis of rotational curves is given by $\\tilde{g}_0=(1.24\\pm 0.14)\\times 10^{-10}\\;m/s^2$ [10], while $\\phi _M$ denotes the Newtonian gravitational potential of ordinary matter, satisfying $\\nabla ^2\\phi _M=-4\\pi \\,G\\,\\rho _M$ .",
"At $g/\\tilde{g}_0\\gg 1$ , we get the Newtonian limit of gravitational forceSub-leading terms are suppressed, so that the force at the Earth and in the Solar system is not distinguishable from the Newtonian one., while at $g/\\tilde{g}_0\\ll 1$ the Tully-Fisher law is satisfied by construction.",
"Note that (REF ) successfully predicts the rotational curves by the given distribution of visible matterThe only parameter of fitting the rotational curves within MOND is the light-to-mass ratio, which strictly correlates with astrophisical expectations for given galaxies.",
"Moreover, in gas-rich galaxies this uncertainty is absent, that means the MOND predicts rotational curves with no adjustment of any parameters, see details in [4].",
"with appropriate imprints of its features, see review in [4].",
"However, the straightforward insertion of (REF ) into the dynamics at cosmological scales would results in an inconsistent distortion of vacuum homogeneity during the evolution of Universe, for instance, i.e.",
"in the vacuum instability as was shown in [18], wherein authors offered to introduce the cosmological behavior of critical acceleration in the form of $\\tilde{g}_0\\mapsto g_0=g_0^\\prime \\,|x|,$ where the distance is determined by comoving coordinate $r$ and scale factor of Universe expansion $a(t)$ , so that $x=a(t)\\,r$ .",
"Then, the homogeneous cosmology with the gravity law modified at accelerations below the critical value of $\\tilde{g}_0$ is consistent, that constitutes the cosmological extrapolation of MOND [18].",
"The cosmological regime is matched to MOND at a size of large scale structure $|x|_\\mathrm {lss}$ , i.e.",
"at the characteristic scale of inhomogeneity, that is related to the acoustic scale and sound horizon in the baryon-electron-photon plasma (see details in [18]).",
"In the framework of cosmological extrapolation of MOND with the interpolation function $\\zeta (y)$ in (REF ) the gravity equations for the evolution of homogeneous and isotropic Universe can be written in the form [18] $\\left(R_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^4=\\left(\\left(R_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2+\\left(K_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2 \\right)\\left(\\bar{R}_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2,$ in terms of Ricci tensor $R_{\\mu \\nu }$ for the metric $\\eta _{\\mu \\nu }$ and 4-vector in the direction of cosmological time $\\xi ^\\mu =(1,0)$ , wherein the matter energy-momentum tensor $T_{\\mu \\nu }$ defines $\\bar{R}_\\mu ^\\nu [\\eta ]=8\\pi G \\left(T_\\mu ^\\nu -\\textstyle {\\frac{1}{2}} \\eta _\\mu ^\\nu T\\right),$ while the extra tensor of curvature $K_\\mu ^\\nu $ is the Ricci tensor of de Sitter space-time being both homogeneous in time and space as well as isotropic $K_\\mu ^\\nu =3g_0^{\\prime } \\eta _\\mu ^\\nu .$ In addition to (REF ) the conservation of energy-momentum $\\nabla _\\mu T^\\mu _\\nu =0$ is hold, of course.",
"Then, in the limit of general relativity we put $\\left(R_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2\\gg \\left(K_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2$ , and we find $R_\\mu ^\\nu \\approx \\bar{R}_\\mu ^\\nu $ , that results in the Einstein equations $R_\\mu ^\\nu =8\\pi G \\left(T_\\mu ^\\nu -\\textstyle {\\frac{1}{2}} \\eta _\\mu ^\\nu T\\right).$ In the limit of $\\left(R_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2\\ll \\left(K_\\mu ^\\nu \\xi ^\\mu \\xi _\\nu \\right)^2$ we get the modified evolution of Universe, effective at present.",
"It is important that parameterizing the size of large scale structure by $|x|_\\mathrm {lss}\\sim \\lambda ^2/H_0$ at $g_0^\\prime \\sim H_0^2/\\lambda $ and moderate value of $\\lambda \\sim \\frac{1}{7}$ , we find the simplest solution for the coincidence problem, because the Milgrom acceleration $\\tilde{g}_0\\sim \\lambda H_0$ becomes close to the scale of cosmological constant $\\Lambda \\sim H_0^2$ .",
"Moreover, we will see that the dark matter fraction of energy is also regulated by the value of $\\lambda $ .",
"Eq.",
"(REF ) successfully fits the evolution of Universe measured by observing the brightness of type Ia supernovae versus the red shift [18].",
"So, the stellar magnitude $\\mu =\\mu _{abs}+5\\log _{10}d_L(z) + 25,$ depends on the photometric distance $d_L$ (in Mpc), determined by the Hubble constant evolution $H(z)=\\dot{a}/a$ , $d_L(z)=(1+z)\\int \\limits _0^z\\frac{dz}{H(z)},$ where $\\mu _{abs}$ is an absolute stellar magnitude of light source at the distance of 10 pc.",
"We show the Hubble diagram for the type Ia Supernovae in Fig.",
"REF .",
"The mean deviation squared per degree of freedom gives $\\chi ^2/\\mbox{d.o.f.",
"}=1.03$ for our fit with the following assignment of parameters: $\\begin{array}{rcl}q_0=-0.853, &\\qquad & \\,\\,z_t=0.375, \\\\\\,\\,h=0.71, & & \\Omega _b=0.115,\\end{array}$ where $q_0$ determines the deceleration parameter at red sift $z=0$ , $q(z)=-\\ddot{a}/aH^2(z)$ , $z_t$ stands for the red shift, when the acceleration is equal to zero, $h$ parameterizes the Hubble rate at $z=0$ via $H_0=h\\cdot 100\\,\\mathrm {km/s\\cdot Mpc^{-1}}$ .",
"Accepting the prescription of $g_0^\\prime =K_0H_0^2,$ we can find that $q_0=\\frac{1}{2}\\,K_0\\Omega _b\\left((1+z_t)^3-1\\right),$ when the energy density determined by the cosmological constant, is given by the energy budget of Universe, $\\Omega _\\Lambda =1-\\Omega _b$ .",
"The same values of deceleration, $q_0$ , and red shift of transition from the deceleration to acceleration, $z_t$ , could be obtained in concordance model of $\\Lambda $ CDMThe $\\Lambda $ CDM parameters are marked by bars.",
"at $\\frac{\\bar{\\Omega }_M}{\\bar{\\Omega }_b}=-\\frac{K_0}{q_0}.$ Therefore, the modified gravity with the critical acceleration, i.e.",
"at nonzero $K_0\\approx 7.9$ , determines the ratio of matter to baryon fractions of energy, so that at $q_0\\sim -1$ we get $\\bar{\\Omega }_M/\\bar{\\Omega }_b\\sim K_0$ , which correlates with the scale of large scale structure, considered above, $K_0\\sim 1/\\lambda $ .",
"Figure: The magnitude of supernova luminosity versus the redshift zz.",
"The data witherror bars are taken from the Union2 collection and the results ofHubble Space Telescope on the type Ia supernovae .",
"The curverepresents our fit in the framework of modified cosmology.Note that the quality of fit is not sensitive to valuable variations of baryonic fraction $\\Omega _b$ , while it was extracted from the appropriate value of sound horizon $r_s(z)=\\int \\limits _0^{t(z)}c_s\\,dt,$ where the speed of sound is determined by the baryon-to-photon ratio of energy $R$ , so that $c_s=\\frac{1}{\\sqrt{3}}\\,\\frac{1}{\\sqrt{1+R}},\\qquad R=\\frac{3}{4}\\,\\frac{\\rho _b}{\\rho _\\gamma }=\\frac{3}{4}\\,\\frac{\\Omega _b}{(1+z)\\Omega _\\gamma }.$ The value of sound horizon at $z=0.2$ and $z=0.35$ was extracted from baryonic acoustic oscillations (BAO) [19], which, in the case of baryon matter only, favor for the enhanced estimate of $\\Omega _b$ shown above.",
"The same conclusion follows from the calculation of acoustic scale in the spectrum of CMBR anisotropy [16], $l_A=\\frac{\\pi d_L(z_*)}{(1+z_*)\\,r_s(z_*)},$ here $z_*$ is the redshift of decoupling, when due to the recombination of electrons with protons the medium becomes transparent for photons (see analytical approximations for $z_*$ in terms of baryonic density, matter density and Hubble constant in [20].).",
"WMAP gives $l_A=302.69 \\pm 0.76$ , while we deduce $l_A=302.5$ [18] compatible with the uncertainty of measurement.",
"Hence, we expect that scale features of CBMR anisotropy could be fitted in the framework of cosmological extrapolation of MOND.",
"In present paper we describe the procedure of fitting the CMBR anisotropy spectrum with the model of modified gravity and point to accepted approximations in Section .",
"Some actual problems associated with the theory and phenomenology of our model of modified gravity, are considered in Section .",
"We present an itemized discussion of model verification and falsification in Section ."
],
[
"Fitting the CMBR anisotropy ",
"The tool for the calculation of multipole spectrum of CMBR anisotropy [21], [22], [23] operates with the Friedmann equation, which is not valid in the framework of cosmological extrapolation of MOND.",
"Nevertheless, we can integrate out the dynamical equations of our model in order to extract the Hubble rate at any red shift and to parameterize it with a mimic dark energy contribution in addition to the fraction of baryonic matter.",
"Indeed, exploring the general relativity in the case of baryonic matter and dark energy in the isotropic homogeneous curved space we get $\\hspace{17.07164pt}\\frac{H^2}{H_0^2} &=& \\frac{\\Omega _b}{a^3}+\\frac{\\Omega _k}{a^2}+\\Omega _X(a), \\\\[2mm]-\\frac{2\\ddot{a}/a}{ H_0^2} &=&\\frac{\\Omega _b}{a^3}+\\Omega _X(a)(1+3w_X(a)),$ where $\\Omega _k$ stands for the contribution of space curvature, and the energy budget holds $\\Omega _b+\\Omega _X(1)+\\Omega _k=1$ .",
"Excluding $\\Omega _X(a)$ , we get the expression for the dark equation of state $w_X(a)=-\\frac{1}{3}\\left(1+\\frac{2\\frac{\\ddot{a}}{aH_0^2}+\\frac{\\Omega _b}{a^3}}{\\frac{H^2}{H_0^2}-\\frac{\\Omega _b}{a^3}-\\frac{\\Omega _k}{a^2}}\\right).$ Here we insert the expression for the acceleration that follows from eq.",
"(REF ) for the modified gravity including the actual values of energy fractions for baryons and vacuum, $\\Omega _b$ and $\\Omega _\\Lambda $ , respectively, $\\frac{\\ddot{a}}{a H_0^2}=\\left(\\Omega _\\Lambda -\\frac{\\Omega _b}{2a^3}\\right)\\frac{1}{\\sqrt{2}}\\sqrt{1+\\sqrt{1+\\frac{(2g_0^\\prime /H_0^2)^2}{\\left(\\Omega _\\Lambda -\\frac{\\Omega _b}{2a^3}\\right)^2}}}.$ Note that the red shift of transition from the acceleration to deceleration of Universe, $z_t$ is related to the fraction of cosmological constant due to $\\Omega _\\Lambda =\\frac{1}{2}\\,\\Omega _b(1+z_t)^3.$ Again, eq.",
"(REF ) clearly shows that putting $g_0^\\prime $ equal to zero, we get the dynamics of general relativity, otherwise near $\\ddot{a}=0$ the dynamics enters the region of strong dominance of modification and the greatest deviation from the general relativity, as it is actual at present.",
"Integrating out (REF ) at the initial condition $\\dot{a}/a({t=t_0})=H_0$ , we obtainIn practice, we use the scaling variable $\\tau =tH_0$ , that completely covers the differential equations under consideration.",
"the scaling quantity $H/H_0$ required for the complete definition of r.h.s.",
"in (REF ).",
"Fig.",
"REF shows the behavior of $w_X$ versus the scale factor $a$ as we have calculated in the cosmological extrapolation of MOND with parameters listed in (REF ) at $\\Omega _k=0$ .",
"Small variations of the spatial curvature in limits $|\\Omega _k|<0.02$ , deceleration parameter $q_0$ , transitional red shift $z_t$ and baryonic fraction $\\Omega _b$ within 10% lead to negligible changes, which are only just visible in the figure.",
"We emphasize that the modification of gravity predicts the very specific dependence of equation of state for the dark energy, that we will discuss in Section .",
"Figure: The equation of state w X w_X versus the scale factor aa in themodel of modified gravity involving the critical acceleration.After the definition of referenced homogeneous evolution of Universe under the modified gravity, we can use the standard tool for the calculation of CMBR anisotropy spectrum [21], [22], [23].",
"However, in this way the propagation and smearing of sound waves would be described in the framework of general relativity with no dark matter, i.e.",
"at $\\Omega _\\mathrm {DM}=0$ , while we have to modify this procedure in accordance with a structure formation under the modified law at accelerations below the critical one.",
"The problem is that such the modification of perturbation transfer function is not linear, and the appropriate machinery of calculation is not yet developed.",
"That is missing point of our consideration, of course.",
"Nevertheless, the extensive usages of tool have shown that the influence of dark matter on the spectrum is reduced to relative enhancement of third peak, whereas this enhancement is due to enforcing the gravity.",
"So, since the modified gravity produces the very similar effect of enforcing the gravity, we can expect that it could results in the same fine feature as concerns for the enlarging the third peak.",
"In this respect, we have to emphasize that a complete axiomatic approach based on a formulation of action for a modified gravity in terms of given extended set of gravitational fields has got the advantage in calculating of various predictions including the CMBR anisotropy.",
"So, we can mention the following fully relativistic schemes (more examples and references find in [4]): Bekenstein's theory of tensor-vector-scalar (TeVeS) gravitational fields [24] involving Maxwellian vector field and reproducing the critical acceleration in the case of isolated gravitational source, equivalent to MOND, Moffat's modified gravity [25], [26], [27], giving the approximation of gravity law similar to the MOND, generalized TeVeS theories [28] with non-Maxwellian vector field.",
"However, first, these theories include dark gravitational fields actually replacing the dark matter that looks like a refinement of problem.",
"Second, all of them have the strict theoretical illness: there are configurations with unlimited, infinite negative energy, that leads to instability of physically sensible solutions (see details and references in [4]).",
"Third, the critical acceleration is still introduced ad hoc with no reasonable connection to the present Hubble rate or the cosmological constant.",
"So, we prefer for the phenomenological approach, which does not introduce new artificial and heuristic notions.",
"In this way, we can investigate the role of critical acceleration in the modified gravity by studying various phenomena step by step in order to find fundamental features and differences from the general relativity.",
"At present, we try to fit the spectrum of CMBR anisotropy by using the modified evolution of homogeneous Universe and optimizing the primary spectrum of inhomogeneity, which develops as the sound smeared by ordinary gravity.",
"So, we suggest that a modification of smearing will not be very crucial for the main features of spectrum.",
"Figure: The spectrum of CMBR anisotropy calculated in the model of modified gravity,l(l+1)C l /2πl(l+1)C_l/2\\pi (in μK 2 \\mu K^2)versus the multipole number ll, in comaprison with WMAP data .The dashed line represents the Harrison–Zeldovich approximation, whilethe solid curve corresponds to the running spectral index (as described in the text).The results of such the fitting are presented in Figs.",
"REF –REF .",
"First, we study the fit of WMAP data [16] with the Harrison–Zeldovich primary spectrum (HS) at $n_s=1$ , which is shown in Fig.",
"REF by dashed line.",
"It is spectacular that the no-scale HS prescription correctly reproduces the angular scale of multipole momentum, i.e.",
"the position and profile of first acoustic peak.",
"This feature is obtained due to the correct adjustment of this scale by the sound horizon in the case of no dark matter [18] as mentioned in the Introduction.",
"Then, we find that the running of spectral index $n_s(k)=n_s^{(0)}+n_s^\\prime \\ln k/k_0$ leads to suitable description of both, first and second acoustic peaks in the modified cosmology with no dark matter.",
"Setting parameters equal to the following valuesWe list the quantities, which we change from default values assigned in the tool [21], [22], [23].",
"Other quantities have been set to its standard prescriptions.",
": $\\begin{array}{rclcrcl}n_s^{(0)} & = & 1.625, & & n_s^\\prime & = & 0.24, \\\\A & = & 5.5\\times 10^{-9}, & & \\Omega _k & = & -0.015, \\\\z_{re} & = & 23, & & k_0 & = & 0.04\\; \\mbox{Mpc}^{-1}, \\\\\\end{array}$ we get the fit shown by solid line in Fig.",
"REF .",
"Set (REF ) needs comments.",
"Figure: The same as in Fig.",
"with addition of BOOMERANG data, (light crosses) and ACBAR data (dark crosses).First, the running of spectral index allows to adjust the relative height of second acoustic peak in the spectrumThe opportunity to fit the second peak in the model with suppressed dark matter was considered in [29]..",
"This running is dynamically essential and numerically significant.",
"Moreover, because of sizable value of slope of spectral index, we expect that the approximation neglecting the higher orders of expansion versus the logarithm of comoving wave vector $\\ln k$ , would be inaccurate at large intervals of multipole moment.",
"Figure: The same as in Fig.",
", but focused at the region ofthird and further peaks (l>600l>600).Second, we introduce a small spatial curvature in order to adjust the position of peaks, which have been displaced under the strong running of spectral index.",
"Then, the spatial curvature essentially improves the quality of fit.",
"Remember, that such the low value of spatial curvature is consistent with the equation of state we have deduce from the modified gravityNote that at $a\\rightarrow 0$ the equation of state for the mimic dark energy tends to $w_X(0)\\approx -\\frac{1}{3}$ , which points to a possibility of nonzero spatial curvature..",
"Moreover, acceleration $\\ddot{a}$ does not involve the spatial curvature because of its specific value of state parameter $w_k=-\\frac{1}{3}$ , giving $\\rho _k+3p_k=0$ .",
"We note also that a nonzero value of spatial curavture obeys the scaling $|K_0^2\\Omega _k|\\sim 1$ , that is a feature consistent with our solution or treatment of coincidence problem (see the Introduction).",
"Third, the red shift of reionization $z_{re}$ , when inhomogeneities of cold gas are contacted due to the gravity in order to form stars, refers to the heating of gas, that transforms it to plasma again.",
"It causes a reduction of intensity of radiation passing through the hot secondary plasma.",
"So, the amplitude of primary CMBR, $A$ correlates with the red shift of reionization.",
"In MOND with the enhanced gravity at the stage of diluted gas, one expects that the star formation starts early than in the general relativity [4], hence, we try to improve the fit quality by enlarging the red shift of reionizationWe set $z_{re}$ via “a sizable change”: the typical value of $z_{re}=11$ in $\\Lambda $ CDM has been enlarged twice for a distinguishability with no strict reasons or prerequisites..",
"The corresponding improvement of mean deviation squared is equal to -0.11 per degree of freedom.",
"The optical depth is shifted from $0.9$ to $0.78$ .",
"So, we refer this adjustment as fine effect beyond a sole significance, as well as slow variation of pivot wave vector $k_0$ from $0.05$ to $0.04\\;\\mbox{Mpc}^{-1}$ .",
"Thus, in the simplest way of modifying the background cosmology by the cosmological extrapolation of MOND, we find that WMAP data [16] can be fitted at the following mean deviation squared per degree of freedom: $\\chi ^2/\\mbox{d.o.f.",
"}=1.34,$ which is just 0.19 worse than the typical value of $\\chi ^2/\\mbox{d.o.f.",
"}$ in the concordance model of $\\Lambda $ CDM.",
"Fig.",
"REF clearly shows that at $l>600$ the WMAP data suffer from huge statistical and systematical uncertainties.",
"Indeed, the inclusion or exclusion of data at $l>600$ , in practice, do not change $\\chi ^2/\\mbox{d.o.f.",
"}$ Therefore, the WMAP data beyond the first and second peaks are not conclusive, at all to the moment.",
"Then, higher multipoles can be studied due to the BOOMERANG and ACBAR data sets [30], [31], [32].",
"We present the comparison in Fig.",
"REF .",
"Evidently, the dark matter indication is reduced to the different form of third peak rise.",
"Therefore, we can hope that a nonlinear smearing of sound waves in MOND could give the same effect in the region of third peak, too, although to the moment we see the clear tension between the data in the region of third acoustic peak and the simplest version of fit based on the modified cosmology of homogeneous Universe only.",
"Thus, we need the development of procedure to calculate the transfer function of inhomogeneity versus the red shift [33] in the case of nonlinear MOND.",
"The same is true as concerns for the simulation of large scale structure formation visible at present time.",
"Nevertheless, we insist on quite the successful fitting of CMBR anisotropy in the framework of cosmological extrapolation of MOND, as the variant of modified gravity involving the critical acceleration in the case of homogeneous matter."
],
[
"Main problems",
"The enhanced value of baryon fraction in the energy budget, $\\Omega _b\\approx 0.115$ implies the enhanced value of baryon-to-photon ratio $\\eta _b$ being the only free parameter of BBN calculation [17].",
"The BBN takes place during the period when the deviations from the general relativity are negligible.",
"Therefore, the primary abundance of light elements in cosmological extrapolation of MOND should differ from the BBN estimated within the concordance model of $\\Lambda $ CDM.",
"At present the data on the helium abundance has large uncertainties which are compatible with both models under consideration.",
"Then, the deuterium and lithium primary abundances are clearly able to discriminate between the enhanced value of $\\eta _b$ and its concordance value.",
"However, these quantities are not measured directly, they are extrapolations from a suggested model of evolution.",
"Indeed, we observe the visible sources of light, that mean the primary matter is contracted in stars with further development of nuclear reactions, and a model of star evolution has to reproduce the primary values of abundances.",
"In this respect, one has to take into account the different red shifts of star formation, i.e.",
"different ages of luminous objects, as considered in the framework of general relativity or MOND, of course.",
"Next, at present the standard model for the extrapolation to primary abundances is not self consistent, because it predicts different ratios of both deuterium to lithium-7 and lithium-7 to lithium-6, whereas the former ratio is in bright tension of prediction with the extracted values, while the later is in a deep contradiction (about three order of magnitude!).",
"Thus, the present BBN status can not be surely conclusive.",
"It is desirable to get more better reliability of procedure for the empirical extraction of primary abundance of light elements.",
"On the other hand, the doubling of baryonic fraction should appear in cosmological effects.",
"So, this doubling could appropriately explain a missing mass in galactic clusters, as found even within MOND.",
"Then, a significant portion of baryonic matter should be in cold form (Jupiter-like objects), for instance.",
"Note that even in $\\Lambda $ CDM the visible matter composes the tenth fraction of all baryonic matter, only, hence, the most of baryons are in cold form.",
"Next, the visible large scale structure should be explained by appropriate propagation of primary spatial inhomogeneities.",
"However, this issue of modified gravity involving the critical acceleration is not yet developed because of nonlinearity of the problem."
],
[
"Discussion and conclusion",
"In this paper, we have shown that the cosmological extrapolation of MOND as the modified gravity involving the critical acceleration, can successfully reproduce main features of multipole spectrum of CMBR anisotropy.",
"Let us list conclusions of our investigation.",
"The modified dynamics adjusted to empirical values of the Hubble rate, sound horizon in the baryon-photon medium, acoustic scale in the multipole spectrum of CMBR anisotropy and magnitudes of type Ia supernovae at red shifts $z<2$ , mimics the dark energy with the very specific equation of state $w_X$ , shown in Fig.",
"REF .",
"This is the falsifiable prediction of cosmological extrapolation of MOND.",
"It can be verified in the nearest future by extensive measuring of type Ia supernova magnitudes versus the red shift.",
"Even $z<2$ , i.e.",
"the scale factor variation within the interval $0.3< a <1$ , would be enough in order to make decision on the direct falsification of cosmological model involving the critical acceleration.",
"Moreover, such the exotic behavior of dark energy state parameter $w_X$ , if would be confirmed, will be marginally artificial for the general relativity, that would mean the straightforward indication of its inadequateness.",
"The spectral index of primary spatial inhomogeneity $n_s(k)=n_s^{(0)}+n_s^\\prime \\ln k/k_0$ , essentially runs.",
"The character of running is model-dependent, and It is very different in the concordance model of general relativity and in the cosmological extrapolation of MOND: in the modified gravity the running is rather fast, and its parameters signalize on the hybrid (multified) inflation, that could generate such the spectrum, while the general relativity gives the slow running, which preferably corresponds to the simplest top-hill inflation due to a single inflaton field [34].",
"The modified gravity gives $n_s^{(0)}>1$ at $n_s^\\prime \\approx \\frac{1}{4}>0$ , while the general relativity results in $n_s^{(0)}<1$ at $n_s^\\prime \\rightarrow 0$ .",
"The fast running probably indicates the need to improve the calculation tool in order to include higher derivatives of spectral index with respect to logarithm of wave vector.",
"The spectral index of primary inhomogeneity define initial conditions for the transfer of inhomogeneity during the evolution, which can be observed in baryonic acoustic oscillations in large scale structure of present Universe [19].",
"The procedure of calculating the transfer function in the framework of modified gravity with the critical acceleration is nonlinear, and it is not still developed, that does not allow us to make a comparison with data, at present.",
"The structure growth is enhanced in MOND, and it can probably need for additional mechanism of smearing the acoustic oscillations [4].",
"The modified gravity in our version results in doubly enhanced fraction of baryons, approximately.",
"This means that baryon-to-photon ratio is twice large, at least.",
"More reliable estimates of primary abundance of light elements is required in order to discriminate the general relativity from the modified gravity by the big bang nucleosynthesis.",
"Therefore, BBN can give the falsification of cosmological extrapolation of MOND.",
"The doubling of baryons in the form of cold baryonic matter (Jupiter-like objects, for instance) should be found in observations.",
"For instance, the mass deficit in galaxy clusters described within MOND, can signalize on the appropriate enhancement of baryons.",
"The multipole spectrum of CMBR anisotropy needs improvements of accuracy in the range of third acoustic peak.",
"If the model of modified gravity will still miss the correct description of third peak after such the improvement, then this would point to the extension of simplest consideration by strict inclusion of inhomogeneity propagation within the modified dynamics.",
"Finally, we have shown that the coincidence problem of general relativity is inherently solved in the framework of cosmological extrapolation of MOND: the critical acceleration is connected to the extra Ricci tensor of de Sitter space, involved in the gravity equations; then, it is naturally correlates with the cosmological constant.",
"In addition, the modified gravity is mostly effective at zero acceleration of Universe expansion.",
"That is why the coincidence notion is actual at present.",
"Keeping in mind soluble problems mentioned, we state that the cosmological extrapolation of MOND is quite successful in cosmology.",
"Moreover, we can falsify it in the nearest future, although the same note on the verification is actual for the general relativity, too.",
"This work was partially supported by the grant of Russian Foundations for Basic Research 10-02-00061.",
"I thank Dr. Timofeev S A for discussions."
]
] | 1204.0869 |
[
[
"On the Spin of the X(3872)"
],
[
"Abstract Whether the much studied X(3872) is an axial or tensor resonance makes an important difference to its interpretation.",
"A recent paper by the BaBar collaboration raised the viable hypothesis that it might be a 2-+ state based on the 3 pions spectrum in the X -> J/psi omega decays.",
"Furthermore, the Belle collaboration published the 2 pions invariant mass and spin-sensitive angular distributions in X -> J/psi rho decays.",
"Starting from a general parametrization of the decay amplitudes for the axial and tensor quantum numbers of the X, we re-analyze the whole set of available data.",
"The level of agreement of the two spin hypotheses with data is interpreted with a rigorous statistical approach based on Monte Carlo simulations in order to be able to combine all the distributions regardless of their different levels of sensitivity to the spin of the X.",
"Our analysis returns a probability of 5.5% and 0.1% for the agreement with data of the 1++ and 2-+ hypotheses, respectively, once we combine the whole information (angular and mass distributions) from both channels.",
"On the other hand, the separate analysis of J/psi rho (angular and mass distributions) and J/psi omega (mass distribution) indicates that the 2-+ assignment is excluded at the 99.9% C.L.",
"by the former case, while the latter excludes at the same level the 1++ hypothesis.",
"There are therefore indications that the two decay modes behave in a different way."
],
[
"Introduction",
"Although the $X(3872)$ resonance is the most studied among the exotic $XYZ$ states, since its discovery in 2003, its quantum numbers have not been definitively identified yet.",
"The CDF collaboration concluded from the analysis of the angular distributions and correlations of the $X$ decay products that the possible quantum numbers are $J^{PC}= 1^{++}$ or $2^{-+}$ [3].",
"Similar results have been found very recently by the BELLE collaboration [2].",
"In the latter paper the $\\pi ^+ \\pi ^-$ invariant mass distribution is analyzed under the $1^{++}$ and $2^{-+}$ hypotheses, finding preference for the former whereas no preferred assignment emerges if an interfering contribution with the isospin-violating decay $\\omega \\rightarrow \\pi ^+ \\pi ^-$ is added to the amplitude.",
"The picture becomes more puzzling as one considers the analysis by the BaBar Collaboration of the decay $X \\rightarrow J/\\psi \\;\\pi ^+ \\pi ^- \\pi ^0$ [1].",
"The expected $3 \\pi $ invariant mass distribution agrees with data slightly better if the $2^{-+}$ signature is assumed.",
"This result on the $3\\pi $ spectrum was later confirmed in [4].",
"In a recent paper [5] the pion invariant masses in the decays $X \\rightarrow J/\\psi \\; 2\\pi $ and $X \\rightarrow J/\\psi \\;3\\pi $ are simultaneously analyzed with a combined fit, concluding that present data favor the $1^{++}$ assignment.",
"In our view, the $2\\pi $ invariant mass distribution is not able to resolve the two hypotheses despite of the high statistics, and the way the fit was performed leads to the dilution of the sensitivity of the $3\\pi $ channel.",
"To improve the analysis of all available data sensitive to the spin of the $X$ $i)$ we write the decay matrix elements for both the $1^{++}$ and $2^{-+}$ hypotheses as given by enforcing Lorentz invariance and parity considerations, $ii)$ we give a functional dependency on the decay momenta to the couplings introducing a length scale $R$ – which is to be related to the finite size of the hadrons participating to the interactions, $iii)$ we use the matrix elements of $\\rho $ and $\\omega $ decays to take into account the appropriate decay waves; we do not pursue the Blatt-Weisskopf description as we find that all spin structure can appropriately be taken into account with no further approximations, $iv)$ we perform a global fit to exploit the information contained in all the distributions available and we adopt a statistical approach appropriate when distributions with different sensitivities to the parameters of interest are combined."
],
[
"Matrix elements",
"The matrix elements describing the amplitudes $X\\rightarrow J/\\psi \\,V$ (where $V=\\rho ,\\omega $ ) are obtained by Lorentz, gauge invariance and parity considerations leading to the formulae reported below [4].",
"In the $X(1^{++})$ case we have $\\langle \\psi (\\epsilon ,p) V(\\eta ,q)|X(\\lambda ,P)\\rangle = g_{1\\psi V} \\;\\epsilon ^{\\mu \\nu \\rho \\sigma }\\;\\lambda _\\mu (P)\\;\\epsilon ^*_\\nu (p)\\;\\eta ^*_\\rho (q)\\;P_\\sigma $ the polarization vectors carry a complex conjugation when referred to final states.",
"In the $2^{-+}$ case we have a more complicated structure $\\langle \\psi (\\epsilon ,p) V(\\eta ,q)|X(\\pi ,P)\\rangle = g_{2\\psi V} \\;T_A + g_{2\\psi V}^\\prime \\;T_B$ where $\\pi $ is the polarization tensor for a spin two particle with massThe sum over polarizations is $\\sum _{\\rm pol}\\pi _{\\mu \\nu }(k)\\pi ^*_{\\alpha \\beta }(k) = \\frac{1}{2} T_{\\mu \\alpha } T_{\\nu \\beta } + \\frac{1}{2} T_{\\mu \\beta } T_{\\nu \\alpha } - \\frac{1}{3} T_{\\mu \\nu } T_{\\alpha \\beta }$ with $T_{\\mu \\nu } = -g_{\\mu \\nu } + k_\\mu k_\\nu / m^2$ and $k^2=m^2$ .",
"and a standard notation is used for the remaining polarization vectors.",
"We find that $T_{A}$ and $T_{B}$ are given by $T_A=\\epsilon ^{*\\alpha }(p)\\:\\pi _{\\alpha \\mu }(P)\\;\\epsilon ^{\\mu \\nu \\rho \\sigma }\\;p_\\nu \\;q_\\rho \\;\\eta ^*_\\sigma (q)- (\\epsilon ,p \\leftrightarrow \\eta , q )$ and $T_B=Q^\\alpha \\:\\pi _{\\alpha \\mu }(P)\\;\\epsilon ^{\\mu \\nu \\rho \\sigma }\\;P_\\nu \\;\\epsilon ^*_\\rho (p)\\;\\eta ^*_\\sigma (q)$ where $Q=p-q$ and $P=p+q$ .",
"The coupling $g_{1\\psi V}$ is real whereas the couplings $g_{2\\psi V}$ and $g_{2\\psi V}^\\prime $ are separately real but can have a complex relative phase.",
"This is due to the fact that, on the basis of Lorentz invariance and parity conservation, we can indeed write three terms $T_{A}^{(\\lambda )},T_{B}^{(\\lambda )}$ and $T_{C}^{(\\lambda )}$ where $\\lambda $ labels one out of the $3\\times 3\\times 5$ polarization combinations which define $T_{A},T_{B},T_{C}$ .",
"$T_{C}$ is the same as $T_{A}$ but with a plus relative sign between the two terms on the rhs of (REF ).",
"Only two out of these three terms are linearly independent (say $T_{A}$ and $T_{B}$ ) for $\\left|\\sum _{\\lambda =1}^{3\\times 3\\times 5} T_{A}^{(\\lambda )}T_{C}^{(\\lambda )*}\\right|^{2}=\\sum _{\\lambda } |T_{A}^{(\\lambda )}|^{2} \\sum _{\\lambda } |T_{C}^{(\\lambda )}|^{2}$ which is the equality of the Schwartz inequality: this holds if $z_{1} T_{A}= z_{2}T_{C}$ where $z_{1}$ and $z_{2}$ are two complex numbers both different from zero.",
"Thus we can exclude $T_{C}$ and retain $T_{B}$ and, in general, $z_{1}T_{A}$ to characterize the decay amplitude.",
"In Refs.",
"[2], [1], [5] the $P$ -wave fit functions contain a Blatt-Weisskopf angular momentum barrier factor of the form $\\left(1+R^2 q^{*2}\\right)^{-1/2}$ , where $q^*$ is the $X$ decay 3-momentum.",
"The value of $R$ cannot be extracted from data since the $P$ -wave distribution will approach the $S$ -wave distribution in the limit $R\\rightarrow \\infty $ , so that if we let $R$ free, the fit will not converge.",
"On the other hand, in our discussion we do not need any barrier factor the decay wave being dictated by the expressions of the matrix elements.",
"We instead take into account the finite size of the $X$ (and of $V$ and $J/\\psi $ as well) introducing a `polar' form factor, namely $g \\rightarrow \\frac{ g }{(1+R^2 q^{*2})^{n}}$ where $g$ stands in general for $g_{1\\psi V}$ , $g_{2\\psi V}$ and $g_{2\\psi V}^\\prime $ .",
"We tested the values $n=1$ and $n=2$ (the latter coinciding with the Fourier transform of a an exponential $g(r)\\sim \\exp (-r/R)$ strong charge distribution).",
"Both the fitting functions turn out to be rather effective at improving our results, with no significative change for the the two choices of $n$ .",
"We also underscore that $g_{1\\psi V}$ (regulating the $S$ -wave decay) is assumed to have the same polar behavior since Eq.",
"(REF ) does not concern any orbital angular momentum considerations.",
"The size parameters $R_{J}$ will eventually be fitted from data.",
"As for the $\\rho $ and $\\omega $ decay amplitudes, we use $\\langle \\pi ^+(p) \\pi ^-(q) |\\rho \\left(\\epsilon ,P\\right)\\rangle = g_{\\rho 2\\pi } \\,\\epsilon \\cdot p$ which describes a $P$ -wave decay (the square modulus of this matrix element is $g^2_{\\rho 2\\pi }$ times the decay momentum squared).",
"For the $\\omega $ we have $\\langle \\pi ^+(p) \\pi ^-(q) \\pi ^0(r) | \\omega \\left(\\epsilon ,P\\right) \\rangle = g_{\\omega 3\\pi }\\, \\epsilon ^{\\mu \\nu \\rho \\sigma } \\epsilon _\\mu p_\\nu q_\\rho r_\\sigma $ The last two couplings will simply be written in terms of the partial widths $\\Gamma (\\rho \\rightarrow 2\\pi )$ and $\\Gamma (\\omega \\rightarrow 3\\pi )$ , as shown in the next section."
],
[
"Decay Widths",
"We have to calculate the partial widths $\\Gamma (X\\rightarrow J/\\psi \\;\\pi ^+\\pi ^-)$ and $\\Gamma (X\\rightarrow J/\\psi \\;\\pi ^+\\pi ^-\\pi ^0)$ .",
"In what follows we will neglect the $\\rho $ -$\\omega $ mixing since we demonstrate in Appendix that it does not alter significantly the results.",
"The partial widths in the narrow width approximation are [4] $\\begin{split}\\Gamma (X\\rightarrow J/\\psi \\;\\pi ^+\\pi ^-) &=\\frac{1}{2J+1}\\frac{1}{48\\pi m^2_X}\\int \\;ds\\;\\sum _{\\begin{array}{c}{\\rm pol}\\end{array}}|\\langle \\psi \\;\\rho (s)|X\\rangle |^2p^*(m^2_X,m^2_\\psi ,s)\\\\&\\quad \\times \\frac{1}{\\pi }\\frac{1}{(s-m_\\rho ^2)^2+(m_\\rho \\Gamma _\\rho )^2} \\int d\\Phi ^{(2)} \\sum _{\\begin{array}{c}{\\rm pol}\\end{array}} |\\langle \\pi ^+ \\pi ^-|\\rho (s) \\rangle |^2\\end{split}$ where by $d\\Phi ^{(2)}$ we mean the 2-body phase space measure.",
"The decay momentum $q^{*}$ in the matrix element $\\langle \\psi \\;\\rho (s)|X\\rangle $ coincides with $q^{*}\\equiv p^*(m^2_X,m^2_\\psi ,s)$ ; similarly for the width in $J/\\psi \\; \\omega $ discussed below.",
"The sum over polarizations in (REF ), simply yields $\\sum _{\\begin{array}{c}{\\rm pol}\\end{array}} |\\langle \\pi ^+ \\pi ^- |\\rho (s) \\rangle |^2 = g_{\\rho 2\\pi }^2\\; p^*(s,m^2_\\pi ,m^2_{\\pi })^2$ Finally, we can eliminate the coupling by evaluating the (REF ) on the mass-shell and by relating it to the partial width $\\Gamma \\left(\\rho \\rightarrow \\pi \\pi \\right)$ $g^2_{\\rho 2\\pi }= 6 m^2_\\rho \\;\\Gamma (\\rho \\rightarrow \\pi \\pi )\\;\\frac{4\\pi }{p^*(m_\\rho ^2,m^2_\\pi ,m^2_\\pi )^3}$ Inserting the above expressions (REF ), (REF ) into (REF ) gives $\\begin{split}\\Gamma (X\\rightarrow J/\\psi \\;\\pi ^+\\pi ^-) &=\\frac{1}{2J+1}\\frac{1}{8\\pi m^2_X}\\int \\;ds\\;\\sum _{\\begin{array}{c}{\\rm pol}\\end{array}}|\\langle \\psi \\;\\rho (s)|X\\rangle |^2p^*(m^2_X,m^2_\\psi ,s)\\\\&\\quad \\times \\frac{1}{\\pi }\\frac{m_\\rho \\Gamma _\\rho \\; \\mathcal {BR}(\\rho \\rightarrow \\pi \\pi )}{(s-m_\\rho ^2)^2+(m_\\rho \\Gamma _\\rho )^2}\\frac{m_\\rho }{\\sqrt{s}}\\left(\\frac{p^*(s,m^2_\\pi ,m^2_{\\pi })}{p^*(m_\\rho ^2,m^2_\\pi ,m^2_\\pi )}\\right)^3\\\\\\end{split}$ Similarly, for the $\\omega $ we obtain $\\begin{split}\\Gamma (X\\rightarrow J/\\psi \\;\\pi ^+\\pi ^-\\pi ^0) &=\\frac{1}{2J+1}\\frac{1}{48\\pi m^2_X}\\int \\;ds\\;\\sum _{\\begin{array}{c}{\\rm pol}\\end{array}}|\\langle \\psi \\;\\omega (s)|X\\rangle |^2p^*(m^2_X,m^2_\\psi ,s)\\\\&\\quad \\times \\frac{1}{\\pi }\\frac{1}{(s-m_\\omega ^2)^2+(m_\\omega \\Gamma _\\omega )^2} \\int d\\Phi ^{(3)} \\sum _{\\begin{array}{c}{\\rm pol}\\end{array}} |\\langle \\pi ^+ \\pi ^- \\pi ^0|\\omega (s) \\rangle |^2\\end{split}$ Summing over the $\\omega $ polarizations $\\sum _{\\begin{array}{c}{\\rm pol}\\end{array}} |\\langle \\pi ^+ \\pi ^- \\pi ^0|\\omega \\left(s\\right)\\rangle |^2 = \\frac{g^2_{\\omega 3\\pi }}{s^3}\\frac{s}{4} \\left[\\left( m^2_0 + s - 2 \\sqrt{s} \\;\\omega \\right)\\left(\\omega ^2-m^2_0-4x^2\\right)- 4 m^2_+ \\left(\\omega ^2 - m^2_0\\right)\\right] \\equiv g^2_{\\omega 3\\pi } \\mathcal {M}\\!\\left(\\sqrt{s}\\right)$ where $\\omega = E_{\\pi ^0}$ , $x = \\frac{1}{2} \\left(E_{\\pi ^+}-E_{\\pi ^-}\\right)$ and $m_0 = m_{\\pi ^0}$ , $m_+ = m_{\\pi ^+} = m_{\\pi ^-}$ .",
"An adimensional coupling has been formed by substituting $g^2_{\\omega 3\\pi } \\rightarrow \\frac{1}{s^3} g^2_{\\omega 3\\pi }$ This rescaling is arbitrary and in part relies on the narrowness of the $\\omega $ .",
"Avoiding the introduction of the $1/s^3$ term, the separate fit of $3\\pi $ does not change significantly (for example, with $n=1$ , the $2^{-+}$ is unchanged, the $1^{++}$ gets worse from $\\chi ^2=9.9 \\rightarrow 11.1$ ).",
"In the combined fit of invariant mass distributions both hypotheses become a bit worse (again for $n=1$ , $2^{-+}$ : $\\chi ^2 = 17.7\\rightarrow 18.4$ ; $1^{++}$ : $\\chi ^2 = 25.2\\rightarrow 26.2$ ) so that the $\\Delta \\chi ^2$ remains almost unchanged.",
"See Sec.",
"REF .. We eliminate the coupling by evaluating (REF ) on the mass-shell $g^2_{\\omega 3\\pi }= 6 m_\\omega \\;\\Gamma (\\omega \\rightarrow 3\\pi )\\left( \\int d\\Phi ^{(3)} \\mathcal {M}\\!\\left(m_\\omega \\right) \\right)^{-1}$ Inserting (REF ), (REF ) into (REF ) we obtain $\\begin{split}\\Gamma (X\\rightarrow J/\\psi \\;\\pi ^+\\pi ^-\\pi ^0) &=\\frac{1}{2J+1}\\frac{1}{8\\pi m^2_X}\\int \\;ds\\;\\sum _{\\begin{array}{c}{\\rm pol}\\end{array}}|\\langle \\psi \\;\\omega (s)|X\\rangle |^2p^*(m^2_X,m^2_\\psi ,s)\\\\&\\quad \\times \\frac{1}{\\pi }\\frac{m_\\omega \\Gamma _\\omega \\; \\mathcal {BR}(\\omega \\rightarrow 3\\pi )}{(s-m_\\omega ^2)^2+(m_\\omega \\Gamma _\\omega )^2} \\frac{{\\Phi ^{(3)\\prime }}(\\sqrt{s},m_0,m_+,m_+)}{{\\Phi ^{(3)\\prime }}(m_{\\omega },m_0,m_+,m_+)}\\end{split}$ where ${\\Phi ^{(3)\\prime }}(\\sqrt{s},m_0,m_+,m_+) =\\frac{1}{32\\pi ^{3}} \\int _{m_0}^{\\omega _m} d\\omega \\int _{x_-}^{x_+} dx \\frac{1}{4s^2} \\left[\\left( m^2_0 + s - 2 \\sqrt{s} \\;\\omega \\right)\\left(\\omega ^2-m^2_0-4x^2\\right) - 4 m^2_+ \\left(\\omega ^2 - m^2_0\\right)\\right]$ with $\\omega _m=(m^2_0-4m^2_++s)/(2 \\sqrt{s})$ and $x_\\pm =\\pm \\frac{1}{2} \\sqrt{\\frac{\\left(\\omega ^2-m_0^2\\right) \\left(\\omega _m-\\omega \\right)\\sqrt{s}}{4m^2_+ +\\left(\\omega _m-\\omega \\right)\\sqrt{s}}}$ to be compared to the notations used in [4].",
"In formulae (REF ) and (REF ) angular correlations are not taken into account for we factorize matrix elements.",
"On the other hand, the only way to consider both off-shellness and angular correlations is to compute the full matrix element for the $1\\rightarrow 5$ decay $\\begin{split}\\Gamma \\left(B\\rightarrow K\\,X \\rightarrow K\\,J/\\psi \\,\\rho \\rightarrow K\\,l^+l^-\\,\\pi ^+\\pi ^-\\right) &=\\frac{1}{2 m_B} \\int \\;\\prod _{i=1}^5 \\frac{d^3p_i}{(2\\pi )^3 2E_i} (2\\pi )^4 \\delta \\left(p_B - \\sum _i p_i\\right) \\\\&\\quad \\times \\sum _{\\begin{array}{c}{\\rm pol}\\end{array}} |\\langle K\\,l^+l^-\\,\\pi ^+\\pi ^-|B \\rangle |^2\\end{split}$ The matrix element can be decomposed as $\\begin{split}\\langle K\\,l^+l^-\\,\\pi ^+\\pi ^-|B \\rangle &= \\langle l^+l^- | \\psi \\rangle \\, \\frac{1}{p^2_\\psi - m^2_\\psi + i m_\\psi \\Gamma _\\psi }\\, \\langle \\pi ^+ \\pi ^- |\\rho \\rangle \\,\\frac{1}{p^2_\\rho - m^2_\\rho + i m_\\rho \\Gamma _\\rho } \\\\&\\quad \\times \\langle \\psi \\,\\rho |X \\rangle \\, \\frac{1}{p^2_X - m^2_X + i m_X \\Gamma _X}\\, \\langle K \\,X |B \\rangle \\end{split}$ We already gave a form to $\\langle \\psi \\,\\rho |X\\rangle $ and $\\langle \\pi ^+ \\pi ^-|\\rho \\rangle $ in Sec.",
"for both $1^{++}$ and $2^{-+}$ .",
"Moreover $\\langle l^+ \\, l^- |\\psi \\rangle &= (\\epsilon _\\psi )_\\beta \\;\\bar{u}(l^-) \\gamma ^\\beta v(l^+)\\\\\\langle \\pi ^+ \\, \\pi ^- |\\rho \\rangle &= \\epsilon _\\rho \\cdot p_{\\pi ^+}\\\\\\langle X \\,K |B \\rangle &= \\left\\lbrace \\begin{aligned}&\\epsilon ^*_X \\cdot p_K & \\mbox{for }&1^{++}\\\\&(\\pi ^*_X)_{\\delta \\varphi } (p_K)^\\delta (p_K)^\\varphi & \\mbox{for }&2^{-+}\\end{aligned}\\right.$ The sum over polarizations of inner legs returns the usual numerators of spin-1 and spin-2 propagators.",
"The expressions obtained are not reported here because of their algebraic complexity These are available upon request in form of Fortran routines.. All matrix elements but the $\\langle \\psi \\,\\rho |X(2^{-+})\\rangle $ can be written in terms of one coupling only times a scalar function of momenta and polarizations.",
"The widths are expressed in terms of sums of products of couplings times scalar functions of momenta.",
"Products of couplings are absorbed within the fit parameters $r_{J}^{\\text{ang}}$ see Sec. .",
"Since the integral in Eq.",
"(REF ) has to be evaluated numerically, we use Monte Carlo techniques.",
"The integration on the 5-body phase space is carried out using importance sampling on the Breit-Wigner peaks.",
"As a further option of the calculation, unweighted decay configurations can be generated according to full matrix element weights.",
"As usual, in the calculations we use the so-called comoving width, i.e.",
"we rescale the width in the denominators of Eqs.",
"(REF ), (REF ) and (REF ) according to the standard prescription $m\\Gamma \\rightarrow (s/m)\\Gamma $ ."
],
[
"Data re-analysis",
"In order to extract information on the spin of the $X$ particle, we re-analyze the $B\\rightarrow X K$ data published in Ref.",
"[2] (for $X\\rightarrow J/\\psi \\; \\pi ^+\\pi ^-$ ) and in Ref.",
"[1] (for $X\\rightarrow J/\\psi \\; \\pi ^+\\pi ^-\\pi ^0$ ).",
"In particular, in the $X \\rightarrow J/\\psi \\; \\pi ^+\\pi ^-$ sample we consider the di-pion invariant mass ($m_{2\\pi }$ ) and the angles defined in the $X$ rest frame in Ref.",
"[7] and described in Fig REF : the angle between the $J/\\psi $ and the direction opposite to $K$ ($\\theta _X$ ), the angle between the $\\pi ^+$ and the direction opposite to $K$ ($\\chi $ ), and the angle between the $l^+$ produced by the decay of the $J/\\psi $ and the $z$ -axis of a coordinate system where the $x$ -axis is the direction opposite to the $K$ and the $y$ -axis is the component of $\\pi ^+$ orthogonal to $K$ ($\\theta _l$ ).",
"In the $X \\rightarrow J/\\psi \\; \\pi ^+\\pi ^-\\pi ^0$ , instead, only the three pions invariant mass is considered ($m_{3\\pi }$ ).",
"Figure: Definition of the angles in X→J/ψπ + π - X \\rightarrow J/\\psi \\; \\pi ^+\\pi ^- .With such distributions at hand the parameters of the model we have sensitivity upon are the radii $R_J$ ($J=1,2$ ) as defined in Eq.",
"(REF ), to which only the invariant mass distributions are sensitive the relative amplitudes and phases of the two contributions in case $J=2$ ($g_{2\\psi V}$ and $g^\\prime _{2\\psi V}$ in Eq.",
"(REF )).",
"We redefine $\\begin{aligned}g_{2\\psi \\xi } &= r_{2\\xi } \\cos \\frac{\\theta _\\xi }{2}, & g^{\\prime }_{2\\psi \\xi } &= r_{2\\xi } \\sin \\frac{\\theta _\\xi }{2}e^{i\\varphi _\\xi }\\\\\\end{aligned}$ where $\\xi $ can be $\\rho ,\\; \\omega $ or $\\text{ang}$ , depending on whether we are fitting the $m_{2\\pi }$ , $m_{3\\pi }$ , or angular distributions respectively.",
"The parameters $\\theta _\\rho $ and $\\varphi _\\rho $ satisfy $\\theta _\\rho =\\theta _{\\text{ang}}$ and $\\varphi _\\rho =\\varphi _{\\text{ang}}$ since in the $\\rho $ channel we have information on both mass spectrum and angular distributions.",
"Only the angular distributions are sensitive to the $\\theta $ and $\\varphi $ parameters the overall normalizations, $r_{2\\xi }$ and $r_{1\\xi }$ ($=g_{1\\psi \\rho }$ in Eq.",
"((REF ))).",
"It has to be noted that the normalization depends on the distributions being studied Given the different sensitivities of the mass and angle distributions and the computational complexity of the angular fits, the combined fits are performed in three steps: $i)$ the invariant mass distributions are fitted letting all parameters float, $ii)$ the fits to the three angular distributions are performed fixing the $R_J$ parameters to the results obtained in the previous fits, $iii)$ the invariant mass fits are repeated by fixing the $\\theta _\\rho $ and $\\varphi _\\rho $ distributions to the results of the angular fits.",
"$\\varphi _\\omega $ and $\\theta _\\omega $ are always set to zero.",
"Mass and widths of the $\\rho $ , $\\omega $ , and $X$ are fixed.",
"To account for the $X$ width in the invariant mass distributions, we extract randomly the values of the mass of the $X$ from a Breit-Wigner centered at $m_X = 3872\\,\\textrm {MeV}$ with $\\Gamma _X = 1.7\\,\\textrm {MeV}$ [8], accepting only those values kinematically consistent with the decay of interest.",
"Finally, since as described in Appendix the invariant mass fits return consistent results with either the $n=1$ or the $n=2$ hypotheses (where $n$ is defined in Eq.",
"(REF )) we will only report results with $n=1$ .",
"The resulting fits are shown in Figs.",
"REF and REF and the fitted parameters are summarized in Tab.",
"REF .",
"It is interesting to note that the radii, which are the fit parameters with a physical content, have reasonably small errors and get values consistent with 1 fm, the size scale of a standard hadron.",
"Here it is hard to judge if we are probing the size of the $X$ , in which case we would say that the results obtained for $R$ disfavor a large ($\\sim 10$ fm) loosely bound molecule [9], or the size of the interaction making the $X$ decay into $J/\\psi +V$ .",
"For sure the loosely bound molecule is requested to have a large wave function at the origin to make such a decay possible Moreover we remind that loosely bound molecules are very difficult to be produced with high cross sections at hadron colliders [10] if not invoking final state interactions effect [11].. On the other hand in any compact multiquark model the decay into $J/\\psi $ does not require special conditions.",
"Table: Fit results for the two J PC J^{PC} assignments.Figure: Fit to the m 2π m_{2\\pi } (left) and m 3π m_{3\\pi } (right) distributions as described in the text, with the n=1n=1 model.",
"The dashed curve refers to the 1 ++ 1^{++} hypothesis whereas the solid one is for the 2 -+ 2^{-+}.Figure: Fit to the angular distributions in X→J/ψπ + π - X \\rightarrow J/\\psi \\, \\pi ^+\\pi ^- decays as described in the text.",
"The dashed curve refers to the 1 ++ 1^{++} hypothesis whereas the solid one is for the 2 -+ 2^{-+}."
],
[
"Statistical interpretation",
"The naïve approach to evaluate the likelihood of the two spin hypotheses would be to consider the $\\chi ^2$ of the fits, obtained by adding the contributions from all the distributions, and to compare them with the number of degrees of freedom.",
"In our case this would return a $\\chi ^2/ \\textrm {DOF}=31.8/36$ for the $1^{++}$ hypothesis and $\\chi ^2/ \\textrm {DOF}=37.3/33$ for the $2^{-+}$ hypothesis.",
"The probability of obtaining a worse value is $P(\\chi ^2)= \\int _{\\chi ^2}^\\infty {\\rm PDF}(x,N_{\\rm DOF}) dx=67\\%$ and $28\\%$ respectively, i.e., both hypotheses seem to fit data well.",
"Nonetheless this conclusion would not consider two effects.",
"First, the overall good agreements is caused by the fact that we are simultaneously considering the distributions which favor the $1^{++}$ hypothesis ($\\theta _X, \\chi , \\theta _l$ ) and the one which favors the $2^{-+}$ hypothesis ($m_{3\\pi }$ ).",
"This can be qualitatively seen in the fit results (Figs.",
"REF and REF ) and in the $P(\\chi ^2)$ values obtained on each fitted distribution as listed in Tab.",
"REF .",
"Then, a large number of degrees of freedom are not sensitive to the spin of the $X$ and therefore wash out the overall $\\chi ^2$ .",
"This is the case for the $m_{2\\pi }$ distribution, which is sensitive only to the radii, but not to the spin.",
"To quantify this latter statement and develop a sounder statistical analysis, we performed a Monte Carlo (MC) study.",
"We generate $N$ data samples with the same number of events as the experimental samples and take into account the background, the statistical fluctuations, and the uncertainties in the model parameters.",
"The simulations can be performed either by making the hypothesis that the $X$ is truly a $1^{++}$ state or a $2^{-+}$ state.",
"As the starting point of the simulation for a given $J$ and $n$ hypothesis we take the corresponding model extracting sample by sample the parameters according to the result of the combined fit to the data.",
"The invariant mass MC samples are generated by filling the bins $b_i$ according to Poisson distributions with mean values $\\mu _i$ corresponding, bin per bin, to the sum of the values expected from the model and the experimental background.",
"The angular pseudodata are generated as MC unweighted events.",
"After the extraction this background is treated in the same way as the data, subtracted in the case of the invariant mass distributions and accounted for in the fits in the case of the angular distributions.",
"The $N$ data samples obtained with this procedure are then analyzed with the same fitting function and statistical analysis as the experimental data.",
"With such a tool, we first show the $P(\\chi ^2)$ values obtained on the $m_{2\\pi }$ fits (Fig.",
"REF ) performed by generating the MC events with both the spin hypotheses and by fitting them either with the correct or the wrong hypothesis.",
"These figures show that even when fitting with the incorrect $X$ spin the fit would return a good $P(\\chi ^2)$ .",
"We therefore conclude that $m_{2\\pi }$ is not sensitive to the $X$ spin.",
"Figure: Distribution of the P(χ 2 )P(\\chi ^2) resulting from the fits to m 2π m_{2\\pi } in MC samples generated assuming 1 ++ 1^{++} (left) and 2 -+ 2^{-+} (right).",
"The solid (dashed) histogram corresponds to the fit to the 2 -+ 2^{-+} (1 ++ 1^{++}) model.Since we need the invariant mass distributions to perform the combined fit to constraint the $R_J$ variables, we developed a more robust method of testing the hypotheses by using as estimator the difference between the $\\chi ^2$ obtained from the combined fits performed under the two $J$ hypotheses [12] $\\Delta \\chi ^2= \\chi ^2\\left(1^{++}\\right) - \\chi ^2\\left(2^{-+}\\right)$ Fig.",
"REF shows that the distribution of such a variable is peaked around a negative value when the MC samples are generated with $1^{++}$ and, viceversa, it is peaked around a positive value when the samples are generated with $2^{-+}$ .",
"These distributions are used to calculate the fraction of samples in which $\\Delta \\chi ^2$ has a value larger (for $1^{++}$ ) or smaller (for $2^{-+}$ ) than the one obtained on data.",
"This fraction, that we call CL, estimates the probability of the hypothesis.",
"Given that in the combined fit $\\Delta \\chi ^2=-5.5$ , the $2^{-+}$ hypothesis is excluded at 99.9% C.L., while the $1^{++}$ hypothesis has a C.L.",
"of 5.5%.",
"The different conclusion with respect to the naïve expectations is due to the fact that individual $\\chi ^2$ values account only for the agreement of the data with the specific hypothesis under test.",
"The $\\Delta \\chi ^2$ approach instead also considers the level of agreement with the other hypothesis, weighting the degrees of freedom with their power to discriminate among the two hypotheses.",
"Figure: Distribution of the Δχ 2 =χ 2 (1 ++ )-χ 2 (2 -+ )\\Delta \\chi ^2=\\chi ^2(1^{++})-\\chi ^2(2^{-+}) resulting from the combined fits to Monte Carlo data samples with n=1n=1.",
"The solid (dashed) histogram corresponds to events generated assuming the XX to be a 2 -+ 2^{-+} (1 ++ 1^{++}) state.",
"We mark with a linethe position of the experimental Δχ 2 \\Delta \\chi ^2."
],
[
"Fits on sub-samples",
"The $\\Delta \\chi ^2$ interpretation of the combined fit shows that even the agreement of the data with the $1^{++}$ hypothesis is marginal and the fit results seem to hint that the disagreement concentrates in the $m_{3\\pi }$ distribution (see Fig.",
"REF ).",
"To quantify this statement we have performed the toy MC analysis on the $\\Delta \\chi ^2$ value obtained separately on the $m_{3\\pi }$ distribution (obtained on the $X\\rightarrow J/\\psi \\; \\pi ^+\\pi ^-\\pi ^0$ sample) and on the combination of the other distributions (obtained on the $X\\rightarrow J/\\psi \\; \\pi ^+\\pi ^-$ sample).",
"The results, listed in Tab.",
"REF , show that the fits to the $X\\rightarrow J/\\psi \\; \\pi ^+\\pi ^-$ distributions (second row) exclude the $2^{-+}$ hypothesis and have an agreement with the $1^{++}$ hypothesis at 23% C.L., while conversely the $X\\rightarrow J/\\psi \\; \\pi ^+\\pi ^-\\pi ^0$ data (third row) favor the $2^{-+}$ at 81% and exclude the $1^{++}$ hypothesis at 99.9% C.L.",
"The two samples return therefore inconsistent answers.",
"Finally, in order to compare directly with Ref.",
"[5] we report in Tab.",
"REF also the results performed using exclusively the two invariant mass distributions or the three angular distributions.",
"Also here the former give a clear indication in favor of the $2^{-+}$ hypothesis, the latter in favor of the $1^{++}$ hypothesis.",
"The different conclusion with respect to Ref.",
"[5] comes both from a different theoretical model and from the appropriate treatment of the degrees of freedom with no sensitivity to the spin of the $X$ .",
"Table: Results of the Toy MC"
],
[
"Conclusions",
"We re-analyzed the $X \\rightarrow J/\\psi \\;\\pi ^+ \\pi ^- \\pi ^0$ and $X \\rightarrow J/\\psi \\;\\pi ^+ \\pi ^- $ invariant mass and angle distributions published by the Belle [2] and BaBar [1] collaborations respectively, with the goal to extract the most information about the spin of the $X$ particle.",
"With respect to the existing analyzes, we have improved two aspects.",
"On one side, the $X$ decay amplitudes are parameterized by effective strong couplings, weighting terms written as products of momenta and polarizations as dictated by Lorentz invariance, and parity considerations in a model independent way.",
"The strong couplings are given a momentum dependency according to the model defined in Eq.",
"(REF ).",
"This requires the introduction of an additional parameter $R$ which can be related to the finite size of hadrons in the interaction region.",
"The results, shown in Tab.",
"REF , are consistent with $R$ of the order of 1 fm.",
"The treatment is fully relativistic and particularly appropriate for angular analyses.",
"On the other side, in order to properly account for the sensitivities to the $X$ spin of the considered distributions, we pursue a statistical approach based on Monte Carlo simulations.",
"We performed $i)$ a global fit based on the whole information available from the $2\\pi $ , $3\\pi $ invariant mass spectra and the angular distributions of the $X\\rightarrow J/\\psi \\; \\rho $ decays, $ii)$ two separated fits relative to the channels $J/\\psi \\;\\rho $ (where we do the combined fit of the angular and invariant mass distributions) and $J/\\psi \\;\\omega $ (fitting only the invariant mass distribution).",
"We also studied the fit to the invariant mass and the angular distributions separately.",
"The combined fit $i)$ excludes the $2^{-+}$ hypothesis at $99.9\\%$ C.L., but returns a probability of only $5.5\\%$ of the $1^{++}$ hypothesis being correct.",
"The separate fits $ii)$ , return a clear a preference for the $1^{++}$ hypothesis in the $J/\\psi \\; \\rho $ channel with a probability of 23% and an 81% preference for the $2^{-+}$ assignment in the $J/\\psi \\; \\omega $ channel, with very strong exclusions of other hypotheses.",
"Such results go in the direction of two different assignments for the two samples, which can lead to a host of interesting considerations."
],
[
"Acknowledgements",
"We would like to thank F.C.",
"Porter for the discussion on the statistical approach, and F. Renga for the analysis of experimental data about the width of the $X$ .",
"We also wish to thank C. Sabelli for her collaboration in the early stages of this work.",
"The work of F. P. was supported by the Research Executive Agency (REA) of the European Union under the Grant Agreement number PITN-GA-2010-264564 (LHCPhenoNet)."
],
[
"$\\rho $ -{{formula:7e5438bb-c3f7-4903-8438-ec7f97f2689f}} mixing",
"As discussed in Sec.",
", the isosping-violating $\\rho $ -$\\omega $ mixing introduces a correction near the pole of the $\\omega $ , without affecting the core of our analysis.",
"Moreover, we found that the mixing improves the quality of the fit of $2^{-+}$ and worsens the $1^{++}$ .",
"First of all, we have to insert the mixing into (REF ) and (REF ).",
"Following Ref.",
"[5], we describe the decay $\\omega \\rightarrow 2\\pi $ , as the oscillation $\\omega \\rightarrow \\rho \\rightarrow 2\\pi $ , as explained in Fig.",
"REF .",
"We call $-\\epsilon $ the coupling of the vertex $\\rho \\,\\omega $ , whose value can be extracted by [5], [13]: $\\epsilon \\approx \\sqrt{m_\\omega \\, m_\\rho \\, \\Gamma _\\rho \\,\\Gamma _\\omega \\, \\mathcal {BR}(\\omega \\rightarrow 2\\pi )} \\approx 3.4 \\cdot 10^{-3} \\,\\textrm {GeV}^2$ .",
"The choice to treat the mixing reproduces naturally the phase of $95^\\circ $ of the complex mixing parameter used in Ref. [2].",
"Under the hypothesis $1^{++}$ we have $\\langle \\psi \\,\\rho (q) |X \\rangle = g_{1\\psi \\rho } \\, T - \\epsilon \\,g_{1\\psi \\omega } \\frac{1}{q^2 - m^2_\\omega + i m_\\omega \\Gamma _\\omega } T$ where $T$ is the S-wave scalar described in (REF ).",
"Similarly for $2^{-+}$ $\\langle \\psi \\,\\rho (q) |X \\rangle = \\left(g_{2\\psi \\rho } \\;T_A + g_{2\\psi \\rho }^\\prime \\;T_B\\right) - \\epsilon \\, \\frac{g_{2\\psi \\omega } \\;T_A + g_{2\\psi \\omega }^\\prime \\;T_B}{q^2 - m_\\omega ^2 + i m_\\omega \\Gamma _\\omega }$ We can repeat the same argument for $\\langle \\psi \\,\\omega (q) |X \\rangle $ by swapping the role of $\\rho $ and $\\omega $ .",
"The rest of Eqs.",
"(REF ) and (REF ) (namely the Breit-Wigner and the decay into pions) is unchanged, because in this picture it is only the $\\rho $ (resp.",
"the $\\omega $ ) which can decay in $2\\pi $ ($3\\pi $ ), being the mixing exhausted in the $\\langle \\psi V |X \\rangle $ matrix element.",
"Figure: Feynman diagrams of X→J/ψρX \\rightarrow J/\\psi \\,\\rho including the ρ\\rho -ω\\omega mixing.",
"The second diagram includes the coupling ϵ\\epsilon and the propagator of the ω\\omega .Moreover, if we take the mixing into account, the parameters $r_{J\\rho }$ and $r_{J\\omega }$ of the fit appear together in both channels; therefore we must impose their correct relative normalization.",
"This can be solved by imposing that $\\Gamma \\left(X \\rightarrow J/\\psi \\,3\\pi \\right) / \\Gamma \\left(X \\rightarrow J/\\psi \\,2\\pi \\right) = 0.8 \\pm 0.3$ [1], [8].",
"The angular distributions are unaffected by this term since they depend exclusively on the spin of the particles involved and not on the invariant mass spectrum.",
"Therefore, to show the impact of the $\\rho $ -$\\omega $ mixing we perform fits to the $m_{2\\pi }$ and $m_{3\\pi }$ distributions including this effect.",
"The results are shown in Fig.",
"REF , with the individual components detailed for the $m_{2\\pi }$ distribution in Fig.",
"REF .",
"The resulting values of the radii vary from $R_1=1.6\\pm 0.3$ to $R_1=1.1\\pm 0.4$ and from $R_2=5.6\\pm 0.8$ to $R_2=4.5\\pm 0.7$ .",
"As far as the hypothesis testing is concerned the $\\chi ^2/\\textrm {DOF}$ changes from $25.2/22$ to $29.3/22$ for the $1^{++}$ hypothesis, and from $17.7/20$ to $16.0/20$ for the $2^{-+}$ one.",
"None of the relevant conclusions is altered by this small effect and we will therefore neglect it in the main analysis.",
"Figure: Fit including ρ\\rho -ω\\omega mixing to the m 2π m_{2\\pi } (left) and m 3π m_{3\\pi } (right) distributions, with the model with n=1n=1.",
"The dashed curve refers to the 1 ++ 1^{++} hypothesis, the solid one to the 2 -+ 2^{-+} one.One could have thought that the presence of the narrow propagator of the $\\omega $ would have constrained the fit to raise a peak at $m_\\omega = 782$ MeV.",
"The smallness of $\\epsilon $ , instead, does not allow this, and the curve stays smooth at the pole, coherently with Refs.",
"[6], [2], [5].",
"Figure: Fit to the m 2π m_{2\\pi } distribution including ρ\\rho -ω\\omega mixing, with the model with n=1n=1, and under the hypothesis J PC =1 ++ J^{PC}=1^{++} (left) and 2 -+ 2^{-+} (right).The solid light curve is the ρ\\rho contribution to the fit; the dotted light curve is the ω\\omega contribution, and the dashed curve is the interference term.",
"The darker curve is the sum of all contributions."
],
[
"$n=2$ case",
"The theoretical model chosen for the fits described in this paper does not specify the value of $n$ in Eq.",
"(REF ), and it is therefore a free parameter of the model itself.",
"Before adopting $n=1$ for the rest of our considerations, we have performed a check on the dependence of the results on the choice of $n$ .",
"Since the angular distributions have shown a scarce dependence on $R$ , we fitted the invariant mass spectra with $n=2$ comparing the result with the $n=1$ case.",
"The results are shown in Fig.",
"REF .",
"The resulting values of the radii vary from $R_1=1.6\\pm 0.3$ to $R_1=1.1\\pm 0.2$ and from $R_2=5.6\\pm 0.8$ to $R_2=2.8\\pm 0.3$ .",
"As far as the hypothesis testing is concerned the $\\chi ^2/\\textrm {DOF}$ changes from $25.2/22$ to $24.8/22$ for the $1^{++}$ hypothesis, and from $17.7/20$ to $18.9/20$ for the $2^{-+}$ one.",
"Also in this case none of the relevant conclusions is altered.",
"Figure: Fit to the m 2π m_{2\\pi } (left) and m 3π m_{3\\pi } (right) distributions as described in the text, with the model with n=2n=2.",
"The dashed curve refers to the 1 ++ 1^{++} hypothesis, the solid one is for the 2 -+ 2^{-+} one."
]
] | 1204.1223 |
[
[
"The Magnon Spectrum in the Domain Ferromagnetic State of Antisite\n Disordered Double Perovskites"
],
[
"Abstract In their ideal structure, double perovskites like Sr_2FeMoO_6 have alternating Fe and Mo along each cubic axes, and a homogeneous ferromagnetic metallic ground state.",
"Imperfect annealing leads to the formation of structural domains.",
"The moments on mislocated Fe atoms that adjoin each other across the domain boundary have an antiferromagnetic coupling between them.",
"This leads to a peculiar magnetic state, with ferromagnetic domains coupled antiferromagnetically.",
"At short distance the system exhibits ferromagnetic correlation while at large lengthscales the net moment is strongly suppressed due to inter-domain cancellation.",
"We provide a detailed description of the spin wave excitations of this complex magnetic state, obtained within a 1/S expansion, for progressively higher degree of mislocation, i.e., antisite disorder.",
"At a given wavevector the magnons propagate at multiple energies, related, crudely, to `domain confined' modes with which they have large overlap.",
"We provide a qualitative understanding of the trend observed with growing antisite disorder, and contrast these results to the much broader spectrum that one obtains for uncorrelated antisites."
],
[
"Introduction",
"Double perovskite (DP) materials with general formula A$_{2}$ BB'O$_{6}$ have generated a great deal of interest [1] both in terms of their basic physics as well as the possibility of technological applications.",
"In particular, Sr$_{2}$ FeMoO$_{6}$ (SFMO) shows high ferromagnetic $T_c, \\sim $ 420K, large electron spin polarisation (half-metallicity) and significant low field magnetoresistance [2], [3].",
"The ferromagnetic coupling between the $S=5/2$ localized magnetic moments in SFMO (Fe$^{3+}$ ion, $3d^5$ state) is driven by a “double exchange” mechanism, where electrons from Mo delocalise over the Mo-O-Fe network.",
"The B (Fe) ions order ferromagnetically while the conduction electrons that mediate the exchange are aligned opposite to the Fe moments, leading to a saturation magnetization of $4 \\mu _B$ per formula unit in ordered SFMO.",
"However, the large entropy gain from disordering promote `antisite disorder' (ASD) whereby some B ions occupy the positions of B' ions and vice versa.",
"There is clear evidence now that B-B' mislocations are not random but spatially correlated [4], [5].",
"While ASD suppresses long range structural order, electron microscopy [4] and XAFS [5] reveal that a high degree of short range order survives.",
"The structural disorder has a direct magnetic impact.",
"If two Fe ions adjoin each other the filled shell $d^5$ configuration leads to antiferromagnetic (AFM) superexchange between them.",
"The result is a pattern of structural domains, with each domain internally ferromagnetic (FM) while adjoining domains are AFM with respect to each other.",
"This naturally leads to a suppression of the bulk magnetisation with growing ASD.",
"Domain structure has been inferred in the low doping manganites as well, due to competing FM and AFM interactions.",
"Inelastic neutron scattering in those materials suggest the presence of FM domains in a predominantly AFM matrix, and allows a rough estimate of the domain size [6], [7].",
"We aim to provide a similar framework for interpreting the magnetic state and domain structure in the DP from spin wave data.",
"Our main results are the following.",
"(i) We compute the dynamical magnetic structure factor, that encodes magnon energy and damping, within a $1/S$ expansion of an effective Heisenberg model chosen to fit the electronic model results.",
"(ii) The magnon data is reminiscent of the clean limit even at maximum ASD (50%), where the bulk magnetisation vanishes due to interdomain cancellation.",
"(iii) We suggest a rough method for inferring the domain size from the magnon data and check its consistency with the ASD configurations used.",
"(iv) We demonstrate that uncorrelated ASD leads to a much greater scattering of magnons and a much broader lineshape.",
"This suggests that in addition to XAFS and microscopy, neutron scattering would be a sensitive probe of the nature of disorder in these materials.",
"The paper is organized as follows: In Sec.II we discuss the generation of the structural motif, the solution of the electronic problem, and the estimation of exchanges for an effective Heisenberg model.",
"In Sec.III we recapitulate the spin-wave formulation for non collinear phases and present the magnon spectrum obtained for the different disordered configurations.",
"In Sec.IV we discuss the results, attempting to analyse the magnon spectrum for correlated antisites in terms of confined spin wave modes, and contrasting the result to magnons in an `uncorrelated' antisite background.",
"Given the similar location of the B and B' ions (at the center of the octahedra) the tendency towards defect formation is more pronounced in the DP's.",
"This tendency of mislocation interplays with the inherent B-B' ordering tendency and creates a spatially correlated pattern of antisites [4], [5] rather than random mislocation.",
"To model this situation we have used a simple “lattice-gas” model [9].",
"On proper annealing it will go to a long range ordered B, B', B, B'... pattern.",
"We frustrate this by using a short annealing time to mimic the situation in the real materials.",
"We encode the atomic positions by defining a binary variable $\\eta _{i}$ , such that $\\eta _{i} =1$ when a site has a B ion, and $\\eta _{i} =0$ when a site has a B' ion.",
"Thus for an ordered case we will get $\\eta $ 's as $1,0,1,0,1,0...$ along each cubic axes.",
"The B-B' patterns that emerge on short annealing are characterised by the structural order parameter $S=1-2x$ , where $x$ is the fraction of B (or B') atoms that are on the wrong sublattice.",
"We have choosen four disordered families with increasing disorder for our study.",
"One structural motif each for these families is shown in the first column of Fig.1, with progressively increasing disorder (from top to bottom).",
"We plot $g({\\bf r}_i)=(\\eta _i-\\frac{1}{2})e^{i\\pi (x_i+y_i)}$ as an indicator of structural order.",
"For a perfectly ordered structure $g(\\bf {r_i})$ is constant.",
"The pattern along the first column are different realisations of ASD with $S = 0.98, 0.88, 0.59, 0.17$ (top to bottom).",
"We solve the electronic-magnetic problem on these structural motifs."
],
[
"Electronic Hamiltonian",
"To study the magnetic order we use the electronic-magnetic Hamiltonian that has the usual couplings of the ordered double perovskite, and an additional antiferromagnetic coupling when two magnetic B ions are nearest neighbour (NN).",
"The Hamiltonian for the microscopic model is: $H = H_{loc} \\lbrace \\eta \\rbrace + H_{kin} \\lbrace \\eta \\rbrace + H_{mag} \\lbrace \\eta \\rbrace ,$ $H_{loc} \\lbrace \\eta \\rbrace =\\epsilon _{B}\\sum _i \\eta _i f_{i\\sigma }^{\\dagger }f_{i\\sigma }+\\epsilon _{B^{\\prime }}\\sum _i (1 - \\eta _i)m_{i\\sigma }^{\\dagger }m_{i\\sigma }$ is the onsite term where $\\epsilon _{B}$ and $\\epsilon _{B^{\\prime }}$ are level energies, respectively, at the B and B' sites.",
"Here $f$ is the electron operator referring to the magnetic B site and $m$ is that of the non-magnetic B' site.",
"The NN hopping term is given by $H_{kin} \\lbrace \\eta \\rbrace =-t_{1}\\sum _{<ij>\\sigma }\\eta _{i}\\eta _{j} f_{i\\sigma }^{\\dagger }f_{j\\sigma }-t_{2}\\sum _{<ij>\\sigma }(1-\\eta _{i}) (1 - \\eta _{j}) m_{i\\sigma }^{\\dagger }m_{j\\sigma }-t_{3}\\sum _{<ij>\\sigma }(\\eta _{i}+\\eta _{j}-2\\eta _{i}\\eta _{j})(f_{i\\sigma }^{\\dagger }m_{j\\sigma }+h.c.",
")$ .",
"For simplicity we set all the NN hopping amplitudes to be same $t_1$ =$t_2$ =$t_3=t$ .",
"The magnetic interaction term consists of the Hund's coupling $J$ on B sites, and AFM superexchange coupling $J^{AF}$ between two NN magnetic B sites.",
"Thus, $H_{mag} \\lbrace \\eta \\rbrace =J\\sum _i \\eta _i {\\bf S}_{i}.f_{i\\alpha }^{\\dagger }\\vec{\\sigma }_{\\alpha \\beta }f_{i\\beta }+ J^{AF} \\sum _{\\langle ij \\rangle } \\eta _i \\eta _j {\\bf S}_i.",
"{\\bf S}_j $ .",
"Here ${\\bf S}_i$ is the classical core spin on the B site at ${\\bf r}_i$ with $\\vert {\\bf S}_i \\vert = 1$ .",
"We take $J/t \\gg 1$ with $J>0$ and $J^{AF}\\vert {\\bf S} \\vert ^2/t = 0.08$ , based on the T$_N$ scale in SrFeO$_3$ .",
"We have ignored orbital degeneracy, coulomb effects, etc, to focus on the essential magnetic model on the disordered structure.",
"We will use a two dimensional model because it already captures the qualitative physics while allowing ease of visualisation and access large system size.",
"The formulation readily carries over to three dimensions as well.",
"We have used a real space exact diagonalisation based Monte Carlo method involving a traveling cluster approximation (TCA)[10] to anneal the spin-fermion system towards its ground state in the disordered background.",
"Figure: (Colour online)First column contains the structural motif for four disorderedfamilies with progressivelyincreasing disorder (from top to bottom).We plot the g(𝐫 𝐢 )=(η 𝐢 -1 2)𝐞 𝐢π(𝐱 𝐢 +𝐲 𝐢 ) g(\\bf {r_i})=(\\eta _i-\\frac{1}{2})e^{i\\pi (x_i+y_i)} .We denote the configurations as C1, C2, C3, C4and the correspondingstructural order parameter has values S=0.98,0.88,0.59,0.17S = 0.98,~0.88,~0.59,~0.17from top to bottom.Second column shows the ground state spin overlap factor,h i =𝐒 0 .𝐒 𝐢 h_i=\\bf {S_0}.\\bf {S_i}, where 𝐒 0 \\bf {S_0}is the left-lower-corner spin in the lattice.In the third column, we have shown the correspondingNN bond configurations.",
"Here we have threedifferent type of NN bonding present betweenB-B, B'-B' and B-B', represented bycolours red, blue and green respectively in the plot.Lattice size is 40×4040 \\times 40.Annealing the electron-spin system down to low temperature on a given structural motif leads to the magnetic ground states shown in the middle column of Fig.1.",
"We plot the spin overlap factor, $h_i=\\bf {S_0}.\\bf {S_i}$ , where $\\bf {S_0}$ is the left-lower-corner spin in the lattice.",
"The comparison of the first and second columns in Fig.1 indicate that the structural and magnetic domains coincide with each other.",
"The third column of Fig.1 shows the NN structural partners.",
"We have three possibilities: B-B, B'-B' and B-B', represented by colours red, blue and green respectively in the plot."
],
[
"Effective Heisenberg Hamiltonian",
"Considering the difficulty in doing a spin-wave analysis on the full electronic-magnetic Hamiltonian (Eq.",
"REF ), we assume that the spin dynamics can be described by an $\\it effective$ Heisenberg model $H_{eff} = \\sum _{\\lbrace ij \\rbrace }J_{ij}\\;{\\bf S}_i.",
"{\\bf S}_j$ where $\\lbrace \\rbrace $ represents the set of NN and next nearest neighbour (NNN) sites.",
"$J_{ij}$ is the effective coupling (FM/AFM) between the local moments at ${\\bf r}_i$ and ${\\bf r}_j$ sites.",
"In our two dimensional ASD configurations $J_{F}$ operates between two local moments when they are at the NNN position and $J_{AF}$ is active when the moments are at the NN position (a B-O-B arrangement).",
"We have estimated the effective coupling $J_{F}$ and $J_{AF}$ as follows.",
"For getting the FM coupling ($J_{F}$ ) we have considered the ordered double perovskite structure.",
"We calculated the order parameter, i.e, the magnetic structure factor S($\\bf {k}$ ) at ${\\bf {k}}=(0,0)$ , as a function of temperature for the full electronic Hamiltonian (Eq.",
"REF ) using Monte Carlo simulation.",
"We then repeated the same procedure for the NNN FM Heisenberg Hamiltonian, defined on only the magnetic sites of the double perovskite.",
"We found that for $J_{F}/t=-0.04$ , two results matches very well.",
"Figure: (Colour online) Comparison between the evolution of thespin structure factor S(𝐤\\bf k) at 𝐤=(0,0){\\bf k} = (0,0)with temperature for the spin configurations of various disorderfamilies (from top to bottom) C1, C2, C3 and C4 obtainedfrom the full electronic Hamiltonian withJ AF S 2 /t=0.08J^{AF}S^2/t = 0.08 and the effective Heisenberg modelwith J F /t=-0.04J_{F}/t=-0.04 and J AF /t=0.065J_{AF}/t=0.065.Lattice size is 40×4040 \\times 40.In order to get the AFM coupling we considered the ordered perovskite where both the B and B' site carry a magnetic moment (mimicking SrFeO$_3$ ) and computed its AFM structure factor peak k=($\\pi ,\\pi $ ).",
"This model involves both electronic kinetic energy and Fe-Fe superexchange.",
"We find that the result can be modelled via a Heisenberg model with $J_{AF}/t=0.065$ .",
"Using the couplings inferred from these limiting cases, $J_{F}/t=-0.04$ and $J_{AF}/t=0.065$ , we studied the bond disordered Heisenberg model for the antisite disordered DP magnet.",
"We compared the FM structure factor peak S($\\bf {k}$ ) at ${\\bf k}$ =(0,0) obtained from the disordered Heisenberg model with that from the full electronic Hamiltonian (Eq.",
"REF ).",
"The Heisenberg result for the FM structure factor S(0,0) as a function of temperature matches very well, Fig.2, with the electronic Hamiltonian result for all ASD configurations.",
"This gives us confidence in the usefulness of the Heisenberg model for spin dynamics."
],
[
"Spin-Wave Excitation",
"In this section we use the spin rotation technique [11] to evaluate the spin-wave modes and dynamic structure factor at zero temperature.",
"The effective Heisenberg model (Eq.",
"REF ) can be cast in a form useful for spin wave analysis by defining a local frame at each site so that the spins point along the $+z$ direction in the ground state.",
"We can use ${\\bar{\\bf S}}_i = U_i{\\bf S}_i$ , where ${\\bar{\\bf S}}_i$ points along its local $z-$ axis in the classical limit.",
"The unitary rotation matrix $U_i$ for site ${\\bf r}_i$ is given by $U_i =\\left| \\begin{array}{ccc}\\cos (\\theta _i)\\cos (\\psi _i) & \\cos (\\theta _i)\\sin (\\psi _i)& -\\sin (\\theta _i) \\\\-\\sin (\\psi _i) & \\cos (\\psi _i)& 0 \\\\\\sin (\\theta _i)\\cos (\\psi _i) & \\sin (\\theta _i)\\sin (\\psi _i)& \\cos (\\theta _i) \\end{array} \\right|,$ where $\\theta $ and $\\psi $ are the Euler rotation angles.",
"Now one can write the generalized Hamiltonian $H_{eff} = \\sum _{\\lbrace ij \\rbrace }J_{ij}{\\bar{\\bf S}}_i.F_{ij}{\\bar{\\bf S}}_j,$ where $F_{ij}= U_iU_j$ is the overall rotation from one reference frame to another and its elements $F_{ij}^{\\alpha \\beta }$ can be obtained from Eq.",
"(REF ).",
"Applying the approximate Holstein-Primakoff (HP) transformation in the large $S$ limit the spin operators in the local reference frame become: $\\bar{S}_i^+ = \\sqrt{2S}\\:b_{i}$ , $\\bar{S}_i^- = \\sqrt{2S}\\:b_{i}^{\\dag }$ and $\\bar{S}_i^z = S -b_{i}^{\\dag } b_{i}$ , where $b_i$ and $b_i^{\\dag }$ are the boson (magnon) annihilation and creation operators respectively.",
"Only retaining the quadratic terms in $b$ and $b^{\\dag }$ , which describe the dynamics of the non-interacting magnons and neglect magnon interaction terms of order $1/S$ , the generalized Hamiltonian (Eq.",
"REF ) reduces to ${\\cal H} = \\sum _{\\lbrace ij \\rbrace }[{\\cal J}_{ij}(G_{ij}^1 b_i^{\\dag }b_j +G_{ij}^2 b_ib_j + h.c.) +f_{ij}(b_i^{\\dag }b_i + b_j^{\\dag }b_j)],$ where ${\\cal J}_{ij} =SJ_{ij}/2$ , $f_{ij} =-SJ_{ij}F_{ij}^{zz}$ and the rotation coefficients $G^{{}^1_2}= (F_{ij}^{xx} \\pm F_{ij}^{yy}) - i(F_{ij}^{xy} \\mp F_{ij}^{yx})$ .",
"The Hamiltonian (REF ) is diagonalized by the transfermation $b_i = \\sum _n (u_n^i c_n + v_n^{i^{*}} c_n^{\\dag }),$ where $c^{\\dag }$ and $c$ are the quasiparticle operators.",
"$u$ and $v$ , which satisfy $\\sum _n (u_n^i u_n^{j^{*}} -v_n^{i^{*}} v_n^j) = \\delta _{ij}$ ensuring the bosonic character of the quasiparticles are obtained from $\\left( \\begin{array}{cc}A_{ij} & B_{ij}^* \\\\B_{ij} & A_{ij}^*\\end{array} \\right)\\left( \\begin{array}{c}u_n^j\\\\v_n^j\\end{array} \\right) = \\omega _n\\left( \\begin{array}{cc}\\delta _{ij} & 0 \\\\0 & -\\delta _{ij}\\end{array} \\right)\\left( \\begin{array}{c}u_n^j\\\\v_n^j\\end{array} \\right),$ where $A_{ij} = {\\cal J}_{ij}(G_{ij}^1 +G_{ji}^{1^{*}}) + \\epsilon _i\\delta _{ij}$ , $B_{ij} = {\\cal J}_{ij}(G_{ij}^2 + G_{ji}^2)$ and $\\epsilon _i = \\sum _j(f_{ij} + f_{ji})$ .",
"Now the spin-spin correlation function can be evaluated using the magnon energies and wavefunctions obtained from Eq.",
"(REF ), where the excitation eigenvalues $\\omega _n \\ge 0$ ."
],
[
"Dynamical Structure Factor",
"A neutron scattering experiment measures the spin-spin correlation function in Fourier and frequency space $S({\\bf k},\\omega )$ to describe the spin dynamics of the magnetic systems on an atomic scale.",
"From ${\\bf S}_i = U_i^{-1}{\\bar{\\bf S}}_i$ one can express $S_i^{\\alpha } = \\sum _{\\mu } U_i^{{\\mu \\alpha }^*} {\\bar{S}_i^{\\mu }},$ where $\\alpha $ and $\\mu $ represents the $x$ , $y$ , and $z$ components.",
"Now applying the approximate HP transformation to the rotated spins one can write $S_i^{\\beta } = p_i^{\\beta }b_i +q_i^{\\beta }b_i^{\\dag } + r_i^{\\beta }(S - b_i^{\\dag }b_i),$ where $\\beta = +$ , $-$ and $z$ , and $p,q$ and $r$ are the rotation coefficients (given in the Appendix).",
"Putting Eq.",
"(REF ) in (REF ) the space time spin-spin correlation function can be written as $S_i^{\\alpha }(t) S_j^{\\beta }(0) =\\sum _{mn}[A_{{}^{mn}_{ij}}^{\\alpha \\beta }c_m^{\\dag }(t) c_n(0) +B_{{}^{mn}_{ij}}^{\\alpha \\beta } c_m(t) c_n^{\\dag }(0)],$ where the coefficients A and B are expressed in the Appendix.",
"In Fourier and frequency space $S^{\\alpha , \\beta }({\\bf k},\\omega ) =\\frac{1}{N}\\int dt e^{- i\\omega t} \\sum _{ij}e^{i{\\bf k}.",
"({\\bf r}_i - {\\bf r}_j)}\\langle S_i^{\\alpha }(t)S_j^{\\beta }(0) \\rangle .$ and the total spin-spin correlation function $S({\\bf k},\\omega ) &=& \\frac{1}{2}[S^{+,-}({\\bf k},\\omega )+S^{-,+}({\\bf k},\\omega )]+ S^{z,z}({\\bf k},\\omega ) \\nonumber \\\\&=& \\sum _{l} W_{\\bf k}^{l}\\delta (\\omega - \\omega _{l}), \\nonumber $ where the coefficient of the delta function $W_{\\bf k}^{l}=\\frac{1}{N}\\sum _{ij}{\\cal B}_{ij}^l e^{i{\\bf k}.",
"({\\bf r}_i - {\\bf r}_j)}$ is the SW weight with ${\\cal B}_{ij}^l =\\frac{1}{2}(B_{{}^{ll}_{ij}}^{+-} +B_{{}^{ll}_{ij}}^{-+}) + B_{{}^{ll}_{ij}}^{zz}$ .",
"$W_{\\bf k}^{l}$ is observed as the intensity of magnon spectrum in the neutron scattering experiment.",
"Figure: (Colour online) Spin-wave spectrum along main symmetrydirections of the Brillouin zone for spin configurations C1, C2, C3 and C4(xx = 0.01, 0.11, 0.21 and 0.41 respectively).shown in Fig.1With increasingASD from C1 to C4 the spectrum becomes broader for a fixed value ofmomentum 𝐤{\\bf k}.",
"Here J F =-0.04J_{F}=-0.04, J AF =0.065J_{AF}=0.065, and lattice size is40×4040 \\times 40.Figure: (Colour online)Mean spin wave energy ω ¯ 𝐤 {\\bar{\\omega }}_{\\bf k} (dots) andspin-wave width Δω 𝐤 \\Delta \\omega _{\\bf k} (bars), defined in thetext,for the correlated antisite configurations C2-C4.",
"The curves arevertically shifted for clarity."
],
[
"Results and Discussion",
"We start by presenting the results for magnons in the configurations C1-C4 shown in Fig.1 and then move to an analysis of the linewidth, the estimation of domain size, and the contrast with uncorrelated disorder."
],
[
"Results for AF coupled domains",
"Fig.3 shows the magnon spectra of C1-C4 with obtained from the Heisenberg model with the FM and AFM couplings discussed earlier.",
"In a model with only FM couplings, i.e., no disorder, we would have obtained only the red curve, $\\omega ^0_{\\bf k}$ , for propagating magnons.",
"The striking feature in all these panels is how closely the mean energy of the magnons follow $\\omega ^0_{\\bf k}$ despite the large degree of mislocation in C2 and C3 and maximal disorder ($ x \\sim 0.5$ ) in C4 (refer to the spatial plots in Fig.1).",
"The broadening, although noticeable in C4, does not obscure the basic dispersion.",
"Fig.4 quantifies the mean energy and broadening by computing: ${\\bar{\\omega }}_{\\bf k} = \\int ~S({\\bf k}, \\omega ) \\omega d \\omega $ $[ {\\Delta \\omega _{\\bf k}} ]^2 =[\\int ~S({\\bf k}, \\omega ) \\omega ^2 d \\omega ] - {\\bar{\\omega }}^2_{\\bf k}$ We have shown these two quantities for the C2-C4 structures in Fig.1.",
"The ${\\bar{\\omega }}_{\\bf k}$ have been vertically shifted for clarity and the ${\\Delta \\omega _{\\bf k}}$ are superposed as `error bars' on these.",
"It is clear that even in the most disordered sample (C4), where the mislocation $x \\sim 0.4$ , the broadening is only a small fraction of the magnon energy.",
"This will be an indicator when we discuss spin waves in an uncorrelated disorder background.",
"Figure: (Colour online)Spin-wave spectra along main symmetry directions of the Brillouin zonefor spin configurations C1, C2, C3 and C4 shown in Fig.1(xx = 0.01, 0.11, 0.21 and 0.41 respectively).Increasing fractional weakly coupleddomain boundary spins from C1 to C4enhances the SW softening near the ZB along [π,0][\\pi ,0] and the spectrum alsobecomes broader for a given 𝐤{\\bf k}.Here J F =-0.04J_{F} =-0.04, J AF =0J_{AF} = 0, and lattice size is 40×4040 \\times 40.Figure: (Colour online)Mean spin wave energy ω ¯ 𝐤 {\\bar{\\omega }}_{\\bf k} (dots) andspin-wave width Δω 𝐤 \\Delta \\omega _{\\bf k} (bars)for C2-C4 now with J AF =0J_{AF}=0, i.e, decoupled domains.The curves are vertically shifted for clarity."
],
[
"Broadening: impact of domain size",
"There are two ingredients responsible for the spectrum that one observes in Fig.3, (i) the domain structure, and (ii) the AF coupling across the domains.",
"To deconvolve these effects and have a strategy for inferring domain size from neutron data, we studied a situation where we set $J_{AF}=0$ in the Heisenberg model defined on the structures C1-C4.",
"In that case we will have decoupled FM domains without any antiparallel spin orientation between them.",
"We think this is a interesting scheme to explore since the AF bonds are limited to the domain boundaries and is not equal to the number of mislocated sites.",
"Fig.5 shows the overall magnon spectrum for this case, using the same convention as in Fig.3, while Fig.6 quantifies the mean energy and broadening in this `decoupled domain' case.",
"The absence of $J_{AF}$ does not seem to make a significant difference to the spectrum as a comparison of Fig.4 and Fig.6 reveal.",
"Figure: (Colour online)Modelling C2 in terms of a domain of size 10×1010 \\times 10 (left)and of seven domains of size 4×44 \\times 4 (right).",
"The correspondingmean energy and broadening are shown below.This correspondence, valid even in C4, suggests the following: (i) most of the spectral features arise from the domain structure, and the associated confinement of spin waves, rather than the AF coupling, and (ii) we can proceed with a much simpler modelling of the spectrum and estimation of domain size without invoking the complicated BdG formulation that AFM coupling requires.",
"Essentially, much can be learnt from `tight binding' models defined on appropriate stuctures, as happens for FM states, without having to invoke the `pairing' terms that arise for AF coupling.",
"A modelling of the full dispersion will require the AF terms as well, but the inference about presence of domains, and an estimate of their size, need not.",
"We proceed with this next.",
"To make some progress in estimating the typical domain size we need a few assumptions; (i) the total degree of mislocation, $x$ , should be known, based on the bulk magnetisation measurement.",
"(ii) If the overall system size in $L \\times L$ (or equivalent in a 3D model), the number of mislocated sites would be $x L^2$ .",
"(iii) If the domain size is $L_d$ then the number of domains within the $L \\times L$ area is $N_d \\sim xL^2/L_d^2$ .",
"In reality domains need not have one single size, as C2-C4 indicate, but we need the assumption to make some headway.",
"(iv) We need to locate these $N_d$ domains randomly, in a non overlapping manner, within the $L \\times L$ system, and average the spectrum obtained over different realisations of domain location.",
"This scheme, carried out for various $L_d$ , can be compared to the full $S({\\bf k}, \\omega )$ data to get a feel for the appropriate $L_d$ .",
"We show the result below for such a tight binding exploration for the C2 configuration, modelled in terms of different domain distributions that respect the same overall mislocation.",
"When we compare the ratio of mean broadening to bandwidth obtained at different values of $L_d$ (and so $N_d$ ) with that for the real data, Fig.4, it turns out that $L_d=10$ provides a best estimate.",
"It also reasonably describes the broadening at stronger disorder, C3 and C4, where of course $N_d$ is larger.",
"An analytic feel for these results can be obtained by considering the modes of a square size $L_d \\times L_d$ under open boundary conditions."
],
[
"Contrast with uncorrelated antisites",
"In modelling the antisite disorder much of the earlier work in the field assume the defect locations to be random.",
"We have followed the experimentally motivated path which suggests that the mislocated sites themselves form an ordered structure separated from the parent (or majority) by an antiphase boundary.",
"The sources of scattering are the boundary between these domains rather than random point defects.",
"Since much of double perovskite modelling has assumed the random antisite situation it is worth exploring the differences in the magnon spectrum between correlated and uncorrelated antisites.",
"We have already seen the results for correlated disorder for different degrees of mislocation, $x$ .",
"We generated uncorrelated antisite configurations with the same $x$ by starting with ordered configurations and randomly exchanging B and B' till the desired degree of disorder is reached.",
"These configurations naturally do not have any structural domains.",
"Annealing the full electronic Hamiltonian on these configurations, call them $C^{\\prime }_1,~C^{\\prime }_2,..$ , etc, down to low $T$ , leads to the magnetic ground states.",
"The ground states are disordered ferromagnets but without any domain pattern.",
"We computed the magnon lineshape in these configurations, and, for illustration, show the results for $C^{\\prime }_2$ and $C^{\\prime }_3$ side by side with their correlated counterparts $C_2$ and $C_3$ .",
"There is a striking increase in the magnon line width (or $\\Delta {\\omega }_{\\bf k}$ ) in the uncorrelated case.",
"There is almost nine fold increase in the magnon line width in $C_2$ and six fold in $C_3$ of the uncorrelated disorder with respect to the correlated disorder case.",
"Figure: The left set of panels correspond to mislocation x=0.11x=0.11where we compare the magnon spectrum for uncorrelated disorder (left)with correlated disorder (C2) right.",
"the top panels refer to thestructural pattern, the middle to the magnetic ground state, andthe bottom to the magnon response.",
"The right set of panels referto x=0.21x= 0.21, and the same indicators as for the left panels.Notice the remarkably broader lineshape for the uncorrelateddisorder case where it is difficult to make much of acorrespondence with the clean dispersion."
],
[
"Conclusion",
"We have studied the dynamical magnetic structure factor of a double perovskite system taking into account the basic ferromagnetic ordering tendency and the defect induced local antiferromagnetic correlations.",
"We used structural motifs that correspond to correlated disorder, obtained from an annealing process.",
"The results on magnon energy and broadening reveal that even at very large disorder, the existence of domain like structure ensures that the response has a strong similarity to the clean case.",
"We tried out a scheme for inferring the domain size from the spin wave damping, so that experimenters can make an estimate of domains without having spatial data, and find it to be reasonably successful.",
"We also highlight how the common assumption about random antisites, that is widely used in modelling these materials, would lead to a gross overestimate of magnon damping.",
"In summary, dynamical neutron scattering can be a direct probe of the unusual ferromagnetic state in these materials and confirm the presence of correlated antisites."
],
[
"Acknowledgement",
"We acknowledge use of the High Performance Computing facility at HRI.",
"PM thanks the DAE-SRC Outstanding Research Investigator grant, and the DST India (Athena) for support."
],
[
"appendix",
"The rotation coefficients are $\\begin{array}{c}p_i^{\\pm } = \\sqrt{\\frac{S}{2}}{(U_i^{xx} \\pm U_i^{yy}) - i(U_i^{yx} \\mp U_i^{xy})} \\\\q_i^{\\pm } = \\sqrt{\\frac{S}{2}}{(U_i^{xx} \\mp U_i^{yy}) + i(U_i^{yx} \\pm U_i^{xy})} \\\\\\hspace{-93.0pt} r_i^{\\pm } = U_i^{zx} \\pm iU_i^{zy} \\\\\\hspace{-93.0pt} p_i^z = U_i^{xz} - iU_i^{yz} \\\\\\hspace{-93.0pt} q_i^z = U_i^{xz} + iU_i^{yz} \\\\\\hspace{-124.0pt} r_i^z = U_i^{zz}.\\end{array}$ And the structure factor coefficients are $\\begin{array}{c}A_{{}^{mn}_{ij}}^{\\alpha \\beta } = q_i^{\\alpha }p_j^{\\beta } u_i^{m^*}u_j^n +p_i^{\\alpha }q_j^{\\beta } v_i^{m^*}v_j^n +p_i^{\\alpha }p_j^{\\beta } v_i^{m^*}u_j^n \\\\+ q_i^{\\alpha }q_j^{\\beta } u_i^{m^*}v_j^n - S\\times r_i^{\\alpha }r_j^{\\beta }(u_i^{m^*}u_i^n + u_j^{m^*}u_j^n) \\\\\\end{array}B_{{}^{mn}_{ij}}^{\\alpha \\beta } = q_i^{\\alpha }p_j^{\\beta } v_i^mv_j^{n^*} +p_i^{\\alpha }q_j^{\\beta } u_i^mu_j^{n^*} +p_i^{\\alpha }p_j^{\\beta } u_i^mv_j^{n^*} \\\\+ q_i^{\\alpha }q_j^{\\beta } v_i^mu_j^{n^*} - S\\times r_i^{\\alpha }r_j^{\\beta }(v_i^mv_i^{n^*} + v_j^mv_j^{n^*}).$"
]
] | 1204.1194 |
[
[
"Dynamic Bayesian diffusion estimation"
],
[
"Abstract The rapidly increasing complexity of (mainly wireless) ad-hoc networks stresses the need of reliable distributed estimation of several variables of interest.",
"The widely used centralized approach, in which the network nodes communicate their data with a single specialized point, suffers from high communication overheads and represents a potentially dangerous concept with a single point of failure needing special treatment.",
"This paper's aim is to contribute to another quite recent method called diffusion estimation.",
"By decentralizing the operating environment, the network nodes communicate just within a close neighbourhood.",
"We adopt the Bayesian framework to modelling and estimation, which, unlike the traditional approaches, abstracts from a particular model case.",
"This leads to a very scalable and universal method, applicable to a wide class of different models.",
"A particularly interesting case - the Gaussian regressive model - is derived as an example."
],
[
"Introduction",
"We deal with the problem of collaborative estimation of unknown environmental parameter from noisy measurements.",
"It naturally arises, e.g., in modern complex wireless systems and distributed sensor networks [1].",
"There exist two principal design schemes how to treat this estimation task: (i) the centralized approach, where the data are transmitted to a designated processing center (sometimes called fusion center) responsible for estimation (e.g., [1] and many others); and (ii) the decentralized concept, where the nodes are responsible for estimation (e.g., [14], [3]).",
"The decentralized methods become very promising, since the increasing complexity of modern networks calls for approaches with low overheads with respect to the time, energy and communication resources.",
"Besides that, the potential single-points of failure (SPOFs) are principally avoided and a good design of the algorithm allows fast spatial reconfigurations of the network.",
"There exist several Bayesian methods treating general tasks with distributed character from the decision making perspective, ranging from [11] to [1].",
"We focus ourselves on a recently formulated diffusion estimation problem, i.e., fully decentralized collaborative estimation in networks allowing the nodes to communicate only with their adjacent neighbours.",
"In this field, a couple of non-Bayesian estimation algorithms were proposed.",
"However, these are mostly single problem oriented, e.g., on least-squares estimation [14], recursive least-squares (RLS, [4]), least mean squares (LMS, [8], [3]), Kalman filters ([4]) etc.",
"We propose a new method called dynamic Bayesian diffusion estimation, which tackles the problem from the consistent and versatile Bayesian viewpoint and yields rather a methodology applicable to a much wider class of models, including, of course, the mentioned traditional ones.",
"A particularly interesting application of the method to a Gaussian linear regressive model results in the so-called diffusion recursive least-squares method, proposed in [4].",
"This demonstrates the generality of the method and advocates its feasibility.",
"Furthermore, it shows that it is possible to shift from the viewpoint of a Bayesian statistician to the traditionalist's one, disregarding the probabilistic treatment of parameters of interest.",
"In this paper, we implicitly assume that the communication among nodes does not violate the bandwidth or other restrictions.",
"The cases of restricted networks would require a specific solution which is behind the scope of this paper.",
"The organization of the paper is as follows: In Section 2, we briefly introduce the basic principle of Bayesian estimation.",
"In Section 3, the dynamic Bayesian diffusion estimation theory is developed.",
"Its application to the Gaussian linear regressive model follows in Section 4.",
"Since we show that it leads to an existing solution, a demonstration example is avoided in the paper.",
"We conclude our work and outline the future research topics in Section ."
],
[
"Bayesian estimation",
"Let us consider a linear stochastic system with a real input variable $u_{t}$ and a real output variable $y_t$ , observed at discrete time instants $t=1,2,\\dots $ Both $u_{t}$ and $y_{t}$ can be scalar or multivariate.",
"We form a data $\\mathbf {d}(t)$ as an ordered set of observations and inputs, $\\mathbf {d}(t) = \\lbrace y_0, u_0, \\dots , y_t, u_t\\rbrace $ .",
"The dependence of the output $y_t$ on the previous data ${\\mathbf {d}(t-1)}$ and the current input $u_{t}$ can be modelled by a conditional probability density function (pdf) $f(y_t|u_t, \\mathbf {d}(t-1), \\mathbf {\\Theta }),$ where $\\mathbf {\\Theta }$ is a random potentially multivariate model parameter.",
"The Bayesian methodology treats the model parameter as an unobservable random variable whose knowledge at time $t$ is carried by past data $\\mathbf {d}(t-1)$ .",
"The Bayesian estimation of $\\mathbf {\\Theta }$ then exploits pdf $g(\\mathbf {\\Theta }|\\mathbf {d}(t-1))$ .",
"By the assumption of natural conditions of control [9] we have $g(\\mathbf {\\Theta }|u_t,\\mathbf {d}(t-1)) = g(\\mathbf {\\Theta }|\\mathbf {d}(t-1)),$ i.e., the information about parameter $\\mathbf {\\Theta }$ at time $t$ is conditionally independent of the current input $u_t$ .",
"The prior knowledge $\\mathbf {d}(0) = \\lbrace y_{0}, u_{0}\\rbrace $ formed by the initial data can be determined by an expert or it follows from past estimation.",
"It is also possible to start from a noninformative (flat) prior pdf.",
"The Bayesian recursive estimation exploits the Bayes' rule to incorporate new data into the prior pdf of $\\mathbf {\\Theta }$ as follows $g(\\mathbf {\\Theta }|\\mathbf {d}(t)) \\propto f(y_t|u_{t},\\mathbf {d}(t-1), \\mathbf {\\Theta })g(\\mathbf {\\Theta }|\\mathbf {d}(t-1)),$ where $\\propto $ denotes equality up to a normalizing constant.",
"At the next time instant, the posterior pdf on the left-hand side of (REF ) is used as the prior pdf.",
"The last relation is also known as the dynamic Bayesian data update."
],
[
"Dynamic Bayesian diffusion estimation",
"Let us now focus on the diffusion estimation task.",
"Let there be a distributed network consisting of a set of nodes interacting with their neighbours, which collectively estimate the common parameter of interest using the same model structure.",
"Furthermore, let us impose the following constraint: the nodes are able to communicate one-to-one only within their closed neighbourhood defined as follows: Given a network represented by an undirected graph consisting of $M \\in \\mathbb {N}$ nodes, the closed neighbourhood $\\mathcal {N}_{k}$ of the $k$ th node, $1 \\le k \\le M$ , is the set consisting of its adjacent nodes and node $k$ .",
"An example of a network including a closed neighbourhood $\\mathcal {N}_{1} = \\left\\lbrace 1,2,3,5 \\right\\rbrace $ of node $k=1$ is depicted in Figure REF .",
"Figure: Closed neighbourhood 𝒩 1 =1,2,3,5\\mathcal {N}_{1}=\\left\\lbrace 1,2,3,5 \\right\\rbrace .The diffusion estimation involves two subsequent steps, the former of which is optional but preferred: Incremental update – also known as the data update, is a diffusion alternative of (REF ).",
"The nodes propagate data within their closed neighbourhood and incorporate them into their local statistical knowledge; Spatial update – the nodes propagate point parameter estimates (i.e.",
"mean values) or posterior pdfs within their closed neighbourhood and correct their local estimates.",
"Figure: Incremental update of node k=1k=1 by data from its adjacent neighbours l∈𝒩 k l\\in \\mathcal {N}_{k}.",
"The spatial update looks similarly, the nodes exchange either whole pdfs (i.e., the hyperparameters) of Θ\\mathbf {\\Theta } or its estimates."
],
[
"Incremental update",
"First, we develop the general theory of the incremental update using the Bayesian decision making paradigm.",
"Let $A$ be a measurable space of decisions, $\\beta = \\dim (\\mathbf {\\Theta })$ and let $L: \\mathbb {R}^{\\beta } \\times A \\rightarrow \\mathbb {R}$ be an $\\mathbb {L}_{1}$ -measurable loss function.",
"The Bayesian decision making problem consists of choosing $a \\in A$ by using a measurable decision rule $\\delta : \\mathbb {R} \\rightarrow A$ after an observation of random variable $X$ being obtained.",
"Therefore, we introduce the risk $R\\left( \\mathbf {\\Theta }, \\delta \\right)= \\mathbb {E}_{X} \\left[ L(\\mathbf {\\Theta }, \\delta (X))|\\mathbf {\\Theta }\\right]$ and the Bayesian risk function $\\rho (g,\\delta )=\\mathbb {E}_{\\mathbf {\\Theta }}\\left[R\\left( \\mathbf {\\Theta }, \\delta \\right)\\right]$ measuring the quality of a decision rule $\\delta $ under ignorance of a parameter $\\mathbf {\\Theta }$ with prior $g(\\mathbf {\\Theta })$ .",
"The Bayes' rule is that one which satisfies the condition $\\mathbb {E}_{\\mathbf {\\Theta }} \\left[ L(\\mathbf {\\Theta }, \\delta (X)) | X=x \\right]= \\inf _{a \\in A} \\mathbb {E}_{\\mathbf {\\Theta }}\\left[ L(\\mathbf {\\Theta }, a)|X=x \\right]$ where the integration is with respect to the posterior pdf of $\\mathbf {\\Theta }$ .",
"Consider now the situation from the $k$ th node's perspective, exploiting the data from its closed neighbourhood.",
"In [10], for any given $a$ and weights $c_{l,k}$ (where $l\\in \\mathcal {N}_{k}$ ), the approximate of the Bayesian inference under ignorance of the prior distribution was proposed in terms of $\\widehat{\\mathbb {E}} \\left[ L_{k}(\\mathbf {\\Theta }, a) | X=x \\right]= \\sum _{l\\in \\mathcal {N}_{k}} c_{l,k} L_{l}(\\mathbf {\\Theta }, a).$ Namely, $c_{l,k}$ represents weight of $l$ th node with respect to the $k$ th one and $\\sum _{l\\in \\mathcal {N}_{k}}c_{l,k} = 1$ .",
"Remind, that the Bayes' rule transforming the prior pdf to the posterior pdf is completely compatible with the maximum entropy principle [6], hence we only need to reflect the fact that for a fixed time, multiple data are at disposal.",
"To stay in the entropy framework, we will exploit the minimum cross-entropy principle (MinXEnt) to find a rule for handling the data.",
"[Kullback Leibler divergence] Let $f$ , $g$ be two pdfs describing random variable $X$ .",
"The Kullback-Leibler divergence (also known as the cross-entropy) of $f$ and $g$ is defined as $\\mathcal {D}(f||g) &= \\int f(x) \\log \\frac{f(x)}{g(x)} \\mathrm {d}x \\\\&= \\int f(x) \\log f(x) \\mathrm {d}x - \\int f(x) \\log g(x) \\mathrm {d}x \\\\&= H(f,g) - H(f) $ where $H(\\cdot )$ denotes entropy and $H(\\cdot ,\\cdot )$ stands for the cross-entropy.",
"Given $f$ , the minimization of the Kullback-Leibler divergence $\\mathcal {D}(f||g)$ is equivalent to the minimization of $H(f,g)$ .",
"Trivial.",
"Instead of operating on nodes' posterior pdfs using a sort of averaging or projection, e.g.",
"[7], we propose to exploit the principle of weighted likelihoods [12], [13].",
"Let $f(x|\\mathbf {\\Theta })$ and $f(x|a)$ denote conditional pdfs with respect to $\\mathbf {\\Theta }$ and $a$ respectively.",
"The Bayesian framework assigns $\\mathcal {D}(f(x|\\mathbf {\\Theta })||f(x|a)) = L(\\mathbf {\\Theta }, a)$ .",
"Under $k$ fixed, (REF ) reads $\\widehat{\\mathbb {E}} \\left[ L_{k}(\\mathbf {\\Theta }, a)|x_{k} \\right]= \\sum _{l\\in \\mathcal {N}_{k}} c_{l,k} \\mathcal {D}\\!\\left(f_{l}(x_l|\\mathbf {\\Theta })\\big |\\big |f_{l}(x_l|a)\\right),$ where $x_{l}$ denotes data from $l$ th node.",
"Since we have just one observation for each node $l\\in \\mathcal {N}_{k}$ , we get $\\widehat{\\mathbb {E}} \\left[ L_{k}(\\mathbf {\\Theta }, a)|x_{k} \\right]= \\sum _{l\\in \\mathcal {N}_{k}} c_{l,k} f_{l}(x_l|\\mathbf {\\Theta })\\log \\frac{f_{l}(x_l|\\mathbf {\\Theta })}{f_{l}(x_l|a)}.$ Under ignorance of $\\mathbf {\\Theta }$ we set, accordingly to maximum entropy principle, $f_{l}(x_l|\\mathbf {\\Theta })=1/\\textrm {card}(\\mathcal {N}_{k})$ where $\\textrm {card}$ denotes set cardinality.",
"Formula (REF ) then looks as follows: $\\sum _{l\\in \\mathcal {N}_{k}} \\frac{c_{l,k}}{\\textrm {card}(\\mathcal {N}_{k})}\\log f_{l}(x_l|\\mathbf {\\Theta })-\\sum _{l\\in \\mathcal {N}_{k}} \\frac{c_{l,k}}{\\textrm {card}(\\mathcal {N}_{k})}\\log f_{l}(x_l|a).$ We see that only the second part of (REF ) should be considered for the minimization \"through\" the set $A$ of possible decisions.",
"Particularly: $\\mathop {\\arg \\,\\min }\\limits _{a\\in A} \\left( -\\sum _{l\\in \\mathcal {N}_{k}} c_{l,k}\\log f_{l}(x_l|a) \\right) \\\\[2mm]= \\mathop {\\arg \\,\\max }\\limits _{a\\in A} \\sum _{l\\in \\mathcal {N}_{k}} c_{l,k}\\log f_{l}(x_l|a) \\\\[2mm]= \\mathop {\\arg \\,\\max }\\limits _{a\\in A} \\prod _{l\\in \\mathcal {N}_{k}} f_{l}(x_l|a)^{c_{l,k}},$ where $c_{l,k}$ denote the previously mentioned weights.",
"The argument (REF ) together with the Bayes' rule (REF ), preserving entropy maximization, yield theoretically consistent incremental update in the form $g_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t)) &\\propto g_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t-1)) \\\\[2mm]&\\times \\prod _{l\\in \\mathcal {N}_{k}} f_{l}(y_{l,t}|u_{l,t}, \\mathbf {d}_{l}(t-1), \\mathbf {\\Theta })^{c_{l,k}},$ where $\\overline{\\mathbf {d}}(t)$ stands for all data available from sources in $\\mathcal {N}_{k}$ ."
],
[
"Spatial update",
"The spatial update follows after the incremental update.",
"In this step, the nodes exchange information about unknown model parameter $\\mathbf {\\Theta }$ , either in the form of its estimates or hyperparameters of its distribution.",
"Formally, for fixed $k$ , the information from all nodes in $\\mathcal {N}_{k}$ describes the finite mixture density $g_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t)) = \\sum _{l\\in \\mathcal {N}_{k}} a_{l,k} g_l(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t)), \\quad \\sum _{l\\in \\mathcal {N}_{k}}a_{l,k} = 1,$ where $0\\le a_{l,k}\\le 1$ is the weight of $l$ th node's estimate from $k$ th node's viewpoint.",
"Here, two possible departure points arise.",
"First, more generally, we may be interested in a “consensus” distribution, i.e., a single distribution best representing the mixture (REF ) at node $k$ .",
"Its pdf can be found as the argument minimizing the Kullback-Leibler divergence, $\\mathop {\\arg \\,\\min }\\limits _{\\tilde{g}_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t))\\in \\mathcal {G}} \\mathcal {D}\\left(g_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t))\\bigg |\\bigg | \\tilde{g}_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t))\\right),$ where $\\mathcal {G}$ is the class of all admissible pdfs.",
"The second possibility emerges if we are interested just in the moment(s) available from $g_{k}(\\mathbf {\\Theta }|\\overline{\\mathbf {d}}(t))$ .",
"Then, e.g., the first moment (the mean value) is given by the convex combination of mean values of the mixture density components, $\\widehat{\\mathbf {\\Theta }}_{k} \\leftarrow \\sum _{l\\in \\mathcal {N}_{k}} a_{l,k} \\widehat{\\mathbf {\\Theta }}_l.$ For other moments see, e.g., [5].",
"The latter approach is of particular interest if the distribution is parameterized by moments (e.g., the Gaussian distribution).",
"Another appealing fact related to these distributions is that (REF ) is often a direct consequence of (REF ).",
"In these cases, it is possible to omit the Kullback-Leibler divergence minimization and benefit directly from (REF ).",
"While (REF ) is a final product at time $t$ , the pdf resulting from (REF ) can be reused as the $k$ 's prior pdf at the next time step.",
"Properties of the diffusion estimator strongly depend on the underlaying particular estimators in a neighbourhood and their weights $a_{l,k}$ and $c_{l,k}$ .",
"In this respect, the need for effective determination of weights is essential."
],
[
"Determination of weights $a_{l,k}$ and {{formula:cc9eb7ff-d701-437b-85f6-0efb7df184e2}}",
"There are several possible strategies how to determine the weights $a_{l,k}$ and $c_{l,k}$ .",
"Besides the relatively unfeasible uniform weights, the user can perform with the aid of Metropolis weights, proposed by [14] and further used in recent literature.",
"Another options are relative degree and yet more sophisticated relative degree-variance weights, based on the cardinality of the node's closed neighbourhood, [3].",
"We only conjecture that a suitable probabilistic method exploiting, e.g., the likelihood of $l$ th data with respect to $k$ th node could be found as well.",
"A substantial advantage of such method would be its suitability for dynamic cases, requiring stable determination of $a_{l,k}$ and $c_{l,k}$ .",
"As a consequence, it would allow to suppress the influence of data and/or estimates from a failing node (sensor) on other nodes.",
"However, such methods are being developed in the meantime."
],
[
"Derivation for Gaussian regressive model",
"In this section, a practical application of the proposed methodology is given.",
"We derive the dynamic Bayesian diffusion estimator of the popular Gaussian linear regressive model.",
"In two following subsections, we shortly present the standard Bayesian estimation of such model and develop its diffusion estimator.",
"This case is just one example of a wide class of possible models, the applicability on which is straightforward.",
"This class includes particularly popular Bayesian models with conjugate priors."
],
[
"Gaussian linear regressive model",
"Given a regression vector $\\mathbf {\\psi }_t \\in \\mathbb {R}^{n}, t=1,2,\\dots $ and a dependent random variable $y_t \\in \\mathbb {R}$ , the Gaussian linear regressive model takes the form $y_t = \\mathbf {\\psi }_{t}{\\theta } + \\varepsilon _{t},$ where $\\mathbf {\\theta }\\in \\mathbb {R}^{n}$ is the regression coefficient and ${\\varepsilon _{t} \\sim \\mathcal {N}(0,\\!",
"\\sigma ^2)}$ is the Gaussian white noise.",
"This makes ${y_t \\sim \\mathcal {N}(\\mathbf {\\psi }_t{\\theta }, \\sigma ^{2})}$ and the regression model (REF ) can be expressed by pdf ${f(y_{t}|\\mathbf {\\psi }_{t}, \\mathbf {\\Theta }})$ .",
"From the Bayesian viewpoint, the model parameters ${\\mathbf {\\Theta } \\equiv \\lbrace \\mathbf {\\theta }, \\sigma ^{2}\\rbrace }$ are also random variables.",
"Under ignorance of their values, the proper conjugate prior distribution is the normal inverse-gamma ($\\mathcal {N}i$ ) one [2].",
"Namely, $\\mathbf {\\theta }$ is normal and $\\sigma ^{2}$ is inverse-gamma.",
"[Normal inverse-gamma pdf] For a variable $\\mathbf {\\Theta } = \\lbrace \\mathbf {\\theta }, \\sigma ^{2}\\rbrace $ , $\\mathbf {\\theta } \\in \\mathbb {R}^{n}$ and ${\\sigma ^{2} \\in \\mathbb {R}}$ , the normal inverse-gamma $\\mathcal {N}i(\\mathbf {V}, \\nu )$ pdf with a symmetric positive definite extended information matrix ${\\mathbf {V}\\in \\mathbb {R}^{N\\times N}}, {N=n+1}$ and the degrees of freedom $\\nu \\in \\mathbb {R}$ has the form $g(\\mathbf {\\theta }, \\sigma ^2|\\mathbf {V},\\nu )= \\frac{\\sigma ^{-(\\nu +n+1)}}{\\mathcal {I}(\\mathbf {V}, \\nu )}\\!\\exp \\!\\left\\lbrace \\!-\\frac{1}{2\\sigma ^{2}}\\begin{bmatrix}-1 \\\\\\mathbf {\\theta }\\end{bmatrix}\\mathbf {V}\\begin{bmatrix}-1 \\\\\\mathbf {\\theta }\\end{bmatrix}\\right\\rbrace $ where $\\mathcal {I}(\\cdot )$ is the normalization term such that $\\int g(\\mathbf {\\theta }, \\sigma ^2|\\mathbf {V},\\nu ) \\mathrm {d}\\mathbf {\\Theta } = 1.$ Both $\\mathbf {V}$ and $\\nu $ are sufficient statistics [2] representing data $\\mathbf {d}(t-1) = \\lbrace y_{t-1}, \\psi _{t-1}, \\dots , y_{0}, \\psi _{0}\\rbrace $ .",
"The Bayesian recursive estimation (REF ) updates the prior pdf by new data according to the following theorem.",
"[Bayesian estimation of a $\\mathcal {N}i$ model] Let $g(\\mathbf {\\theta }, \\sigma ^{2}|\\mathbf {V}, \\nu )$ be a $\\mathcal {N}i$ pdf, $t=1,2,\\dots $ The Bayesian estimation (REF ) updates the sufficient statistics $\\mathbf {V}\\in \\mathbb {R}^{N\\times N}$ and $\\nu \\in \\mathbb {R}$ by real scalar realization $y_{t}$ and regression vector $\\mathbf {\\psi }_{t} \\in \\mathbb {R}^{N-1}$ as follows: $\\mathbf {V}_{t} &= \\mathbf {V}_{t-1}+\\begin{bmatrix}y_{t} \\\\\\mathbf {\\psi }_{t}\\end{bmatrix}\\begin{bmatrix}y_{t} \\\\\\mathbf {\\psi }_{t}\\end{bmatrix}{E:update-v} \\\\\\nu _{t} &= \\nu _{t-1} + 1 $ The multivariate point estimator $\\widehat{\\mathbf {\\theta }}_t \\in \\mathbb {R}^{N-1}$ of regression coefficient is the mean value of the $\\mathcal {N}i$ distribution given by $\\widehat{\\mathbf {\\theta }}_{t}=\\begin{bmatrix}V_{22} & \\ldots & V_{2N} \\\\\\vdots & \\ddots & \\vdots \\\\V_{N2} & \\ldots & V_{NN}\\end{bmatrix}^{-1}_{t}\\begin{bmatrix}V_{21} \\\\ \\vdots \\\\ V_{N1}\\end{bmatrix}_{t}$ The update of statistics $\\mathbf {V}$ and $\\nu $ follows directly from multiplication of Gaussian models (likelihoods), see, e.g., [9].",
"The point estimator is the well-known ordinary least squares estimator."
],
[
"Diffusion estimation of the Bayesian regressive model",
"In order to derive the dynamic Bayesian diffusion estimator of $\\mathbf {\\Theta }$ , we follow the principles given in Section .",
"Let us consider a network of $M\\in \\mathbb {N}$ distributed nodes.",
"Each node ${k\\in \\lbrace 1,\\dots ,M\\rbrace }$ evaluates a model $f(y_{k;t}|\\mathbf {\\psi }_{k;t}, \\mathbf {\\Theta }, \\mathbf {V}_{k;t-1}, \\nu _{k;t-1})$ and runs the diffusion Bayesian estimation (REF ) of its parameters in the form $g_{k}(\\mathbf {\\Theta }|\\mathbf {V}_{k;t}, \\nu _{k;t})&\\propto g_{k}(\\mathbf {\\Theta }|\\mathbf {V}_{k;t-1}, \\nu _{k;t-1}) \\\\[2mm]&\\!\\!\\!",
"\\!\\!\\!\\times \\prod _{l\\in \\mathcal {N}_{k}} f_{l}(y_{l;t}|\\mathbf {\\psi }_{l;t}, \\mathbf {\\Theta }, \\mathbf {V}_{l;t-1}, \\nu _{l;t-1})^{c_{l,k}}.",
"$ Here $0 \\le c_{l,k} \\le 1$ weights $l$ th node's data with respect to $k$ th node, $l\\in \\mathcal {N}_{k}$ , where $\\sum _{l\\in \\mathcal {N}_{k}} c_{l,k} = 1$ .",
"Simply put, the $k$ th node updates its prior pdf of $\\mathbf {\\Theta }$ by data from its closed neighbourhood $\\mathcal {N}_{k}$ .",
"Since we deal with the $\\mathcal {N}i$ pdf, this update takes the form expressed by the following proposition.",
"[Incremental update of $\\mathcal {N}i$ pdf] Given a $k$ th node, $k\\in \\lbrace 1,\\dots ,M\\rbrace $ , the incremental version of the Bayesian estimation (Theorem REF ) updates the $k$ th node's prior $\\mathcal {N}i$ pdf of $\\mathbf {\\Theta }$ by data $[y_{l;t}, \\mathbf {\\psi }_{l,t}], weighted by $ cl,k$, from its adjacent neighbours $ lNk$ according to the following rules:{\\begin{@align}{1}{-1}\\mathbf {V}_{k;t} &= \\mathbf {V}_{k;t-1}+\\sum _{l \\in \\mathcal {N}_{k}}c_{l,k}\\begin{bmatrix}y_{l;t} \\\\\\mathbf {\\psi }_{l;t}\\end{bmatrix}\\begin{bmatrix}y_{l;t} \\\\\\mathbf {\\psi }_{l;t}\\end{bmatrix}{E:diff-update-v} \\\\\\nu _{k;t} &= \\nu _{k;t-1} + 1, \\end{@align}}where\\begin{equation}0 \\le c_{l,k} \\le 1, \\qquad \\sum _{l\\in \\mathcal {N}_{k}} c_{l,k} = 1, \\quad l\\in \\mathcal {N}_{k}.\\end{equation}$ Let $\\kappa = \\mathrm {card}(\\mathcal {N}_{k})$ .",
"The formula () following from (REF ) is equivalent to $\\kappa $ updates () of $\\mathbf {V}_{k,t-1}$ by data $[y_{l,t}, \\mathbf {\\psi }_{l,t}] weighted by $ cl,k$.",
"Formula (\\ref {E:diff-update-nu}) is a direct equivalent of (\\ref {E:update-nu}).",
"\\phantom{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx} \\Box $ In linear regression, we are particularly interested in point estimation of the regression coefficient $\\mathbf {\\theta }$ .",
"[Spatial update of $\\widehat{\\mathbf {\\theta }}$ ] Given a $k$ th node, $k\\in \\lbrace 1,\\dots ,M\\rbrace $ .",
"The spatial update (REF ) of the estimate $\\hat{\\mathbf {\\theta }}_{k;t}$ has the form $\\hat{\\mathbf {\\theta }}_{k;t} = \\sum _{l\\in \\mathcal {N}_{k}} a_{l,k} \\hat{\\mathbf {\\theta }}_{l;t},$ where $0 \\le a_{l,k} \\le 1, \\qquad \\sum _{l\\in \\mathcal {N}_{k}} a_{l,k} = 1.$ $a_{l,k}$ denotes the weight of $l$ th node's point estimate with respect to $k$ th node.",
"This is a straightforward use of (REF ).xxxxxxx $\\Box $ Similar procedure applies to estimation of $\\sigma ^{2}$ .",
"The summary of the derived steps is in Algorithm 1."
],
[
"Dynamic Bayesian diffusion regressive model and RLS",
"Let us demonstrate the simplicity of transition from the dynamic Bayesian diffusion estimation to its non-Bayesian counterpart.",
"For simplicity, consider $y$ scalar and partition the extended information matrix $\\mathbf {V}$ as follows: $\\mathbf {V} =\\left[\\begin{array}{c|c}V_{y} & \\mathbf {V}_{y\\psi } \\hline \\mathbf {V}_{y\\psi } & \\mathbf {V}_{\\psi }\\end{array}\\right],\\text{ where }V_{y} \\in \\mathbb {R}, \\mathbf {V}_{\\psi } \\in \\mathbb {R}^{n\\times n}.$ Furthermore, let us denote $\\mathbf {C} = \\mathbf {V}_{\\psi }^{-1}$ and see, how the update – Proposition REF – performs on reparameterized $\\mathcal {N}i$ pdf.",
"[Reparametrization of $\\mathcal {N}i$ pdf] Given pdf $\\mathcal {N}i(\\mathbf {V},\\nu )$ of $\\mathbf {\\Theta } = \\lbrace \\mathbf {\\theta }, \\sigma ^{2}\\rbrace $ .",
"The statistic $\\mathbf {V}\\in \\mathbb {R}^{N\\times N}$ can be decomposed into the lower-dimensional statistics $\\mathbf {C}\\in \\mathbb {R}^{n\\times n}, \\widehat{\\mathbf {\\theta }} \\in \\mathbb {R}^{n}$ and $\\Lambda \\in \\mathbb {R}$ where $n = N-1$ , yielding the reparametrized pdf $\\mathcal {N}i(\\mathbf {C}, \\widehat{\\mathbf {\\theta }}, \\Lambda , \\nu )$ as follows: $g&(\\mathbf {\\theta }, \\sigma ^2|\\mathbf {C},\\widehat{\\mathbf {\\theta }}, \\Lambda , \\nu )= \\frac{\\sigma ^{-(\\nu +n+1)}}{\\mathcal {I}(\\mathbf {C},\\widehat{\\mathbf {\\theta }}, \\Lambda , \\nu )} \\times \\\\&\\times \\exp \\left\\lbrace -\\frac{1}{2\\sigma ^{2}}\\left[(\\mathbf {\\theta } - \\widehat{\\mathbf {\\theta }}){C}^{-1}(\\mathbf {\\theta } - \\widehat{\\mathbf {\\theta }})+ \\Lambda \\right]\\right\\rbrace $ where $\\widehat{\\mathbf {\\theta }} &= \\mathbf {C} \\mathbf {V}_{y\\psi } , \\\\\\Lambda &= V_{y} - \\mathbf {V}_{y\\psi }{C} \\mathbf {V}_{y\\psi } $ and where $\\mathcal {I}(\\mathbf {C},\\widehat{\\mathbf {\\theta }}, \\Lambda , \\nu )$ is the normalization term such that $\\int g(\\mathbf {\\theta }, \\sigma ^2|\\mathbf {C},\\widehat{\\mathbf {\\theta }}, \\Lambda , \\nu ) \\mathrm {d}\\mathbf {\\Theta } = 1.",
"$ By completion of squares $\\begin{bmatrix}-1 \\\\\\mathbf {\\theta }\\end{bmatrix}[\\begin{array}{cc}V_{y} & \\mathbf {V}_{y\\psi }\\mathbf {V}_{y\\psi } & \\mathbf {V}_{\\psi }\\end{array}\\begin{bmatrix}-1 \\\\\\mathbf {\\theta }\\end{bmatrix}=V_{y} - 2 \\mathbf {\\theta }{V}_{y\\psi } + \\mathbf {\\theta }{V}_{\\psi } \\mathbf {\\theta }\\\\[2mm]=\\left( \\mathbf {\\theta } - \\mathbf {C} \\mathbf {V}_{y\\mathbf {\\psi }} \\right)^{\\!\\mathrm {T}}\\!\\!\\mathbf {C}^{-1}\\!\\left( \\mathbf {\\theta } - \\mathbf {C} \\mathbf {V}_{y\\mathbf {\\psi }} \\right)\\!+\\!\\left( V_{y} - \\mathbf {V}_{y\\psi }{C} \\mathbf {V}_{y\\psi } \\right).$ $\\Box $ Now, we focus on the recursive update of $k$ th node's reparameterized $\\mathcal {N}i$ pdf statistics.",
"First note, that the right-hand side of formula () can be viewed as a sequential (one-by-one) update of $k$ th nodes' $\\mathbf {V}_{k,t}$ by data $[y_{l;t}, \\mathbf {\\psi }_{l,t}] with weights $ cl,k$ where $ lNk$.",
"This means, that when the transition $ (t-1) t$ occurs, the assignment\\begin{equation}\\mathbf {V}_{k;t} := \\mathbf {V}_{k;t-1}\\end{equation}is made, followed by the updates\\begin{equation}\\mathbf {V}_{k;t}\\leftarrow \\mathbf {V}_{k;t} + c_{l,k}\\begin{bmatrix}y_{l;t} \\\\\\mathbf {\\psi }_{l,t}\\end{bmatrix}\\begin{bmatrix}y_{l;t} \\\\\\mathbf {\\psi }_{l;t}\\end{bmatrix}\\text{for all}\\quad l\\in \\mathcal {N}_{k}.\\end{equation}Therefore, we can take advantage of deriving the update of $ k$th reparameterized pdf by data from $ l$th node.",
"The reparameterized equivalent of (\\ref {E:diff-update-v}) then results from (\\ref {E:V-update1}) for all $ lNk$ and $ t$ fixed.",
"This sequential update procedure describes the following proposition.$ [Update of reparameterized $\\mathcal {N}i$ pdf] Given a pdf $g(\\mathbf {\\theta }, \\sigma ^{2}|\\mathbf {C}, \\widehat{\\mathbf {\\theta }}, \\Lambda , \\nu )$ of $k$ th node at fixed time $t$ .",
"After initialization $\\mathbf {C}_{k,t}&:=\\mathbf {C}_{k,t-1},&\\widehat{\\mathbf {\\theta }}_{k;t}&:=\\widehat{\\mathbf {\\theta }}_{k;t-1}, \\\\\\Lambda _{k;t}&:= \\Lambda _{k;t-1},&\\nu _{k;t}&:=\\nu _{k;t-1}, $ the update by data $y_{l;t}, \\mathbf {\\psi }_{l;t}$ , weighted by $c_{l,k}$ for all $l\\in \\mathcal {N}_{k}$ reads $\\mathbf {C}_{k;t} &\\leftarrow \\mathbf {C}_{k;t}- \\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}\\mathbf {\\psi }_{l;t}{C}_{k;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}} \\\\\\widehat{\\mathbf {\\theta }}_{k;t} &\\leftarrow \\widehat{\\mathbf {\\theta }}_{k;t}+\\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}[y_{l;t}-\\mathbf {\\psi }_{l;t}{\\mathbf {\\theta }}_{k;t}] \\\\\\Lambda _{k;t} &\\leftarrow \\Lambda _{k;t}+\\frac{\\left(c_{l,k}y_{l;t}+c_{l,k}\\mathbf {\\psi }_{k;t}{\\mathbf {\\theta }}_{k;t}\\right)^2}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}} \\\\\\nu _{k;t} &\\leftarrow \\nu _{k;t} + c_{l,k} $ Fix $t$ and rewrite the update of blocks of $\\mathbf {V}_{k;t}$ of $k$ th node by $y_{l;t}$ and $\\mathbf {\\psi }_{l;t}$ from its adjacent neighbour $l\\in \\mathcal {N}_{k}$ .",
"The initialization (REF ) is equivalent to $\\mathbf {V}_{k;t} \\leftarrow \\mathbf {V}_{k;t-1}, \\qquad \\nu _{k;t} \\leftarrow \\nu _{k;t-1}.",
"$ The blocks of $\\mathbf {V}_{k;t}$ are updated as follows: $V_{k;y;t} &\\leftarrow V_{k;y;t} + c_{l,k} y_{l;t}^{2} \\\\\\mathbf {V}_{k;\\psi ;t} &\\leftarrow \\mathbf {V}_{k;\\psi ;t} + c_{l,k} \\mathbf {\\psi }_{l;t} \\mathbf {\\psi }_{l;t}{E:rls-Vpsi} \\\\\\mathbf {V}_{k;y\\psi ;t} &\\leftarrow \\mathbf {V}_{k;y\\psi ;t} + c_{l,k} \\mathbf {\\psi }_{l;t} y_{l;t} $ Notice, that () is equivalent to $\\mathbf {C}_{k;t}^{-1} \\leftarrow \\mathbf {C}_{k;t}^{-1} + c_{l,k} \\mathbf {\\psi }_{l;t} \\mathbf {\\psi }_{l;t}$ By application of the Sherman-Morrison formula, Proposition in Appendix, we obtain $\\mathbf {C}_{k;t} \\leftarrow \\mathbf {C}_{k;t}- \\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}\\mathbf {\\psi }_{l;t}{C}_{k;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}, $ which proves (REF ).",
"The substitution of (REF ) and () into (REF ) yields $\\widehat{\\mathbf {\\theta }}_{k;t} &\\leftarrow \\left(\\!\\!\\mathbf {C}_{k;t}- \\frac{c_{l,k}\\!",
"\\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}\\mathbf {\\psi }_{l;t}{C}_{k;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}\\right)\\!\\!\\left(\\mathbf {V}_{k;y\\psi ;t} + c_{l,k} \\mathbf {\\psi }_{l;t} y_{l;t}\\right) \\\\[1mm]&\\leftarrow {\\mathbf {C}_{k;t}\\mathbf {V}_{k;y\\psi ;t}} + c_{l,k}\\mathbf {C}_{k;t}\\mathbf {\\psi }_{l;t} y_{l;t}-\\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}\\mathbf {\\psi }_{l;t}{C}_{k;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}} \\\\[1mm]&\\times \\mathbf {V}_{k;y\\psi ;t}- \\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}\\mathbf {\\psi }_{l;t}{C}_{k;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}c_{l,k} \\mathbf {\\psi }_{l;t} y_{l;t} \\\\[1mm]{} & \\leftarrow \\widehat{\\mathbf {\\theta }}_{k;t}+\\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}[y_{l;t}-\\mathbf {\\psi }_{l;t}]\\\\[1mm]{} & \\leftarrow \\widehat{\\mathbf {\\theta }}_{k;t}+\\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}[y_{l;t}-\\mathbf {\\psi }_{l;t}{\\mathbf {\\theta }}_{k;t}] $ proving ().",
"Similarly obtained Formula for $\\Lambda $ : $\\Lambda _{k;t} & \\leftarrow V_{k;y;t}+c_{l,k}y_{l;t}^2-\\left(\\mathbf {V}_{k;y\\psi ;t}+c_{l,k} \\mathbf {\\psi }_{l;t} y_{l;t}\\right)\\\\[1mm]{} &\\times \\left(\\!\\!\\mathbf {C}_{k;t}\\!-\\!",
"\\frac{c_{l,k} \\mathbf {C}_{k;t} \\mathbf {\\psi }_{l;t}\\mathbf {\\psi }_{l;t}{C}_{k;t}}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}\\!\\right)\\!\\!\\left( \\mathbf {V}_{k;y\\psi ;t}+c_{l,k} \\mathbf {\\psi }_{l;t} y_{l;t}\\right)\\\\[1mm]{} &\\leftarrow \\Lambda _{k;t}+ \\frac{\\left(c_{l,k}y_{l;t}-c_{l,k}\\mathbf {\\psi }_{k;t}{\\mathbf {\\theta }}_{k;t}\\right)^2}{1 + c_{l,k} \\mathbf {\\psi }_{l;t}{C}_{k;t} \\mathbf {\\psi }_{l;t}}$ proves ().",
"Finally, the fact that $\\sum _{l\\in \\mathcal {N}_{k}} c_{l,k} = 1 $ proves ().",
"xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx$\\Box $ Obviously, since $c_{l,k}$ sum to unity, it is sufficient to increment $\\nu _{k;t}$ at each time step by 1.",
"The well-known recursive least-squares evaluate a covariance matrix and the regression coefficients estimates, which is the same as $\\mathbf {C}$ and $\\widehat{\\mathbf {\\theta }}$ in the reparameterized $\\mathcal {N}i$ pdf.",
"In this respect, the dynamic Bayesian diffusion estimation of the Bayesian regressive model is completely equivalent to the diffusion (unweighted) RLS, cf.",
"[4].",
"This proves the feasibility of the method.",
"However, the exploited probabilistic framework allows to use the very general principles given in Section with a wider class of various models.",
"Diffusion Bayesian regressive model Initialization: $k\\in \\lbrace 1,\\dots ,M\\rbrace $ Set prior statistics $\\mathbf {V}_{k;0}$ and $\\nu _{k;0}$ .",
"Set weights $c_{l,k}$ and $a_{l,k}$ , $l\\in \\mathcal {N}_{k}$ .",
"Online steps: $t=1,2,\\ldots $ Incremental update: $k\\in \\lbrace 1,\\dots ,M\\rbrace $ Gather data $[y_{l;t}, \\mathbf {\\psi }_{l;t}] for all $ lNk$.",
"\\;Perform the updates of $ Vk,t-1, k;t-1$, Prop.",
"\\ref {P:gauss-incremental-update}.",
"\\;Calculate point estimates $ k;t$, Prop.",
"\\ref {T:bayesian-estimation-nig}.",
"\\;$ Spatial update: $k\\in \\lbrace 1,\\dots ,M\\rbrace $ Gather point estimates $\\widehat{\\mathbf {\\theta }}_{l;t}$ for all $l\\in \\mathcal {N}_{k}$ .",
"Perform the update of $\\widehat{\\mathbf {\\theta }}_{k;t}$ , Prop.",
"REF ."
],
[
"Conclusions",
"The dynamic Bayesian diffusion estimation methodology provides a way to solving the decentralized estimation problems in the modern complex distributed systems, e.g., the sensor and ad-hoc networks.",
"The theoretical aspects of the method are advocated by the maximum entropy and minimum cross-entropy principles.",
"Being developed in the Bayesian framework, it is directly applicable to a wide class of different models.",
"As a special case, the application of the methodology to the dynamic Bayesian linear regression yields particularly useful diffusion recursive least squares.",
"This aspect also supports the assumption of validity of the method.",
"In addition, it demonstrates that for practical purposes it is possible to leave the distribution-oriented perspective in favor of the traditional non-Bayesian reasoning.",
"The foreseen research activities comprise, among others, the analysis of properties of the diffusion estimator, the Bayesian estimation under specific constraints related, e.g., to bandwidth etc.",
"Also, a probabilistic method for dynamic determination of the weighting coefficients $a_{l,k}$ and $c_{l,k}$ is of particular interest."
],
[
"appendix",
"[Sherman-Morrison formula] Let $\\mathbf {A} \\in \\mathbb {R}^{n\\times n}$ be an invertible matrix and $\\mathbf {u},\\mathbf {v} \\in \\mathbb {R}^{n}$ two vectors.",
"Then, the following equality holds, $\\left( \\mathbf {A} + \\mathbf {u}\\mathbf {v})^{-1}= \\mathbf {A}^{-1}- \\frac{\\mathbf {A}^{-1} \\mathbf {u}\\mathbf {v}{A}^{-1}}{1 + \\mathbf {v}{A}^{-1} \\mathbf {u}}.\\right.$ Trivial.xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx$\\Box $"
]
] | 1204.1158 |
[
[
"Unified analytical treatments to qubit-oscillator systems"
],
[
"Abstract An effective scheme within two displaced bosonic operators with equal positive and negative displacements is extended to study qubit-oscillator systems analytically in an unified way.",
"Many previous analytical treatments, such as generalized rotating-wave approximation (GRWA) [Phys.",
"Rev.",
"Lett.",
"99, 173601 (2007)] and an expansion in the qubit tunneling matrix element in the deep strong coupling regime [Phys.",
"Rev.",
"Lett.",
"105, 263603 (2010)] can be recovered straightforwardly in the present scheme.",
"Moreover, further improving GRWA and extension to the finite-bias case are implemented easily.",
"The analytical expressions are then derived explicitly and uniquely, which work well in a wide range of the coupling strengthes, detunings, and static bias including the recent experimentally accessible parameters."
],
[
"introduction",
"Matter-matter interaction is fundamental and ubiquitous in modern physics ranging from quantum optics, quantum information science to condensed matter physics.",
"The simplest paradigm is a two-level atom (qubit) coupled to the electromagnetic mode of a cavity (oscillator).",
"In the strong coupling regime where the coupling strength $ g/\\omega $ ($\\omega $ is the cavity frequency) between the atom and the cavity mode exceeds the loss rates, the atom and the cavity can repeatedly exchange excitations before coherence is lost.",
"The Rabi oscillations can be observed in this strong coupling atom-cavity system, which is usually called as cavity quantum electrodynamics (QED) [1].",
"Typically, the coupling strength in cavity QED reaches $g/\\omega \\sim 10^{-6}$ .",
"It can be described by the well-known Jaynes-Cummings (JC) model [2] without the rotating-wave approximation (RWA).",
"Recently, for superconducting qubits, a one-dimensional (1D) transmission line resonator [3] or a LC circuit [4], [5], [6], [7] can play a role of the cavity, which is known today as circuit QED.",
"More recently, LC resonator inductively coupled to a superconducting qubit [8], [9], [10] has been realized experimentally.",
"The qubit-resonator coupling has been strengthened from $g/\\omega \\sim 10^{-3}$ in the earlier realization [3], a few percentage later [7], [6] , to most recent ten percentages [8], [9], [10].",
"Due to the ultra-strong coupling strength $g/\\omega \\sim 0.1$ , evidence for the breakdown of the RWA has been provided [8].",
"Recently, some works have been devoted to this qubit-oscillator system in the ultra-strong coupling regime [11], [12], [13], [14], [15].",
"Actually, the JC model without the RWA in a wide coupling regime has been studied extensively for more than 40 years.",
"By polaronic-like transformations or displaced operators, various analytical and numerical approaches have been developed in recent years, an incomplete list is given by Refs [16], [17], [18], [19], [20], [21], [22], [23].",
"Very accurate or exact solutions have been obtained.",
"Various forms of RWA energies have been developed [24], [25] and extensions to the N-level case have been performed recently[26].",
"Most works have been mainly devoted to the zero static bias.",
"The analytical expression with high accuracy for the qubit-Oscillator systems with both zero and finite static bias should be of practical interest.",
"In this paper, by using two displaced bosonic operators with equal positive and negative displacements, we can recover many previous analytical treatments for both zero and finite static bias within the same scheme.",
"What is more, we can extend analytically the previous generalized RWA (GRWA) to the finite static bias.",
"Beyond the GRWA for the zero bias is also performed.",
"The expression is uniquely given and the results are very close to exact ones for wide range of the model parameters which cover the present-day experimentally accessible parameters."
],
[
"Model and exact solution",
"The Hamiltonian for a superconducting qubit coupled to a harmonic oscillator in circuit QED consists of three parts[13], [14].",
"The first one is the interaction between the qubit and the LC resonator, which is described by $H_{int}=\\hbar g(a^{\\dagger }+a)\\sigma _z,$ where $a^{\\dagger }$ , $a$ are the photon creation and annihilation operators in the basis of Fock states of the LC resonator, $g$ is the qubit-cavity coupling constant.",
"The RWA has not been employed here.",
"The effective Hamiltonian for the qubit can be written as the standard one for a two-level system $H=-\\frac{1}{2}\\left( \\varepsilon \\sigma _z+\\Delta \\sigma _x\\right),$ where $\\varepsilon $ and $\\Delta $ are qubit static bias and tunneling matrix element.",
"In the recent circuit QED [8], [9] operating in the ultra-strong coupling regime, they describe the transition frequency of the flux qubit and the tunneling coupling between the two persistent current states.",
"$\\varepsilon =I_p(\\Phi -\\Phi _0/2)$ with $I_p$ the persistent current in the qubit loop, $\\Phi $ an externally applied magnetic flux, and $\\Phi _0$ the flux quantum.",
"In contrast to atomic cavity QED systems, $\\varepsilon $ is easily tunable in circuit QED systems using superconducting qubit.",
"In the above two equations, the Pauli matrix notations $\\sigma _k(k=x,y,z)$ $\\ $ are used in the basis of the two persistent current states.",
"The third one is LC resonator $\\omega a^{\\dagger }a$ with single mode frequency $\\omega $ .",
"Then the Hamiltonian for the whole system reads $\\left( \\hbar =\\omega =1\\right) $ $H=-\\frac{1}{2}\\left( \\varepsilon \\sigma _z+\\Delta \\sigma _x\\right) +a^{\\dagger }a+g(a^{\\dagger }+a)\\sigma _z .$ Motivated by the work in the Dicke model[17], we have introduced two displaced bosonic operators with equal positive and negative displacements in this system[15] $A=a+g,B=a-g ,$ then the Hamiltonian can be written in the following matrix form $H=\\left(\\begin{array}{cc}A^{\\dagger }A-g^2-\\frac{\\varepsilon }{2} & -\\Delta /2 \\\\-\\Delta /2 & B^{\\dagger }B-g^2+\\frac{\\varepsilon }{2}\\end{array}\\right) .$ Note that the linear term for the original bosonic operator $a^{\\dagger }(a)$ is removed, and only the number operators $A^{+}A$ and $B^{+}B$ are left.",
"Therefore the wavefunction can be expanded in terms of these new operators as $\\left| {}\\right\\rangle =\\left(\\begin{array}{l}\\sum _{n=0}^{N_{tr}}c_n\\left| n\\right\\rangle _A \\\\\\sum _{n=0}^{N_{tr}}(-1)^nd_n\\left| n\\right\\rangle _B,\\end{array}\\right) ,$ where $N_{tr}$ is the truanted number.",
"For $A$ operator, we have $\\left| n\\right\\rangle _A &=&\\frac{\\left( A^{\\dagger }\\right)^n}{\\sqrt{n!}}",
"\\left| 0\\right\\rangle _A=\\frac{\\left( a^{\\dagger }+g\\right) ^n}{\\sqrt{n!}}",
"\\left| 0\\right\\rangle _A, \\\\ \\left|0\\right\\rangle _A &=&e^{-\\frac{1}{2}g^2-ga^{\\dagger })}\\left|0\\right\\rangle _a,$ $B$ operator has the same properties.",
"Inserting Eqs.",
"(6) and (7) into the Schr$\\stackrel{..}{o}$ dinger equation, we have $\\left( m-g^2-\\frac{\\varepsilon }{2}\\right) c_m-\\sum _{n=0\\ }D_{mn}d_n&=&Ec_m, \\\\\\left( m-g^2+\\frac{\\varepsilon }{2}\\right) d_m-\\sum _{n=0}D_{mn}c_n&=&Ed_m ,$ where $D_{mn} &=&\\frac{\\Delta }{2}\\left( -1\\right) ^m{}_B\\left\\langle m\\right| \\left|n\\right\\rangle _A, \\\\_B\\left\\langle m\\right| \\left| n\\right\\rangle _A &=&(2g)^{n-m}\\exp (-2g^2)\\sqrt{\\frac{m!}{n!",
"}}L_m^{n-m}\\left( 4g^2\\right),$ for $n\\ge m$ , $L_m^{n-m}(x)$ is Laguerre polynomial, $D_{mn}=D_{nm}$ .",
"Based on Eqs.",
"(REF ) and (), we have given numerically exact solutions to the qubit-oscillator system with any finite static bias $ \\varepsilon $[15].",
"In this paper, we alternatively present some analytical results in the framework of the above formalism.",
"One can see that some recent analytical results by other authors are explicitly covered in the present framework.",
"Moreover, the present scheme is more convenient to perform further analytical studies."
],
[
"Variational study for $\\epsilon =0$",
"To have a sense of two displaced bosonic operators Eq.",
"(REF ), we relax the displacement to be a variational parameter $\\alpha $ , $A=a+\\alpha , B=a-\\alpha ,$ then study the unbiased Hamiltonian ($\\varepsilon =0$ ) variationally.",
"Suppose that the trial state is the vacuum state in these displaced operators as the following $|\\Phi _0\\rangle =\\left( \\ \\begin{array}{l}\\left| 0\\right\\rangle _A \\\\\\left| 0\\right\\rangle _B\\end{array}\\right).$ The energy expectation is derived as $E_0=-\\frac{\\Delta }{2}e^{-2\\alpha ^2}-2g\\alpha +\\alpha ^2 .$ Minimizing the energy gives $\\Delta \\alpha e^{-2\\alpha ^2}-g+\\alpha =0.",
"$ In the weak coupling limit, we can obtain the variational parameter and the ground state (GS) energy respectively $\\alpha =\\frac{g}{1+\\Delta },$ $E_0=-\\frac{\\Delta }{2}\\exp \\left[ -2\\left( \\frac{g}{1+\\Delta }\\right)^2\\right] -g^2\\frac{1+2\\Delta }{\\left( 1+\\Delta \\right) ^2},$ which are exactly the same as Eqs.",
"(7) and (8) obtained in Ref.",
"[27].",
"In the strong-coupling limit, the first term in Eq.",
"(REF ), which is originated from the qubit tunneling, is too small and can be neglected, then we simply have $\\alpha =g,$ and the GS energy $E^{SCL}=-g^2.$ For the arbitrary coupling, one can solve Eq.",
"(REF ) consistently and the reasonable GS energy will be derived, which is not shown here."
],
[
"Perturbation theory based on the exact solution in the strong\ncoupling limit",
"Note above that in the strong coupling limit, the variational parameter is just exactly $\\alpha =g$ .",
"It can be also readily obtained by neglecting the qubit tunneling term $-\\frac{1}{2}\\Delta \\sigma _x$ in Hamiltonian (REF ) with zero static-bias.",
"In this case, based on the displaced operators $A$ and $B$ , the eigenstates are easily obtained as $\\left| m\\right\\rangle ^{(0)}=\\left(\\begin{array}{l}\\left| m\\right\\rangle _A \\\\\\pm (-1)^m\\left| m\\right\\rangle _B\\end{array}\\right),$ and the eigenvalues are $E_m^{\\pm 0}=m-g^2$ for the $m$ state.",
"Note that the eigenstates are twofold degenerate.",
"Next, considering $H^{\\prime }=-\\frac{1}{2}\\Delta \\sigma _x$ as a perturbation, within the second-order perturbation theory, we can readily derive the eigenenergy with even (odd) parity for zero qubit static bias as $E_m^{\\pm }=m-g^2\\mp (-1)^mD_{mm}+\\sum _{m\\ne n}\\frac{\\left|D_{mn}\\right| ^2 }{m-n} .$ It is interesting to note that it is just the same as Eq.",
"(5) in Ref.",
"[21] obtained by Casanova et al.",
"in the deep strong coupling (DSC) regime of the JC model.",
"The energy levels by Eq.",
"(REF ) against the qubit-oscillator detuning $ \\delta =\\Delta -1\\;$ for $g=0.1,0.5,1.0$ , and $1.5$ , ranging from weak to deep strong coupling, are displayed in Fig.",
"REF .",
"The numerically exact results from Eq.",
"(REF ) and () are also collected as a benchmark.",
"It is found that the DSC results are especially suited to the DSC regime or small detunings.",
"Note that Casanova et al just focused on the investigation in the DSC regime ($g/\\omega =2$ ) or small detunings ($\\Delta \\le 0.5$ ).",
"At weak coupling $g=0.1$ , it is shown in Fig.",
"REF (a) that, even for the negative detuning $\\delta $ , the DSC deviates from the exact ones.",
"However, in the present experimentally accessible systems, the maximum value for the coupling strength is generally realized in the superconducting flux qubit coupled to a circuit resonant [8], which is only around $ g=0.1$ , to our knowledge.",
"So it should be practically interesting to find a good solution in this coupling regime.",
"Figure: The energy levels for zero bias ϵ=0\\epsilon =0 at (a)g=0.1g=0.1, (b) g=0.5 g=0.5 , (c) g=1.0g=1.0, and (d) g=1.5g=1.5.",
"The presentBRWA results (black filled circles) are compared to the exact (solidlines), DSC(red dashed lines), and GRWA (blue dashed lines) ones.For any value of the qubit bias $\\varepsilon $ , the Hamiltonian (REF ) with a vanishing tunneling element $\\Delta =0$ can be diagonalized in terms of two eigenstates $\\left| \\uparrow ,m\\right\\rangle _A\\ $ and $\\left| \\downarrow ,m\\right\\rangle _B$ with $\\left| \\uparrow \\right\\rangle $ ($\\left| \\downarrow \\right\\rangle $ ) the eigenstate of $ \\sigma _z$ , the corresponding eigenvalues are $E_{\\uparrow /\\downarrow ,m}^{(0)}=\\pm \\frac{1}{2}\\varepsilon +m-g^2,$ For finite $\\Delta $ , the perturbative matrix elements becomes $-\\frac{1}{2}\\;_B\\left\\langle \\downarrow ,m\\right| \\Delta \\sigma _x\\left| \\uparrow ,n\\right\\rangle _A=-\\left( -1\\right) ^mD_{mn}.$ Note that these two euqations are exactly the same as Eqs.",
"(7), (8) in Ref.",
"[14].",
"Then the full Hamiltonian can be diagonalized perturbatively to second-order in $\\Delta $ by using Van Vleck perturbation (VVP) theory as outlined in Ref.",
"[14].",
"The eigenvalue is given by $E_m^{\\pm } &=&m+\\frac{l}{2}-g^2+\\frac{1}{2}\\sum _{k=0,k\\ne m\\pm l}^\\infty \\left(\\frac{D_{mk}^2}{\\varepsilon +m-k}-\\frac{D_{nk}^2}{\\varepsilon +k-n}\\right)\\nonumber \\\\&&\\mp \\frac{1}{2}\\sqrt{\\left[ \\varepsilon -l+\\sum _{k=0,k\\ne m\\pm l}^\\infty \\left( \\frac{D_{mk}^2}{\\varepsilon +m-k}-\\frac{D_{nk}^2}{\\varepsilon +k-n}\\right) \\right] ^2+4D_{mn}^2}, \\nonumber \\\\n &=&m+l(l\\ge 0),m=0,1,2,... $ which is the same as Eqs .",
"(12) for VVP in Ref.",
"[14].",
"So the VVP for finite bias can be also recovered easily in the present scheme.",
"It can be reduced to the zero-bias case Eq.",
"(REF ) by set $\\varepsilon =0\\;(m=n)$ .",
"It has been shown [14] that VVP works very well in the deep strong coupling or large static bias.",
"It is consistent with the fact that the unperturbative Hamiltonian includes the qubit-oscillator interaction and qubit bias.",
"What happen for the accessible parameters of the present-day experiments?",
"In addition, VVP at small static bias $\\varepsilon \\le 1$ has not been discussed either so far, which might however be more important.",
"Here, we calculate energy levels in the VVP for different static bias $\\varepsilon \\le 1$ , which are exhibited in Fig.",
"REF .",
"Compare to the exact ones, one can find that VVP deviates strongly with the increase of the tunneling parameters $\\Delta $ in the a wide coupling regime $g<0.5$ , and become more pronounced at small static bias.",
"Especially, around the experimentally accessible coupling strength around $g=0.1$ , VVP becomes worse considerably.",
"Therefore a new analytical treatment is highly desirable."
],
[
"Analytical approximations at different levels",
"In the framework of Eqs.",
"(REF ) and (), analytical approximations can be performed systematically.",
"First, as a zero-order approximation (ZOA), we omit the off-diagonal terms and have $\\left( m-g^2-\\frac{\\varepsilon }{2}-E\\right) c_m-\\;D_{m,m}d_m &=&0, \\\\-D_{m,m}c_m+\\left( m-g^2+\\frac{\\varepsilon }{2}-E\\right) d_m &=&0.$ Nonzero coefficients will give the following equation $\\left( m-g^2-\\frac{\\varepsilon }{2}-E\\right) \\left( m-g^2+\\frac{\\varepsilon }{2}-E\\right) -D_{m,m}^2=0,$ The eigenvalues are then given by $E_{\\pm }=m-g^2\\mp \\frac{1}{2}\\sqrt{\\varepsilon ^2+4D_{m,m}^2},$ The corresponding eigenstate is $\\left| m\\right\\rangle _{\\pm }\\left(\\begin{array}{l}\\left( -1\\right) ^mD_{mm}\\left| m\\right\\rangle _A \\\\\\left( m-g^2-\\frac{\\varepsilon }{2}-E_{\\pm }\\right) \\left| m\\right\\rangle _B\\end{array}\\right).$ The ZOA energies with zero static bias are just the three terms obtained in Eq.",
"(REF ).",
"In Fig.",
"REF , we also plot the ZOA energy levels against the coupling constant $g$ for several static bias.",
"It is demonstrated from the upper and middle panel that for small static bias ( $ \\varepsilon \\leqslant 0.5$ ), ZOA is almost equivalent to the VVP in all parameters.",
"If the high accuracy is not required, the simple expression of the eigensolutions in the ZOA should be practically very useful, at least as a preliminary estimate of some physical quantities .",
"The approximation can be easily improved step by step with the consideration of more off-diagonal elements in the present formalism.",
"The first-order approximation (FOA) is performed by selecting two coefficients $\\begin{array}{llll}c_m & d_m & c_{m+1} & d_{m+1}\\end{array}$ .",
"The determinants for any $m$ is given by $\\left|\\begin{array}{llll}\\Omega _m^{-}(E) & -D_{mm} & 0 & -D_{m,m+1} \\\\-D_{mm} & \\Omega _m^{+}(E) & -D_{m,m+1} & 0 \\\\0 & -D_{m+1,m} & \\Omega _{m+1}^{-}(E) & -D_{m+1,m+1} \\\\-D_{m+1,m} & 0 & -D_{m+1,m+1} & \\Omega _{m+1}^{+}(E)\\end{array}\\right| =0 ,$ where $\\Omega _m^{\\_}(E)=m-g^2-E-\\frac{\\varepsilon }{2},\\;\\Omega _m^{+}(E)=m-g^2-E+\\frac{\\varepsilon }{2}.$ Some roots of this quartic equation will give the energy levels.",
"The analytical expression might be a little bit complicate but should be given unambiguously.",
"We first revisit the zero-bias case $\\varepsilon =0$ .",
"In this case, due to the parity symmetry, we can set $d_n=\\pm c_n$ , then both equations give $ \\left( m-g^2\\right) c_m\\mp \\sum _{n=0\\ }\\;D_{mn}c_n=Ec_m$ .",
"In the FOA, the determinant takes the following 2-by-2 block form $\\left|\\begin{array}{ll}\\left( m-g^2-E\\mp D_{m,m}\\right) & \\mp D_{m,m+1} \\\\\\mp D_{m+1,m} & \\left( m+1-g^2-E\\mp D_{m+1,m+1}\\right)\\end{array}\\right| =0,$ where the sign $-(+)\\;$ is for even (odd) parity.",
"We can readily have two roots for even parity $E_m^{(1,2)}=m-g^2+\\frac{1}{2}-\\frac{1}{2}\\left( D_{m+1,m+1}+D_{m,m}\\right)\\pm \\frac{1}{2}\\sqrt{\\left[ 1+\\left( D_{m,m}-D_{m+1,m+1}\\right) \\right]^2+4D_{m,m+1}^2} ,$ and other two roots for odd parity $E_m^{(3,4)}=m-g^2+\\frac{1}{2}+\\frac{1}{2}\\left( D_{m+1,m+1}+D_{m,m}\\right)\\pm \\frac{1}{2}\\sqrt{\\left[ 1-\\left( D_{m,m}-D_{m+1,m+1}\\right) \\right]^2+4D_{m,m+1}^2} .$ In the ansatz of the wavefunction (REF ), the dimensions of the Hilbert space is only $2(N_{tr}+1)$ .",
"So for each $m$ , we only have two eigenvalues for excited states.",
"The other two roots for each $m$ should be omitted.",
"Note that at weak coupling, the parity for each eigenstate is fixed and arranged from bottom to above with the order as the first even state, then followed by two odd states, two even states, two odd states, and so on.",
"Therefore, the excited states 1 and 2 are of odd parity, which should be given by $E_0^{(3,4)}$ , the excited states 3 and $4\\;$ are of even parity, then given by $E_1^{(1,2)}$ , the excited states 5 and $6\\;$ are of odd parity, then given by $E_{2}^{(3,4)}$ , $\\;$ and so on.",
"In this way, two eigenvalues for excited states for any $m$ can be summarized as $E_m=m-g^2+\\frac{1}{2}+\\frac{(-1)^m}{2}\\left( D_{m+1,m+1}+D_{m,m}\\right)\\pm \\frac{1}{2}\\sqrt{\\left[ 1-(-1)^m\\left( D_{m,m}-D_{m+1,m+1}\\right)\\right] ^2+4D_{m,m+1}^2} .$ Besides, the GS energy is given by $E_0^{(1)}$ $E_{GS}=\\frac{1}{2}-g^2-\\frac{1}{2}\\left( D_{1,1}+D_{0,0}\\right) -\\frac{1}{2}\\sqrt{ \\left[ 1+\\left( D_{0,0}-D_{1,1}\\right) \\right]^2+4D_{0,1}^2} .$ The FOA results in Eqs.",
"(REF ) and (REF ) have been given directly by the determinant with 2-by-2 block form in Ref.",
"[18] by two present authors and one collaborator previously.",
"We here display the derivation in detail.",
"Especially we rule out two pseud solutions for each $m$ by taking the fixed parity of the eigenstates into account.",
"Surprisingly Eq.",
"(REF ) is exactly the same as the previous GRWA result Eq.",
"(20) in Ref.",
"[16] by Irish.",
"We now become aware that the previous GRWA, which were obtained in an alternative way within a lengthy derivation, is just FOA in the present scheme.",
"Actually this expression has been derived much earlier within substantially different approaches[28].",
"What is more, we can straightforwardly perform the second-order approximation for the further improvement, and extension to the biased case in the present framework, which is however not so easy to operate within Irish's approach.",
"To the best of our knowledge, GRWA with finite static bias does not exist in the literature.",
"Figure: The energy levels as a function of coupling constant forfor different qubit bias ϵ=0.1\\epsilon =0.1 (upper panel), 0.50.5(middlepanel), and1.0 1.0 (down panel).",
"The values of Δ\\Delta are 0.5,1.00.5,1.0, and 1.51.5 from left to right column.",
"The present GRWA results(black open circles) are compared to the exact (black solid lines),VVP (red dashed lines), and ZOA (blue filled circles) ones.For the finite bias $\\varepsilon \\ne 0$ , the parity symmetry is broken with $\\varepsilon ,\\;$ with the following notation $E &=&x+m-g^2-\\frac{\\varepsilon }{2} \\\\u &=&-D_{mm},v=-D_{m,m+1},w=-D_{m+1,m+1} \\nonumber ,$ The determinant can be reduced to $\\left|\\begin{array}{llll}-x & u & 0 & v \\\\u & -x+\\varepsilon & v & 0 \\\\0 & v & 1-x & w \\\\v & 0 & w & 1-x+\\varepsilon \\end{array}\\right| =0.$ The corresponding quartic equation is $x^4+bx^3+cx^2+dx+e=0,$ where $b &=&-2-2\\varepsilon , \\\\c &=&1+3\\varepsilon +\\varepsilon ^2-\\left(2v^2+u^2+w^2\\right), \\\\ d &=&\\left( 2v^2+u^2+w^2-\\varepsilon -1\\right) \\varepsilon +2\\left( u^2+v^2\\right), \\\\e &=&\\left(uw-v^2\\right) ^2-u^2\\left( 1+\\varepsilon \\right) -v^2\\varepsilon .$ The solutions to this quartic equation are given in the Appendix A.",
"Compared to the exact solutions, we find that the second and third roots $x_2$ and $x_3$ in Eqs.",
"() and (REF ) are generally the true solutions.",
"The GS energy is given by the first root $x_1$ in Eq.",
"() for $m=0$ .",
"We also call the FOA with finite static bias as GRWA.",
"In this way, we can calculate the eigenenergies uniquely and straightforwardly, which are shown in Fig.",
"REF with black circles.",
"It is very interesting to find that the present GRWA results are very close to the exact ones in the whole coupling regime for wide range of the static bias.",
"Compare to the VVP at static bias $\\varepsilon \\le 1$ , the present GRWA is obviously much better.",
"As stated above, for zero-static bias case $\\varepsilon =0$ , there is still room to improve by performing the higher order approximation.",
"In Ref.",
"[16], after a unitary transformation, only the \"energy-conserving\" one excitation terms like their Eq.",
"(18), a generalization of the energy-conserved term in the usual RWA, is kept in their Eqs.",
"(13) and (14), so it is called GRWA.",
"Because the present FOA is equivalent to GRWA, so in the second-order approximation, the terms beyond their Eq.",
"(18) must be included, so we term this improvement to GRWA as beyond the RWA (BRWA).",
"In other words, BRWA can not be implemented within any renormalized RWA form like in Ref.",
"[16].",
"In the BRWA, the analytical expression can be uniquely and clearly derived within the following procedure.",
"The determinant is $\\left|\\begin{array}{lll}\\left( m-g^2-E\\mp D_{m,m}\\right) & \\mp D_{m,m+1} & \\mp D_{m,m+2} \\\\\\mp D_{m,m+1} & \\left( m+1-g^2-E\\mp D_{m+1,m+1}\\right) & \\mp D_{m+1,m+2} \\\\\\mp D_{m,m+2} & \\mp D_{m+1,m+2} & \\left( m+2-g^2-E\\mp D_{m+2,m+2}\\right)\\end{array}\\right| =0,$ where $-(+)$ for even(odd) parity, which can be simplified as $\\left|\\begin{array}{lll}-X\\mp u & \\mp x & \\mp y \\\\\\mp x & \\left( 1-X\\mp v\\right) & \\mp z \\\\\\mp y & \\mp z & \\left( 2-X\\mp w\\right)\\end{array}\\right| =0,$ where $E &=&X+m-g^2, \\\\u &=&D_{m,m},v=D_{m+1,m+1},w=D_{m+2,m+2}, \\\\x &=&D_{m,m+1},y=D_{m,m+2},z=D_{m+1,m+2},$ which gives the following cubic equation $X^3+bX^2+cX+d=0 ,$ where $b &=&\\left( u+v+w\\right) -3, \\\\c &=&-\\left( x^2+y^2+z^2\\right) +u\\left( v-1\\right) +\\left( u+v-1\\right)\\left( w-2\\right), \\\\d &=&\\left( 2u-uw+y^2\\right) \\left( 1-v\\right)-z^2u+x^2\\left( 2-w\\right) +2xyz,$ for even parity, and $b &=&-\\left( u+v+w\\right) -3, \\\\c &=&-\\left( x^2+y^2+z^2\\right) +u\\left( 1+v\\right) +\\left( u+v+1\\right)\\left( 2+w\\right), \\\\d &=&\\left( y^2-2u-uw\\right) \\left( 1+v\\right)+z^2u+x^2\\left( 2+w\\right) -2xyz,$ for odd parity.",
"The three different roots to the cubic equation can be found in the Appendix A.",
"In this approximation, we have more than one eigenvalues for each eigenstate with fixed parity, which are all true solutions physically, but only some of them would be selected.",
"The criterion for the unique formulae for the BRWA is that the solutions are the most close to the exact results in the whole parameter regime.",
"In this way, we find that, for even (odd) number $m$ , two roots $y_1$ and $y_2$ in Eqs.",
"(REF ) and () of the determinant with odd (even) parity would generally give the best eigenvalues for the excited states.",
"The GS state is given by the first root $y_1$ of the $m=0$ determinant with even parity.",
"Actually, Eq.",
"(REF ) has been written out in Ref.",
"[18] by two present authors and one collaborator.",
"But the detailed expression for the eigenvalues was not presented.",
"Even the further third approximation was also performed in Ref.",
"[18].",
"The direct comparisons between these different order approximations to the GRWA[16], [28] have not been given, which however could reveal the advantage of this scheme.",
"We examine the BRWA energy levels against the qubit-oscillator detuning $ \\delta $ for fixed couplings $g=0.1,0.5,1.0$ , and $1.5$ , respectively in Fig.",
"REF , where the GRWA results have been also collected.",
"It is interesting to note that BRWA result is always more close to the exact one than GRWA one in all values of the coupling strength, which becomes more pronounced with increasing $\\delta $ .",
"Due to the counter-rotating wave terms, the eigenfunctions and eigenvalues of the JC model without the RWA present an open problem because they are not known in anything like a closed form, even given the exact solutions reported recently[22], [23].",
"No analytical expressions for the exact eigenvalues are available in the literature, to the best of our knowledge.",
"The analytical expressions presented in this paper, which is not exact but work well, might be practically useful."
],
[
"summary",
"In this paper, by an effective scheme within two displaced bosonic operators with equal positive and negative displacements, we study the qubit-oscillator systems analytically in an unified way.",
"Many previous analytical treatments, such as GRWA, an expansion in the qubit tunneling matrix element in the deep strong coupling regime can be recovered in the present scheme.",
"More over, we extend the GRWA to the finite-bias case.",
"The results is much better than VVP in the weak and intermediate coupling regime, which is more experimentally interesting.",
"For the zero static bias, the GRWA is further improved to BRWA, which is more close to the exact ones at large detuning while the GRWA deviates strongly.",
"The analytical expression is explicitly given for future applications."
],
[
"ACKNOWLEDGEMENTS",
"This work was supported by National Natural Science Foundation of China, National Basic Research Program of China (Grant Nos.",
"2011CBA00103 and 2009CB929104)."
],
[
"Solutions to univariate cubic and quartic equations",
"The univariate cubic equation can be generally expressed as $x^3+bx^2+cx+d=0.",
"\\nonumber $ Its solutions can be found in any Mathematics manual.",
"If $\\Gamma =B^2-4AC<0,$ with $A=b^2-3c,B=bc-9d,C=c^2-3bd,$ there are three different real roots $y_1 &=&\\frac{-b-2\\sqrt{A}\\cos \\theta }{3}, \\\\ y_2&=&\\frac{-b-2\\sqrt{A}\\cos \\left( \\theta -\\frac{2\\pi }{3}\\right) }{3}, \\\\ y_3 &=&\\frac{-b-2\\sqrt{A}\\cos \\left( \\theta +\\frac{2\\pi }{3}\\right) }{3}, $ where $\\theta =\\frac{1}{3}\\arccos \\left( \\frac{2Ab-3B}{2\\sqrt{A^3}}\\right).$ The univariate quartic equation can be generally expressed as $x^4+bx^3+cx^2+dx+e=0, \\nonumber $ Its four solutions are exactly the four solutions of the following two quadratic equations $x^2+\\frac{b+z}{2}x+\\left( y+\\frac{by-d}{z}\\right) &=&0, \\\\x^2+\\frac{b-z}{2}x+\\left( y-\\frac{by-d}{z}\\right) &=&0,$ where $z=\\sqrt{8y+b^2-4c}$ and $y$ is the third root $y_3$ in Eq.",
"() of the following cubic equation $y^3-\\frac{c}{2}y^2+\\left( \\frac{bd}{4}-e\\right) y+\\frac{e\\left(4c-b^2\\right) -d^2 }{8}=0.$ We haver checked that $\\Gamma <0$ in all parameters in the present case.",
"Therefore the four roots are $ x_1 &=&-\\frac{1}{4}\\left( b+z\\right) -\\frac{1}{4}\\sqrt{\\left( b+z\\right)^2-\\frac{ 16y\\left( b+z\\right) -16d}{z}}, \\\\ x_2 &=&-\\frac{1}{4}\\left( b+z\\right) +\\frac{1}{4}\\sqrt{\\left( b+z\\right) ^2-\\frac{16y\\left( b+z\\right) -16d}{z}}, \\\\ x_3 &=&-\\frac{1}{4}\\left(b-z\\right) -\\frac{1}{4}\\sqrt{\\left( b-z\\right) ^2+\\frac{ 16y\\left(b-z\\right) -16d}{z}}, \\\\ x_4 &=&-\\frac{1}{4}\\left(b-z\\right) +\\frac{1}{4}\\sqrt{\\left( b-z\\right) ^2+\\frac{ 16y\\left(b-z\\right) -16d}{z}}.$ J. M. Raimond, M. Brune, and S. Haroche, Rev.",
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[
[
"Planar Pixel Sensors for the ATLAS Upgrade: Beam Tests results"
],
[
"Abstract Results of beam tests with planar silicon pixel sensors aimed towards the ATLAS Insertable B-Layer and High Luminosity LHC (HL-LHC) upgrades are presented.",
"Measurements include spatial resolution, charge collection performance and charge sharing between neighbouring cells as a function of track incidence angle for different bulk materials.",
"Measurements of n-in-n pixel sensors are presented as a function of fluence for different irradiations.",
"Furthermore p-type silicon sensors from several vendors with slightly differing layouts were tested.",
"All tested sensors were connected by bump-bonding to the ATLAS Pixel read-out chip.",
"We show that both n-type and p-type tested planar sensors are able to collect significant charge even after integrated fluences expected at HL-LHC."
],
[
"Introduction",
"The ATLAS collaboration will upgrade the current Pixel Detector [1] in two phases.",
"A first upgrade will be realized during the shut-down in 2013, by inserting a fourth detection layer (Insertable B-Layer - IBL) at a radius of 3.2 cm from the beam line.",
"The IBL will improve the tracking and vertexing performance of the current pixel detector significantly during operation of the LHC at its nominal centre-of-mass energy ($\\sqrt{s}$ = 14 TeV) [2].",
"The close proximity to the interaction point imposes a very harsh radiation environment on the IBL.",
"At the end of Phase-I operation of the LHC, foreseen around 2020, the IBL must sustain an estimated fluence of $5\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ , including a 60% safety factor at r = 3.2 cm.",
"The Phase-II luminosity upgrade for the LHC (beyond 2020) aims to increase the instantaneous luminosity to $5\\times 10^{34}$ cm$^{-2}$ s$^{-1}$ , posing a serious challenge to the technology for the ATLAS tracker in the High Luminosity era (HL-LHC): the lifetime fluence for the innermost layer, including safety factors, is estimated to be on the order of $2\\times 10^{16}$ n$_{\\rm eq}$ /cm$^2$ [3].",
"Hence, in view of a possible pixel system replacement after 2020, new pixel sensors are under study.",
"Within the Planar Pixel Sensor collaboration (PPS) [7] several optimizations of this well-known technology are under investigation, to address issues arising from the LHC upgrades.",
"The PPS collaboration investigates the suitability of different materials (p- and n-bulk, diffusion oxygenated float zone, magnetic Czochralski), different geometries (slim edge design, number and width of guard rings) and different biasing / isolation choices (punch-through, polysilicon resistance / p-spray, p-stop), for a new generation of planar pixel sensors.",
"Data taken during beam tests complement tests under laboratory conditions in assessing the performance of various sensor prototypes.",
"In this paper results from two beam test campaigns in 2010 are presented.",
"The paper is organized as follows.",
"After a description of the experimental setup of the beam tests in Section and some details of the data analysis in Section , beam test results are presented on three areas of the scientific program of the PPS collaboration: Sensors implemented in p-type silicon are being studied by several groups within the PPS collaboration.",
"Section shows results from the first beam test operation of various prototype sensors.",
"We will show that the performance of p-type sensors in terms of collected charge collected charge is always presented as Most Probable Value (MPV), if not stated otherwise, charge sharing probability, and spatial resolution is very similar to that of n-type sensors.",
"In Section the radiation-hardness of sensors implemented in n-type diffusion oxygenated float-zone silicon are studied.",
"We will show that n-in-n detectors are operable at the nominal bias voltage of 1000 V after a fluence comparable with that expected for IBL ($5\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ ).",
"Noise occupancy, charge collection performance, charge sharing probability and spatial resolution will be presented.",
"The new pixel sensors will not only have to sustain the harsher environment, but also have to show high geometrical acceptance without overlapping adjacent modules.",
"Hence the inactive areas of the future pixel sensor have to be reduced significantly.",
"For this reason, efforts were devoted to design detectors with reduced dead area.",
"The “slim edge” detector-concept will be presented in Section , together with its performance in terms of charge collection at the detectors' edge."
],
[
"Beam test setup",
"Beam tests are crucial for characterizing the performance of any particle detector.",
"Planar silicon sensors for the ATLAS upgrade have been evaluated in several beam tests in 2009 and 2010.",
"Data presented in this paper were taken in two different periods in 2010 at the CERN SPS beamline H6.",
"In both periods pion beams of 120 GeV/c were used.",
"The high momentum of the beam particles minimizes the influence of multiple scattering, enabling high precision tracking using the EUDET beam telescope [8]."
],
[
"The EUDET telescope",
"The telescope consists of six equal planes, divided into two groups (arms) of three planes each.",
"The Devices Under Test (DUTs) are mounted in between these two arms of the telescope as well as downstream of the last telescope plane.",
"The increase of the track extrapolation error downstream of the telescope was minimized by mounting the DUTs as close as possible to the last telescope plane.",
"The sensitive elements of the telescope planes are Mimosa26 [25] active pixel sensors with a pixel pitch of 18.4 $\\mu $ m. Each plane consists of 1152 $\\times $ 576 pixels covering an active area of 21.2$\\times $ 10.6 mm$^2$ .",
"The tracking resolution between the telescope arms is estimated to be 2 $\\mu $ m, at the position of the samples downstream of the telescope it is approximately 10 $\\mu $ m. [26] A coincidence of four scintillators (two upstream and two downstream of the telescope) was used for triggering, which resulted in an effective sensitive area of 2$\\times $ 1 cm$^2$ ."
],
[
"Devices Under Test",
"All DUTs were read out using the current ATLAS Pixel readout chip (FE-I3) [9].",
"The FE-I3 chip is an array of 160 rows $\\times $ 18 columns of 50 $\\mu $ m $\\times $ 400 $\\mu $ m read-out cells.",
"In each readout cell the sensor charge signal is amplified and compared to a programmable threshold by a discriminator.",
"The information on the collected signal is encoded through a digital time over threshold (ToT) [9] measured in units of 25 ns, which is the nominal LHC bunch crossing rate.",
"The ToT to charge conversion was tuned for each individual pixel to 60 ToT for a deposited charge of 20 ke.",
"Discriminator thresholds were tuned to a charge of 3.2 ke.",
"Prior to the beam test, tuning was performed for every readout chip in realistic conditions, designed to closely resemble those at the beam test.",
"Since the ToT-tuning is particularly temperature dependent, the ToT was calibrated on each sample after installation in the beam test setup to ensure a proper charge conversion.",
"For each DUT a fiducial region was defined, based on geometrical and operability considerations.",
"For most studies only the performance of central pixels is of interest.",
"Therefore in many analyses the pixels at the edges of the sensors were masked.",
"In addition, all pixels that were found to have disconnected or merged bump-bonds in laboratory measurements were masked, as were pixels with high noise occupancy.",
"Devices were irradiated to different fluences using 25 MeV energy protons at the Irradiation Center in Karlsruhe [19], 24 ${\\rm GeV/c}$ momentum protons at the CERN PS irradiation facility [12] we observed FE-I3 chip stopped working after being CERN PS irradiated, and reactor neutrons at the TRIGA reactor of the Jožef Stefan Institute, Ljubljana [18].",
"The radiation damage from the different irradiations is scaled to the equivalent damage from 1 MeV neutrons using the NIEL hypothesis [30].",
"During the irradiations the devices were neither powered nor cooled.",
"No standardized annealing procedure was used, but samples were stored below 0 $^{\\circ }$ C to avoid uncontrolled annealing.",
"The FE-I3 was designed for a lifetime irradiation dose of 1 kGy.",
"Some of the DUTs were irradiated to significantly higher doses, leading to increasing numbers of non-working pixels.",
"For the beam tests, irradiated DUTs were cooled via a strip of copper tape thermally connecting the backside of the DUTs with the base plate of the thermal enclosure.",
"The base plate was in turn cooled using dry ice (CO$_2$ ) [17].",
"Due to this setup, the temperature of the sensors varied over time as the dry ice evaporated and needed to be closely monitored.",
"Temperatures were recorded, ranging between -45 $^{\\circ }$ C shortly after filling the coldbox with dry ice and -15 $^{\\circ }$ C towards the end of a data taking period."
],
[
"Track reconstruction",
"The tracks of particles traversing the EUDET telescope are reconstructed from raw hit positions by a sequential algorithm.",
"In the first step, the hits recorded in all telescope planes and DUTs are converted into the EUTelescope [23] internal data format.",
"A time stamp issued by the Trigger Logic Unit (TLU) [24] is attached to each hit, enabling recovery from any loss of synchronisation between telescope and DUTs during this step, provided the desynchronisation is not too severe.",
"As the Mimosa26 sensors of the telescope plane use the rolling shutter read-out technique, the telescope integrates hits for $112\\,\\mu {\\rm s}$ after the arrival of the trigger signal.",
"This is much longer than the 400 ns hit-buffer of the DUTs, so some tracks will be recorded by the telescope, but not by the DUTs.",
"To correct for this effect, only hits that were recorded within the sensitive time of the DUTs are retained for further analysis.",
"This is done by requesting hits spatially associated to the track in one or more of the other DUTs (in-time tracks).",
"A clustering algorithm is then executed searching for clusters in all planes.",
"Hits are then transformed from the local coordinate system of each plane to a global coordinate system, where the z-axis gives the beam direction.",
"During this coordinate transformation the pixel sizes in x- and y-directions, the specified z-position of all sensors, and the rotations of the DUTs about all axes are taken into account.",
"Based on correlations between hit positions in different planes in the global frame, a coarse pre-alignment is calculated.",
"Using this information, the alignment processor (see also REF ) tries to fit tracks through all planes in the setup, taking into account the different spatial resolutions of the planes and individual track selection criteria for each plane.",
"Individual selection criteria are especially necessary, since the pixel size of the telescope-planes is $18.4\\,\\mu {\\rm m}$ in both directions, whereas the investigated FE-I3 based samples have a pixel size of $400\\,\\mu {\\rm m}$ x $50\\,\\mu {\\rm m}$ .",
"The final step is the track-fitting, which is based on a Kalman filter [29].",
"The track fits are unbiased, requiring a hit in at least four out of the six telescope planes and in at least one DUT.",
"Also in this step, different track selection criteria can be applied.",
"The parameters of all reconstructed tracks are finally stored in a ROOT file [31] for further analysis (see Section REF )."
],
[
"Detector alignment",
"The alignment for the telescope planes and the DUTs in the EUTelescope track-reconstruction uses the MILLIPEDE tool [27].",
"In the algorithm the alignment constants are calculated such that the uncertainties of the fitted track parameters, as well as the $\\chi ^2$ of the track residuals, are minimized.",
"Straight line fits to the hit positions in all active planes are performed independently for the x- and y-directions.",
"Individual criteria can be applied to the resulting residual distributions to suppress fake tracks.",
"In the alignment process the pre-alignment constants, calculated in the previous hitmaker step, are taken into account.",
"This enables alignment of all telescope-planes and DUTs in one step, where just the first telescope-plane is fixed in its position and orientation.",
"The alignment constants are applied in the final track fitting process."
],
[
"Data analysis",
"The analysis of the reconstructed tracks is conducted in several steps, using a dedicated data analysis framework (tbmon) [28].",
"Firstly, unresponsive and noisy pixels are identified and masked.",
"A pixel is unresponsive if it registers no hit during the full data taking period and noisy if more than $5\\times 10^{-4}$ of all hits registered in this pixel are not correlated with a beam particle.",
"Typically, less than 1 % of pixels are masked in non-irradiated modules; for irradiated devices the fraction fluctuates between samples.",
"On average, roughly 10 % of the pixels have to be masked due to problems with settings of the readout chip or increased noise due to high leakage current of the sensor.",
"In the following analyses tracks extrapolated from the telescope are “matched” to a hit if the hit and the extrapolated track impact point in the DUT plane containing the hit are closer than 400 $\\mu $ m in the long pixel direction and 150 $\\mu $ m in the short pixel direction.",
"The hit position is defined as the $\\eta $ corrected ToT weighted position of all pixels in a cluster [13].",
"To estimate the intrinsic spatial resolution of the DUTs the distribution of hit residuals is studied.",
"The hit residual is defined as the distance between the reconstructed hit position on the DUT and the extrapolation of the fitted track to the DUT plane.",
"The intrinsic spatial resolution is estimated by the RMS of the residual distribution for clusters of all sizes, while the residual distribution of 2-pixel clusters is used to estimate the width of the area between pixels, where charge sharing occurs.",
"The distribution is fitted with the sum of two Gaussian functions, where one accounts for mis-reconstructed hits, resulting in large residual values (equal to 2 times the pixel pitch or more), and the other for correctly reconstructed hits.",
"The width of this “core” Gaussian gives the width of the charge sharing region.",
"To calculate the charge sharing probability for each hit within a cluster, it is determined whether a hit is found in a pixel cell adjacent to the one matched to a track.",
"This probability increases towards the edge of the pixel since charge carriers are more likely to drift to the neighbouring pixel.",
"The corresponding plot, referred to as a charge sharing map, is centred on one pixel, also showing half of the adjacent pixel in each direction.",
"The overall charge sharing is defined as the number of tracks with at least one hit in a neighbouring pixel divided by the number of all tracks.",
"Due to a problem in the readout system, random DUTs stopped sending data for random short intervals.",
"Therefore, the availability of a reference plane for the selection of in-time tracks cannot be ensured at all times.",
"As this selection is crucial for the measurement of the hit efficiency, this analysis could not be done with the available data, while charge collection and spatial resolution measurements are unaffected.",
"Most of the hits registered by the DUTs were anyway associated with tracks; this can be seen from the LVL1 distribution.",
"The LVL1 distribution (see Figure REF ) shows the arrival time of every recorded hit with respect to the external trigger signal.",
"The very pronounced peak shows that most hits have a strong correlation with the timing of the external trigger signal, meaning that they are indeed generated by the triggered particle traversing the DUT.",
"By applying cuts to the LVL1 distribution, we can therefore suppress most hits that are not associated to a track.",
"Figure: Example of LVL1 distribution."
],
[
"The n-in-p demonstrator program",
"While n-type bulk sensors require patterned guard rings on the back side of the sensor, for p-type material these can be moved to the pixelated side of the sensor (front side); then metallization is the only process for the back side.",
"This makes it a very cost-effective material for future pixel detectors.",
"On the other hand the high voltage, which is applied to the back side of the sensor, is also present on the edges of the front side of the sensor facing the read-out chip, which is at ground potential.",
"Thus spark discharges may occur, posing a risk to the readout electronics itself.",
"This can be limited by the deposition of an insulating coating on the edge of the sensor; more details are given in the next section.",
"Sensors in p-type technology tested in 2010 were produced at CiSForschungsinstitut für Mikrosensorik und Photovoltaik GmbH and at HPKHamamatsu Photonics K. K.. Table REF summarizes the relevant quantities for the devices studied in the beam test.",
"Further details of each sample are described below.",
"Table: Relevant quantities of the n-in-p samples."
],
[
"CiS sensors",
"In the following the p-type sensors produced at CiS will be introduced and their beam test results discussed.",
"The n-in-p pixel sensors labelled as MPP1–MPP5 were produced at CiS with a geometry compatible with FE-I3, in the framework of a common RD50 production [20].",
"They were made from Diffusion Oxygenated Float Zone (DOFZ) 285 $\\mu $ m thick wafer, with $<$ 100$>$ crystal orientation and a wafer resistivity of 15 k$\\Omega $ cm.",
"The depletion voltage before irradiation was nominally 60 V. Two guard rings structures with differing widths have been implemented and tested.",
"One design has the standard inactive area of 1 mm per side of normal ATLAS pixel sensors, while the other has a reduced inactive area, as illustrated in Figure REF .",
"MPP1, MPP3 and MPP5 have 19 guard-rings, with the standard total inactive width of 1 mm on each side.",
"The samples MPP2 and MPP4 (see Table REF ) have 15 guard-rings with a total inactive area of 610 $\\mu $ m on each side.",
"For both guard rings designs the bias ring and the inner guard ring are wider than the other ones, enabling tests of the sensors before connection to the read-out chips.",
"The external guard ring widths grows with promixity to the cutting edge from 17 $\\mu $ m to 22 $\\mu $ m, with the gap between the rings from 5 to 8 $\\mu $ m. The distance between the last ring and the dicing edge is 400 $\\mu $ m for the 19 guard rings design while it is 100 $\\mu $ m in the 15 guard rings design.",
"The inter-pixel isolation is achieved by means of a homogenous p-spray implantation.",
"Figure: Left: View of a corner of an n-in-p sensor of the CiS production with 15 guard rings.",
"Right: View of a corner of an n-in-p sensor of the CiS production,with 19 guard rings.A BCB (Benzocyclobutene) coating has been applied to the pixelated side of the n-in-p pixel sensors, to prevent sparks between the area outside the guard ring area, that is at the same high potential as the back side, and the chip, at ground potential (Figure REF ).",
"The interconnection to the chips has been performed via bump-bonding at IZM-Berlin Fraunhofer-Institut für Zuverlässigkeit und Microintegration, Berlin.",
"Figure: Schematics of a CiS n-in-p pixel assembly.",
"The potential of the different parts is given.",
"The BCB layer is indicated in orange."
],
[
"Beam test results",
"As documented in Table REF , during the first beam test period (July 2010) none of the CiS n-in-p sensors (namely MPP1, MPP2) were irradiated.",
"In the second period (October 2010) MPP4 was tested after irradiation with reactor neutrons [18] to a fluence of $1\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ and MPP5 was irradiated with low energy protons (25 MeV) at the cyclotron of the Karlsruhe Institute of Technology (KIT) [19] to the same equivalent fluence.",
"MPP3 was kept as an unirradiated reference.",
"The performance of these five samples is presented as follows."
],
[
"Cluster size",
"The cluster size distribution was studied for all sensors as a function of the bias voltage.",
"A detailed breakdown is reported in Table REF .",
"For the non-irradiated sample roughly 70 % of the clusters consisted of a single hit.",
"A further 25 % were two-hit clusters, while the remainder were three or more hit clusters.",
"Due to trapping effects and lower overall charges for irradiated sensors, one hit clusters are more often observed than in unirradiated samples.",
"With increasing bias voltage the number of two hit clusters rises since it becomes more likely that a neighbouring pixel is above threshold as well.",
"This behaviour can be clearly seen in Figure REF .",
"Table: Cluster size (CS) composition for CiS modules measured at different bias voltages during beam tests;clusters were matched to a track.Charge sharing probability is also reported.Figure: Relative cluster size abundance as a function of bias voltage for irradiated MPP4 and MPP5 devices.MPP3 (non-irradiated) are added for comparison.",
"Particles were impinging at normal incidence.",
"Errors on fractions are negligible and so not visible in the plot."
],
[
"Collected charge",
"The collected charge was measured as a function of the bias voltage for the different sensors.",
"In Figure REF the sub-pixel resolved charge collection profile is shown for MPP3 at $V_{\\mathrm {bias}}$ of 150 V. The most prominent feature is the lower collected charge value on the right hand side, corresponding to the bias dot region where the pixel implant is connected to the bias grid.",
"The same effect is evident in n-in-n devices with the same design (Section ).",
"In this region the collected charge is still well above the threshold.",
"Figure REF shows lower collected charge along the edges of a pixel, due to charge sharing with the neighboring pixels.",
"As charge sharing occurs, less charge is available for the pixel traversed by the particle, decreasing the probability to pass the electronics threshold.",
"This effect is especially pronounced in the corners of the pixel, as charge can be shared among four pixels.",
"However, in these regions the deposited charge is still high enough for an efficient operation of the device.",
"Figure: Charge collection within a single pixel by track position for MPP3 at V bias V_{\\mathrm {bias}} of 150 VFigure: Charge collection within a single pixel by track position for MPP5 at V bias =500V_{\\mathrm {bias}}=500 VFigure REF shows the most probable charge for all samples.",
"For comparison the charge collected by unirradiated devices (MPP1 and MPP2) is included and a typical discriminator threshold of 3200 e is indicated.",
"A systematic error on the collected charge of 400 e is assumed, due to the finite charge resolution of the ToT mechanism; a 5% systematic uncertainty is taken into account, due to non-uniformity in the injection capacitances.",
"Although the irradiated samples do not show saturation of the collected charge up to 700 V, already at low bias voltages the collected charge exceeds the electronics threshold by more than a factor of two and can thus be considered safe for tracking applications.",
"Figure: Charge collected in a cluster:most probable value of the charge distribution fitted to a Landau function convoluted with a gaussian as a function of thebias voltage.",
"See text for the discussion on the assigned systematic uncertainty.",
"The threshold value is also depicted as a dashed line."
],
[
"Charge sharing",
"In Figure REF (top) the charge sharing probability within one pixel for MPP3 is shown.",
"At normal track incidence increased charge collection probability is evident at the edges and corners of the pixel.",
"The situation after irradiation is shown in Figure REF (bottom).",
"Here the charge sharing especially on the side of the punch through biasing is reduced, since there is a higher probability for the neighbouring pixel to be below threshold.",
"This is also reflected in the average charge sharing probability given in Table REF for the n-in-p CiS detectors in all states.",
"With increasing bias voltage an increase of the charge sharing is observed for the irradiated sensors (MPP4, MPP5) due to the increase in the collected charge.",
"Figure: Charge sharing map for MPP3 detector, biased at 150 V (top) and for MPP4 detector, biased at 700 V."
],
[
"Residuals",
"Figure REF shows the cluster position residual distribution for the irradiated module MPP5, biased at 500 V. The spatial resolution is compatible with the digital resolution, as one-hit clusters are dominant.",
"For comparison, the residual distribution for MPP3 at 150 V is shown in Figure REF .",
"No difference is appreciable between the two samples.",
"Figure: Residual distribution for irradiated sample MPP5 at V bias _{bias} of 500 V. Left: long pixel projection; right: short pixel projection.Figure: Residual distribution for non-irradiated MPP3 biased at 150 Volts.",
"Left: long pixel projection; right: short pixel projection.Figure: Residual distribution for irradiated sample MPP5 biased at V bias _{bias} of 500 V, for two-hit clusters.",
"Left: long pixel projection; right: short pixel projection.For the irradiated assembly MPP5 Figure REF shows in case of two hit clusters a resolution of $(7.2\\pm 2.5)\\mu $ m in the core gaussian along the short pixel direction.",
"For comparison, Figure REF shows the two-hit cluster residuals for MPP3 at 150 V. After irradiation there are more noise related hits, clearly visible in Figure REF .",
"The fitted core fraction indeed decreases with respect to unirradiated sample (Figure REF ).",
"Nonetheless, the tracking capabilities of irradiated DUTs, in terms of spatial resolution, are still satisfactory.",
"Figure: Residual distribution for unirradiated sample MPP3 at V bias _{bias} of 150 V, for two-hit clusters.",
"Analysis restricted to clusters with 2 pixels.",
"Left: long pixel projection; right: short pixel projection."
],
[
"HPK sensors",
"In the following the p-type sensors produced at HPK will be introduced and their beam test results discussed."
],
[
"Sensors design",
"Two modules with different sensor n-in-p layouts were subject to beam tests; one with a polysilicon bias resistor and a common p-stop isolation (KEK1), and the other with a polysilicon bias resistor and an individual p-stop isolation (KEK2).",
"Figure REF shows a sketch of the pixel cell design for these two samples.",
"The sensors came from Float Zone (FZ) wafers with $<$ 100$>$ crystal orientation; the wafer thickness was 320 $\\mu $ m. The measured wafer resistivity was approximately 6 k$\\Omega $ cm.",
"Figure: Pixel cell design details for KEK1 (top) and KEK2 (bottom) samples.A metal bias rail runs along each pixel double-column; there is no bias rail implant underneath it [22].",
"A parylene coating has been applied to the whole body of the pixel modules, after mounted on and connected to the single-chip test card (SCC) with wire-bonding.",
"These modules were beam-tested before any irradiation in 2010, irradiated afterward and tested in the beamtests in 2011.",
"The results before irradiation are reported in this paper; those after irradiation are being analyzed and they will be presented in a different communication.",
"The depletion voltage before irradiation was about 180 V."
],
[
"Beam test results",
"The HPK samples characterization has been carried out by measuring the cluster size, collected charge, charge sharing and spatial resolution as a function of the bias voltage."
],
[
"Cluster Size",
"The KEK1 and KEK2 samples were biased at 100 V and 200 V. The two samples perform in a similar way in terms of cluster size for particles at normal track incidence.",
"As shown in Table REF (see also Figure REF ) more than 80% of the clusters have just one hit pixel at 100 V bias voltage.",
"As the bias voltage increases, the fraction of 2-pixel clusters also increases, as more charge is collected.",
"Therefore the charge fluctuations are small compared to the threshold, which reduces the probability of a pixel collecting a signal below threshold.",
"Table: Cluster composition for HPK detectors for different bias voltages; charge sharing probabilityis reported in the last column.",
"Clusters were matched to a track.Figure: Fraction of cluster sizes as function of the bias voltage for HPK samples; particles were impinging at diffent angles too."
],
[
"Collected charge",
"In Figure REF the collected charge per cluster is shown as a function of bias voltage for the KEK sensors.",
"The charge collection improves with bias voltage, but already at 100 V the signal is more than 4 times the threshold.",
"At 200 V the expected full charge is collected.",
"Figure: Collected charge as a function of the bias voltage for HPK samples; particles were impinging at diffent angles too.",
"A threshold of 3200 e is indicated."
],
[
"Charge sharing",
"Figure REF shows the charge sharing map for KEK1.",
"At normal incidence the fraction of charge sharing is more than 25 % for a sensor biased at 200 V. Results for KEK2 show that the charge sharing is less effective (22 %): this can be related to the different layout between the two sensors.",
"In the bottom figure the combined effect of the bias metal rail and the individual p-stop is visible.",
"Results are summarized in Table REF .",
"Figure: Charge sharing map for KEK1 (top) and KEK2 (bottom) at V bias _{bias} of 200 V."
],
[
"Residuals",
"Figure REF shows the residual distribution for all clusters at normal track incidence.",
"The spatial resolution is about 16 $\\mu $ m along the short pixel direction, which is comparable with the digital resolution of pitch/$\\sqrt{12}$ ; the same is true for the long pixel direction (RMS about 116 $\\mu $ m).",
"No difference is noticeable between KEK1 and KEK2 sensors.",
"Figure: Cluster position residual distribution for non-irradiated sample KEK1 at V bias _{bias} of 200 V at normal incidence.Left: long pixel projection; right: short pixel projection.Figure REF shows the residual distribution for two-hit clusters.",
"For these, the spatial resolution is found to be around 7 $\\mu $ m in the short pixel direction and around 9 $\\mu $ m for the long one (see also Table REF ).",
"The spatial resolution when the cluster contains two hits is larger than the telescope pointing-resolution and gives an estimate of the charge sharing region between neighbouring pixels.",
"Figure: Residual distribution for non-irradiated KEK1 biased at 200 Volts clusters with 2 pixels.",
"Left: long pixel projection; right: short pixel projection.Table: Summary of residual results for KEK1 and KEK2 samples.",
"Core σ\\sigma and fraction are evaluated for 2-pixels clusters only."
],
[
"Radiation hardness of n-in-n sensors",
"The n-in-n sensor technology used in the current ATLAS Pixel detector have been tested to fluences up to $1.1\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ [11].",
"To evaluate the usability of n-type bulk material for IBL and future detector upgrades, sensors have been irradiated with fluences as high as $2\\times 10^{16}$ n$_{\\rm eq}$ /cm$^2$ using both reactor neutrons (Jo$\\check{\\mathrm {z}}$ ef-Stefan Institute, Ljubljana) [18] and protons of 25 MeV (Karlsruhe Institute of Technology - KIT) [19] or 24 GeV (CERN) [12].",
"Most of the sensors follow the ATLAS Pixel detector sensor design with 16 guard rings and a thickness of 250 $\\mu $ m. DO6 is a special sensor with only 11 guard rings overlapping the pixel region, designed to study possibilities to reduce the inactive area at the edge of the sensor (see section ) and produced on 285$\\mu $ m thick bulk material.",
"The n-type sensors were produced at CiS, from Diffusion Oxygenized Float Zone (DOFZ), $<$ 111$>$ oriented wafers; the wafer resistivity was in the range between 2 and 5 k$\\Omega $ .",
"The depletion voltage was in the range between 40 and 100 V for 250 $\\mu $ m thick sensors and between 50 and 140 V for 285 $\\mu $ m thick sensors.",
"The inter-pixel isolation is achieved by means of a “moderated” p-spray implantation [1].",
"A total of 5 irradiated n-in-n pixel sensors were tested.",
"Table REF summarizes the fluences to which the sensors were irradiated.",
"See also [16].",
"Table: Summary of irradiated n-in-n samples in the testbeams.",
"KIT stands for 25 MeV energy proton irradiation."
],
[
"Results",
"Measurements on n-in-n samples have been carried out at temperatures well below 0$^{\\circ }$ C to reduce the large leakage current from irradiated sensors.",
"As an example, we measured a leakage current of 24 ${\\rm \\mu A}$ (10 ${\\rm \\mu A}$ ) for DO10 (DO9), at a bias voltage of 1200 V and at -47$^{\\circ }$ C."
],
[
"Charge collection",
"One of the main effects of irradiation is the increased trapping, which leads to a reduced signal amplitude.",
"As the trapping probability depends on the charge carrier velocity, the collected charge was measured as a function of the bias voltage.",
"Figure REF shows the results for all irradiated n-in-n samples in the two beam test periods; see also Table REF .",
"A systematic error on the collected charge of 400 e is assumed, due to the finite charge resolution of the ToT mechanism; a 5% systematic uncertainty is also taken into account, due to non-uniformity in the injection capacitances.",
"After $5\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ , the collected charge still exceeds 10 ke at a bias voltage of 1000 V. Even if the collected charge is shared equally between two neighboring pixels, this charge is sufficient to detect the hit with FE-I3.",
"Figure: Collected charge as a function of bias voltage for n-in-n samples irradiated to different fluences(see details in the text).A threshold of 3200 e is indicated.Figure REF top, shows that charge is predominantly lost in the region of the punch-through bias grid system.",
"At very high fluences ($2\\times 10^{16}$ n$_{\\rm eq}$ /cm$^2$ , DO10 sample) it is no longer possible to say which region is less efficient than the others, using the charge collection method (Figure REF , bottom).",
"Figure: Charge collection within a pixel.",
"Top: DO9 at V bias _{bias}=1200 V. Bottom: DO10 at V bias _{bias}=1000 V."
],
[
"Charge sharing",
"Figure REF shows the charge sharing probability for DO9 at a bias voltage of 1200 V. Reduced charge sharing probability is visible in the region of the bias dot and the bias grid network.The bias grid network is an aluminum trace arranged on top of the intermediate pixel region connecting all bias dots.",
"Less charge is deposited here, so there is a higher probability for the second pixel in a two-pixel cluster to be below threshold.",
"As only the bias trace makes the difference between both pixel sides, it might cause the lower charge sharing probability.",
"Furthermore, one can see that the region of the bias dot is not affected.",
"While for DO9 a clear increase in charge sharing probability towards the edges of the pixel is visible, at higher fluence the collected charge becomes too small for any significant charge sharing to be observable.",
"This can also be seen in the fractions of clusters with one, two, and more pixels.",
"Figure: Top: Design of the sample of the region shown in the plot below.",
"Bottom: Charge sharing probability for DO9 at V bias _{bias}=1200V.",
"Note the reduced charge sharing in the bias grid region on the right-hand side of the central pixel.Figure REF shows the fractions of one-pixel, two-pixel, and larger clusters as a function of the bias voltage.",
"It is evident, that with increasing bias voltage the cluster size increases, due to the reduced trapping.",
"At a given voltage the fraction of 1-pixel clusters increases with fluence, as more charge is lost due to trapping.",
"For samples irradiated up to $5\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ the cluster size increases slightly with bias voltage, while at $2\\times 10^{16}$ n$_{\\rm eq}$ /cm$^2$ the fraction of clusters with two or more hit pixels is very small and stays nearly constant over the accessible voltage range.",
"Figure: Fractions of 1-, 2-, and 3-hit clusters as function of bias voltage for irradiated n-in-n samples; see text for details.Error bars are too small to be visible.Plotting the residual distribution for two-pixel clusters only allows the width of the charge sharing region between pixels to be determined.",
"Figure REF shows the distributions for DO9 ($5\\times 10^{15}$ n$_{\\rm eq}$ /cm$^2$ ) and DO10 ($2\\times 10^{16}$ n$_{\\rm eq}$ /cm$^2$ ).",
"After correcting for the telescope resolution, the widths of the charge sharing regions are 7.1 $\\mu $ m and 7.7 $\\mu $ m. These values correspond very well with the width found for an unirradiated sample of 6.4 $\\mu $ m. This indicates that the lateral diffusion of the charge cloud does not change significantly with irradiation.",
"Figure: Residual distributions for 2-pixel clusters only.",
"Shown are distributions samples irradiated to 5×10 15 5\\times 10^{15} n eq _{\\rm eq}/cm 2 ^2 (left: DO9, bias voltage 1000 V) and 2×10 16 2\\times 10^{16} n eq _{\\rm eq}/cm 2 ^2 (right, DO10, bias voltage 1200 V), respectively."
],
[
"Residuals",
"Figure REF shows the residual distributions in the 50 $\\mu $ m pixel direction for the unirradiated sample (DO6) and the sample irradiated to $2\\times 10^{16}\\,{\\rm n_{eq}}/{\\rm cm}^2$, respectively.",
"The widths of the distributions are 16 $\\mu $ m and 15.4 $\\mu $ m, comparable with the expected digital resolution of 14.4 $\\mu $ m. Thus, no influence of radiation damage on the spatial resolution can be observed.",
"Figure: Residual distributions in the short pixel direction for an unirradiated sample (DO6, left) and a sample irradiated to 2×10 16 2\\times 10^{16} n eq _{\\rm eq}/cm 2 ^2 operated at a bias voltage of 1000 V (DO10, right).",
"No deterioration of the spatial distribution with irradiation is visible."
],
[
"Slim Edge",
"For slim edge studies the outermost pixels of a sample are of special interest.",
"Therefore, the samples were mounted such that the edge of the sensor was well within the trigger acceptance window.",
"Special analysis classes were written to investigate the characteristics of the edge pixels.",
"The basic principle is the same as for the charge collection analysis but instead of overlaying all pixels onto one single pixel, only pixels at the sensor edge are used and the special geometry is conserved in the overlay process.",
"For the IBL sensors the width of the inactive region at the edge of each sensor tile has to be reduced significantly with respect to the approximately 1 mm wide region on each side of the current ATLAS Pixel detector sensors.",
"One approach is to shift the guard-rings on the p$^+$ -side inwards.",
"Two specially designed DUTs were tested to study the impact of an overlap between the active pixel region with the guard ring region, where the electric field in the sensor is inhomogeneous.",
"In the DO6 sample, the overlap between active pixel region and guard ring region is 210 $\\mu $ m, with the number of guard rings reduced to 11.",
"In the DO3 sample groups of 10 pixels are shifted towards the edge of the sensor in steps of 25 $\\mu $ m, increasing the area in which the pixels overlap with the guard-rings (see Figure REF ).",
"Figure: Test structures for slim edge studies.",
"Top: DO6, the active pixel region overlaps the guard rings by 210 μ\\mu m. Bottom: DO3, groups of 10 pixels shifted towards the edge of the sensor in steps of 25 μ\\mu m.The test structures were mounted such that the edge of the sensor was well in the center of the trigger window, allowing to study charge collection in the shifted pixels in some detail.",
"Figure REF shows the collected charge in the overlap region of DO6.",
"With increasing distance from the bias voltage pad the collected charge decreases, due to the inhomogeneously formed depletion zone.",
"It is evident that the collected charge is sufficient to ensure good hit efficiency up to about 200 $\\mu $ m from the edge of the bias voltage pad.",
"Figure: Charge collection in the pixels shifted underneath the guard rings.Figure REF shows the collected charge in the overlap region for the DO3 sample.",
"Data from pixels with the same shift with respect to the edge of the bias voltage pad are plotted into one pixel.",
"The drop in collected charge systematically occurs at the same distance from the bias voltage pad, regardless of the shift of the pixel.",
"This indicates, that the loss of collected charge is indeed due to the depletion zone which is expected to be inhomogeneous along the x-axis (orthogonal to the bias voltage pad) but homogeneous along the y-axis (parallel to the bias voltage pad).",
"Further studies of this kind can be found in [32].",
"Figure: Charge collection in the pixels shifted underneath the guard rings."
],
[
"Conclusion",
"Planar silicon sensors, have been tested in high energy pion beams at the CERN SPS North Area in 2010 by the ATLAS Planar Pixel Sensors (PPS) collaboration.",
"Different bulk materials, geometries, especially for the guard ring regions, biasing and isolation structures were examined.",
"The goals of the measurement program were threefold: to demonstrate the suitability of p-bulk sensors for tracking purposes, to prove the radiation hardness of n-bulk sensors and to realize pixel sensors with reduced inactive edge area.",
"Pixelated p-bulk sensors produced by different vendors were tested to evaluate their performance, after irradiation too.",
"In terms of the collected charge, charge sharing, and spatial resolution the performance of the p-bulk sensors was very good and comparable to that of n-bulk sensors.",
"The issue of the high potential on the pixelated side of the sensor was tested and operation of the sensors was proven to be very stable.",
"The radiation hardness of n-bulk sensors was tested up to unprecedented fluences, with a maximum of $20\\times 10^{15}\\,{\\rm n_{eq}}/{\\rm cm}^{2}$ .",
"At a bias voltage of 1.2 kV a collected charge of about 6 ke was observed, corresponding to about one third of the collected charge before irradiation.",
"Despite the rather small collected charge and the reduced charge sharing between pixels, no significant deterioration of the spatial resolution was observed.",
"In order to reduce the inactive area at the edge of n-bulk sensors, several modified sensor layouts were tested.",
"The influence of a reduction of the number of guard rings and an increasing overlap between the active pixel region and the guard ring region on the backside of the sensor were studied.",
"It was found that the charge collection efficiency reduces with increasing distance from the edge of the bias voltage pad due to the inhomogeneously formed depletion zone in the sensor.",
"However, the collected charge is sufficient for reliable particle detection up to a distance of about 200 $\\mu $ m from the bias voltage pad.",
"This was very encouraging for the planar ATLAS IBL candidate design, which was finally designed employing the methods evaluated by the beam test measurements described in this paper."
],
[
"Acknowledgements",
"The authors would like to express their gratitude to V. Cindro, G. Kramberger and I. Mandić for their valuable support with irradiations at the TRIGA reactor of the Jožef Stefan Institute, Ljubljana, to A. Dierlamm for his help at the Irradiation Center, Karlsruhe, and to M. Glaser for his help at the CERN PS irradiation facility.",
"The work has been partially performed in the framework of the CERN RD50 Collaboration.",
"This work is supported by the Commission of the European Communities under the 6th Framework Programme “Structuring the European Research Area”, contract number RII3-026126.",
"We gratefully acknowledge the financial support of the German Federal Ministry of Science and Education (BMBF) within their excellence program, in particular as part of the collaborative research center “FSP 101-ATLAS, Physics on the TeV-scale at the Large Hadron Collider”.",
"We acknowledge the support of the Initiative and Networking Fund of the Helmholtz Association, contract HA-101 (“Physics at the Terascale”)."
]
] | 1204.1266 |
[
[
"Search for heavy bottom-like quarks in 4.9 inverse femtobarns of pp\n collisions at sqrt(s) = 7 TeV"
],
[
"Abstract Results are presented from a search for heavy bottom-like quarks, pair-produced in pp collisions at sqrt(s) = 7 TeV, undertaken with the CMS experiment at the LHC.",
"The b' quarks are assumed to decay exclusively to tW.",
"The b' anti-b' to t W(+) anti-t W(-) process can be identified by its distinctive signatures of three leptons or two leptons of same charge, and at least one b-quark jet.",
"Using a data sample corresponding to an integrated luminosity of 4.9 inverse femtobarns, observed events are compared to the standard model background predictions, and the existence of b' quarks having masses below 611 GeV is excluded at 95% confidence level."
],
[
"Introduction",
"The total number of fermion generations is assumed to be three in the standard model (SM), though the model does not provide an explanation of why this should be the case.",
"Thus the possible existence of a fourth generation remains an important subject for experimental investigation.",
"Adding a fourth generation of massive fermions to the model may strongly affect the Higgs and flavour sectors [1], [2], [3], [4], [5].",
"A fourth generation of heavy quarks would enhance the production of Higgs bosons [6], while the indirect bound from electroweak precision data on the Higgs mass would be relaxed [7], [8].",
"Additional massive quarks may provide a key to understanding the matter-antimatter asymmetry in the universe [9].",
"Various searches for fourth-generation fermions have already been reported.",
"Experiments have shown that the number of light neutrino flavours is equal to three [10], [11], [12], [13], but the possibility of additional heavier neutrinos has not been excluded.",
"A search for pair-produced bottom-like quarks (${b}^\\prime $ ) by the ATLAS collaboration excludes a ${b}^\\prime $ -quark mass of less than 480 [14].",
"Earlier studies setting mass limits on possible fourth-generation quarks, from experiments at the Tevatron and the Large Hadron Collider (LHC), can be found in Ref.",
"[15], [16], [17], [18], [19], [20], [21].",
"Using the Compact Muon Solenoid (CMS) detector, we have searched for a heavy ${b}^\\prime $ quark that is pair-produced in collisions at a centre-of-mass energy of 7at the LHC.",
"We assume that the mass of the ${b}^\\prime $ quark ($M_{{b}^\\prime }$ ) is larger than the sum of the top quark and the W-boson masses.",
"If the ${b}^\\prime $ quark couples principally to the top quark, the decay chain ${b}^\\prime \\overline{{b}}{}^\\prime \\rightarrow \\rightarrow $ will dominate [22].",
"Given the 11% branching fraction for a W-boson to each lepton, distinctive signatures of ${b}^\\prime \\overline{{b}}{}^\\prime $ production are expected, specifically those of two isolated leptons with the same charge (“same-charge dileptons\") or three isolated leptons (“trileptons\").",
"Although occurring very rarely in the standard model, these two signatures may be present in 7.3% of the ${b}^\\prime \\overline{{b}}{}^\\prime $ events.",
"An earlier search by CMS [17] in the same-charge dilepton and the trilepton channels, utilizing a data set corresponding to an integrated luminosity of 34, set a lower limit on the mass of the ${b}^\\prime $ quark of 361at the 95% confidence level (CL).",
"Here we present an update of this search using a much larger data set, corresponding to an integrated luminosity of 4.9."
],
[
"CMS detector and trigger",
"This analysis is based on the data recorded by the CMS experiment in 2011.",
"The central feature of the CMS detector is a superconducting solenoid, 13 m in length and 6 m in diameter, which provides an axial magnetic field of 3.8T.",
"Charged-particle trajectories are determined using silicon pixel and silicon strip tracker measurements.",
"A crystal electromagnetic calorimeter, including lead-silicon preshower detectors in the forward directions, together with a surrounding brass/scintillator hadronic calorimeter, encloses the tracking volume and provides energy measurements of electrons and hadronic jets.",
"Muons are identified and measured in the tracker and in gas-ionization detectors embedded in the steel return yoke outside the solenoid.",
"The detector is nearly hermetic, providing measurements of any imbalance of momentum in the plane transverse to the beam direction.",
"A more detailed description of the CMS detector can be found in Ref. [23].",
"A two-level trigger system [24] selects events for further analysis.",
"The events analyzed in this search are collected with the requirement that the trigger system detects at least two lepton candidates.",
"Efficiencies for these dilepton triggers are determined using events that pass a jet trigger, have two reconstructed electrons or muons, and that also pass the full selection criteria described in the next section.",
"For these selected events, the dilepton trigger efficiencies are estimated to be 91%, 96%, and $>$ 99%, for events with two muons, one electron and one muon, and two electrons, respectively."
],
[
"Selection criteria",
"The use of the CMS particle-flow global event reconstruction procedure [25], [26], [27], [28] has been extended beyond its application in Ref. [17].",
"In the present analysis, all physics objects – leptons, jets, and missing transverse energy - are reconstructed with this procedure.",
"The reconstruction and selection criteria for each physics object used in this analysis are described below.",
"Candidate muons are reconstructed through a global fit to trajectories, using hit signals in the inner tracker and in the muon system.",
"Muons are required to have transverse momenta $> 20$ and ${\\eta }<2.4$ , where the pseudorapidity $\\eta = -\\ln [\\tan \\theta /2]$ and $\\theta $ is the polar angle relative to the anticlockwise beam direction.",
"The muon candidate must be associated with hits in the silicon pixel and strip detectors, have segments in the muon chambers, and provide a high-quality global fit to the track segments.",
"The efficiency for these muon selection criteria is $>$ 99% from Z decays [29].",
"In addition, the muon track is required to be consistent with originating from the principal primary interaction vertex, which is defined by the one associated with tracks yielding the largest value for the sum of their $p_\\mathrm {T}^2$ .",
"Reconstruction of electron candidates starts from clusters of energy deposits in the ECAL, which are then matched to hits in the silicon tracker.",
"Electron candidates are required to have $> 20$ .",
"Candidates are required to be reconstructed in the fiducial volume of the barrel ($|\\eta | < 1.44$ ) or in the end-caps ($1.57 < |\\eta | < 2.4$ ).",
"The electron candidate track is required to be consistent with originating from the principal primary interaction vertex.",
"Electrons are identified using variables which include the ratio between the energies deposited in the HCAL and the ECAL, the shower width in $\\eta $ , and the distance between the calorimeter shower and the particle trajectory in the tracker, measured in both $\\eta $ and azimuthal angle ($\\phi $ ).",
"The selection criteria are optimized [30] to reject the background from hadronic jets while maintaining an efficiency of 80% for the electrons from $\\rm W$ or $\\rm Z$ decays.",
"Jets are reconstructed by an anti-$k_\\mathrm {T}$ jet-clustering algorithm with a distance parameter $R=0.5$ [31].",
"Particle energies are calibrated [32] separately for each particle type, and resulting jet energies therefore require only small corrections that account for thresholds and residual inefficiencies.",
"All jet candidates must have $>$ 25and be within $|\\eta |<2.4$ .",
"Neutrinos from $$ boson decays escape the detector, and thereby give rise to a significant imbalance in the net transverse momentum measured for each event.",
"This missing transverse momentum, expressed as the quantity , is defined as the absolute value of the vector sum of the transverse momenta of all reconstructed particles [33].",
"In contrast to the earlier analysis of Ref.",
"[17], b-tagging is now used to reject events from backgrounds that do not include a top-quark decay.",
"The b-tagging algorithm applied in this analysis generates a list of tracks associated with each jet, and calculates the significance of each track's impact-parameter (IP), as determined by the ratio of the IP to its uncertainty.",
"For the jet to be tagged as a b-jet, the IP significance of at least three of its listed tracks must exceed a threshold value, chosen to give an identification efficiency of 50% for b-jets and a misidentification rate of 1% for other particle jets [34].",
"Electrons and muons from $\\rightarrow \\ell \\nu $ ($\\ell =,\\mu $ ) decays are expected to be isolated from other particles in the detector.",
"A cone of $\\Delta R < 0.3$ , where $\\Delta R \\equiv \\sqrt{(\\Delta \\eta )^2 + (\\Delta \\phi )^2}$ , is constructed around each lepton-candidate's direction, and if the scalar sum of the transverse momenta of the particles inside the cone, excluding contributions from the lepton candidate, exceeds 15% of the candidate $$ , then the lepton candidate is rejected.",
"Electron candidates are required to be separated from any selected muon candidates by $\\Delta R > 0.1$ to remove misidentified electrons due to muon bremsstrahlung.",
"Electron candidates identified as originating from photon conversions are also rejected.",
"Events are required to have at least one well-reconstructed interaction vertex [35].",
"Events with two leptons of the same electric charge, or with three leptons (two of which must be oppositely charged), are selected.",
"For the same-charge dilepton (trilepton) channel, events with fewer than four (two) jets are rejected.",
"At least one jet must be identified as a b-jet.",
"In addition, events that have any two muons or electrons whose invariant mass $M_{\\ell \\ell }$ is within 10of the Z-mass ($|M_{\\ell \\ell }-M_| < 10$ ) are rejected, in order to suppress the background from $\\rightarrow \\ell ^+\\ell ^-$ decays.",
"For each event, the scalar quantity $S_\\mathrm {T} = \\sum |\\vec{p_\\mathrm {T}} (\\text{jets})| +\\sum |\\vec{p_\\mathrm {T}} (\\text{leptons})| + $ is required to satisfy the condition $S_\\mathrm {T} > 500$ .",
"The selection criteria described above are not fully optimized in terms of discovery reach, but in fact they are more robust because they have a single background component in the background estimation with data.",
"Signal selection efficiencies are estimated using simulated event samples.",
"Fourth generation quarks production is implemented as a straightforward extension to the standard model configuration of the MadGraph/MadEvent generator version 5.131 [36].",
"Parton showering and hadronization are provided by 6.424 [37] using the matching prescription described in Ref. [38].",
"Finally, these generated signal events are passed through the CMS detector simulation based on [39].",
"Table REF shows the expected efficiencies for a ${b}^\\prime $ signal, for $450 \\le M_{{b}^\\prime } \\le 650$ .",
"The efficiencies vary between 1.5% and 1.7% for the same-charge dilepton channel, and between 0.47% and 0.63% for the trilepton events, in the chosen range of $M_{{b}^\\prime }$ .",
"These efficiencies include the branching fractions for W-decay and the b-tagging performance [34].",
"Jet multiplicities for the same-charge dilepton and the trilepton channels are shown in Fig.",
"REF , and the $S_\\mathrm {T}$ distributions are presented in Fig.",
"REF .",
"The expected distributions for a ${b}^\\prime $ signal having $M_{{b}^\\prime }=500$ are normalized to the production cross sections from Ref.",
"[40] that include approximate next-to-next-to-leading-order perturbative QCD corrections, and standard QCD couplings are assumed.",
"Table: Summary of expected b ' b ¯ ' {b}^\\prime \\overline{{b}}{}^\\prime cross sections ,selection efficiencies, and yields for the two signal channels as a function of the b ' {b}^\\prime mass.Figure: Jet multiplicity distributions for the same-charge dilepton channel (left), and the trilepton channel (right).The open histogram shows the contribution expected from a b ' {b}^\\prime having M b ' =M_{\\rm b^\\prime } = 500.The contributions from standard model processes are normalized to the total estimated background.All selection criteria are applied except the one corresponding to the plotted variable.The vertical dotted lines indicate the minimum number of jets required in events selected for each of the channels.Figure: Distributions in S T S_\\mathrm {T}, the scalar sum of the transverse momenta of objects, inthe same-charge dilepton channel (left), and the trilepton channel (right).The open histogram is the contribution expected from a b ' {b}^\\prime having M b ' =500M_{{b}^\\prime } =500.The histograms for standard model processes are normalized to the total expected background.All selection criteria are applied except the one corresponding to the plotted variable.The vertical dotted line indicates the lower bound on S T S_\\mathrm {T} used in the analysis."
],
[
"Background estimation",
"Because of the b-tagging requirement, 98% of the expected background events in the same-charge dilepton channel have at least one top quark from $$ , $+/$ , or single-top processes.",
"These backgrounds are categorized into three sources: (i) true $\\ell ^+\\ell ^-$ events with a electron of misidentified charge, (ii) single-lepton events with an extra misidentified or non-isolated lepton candidate, and (iii) events with two prompt leptons of the same charge.",
"The contribution due to the charge misidentification of electrons is determined using a control sample that, while keeping the remaining signal selection criteria, has oppositely-charged electron pairs or electrons and muons.",
"The charge misidentification rate (0.03% and 0.31% for barrel and endcap candidates, respectively) is determined by counting the events containing two same-charge electron candidates, whose invariant mass is consistent with that of a $$ -boson, relative to the yield of $\\rightarrow $ events.",
"Background from source (ii) is estimated as follows.",
"Leptons passing the selection criteria described in Section 3 for signal are denoted as “tight”, while muon candidates passing relaxed isolation thresholds and track-fit quality requirements, or electron candidates passing relaxed identification and isolation requirements, are referred to as “loose”.",
"Tight lepton candidates are excluded from the selection of loose lepton candidates.",
"The background from events containing a false or non-isolated lepton candidate is estimated using another data control sample containing one tight lepton candidate and one loose lepton candidate, with the remaining selection criteria kept identical to those used for the signal sample.",
"By definition, this control sample excludes events in the signal sample.",
"The contributions of the backgrounds in the selected events are calculated using the yields observed in the control sample multiplied by the ratio of the number of lepton candidates passing tight selection criteria to those passing the loose criteria.",
"This ratio, also determined in data, is calculated as the number of events containing one loose and one tight lepton candidate divided by the number of those containing two loose lepton candidates.",
"Applying the above methods to data, a background yield of $7.8\\pm 2.8$ events is estimated to originate from sources (i) and (ii).",
"The estimated yield to the same-charge dilepton channel from processes that produce prompt same-charge dileptons, including $+$ , $+$ , and diboson channels ($$ , $$ , and same-charge $^\\pm ^\\pm +$ jets), is determined using simulations of these processes.",
"The contribution in the signal region is estimated to be $3.6\\pm 0.6$ events.",
"For the trilepton channel, the background is an order of magnitude smaller than for the same-charge dilepton channel, and is dominated by processes that produce three prompt leptons, such as $+/$ .",
"The yield in the signal region, which is only $0.78\\pm 0.21$ events, is estimated using simulated samples.",
"Contributions from $\\rightarrow $ and $/$ processes are normalized to the cross sections measured by CMS [41], [42].",
"The single-top contributions are normalized to the next-to-next-to-leading-logarithm cross sections [43], [44].",
"Production rates for dibosons are estimated from the next-to-leading-order cross sections given by mcfm [45].",
"The $+/$ and same-charge $^\\pm ^\\pm +$ jets processes are normalized to the next-to-leading-order cross sections given in Ref. [46].",
"The multijet background contribution is estimated using a control sample of events containing two (three) loose lepton candidates for the same-charge dilepton (trilepton) channel, maintaining other selection criteria.",
"The yield of multijet events in the signal region is calculated by multiplying the yield observed in the control sample by the ratio squared (cubed) of the number of lepton candidates passing tight selection to the number passing loose selection.",
"The contribution of multijet events to the signal region is estimated to be smaller than 0.12 (0.001) events for the same-charge dilepton (trilepton) channel, and thus is negligible compared to contributions from the other background processes."
],
[
"Systematic uncertainties",
"To validate the procedure for estimating background, and to assign a proper systematic uncertainty, the study in the same-charge dilepton channel is repeated using a mixture of simulated samples representing the potential background sources.",
"The full estimation procedure is then applied to the simulated samples, and results are compared to the input values.",
"The observed difference ($2.7\\pm 0.9$ events) is included as a systematic uncertainty.",
"The statistical uncertainties in the control samples are also included in the systematic uncertainties.",
"The following uncertainties are included in both dilepton and trilepton channels.",
"The b-tagging efficiency as measured in data has a precision of 10% per b-jet [34], resulting in a 6.7% uncertainty in the efficiency of signal samples.",
"The effect of this uncertainty on the background contributions determined using simulated samples is estimated to be 0.35 (0.08) events for the dilepton (trilepton) channel.",
"Lepton selection efficiencies are measured using inclusive $\\rightarrow \\ell ^+\\ell ^-$ data, and the difference between efficiencies measured in data and simulation is taken as a systematic uncertainty.",
"An additional systematic uncertainty of 50% of the difference in efficiency between simulated $$ and ${b}^\\prime $ samples is included, to cover the effects of different event topologies.",
"This estimation yields uncertainties of 1.7% and 2.7% for electrons and muons, respectively.",
"The uncertainty in signal efficiency, calculated using appropriate weighting of the electron and muon contributions, is 3.3% (5.0%) for the dilepton (trilepton) channel.",
"The uncertainties in the background normalization are estimated to be 0.74 and 0.12 events for dilepton and trilepton channels, respectively, and the uncertainties for each of the individual processes are included as follows: $\\pm 11$ % for $$ [41], $\\pm 3$ % ($\\pm 4$ %) for $$ ($$ ) [42], $\\pm 30$ % for single top processes, $\\pm 26$ % for $$ , $\\pm 30$ % for $$ , $\\pm 21$ % for $$ , $\\pm 30$ % for $$ , $\\pm 30$ % for $$ , $\\pm 49$ % for $^\\pm ^\\pm +$ jets, and $\\pm 100$ % for multijet.",
"The uncertainties in the normalization of diboson, $$ , $$ , and $^\\pm ^\\pm +$ jets processes are taken from a comparison of next-to-leading-order and leading-order predictions.",
"The uncertainty related to the presence of additional interactions (pile-up) in the same beam crossing interval as an event is examined by varying the number of such interactions included in the simulations.",
"The systematic effects of the uncertainties in jet-energy-scale, jet resolution, resolution, pile-up events, and trigger efficiency are found to be small [32], [33].",
"Uncertainty sets given by CTEQ6 [47] are used to determine the uncertainties from the choice of parton distribution functions.",
"The relative uncertainty in the integrated luminosity measurement is estimated to be 2.2% [48], and is included in the calculation of limits.",
"The details of uncertainties in the signal selection efficiency and in the background estimation are presented in Table REF .",
"Table: Summary of relative systematic uncertainties in signal selection efficiencies (Δϵ/ϵ\\Delta \\epsilon /\\epsilon ) andthe absolute systematic uncertainties in the number of expected background events (ΔB\\Delta B).The ranges given below represent the dependence on M b ' \\rm M_{{b}^\\prime }, varying from 450to 650."
],
[
"Results",
"There are 12 (1) events found in the signal region for the dilepton (trilepton) channel, to be compared with an estimated background of $11.4\\pm 2.9$ ($0.78\\pm 0.21$ ) (Table REF ).",
"Table: Summary of the estimated background contributions to the same-charge dilepton channel andthe trilepton channel, and the observed event yield in data.The given uncertainties are systematic.Most of the background sources contain at least one top quark in the final state, with a b-quark produced in the top quark decay.",
"Therefore, modifying the required number of b-tagged jets, in a separate study, provides a good check of the analysis.",
"The observed yields when requiring $\\ge 0$ , $\\ge 1$ , or $\\ge 2$ b-tagged jets are consistent with the estimated background, and in agreement with the expected dominance of background from top quarks.",
"For each ${b}^\\prime $ mass hypothesis, cross sections, selection efficiencies, and associated uncertainties are estimated (Tables REF and REF ).",
"From these values, the estimated background yield, and the number of observed events, upper limits on ${b}^\\prime \\overline{{b}}{}^\\prime $ pair production cross sections at 95% CL are derived, using a modified frequentist approach ($CL_s$ ) [49].",
"These limits are plotted as the solid line in Fig.",
"REF , while the dotted line represents the limits expected with the available integrated luminosity, assuming the presence of standard model processes alone.",
"By comparing to the theoretical production cross section for $\\rightarrow {b}^\\prime \\overline{{b}}{}^\\prime $ , a lower limit of 611is extracted for the mass of the ${b}^\\prime $ quark, at 95% CL, while a limit of 619is expected for a background-only hypothesis.",
"Figure: Exclusion limits at 95% CL on the →b ' b ¯ ' \\rightarrow {b}^\\prime \\overline{{b}}{}^\\prime production cross section (σ\\sigma ).The solid line represents the observed limits,while the dotted line representsthe limits expected for the available integrated luminosity,assuming the presence of standard model processes alone.A comparison with the production cross-sections excludes b ' {b}^\\prime massesM b ' <611M_{{b}^\\prime } < 611at 95% CLfor a 100% b ' →{b}^\\prime \\rightarrow decay branching fraction."
],
[
"Summary",
"Results have been presented from a search for heavy bottom-like quarks pair-produced in proton-proton collisions at $\\sqrt{s} = 7$ .",
"The process of $\\rightarrow {b}^\\prime \\overline{{b}}{}^\\prime \\rightarrow $ has been studied in data corresponding to an integrated luminosity of 4.9, collected with the CMS detector.",
"Estimated background contributions have been found to be small, since final states containing the signatures of trileptons or same-charge dileptons are produced rarely in standard model processes.",
"Assuming a branching fraction of 100% for the decay ${b}^\\prime \\rightarrow $ , ${b}^\\prime $ quarks with masses below 611are excluded at 95% CL.",
"This is the most stringent limit to date."
],
[
"Acknowledgment",
"We wish to congratulate our colleagues in the CERN accelerator departments for the excellent performance of the LHC machine.",
"We thank the technical and administrative staff at CERN and other CMS institutes, and acknowledge support from: FMSR (Austria); FNRS and FWO (Belgium); CNPq, CAPES, FAPERJ, and FAPESP (Brazil); MES (Bulgaria); CERN; CAS, MoST, and NSFC (China); COLCIENCIAS (Colombia); MSES (Croatia); RPF (Cyprus); Academy of Sciences and NICPB (Estonia); Academy of Finland, ME, and HIP (Finland); CEA and CNRS/IN2P3 (France); BMBF, DFG, and HGF (Germany); GSRT (Greece); OTKA and NKTH (Hungary); DAE and DST (India); IPM (Iran); SFI (Ireland); INFN (Italy); NRF and WCU (Korea); LAS (Lithuania); CINVESTAV, CONACYT, SEP, and UASLP-FAI (Mexico); PAEC (Pakistan); SCSR (Poland); FCT (Portugal); JINR (Armenia, Belarus, Georgia, Ukraine, Uzbekistan); MST and MAE (Russia); MSTD (Serbia); MICINN and CPAN (Spain); Swiss Funding Agencies (Switzerland); NSC (Taipei); TUBITAK and TAEK (Turkey); STFC (United Kingdom); DOE and NSF (USA)."
],
[
"The CMS Collaboration ",
"Yerevan Physics Institute, Yerevan, Armenia S. Chatrchyan, V. Khachatryan, A.M. Sirunyan, A. Tumasyan Institut für Hochenergiephysik der OeAW, Wien, Austria W. Adam, T. Bergauer, M. Dragicevic, J. Erö, C. Fabjan, M. Friedl, R. Frühwirth, V.M.",
"Ghete, J. Hammer1, N. Hörmann, J. Hrubec, M. Jeitler, W. Kiesenhofer, M. Krammer, D. Liko, I. Mikulec, M. Pernicka$^{\\textrm {\\dag }}$ , B. Rahbaran, C. Rohringer, H. Rohringer, R. Schöfbeck, J. Strauss, A. Taurok, F. Teischinger, P. Wagner, W. Waltenberger, G. Walzel, E. Widl, C.-E. Wulz National Centre for Particle and High Energy Physics, Minsk, Belarus V. Mossolov, N. Shumeiko, J. Suarez Gonzalez Universiteit Antwerpen, Antwerpen, Belgium S. Bansal, K. Cerny, T. Cornelis, E.A.",
"De Wolf, X. Janssen, S. Luyckx, T. Maes, L. Mucibello, S. Ochesanu, B. Roland, R. Rougny, M. Selvaggi, H. Van Haevermaet, P. Van Mechelen, N. Van Remortel, A.",
"Van Spilbeeck Vrije Universiteit Brussel, Brussel, Belgium F. Blekman, S. Blyweert, J.",
"D'Hondt, R. Gonzalez Suarez, A. Kalogeropoulos, M. Maes, A. Olbrechts, W. Van Doninck, P. Van Mulders, G.P.",
"Van Onsem, I. Villella Université Libre de Bruxelles, Bruxelles, Belgium O. Charaf, B. Clerbaux, G. De Lentdecker, V. Dero, A.P.R.",
"Gay, T. Hreus, A. Léonard, P.E.",
"Marage, T. Reis, L. Thomas, C. Vander Velde, P. Vanlaer Ghent University, Ghent, Belgium V. Adler, K. Beernaert, A. Cimmino, S. Costantini, G. Garcia, M. Grunewald, B. Klein, J. Lellouch, A. Marinov, J. Mccartin, A.A. Ocampo Rios, D. Ryckbosch, N. Strobbe, F. Thyssen, M. Tytgat, L. Vanelderen, P. Verwilligen, S. Walsh, E. Yazgan, N. Zaganidis Université Catholique de Louvain, Louvain-la-Neuve, Belgium S. Basegmez, G. Bruno, L. Ceard, C. Delaere, T. du Pree, D. Favart, L. Forthomme, A. Giammanco2, J. Hollar, V. Lemaitre, J. Liao, O. Militaru, C. Nuttens, D. Pagano, A. Pin, K. Piotrzkowski, N. Schul Université de Mons, Mons, Belgium N. Beliy, T. Caebergs, E. Daubie, G.H.",
"Hammad Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, Brazil G.A.",
"Alves, M. Correa Martins Junior, D. De Jesus Damiao, T. Martins, M.E.",
"Pol, M.H.G.",
"Souza Universidade do Estado do Rio de Janeiro, Rio de Janeiro, Brazil W.L.",
"Aldá Júnior, W. Carvalho, A. Custódio, E.M. Da Costa, C. De Oliveira Martins, S. Fonseca De Souza, D. Matos Figueiredo, L. Mundim, H. Nogima, V. Oguri, W.L.",
"Prado Da Silva, A. Santoro, S.M.",
"Silva Do Amaral, L. Soares Jorge, A. Sznajder Instituto de Fisica Teorica, Universidade Estadual Paulista, Sao Paulo, Brazil T.S.",
"Anjos3, C.A.",
"Bernardes3, F.A.",
"Dias4, T.R.",
"Fernandez Perez Tomei, E. M. Gregores3, C. Lagana, F. Marinho, P.G.",
"Mercadante3, S.F.",
"Novaes, Sandra S. Padula Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria V. Genchev1, P. Iaydjiev1, S. Piperov, M. Rodozov, S. Stoykova, G. Sultanov, V. Tcholakov, R. Trayanov, M. Vutova University of Sofia, Sofia, Bulgaria A. Dimitrov, R. Hadjiiska, V. Kozhuharov, L. Litov, B. Pavlov, P. Petkov Institute of High Energy Physics, Beijing, China J.G.",
"Bian, G.M.",
"Chen, H.S.",
"Chen, C.H.",
"Jiang, D. Liang, S. Liang, X. Meng, J. Tao, J. Wang, J. Wang, X. Wang, Z. Wang, H. Xiao, M. Xu, J. Zang, Z. Zhang State Key Lab.",
"of Nucl. Phys.",
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"Ban, S. Guo, Y. Guo, W. Li, S. Liu, Y. Mao, S.J.",
"Qian, H. Teng, S. Wang, B. Zhu, W. Zou Universidad de Los Andes, Bogota, Colombia C. Avila, B. Gomez Moreno, A.F.",
"Osorio Oliveros, J.C. Sanabria Technical University of Split, Split, Croatia N. Godinovic, D. Lelas, R. Plestina5, D. Polic, I. Puljak1 University of Split, Split, Croatia Z. Antunovic, M. Dzelalija, M. Kovac Institute Rudjer Boskovic, Zagreb, Croatia V. Brigljevic, S. Duric, K. Kadija, J. Luetic, S. Morovic University of Cyprus, Nicosia, Cyprus A. Attikis, M. Galanti, G. Mavromanolakis, J. Mousa, C. Nicolaou, F. Ptochos, P.A.",
"Razis Charles University, Prague, Czech Republic M. Finger, M. Finger Jr. Academy of Scientific Research and Technology of the Arab Republic of Egypt, Egyptian Network of High Energy Physics, Cairo, Egypt Y. Assran6, S. Elgammal, A. Ellithi Kamel7, S. Khalil8, M.A.",
"Mahmoud9, A. Radi8$^{, }$ 10 National Institute of Chemical Physics and Biophysics, Tallinn, Estonia M. Kadastik, M. Müntel, M. Raidal, L. Rebane, A. Tiko Department of Physics, University of Helsinki, Helsinki, Finland V. Azzolini, P. Eerola, G. Fedi, M. Voutilainen Helsinki Institute of Physics, Helsinki, Finland J. Härkönen, A. Heikkinen, V. Karimäki, R. Kinnunen, M.J. Kortelainen, T. Lampén, K. Lassila-Perini, S. Lehti, T. Lindén, P. Luukka, T. Mäenpää, T. Peltola, E. Tuominen, J. Tuominiemi, E. Tuovinen, D. Ungaro, L. Wendland Lappeenranta University of Technology, Lappeenranta, Finland K. Banzuzi, A. Korpela, T. Tuuva DSM/IRFU, CEA/Saclay, Gif-sur-Yvette, France M. Besancon, S. Choudhury, M. Dejardin, D. Denegri, B. Fabbro, J.L.",
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"Physikalisches Institut A, Aachen, Germany M. Ata, J. Caudron, E. Dietz-Laursonn, D. Duchardt, M. Erdmann, A. Güth, T. Hebbeker, C. Heidemann, K. Hoepfner, T. Klimkovich, D. Klingebiel, P. Kreuzer, D. Lanske$^{\\textrm {\\dag }}$ , J. Lingemann, C. Magass, M. Merschmeyer, A. Meyer, M. Olschewski, P. Papacz, H. Pieta, H. Reithler, S.A. Schmitz, L. Sonnenschein, J. Steggemann, D. Teyssier, M. Weber RWTH Aachen University, III.",
"Physikalisches Institut B, Aachen, Germany M. Bontenackels, V. Cherepanov, M. Davids, G. Flügge, H. Geenen, M. Geisler, W. Haj Ahmad, F. Hoehle, B. Kargoll, T. Kress, Y. Kuessel, A. Linn, A. Nowack, L. Perchalla, O. Pooth, J. Rennefeld, P. Sauerland, A. Stahl Deutsches Elektronen-Synchrotron, Hamburg, Germany M. Aldaya Martin, J. Behr, W. Behrenhoff, U. Behrens, M. Bergholz14, A. Bethani, K. Borras, A. Burgmeier, A. Cakir, L. Calligaris, A. Campbell, E. Castro, F. Costanza, D. Dammann, G. Eckerlin, D. Eckstein, D. Fischer, G. Flucke, A. Geiser, I. Glushkov, S. Habib, J. Hauk, H. Jung1, M. Kasemann, P. Katsas, C. Kleinwort, H. Kluge, A. Knutsson, M. Krämer, D. Krücker, E. Kuznetsova, W. Lange, W. Lohmann14, B. Lutz, R. Mankel, I. Marfin, M. Marienfeld, I.-A.",
"Melzer-Pellmann, A.B.",
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"Ziebarth Institute of Nuclear Physics \"Demokritos\", Aghia Paraskevi, Greece G. Daskalakis, T. Geralis, S. Kesisoglou, A. Kyriakis, D. Loukas, I. Manolakos, A. Markou, C. Markou, C. Mavrommatis, E. Ntomari University of Athens, Athens, Greece L. Gouskos, T.J. Mertzimekis, A. Panagiotou, N. Saoulidou University of Ioánnina, Ioánnina, Greece I. Evangelou, C. Foudas1, P. Kokkas, N. Manthos, I. Papadopoulos, V. Patras KFKI Research Institute for Particle and Nuclear Physics, Budapest, Hungary G. Bencze, C. Hajdu1, P. Hidas, D. Horvath15, K. Krajczar16, B. Radics, F. Sikler1, V. Veszpremi, G. Vesztergombi16 Institute of Nuclear Research ATOMKI, Debrecen, Hungary N. Beni, S. Czellar, J. Molnar, J. Palinkas, Z. Szillasi University of Debrecen, Debrecen, Hungary J. Karancsi, P. Raics, Z.L.",
"Trocsanyi, B. Ujvari Panjab University, Chandigarh, India S.B.",
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"Singh University of Delhi, Delhi, India S. Ahuja, A. Bhardwaj, B.C.",
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"Jain, Sh.",
"Jain, R. Khurana, S. Sarkar Bhabha Atomic Research Centre, Mumbai, India A. Abdulsalam, R.K. Choudhury, D. Dutta, S. Kailas, V. Kumar, A.K.",
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"Mohanty, B. Parida, K. Sudhakar, N. Wickramage Tata Institute of Fundamental Research - HECR, Mumbai, India S. Banerjee, S. Dugad Institute for Research in Fundamental Sciences (IPM), Tehran, Iran H. Arfaei, H. Bakhshiansohi20, S.M.",
"Etesami21, A. Fahim20, M. Hashemi, H. Hesari, A. Jafari20, M. Khakzad, A. Mohammadi22, M. Mohammadi Najafabadi, S. Paktinat Mehdiabadi, B. Safarzadeh23, M. Zeinali21 INFN Sezione di Bari $^{a}$ , Università di Bari $^{b}$ , Politecnico di Bari $^{c}$ , Bari, Italy M. Abbrescia$^{a}$$^{, }$$^{b}$ , L. Barbone$^{a}$$^{, }$$^{b}$ , C. Calabria$^{a}$$^{, }$$^{b}$$^{, }$ 1, S.S. Chhibra$^{a}$$^{, }$$^{b}$ , A. Colaleo$^{a}$ , D. Creanza$^{a}$$^{, }$$^{c}$ , N. De Filippis$^{a}$$^{, }$$^{c}$$^{, }$ 1, M. De Palma$^{a}$$^{, }$$^{b}$ , L. Fiore$^{a}$ , G. Iaselli$^{a}$$^{, }$$^{c}$ , L. Lusito$^{a}$$^{, }$$^{b}$ , G. Maggi$^{a}$$^{, }$$^{c}$ , M. Maggi$^{a}$ , B. Marangelli$^{a}$$^{, }$$^{b}$ , S. My$^{a}$$^{, }$$^{c}$ , S. Nuzzo$^{a}$$^{, }$$^{b}$ , N. Pacifico$^{a}$$^{, }$$^{b}$ , A. Pompili$^{a}$$^{, }$$^{b}$ , G. Pugliese$^{a}$$^{, }$$^{c}$ , G. Selvaggi$^{a}$$^{, }$$^{b}$ , L. Silvestris$^{a}$ , G. Singh$^{a}$$^{, }$$^{b}$ , G. Zito$^{a}$ INFN Sezione di Bologna $^{a}$ , Università di Bologna $^{b}$ , Bologna, Italy G. Abbiendi$^{a}$ , A.C. Benvenuti$^{a}$ , D. Bonacorsi$^{a}$$^{, }$$^{b}$ , S. Braibant-Giacomelli$^{a}$$^{, }$$^{b}$ , L. Brigliadori$^{a}$$^{, }$$^{b}$ , P. Capiluppi$^{a}$$^{, }$$^{b}$ , A. Castro$^{a}$$^{, }$$^{b}$ , F.R.",
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"Carrillo Montoya$^{a}$$^{, }$ 1, N. Cavallo$^{a}$$^{, }$ 25, A.",
"De Cosa$^{a}$$^{, }$$^{b}$ , O. Dogangun$^{a}$$^{, }$$^{b}$ , F. Fabozzi$^{a}$$^{, }$ 25, A.O.M.",
"Iorio$^{a}$$^{, }$ 1, L. Lista$^{a}$ , S. Meola$^{a}$$^{, }$ 26, M. Merola$^{a}$$^{, }$$^{b}$ , P. Paolucci$^{a}$ INFN Sezione di Padova $^{a}$ , Università di Padova $^{b}$ , Università di Trento (Trento) $^{c}$ , Padova, Italy P. Azzi$^{a}$ , N. Bacchetta$^{a}$$^{, }$ 1, P. Bellan$^{a}$$^{, }$$^{b}$ , A. Branca$^{a}$$^{, }$ 1, R. Carlin$^{a}$$^{, }$$^{b}$ , P. Checchia$^{a}$ , T. Dorigo$^{a}$ , U. Dosselli$^{a}$ , F. Gasparini$^{a}$$^{, }$$^{b}$ , A. Gozzelino$^{a}$ , K. Kanishchev$^{a}$$^{, }$$^{c}$ , S. Lacaprara$^{a}$ , I. Lazzizzera$^{a}$$^{, }$$^{c}$ , M. Margoni$^{a}$$^{, }$$^{b}$ , A.T. Meneguzzo$^{a}$$^{, }$$^{b}$ , M. Nespolo$^{a}$$^{, }$ 1, L. Perrozzi$^{a}$ , N. Pozzobon$^{a}$$^{, }$$^{b}$ , P. Ronchese$^{a}$$^{, }$$^{b}$ , F. Simonetto$^{a}$$^{, }$$^{b}$ , E. Torassa$^{a}$ , M. Tosi$^{a}$$^{, }$$^{b}$$^{, }$ 1, S. Vanini$^{a}$$^{, }$$^{b}$ , P. Zotto$^{a}$$^{, }$$^{b}$ , G. Zumerle$^{a}$$^{, }$$^{b}$ INFN Sezione di Pavia $^{a}$ , Università di Pavia $^{b}$ , Pavia, Italy M. Gabusi$^{a}$$^{, }$$^{b}$ , S.P.",
"Ratti$^{a}$$^{, }$$^{b}$ , C. Riccardi$^{a}$$^{, }$$^{b}$ , P. Torre$^{a}$$^{, }$$^{b}$ , P. Vitulo$^{a}$$^{, }$$^{b}$ INFN Sezione di Perugia $^{a}$ , Università di Perugia $^{b}$ , Perugia, Italy G.M.",
"Bilei$^{a}$ , L. Fanò$^{a}$$^{, }$$^{b}$ , P. Lariccia$^{a}$$^{, }$$^{b}$ , A. Lucaroni$^{a}$$^{, }$$^{b}$$^{, }$ 1, G. Mantovani$^{a}$$^{, }$$^{b}$ , M. Menichelli$^{a}$ , A. Nappi$^{a}$$^{, }$$^{b}$ , F. Romeo$^{a}$$^{, }$$^{b}$ , A. Saha, A. Santocchia$^{a}$$^{, }$$^{b}$ , S. Taroni$^{a}$$^{, }$$^{b}$$^{, }$ 1 INFN Sezione di Pisa $^{a}$ , Università di Pisa $^{b}$ , Scuola Normale Superiore di Pisa $^{c}$ , Pisa, Italy P. Azzurri$^{a}$$^{, }$$^{c}$ , G. Bagliesi$^{a}$ , T. Boccali$^{a}$ , G. Broccolo$^{a}$$^{, }$$^{c}$ , R. Castaldi$^{a}$ , R.T. D'Agnolo$^{a}$$^{, }$$^{c}$ , R. Dell'Orso$^{a}$ , F. Fiori$^{a}$$^{, }$$^{b}$$^{, }$ 1, L. Foà$^{a}$$^{, }$$^{c}$ , A. Giassi$^{a}$ , A. Kraan$^{a}$ , F. Ligabue$^{a}$$^{, }$$^{c}$ , T. Lomtadze$^{a}$ , L. Martini$^{a}$$^{, }$ 27, A. Messineo$^{a}$$^{, }$$^{b}$ , F. Palla$^{a}$ , F. Palmonari$^{a}$ , A. Rizzi$^{a}$$^{, }$$^{b}$ , A.T. Serban$^{a}$$^{, }$ 28, P. Spagnolo$^{a}$ , P. Squillacioti1, R. Tenchini$^{a}$ , G. Tonelli$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Venturi$^{a}$$^{, }$ 1, P.G.",
"Verdini$^{a}$ INFN Sezione di Roma $^{a}$ , Università di Roma \"La Sapienza\" $^{b}$ , Roma, Italy L. Barone$^{a}$$^{, }$$^{b}$ , F. Cavallari$^{a}$ , D. Del Re$^{a}$$^{, }$$^{b}$$^{, }$ 1, M. Diemoz$^{a}$ , C. Fanelli$^{a}$$^{, }$$^{b}$ , M. Grassi$^{a}$$^{, }$ 1, E. Longo$^{a}$$^{, }$$^{b}$ , P. Meridiani$^{a}$$^{, }$ 1, F. Micheli$^{a}$$^{, }$$^{b}$ , S. Nourbakhsh$^{a}$ , G. Organtini$^{a}$$^{, }$$^{b}$ , F. Pandolfi$^{a}$$^{, }$$^{b}$ , R. Paramatti$^{a}$ , S. Rahatlou$^{a}$$^{, }$$^{b}$ , M. Sigamani$^{a}$ , L. Soffi$^{a}$$^{, }$$^{b}$ INFN Sezione di Torino $^{a}$ , Università di Torino $^{b}$ , Università del Piemonte Orientale (Novara) $^{c}$ , Torino, Italy N. Amapane$^{a}$$^{, }$$^{b}$ , R. Arcidiacono$^{a}$$^{, }$$^{c}$ , S. Argiro$^{a}$$^{, }$$^{b}$ , M. Arneodo$^{a}$$^{, }$$^{c}$ , C. Biino$^{a}$ , C. Botta$^{a}$$^{, }$$^{b}$ , N. Cartiglia$^{a}$ , R. Castello$^{a}$$^{, }$$^{b}$ , M. Costa$^{a}$$^{, }$$^{b}$ , N. Demaria$^{a}$ , A. Graziano$^{a}$$^{, }$$^{b}$ , C. Mariotti$^{a}$$^{, }$ 1, S. Maselli$^{a}$ , E. Migliore$^{a}$$^{, }$$^{b}$ , V. Monaco$^{a}$$^{, }$$^{b}$ , M. Musich$^{a}$$^{, }$ 1, M.M.",
"Obertino$^{a}$$^{, }$$^{c}$ , N. Pastrone$^{a}$ , M. Pelliccioni$^{a}$ , A. Potenza$^{a}$$^{, }$$^{b}$ , A. Romero$^{a}$$^{, }$$^{b}$ , M. Ruspa$^{a}$$^{, }$$^{c}$ , R. Sacchi$^{a}$$^{, }$$^{b}$ , V. Sola$^{a}$$^{, }$$^{b}$ , A. Solano$^{a}$$^{, }$$^{b}$ , A. Staiano$^{a}$ , A. Vilela Pereira$^{a}$ INFN Sezione di Trieste $^{a}$ , Università di Trieste $^{b}$ , Trieste, Italy S. Belforte$^{a}$ , F. Cossutti$^{a}$ , G. Della Ricca$^{a}$$^{, }$$^{b}$ , B. Gobbo$^{a}$ , M. Marone$^{a}$$^{, }$$^{b}$$^{, }$ 1, D. Montanino$^{a}$$^{, }$$^{b}$$^{, }$ 1, A. Penzo$^{a}$ , A. Schizzi$^{a}$$^{, }$$^{b}$ Kangwon National University, Chunchon, Korea S.G. Heo, T.Y.",
"Kim, S.K.",
"Nam Kyungpook National University, Daegu, Korea S. Chang, J. Chung, D.H. Kim, G.N.",
"Kim, D.J.",
"Kong, H. Park, S.R.",
"Ro, D.C.",
"Son, T. Son Chonnam National University, Institute for Universe and Elementary Particles, Kwangju, Korea J.Y.",
"Kim, Zero J. Kim, S. Song Konkuk University, Seoul, Korea H.Y.",
"Jo Korea University, Seoul, Korea S. Choi, D. Gyun, B. Hong, M. Jo, H. Kim, T.J. Kim, K.S.",
"Lee, D.H.",
"Moon, S.K.",
"Park, E. Seo University of Seoul, Seoul, Korea M. Choi, S. Kang, H. Kim, J.H.",
"Kim, C. Park, I.C.",
"Park, S. Park, G. Ryu Sungkyunkwan University, Suwon, Korea Y. Cho, Y. Choi, Y.K.",
"Choi, J. Goh, M.S.",
"Kim, E. Kwon, B. Lee, J. Lee, S. Lee, H. Seo, I. Yu Vilnius University, Vilnius, Lithuania M.J. Bilinskas, I. Grigelionis, M. Janulis, A. Juodagalvis Centro de Investigacion y de Estudios Avanzados del IPN, Mexico City, Mexico H. Castilla-Valdez, E. De La Cruz-Burelo, I. Heredia-de La Cruz, R. Lopez-Fernandez, R. Magaña Villalba, J. Martínez-Ortega, A. Sánchez-Hernández, L.M.",
"Villasenor-Cendejas Universidad Iberoamericana, Mexico City, Mexico S. Carrillo Moreno, F. Vazquez Valencia Benemerita Universidad Autonoma de Puebla, Puebla, Mexico H.A.",
"Salazar Ibarguen Universidad Autónoma de San Luis Potosí, San Luis Potosí, Mexico E. Casimiro Linares, A. Morelos Pineda, M.A.",
"Reyes-Santos University of Auckland, Auckland, New Zealand D. Krofcheck University of Canterbury, Christchurch, New Zealand A.J.",
"Bell, P.H.",
"Butler, R. Doesburg, S. Reucroft, H. Silverwood National Centre for Physics, Quaid-I-Azam University, Islamabad, Pakistan M. Ahmad, M.I.",
"Asghar, H.R.",
"Hoorani, S. Khalid, W.A.",
"Khan, T. Khurshid, S. Qazi, M.A.",
"Shah, M. Shoaib Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Warsaw, Poland G. Brona, K. Bunkowski, M. Cwiok, W. Dominik, K. Doroba, A. Kalinowski, M. Konecki, J. Krolikowski Soltan Institute for Nuclear Studies, Warsaw, Poland H. Bialkowska, B. Boimska, T. Frueboes, R. Gokieli, M. Górski, M. Kazana, K. Nawrocki, K. Romanowska-Rybinska, M. Szleper, G. Wrochna, P. Zalewski Laboratório de Instrumentação e Física Experimental de Partículas, Lisboa, Portugal N. Almeida, P. Bargassa, A. David, P. Faccioli, P.G.",
"Ferreira Parracho, M. Gallinaro, P. Musella, J. Seixas, J. Varela, P. Vischia Joint Institute for Nuclear Research, Dubna, Russia I. Belotelov, P. Bunin, I. Golutvin, I. Gorbunov, A. Kamenev, V. Karjavin, G. Kozlov, A. Lanev, A. Malakhov, P. Moisenz, V. Palichik, V. Perelygin, M. Savina, S. Shmatov, V. Smirnov, A. Volodko, A. Zarubin Petersburg Nuclear Physics Institute, Gatchina (St Petersburg), Russia S. Evstyukhin, V. Golovtsov, Y. Ivanov, V. Kim, P. Levchenko, V. Murzin, V. Oreshkin, I. Smirnov, V. Sulimov, L. Uvarov, S. Vavilov, A. Vorobyev, An.",
"Vorobyev Institute for Nuclear Research, Moscow, Russia Yu.",
"Andreev, A. Dermenev, S. Gninenko, N. Golubev, M. Kirsanov, N. Krasnikov, V. Matveev, A. Pashenkov, D. Tlisov, A. Toropin Institute for Theoretical and Experimental Physics, Moscow, Russia V. Epshteyn, M. Erofeeva, V. Gavrilov, M. Kossov1, N. Lychkovskaya, V. Popov, G. Safronov, S. Semenov, V. Stolin, E. Vlasov, A. Zhokin Moscow State University, Moscow, Russia A. Belyaev, E. Boos, M. Dubinin4, L. Dudko, A. Ershov, A. Gribushin, V. Klyukhin, O. Kodolova, I. Lokhtin, A. Markina, S. Obraztsov, M. Perfilov, S. Petrushanko, L. Sarycheva$^{\\textrm {\\dag }}$ , V. Savrin, A. Snigirev P.N.",
"Lebedev Physical Institute, Moscow, Russia V. Andreev, M. Azarkin, I. Dremin, M. Kirakosyan, A. Leonidov, G. Mesyats, S.V.",
"Rusakov, A. Vinogradov State Research Center of Russian Federation, Institute for High Energy Physics, Protvino, Russia I. Azhgirey, I. Bayshev, S. Bitioukov, V. Grishin1, V. Kachanov, D. Konstantinov, A. Korablev, V. Krychkine, V. Petrov, R. Ryutin, A. Sobol, L. Tourtchanovitch, S. Troshin, N. Tyurin, A. Uzunian, A. Volkov University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia P. Adzic29, M. Djordjevic, M. Ekmedzic, D. Krpic29, J. Milosevic Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT), Madrid, Spain M. Aguilar-Benitez, J. Alcaraz Maestre, P. Arce, C. Battilana, E. Calvo, M. Cerrada, M. Chamizo Llatas, N. Colino, B.",
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"Josa, G. Merino, J. Puerta Pelayo, I. Redondo, L. Romero, J. Santaolalla, M.S.",
"Soares, C. Willmott Universidad Autónoma de Madrid, Madrid, Spain C. Albajar, G. Codispoti, J.F.",
"de Trocóniz Universidad de Oviedo, Oviedo, Spain J. Cuevas, J. Fernandez Menendez, S. Folgueras, I. Gonzalez Caballero, L. Lloret Iglesias, J. Piedra Gomez30, J.M.",
"Vizan Garcia Instituto de Física de Cantabria (IFCA), CSIC-Universidad de Cantabria, Santander, Spain J.A.",
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"Rodríguez-Marrero, A. Ruiz-Jimeno, L. Scodellaro, M. Sobron Sanudo, I. Vila, R. Vilar Cortabitarte CERN, European Organization for Nuclear Research, Geneva, Switzerland D. Abbaneo, E. Auffray, G. Auzinger, P. Baillon, A.H. Ball, D. Barney, C. Bernet5, G. Bianchi, P. Bloch, A. Bocci, A. Bonato, H. Breuker, T. Camporesi, G. Cerminara, T. Christiansen, J.A.",
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"Mozer, M. Mulders, E. Nesvold, M. Nguyen, T. Orimoto, L. Orsini, E. Palencia Cortezon, E. Perez, A. Petrilli, A. Pfeiffer, M. Pierini, M. Pimiä, D. Piparo, G. Polese, L. Quertenmont, A. Racz, W. Reece, J. Rodrigues Antunes, G. Rolandi32, T. Rommerskirchen, C. Rovelli33, M. Rovere, H. Sakulin, F. Santanastasio, C. Schäfer, C. Schwick, I. Segoni, S. Sekmen, A. Sharma, P. Siegrist, P. Silva, M. Simon, P. Sphicas34, D. Spiga, M. Spiropulu4, M. Stoye, A. Tsirou, G.I.",
"Veres16, J.R. Vlimant, H.K.",
"Wöhri, S.D.",
"Worm35, W.D.",
"Zeuner Paul Scherrer Institut, Villigen, Switzerland W. Bertl, K. Deiters, W. Erdmann, K. Gabathuler, R. Horisberger, Q. Ingram, H.C. Kaestli, S. König, D. Kotlinski, U. Langenegger, F. Meier, D. Renker, T. Rohe, J. Sibille36 Institute for Particle Physics, ETH Zurich, Zurich, Switzerland L. Bäni, P. Bortignon, M.A.",
"Buchmann, B. Casal, N. Chanon, Z. Chen, A. Deisher, G. Dissertori, M. Dittmar, M. Dünser, J. Eugster, K. Freudenreich, C. Grab, P. Lecomte, W. Lustermann, A.C. Marini, P. Martinez Ruiz del Arbol, N. Mohr, F. Moortgat, C. Nägeli37, P. Nef, F. Nessi-Tedaldi, L. Pape, F. Pauss, M. Peruzzi, F.J. Ronga, M. Rossini, L. Sala, A.K.",
"Sanchez, A. Starodumov38, B. Stieger, M. Takahashi, L. Tauscher$^{\\textrm {\\dag }}$ , A. Thea, K. Theofilatos, D. Treille, C. Urscheler, R. Wallny, H.A.",
"Weber, L. Wehrli Universität Zürich, Zurich, Switzerland E. Aguilo, C. Amsler, V. Chiochia, S. De Visscher, C. Favaro, M. Ivova Rikova, B. Millan Mejias, P. Otiougova, P. Robmann, H. Snoek, S. Tupputi, M. Verzetti National Central University, Chung-Li, Taiwan Y.H.",
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"Liu, Y.J.",
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"Singh, R. Volpe, S.S. Yu National Taiwan University (NTU), Taipei, Taiwan P. Bartalini, P. Chang, Y.H.",
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"Chen, C. Dietz, U. Grundler, W.-S. Hou, Y. Hsiung, K.Y.",
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"Shiu, Y.M.",
"Tzeng, M. Wang Cukurova University, Adana, Turkey A. Adiguzel, M.N.",
"Bakirci39, S. Cerci40, C. Dozen, I. Dumanoglu, E. Eskut, S. Girgis, G. Gokbulut, I. Hos, E.E.",
"Kangal, G. Karapinar, A. Kayis Topaksu, G. Onengut, K. Ozdemir, S. Ozturk41, A. Polatoz, K. Sogut42, D. Sunar Cerci40, B. Tali40, H. Topakli39, L.N.",
"Vergili, M. Vergili Middle East Technical University, Physics Department, Ankara, Turkey I.V.",
"Akin, T. Aliev, B. Bilin, S. Bilmis, M. Deniz, H. Gamsizkan, A.M. Guler, K. Ocalan, A. Ozpineci, M. Serin, R. Sever, U.E.",
"Surat, M. Yalvac, E. Yildirim, M. Zeyrek Bogazici University, Istanbul, Turkey M. Deliomeroglu, E. Gülmez, B. Isildak, M. Kaya43, O. Kaya43, S. Ozkorucuklu44, N. Sonmez45 Istanbul Technical University, Istanbul, Turkey K. Cankocak National Scientific Center, Kharkov Institute of Physics and Technology, Kharkov, Ukraine L. Levchuk University of Bristol, Bristol, United Kingdom F. Bostock, J.J. Brooke, E. Clement, D. Cussans, H. Flacher, R. Frazier, J. Goldstein, M. Grimes, G.P.",
"Heath, H.F. Heath, L. Kreczko, S. Metson, D.M.",
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"Poll, S. Senkin, V.J.",
"Smith, T. Williams Rutherford Appleton Laboratory, Didcot, United Kingdom L. Basso46, K.W.",
"Bell, A. Belyaev46, C. Brew, R.M.",
"Brown, D.J.A.",
"Cockerill, J.A.",
"Coughlan, K. Harder, S. Harper, J. Jackson, B.W.",
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"Radburn-Smith, C.H.",
"Shepherd-Themistocleous, I.R.",
"Tomalin, W.J.",
"Womersley Imperial College, London, United Kingdom R. Bainbridge, G. Ball, R. Beuselinck, O. Buchmuller, D. Colling, N. Cripps, M. Cutajar, P. Dauncey, G. Davies, M. Della Negra, W. Ferguson, J. Fulcher, D. Futyan, A. Gilbert, A. Guneratne Bryer, G. Hall, Z. Hatherell, J. Hays, G. Iles, M. Jarvis, G. Karapostoli, L. Lyons, A.-M. Magnan, J. Marrouche, B. Mathias, R. Nandi, J. Nash, A. Nikitenko38, A. Papageorgiou, J. Pela1, M. Pesaresi, K. Petridis, M. Pioppi47, D.M.",
"Raymond, S. Rogerson, N. Rompotis, A.",
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"Cole, P.R.",
"Hobson, A. Khan, P. Kyberd, D. Leggat, D. Leslie, W. Martin, I.D.",
"Reid, P. Symonds, L. Teodorescu, M. Turner Baylor University, Waco, USA K. Hatakeyama, H. Liu, T. Scarborough The University of Alabama, Tuscaloosa, USA C. Henderson, P. Rumerio Boston University, Boston, USA A. Avetisyan, T. Bose, C. Fantasia, A. Heister, J. St. John, P. Lawson, D. Lazic, J. Rohlf, D. Sperka, L. Sulak Brown University, Providence, USA J. Alimena, S. Bhattacharya, D. Cutts, A. Ferapontov, U. Heintz, S. Jabeen, G. Kukartsev, G. Landsberg, M. Luk, M. Narain, D. Nguyen, M. Segala, T. Sinthuprasith, T. Speer, K.V.",
"Tsang University of California, Davis, Davis, USA R. Breedon, G. Breto, M. Calderon De La Barca Sanchez, S. Chauhan, M. Chertok, J. Conway, R. Conway, P.T.",
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"Mall, T. Miceli, R. Nelson, D. Pellett, B. Rutherford, M. Searle, J. Smith, M. Squires, M. Tripathi, R. Vasquez Sierra University of California, Los Angeles, Los Angeles, USA V. Andreev, D. Cline, R. Cousins, J. Duris, S. Erhan, P. Everaerts, C. Farrell, J. Hauser, M. Ignatenko, C. Plager, G. Rakness, P. Schlein$^{\\textrm {\\dag }}$ , J. Tucker, V. Valuev, M. Weber University of California, Riverside, Riverside, USA J. Babb, R. Clare, M.E.",
"Dinardo, J. Ellison, J.W.",
"Gary, F. Giordano, G. Hanson, G.Y.",
"Jeng48, H. Liu, O.R.",
"Long, A. Luthra, H. Nguyen, S. Paramesvaran, J. Sturdy, S. Sumowidagdo, R. Wilken, S. Wimpenny University of California, San Diego, La Jolla, USA W. Andrews, J.G.",
"Branson, G.B.",
"Cerati, S. Cittolin, D. Evans, F. Golf, A. Holzner, R. Kelley, M. Lebourgeois, J. Letts, I. Macneill, B. Mangano, J. Muelmenstaedt, S. Padhi, C. Palmer, G. Petrucciani, M. Pieri, R. Ranieri, M. Sani, V. Sharma, S. Simon, E. Sudano, M. Tadel, Y. Tu, A. Vartak, S. Wasserbaech49, F. Würthwein, A. Yagil, J. Yoo University of California, Santa Barbara, Santa Barbara, USA D. Barge, R. Bellan, C. Campagnari, M. D'Alfonso, T. Danielson, K. Flowers, P. Geffert, J. Incandela, C. Justus, P. Kalavase, S.A. Koay, D. Kovalskyi1, V. Krutelyov, S. Lowette, N. Mccoll, V. Pavlunin, F. Rebassoo, J. Ribnik, J. Richman, R. Rossin, D. Stuart, W. To, C. West California Institute of Technology, Pasadena, USA A. Apresyan, A. Bornheim, Y. Chen, E. Di Marco, J. Duarte, M. Gataullin, Y. Ma, A. Mott, H.B.",
"Newman, C. Rogan, V. Timciuc, P. Traczyk, J. Veverka, R. Wilkinson, Y. Yang, R.Y.",
"Zhu Carnegie Mellon University, Pittsburgh, USA B. Akgun, R. Carroll, T. Ferguson, Y. Iiyama, D.W. Jang, Y.F.",
"Liu, M. Paulini, H. Vogel, I. Vorobiev University of Colorado at Boulder, Boulder, USA J.P. Cumalat, B.R.",
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"Wagner Cornell University, Ithaca, USA L. Agostino, J. Alexander, A. Chatterjee, N. Eggert, L.K.",
"Gibbons, B. Heltsley, W. Hopkins, A. Khukhunaishvili, B. Kreis, N. Mirman, G. Nicolas Kaufman, J.R. Patterson, A. Ryd, E. Salvati, W. Sun, W.D.",
"Teo, J. Thom, J. Thompson, J. Vaughan, Y. Weng, L. Winstrom, P. Wittich Fairfield University, Fairfield, USA D. Winn Fermi National Accelerator Laboratory, Batavia, USA S. Abdullin, M. Albrow, J. Anderson, L.A.T.",
"Bauerdick, A. Beretvas, J. Berryhill, P.C.",
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"Marraffino, S. Maruyama, D. Mason, P. McBride, K. Mishra, S. Mrenna, Y. Musienko50, C. Newman-Holmes, V. O'Dell, O. Prokofyev, E. Sexton-Kennedy, S. Sharma, W.J.",
"Spalding, L. Spiegel, P. Tan, L. Taylor, S. Tkaczyk, N.V. Tran, L. Uplegger, E.W.",
"Vaandering, R. Vidal, J. Whitmore, W. Wu, F. Yang, F. Yumiceva, J.C. Yun University of Florida, Gainesville, USA D. Acosta, P. Avery, D. Bourilkov, M. Chen, S. Das, M. De Gruttola, G.P.",
"Di Giovanni, D. Dobur, A. Drozdetskiy, R.D.",
"Field, M. Fisher, Y. Fu, I.K.",
"Furic, J. Gartner, J. Hugon, B. Kim, J. Konigsberg, A. Korytov, A. Kropivnitskaya, T. Kypreos, J.F.",
"Low, K. Matchev, P. Milenovic51, G. Mitselmakher, L. Muniz, R. Remington, A. Rinkevicius, P. Sellers, N. Skhirtladze, M. Snowball, J. Yelton, M. Zakaria Florida International University, Miami, USA V. Gaultney, L.M.",
"Lebolo, S. Linn, P. Markowitz, G. Martinez, J.L.",
"Rodriguez Florida State University, Tallahassee, USA T. Adams, A. Askew, J. Bochenek, J. Chen, B. Diamond, S.V.",
"Gleyzer, J. Haas, S. Hagopian, V. Hagopian, M. Jenkins, K.F.",
"Johnson, H. Prosper, V. Veeraraghavan, M. Weinberg Florida Institute of Technology, Melbourne, USA M.M.",
"Baarmand, B. Dorney, M. Hohlmann, H. Kalakhety, I. Vodopiyanov University of Illinois at Chicago (UIC), Chicago, USA M.R.",
"Adams, I.M.",
"Anghel, L. Apanasevich, Y. Bai, V.E.",
"Bazterra, R.R.",
"Betts, J. Callner, R. Cavanaugh, C. Dragoiu, O. Evdokimov, E.J.",
"Garcia-Solis, L. Gauthier, C.E.",
"Gerber, D.J.",
"Hofman, S. Khalatyan, F. Lacroix, M. Malek, C. O'Brien, C. Silkworth, D. Strom, N. Varelas The University of Iowa, Iowa City, USA U. Akgun, E.A.",
"Albayrak, B. Bilki52, K. Chung, W. Clarida, F. Duru, S. Griffiths, C.K.",
"Lae, J.-P. Merlo, H. Mermerkaya53, A. Mestvirishvili, A. Moeller, J. Nachtman, C.R.",
"Newsom, E. Norbeck, J. Olson, Y. Onel, F. Ozok, S. Sen, E. Tiras, J. Wetzel, T. Yetkin, K. Yi Johns Hopkins University, Baltimore, USA B.A.",
"Barnett, B. Blumenfeld, S. Bolognesi, D. Fehling, G. Giurgiu, A.V.",
"Gritsan, Z.J.",
"Guo, G. Hu, P. Maksimovic, S. Rappoccio, M. Swartz, A. Whitbeck The University of Kansas, Lawrence, USA P. Baringer, A. Bean, G. Benelli, O. Grachov, R.P.",
"Kenny Iii, M. Murray, D. Noonan, V. Radicci, S. Sanders, R. Stringer, G. Tinti, J.S.",
"Wood, V. Zhukova Kansas State University, Manhattan, USA A.F.",
"Barfuss, T. Bolton, I. Chakaberia, A. Ivanov, S. Khalil, M. Makouski, Y. Maravin, S. Shrestha, I. Svintradze Lawrence Livermore National Laboratory, Livermore, USA J. Gronberg, D. Lange, D. Wright University of Maryland, College Park, USA A. Baden, M. Boutemeur, B. Calvert, S.C. Eno, J.A.",
"Gomez, N.J. Hadley, R.G.",
"Kellogg, M. Kirn, T. Kolberg, Y. Lu, M. Marionneau, A.C. Mignerey, A. Peterman, K. Rossato, A. Skuja, J. Temple, M.B.",
"Tonjes, S.C. Tonwar, E. Twedt Massachusetts Institute of Technology, Cambridge, USA G. Bauer, J. Bendavid, W. Busza, E. Butz, I.A.",
"Cali, M. Chan, V. Dutta, G. Gomez Ceballos, M. Goncharov, K.A.",
"Hahn, Y. Kim, M. Klute, Y.-J.",
"Lee, W. Li, P.D.",
"Luckey, T. Ma, S. Nahn, C. Paus, D. Ralph, C. Roland, G. Roland, M. Rudolph, G.S.F.",
"Stephans, F. Stöckli, K. Sumorok, K. Sung, D. Velicanu, E.A.",
"Wenger, R. Wolf, B. Wyslouch, S. Xie, M. Yang, Y. Yilmaz, A.S. Yoon, M. Zanetti University of Minnesota, Minneapolis, USA S.I.",
"Cooper, P. Cushman, B. Dahmes, A.",
"De Benedetti, G. Franzoni, A. Gude, J. Haupt, S.C. Kao, K. Klapoetke, Y. Kubota, J. Mans, N. Pastika, R. Rusack, M. Sasseville, A. Singovsky, N. Tambe, J. Turkewitz University of Mississippi, University, USA L.M.",
"Cremaldi, R. Kroeger, L. Perera, R. Rahmat, D.A.",
"Sanders University of Nebraska-Lincoln, Lincoln, USA E. Avdeeva, K. Bloom, S. Bose, J.",
"Butt, D.R.",
"Claes, A. Dominguez, M. Eads, P. Jindal, J. Keller, I. Kravchenko, J. Lazo-Flores, H. Malbouisson, S. Malik, G.R.",
"Snow State University of New York at Buffalo, Buffalo, USA U. Baur, A. Godshalk, I. Iashvili, S. Jain, A. Kharchilava, A. Kumar, S.P.",
"Shipkowski, K. Smith Northeastern University, Boston, USA G. Alverson, E. Barberis, D. Baumgartel, M. Chasco, J. Haley, D. Trocino, D. Wood, J. Zhang Northwestern University, Evanston, USA A. Anastassov, A. Kubik, N. Mucia, N. Odell, R.A. Ofierzynski, B. Pollack, A. Pozdnyakov, M. Schmitt, S. Stoynev, M. Velasco, S. Won University of Notre Dame, Notre Dame, USA L. Antonelli, D. Berry, A. Brinkerhoff, M. Hildreth, C. Jessop, D.J.",
"Karmgard, J. Kolb, K. Lannon, W. Luo, S. Lynch, N. Marinelli, D.M.",
"Morse, T. Pearson, R. Ruchti, J. Slaunwhite, N. Valls, J. Warchol, M. Wayne, M. Wolf, J. Ziegler The Ohio State University, Columbus, USA B. Bylsma, L.S.",
"Durkin, C. Hill, R. Hughes, P. Killewald, K. Kotov, T.Y.",
"Ling, D. Puigh, M. Rodenburg, C. Vuosalo, G. Williams, B.L.",
"Winer Princeton University, Princeton, USA N. Adam, E. Berry, P. Elmer, D. Gerbaudo, V. Halyo, P. Hebda, J. Hegeman, A.",
"Hunt, E. Laird, D. Lopes Pegna, P. Lujan, D. Marlow, T. Medvedeva, M. Mooney, J. Olsen, P. Piroué, X. Quan, A. Raval, H. Saka, D. Stickland, C. Tully, J.S.",
"Werner, A. Zuranski University of Puerto Rico, Mayaguez, USA J.G.",
"Acosta, X.T.",
"Huang, A. Lopez, H. Mendez, S. Oliveros, J.E.",
"Ramirez Vargas, A. Zatserklyaniy Purdue University, West Lafayette, USA E. Alagoz, V.E.",
"Barnes, D. Benedetti, G. Bolla, D. Bortoletto, M. De Mattia, A. Everett, Z. Hu, M. Jones, O. Koybasi, M. Kress, A.T. Laasanen, N. Leonardo, V. Maroussov, P. Merkel, D.H. Miller, N. Neumeister, I. Shipsey, D. Silvers, A. Svyatkovskiy, M. Vidal Marono, H.D.",
"Yoo, J. Zablocki, Y. Zheng Purdue University Calumet, Hammond, USA S. Guragain, N. Parashar Rice University, Houston, USA A. Adair, C. Boulahouache, V. Cuplov, K.M.",
"Ecklund, F.J.M.",
"Geurts, B.P.",
"Padley, R. Redjimi, J. Roberts, J. Zabel University of Rochester, Rochester, USA B. Betchart, A. Bodek, Y.S.",
"Chung, R. Covarelli, P. de Barbaro, R. Demina, Y. Eshaq, A. Garcia-Bellido, P. Goldenzweig, Y. Gotra, J. Han, A. Harel, S. Korjenevski, D.C.",
"Miner, D. Vishnevskiy, M. Zielinski The Rockefeller University, New York, USA A. Bhatti, R. Ciesielski, L. Demortier, K. Goulianos, G. Lungu, S. Malik, C. Mesropian Rutgers, the State University of New Jersey, Piscataway, USA S. Arora, A. Barker, J.P. Chou, C. Contreras-Campana, E. Contreras-Campana, D. Duggan, D. Ferencek, Y. Gershtein, R. Gray, E. Halkiadakis, D. Hidas, D. Hits, A. Lath, S. Panwalkar, M. Park, R. Patel, V. Rekovic, A. Richards, J. Robles, K. Rose, S. Salur, S. Schnetzer, C. Seitz, S. Somalwar, R. Stone, S. Thomas University of Tennessee, Knoxville, USA G. Cerizza, M. Hollingsworth, S. Spanier, Z.C.",
"Yang, A. York Texas A&M University, College Station, USA R. Eusebi, W. Flanagan, J. Gilmore, T. Kamon54, V. Khotilovich, R. Montalvo, I. Osipenkov, Y. Pakhotin, A. Perloff, J. Roe, A. Safonov, T. Sakuma, S. Sengupta, I. Suarez, A. Tatarinov, D. Toback Texas Tech University, Lubbock, USA N. Akchurin, J. Damgov, P.R.",
"Dudero, C. Jeong, K. Kovitanggoon, S.W.",
"Lee, T. Libeiro, Y. Roh, I. Volobouev Vanderbilt University, Nashville, USA E. Appelt, D. Engh, C. Florez, S. Greene, A. Gurrola, W. Johns, P. Kurt, C. Maguire, A. Melo, P. Sheldon, B. Snook, S. Tuo, J. Velkovska University of Virginia, Charlottesville, USA M.W.",
"Arenton, M. Balazs, S. Boutle, B. Cox, B. Francis, J. Goodell, R. Hirosky, A. Ledovskoy, C. Lin, C. Neu, J.",
"Wood, R. Yohay Wayne State University, Detroit, USA S. Gollapinni, R. Harr, P.E.",
"Karchin, C. Kottachchi Kankanamge Don, P. Lamichhane, A. Sakharov University of Wisconsin, Madison, USA M. Anderson, M. Bachtis, D. Belknap, L. Borrello, D. Carlsmith, M. Cepeda, S. Dasu, L. Gray, K.S.",
"Grogg, M. Grothe, R. Hall-Wilton, M. Herndon, A. Hervé, P. Klabbers, J. Klukas, A. Lanaro, C. Lazaridis, J. Leonard, R. Loveless, A. Mohapatra, I. Ojalvo, G.A.",
"Pierro, I. Ross, A. Savin, W.H.",
"Smith, J. Swanson †: Deceased 1: Also at CERN, European Organization for Nuclear Research, Geneva, Switzerland 2: Also at National Institute of Chemical Physics and Biophysics, Tallinn, Estonia 3: Also at Universidade Federal do ABC, Santo Andre, Brazil 4: Also at California Institute of Technology, Pasadena, USA 5: Also at Laboratoire Leprince-Ringuet, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France 6: Also at Suez Canal University, Suez, Egypt 7: Also at Cairo University, Cairo, Egypt 8: Also at British University, Cairo, Egypt 9: Also at Fayoum University, El-Fayoum, Egypt 10: Now at Ain Shams University, Cairo, Egypt 11: Also at Soltan Institute for Nuclear Studies, Warsaw, Poland 12: Also at Université de Haute-Alsace, Mulhouse, France 13: Also at Moscow State University, Moscow, Russia 14: Also at Brandenburg University of Technology, Cottbus, Germany 15: Also at Institute of Nuclear Research ATOMKI, Debrecen, Hungary 16: Also at Eötvös Loránd University, Budapest, Hungary 17: Also at Tata Institute of Fundamental Research - HECR, Mumbai, India 18: Now at King Abdulaziz University, Jeddah, Saudi Arabia 19: Also at University of Visva-Bharati, Santiniketan, India 20: Also at Sharif University of Technology, Tehran, Iran 21: Also at Isfahan University of Technology, Isfahan, Iran 22: Also at Shiraz University, Shiraz, Iran 23: Also at Plasma Physics Research Center, Science and Research Branch, Islamic Azad University, Teheran, Iran 24: Also at Facoltà Ingegneria Università di Roma, Roma, Italy 25: Also at Università della Basilicata, Potenza, Italy 26: Also at Università degli Studi Guglielmo Marconi, Roma, Italy 27: Also at Università degli studi di Siena, Siena, Italy 28: Also at University of Bucharest, Bucuresti-Magurele, Romania 29: Also at Faculty of Physics of University of Belgrade, Belgrade, Serbia 30: Also at University of Florida, Gainesville, USA 31: Also at University of California, Los Angeles, Los Angeles, USA 32: Also at Scuola Normale e Sezione dell' INFN, Pisa, Italy 33: Also at INFN Sezione di Roma; Università di Roma \"La Sapienza\", Roma, Italy 34: Also at University of Athens, Athens, Greece 35: Also at Rutherford Appleton Laboratory, Didcot, United Kingdom 36: Also at The University of Kansas, Lawrence, USA 37: Also at Paul Scherrer Institut, Villigen, Switzerland 38: Also at Institute for Theoretical and Experimental Physics, Moscow, Russia 39: Also at Gaziosmanpasa University, Tokat, Turkey 40: Also at Adiyaman University, Adiyaman, Turkey 41: Also at The University of Iowa, Iowa City, USA 42: Also at Mersin University, Mersin, Turkey 43: Also at Kafkas University, Kars, Turkey 44: Also at Suleyman Demirel University, Isparta, Turkey 45: Also at Ege University, Izmir, Turkey 46: Also at School of Physics and Astronomy, University of Southampton, Southampton, United Kingdom 47: Also at INFN Sezione di Perugia; Università di Perugia, Perugia, Italy 48: Also at University of Sydney, Sydney, Australia 49: Also at Utah Valley University, Orem, USA 50: Also at Institute for Nuclear Research, Moscow, Russia 51: Also at University of Belgrade, Faculty of Physics and Vinca Institute of Nuclear Sciences, Belgrade, Serbia 52: Also at Argonne National Laboratory, Argonne, USA 53: Also at Erzincan University, Erzincan, Turkey 54: Also at Kyungpook National University, Daegu, Korea"
]
] | 1204.1088 |
[
[
"Diffractive production of c \\bar c"
],
[
"Abstract At high-energies the gluon-gluon fusion is the dominant mechanism of c \\bar c production.",
"This process was calculated in the NLO collinear as well as in the kt-factorization approaches in the past.",
"In this presentation we concentrate on production of c \\bar c pairs including several subleading mechanisms.",
"In this context we use MRST-QED parton distributions which include photon as a parton in the proton as well as elastic photon distributions calculated in the equivalent photon approximation.",
"We present distributions in the c quark (c antiquark) rapidity and transverse momenta and compare them to the dominant gluon-gluon fusion contribution.valent photon approximation.",
"We discuss also inclusive single and central diffractive processes using diffractive parton distribution found from the analysis of HERA diffractive data.",
"As in the previous case we present distribution in c \\bar c rapidity and transverse momentum.",
"Next we present results for exclusive central diffractive mechanism discussed recently in the literature.",
"We show corresponding differential distributions and compare them with corresponding distributions for single and central diffractive components.",
"Finally we discuss production of two pairs of c \\bar c within a simple formalism of double-parton scattering (DPS).",
"Surprisingly very large cross sections, comparable to single-parton scattering (SPS) contribution, are predicted for LHC energies."
],
[
"Introduction",
"In this presentation we discuss contributions of some subleading mechanisms neglected in the analysis of $c \\bar{c}$ production.",
"We include contributions of photon-gluon (gluon-photon) as well as purely electromagnetic photon-photon fusion.",
"Here we present only some selective results.",
"The formalism and more details has been shown and discussed elsewhere [1].",
"We discuss also diffractive processes (single and central) in the framework of Ingelman-Schlein model corrected for absorption.",
"Such a model was used in the estimation of several diffractive processes [2], [3], [4], [5].",
"The absorption corrections are necessary to understand a huge Regge-factorization breaking observed in single and central production at Tevatron."
],
[
"Production of heavy quarks",
"The cross section for the $c \\bar{c}$ production, assuming gluon-gluon fusion, was calculated both in collinear and $k_t$ factorization approaches.",
"Our group has done detailed calculations in the second approach (see e.g.",
"[6], [7]).",
"In the leading-order (LO) approximation within the $k_t$ -factorization approach the quadruply differential cross section in the rapidity of $Q$ ($y_1$ ), in the rapidity of $\\bar{Q}$ ($y_2$ ) and in the transverse momentum of $Q$ ($p_{1,t}$ ) and $Q$ ($p_{2,t}$ ) can be written as [6], [7] $\\nonumber \\frac{d \\sigma }{d y_1 d y_2 d^2p_{1,t} d^2p_{2,t}} =\\sum _{i,j} \\; \\int \\frac{d^2 \\kappa _{1,t}}{\\pi } \\frac{d^2 \\kappa _{2,t}}{\\pi }\\frac{1}{16 \\pi ^2 (x_1 x_2 s)^2} \\;\\\\\\nonumber \\overline{ | {\\cal M}_{ij \\rightarrow Q \\bar{Q}} |^2}\\delta ^{2} \\left( \\vec{\\kappa }_{1,t} + \\vec{\\kappa }_{2,t}- \\vec{p}_{1,t} - \\vec{p}_{2,t} \\right) \\;\\\\\\nonumber {\\cal F}_i(x_1,\\kappa _{1,t}^2) \\; {\\cal F}_j(x_2,\\kappa _{2,t}^2) \\; ,$ where ${\\cal F}_i(x_1,\\kappa _{1,t}^2)$ and ${\\cal F}_j(x_2,\\kappa _{2,t}^2)$ are the so-called unintegrated gluon (parton) distributions.",
"The unintegrated parton distributions must be evaluated at: $x_1 &=& \\frac{m_{1,t}}{\\sqrt{s}}\\exp ( y_1)+ \\frac{m_{2,t}}{\\sqrt{s}}\\exp ( y_2),\\nonumber \\\\x_2 &=& \\frac{m_{1,t}}{\\sqrt{s}}\\exp (-y_1)+ \\frac{m_{2,t}}{\\sqrt{s}}\\exp (-y_2),\\nonumber $ where $m_{i,t} = \\sqrt{p_{i,t}^2 + m_Q^2}$ ."
],
[
"Photon induced production of heavy quarks",
"The dominant contributions of heavy quark-antiquark production are initiated by gluon-gluon fusion or quark-antiquark annihilation.",
"In general, even photon can be a constituent of the proton.",
"This idea was considered by Martin, Roberts, Stirling and Thorne in Ref.[9].",
"If the photon is a constituent of the nucleon then other mechanisms of $c \\bar{c}$ production presented in Fig.REF are possible.",
"Figure: Diagrams representing photon-induced mechanismsof heavy quark production."
],
[
"Gluon-gluon fusion",
"Before we go to the new mechanisms we will present results for the dominant gluon-gluon fusion.",
"In Fig.REF we show distributions in transverse momentum of $c$ (or $\\bar{c}$ ) for the gluon-gluon fusion mechanism for different popular choices of scales ($\\mu ^2 = 4 m_c^2, M_{c \\bar{c}}^2,p_t^2 + m_c^2$ ).",
"We show our results for $\\sqrt{s}$ = 500 GeV (left panel) and $\\sqrt{s}$ = 14 TeV (right panel).",
"In this calculation we have used GRV [8] PDFs.",
"The figure shows typical uncertainties due to the choice of the scale.",
"We wish to stress here that at the higher energies the results of the calculations depend on the gluon distributions at small values of $x$ .",
"Figure: Distribution in quark/antiquarktransverse momentum at s\\sqrt{s} = 500 GeV (left panel) and fors\\sqrt{s} = 14 TeV (right panel) for different choices ofscales and for GRV gluon distribution."
],
[
"$\\gamma g$ and {{formula:d5ce58fc-43c5-4212-b7aa-bf6e41a68079}} subprocesses",
"In Fig.REF we show transverse momentum distributions for the dominant gluon-gluon as well as for the subleading photon-gluon (gluon-photon) and photon-photon components for different gluon distribution functions [8], [9], [10] for the RHIC energy $\\sqrt{s}$ = 500 GeV and for the nominal LHC energy $\\sqrt{s}$ = 14 TeV, respectively.",
"At the LHC energy the results for different GPDs differ considerably which is a consequence of the poorly known small-x region.",
"The differences at the energy $\\sqrt{s}$ = 14 TeV are particularly large which can be explained by the fact that a product of gluon distributions (both at small x) enters the cross section formula.",
"New measurement of $c \\bar{c}$ at the nominal LHC energy will be therefore a severe test of gluon distributions at small $x$ and not too high factorization scales not tested so far.",
"Similar uncertainties for the $\\gamma g$ and $g \\gamma $ are smaller as here only one gluon distribution appears in the corresponding cross section formula.",
"The uncertainties for the photon distributions are not yet quantified.",
"Figure: Transverse momentum distribution for the standard gluon-gluonmixed gluon-photon and photon-gluon as well as for photon-photoncontributions for RHIC (left) and LHC (right).It is very difficult to quantify uncertainties related to photon PDFs as only one set of PDFs includes photon as a parton of the proton.",
"Here the isospin symmetry violation (not well known at present) would be an useful limitation.",
"Our collection of the results for the photon induced mechanisms show that they are rather small and their identification would be rather difficult as the different distributions are very similar to those for the gluon-gluon fusion.",
"Our intension is to document all the subleading terms.",
"Our etimation shows that the sum of all the photon induced terms is less than 0.5 % and is by almost 2 orders of magnitude smaller than the uncertainties of the dominant leading-order gluon-gluon term."
],
[
"Formalism",
"The mechanisms of the diffractive production of heavy quarks ($c \\bar{c}$ ) are shown in Figs.REF , REF .",
"The formalism how to calculate respective cross section has been presented elsewhere [1].",
"Figure: The mechanism of single-diffractive production of cc ¯c \\bar{c}.Figure: The mechanism of central-diffractive production of cc ¯c \\bar{c}."
],
[
"Results",
"In Fig.REF we show transverse momentum distributions of charm quarks (or antiquarks).",
"The distribution for single diffractive component is smaller than that for the inclusive gluon-gluon fusion by almost 2 orders of magnitude.",
"Our results include gap survival factor [1].",
"The cross section for central diffractive component is smaller by additional order of magnitude.",
"Figure: Transverse momentum distribution of cc quarks (antiquarks)for RHIC energy s=\\sqrt{s} = 500 GeV (left panel) and for LHC energys\\sqrt{s} = 14 TeV (right panel) for the GRV94 gluon distributions.The result for single diffractive (0d or d0), central diffractive (dd)mechanisms are compared with that for the standard gluon-gluon fusion (00).In Fig.REF we show distributions in quark (antiquark) rapidity.",
"We show separately contributions of two different single-diffractive components (which give the same distributions in transverse momentum) and the contribution of central-diffractive component in Fig.REF .",
"When added together the single-diffractive components produce a distribution in rapidity similar in shape to that for the standard inclusive case.",
"Figure: Rapidity distribution of cc quarks (antiquarks)for RHIC energy s=\\sqrt{s} = 500 GeV (left panel) andfor LHC energy s\\sqrt{s} = 14 TeV (right panel) for the GRV94 gluondistributions.",
"The result for single diffractive (0d or d0),central diffractive (dd) mechanismsare compared with that for the standard gluon-gluon fusion (00).The cross section for single and central diffraction is rather small compared to the dominant gluon-gluon fusion component.",
"However, a very specific final state should allow its identification by imposing special conditions on the one-side (single-diffractive process) and both-side (central diffractive process) rapidity gaps.",
"We hope that such an analysis is possible at LHC.",
"Special care should be devoted to the observation of the exclusive $c \\bar{c}$ production.",
"Without a special analysis of the final state multiplicity the exclusive $c \\bar{c}$ production may look like an inclusive central diffraction."
],
[
"Production of two $c \\bar{c}$ pairs in double-parton scattering",
"The general formula for the cross section in terms of double-parton distributions can be written [11]: $d \\sigma ^{DPS} &=& \\frac{1}{2 \\sigma _{eff}}F_{gg}(x_1,x_2,\\mu _1^2,\\mu _2^2) F_{gg}(x^{\\prime }_{1}x^{\\prime }_{2},\\mu _1^2,\\mu _2^2)\\nonumber \\\\&&d \\sigma _{gg \\rightarrow c \\bar{c}}(x_1,x^{\\prime }_{1},\\mu _1^2)d \\sigma _{gg \\rightarrow c \\bar{c}}(x_2,x^{\\prime }_{2},\\mu _2^2) \\; dx_1 dx_2 dx^{\\prime }_1 dx^{\\prime }_2 \\, .$ In Fig.",
"REF we compare cross sections for the single and double-parton scattering as a function of proton-proton center-of-mass energy.",
"At low energies the conventional single-parton scattering dominates.",
"At low energy the $c \\bar{c}$ or $ c \\bar{c} c \\bar{c}$ cross sections are much smaller than the total cross section.",
"At higher energies the contributions dangerously approach the expected total cross sectionNew experiments at LHC will provide new input for parametrizations of the total cross section..",
"This shows that inclusion of unitarity effect and/or saturation of parton distributions may be necessary.",
"The effect of saturation in $c\\bar{c}$ production has been included but not checked versus experimental data.",
"Presence of double-parton scattering changes the situation.",
"At LHC energies the cross section for both terms become comparableIf inclusive cross section for $c$ or $\\bar{c}$ was shown the cross section should be multiplied by a factor of two – two $c$ or two $\\bar{c}$ in each event..",
"This is a completely new situation when the double-parton scattering gives a huge contribution to inclusive charm production.",
"Figure: Total LO cross section for single-parton and double-partonscattering as a function of center-of-mass energy (left panel) anduncertainties due to the choice of (factorization, renormalization) scales (right panel).We show in addition a parametrization of the total cross section in the left panel.In Fig.",
"REF , we present single $c$ ($\\bar{c}$ ) distributions.",
"Within approximations made in this paper the distributions are identical in shape to single-parton scattering distributions.",
"This means that double-scattering contribution produces naturally an extra center-of-mass energy dependent $K$ factor to be contrasted with approximately energy-independent $K$ -factor due to next-to-leading order corrections.",
"One can see a strong dependence on the factorization and renormalization scales which produces almost order-of-magnitude uncertainties and precludes a more precise estimation.",
"A better estimate could be done when LHC charm data are published and the theoretical distributions are somewhat adjusted to experimental data.",
"Figure: Distribution in rapidity (left panel) and transverse momentum (right panel) ofcc or c ¯\\bar{c} quarks at s\\sqrt{s} = 7 TeV."
]
] | 1204.1557 |
[
[
"Continuous time Boolean modeling for biological signaling: application\n of Gillespie algorithm"
],
[
"Abstract This article presents an algorithm that allows modeling of biological networks in a qualitative framework with continuous time.",
"Mathematical modeling is used as a systems biology tool to answer biological questions, and more precisely, to validate a network that describes biological observations and to predict the effect of perturbations.",
"We propose a modeling approach that is intrinsically continuous in time.",
"The algorithm presented here fills the gap between qualitative and quantitative modeling.",
"It is based on continuous time Markov process applied on a Boolean state space.",
"In order to describe the temporal evolution, we explicitly specify the transition rates for each node.",
"For that purpose, we built a language that can be seen as a generalization of Boolean equations.",
"The values of transition rates have a natural interpretation: it is the inverse of the time for the transition to occur.",
"Mathematically, this approach can be translated in a set of ordinary differential equations on probability distributions; therefore, it can be seen as an approach in between quantitative and qualitative.",
"We developed a C++ software, MaBoSS, that is able to simulate such a system by applying Kinetic Monte-Carlo (or Gillespie algorithm) in the Boolean state space.",
"This software, parallelized and optimized, computes temporal evolution of probability distributions and can also estimate stationary distributions.",
"Applications of Boolean Kinetic Monte-Carlo have been demonstrated for two qualitative models: a toy model and a published p53/Mdm2 model.",
"Our approach allows to describe kinetic phenomena which were difficult to handle in the original models.",
"In particular, transient effects are represented by time dependent probability distributions, interpretable in terms of cell populations."
],
[
"Background",
"Mathematical models of signaling pathways can be seen as tools to answer biological questions.",
"The most widely used mathematical formalisms to answer these questions are ordinary differential equations (ODEs) and Boolean modeling.",
"Ordinary differential equations (ODEs) have been widely used to model signaling pathways.",
"It is the most natural formalism for translating detailed reaction networks into a mathematical model.",
"Indeed, equations can be directly derived using mass action laws, Michaelis-Menten kinetics or Hill functions for each reaction in order to account for the observed behaviors.",
"This framework has limitations, though.",
"The first one concerns the difficulty of assigning the kinetic parameter values in the model.",
"Ideally, these parameters would be extracted from experimental data.",
"However, they are often chosen so as to fit qualitatively the expected phenotypes.",
"The second limitation arises when studying cell population heterogeneity.",
"In this case, ODEs are no longer appropriate since the approach is deterministic and thus focuses on the average behavior.",
"To include non-determinism, an ODE model needs to be transformed into stochastic chemical model.",
"In this formalism, a master equation is written on the probabilities of number of molecules for each species (instead of being written on ODEs of continuous concentrations).",
"In the translation process, the same parameters used in ODEs (more particularly in ODEs written with mass action law) can be used in the master equation, but in this case, the number of initial conditions explodes along with the computation time.",
"Boolean formalism is another formalism used to model signaling pathways where genes/proteins are parameterized by 0s and 1s only.",
"It is the most natural formalism to translate an influence network into a mathematical model.",
"In such networks, each node corresponds to a species and each arrow to an interaction or an influence (positive or negative).",
"In a Boolean model, a logical rule integrating the signs of the input links is assigned to each node.",
"As a result, there are no real parameter values to adjust besides choosing the appropriate logical rules that best describe the system.",
"In this paper, we will refer to a network state as a state in which each node of the influence network has a Boolean value.",
"The set of all possible transitions between the network states is defined as a transition graph.",
"There are two types of transition graphs, one deduced from the synchronous update strategy [1], for which all the nodes that can be updated are updated in one transition, and another one deduced from the asynchronous update strategy [2], for which only one node is updated in one transition.",
"In the Boolean formalism, each transition can be interpreted as a time step.",
"Real characterization of biological time is not taken into consideration.",
"However, in many biological problems, time plays a crucial role.",
"As mentioned for ODE, stochasticity is an important aspect when studying cell populations, It can be done: on nodes (by randomly flipping the node state [3], [4]), on the logical rules (by allowing to change an AND gate by an OR gate [5]), and on the update rules (by defining the probability and the priority of changing one Boolean value in an asynchronous strategy [6] or by adding noise to the whole system in a synchronous strategy [7]).",
"One of the main drawbacks of the Boolean approach is the explosion of solutions.",
"In an asynchronous update strategy, the transition graph can reach $2^\\text{\\#nodes}$ .",
"Both discrete and continuous frameworks have advantages and disadvantages above-mentioned.",
"We propose here to combine some of the advantages of both approaches with what we call the “Boolean Kinetic Monte-Carlo” algorithm (BKMC).",
"It consists of a natural generalization of the asynchronous Boolean dynamics [2], with a direct probabilistic interpretation.",
"In BKMC framework, the dynamics is parameterized by a biological time, the order of update is noisy (less strict than priority classes introduced in GINsim [8]).",
"A BKMC model is specified by logical rules as in regular Boolean models but with a more precise information: a numerical rate is added for each transition.",
"BKMC is best suited to model signaling pathways in the following cases: The model is based on an influence network, because BKMC is a generalization of the asynchronous Boolean dynamics.",
"Typical examples are published models of cell cycle [6].",
"The model describes processes for which information about biological time is known (relative rate, order, etc.",
"), because in BKMC, time is parameterized by a real number.",
"This is typically the case when studying developmental biology, where animal models provide time changes of gene/protein activities [9].",
"The model describes heterogeneous cell population behavior, because BKMC has a probabilistic interpretation.",
"For example, modeling heterogeneous cell population can help to understand tissu formation based on cell differentiation [10].",
"The model can contain many nodes (up to 64 in the present implementation), because BKMC is a simulation algorithm that converges fast.",
"This can be useful for big models that have already been modeled with a discrete time Boolean method [11], in order to obtain a finer description of transient effects.",
"Previous published works have also introduced a continuous time approach in the Boolean framework([12], [13], [14], [15], [16], [17], [18]).",
"We will review them and present BKMC approach.",
"We will describe the C++ software, MaBoSS, developed to implement BKMC algorithm and illustrate its use with two examples, a toy model and a published model of p53-MDM2 interaction.",
"All abbreviations, definitions and algorithms used in this article can be found in Supplementary Material.",
"Throughout the article, all terms that are italicized are defined in the Supplementary Material (Part 3).",
"In our context, we briefly recall that, in Boolean approaches for modeling networks, we define the state of each node of the network by a Boolean value (node state) and the network state by the set of node states.",
"Any dynamics in the transition graph is represented by sequences of network states.",
"A node state is based on the sign of the input arrows and the logic that links them.",
"The dynamics can be deterministic (in the case of synchronized update [1]) or non-deterministic (in the case of asynchronized update [2] or probabilistic Boolean networks [7]).",
"The difficulty to interpret a dynamics in terms of biological time has led to several works that have generalized Boolean approaches.",
"These approaches can be divided in two classes that we call explicit and implicit time for discrete steps.",
"The explicit time for discrete steps consists of adding a real parameter to each node.",
"These parameters correspond to the time associated to each node state before it flips to another one.",
"They need to be set for each node state ([13], [12]).",
"Because data about these time lengths are difficult to extract from experimental studies, some works have included noise in the definition of these time parameters [18].",
"The drawback of this method is that the computation of the Boolean model becomes sensitive to both the type of noise and the initial conditions.",
"As a result, these time parameters become new parameters that need to be tuned carefully and thus add complexity to the modeling.",
"The implicit time for discrete steps consists of adding probabilities to each transition of the transition graph, in the case of non-deterministic transitions.",
"It is argued that these probabilities could be interpreted as adding time to a biological process.",
"As an illustration, let us assume a small network of two nodes, A and B.",
"At time t, A and B are inactive: [AB] = [00].",
"In the transition graph, there exist two possible transitions at t+1: [00] $\\rightarrow $ [01] and [00] $\\rightarrow $ [10].",
"If the first transition has a significant higher probability than the second one, then we can conclude that B has a higher tendency to activate before A.",
"Therefore, it is equivalent to say that the activation of B is faster than the activation of A.",
"Thus, in this case, the notion of time is implicitly modeled by setting probability transitions.",
"In particular, priority rules, in the asynchronous strategy, consist of putting some of these probabilities to zero [6].",
"In our example, if B is faster than A then the probability of the transition [00] $\\rightarrow $ [10] is zero.",
"As a result, the prioritized nodes always activate before the others.",
"From a different perspective but keeping the same idea, Vahedi and colleagues [14] have set up a method to deduce explicitly these probabilities from the duration of each discrete step.",
"With the implementation of implicit time in a Boolean model, the dynamics remains difficult to interpret in terms of biological time.",
"As an alternative to these approaches, we propose BKMC algorithm.",
"BKMC algorithm was built such as to meet the following principles: The state of each node is given by a Boolean number (0 or 1) (referred to as node state); The state of the network is given by the set of node states (referred to as network state); The update of a node state is based on the signs linking the incoming arrows of this node and the logic; Time is represented by a real number; Evolution is stochastic.",
"For that, we choose to describe the time evolution of network states by a Markov process with continuous time.",
"Therefore, the dynamics is defined by transition rates inserted in a master equation (see Supplementary Material, section 1.1).",
"Transitions for which a rate is specified are based on asynchronous Boolean dynamics.",
"In BKMC, we adapt the Markov process to the Boolean approach.",
"Consider a network of $n$ nodes (or agents, that can represent any species, i.e.",
"mRNA, proteins, complexes, etc.).",
"In a Boolean framework, the network state of the system is described by a vector ${\\bf S}$ of Boolean values, i.e.",
"$S_i \\in \\lbrace 0,1\\rbrace , i=1,\\ldots ,n$ where $S_i$ is the state of the node $i$ .",
"The set of all possible network states, also referred to as the network state space, will be called $\\Sigma $ .",
"A stochastic description of state evolution is represented by a stochastic process ${s}(t)$ defined on $t\\in I \\subset \\mathbb {R}$ to the network state space, where $I$ is an interval: for each time $t \\in I \\subset \\mathbb {R}$ , ${s}(t)$ represents a random variable to the network state space: $\\mathbf {P}\\left[{s}(t) = {\\bf S}\\right] & \\in & [0,1] \\text{ for any state }{\\bf S}\\in \\Sigma \\nonumber \\\\\\sum _{{\\bf S}\\in \\Sigma }\\mathbf {P}\\left[{s}(t) = {\\bf S}\\right] & = & 1$ Notice that the random variables ${s}(t)$ are not independent, therefore $\\mathbf {P}\\left[{s}(t) = {\\bf S},{s}(t^{\\prime }) = {\\bf S}^{\\prime }\\right] \\ne \\mathbf {P}\\left[{s}(t) = {\\bf S}\\right]\\mathbf {P}\\left[{s}(t^{\\prime }) = {\\bf S}^{\\prime }\\right]$ .",
"From now on, we define $\\mathbf {P}\\left[{s}(t) = {\\bf S}\\right]$ as instantaneous probabilities (or first order probabilities).",
"Since the instantaneous probabilities do not define the full stochastic process, all possible joint probabilities should also be defined.",
"In order to simplify the stochastic process, Markov property is imposed.",
"It can be expressed in the following way: “the conditional probabilities in the future, related to the present and the past, depend only on the present” (see Supplementary Material, section 1.1 for the mathematical definition).",
"A stochastic process with the Markov property is called a Markov process.",
"Any Markov process can be defined by (see Van Kampen [19], chapter IV): an initial condition: $\\mathbf {P}\\left[{s}(0) = {\\bf S}\\right]\\;; \\forall {\\bf S}\\in \\Sigma $ conditional probabilities (of a single condition): $\\mathbf {P}\\left[{s}(t) = {\\bf S}|{s}(t^{\\prime }) = {\\bf S}^{\\prime }\\right]\\; ; \\forall {\\bf S},{\\bf S}^{\\prime } \\in \\Sigma \\; ; \\forall t^{\\prime },t \\in I; t^{\\prime }<t$ Concerning time, two cases can be considered: Time is discrete: $t\\in I=\\lbrace t_0,t_1,\\cdots \\rbrace $ .",
"In that case, it can be shown[20] that all possible conditional probabilities are function of transition probabilities: $\\mathbf {P}\\left[{s}(t_i) = {\\bf S}|{s}(t_{i-1}) = {\\bf S}^{\\prime }\\right]$ .",
"In that case, a Markov process is often named a Markov chain.",
"Time is continuous: $t\\in I=[a,b]$ .",
"In that case, it can be shown[19] that all possible conditional probabilities are function of transition rates: $\\rho _{({\\bf S}^{\\prime }\\rightarrow {\\bf S})}(t) \\in [0,\\infty [$ (see Supplementary Material, section 1.1 for the definition of transition rates).",
"Notice that a discrete time Markov process can be derived from continuous time Markov process, called Jump Process, with the following transition probabilities: $\\mathbf {P}_{{\\bf S}\\rightarrow {\\bf S^{\\prime }}} \\equiv \\frac{\\rho _{{\\bf S}\\rightarrow {\\bf S^{\\prime }}}}{\\sum _{{\\bf S}^{\\prime \\prime }} \\rho _{{\\bf S}\\rightarrow {\\bf S}^{\\prime \\prime }}} \\nonumber $ If the transition probabilities or transition rates are time independent, the Markov process is called a time independent Markov process.",
"In BKMC, only this case will be considered.",
"For a time independent Markov process, the transition graph (often called Boolean state graph in the Boolean framework) can be defined as follows: a transition graph is a graph in $\\Sigma $ , with an edge between ${\\bf S}$ and ${\\bf S}^{\\prime }$ if and only if $\\rho _{{\\bf S}\\rightarrow {\\bf S}^{\\prime }}>0$ (or $\\mathbf {P}\\left[{s}(t_i) = {\\bf S}|{s}(t_{i-1}) = {\\bf S}^{\\prime }\\right]>0$ if time is discrete).",
"Asynchronous Boolean dynamics[2] is widely used in Boolean modeling.",
"It can be easily interpreted as discrete time Markov process[21], [22] as shown below.",
"In the case of asynchronous Boolean dynamics, the system is given by $n$ nodes (or agents), with a set of directed arrows linking these nodes and defining a network.",
"For each node $i$ , a Boolean logic $B_i({\\bf S})$ is specified that depends only on the nodes $j$ for which there exists an arrow from node $j$ to $i$ (e.g.",
"$B_1=S_3 \\text{ AND} \\text{ NOT } S_4$ , where $S_3$ and $S_4$ are the Boolean values of nodes 3 and 4 respectively).",
"The notion of asynchronous transition (AT) can be defined as a pair of network states $({\\bf S},{\\bf S}^{\\prime }) \\in \\Sigma $ , written $({\\bf S} \\rightarrow {\\bf S}^{\\prime })$ such that $S^{\\prime }_j & = & B_j({\\bf S})\\text{ for a given }j \\nonumber \\\\S^{\\prime }_i & = & S_i\\text{ for }i\\ne j$ To define a Markov process, the transition probabilities $\\mathbf {P}\\left[{s}(t_i)={\\bf S}| {s}(t_{i-1})={\\bf S}^{\\prime }\\right]$ can be defined such that: given two network states ${\\bf S}$ and ${\\bf S}^{\\prime }$ , let $\\gamma ({\\bf S})$ be the number of asynchronous transitions from ${\\bf S}$ to all possible states ${\\bf S}^{\\prime }$ .",
"Then $\\mathbf {P}\\left[{s}(t_i)={\\bf S}^{\\prime }| {s}(t_{i-1}) = {\\bf S}\\right] & = &1/ \\gamma ({\\bf S})\\text{ if }({\\bf S}\\rightarrow {\\bf S}^{\\prime })\\text{ is an AT} \\nonumber \\\\\\mathbf {P}\\left[{s}(t_i)={\\bf S}^{\\prime }| {s}(t_{i-1}) = {\\bf S}\\right] & = & 0 \\text{ if }({\\bf S}\\rightarrow {\\bf S}^{\\prime })\\text{ is not an AT}$ In this formalism, asynchronous Boolean dynamics completely defines a discrete time Markov process when the initial condition is specified.",
"Notice that here the transition probabilities are time independent, i.e.",
"$\\mathbf {P}\\left[{s}(t_i)={\\bf S}| {s}(t_{i-1})={\\bf S}^{\\prime }\\right]=\\mathbf {P}\\left[{s}(t_{i+1})={\\bf S}| {s}(t_i)={\\bf S}^{\\prime }\\right]$ .",
"Other definition of $\\gamma ({\\bf S})$ can be used, defining another Markov process that have the same transition graph.",
"Therefore, the approaches mention above, that introduce time implicitly by adding probabilities to each transition of the transition graph, can be seen as a generalization of the definition of $\\gamma ({\\bf S})$ .",
"To transform the discrete time Markov process described above in a continuous time Markov process, transition probabilities should be replaced by transition rates $\\rho _{({\\bf S} \\rightarrow {\\bf S}^{\\prime })}$ .",
"In that case, conditional probabilities are computed by solving a master equation (equation 2 in Supplementary Material, section 1.1).",
"We present below the corresponding numerical algorithm (Kinetic Monte-Carlo [23]).",
"Because we want a generalization of asynchronous Boolean dynamics, transition rates $\\rho _{({\\bf S} \\rightarrow {\\bf S}^{\\prime })}$ are non-zero only if ${\\bf S}$ and ${\\bf S}^{\\prime }$ differ by only one node.",
"In that case, each Boolean logic $B_i({\\bf S)}$ is replaced by two functions $R_i^\\text{up/down}({\\bf S})\\in [0,\\infty [$ .",
"The transition rates are defined as follow: if $i$ is the node that differs from ${\\bf S}$ and ${\\bf S}^{\\prime }$ , $\\rho _{({\\bf S} \\rightarrow {\\bf S}^{\\prime })} & = & R^\\text{up}_i({\\bf S})\\text{ if }S_i=0 \\nonumber \\\\\\rho _{({\\bf S} \\rightarrow {\\bf S}^{\\prime })} & = & R^\\text{down}_i({\\bf S})\\text{ if }S_i=1 $ $R^\\text{up}_i$ corresponds to the activation rate of node $i$ , $R^\\text{down}_i$ corresponds to the inhibition rate of node $i$ .",
"Therefore, the continuous Markov process is completely defined by all these $R^\\text{up/down}$ and an initial condition.",
"In the case of continuous time Markov process, instantaneous probabilities always converge to a stationary distribution (see Supplementary Material, corollary 2, section 1.2).",
"A stationary distribution of a given Markov process corresponds to the set of instantaneous probabilities of a stationary Markov process which has the same transition probabilities (or transition rates) of the given discrete (or continuous) time Markov process.",
"A stationary Markov process has the following property: for every joint probability $\\mathbf {P}\\left[{s}(t_1)={\\bf S}^{(1)},{s}(t_2)={\\bf S}^{(2)},\\ldots \\right]$ and $\\forall \\tau $ , $\\mathbf {P}\\left[{s}(t_1)={\\bf S}^{(1)},{s}(t_2)={\\bf S}^{(1)},\\ldots \\right]=\\mathbf {P}\\left[{s}(t_1+\\tau )={\\bf S}^{(1)},{s}(t_2+\\tau )={\\bf S}^{(1)},\\ldots \\right]$ Notice that instantaneous probabilities $\\mathbf {P}\\left[{s}(t)={\\bf S}\\right]$ of a stationary stochastic process are time independent.",
"The asymptotic behavior of a continuous time Markov process can be detailed by using the concept of indecomposable stationary distributions: indecomposable stationary distributions are stationary distributions that cannot be expressed as linear combination of different stationary distributions.",
"Notice that a linear combination of stationary distributions is also a stationary distribution, up to a constant.",
"This comes from the fact that instantaneous probabilities are solutions of a master equation, which is linear (see Supplementary Material, equation 2, section 1.1).",
"Therefore, a complete description of the asymptotic behavior is given by the linear combination of indecomposable stationary distributions, to which the Markov process converges.",
"In order to describe a periodic behavior, the notion of cycle and oscillation for a continuous time Markov process is defined precisely.",
"A cycle is a loop in the transition graph.",
"This is a topological characterization that does not depend on the exact value of transition rates.",
"It can be shown that a cycle with no outcoming edge corresponds to an indecomposable stationary distribution (see Supplementary Material, corollary 1, section 1.2).",
"The question is then to link the notion of cycle to that of periodic behavior of instantaneous probabilities.",
"Instantaneous probabilities cannot be perfectly periodic; at most, they have a damped ocscillating behavior (see Supplementary Material, section 1.3).",
"Let us define formally a damped oscillatory Markov process as a continuous time process that has at least one instantaneous probability with an infinite number of extrema.",
"According to theorems described in Supplementary Material (theorems 6-8 and Corollary 3, section 1.3), a necessary condition for having damped oscillation is that the transition matrix (see Supplementary Material, equation 4, section 1.1) has at least one non-real eigenvalue.",
"In that case, there always exists an initial condition that produces damped oscillations.",
"For the transition matrix to have a non-real eigenvalue, a Markov process needs to have a cycle.",
"However, the reverse is not true.",
"In the toy model of single cycle, presented in the section of examples, non-real eigenvalues may or may not exist, according to different sets of transition rates, although the transition graph remains the same.",
"It has been previously stated that a continuous time Markov process is completely defined by its initial condition and its transition rates.",
"For computing any conditional probability (and any joint probability), a set of linear differential equations has to be solved (the master equation).",
"Theoretically, the master equation can be solved exactly by computing the exponential of the transition matrix (see Supplementary Material, equation 5, section 1.1).",
"However, because the size of this transition matrix is $2^n\\times 2^n$ , practical computation soon becomes impossible if $n$ is large.",
"To remedy this problem, it is possible to use a simulation algorithm that samples the probability space by computing time trajectories in the network state space.",
"The Kinetic Monte-Carlo[23] (or Gillespie algorithm[24]) is a simple algorithm for exploring the probability space of a Markov process defined by a set of transition rates.",
"In fact, it can be understood as a formal definition of a continuous time Markov process.",
"This algorithm produces a set of realizations or stochastic trajectories of the Markov process, given a set of uniform random numbers in $[0,1[$ .",
"By definition, a trajectory $\\hat{\\bf S}(t)$ is a function from a time window $[0,t_\\text{max}]$ to $\\Sigma $ .",
"The set of realizations or stochastic trajectories represents the given Markov process in the sense that these trajectories can be used to computed probabilities.",
"Practically, a finite set of these trajectories is produced, then probabilities are estimated from this finite set (as described below).",
"The algorithm is based on an iterative step: from a state ${\\bf S}$ at time $t_0$ (given two uniform random numbers), it produces a transition time $\\delta t$ and a new state ${\\bf S^{\\prime }}$ , with the following interpretation: the trajectory $\\hat{\\bf S}(t)$ is such that $\\hat{\\bf S}(t)={\\bf S}$ for $t \\in [t_0,t_0+\\delta t[$ and $\\hat{\\bf S}(t_0+\\delta t)={\\bf S}^{\\prime }$ .",
"Iteration of this step is done until a specified maximum time is reached.",
"The initial state of each trajectory is based on the (probabilistic) initial condition, that needs also to be specified.",
"The exact iterative step is the following, given ${\\bf S}$ and two uniform random number $u,u^{\\prime }\\in [0,1[$ : Compute the total rate of possible transitions for leaving ${\\bf S}$ state: $\\rho _\\text{tot}\\equiv \\sum _{{\\bf S^{\\prime }}}\\rho _{({\\bf S}\\rightarrow {\\bf S^{\\prime }})}$ .",
"Compute the time of the transition: $\\delta t \\equiv -\\log (u)/\\rho _\\text{tot}$ Order the possible new states ${\\bf S}^{\\prime (j)}, j=1 \\dots $ and their respective transition rates $\\rho ^{(j)}=\\rho _{({\\bf S}\\rightarrow {\\bf S^{\\prime (j)}})}$ .",
"Compute the new state ${\\bf S}^{\\prime (k)}$ such that $\\sum _{j=0}^{k-1}\\rho _j<(u^{\\prime }\\rho _\\text{tot})\\le \\sum _{j=0}^k\\rho _j$ (by convention, $\\rho ^{(0)}=0$ ).",
"The application of this algorithm to continuous time Markov process in network state space will be referred to as Boolean Kinetic Monte-Carlo or BKMC.",
"Biological data are summarized into an influence network with logical rules associated to each node of the network.",
"The value of one node depends on the value of the input nodes.",
"For BKMC, another layer of information is provided when compared to the standard definition of Boolean models: transition rates are provided for all nodes, specifying the rates at which the node turns on and off based on their logic for both the on and off rules.",
"This refinement conserves the simplicity of Boolean description but allows to reproduce the observed biological dynamics.",
"The parameters do not need to be exact as in nonlinear ordinary differential equation models but they can be used to illustrate the relative speed of reactions.",
"For that purpose, we developed a software tool, MaBoSS, that applies BKMC algorithm.",
"MaBoSS stands for Markov Boolean Stochastic Simulator.",
"Practically, MaBoSS needs two input files: one describing the network and its transition rates, one describing the parameters controlling the different estimates described in the Methods section.",
"Source code, reference card and examples are available on the web: https://maboss.curie.fr.",
"MaBoSS defines transition rates $\\rho _{({\\bf S} \\rightarrow {\\bf S}^{\\prime })}$ by the functions $R^\\text{up/down}_j({\\bf S})$ (see equation REF ).",
"The format of these functions is very flexible.",
"It includes all Boolean operators (AND, OR, NOT, XOR), arithmetic operators (+,-,* /), external variables, node variables, comparison operators and the conditional operator (?:).",
"Examples of the use of the language are given below to illustrate three different cases: different speeds for different inputs, buffering effect and the translation of discrete variables (with more than the 2 values, 0 and 1) in MaBoSS.",
"Modeling different speeds of incoming influences: suppose that C is activated by A and B, but that B can activate C faster than A.",
"In this case, we write: node C { rate_up= B ? $kb : (A ?",
"$ka : 0.0); rate_down= (A | B ) ?",
"0.0 : 1.0} When C is off (equal to 0), C is activated by B at a speed kb.",
"If B is absent, then C is activated by A at a speed $ka.",
"If both are absent, C is not activated.",
"Note that if both A and B are present, because of the way the logic is written in this particular case, C is activated at a speed $kb.",
"When C is on (equal to 1), C is inactivated at a rate equal to 1 if A and B are both absent.",
"To implement the synergetic effect of A and B, i.e.",
"when both A and B are on, C activates at a rate $kab, then we can write: node C { rate_up= (A & !B ? $ka : 0)+(B & !A ? $kb : 0) + (A & B ?",
"$kab : 0.0); rate_down= (A | B ) ?",
"0.0 : 1.0} Modeling buffering effect: suppose that B is activated by A, but that B can remain active a long time after A has shut down.",
"For that, it is enough to define different speeds of activation and inhibition: node B { rate_up= A ?",
"2.0 : 0.0; rate_down= A ?",
"0.0 : 0.001;} B is activated by A at a rate equal to 2.",
"When A is turned off, B is inactivated more slowly at a rate equal to 0.001.",
"Modeling more than two discrete states for a given node: Suppose that B is activated by A, but if the activity of A is maintained, B can reach a second level.",
"For this, we define a second node B_h (for “B high”) with the following rules: node B { rate_up= A ?",
"1.0 : 0.0; rate_down= (A | B_h) ?",
"0.0 : 1.0;} node B_h { rate_up= (A & B) ?",
"1.0 : 0.0; rate_down= (A) ?",
"0.0 : 1.0;} In this example, B is separated in two variables: B which corresponds to the first level of B and B_h which corresponds to the higher level of B.",
"B is activated by A at a rate equal to 1.",
"If A disappears before B has reached its second level B_h then B is turned off at a rate equal to 1.",
"If A is maintained and B is active, then B_h is activated at a rate equal to 1.",
"When A is turned off, B_h is inactivated at a rate equal to 1.",
"To simulate a process in MaBoSS, a set of parameters need to be adjusted (see simulation parameters in reference card).",
"MaBoSS assigns default values, however, they need to be tuned for each model to achieve optimal performances: best balance between the convergence of estimates and the computation time.",
"Therefore, several simulations should be run with different set of parameters for best tuning.",
"Internal nodes: node.is_internal As explained in Methods (“Initial conditions and outputs”), internal nodes correspond to species that are not measured explicitly.",
"Practically, the higher the number of internal nodes, the better the convergence of the BKMC algorithm.",
"Time window for probabilities: timetick This parameter is used to compute estimates of network state probabilities (see “Network state probabilities on time window” in Methods).",
"A time window can be set as the minimum time needed for nodes to change their states.",
"This parameter also controls the convergence of probability estimates.",
"The larger the time window parameter, the better the convergence of probability estimates.",
"With practice, the tradeoff between timetick parameter value and the convergence speed will be defined.",
"Maximum time: max_time MaBoSS produces trajectories for a predefined amount of time set by the parameter: max_time.",
"This maximum time needs to be specified.",
"If the time of the biological process is known, then the maximum time parameter can be explicitly set.",
"If the time of the biological process is not known, then there exists a more empirical way to set the maximum time.",
"It is advised to choose a maximum time parameter that is slightly bigger than the inverse of the smallest transition rate.",
"Note that the computing time in MaBoSS is proportional to this maximum time.",
"Moreover, the choice of the maximum time impacts the stationary distribution estimates (see below): a longer maximum time increases the quality of these estimates.",
"Number of trajectories: sample_count This parameter directly controls the quality of BKMC estimation algorithm.",
"Practically, the convergence of the estimates increases as the number of trajectories is increased.",
"Number of trajectories (statdist_traj_count) and similarity threshold (statsdist_cluster_threshold) for stationary distribution estimates The statdist_traj_count parameter corresponds to a subset of trajectories use only for stationary distribution estimates (the statdist_traj_count first trajectories are chosen by the algorithm).",
"To avoid explosion of computing time, this parameter needs to be lower than the number of trajectories (rather than equal to).",
"The statsdist_cluster_threshold parameter corresponds to the threshold for constructing the clusters of stationary distribution estimates.",
"Ideally, it should be set to a high value (close to 1).",
"However, if the threshold is too high then the clustering algorithm might not be efficient.",
"For optimal results, the identification of the full set of indecomposable stationary distributions should be done in the following way: Run a simulation with an initial condition and maximum time, with a reasonable similarity threshold (around 0.8) and a number of trajectories (around 1000).",
"As a response, MaBoSS provides a set of indecomposable stationary distributions (it corresponds to the stationary distributions associated to each cluster).",
"Select the states with non-zero probability in the set of indecomposable stationary distributions and set them as initial conditions, increase the maximum time (max_time), the number of trajectories (statdist_traj_count) and/or the similarity threshold (statsdist_cluster_threshold).",
"Run the simulations for these new parameters.",
"If new indecomposable stationary distributions appear, start again the two previous steps.",
"Stop when the indecomposable stationary distributions remain stable with respect to the simulation parameters, i.e.",
"after several rounds.",
"The relationship between a model and experimental data is strongly dependent on the type of model and the question that needs to be solved.",
"Because MaBoSS is based on Boolean modeling, the biological data need to be discretized.",
"Each node of the model should represent discrete levels of the respective species (mRNA, protein, protein complex, etc.).",
"It is possible to have more than two discrete levels in a model, as shown in the example “Modeling more than two discrete states for a given node”.",
"The transitions rates are positive numbers that should be introduced in a model; it is possible to extract them from experimental data, using the following property: the rate of a given transition is the inverse of the mean time, for this transition to happen.",
"It should be noticed than BKMC is an algorithm based on a linear equation (Supplementary Material, equation 2, section 1.1); therefore, small variations of transition rates won't affect qualitative behavior of a model.",
"The basic outputs of BKMC algorithm are network state probabilities over time.",
"These can be interpreted in terms of a cell population, once experimental data are discretized.",
"The asymptotic behavior of a model, represented by a linear combination of indecomposable stationary distributions, can be interpreted as a combination of cell sub-populations.",
"More precisely, consider an indecomposable stationary distribution and the associated set of network states with non-zero probability.",
"A cell in such a network state can only evolve in other network states with non-zero probability, within the same indecomposable stationary distribution (Supplementary Material, corollary 1, section 1.2 and by using the definition of strongly connected component with no outcoming edge).",
"Therefore, the set of network states with non-zero probability can be interpreted as a sub-population whose cells evolve only within the sub-population.",
"Two models using BKMC applied to Boolean networks are given as examples.",
"The first one is a toy model, illustrating the dynamics of a single cycle.",
"The second one uses a published Boolean model of p53-Mdm2 response to DNA damage.",
"Note that MaBoSS has been used for these two examples, but Markov process can be computed directly, without our BKMC algorithm because the model is small enough (by computing exponential of transition matrix, see Supplementary Material, section 1.1), as proposed in [16].",
"BKMC is best suited for larger networks, when the network state space is too large to be computed with standard existing tools ($ >\\sim 2^{10}$ ).",
"These examples were chosen for their simplicity, and because they illustrate how global characterizations (entropy and transition entropy, see “Entropies” in Methods) can be used.",
"For the purpose of this article, we built the transition graphs for both examples (with GINsim [8]) in order to help the reasoning.",
"However, BKMC algorithm does not construct the transition graph explicitly.",
"We consider three species, A, B and C, where A is activated by C and inhibited by B, B is activated by A and C is activated by A or B (Figure REF ).",
"Figure: Toy model of a single cycle.",
"(A) Influence network.",
"(B) Logic and transition rates of the model.",
"(C) Simulation parametersThe model is defined within the language of MaBoSS (defined in the web page, https://maboss.curie.fr).",
"The associated transition graph is shown in figure REF .",
"In this model, the only stationary distribution is the fixed point [ABC]=[000].",
"We studied two cases: when the rate of the transition from state [001] to state [000] is either fast or slow (inactivation of C).",
"We will refer to this transition rate as the escape rate.",
"Figure: Transition graph for the toy model.",
"The node state should be read as [ABC] = [***].",
"[ABC]=[100] corresponds to a state in which only A is active.",
"The nodes in green belong to a cycle, the node in red is the fixed point and the other state nodes are in blue.In the first case, when the escape rate is fast, we set the parameter for the transition to a high value (rate_up = 10).",
"In figure REF , we notice that the probability that [ABC] is equal to [000] converges to 1.",
"We can conclude that [ABC]=[000] is a fixed point.",
"In addition, the entropy and the transition entropy converge to zero.",
"With BKMC, these properties correspond to a signature of a fixed point.",
"The peak in the entropy (between times 0 and 0.6) corresponds to a set of transiently activated states before reaching the fixed point.",
"Figure: BKMC algorithm applied to the toy model, with a fast escape rate.",
"Time trajectory of probabilities ([ABC]=[000] and [ABC]=[1**] where * can be either 0 or 1), the entropy (HH) and the transition entropy (THTH) are plotted.",
"Because the probability of [ABC]=[000] converges to 1, [ABC]=[000] is a fixed point.",
"The asymptotic behavior of both the entropy and the transition entropy is also the signature of a fixed point.In the second case, when the escape rate is slow, we set the parameter for the transition to a low value (rate_down = $10^{-5}$ ).",
"The model seems to show another stationary distribution.",
"As illustrated in figure REF , the transition entropy is and remains close to zero but the entropy does not converge to zero, which is the signature of a cyclic stationary distribution (see “Entropies” in Methods).",
"This corresponds to the cycle [111] $\\rightarrow $ [011] $\\rightarrow $ [001] $\\rightarrow $ [101] in the transition graph (REF ).",
"However, as seen in the transition graph, one state in the cycle will eventually lead the fixed point (through the transition [001] $\\rightarrow $ [000] in figure REF ).",
"Therefore, if the temporal evolution is plotted on a larger time scale (Figure REF ), it looks similar to the case of fast escape rate.",
"This case can be anticipated.",
"Indeed, the value of the transition entropy of figure REF is not exactly zero, but $10^{-4}$ .",
"Therefore, the cyclic behavior is not stable.",
"We can conclude on stable cyclic behaviors only when the transition entropy is exactly zero.",
"By considering the spectrum of the transition matrix (see Supplementary Material, section 1.1 and proof of theorem 4), it can be proven that the model with a slow escape rate is a damped oscillatory process whereas the model with a large escape rate is not a damped oscillatory process.",
"As mentioned previously, a cycle in the transition graph may or may not lead to an oscillatory behavior.",
"Moreover, if the transition entropy seems to converge to a small value on a small time scale, and the entropy does not, this behavior illustrates the case of a transient cycle in the transition graph.",
"Figure: BKMC algorithm applied to the toy model, with a slow escape rate.",
"Time trajectory of probabilities ([ABC]=[000] and [ABC]=[1**]), the entropy (HH) and the transition entropy (THTH) are plotted.",
"The asymptotic behavior of both the entropy and the transition entropy seems to be the signature of a cycle.Figure: BKMC algorithm applied to the toy model, with a slow escape rate, plotted on a larger time scale.",
"Time trajectory of probabilities ([ABC]=[000] and [ABC]=[1**]), the entropy (HH) and the transition entropy (THTH) are plotted.",
"On a large time scale, the asymptotic behavior of both the entropy and the transition entropy is similar to the case of the fast escape rate (figure ).We consider a model of p53 response to DNA damage [18].",
"p53 interacts with Mdm2, which appears in two forms, cytoplasmic and nuclear.",
"On one hand, p53 upregulates the level of cytoplasmic Mdm2 which is then transported into the nucleus and inhibits the export of nuclear Mdm2.",
"On the other hand, Mdm2 facilitates the degradation of p53 through ubiquitination.",
"In the model, stress regulates the level of DNA damage, which in turn participates in the degradation process of Mdm2.",
"p53 inhibits the DNA damage signal by promoting DNA repair.",
"Here, stress is not shown explicitly (Figure REF ).",
"Figure: Boolean model of p53 response to DNA damageThe model is defined within the language of MaBoSS, with two levels of p53 as it is done in Abou-Jaoudé et al.",
"[18].",
"The model is implemented in MaBoSS (provided in the web page (https://maboss.curie.fr) along with the simulation parameters).",
"The associated transition graph is given in figure REF .",
"It shows the existence of two cycles and of a fixed point [p53 Mdm2C Mdm2N Dam] = [0010].",
"Figure: Transition graph of the p53 model.The node states should be read as [p53 Mdm2C Mdm2N Dam] = [****] (where * can be either 0 or 1).",
"For instance, [p53 Mdm2C Mdm2N Dam]=[1000] corresponds to a state in which only p53 (at its level 1) is active.",
"The nodes in green and the nodes in light blue belong to two cycles, the node in red is the fixed point and the other state nodes are in dark blue.In order to represent the activity of p53, time evolution of its expectation value is shown in figure REF , with the initial condition: [p53 Mdm2C Mdm2N Dam] = [0*11].",
"The activation of p53 seems to be transient.",
"Figure: Time trajectories of p53 expectation value and standard deviationThe qualitative results obtained with MaBoSS are similar to those of Abou-Jaoudé and colleagues.",
"However, at the level of cell population, some discrepancies appear: in figure REF , no damped oscillations can be seen as opposed to figure 8 of their article.",
"The reason is that in their computations, noise imposed on time is defined by a square distribution on a limited time frame, whereas in BKMC, Markovian hypotheses imply that the noise distribution is more spread out from 0 to infinity.",
"The consequence is that synchronization is lost very fast.",
"Damped oscillations could be observed with BKMC with a particular set of parameters: fast activation of p53 and slow degradation of p53 (results not shown).",
"With MaBoSS, we clearly interpret the system as a population and not as a single cell.",
"In addition, we can simulate different contexts, presented in the initial article as different models, within one model that used different parameters to account for these contexts (see web page https://maboss.curie.fr for more details).",
"Figure: Time trajectories of the entropy (HH) and the transition entropy (THTH)Note that the existence of transient cycles, as shown in the toy model, can be deduced from the time trajectory of the entropy that is significantly higher than the time trajectory of the transition entropy (which is non zero therefore the transient cycles are not stable) (Figure REF ).",
"An other indirect effect of transient cycles is the flat part in p53 standard deviation, in figure REF .",
"In this work, we present a new algorithm, Boolean Kinetic Monte-Carlo or BKMC, applicable to dynamical simulation of signaling networks based on continuous time in the Boolean framework.",
"BKMC algorithm is a natural generalization of asynchronous Boolean dynamics [2], with time trajectories that can be interpreted in terms of biological time.",
"The variables of the Boolean model are biological species and the parameters are rates of activation or inhibition of these species.",
"These parameters can be deduced from experimental data.We applied this algorithm to two different models: a toy model that illustrates a simple cyclic behavior, and a published model of p53 response to DNA damage.",
"This algorithm is provided within a freely available software, MaBoSS, that can run BKMC algorithm on networks up to 64 nodes, in the present version.",
"The construction of a model uses a specific grammar that allows to introduce logical rules and rates of node activation/inhibition in a flexible manner.",
"This software provides global and semi-global outputs of the model dynamics that can be interpreted as signatures of the dynamical behaviors.",
"These interpretations become particularly useful when the network state space is too large to be handled.",
"The convergence of BKMC algorithm can be controlled by tuning different parameters: maximum time of the simulation, number of trajectories, length of a time window on which the average of probabilities is performed, and the threshold for the definition of stationary distributions clusters.",
"The next step is to apply BKMC algorithm with MaBoSS on large signaling networks, e.g.",
"EGFR pathway, cell cycle signaling, apoptosis pathway, etc.",
"The translation of existing Boolean models in MaBoSS is straightforward but requires the addition of transition rates.",
"In these future works, we expect to illustrate the advantage of BKMC on other simulation algorithms.",
"Moreover, in future developments of MaBoSS, we plan to introduce methods for sensitivity analyses and to refine approximation methods used in BKMC, and generalize Markov property.",
"BKMC generates stochastic trajectories.",
"Here, we describe how we use and interpret them.",
"To relate continuous time probabilities to real processes, an observable time window $\\Delta t$ is defined.",
"A discrete time ($\\tau \\in \\mathbb {N}$ ) stochastic process ${s}(\\tau )$ (that is not necessary Markovian) can be extracted from the continuous time Markov process: $\\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right] \\equiv \\frac{1}{\\Delta t} \\int _{\\tau \\Delta t}^{(\\tau +1)\\Delta t} dt\\; \\mathbf {P}\\left[{s}(t)={\\bf S}\\right]$ BKMC is used for estimating $\\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right]$ as follows: For each trajectory $j$ , compute the time for which the system is in state ${\\bf S}$ , in the window $[\\tau \\Delta t,(\\tau +1)\\Delta t]$ .",
"Divide this time by $\\Delta t$ .",
"Obtain an estimate of $\\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right]$ for trajectory $j$ , i.e.",
"$\\hat{\\mathbf {P}}_j\\left[{s}(\\tau )={\\bf S}\\right]$ .",
"Compute the average over $j$ of all $\\hat{\\mathbf {P}}_j\\left[{s}(\\tau )={\\bf S}\\right]$ to obtain $\\hat{\\mathbf {P}}\\left[{s}(\\tau )={\\bf S}\\right]$ .",
"Compute the error of this average ($\\sqrt{\\text{Var}(\\hat{\\mathbf {P}}\\left[{s}(\\tau )={\\bf S}\\right])/\\text{\\# trajectories}}$ ).",
"Once $\\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right]$ is computed, the entropy $H(\\tau )$ can be estimated $H(\\tau )=-\\sum _{{\\bf S}}\\log _2\\left(\\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right]\\right) \\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right]$ The entropy measures the disorder of the system.",
"A maximum entropy means that all states have the same probability, a zero entropy means that one of the states has a probability of one.",
"The estimation of the entropy can be seen as a global characterization (by a single real number) of a full probability distribution.",
"The choice of $\\log _2$ allows to interpret $H(\\tau )$ in an easier manner: $2^{H(\\tau )}$ is an estimate of the number of states that have a non-negligible probability in the time window $[\\tau \\Delta t,(\\tau +1)\\Delta t]$ .",
"A more computer-like interpretation of $H(\\tau )$ is the number of bits that is necessary for describing states of non-negligible probability.",
"The Transition Entropy $TH$ is a finer measure that characterizes the system at the single trajectory level; it can be computed in the following way: for each state ${\\bf S}$ , there exists a set of possible transitions ${\\bf S}\\rightarrow {\\bf S}^{\\prime }$ .",
"For each of these transitions, a probability is associated: $\\mathbf {P}_{{\\bf S}\\rightarrow {\\bf S^{\\prime }}} \\equiv \\frac{\\rho _{{\\bf S}\\rightarrow {\\bf S^{\\prime }}}}{\\sum _{{\\bf S}^{\\prime }} \\rho _{{\\bf S}\\rightarrow {\\bf S}^{\\prime }}} .$ By convention, $\\mathbf {P}_{{\\bf S}\\rightarrow {\\bf S^{\\prime }}}=0$ if there is no transition from ${\\bf S}$ to any other state.",
"Therefore, the transition entropy $TH$ can be associated to each state ${\\bf S}$ : $TH({\\bf S})=-\\sum _{{\\bf S}^{\\prime }} \\log _2(\\mathbf {P}_{{\\bf S}\\rightarrow {\\bf S^{\\prime }}})\\mathbf {P}_{{\\bf S}\\rightarrow {\\bf S^{\\prime }}} $ Similarly, $TH({\\bf S})=0$ if there is no transition from ${\\bf S}$ to any other state.",
"The discrete time transition entropy $TH(\\tau )$ is defined as: $TH(\\tau )=\\sum _{{\\bf S}}\\mathbf {P}\\left[{s}(\\tau )={\\bf S}\\right]TH({\\bf S}) \\nonumber $ This transition entropy is estimated in the following way: For each trajectory $j$ , compute the set ($\\Phi $ ) of visited states (${\\bf S}$ ) in time window $[\\tau \\Delta t,(\\tau +1)\\Delta t]$ and their respective duration ($\\mu _{\\bf S})$ .",
"The estimated transition entropy is $\\hat{TH(\\tau )}_j=\\sum _{{\\bf S}\\in \\Phi }TH({\\bf S})\\frac{ \\mu _{\\bf S}}{\\Delta t}$ Compute the average over $j$ of all $\\hat{TH(\\tau )}_j$ to obtain $\\hat{TH(\\tau )}$ .",
"Compute the error of that average ($\\sqrt{\\text{Var}(\\hat{TH(\\tau )})/\\text{\\# trajectories}}$ ).",
"This transition entropy is a way to measure how deterministic the dynamics is.",
"If the transition entropy is always zero, the system can only make a transition to a given state (although time of transitions remains stochastic).",
"If probability distributions on time window tend to constant values (or tend to a stationary distribution), the entropy and the transition entropy can help characterize this stationary distribution such that: A fixed point has zero entropy and zero transition entropy, A cyclic stationary distribution has non-zero entropy and zero transition entropy.",
"Entropy and transition entropy can be considered as “global characterization” of time evolution model: for a given time window, they always consist of two real numbers, whatever the size of the network.",
"The Hamming Distance between two states ${\\bf S}$ and ${\\bf S}^{\\prime }$ is the number of nodes that have different node states between ${\\bf S}$ and ${\\bf S}^{\\prime }$ : $HD({\\bf S},{\\bf S}^{\\prime }) \\equiv \\sum _i (1-\\delta _{S_i,S^{\\prime }_i})$ where $\\delta $ is the Kronecker delta ($\\delta _{S_i,S^{\\prime }_i}=1$ if $S_i=S^{\\prime }_i$ , $\\delta _{S_i,S^{\\prime }_i}=0$ if $S_i\\ne S^{\\prime }_i$ ).",
"Given a reference state ${\\bf S}_\\text{ref}$ , the Hamming distance distribution (over time) is given by $\\mathbf {P}(HD,t)=\\sum _{\\bf S}\\mathbf {P}\\left[{s}(t)={\\bf S}\\right]\\delta _{HD,HD({\\bf S},{\\bf S}_\\text{ref})}$ The estimation of discrete time Hamming distance distribution $\\mathbf {P}(HD,\\tau )$ is similar to that of stochastic probabilities on time window.",
"The Hamming distance distribution is a useful characterization when the set of instantaneous probabilities is compared to a reference state (${\\bf S}_\\text{ref}$ ).",
"In that case, Hamming distance distribution describes how far this set is to this reference state.",
"The Hamming distance distribution can be considered as a “semi-global” characterization of time evolution: for a given time window, the size of this characterization is the number of nodes (to be compared with probabilities on time window, whose size is $2^\\text{\\#nodes}$ ).",
"Inputs Nodes are defined as the nodes for which the initial condition is fixed.",
"Therefore, each trajectory of BKMC starts with fixed values of input nodes and random values of other nodes.",
"Internal nodes are nodes that are not considered for computing probability distributions, entropies and transition entropies.",
"Output nodes are nodes that are not internal.",
"Technically, probabilities are sum up over network states that differ only by the state of internal nodes.",
"These internal nodes are only used for generating time trajectories through BKMC algorithm.",
"Usually, nodes are chosen to be internal when the corresponding species is not measured experimentally.",
"Mathematically, it is equivalent to transform the original Markov process to a new stochastic process (that is not necessary Markovian) defined to a new network state space; this new state space is defined by states of output nodes.",
"This raises the question of the transition entropy $TH$ : formally, this notion has only a sense within Markovian processes, i.e.",
"when there are no internal nodes.",
"Here, we generalize the notion of transition entropy even in the case of internal nodes.",
"Suppose that the system is in state ${\\bf S}$ : If the only possible transitions from state ${\\bf S}$ to any other state consist of flipping an internal node, the transition entropy is zero.",
"If there is, at least, one transition from state ${\\bf S}$ to another state that flips a non-internal node, then only the non-internal nodes will be considered for computing probabilities in equations REF .",
"In particular, $\\sum _{{\\bf S}^{\\prime }}\\rho _{{\\bf S}\\rightarrow {\\bf S}^{\\prime }}$ is computed only on non-internal node flipping events.",
"Reference nodes are nodes for which a reference node state is specified and for which the Hamming distance is computed.",
"In this framework, a reference state is composed of reference nodes for which the node state is known and non-reference nodes for which the node state is unknown.",
"Note that non-reference nodes may differ from internal nodes.",
"It can be shown (see Supplementary Material, corollary 2, section 1.2) that instantaneous probabilities of a continuous time Markov process converge to a stationary distribution.",
"Fixed points and cycles are two special cases of stationary distributions.",
"They can be identified by the asymptotic behavior of entropy and transition entropy (this works only if no nodes are internal): if the transition entropy and the entropy converge to zero, then the process converges to a fixed point if the transition entropy and the entropy does not converge to zero, then the process converges to a cycle More generally, the complete description of the Markov process asymptotic behavior can be expressed as a linear combination of the indecomposable stationary distributions.",
"A set of finite trajectories, produced by BKMC, can be used to estimate the set of indecomposable stationary distributions.",
"Consider a trajectory $\\hat{\\bf S}(t), t\\in [0,T], i=1,\\cdots , n$ .",
"Let $I_{\\bf S}(t) \\equiv \\delta _{{\\bf S},\\hat{\\bf S}(t)}$ .",
"The estimation of the associated indecomposable stationary probability distribution (${s}_0$ ) is done by averaging over the whole trajectory: $\\hat{\\mathbf {P}}\\left[{s}_0={\\bf S}\\right]=\\frac{1}{T}\\int _0^TdtI_{\\bf S}(t)$ Therefore, a set of indecomposable stationary distribution estimates can be obtained by a set of trajectories.",
"These indecomposable stationary distribution estimates should be clustered in groups, where each group consists of estimates for the same indecomposable stationary distribution.",
"For that, we use the fact that two indecomposable stationary distributions are identical if they have the same support, i.e.",
"the same set of network states with non-zero probabilities (shown in Supplementary Material, theorem 2).",
"Therefore, it is possible to quantify how similar two indecomposable stationary distribution estimates are.",
"A similarity coefficient $D({s}^{(i)}_0,{s}^{(j)}_0) \\in [0,1]$ , given two stationary distribution estimates ${s}^{(i)}_0$ and ${s}^{(j)}_0$ , is defined: $D({s}^{(i)}_0,{s}^{(j)}_0)\\equiv \\left(\\sum _{{\\bf S}\\in \\text{support}({s}^{(i)}_0,{s}^{(j)}_0)}\\hat{\\mathbf {P}}\\left[{s}^{(i)}_0={\\bf S}\\right]\\right)\\left(\\sum _{{\\bf S}^{\\prime }\\in \\text{support}({s}^{(i)}_0,{s}^{(j)}_0)}\\hat{\\mathbf {P}}\\left[{s}^{(j)}_0={\\bf S}^{\\prime }\\right]\\right)$ where $\\text{support}({s}^{(i)}_0,{s}^{(j)}_0)\\equiv \\left\\lbrace {\\bf S}\\text{ such that }\\hat{\\mathbf {P}}\\left[{s}^{(i)}_0={\\bf S}\\right]\\hat{\\mathbf {P}}\\left[{s}^{(j)}_0={\\bf S}\\right]>0\\right\\rbrace $ Clusters can be constructed when a similarity threshold $\\alpha $ is provided.",
"A cluster of stationary distributions is defined as follows: ${\\cal C}=\\left\\lbrace {s}_0|\\;\\exists {s}^{\\prime }_0\\in {\\cal C} \\text{ s. t. }D({s}_0,{s}^{\\prime }_0)\\ge \\alpha \\right\\rbrace $ For each cluster ${\\cal C}$ , an associated distribution ${s}_{\\cal C}$ , that estimates an indecomposable stationary distribution, can be defined ${\\mathbf {P}}\\left[{s}_{\\cal C}={\\bf S}\\right]=\\frac{1}{|{\\cal C}|}\\sum _{{s}\\in {\\cal C}}{\\mathbf {P}}\\left[{s}={\\bf S}\\right]$ Errors on this estimate can be computed by $\\text{Err}\\left({\\mathbf {P}}\\left[{s}_{\\cal C}={\\bf S}\\right]\\right)=\\sqrt{\\text{Var}({\\mathbf {P}}\\left[{s}={\\bf S}\\right],{s}\\in {\\cal C})/|{\\cal C}|}$ Notice that this clustering procedure has no sense if the process is not Markovian; therefore, no node will be considered as internal.",
"BKMC: Boolean Kinetic Monte-Carlo AT: Asynchronous transition ODEs: Ordinary Differential Equations MaBoSS: Markov Boolean Stochastic Simulator G. Stoll organized the project, set up the algorithms, participated in writing the software, set up the examples and wrote the article.",
"E. Viara wrote the software and participated in setting up the algorithms.",
"E. Barillot participated in writing the article.",
"L. Calzone organized the project, set up the examples and wrote the article.",
"This project was supported by the Institut National du Cancer (SybEwing project), the Agence National de la Recherche (Calamar project).",
"The research leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement nb HEALTH-F4-2007-200767 for APO-SYS and nb FP7-HEALTH-2010-259348 for ASSET.",
"GS, EB and LC are members of the team ”Computational Systems Biology of Cancer”, Equipe labellisée par la Ligue Nationale Contre le Cancer.",
"We'd like to thank Camille Sabbah, Jacques Rougemont, Denis Thieffry, Elisabeth Remy, Luca Grieco and Andrei Zinovyev."
]
] | 1204.0997 |
[
[
"Sharp lower bound on the curvatures of ASD connections over the cylinder"
],
[
"Abstract We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder."
],
[
"Introduction",
"The purpose of this note is to calculate explicitly a universal lower bound on the curvatures of non-flat ASD connections over the cylinder $\\mathbb {R}\\times S^3$ .",
"First we fix our conventions.",
"Let $S^3 = \\lbrace x_1^2+x_2^2+x_3^2+x_1^4=1\\rbrace \\subset \\mathbb {R}^4$ be the 3-sphere equipped with the Riemannian metric induced by the Euclidean metric on $\\mathbb {R}^4$ .",
"Set $X:=\\mathbb {R}\\times S^3$ .",
"We give the standard metric on $\\mathbb {R}$ , and $X$ is equipped with the product metric.",
"Let $\\mathbb {H}$ be the space of quaternions.",
"Consider $SU(2) = \\lbrace x\\in \\mathbb {H}|\\, |x|=1\\rbrace $ with the Riemannian metric induced by the Euclidean metric on $\\mathbb {H}$ .",
"(Hence it is isometric to $S^3$ above.)",
"We naturally identify $su(2) := T_1 SU(2)$ with the imaginary part $\\mathrm {Im} \\mathbb {H} := \\mathbb {R} i + \\mathbb {R} j + \\mathbb {R} k$ .",
"Here $i$ , $j$ and $k$ have length 1.",
"Let $E := X\\times SU(2)$ be the product $SU(2)$ -bundle.",
"Let $A$ be a connection on $E$ , and let $F_A$ be its curvature.",
"$F_A$ is a $su(2)$ -valued 2-form on $X$ .",
"Hence for each point $p\\in X$ the curvature $F_A$ can be considered as a linear map $ F_{A,p}: \\Lambda ^2(T_pX)\\rightarrow su(2).$ We denote by $|F_{A,p}|_{\\mathrm {op}}$ the operator norm of this linear map.",
"The explicit formula is as follows: Let $x_1,x_2,x_3,x_4$ be the normal coordinate system on $X$ centered at $p$ .",
"Let $A=\\sum _{i=1}^4 A_i dx_i$ .",
"Each $A_i$ is a $su(2)$ -valued function.",
"Then $F(A)_{ij} := F_A(\\partial /\\partial x_i, \\partial /\\partial x_j)= \\partial _i A_j - \\partial _j A_i + [A_i,A_j]$ .",
"Since $\\partial /\\partial x_i \\wedge \\partial /\\partial x_j$ $(1\\le i<j\\le 4)$ become a orthonormal basis of $\\Lambda ^2(TX)$ at $p$ , the norm $|F_{A,p}|_{\\mathrm {op}}$ is equal to $ \\sup \\left\\lbrace \\left|\\sum _{1\\le i<j\\le 4}a_{ij}F(A)_{ij,p}\\right||\\, a_{ij}\\in \\mathbb {R},\\sum _{1\\le i<j\\le 4} a_{ij}^2=1 \\right\\rbrace .",
"$ Let $\\left|\\!\\left|F_A\\right|\\!\\right|_{\\mathrm {op}}$ be the supremum of $|F_{A,p}|_{\\mathrm {op}}$ over $p\\in X$ .",
"The main result is the following.",
"Theorem 1.1 The minimum of $\\left|\\!\\left|F_A\\right|\\!\\right|_{\\mathrm {op}}$ over non-flat ASD connections $A$ on $E$ is equal to $1/\\sqrt{2}$ .",
"The above minimum value $1/\\sqrt{2}$ is attained by the following BPST instanton ([1]).",
"Example 1.2 We define a $SU(2)$ instanton $A$ on $\\mathbb {R}^4$ by $ A := \\mathrm {Im}\\left(\\frac{\\bar{x}dx}{1+|x|^2}\\right), \\quad (x= x_1+x_2i+x_3j+x_4k).$ By the conformal map $ \\mathbb {R}\\times S^3\\rightarrow \\mathbb {R}^4\\setminus \\lbrace 0\\rbrace ,\\quad (t, \\theta ) \\mapsto e^t \\theta ,$ the connection $A$ is transformed into an ASD connection $A^{\\prime }$ on $E$ over $\\mathbb {R}\\times S^3$ .",
"Then $ |F_{A^{\\prime },(t,\\theta )}|_{\\mathrm {op}} = \\frac{2\\sqrt{2}}{(e^t+e^{-t})^2}.$ Hence $ \\left|\\!\\left|F_{A^{\\prime }}\\right|\\!\\right|_{\\mathrm {op}} = \\frac{1}{\\sqrt{2}}.$ Theorem REF is a Yang-Mills analogy of the classical result of Lehto [7] in complex analysis.",
"(The formulation below is due to Eremenko [4].",
"See also Lehto-Virtanen [8].)",
"Consider $\\mathbb {C}^* := \\mathbb {C}\\setminus \\lbrace 0\\rbrace $ with the length element $|dz|/|z|$ .",
"We give a metric on $\\mathbb {C}P^1 = \\mathbb {C}\\cup \\lbrace \\infty \\rbrace $ by (naturally) identifying it with the unit 2-sphere $\\lbrace x_1^2+x_2^2+x_3^2=1\\rbrace $ .",
"For a map $f:\\mathbb {C}^*\\rightarrow \\mathbb {C}P^1$ we denote its Lipschitz constant by $\\mathrm {Lip}(f)$ .",
"Then Lehto [7] proved that the minimum of $\\mathrm {Lip}(f)$ over non-constant holomorphic maps $f:\\mathbb {C}^*\\rightarrow \\mathbb {C}P^1$ is equal to 1.",
"The function $f(z) = z$ attains the minimum.",
"Eremenko [4] discussed the relation between this Lehto's result and a quantitative homotopy argument of Gromov [6].",
"Our proof of Theorem REF is inspired by this idea."
],
[
"Preliminaries: Connections over $S^3$",
"In this section we study the method of choosing good gauges for some connections over $S^3$ .",
"The argument below is a careful study of [5].",
"Set $N:=(1,0,0,0)\\in S^3$ and $S:=(-1,0,0,0)\\in S^3$ .",
"Let $P:=S^3\\times SU(2)$ be the product $SU(2)$ -bundle over $S^3$ .",
"For a connection $B$ on $P$ we define the operator norm $\\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}$ in the same way as in Section .",
"Let $v_1,v_2\\in T_NS^3$ be two unit tangent vectors at $N$ .",
"($|v_1|=|v_2|=1$ .)",
"Let $\\exp _N:T_NS^3\\rightarrow S^3$ be the exponential map at $N$ .",
"Since $|v_1|=|v_2|=1$ , we have $\\exp _N(\\pi v_1) = \\exp _N(\\pi v_2) =S$ .",
"We define a loop $l:[0,2\\pi ]\\rightarrow S^3$ by $l(t) := {\\left\\lbrace \\begin{array}{ll}\\exp _N(tv_1) \\quad &(0\\le t\\le \\pi ) \\\\\\exp _N((2\\pi -t)v_2) \\quad &(\\pi \\le t\\le 2\\pi ).\\end{array}\\right.",
"}$ Lemma 2.1 Let $B$ be a connection on $P$ .",
"Let $\\mathrm {Hol}_l(B)\\in SU(2)$ be the holonomy of $B$ along the loop $l$ .",
"Then $ d(\\mathrm {Hol}_l(B), 1)\\le 2\\pi \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}.$ Here $d(\\cdot ,\\cdot )$ is the distance on $SU(2)$ defined by the Riemannian metric.",
"This follows from the standard fact that curvature is an infinitesimal holonomy [3].",
"($2\\pi $ is half the area of the unit 2-sphere.)",
"The explicit proof is as follows: Take a unit tangent vector $v_3\\in T_NS^3$ orthogonal to $v_1$ such that there is $\\alpha \\in [0,\\pi ]$ satisfying $v_2= v_1\\cos \\alpha + v_3\\sin \\alpha $ .",
"Consider (the spherical polar coordinate of the totally geodesic $S^2\\subset S^3$ tangent to $v_1$ and $v_3$ ): $\\Phi : [0,\\alpha ]\\times [0,\\pi ]\\rightarrow S^3, \\quad (\\theta _1, \\theta _2)\\mapsto \\exp _N\\lbrace \\theta _2(v_1\\cos \\theta _1 + v_3\\sin \\theta _1)\\rbrace .$ Let $Q$ be the pull-back of the bundle $P$ by $\\Phi $ .",
"Since $\\Phi ([0,\\alpha ]\\times \\lbrace 0\\rbrace )=\\lbrace N\\rbrace $ and $\\Phi ([0,\\alpha ]\\times \\lbrace \\pi \\rbrace )=\\lbrace S\\rbrace $ , $Q$ admits a trivialization under which the pull-back connection $\\Phi ^*B$ is expressed as $\\Phi ^*B=B_1d\\theta _1+B_2\\theta _2$ with $B_1=0$ on $[0,\\alpha ]\\times \\lbrace 0,\\pi \\rbrace $ .",
"We take a smooth map $g:[0,\\alpha ]\\times [0,\\pi ]\\rightarrow SU(2)$ satisfying $ g(\\theta _1,0)= 1 \\quad (\\forall \\theta _1\\in [0,\\alpha ]),\\quad (\\partial _2+B_2) g=0.$ We have $\\mathrm {Hol}_l(B) = g(\\alpha ,\\pi )^{-1}g(0,\\pi )$ .",
"Then $F_{\\Phi ^*B}(\\partial _1, \\partial _2)g =[\\partial _1+B_1,\\partial _2+B_2]g=-(\\partial _2+B_2)(\\partial _1+B_1)g$ .",
"Since $B_1=0$ on $[0,\\alpha ]\\times \\lbrace 0,\\pi \\rbrace $ , $ |\\partial _1g(\\theta _1, \\pi )|\\le \\int _{\\lbrace \\theta _1\\rbrace \\times [0,\\pi ]}|F_{\\Phi ^*B}(\\partial _1,\\partial _2)|d\\theta _2.$ Then $ d(\\mathrm {Hol}_l(B),1)= d(g(0,\\pi ),g(\\alpha ,\\pi )) \\le \\int _{[0,\\alpha ]\\times [0,\\pi ]}|F_{\\Phi ^*B}(\\partial _1, \\partial _2)|d\\theta _1 d\\theta _2.$ $F_{\\Phi ^*B}(\\partial _1,\\partial _2) =F_B(d\\Phi (\\partial /\\partial \\theta _1),d\\Phi (\\partial /\\partial \\theta _2))$ .",
"The vectors $d\\Phi (\\partial /\\partial \\theta _1)$ and $d\\Phi (\\partial /\\partial \\theta _2)$ are orthogonal to each other, and $|d\\Phi (\\partial /\\partial \\theta _1)| = \\sin \\theta _2$ and $|d\\Phi (\\partial /\\partial \\theta _2)|=1$ .",
"Hence $|F_{\\Phi ^*B}(\\partial _1,\\partial _2)|\\le \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}} \\sin \\theta _2$ .",
"Thus, from $0\\le \\alpha \\le \\pi $ , $ d(\\mathrm {Hol}_l(B),1)\\le \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}\\int _{[0,\\alpha ]\\times [0,\\pi ]}\\sin \\theta _2 \\, d\\theta _1 d\\theta _2= 2\\alpha \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}\\le 2\\pi \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}.$ Let $\\tau <1/2$ .",
"Let $B$ be a connection on $P$ satisfying $\\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}\\le \\tau $ .",
"We will construct a good connection matrix of $B$ .",
"Fix $v\\in T_NS^3$ .",
"By the parallel translation along the geodesic $\\exp _N (tv)$ $(0\\le t\\le \\pi )$ we identify the fiber $P_S$ with the fiber $P_N$ .",
"Let $g_N$ and $g_S$ be the exponential gauges (see [5] or [3]) centered at $N$ and $S$ respectively: $ g_N:P|_{S^3\\setminus \\lbrace S\\rbrace }\\rightarrow (S^3\\setminus \\lbrace S\\rbrace )\\times P_N, \\quad g_S:P|_{S^3\\setminus \\lbrace N\\rbrace }\\rightarrow (S^3\\setminus \\lbrace N\\rbrace )\\times P_N.$ (In the definition of $g_S$ we identify $P_S$ with $P_N$ as in the above.)",
"By Lemma REF , for $x\\in S^3\\setminus \\lbrace N,S\\rbrace $ , $ d(g_N(x),g_S(x))\\le 2\\pi \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}} \\le 2\\pi \\tau < \\pi .$ The injectivity radius of $SU(2) = S^3$ is $\\pi $ (this is a crucial point of the argument).",
"Hence there uniquely exists $u(x)\\in \\mathrm {ad} P_N (\\cong su(2))$ satisfying $ |u(x)|\\le 2\\pi \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}, \\quad g_S(x) = e^{u(x)}g_N(x).$ We take and fix a cut-off function $\\varphi :S^3\\rightarrow [0,1]$ such that $\\varphi (x_1,x_2,x_3,x_4)$ is equal to 0 over $\\lbrace x_1>1/2\\rbrace $ and equal to 1 over $\\lbrace x_1<-1/2\\rbrace $ .",
"We can define a bundle trivialization $g$ of $P$ all over $S^3$ by $g := e^{\\varphi u}g_N$ Then the connection matrix $g(B)$ satisfies $ |g(B)|\\le C_\\tau \\left|\\!\\left|F_B\\right|\\!\\right|_{\\mathrm {op}}.$ Here $C_\\tau $ is a positive constant depending on $\\tau $ ."
],
[
"Proof of Theorem ",
"In this section we denote by $t$ the standard coordinate of $\\mathbb {R}$ .",
"Let $A$ be an ASD connection on $E$ satisfying $\\left|\\!\\left|F_A\\right|\\!\\right|_{\\mathrm {op}} < 1/\\sqrt{2}$ .",
"We will prove that $A$ must be flat.",
"Set $\\tau :=\\left|\\!\\left|F_A\\right|\\!\\right|_{\\mathrm {op}}/\\sqrt{2} <1/2$ .",
"The ASD equation implies that $F_A$ has the following form: $ F_A = -dt\\wedge (*_3F(A|_{\\lbrace t\\rbrace \\times S^3})) + F(A|_{\\lbrace t\\rbrace \\times S^3}), $ where $A|_{\\lbrace t\\rbrace \\times S^3}$ is the restriction of $A$ to $\\lbrace t\\rbrace \\times S^3$ and $*_3$ is the Hodge star on $\\lbrace t\\rbrace \\times S^3$ .",
"Hence $ |F_{A,(t,\\theta )}|_{\\mathrm {op}} = \\sqrt{2}|F(A|_{\\lbrace t\\rbrace \\times S^3})_\\theta |_{\\mathrm {op}}.$ Therefore $ \\left|\\!\\left|F(A|_{\\lbrace t\\rbrace \\times S^3})\\right|\\!\\right|_{\\mathrm {op}} \\le \\tau < \\frac{1}{2}\\quad (\\forall t\\in \\mathbb {R}).$ Thus we can apply the construction of Section to $A|_{\\lbrace t\\rbrace \\times S^3}$ .",
"Fix a bundle trivialization of $E$ over $\\mathbb {R}\\times \\lbrace N\\rbrace $ .",
"(Any choice will do.)",
"Then the construction in Section gives a bundle trivialization $g$ of $E$ over $X$ satisfying $ |g(A)|_{\\lbrace t\\rbrace \\times S^3}|\\le C_\\tau \\left|\\!\\left|F(A|_{\\lbrace t\\rbrace \\times S^3})\\right|\\!\\right|_{\\mathrm {op}} \\quad (\\forall t\\in \\mathbb {R}).$ Set $A^{\\prime } := g(A)$ .",
"We consider the Chern-Simons functional $ cs(A^{\\prime }) := \\mathrm {tr}(A^{\\prime }\\wedge F_{A^{\\prime }} - \\frac{1}{3}A^{\\prime 3}).$ For $R>0$ $ \\begin{split}\\int _{[-R,R]\\times S^3}|F_A|^2d\\mathrm {vol}&= \\int _{\\lbrace R\\rbrace \\times S^3}cs(A^{\\prime })-\\int _{\\lbrace -R\\rbrace \\times S^3} cs(A^{\\prime }) \\\\&\\le \\mathrm {const}_\\tau \\left(\\left|\\!\\left|F(A|_{\\lbrace R\\rbrace \\times S^3})\\right|\\!\\right|_{\\mathrm {op}}+ \\left|\\!\\left|F(A|_{\\lbrace -R\\rbrace \\times S^3})\\right|\\!\\right|_{\\mathrm {op}}\\right).\\end{split}$ $\\left|\\!\\left|F(A|_{\\lbrace \\pm R\\rbrace \\times S^3})\\right|\\!\\right|_{\\mathrm {op}}$ are bounded as $R\\rightarrow +\\infty $ .",
"Thus $ \\int _X |F_A|^2 d\\mathrm {vol}< +\\infty .$ This implies that the curvature $F_A$ has an exponential decay at the ends (see [2]).",
"In particular $ \\left|\\!\\left|F(A|_{\\lbrace \\pm R\\rbrace \\times S^3})\\right|\\!\\right|_{\\mathrm {op}} \\rightarrow 0\\quad (R\\rightarrow +\\infty ).$ By the above (REF ) $ \\int _X |F_A|^2 d\\mathrm {vol}= 0.$ This shows $F_A \\equiv 0$ .",
"So $A$ is flat.",
"E-mail address: [email protected]"
]
] | 1204.1143 |
[
[
"Surfaces in three-dimensional space forms with divergence-free\n stress-bienergy tensor"
],
[
"Abstract We introduce the notion of biconservative hypersurfaces, that is hypersurfaces with conservative stress-energy tensor with respect to the bienergy.",
"We give the (local) classification of biconservative surfaces in 3-dimensional space forms."
],
[
"Introduction",
"A hypersurface $M^m$ in an $(m+1)$ -dimensional Riemannian manifold $N^{m+1}$ is called biconservative if $2 A(\\operatorname{grad}f)+ f \\operatorname{grad}f=2 f \\operatorname{Ricci}^N(\\eta )^{\\top }\\,,$ where $A$ is the shape operator, $f=\\operatorname{trace}A$ is the mean curvature function and $\\operatorname{Ricci}^N(\\eta )^{\\top }$ is the tangent component of the Ricci curvature of $N$ in the direction of the unit normal $\\eta $ of $M$ in $N$ .",
"The name biconservative, as we shall describe in Section , comes from the fact that condition (REF ) is equivalent to the conservativeness of a certain stress-energy tensor $S_2$ , that is $\\operatorname{div}S_2=0$ if and only if the hypersurface is biconservative.",
"The tensor $S_2$ is associated to the bienergy functional.",
"In general, a submanifold is called biconservative if $\\operatorname{div}S_2=0$ .",
"Moreover, the class of biconservative submanifolds includes that of biharmonic submanifolds, which have been of large interest in the last decade (see, for example, [1], [2], [3], [4], [9], [19], [20]).",
"Biharmonic submanifolds are characterized by the vanishing of the bitension field and they represent a generalization of harmonic (minimal) submanifolds.",
"In fact, as detailed in Section , a submanifold is biconservative if the tangent part of the bitension field vanishes.",
"It is worth to point out that, thinking at the energy functional instead of the bienergy functional, the notion of conservative submanifolds is not useful as all submanifolds are conservative (see Remark REF ).",
"We also would like to point out that submanifolds with vanishing tangent part of the bitension field have been considered by Sasahara in [22] where he studied certain 3-dimensional submanifolds in $\\mbox{${\\mathbb {R}}$}^6$ .",
"In this paper we consider biconservative surfaces in a 3-dimensional space form $N^3(c)$ of constant sectional curvature $c$ .",
"In this case (REF ) becomes $2 A(\\operatorname{grad}f)+ f \\operatorname{grad}f=0\\,.$ From (REF ) we see that CMC surfaces, i.e.",
"surfaces with constant mean curvature, in space forms are biconservative.",
"Thus our interest will be on NON CMC biconservative surfaces.",
"As a general fact, we first prove that the mean curvature function $f$ of a biconservative surface in a 3-dimensional space form satisfies the following PDE $f \\Delta f+|\\operatorname{grad}f|^2-\\frac{16}{9} K(K-c)=0\\,,$ where $K$ denotes the Gauss curvature of the surface, while $\\Delta $ is the Laplace-Beltrami operator on $M$ .",
"Then the paper is completely devoted to the local classification of biconservative surfaces in 3-dimensional space forms.",
"This is done in three sections where we examine, separately, the cases of: surfaces in the 3-dimensional euclidean space; surfaces in the 3-dimensional sphere; surfaces in the 3-dimensional hyperbolic space.",
"For biconservative surfaces in $\\mbox{${\\mathbb {R}}$}^3$ , we shall reprove a result of Hasanis and Vlachos contained in [15], where they call $H$ -surfaces the biconservative surfaces.",
"Theorem REF .",
"Let $M^2$ be a biconservative surface in $\\mbox{${\\mathbb {R}}$}^3$ with $f(p)>0$ and $\\operatorname{grad}f(p)\\ne 0$ for any $p\\in M$ .",
"Then, locally, $M^2$ is a surface of revolution.",
"In fact, we give the explicit parametrization of the profile curve of a biconservative surface of revolution (see Proposition REF ), which is not in [15].",
"In their paper, the authors also studied the case of biconservative hypersurfaces in $\\mbox{${\\mathbb {R}}$}^4$ obtaining a similar result to Theorem REF .",
"Our approach is slightly different and allow us to go further and classify the biconservative surfaces in ${\\mathbb {S}}^3$ and in $\\mathbb {H}^3$ .",
"Moreover, the notion of biconservative submanifolds is more general than the notion of $H$ -hypersurfaces in $\\mbox{${\\mathbb {R}}$}^n$ .",
"Considering ${\\mathbb {S}}^3$ as a submanifold of $\\mbox{${\\mathbb {R}}$}^4$ , the biconservative surfaces in ${\\mathbb {S}}^3$ are characterized by the following Theorem REF .",
"Let $M^2$ be a biconservative surface in ${\\mathbb {S}}^3$ with $f(p)>0$ and $\\operatorname{grad}f(p)\\ne 0$ at any point $p\\in M$ .",
"Then, locally, $M^2\\subset \\mbox{${\\mathbb {R}}$}^4$ can be parametrized by $X_C(u,v)=\\sigma (u)+\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\big (C_1 (\\cos v-1) +C_2 \\sin v \\big ),$ where $C$ is a positive constant of integration, $C_1,C_2\\in \\mbox{${\\mathbb {R}}$}^4$ are two constant orthonormal vectors such that $\\langle \\sigma (u), C_1\\rangle =\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\,,\\quad \\langle \\sigma (u), C_2\\rangle =0\\,,$ while $\\sigma =\\sigma (u)$ is a curve lying in the totally geodesic ${\\mathbb {S}}^2={\\mathbb {S}}^3\\cap \\Pi $ ($\\Pi $ the linear hyperspace of $\\mbox{${\\mathbb {R}}$}^4$ orthogonal to $C_2$ ), whose geodesic curvature $k=k(u)$ is a positive non constant solution of the following ODE $k^{\\prime \\prime } k =\\frac{7}{4} (k^{\\prime })^2+\\frac{4}{3} k^2-4 k^4\\,.$ Geometrically Theorem REF means that, locally, the surface $M^2$ is given by a family of circles of $\\mbox{${\\mathbb {R}}$}^4$ , passing through the curve $\\sigma $ , and belonging to a pencil of planes which are parallel to the linear space spanned by $C_1$ and $C_2$ .",
"Now, these circles must be the intersection of the pencil with the sphere ${\\mathbb {S}}^3$ .",
"Let $G$ be the 1-parameter group of isometries of $\\mbox{${\\mathbb {R}}$}^4$ generated by the Killing vector field $T=\\langle {\\mathbf {r}}, C_2\\rangle C_1+\\langle {\\mathbf {r}}, C_1\\rangle C_2\\,,$ where ${\\mathbf {r}}$ represents the position vector of a point in $\\mbox{${\\mathbb {R}}$}^4$ .",
"Then $G$ acts also on ${\\mathbb {S}}^3$ by isometries and it can be identified with the group $SO(2)$ .",
"Since the orbits of $G$ are circles of ${\\mathbb {S}}^3$ we deduce that $X(u,v)$ , in Theorem REF , describes an $SO(2)$ invariant surface of ${\\mathbb {S}}^3$ obtained by the action of $G$ on the curve $\\sigma $ .",
"Moreover, as we shall explain in Remark REF , there exist solutions of the ODE in Theorem REF for the corresponding profile curve $\\sigma $ .",
"Although we are not able to give explicit solutions for $\\sigma $ , as we have done for the biconservative surfaces in $\\mbox{${\\mathbb {R}}$}^3$ , using Mathematica we give a plot of a numerical solution of the ODE in Theorem REF , which describes the behavior of the curvature of $\\sigma $ .",
"Let consider the following model for the hyperbolic space $\\mathbb {H}^3 = \\lbrace (x_1,x_2,x_3,x_4)\\in {\\mathbb {L}}^4\\,:\\, x_1^2 +x_2^2 +x_3^2 -x_4^2 = -1,\\, x_4 > 0\\rbrace ,$ where ${\\mathbb {L}}^4$ is the 4-dimensional Lorentz-Minkowski space.",
"Then we have the following description of biconservative surfaces in $\\mathbb {H}^3$ .",
"Theorem REF .",
"Let $M^2$ be a biconservative surface in $\\mathbb {H}^3$ with $f(p)>0$ and $\\operatorname{grad}f(p)\\ne 0$ at any point $p\\in M$ .",
"Put $W={9|\\operatorname{grad}f|^2}/({16f^2})+{9f^2}/{4}-1$ .",
"Then, locally, $M^2\\subset {\\mathbb {L}}^4$ can be parametrized by: if $W>0$ $X_C(u,v)=\\sigma (u)+\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\big (C_1 (\\cos v-1) +C_2 \\sin v \\big ),$ where $C$ is a positive constant of integration, $C_1,C_2\\in {\\mathbb {L}}^4$ are two constant vectors such that $\\langle C_i,C_j\\rangle =\\delta _{ij}\\,,\\quad \\langle \\sigma (u), C_1\\rangle =\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\,,\\quad \\langle \\sigma (u), C_2\\rangle =0\\,,$ while $\\sigma =\\sigma (u)$ is a curve lying in the totally geodesic $\\mathbb {H}^2=\\mathbb {H}^3\\cap \\Pi $ ($\\Pi $ the linear hyperspace of ${\\mathbb {L}}^4$ defined by $\\langle {\\mathbf {r}},C_2\\rangle =0$ ), whose geodesic curvature $k=k(u)$ is a positive non constant solution of the following ODE $k^{\\prime \\prime } k =\\frac{7}{4} (k^{\\prime })^2-\\frac{4}{3} k^2-4 k^4\\,.$ if $W<0$ $X_C(u,v)=\\sigma (u)+\\frac{4}{3 \\sqrt{-C} k(u)^{3/4}}\\big (C_1 (e^v-1) +C_2 (e^{-v} -1)\\big ),$ where $C$ is a negative constant of integration, $C_1,C_2\\in {\\mathbb {L}}^4$ are two constant vectors such that $\\langle C_i,C_i\\rangle =0\\,,\\quad \\langle C_1,C_2\\rangle =-1\\,,\\quad \\langle \\sigma (u), C_1\\rangle =\\langle \\sigma (u), C_2\\rangle =-\\frac{2\\sqrt{2}}{3 \\sqrt{-C} k(u)^{3/4}}\\,,$ while $\\sigma =\\sigma (u)$ is a curve lying in the totally geodesic $\\mathbb {H}^2=\\mathbb {H}^3\\cap \\Pi $ ($\\Pi $ the linear hyperspace of ${\\mathbb {L}}^4$ orthogonal to $C_1-C_2$ ), whose geodesic curvature $k=k(u)$ is a positive non constant solution of the same ODE in (a).",
"We note that a surface in a 3-dimensional space form for which both tangent and normal part of its bitension field vanish, i.e.",
"a biharmonic surface, must be CMC (see [6], [8]).",
"Therefore, the assumption that only the tangent part of the bitension field vanishes does not imply that the surface is CMC.",
"Conventions.",
"Throughout this paper all manifolds, metrics, maps are assumed to be smooth, i.e.",
"of class $C^\\infty $ .",
"All manifolds are assumed to be connected.",
"The following sign conventions are used $\\Delta ^\\varphi V=-\\operatorname{trace}\\nabla ^2 V\\,,\\qquad R^N(X,Y)=[\\nabla _X,\\nabla _Y]-\\nabla _{[X,Y]},$ where $V\\in C(\\varphi ^{-1}(TN))$ and $X,Y\\in C(TN)$ .",
"By a submanifold $M$ in a Riemannian manifold $(N,h)$ we mean an isometric immersion $\\varphi :M\\rightarrow (N,h)$ .",
"Acknowledgement.",
"The authors would like to thank Ye-Lin Ou for reading a first draft of the paper and making some helpful suggestions."
],
[
"Biharmonic maps and the stress-energy tensor",
"As described by Hilbert in [16], the stress-energy tensor associated to a variational problem is a symmetric 2-covariant tensor $S$ conservative at critical points, i.e.",
"with $\\operatorname{div}S=0$ .",
"In the context of harmonic maps $\\varphi :(M,g)\\rightarrow (N,h)$ between two Riemannian manifolds, that by definition are critical points of the energy $E(\\varphi )=\\frac{1}{2}\\int _M\\vert d\\varphi \\vert ^2 \\ v_g,$ the stress-energy tensor was studied in detail by Baird and Eells in [5] and Sanini in [21].",
"Indeed, the Euler-Lagrange equation associated to the energy is equivalent to the vanishing of the tension field $\\tau (\\varphi )=\\operatorname{trace}\\nabla d\\varphi $ (see [11]), and the tensor $S=\\frac{1}{2}\\vert d\\varphi \\vert ^2 g - \\varphi ^{\\ast }h$ satisfies $\\operatorname{div}S=-\\langle \\tau (\\varphi ),d\\varphi \\rangle $ .",
"Therefore, $\\operatorname{div}S=0$ when the map is harmonic.",
"Remark 2.1 We point out that, in the case of isometric immersions, the condition $\\operatorname{div}S=0$ is always satisfied, since $\\tau (\\varphi )$ is normal.",
"A natural generalization of harmonic maps, first proposed in [12], can be obtained considering the bienergy of $\\varphi :(M,g)\\rightarrow (N,h)$ which is defined by $E_2(\\varphi )=\\frac{1}{2}\\int _M\\vert \\tau (\\varphi )\\vert ^2 \\ v_g.$ The map $\\varphi $ is biharmonic if it is a critical point of $E_2$ or, equivalently, if it satisfies the associated Euler-Lagrange equation $\\tau _2(\\varphi )=-\\Delta \\tau (\\varphi ) -\\operatorname{trace}R^N(d\\varphi ,\\tau (\\varphi ))d\\varphi = 0.$ The study of the stress-energy tensor for the bienergy was initiated in [17] and afterwards developed in [14].",
"Its expression is $S_2(X,Y)&=&\\frac{1}{2}\\vert \\tau (\\varphi )\\vert ^2\\langle X,Y\\rangle +\\langle d\\varphi ,\\nabla \\tau (\\varphi )\\rangle \\langle X,Y\\rangle \\\\\\nonumber && -\\langle d\\varphi (X), \\nabla _Y\\tau (\\varphi )\\rangle -\\langle d\\varphi (Y), \\nabla _X\\tau (\\varphi )\\rangle ,$ and it satisfies the condition $\\operatorname{div}S_2=-\\langle \\tau _2(\\varphi ),d\\varphi \\rangle ,$ thus conforming to the principle of a stress-energy tensor for the bienergy.",
"If $\\varphi :(M,g)\\rightarrow (N,h)$ is an isometric immersion then (REF ) becomes $\\operatorname{div}S_2=-\\tau _2(\\varphi )^{\\top }.$ This means that isometric immersions with $\\operatorname{div}S_2=0$ correspond to immersions with vanishing tangent part of the corresponding bitension field.",
"The decomposition of the bitension field with respect to its normal and tangent components was obtained with contributions of [1], [7], [13], [18], [19] and for hypersurfaces it can be summarized in the following theorem.",
"Theorem 2.2 Let $\\varphi :M^m\\rightarrow N^{m+1}$ be an isometric immersion with mean curvature vector field $H=f\\eta $ .",
"Then, $\\varphi $ is biharmonic if and only if the normal and the tangent components of $\\tau _2(\\varphi )$ vanish, i.e.",
"respectively $\\Delta {f}-f |A|^2+ f\\operatorname{Ricci}^N(\\eta ,\\eta )=0,$ and $2 A(\\operatorname{grad}f)+ f \\operatorname{grad}f-2 f \\operatorname{Ricci}^N(\\eta )^{\\top }=0$ where $A$ is the shape operator, $f=\\operatorname{trace}A$ is the mean curvature function and $\\operatorname{Ricci}^N(\\eta )^{\\top }$ is the tangent component of the Ricci curvature of $N$ in the direction of the unit normal $\\eta $ of $M$ in $N$ .",
"Finally, from (REF ), an isometric immersion $\\varphi :M^m\\rightarrow N^{m+1}$ satisfies $\\operatorname{div}S_2=0$ , i.e.",
"it is biconservative, if and only if $2 A(\\operatorname{grad}f)+ f \\operatorname{grad}f-2 f \\operatorname{Ricci}^N(\\eta )^{\\top }=0$ which is Equation (REF ) given in the introduction."
],
[
"Biconservative surfaces in the 3-dimensional space forms",
"In this section we consider the case of biconservative surfaces $M^2$ in a 3-dimensional space form $N^3(c)$ of sectional curvature $c$ .",
"In this setting (REF ) becomes $A(\\operatorname{grad}f)=-\\frac{f}{2} \\operatorname{grad}f\\,.$ If $M^2$ is a CMC surface, that is $f=\\operatorname{constant}$ , then $\\operatorname{grad}f=0$ and (REF ) is automatically satisfied.",
"Thus biconservative surfaces include the class CMC surfaces whether compact or not.",
"We now assume that $\\operatorname{grad}f\\ne 0$ at a point $p\\in M$ and, therefore, there exists a neighbourhood $U$ of $p$ such that $\\operatorname{grad}f\\ne 0$ at any point of $U$ .",
"On the set $U$ we can define an orthonormal frame $\\lbrace X_1,X_2\\rbrace $ of vector fields by $X_1=\\frac{\\operatorname{grad}f}{|\\operatorname{grad}f|}\\,,\\quad X_2\\perp X_1\\,,\\quad |X_2|=1\\,.$ From (REF ) we have $A(X_1)=-\\frac{f}{2}X_1\\,,$ thus $X_1$ is a principal direction corresponding to the principal curvature $\\lambda _1=-f/2$ .",
"Since $X_2\\perp X_1$ , $X_2$ is a principal direction with eigenvalue $\\lambda _2$ such that $f=\\operatorname{trace}A=\\lambda _1+\\lambda _2=-\\frac{f}{2}+\\lambda _2$ and therefore $\\lambda _2=3f/2$ .",
"From this, using the Weingarten equation, we immediately see that the Gauss curvature of the surface is $K=\\det A+c=- 3f^2/4+c$ and the norm of the shape operator is $|A|^2=5f^2/2$ .",
"Moreover, by the definition of $X_1$ , we obtain $(X_1 f) X_1=\\langle \\operatorname{grad}f, X_1 \\rangle X_1 =\\operatorname{grad}f.$ Thus, $\\operatorname{grad}f=(X_1 f)X_1 +(X_2 f) X_2=\\operatorname{grad}f + (X_2 f) X_2,$ which implies that $X_2 f=0.$ We are now in the right position to state the main result of this section.",
"Theorem 3.1 Let $M^2$ be a biconservative surface in $N^3(c)$ which is not CMC.",
"Then, there exists an open subset $U$ of $M$ , such that the restriction of $f$ in $U$ satisfies the following equations $K=\\det A+c=- 3f^2/4+c$ and $f \\Delta f+|\\operatorname{grad}f|^2-\\frac{16}{9} K(K-c)=0,$ where $\\Delta $ is the Laplace-Beltrami operator on $M$ .",
"Since $M^2$ is not CMC, there exists a point $p$ with $\\operatorname{grad}f(p)\\ne 0$ .",
"Thus $\\operatorname{grad}f \\ne 0$ in a neighborhood $V$ of $p$ .",
"Now, since $f$ cannot be zero for all $q\\in V$ , there exists an open set $U\\subset V$ with $f(q)\\ne 0$ for all $q\\in U$ .",
"Let us define on $U$ the local orthonormal frame $\\lbrace X_1,X_2\\rbrace $ as in (REF ) and let $\\lbrace \\omega ^1,\\omega ^2\\rbrace $ be the dual 1-forms of $\\lbrace X_1,X_2\\rbrace $ with $\\omega ^j_i$ the connection 1-forms given by $\\nabla X_i=\\omega ^j_iX_j$ .",
"Since $f\\ne 0$ on $U$ , we can assume that $f>0$ on $U$ .",
"Equation (REF ) is just (REF ).",
"We shall prove (REF ).",
"Since $A(X_1)=-(f/2) X_1$ and $A(X_2)=(3f/2) X_2$ , from the Codazzi equation $\\nabla _{X_1}A(X_2)-\\nabla _{X_2}A(X_1)=A([X_1,X_2])$ we obtain $\\left(4f \\omega ^1_2(X_1)+X_2f \\right)X_1+\\left(3X_1f +4f \\omega ^2_1(X_2)\\right)X_2=0.$ Since $X_2f=0$ and $f(p)\\ne 0$ for all $p\\in U$ , we deduce that ${\\left\\lbrace \\begin{array}{ll}\\omega ^1_2(X_1)=0\\\\\\\\\\omega ^1_2(X_2)=\\dfrac{3}{4} \\dfrac{X_1f}{f}\\,.\\end{array}\\right.",
"}$ Next, using (REF ), the Gauss curvature of $M^2$ is $K=\\langle R(X_1,X_2)X_2,X_1\\rangle =X_1(\\omega ^1_2(X_2))-(\\omega ^1_2(X_2))^2,$ that, together with (REF ), gives $-\\frac{3f^2}{4}+c=X_1(\\omega ^1_2(X_2))-(\\omega ^1_2(X_2))^2$ which is equivalent, taking into account (REF ), to $(X_1X_1f) f = \\frac{7}{4} (X_1f)^2+ \\frac{4c}{3}f^2-f^4\\,.$ Now, a straightforward computation gives $-\\Delta f= X_1X_1f - \\frac{3}{4f} (X_1f)^2\\,,$ that, substituted in (REF ), taking into account (REF ), yields the desired equation $f \\Delta f+|\\operatorname{grad}f|^2-\\frac{16}{9} K(K-c)=0.$"
],
[
"Biconservative surfaces in $\\mbox{${\\mathbb {R}}$}^3$",
"We shall now consider the case of biconservative surfaces in $\\mbox{${\\mathbb {R}}$}^3$ .",
"We start our study investigating in detail the case of surfaces of revolution.",
"Without loss of generality we can assume that the surface is (locally) parametrised by $X(u,v)=(\\rho (u) \\cos v, \\rho (u) \\sin v, u)$ where the real valued function $\\rho $ is assumed to be positive.",
"The induced metric is $ds^2=(1+\\rho ^{\\prime 2})du^2+\\rho ^2dv^2$ , and a routine calculation gives $A=\\left(\\begin{array}{cc}-\\dfrac{\\rho ^{\\prime \\prime }}{(1+\\rho ^{\\prime 2})^{3/2}}&0\\\\0& \\dfrac{1}{\\rho (1+\\rho ^{\\prime 2})^{1/2}}\\end{array}\\right)\\,.$ Thus $f= \\dfrac{1}{(1+\\rho ^{\\prime 2})^{1/2}}\\left(\\frac{1}{\\rho }-\\dfrac{\\rho ^{\\prime \\prime }}{(1+\\rho ^{\\prime 2})}\\right)\\,,$ and $\\operatorname{grad}f=\\dfrac{1}{(1+\\rho ^{\\prime 2})} f^{\\prime } \\frac{\\partial }{\\partial u}.$ Then (REF ) becomes $\\frac{f^{\\prime }}{2(1+\\rho ^{\\prime 2})^{3/2}} \\left(\\dfrac{3\\rho ^{\\prime \\prime }}{1+\\rho ^{\\prime 2}} - \\dfrac{1}{\\rho } \\right)= 0\\,.$ Proposition 4.1 Let $M^2$ be a biconservative surface of revolution in $\\mbox{${\\mathbb {R}}$}^3$ with non constant mean curvature.",
"Then, locally, the surface can be parametrized by $X_C(\\rho ,v)=(\\rho \\cos v, \\rho \\sin v, u(\\rho ))$ where $u(\\rho )=\\frac{3}{2 C}\\left(\\rho ^{1/3} \\sqrt{C\\rho ^{2/3}-1}+\\frac{1}{\\sqrt{C}} \\ln \\left[2(C \\rho ^{1/3} + \\sqrt{C^2\\rho ^{2/3}-C})\\right]\\right)\\,,$ with $C$ a positive constant and $\\rho \\in (C^{-3/2},\\infty )$ .",
"The parametrization $X_C$ consists of a family of biconservative surfaces of revolution any two of which are not locally isometric.",
"If $f$ is not constant, then from (REF ) we must have that $\\rho $ is a solution of the following ODE $3\\rho \\, \\rho ^{\\prime \\prime }=1+(\\rho ^{\\prime })^2\\,.$ We shall now integrate (REF ).",
"Using the change of variables $y=\\rho ^{\\prime 2}$ we get $3 \\dfrac{dy}{1+y}=2\\dfrac{d\\rho }{\\rho }\\,.$ Integration yields $\\rho ^{\\prime 2}=C \\rho ^{2/3}-1\\,,$ where $C$ is a positive constant.",
"Thus $\\frac{d\\rho }{\\sqrt{C \\rho ^{2/3}-1}}=\\pm du\\,.$ Now, using the change of variable $y=\\rho ^{1/3}$ , we obtain $\\frac{3 y^2}{\\sqrt{C y^2-1}}dy= \\pm du\\,.$ The latter equation can be integrated and, up to a symmetry with respect to the $xy$ -plane, followed by a translation along the vertical $z$ -axis, gives the following solution $u=u(\\rho )=\\frac{3}{2 C}\\left(\\rho ^{1/3} \\sqrt{C\\rho ^{2/3}-1}+\\frac{1}{\\sqrt{C}} \\ln \\left[2(C \\rho ^{1/3} + \\sqrt{C^2\\rho ^{2/3}-C})\\right]\\right)\\,,$ where $\\rho \\in (C^{-3/2},\\infty )$ .",
"Since the derivative of $u(\\rho )$ is $u^{\\prime }(\\rho )=\\frac{1}{\\sqrt{C\\rho ^{2/3}-1}}$ we deduce that $u(\\rho )$ is invertible for $\\rho \\in (C^{-3/2},\\infty )$ and its inverse function produces the desired solution of (REF ).",
"For a plot of the function $u(\\rho )$ see Figure REF .",
"Remark 4.2 If we denote by $\\sigma (u)=(\\rho (u),0,u)$ the profile curve of the surface described in Proposition REF and we reparametrize it by arclength, then its curvature function $k$ satisfies the ODE $k k^{\\prime \\prime }=\\frac{7}{4} (k^{\\prime })^2-4 k^4\\,.$ Moreover, the Gauss curvature and mean curvature functions of the surface are $K(\\rho ,v)=-\\frac{1}{3 C \\rho ^{8/3}}\\,,\\quad f(\\rho ,v)=\\frac{2}{3 \\sqrt{C} \\rho ^{4/3}}\\,.$ It is worth remarking that $f$ is non constant (as assumed in the Proposition REF ) and that the values of $K$ and $f$ are in accord with (REF ).",
"Figure: Plots of the function u(ρ)u(\\rho ) for C=1C=1, C=1.5C=1.5 and C=2C=2."
],
[
"The general case",
"We shall now prove that, essentially, the family described in Proposition REF gives, locally, all non CMC biconservative surfaces.",
"To achieve this we assume that $f$ is strictly positive and that $\\operatorname{grad}f\\ne 0$ at any point.",
"We define the local orthonormal frame $\\lbrace X_1,X_2\\rbrace $ as in (REF ) and from the calculations in the proof of Theorem REF we have $\\left\\lbrace \\begin{array}{ll}\\nabla _{X_1}X_1=0,&\\quad \\nabla _{X_1}X_2=0,\\\\&\\\\\\nabla _{X_2}X_1=-\\dfrac{3(X_1f)}{4f}X_2,&\\quad \\nabla _{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1\\,.\\end{array}\\right.$ Let $\\eta $ be a unit vector field normal to the surface $M$ .",
"Then, if we denote by $\\overline{\\nabla }$ the connection of $\\mbox{${\\mathbb {R}}$}^3$ , a straightforward computation gives $\\left\\lbrace \\begin{array}{ll}\\overline{\\nabla }_{X_1}X_1=-\\dfrac{f}{2}\\eta ,&\\quad \\overline{\\nabla }_{X_1}X_2=0,\\\\&\\\\\\overline{\\nabla }_{X_2}X_1=-\\dfrac{3(X_1f)}{4f}X_2,&\\quad \\overline{\\nabla }_{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta ,\\\\&\\\\\\overline{\\nabla }_{X_1}\\eta =\\dfrac{f}{2}X_1,&\\quad \\overline{\\nabla }_{X_2}\\eta =-\\dfrac{3f}{2}X_2\\,.\\end{array}\\right.$ Put $\\kappa _2\\, \\xi =\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta =\\overline{\\nabla }_{X_2}X_2$ where $\\kappa _2=\\sqrt{\\dfrac{9(X_1f)^2}{16f^2}+\\dfrac{9f^2}{4}}.$ We have the following lemma.",
"Lemma 4.3 The function $\\kappa _2$ and the vector field $\\xi $ satisfy (a) $X_2\\kappa _2=0$ ; (b) $\\overline{\\nabla }_{X_2}\\xi =-\\kappa _2\\,X_2$ ; (c) $4 (X_1\\kappa _2)/\\kappa _2=3 (X_1f)/f$ ; (d) $\\overline{\\nabla }_{X_1}\\xi =0$ .",
"From $X_2f=0$ and $[X_1,X_2]=3 (X_1f)X_2/(4f)$ , if follows that $X_2X_1f=X_1X_2f-[X_1,X_2]f=0.$ Since $\\kappa _2$ depends only on $f$ and $X_1f$ , (a) follows.",
"To prove (b), using (a) and (REF ), we have $\\overline{\\nabla }_{X_2}\\xi &=&\\frac{1}{\\kappa _2}\\overline{\\nabla }_{X_2}\\left(\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta \\right)\\\\&=& \\frac{1}{\\kappa _2}\\left( -\\dfrac{9(X_1f)^2}{16f^2} X_2- \\frac{9f^2}{4} X_2\\right)\\\\&=&-\\frac{1}{\\kappa _2}\\, \\kappa _2^2\\, X_2=-\\kappa _2\\,X_2\\,.$ To prove (c), first observe that a direct computation gives $4\\frac{X_1\\kappa _2}{\\kappa _2}=\\frac{1}{4f^4}\\frac{9f^2 (X_1f)(X_1X_1f)-9f (X_1f)^3+36 f^5 (X_1f)}{\\dfrac{9(X_1f)^2}{16f^2}+\\dfrac{9f^2}{4}}\\,.$ Then (c) is equivalent to $3\\frac{X_1f}{f}=\\frac{1}{4f^4}\\frac{9f^2 (X_1f)(X_1X_1f)-9f (X_1f)^3+36 f^5 (X_1f)}{\\dfrac{9(X_1f)^2}{16f^2}+\\dfrac{9f^2}{4}}$ which is itself equivalent to $f(X_1X_1f)-\\frac{7}{4}(X_1f)^2+f^4=0\\,.$ Now, the latter equation is (REF ) with $c=0$ (see also (REF )).",
"We now prove (d).",
"First, from a direct computation, taking into account (REF ), we have $\\overline{\\nabla }_{X_1}\\xi =\\frac{3}{4}\\left(X_1\\left(\\frac{X_1f}{f\\kappa _2}\\right)+\\frac{f^2}{\\kappa _2}\\right)X_1+\\frac{3}{2}\\left(X_1\\left(\\frac{f}{\\kappa _2}\\right)-\\frac{1}{4}\\frac{X_1f}{\\kappa _2}\\right)\\eta \\,.$ We have to show that both components are zero.",
"First $X_1\\left(\\frac{f}{\\kappa _2}\\right)-\\frac{1}{4}\\frac{X_1f}{\\kappa _2}=0$ if and only if $4 \\frac{X_1\\kappa _2}{\\kappa _2}=3 \\frac{X_1f}{f},$ which is identity (c).",
"Similarly, using (c), $X_1\\left(\\frac{X_1f}{f\\kappa _2}\\right)+\\frac{f^2}{\\kappa _2}=0$ if and only if $f(X_1X_1f)-\\frac{7}{4}(X_1f)^2+f^4=0,$ which is identity (REF ).",
"Remark 4.4 It is useful to observe that, from Lemma REF , (a)-(b), the integral curves of the vector field $X_2$ are circles in $\\mbox{${\\mathbb {R}}$}^3$ with curvature $\\kappa _2$ .",
"We are now in the right position to state the main result of this section.",
"Theorem 4.5 (See also Proposition 3.1 in [15]) Let $M^2$ be a biconservative surface in $\\mbox{${\\mathbb {R}}$}^3$ with $f(p)>0$ and $\\operatorname{grad}f(p)\\ne 0$ for any $p\\in M$ .",
"Then, locally, $M^2$ is a surface of revolution.",
"Let $\\gamma $ be an integrable curve of $X_2$ parametrized by arc-length.",
"From Lemma REF , (a)-(b), $\\gamma $ is a circle in $\\mbox{${\\mathbb {R}}$}^3$ with curvature $\\kappa _2$ , that can be parametrized by $\\gamma (s)=c_0+c_1 \\cos (\\kappa _2 s)+c_2 \\sin (\\kappa _2 s),\\quad c_0,c_1,c_2\\in \\mbox{${\\mathbb {R}}$}^3$ with $|c_1|=|c_2|=\\frac{1}{\\kappa _2}\\,,\\quad \\langle c_1,c_2\\rangle =0\\,.$ Let $p_0\\in M$ be an arbitrary point and let $\\sigma (u)$ be an integral curve of $X_1$ with $\\sigma (0)=p_0$ .",
"Consider the flow $\\phi $ of the vector field $X_2$ near the point $p_0$ .",
"Then, for all $u\\in (-\\delta ,\\delta )$ and for all $s\\in (-\\varepsilon ,\\varepsilon )$ , $\\phi _{\\sigma (u)}(s)=c_0(u)+c_1(u) \\cos (\\kappa _2(u) s)+c_2(u) \\sin (\\kappa _2(u) s),$ where the vectorial functions $c_0(u),c_1(u),c_2(u)$ , which are uniquely determined by their initial conditions, satisfy $\\sigma (u)=c_0(u)+c_1(u)\\,,\\quad |c_1(u)|=|c_2(u)|=\\frac{1}{\\kappa _2(u)}\\,,\\quad \\langle c_1(u),c_2(u)\\rangle =0\\,,$ while $\\kappa _2(u)=\\kappa _2(\\sigma (u))$ .",
"Thus, locally, the surface can be parametrized by $X(u,s)=\\phi _{\\sigma (u)}(s)\\,.$ Now, since $\\kappa _2(0)>0$ , there exists $\\delta ^{\\prime }>0$ such that for $u\\in (-\\delta ^{\\prime },\\delta ^{\\prime })$ , we have $\\kappa _2(u)>\\kappa _2(0)/2$ .",
"Then we can reparametrize $X(u,s)$ using the change of parameter $(u,s)\\rightarrow (u,v=\\kappa _2(u) s),$ where $v$ is defined in a interval which includes $(-\\kappa _2(0)\\varepsilon /2,\\kappa _2(0)\\varepsilon /2)$ .",
"With respect to the above change of parameters, the parametrization of the surface becomes $X(u,v)=c_0(u)+\\frac{1}{\\kappa _2(u)}\\left(C_1(u) \\cos (v)+C_2(u) \\sin (v)\\right),$ where $C_1(u)=\\kappa _2(u) c_1(u)\\,, \\quad C_2(u)=\\kappa _2(u) c_2(u)\\,.$ Since the integral curves of $X_2$ start (at $v=0$ ) from $\\sigma $ , we have $\\sigma (u)=X(u,0)=c_0(u)+\\frac{1}{\\kappa _2(u)} C_1(u).$ From this $X(u,v)=\\sigma (u)+\\frac{1}{\\kappa _2(u)}\\big (C_1(u) (\\cos v -1)+C_2(u) \\sin v \\big ).$ Using (REF ) we find $C_2=\\kappa _2 c_2=\\gamma ^{\\prime }(0)=X_2(\\gamma (0)),$ which implies that $C_2(u)=X_2(\\sigma (u))$ .",
"Using (REF ) again, we get $- \\kappa _2^2\\, c_1=\\gamma ^{\\prime \\prime }(0)=\\kappa _2(\\gamma (0))\\, \\xi (\\gamma (0))=\\kappa _2(u)\\, \\xi (\\sigma (u)),$ which implies that $C_1(u)=-\\xi (\\sigma (u))$ .",
"Now we shall prove that $C_1(u)$ and $C_2(u)$ are, in fact, constant vectors.",
"Indeed, taking into account Lemma REF ,(d), $\\frac{d C_1}{du}=-\\overline{\\nabla }_{\\sigma ^{\\prime }}\\xi =-\\overline{\\nabla }_{X_1}\\xi =0.$ Moreover, using (REF ), $\\frac{d C_2}{du}=\\overline{\\nabla }_{\\sigma ^{\\prime }}X_2=\\overline{\\nabla }_{X_1}X_2=0.$ Thus the image of the parametrization (REF ) is given by a 1-parameter family of circles passing through the points of $\\sigma (u)$ lying in affine planes parallel to the space spanned by $C_1$ and $C_2$ .",
"To finish the proof we need to show that the curve of the centers of the circles is a line orthogonal to $C_1$ and $C_2$ .",
"The parametrization (REF ) can be written as $X(u,v)=\\beta (u)+\\frac{1}{\\kappa _2(u)}\\big (C_1\\cos v +C_2 \\sin v \\big ),$ where $\\beta (u)=\\sigma (u)-\\frac{C_1}{\\kappa _2(u)}$ is the curve of the centers.",
"Let show that $\\beta $ is a line.",
"For this we prove that $\\beta ^{\\prime }\\wedge \\beta ^{\\prime \\prime }=0$ .",
"Since $\\sigma ^{\\prime \\prime }(u)=-\\frac{f(u)}{2}\\, \\eta (\\sigma (u)),$ where $f(u)=f(\\sigma (u))$ and $X_1\\wedge X_2=\\eta $ , we have $\\beta ^{\\prime }\\wedge \\beta ^{\\prime \\prime }&=& \\left(\\sigma ^{\\prime }-\\left(\\frac{1}{\\kappa _2}\\right)^{\\prime } C_1\\right)\\wedge \\left(\\sigma ^{\\prime \\prime }-\\left(\\frac{1}{\\kappa _2}\\right)^{\\prime \\prime }C_1\\right)\\\\&=& -\\frac{f}{2} X_1\\wedge \\eta + \\left(\\frac{1}{\\kappa _2}\\right)^{\\prime \\prime } X_1\\wedge \\xi -\\frac{f}{2} \\left(\\frac{1}{\\kappa _2}\\right)^{\\prime } \\xi \\wedge \\eta \\\\\\rm (using\\; (\\ref {eq-defxi}))&=&\\left(\\frac{f}{2}-3\\frac{f}{2} \\left(\\frac{1}{\\kappa _2}\\right)^{\\prime \\prime } \\left(\\frac{1}{\\kappa _2}\\right)+\\frac{3}{4} \\frac{X_1f}{2}\\left(\\frac{1}{\\kappa _2}\\right)\\left(\\frac{1}{\\kappa _2}\\right)^{\\prime } \\right)X_2\\,.$ Now, replacing (REF ) in $\\left(\\frac{f}{2}-3\\frac{f}{2} \\left(\\frac{1}{\\kappa _2}\\right)^{\\prime \\prime } \\left(\\frac{1}{\\kappa _2}\\right)+\\frac{3}{4} \\frac{X_1f}{2}\\left(\\frac{1}{\\kappa _2}\\right)\\left(\\frac{1}{\\kappa _2}\\right)^{\\prime } \\right)$ and using the identities (REF ) and Lemma REF , (c), we find zero.",
"Finally, $\\beta ^{\\prime }$ is clearly orthogonal to $C_2$ and $\\langle \\beta ^{\\prime },C_1\\rangle &=&\\langle X_1, C_1\\rangle - \\left(\\frac{1}{\\kappa _2}\\right)^{\\prime }\\\\&=& -\\langle X_1, \\xi \\rangle - \\left(\\frac{1}{\\kappa _2}\\right)^{\\prime }\\\\\\rm {(using\\; (\\ref {eq-defxi}))}&=& -\\frac{1}{\\kappa _2}\\left(\\frac{3}{4} \\frac{X_1f}{f}-\\frac{\\kappa _2^{\\prime }}{\\kappa _2}\\right)\\\\\\rm {(using\\; Lemma~\\ref {lem-nablax2xi}\\, (c))}&=&0\\,.$"
],
[
"Biconservative surfaces in ${\\mathbb {S}}^3$",
"In this section we consider biconservative surfaces in 3-dimensional sphere ${\\mathbb {S}}^3$ .",
"We assume that the surface is not CMC and thus we can choose $f$ to be positive and $\\operatorname{grad}f\\ne 0$ at any point of the surface.",
"We define the local orthonormal frame $\\lbrace X_1,X_2\\rbrace $ as in (REF ) and we look at ${\\mathbb {S}}^3$ as a submanifold of $\\mbox{${\\mathbb {R}}$}^4$ .",
"With this in mind and denoting by $\\nabla , \\nabla ^{{\\mathbb {S}}^3}$ and $\\overline{\\nabla }$ the connections of $M$ , ${\\mathbb {S}}^3$ and $\\mbox{${\\mathbb {R}}$}^4$ , respectively, we have at a point $\\mathbf {r} \\in M \\subset {\\mathbb {S}}^3 \\subset \\mbox{${\\mathbb {R}}$}^4$ $\\left\\lbrace \\begin{array}{ll}\\nabla ^{{\\mathbb {S}}^3}_{X_1}X_1=-\\dfrac{f}{2}\\eta ,&\\quad \\nabla ^{{\\mathbb {S}}^3}_{X_1}X_2=0,\\\\&\\\\\\nabla ^{{\\mathbb {S}}^3}_{X_2}X_1=-\\dfrac{3(X_1f)}{4f}X_2,&\\quad \\nabla ^{{\\mathbb {S}}^3}_{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta ,\\\\\\end{array}\\right.$ and $\\left\\lbrace \\begin{array}{ll}\\overline{\\nabla }_{X_1}X_1=-\\dfrac{f}{2}\\eta -{{\\mathbf {r}}},&\\quad \\overline{\\nabla }_{X_1}X_2=0,\\\\&\\\\\\overline{\\nabla }_{X_2}X_1=-\\dfrac{3(X_1f)}{4f}X_2,&\\quad \\overline{\\nabla }_{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta -{{\\mathbf {r}}},\\\\&\\\\\\overline{\\nabla }_{X_1}\\eta =\\dfrac{f}{2}X_1,&\\quad \\overline{\\nabla }_{X_2}\\eta =-\\dfrac{3f}{2}X_2\\,,\\end{array}\\right.$ where $\\eta $ is a unit vector field normal to the surface $M$ in ${\\mathbb {S}}^3$ .",
"Put $\\kappa _2\\, \\xi =\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta -{\\mathbf {r}}=\\overline{\\nabla }_{X_2}X_2$ where $\\kappa _2=\\sqrt{\\dfrac{9(X_1f)^2}{16f^2}+\\dfrac{9f^2}{4}+1}.$ We have the following analogue of Lemma REF .",
"Lemma 5.1 The function $\\kappa _2$ and the vector field $\\xi $ satisfy (a) $X_2\\kappa _2=0$ ; (b) $\\overline{\\nabla }_{X_2}\\xi =-\\kappa _2\\,X_2$ ; (c) $4 (X_1\\kappa _2)/\\kappa _2=3 (X_1f)/f$ ; (d) $\\overline{\\nabla }_{X_1}\\xi =0$ .",
"Now, let $M^2$ be a biconservative surface in ${\\mathbb {S}}^3$ with $f>0$ and $\\operatorname{grad}f\\ne 0$ at any point.",
"Then, using the same argument as in the proof of Theorem REF , we find that, locally, $M^2\\subset \\mbox{${\\mathbb {R}}$}^4$ can be parametrized by $X(u,v)=\\sigma (u)+\\frac{1}{\\kappa _2(u)}\\big (C_1(u) (\\cos v-1) +C_2(u) \\sin v \\big ),$ where $\\sigma (u)$ is an integral curve of $X_1$ , $\\kappa _2(u)=\\kappa _2(\\sigma (u))$ is the curvature of the integral curves of $X_2$ , which are circles in $\\mbox{${\\mathbb {R}}$}^4$ , and $C_1,C_2$ are two vector functions such that $|C_1|=|C_2|=1$ and $\\langle C_1,C_2\\rangle =0$ .",
"Moreover, $C_1(u)=-\\xi (\\sigma (u)),\\quad C_2(u)=X_2(\\sigma (u))\\,.$ Further, it is easy to see that $C_1$ and $C_2$ are constant vectors.",
"Then, it is clear from (REF ) that locally the surface $M^2$ is given by a family of circles of $\\mbox{${\\mathbb {R}}$}^4$ , passing through the curve $\\sigma $ , and belonging to a pencil of planes which are parallel to the linear space spanned by $C_1$ and $C_2$ .",
"Now, these circles must be the intersection of the pencil with the sphere ${\\mathbb {S}}^3$ .",
"Let $G$ be the 1-parameter group of isometries of $\\mbox{${\\mathbb {R}}$}^4$ generated by the Killing vector field $T=\\langle {\\mathbf {r}}, C_2\\rangle C_1+\\langle {\\mathbf {r}}, C_1\\rangle C_2\\,.$ Then $G$ acts also on ${\\mathbb {S}}^3$ by isometries and it can be identified with the group $SO(2)$ .",
"Since the orbits of $G$ are circles of ${\\mathbb {S}}^3$ we deduce that $X(u,v)$ , in (REF ), describes an $SO(2)$ invariant surface of ${\\mathbb {S}}^3$ obtained by the action of $G$ on the curve $\\sigma $ .",
"Moreover, we can give the following explicit construction.",
"Theorem 5.2 Let $M^2$ be a biconservative surface in ${\\mathbb {S}}^3$ with $f>0$ and $\\operatorname{grad}f\\ne 0$ at any point.",
"Then, locally, $M^2\\subset \\mbox{${\\mathbb {R}}$}^4$ can be parametrized by $X_C(u,v)=\\sigma (u)+\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\big (C_1 (\\cos v-1) +C_2 \\sin v \\big ),$ where $C$ is a positive constant of integration, $C_1,C_2\\in \\mbox{${\\mathbb {R}}$}^4$ are two constant orthonormal vectors such that $\\langle \\sigma (u), C_1\\rangle =\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\,,\\quad \\langle \\sigma (u), C_2\\rangle =0\\,,$ while $\\sigma =\\sigma (u)$ is a curve lying in the totally geodesic ${\\mathbb {S}}^2={\\mathbb {S}}^3\\cap \\Pi $ ($\\Pi $ the linear hyperspace of $\\mbox{${\\mathbb {R}}$}^4$ orthogonal to $C_2$ ), whose geodesic curvature $k=k(u)$ is a positive non constant solution of the following ODE $k^{\\prime \\prime } k =\\frac{7}{4} (k^{\\prime })^2+\\frac{4}{3} k^2-4 k^4\\,.$ From (REF ) we know that $X(u,v)=\\sigma (u)+\\frac{1}{\\kappa _2(u)}\\big (C_1 (\\cos v-1) +C_2 \\sin v \\big )\\,,$ Since $\\langle \\sigma (u), C_2\\rangle =\\langle \\sigma (u), X_2(\\sigma (u))\\rangle =0,$ we deduce that $\\sigma \\subset \\Pi $ , where $\\Pi $ is the hyperplane of $\\mbox{${\\mathbb {R}}$}^4$ defined by the equation $\\langle {\\mathbf {r}},C_2\\rangle =0$ .",
"Thus $\\sigma $ is a curve in ${\\mathbb {S}}^3\\cap \\Pi ={\\mathbb {S}}^2$ , where ${\\mathbb {S}}^2$ is a totally geodesic 2-sphere of ${\\mathbb {S}}^3$ .",
"Now, let $k$ denote the geodesic curvature of $\\sigma $ in ${\\mathbb {S}}^2$ .",
"Then, taking into account (REF ), we have $\\nabla ^{{\\mathbb {S}}^2}_{\\sigma ^{\\prime }}\\sigma ^{\\prime }=\\nabla ^{{\\mathbb {S}}^3}_{\\sigma ^{\\prime }}\\sigma ^{\\prime }=-\\frac{f(u)}{2}\\, \\eta (\\sigma (u))\\,,$ where $f(u)=f\\circ \\sigma (u)$ .",
"We deduce that $k(u)=|\\nabla ^{{\\mathbb {S}}^2}_{\\sigma ^{\\prime }}\\sigma ^{\\prime }|=f(u)/2$ .",
"From (REF ), with $c=1$ , we know that $f=f(u)$ is a solution of $f^{\\prime \\prime } f = \\frac{7}{4} (f^{\\prime })^2+ \\frac{4}{3}f^2-f^4\\,,$ which implies that $k=k(u)$ is a solution of (REF ).",
"To finish we have to compute $\\kappa _2(u)$ as a function of $k(u)$ .",
"First, by a standard argument, we find that (REF ) has the prime integral, $(k^{\\prime })^2=-\\frac{16}{9} k^2-16 k^4+Ck^{7/2}\\,,\\quad C\\in \\mbox{${\\mathbb {R}}$},\\, C>0\\,.$ Substituting (REF ) in (REF ) we find $\\kappa _2(u)=\\frac{3}{4} \\sqrt{C} k(u)^{3/4}\\,.$ Finally, using the value of $C_1$ in (REF ) and that of $\\xi $ in (REF ), we get $\\langle \\sigma (u), C_1 \\rangle =\\langle \\sigma (u), -\\xi (\\sigma (u)) \\rangle =\\frac{1}{\\kappa _2(u)}=\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\,.$ Remark 5.3 Theorem REF asserts that if $M^2$ is a biconservative surface of ${\\mathbb {S}}^3$ , then, locally, it is an $SO(2)$ -invariant surface whose profile curve $\\sigma $ satisfies (REF ) and (REF ).",
"It is worth to show that such a curve exists.",
"First, the condition in Theorem REF that $k$ is a positive non constant solution of (REF ) is not restrictive.",
"In fact, choosing the initial condition $k(u_0)>0$ and $k^{\\prime }(u_0)>0$ , from Picard's theorem there is a unique solution of (REF ) which is positive and non constant in an open interval containing $u_0$ .",
"Next, let assume that $C_1=e_3$ and $C_2=e_4$ , where $\\lbrace e_1,\\ldots ,e_4\\rbrace $ is the canonical basis of $\\mbox{${\\mathbb {R}}$}^4$ .",
"Then, using (REF ), $\\sigma $ can be explicitly described as $\\sigma (u)=(x(u),y(u),\\frac{4}{3 \\sqrt{C}}\\, k(u)^{-3/4},0)\\,,$ for some functions $x(u)$ and $y(u)$ .",
"Since $\\sigma $ is parametrized by arc-length and its curvature must be the given function $k$ (i.e.",
"$\\sigma ^{\\prime \\prime }=-k\\,\\eta -{\\mathbf {r}}$ ), the functions $x=x(u)$ and $y=y(u)$ must satisfy the system ${\\left\\lbrace \\begin{array}{ll}x^2+y^2+\\dfrac{16}{9 C}\\, k^{-3/2}=1\\vspace{8.53581pt}\\\\(x^{\\prime })^2+(y^{\\prime })^2+\\dfrac{16}{9 C}\\,\\left(\\left( k^{-3/4}\\right)^{\\prime }\\right)^2=1\\vspace{8.53581pt}\\\\(x^{\\prime \\prime })^2+(y^{\\prime \\prime })^2+\\dfrac{16}{9 C}\\,\\left(\\left( k^{-3/4}\\right)^{\\prime \\prime }\\right)^2=1+k^2\\,.\\end{array}\\right.",
"}$ Taking the derivative and using (REF )-(REF ), system (REF ) becomes ${\\left\\lbrace \\begin{array}{ll}x^2+y^2+\\dfrac{16}{9 C}\\, k^{-3/2}=1\\vspace{8.53581pt}\\\\(x^{\\prime })^2+(y^{\\prime })^2=\\dfrac{16}{9 C}\\,(1+9k^2)\\, k^{-3/2}\\vspace{8.53581pt}\\\\(x^{\\prime \\prime })^2+(y^{\\prime \\prime })^2+\\dfrac{16}{9 C}\\,(1-3k^2)^2\\, k^{-3/2}=1+k^2\\,.\\end{array}\\right.",
"}$ Now, since $k^{\\prime }\\ne 0$ , we can locally invert the function $k=k(u)$ and write $u=u(k)$ .",
"Then System (REF ) becomes ${\\left\\lbrace \\begin{array}{ll}x^2+y^2+\\dfrac{16}{9 C}\\, k^{-3/2}=1\\vspace{8.53581pt}\\\\(k^{\\prime })^2\\left(\\dfrac{dx}{dk}\\right)^2+(k^{\\prime })^2\\left(\\dfrac{dy}{dk}\\right)^2=\\dfrac{16}{9 C}\\,(1+9k^2)\\, k^{-3/2}\\vspace{8.53581pt}\\\\\\left(\\dfrac{d^2x}{dk^2}(k^{\\prime })^2+\\dfrac{dx}{dk}k^{\\prime \\prime }\\right)^2+\\left(\\dfrac{d^2y}{dk^2}(k^{\\prime })^2+\\dfrac{dy}{dk}k^{\\prime \\prime }\\right)^2+\\dfrac{16}{9 C}\\dfrac{(1-3k^2)^2}{k^{3/2}}=1+k^2\\,,\\end{array}\\right.",
"}$ where, according to (REF ), $(k^{\\prime })^2=-\\frac{16}{9} k^2-16 k^4+Ck^{7/2}\\,,\\quad k^{\\prime \\prime }=-\\frac{16}{9} k-32 k^3+\\frac{7}{4}Ck^{5/2}\\,.$ From the first equation of (REF ), we get $y(k)=\\pm \\sqrt{1-x(k)^2-\\dfrac{16}{9 C}\\, k^{-3/2}}\\,,$ that substituted in the second gives $\\dfrac{dx}{dk}&=&\\dfrac{12 x(k)}{k (9 Ck^{3/2}-16)}\\nonumber \\\\&&\\pm \\dfrac{36\\sqrt{-9 C k^{3/2} x(k)^2+9 C k^{3/2}-16}}{(9 Ck^{3/2}-16) \\sqrt{9 C k^{3/2}-144 k^2-16}}\\,.$ We note that $dx/dk\\ne 0$ .",
"In fact, if it were zero, from (REF ), we should have $x(k)=\\pm 3k/\\sqrt{1+9k^2}$ which is not constant.",
"Taking the derivative of (REF ) with respect to $k$ and replacing in it the value $dx/dk$ given in (REF ) we find that $d^2x/dk^2$ depends only on $x(k)$ and $k$ .",
"In the same way we find that $dy/dk$ and $d^2y/dk^2$ depend only on $x(k)$ and $k$ .",
"Finally, substituting in the third equation of system (REF ) the values of $dx/dk$ , $dy/dk$ , $d^2x/dk^2$ , $d^2y/dk^2$ , $k^{\\prime }$ and $k^{\\prime \\prime }$ we find an identity.",
"This means that the solution $x(k)$ of (REF ) and the corresponding $y(k)$ give a curve $\\sigma $ , as described in (REF ), which satisfies all the desired conditions.",
"Now, although we could not find an explicit solution of (REF ), which would give the curvature of the profile curve $\\sigma $ , using Mathematica we were able to plot a numerical solution as shown in Figure REF .",
"Figure: Plot of a numerical solution of ()with k(0)=1k(0)=1 and k ' (0)=1k^{\\prime }(0)=1.",
"The constant of integration is, in this case, C=169/9C=169/9."
],
[
"Biconservative surfaces in the hyperbolic space",
"Let ${\\mathbb {L}}^4$ be the 4-dimensional Lorentz-Minkowski space, that is, the real vector space $\\mbox{${\\mathbb {R}}$}^4$ endowed with the Lorentzian metric tensor $\\langle ,\\rangle $ given by $\\langle ,\\rangle = dx_1^2 +dx_2^2 +dx_3^2 -dx_4^2,$ where $(x_1,x_2,x_3,x_4)$ are the canonical coordinates of $\\mbox{${\\mathbb {R}}$}^4$ .",
"The 3-dimensional unitary hyperbolic space is given as the following hyperquadric of ${\\mathbb {L}}^4$ , $\\mathbb {H}^3 = \\lbrace (x_1,x_2,x_3,x_4)\\in {\\mathbb {L}}^4\\,:\\, x_1^2 +x_2^2 +x_3^2 -x_4^2 = -1,\\, x_4 > 0\\rbrace .$ As it is well known, the induced metric on $\\mathbb {H}^3$ from ${\\mathbb {L}}^4$ is Riemannian with constant sectional curvature $-1$ .",
"In this section we shall use this model of the hyperbolic space.",
"For convenience we shall recall that, if $X,Y$ are tangent vector fields to $\\mathbb {H}^3$ , then $\\overline{\\nabla }_{X}Y=\\nabla ^{\\mathbb {H}^3}_XY+\\langle X,Y\\rangle {\\mathbf {r}}$ where $\\overline{\\nabla }$ is the connection on ${\\mathbb {L}}^4$ , $\\nabla ^{\\mathbb {H}^3}$ is that of $\\mathbb {H}^3$ , while ${\\mathbf {r}}$ is the position vector of a point $\\mathbf {r} \\in M \\subset \\mathbb {H}^3 \\subset {\\mathbb {L}}^4$ .",
"Let $M^2$ be a biconservative surface in the 3-dimensional hyperbolic space $\\mathbb {H}^3$ .",
"We assume that the surface is not CMC and thus we can choose $f$ to be positive and $\\operatorname{grad}f\\ne 0$ at any point of the surface.",
"We define again the local orthonormal frame $\\lbrace X_1,X_2\\rbrace $ as in (REF ).",
"We have $\\left\\lbrace \\begin{array}{ll}\\nabla ^{\\mathbb {H}^3}_{X_1}X_1=-\\dfrac{f}{2}\\eta ,&\\quad \\nabla ^{\\mathbb {H}^3}_{X_1}X_2=0,\\\\&\\\\\\nabla ^{\\mathbb {H}^3}_{X_2}X_1=-\\dfrac{3(X_1f)}{4f}X_2,&\\quad \\nabla ^{\\mathbb {H}^3}_{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta ,\\\\\\end{array}\\right.$ and $\\left\\lbrace \\begin{array}{ll}\\overline{\\nabla }_{X_1}X_1=-\\dfrac{f}{2}\\eta +{{\\mathbf {r}}},&\\quad \\overline{\\nabla }_{X_1}X_2=0,\\\\&\\\\\\overline{\\nabla }_{X_2}X_1=-\\dfrac{3(X_1f)}{4f}X_2,&\\quad \\overline{\\nabla }_{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta +{{\\mathbf {r}}},\\\\&\\\\\\overline{\\nabla }_{X_1}\\eta =\\dfrac{f}{2}X_1,&\\quad \\overline{\\nabla }_{X_2}\\eta =-\\dfrac{3f}{2}X_2\\,,\\end{array}\\right.$ where $\\eta $ is a unit vector field normal to the surface $M$ tangent to $\\mathbb {H}^3$ .",
"Put $\\kappa _2\\, \\xi =\\overline{\\nabla }_{X_2}X_2=\\dfrac{3(X_1f)}{4f}X_1+\\dfrac{3f}{2}\\eta +{\\mathbf {r}}$ where $\\kappa _2=\\sqrt{\\left|\\dfrac{9(X_1f)^2}{16f^2}+\\dfrac{9f^2}{4}-1\\right|}.$ Differently from the case of surfaces in $\\mbox{${\\mathbb {R}}$}^3$ or in ${\\mathbb {S}}^3$ , in this case the quantity $W=\\dfrac{9(X_1f)^2}{16f^2}+\\dfrac{9f^2}{4}-1=\\dfrac{9|\\operatorname{grad}f|^2}{16f^2}+\\dfrac{9f^2}{4}-1$ can take both positive and negative values.",
"Taking this in consideration, we have the following analogue of Lemma REF .",
"Lemma 6.1 The function $\\kappa _2$ and the vector field $\\xi $ satisfy (a) $X_2\\kappa _2=0$ ; (b) $\\overline{\\nabla }_{X_2}\\xi =-\\varepsilon \\kappa _2\\,X_2$ ; (c) $4 (X_1\\kappa _2)/\\kappa _2=3 (X_1f)/f$ ; (d) $\\overline{\\nabla }_{X_1}\\xi =0$ , where $\\varepsilon $ is 1 when $W>0$ and is $-1$ when $W<0$ .",
"As in the case of biconservative surfaces in ${\\mathbb {S}}^3$ , we can give the following explicit construction.",
"Theorem 6.2 Let $M^2$ be a biconservative surface in $\\mathbb {H}^3$ with $f>0$ and $\\operatorname{grad}f\\ne 0$ at any point.",
"Then, locally, $M^2\\subset {\\mathbb {L}}^4$ can be parametrized by: if $W>0$ , $X_C(u,v)=\\sigma (u)+\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\big (C_1 (\\cos v-1) +C_2 \\sin v \\big ),$ where $C$ is a positive constant of integration, $C_1,C_2\\in {\\mathbb {L}}^4$ are two constant vectors such that $\\langle C_i,C_j\\rangle =\\delta _{ij}\\,,\\quad \\langle \\sigma (u), C_1\\rangle =\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\,,\\quad \\langle \\sigma (u), C_2\\rangle =0\\,,$ while $\\sigma =\\sigma (u)$ is a curve lying in the totally geodesic $\\mathbb {H}^2=\\mathbb {H}^3\\cap \\Pi $ ($\\Pi $ the linear hyperspace of ${\\mathbb {L}}^4$ defined by $\\langle {\\mathbf {r}},C_2\\rangle =0$ ), whose geodesic curvature $k=k(u)$ is a positive non constant solution of the following ODE $k^{\\prime \\prime } k =\\frac{7}{4} (k^{\\prime })^2-\\frac{4}{3} k^2-4 k^4\\,;$ if $W<0$ , $X_C(u,v)=\\sigma (u)+\\frac{4}{3 \\sqrt{-C} k(u)^{3/4}}\\big (C_1 (e^v-1) +C_2 (e^{-v} -1)\\big ),$ where $C$ is a negative constant of integration, $C_1,C_2\\in {\\mathbb {L}}^4$ are two constant vectors such that $\\langle C_i,C_i\\rangle =0\\,,\\quad \\langle C_1,C_2\\rangle =-1\\,,\\quad \\langle \\sigma (u), C_1\\rangle =\\langle \\sigma (u), C_2\\rangle =-\\frac{2\\sqrt{2}}{3 \\sqrt{-C} k(u)^{3/4}}\\,,$ while $\\sigma =\\sigma (u)$ is a curve lying in the totally geodesic $\\mathbb {H}^2=\\mathbb {H}^3\\cap \\Pi $ ($\\Pi $ the linear hyperspace of ${\\mathbb {L}}^4$ defined by $\\langle {\\mathbf {r}},C_1-C_2\\rangle =0$ ), whose geodesic curvature $k=k(u)$ is a positive non constant solution of (REF ).",
"(a).",
"In this case $W>0$ .",
"Define the local orthonormal frame $\\lbrace X_1,X_2\\rbrace $ as in (REF ).",
"Let $\\gamma (s)$ be an integral curve of $X_2$ parametrized by arc-length.",
"Then from $\\gamma ^{\\prime \\prime }(s)=\\overline{\\nabla }_{\\gamma ^{\\prime }}\\gamma ^{\\prime }=\\kappa _2(s) \\xi (s)$ and $\\gamma ^{\\prime \\prime \\prime }(s)=\\overline{\\nabla }_{\\gamma ^{\\prime }}\\gamma ^{\\prime \\prime }=-\\kappa _2^2(s) \\gamma ^{\\prime }(s)$ it follows that the parametrization $\\gamma (s)$ satisfies the following ODE $\\gamma ^{\\prime \\prime \\prime }+\\kappa _2^2\\gamma ^{\\prime }=0\\,.$ Then, as we have proceeded in the proof of Theorem REF , we find that, locally, $M^2\\subset {\\mathbb {L}}^4$ can be parametrized by $X(u,v)=\\sigma (u)+\\frac{1}{\\kappa _2(u)}\\big (C_1 (\\cos v-1) +C_2 \\sin v \\big ),$ where $\\sigma (u)$ is and integral curve of $X_1$ , $\\kappa _2(u)=\\kappa _2(\\sigma (u))$ is the curvature of the integral curves of $X_2$ and $C_1,C_2\\in {\\mathbb {L}}^4$ are two constant vectors such that $\\langle C_i,C_j\\rangle =\\delta _{ij}\\,,\\quad C_1=-\\xi (\\sigma (u))\\,,\\quad C_2=X_2(\\sigma (u))\\,.$ Since $\\langle \\sigma (u), C_2\\rangle =\\langle \\sigma (u), X_2(\\sigma (u))\\rangle =0,$ we deduce that $\\sigma \\subset \\Pi $ , where $\\Pi $ is the hyperspace of ${\\mathbb {L}}^4$ defined by the equation $\\langle {\\mathbf {r}},C_2\\rangle =0$ .",
"Thus $\\sigma $ is a curve in $\\mathbb {H}^3\\cap \\Pi =\\mathbb {H}^2$ , where $\\mathbb {H}^2$ is totally geodesic in $\\mathbb {H}^3$ .",
"Now, let $k=k(u)$ denote the geodesic curvature of $\\sigma $ in $\\mathbb {H}^2$ .",
"Then, as in the proof of Theorem REF , we find that $k$ is a solution of (REF ).",
"In order to conclude, we have to compute $\\kappa _2(u)$ as a function of $k(u)$ .",
"First, by a standard argument, we find that (REF ) has the prime integral $(k^{\\prime })^2=\\frac{16}{9} k^2-16 k^4+Ck^{7/2}\\,,\\quad C\\in \\mbox{${\\mathbb {R}}$},\\, C>0\\,.$ Substituting (REF ) in (REF ) and recalling that $k(u)=|\\nabla ^{\\mathbb {H}^3}_{\\sigma ^{\\prime }}\\sigma ^{\\prime }|=f(u)/2$ , we find $\\kappa _2(u)=\\frac{3}{4} \\sqrt{C} k(u)^{3/4}\\,.$ Finally, by using the value of $C_1$ in (REF ) and that of $\\xi $ in (REF ), we get $\\langle \\sigma (u), C_1 \\rangle =\\langle \\sigma (u), -\\xi (\\sigma (u)) \\rangle =\\frac{1}{\\kappa _2(u)}=\\frac{4}{3 \\sqrt{C} k(u)^{3/4}}\\,.$ (b).",
"In this case $W<0$ and the curve $\\gamma (s)$ satisfies the following ODE $\\gamma ^{\\prime \\prime \\prime }- \\kappa _2^2\\gamma ^{\\prime }=0.$ Thus $\\gamma (s)=c_o +c_1\\, e^{\\kappa _2 s} +c_2\\, e^{-\\kappa _2 s}$ , where, since $\\langle \\gamma ^{\\prime },\\gamma ^{\\prime }\\rangle =1$ , $c_1$ and $c_2$ are vectorial functions such that $\\langle c_1, c_1\\rangle =\\langle c_2, c_2\\rangle =0$ and $\\langle c_1, c_2\\rangle =-1/(2 \\kappa _2^2)$ .",
"It follows that, locally, $M^2\\subset {\\mathbb {L}}^4$ can be parametrized by $X(u,s)=c_0(u)+c_1(u)\\, e^{\\kappa _2(u) s} +c_2(u)\\, e^{-\\kappa _2(u) s},$ where $\\kappa _2(u)=\\kappa _2(\\sigma (u))$ , $\\sigma =\\sigma (u)$ being an integral curve of $X_1$ .",
"Now, if we perform the change of variables $v=\\kappa _2(u) s$ and use the condition $X(u,0)=\\sigma (u)$ , we obtain that the parametrization of $M^2$ in ${\\mathbb {L}}^4$ is $X(u,v)=\\sigma (u)+\\frac{1}{\\sqrt{2} \\kappa _2(u)}\\big (C_1 (e^v-1) +C_2(e^{-v}-1) \\big ),$ where $C_1,C_2\\in \\mbox{${\\mathbb {R}}$}^4$ are two constant vectors such that $\\langle C_i,C_i\\rangle =0\\,,\\quad \\langle C_1,C_2\\rangle =-1\\,,\\quad C_1+C_2=\\sqrt{2}\\,\\xi (\\sigma (u))\\,,\\quad C_1-C_2=\\sqrt{2}\\,X_2(\\sigma (u)).$ Since $\\langle \\sigma (u), C_1-C_2\\rangle =\\sqrt{2}\\langle \\sigma (u), X_2(\\sigma (u))\\rangle =0$ we deduce that $\\sigma \\subset \\Pi $ , where $\\Pi $ is the hyperspace of ${\\mathbb {L}}^4$ defined by the equation $\\langle {\\mathbf {r}},C_1-C_2\\rangle =0$ .",
"Thus $\\sigma $ is a curve in $\\mathbb {H}^3\\cap \\Pi =\\mathbb {H}^2$ , where $\\mathbb {H}^2$ is totally geodesic in $\\mathbb {H}^3$ .",
"Now, let $k(u)$ denote the geodesic curvature of $\\sigma (u)$ in $\\mathbb {H}^2$ .",
"Then $k=k(u)$ is a solution of (REF ) and, in this case, we find the same prime integral (REF ) but with the constant $C<0$ .",
"Next, as we have done in case (a), we get the value of $\\kappa _2(u)$ as a function of $k(u)$ as well as $\\langle \\sigma (u), C_1\\rangle $ and $\\langle \\sigma (u), C_2\\rangle $ as indicated in (REF ).",
"Remark 6.3 If we assume that $C_1=e_2$ and $C_2=e_1$ , where $\\lbrace e_1,\\ldots ,e_4\\rbrace $ is the canonical basis of ${\\mathbb {L}}^4$ , using an argument as in Remark REF , we can check that the curve $\\sigma (u)$ in Theorem REF (a) must be of the form $\\sigma (u)=(0,\\frac{4}{3 \\sqrt{C}}\\, k(u)^{-3/4},x(u),y(u))\\,,$ for some functions $x(u)$ and $y(u)$ which are solution of the system ${\\left\\lbrace \\begin{array}{ll}x^2-y^2+\\dfrac{16}{9 C}\\, k^{-3/2}=-1\\vspace{8.53581pt}\\\\(x^{\\prime })^2-(y^{\\prime })^2=\\dfrac{16}{9 C}\\,(9k^2-1)\\, k^{-3/2}\\vspace{8.53581pt}\\\\(x^{\\prime \\prime })^2-(y^{\\prime \\prime })^2+\\dfrac{16}{9 C}\\,(1+3k^2)^2\\, k^{-3/2}=k^2-1\\,.\\end{array}\\right.",
"}$ By a direct computation one can show that this system has a solution.",
"For the curve $\\sigma (u)$ in Theorem REF (b) we have that, choosing $C_1=e_1+e_4$ and $C_2=e_2+e_4$ , $\\sigma (u)=(y(u)-\\frac{\\sqrt{2}}{2 \\kappa _2(u)},y(u) -\\frac{\\sqrt{2}}{2 \\kappa _2(u)},x(u),y(u))\\,,$ where, in this case, $x(u)$ and $y(u)$ are solution of the system ${\\left\\lbrace \\begin{array}{ll}2 \\left(y-\\frac{\\sqrt{2}}{2 \\kappa _2}\\right)^2 +x^2-y^2=-1\\vspace{8.53581pt}\\\\2 \\left(\\left(y-\\frac{\\sqrt{2}}{2 \\kappa _2}\\right)^{\\prime }\\right)^2 +(x^{\\prime })^2-(y^{\\prime })^2=1\\vspace{8.53581pt}\\\\2 \\left(\\left(y-\\frac{\\sqrt{2}}{2 \\kappa _2}\\right)^{\\prime \\prime }\\right)^2 +(x^{\\prime \\prime })^2-(y^{\\prime \\prime })^2=k^2-1\\,.\\end{array}\\right.",
"}$ Again, using the same machineries as in Remark REF , we can check that this system has a solution.",
"Moreover, also in this case, as we have noticed in Remark REF , we can plot a numerical solution of (REF ) as shown in Figure REF .",
"Figure: Plot of a numerical solution of ()with k(0)=1k(0)=1 and k ' (0)=1k^{\\prime }(0)=1 and integration constant C=137/9C=137/9.",
"Choosing k(0)=1/4k(0)=1/4 and k ' (0)=1/5k^{\\prime }(0)=1/5we obtain a negative integration constant C=-248/225C=-248/225 (thus a solution to the case (b) of Theorem )but the qualitative behavior of kk is similar to the case C>0C>0.Remark 6.4 We have the following geometric interpretation of the surfaces described in Theorem REF (a).",
"As we have already observed, choosing $C_1=e_2$ and $C_2=e_1$ , where $\\lbrace e_1,\\ldots ,e_4\\rbrace $ is the canonical basis of ${\\mathbb {L}}^4$ , the curve $\\sigma (u)$ is of the form $\\sigma (u)=(0,\\frac{1}{\\kappa _2(u)},x(u),y(u))\\,,$ and the corresponding biconservative surface is parametrized by $X(u,v)=(\\frac{1}{\\kappa _2(u)} \\sin v, \\frac{1}{\\kappa _2(u)} \\cos v, x(u), y(u))\\,.$ Therefore, the surface is clearly given by the action, on the curve $\\sigma $ , of the group of isometries of ${\\mathbb {L}}^4$ which leaves the plane $P^2$ generated by $e_3$ and $e_4$ fixed.",
"These surfaces, following the terminology given by do Carmo and Dajczer (see [10]), are called rotational surfaces of spherical type.",
"In fact, the metric of ${\\mathbb {L}}^4$ restricted on $P^2$ is Lorentzian and when this happens, as described in [10], the orbits are circles."
]
] | 1204.1484 |
[
[
"Principal Series Representations of Infinite Dimensional Lie Groups, I:\n Minimal Parabolic Subgroups"
],
[
"Abstract We study the structure of minimal parabolic subgroups of the classical infinite dimensional real simple Lie groups, corresponding to the classical simple direct limit Lie algebras.",
"This depends on the recently developed structure of parabolic subgroups and subalgebras that are not necessarily direct limits of finite dimensional parabolics.",
"We then discuss the use of that structure theory for the infinite dimensional analog of the classical principal series representations.",
"We look at the unitary representation theory of the classical lim--compact groups $U(\\infty)$, $SO(\\infty)$ and $Sp(\\infty)$ in order to construct the inducing representations, and we indicate some of the analytic considerations in the actual construction of the induced representations."
],
[
"Introduction",
"This paper reports on some recent developments in a program of extending aspects of real semisimple group representation theory to infinite dimensional real Lie groups.",
"The finite dimensional theory is entwined with the structure of parabolic subgroups, and that structure has recently been worked out for the classical direct limit groups such as $SL(\\infty ,\\mathbb {R})$ and $Sp(\\infty ;\\mathbb {R})$ .",
"Here we explore the consequences of that structure theory for the construction of the counterpart of various Harish–Chandra series of representations, specifically the principal series.",
"The representation theory of finite dimensional real semisimple Lie groups is based on the now–classical constructions and Plancherel Formula of Harish–Chandra.",
"Let $G$ be a real semisimple Lie group, e.g.",
"$SL(n;\\mathbb {R})$ , $SU(p,q)$ , $SO(p,q)$ , .... Then one associates a series of representations to each conjugacy class of Cartan subgroups.",
"Roughly speaking this goes as follows.",
"Let $Car(G)$ denote the set of conjugacy classes $[H]$ of Cartan subgroups $H$ of $G$ .",
"Choose $[H] \\in Car(G)$ , $H \\in [H]$ , and an irreducible unitary representation $\\chi $ of $H$ .",
"Then we have a “cuspidal” parabolic subgroup $P$ of $G$ constructed from $H$ , and a unitary representation $\\pi _\\chi $ of $G$ constructed from $\\chi $ and $P$ .",
"Let $\\Theta _{\\pi _\\chi }$ denote the distribution character of $\\pi _\\chi $ .",
"The Plancherel Formula: if $f\\in \\mathcal {C}(G)$ , the Harish-Chandra Schwartz space, then $f(x) = \\sum _{[H] \\in Car(G)} \\,\\,\\int _{\\widehat{H}}\\Theta _{\\pi _\\chi }(r_xf) d\\mu _{[H]}(\\chi )\\qquad \\mathrm {(1.2)}$ where $r_x$ is right translation and $\\mu _{[H]}$ is Plancherel measure on the unitary dual $\\widehat{H}$ .",
"In order to consider any elements of this theory in the context of real semisimple direct limit groups, we have to look more closely at the construction of the Harish–Chandra series that enter into (REF ).",
"Let $H$ be a Cartan subgroup of $G$ .",
"It is stable under a Cartan involution $\\theta $ , an involutive automorphism of $G$ whose fixed point set $K = G^\\theta $ is a maximal compactly embeddedA subgroup of $G$ is compactly embedded if it has compact image under the adjoint representation of $G$ .",
"subgroup.",
"Then $H$ has a $\\theta $ –stable decomposition $T \\times A$ where $T = H \\cap K$ is the compactly embedded part and (using lower case Gothic letters for Lie algebras) $\\exp : \\mathfrak {a}\\rightarrow A$ is a bijection.",
"Then $\\mathfrak {a}$ is commutative and acts diagonalizably on $\\mathfrak {g}$ .",
"Any choice of positive $\\mathfrak {a}$ –root system defines a parabolic subalgebra $\\mathfrak {p}= \\mathfrak {m}+ \\mathfrak {a}+ \\mathfrak {n}$ in $\\mathfrak {g}$ and thus defines a parabolic subgroup $P = MAN$ in $G$ .",
"If $\\tau $ is an irreducible unitary representation of $M$ and $\\sigma \\in \\mathfrak {a}^*$ then $\\eta _{\\tau ,\\sigma }: man \\mapsto e^{i\\sigma (\\log a)}\\tau (m)$ is a well defined irreducible unitary representation of $P$ .",
"The equivalence class of the unitarily induced representation $\\pi _{\\tau ,\\sigma } ={\\rm Ind\\,}_P^G(\\eta _{\\tau ,\\sigma })$ is independent of the choice of positive $\\mathfrak {a}$ –root system.",
"The group $M$ has (relative) discrete series representations, and $\\lbrace \\pi _{\\tau ,\\sigma } \\mid \\tau \\text{ is a discrete series rep of } M\\rbrace $ is the series of unitary representations associated to $\\lbrace {\\rm Ad}(g)H \\mid g \\in G\\rbrace $ .",
"One of the most difficult points here is dealing with the discrete series.",
"In fact the possibilities of direct limit representations of direct limit groups are somewhat limited except in cases where one can pass cohomologies through direct limits without change of cohomology degree.",
"See [14] for limits of holomorphic discrete series, [15] for Bott–Borel–Weil theory in the direct limit context, [11] for some nonholomorphic discrete series cases, and [24] for principal series of classical type.",
"The principal series representations in (REF ) are those for which $M$ is compactly embedded in $G$ , equivalently the ones for which $P$ is a minimal parabolic subgroup of $G$ .",
"Here we work out the structure of the minimal parabolic subgroups of the finitary simple real Lie groups and discuss construction of the associated principal series representations.",
"As in the finite dimensional case, a minimal parabolic has structure $P = MAN$ .",
"Here $M = P \\cap K$ is a (possibly infinite) direct product of torus groups, compact classical groups such as $Spin(n)$ , $SU(n)$ , $U(n)$ and $Sp(n)$ , and their classical direct limits $Spin(\\infty )$ , $SU(\\infty )$ , $U(\\infty )$ and $Sp(\\infty )$ (modulo intersections and discrete central subgroups).",
"Since this setting is not standard we must start by sketching the background.",
"In Section we recall the classical simple real direct limit Lie algebras and Lie groups.",
"There are no surprises.",
"Section sketches their relatively recent theory of complex parabolic subalgebras.",
"It is a little bit complicated and there are some surprises.",
"Section carries those results over to real parabolic subalgebras.",
"There are no new surprises.",
"Then in Sections and we deal with Levi components and Chevalley decompositions.",
"That completes the background.",
"In Section we examine the structure real group structure of Levi components of real parabolics.",
"Then we specialize this to minimal self–normalizing parabolics in Section .",
"There the Levi components are locally isomorphic to direct sums in an explicit way of subgroups that are either the compact classical groups $SU(n)$ , $SO(n)$ or $Sp(n)$ , or their limits $SU(\\infty )$ , $SO(\\infty )$ or $Sp(\\infty )$ .",
"The Chevalley (maximal reductive part) components are slightly more complicated, for example involving extensions $1 \\rightarrow SU(*) \\rightarrow U(*) \\rightarrow T^1 \\rightarrow 1$ as well as direct products with tori and vector groups.",
"The main result is Theorem REF , which gives the structure of the minimal self–normalizing parabolics in terms similar to those of the finite dimensional case.",
"Proposition REF then gives an explicit construction for minimal parabolics with a given Levi factor.",
"In Section we discuss the various possibilities for the inducing representation.",
"There are many good choices, for example tame representations or more generally representations that are factors of type $II$ .",
"The theory is at such an early stage that the best choice is not yet clear.",
"Finally, in Section we indicate construction of the induced representations in our infinite dimensional setting.",
"Smoothness conditions do not introduce surprises, but unitarity is a problem, and we defer details of that construction to [26] and applications to [27].",
"I thank Elizabeth Dan–Cohen and Ivan Penkov for many very helpful discussions on parabolic subalgebras and Levi components."
],
[
"The Classical Simple Real Groups",
"In this section we recall the real simple countably infinite dimensional locally finite (“finitary”) Lie algebras and the corresponding Lie groups.",
"This material follows from results in [1], [2] and [6].",
"We start with the three classical simple locally finite countable–dimensional Lie algebras $\\mathfrak {g}_\\mathbb {C}= \\varinjlim \\mathfrak {g}_{n,\\mathbb {C}}$ , and their real forms $\\mathfrak {g}_\\mathbb {R}$ .",
"The Lie algebras $\\mathfrak {g}_\\mathbb {C}$ are the classical direct limits, $\\begin{aligned}& \\mathfrak {sl}(\\infty ,\\mathbb {C}) = \\varinjlim \\mathfrak {sl}(n;\\mathbb {C}),\\\\& \\mathfrak {so}(\\infty ,\\mathbb {C}) = \\varinjlim \\mathfrak {so}(2n;\\mathbb {C}) = \\varinjlim \\mathfrak {so}(2n+1;\\mathbb {C}), \\\\& \\mathfrak {sp}(\\infty ,\\mathbb {C}) = \\varinjlim \\mathfrak {sp}(n;\\mathbb {C}),\\end{aligned}\\qquad \\mathrm {(2.2)}$ where the direct systems are given by the inclusions of the form $A \\mapsto ({\\begin{matrix} A & 0 \\\\ 0 & 0 \\end{matrix}} )$ .",
"We will also consider the locally reductive algebra $\\mathfrak {gl}(\\infty ;\\mathbb {C}) = \\varinjlim \\mathfrak {gl}(n;\\mathbb {C})$ along with $\\mathfrak {sl}(\\infty ;\\mathbb {C})$ .",
"The direct limit process of (REF ) defines the universal enveloping algebras $\\begin{aligned}& \\mathcal {U}(\\mathfrak {sl}(\\infty ,\\mathbb {C})) = \\varinjlim \\mathcal {U}(\\mathfrak {sl}(n;\\mathbb {C})) \\text{ and }\\mathcal {U}(\\mathfrak {gl}(\\infty ,\\mathbb {C})) = \\varinjlim \\mathcal {U}(\\mathfrak {gl}(n;\\mathbb {C})), \\\\& \\mathcal {U}(\\mathfrak {so}(\\infty ,\\mathbb {C})) = \\varinjlim \\mathcal {U}(\\mathfrak {so}(2n;\\mathbb {C})) = \\varinjlim \\mathcal {U}(\\mathfrak {so}(2n+1;\\mathbb {C})),\\text{ and }\\\\& \\mathcal {U}(\\mathfrak {sp}(\\infty ,\\mathbb {C})) = \\varinjlim \\mathcal {U}(\\mathfrak {sp}(n;\\mathbb {C})),\\end{aligned}\\qquad \\mathrm {(2.4)}$ Of course each of these Lie algebras $\\mathfrak {g}_\\mathbb {C}$ has the underlying structure of a real Lie algebra.",
"Besides that, their real forms are as follows ([1], [2], [6]).",
"If $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {sl}(\\infty ;\\mathbb {C})$ , then $\\mathfrak {g}_\\mathbb {R}$ is one of $\\mathfrak {sl}(\\infty ;\\mathbb {R}) = \\varinjlim \\mathfrak {sl}(n;\\mathbb {R})$ , the real special linear Lie algebra; $\\mathfrak {sl}(\\infty ;\\mathbb {H}) = \\varinjlim \\mathfrak {sl}(n;\\mathbb {H})$ , the quaternionic special linear Lie algebra, given by $\\mathfrak {sl}(n;\\mathbb {H}) := \\mathfrak {gl}(n;\\mathbb {H}) \\cap \\mathfrak {sl}(2n;\\mathbb {C})$ ; $\\mathfrak {s}\\mathfrak {u}(p,\\infty ) = \\varinjlim \\mathfrak {s}\\mathfrak {u}(p,n)$ , the complex special unitary Lie algebra of real rank $p$ ; or $\\mathfrak {s}\\mathfrak {u}(\\infty ,\\infty ) = \\varinjlim \\mathfrak {s}\\mathfrak {u}(p,q)$ , complex special unitary algebra of infinite real rank.",
"If $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(\\infty ;\\mathbb {C})$ , then $\\mathfrak {g}_\\mathbb {R}$ is one of $\\mathfrak {so}(p,\\infty ) = \\varinjlim \\mathfrak {so}(p,n)$ , the real orthogonal Lie algebra of finite real rank $p$ ; $\\mathfrak {so}(\\infty ,\\infty ) = \\varinjlim \\mathfrak {so}(p,q)$ , the real orthogonal Lie algebra of infinite real rank; or $\\mathfrak {so}^*(2\\infty ) = \\varinjlim \\mathfrak {so}^*(2n)$ If $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {sp}(\\infty ;\\mathbb {C})$ , then $\\mathfrak {g}_\\mathbb {R}$ is one of $\\mathfrak {sp}(\\infty ;\\mathbb {R}) = \\varinjlim \\mathfrak {sp}(n;\\mathbb {R})$ , the real symplectic Lie algebra; $\\mathfrak {sp}(p,\\infty ) = \\varinjlim \\mathfrak {sp}(p,n)$ , the quaternionic unitary Lie algebra of real rank $p$ ; or $\\mathfrak {sp}(\\infty ,\\infty ) = \\varinjlim \\mathfrak {sp}(p,q)$ , quaternionic unitary Lie algebra of infinite real rank.",
"If $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {gl}(\\infty ;\\mathbb {C})$ , then $\\mathfrak {g}_\\mathbb {R}$ is one $\\mathfrak {gl}(\\infty ;\\mathbb {R}) = \\varinjlim \\mathfrak {gl}(n;\\mathbb {R})$ , the real general linear Lie algebra; $\\mathfrak {gl}(\\infty ;\\mathbb {H}) = \\varinjlim \\mathfrak {gl}(n;\\mathbb {H})$ , the quaternionic general linear Lie algebra; $\\mathfrak {u}(p,\\infty ) = \\varinjlim \\mathfrak {u}(p,n)$ , the complex unitary Lie algebra of finite real rank $p$ ; or $\\mathfrak {u}(\\infty ,\\infty ) = \\varinjlim \\mathfrak {u}(p,q)$ , the complex unitary Lie algebra of infinite real rank.",
"As in (REF ), given one of these Lie algebras $\\mathfrak {g}_\\mathbb {R}= \\varinjlim \\mathfrak {g}_{n,\\mathbb {R}}$ we have the universal enveloping algebra.",
"We will need it for the induced representation process.",
"As in the finite dimensional case, we use the universal enveloping algebra of the complexification.",
"Thus when we write $\\mathcal {U}(\\mathfrak {g}_\\mathbb {R})$ it is understood that we mean $\\mathcal {U}(\\mathfrak {g}_\\mathbb {C})$ .",
"The reason for this is that we will want our representations of real Lie groups to be representations on complex vector spaces.",
"The corresponding Lie groups are exactly what one expects.",
"First the complex groups, viewed either as complex groups or as real groups, $\\begin{aligned}& SL(\\infty ;\\mathbb {C}) = \\varinjlim SL(n;\\mathbb {C}) \\text{ and } GL(\\infty ;\\mathbb {C}) =\\varinjlim GL(n;\\mathbb {C}), \\\\& SO(\\infty ;\\mathbb {C}) = \\varinjlim SO(n;\\mathbb {C}) = \\varinjlim SO(2n;\\mathbb {C})= \\varinjlim SO(2n+1;\\mathbb {C}), \\\\& Sp(\\infty ;\\mathbb {C}) = \\varinjlim Sp(n;\\mathbb {C}).\\end{aligned}\\qquad \\mathrm {(2.6)}$ The real forms of the complex special and general linear groups $SL(\\infty ;\\mathbb {C})$ and $GL(\\infty ;\\mathbb {C})$ are $\\begin{aligned}& SL(\\infty ;\\mathbb {R}) \\text{ and } GL(\\infty ;\\mathbb {R}):\\text{ real special/general linear groups, } \\\\& SL(\\infty ;\\mathbb {H}): \\text{ quaternionic special linear group, } \\\\& (S)U(p,\\infty ): \\text{ (special) unitary groupsof real rank } p < \\infty , \\\\& (S)U(\\infty ,\\infty ): \\text{ (special)unitary groups of infinite real rank.",
"}\\end{aligned}\\qquad \\mathrm {(2.8)}$ The real forms of the complex orthogonal and spin groups $SO(\\infty ;\\mathbb {C})$ and $Spin(\\infty ;\\mathbb {C})$ are $\\begin{aligned}& SO(p,\\infty ) \\text{, } Spin(p;\\infty ): \\text{ real orth./spingroups of real rank } p < \\infty , \\\\& SO(\\infty ,\\infty ) \\text{, } Spin(\\infty ,\\infty ): \\text{ realorthog./spin groups of real rank } \\infty , \\\\& SO^*(2\\infty ) = \\varinjlim SO^*(2n), \\text{ which doesn't have astandard name}\\end{aligned}\\qquad \\mathrm {(2.10)}$ Here $SO^*(2n) = \\lbrace g \\in SL(n;\\mathbb {H}) \\mid g \\text{ preserves the form }\\kappa (x,y) := \\sum x^\\ell i \\bar{y}^\\ell = {^tx} i \\bar{y}\\rbrace .$ Alternatively, $SO^*(2n) = SO(2n;\\mathbb {C}) \\cap U(n,n)$ with $SO(2n;C) \\text{ defined by }(u,v) = \\sum (u_j v_{n+jr} + v_{n+j}w_j).$ Finally, the real forms of the complex symplectic group $Sp(\\infty ;\\mathbb {C})$ are $\\begin{aligned}& Sp(\\infty ;\\mathbb {R}): \\text{ real symplectic group,} \\\\& Sp(p,\\infty ): \\text{ quaternion unitary group of real rank } p < \\infty , \\text{ and }\\\\& Sp(\\infty ,\\infty ): \\text{ quaternion unitary group of infinite real rank.",
"}\\end{aligned}\\qquad \\mathrm {(2.12)}$"
],
[
"Complex Parabolic Subalgebras",
"In this section we recall the structure of parabolic subalgebras of $\\mathfrak {gl}(\\infty ;\\mathbb {C})$ , $\\mathfrak {sl}(\\infty );\\mathbb {C})$ , $\\mathfrak {so}(\\infty ;\\mathbb {C})$ and $\\mathfrak {sp}(\\infty ;\\mathbb {C})$ .",
"We follow Dan–Cohen and Penkov ([3], [4]).",
"We first describe $\\mathfrak {g}_\\mathbb {C}$ in terms of linear spaces.",
"Let $V$ and $W$ be nondegenerately paired countably infinite dimensional complex vector spaces.",
"Then $\\mathfrak {gl}(\\infty ,\\mathbb {C}) = \\mathfrak {gl}(V,W) := V\\otimes W$ consists of all finite linear combinations of the rank 1 operators $v\\otimes w: x \\mapsto \\langle w, x\\rangle v$ .",
"In the usual ordered basis of $V = \\mathbb {C}^\\infty $ , parameterized by the positive integers, and with the dual basis of $W = V^* = (\\mathbb {C}^\\infty )^*$ , we can view $\\mathfrak {gl}(\\infty ,\\mathbb {C})$ can be viewed as infinite matrices with only finitely many nonzero entries.",
"However $V$ has more exotic ordered bases, for example parameterized by the rational numbers, where the matrix picture is not intuitive.",
"The rank 1 operator $v\\otimes w$ has a well defined trace, so trace is well defined on $\\mathfrak {gl}(\\infty ,\\mathbb {C})$ .",
"Then $\\mathfrak {sl}(\\infty ,\\mathbb {C})$ is the traceless part, $\\lbrace g \\in \\mathfrak {gl}(\\infty ;\\mathbb {C}) \\mid {\\rm trace\\,\\,}g = 0\\rbrace $ .",
"In the orthogonal case we can take $V = W$ using the symmetric bilinear form that defines $\\mathfrak {so}(\\infty ;\\mathbb {C})$ .",
"Then $\\mathfrak {so}(\\infty ;\\mathbb {C}) = \\mathfrak {so}(V,V) = \\Lambda \\mathfrak {gl}(\\infty ;\\mathbb {C}) \\text{ where }\\Lambda (v\\otimes v^{\\prime }) = v\\otimes v^{\\prime } - v^{\\prime }\\otimes v.$ In other words, in an ordered orthonormal basis of $V = \\mathbb {C}^\\infty $ parameterized by the positive integers, $\\mathfrak {so}(\\infty ;\\mathbb {C})$ can be viewed as the infinite antisymmetric matrices with only finitely many nonzero entries.",
"Similarly, in the symplectic case we can take $V = W$ using the antisymmetric bilinear form that defines $\\mathfrak {sp}(\\infty ;\\mathbb {C})$ , and then $\\mathfrak {sp}(\\infty ;\\mathbb {C}) = \\mathfrak {sp}(V,V) = S\\mathfrak {gl}(\\infty ;\\mathbb {C}) \\text{ where }S(v\\otimes v^{\\prime }) = v\\otimes v^{\\prime } + v^{\\prime }\\otimes v.$ In an appropriate ordered basis of $V = \\mathbb {C}^\\infty $ parameterized by the positive integers, $\\mathfrak {sp}(\\infty ;\\mathbb {C})$ can be viewed as the infinite symmetric matrices with only finitely many nonzero entries.",
"In the finite dimensional setting, Borel subalgebra means a maximal solvable subalgebra, and parabolic subalgebra means one that contains a Borel.",
"It is the same here except that one must use locally solvable to avoid the prospect of an infinite derived series.",
"Definition 3.1 A Borel subalgebra of $\\mathfrak {g}_\\mathbb {C}$ is a maximal locally solvable subalgebra.",
"A parabolic subalgebra of $\\mathfrak {g}_\\mathbb {C}$ is a subalgebra that contains a Borel subalgebra.",
"$\\diamondsuit $ In the finite dimensional setting a parabolic subalgebra is the stabilizer of an appropriate nested sequence of subspaces (possibly with an orientation condition in the orthogonal group case).",
"In the infinite dimensional setting here, one must be very careful as to which nested sequences of subspaces are appropriate.",
"If $F$ is a subspace of $V$ then $F^\\perp $ denotes its annihilator in $W$ .",
"Similarly if ${^{\\prime }F}$ is a subspace of $W$ the ${^{\\prime }F}^\\perp $ denotes its annihilator in $V$ .",
"We say that $F$ (resp.",
"${^{\\prime }F}$ ) is closed if $F = F^{\\perp \\perp }$ (resp.",
"${^{\\prime }F} = {^{\\prime }F}^{\\perp \\perp }$ ).",
"This is the closure relation in the Mackey topology [13], i.e.",
"the weak topology for the functionals on $V$ from $W$ and on $W$ from $V$ .",
"In order to avoid repeating the following definitions later on, we make them in somewhat greater generality than we need just now.",
"Definition 3.2 Let $V$ and $W$ be countable dimensional vector spaces over a real division ring $\\mathbb {D}= \\mathbb {R}, \\mathbb {C}\\text{ or } \\mathbb {H}$ , with a nondegenerate bilinear pairing $\\langle \\cdot , \\cdot \\rangle : V \\times W \\rightarrow \\mathbb {D}$ .",
"A chain or $\\mathbb {D}$ –chain in $V$ (resp.",
"$W$ ) is a set of $\\mathbb {D}$ –subspaces totally ordered by inclusion.",
"An generalized $\\mathbb {D}$ –flag in $V$ (resp.",
"$W$ ) is an $\\mathbb {D}$ –chain such that each subspace has an immediate predecessor or an immediate successor in the inclusion ordering, and every nonzero vector of $V$ (or $W$ ) is caught between an immediate predecessor successor (IPS) pair.",
"An generalized $\\mathbb {D}$ –flag $\\mathcal {F}$ in $V$ (resp.",
"${^{\\prime }\\mathcal {F}}$ in $W$ ) is semiclosed if $F \\in \\mathcal {F}$ with $F \\ne F^{\\perp \\perp }$ implies $\\lbrace F,F^{\\perp \\perp }\\rbrace $ is an IPS pair (resp.",
"$^{\\prime }F \\in {^{\\prime }\\mathcal {F}}$ with $^{\\prime }F \\ne ^{\\prime }F^{\\perp \\perp }$ implies $\\lbrace ^{\\prime }F,^{\\prime }F^{\\perp \\perp }\\rbrace $ is an IPS pair).",
"$\\diamondsuit $ Definition 3.3 Let $\\mathbb {D}$ , $V$ and $W$ be as above.",
"Generalized $\\mathbb {D}$ –flags $\\mathcal {F}$ in $V$ and ${^{\\prime }\\mathcal {F}}$ in $W$ form a taut couple when (i) if $F \\in \\mathcal {F}$ then $F^\\perp $ is invariant by the $\\mathfrak {gl}$ –stabilizer of $^{\\prime }\\mathcal {F}$ and (ii) if $^{\\prime }F \\in {^{\\prime }\\mathcal {F}}$ then its annihilator $^{\\prime }F^\\perp $ is invariant by the $\\mathfrak {gl}$ –stabilizer of $\\mathcal {F}$ .",
"$\\diamondsuit $ In the $\\mathfrak {so}$ and $\\mathfrak {sp}$ cases one can use the associated bilinear form to identify $V$ with $W$ and $\\mathcal {F}$ with ${^{\\prime }\\mathcal {F}}$ .",
"Then we speak of a generalized flag $\\mathcal {F}$ in $V$ as self–taut.",
"If $\\mathcal {F}$ is a self–taut generalized flag in $V$ then [6] every $F \\in \\mathcal {F}$ is either isotropic or co–isotropic.",
"Example 3.4 Here is a quick peek at an obvious phenomenon introduced by infinite dimensionality.",
"Enumerate bases of $V = \\mathbb {C}^\\infty $ and $W = \\mathbb {C}^\\infty $ by $(\\mathbb {Z}^+)^2$ , say $\\lbrace v_i = v_{i_1,i_2}\\rbrace $ and $\\lbrace w_j = w_{j_1,j_2}\\rbrace $ , with $\\langle v_i,w_j\\rangle = 1$ if both $i_1 = j_1$ and $i_2 = j_2$ and $\\langle v_i,w_j\\rangle = 0$ otherwise.",
"Define $\\mathcal {F}= \\lbrace F_i\\rbrace $ ordered by inclusion where one builds up bases of the $F_i$ first with the $v_{i_1,1},\\ i_1 \\geqq 1$ and then the $v_{i_1,2},\\ i_1 \\geqq 1$ and then the $v_{i_1,3},\\ i_1 \\geqq 1$ , and so on.",
"One does the same for $^{\\prime }\\mathcal {F}$ using the $\\lbrace w_j\\rbrace $ .",
"Now these form a taut couple of semiclosed generalized flags whose ordering involves an infinite number of limit ordinals.",
"That makes it hard to use matrix methods.",
"$\\diamondsuit $ Theorem 3.5 The self–normalizing parabolic subalgebras of the Lie algebras $\\mathfrak {sl}(V,W)$ and $\\mathfrak {gl}(V,W)$ are the normalizers of taut couples of semiclosed generalized flags in $V$ and $W$ , and this is a one to one correspondence.",
"The self–normalizing parabolic subalgebras of $\\mathfrak {sp}(V)$ are the normalizers of self–taut semiclosed generalized flags in $V$ , and this too is a one to one correspondence.",
"Theorem 3.6 The self–normalizing parabolic subalgebras of $\\mathfrak {so}(V)$ are the normalizers of self–taut semiclosed generalized flags $\\mathcal {F}$ in $V$ , and there are two possibilities: the flag $\\mathcal {F}$ is uniquely determined by the parabolic, or there are exactly three self–taut generalized flags with the same stabilizer as $\\mathcal {F}$ .",
"The latter case occurs precisely when there exists an isotropic subspace $L \\in \\mathcal {F}$ with $\\dim _\\mathbb {C}L^\\perp / L = 2$ .",
"The three flags with the same stabilizer are then $\\lbrace F \\in \\mathcal {F}\\mid F \\subset L \\textrm { or } L^\\perp \\subset F \\rbrace $ $\\lbrace F \\in \\mathcal {F}\\mid F \\subset L \\textrm { or } L^\\perp \\subset F \\rbrace \\cup M_1$ $\\lbrace F \\in \\mathcal {F}\\mid F \\subset L \\textrm { or } L^\\perp \\subset F \\rbrace \\cup M_2$ where $M_1$ and $M_2$ are the two maximal isotropic subspaces containing $L$ .",
"Example 3.7 Before proceeding we indicate an example which shows that not all parabolics are equal to their normalizers.",
"Enumerate bases of $V = \\mathbb {C}^\\infty $ and $W = \\mathbb {C}^\\infty $ by rational numbers with pairing $\\langle v_q , w_r \\rangle = 1 \\text{ if } q > r, \\, \\, \\, \\,= 0 \\text{ if } q \\leqq r$ Then Span$\\lbrace v_q \\otimes w_r \\mid q \\leqq r \\rbrace $ is a Borel subalgebra of $\\mathfrak {gl}(\\infty )$ contained in $\\mathfrak {sl}(\\infty )$ .",
"This shows that $\\mathfrak {sl}(\\infty )$ is parabolic in $\\mathfrak {gl}(\\infty )$ .",
"$\\diamondsuit $ One pinpoints this situation as follows.",
"If $\\mathfrak {p}$ is a (real or complex) subalgebra of $\\mathfrak {g}_\\mathbb {C}$ and $\\mathfrak {q}$ is a quotient algebra isomorphic to $\\mathfrak {gl}(\\infty ;\\mathbb {C})$ , say with quotient map $f : \\mathfrak {p}\\rightarrow \\mathfrak {q}$ , then we refer to the composition $trace\\circ f : \\mathfrak {p}\\rightarrow \\mathbb {C}$ as an infinite trace on $\\mathfrak {g}_\\mathbb {C}$ .",
"If $\\lbrace f_i\\rbrace $ is a finite set of infinite traces on $\\mathfrak {g}_\\mathbb {C}$ and $\\lbrace c_i\\rbrace $ are complex numbers, then we refer to the condition $\\sum c_if_i = 0$ as an infinite trace condition on $\\mathfrak {p}$ .",
"These quotients can exist.",
"In Example REF we can take $V_a$ to be the span of the $v_{i_1,a}$ and $W_a$ the span of the the dual $w_{i_1,a}$ for $a = 1, 2, ...$ and then the normalizer of the taut couple $(\\mathcal {F},{^{\\prime }\\mathcal {F}})$ has infinitely many quotients $\\mathfrak {gl}(V_a,W_a)$ .",
"Theorem 3.8 The parabolic subalgebras $\\mathfrak {p}$ in $\\mathfrak {g}_\\mathbb {C}$ are the algebras obtained from self normalizing parabolics $\\widetilde{\\mathfrak {p}}$ by imposing infinite trace conditions.",
"As a general principle one tries to be explicit by constructing representations that are as close to irreducible as feasible.",
"For this reason we will be constructing principal series representations by inducing from parabolic subgroups that are minimal among the self–normalizing parabolic subgroups.",
"Still, one should be aware of the phenomenon of Example REF and Theorem REF ."
],
[
"Real Parabolic Subalgebras and Subgroups",
"In this section we discuss the structure of parabolic subalgebras of real forms of the classical $\\mathfrak {sl}(\\infty ,\\mathbb {C})$ , $\\mathfrak {so}(\\infty ,\\mathbb {C})$ , $\\mathfrak {sp}(\\infty ,\\mathbb {C})$ and $\\mathfrak {gl}(\\infty ,\\mathbb {C})$ .",
"In this section $\\mathfrak {g}_\\mathbb {C}$ will always be one of them and $G_\\mathbb {C}$ will be the corresponding connected complex Lie group.",
"Also, $\\mathfrak {g}_\\mathbb {R}$ will be a real form of $\\mathfrak {g}_\\mathbb {C}$ , and $G_\\mathbb {R}$ will be the corresponding connected real subgroup of $G_\\mathbb {C}$ .",
"Definition 4.1 Let $\\mathfrak {g}_\\mathbb {R}$ be a real form of $\\mathfrak {g}_\\mathbb {C}$ .",
"Then a subalgebra $\\mathfrak {p}_\\mathbb {R}\\subset \\mathfrak {g}_\\mathbb {R}$ is a parabolic subalgebra if its complexification $\\mathfrak {p}_\\mathbb {C}$ is a parabolic subalgebra of $\\mathfrak {g}_\\mathbb {C}$ .",
"$\\diamondsuit $ When $\\mathfrak {g}_\\mathbb {R}$ has two inequivalent defining representations, in other words when $\\mathfrak {g}_\\mathbb {R}= \\mathfrak {sl}(\\infty ;\\mathbb {R}),\\,\\, \\mathfrak {gl}(\\infty ;\\mathbb {R}),\\,\\,\\mathfrak {su}(*,\\infty ),\\,\\, \\mathfrak {u}(*,\\infty ),\\,\\,\\text{ or } \\,\\,\\mathfrak {sl}(\\infty ;\\mathbb {H})$ we denote them by $V_\\mathbb {R}$ and $W_\\mathbb {R}$ , and when $\\mathfrak {g}_\\mathbb {R}$ has only one defining representation, in other words when $\\mathfrak {g}_\\mathbb {R}= \\mathfrak {so}(*,\\infty ),\\,\\, \\mathfrak {sp}(*,\\infty ),\\,\\,\\mathfrak {sp}(\\infty ;\\mathbb {R}),\\,\\, \\text{ or } \\,\\,\\mathfrak {so}^*(2\\infty )\\text{ as quaternion matrices,}$ we denote it by $V_\\mathbb {R}$ .",
"The commuting algebra of $\\mathfrak {g}_\\mathbb {R}$ on $V_\\mathbb {R}$ is a real division algebra $\\mathbb {D}$ .",
"The main result of [6] is Theorem 4.2 Suppose that $\\mathfrak {g}_\\mathbb {R}$ has two inequivalent defining representations.",
"Then a subalgebra of $\\mathfrak {g}_\\mathbb {R}$ (resp.",
"subgroup of $G_\\mathbb {R}$ ) is parabolic if and only if it is defined by infinite trace conditions (resp.",
"infinite determinant conditions) on the $\\mathfrak {g}_\\mathbb {R}$ –stabilizer (resp.",
"$G_\\mathbb {R}$ –stabilizer) of a taut couple of generalized $\\mathbb {D}$ –flags $\\mathcal {F}$ in $V_\\mathbb {R}$ and $^{\\prime }\\mathcal {F}$ in $W_\\mathbb {R}$ .",
"Suppose that $\\mathfrak {g}_\\mathbb {R}$ has only one defining representation.",
"A subalgebra of $\\mathfrak {g}_\\mathbb {R}$ (resp.",
"subgroup) of $G_\\mathbb {R}$ is parabolic if and only if it is defined by infinite trace conditions (resp.",
"infinite determinant conditions) on the $\\mathfrak {g}_\\mathbb {R}$ –stabilizer (resp.",
"$G_\\mathbb {R}$ –stabilizer) of a self–taut generalized $\\mathbb {D}$ –flag $\\mathcal {F}$ in $V_\\mathbb {R}$ ."
],
[
"Levi Components of Complex Parabolics",
"In this section we discuss Levi components of complex parabolic subalgebras, recalling results from [8], [9], [4], [10], [5] and [25].",
"We start with the definition.",
"Definition 5.1 Let $\\mathfrak {p}$ be a locally finite Lie algebra and $\\mathfrak {r}$ its locally solvable radical.",
"A subalgebra $\\mathfrak {l}\\subset \\mathfrak {p}$ is a Levi component if $[\\mathfrak {p},\\mathfrak {p}]$ is the semidirect sum $(\\mathfrak {r}\\cap [\\mathfrak {p},\\mathfrak {p}]) \\mathfrak {l}$ .$\\diamondsuit $ Every finitary Lie algebra has a Levi component.",
"Evidently, Levi components are maximal semisimple subalgebras, but the converse fails for finitary Lie algebras.",
"In any case, parabolic subalgebras of our classical Lie algebras $\\mathfrak {g}_\\mathbb {C}$ have maximal semisimple subalgebras, and those are their Levi components.",
"Definition 5.2 Let $X \\subset V$ and $Y \\subset W$ be paired subspaces, isotropic in the orthogonal and symplectic cases.",
"The subalgebras $\\begin{aligned}&\\mathfrak {gl}(X,Y) \\subset \\mathfrak {gl}(V,W) \\phantom{an}\\text{ and } \\mathfrak {sl}(X,Y) \\subset \\mathfrak {sl}(V,W),\\\\&\\Lambda \\mathfrak {gl}(X,Y) \\subset \\Lambda \\mathfrak {gl}(V,V) \\text{ and }S\\mathfrak {gl}(X,Y) \\subset S\\mathfrak {gl}(V,V)\\end{aligned}$ are called standard.$\\diamondsuit $ Proposition 5.3 A subalgebra $\\mathfrak {l}_\\mathbb {C}\\subset \\mathfrak {g}_\\mathbb {C}$ is the Levi component of a parabolic subalgebra of $\\mathfrak {g}_\\mathbb {C}$ if and only if it is the direct sum of standard special linear subalgebras and at most one subalgebra $\\Lambda \\mathfrak {gl}(X,Y)$ in the orthogonal case, at most one subalgebra $S\\mathfrak {gl}(X,Y)$ in the symplectic case.",
"The occurrence of “at most one subalgebra” in Proposition REF is analogous to the finite dimensional case, where it is seen by deleting some simple root nodes from a Dynkin diagram.",
"Let $\\mathfrak {p}$ be the parabolic subalgebra of $\\mathfrak {sl}(V, W)$ or $\\mathfrak {gl}(V, W)$ defined by the taut couple $(\\mathcal {F}, {^{\\prime }\\mathcal {F}})$ of semiclosed generalized flags.",
"Denote $\\begin{aligned}&J = \\lbrace (F^{\\prime },F^{\\prime \\prime }) \\text{ IPS pair in } \\mathcal {F}\\mid F^{\\prime } = (F^{\\prime })^{\\perp \\perp }\\text{ and } \\dim F^{\\prime \\prime }/F^{\\prime } > 1\\rbrace ,\\\\&^{\\prime }J = \\lbrace (^{\\prime }F^{\\prime },{^{\\prime }F^{\\prime \\prime }}) \\text{ IPS pair in } {^{\\prime }\\mathcal {F}} \\mid {^{\\prime }F}^{\\prime } = (^{\\prime }F^{\\prime })^{\\perp \\perp },\\dim {^{\\prime }F^{\\prime \\prime }}/{^{\\prime }F^{\\prime }} > 1\\rbrace .\\end{aligned}\\qquad \\mathrm {(5.5)}$ Since $V \\times W \\rightarrow \\mathbb {C}$ is nondegenerate the sets $J$ and $^{\\prime }J$ are in one to one correspondence by: $(F^{\\prime \\prime }/F^{\\prime }) \\times ({^{\\prime }F^{\\prime \\prime }}/{^{\\prime }F^{\\prime }}) \\rightarrow \\mathbb {C}$ is nondegenerate.",
"We use this to identify $J$ with $J^{\\prime }$ , and we write $(F_j^{\\prime },F_j^{\\prime \\prime })$ and $(^{\\prime }F_j^{\\prime },{^{\\prime }F_j^{\\prime \\prime }})$ treating $J$ as an index set.",
"Theorem 5.6 Let $\\mathfrak {p}$ be the parabolic subalgebra of $\\mathfrak {sl}(V, W)$ or $\\mathfrak {gl}(V, W)$ defined by the taut couple $\\mathcal {F}$ and $^{\\prime }\\mathcal {F}$ of semiclosed generalized flags.",
"For each $j \\in J$ choose a subspace $X_j \\subset V$ and a subspace $Y_j \\subset W$ such that $F_j^{\\prime \\prime } = X_j + F_j^{\\prime }$ and $^{\\prime }F_j^{\\prime \\prime } = Y_j + {^{\\prime }F_j}^{\\prime }$ Then $\\bigoplus _{j \\in J}\\, \\mathfrak {sl}(X_j,Y_j)$ is a Levi component of $\\mathfrak {p}$ .",
"The inclusion relations of $\\mathcal {F}$ and $^{\\prime }\\mathcal {F}$ induce a total order on $J$ .",
"Conversely, if $\\mathfrak {l}$ is a Levi component of $\\mathfrak {p}$ then there exist subspaces $X_j \\subset V$ and $Y_j \\subset W$ such that $\\mathfrak {l}= \\bigoplus _{j \\in J}\\, \\mathfrak {sl}(X_j,Y_j)$ .",
"Now the idea of finite matrices with blocks down the diagonal suggests the construction of $\\mathfrak {p}$ from the totally ordered set $J$ and the direct sum $\\mathfrak {l}= \\bigoplus _{j \\in J}\\, \\mathfrak {sl}(X_j,Y_j)$ of standard special linear algebras.",
"We outline the idea of the construction; see [5].",
"First, $\\langle X_j, Y_{j^{\\prime }}\\rangle = 0$ for $j \\ne j^{\\prime }$ because the $\\mathfrak {s}_j = \\mathfrak {sl}(X_j,Y_j)$ commute with each other.",
"Define $U_j := (( \\bigoplus _{k \\leqq j}\\, X_k)^\\perp \\oplus Y_j)^\\perp $ .",
"Then one proves $U_j = ((U_j \\oplus X_j)^\\perp \\oplus Y_j)^\\perp $ .",
"From that, one shows that there is a unique semiclosed generalized flag $\\mathcal {F}_{min}$ in $V$ with the same stabilizer as the set $\\lbrace U_j, U_j \\oplus X_j\\, |\\, j \\in J\\rbrace $ .",
"One constructs similar subspaces $^{\\prime }U_j \\subset W$ and shows that there is a unique semiclosed generalized flag $^{\\prime }\\mathcal {F}_{min}$ in $W$ with the same stabilizer as the set $\\lbrace ^{\\prime }U_j, {^{\\prime }U}_j \\oplus Y_j \\, |\\, j \\in J\\rbrace $ .",
"In fact $(\\mathcal {F}_{min} , {^{\\prime }\\mathcal {F}}_{min})$ is the minimal taut couple with IPS pairs $U_j \\subset (U_j \\oplus X_j)$ in $\\mathcal {F}_0$ and $(U_j \\oplus X_j)^\\perp \\subset ((U_j \\oplus X_j)^\\perp \\oplus Y_j)$ in ${^{\\prime }\\mathcal {F}}_0$ for $j \\in J$ .",
"If $(\\mathcal {F}_{max}, {^{\\prime }\\mathcal {F}}_{max})$ is maximal among the taut couples of semiclosed generalized flags with IPS pairs $U_j \\subset (U_j \\oplus X_j)$ in $\\mathcal {F}_{max}$ and $(U_j \\oplus X_j)^\\perp \\subset ((U_j \\oplus X_j)^\\perp \\oplus Y_j)$ in ${^{\\prime }\\mathcal {F}}_{max}$ then the corresponding parabolic $\\mathfrak {p}$ has Levi component $\\mathfrak {l}$ .",
"The situation is essentially the same for Levi components of parabolic subalgebras of $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(\\infty ;\\mathbb {C}) \\text{ or }\\mathfrak {sp}(\\infty ;\\mathbb {C})$ , except that we modify the definition (REF ) of $J$ to add the condition that $F^{\\prime \\prime }$ be isotropic, and we add the orientation aspect of the $\\mathfrak {so}$ case.",
"Theorem 5.7 Let $\\mathfrak {p}$ be the parabolic subalgebra of $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(V)$ or $\\mathfrak {sp}(V)$ , defined by the self–taut semiclosed generalized flag $\\mathcal {F}$ .",
"Let $\\widetilde{F}$ be the union of all subspaces $F^{\\prime \\prime }$ in IPS pairs $(F^{\\prime },F^{\\prime \\prime })$ of $\\mathcal {F}$ for which $F^{\\prime \\prime }$ is isotropic.",
"Let $\\widetilde{^{\\prime }F}$ be the intersection of all subspaces $F^{\\prime }$ in IPS pairs for which $F^{\\prime }$ is closed ($F^{\\prime } = (F^{\\prime })^{\\perp \\perp }$ ) and coisotropic.",
"Then $\\mathfrak {l}$ is a Levi component of $\\mathfrak {p}$ if and only if there are isotropic subspaces $X_j, Y_j$ in $V$ such that $\\text{$F^{\\prime \\prime }_j =F^{\\prime }_j + X_j$ and ${^{\\prime }F^{\\prime \\prime }_j} ={^{\\prime }F_j} +Y_j$ for every $j \\in J$}$ and a subspace $Z$ in $V$ such that $\\widetilde{F} = Z + \\widetilde{^{\\prime }F}$ , where $Z = 0$ in case $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(V)$ and $\\dim \\widetilde{F}/\\widetilde{^{\\prime }F} \\leqq 2$ , such that $\\begin{aligned}&\\mathfrak {l}= \\mathfrak {sp}(Z) \\oplus {\\bigoplus }_{j \\in J}\\ \\mathfrak {sl}(X_j,Y_j) \\text{ if }\\mathfrak {g}_\\mathbb {C}= \\mathfrak {sp}(V),\\\\&\\mathfrak {l}= \\mathfrak {so}(Z) \\oplus {\\bigoplus }_{j \\in J}\\ \\mathfrak {sl}(X_j,Y_j) \\text{ if }\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(V).\\end{aligned}$ Further, the inclusion relations of $\\mathcal {F}$ induce a total order on $J$ which leads to a construction of $\\mathfrak {p}$ from $\\mathfrak {l}$ ."
],
[
"Chevalley Decomposition",
"In this section we apply the extension [4] to our parabolic subalgebras, of the Chevalley decomposition for a (finite dimensional) algebraic Lie algebra.",
"Let $\\mathfrak {p}$ be a locally finite linear Lie algebra, in our case a subalgebra of $\\mathfrak {gl}(\\infty )$ .",
"Every element $\\xi \\in \\mathfrak {p}$ has a Jordan canonical form, yielding a decomposition $\\xi = \\xi _{ss} + \\xi _{nil}$ into semisimple and nilpotent parts.",
"The algebra $\\mathfrak {p}$ is splittable if it contains the semisimple and the nilpotent parts of each of its elements.",
"Note that $\\xi _{ss}$ and $\\xi _{nil}$ are polynomials in $\\xi $ ; this follows from the finite dimensional fact.",
"In particular, if $X$ is any $\\xi $ –invariant subspace of $V$ then it is invariant under both $\\xi _{ss}$ and $\\xi _{nil}$ .",
"Conversely, parabolic subalgebras (and many others) of our classical Lie algebras $\\mathfrak {g}$ are splittable.",
"The linear nilradical of a subalgebra $\\mathfrak {p}\\subset \\mathfrak {g}$ is the set $\\mathfrak {p}_{nil}$ of all nilpotent elements of the locally solvable radical $\\mathfrak {r}$ of $\\mathfrak {p}$ .",
"It is a locally nilpotent ideal in $\\mathfrak {p}$ and satisfies $\\mathfrak {p}_{nil} \\cap [\\mathfrak {p}, \\mathfrak {p}] = \\mathfrak {r}\\cap [\\mathfrak {p}, \\mathfrak {p}]$ .",
"If $\\mathfrak {p}$ is splittable then it has a well defined maximal locally reductive subalgebra $\\mathfrak {p}_{red}$ .",
"This means that $\\mathfrak {p}_{red}$ is an increasing union of finite dimensional reductive Lie algebras, each reductive in the next.",
"In particular $\\mathfrak {p}_{red}$ maps isomorphically under the projection $\\mathfrak {p}\\rightarrow \\mathfrak {p}/\\mathfrak {p}_{nil}$ .",
"That gives a semidirect sum decomposition $\\mathfrak {p}= \\mathfrak {p}_{nil} \\mathfrak {p}_{red}$ analogous to the Chevalley decomposition mentioned above.",
"Also, here, $\\mathfrak {p}_{red} = \\mathfrak {l}\\mathfrak {t}\\quad \\text{ and } \\quad [\\mathfrak {p}_{red},\\mathfrak {p}_{red}] = \\mathfrak {l}$ where $\\mathfrak {t}$ is a toral subalgebra and $\\mathfrak {l}$ is the Levi component of $\\mathfrak {p}$ .",
"A glance at $\\mathfrak {u}(\\infty )$ or $\\mathfrak {gl}(\\infty ;\\mathbb {C})$ shows that the semidirect sum decomposition of $\\mathfrak {p}_{red}$ need not be direct."
],
[
"Levi and Chevalley Components of Real Parabolics",
"Now we adapt the material of Sections and to study Levi and Chevalley components of real parabolic subalgebras in the real classical Lie algebras.",
"Let $\\mathfrak {g}_\\mathbb {R}$ be a real form of a classical locally finite complex simple Lie algebra $\\mathfrak {g}_\\mathbb {C}$ .",
"Consider a real parabolic subalgebra $\\mathfrak {p}_\\mathbb {R}$ .",
"It has form $\\mathfrak {p}_\\mathbb {R}= \\mathfrak {p}_\\mathbb {C}\\cap \\mathfrak {g}_\\mathbb {R}$ where its complexification $\\mathfrak {p}_\\mathbb {C}$ is parabolic in $\\mathfrak {g}_\\mathbb {C}$ .",
"Let $\\tau $ denote complex conjugation of $\\mathfrak {g}_\\mathbb {C}$ over $\\mathfrak {g}_\\mathbb {R}$ .",
"Then the locally solvable radical $\\mathfrak {r}_\\mathbb {C}$ of $\\mathfrak {p}_\\mathbb {C}$ is $\\tau $ –stable because $\\mathfrak {r}_\\mathbb {C}+ \\tau \\mathfrak {r}_\\mathbb {C}$ is a locally solvable ideal, so the locally solvable radical $\\mathfrak {r}_\\mathbb {R}$ of $\\mathfrak {p}_\\mathbb {R}$ is a real form of $\\mathfrak {r}_\\mathbb {C}$ .",
"Let $\\mathfrak {l}_\\mathbb {R}$ be a maximal semisimple subalgebra of $\\mathfrak {p}_\\mathbb {R}$ .",
"Its complexification $\\mathfrak {l}_\\mathbb {C}$ is a maximal semisimple subalgebra, hence a Levi component, of $\\mathfrak {p}_\\mathbb {C}$ .",
"Thus $[\\mathfrak {p}_\\mathbb {C},\\mathfrak {p}_\\mathbb {C}]$ is the semidirect sum $(\\mathfrak {r}_\\mathbb {C}\\cap [\\mathfrak {p}_\\mathbb {C},\\mathfrak {p}_\\mathbb {C}]) \\mathfrak {l}_\\mathbb {C}$ .",
"The elements of this formula all are $\\tau $ –stable, so we have proved Lemma 7.1 The Levi components of $\\mathfrak {p}_\\mathbb {R}$ are real forms of the Levi components of $\\mathfrak {p}_\\mathbb {C}$ .",
"If $\\mathfrak {g}_\\mathbb {C}$ is $\\mathfrak {sl}(V,W)$ or $\\mathfrak {gl}(V,W)$ as in Theorem REF , then $\\mathfrak {l}_\\mathbb {C}= \\bigoplus _{j \\in J}\\, \\mathfrak {sl}(X_j,Y_j)$ as indicated there.",
"Initially the possibilities for the action of $\\tau $ are $\\tau $ preserves $\\mathfrak {sl}(X_j,Y_j)$ with fixed point set $\\mathfrak {sl}(X_{j,\\mathbb {R}},Y_{j,\\mathbb {R}}) \\cong \\mathfrak {sl}(*;\\mathbb {R})$ , $\\tau $ preserves $\\mathfrak {sl}(X_j,Y_j)$ with fixed point set $\\mathfrak {sl}(X_{j,\\mathbb {H}},Y_{j,\\mathbb {H}}) \\cong \\mathfrak {sl}(*;\\mathbb {H})$ , $\\tau $ preserves $\\mathfrak {sl}(X_j,Y_j)$ with f.p.",
"set $\\mathfrak {su}(X^{\\prime }_j,X_j^{\\prime \\prime }) \\cong \\mathfrak {su}(*,*)$ , $X_j = X^{\\prime }_j + X_j^{\\prime \\prime }$ , and $\\tau $ interchanges two summands $\\mathfrak {sl}(X_j,Y_j)$ and $\\mathfrak {sl}(X_{j^{\\prime }},Y_{j^{\\prime }})$ of $\\mathfrak {l}_\\mathbb {C}$ , with fixed point set the diagonal ($\\cong \\mathfrak {sl}(X_j,Y_j)$ ) of their direct sum.",
"If $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(V)$ as in Theorem REF , $\\mathfrak {l}_\\mathbb {C}$ can also have a summand $\\mathfrak {so}(Z)$ , or if $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {sp}(V)$ it can also have a summand $\\mathfrak {sp}(V)$ .",
"Except when $A_4 = D_3$ occurs, these additional summands must be $\\tau $ –stable, resulting in fixed point sets when $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {so}(V)$ : $\\mathfrak {so}(Z)^\\tau $ is $\\mathfrak {so}(*,*)$ or $\\mathfrak {so}^*(2\\infty )$ , when $\\mathfrak {g}_\\mathbb {C}= \\mathfrak {sp}(V)$ : $\\mathfrak {sp}(Z)^\\tau $ is $\\mathfrak {sp}(*,*)$ or $\\mathfrak {sp}(*;\\mathbb {R})$ ."
],
[
"Minimal Parabolic Subgroups",
"We describe the structure of minimal parabolic subgroups of the classical real simple Lie groups $G_\\mathbb {R}$ .",
"Proposition 8.1 Let $\\mathfrak {p}_\\mathbb {R}$ be a parabolic subalgebra of $\\mathfrak {g}_\\mathbb {R}$ and $\\mathfrak {l}_\\mathbb {R}$ a Levi component of $\\mathfrak {p}_\\mathbb {R}$ .",
"If $\\mathfrak {p}_\\mathbb {R}$ is a minimal parabolic subalgebra then $\\mathfrak {l}_\\mathbb {R}$ is a direct sum of finite dimensional compact algebras $\\mathfrak {su}(p)$ , $\\mathfrak {so}(p)$ and $\\mathfrak {sp}(p)$ , and their infinite dimensional limits $\\mathfrak {su}(\\infty )$ , $\\mathfrak {so}(\\infty )$ and $\\mathfrak {sp}(\\infty )$ .",
"If $\\mathfrak {l}_\\mathbb {R}$ is a direct sum of finite dimensional compact algebras $\\mathfrak {su}(p)$ , $\\mathfrak {so}(p)$ and $\\mathfrak {sp}(p)$ and their limits $\\mathfrak {su}(\\infty )$ , $\\mathfrak {so}(\\infty )$ and $\\mathfrak {sp}(\\infty )$ , then $\\mathfrak {p}_\\mathbb {R}$ contains a minimal parabolic subalgebra of $\\mathfrak {g}_\\mathbb {R}$ with the same Levi component $\\mathfrak {l}_\\mathbb {R}$ .",
"Suppose that $\\mathfrak {p}_\\mathbb {R}$ is a minimal parabolic subalgebra of $\\mathfrak {g}_\\mathbb {R}$ .",
"If a direct summand $\\mathfrak {l}^{\\prime }_\\mathbb {R}$ of $\\mathfrak {l}_\\mathbb {R}$ has a proper parabolic subalgebra $\\mathfrak {q}_\\mathbb {R}$ , we replace $\\mathfrak {l}^{\\prime }_\\mathbb {R}$ by $\\mathfrak {q}_\\mathbb {R}$ in $\\mathfrak {l}_\\mathbb {R}$ and $\\mathfrak {p}_\\mathbb {R}$ .",
"In other words we refine the flag(s) that define $\\mathfrak {p}_\\mathbb {R}$ .",
"The refined flag defines a parabolic $\\mathfrak {q}_\\mathbb {R}\\subsetneqq \\mathfrak {p}_\\mathbb {R}$ .",
"This contradicts minimality.",
"Thus no summand of $\\mathfrak {l}_\\mathbb {R}$ has a proper parabolic subalgebra.",
"Theorems REF and REF show that $\\mathfrak {su}(p)$ , $\\mathfrak {so}(p)$ and $\\mathfrak {sp}(p)$ , and their limits $\\mathfrak {su}(\\infty )$ , $\\mathfrak {so}(\\infty )$ and $\\mathfrak {sp}(\\infty )$ , are the only possibilities for the simple summands of $\\mathfrak {l}_\\mathbb {R}$ .",
"Conversely suppose that the summands of $\\mathfrak {l}_\\mathbb {R}$ are $\\mathfrak {su}(p)$ , $\\mathfrak {so}(p)$ and $\\mathfrak {sp}(p)$ or their limits $\\mathfrak {su}(\\infty )$ , $\\mathfrak {so}(\\infty )$ and $\\mathfrak {sp}(\\infty )$ .",
"Let $(\\mathcal {F}, {^{\\prime }\\mathcal {F}})$ or $\\mathcal {F}$ be the flag(s) that define $\\mathfrak {p}_\\mathbb {R}$ .",
"In the discussion between Theorems REF and REF we described a a minimal taut couple $(\\mathcal {F}_{min}, {^{\\prime }\\mathcal {F}}_{min})$ and a maximal taut couple $(\\mathcal {F}_{max}, {^{\\prime }\\mathcal {F}}_{max})$ (in the $\\mathfrak {sl}$ and $\\mathfrak {gl}$ cases) of semiclosed generalized flags which define parabolics that have the same Levi component $\\mathfrak {l}_\\mathbb {C}$ as $\\mathfrak {p}_\\mathbb {C}$ .",
"By construction $(\\mathcal {F}, {^{\\prime }\\mathcal {F}})$ refines $(\\mathcal {F}_{min}, {^{\\prime }\\mathcal {F}}_{min})$ and $(\\mathcal {F}_{max}, {^{\\prime }\\mathcal {F}}_{max})$ refines $(\\mathcal {F}, {^{\\prime }\\mathcal {F}})$ .",
"As $(\\mathcal {F}_{min}, {^{\\prime }\\mathcal {F}}_{min})$ is uniquely defined from $(\\mathcal {F}, {^{\\prime }\\mathcal {F}})$ it is $\\tau $ –stable.",
"Now the maximal $\\tau $ –stable taut couple $(\\mathcal {F}^*_{max}, {^{\\prime }\\mathcal {F}}^*_{max})$ of semiclosed generalized flags defines a $\\tau $ –stable parabolic $\\mathfrak {q}_\\mathbb {C}$ with the same Levi component $\\mathfrak {l}_\\mathbb {C}$ as $\\mathfrak {p}_\\mathbb {C}$ , and $\\mathfrak {q}_\\mathbb {R}:= \\mathfrak {q}_\\mathbb {C}\\cap \\mathfrak {g}_\\mathbb {R}$ is a minimal parabolic subalgebra of $\\mathfrak {g}_\\mathbb {R}$ with Levi component $\\mathfrak {l}_\\mathbb {R}$ .",
"The argument is the same when $\\mathfrak {g}_\\mathbb {C}$ is $\\mathfrak {so}$ or $\\mathfrak {sp}$ .",
"Proposition REF says that the Levi components of the minimal parabolics are the compact real forms, in the sense of [21], of the complex $\\mathfrak {sl}$ , $\\mathfrak {so}$ and $\\mathfrak {sp}$ .",
"We extend this notion.",
"The group $G_\\mathbb {R}$ has the natural Cartan involution $\\theta $ such that $d\\theta ((\\mathfrak {p}_\\mathbb {R})_{red}) = (\\mathfrak {p}_\\mathbb {R})_{red}$ , defined as follows.",
"Every element of $\\mathfrak {l}_\\mathbb {R}$ is elliptic, and $(\\mathfrak {p}_\\mathbb {R})_{red} = \\mathfrak {l}_\\mathbb {R}\\mathfrak {t}_\\mathbb {R}$ where $\\mathfrak {t}_\\mathbb {R}$ is toral, so every element of $(\\mathfrak {p}_\\mathbb {R})_{red}$ is semisimple.",
"(This is where we use minimality of the parabolic $\\mathfrak {p}_\\mathbb {R}$ .)",
"Thus $(\\mathfrak {p}_\\mathbb {R})_{red}\\cap \\mathfrak {g}_{n,\\mathbb {R}}$ is reductive in $\\mathfrak {g}_{m,\\mathbb {R}}$ for every $m \\geqq n$ .",
"Consequently we have Cartan involutions $\\theta _n$ of the groups $G_{n,\\mathbb {R}}$ such that $\\theta _{n+1}|_{G_{n,\\mathbb {R}}} = \\theta _n$ and $d\\theta _n((\\mathfrak {p}_\\mathbb {R})_{red}\\cap \\mathfrak {g}_{n,\\mathbb {R}}) =(\\mathfrak {p}_\\mathbb {R})_{red}\\cap \\mathfrak {g}_{n,\\mathbb {R}}$ .",
"Now $\\theta = \\varinjlim \\theta _n$ (in other words $\\theta |_{G_{n,\\mathbb {R}}} = \\theta _n$ ) is the desired Cartan involution of $\\mathfrak {g}_\\mathbb {R}$ .",
"Note that $\\mathfrak {l}_\\mathbb {R}$ is contained in the fixed point set of $d\\theta $ .",
"The Lie algebra $\\mathfrak {g}_\\mathbb {R}= \\mathfrak {k}_\\mathbb {R}+ \\mathfrak {s}_\\mathbb {R}$ where $\\mathfrak {k}_\\mathbb {R}$ is the $(+1)$ –eigenspace of $d\\theta $ and $\\mathfrak {s}_\\mathbb {R}$ is the $(-1)$ –eigenspace.",
"The fixed point set $K_\\mathbb {R}= G_\\mathbb {R}^\\theta $ is the direct limit of the maximal compact subgroups $K_{n,\\mathbb {R}} = G_{n,\\mathbb {R}}^{\\theta _n}$ .",
"We will refer to $K_\\mathbb {R}$ as a maximal lim–compact subgroup of $G_\\mathbb {R}$ and to $\\mathfrak {k}_\\mathbb {R}$ as a maximal lim–compact subalgebra of $\\mathfrak {g}_\\mathbb {R}$ .",
"By construction $\\mathfrak {l}_\\mathbb {R}\\subset \\mathfrak {k}_\\mathbb {R}$ , as in the case of finite dimensional minimal parabolics.",
"Also as in the finite dimensional case (and using the same proof), $[\\mathfrak {k}_\\mathbb {R},\\mathfrak {k}_\\mathbb {R}] \\subset \\mathfrak {k}_\\mathbb {R}$ , $[\\mathfrak {k}_\\mathbb {R},\\mathfrak {s}_\\mathbb {R}] \\subset \\mathfrak {s}_\\mathbb {R}$ and $[\\mathfrak {s}_\\mathbb {R},\\mathfrak {s}_\\mathbb {R}] \\subset \\mathfrak {k}_\\mathbb {R}$ .",
"Lemma 8.2 Decompose $(\\mathfrak {p}_\\mathbb {R})_{red} = \\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}$ where $\\mathfrak {m}_\\mathbb {R}= (\\mathfrak {p}_\\mathbb {R})_{red}\\cap \\mathfrak {k}_\\mathbb {R}$ and $\\mathfrak {a}_\\mathbb {R}= (\\mathfrak {p}_\\mathbb {R})_{red}\\cap \\mathfrak {s}_\\mathbb {R}$ .",
"Then $\\mathfrak {m}_\\mathbb {R}$ and $\\mathfrak {a}_\\mathbb {R}$ are ideals in $(\\mathfrak {p}_\\mathbb {R})_{red}$ with $\\mathfrak {a}_\\mathbb {R}$ commutative.",
"In particular $(\\mathfrak {p}_\\mathbb {R})_{red} = \\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}_\\mathbb {R}$ , direct sum of ideals.",
"Since $\\mathfrak {l}_\\mathbb {R}= [(\\mathfrak {p}_\\mathbb {R})_{red}, (\\mathfrak {p}_\\mathbb {R})_{red}]$ we compute $[\\mathfrak {m}_\\mathbb {R},\\mathfrak {a}_\\mathbb {R}] \\subset \\mathfrak {l}_\\mathbb {R}\\cap \\mathfrak {a}_\\mathbb {R}= 0$ .",
"In particular $[[\\mathfrak {a}_\\mathbb {R}, \\mathfrak {a}_\\mathbb {R}],\\mathfrak {a}_\\mathbb {R}] = 0$ .",
"So $[\\mathfrak {a}_\\mathbb {R}, \\mathfrak {a}_\\mathbb {R}]$ is a commutative ideal in the semisimple algebra $\\mathfrak {l}_\\mathbb {R}$ , in other words $\\mathfrak {a}_\\mathbb {R}$ is commutative.",
"The main result of this section is the following generalization of the standard decomposition of a finite dimensional real parabolic.",
"We have formulated it to emphasize the parallel with the finite dimensional case.",
"However some details of the construction are rather different; see Proposition REF and the discussion leading up to it.",
"Theorem 8.3 The minimal parabolic subalgebra $\\mathfrak {p}_\\mathbb {R}$ of $\\mathfrak {g}_\\mathbb {R}$ decomposes as $\\mathfrak {p}_\\mathbb {R}= \\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}+ \\mathfrak {n}_\\mathbb {R}=\\mathfrak {n}_\\mathbb {R}(\\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}_\\mathbb {R})$ , where $\\mathfrak {a}_\\mathbb {R}$ is commutative, the Levi component $\\mathfrak {l}_\\mathbb {R}$ is an ideal in $\\mathfrak {m}_\\mathbb {R}$ , and $\\mathfrak {n}_\\mathbb {R}$ is the linear nilradical $(\\mathfrak {p}_\\mathbb {R})_{nil}$ .",
"On the group level, $P_\\mathbb {R}= M_\\mathbb {R}A_\\mathbb {R}N_\\mathbb {R}= N_\\mathbb {R}\\ltimes (M_\\mathbb {R}\\times A_\\mathbb {R})$ where $N_\\mathbb {R}= \\exp (\\mathfrak {n}_\\mathbb {R})$ is the linear unipotent radical of $P_\\mathbb {R}$ , $A_\\mathbb {R}= \\exp (\\mathfrak {a}_\\mathbb {R})$ is isomorphic to a vector group, and $M_\\mathbb {R}= P_\\mathbb {R}\\cap K_\\mathbb {R}$ is limit–compact with Lie algebra $\\mathfrak {m}_\\mathbb {R}$ .",
"The algebra level statements come out of Lemma REF and the semidirect sum decomposition $\\mathfrak {p}_\\mathbb {R}= (\\mathfrak {p}_\\mathbb {R})_{nil} (\\mathfrak {p}_\\mathbb {R})_{red}$ .",
"For the group level statements, we need only check that $K_\\mathbb {R}$ meets every topological component of $P_\\mathbb {R}$ .",
"Even though $P_\\mathbb {R}\\cap G_{n,\\mathbb {R}}$ need not be parabolic in $G_{n,\\mathbb {R}}$ , the group $P_\\mathbb {R}\\cap \\theta P_\\mathbb {R}\\cap G_{n,\\mathbb {R}}$ is reductive in $G_{n,\\mathbb {R}}$ and $\\theta _n$ –stable, so $K_{n,\\mathbb {R}}$ meets each of its components.",
"Now $K_\\mathbb {R}$ meets every component of $P_\\mathbb {R}\\cap \\theta P_\\mathbb {R}$ .",
"The linear unipotent radical of $P_\\mathbb {R}$ has Lie algebra $\\mathfrak {n}_\\mathbb {R}$ and thus must be equal to $\\exp (\\mathfrak {n}_\\mathbb {R})$ , so it does not effect components.",
"Thus every component of $P_{red}$ is represented by an element of $K_\\mathbb {R}\\cap P_\\mathbb {R}\\cap \\theta P_\\mathbb {R}= K_\\mathbb {R}\\cap P_\\mathbb {R}= M_\\mathbb {R}$ .",
"That derives $P_\\mathbb {R}= M_\\mathbb {R}A_\\mathbb {R}N_\\mathbb {R}= N_\\mathbb {R}\\ltimes (M_\\mathbb {R}\\times A_\\mathbb {R})$ from $\\mathfrak {p}_\\mathbb {R}= \\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}+ \\mathfrak {n}_\\mathbb {R}= \\mathfrak {n}_\\mathbb {R}(\\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}_\\mathbb {R})$ .",
"The reductive part of the group $\\mathfrak {p}_\\mathbb {R}$ can be constructed explicitly.",
"We do this for the cases where $\\mathfrak {g}_\\mathbb {R}$ is defined by a hermitian form $f: V_\\mathbb {F}\\times V_\\mathbb {F}\\rightarrow \\mathbb {F}$ where $\\mathbb {F}$ is $\\mathbb {R}$ , $\\mathbb {C}$ or $\\mathbb {H}$ .",
"The idea is the same for the other cases.",
"See Proposition REF below.",
"Write $V_\\mathbb {F}$ for $V_\\mathbb {R}$ , $V_\\mathbb {C}$ or $V_\\mathbb {H}$ , as appropriate, and similarly for $W_\\mathbb {F}$ .",
"We use $f$ for an $\\mathbb {F}$ –conjugate–linear identification of $V_\\mathbb {F}$ and $W_\\mathbb {F}$ .",
"We are dealing with a minimal Levi component $\\mathfrak {l}_\\mathbb {R}=\\bigoplus _{j \\in J}\\, \\mathfrak {l}_{j,\\mathbb {R}}$ where the $\\mathfrak {l}_{j,\\mathbb {R}}$ are simple.",
"Let $X_\\mathbb {F}$ denote the sum of the corresponding subspaces $(X_j)_\\mathbb {F}\\subset V_\\mathbb {F}$ and $Y_\\mathbb {F}$ the analogous sum of the $(Y_j)_\\mathbb {F}\\subset W_\\mathbb {F}$ .",
"Then $X_\\mathbb {F}$ and $Y_\\mathbb {F}$ are nondegenerately paired.",
"Of course they may be small, even zero.",
"In any case, $\\begin{aligned}&V_\\mathbb {F}= X_\\mathbb {F}\\oplus Y_\\mathbb {F}^\\perp \\, ,W_\\mathbb {F}= Y_\\mathbb {F}\\oplus X_\\mathbb {F}^\\perp , \\text{ and } \\\\&X_\\mathbb {F}^\\perp \\text{ and } Y_\\mathbb {F}^\\perp \\text{ are nondegenerately paired.",
"}\\end{aligned}\\qquad \\mathrm {(8.5)}$ These direct sum decompositions (REF ) now become $V_\\mathbb {F}= X_\\mathbb {F}\\oplus X_\\mathbb {F}^\\perp \\quad \\text{ and } \\quad f\\text{ is nondegenerate on each summand.",
"}\\qquad \\mathrm {(8.7)}$ Let $X^{\\prime }$ and $X^{\\prime \\prime }$ be paired maximal isotropic subspaces of $X_\\mathbb {F}^\\perp $ .",
"Then $V_\\mathbb {F}= X_\\mathbb {F}\\oplus (X^{\\prime }_\\mathbb {F}\\oplus X^{\\prime \\prime }_\\mathbb {F}) \\oplus Q_\\mathbb {F}\\text{ where }Q_\\mathbb {F}:= (X_\\mathbb {F}\\oplus (X^{\\prime }_\\mathbb {F}\\oplus X^{\\prime \\prime }_\\mathbb {F}))^\\perp .\\qquad \\mathrm {(8.9)}$ The subalgebra $\\lbrace \\xi \\in \\mathfrak {g}_\\mathbb {R}\\mid \\xi (X_\\mathbb {F}\\oplus Q_\\mathbb {F}) = 0\\rbrace $ of $\\mathfrak {g}_\\mathbb {R}$ has a maximal toral subalgebra $\\mathfrak {a}\\dagger _\\mathbb {R}$ , contained in $\\mathfrak {s}_\\mathbb {R}$ , in which every element has all eigenvalues real.",
"One example, which is diagonalizable (in fact diagonal) over $\\mathbb {R}$ , is $\\begin{aligned}\\mathfrak {a}^\\dagger _\\mathbb {R}= {\\bigoplus }_{\\ell \\in C}\\, \\mathfrak {gl}(x_\\ell ^{\\prime }\\mathbb {R},x_\\ell ^{\\prime \\prime }\\mathbb {R})&\\text{ where }\\\\& \\lbrace x^{\\prime }_\\ell \\mid \\ell \\in C\\rbrace \\text{ is a basis of } X^{\\prime }_\\mathbb {F}\\text{ and } \\\\& \\lbrace x^{\\prime \\prime }_\\ell \\mid \\ell \\in C\\rbrace \\text{ is the dual basis of } X^{\\prime \\prime }_\\mathbb {F}.\\end{aligned}\\qquad \\mathrm {(8.11)}$ We interpolate the self–taut semiclosed generalized flag $\\mathcal {F}$ defining $\\mathfrak {p}$ with the subspaces $x_\\ell ^{\\prime }\\mathbb {R}\\oplus x_\\ell ^{\\prime \\prime }\\mathbb {R}$ .",
"Any such interpolation (and usually there will be infinitely many) gives a self–taut semiclosed generalized flag $\\mathcal {F}^\\dagger $ and defines a minimal self–normalizing parabolic subalgebra $\\mathfrak {p}^\\dagger _\\mathbb {R}$ of $\\mathfrak {g}_\\mathbb {R}$ with the same Levi component as $\\mathfrak {p}_\\mathbb {R}$ .",
"The decompositions corresponding to (REF ), (REF ) and (REF ) are given by $X_\\mathbb {F}^\\dagger = X_\\mathbb {F}\\oplus (X^{\\prime }_\\mathbb {F}\\oplus X^{\\prime \\prime }_\\mathbb {F})$ and $Q_\\mathbb {F}^\\dagger = Q_\\mathbb {F}$ .",
"In addition, the subalgebra $\\lbrace \\xi \\in \\mathfrak {p}_\\mathbb {R}\\mid \\xi (X_\\mathbb {F}\\oplus (X^{\\prime }_\\mathbb {F}\\oplus X^{\\prime \\prime }_\\mathbb {F})) = 0\\rbrace $ has a maximal toral subalgebra $\\mathfrak {t}^{\\prime }_\\mathbb {R}$ in which every eigenvalue is pure imaginary, because $f$ is definite on $Q_\\mathbb {F}$ .",
"It is unique because it has derived algebra zero and is given by the action of the $\\mathfrak {p}_\\mathbb {R}$ –stabilizer of $Q_\\mathbb {F}$ on the definite subspace $Q_\\mathbb {F}$ .",
"This uniqueness tell us that $\\mathfrak {t}^{\\prime }_\\mathbb {R}$ is the same for $\\mathfrak {p}_\\mathbb {R}$ and $\\mathfrak {p}^\\dagger _\\mathbb {R}$ .",
"Let $\\mathfrak {t}^{\\prime \\prime }_\\mathbb {R}$ denote the maximal toral subalgebra in $\\lbrace \\xi \\in \\mathfrak {p}_\\mathbb {R}\\mid \\xi (X_\\mathbb {F}\\oplus Q_\\mathbb {F})) = 0\\rbrace $ .",
"It stabilizes each Span($x^{\\prime }_\\ell ,x^{\\prime \\prime }_\\ell $ ) in (REF ) and centralizes $\\mathfrak {a}^\\dagger _\\mathbb {R}$ , so it vanishes if $\\mathbb {F}\\ne \\mathbb {C}$ .",
"The $\\mathfrak {p}_\\mathbb {R}^\\dagger $ analog of $\\mathfrak {t}^{\\prime \\prime }_\\mathbb {R}$ is 0 because $X^\\dagger _\\mathbb {F}\\oplus Q_\\mathbb {F}= 0$ .",
"In any case we have $\\mathfrak {t}_\\mathbb {R}= \\mathfrak {t}_\\mathbb {R}^\\dagger := \\mathfrak {t}^{\\prime }_\\mathbb {R}\\oplus \\mathfrak {t}^{\\prime \\prime }_\\mathbb {R}\\,.\\qquad \\mathrm {(8.13)}$ For each $j \\in J$ we define an algebra that contains $\\mathfrak {l}_{j,\\mathbb {R}}$ and acts on $(X_j)_\\mathbb {F}$ by: if $\\mathfrak {l}_{j,\\mathbb {R}} = \\mathfrak {su}(*)$ then $\\widetilde{\\mathfrak {l}_{j,\\mathbb {R}}} = \\mathfrak {u}(*)$ (acting on $(X_j)_\\mathbb {C}$ ); otherwise $\\widetilde{\\mathfrak {l}_{j,\\mathbb {R}}} = \\mathfrak {l}_{j,\\mathbb {R}}$ .",
"Define $\\widetilde{\\mathfrak {l}_\\mathbb {R}} =\\bigoplus _{j \\in J}\\, \\widetilde{\\mathfrak {l}_{j,\\mathbb {R}}} \\quad \\text{ and } \\quad \\mathfrak {m}^\\dagger _\\mathbb {R}= \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {t}_\\mathbb {R}\\, .\\qquad \\mathrm {(8.15)}$ Then, by construction, $\\mathfrak {m}^\\dagger _\\mathbb {R}= \\mathfrak {m}_\\mathbb {R}$ .",
"Thus $\\mathfrak {p}^\\dagger _\\mathbb {R}$ satisfies $\\mathfrak {p}^\\dagger _\\mathbb {R}:= \\mathfrak {m}_\\mathbb {R}+\\mathfrak {a}^\\dagger _\\mathbb {R}+ \\mathfrak {n}^\\dagger _\\mathbb {R}=\\mathfrak {n}^\\dagger _\\mathbb {R}(\\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}^\\dagger _\\mathbb {R}).\\qquad \\mathrm {(8.17)}$ Let $\\mathfrak {z}_\\mathbb {R}$ denote the centralizer of $\\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}_\\mathbb {R}$ in $\\mathfrak {g}_\\mathbb {R}$ and let $\\mathfrak {z}^\\dagger _\\mathbb {R}$ denote the centralizer of $\\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}^\\dagger _\\mathbb {R}$ in $\\mathfrak {g}_\\mathbb {R}$ .",
"We claim $\\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}= \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {z}_\\mathbb {R}\\text{ and }\\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}^\\dagger _\\mathbb {R}= \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {z}^\\dagger _\\mathbb {R}\\qquad \\mathrm {(8.19)}$ For by construction $\\mathfrak {m}_\\mathbb {R}\\oplus \\mathfrak {a}_\\mathbb {R}=\\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {t}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}\\subset \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {z}_\\mathbb {R}$ .",
"Conversely if $\\xi \\in \\mathfrak {z}_\\mathbb {R}$ it preserves each $X_{j,\\mathbb {F}}$ , each joint eigenspace of $\\mathfrak {a}_\\mathbb {R}$ on $X^{\\prime }_\\mathbb {F}\\oplus X^{\\prime \\prime }_\\mathbb {F}$ , and each joint eigenspace of $\\mathfrak {t}_\\mathbb {R}$ , so $\\xi \\subset \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {t}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}$ .",
"Thus $\\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}= \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {z}_\\mathbb {R}$ .",
"The same argument shows that $\\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}^\\dagger _\\mathbb {R}= \\widetilde{\\mathfrak {l}_\\mathbb {R}} + \\mathfrak {z}^\\dagger _\\mathbb {R}$ .",
"If $\\mathfrak {a}_\\mathbb {R}$ is diagonalizable as in the definition (REF ) of $\\mathfrak {a}^\\dagger _\\mathbb {R}$ , in other words if it is a sum of standard $\\mathfrak {gl}(1;\\mathbb {R})$ 's, then we could choose $\\mathfrak {a}_\\mathbb {R}^\\dagger = \\mathfrak {a}_\\mathbb {R}$ , hence could construct $\\mathcal {F}^\\dagger $ equal to $\\mathcal {F}$ , resulting in $\\mathfrak {p}_\\mathbb {R}= \\mathfrak {p}_\\mathbb {R}^\\dagger $ .",
"In summary: Proposition 8.20 Let $\\mathfrak {g}_\\mathbb {R}$ be defined by a hermitian form and let $\\mathfrak {p}_\\mathbb {R}$ be a minimal self–normalizing parabolic subalgebra.",
"In the notation above, $\\mathfrak {p}^\\dagger _\\mathbb {R}$ is a minimal self–normalizing parabolic subalgebra of $\\mathfrak {g}_\\mathbb {R}$ with $\\mathfrak {m}^\\dagger _\\mathbb {R}= \\mathfrak {m}_\\mathbb {R}$ .",
"In particular $\\mathfrak {p}^\\dagger _\\mathbb {R}$ and $\\mathfrak {p}_\\mathbb {R}$ have the same Levi component.",
"Further we can take $\\mathfrak {p}_\\mathbb {R}= \\mathfrak {p}^\\dagger _\\mathbb {R}$ if and only if $\\mathfrak {a}_\\mathbb {R}$ is the sum of commuting standard $\\mathfrak {gl}(1;\\mathbb {R})$ 's.",
"Similar arguments give the construction behind Proposition REF for the other real simple direct limit Lie algebras."
],
[
"The Inducing Representation",
"In this section $P_\\mathbb {R}$ is a self normalizing minimal parabolic subgroup of $G_\\mathbb {R}$ .",
"We discuss representations of $P_\\mathbb {R}$ and the induced representations of $G_\\mathbb {R}$ .",
"The latter are the principal series representations of $G_\\mathbb {R}$ associated to $\\mathfrak {p}_\\mathbb {R}$ , or more precisely to the pair $(\\mathfrak {l}_\\mathbb {R},J)$ where $\\mathfrak {l}_\\mathbb {R}$ is the Levi component and $J$ is the ordering on the simple summands of $\\mathfrak {l}_\\mathbb {R}$ .",
"We must first choose a class $\\mathcal {C}_{M_\\mathbb {R}}$ of representations of $M_\\mathbb {R}$ .",
"Reasonable choices include various classes of unitary representations (we will discuss this in a moment) and continuous representations on nuclear Fréchet spaces, but “tame” (essentially the same as $II_1$ ) may be the best with which to start.",
"In any case, given a representation $\\kappa $ in our chosen class and a linear functional $\\sigma : \\mathfrak {a}_\\mathbb {R}\\rightarrow \\mathbb {R}$ we have the representation $\\kappa \\otimes e^{i\\sigma }$ of $M_\\mathbb {R}\\times A_\\mathbb {R}$ .",
"Here $e^{i\\sigma }(a)$ means $e^{i\\sigma (\\log a)}$ where $\\log : A_\\mathbb {R}\\rightarrow \\mathfrak {a}_\\mathbb {R}$ inverts $\\exp : \\mathfrak {a}_\\mathbb {R}\\rightarrow A_\\mathbb {R}$ .",
"We write $E_\\kappa $ for the representation space of $\\kappa $ .",
"We discuss some possibilities for $\\mathcal {C}_{M_\\mathbb {R}}$ .",
"Note that $\\mathfrak {l}_\\mathbb {R}= [(\\mathfrak {p}_\\mathbb {R})_{red},(\\mathfrak {p}_\\mathbb {R})_{red}]= [\\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}, \\mathfrak {m}_\\mathbb {R}+ \\mathfrak {a}_\\mathbb {R}] = [\\mathfrak {m}_\\mathbb {R},\\mathfrak {m}_\\mathbb {R}]$ .",
"Define $L_\\mathbb {R}= [M_\\mathbb {R},M_\\mathbb {R}] \\text{ and } T_\\mathbb {R}= M_\\mathbb {R}/L_\\mathbb {R}\\,.$ Then $T_\\mathbb {R}$ is a real toral group with all eigenvalues pure imaginary, and $M_\\mathbb {R}$ is an extension $1 \\rightarrow L_\\mathbb {R}\\rightarrow M_\\mathbb {R}\\rightarrow T_\\mathbb {R}\\rightarrow 1 \\,.$ Examples indicate that $M_\\mathbb {R}$ is the product of a closed subgroup $T^{\\prime }_\\mathbb {R}$ of $T_\\mathbb {R}$ with factors of the group $L^{\\prime }_\\mathbb {R}$ indicated in the previous section.",
"That was where we replaced summands $\\mathfrak {su}(*)$ of $\\mathfrak {l}_\\mathbb {R}$ by slightly larger algebras $\\mathfrak {u}(*)$ , hence subgroups $SU(*)$ of $L_\\mathbb {R}$ by slightly larger groups $U(*)$ .",
"There is no need to discuss the representations of the classical finite dimensional $U(n)$ , $SO(n)$ or $Sp(n)$ , where we have the Cartan highest weight theory and other classical combinatorial methods.",
"So we look at $U(\\infty )$ .",
"Tensor Representations of $U(\\infty )$.",
"In the classical setting, one can use the action of the symmetric group $\\mathfrak {S}_n$ , permuting factors of $\\otimes ^n(\\mathbb {C}^p)$ .",
"This gives a representation of $U(p) \\times \\mathfrak {S}_n$ .",
"Then we have the action of $U(p)$ on tensors picked out by an irreducible summand of that action of $\\mathfrak {S}_n$ .",
"These summands occur with multiplicity 1.",
"See Weyl's book [23].",
"Segal [17], Kirillov [12], and Strătilă & Voiculescu [18] developed and proved an analog of this for $U(\\infty )$ .",
"However those “tensor representations” form a small class of the continuous unitary representations of $U(\\infty )$ .",
"They are factor representations of type $II_\\infty $ , but they are somewhat restricted in that they do not even extend to the class of unitary operators of the form $1 + \\text{(compact)}$ .",
"See [19] for a summary of this topic.",
"Because of this limitation one may also wish to consider other classes of factor representations of $U(\\infty )$ .",
"Type $II_1$ Representations of $U(\\infty )$.",
"Let $\\pi $ be a continuous unitary finite factor representation of $U(\\infty )$ .",
"It has a character $\\chi _\\pi (x) = {\\rm trace\\,\\,}\\pi (x)$ (normalized trace).",
"Voiculescu [22] worked out the parameter space for these finite factor representations.",
"It consists of all bilateral sequences $\\lbrace c_n\\rbrace _{-\\infty < n < \\infty }$ such that (i) $\\det ((c_{m_i + j - i})_{1 \\leqq i, j \\leqq N} \\geqq 0$ for $m_i \\in \\mathbb {Z}$ and $N \\geqq 0$ and (ii) $\\sum c_n = 1$ .",
"The character corresponding to $\\lbrace c_n\\rbrace $ and $\\pi $ is $\\chi _\\pi (x) = \\prod _i p(z_i)$ where $\\lbrace z_i\\rbrace $ is the multiset of eigenvalues of $x$ and $p(z) = \\sum c_nz^n$ .",
"Here $\\pi $ extends to the group of all unitary operators $X$ on the Hilbert space completion of $\\mathbb {C}^\\infty $ such that $X - 1$ is of trace class.",
"See [19] for a more detailed summary.",
"This may be the best choice of class $\\mathcal {C}_{M_\\mathbb {R}}$ .",
"It is closely tied to the Olshanskii–Vershik notion (see [16]) of tame representation.",
"Other Factor Representations of $U(\\infty )$.",
"Let $\\mathcal {H}$ be the Hilbert space completion of $\\varinjlim \\mathcal {H}_n$ where $\\mathcal {H}_n$ is the natural representation space of $U(n)$ .",
"Fix a bounded hermitian operator $B$ on $\\mathcal {H}$ with $0 \\leqq B \\leqq I$ .",
"Then $\\psi _B : U(\\infty ) \\rightarrow \\mathbb {C}\\,, \\text{ defined by }\\psi _B(x) = \\det ((1 - B) + B x)$ is a continuous function of positive type on $U(\\infty )$ .",
"Let $\\pi _B$ denote the associated cyclic representation of $U(\\infty )$ .",
"Then ([20], or see [19]), (1) $\\,\\,\\psi _B$ is of type $I$ if and only if $B(I-B)$ is of trace class.",
"In that case $\\pi _B$ is a direct sum of irreducible representations.",
"(2) $\\,\\,\\psi _B$ is factorial and type $I$ if and only if $B$ is a projection.",
"In that case $\\pi _B$ is irreducible.",
"(3) $\\,\\,\\psi _B$ is factorial but not of type $I$ if and only if $B(I-B)$ is not of trace class.",
"In that case (i) $\\,\\,\\psi _B$ is of type $II_1$ if and only if $B-tI$ is Hilbert–Schmidt where $0 < t < 1$ ; then $\\pi _B$ is a factor representation of type $II_1$ .",
"(ii) $\\,\\,\\psi _B$ is of type $II_\\infty $ if and only if (a) $B(I-B)(B-pI)^2$ is trace class where $0 < t < 1$ and (b) the essential spectrum of $B$ contains 0 or 1; then $\\pi _B$ is a factor representation of type $II_\\infty $ .",
"(iii) $\\,\\,\\psi _B$ is of type $III$ if and only if $B(I-B)(B-pI)^2$ is not of trace class whenever $0 < t < 1$ ; then $\\pi _B$ is a factor representation of type $III$ .",
"Similar considerations hold for $SU(\\infty )$ , $SO(\\infty )$ and $Sp(\\infty )$ .",
"This gives an indication of the delicacy in choice of type of representations of $M_\\mathbb {R}$ .",
"Clearly factor representations of type $I$ and $II_1$ will be the easiest to deal with.",
"It is worthwhile to consider the case where the inducing representation $\\kappa \\otimes e^{i\\sigma }$ is trivial on $M_\\mathbb {R}$ , in other words is a unitary character on $P_\\mathbb {R}$ .",
"In the finite dimensional case this leads to a $K_\\mathbb {R}$ –fixed vector, spherical functions on $G_\\mathbb {R}$ and functions on the symmetric space $G_\\mathbb {R}/K_\\mathbb {R}$ .",
"In the infinite dimensional case it leads to open problems, but there are a few examples ([7], [24]) that may give accurate indications."
],
[
"Parabolic Induction",
"We view $\\kappa \\otimes e^{i\\sigma }$ as a representation $man \\mapsto e^{i\\sigma }(a)\\kappa (m)$ of $P_\\mathbb {R}= M_\\mathbb {R}A_\\mathbb {R}N_\\mathbb {R}$ on $E_\\kappa $ .",
"It is well defined because $N_\\mathbb {R}$ is a closed normal subgroup of $P_\\mathbb {R}$ .",
"Let $\\mathcal {U}(\\mathfrak {g}_\\mathbb {C})$ denote the universal enveloping algebra of $\\mathfrak {g}_\\mathbb {C}$ .",
"The algebraically induced representation is given on the Lie algebra level as the left multiplication action of $\\mathfrak {g}_\\mathbb {C}$ on $\\mathcal {U}(\\mathfrak {g}_\\mathbb {C}) \\otimes _{\\mathfrak {p}_\\mathbb {R}} E_\\kappa $ , $d\\pi _{\\kappa ,\\sigma ,alg}(\\xi ): \\mathcal {U}(\\mathfrak {g}_\\mathbb {C}) \\otimes _{\\mathfrak {p}_\\mathbb {R}} E_\\kappa \\rightarrow \\mathcal {U}(\\mathfrak {g}_\\mathbb {C}) \\otimes _{\\mathfrak {p}_\\mathbb {R}} E_\\kappa \\text{ by }\\eta \\otimes e \\mapsto (\\xi \\eta )\\otimes e.$ If $\\xi \\in \\mathfrak {p}_\\mathbb {R}$ then $d\\pi _{\\kappa ,\\sigma ,alg}(\\xi )(\\eta \\otimes e) = {\\rm Ad}(\\xi )\\eta \\otimes e +\\eta \\otimes d(\\kappa \\otimes e^{i\\sigma })(\\xi )e\\, .$ To obtain the associated representation $\\pi _{\\kappa ,\\sigma }$ of $G_\\mathbb {R}$ we need a $G_\\mathbb {R}$ –invariant completion of $\\mathcal {U}(\\mathfrak {g}_\\mathbb {C}) \\otimes _{\\mathfrak {p}_\\mathbb {R}} E_\\kappa $ so that the $\\pi _{\\kappa ,\\sigma ,alg}(\\exp (\\xi )) := \\exp (d\\pi _{\\kappa ,\\sigma ,alg}(\\xi ))$ are well defined.",
"For example we could use a $C^k$ completion, $k \\in \\lbrace 0, 1, 2, \\dots , \\infty , \\omega \\rbrace $ , representation of $G_\\mathbb {R}$ on $C^k$ sections of the vector bundle $\\mathbb {E}_{\\kappa \\otimes e^{i\\sigma }} \\rightarrow G_\\mathbb {R}/P_\\mathbb {R}$ associated to the action $\\kappa \\otimes e^{i\\sigma }$ of $P_\\mathbb {R}$ on $E_\\kappa $ .",
"The representation space is $\\lbrace \\varphi : G_\\mathbb {R}\\rightarrow E_\\kappa \\mid \\varphi \\text{ is } C^k \\text{ and }\\varphi (xman) =e^{i\\sigma }(a)^{-1}\\kappa (m)^{-1}f(x)\\rbrace $ where $m \\in M_\\mathbb {R}$ , $a \\in A_\\mathbb {R}$ and $n \\in N_\\mathbb {R}$ , and the action of $G_\\mathbb {R}$ is $[\\pi _{\\kappa ,\\sigma ,C^k}(x)(\\varphi )](z) = \\varphi (x^{-1}z)$ .",
"In some cases one can unitarize $d\\pi _{\\kappa ,\\sigma ,alg}$ by constructing a Hilbert space of sections of $\\mathbb {E}_{\\kappa \\otimes e^{i\\sigma }} \\rightarrow G_\\mathbb {R}/P_\\mathbb {R}$ .",
"This has been worked out explicitly when $P_\\mathbb {R}$ is a direct limit of minimal parabolic subgroups of the $G_{n,\\mathbb {R}}$ [24], and more generally it comes down to transitivity of $K_\\mathbb {R}$ on $G_\\mathbb {R}/P_\\mathbb {R}$ [26].",
"In any case the resulting representations of $G_\\mathbb {R}$ depend on the choice of class $\\mathcal {C}_{M_\\mathbb {R}}$ of representations of $M_\\mathbb {R}$ .",
"Department of Mathematics, University of California,Berkeley, CA 94720–3840, USA e-mail: [email protected]"
]
] | 1204.1357 |
[
[
"Generalized form of optimal teleportation witnesses"
],
[
"Abstract We propose a generalized form of optimal teleportation witness to demonstrate their importance in experimental detection of the larger set of entangled states useful for teleportation in higher dimensional systems.",
"The interesting properties of our witness reveal that teleportation witness can be used to characterize mixed state entanglement using Schmidt numbers.",
"Our results show that while every teleportation witness is also a entanglement witness, the converse is not true.",
"Also, we show that a hermitian operator is a teleportation witness iff it is a decomposable entanglement witness.",
"In addition, we analyze the practical significance of our study by decomposing our teleportation witness in terms of Pauli and Gell-Mann matrices, which are experimentally measurable quantities."
],
[
"Introduction",
"Quantum entanglement [1], [2] is an essential feature of quantum mechanics which has no classical analogues.",
"The existence of long range quantum correlations between entangled particles allows the use of entangled systems as resources for efficient information transfer through protocols such as quantum computing [3], cryptography [4], teleportation [5] and dense coding [6], [7].",
"Although entangled states are used for various theoretical applications in quantum information processing, the practical use of an entangled resource is restricted to the successful experimental realization of the resource.",
"In real experimental set-ups, it is always a challenge to create and detect entangled states.",
"Also, the prepared entangled channel may not be robust enough to preserve the necessary quantum correlations and thus may or may not be entangled.",
"Hence, successful generation and detection of entanglement are essential features in any quantum information processing protocol.",
"For this reason, it is important to develop efficient and easy to implement experimental set-ups to create and detect quantum entanglement.",
"Interestingly, class of entanglement witnesses is necessary and sufficient to detect entanglement [8].",
"The importance of such witness becomes even more significant due to their decomposition in terms of Pauli spin matrices (for qubits) and Gell-Mann matrices (for qutrits and other higher dimensions) which are experimentally measurable quantities [9].",
"Recently, a teleportation witness was proposed to demonstrate whether an underlying quantum state can be used for teleportation or not [10].",
"The measurement of the expectation value of the witness for unknown states reveals which states are useful as resource for performing teleportation.",
"In this article, we further address the question of efficiently distinguishing entangled states for quantum teleportation using teleportation witness.",
"For this, we propose a general Hermitian operator and demonstrate that it can be successfully used as a teleportation witness to differentiate between entangled classes that can or cannot be used for teleportation.",
"The interesting properties of our witness show that it is also an optimal witness of Schmidt number $r~(\\ge 2)$ as compared to the previous work [10] where the witness is an optimal witness of Schmidt number 2 only.",
"As the witness proposed here is optimal, it therefore detects a much larger set of entangled states in comparison to other witnesses that are not optimal.",
"In addition, we also prove that all teleportation witnesses are also entanglement witnesses but every entanglement witness may not be a teleportation witness.",
"Our results are of importance not only for practical detection of a larger set of entangled states useful for quantum teleportation, they also shed light on the characterization of mixed state entanglement using Schmidt number.",
"The experimental realization of the witness by decomposing it into the form of Pauli matrices for $r=2$ and in the form of Gell-Mann matrices for $r >2$ adds another important dimension to our study.",
"In section 2, we provide an introduction to witnesses and then propose our generalized optimal teleportation witness for the detection of entangled states useful for the successful completion of teleportation and demonstrate its properties through theorems and illustrations.",
"Section 3 is devoted to the application of our teleportation witness in determining the Schmidt number for the entangled systems.",
"We discuss the decomposition of teleportation witness in terms of Pauli and Gell-mann matrices for the experimental detection of entanglement in section 4.",
"This is followed by the conclusion."
],
[
"Entanglement and Teleportation Witnesses",
"Quantum communication protocols use entanglement between particles for efficient information transfer from one remote location to another with reliability.",
"Hence, for an unknown quantum state to be used as a resource in communication protocols, one of the most significant questions we need to answer is whether the underlying quantum state is useful for communication purposes or not.",
"For this, one needs to detect whether the unknown quantum state is entangled or separable.",
"In lower dimensions, Peres-Horodecki criterion provides a necessary and sufficient condition for separability based on the fact that separable states have a positive-partial transpose (PPT) [11].",
"However in higher dimensions, the complex nature of quantum entanglement does not allow for a necessary and sufficient condition for its detection.",
"For example, in higher dimensions, one can find entangled states with negative (NPT) as well as positive partial transpose.",
"One solution to this problem is entanglement witnesses [12], [13], which provide a way to detect whether an unknown quantum state is entangled or not.",
"Such witnesses are Hermitian operators with at least one negative eigenvalue; acting as a hyperplane separating entangled states and separable states.",
"The notion of an entanglement witness was extended to Schmidt number witness, which detects the Schmidt number of quantum states [14].",
"Moreover, witnesses can be separated into two different classes- decomposable and non-decomposable witnesses [15].",
"Although non-decomposable witnesses detect both NPT and PPT entangled states, the decomposable witnesses can only detect NPT entangled states.",
"In this article, we provide a necessary and sufficient condition to show that a witness is a teleportation witness if it is decomposable.",
"Quantum teleportation allows for the transmission of arbitrary information from a sender to a receiver using a shared entangled resource between the two.",
"The significant question in this context is the usefulness of the entangled resource for the communication process.",
"We propose a generalized teleportation witness to detect whether an unknown entangled state is useful for teleportation or not.",
"In order to facilitate the discussion of our results, we first briefly describe the fully entangled fraction (FEF) [16], a property of entangled states which is related to the efficacy of quantum teleportation.",
"The fully entangled fraction of a composite system defined by a density matrix $\\rho $ can be represented as $F(\\rho ) &=& max_{U}\\langle \\phi ^{+}|U^{\\dagger }\\otimes I\\rho U\\otimes I |\\phi ^{+}\\rangle $ where $|\\phi ^{+}\\rangle =\\frac{1}{\\sqrt{d}}\\sum _{i=0}^{d-1}|ii\\rangle $ .",
"The FEF provides a sufficient condition for the determination of states useful for the teleportation process.",
"For example, states in $d \\otimes d$ dimensions can always be used as entanglement resources for successful teleportation if the FEF of states is $>$ $1/d$ .",
"We propose a Hermitian operator that successfully distinguishes between the states having FEF $>$ or $\\le $ 1/d.",
"Our teleportation witness is optimal and serves as a hyperplane that distinguishes between separable and a larger set of entangled states.",
"We now proceed to propose our generalized teleportation witness as $T_{W} &= &\\frac{1}{d}I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|-\\langle \\rho _{0},\\frac{1}{d}I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|\\rangle I \\nonumber \\\\& = & \\left[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|\\right]$ where $\\rho _{0}$ is any reference physical state.",
"In order to show that the Hermitian operator $T_{W}$ is a teleportation witness, we need to prove that the Hermitian operator $T_{W}$ satisfies the following two conditions $(i) \\langle \\sigma , T_{W}\\rangle \\ge 0,~~ \\textrm {for~~ all~~states}~~ \\sigma ~~\\textrm {which~~ are~~ not~~ useful~~ for~~teleportation}.",
"$ $(ii) \\langle \\varrho ,T_{W}\\rangle <0,~~ \\textrm {for~~ at~~ least~~one~~ state}~~ \\varrho ~~ \\textrm {which~~ is~~ useful~~ for~~teleportation}.",
"$ where, $\\langle \\sigma , T_{W}\\rangle = {\\rm Tr}(\\sigma T_{W})$ .",
"From Eq.",
"(2), it is clear that our teleportation witness is equivalent to the teleportation witness [10] $W=\\frac{1}{d}I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|$ if $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle =\\frac{1}{d}$ .",
"The similarity and differences between the two teleportation witnesses $T_{W}$ and $W$ will be discussed in detail as we move further.",
"For now, we focus on proving that our witness $T_{W}$ is indeed a teleportation witness.",
"For this, we first demonstrate that the operator $T_{W}$ gives a non-negative expectation over all states which are not useful for teleportation.",
"Theorem 1: The hermitian operator $T_{W}$ is a teleportation witness if $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle \\ge \\frac{1}{d}$ .",
"Proof: Let us choose a state $\\rho _{0}$ in such a way that $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle \\ge \\frac{1}{d}$ .",
"The operator $T_{W}$ would be a teleportation witness if it satisfies the conditions (REF ) and (REF ).",
"(i) Let $\\sigma $ be an arbitrary state chosen from the set which is not useful for teleportation, i.e.",
"$F(\\sigma )\\le \\frac{1}{d}$ .",
"Hence, we have $\\langle \\sigma ,T_{W}\\rangle &&= \\langle \\sigma ,\\left[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|\\right]\\rangle \\nonumber \\\\&& =\\left[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle - \\langle \\phi ^{+}|\\sigma |\\phi ^{+}\\rangle \\right] \\nonumber \\\\&&\\ge \\left[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle -max_{U}\\langle \\phi ^{+}|U^{\\dagger }\\otimes I\\sigma U\\otimes I |\\phi ^{+}\\rangle \\right] \\nonumber \\\\&&=\\left[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle - F(\\sigma )\\right] \\nonumber \\\\&&\\ge 0$ Therefore, the expectation value of the Hermitian operator $T_{W}$ is non-negative for all bipartite $d$ -dimensional states which are not useful for teleportation.",
"Hence, the Hermitian operator $T_{W}$ satisfies (REF ).",
"(ii) To prove that our witness detects at least one entangled state $\\varrho $ that is useful for teleportation, we fix $\\rho _{0}$ and $\\varrho $ as $\\rho _{0}=k|\\phi ^{+}\\rangle \\langle \\phi ^{+}|+\\frac{1-k}{d^{2}}I,~~~0\\le k\\le 1$ and $\\varrho =\\beta |\\phi ^{+}\\rangle \\langle \\phi ^{+}|+\\frac{1-\\beta }{d^{2}}I,~~~\\frac{-1}{d^{2}-1}\\le \\beta \\le 1$ where $\\varrho $ is an isotropic state and is entangled $\\forall \\, \\, \\beta > \\frac{1}{{d + 1}}$ .",
"Moreover, the condition $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle \\ge \\frac{1}{d}$ shows that $k \\ge \\frac{1}{d+1}$ Hence, $\\langle \\varrho ,T_{w}\\rangle $ can be rewritten as $\\langle \\varrho ,T_{W}\\rangle &&= \\langle \\varrho ,\\left[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|\\right]\\rangle \\nonumber \\\\&&=[\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle )-\\langle \\phi ^{+}|\\varrho |\\phi ^{+}\\rangle ]{}\\nonumber \\\\&&=\\frac{(k-\\beta )(d^{2}-1)}{d^{2}}<0,~~~{\\rm when}~~k<\\beta $ Therefore, our witness $T_{W}$ detects at least one state useful for teleportation when $\\frac{1}{d+1} \\le k<\\beta $ and thus satisfies (REF ).",
"Our analysis shows that all entangled isotropic states can be successfully used for teleportation, which is a well known result [17].",
"This completes the proof that our witness satisfies both the required conditions (REF ) and (REF ) and thus is a teleportation witness.",
"The form of teleportation witness $T_{W}$ suggests that one can also replace $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle = s$ ; a real constant, but for reasons to be described below, we would like to retain the original form of our operator $T_{W}$ = $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|$ .",
"For example, if one uses the form $T^{\\prime }_{W}$ = $s I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|$ , then the condition $\\langle \\varrho ,T^{\\prime }_{W}\\rangle < 0$ requires that $\\beta > s$ .",
"Hence, the operator detects states useful for teleportation when $\\frac{1}{d} \\le s<\\beta $ which shows that the operator $T^{\\prime }_{W}$ detects a smaller set of entangled states with respect to $T_{W}$ , at least in this particular case.",
"Theorem 2: Every teleportation witness in $d \\otimes d$ dimensions $(d > 2)$ is an entanglement witness, but every entanglement witness in $d \\otimes d$ dimensions $(d > 2)$ may not be a teleportation witness.",
"Proof: A Hermitian operator $A$ is an entanglement witness if it satisfies the following two conditions: (i) The expectation value of the operator $A$ must always be non-negative for all separable states in $d \\otimes d$ dimensions i.e.",
"for states with FEF $\\le $ $\\frac{1}{d}$ .",
"(ii) The expectation value of the operator $A$ must be negative for at least one entangled state in $d \\otimes d$ dimensions i.e.",
"for a state with FEF $>$ $\\frac{1}{d}$ .",
"Teleportation witnesses also satisfy above two conditions as required by the conditions (REF ) and (REF ).",
"Hence every teleportation witness is also an entanglement witness.",
"Conversely, we can always show that there exists an indecomposable entanglement witness which would not satisfy the condition (REF ) and therefore cannot be termed as a teleportation witness.",
"For example, one can always find at least one bound entangled state for which the expectation value of non-decomposable entanglement witnesses would be negative [18].",
"Thus, non-decomposable entanglement witnesses detect at least one bound entangled state which is not useful for teleportation (since the singlet fraction of a bound entangled state is equal to $\\frac{1}{d}$ ).",
"Hence, non-decomposable entanglement witnesses do not satisfy the required condition (REF ) to be a teleportation witness.",
"This completes the proof that in $d \\otimes d$ dimensions $(d > 2)$ , every teleportation witness is also an entanglement witness, however the converse is not true.",
"Corollary 1: In $2 \\otimes 2$ dimensions, every teleportation witness is an entanglement witness and vice versa.",
"Proof: In $2 \\otimes 2$ dimensions, every entangled bi-partite state can be made useful for teleportation up to stochastic local operation and classical communication (SLOCC) [19].",
"Thus teleportation and entanglement witnesses will always satisfy the conditions (REF ) and (REF ).",
"Hence, every teleportation witness in $2 \\otimes 2$ dimensions will also be an entanglement witness and vice versa.",
"Corollary 2: A witness is a teleportation witness iff it is decomposable.",
"Proof: By definition, the expectation value of teleportation witnesses for all bound entangled states is always non-negative.",
"Using theorem 2, we have shown that every teleportation witness is an entanglement witness and since teleportation witnesses cannot detect bound entangled states, a teleportation witness can only be a decomposable entanglement witness.",
"Conversely, a decomposable entanglement witness can only detect NPT states i.e.",
"states for which the expectation value of the decomposable entanglement witness would be negative (REF ).",
"As it also satisfies the condition (REF ), a decomposable entanglement witness is also a teleportation witness."
],
[
"Optimal Teleportation Schmidt Witness",
"In this section, we show that our generalized teleportation witness is also an optimal witness.",
"In addition, we analyze the Schmidt number of an arbitrary mixed state useful for teleportation by constructing a Schmidt number teleportation witness.",
"The results obtained in this section provide a way to characterize the mixed state entanglement in bi-partite systems.",
"For example, we propose a form of teleportation Schmidt witness to demonstrate its significance in calculating the Schmidt number of mixed entangled states for a given range of parameters.",
"For bipartite systems, Schmidt number indicates the number of degrees of freedom that are entangled between two subsystems.",
"Theorem 3: Teleportation witness $T_{W}$ with $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle \\ge \\frac{1}{d}$ in $d \\otimes d$ dimensions is an optimal teleportation witness of Schmidt number $r \\ge 2$ .",
"Proof: An optimal entanglement witness of Schmidt number r in $d \\otimes d$ dimensions is given by [20] $W_{opt}=I-\\frac{d}{r-1}|\\phi ^{+}\\rangle \\langle \\phi ^{+}|$ If $r=2$ , we have $W_{opt}&=&I-d|\\phi ^{+}\\rangle \\langle \\phi ^{+}|{}\\nonumber \\\\&=&d(\\frac{1}{d}I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}|){}\\nonumber \\\\&\\propto & W ~~~{\\rm from [Eq.",
"(5)]}~~ \\nonumber \\\\&= & T_{W} ~~~{\\rm if}~~ \\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle = \\frac{1}{d}$ Hence, teleportation witnesses $W$ and $T_{W}$ are optimal teleportation witnesses of Schmidt number 2.",
"For $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle > \\frac{1}{d}$ , the teleportation witness $T_{W}$ can be re-expressed as $T_{W} &=&\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle I-|\\phi ^{+}\\rangle \\langle \\phi ^{+}| \\nonumber \\\\&=&\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle \\left[I-\\frac{1}{\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle }|\\phi ^{+}\\rangle \\langle \\phi ^{+}|\\right]$ Comparing Eqs.",
"(REF ) and (REF ) shows that the teleportation witness $T_{W}$ is proportional to the optimal witness $W_{opt}$ if $r &= & d\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle +1 {}\\nonumber \\\\&>&2,~~{\\rm since}~~\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle >\\frac{1}{d}$ This completes the proof that our teleportation witness $T_{W}$ is an optimal witness of Schmidt number $r \\ge 2$ .",
"Illustration: Let us consider a family of states for $d\\times d$ dimensional systems $\\chi _{\\beta }=\\beta |\\phi ^{+}\\rangle \\langle \\phi ^{+}|+\\frac{1-\\beta }{d^{2}}I,~~\\frac{-1}{d^2-1}\\le \\beta \\le 1$ where the state $\\chi _{\\beta }$ is entangled $\\forall \\, \\,\\beta >\\frac{1}{d+1}$ .",
"In this illustration, we propose the form of our teleportation Schmidt witness to demonstrate that it identifies the ranges of parameters for which the state in Eq.",
"(15) is of Schmidt number $r \\ge 2$ .",
"For this, we fix the reference state $\\rho _{0}$ as $\\rho _{0}=\\frac{1-f_{0}}{d^{2}-1}I+\\frac{d^2f_{0}-1}{d^{2}-1}|\\phi ^{+}\\rangle \\langle \\phi ^{+}|$ which gives $f_{0}=\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle $ , satisfying $\\frac{1}{d}\\le f_{0}\\le 1$ .",
"Also, the singlet fraction of $\\rho _{0}$ is given by [17] $F(\\rho _{0})= f_{0}, ~~~~\\frac{1}{d}\\le f_{0}\\le 1$ Thus, the expectation value of the teleportation witness $T_{W}$ in the state $\\chi _{\\beta }$ is $Tr(T_{W}\\chi _{\\beta })=\\frac{(d^2f_{0}-1)-\\beta (d^{2}-1)}{d^{2}}$ If we take $f_{0}=\\frac{1}{d}$ then the teleportation witness $T_{W}$ assures that the state $\\chi _{\\beta }$ , useful for teleportation, is of Schmidt number 2 when $\\beta \\in \\left(\\frac{1}{d+1},\\frac{2d-1}{d^{2}-1}\\right]$ .",
"In general, if we take $f_{0}=\\frac{r-1}{d}, (r=2,3,....d)$ then the corresponding teleportation witness $(T_{W})_{r}$ assures that the state $\\chi _{\\beta }$ , useful for teleportation, is of Schmidt number $r$ when $\\beta \\in \\left(\\frac{d(r-1)-1}{d^2-1},\\frac{dr-1}{d^{2}-1}\\right]$ .",
"For example, in $3 \\otimes 3$ dimensions the state $\\chi _{\\beta }$ is entangled for $\\forall \\, \\, \\beta >\\frac{1}{4}$ and the range of $\\beta $ confirms that $2 < r \\le 3$ .",
"Therefore, the Schmidt number of the state $\\chi _{\\beta }$ must be 3 for the given range of $\\beta $ .",
"Similarly, one can calculate the Schmidt number for higher dimensional systems for the given range of parameters.",
"Hence, our teleportation Schmidt witness can also be used to characterize mixed state entanglement in terms of Schmidt numbers."
],
[
"Experimental determination of $T_{W}$",
"In this section, we analyze the decomposition of our teleportation witness in terms of Pauli spin matrices for $2 \\otimes 2$ systems, and Gell-Mann matrices for higher dimensional systems.",
"Such decompositions allow for the experimental detection of entanglement in an unknown state through the measurements of expectation values of teleportation witness.",
"For experimental realization of teleportation witnesses, we need to decompose them into projectors of the form [13] ${T_W} = \\sum \\limits _{i = 1}^j {{k_i}\\left| {{e_i}} \\right\\rangle } \\left\\langle {{e_i}} \\right| \\otimes \\left| {{f_i}} \\right\\rangle \\left\\langle {{f_i}} \\right|$ If $\\langle \\phi ^{+}|\\rho _{0}|\\phi ^{+}\\rangle =\\frac{1}{2}$ then the form of $T_{W}$ is equivalent to the teleportation witness $W$ in Eq.",
"(5) and thus can be decomposed in form of Pauli spin matrices such that ${T_W} = \\frac{1}{2}[I \\otimes I - {\\sigma _x} \\otimes {\\sigma _x} + {\\sigma _y} \\otimes {\\sigma _y} - {\\sigma _z} \\otimes {\\sigma _z}]$ Eq.",
"(20) indicates that the total number of measurements required to estimate our witness is limited to 3 as compared to the measurement of 15 parameters required for full state tomography.",
"This difference in number of measurements is even more evident for higher dimensional systems providing a practical utility to our results when compared to full state tomography in distinguishing the useful states for quantum teleportation.",
"Similarly, for $d\\otimes d$ systems our teleportation witness can be decomposed as $T_{W}^{d} = \\left( {\\left\\langle {{\\phi ^ + }} \\right|{\\rho _0}\\left| {{\\phi ^ + }} \\right\\rangle - \\frac{1}{{{d^2}}}} \\right)I - \\frac{1}{{2d}}\\Lambda $ where $\\Lambda = \\sum \\limits _{i < j} {\\Lambda _s^{ij} \\otimes } \\Lambda _s^{ij} - \\sum \\limits _{i < j} {\\Lambda _a^{ij} \\otimes } \\Lambda _a^{ij} + \\sum \\limits _{m = 1}^{d-1} {{\\Lambda ^m}} \\otimes {\\Lambda ^m}$ The suffix s, a and superscript m represent the symmetric, antisymmetric and diagonal Gell-Mann matrices [21], respectively.",
"For example, in case of qutrits i.e.",
"$3 \\otimes 3$ dimensional systems the teleportation witness $T_{W}$ can be represented as $T_{W}^{3} = \\left( {\\left\\langle {{\\phi ^ + }} \\right|{\\rho _0}\\left| {{\\phi ^ + }} \\right\\rangle - \\frac{1}{{{9}}}} \\right)I - \\frac{1}{{6}}\\Lambda $ where $\\Lambda &=& \\Lambda _s^{12} \\otimes \\Lambda _s^{12} + \\Lambda _s^{13} \\otimes \\Lambda _s^{13} + \\Lambda _s^{23} \\otimes \\Lambda _s^{23} - \\Lambda _a^{12} \\otimes \\Lambda _a^{12} - \\Lambda _a^{13} \\otimes \\Lambda _a^{13} - \\Lambda _a^{23} \\otimes \\Lambda _a^{23} \\nonumber \\\\&+& {\\Lambda ^1} \\otimes {\\Lambda ^1}{\\rm { + }}{\\Lambda ^2} \\otimes {\\Lambda ^2}{\\rm { }}$ and $\\Lambda _s^{12} = \\left( {\\begin{array}{*{20}{c}}0&1&0\\\\1&0&0\\\\0&0&0\\end{array}} \\right)$ , $\\Lambda _s^{13} = \\left( {\\begin{array}{*{20}{c}}0&0&1\\\\0&0&0\\\\1&0&0\\end{array}} \\right)$ , $\\Lambda _s^{23} = \\left( {\\begin{array}{*{20}{c}}0&0&0\\\\0&0&1\\\\0&1&0\\end{array}} \\right)$ , $\\Lambda _a^{12} = \\left( {\\begin{array}{*{20}{c}}0&-i&0\\\\i&0&0\\\\0&0&0\\end{array}} \\right)$ , $\\Lambda _a^{13} = \\left( {\\begin{array}{*{20}{c}}0&0&-i\\\\0&0&0\\\\i&0&0\\end{array}} \\right)$ , $\\Lambda _a^{23} = \\left( {\\begin{array}{*{20}{c}}0&0&0\\\\0&0&-i\\\\0&i&0\\end{array}} \\right)$ , $\\Lambda ^{1} = \\left( {\\begin{array}{*{20}{c}}1&0&0\\\\0&-1&0\\\\0&0&0\\end{array}} \\right)$ , $\\Lambda ^{2} = \\frac{1}{\\sqrt{3}}\\left( {\\begin{array}{*{20}{c}}1&0&0\\\\0&1&0\\\\0&0&-2\\end{array}} \\right).$ Hence, the decomposition of our teleportation witness in terms of measurable quantities not only allows for the experimental detection of entanglement but also provides a way to decrease the requirement of number of measurements when compared to full state tomography."
],
[
"Conclusion",
"We have proposed an optimal teleportation witness and demonstrated its practical utility in distinguishing the entangled states useful for teleportation.",
"The form of teleportation witness $T_{W}$ studied here is a generalized form of the witness $W$ discussed in [10].",
"The results obtained in this article are different from the previous study [10] in a sense that the witness $W$ is an optimal witness of Schmidt number 2, but our witness $T_{W}$ is an optimal witness of Schmidt number $\\ge 2$ .",
"This allows for the detection of larger sets of mixed entangled states useful for teleportation in higher dimensions as well.",
"We found that all the teleportation witnesses are also entanglement witnesses, however the converse is not true.",
"It also turned out that a teleportation witness is always a decomposable entanglement witness, which is a necessary and sufficient condition.",
"The experimental determination of such teleportation witnesses decreases the number of parameters to be measured in comparison to the full state tomography of an unknown state useful for teleportation, indicating the practical significance of our study.",
"In future, we would like to study the utility of such witnesses for other communication protocols as well.",
"Another question of particular interest would be the form and characteristics of teleportation witnesses in multiqubit systems."
]
] | 1204.0983 |
[
[
"Numerical simulations of chromospheric hard X-ray source sizes in solar\n flares"
],
[
"Abstract X-ray observations are a powerful diagnostic tool for transport, acceleration, and heating of electrons in solar flares.",
"Height and size measurements of X-ray footpoints sources can be used to determine the chromospheric density and constrain the parameters of magnetic field convergence and electron pitch-angle evolution.",
"We investigate the influence of the chromospheric density, magnetic mirroring and collisional pitch-angle scattering on the size of X-ray sources.",
"The time-independent Fokker-Planck equation for electron transport is solved numerically and analytically to find the electron distribution as a function of height above the photosphere.",
"From this distribution, the expected X-ray flux as a function of height, its peak height and full width at half maximum are calculated and compared with RHESSI observations.",
"A purely instrumental explanation for the observed source size was ruled out by using simulated RHESSI images.",
"We find that magnetic mirroring and collisional pitch-angle scattering tend to change the electron flux such that electrons are stopped higher in the atmosphere compared with the simple case with collisional energy loss only.",
"However, the resulting X-ray flux is dominated by the density structure in the chromosphere and only marginal increases in source width are found.",
"Very high loop densities (>10^{11} cm^{-3}) could explain the observed sizes at higher energies, but are unrealistic and would result in no footpoint emission below about 40 keV, contrary to observations.",
"We conclude that within a monolithic density model the vertical sizes are given mostly by the density scale-height and are predicted smaller than the RHESSI results show."
],
[
"Introduction",
"The transport of flare-accelerated electrons and the generation of hard X-ray (HXR) emission in the solar atmosphere is one of the most important and widely used diagnostics of flare accelerated electrons.",
"These supra-thermal particles precipitate along the field lines of a magnetic loop from the acceleration site towards the denser regions of the chromosphere.",
"They undergo Coulomb collisions with electrons and ions in the ambient plasma, and can be observed via their bremsstrahlung emission.",
"The bulk of the observed HXR emission comes from the footpoints of magnetic structures where the density is high and electrons lose their energy completely, with the electron stopping location (depth) determined by the initial electron energy and the ambient density.",
"Using this energy dependency in combination with observations from RHESSI [19], it has become possible to characterize the structure of X-ray sources.",
"Assuming collisional transport, the chromospheric density could be inferred [3], [15], [14], [26], [5].",
"These observations suggest that the bulk of HXRs in the range 30-100 keV is produced at heights of $\\sim 0.7-1.2$ Mm.",
"The density scale heights generally agree well with hydrostatic chromospheric density models such as [29].",
"However, using RHESSI visibility techniques [11], [27], it is also possible to infer the energy-dependent source sizes with better than arcsecond accuracy as shown by [5], [15], [14].",
"The source sizes in the direction along the magnetic field inferred by these authors are in the range from 2 arcsec to 6 arcsec [5] which is up to a factor of 4 larger than what would be expected from collisional transport in the same density profile.",
"To explain this discrepancy, [14] suggested that the electrons propagate in a multi-thread loop with different density profiles along each thread, so that the X-ray source positions are the same as in the case of collisional transport in a single density loop, but the vertical size is enlarged.",
"However, even within a single monolithic loop, [5] indicated a number of processes which might increase the vertical extent of HXR sources.",
"To investigate those effects quantitatively one has to solve the equations for electron transport in the footpoints.",
"The electron transport problem has been considered in the past by several authors to investigate various aspects of electron transport, electron trapping and energy losses in the solar atmosphere.",
"Both semi-analytical and full numerical solutions can be found in the literature.",
"[17], [16], [4], [22], [23], and [12] use numerical methods to investigate the effect of the magnetic field geometry and magnetic trapping on the electron spectrum while [21] use a test-particle method of solution, applying it to a situation without magnetic mirroring.",
"Many of those studies focused on the electron transport and the effect of scattering and magnetic mirroring on the electron distribution.",
"To compare these models with actual observations, one has to go a step further and model the resulting X-ray flux, as was done by e.g.",
"[18].",
"With the observational capabilities of instruments such as Yohkoh and RHESSI it became possible to directly compare the models with observations.",
"[25] investigate the conditions for formation of loop-top sources and compare predicted time-profiles of the X-ray emission with time-profiles observed by Yohkoh.",
"[24] use a trap-plus-precipitation model to explain RHESSI observations.",
"While those studies all focused on spectra, [8] investigated the height of X-ray sources obtained from numerical simulations and compared them with Yohkoh observations, finding that they are consistent with partial electron trapping in a magnetic trap.",
"However, there are no quantitative studies of the HXR source sizes.",
"In this paper, we focus on the size of HXR sources produced by non-thermal electrons ($E\\gtrsim 30$ keV) in Coulomb collisions with the ambient plasma.",
"Within the assumption of a single monolithic loop we analyze a range of processes which could increase the vertical extent (along the magnetic field lines) of the sources and explain the observations.",
"Thus we consider: i) density variations in a single loop; ii) the role of the initial pitch angle distribution iii) the effect of pitch-angle scattering; iv) magnetic mirroring, and v) instrumental effects related to RHESSI observations."
],
[
"Transport of energetic electrons in X-ray footpoints",
"The evolution of an initial electron flux distribution $F_0(E_0,z)$ as a function of distance $z$ along the magnetic field lines is described by the time independent Fokker-Planck equation [10], including magnetic mirroring and collisional pitch-angle scattering and energy loss: $ \\frac{\\partial F}{\\partial z}-\\frac{1-\\mu ^2}{2\\mu }\\frac{\\mathrm {d}(\\ln B)}{\\mathrm {dz}}\\frac{\\partial F}{\\partial \\mu }-\\frac{Kn(z)}{\\mu E}\\frac{\\partial F}{\\partial E}+\\frac{n(z)K}{2E^2\\mu }\\frac{\\partial }{\\partial \\mu }\\left[(1-\\mu ^2)\\frac{\\partial F}{\\partial \\mu }\\right]=-\\frac{Kn(z)}{\\mu E^2}F,$ where $n(z)$ is the plasma density at distance $z$ , $K=2\\pi e^4\\Lambda $ , with $\\Lambda $ the Coulomb logarithm, $B$ is the magnetic field strength, and $\\mu $ is the cosine of the electron pitch angle relative to the magnetic field direction.",
"In the case of purely collisional energy loss in a uniform magnetic field and an injected electron power-law flux distribution $F_0(E_0)\\sim E_0^{-\\delta }$ this leads to: $F(E,z)=F_0[E_0(E,z)]E{\\sqrt{E^2+2KN(z)}}^{(-\\delta -1)},$ where $E_0(E,z)=(E^2+2KN(z))^{1/2}$ and $N(z)=\\int n(z)dz$ is the column depth.",
"This we will refer to as “simple thick-target” in this paper.",
"Equation REF can also be solved analytically by neglecting collisions but including the term for the magnetic field [17], [31].",
"However, this analytical solution is only applicable as long as the electron pitch angle cosine is larger than about 0.7 because of a factor $\\sim 1/\\mu $ that is introduced in Eq.",
"REF .",
"Thus, if one wants to include all physical effects one has to use test-particle simulations.",
"Equation REF can be written as a set of stochastic differential equations [16], [8].",
"Since the typical integration time necessary for reliable imaging of footpoints with RHESSI is of the order of 30 - 60 seconds, which is very large compared to the electron loop transit time and collisional loss time in the chromosphere, a time-independent treatment is a good approximation.",
"Therefore we consider time-independent equations along the particle path: $\\frac{dz}{ds}&=&\\mu \\\\\\frac{dE}{ds }&=&-\\frac{K}{E}n(z)\\\\\\frac{d\\mu }{ds}&=&-(1-\\mu ^2)\\frac{d\\ln B}{2dz}-\\frac{\\mu Kn(z)}{E^2}+\\left[(1-\\mu ^2)\\frac{Kn(z)}{E^2}\\right]^{1/2}W, $ where $z$ is the distance from the point of injection along the magnetic field lines, $s$ is the path of the electron, and $W(s)$ denotes a standard Wiener process.",
"Equations REF - are solved using a numerical scheme: $z_{j+1}&=&z_j+\\mu _ j\\Delta s\\\\ E_{j+1}&=&\\left[E_j^2-2{K}n(z_j)\\Delta s\\right]^{1/2}\\\\\\mu _{j+1}&=& \\mu _{j}-\\left.\\frac{1}{2}(1-\\mu _{j}^2)\\frac{d\\ln B}{dz}\\right| _{z=z_j}\\Delta s-\\frac{\\mu _j Kn(z_j)}{E_j^2}\\Delta s+\\left[(1-\\mu _j^2)\\frac{Kn(z_j)}{E_j^2}\\Delta s\\right]^{1/2}\\xi , $ where $\\xi $ is a random variable taken from the normal distribution $p(\\xi )=1/\\sqrt{2\\pi }\\exp {(-\\xi ^2/2)}$ for each step $\\Delta s$ .",
"The scheme proved to be reliable in modelling the effect of collisional scattering on pitch angle [4], [21].",
"Equations (REF -) are solved using a power-law electron distribution $F_0(E_0,h=h_{loop})\\sim E_0^{-\\delta }$ injected at the top of the loop with high energy cutoff energy at 500 keV and injection height $h_{loop}=12.5$ Mm.",
"Equation REF was used to test the simulations.",
"Both numerical and analytical solutions result in the number of electrons as a function of distance $z$ from the injection point, or, equivalently, as a function of height $h$ above the photosphere $F(E,h)$ where $h=h_{loop}-z$ .",
"Finally, calculating the corresponding X-ray bremsstrahlung intensity $I(\\epsilon ,h)$ , and determining the X-ray source vertical profiles, this is compared to observations made with RHESSI."
],
[
"Definition of size and position",
"In the observations of limb events presented in [15], [14] and [5], we found the position and size by forward fitting X-ray visibilities (2D spatial Fourier components) with a circular or elliptical Gaussian source model.",
"The size of a source is thereby defined as the full-width at half maximum (FWHM) of the fitted Gaussian.",
"$I(x,y; \\epsilon )=\\frac{I_0(\\epsilon )}{2\\pi \\sigma _x\\sigma _y}\\exp \\left(-\\frac{(x-x_0(\\epsilon ))^2}{2\\sigma _x^2}-\\frac{(y-y_0(\\epsilon ))^2}{2\\sigma _y^2}\\right),$ where $2\\sqrt{2\\ln 2}\\sigma _x$ and $2\\sqrt{2\\ln 2}\\sigma _y$ are FWHMs of an elliptical Gaussian source in the $x$ and $y$ direction respectively.",
"For HXR sources observed at the solar limb, the size along the radial direction represents the vertical FWHM size of the source, while the perpendicular size (along the solar limb) is equivalent to the size of a HXR footpoint parallel to the solar surface.",
"The simulations performed here provide the electron flux as a function of height $F(E,h)$ (one dimension) and hence the vertical extent of HXR sources.",
"The X-ray flux $I(\\epsilon ,h)$ per unit distance is then given as: $ I(\\epsilon ,h)=\\frac{n(h)A(h)}{4\\pi R^2}\\int _\\epsilon ^\\infty F(E,h)\\sigma (\\epsilon ,E)\\mathrm {dE}$ where $A(h)$ is the area of the magnetic flux tube at height $h$ , $R$ is the distance Sun-Earth and $\\sigma (\\epsilon ,E)$ the bremsstrahlung cross-section.",
"We define the height of a source as the first moment of the X-ray flux profile and the FWHM as the second moment.",
"Only emission larger than 10% of the maximum flux was used to compute the moments, to emulate the fact that RHESSI images have a limited dynamic range [11].",
"Figure REF illustrates the photon flux as a function of height for two photon energies in the analytical solution of the simple thick-target case.",
"The right-hand side of the figure shows the observed FWHM in the event analyzed in [14] compared with the expected FWHM in a simple thick-target model given by the analytical solution of Eqs.",
"(REF ,REF ) .",
"Figure: Left: Normalized photon flux as a function of height above the photosphere at 30 keV and 70 keV (red), using a density profile of n(h)=10 10 +1.16×10 17 ×exp(-h/h 0 )n(h)=10^{10}+1.16\\times 10^{17}\\times exp(-h/h_0) (comp.",
"Eq. )",
"with a scale-height h 0 h_0=130 km.",
"The vertical dashed lines give the first moment (maximum position) of the emission above 10 % (horizontal dashed line).",
"Right: Observed FWHM in the event of January 6th 2004 .",
"The black line is the expected source FWHM in a simple thick target, using Eqs.",
"and ."
],
[
"Analytical and numerical results",
"We first investigate the effect of the chromospheric density function on the resulting source size in the simple thick target (analytically), using Eq.",
"REF .",
"Then, the influence of the initial pitch-angle distribution, collisional pitch-angle scattering and magnetic mirroring will be explored using test-particle simulations (Eq.",
"REF - ).",
"Finally, in Section REF we will discuss in more detail how the sources would be observed by RHESSI and what results we would expect using visibility forward fitting."
],
[
"Chromospheric density structure",
"The deposition of large amounts of energy into the chromosphere by energetic electrons could lead to a number of processes, including heating and expansion of the chromosphere.",
"This process of chromospheric evaporation generally leads to redistribution of plasma density in the flaring atmosphere, increasing the density of plasma in the flaring loop [9], [2].",
"At the same time, the density structure in a flaring loop can strongly affect the source size even in the case of purely collisional energy loss.",
"In previous work, a single scale-height exponential density [15], [14], [5], multiple scale-height density [26] or a power-law density function [3] have been investigated.",
"Although the positions of HXR sources are in agreement with the single scale-height exponential density profiles with scale heights of 130-200 km, the predicted vertical source sizes are up to a factor of 4 smaller than observed [5] ([14] suggested that a multi-threaded density structure with vertical strands of different density could increase the source size significantly).",
"The exponential density function $n(h)^{exp}=n_{l}+n_{phot}\\times exp(-h/h_0)$ is used by [5] where $n_{l}$ is the (constant) loop density, $n_{phot}=1.16\\times 10^{17}$ $\\rm {cm^{-3}}$ the photospheric density following [29], and $h_0$ is the density scale height.",
"Figure REF illustrates such an exponential density model with scale-height $h_0$ =130 km, and also a density model of the shape of a $\\kappa $ -function $n(h)^{\\kappa }=n_{phot}\\times \\left(1+\\frac{h}{\\kappa h_0}\\right)^{(-\\kappa +1)},$ for a $\\kappa $ factor of 10 and two different scale heights $h_0=130$ km and $h_0=290$ km (dark blue and light blue curves in Fig.",
"REF ).",
"In addition, densities that combine the exponential function near the photosphere with either a quadratic function or a $\\kappa $ -function at larger heights were used, where the exponential plus quadratic function was given as $n(h)=n(h)^{exp}+1.16\\times 10^{a}(h_{max}-h)^2$ , where $h_{max}$ is the loop height and $a=[10,11,12]$ (yellow, dark red, and red curves in Fig.",
"REF ).",
"The exponential plus $\\kappa $ -function was given as $n(h)=n(h)^{exp}+n(h)^{\\kappa }$ (green curve in Fig.",
"REF ).",
"These latter models account for the fact that the high density in the lower chromosphere, below about 1 Mm, will not be affected significantly by processes such as evaporation [20], while the loop density may be considerably different from quiet Sun density models.",
"As Fig.",
"REF demonstrates, a notable effect on the FWHM is only found if the loop density reaches extreme values of more than $10^{11}$ $\\rm {cm^{-3}}$ .",
"Such coronal densities are rather non-typically high although are observed in some flares [30].",
"Such high densities will lead to energetic electrons of energy $\\sim 20$ keV and even higher being collisionally stopped in the coronal part of the loop before reaching the chromospheric footpoints.",
"All these density functions result in an effective increase of the loop density compared with the single scale-height exponential and the resulting HXR source FWHM is larger by up to a factor of 4 than in the case of an exponential density (Fig.",
"REF ).",
"However, the location of the peak of the emission is also found to be a factor of 4 higher, and the bulk of the emission below 40 keV comes from the top of the coronal loop, therefore no footpoints would be observed below 40 keV, which is not the case in the observations [5], where footpoints are observed at energies as low as 20 keV.",
"Figure: Effect of different density models on the height and FWHM of HXR sources.",
"Left: Density models.The black curve is an exponential density with a single scale-height of 130 km.",
"Dark blue and light blue are κ\\kappa -functionswith different scale heights.",
"Green is an exponential density combined with a κ\\kappa -function.",
"Yellow, red and darkred represent an exponential density combined with a ∼h 2 \\sim h^2 function (see text for details).Middle: Height of peak emission as a function of energy.",
"Right: FWHM as a function of energy.Due to the relatively modest increase of the chromospheric temperature at heights below 1 Mm, we assumed a neutral atmosphere.",
"However, the plasma in the transition region and lower corona will be partially or completely ionized with a change in the ionization fraction at some height in the chromosphere.",
"The ionization state of the medium affects the Coulomb logarithm, so that $\\ln \\Lambda _{eH}\\simeq 7.1$ in neutral media and $\\ln \\Lambda _{ee} \\simeq 20$ in fully ionized plasma, where $\\Lambda _{ee}$ and $\\Lambda _{eH}$ are the electron-electron and electron-hydrogen Coulomb logarithms.",
"The effect of this on HXR spectra has been discussed by [6] and in the context of RHESSI spectroscopy by [13].",
"To investigate the influence of ionization change on the height and the FWHM of HXR sources, we introduced an effective Coulomb logarithm $\\Lambda =(\\Lambda _{ee}-\\Lambda _{eH})\\left(x(h)+\\frac{\\Lambda _{eH}}{\\Lambda _{ee}-\\Lambda _{eH}}\\right),$ The ionization fraction $x(h)$ was assumed to be a step function with $x(h)=1$ for $h> 0.9$ Mm and $x(h)=0$ for $h<0.9$ Mm.",
"The effect of non-uniform density is illustrated in Fig.",
"REF along with a comparison with the case of constant Coulomb logarithms.",
"The case of a more realistic atmosphere with partial ionization would lead to results within the extremes illustrated in Fig.",
"REF and will not change the source size noticeably.",
"Thus, the use of an idealized ionization structure is justified.",
"Figure: Left: Total photon spectra for completely neutral target (black), completely ionized target (blue) and ionization step-change (red).",
"Middle and right: Height and FWHM of HXR sources as a function of energy in the three cases.The simulations show that although different chromospheric and loop density models can increase the source size by up to a factor of 4 to fit the observations, this will also change the height of the sources, contrary to the observations.",
"It has to be noted that in order to produce a noticeable change of the HXR source size, the density structure of the whole atmosphere, including the transition region and corona needs to be changed by a few orders of magnitude.",
"This seems in contradiction with both theoretical models of the flaring atmosphere [1] and observations."
],
[
"Pitch-angle distribution",
"The height and the width of a source is also likely to depend on the initial pitch-angle distribution of the injected electron beam.",
"In the simple thick-target model, injection and propagation of energetic electrons is assumed parallel to the magnetic field (i.e.",
"cosine of initial pitch angle $\\mu _0=1$ ), but in the other extreme case of injection perpendicular to the field ($\\mu _0=0$ ) or strong trapping, the electrons would lose energy near the acceleration region, the height would be constant as a function of energy, and the FWHM would depend on the extent of the acceleration region.",
"In an intermediate situation $0<\\mu _0<1$ , energetic electrons are expected to lose energy at different heights depending on their initial pitch-angle distribution.",
"The effect of an initial pitch-angle distribution is investigated including collisional energy loss, but no change in the initial pitch-angle distribution is assumed, i.e.",
"Eq.",
"is $d\\mu /ds=0$ .",
"In this and the subsequent sections, an exponential density profile $n(h)=n_{l}+n_{phot}\\exp (-h/h_0)$ with a scale height of 144 km and a constant Coulomb logarithm of $\\Lambda =7.1$ is used.",
"This density profile is consistent with the source heights measured with RHESSI and the theoretical modeling of the low atmosphere at heights $h\\lesssim 1$ Mm.",
"Figure REF illustrates how the initial pitch-angle distribution affects the height and the FWHM.",
"Three different cases are presented: $\\mu _0=1$ (as a reference), $0.9<\\mu _0\\le 1$ uniform (strongly beamed) and $0.1<\\mu _0\\le 1$ uniform (broad distribution).",
"The broad distribution leads, as expected, to a larger source size as a function of energy, as well as a larger FWHM.",
"However, the maximum change in both height and FWHM is about 10% in the case of the broad distribution.",
"Therefore, this effect alone cannot account lead to the observed vertical sizes of HXR sources.",
"Figure: Normalized photon flux as a function of height at 30 keV and 70 keV (solid and dashed lines, left), height of maximum emission and FWHM as a function of photon energy (middle and left), for three cases of initial pitch-angle distribution: μ 0 \\mu _0=1 (black), 0.9<μ 0 ≤10.9<\\mu _0\\le 1 (red), 0.1<μ 0 ≤10.1<\\mu _0\\le 1 (blue)."
],
[
"Collisional pitch-angle scattering",
"In collisional interactions with the ambient plasma, electrons do not only lose energy, but are pitch-angle scattered with a similar rate.",
"Therefore, their pitch-angle distribution changes as the particles propagate downwards towards dense regions of the atmosphere.",
"Figure REF compares the standard case (collisional energy loss only), with the outcome of a situation with collisional pitch-angle scattering included, i.e.",
"Eq.",
"becomes $\\frac{d\\mu }{ds}=-\\frac{\\mu Kn(z)}{E^2}+\\left[(1-\\mu ^2)\\frac{Kn(z)}{E^2}\\right]^{1/2}W.$ In this case, the maximum emission is located up to 20 % higher than in the no-scattering case.",
"However, the effect on the FWHM is small, at the level of about 5%.",
"As expected, the effect is qualitatively similar to the injection of a broad initial pitch-angle distribution.",
"Figure: Height of maximum emission and FWHM as a function of photon energy (middle and left), for two cases of initial pitch-angle distribution: 0.9<μ 0 ≤10.9<\\mu _0\\le 1 (red), 0.1<μ 0 ≤10.1<\\mu _0\\le 1 (blue), and including collisional pitch-angle scattering.",
"The black line indicates the result for collisional energy loss only."
],
[
"Magnetic mirroring",
"A converging magnetic field at the loop footpoints also changes the pitch-angle distribution of energetic electrons, causing electrons with large pitch angle (small $\\cos \\theta $ ) to mirror upwards from the footpoints before they are collisionally stopped.",
"This might further contribute to an increase in source size.",
"In this Section we consider collisional energy loss, and pitch-angle change due to magnetic field convergence, but not collisional pitch-angle change, i.e.",
"Eq.",
"becomes $d\\mu /ds=-(1-\\mu ^2)\\frac{d\\ln B}{2dz}$ .",
"The magnetic field strength is modeled as $B(h)=B_0+B_1 \\tanh (-(h-h_M)/h_M)$ , which adequately represents a converging magnetic field in the chromosphere [7].",
"This model gives a magnetic field $B(h>>h_M)\\rightarrow B_0-B_1$ at coronal heights and $B(h=0)\\rightarrow B_0+B_1\\tanh (1)$ at the photospheric level.",
"The increase in field strength relative to the ambient density is illustrated in Fig.",
"REF .",
"The field convergence (and the electron pitch angle) defines the depth of the mirroring point.",
"If the magnetic field converged higher up in the loop, the sources would be observed higher up.",
"In the extreme case, one could simulate a coronal source caused by electron trapping.",
"Figure: Left: Density and magnetic field strength as a function of height.",
"Height of maximum emission and FWHM as a function of photon energy (middle and right), for two cases of initial pitch-angle distribution: 0.9<μ 0 ≤10.9<\\mu _0\\le 1 (red), 0.1<μ 0 ≤10.1<\\mu _0\\le 1 (blue), and including magnetic mirroring.",
"The black line indicates the result for collisional energy loss only.",
"B(h)=B 0 +B 1 tanh(-(h-h M )/h M )B(h)=B_0+B_1\\tanh (-(h-h_M)/h_M) with h M =1h_M=1 Mm, B 0 =600B_0=600 Gauss, B 1 =500B_1=500 Gauss.In the case presented in Fig.",
"REF , the source height is increased by a factor of $\\sim $ 1.6, while the FWHM increases by a factor of 1.7 - 1.3, depending on energy.",
"Although the size increase is larger than in the case of collisional scattering, it is still not strong enough to explain the FWHM observations.",
"In addition, a nearly isotropic initial distribution of electrons (Fig.",
"REF ) leads to a larger source FWHM at higher energies, contrary to X-ray observations."
],
[
"Magnetic mirroring and collisional pitch-angle scattering",
"Finally, the effects described in the above two sections are combined and the full Eqs.",
"REF - are solved numerically (Fig.",
"REF ).",
"Since the effect of pitch-angle scattering itself is small compared to the effects of magnetic mirroring, this case is dominated by the effect of the magnetic field and the result very similar to the case of magnetic mirroring (Fig.",
"REF ).",
"Figure: Height of maximum emission and FWHM as a function of photon energy (middle and left),for three cases of initial pitch-angle distribution: μ 0 \\mu _0=1 (black), 0.9<μ 0 ≤10.9<\\mu _0\\le 1 (red), 0.1<μ 0 ≤10.1<\\mu _0\\le 1 (blue),and including both collisional pitch-angle scattering and magnetic mirroring."
],
[
"Constructing an electron distribution as a function of height",
"The processes described in Sections REF - REF all influence the electron distribution as a function of height, e.g.",
"pitch-angle scattering causes electrons to be stopped higher up in the loop.",
"However, as illustrated in Fig.",
"REF - REF , this has a rather small effect on the X-ray flux profile and thus the source FWHM.",
"The FWHM of an X-ray source is proportional to the product of electron flux density and plasma density $I(\\epsilon , h) \\sim F(E,h) n(h)$ .",
"In the extreme case of $F(E,h)\\sim 1/n(h)$ , the resulting $I(\\epsilon , h)$ is independent of height and this constant value could extend vertically over all $h$ where $F(E,h)\\sim 1/n(h)$ .",
"We can therefore ask how $F(E,h)$ should look in order to make the product of $F(E,h)n(h)$ independent of $h$ , hence increasing the size of the X-ray source.",
"Starting with $F(E,h)$ , as found in the simple thick target case, we modified the shape of $F(E,h)$ for every energy, so that the slope of the curve was close to $1/n(h)$ , as shown in Fig.",
"REF (top left), then we computed the height of the maximum emission and FWHM.",
"As $F(h)$ approaches $1/n(h)$ , the FWHM of the resulting X-ray flux increases up to 5 arcsec.",
"Figure: Top left: Electron flux as a function of height for 24 keV electrons.",
"The solid line represents the simple thick target case.The dashed and dotted lines are “artificial” distributions, constructed so that the slope nears that of 1/n(h)1/n(h).",
"The blue dash-dotted lineillustrates 1/n(h)1/n(h).",
"Top right: normalized X-ray flux.",
"Bottom left: height and FWHM (red curves) for the three different electrondistributions where the line styles correspond to the line styles in the top left panel.",
"Bottom right: Injected electronspectrum (black line) and electron spectrum at a height of 0.8 Mm.However, the height of the resulting HXR maximum as a function of energy is constant, as is the FWHM.",
"Further, the electron spectrum at low heights is completely different from a thick target spectrum (Fig.",
"REF , bottom right).",
"Most importantly, such an electron distribution would have to be extremely “fine tuned” to the ambient density."
],
[
"Instrumental effects and method",
"Comparing the modeling of the main electron transport effects with observations and finding physical explanations for the observed source sizes, we assumed that the observed difference is not entirely due to instrumental effects.",
"This was based on there being no modulation in the finest RHESSI grids (Grid 1 has a spatial resolution of $\\sim 2.3$ arcsec and grid 2 has spatial resolution $\\sim 3.9$ arcsec) in the observed events [5], indicating that the source dimensions must be of the order of $\\gtrsim 4$ arcsec.",
"Here we perform a more quantitative study of the instrumental effects, using the simulation software developed by Richard Schwartz (private communication).",
"The simulation software uses an arbitrary 2-dimensional map as input and creates a corresponding calibrated event list [28].",
"The standard RHESSI imaging algorithms are then used to construct the image.",
"We used several source models (circular Gaussian, elliptical Gaussian) as the initial map and forward-fitted the visibilities from the corresponding calibrated eventlist to compare the fitted FWHM with that of the original map.",
"We find that circular Gaussians are correctly recovered within the uncertainties, e.g.",
"a circular Gaussian with FWHM 1 arcsec is fitted with a FWHM of $0.93 \\pm 0.1$ .",
"In the case of an elliptical Gaussian, the fitted major and minor axes tend to be larger than the original ellipse, e.g.",
"an input elliptical source with major and minor axes of 3 and 1 arcsec respectively is fitted with 3.1 and 1.5 arcsec.",
"Thus the fitted minor axis is 50% larger than the input minor axis.",
"This last example shows that, while there are instrumental effects, especially in the case of elliptical sources, these effects cannot entirely account for the observed source sizes.",
"The FWHM found in the previous sections are all in terms of the second moment of the 1-dimensional height distribution of X-rays $I(\\epsilon , h)$ .",
"However, the limited dynamic range of RHESSI influences the accuracy of the measured moments.",
"X-ray flux in RHESSI images which is around or less than 10% of the brightest part of the image is dominated by an error from the brightest source.",
"However, using the simulations we can address the question of how such a source would be observed with RHESSI, and how the second moment found in the simulations relates to an observed RHESSI image.",
"As input we used the shape of the photon-flux as a function of height found in case of a density $n(h)=n(h)^{exp}+n(h)^{\\kappa }$ (green line in Fig.",
"REF ) with 6 arcsec width.",
"This map was used as input for the simulation software and Clean and Pixon images (using software defaults and grids 1-8) were reconstructed from the calibrated event lists.",
"Figure REF displays the profile of the photon flux as a function of height, the input map, and the Clean and Pixon maps.",
"The resulting fitted FWHM are $4.9^{\\prime \\prime }$ and $2.7^{\\prime \\prime }$ for the major and minor axes.",
"The second moment of the 1-dimensional X-ray flux distribution are is $2.8^{\\prime \\prime }$ and becomes $2.1^{\\prime \\prime }$ if computed only for the emission exceeding the 10% level.",
"Therefore, the second moment of the full distribution overestimates the size, while the moment of the flux $> 10\\%$ underestimates the size, compared to visibility forward fitting.",
"It has to be added that the source model was very simple and there is no unmodulated background added to calibrated event lists.",
"Figure: Top left: X-ray (black) profile for simulations with high loop density (blue).",
"Top right: Simulated map of a loop with ∼\\sim 6 arcsecwidth and the height profile of the simulated X-ray profile.",
"Bottom: RHESSI CLEAN and Pixon image representationof the simulated map.",
"The contours represent the 50%, 70%, and 90% levels of the elliptical Gaussian found with visibility forward fitting."
],
[
"Summary and conclusions",
"In the simple collisional thick-target model, both the height of the maximum HXR emission and the vertical HXR source sizes are determined by the density scale height.",
"RHESSI observations suggest that the HXR source positions can be well fitted with a single exponential scale-height density model, assuming a simple collisional thick target.",
"This results in scale heights between about 130 km and 200 km [15], [14], [5], consistent with chromospheric models [20], [29].",
"However, the observed HXR sizes are about 4 times larger than expected from the simple collisional transport model.",
"Here we have quantitatively investigated how the density profile, collisional pitch-angle scattering, magnetic mirroring, as well as instrumental effects affect the source sizes.",
"In [5] we showed that projection effects and source motion over the RHESSI image time interval cannot account for the observed source sizes.",
"In the present work, applying RHESSI visibility forward fitting on simulated HXR source maps we demonstrate that the source size cannot be due to instrumental effects, alone.",
"This leaves the physical effects of magnetic mirroring and collisional pitch-angle scattering which we investigate by solving the Fokker-Planck equation both numerically and analytically.",
"While pitch angle and magnetic mirroring effects change the electron flux distribution, these effects tend to increase the FWHM of the X-ray source profile by only up to a factor $1.5$ , which is not enough to explain the observations.",
"The dominating factor that determines the X-ray source size is the atmospheric density structure.",
"In the case of an exponential density model with a single scale height and a constant coronal loop density of around $10^{10}$ cm$^{-3}$ , the X-ray emission will originate predominantly from the region of highest density.",
"Thus, even though the effects described above alter the electron distribution as a function of height, emission by electrons higher up in the loop will always be faint compared to the emission from the denser chromosphere.",
"Source sizes of around 4 arcseconds can only be achieved by unlikely loop densities of the order of $10^{13}$ cm$^{-3}$ .",
"Such high densities will also cause the HXR sources to appear at larger heights, well above typical chromospheric heights.",
"Thus, within the traditional thick target model, the only plausible explanation for the observed HXR source sizes remains a multi-threaded density structure.",
"This work is supported by the Leverhulme Trust (M.B., E.P.K., L.F.) and STFC grant ST/I001808 (E.P.K., L.F., A.L.M).",
"Financial support by the European Commission through the FP7 HESPE network (FP7-2010-SPACE-263086) is gratefully acknowledged.",
"We thank Richard Schwartz for providing and helping with the RHESSI image simulation software and Gordon Hurford for discussions."
]
] | 1204.1151 |
[
[
"Sharp Bounds on Davenport-Schinzel Sequences of Every Order"
],
[
"Abstract One of the longest-standing open problems in computational geometry is to bound the lower envelope of $n$ univariate functions, each pair of which crosses at most $s$ times, for some fixed $s$.",
"This problem is known to be equivalent to bounding the length of an order-$s$ Davenport-Schinzel sequence, namely a sequence over an $n$-letter alphabet that avoids alternating subsequences of the form $a \\cdots b \\cdots a \\cdots b \\cdots$ with length $s+2$.",
"These sequences were introduced by Davenport and Schinzel in 1965 to model a certain problem in differential equations and have since been applied to bounding the running times of geometric algorithms, data structures, and the combinatorial complexity of geometric arrangements.",
"Let $\\lambda_s(n)$ be the maximum length of an order-$s$ DS sequence over $n$ letters.",
"What is $\\lambda_s$ asymptotically?",
"This question has been answered satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and Nivasch) when $s$ is even or $s\\le 3$.",
"However, since the work of Agarwal, Sharir, and Shor in the mid-1980s there has been a persistent gap in our understanding of the odd orders.",
"In this work we effectively close the problem by establishing sharp bounds on Davenport-Schinzel sequences of every order $s$.",
"Our results reveal that, contrary to one's intuition, $\\lambda_s(n)$ behaves essentially like $\\lambda_{s-1}(n)$ when $s$ is odd.",
"This refutes conjectures due to Alon et al.",
"(2008) and Nivasch (2010)."
],
[
"Introduction",
"Consider the problem of bounding the complexity of the lower envelope of $n$ continuous univariate functions $f_1,\\ldots ,f_n$ , each pair of which cross at most $s$ times.",
"In other words, how many maximal connected intervals of the $\\lbrace f_i\\rbrace $ are contained in the graph of the function $f_{\\min }(x) = \\min \\lbrace f_1(x),\\ldots ,f_n(x)\\rbrace $ ?",
"In the absence of any constraints on $\\lbrace f_i\\rbrace $ this problem can be completely stripped of its geometry by transcribing the lower envelope $f_{\\min }$ as a Davenport-Schinzel (DS) sequence of order $s$ , namely, a repetition-free sequence over the alphabet $\\lbrace 1,\\ldots ,n\\rbrace $ that does not contain any alternating subsequences of the form $\\cdots a\\cdots b \\cdots a \\cdots b \\cdots $ with length $s+2$ , for any $a,b\\in \\lbrace 1,\\ldots ,n\\rbrace $ .If the sequence corresponding to the lower envelope contained an alternating subsequence $abab\\cdots $ with length $s+2$ then the functions $f_a$ and $f_b$ must have crossed at least $s+1$ times, a contradiction.",
"Although Davenport and Schinzel [27] introduced this problem nearly 50 years ago, it was only in the early 1980s that DS sequences became well known in the computational geometry community [14], [76].",
"Since then DS sequences have found a startling number of geometric applications, with a growing number [72], [59], [9], [21], [48], [65] that are not overtly geometric.To cite a fraction of the literature, DS sequences/lower envelopes are routinely applied to problems related to geometric arrangements [61], [13], [12], [31], [69], [29], [30], [19], [42], [44], [54], [40], [32], [55], [80], [3], [46], in kinetic data structures and dynamic geometric algorithms [6], [39], [47], [1], [5], [14], [43], [85], in visibility [25], [76], [62], motion planning [76], [58], and geometric containment problems [10], [76], [77], [15], as well as variations on classical problems such as computing shortest paths [17], [11], [18] and convex hulls [33], [16].",
"They have also been used in some industrial applications [45], [20].",
"Refer to Sharir and Agarwal [75] for a survey of DS sequences and their early applications in computational geometry and to Klazar [52] for a survey of DS sequences and related problems in extremal combinatorics.",
"In each of these applications some quantity (e.g., running time, combinatorial complexity) is expressed in terms of $\\lambda _{s}(n)$ , the maximum length of an order-$s$ DS sequence over an $n$ -letter alphabet.",
"To improve bounds on $\\lambda _{s}$ is, therefore, to improve our understanding of numerous problems in algorithms, data structures, and discrete geometry.",
"Davenport and Schinzel [27] established $n^{1+o(1)}$ upper bounds on $\\lambda _{s}(n)$ for every order $s$ .",
"In order to properly survey the improvements that followed [26], [79], [41], [73], [74], [56], [4], [51], [63] we must define some notation for forbidden sequences and their extremal functions."
],
[
"Sequence Notation and Terminology",
"Let $|\\sigma |$ be the length of a sequence $\\sigma = (\\sigma (i))_{1\\le i\\le |\\sigma |}$ and let $\\Vert \\sigma \\Vert $ be the size of its alphabet $\\Sigma (\\sigma ) = \\lbrace \\sigma (i)\\rbrace $ .",
"Two equal length sequences are isomorphic if they are the same up to a renaming of their alphabets.",
"We say $\\sigma $ is a subsequence of $\\sigma ^{\\prime }$ , written $\\sigma \\;\\hbox{\\vbox {\\hrule height 0.5pt \\hspace{1.27501pt}\\hbox{\\hspace{-1.00006pt}$\\prec $\\hspace{-1.00006pt}}}}\\;\\sigma ^{\\prime }$ , if $\\sigma $ can be obtained by deleting symbols from $\\sigma ^{\\prime }$ .",
"The predicate $\\sigma \\prec \\sigma ^{\\prime }$ asserts that $\\sigma $ is isomorphic to a subsequence of $\\sigma ^{\\prime }$ .",
"If $\\sigma \\nprec \\sigma ^{\\prime }$ we say $\\sigma ^{\\prime }$ is $\\sigma $ -free.",
"If $P$ is a set of sequences, $P\\nprec \\sigma ^{\\prime }$ holds if $\\sigma \\nprec \\sigma ^{\\prime }$ for every $\\sigma \\in P$ .",
"The assertion that $\\sigma $ appears in or occurs in or is contained in $\\sigma ^{\\prime }$ means either $\\sigma \\prec \\sigma ^{\\prime }$ or $\\sigma \\;\\hbox{\\vbox {\\hrule height 0.5pt \\hspace{1.27501pt}\\hbox{\\hspace{-1.00006pt}$\\prec $\\hspace{-1.00006pt}}}}\\;\\sigma ^{\\prime }$ , which one being clear from context.",
"The projection of a sequence $\\sigma $ onto $G\\subseteq \\Sigma (\\sigma )$ is obtained by deleting all non-$G$ symbols from $\\sigma $ .",
"A sequence $\\sigma $ is $k$ -sparse if whenever $\\sigma (i) = \\sigma (j)$ and $i\\ne j$ , then $|i-j| \\ge k$ .",
"A block is a sequence of distinct symbols.",
"If $\\sigma $ is understood to be partitioned into a sequence of blocks, $\\llbracket \\sigma \\rrbracket $ is the number of blocks.",
"The predicate $\\llbracket \\sigma \\rrbracket =m$ asserts that $\\sigma $ can be partitioned into at most $m$ blocks.",
"The extremal functions for generalized Davenport-Schinzel sequences are defined to be $\\operatorname{Ex}(\\sigma ,n,m) &= \\max \\lbrace |S| \\;:\\; \\sigma \\nprec S, \\; \\Vert S\\Vert =n, \\mbox{ and } \\llbracket S \\rrbracket \\le m\\rbrace \\\\\\operatorname{Ex}(\\sigma ,n) &= \\max \\lbrace |S| \\;:\\; \\sigma \\nprec S, \\; \\Vert S\\Vert =n, \\mbox{ and } S \\mbox{ is } \\Vert \\sigma \\Vert \\mbox{-sparse}\\rbrace \\\\\\multicolumn{2}{l}{\\text{where $\\sigma $ may be a single sequence or a set of sequences.The conditions ``$\\llbracket S \\rrbracket \\le m$'' and ``$S$ is $\\Vert \\sigma \\Vert $-sparse'' guarantee that the extremal functions are finite.For example, if $\\Vert \\sigma \\Vert =2$ the sparsity criterion forbids immediate repetitions andsuch infinite degenerate sequences as $aaaaa\\cdots $.Blocked sequences, on the other hand, have no sparsity criterion.The extremal functions for (standard) Davenport-Schinzel sequences are defined to be}}\\\\\\lambda _{s}(n,m) &= \\operatorname{Ex}(\\overbrace{ababa\\cdots }^{\\mbox{\\scriptsize length $s+2$}},n,m)\\hspace*{11.38092pt}\\mbox{ and }\\hspace*{11.38092pt}\\lambda _{s}(n) = \\operatorname{Ex}(\\overbrace{ababa\\cdots }^{\\mbox{\\scriptsize length $s+2$}},n)$ Bounds on generalized Davenport-Schinzel sequences are expressed as a function of the inverse-Ackermann function $\\alpha $ , yet there is no universally agreed-upon definition of Ackermann's function or its inverse.",
"All definitions in the literature differ by at most a constant, which usually obviates the need for more specificity.",
"In this article we use the following definition of Ackermann's function.",
"$a_{1,j} &\\;\\makebox{[}0mm][l]{= 2^j} & j\\ge 1\\\\a_{i,1} &\\;\\makebox{[}0mm][l]{= 2} & i\\ge 2\\\\ a_{i,j} &\\;\\makebox{[}0mm][l]{= w\\cdot a_{i-1,w}} & i,j\\ge 2\\\\& \\hspace*{14.22636pt}\\mbox{where $w=a_{i,j-1}$}\\multicolumn{2}{l}{\\text{Note that in the table of $\\lbrace a_{i,j}\\rbrace $ values, the first column is constant($a_{i,1} = 2$) and the second merely exponential ($a_{i,2} = 2^i$), so we haveto look to the third column to find Ackermann-type growth.",
"We define the double- and single-argument versions of the inverse-Ackermann function to be}}\\\\\\alpha (n,m) &= \\makebox{[}0mm][l]{\\min \\lbrace i \\;|\\; a_{i,j} \\ge m,\\, \\mbox{ where } j = \\max \\lbrace \\lceil n/m \\rceil ,3\\rbrace \\rbrace }\\\\\\alpha (n) &= \\alpha (n,n)$ We could have defined $\\alpha (n,m)$ without direct reference to Ackermann's function.",
"Note that $j = \\log (a_{1,j})$ .",
"One may convince oneself that $j = \\log ^\\star (a_{2,j}) - O(1)$ , $j = \\log ^{\\star \\star }(a_{3,j})-O(1)$ , and in general, that $j = \\log ^{[i-1]}(a_{i,j}) - O(1)$ , where $[i-1]$ is short for $i-1$ $\\star $ s.If $f : \\mathbb {N}\\backslash \\lbrace 0\\rbrace \\rightarrow \\mathbb {N}$ is a decreasing function, $f^\\star (m)$ is, by definition, $\\min \\lbrace \\ell \\;|\\; f^{(\\ell )}(m) \\le 1\\rbrace $ , where $f^{(0)}(m)=m$ and $f^{(\\ell )}(m) = f(f^{(\\ell -1)}(m))$ .",
"Thus, up to $O(1)$ differences $\\alpha (n,m)$ could be defined as $\\min \\left\\lbrace i \\;\\left|\\; \\log ^{[i-1]}(m) \\le \\max \\lbrace \\lceil n/m \\rceil ,3\\rbrace \\right.\\right\\rbrace $ .",
"We state previous results in terms of the single argument version of $\\alpha $ .",
"However, they all generalize to the two-argument version by replacing $\\lambda _{s}(n)$ with $\\lambda _{s}(n,m)$ and $\\alpha (n)$ with $\\alpha (n,m)$ ."
],
[
"A Brief History of $\\lambda _{s}$",
"After introducing the problem in 1965, Davenport and Schinzel [27] proved that $\\lambda _{1}(n) = n, \\lambda _{2}(n)=2n-1,\\lambda _{3}(n) = O(n\\log n)$ , and for all $s\\ge 4$ , that $\\lambda _{s}(n) = n\\cdot 2^{O(\\sqrt{\\log n})}$ , where the leading constant in the exponent depends on $s$ .",
"Shortly thereafter Davenport [26] improved the bound on $\\lambda _{3}(n)$ to $O(n\\log n/\\log \\log n)$ .",
"In 1973 Szemerédi [79] dramatically improved the upper bounds for all $s\\ge 3$ , showing that $\\lambda _{s}(n) = O(n\\log ^\\star n)$ , where the leading constant depends on $s$ .",
"From a purely numerical perspective Szemerédi's bound settled the problem for all values of $n$ one might encounter in nature (the log-star function being at most 5 for $n$ less than $10^{19,000}$ ), so why should any thoughtful mathematician continue to work on the problem?",
"In our view, the problem of quantitatively estimating $\\lambda _{s}(n)$ has always been a proxy for several qualitative questions: is $\\lambda _{s}(n)$ linear or nonlinear?",
"what is the structure of extremal sequences realizing $\\lambda _{s}(n)$ ?",
"and does it even matter what $s$ is?",
"In 1984 Hart and Sharir [41] answered the first two questions for order-3 DS sequences.",
"They gave a bijection between order-3 ($ababa$ -free) DS sequences and so-called generalized postorder path compression schemes.",
"Although these schemes resembled the path compressions found in set-union data structures, Tarjan's analysis [82] did not imply any non-trivial upper or lower bounds on their length.",
"Hart and Sharir proved that such path compression schemes have length $\\Theta (n\\alpha (n))$ , thereby settling the asymptotics of $\\lambda _{3}(n)$ .",
"In 1989 Agarwal, Sharir, and Shor [4] (improving on [73], [74]) gave asymptotically tight bounds on order-4 DS sequences and reasonably tight bounds on higher order sequences.",
"$\\lambda _{4}(n) &= \\Theta (n\\cdot 2^{\\alpha (n)})\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{7 mm}\\lambda _{s}(n) & \\; \\left\\lbrace \\begin{array}{l}> n\\cdot \\makebox{[}0mm][l]{2^{(1+o(1))\\alpha ^t(n)\\,/\\, t!}}",
"\\\\< n\\cdot \\makebox{[}0mm][l]{2^{(1+o(1))\\alpha ^t(n)} } \\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm}\\\\\\end{array}\\right.& \\raisebox {0 mm}[0mm][0mm]{\\mbox{for even $s \\ge 6, t=\\lfloor {2} \\rfloor $.",
"}}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{7.5 mm}\\lambda _{s}(n) & \\; \\left\\lbrace \\begin{array}{l}> \\lambda _{s-1}(n)\\\\< n\\cdot \\makebox{[}0mm][l]{(\\alpha (n))^{(1+o(1))\\alpha ^t(n)}} \\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm}\\\\\\end{array}\\right.& \\raisebox {0 mm}[0mm][0mm]{\\mbox{for odd $s \\ge 5$, $t=\\lfloor {2} \\rfloor $.",
"}}$ For even $s$ their bounds were tight up to the constant in the exponent: 1 for the upper bound and $1/t!$ for the lower bound.",
"Moreover, their lower bound construction gave a qualitatively satisfying answer to the question of how extremal sequences are structured when $s$ is even.",
"For odd $s$ the gap between upper and lower bounds was wider, the base of the exponent being 2 at the lower bound and $\\alpha (n)$ at the upper bound.",
"Remark 1.1 The results of Agarwal, Sharir, and Shor [4] force us to confront another question, namely, when is it safe to declare victory and call the problem closed?",
"As Nivasch [63] observed, the “$+o(1)$ ” in the exponent necessarily hides a $\\pm \\, \\Omega (\\alpha ^{t-1}(n))$ term if we express the bound in an “Ackermann-invariant” fashion, that is, in terms of the generic $\\alpha (n)$ , without specifying the precise variant of Ackermann's function for which it is the inverse.",
"Furthermore, under any of the definitions in the literature $\\alpha (n)$ is an integer-valued function whereas $\\lambda _{s}(n)/n$ must increase fairly smoothly with $n$ , that is, an estimate of $\\lambda _{s}(n)$ that is expressed as a function of any integer-valued $\\alpha (n)$ must be off by at least a $2^{\\Omega (\\alpha ^{t-1}(n))}$ factor.",
"A reasonable definition of sharp bound (when dealing with generalized Davenport-Schinzel sequences) is an expression that cannot be improved, given $\\pm \\, \\Theta (1)$ uncertainty in the definition of $\\alpha (n)$ .",
"For example, $\\lambda _{4}(n) = \\Theta (n2^{\\alpha (n)})$ is sharp in this sense since the constant hidden by $\\Theta $ reflects this uncertainty.",
"In contrast, $\\lambda _{3}(n) = \\Theta (n\\alpha (n))$ is not sharp in an Ackermann-invariant sense.",
"See the tighter bounds on $\\lambda _{3}(n)$ cited below and in Theorem REF .",
"In 2009 Nivasch [63] presented a superior method for upper bounding $\\lambda _{s}(n)$ .",
"In addition, he provided a new construction of order-3 DS sequences that matched an earlier upper bound of Klazar [51] up to the leading constant.",
"$\\lambda _{s}(n) =& \\,\\left\\lbrace \\begin{array}{l@{\\hspace*{46.94687pt}}l}2n\\alpha (n)+O(n\\sqrt{\\alpha (n)}) & \\mbox{for $s=3$; upper bound is due to~\\cite {Klazar99}.",
"}\\\\\\Theta (n\\cdot 2^{\\alpha (n)}) & \\rule [- 3 mm]{0mm}{3 mm}\\rule {0mm}{5 mm}\\mbox{for $s=4$.",
"}\\\\n\\cdot 2^{(1 + o(1))\\alpha ^t(n)\\,/\\, t!}",
"& \\mbox{for even $s \\ge 6,$ $t=\\lfloor {2} \\rfloor $.",
"}\\end{array}\\right.\\nonumber \\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{9 mm}\\lambda _{s}(n) & \\; \\left\\lbrace \\begin{array}{l@{\\hspace*{28.45274pt}}l}> \\lambda _{s-1}(n)\\\\< n\\cdot (\\alpha (n))^{(1+o(1))\\alpha ^t(n)\\,/\\, t!}",
"\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm} & \\raisebox {2 mm}[0mm][0mm]{\\mbox{for odd $s \\ge 5$, $t=\\lfloor {2} \\rfloor $.",
"}}\\\\\\end{array}\\right.$ This closed the problem for even $s\\ge 6$ (the leading constant in the exponent being precisely $1/t!$ ) but left the odd case open.",
"Alon, Kaplan, Nivasch, Sharir, and Smorodinsky [8], [63] conjectured that the upper bounds (REF ) for odd orders are tight, that is, the base of the exponent is, in fact, $\\alpha (n)$ .",
"This conjecture was spurred by their discovery of similar functions that arose in an apparently unrelated combinatorial problem, stabbing interval chains with $j$ -tuples [8]."
],
[
"New Results",
"We provide new upper and lower bounds on the length of Davenport-Schinzel sequences and in the process refute conjectures due to Alon et al.",
"[8], Nivasch [63], and Pettie [68].",
"Theorem 1.2 Let $\\lambda _{s}(n)$ be the maximum length of a repetition-free sequence over an $n$ -letter alphabet avoiding subsequences isomorphic to $abab\\cdots $ (length $s+2$ ), or, equivalently, the maximum complexity of the lower envelope of $n$ continuous univariate functions, each pair of which coincide at most $s$ times.",
"For any $s\\ge 1$ , $\\lambda _{s}$ satisfies: $\\lambda _{s}(n) = \\left\\lbrace \\begin{array}{l@{\\hspace*{28.45274pt}}l@{\\rule [- 3 mm]{0mm}{3 mm}\\rule {0mm}{0 mm}}}n & s=1\\\\2n-1 & s=2\\\\2n\\alpha (n) + O(n) & s=3\\\\\\Theta (n2^{\\alpha (n)}) & s=4\\\\\\Theta (n\\alpha (n)2^{\\alpha (n)}) & s=5\\\\n\\cdot 2^{(1 + o(1))\\alpha ^t(n)/t!}",
"& \\mbox{both even and odd } s\\ge 6, \\; t = \\lfloor \\frac{s-2}{2} \\rfloor .\\end{array}\\right.$ Theorem REF is optimal in that it provides the tightest bounds that can be expressed in an Ackermann-invaraint fashion (see Remark REF ), and in this sense closes the Davenport-Schinzel problem.The exponent $(1+o(1))\\alpha ^t(n)/t!$ is the Ackermann-invariant expression $\\alpha ^t(n)/t!",
"+ O(\\alpha ^{t-1}(n))$ .",
"However, we believe our primary contributions are not the tight asymptotic bounds per se but the structural differences they reveal between even and odd $s$ .",
"We can now give a cogent explanation for why odd orders $s\\ge 5$ behave essentially like the preceding even orders and yet why they are intrinsically more difficult to understand."
],
[
"Generalizations of Davenport-Schinzel Sequences",
"The (REF ) bounds are actually corollaries of a more general theorem in [63] concerning the length of sequences avoiding catenated permutations,Nivasch called these formation-free sequences.",
"which were introduced by Klazar [50].",
"Define $\\operatorname{Perm}(r,s+1)$ to be the set of all sequences obtained by concatenating $s+1$ permutations over an $r$ -letter alphabet.",
"For example, $abcd\\; cbad\\; badc\\; abcd\\; dcba\\in \\operatorname{Perm}(4,5)$ .",
"Define the extremal function of $\\operatorname{Perm}(r,s+1)$ -free sequences to be $\\Lambda _{r,s}(n) &= \\operatorname{Ex}(\\operatorname{Perm}(r,s+1),n)$ The “$s+1$ ” here is chosen to highlight the parallels with order-$s$ DS sequences.",
"Every $\\sigma \\in \\operatorname{Perm}(2,s+1)$ contains an alternating sequence $abab\\cdots $ with length $s+2$ ,The first permutation contributes two symbols and every subsequent permutation contributes at least one.",
"so order-$s$ DS sequences are also $\\operatorname{Perm}(2,s+1)$ -free, implying that $\\lambda _{s}(n) \\le \\Lambda _{2,s}(n)$ .",
"Nivasch [63] proved that $\\Lambda _{r,s}(n)$ obeys all the upper bounds of (REF ), as well as its lower bounds when $s\\ge 4$ is even or $s\\le 3$ .",
"There are other natural ways to generalize standard Davenport-Schinzel sequences.",
"Doubled Davenport-Schinzel sequences were studied in [28], [2], [53], [68].",
"Define $\\lambda _{{s}}^{\\scriptscriptstyle \\operatorname{dbl}}(n)$ to be the extremal function of $\\operatorname{dbl}(abab\\cdots )$ -free sequences, where the alternating sequence has length $s+2$ and $\\operatorname{dbl}(\\sigma )$ is obtained by doubling every symbol in $\\sigma $ save the first and last.",
"For example, $\\operatorname{dbl}(abab) = abbaab$ .Why not consider higher multiplicities?",
"It is fairly easy to show that repeating symbols more than twice, or repeating the first and last at all, affects the extremal function by at most a constant factor.",
"See Adamec, Klazar, and Valtr [2].",
"Davenport and Schinzel [28] noted that $\\lambda _{{1}}^{\\scriptscriptstyle \\operatorname{dbl}}(n)=O(\\lambda _{1}(n))=O(n)$ (see [52]) and Adamec, Klazar, and Valtr [2] proved that $\\lambda _{{2}}^{\\scriptscriptstyle \\operatorname{dbl}}(n)=O(\\lambda _{2}(n)) = O(n)$ .",
"Pettie proved that $\\lambda _{{3}}^{\\scriptscriptstyle \\operatorname{dbl}}(n) = O(n\\alpha ^2(n))$ and that $\\lambda _{{s}}^{\\scriptscriptstyle \\operatorname{dbl}}(n)$ obeys all the upper bounds of (REF ) for $s\\ge 4$ .",
"If one views alternating sequences as forming a zigzagging pattern, an obvious generalization is to extend the length of each zig and zag to include a larger alphabet.",
"For example, the $N$ -shaped sequences $N_{k} = 12\\cdots k(k+1)\\cdots 212 \\cdots k(k+1)$ generalize $abab=N_1$ and the $M$ -shaped sequences $M_{k} = 12\\cdots k(k+1)k\\cdots 212\\cdots k(k+1)k\\cdots 21$ generalize $ababa=M_1$ .",
"Klazar and Valtr [53], [69] proved that $\\operatorname{Ex}(\\operatorname{dbl}(N_k),n) = O(\\lambda _{2}(n)) = O(n)$ and Pettie [69] proved that $\\operatorname{Ex}(\\lbrace M_k,ababab\\rbrace ,n) = O(\\lambda _{3}(n))$ .",
"Sequences avoiding $N$ - and $M$ -shaped sequences have proved very useful in bounding the complexity of geometric graphs [84], [78], [34], [69].",
"In a companion paper to be published separately we provide new upper and lower bounds on doubled DS sequences, $M_k$ -free sequences, and $\\operatorname{Perm}(r,s+1)$ -free sequences.",
"The strangest of these results is that $\\Lambda _{r,s}$ is very sensitive to the alphabet size $r$ , but only when $s$ is odd and at least 5.",
"In particular $\\Lambda _{2,s}(n)=\\Theta (\\lambda _{{s}}^{\\scriptscriptstyle \\operatorname{dbl}}(n)) = \\Theta (\\lambda _{s}(n))$ but this is not true for general $r\\ne 2$ .",
"Theorem 1.3 The following bounds hold for all $r\\ge 2, s\\ge 1$ , where $t = \\lfloor \\frac{s-2}{2} \\rfloor $ .",
"$\\Lambda _{r,s}(n) &= \\left\\lbrace \\begin{array}{l@{\\hspace*{11.38092pt}\\rule [- 2.5 mm]{0mm}{2.5 mm}\\rule {0mm}{0 mm}}l}\\Theta (n) & \\mbox{for $s\\in \\lbrace 1,2\\rbrace $ and all $r\\ge 2$}\\\\\\Theta (n\\alpha (n)) & \\mbox{for $s=3$ and all $r\\ge 2$}\\\\\\Theta (n2^{\\alpha (n)}) & \\mbox{for $s=4$ and all $r\\ge 2$}\\\\\\Theta (n\\alpha (n)2^{\\alpha (n)}) & \\rule [- 3 mm]{0mm}{3 mm}\\rule {0mm}{0 mm}\\mbox{for $s=5$ and $r=2$}\\\\n\\cdot (\\alpha (n))^{(1+o(1))\\,\\alpha (n)} & \\mbox{for $s=5$ and all $r\\ge 3$}\\\\n\\cdot 2^{(1+o(1))\\,\\alpha ^t(n)/t!}",
"& \\mbox{for even $s\\ge 6$ and all $r\\ge 2$}\\\\n\\cdot 2^{(1+o(1))\\,\\alpha ^t(n)/t!}",
"& \\mbox{for odd $s\\ge 7$ and $r=2$}\\\\n\\cdot (\\alpha (n))^{(1+o(1))\\,\\alpha ^t(n)/t!}",
"\\;\\;\\; & \\mbox{for odd $s\\ge 7$ and all $r\\ge 3$}\\end{array}\\right.$ Theorem REF is rather surprising, even given Theorem REF and even in retrospect.",
"One consequence of Theorem REF is that Cibulka and Kynčl's [24], [71] upper bounds on the size of sets of permutations with fixed VC-dimension are tight."
],
[
"Organization.",
"In Section we present an informal discussion of the method of Agarwal, Sharir, and Shor [4] and Nivasch [63], its limitations for dealing with odd-order DS sequences, and the key ideas behind the proof of Theorem REF .",
"Section reviews Nivasch's recurrence for $\\lambda _{s}$ as well as some basic upper bounds on $\\lambda _{s}$ .",
"The critical structure in our analysis is the derivation tree of a DS sequence.",
"Its properties are analyzed in Section .",
"In Section we use the derivation tree to obtain a new recurrence for odd-order DS sequences.",
"The recurrences for even- and odd-order DS sequences are solved in Section REF .",
"In Section REF we complete the proof of the upper bounds of Theorem REF for all orders except $s=5$ .",
"In Sections and we establish Theorem REF 's lower bounds and upper bounds on order-5 DS sequences.",
"We discuss several open problems in Section .",
"Some proofs appear in Appendices –."
],
[
"A Tour of the Proof",
"The proof of Theorem REF diverges sharply from previous analyses [73], [4], [63] in that it treats even and odd orders as fundamentally different beasts.",
"To understand why all orders cannot be analyzed in a uniform fashion we must review the method of Agarwal, Sharir, and Shor [4] and Nivasch [63].",
"The basic inductive hypothesis of [63] is that there are values $\\lbrace \\mu _{s,i}\\rbrace $ , increasing in both $s$ and $i$ , for which $\\lambda _{s}(n,m) < \\mu _{s,i}{\\left( n + m\\operatorname{poly}(\\log ^{[i-1]}(m)) \\right)}$ , for any choice of $i$ .Recall that $\\log ^{[i-1]}(m)$ is the $\\log ^{\\star \\cdots \\star }(m)$ function, with $i-1$ $\\star $ s. In other words, the multiplicity of symbols is at most $\\mu _{s,i}$ , up to an additive term that depends on the block count $m$ , which may be the dominant term if $i$ is too small.",
"One would, ultimately, choose $i$ to make $m\\operatorname{poly}(\\log ^{[i-1]}(m)) = O(n)$ (and $\\alpha (n,m)$ is a good choice) but for the sake of simplifying the discussion we ignore the dependence on $m$ .",
"Given a sequence $S$ with parameters $s,n,m$ , to invoke the inductive hypothesis with parameter $i$ means to upper bound $|S|$ by $\\mu _{s,i}n$ , with the understanding that $i$ is not chosen to be too small.",
"Suppose $S$ is an order-$s$ , $m$ -block DS sequence over an $n$ -letter alphabet.",
"The analysis of [4], [63] begins by partitioning $S$ into $\\hat{m}$ intervals of consecutive blocks, $\\hat{m}$ typically being much smaller than $m$ .",
"Write $S$ as $S_1\\cdots S_{\\hat{m}}$ .",
"We can put symbols into two categories: local symbols are those that appear exclusively in one interval $S_q$ and global symbols are those that appear in multiple intervals.",
"Let $\\check{S}=\\check{S}_1\\cdots \\check{S}_{\\hat{m}}$ be the subsequence of $S$ consisting of local symbols and $\\hat{S}= \\hat{S}_1\\cdots \\hat{S}_{\\hat{m}}$ the subsequence of $S$ consisting of global symbols, so $|S| = |\\check{S}| + |\\hat{S}|$ .",
"On each $\\check{S}_q$ (for $1\\le q\\le \\hat{m}$ ) we invoke the inductive hypothesis with parameter $i$ and deduce that $|\\check{S}| \\le \\mu _{s,i}\\Vert \\check{S}\\Vert $ .",
"What remains is to bound the length of $\\hat{S}$ .",
"The next step is to form a contracted sequence from $\\hat{S}$ that has a much higher alphabet-to-block count ratio, thereby allowing us to invoke the inductive hypothesis with a smaller `$i$ ' parameter.",
"Let $\\hat{S}^{\\prime }$ be obtained from $\\hat{S}$ by replacing each interval $\\hat{S}_q$ with a single block $\\beta _q$ containing the first occurrence of each distinct symbol in $\\hat{S}_q$ .",
"Thus, the alphabet of $\\hat{S}^{\\prime }$ is the same as $\\hat{S}$ but it consists of just $\\hat{m}$ blocks $\\beta _1\\beta _2\\cdots \\beta _{\\hat{m}}$ .",
"On $\\hat{S}^{\\prime }$ we invoke the inductive hypothesis with parameter $i-1$ and conclude that $|\\hat{S}^{\\prime }| \\le \\mu _{s,i-1}\\Vert \\hat{S}\\Vert $ .",
"One cannot immediately deduce any bound on $\\hat{S}$ from a bound on $\\hat{S}^{\\prime }$ since each interval $\\hat{S}_q$ could contain numerous copies of a symbol, only one of which is retained in $\\beta _q$ .",
"Imagine reversing the contraction operation.",
"We replace each block $\\beta _q$ with a sequence $\\hat{S}_q$ , thereby reconstructing $\\hat{S}$ .",
"To bound the length of $\\hat{S}_q$ in terms of $|\\beta _q| = \\Vert \\hat{S}_q\\Vert $ we will invoke the inductive hypothesis three more times.",
"Put the symbols of $\\beta _q$ in three categories: those that make their first appearance (in $\\hat{S}^{\\prime }$ ) in $\\beta _q$ , those that make their last appearance in $\\beta _q$ , and those that make a middle (non-first, non-last) appearance in $\\beta _q$ .",
"Discard from $\\hat{S}_q$ all symbols not classified as first in $\\beta _q$ and call the resulting sequence $\\acute{S}_q$ .",
"Every symbol in $\\acute{S}_q$ appears at least once after $\\hat{S}_q$ (by virtue of being categorized as first in $\\beta _q$ ), which implies that $\\acute{S}=\\acute{S}_1\\cdots \\acute{S}_{\\hat{m}}$ is an order-$(s-1)$ DS sequence.",
"See the diagram below.",
"Figure: NO_CAPTION An occurrence of $\\sigma _{s+1} = abab\\cdots $ (length $s+1$ ) in $\\acute{S}_q$ , together with an $a$ or $b$ following $\\hat{S}_q$ (depending on whether $\\sigma _{s+1}$ ends in $b$ or $a$ ) gives an occurrence of $\\sigma _{s+2}$ in $\\hat{S}$ , contradicting the fact that it has order $s$ .",
"The same argument applies in a symmetric fashion to the subsequence of $\\hat{S}_q$ formed by symbols making their last appearance in $\\beta _q$ , call it $\\grave{S}_q$ .",
"By invoking the inductive hypothesis with parameter $i$ on $\\acute{S}_1\\cdots \\acute{S}_{\\hat{m}}$ and $\\grave{S}_1\\cdots \\grave{S}_{\\hat{m}}$ we can conclude the contribution of first and last symbols to $S$ is $2\\mu _{s-1,i}\\Vert \\hat{S}\\Vert $ .",
"The length of the subsequence of middle symbols in $\\hat{S}_q$ , call it $\\bar{S}_q$ , is bounded with the same argument, except now there are, by definition of middle, occurrences of both $a,b\\in \\Sigma (\\bar{S}_q)$ both before and after $\\hat{S}_q$ .",
"That is, if $\\sigma _{s} = baba\\cdots $ (length $s$ ) appeared in $\\bar{S}_q$ then, together with an $a$ preceding $\\hat{S}_q$ and either an $a$ or $b$ following $\\hat{S}_q$ (depending on whether $\\sigma _{s}$ ends in $b$ or $a$ ) there would be an occurrence of $\\sigma _{s+2}$ in $\\hat{S}$ , contradicting the fact that it has order $s$ .",
"We invoke the inductive hypothesis one last time, with parameter $i$ , on each $\\bar{S}_1,\\ldots ,\\bar{S}_{\\hat{m}}$ , which implies that $|\\bar{S}_q| \\le \\mu _{s-2,i}\\Vert \\bar{S}_q\\Vert $ .",
"Recall that each symbol in $\\hat{S}^{\\prime }$ appeared $\\mu _{s,i-1}$ times, $\\mu _{s,i-1}-2$ times in blocks where it was categorized as middle.",
"Thus, the contribution of middle symbols to $|\\hat{S}|$ is $\\mu _{s-2,i}(\\mu _{s,i-1}-2)$ .",
"In order for every symbol, local and global alike, to appear in $S$ with multiplicity at most $\\mu _{s,i}$ , we must have $\\mu _{s,i} &\\ge 2\\mu _{s-1,i} + \\mu _{s-2,i}(\\mu _{s,i-1}-2)$ When $s=3$ we do not need to use an inductive hypothesis to determine $\\mu _{1,i}$ and $\\mu _{2,i}$ .",
"They are just 1 and 2; the $i$ parameter does not come into play.It is easy to show that $\\lambda _{1}(n,m) < \\underline{1}\\cdot n + m$ and $\\lambda _{2}(n,m) < \\underline{2}\\cdot n + m$ .",
"See Lemma REF .",
"This leads to a bound of $\\mu _{3,i} = 2i+O(1)$ .We have not said what to do in the base case when $i=1$ , which determines the $O(1)$ term.",
"Although the contribution of first and last symbols is significant at $s=3$ , entertain the idea that their contribution becomes negligible at higher orders, so we can further simplify (REF ) as follows $\\mu _{s,i} &\\ge \\mu _{s-2,i}\\mu _{s,i-1} $ Inequality (REF ) is satisfied when $\\mu _{s,i} = g^{i+t\\atopwithdelims ()t}$ for any base $g$ ; recall that $t=\\lfloor \\frac{s-2}{2} \\rfloor $ by definition.",
"By Pascal's identity $g^{i+t\\atopwithdelims ()t} = g^{i+(t-1)\\atopwithdelims ()(t-1)}\\cdot g^{(i-1) + t\\atopwithdelims ()t}$ .",
"The correct base depends on where the inductively defined inequality (REF ) bottoms out: at order 2 when $s\\ge 4$ is even and at order 3 when $s\\ge 5$ is odd.",
"When $s$ is even the correct base is $2 = \\mu _{2,i}$ .",
"When $s$ is odd the calculations are less clean since $\\mu _{3,i} = 2i+ O(1)$ is not constant but depends on $i$ .",
"Nonetheless, the correct base is on the order of $i$ , that is, $\\mu _{s,i} = \\Theta (i)^{i+t\\atopwithdelims ()t}$ satisfies (REF ) at the odd orders.",
"Plugging in $\\alpha (n,m)$ for $i$ ultimately leads to Nivasch's bounds (REF ), since ${i+t\\atopwithdelims ()t} = i^t/t!",
"+ O(i^{t-1}) = (1+o(1))i^t/t!$ .",
"To obtain a construction of order-$s$ sequences realizing the (REF ) bounds one should start by attempting to reverse-engineer the argument above.",
"To form an order-$s$ sequence $S$ with certain alphabet and block parameters, start by generating (inductively) local order-$s$ sequences $\\check{S}_1\\cdots \\check{S}_{\\hat{m}}$ over disjoint alphabets, and a single global order-$s$ sequence $\\hat{S}^{\\prime }$ having $\\hat{m}$ blocks.",
"Take some block $\\beta _q$ in $\\hat{S}^{\\prime }$ and suppose for the sake of simplicity that $\\beta _q$ consists solely of middle symbols.",
"We need to substitute for $\\beta _q$ an order-$(s-2)$ DS sequence $\\bar{S}_q$ and then somehow merge it with $\\check{S}_q$ , in a way that does not introduce into $S$ an alternating sequence with length $s+2$ .",
"This is the point at which the even and odd orders diverge.",
"If $s$ is even the longest alternating sequence $baba\\cdots b$ in $\\bar{S}_q$ has length $s-1$ and therefore begins and ends with $b$ .",
"We can only afford to introduce one alternation at each boundary of $\\bar{S}_q$ , so the pattern of $a$ s and $b$ s on either side of $\\beta _q$ must look like $a^* \\; b^* \\;\\, \\beta _q \\;\\, b^* \\; a^*$ , as in the diagram below.",
"We will call $a$ and $b$ nested in $\\beta _q$ if the sequence contains $a\\, b\\, \\beta _q\\, b\\, a$ or the equivalent $b\\, a\\, \\beta _q\\, a\\, b$ .",
"See the diagram below.",
"Figure: NO_CAPTION On the other hand, if $s$ is odd then the longest alternating sequence $baba\\cdots a$ in $\\bar{S}_q$ has length $s-1$ , begins with $b$ and ends with $a$ , so the pattern of $a$ s and $b$ s in $\\hat{S}^{\\prime }$ looks like $a^* \\; b^* \\; \\beta _q \\; a^* \\; b^*$ .",
"A pair of middle symbols that are not nested in $\\beta _q$ are called interleaved in $\\beta _q$.",
"Figure: NO_CAPTION If the (REF ) bounds prove to be tight, there must be two systems for generating sequences: one where nesting is the norm, when $s$ is even, and one where interleaving is the norm, when $s$ is odd.",
"If interleaving were somehow outlawed then to avoid creating an alternating sequence with length $s+2$ , the sequence $\\bar{S}_q$ substituted for $\\beta _q$ would have to be an order-$(s-3)$ DS sequence rather than an order-$(s-2)$ one.",
"However, it is clearly impossible to claim that interleaving simply cannot exist.",
"What makes the argument of [4], [63] brilliantly simple is how little it leaves to direct calculation.",
"The length of every sequence ($\\check{S}_q,\\hat{S}^{\\prime },\\acute{S}_q,\\bar{S}_q,$ etc.)",
"is bounded by delegation to an inductive hypothesis.",
"However, such useful notions as nearly all middle symbols in a block are mutually nested are difficult to capture in a strengthened inductive hypothesis.",
"We need to understand and characterize the phenomenon of nestedness to improve on [4], [63].",
"This requires a deeper understanding of the structure of Davenport-Schinzel sequences."
],
[
"The Derivation Tree.",
"Inductively defined objects can be apprehended inductively or, alternatively, apprehended holistically by completely “unrolling” the induction.",
"From the first perspective $S$ is the merger of $\\check{S}$ and $\\hat{S}$ , which is derived from $\\hat{S}^{\\prime }$ , all of which are analyzed inductively.",
"By iteratively unrolling the decomposition of $\\check{S}$ and $\\hat{S}^{\\prime }$ we obtain a derivation tree $\\mathcal {T}$ whose nodes represent every block in every sequence encountered in the recursive decomposition of $S$ .",
"Whereas $S$ occupies the leaves of $\\mathcal {T}$ , derived sequences such as $\\hat{S}^{\\prime }$ occupy levels higher in $\\mathcal {T}$ .",
"Whereas $S$ (and every sequence) is a static object, $\\mathcal {T}$ can be thought of a process for generating $S$ whose history can be reasoned about explicitly.",
"But how does $\\mathcal {T}$ let us deduce something about the nestedness and non-nestedness of symbols in a common block?",
"Suppose we are interested in the nestedness of middle symbols $a,b$ in block $\\beta $ , which corresponds to a leaf-node in $\\mathcal {T}$ .",
"Imagine taking $\\mathcal {T}$ and deleting every node whose block does not contain $b$ , that is, projecting $\\mathcal {T}$ onto the symbol $b$ .",
"What remains, $\\mathcal {T}_{|b}$ , is a tree rooted at the location in $\\mathcal {T}$ where $b$ is “born” and represents how occurrences of $b$ have proliferated during the process that culminates in the construction of $S$ .",
"The block/node $\\beta $ occupies a location in $\\mathcal {T}_{|b}$ and a location in $\\mathcal {T}_{|a}$ , whose node sets are only guaranteed to intersect at $\\beta $ .",
"Some locations in $\\mathcal {T}_{|a}$ and $\\mathcal {T}_{|b}$ are intrinsically bad—these are called feathers in Section .",
"(Whether a node is a feather in $\\mathcal {T}_{|a}$ depends solely on the structure of $\\mathcal {T}_{|a}$ , not how it is embedded in $\\mathcal {T}$ nor its relationship to a different $\\mathcal {T}_{|b}$ .)",
"We show that if $\\beta $ is not a feather in $\\mathcal {T}_{|a}$ and not a feather in $\\mathcal {T}_{|b}$ , then $a$ and $b$ are nested in $\\beta $ .",
"In other words, the middle symbols in $\\beta $ are partitioned into two equivalence classes, depending on whether or not they appear at feathers in their respective derivation trees.",
"We could not outlaw interleavedness in general, yet we managed to outlaw it within one equivalence class!",
"The question is, what are the relative sizes of these two equivalence classes, and in particular, how many feathers can a $\\mathcal {T}_{|b}$ have?",
"Our aim is to get stronger asymptotic bounds on $\\lambda _{s}$ for odd $s$ , which means the number of feathers should be a negligible ($o(1)$ fraction) of the size of $\\mathcal {T}_{|b}$ .",
"In the same way that the multiplicities $\\mu _{s,i}$ are bounded inductively, as in (REF ) for example, we are able to bound the number of feathers in $\\mathcal {T}_{|b}$ inductively, call it $\\nu _{s,i}$ , in terms of $\\nu _{s,i-1}$ and $\\mu _{s-1,i}$ .",
"However, now $\\mu _{s,i}$ is bounded in terms of $\\mu _{s,i-1}$ (the multiplicity of symbols in the contracted sequence $\\hat{S}^{\\prime }$ ), $\\mu _{s-1,i}$ (the multiplicity of symbols in $\\hat{S}$ begat by first and last occurrences in $\\hat{S}^{\\prime }$ ), $\\mu _{s-3,i}$ (the multiplicity of middle occurrences in $\\hat{S}$ begat by non-feathers in $\\hat{S}^{\\prime }$ ), and both $\\nu _{s,i-1}$ and $\\mu _{s-2,i}$ , which count the number of feathers in $\\hat{S}^{\\prime }$ and the multiplicity of middle occurrences in $\\hat{S}$ begat by feathers in $\\hat{S}^{\\prime }$ .",
"This leads to a system of three interconnected recurrences: one for $\\mu _{s,i}$ at odd $s$ , one for $\\mu _{s,i}$ at even $s$ , and one for the feather count $\\nu _{s,i}$ .",
"An elementary (though necessarily detailed) proof by induction gives solutions for $\\mu _{s,i}$ and $\\nu _{s,i}$ that ultimately lead to the upper bounds of Theorem REF , with one exception.",
"At order $s=5$ this method only gives us an $O(n\\alpha ^2(n)2^{\\alpha (n)})$ upper bound on $\\lambda _{5}(n)$ .",
"To obtain a sharp $O(n\\alpha (n)2^{\\alpha (n)})$ bound we are forced to analyze not just one derivation tree $\\mathcal {T}$ (of an order-5 DS sequence) but a system of derivation trees of order-4 DS sequences associated with all the sequences $\\acute{S}$ and $\\grave{S}$ (of global first and last occurrences) encountered in the construction of $\\mathcal {T}$ ."
],
[
"Basic Upper Bounds",
"In Section REF we review and expand on the notation introduced informally in Section .",
"It will be used repeatedly throughout Sections –."
],
[
"Sequence Decomposition",
"Let $S$ be a sequence over an $n=\\Vert S\\Vert $ letter alphabet consisting of $m=\\llbracket S \\rrbracket $ blocks.",
"Suppose we partition $S$ into $\\hat{m}$ intervals of consecutive blocks $S_1S_2\\cdots S_{\\hat{m}}$ , where $m_q = \\llbracket S_q \\rrbracket $ is the number of blocks in interval $q$ .",
"Let $\\check{\\Sigma }_q$ be the alphabet of symbols local to $S_q$ (that do not appear in any $S_p$ , $p\\ne q$ ) and let $\\hat{\\Sigma }= \\Sigma (S) \\backslash \\bigcup _{q}\\check{\\Sigma }_q$ be the alphabet of all other global symbols.",
"The cardinalities of $\\check{\\Sigma }_q$ and $\\hat{\\Sigma }$ are $\\check{n}_q$ and $\\hat{n}$ , thus $n = \\hat{n}+ \\sum _{q=1}^{\\hat{m}} \\check{n}_q$ .",
"A global symbol in $S_q$ is called first, last, or middle if it appears in no earlier interval, no later interval, or appears in both earlier and later intervals, respectively.",
"Let $\\acute{\\Sigma }_q,\\grave{\\Sigma }_q,\\bar{\\Sigma }_q,\\hat{\\Sigma }_q$ be the subset of $\\Sigma (S_q)$ consisting of, respectively, first, last, middle, and all global symbols, and let $\\acute{n}_q,\\grave{n}_q,\\bar{n}_q,$ and $\\hat{n}_q$ be their cardinalities.",
"Let $\\check{S}_q,\\hat{S}_q,\\acute{S}_q,\\grave{S}_q,\\bar{S}_q$ be the projection of $S_q$ onto $\\check{\\Sigma }_q,\\hat{\\Sigma }_q,\\acute{\\Sigma }_q,\\grave{\\Sigma }_q,$ and $\\bar{\\Sigma }_q$ .",
"Note that $\\hat{S}_1$ consists solely of first occurrences; if the last occurrence of a symbol appeared in $\\hat{S}_1$ the symbol would be classified as local to $S_1$ , not global.",
"The same argument shows that $\\hat{S}_{\\hat{m}}$ consists solely of last occurrences.",
"Let $\\check{S},\\hat{S},\\acute{S},\\grave{S},$ and $\\bar{S}$ be the subsequences of local, global, first, last, and middle occurrences, respectively, that is, $\\check{S}= \\check{S}_1\\cdots \\check{S}_{\\hat{m}}$ , $\\hat{S}= \\hat{S}_1\\cdots \\hat{S}_{\\hat{m}}$ , $\\acute{S}= \\acute{S}_1\\cdots \\acute{S}_{\\hat{m}-1}$ , $\\grave{S}= \\grave{S}_2\\cdots \\grave{S}_{\\hat{m}}$ , and $\\bar{S}= \\bar{S}_2\\cdots \\bar{S}_{\\hat{m}-1}$ , the last of which would be empty if $\\hat{m}=2$ .",
"Let $\\hat{S}^{\\prime } = \\beta _1\\cdots \\beta _{\\hat{m}}$ be an $\\hat{m}$ -block sequence obtained from $\\hat{S}$ by replacing each $\\hat{S}_q$ with a single block $\\beta _q$ containing its alphabet $\\hat{\\Sigma }_q$ , listed in order of first appearance in $\\hat{S}_q$ ."
],
[
"2-Sparse vs. Blocked Sequences",
"Every analysis of Davenport-Schinzel sequences since [41] uses Lemma REF (REF ) to reduce the problem of bounding 2-sparse DS sequences to bounding $m$ -block DS sequences, that is, expressing $\\lambda _{s}(n)$ in terms of $\\lambda _{s}(n,m)$ , where $m=O(n)$ .",
"Lemma 3.1 Let $\\gamma _s(n) : \\mathbb {N}\\rightarrow \\mathbb {N}$ be a non-decreasing function such that $\\lambda _{s}(n) \\le \\gamma _s(n)\\cdot n$ .",
"(Trivial) For $s\\ge 1$ , $\\lambda _{s}(n,m) \\le m-1 + \\lambda _{s}(n)$ .",
"(Sharir [73]) For $s\\ge 3$ , $\\lambda _{s}(n) \\le \\gamma _{s-2}(n)\\cdot \\lambda _{s}(n,2n-1)$ .",
"(This generalizes Hart and Sharir's proof [41] for $s=3$ .)",
"(Sharir [73]) For $s\\ge 2$ , $\\lambda _{s}(n) \\le \\gamma _{s-1}(n)\\cdot \\lambda _{s}(n,n)$ .",
"(New) For $s\\ge 3$ , $\\lambda _{s}(n) = \\gamma _{s-2}(\\gamma _s(n))\\cdot \\lambda _{s}(n,3n-1)$ .",
"Lemma REF (REF ) is obtained by synthesizing ideas from Sharir [73] and Füredi and Hajnal [36].",
"Refer to Appendix for the proof of Lemma REF ."
],
[
"Orders 1 and 2",
"In the interest of completeness we shall reestablish the known bounds on order-1 and order-2 DS sequences, in both their 2-sparse and blocked forms.",
"Lemma 3.2 (Davenport and Schinzel [27]) The extremal functions for order-1 and order-2 DS sequences are $\\lambda _{1}(n) &= n\\\\\\lambda _{2}(n) &= 2n-1\\\\\\lambda _{1}(n,m) &= n+ m-1\\\\\\lambda _{2}(n,m) &= 2n+m-2 & \\mbox{for $m\\ge 2$}$ Proof: Let $S$ be a 2-sparse sequence with $n=\\Vert S\\Vert $ .",
"If $|S| > n$ then there are two copies of some symbol, say $a$ .",
"The $a$ s cannot be adjacent, due to 2-sparseness, so $S$ must contain a subsequence $aba$ , for some $b\\ne a$ .",
"Such an $S$ is not an order-1 DS sequence, hence $\\lambda _{1}(n) \\le n$ .",
"If $S$ has order 2 then some symbol must appear exactly once.",
"To see this, consider the closest pair of occurrences of some symbol, say $a$ .",
"If every symbol $b$ appearing between this pair of $a$ s occurred twice in $S$ then $S$ would contain $baba, abab,$ or $abba$ .",
"The first two are precluded since $S$ has order 2 and the third violates the fact that the two $a$ s are the closest such pair.",
"Thus, every symbol $b$ between the two $a$ s occurs once.",
"Remove one such $b$ ; if this causes the two $a$ s to become adjacent, remove one of the $a$ s. What remains is a 2-sparse sequence over an $(n-1)$ -letter alphabet, so $\\lambda _{2}(n) \\le \\lambda _{2}(n-1) + 2$ .",
"Since $\\lambda _{2}(1)=1$ we have $\\lambda _{2}(n) \\le 2n-1$ .",
"Lemma REF (REF ) and the bounds established above imply $\\lambda _{1}(n,m) \\le n + m-1$ and $\\lambda _{2}(n,m) \\le 2n + m - 2$ .",
"All these upper bounds are tight.",
"The unique extremal order-1, 2-sparse DS sequence is $123\\cdots n$ , which can be converted into an extremal $m$ -block sequence $[123\\cdots n][n]^{m-1}$ .",
"Brackets mark block boundaries.",
"There are exponentially many extremal DS sequences of order 2, each corresponding to an Euler tour around a rooted tree with vertex labels from $\\lbrace 1,\\ldots ,n\\rbrace $ .",
"For example, $123\\cdots (n-1)n(n-1)\\cdots 321$ and $1213141\\cdots 1(n-1)1n1$ are extremal 2-sparse, order-2 DS sequences.",
"The first corresponds to an Euler tour around a path, the second an Euler tour around a star.",
"The first sequence can be converted into an extremal $m$ -block, order-2 DS sequence $[12\\cdots (n-1)n][n(n-1)\\cdots 21][1]^{m-2}$ , assuming that $m\\ge 2$ .",
"When there is only 1 block we have $\\lambda _{s}(n,1) = n$ , regardless of the order $s$ .",
"$\\Box $"
],
[
"Nivasch's Recurrence",
"Nivasch's [63] upper bounds (REF ) are a consequence of a recurrence for $\\lambda _{s}$ that is stronger than one of Agarwal, Sharir, and Shor [4].",
"Here we present a streamlined version of Nivasch's recurrence.",
"Recurrence 3.3 Let $m,n,$ and $s\\ge 3$ be the block count, alphabet size, and order parameters.",
"For any $\\hat{m}<m$ , any block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ , and any alphabet partition $\\lbrace \\hat{n}\\rbrace \\cup \\lbrace \\check{n}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , where $m = \\sum _q m_q$ and $n=\\hat{n}+ \\sum _q \\check{n}_q$ , we have $\\lambda _{s}(n,m) \\;\\le \\; \\sum _{q=1}^{\\hat{m}} \\lambda _{s}(\\check{n}_q,m_q) \\;+\\; 2\\cdot \\lambda _{s-1}(\\hat{n},m) \\;+\\; \\lambda _{s-2}(\\lambda _{s}(\\hat{n},\\hat{m}) - 2\\hat{n}, m)\\nonumber $ Proof: We adopt the notation and definitions from Section REF , where $S$ is an extremal order-$s$ DS sequence with $\\Vert S\\Vert =n$ and $\\llbracket S \\rrbracket =m$ .",
"We shall bound $|S|$ by considering its four constituent subsequences $\\check{S},\\acute{S},\\grave{S},$ and $\\bar{S}$ .",
"Each $\\check{S}_q$ is an order-$s$ DS sequence, therefore the contribution of local symbols is $|\\check{S}|\\le \\sum _{q=1}^{\\hat{m}} \\lambda _{s}(\\check{n}_q,m_q)$ .",
"We claim each $\\acute{S}_q$ is an order-$(s-1)$ DS sequence.",
"By virtue of being categorized as first in $\\hat{S}_q$ , every symbol in $\\acute{S}_q$ appears at least once after $\\acute{S}_q$ .",
"Therefore an occurrence of an alternating sequence $\\sigma _{s+1} = abab\\cdots $ (length $s+1$ ), in $\\acute{S}_q$ would imply an occurrence of $\\sigma _{s+2}$ in $S$ , a contradiction.",
"By symmetry it also follows that $\\grave{S}_q$ is an order-$(s-1)$ DS sequence, hence $|\\acute{S}| = \\sum _{q=1}^{\\hat{m}-1} \\lambda _{s-1}(\\acute{n}_q,m_q)$ and $|\\grave{S}| = \\sum _{q=2}^{\\hat{m}} \\lambda _{s-1}(\\grave{n}_q,m_q)$ .",
"Since $\\lambda _{s}$ is clearly superadditiveIt is straightforward to show that $\\lambda _{s}(n^{\\prime },m^{\\prime }) + \\lambda _{s}(n^{\\prime \\prime },m^{\\prime \\prime }) \\le \\lambda _{s}(n^{\\prime }+n^{\\prime \\prime },m^{\\prime }+m^{\\prime \\prime }-1)$ , for all $n^{\\prime },n^{\\prime \\prime },m^{\\prime },m^{\\prime \\prime }$ .",
"we can bound these sums by $\\lambda _{s-1}(\\hat{n},m-m_{\\hat{m}})$ and $\\lambda _{s-1}(\\hat{n},m-m_1)$ .",
"(Note that $\\sum _q \\acute{n}_q = \\hat{n}$ and $\\sum _q \\grave{n}_q = \\hat{n}$ as each sum counts each global symbol exactly once.)",
"The contribution of first and last symbols is therefore upper bounded by $2\\cdot \\lambda _{s-1}(\\hat{n},m)$ .",
"The same argument shows that $\\bar{S}_q$ is an order-$(s-2)$ DS sequence.",
"Symbols in $\\bar{S}_q$ were categorized as middle, so an alternating subsequence $\\sigma _s = baba\\cdots $ (length $s$ ) in $\\bar{S}_q$ , together with an $a$ preceding $\\bar{S}_q$ and either an $a$ or $b$ following $\\bar{S}_q$ (depending on whether $s$ is even or odd), yields an instance of $\\sigma _{s+2}$ in $S$ , a contradiction.",
"Thus the contribution of middle symbols is $|\\bar{S}| &\\le \\sum _{q=2}^{\\hat{m}-1} \\lambda _{s-2}(\\bar{n}_q,m_q)\\nonumber \\\\&\\le \\lambda _{s-2}{\\left( \\sum _{q=2}^{\\hat{m}-1} \\bar{n}_q, m-m_1-m_{\\hat{m}} \\right)} & \\mbox{\\lbrace superadditivity of $\\lambda _{s-2}$\\rbrace }\\nonumber \\\\&\\le \\lambda _{s-2}(|\\hat{S}^{\\prime }|-2\\hat{n},m-m_1-m_{\\hat{m}})\\\\&\\le \\lambda _{s-2}(\\lambda _{s}(\\hat{n},\\hat{m})-2\\hat{n},m)$ Inequality (REF ) follows from the fact that $\\sum _q \\bar{n}_q$ counts the length of $\\hat{S}^{\\prime }$ , save the first and last occurrence of each global symbol, that is, $2\\hat{n}$ occurrences in total.",
"Since $\\hat{S}^{\\prime }$ is a subsequence of $S$ , it too is an order-$s$ DS sequence, so $|\\hat{S}^{\\prime }| \\le \\lambda _{s}(\\hat{n},\\hat{m})$ .",
"Inequality () follows.",
"$\\Box $ Recurrence REF offers us the freedom to choose the block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ but it does not suggest what the optimal partition might look like.",
"One natural starting place [41], [4], [63] is to always choose $\\hat{m}=2$ , partitioning the sequence into 2 intervals each containing $m/2$ blocks.",
"This choice leads to $O(n + m\\log ^{s-2}m)$ upper bounds on $\\lambda _{s}(n,m)$ , which is $O(n+m)$ if the alphabet/block density $n/m=\\Omega (\\log ^{s-2}m)$ .",
"Call this Analysis (1).",
"Given Analysis (1) we can conduct a stronger Analysis (2) by selecting $\\hat{m}= m/\\log ^{s-2} m$ , so each interval contains $\\log ^{s-2} m$ blocks.",
"The $\\lambda _{s}(\\hat{n},\\hat{m})$ term is bounded via Analysis (1) (that is, $\\lambda _{s}(\\hat{n},\\hat{m}) = O(\\hat{n}+ \\hat{m}\\log ^{s-2}\\hat{m}) = O(\\hat{n}+ m)$ ) and the remaining terms bounded inductively via analysis Analysis (2).",
"This leads to bounds of the form $\\lambda _{s}(n,m) = O(n + m\\operatorname{poly}(\\log ^* m))$ .",
"By iterating this process, Analysis ($i$ ) gives bounds of the form $O(n + m\\operatorname{poly}(\\log ^{[i-1]} m))$ .",
"(the $[i-1]$ here being short for $i-1$ $\\star $ s) We cannot conclude that $\\lambda _{s}(n,m) = O(n+m)$ since the constant hidden by the asymptotic notation, call it $\\mu _{s,i}$ , increases with $i$ and $s$ .",
"The discussion above is merely meant to foreshadow the analysis of Recurrence REF and subsequent Recurrences REF , REF , REF , and REF ; see Appendices and .",
"We have made every attempt to segregate recurrences and structural arguments from their quantitative analyses, which are important but nonetheless rote.",
"As a consequence, Ackermann's function, its various inverses, and quantities such as $\\lbrace \\mu _{s,i}\\rbrace $ will be introduced as late as possible."
],
[
"The Evolution of Recurrence ",
"The statement of Recurrence REF is simple, and arguably cannot be made simpler.",
"We feel it is worthwhile to recount how it was assembled over the years in the works of [41], [73], [4], [51], [63].",
"When $s$ is fixed the function $\\lambda _{s}(n)$ depends only on one parameter, $n$ , a situation that would not ordinarily lead to expressions involving “$\\alpha $ ”, which is most naturally expressed as a function of two independent parameters.In graph algorithms these parameters typically correspond to nodes and edges [83], [57], [22], in matrix problems [49], [48] to rows and columns, and in data structures they may correspond to elements and queries [82], [37], query time and preprocessing time [64], or input size and storage space [87], [7], [23].",
"Hart and Sharir's [41] insight was to recognize an additional parameter $m$ (the block count) and obtain bounds on $\\lambda _{s}(n)$ via bounds on $\\lambda _{s}(n,m)$ .",
"See Lemma REF .",
"Implicit in Hart and Sharir's analysis is a classification of symbols into local and global, and of global occurrences into first, middle, and last.This part of their analysis is ostensibly about nodes and path compressions, not blocks and symbols.",
"Agarwal, Sharir, and Shor [4] made this local/global and first/middle/last classification explicit, and arrived at a recurrence very close to Recurrence REF .Sharir [73] split global occurrences into two categories—first and non-first—which leads to a near-linear upper bound of $\\lambda _{s}(n) < n\\cdot \\alpha (n)^{O(\\alpha (n))^{s-3}}$ .",
"However, they did not bound the contribution of global middle occurrences in the same way.",
"Whereas $\\bar{S}_q$ is an $m_q$ -block sequence, it can be converted to 2-sparse one by removing up to $m_q-1$ repeated symbols at block boundaries.",
"By Lemma REF (REF ,REF ) $|\\bar{S}_q| < m_q + \\lambda _{s-2}(\\bar{n}_q) \\;\\le \\; m_q + \\gamma _{s-2}(\\bar{n}_q)\\cdot \\bar{n}_q \\;\\le \\; m_q + \\gamma _{s-2}(n)\\cdot \\bar{n}_q$ In other words, when “contracting” $\\bar{S}$ to form $\\bar{S}^{\\prime }$ , the shrinkage factor is at most $\\gamma _{s-2}(n)$ .",
"A similar statement holds for first and last occurrences, where the shrinkage factor is at most $\\gamma _{s-1}(n)$ .",
"This leads to a recurrence [4] that forgets the role of $m$ when analyzing global occurrences.",
"$\\lambda _{s}(n,m) \\;\\le \\; \\sum _{q=1}^{\\hat{m}} \\lambda _{s}(\\check{n}_q,m_q) \\:+\\: 2\\cdot \\gamma _{s-1}(n)\\cdot n \\:+\\: \\gamma _{s-2}(n)\\cdot \\lambda _{s}(\\hat{n},\\hat{m}) \\:+\\: O(m)$ Nivasch's recurrence [63] improves that of Agarwal, Sharir, and Shor [4] by not forgetting that $\\bar{S}$ is an $m$ -block sequence.",
"In particular, $|\\bar{S}| \\le \\sum _q \\lambda _{s-2}(\\hat{n}_q,m_q)$ where $|\\bar{S}^{\\prime }| < \\sum _q \\hat{n}_q \\le \\lambda _{s}(\\hat{n},\\hat{m})$ .",
"Recurrence REF is substantively no different than that of [63] but it is more succinct, for two reasons.",
"First, the superadditivity of $\\lambda _{s}$ lets us bound the number of middle occurrences with the single term $\\lambda _{s-2}(\\lambda _{s}(\\hat{n},\\hat{m})-2\\hat{n},m)$ .One might think it would be dangerous to bound middle occurrence with one aggregated term since we “forget” that $\\bar{S}$ is partitioned into $\\hat{m}-2$ order-$(s-2)$ DS sequences.",
"Doing this does not affect the solution of $\\lambda _{s}(n,m)$ asymptotically.",
"Second, the function equivalent to $\\lambda _{s}(n,m)$ from [4], [63] is the extremal function of order-$s$ DS sequences that are both 2-sparse and have $m$ blocks.",
"This small change introduces $O(m)$ terms in [63] and [4] since the derived sequences $\\hat{S},\\hat{S}^{\\prime },$ and $\\lbrace \\check{S}_q,\\acute{S}_q,\\bar{S}_q,\\grave{S}_q\\rbrace _{1\\le q\\le \\hat{m}}$ are not necessarily 2-sparse, and must be made 2-sparse by removing $O(m)$ symbols at block boundaries.",
"Recurrence REF could be made yet more succinct by removing the “$-2\\hat{n}$ ” from the estimation of global middle occurrences.",
"This would not affect the solution asymptotically, but keeping it is essential for obtaining bounds on $\\lambda _{3}(n)$ tight to the leading constant."
],
[
"Derivation Trees",
"A derivation tree $\\mathcal {T}(S)$ for an $m$ -block sequence $S$ is a rooted, ordered tree whose nodes are identified with the blocks encountered in recursively decomposing $S$ , as in Section REF and Recurrence REF .",
"Let $\\mathcal {B}(u)$ be the block associated with $u\\in \\mathcal {T}(S)$ .",
"The leaf level of $\\mathcal {T}(S)$ coincides with $S$ , that is, the $p$ th leaf of $\\mathcal {T}(S)$ holds the $p$ th block of $S$ .",
"As we are sometimes indifferent to the order of symbols within a block, $\\mathcal {B}(v)$ is often treated as a set.",
"We assume without much loss in generality that no symbol appears just once in $S$ .",
"As usual, we adopt the sequence decomposition notation from Section REF ."
],
[
"Base Case.",
"Suppose $S=\\beta _1\\beta _2$ is a two block sequence, where each block contains the whole alphabet $\\Sigma (S)$ .",
"The tree $\\mathcal {T}(S)$ consists of three nodes $u,u_1,$ and $u_2$ , where $u$ is the parent of $u_1$ and $u_2$ , $\\mathcal {B}(u_1)=\\beta _1$ , $\\mathcal {B}(u_2)=\\beta _2$ , and $\\mathcal {B}(u)$ does not exist.",
"For every $a\\in \\Sigma (S)$ call $u$ its crown and $u_1$ and $u_2$ its left and right heads, respectively.",
"These nodes are denoted $\\operatorname{cr}_{|a}, \\operatorname{lhe}_{|a},$ and $\\operatorname{rhe}_{|a}$ ."
],
[
"Inductive Case.",
"If $S$ contains $m>2$ blocks, choose an $\\hat{m}< m$ and an arbitrary block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ .",
"Inductively construct derivation trees $\\hat{\\mathcal {T}}= \\mathcal {T}(\\hat{S}^{\\prime })$ and $\\lbrace \\check{\\mathcal {T}}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , where $\\check{\\mathcal {T}}_q = \\mathcal {T}(\\check{S}_q)$ , then identify the root of $\\check{\\mathcal {T}}_q$ (which has no block) with the $q$ th leaf of $\\hat{\\mathcal {T}}$ .",
"Finally, place the blocks of $S$ at the leaves of $\\mathcal {T}$ .",
"This last step is necessary since only local symbols appear in the blocks of $\\lbrace \\check{\\mathcal {T}}_q\\rbrace $ whereas the leaves of $\\mathcal {T}$ must be identified with the blocks of $S$ .",
"Note that nodes at or above the leaf level of $\\hat{\\mathcal {T}}$ carry only global symbols in their blocks and that internal nodes in $\\lbrace \\check{\\mathcal {T}}_q\\rbrace $ carry only local symbols in their blocks.",
"Local and global symbols only mingle at the leaf level of $\\mathcal {T}$ .",
"The crown and heads of each symbol $a\\in \\Sigma (S)$ are inherited from $\\hat{\\mathcal {T}}$ , if $a$ is global, or some $\\check{\\mathcal {T}}_q$ if $a$ is local to $S_q$ .",
"See Figure REF for an illustration.",
"Figure: A derivation tree 𝒯(S)\\mathcal {T}(S) for a 32-block sequenceSS.",
"The tree is generated by always choosing m ^=2\\hat{m}=2 and the uniform block partitionm 1 =m 2 =m/2m_1=m_2=m/2, where m>2m>2 is the number of blocks in the given sequence.The frames isolate the base case derivation treesthat assign the crown and heads for symbols a,b∈Σ(S)a,b\\in \\Sigma (S)."
],
[
"Anatomy of the Tree",
"The projection of $\\mathcal {T}$ onto $a\\in \\Sigma (S)$, denoted $\\mathcal {T}_{|a}$ , is the tree on the node set $\\lbrace \\operatorname{cr}_{|a}\\rbrace \\cup \\lbrace v \\in \\mathcal {T}\\:|\\: a\\in \\mathcal {B}(v)\\rbrace $ that inherits the ancestor/descendant relation from $\\mathcal {T}$ , that is, the parent of $v$ in $\\mathcal {T}_{|a}$ , where $v\\notin \\lbrace \\operatorname{cr}_{|a},\\operatorname{lhe}_{|a},\\operatorname{rhe}_{|a}\\rbrace $ , is $v$ 's nearest strict ancestor $u$ for which $a\\in \\mathcal {B}(u)$ .",
"For example, in Figure REF $\\mathcal {T}_{|a}$ consists of $\\operatorname{cr}_{|a}$ , its children $\\operatorname{lhe}_{|a},\\operatorname{rhe}_{|a}$ , and four grandchildren at the leaf level of $\\mathcal {T}$ .",
"Definition 4.1 (Anatomy) The leftmost and rightmost leaves of $\\mathcal {T}_{|a}$ are wingtips, denoted $\\operatorname{lwt}_{|a}$ and $\\operatorname{rwt}_{|a}$ .",
"The left and right wings are those paths in $\\mathcal {T}_{|a}$ extending from $\\operatorname{lhe}_{|a}$ to $\\operatorname{lwt}_{|a}$ and from $\\operatorname{rhe}_{|a}$ to $\\operatorname{rwt}_{|a}$ .",
"Descendants of $\\operatorname{lhe}_{|a}$ and $\\operatorname{rhe}_{|a}$ in $\\mathcal {T}_{|a}$ are called doves and hawks, respectively.",
"A child of a wing node that is not itself on the wing is called a quill.",
"A leaf is called a feather if it is the rightmost descendant of a dove quill or leftmost descendant of a hawk quill.",
"Suppose $v$ is a node in $\\mathcal {T}_{|a}$ .",
"Let $\\operatorname{he}_{|a}(v)$ be the head ancestral to $v$ and $\\overline{\\operatorname{he}}_{|a}(v)$ be the other head.",
"Let $\\operatorname{wt}_{|a}(v)$ and $\\overline{\\operatorname{wt}}_{|a}(v)$ be the wingtips descending from $\\operatorname{he}_{|a}(v)$ and $\\overline{\\operatorname{he}}_{|a}(v)$ .",
"Let $\\operatorname{wi}_{|a}(v)$ be the nearest wing node ancestor of $v$ , $\\operatorname{qu}_{|a}(v)$ the quill ancestral to $v$ , and $\\operatorname{fe}_{|a}(v)$ the feather descending from $\\operatorname{qu}_{|a}(v)$ .",
"See Figure REF for an illustration.",
"Once $a\\in \\Sigma (S)$ is known or specified, we will use these terms (feather, wingtip, etc.)",
"to refer to nodes in $\\mathcal {T}_{|a}$ or to the occurrences of $a$ within those blocks.",
"For example, an occurrence of $a$ in $S$ would be a feather if it appears in a block $\\mathcal {B}(v)$ in $S$ , where $v$ is a feather in $\\mathcal {T}_{|a}$ .",
"Note that the nodes $\\operatorname{he}_{|a}(v),\\operatorname{wi}_{|a}(v),\\operatorname{qu}_{|a}(v),\\operatorname{wt}_{|a}(v),$ and $\\operatorname{fe}_{|a}(v)$ are not necessarily distinct.",
"It may be that $\\operatorname{he}_{|a}(v)=\\operatorname{wi}_{|a}(v)$ , and it may be that $v = \\operatorname{qu}_{|a}(v) = \\operatorname{fe}_{|a}(v)$ if $v$ 's parent in $\\mathcal {T}_{|a}$ is $\\operatorname{wi}_{|a}(v)$ .",
"Figure: In this example vv is a hawk leaf in 𝒯 |a \\mathcal {T}_{|a} since its head he |a (v)=rhe |a \\operatorname{he}_{|a}(v) = \\operatorname{rhe}_{|a} is the right child of cr |a \\operatorname{cr}_{|a}.Its wing node wi |a (v)\\operatorname{wi}_{|a}(v), wingtip wt |a (v)\\operatorname{wt}_{|a}(v), quill qu |a (v)\\operatorname{qu}_{|a}(v), and feather fe |a (v)\\operatorname{fe}_{|a}(v) are indicated.Lemma REF identifies one property of $\\mathcal {T}$ used in the proof of Lemma REF .",
"Lemma 4.2 Suppose that on a leaf-to-root path in $\\mathcal {T}$ we encounter nodes $u,v,x,$ and $y$ (the last two possibly identical), where $u,x\\in \\mathcal {T}_{|a}$ and $v,y \\in \\mathcal {T}_{|b}$ .",
"It must be that $a\\in \\mathcal {B}(v)$ and therefore $v\\in \\mathcal {T}_{|a}$ .",
"Proof: Consider the decomposition of $\\mathcal {T}$ into a global derivation tree $\\hat{\\mathcal {T}}$ and local derivation trees $\\lbrace \\check{\\mathcal {T}}_q\\rbrace $ .",
"If $v$ were an internal node in some $\\check{\\mathcal {T}}_q$ then $b$ would be classified as local.",
"This implies $y\\in \\check{\\mathcal {T}}_q$ as well and the claim follows by induction on the construction of $\\check{\\mathcal {T}}_q$ .",
"If $v$ were an internal node in $\\hat{\\mathcal {T}}$ then let $u^{\\prime }$ be the leaf of $\\hat{\\mathcal {T}}$ ancestral to $u$ .",
"The nodes $u^{\\prime },v,x,y\\in \\hat{\\mathcal {T}}$ also satisfy the criteria of the lemma; the claim follows by induction on the construction of $\\hat{\\mathcal {T}}$ .",
"Thus, we can assume $v$ is a leaf of $\\hat{\\mathcal {T}}$ and $u$ is a leaf of $\\mathcal {T}$ .",
"See Figure REF .",
"By construction all global symbols in $\\mathcal {B}(u)$ also appear in $\\mathcal {B}(v)$ .",
"Since $x\\in \\hat{\\mathcal {T}}$ , the symbol $a$ is classified as global and must appear in $\\mathcal {B}(v)$ .",
"Figure: The case where vv is a leaf of 𝒯 ^\\hat{\\mathcal {T}}.",
"Both xx and yyare necessarily in 𝒯 ^\\hat{\\mathcal {T}}, which implies that aa and bb are global, and further impliesthat uu is a leaf of 𝒯 ˇ q \\check{\\mathcal {T}}_q since global symbols do not appear in the internal nodesof 𝒯 ˇ q \\check{\\mathcal {T}}_q.",
"All global symbols of uu also appear in vv.$\\Box $"
],
[
"Habitual Nesting",
"Suppose a block $\\beta $ in $S$ contains two symbols $a,b$ that are not wingtips, that is, they make neither their first nor last appearance in $\\beta $ .",
"We call $a$ and $b$ nested in $\\beta $ if $S$ contains either $ab\\,\\beta \\, ba$ or $ba\\, \\beta \\, ab$ and call them interleaved in $\\beta $ otherwise, that is, if the occurrences of $a$ and $b$ in $S$ take the form $a^* b^* \\,\\beta \\; a^* b^*$ or $b^*a^*\\,\\beta \\; b^*a^*$ .",
"Lemma REF is the critical structural lemma used in our analysis.",
"It provides us with simple criteria for nestedness.",
"Lemma 4.3 Suppose that $v\\in \\mathcal {T}(S)$ is a leaf and $a,b$ are symbols in a block $\\mathcal {B}(v)$ of $S$ .",
"If the following two criteria are satisfied then $a$ and $b$ are nested in $\\mathcal {B}(v)$ .",
"$v$ is not a wingtip in either $\\mathcal {T}_{|a}$ or $\\mathcal {T}_{|b}$ .",
"$v$ is not a feather in either $\\mathcal {T}_{|a}$ or $\\mathcal {T}_{|b}$ .",
"Proof: Without loss of generality we can assert two additional criteria.",
"$\\operatorname{cr}_{|b}$ is equal to or strictly ancestral to $\\operatorname{cr}_{|a}$ .",
"$v$ is a dove in $\\mathcal {T}_{|a}$ .",
"By Criterion (REF ) the leftmost leaf descendant of $\\operatorname{wi}_{|a}(v)$ is $\\operatorname{wt}_{|a}(v)$ .",
"Let $u$ be its rightmost leaf descendant.",
"According to Criteria (REF ,REF ) $v$ is distinct from both $\\operatorname{wt}_{|a}(v)$ and $u$ since $u$ must be a feather.",
"We partition the sequence $S$ outside of $\\mathcal {B}(v)$ into the following four intervals.",
"everything preceding the $a$ in $\\mathcal {B}(\\operatorname{wt}_{|a}(v))$ , everything from the end of $I_1$ to $\\mathcal {B}(v)$ , everything from $\\mathcal {B}(v)$ to the $a$ in $\\mathcal {B}(u)$ , and everything following $I_3$ .",
"Since $v$ is not a wingtip of $\\mathcal {T}_{|b}$ , by Criterion (REF ), there must be occurrences of $b$ in $S$ both before and after $\\mathcal {B}(v)$ .",
"If, contrary to the claim, $a$ and $b$ are not nested in $\\mathcal {B}(v)$ , all other occurrences of $b$ must appear exclusively in $I_1$ and $I_3$ or exclusively in $I_2$ and $I_4$ .",
"We show that both possibilities lead to contradictions.",
"Figures REF and REF illuminate the proof."
],
[
"Case 1: $b$ does not appear in {{formula:613ef7f7-9438-4190-9f0d-a1d17b56d236}} or {{formula:62a2fa8a-cfb3-495a-949c-810c54656547}}",
"According to Criterion (REF ) the left wingtip $\\operatorname{lwt}_{|b}$ of $\\mathcal {T}_{|b}$ is distinct from $v$ , and therefore appears in interval $I_2$ .",
"Since $\\operatorname{lwt}_{|b}$ and $v$ are descendants of $\\operatorname{wi}_{|a}(v)$ , which is a strict descendant of $\\operatorname{cr}_{|a}$ , which, by Criterion (REF ), is a descendant of $\\operatorname{cr}_{|b}$ , it must also be that $\\operatorname{lwt}_{|b}$ and $v$ descend from the same child of $\\operatorname{cr}_{|b}$ , that is, $v$ is a dove in $\\mathcal {T}_{|b}$ and therefore $\\operatorname{wt}_{|b}(v) = \\operatorname{lwt}_{|b}$ .",
"We shall argue below that Figure: Boxes represent nodes in 𝒯(S)\\mathcal {T}(S) and their associated blocks.The blocks at the leaf-level correspond to those in SS.In Case 1 all occurrences of bb outside of ℬ(v)\\mathcal {B}(v) appear in intervals I 2 I_2 and I 4 I_4.Contrary to the depiction, it may be that cr |a \\operatorname{cr}_{|a} and cr |b \\operatorname{cr}_{|b} are identical, thatwt |a (v)\\operatorname{wt}_{|a}(v) and wt |b (v)\\operatorname{wt}_{|b}(v) are identical,that uu and fe |b (v)\\operatorname{fe}_{|b}(v) are identical, and that wi |b (v)\\operatorname{wi}_{|b}(v) is not a strict ancestor of wi |a (v)\\operatorname{wi}_{|a}(v).Figure: In Case 2 all occurrences of bb outside of ℬ(v)\\mathcal {B}(v) appear in intervals I 1 I_1 and I 3 I_3.",
"In $\\mathcal {T}$ , $\\operatorname{qu}_{|b}(v)$ is a strict descendant of $\\operatorname{wi}_{|a}(v)$ and a strict ancestor of $u$ , and $\\operatorname{fe}_{|b}(v)$ lies in interval $I_4$ .",
"The least common ancestor of $v$ and $\\operatorname{wt}_{|b}(v)$ in $\\mathcal {T}_{|b}$ is by definition $\\operatorname{wi}_{|b}(v)$ .",
"The quill $\\operatorname{qu}_{|b}(v)$ is a child of $\\operatorname{wi}_{|b}(v)$ not on a wing, hence $\\operatorname{qu}_{|b}(v)$ cannot be ancestral to $\\operatorname{wt}_{|b}(v)$ , and hence $\\operatorname{qu}_{|b}(v)$ must be a strict descendant of $\\operatorname{wi}_{|a}(v)$ .",
"By Criterion (REF ) and Inference (REF ), $\\operatorname{fe}_{|b}(v)$ is the rightmost leaf descendant of $\\operatorname{qu}_{|b}(v)$ and distinct from $v$ .",
"However, by supposition $I_3$ contains no occurrences of $b$ , so $\\operatorname{fe}_{|b}(v)$ must lie in interval $I_4$ .",
"For $\\operatorname{qu}_{|b}(v)$ to have descendants in both $I_2$ and $I_4$ it must be a strict ancestor of $u$ in $\\mathcal {T}$ .",
"As we explain below, a consequence of Inference (REF ) is that $\\overline{\\operatorname{wt}}_{|a}(v)$ lies to the right of $\\operatorname{fe}_{|b}(v)$ .",
"According to Inference (REF ) $\\operatorname{qu}_{|b}(v)$ is a descendant of $\\operatorname{wi}_{|a}(v)$ , which is a descendant of $\\operatorname{he}_{|a}(v)$ .",
"According to Criterion (REF ) $\\operatorname{he}_{|a}(v)$ is the left head of $\\mathcal {T}_{|a}$ .",
"Since $\\overline{\\operatorname{wt}}_{|a}(v)$ is a descendant of $\\overline{\\operatorname{he}}_{|a}(v)$ , the right sibling of $\\operatorname{he}_{|a}(v)$ , $\\overline{\\operatorname{wt}}_{|a}(v)$ must lie to the right of $\\operatorname{fe}_{|b}(v)$ .",
"Let us review the situation.",
"Scanning the leaves from left to right we see the blocks $\\operatorname{wt}_{|a}(v)$ , $\\operatorname{wt}_{|b}(v)$ , $v$ , $u$ , $\\operatorname{fe}_{|b}(v)$ , and $\\overline{\\operatorname{wt}}_{|a}(v)$ .",
"It may be that $\\operatorname{wt}_{|a}(v)$ and $\\operatorname{wt}_{|b}(v)$ are equal and it may be that $u$ and $\\operatorname{fe}_{|b}(v)$ are equal.",
"If either of these cases hold then the $a$ precedes the $b$ in the given block.",
"The blocks $\\operatorname{wt}_{|a}(v),\\operatorname{wt}_{|b}(v),v,\\operatorname{fe}_{|b}(v),\\overline{\\operatorname{wt}}_{|a}(v)$ certify that $a$ and $b$ are nested in $\\mathcal {B}(v)$ ."
],
[
"Case 2: $b$ does not appear in {{formula:931677c9-3f7f-4975-b08b-6294873390b8}} or {{formula:6bcbee22-abfe-4b41-84fa-d2731c8111df}}",
"By Criterion (REF ) the right wingtip $\\operatorname{rwt}_{|b}$ is distinct from $v$ and must therefore lie in $I_3$ .",
"Following the same reasoning from Case 1 we can deduce that $v$ is a hawk in $\\mathcal {T}_{|b}$ .",
"In $\\mathcal {T}$ , $\\operatorname{qu}_{|b}(v)$ is strict descendant of $\\operatorname{wi}_{|a}(v)$ and a strict ancestor of $\\operatorname{wt}_{|a}(v)$ .",
"Inference (REF ) follows since $v$ and $\\operatorname{rwt}_{|b}$ must be descendants of the same head in $\\mathcal {T}_{|b}$ .",
"This implies that $\\operatorname{fe}_{|b}(v)$ is the leftmost leaf descendant of $\\operatorname{qu}_{|b}(v)$ .",
"Since $\\operatorname{fe}_{|b}(v)$ is distinct from $v$ and interval $I_2$ is free of $b$ s, it must be that $\\operatorname{fe}_{|b}(v)$ lies in $I_1$ and that $\\operatorname{qu}_{|b}(v)$ is a strict descendant of $\\operatorname{wi}_{|a}(v)$ and a strict ancestor of $\\operatorname{wt}_{|a}(v)$ .",
"Inference (REF ) follows.",
"See Figure REF .",
"It follows from Criterion (REF ) and Inference (REF ) that on a leaf-to-root path one encounters the nodes $\\operatorname{wt}_{|a}(v)$ , $\\operatorname{qu}_{|b}(v)$ , $\\operatorname{wi}_{|a}(v)$ , and $\\operatorname{cr}_{|b}$ , in that order.",
"Lemma REF implies that $a\\in \\mathcal {B}(\\operatorname{qu}_{|b}(v))$ .",
"We have deduced that $\\operatorname{qu}_{|b}(v)$ is in $\\mathcal {T}_{|a}$ , is a strict descendant of $\\operatorname{wi}_{|a}(v)$ , and is a common ancestor of $\\operatorname{wt}_{|a}(v)$ and $v$ .",
"This contradicts the fact that $\\operatorname{wi}_{|a}(v)$ is the least common ancestor of $v$ and $\\operatorname{wt}_{|a}(v)$ in $\\mathcal {T}_{|a}$ .",
"$\\Box $ Note that Lemma REF applies to any blocked sequence and an associated derivation tree.",
"It has nothing to do with Davenport-Schinzel sequences as such."
],
[
"A Recurrence for Odd Orders",
"Lemma REF may be rephrased as follows.",
"Every blocked sequence $S$ is the union of four sequences: two comprising wingtips (first occurrences and last occurrences, each of length $n$ ), one comprising all feathers, and one comprising non-wingtip non-feathers.",
"The last sequence is distinguished by the property that each pair of symbols in any block is nested with respect to $S$ , which is a “good” thing if we are intent on giving strong upper bounds on odd-order sequences.",
"The sequence comprising feathers is “bad” in this sense, therefore we must obtain better-than-trivial upper bounds on its length if this strategy is to bear fruit.",
"Recall that feather is a term that can be applied to nodes in some $\\mathcal {T}_{|b}$ or the corresponding occurrences of $b$ in the given sequence $S$ .",
"This definition is with respect to one derivation tree $\\mathcal {T}$ for $S$ , which is not necessarily the best one.",
"In Recurrences REF and REF it is useful to reason about the optimal derivation tree.",
"Let $\\mathcal {T}^*(S)$ be the derivation tree for $S$ that minimizes the number of occurrences in $S$ classified as feathers.",
"Recurrence 5.1 Define $\\Phi _{s}(n,m)$ to be the maximum number of feathers in any order-$s$ , $m$ -block DS sequence $S$ over an $n$ -letter alphabet, with respect to the optimal derivation tree $\\mathcal {T}^*(S)$ .",
"When $m\\le 2$ we have $\\Phi _{s}(n,m)=0$ .",
"For any $\\hat{m}<m$ , any block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ , and any alphabet partition $\\lbrace \\hat{n}\\rbrace \\cup \\lbrace \\check{n}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , we have $\\Phi _{s}(n,m)&\\le \\; \\sum _{q=1}^{\\hat{m}} \\Phi _{s}(\\check{n}_q,m_q)\\,+\\, \\Phi _{s}(\\hat{n},\\hat{m}) \\,+\\, 2\\cdot \\lambda _{s-1}(\\hat{n},m)$ Proof: When $m\\le 2$ , $\\Phi _{s}(n,m)$ is trivially 0 since every occurrence in $S$ is a wingtip, and feathers are not wingtips.",
"When $m>2$ we shall decompose $S$ as in Section REF .",
"The choice of $\\hat{m}$ and the block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ are not necessarily those of the optimal derivation tree, but we do not need them to be.",
"We are only interested in an upper bound on $\\Phi _{s}(n,m)$ .",
"Let $\\hat{\\mathcal {T}}^*$ and $\\lbrace \\check{\\mathcal {T}}_q^*\\rbrace _{1\\le q\\le \\hat{m}}$ be the optimal derivation trees for $\\hat{S}^{\\prime }$ and $\\lbrace \\check{S}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , and let $\\mathcal {T}$ be their composition, with the blocks of $S$ placed at $\\mathcal {T}$ 's leaves.",
"The number of occurrences of local feathers with respect to $\\lbrace \\check{\\mathcal {T}}^*_q\\rbrace $ is at most $\\sum _q \\Phi _{s}(\\check{n}_q,m_q)$ .",
"An occurrence of $a\\in \\mathcal {B}(v)$ in $\\hat{S}$ will be a dove feather in $\\mathcal {T}$ if either (i) $v$ is the rightmost child of a dove feather in $\\hat{\\mathcal {T}}^*_{|a}$ or (ii) $v$ is a non-wingtip child of the left wingtip in $\\hat{\\mathcal {T}}^*_{|a}$ , which corresponds to an occurrence of $a$ in $\\acute{S}$ .",
"The same statement is true of hawk feathers, swapping the roles of left and right and substituting $\\grave{S}$ for $\\acute{S}$ .",
"There are at most $\\Phi _{s}(\\hat{n},\\hat{m})$ feathers of type (i) and, since $\\acute{S}$ and $\\grave{S}$ are order-$(s-1)$ DS sequences, less than $2\\cdot \\lambda _{s-1}(\\hat{n},m)$ of type (ii).",
"$\\Box $ We now have all the elements in place to provide a recurrence for odd-order Davenport-Schinzel sequences.",
"Recurrence 5.2 Let $m,n,$ and $s$ be the block count, alphabet size, and order parameters, where $s\\ge 5$ is odd.",
"For any $\\hat{m}<m$ , any block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ , and any alphabet partition $\\lbrace \\hat{n}\\rbrace \\cup \\lbrace \\check{n}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , we have $\\lambda _{s}(n,m) &\\le \\sum _{q=1}^{\\hat{m}} \\lambda _{s}(\\check{n}_q,m_q) \\,+\\, 2\\cdot \\lambda _{s-1}(\\hat{n},m)\\,+\\, \\lambda _{s-2}(\\Phi _{s}(\\hat{n},\\hat{m}),m) \\,+\\, \\lambda _{s-3}(\\lambda _{s}(\\hat{n},\\hat{m}), m)$ Proof: As always, we adopt the notation from Section REF .",
"Define $\\hat{\\mathcal {T}}^*, \\lbrace \\check{\\mathcal {T}}^*_q\\rbrace ,$ and $\\mathcal {T}$ as in the proof of Recurrence REF .",
"In Recurrence REF we partitioned $S$ into local and global symbols and partitioned the occurrences of global symbols into first, middle, and last.",
"We now partition the middle occurrences one step further.",
"Define $\\tilde{S}^{\\prime }$ and $\\breve{S}^{\\prime }$ to be the subsequences of $\\hat{S}^{\\prime }$ consisting of feathers (according to $\\hat{\\mathcal {T}}^*$ ) and non-feather, non-wingtips, respectively.",
"That is, $|\\hat{S}^{\\prime }| = |\\tilde{S}^{\\prime }| + |\\breve{S}^{\\prime }| + 2\\hat{n}$ .",
"In an analogous fashion define $\\tilde{S}$ and $\\breve{S}$ to be the subsequences of $\\hat{S}$ consisting of children of occurrences in $\\tilde{S}^{\\prime }$ and $\\breve{S}^{\\prime }$ .",
"The sequences $\\acute{S}$ and $\\grave{S}$ represent the children of dove and hawk wingtips in $\\hat{\\mathcal {T}}^*$ .",
"Thus, $|S| = \\sum _{q} |\\check{S}_q| + |\\acute{S}| + |\\grave{S}| + |\\tilde{S}| + |\\breve{S}|$ .",
"The local sequences $\\lbrace \\check{S}_q\\rbrace $ are order-$s$ DS sequences.",
"According to the standard argument $\\acute{S}$ and $\\grave{S}$ are order-$(s-1)$ DS sequences and $\\tilde{S}= \\tilde{S}_1\\cdots \\tilde{S}_{\\hat{m}}$ is obtained from $\\tilde{S}^{\\prime }$ by substituting for its $q$ th block an order-$(s-2)$ DS sequence $\\tilde{S}_q$ .",
"From the superadditivity of $\\lambda _{s-2}$ it follows that $|\\tilde{S}| \\le \\lambda _{s-2}(|\\tilde{S}^{\\prime }|,m) \\le \\lambda _{s-2}(\\Phi _{s}(\\hat{n},\\hat{m}), m)$ .",
"We claim that $\\breve{S}=\\breve{S}_1\\cdots \\breve{S}_{\\hat{m}}$ is obtained from $\\breve{S}^{\\prime }$ by substituting for its $q$ th block an order-$(s-3)$ DS sequence $\\breve{S}_q$ , which, if true, would imply that $|\\breve{S}| \\le \\lambda _{s-3}(|\\breve{S}^{\\prime }|,m) < \\lambda _{s-3}(\\lambda _{s}(\\hat{n},\\hat{m}),m)$ .",
"Suppose for the purpose of obtaining a contradiction that the $q$ th block $\\beta $ in $\\breve{S}^{\\prime }$ contains $a,b\\in \\hat{\\Sigma }$ , and that $\\breve{S}_q$ is not an order-$(s-3)$ DS sequence, that is, it contains an alternating subsequence $ab\\cdots ab$ of length $s-1$ .",
"Note that $s-1$ is even.",
"By definition $\\beta $ is a non-feather, non-wingtip in both $\\hat{\\mathcal {T}}^*_{|a}$ and $\\hat{\\mathcal {T}}^*_{|b}$ .",
"According to Lemma REF , $a$ and $b$ must be nested in $\\beta $ , which implies that $S$ contains a subsequence of the form $&a \\cdots b \\cdots \\left| \\cdots \\overbrace{a \\cdots b \\cdots a \\cdots b}^{s-1} \\cdots \\right| \\cdots b \\cdots a\\\\\\mbox{ or } \\; \\; \\; \\; \\; \\; \\; &b \\cdots a \\cdots \\left| \\cdots \\overbrace{a \\cdots b \\cdots a \\cdots b}^{s-1} \\cdots \\right| \\cdots a \\cdots b$ where the portion between bars is in $S_q$ .",
"In either case $S$ contains an alternating subsequence with length $s+2$ , contradicting the fact that $S$ is an order-$s$ DS sequence.",
"$\\Box $"
],
[
"Analysis of the Recurrences",
"The dependencies between $\\lambda _{}$ and $\\Phi $ established by Recurrences REF , REF , and REF are rather intricate.",
"For even $s$ , $\\lambda _{s}$ is a function of $\\lambda _{s}, \\lambda _{s-1}$ and $\\lambda _{s-2}$ , and for odd $s$ , $\\lambda _{s}$ is a function of $\\lambda _{s}, \\lambda _{s-1},\\lambda _{s-2},\\lambda _{s-3}$ , and $\\Phi _{s}$ while $\\Phi _{s}$ is a function of $\\Phi _{s}$ and $\\lambda _{s-1}$ .",
"The proof of Lemma REF is by induction over parameters: $s,n,c,i,$ and $j$ , where $s$ is the order, $n$ the alphabet size, $c\\ge s-2$ a constant that determines how $\\hat{m}$ and the block partition is chosen, $i\\ge 1$ is an integer, and $j$ is minimal such that the block count $m\\le a_{i,j}^c$ .",
"Some level of complexity is therefore unavoidable.",
"Furthermore, when $s\\ge 5$ is odd, $\\lambda _{s}$ is so sensitive to approximations of $\\lambda _{s-3}$ that we must treat $s\\in \\lbrace 1,2,3,4,5\\rbrace $ as distinct base cases, and treat even and odd $s\\ge 6$ as separate inductive cases.",
"Given these constraints we feel our analysis is reasonably simple.",
"Lemma 5.3 Let $s\\ge 1$ be the order parameter, $c\\ge s-2$ be a constant, and $i\\ge 1$ be an arbitrary integer.",
"The following upper bounds on $\\lambda _{s}$ and $\\Phi _{s}$ hold for all $s\\ge 1$ and all odd $s\\ge 5$ , respectively.",
"Define $j$ to be maximum such that $m\\le a_{i,j}^c$ .",
"$\\lambda _{1}(n,m) &= n + m-1 & \\mbox{$s=1$}\\\\\\lambda _{2}(n,m) &= 2n + m-2 & \\mbox{$s=2$}\\\\\\lambda _{3}(n,m) &\\le (2i+2)n + (3i-2)cj(m-1) & \\mbox{$s=3$}\\\\\\lambda _{s}(n,m) &\\le \\mu _{s,i}{\\left( n + (cj)^{s-2}(m-1) \\right)} & \\mbox{all $s\\ge 4$} \\\\\\Phi _{s}(n,m) &\\le \\nu _{s,i}{\\left( n + (cj)^{s-2}(m-1) \\right)} & \\mbox{odd $s\\ge 5$}\\multicolumn{2}{l}{\\text{The values $\\lbrace \\mu _{s,i},\\nu _{s,i}\\rbrace $ are defined as follows, where $t=\\lfloor \\frac{s-2}{2} \\rfloor $.",
"}}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{5 mm}\\mu _{s,i} &= \\left\\lbrace \\begin{array}{l}2^{{i+t+3}\\atopwithdelims ()t} - 3(2(i+t+1))^t \\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm}\\mbox{$\\frac{3}{2}$}(2(i+t+1))^{t+1}2^{{i+t+3}\\atopwithdelims ()t}\\end{array}\\right.&\\begin{array}{r}\\mbox{even $s\\ge 4$\\hspace*{-5.12128pt}}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm}\\mbox{odd $s\\ge 5$\\hspace*{-5.12128pt}}\\end{array}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{8 mm}\\nu _{s,i} &= 4\\cdot 2^{{i+t+3}\\atopwithdelims ()t} &\\begin{array}{r}\\mbox{odd $s\\ge 5$\\hspace*{-5.12128pt}}\\end{array}$ One may want to keep in mind that we will eventually substitute $\\alpha (n,m)+ O(1)$ for the parameter $i$ , and that ${i+t+O(1) \\atopwithdelims ()t} = i^t/t!",
"+ O(i^{t-1})$ .",
"Lemma REF will, therefore, imply bounds on $\\lambda _{s}(n,m)$ analogous to those claimed for $\\lambda _{s}(n)$ in Theorem REF .",
"The proof of Lemma REF appears in Appendix ."
],
[
"The Upper Bounds of Theorem ",
"Fix $s\\ge 3, n,m$ and let $c=s-2$ .",
"For $i \\ge 1$ let $j_i$ be minimum such that $m\\le a_{i,j_i}^{c}$ .",
"Lemma REF implies that an order-$s$ DS sequence has length at most $\\mu _{s,i}(n + (cj_i)^{s-2} m)$ .",
"Choose $\\iota $ to be minimum such thatWe want $(cj_{\\iota })^{s-2}m$ not to be the dominant term, so $(cj_{\\iota })^{s-2}$ should be less than $\\lceil n/m \\rceil $ .",
"On the other hand, the first and second columns of Ackermann's function ($a_{i,1}$ and $a_{i,2}$ ) do not exhibit sufficient growth, so $j_\\iota $ must also be at least 3.",
"$(cj_\\iota )^{s-2} \\le \\max \\lbrace n̑{m}, (c\\cdot 3)^{s-2}\\rbrace $ .",
"One can show that $\\iota = \\alpha (n,m) + O(1)$ .",
"By choice of $\\iota $ it follows that $(cj_\\iota )^{s-2} m = O(m+n)$ , so $\\lambda _{s}(n,m) = O((n+m)\\mu _{s,\\iota })$ .",
"According to Lemma REF 's definition of $\\mu _{s,\\iota }$ , we have $\\lambda _{3}(n,m) &= O((n+m)\\alpha (n,m))\\\\\\lambda _{4}(n,m) &= O{\\left( (n+m)2^{\\alpha (n,m)} \\right)}\\\\\\lambda _{5}(n,m) &= O{\\left( (n+m)\\alpha ^2(n,m)2^{\\alpha (n,m)} \\right)}\\\\\\lambda _{s}(n,m) &= (n+m)\\cdot 2^{\\alpha ^t(n,m)/t!",
"\\,+\\, O(\\alpha ^{t-1}(n,m))} & \\mbox{both even and odd $s\\ge 6$, where $t=\\lceil \\frac{s-2}{2} \\rceil $.",
"}$ The bound on $\\lambda _{5}(n,m)$ follows since $\\mu _{5,\\iota } = O(\\iota ^2 2^{\\iota })$ .",
"When $s\\ge 6$ and $t\\ge 2$ , $\\mu _{s,\\iota } < \\iota ^{t+1}2^{\\iota + t+ O(1) \\atopwithdelims ()t} = 2^{\\iota ^t/t!",
"\\,+\\, O(\\iota ^{t-1})}$ .",
"Theorem REF stated bounds on $\\lambda _{s}(n)$ rather than $\\lambda _{s}(n,m)$ .",
"If it were known that extremal order-$s$ DS sequences consisted of $m=O(n)$ blocks we could simply substitute $\\alpha (n)$ for $\\alpha (n,m)$ in the bounds above, but this is not known to be true.",
"According to Lemma REF (REF ,REF ), if $\\gamma _s$ is such that $\\lambda _{s}(n)\\le \\gamma _{s}(n)\\cdot n$ then $\\lambda _{s}(n) \\le \\gamma _{s-2}(n)\\cdot \\lambda _{s}(n,2n-1)$ and $\\lambda _{s}(n) \\le \\gamma _{s-2}(\\gamma _s(n))\\cdot \\lambda _{s}(n,3n-1)$ .",
"Applying Lemma REF when $s\\in \\lbrace 3,4\\rbrace $ has no asymptotic affect since $\\gamma _1=1$ and $\\gamma _2=2$ .",
"It has no perceptible effect when $s\\ge 6$ since $\\gamma _{s-2}(n)$ or $\\gamma _{s-2}(\\gamma _s(n))$ is dwarfed by the lower order terms in the exponent.",
"However, for $s\\in \\lbrace 3,5\\rbrace $ these reductions only show that $\\lambda _{3}(n) = O(n\\alpha (n))$ and that $\\lambda _{5}(n) = O(n\\alpha (\\alpha (n))\\alpha ^2(n)2^{\\alpha (n)})$ , which are weaker than the bounds claimed in Theorem REF .",
"In Section REF we prove $\\lambda _{3}(n) = 2n\\alpha (n) + O(n)$ , which is a tiny improvement over Klazar's bound [51], [63], though it is within $O(n)$ of Nivasch's construction [63] and is therefore optimal in the Ackermann-invariant sense.",
"See Remark REF .",
"To prove $\\lambda _{5}(n) = \\Theta (n\\alpha (n)2^{\\alpha (n)})$ we require a significant generalization of the derivation tree method.",
"Sections and give the matching lower and upper bounds on order-5 DS sequences."
],
[
"Order $s=3$",
"Let $S$ be an order-3 DS sequence over an $n$ -letter alphabet.",
"According to Lemma REF ([41]), $|S| \\le \\lambda _{3}(n) \\le \\lambda _{3}(n,m)$ , where $m=2n-1$ .",
"Letting $\\iota $ be minimum such that $m \\le a_{\\iota ,3}$ , Lemma REF implies that $\\lambda _{3}(n,m) < (2 \\iota +2)n + (3\\iota -2)m < (8\\iota -2)n$ .",
"It is straightforward to show that $\\iota \\le \\alpha (n)+O(1)$ .",
"The problem is clearly that there are too many blocks.",
"Were there less than $(2n-1)/\\iota $ blocks, Lemma REF would give a bound of $(2\\iota +2)n + O(\\iota m/\\iota )=2n\\alpha (n) + O(n)$ .",
"We can invoke Recurrence REF to divide $S$ into a global $\\hat{S}$ and local $\\check{S}=\\check{S}_1\\cdots \\check{S}_{\\hat{m}}$ , where $\\hat{m}= m/\\iota \\le (2n-1)/\\iota $ , that is, each $\\check{S}_q$ is an $\\iota $ -block sequence.",
"Using Lemma REF we will bound $\\hat{S}$ with $i=\\iota $ and each of the $\\lbrace \\check{S}_q\\rbrace _q$ with $i=1$ .",
"$|S| &\\le \\lambda _{3}(n) \\le \\lambda _{3}(n,m) & \\mbox{\\lbrace where $m=2n-1$\\rbrace }\\\\&\\le \\sum _{q=1}^{\\hat{m}} \\lambda _{3}(\\check{n}_q,\\iota ) \\;+\\; 2\\cdot \\lambda _{2}(\\hat{n},m) \\;+\\; \\lambda _{1}(\\lambda _{3}(\\hat{n},\\hat{m})-2\\hat{n},m) & \\mbox{\\lbrace Recurrence~\\ref {thm:recurrence-even}\\rbrace }\\\\&< \\sum _{q=1}^{\\hat{m}} \\Big [ 4\\check{n}_q \\;+\\; \\min \\Big \\lbrace \\iota \\lceil \\log \\iota \\rceil , \\;\\, (\\iota -1) + (2\\check{n}_q-1)\\lceil \\log (2\\check{n}_q-1) \\rceil \\Big \\rbrace \\Big ] & \\mbox{(*)}\\\\&\\hspace*{28.45274pt}+\\; [4\\hat{n}+ 2m] \\;+\\; [2\\iota \\hat{n}\\,+\\, (3\\iota -2)\\hat{m}\\,+\\, m] & \\mbox{\\lbrace Lemmas~\\ref {lem:DS12}, \\ref {lem:agreeable}\\rbrace }\\\\&< \\Big [ m + (n-\\hat{n})(4 + 2\\lceil \\log (2\\iota -1) \\rceil ) \\Big ] + (2\\iota +4)\\hat{n}+ (3\\iota -2)m/\\iota + 3m & \\mbox{\\lbrace $\\hat{m}= m/\\iota \\rbrace $}\\\\&< (2\\iota +4)n + 7m & \\mbox{\\lbrace worst case if $\\hat{n}=n$\\rbrace }\\\\&\\le 2n\\alpha (n) + O(n) & \\mbox{\\lbrace $\\iota = \\alpha (n)+O(1)$\\rbrace }$ The bound on local symbols in line (*) follows from Lemma REF and Hart and Sharir's [41] observation that $\\lambda _{3}(n)\\le \\lambda _{3}(n,2n-1)$ .",
"When $i=1$ and $j=\\lceil \\log \\iota \\rceil $ , Lemma REF gives us a bound of $\\lambda _{3}(\\check{n}_q,\\iota ) \\le 4\\check{n}_q + \\iota \\lceil \\log \\iota \\rceil $ .",
"Alternatively, we could make $\\check{S}_q$ 2-sparse by removing up to $\\iota -1$ duplicated symbols at block boundaries, then partitioning the remaining sequence into $2\\check{n}_q-1$ blocks, hence $\\lambda _{3}(\\check{n}_q,\\iota ) \\le \\iota -1 + \\lambda _{3}(\\check{n}_q,2\\check{n}_q-1) \\le \\iota -1 + 4\\check{n}_q + (2\\check{n}_q-1)\\lceil \\log (2\\check{n}_q-1) \\rceil $ .",
"This matches Nivasch's lower bound [63] on $\\lambda _{3}(n)$ to within $O(n)$ ."
],
[
"Lower Bounds on Fifth-Order Sequences",
"We have established every bound claimed in Theorem REF except for those on order-5 DS sequences.",
"In this section we give a construction that yields bounds of $\\lambda _{5}(n,m) = \\Omega (n\\alpha (n,m)2^{\\alpha (n,m)})$ and $\\lambda _{5}(n) = \\Omega (n\\alpha (n)2^{\\alpha (n)})$ .",
"This is the first construction that is asymptotically longer than the order-4 DS sequences of [4] having length $\\Theta (n2^{\\alpha (n)})$ .",
"Our construction is based on generalized forms of sequence composition and shuffling used by Agarwal, Sharir, and Shor [4], Nivasch [63], and Pettie [68].",
"Recall from Section REF that $\\Vert S\\Vert =|\\Sigma (S)|$ is the alphabet size of $S$ and, if $S$ is partitioned into blocks, $\\llbracket S \\rrbracket $ is its block count."
],
[
"Composition and Shuffling",
"In its generic form, a sequence $S$ is assumed to be over the alphabet $\\lbrace 1,\\ldots ,\\Vert S\\Vert \\rbrace $ , that is, any totally ordered set with size $\\Vert S\\Vert $ .",
"To substitute $S$ for a block $\\beta = [a_1\\ldots a_{|\\beta |}]$ means to replace $\\beta $ with a copy $S(\\beta )$ under the alphabet mapping $k \\mapsto a_k$ , where $|\\beta | \\le \\Vert S\\Vert $ .",
"If $|\\beta |$ is strictly smaller than $\\Vert S\\Vert $ , any occurrences of the $\\Vert S\\Vert -|\\beta |$ unused symbols of $\\Sigma (S)$ do not appear in $S(\\beta )$ .",
"We always assume that $S$ is in canonical form: the symbols are ordered according to the position of their first appearance in $S$ ."
],
[
"Composition.",
"If $S_{\\operatorname{mid}}$ is a sequence in canonical form with $\\Vert S_{\\operatorname{mid}}\\Vert = j$ and $S_{\\operatorname{top}}$ a sequence partitioned into blocks with length at most $j$ , $S_{\\operatorname{sub}}= S_{\\operatorname{top}}\\operatorname{\\circ }S_{\\operatorname{mid}}$ is obtained by substituting for each block $\\beta $ in $S_{\\operatorname{top}}$ a copy $S_{\\operatorname{mid}}(\\beta )$ .",
"Clearly $\\llbracket S_{\\operatorname{sub}} \\rrbracket = \\llbracket S_{\\operatorname{top}} \\rrbracket \\cdot \\llbracket S_{\\operatorname{mid}} \\rrbracket $ .",
"If $S_{\\operatorname{mid}}$ and $S_{\\operatorname{top}}$ contain $\\mu $ and $\\mu ^{\\prime }$ occurrences of each symbol, respectively, then $S_{\\operatorname{sub}}$ contains $\\mu \\mu ^{\\prime }$ occurrences of each symbol.",
"Composition preserves canonical form, that is, if $S_{\\operatorname{mid}}$ and $S_{\\operatorname{top}}$ are in canonical form, so is $S_{\\operatorname{sub}}$ ."
],
[
"Shuffling.",
"If $S_{\\operatorname{bot}}$ is a $j^{\\prime }$ -block sequence and $S_{\\operatorname{sub}}$ is partitioned into blocks of length at most $j^{\\prime }$ , we can form the shuffle $S_{\\operatorname{sh}}= S_{\\operatorname{sub}}\\operatorname{\\diamond }S_{\\operatorname{bot}}$ as follows.",
"First create a sequence $S_{\\operatorname{bot}}^*$ consisting of the concatenation of $\\llbracket S_{\\operatorname{sub}} \\rrbracket $ copies of $S_{\\operatorname{bot}}$ , each copy being over an alphabet disjoint from the other copies and disjoint from that of $S_{\\operatorname{sub}}$ .",
"By design the length of $S_{\\operatorname{sub}}$ is at most the number of blocks in $S_{\\operatorname{bot}}^*$ , and precisely the same if all blocks in $S_{\\operatorname{sub}}$ have their maximum length $j^{\\prime }$ .",
"The sequence $S_{\\operatorname{sub}}\\operatorname{\\diamond }S_{\\operatorname{bot}}$ is obtained by shuffling the $j^{\\prime }$ symbols of the $l$ th block of $S_{\\operatorname{sub}}$ into the $j^{\\prime }$ blocks of the $l$ th copy of $S_{\\operatorname{bot}}$ in $S_{\\operatorname{bot}}^*$ .",
"Specifically, the $k$ th symbol of the $l$ th block is inserted at the end of the $k$ th block of the $l$ th copy of $S_{\\operatorname{bot}}$ ."
],
[
"Three-Fold Composition.",
"Our construction of order-5 DS sequences uses a generalized form of composition that treats symbols in $\\beta $ differently based on context.",
"Suppose $S_{\\operatorname{top}}$ is partitioned into blocks with length at most $j$ and $S_{\\operatorname{mid}}^{\\operatorname{f}},S_{\\operatorname{mid}}^{\\operatorname{m}},$ and $S_{\\operatorname{mid}}^{\\operatorname{l}}$ are sequences with alphabet size $\\Vert S_{\\operatorname{mid}}^{\\operatorname{f}}\\Vert =\\Vert S_{\\operatorname{mid}}^{\\operatorname{m}}\\Vert =\\Vert S_{\\operatorname{mid}}^{\\operatorname{l}}\\Vert =j$ .",
"The 3-fold composition $S_{\\operatorname{top}}\\operatorname{\\circ }\\left< S_{\\operatorname{mid}}^{\\operatorname{f}},S_{\\operatorname{mid}}^{\\operatorname{m}},S_{\\operatorname{mid}}^{\\operatorname{l}} \\right>$ is formed as follows.",
"For each block $\\beta $ in $S_{\\operatorname{top}}$ , categorize its symbols as first if they occur in no earlier block, last if they occur in no later block, and middle otherwise.",
"Let $\\beta ^{\\operatorname{f}},\\beta ^{\\operatorname{m}},$ and $\\beta ^{\\operatorname{l}}$ be the subsequences of $\\beta $ consisting of first, middle, and last symbols.",
"Note that these three sequences do not necessarily occur contiguously in $\\beta $ , but each is nonetheless a subsequence of $\\beta $ .",
"Substitute for $\\beta $ the concatenation of $S_{\\operatorname{mid}}^{\\operatorname{f}}(\\beta ^{\\operatorname{f}}), S_{\\operatorname{mid}}^{\\operatorname{m}}(\\beta ^{\\operatorname{m}}),$ and $S_{\\operatorname{mid}}^{\\operatorname{l}}(\\beta ^{\\operatorname{l}})$ .",
"Note that if $S_{\\operatorname{top}},S_{\\operatorname{mid}}^{\\operatorname{f}},S_{\\operatorname{mid}}^{\\operatorname{m}},$ and $S_{\\operatorname{mid}}^{\\operatorname{l}}$ contain $\\mu \\ge 2,\\mu ^{\\operatorname{f}},\\mu ^{\\operatorname{m}},$ and $\\mu ^{\\operatorname{l}}$ occurrences of each symbol then $S_{\\operatorname{top}}\\operatorname{\\circ }\\left< S_{\\operatorname{mid}}^{\\operatorname{f}},S_{\\operatorname{mid}}^{\\operatorname{m}},S_{\\operatorname{mid}}^{\\operatorname{l}} \\right>$ contains $\\mu ^{\\operatorname{f}} + \\mu ^{\\operatorname{l}} + (\\mu -2)\\mu ^{\\operatorname{m}}$ occurrences of each symbol.",
"Figure REF gives a schematic of the generation of the sequence ${\\left( S_{\\operatorname{top}}\\operatorname{\\circ }\\left< S_{\\operatorname{mid}}^{\\operatorname{f}},S_{\\operatorname{mid}}^{\\operatorname{m}},S_{\\operatorname{mid}}^{\\operatorname{l}} \\right> \\right)}\\operatorname{\\diamond }S_{\\operatorname{bot}}$ .",
"Figure: Three-fold composition followed by shuffling.Each block β\\beta in S top S_{\\operatorname{top}} is replaced with the concatenation ofS mid f (β f ),S mid m (β m )S_{\\operatorname{mid}}^{\\operatorname{f}}(\\beta ^{\\operatorname{f}}), S_{\\operatorname{mid}}^{\\operatorname{m}}(\\beta ^{\\operatorname{m}}),and S mid l (β l )S_{\\operatorname{mid}}^{\\operatorname{l}}(\\beta ^{\\operatorname{l}})and each block of that sequence is shuffled with a single copy of S bot S_{\\operatorname{bot}} in S bot * S_{\\operatorname{bot}}^*.In general, blocks in S mid f (β f ),S mid m (β m )S_{\\operatorname{mid}}^{\\operatorname{f}}(\\beta ^{\\operatorname{f}}), S_{\\operatorname{mid}}^{\\operatorname{m}}(\\beta ^{\\operatorname{m}}),and S mid l (β l )S_{\\operatorname{mid}}^{\\operatorname{l}}(\\beta ^{\\operatorname{l}}) will not attain their maximum length j ' j^{\\prime }."
],
[
"Sequences of Orders 4 and 5",
"The sequences $S_4(i,j)$ and $S_5(i,j)$ are defined inductively below.",
"As we will prove, $S_4(i,j)$ is an order-4 DS sequence partitioned into blocks of length precisely $j$ in which each symbol appears $2^i$ times, whereas $S_5(i,j)$ is an order-5 DS sequence partitioned into blocks of length at most $j$ in which each symbol appears $(2i-3)2^i+4$ times.",
"Let $B_s(i,j) = \\llbracket S_s(i,j) \\rrbracket $ and $N_s(i,j) = \\Vert S_s(i,j)\\Vert $ be, respectively, the number of blocks in $S_s(i,j)$ and the alphabet size of $S_s(i,j)$ .",
"By definition $|S_4(i,j)| = 2^i\\cdot N_4(i,j) = j\\cdot B_4(i,j)$ and $|S_5(i,j)| = ((2i-3)2^i+4)\\cdot N_5(i,j) \\le j\\cdot B_5(i,j)$ .",
"The construction of $S_4$ is the same as Nivasch's [63] and similar to that of Agarwal et al. [4].",
"The base cases for our sequences are given below, where square brackets indicate blocks: $S_2(j) &= [12\\cdots (j-1)j] \\:\\: \\makebox{[}0mm][l]{[j(j-1)\\cdots 21]} & \\mbox{ two blocks with length $j$}\\\\S_4(1,j) &= S_5(1,j) = S_2(j)\\\\S_4(i,1) &= [1]^{2^i} & \\mbox{ $2^i$ identical blocks}\\\\S_5(i,1) &= [1]^{(2i-3)2^i+4} & \\mbox{ $(2i-3)2^i + 4$ identical blocks}\\\\\\multicolumn{2}{l}{\\text{Observe that these base cases satisfy the property that symbols appear precisely $2^i$times in $S_4(i,\\cdot )$ and $(2i-3)2^i + 4$ times in $S_5(i,\\cdot )$.",
"Define $S_4(i,j)$ as}}\\\\S_4(i,j) &= \\makebox{[}0mm][l]{\\Big ( S_4(i-1, y)\\operatorname{\\circ }S_2(y) \\Big ) \\operatorname{\\diamond }S_4(i,j-1),} & \\mbox{ where $y=B_4(i,j-1)$}\\\\\\multicolumn{2}{l}{\\text{and $S_5(i,j)$ as}}\\\\S_5(i,j) &= \\makebox{[}0mm][l]{\\left(S_{\\operatorname{top}}\\operatorname{\\circ }\\Big < S_{\\operatorname{mid}}^{\\operatorname{f}},S_{\\operatorname{mid}}^{\\operatorname{m}},S_{\\operatorname{mid}}^{\\operatorname{l}} \\Big >\\right)\\operatorname{\\diamond }S_{\\operatorname{bot}}}\\\\\\mbox{where \\ } & S_{\\operatorname{bot}}= S_5(i,j-1), & \\mbox{ $z=B_5(i,j-1)$}\\\\& S_{\\operatorname{mid}}^{\\operatorname{f}} = S_{\\operatorname{mid}}^{\\operatorname{l}} = S_4(i,z), \\\\& S_{\\operatorname{mid}}^{\\operatorname{m}} = S_2(N_4(i,z)),\\\\\\mbox{and \\ } & S_{\\operatorname{top}}= S_5(i-1, N_4(i,z)),\\\\$ By definition $S_{\\operatorname{mid}}^{\\operatorname{f}}$ and $S_{\\operatorname{mid}}^{\\operatorname{l}}$ are partitioned into blocks with length $z$ .",
"In the three-fold composition operation we also interpret $S_2(N_4(i,z))$ as a sequence of blocks of length precisely $z$ .",
"It will be shown shortly that $N_4(i,z)$ is, in fact, a multiple of $z$ .",
"We argue by induction that symbols appear with the correct multiplicity in $S_4$ and $S_5$ .",
"In the case of $S_4$ each symbol appears $2^{i-1}$ times in $S_4(i-1,y)$ (by the inductive hypothesis), twice in $S_2(y)$ , and therefore $2^i$ times in $S_4(i-1,y)\\operatorname{\\circ }S_2(y)$ .",
"Symbols in copies of $S_4(i,j-1)$ already appear $2^i$ times, by the inductive hypothesis.",
"In $S_5(i-1,N_4(i,z))$ each symbol appears $(2i-5)2^{i-1} + 4$ times.",
"The 3-fold composition operation increases the multiplicity of such symbols to $2\\left((2i-5)2^{i-1} + 2\\right) + 2\\left(2^{i}\\right) = (2i-3)2^i + 4$ , where the first term accounts for the blowup in middle occurrences and the second term for the blowup in first and last occurrences.",
"It follows that $B$ and $N$ are defined inductively as follows.",
"$B_4(1,j) &= B_5(1,j) = B_2(j) = 2\\\\B_4(i,1) &= 2^i\\\\B_5(i,1) &= (2i-3)2^i+4\\\\B_4(i,j) &= B_4(i-1,y) \\cdot 2 \\cdot y & \\mbox{where $y=B_4(i,j-1)$}\\\\B_5(i,j) &= B_5(i-1, N_4(i,z)) \\cdot (2+2^{-i+1}) B_4(i,z) \\cdot z & \\mbox{where $z=B_5(i,j-1)$} \\\\N_4(1,j) &= N_5(1,j) = N_2(j) = j\\\\N_4(i,1) &= N_5(i,1) = 1\\\\N_4(i,j) &= N_4(i-1,y) \\; + \\; B_4(i-1,y) \\cdot 2 \\cdot N_4(i,j-1)\\\\N_5(i,j) &= \\makebox{[}0mm][l]{N_5(i-1,N_4(i,z)) \\; + \\; B_5(i-1,N_4(i,z)) \\cdot (2+2^{-i+1}) B_4(i,z) \\cdot N_5(i,j-1)}$ The $2+2^{-i+1}$ factor in the definition of $B_5(i,j)$ and $N_5(i,j)$ comes from the fact that in the shuffling step, $S_2(N_4(i,z))$ is interpreted as having $|S_2(N_4(i,z))|/z$ blocks of length $z$ , where ${z} = {z} = {z\\cdot 2^{i}} = 2^{-i+1}B_4(i,z)$ Lemma 6.1 For $s\\in \\lbrace 4,5\\rbrace $ , $S_s(i,j)$ is an order-$s$ Davenport-Schinzel sequence.",
"Proof: We use brackets to indicate block boundaries in (forbidden) patterns, e.g., $[ba]ba$ is a pattern where the first $ba$ appears in one block and the last $ba$ appears outside that block.",
"One can easily show by induction that $ba[ba] \\nprec S_s(i,j)$ and $[ba]ab\\nprec S_s(i,j)$ for all $s\\in \\lbrace 4,5\\rbrace ,i>1,j\\ge 1$ .",
"The base cases are trivial.",
"When $a$ is shuffled into the indicated block in a copy of $S_s(i,j-1)$ , all $b$ s appear in that copy and all other $a$ s are shuffled into different copies, hence $[ba]$ cannot be preceded by $ba$ or followed by $ab$ .",
"This also implies that two symbols cannot both appear in two blocks of $S_s(i,j)$ , for all $i>1$ .",
"It follows that the patterns $ababab$ (and $abababa$ ) cannot be introduced into $S_4$ (and $S_5$ ) by the shuffling operation but must come from the composition (and 3-fold composition) operation.",
"Suppose $ba\\prec \\beta $ for some block $\\beta $ in $S_4(i-1, y)$ .",
"It follows that composing $\\beta $ with $S_2(y)$ (a $baba$ -free sequence) does not introduce an $ababab$ pattern.",
"(Substituting $\\underline{bab}$ for $ba\\prec \\beta $ and projecting onto $\\lbrace a,b\\rbrace $ yields sequences of the form $a^* b^* \\underline{bab} b^* a^*$ .)",
"Turning to $S_5$ , suppose $ab\\prec \\beta $ for some block $\\beta $ in $S_{\\operatorname{top}}$ .",
"If $a$ and $b$ are both middle symbols in $\\beta $ then, by the same argument, composing $\\beta $ with $S_{\\operatorname{mid}}^{\\operatorname{m}} = S_2(N_4(i,z))$ does not introduce an $ababab$ pattern much less an $abababa$ pattern.",
"If both $a,b$ are first then composing $\\beta $ with an order-4 DS sequence $S_{\\operatorname{mid}}^{\\operatorname{f}} = S_4(i,z)$ and projecting onto $\\lbrace a,b\\rbrace $ yields patterns of the form $\\underline{a^*b^*a^*b^*a^*} a^*b^*$ , where the underlined portion originated from $\\beta $ .",
"The case when $a$ and $b$ are last is symmetric.",
"The cases when $a$ and $b$ are of different types (first-middle, first-last, last-middle) are handled similarly.",
"$\\Box $ We have shown that $\\lambda _{4}(N_4(i,j), B_4(i,j)) \\ge 2^i N_4(i,j)$ and $\\lambda _{5}(N_5(i,j), B_5(i,j)) \\ge ((2i-3)2^i + 4) N_5(i,j)$ .",
"Since any blocked sequence can be turned into a 2-sparse sequence by removing duplicates at block boundaries this also implies that $\\lambda _{4}(N_4(i,j)) \\ge 2^i N_4(i,j) - B_4(i,j) > (1-1/j)2^iN_4(i,j)$ .",
"Remember that all blocks in $S_4(i,j)$ have length exactly $j$ .",
"There is no such guarantee for $S_5$ , however.",
"It is conceivable that it consists largely of long runs of identical symbols (each in a block of length 1), nearly all of which would be removed when converting it to a 2-sparse sequence.",
"That is, statements of the form $\\lambda _{5}(N_5(i,j)) \\ge ((2i-3)2^i+4)N_5(i,j) - B_5(i,j)$ become trivial if the $B_5(i,j)$ term dominates.",
"Lemma REF shows that for $j$ sufficiently large this does not occur and therefore removing duplicates at block boundaries does not affect the length of $S_5(i,j)$ asymptotically.",
"Lemma 6.2 $N_5(i,j) \\ge j\\cdot B_5(i,j) / \\xi (i)$ , where $\\xi (i) = 3^i 2^{i+1\\atopwithdelims ()2}$ .",
"Proof: When $i=1$ we have $N_5(1,j) = j \\ge j\\cdot B_5(1,j) / \\xi (1) = 2j/6$ .",
"When $j=1$ we have $N_5(i,1) = 1 \\ge B_5(i,1)/\\xi (i) = \\left((2i-3)2^i+4\\right)/ 3^i2^{i+1 \\atopwithdelims ()2}$ .",
"Assuming the claim holds for all $(i^{\\prime },j^{\\prime }) < (i,j)$ lexicographically, ${N_5(i,j)}\\\\&= N_5(i-1,N_4(i,z)) + B_5(i-1,N_4(i,z)) \\cdot (2+2^{-i+1}) B_4(i,z)\\cdot N_5(i,j-1) \\hspace*{-22.76228pt} & \\mbox{\\lbrace defn.",
"of $N_5$\\rbrace }\\\\&\\ge N_5(i-1,N_4(i,z)) + {\\xi (i)} B_5(i-1,N_4(i,z)) \\cdot (2+2^{-i+1}) B_4(i,z)\\cdot (j-1) z \\hspace*{-22.76228pt}&\\mbox{\\lbrace ind., defn.",
"of $z$\\rbrace }\\\\&= N_5(i-1,N_4(i,z)) \\,+\\, {\\xi (i)} B_5(i,j) & \\mbox{\\lbrace defn.",
"of $B_5$\\rbrace }\\\\&\\ge {\\xi (i-1)} N_4(i,z)\\cdot B_5(i-1, N_4(i,z)) \\,+\\, {\\xi (i)} B_5(i,j) & \\mbox{\\lbrace ind.",
"hyp.\\rbrace }\\\\&\\ge {\\xi (i-1)\\cdot 2^{i}}\\cdot z\\cdot B_4(i,z) \\cdot B_5(i-1, N_4(i,z)) + {\\xi (i)} B_5(i,j) & \\mbox{\\lbrace $N_4(i,z) = \\mbox{$\\frac{z}{2^i}$}B_4(i,z)$\\rbrace }\\\\&\\ge {\\xi (i-1)\\cdot 2^{i}\\cdot 3} \\cdot (2+2^{-i+1}) \\cdot z \\cdot B_4(i,z) \\cdot B_5(i-1,N_4(i,z)) + {\\xi (i)} B_5(i,j) \\hspace*{-22.76228pt}& \\mbox{\\lbrace $2+2^{-i+1} \\le 3$\\rbrace }\\\\&= {\\xi (i)}B_5(i,j) + {\\xi (i)} B_5(i,j) \\; = \\; j̑{\\xi (i)}B_5(i,j) & \\mbox{\\lbrace defn.",
"of $B_5$, $\\xi $\\rbrace }$ $\\Box $ Theorem 6.3 For any $n$ and $m$ , $\\lambda _{5}(n,m) = \\Omega (n \\alpha (n,m)2^{\\alpha (n,m)})$ and $\\lambda _{5}(n) = \\Omega (n\\alpha (n)2^{\\alpha (n)})$ .",
"Proof: Consider the sequence $S_5 = S_5(i,j)$ , where $j \\ge \\xi (i)$ , and let $S_5^{\\prime }$ be obtained by removing duplicates at block boundaries.",
"It follows that $S_5^{\\prime }$ is 2-sparse and, from Lemma REF , that $|S_5^{\\prime }| \\ge ((2i-3)2^i + 3)N_5(i,j)$ .",
"It is straightforward to prove that $i = \\alpha (N_5(i,j), B_5(i,j)) + O(1)$ and that $i = \\alpha (N_5(i,j)) + O(1)$ when $j = \\xi (i)$ .",
"$\\Box $"
],
[
"Upper Bounds on Fifth-Order Sequences",
"Recall from Section that a derivation tree $\\mathcal {T}(S)$ for a blocked sequence $S$ is the composition of a global tree $\\hat{\\mathcal {T}}= \\hat{\\mathcal {T}}(\\hat{S}^{\\prime })$ and local trees $\\lbrace \\check{\\mathcal {T}}_q\\rbrace _{q\\le \\hat{m}}$ , where $\\check{\\mathcal {T}}_q = \\mathcal {T}(\\check{S}_q)$ .",
"The composition is effected by identifying the $q$ th leaf of $\\hat{\\mathcal {T}}$ , call it $x_q$ , with the root of $\\check{\\mathcal {T}}_q$ , then populating the leaves of $\\mathcal {T}$ with the blocks of $S$ ."
],
[
"Superimposed Derivation Trees",
"One can view $\\mathcal {T}(S)$ as representing a hypothetical process for generating the sequence $S$ , but it only represents this process at one granularity.",
"For example, $\\hat{S}_q$ is the portion of $\\hat{S}$ at the leaf descendants of $x_q$ in $\\mathcal {T}$ .",
"The derivation tree $\\mathcal {T}$ does not let us inspect the structure of $\\hat{S}_q$ and provides no explanation for how $\\hat{S}_q$ came to be.",
"To reason about $\\hat{S}_q$ we could, of course, build a new derivation tree $\\hat{\\mathcal {T}}_q = \\mathcal {T}(\\hat{S}_q)$ just for $\\hat{S}_q$ .",
"One can see that $\\check{\\mathcal {T}}_q$ and $\\hat{\\mathcal {T}}_q$ will be structurally identical if, in their inductive construction, we always choose block partitions in the same way.",
"One can imagine superimposing $\\hat{\\mathcal {T}}_q$ onto $\\check{\\mathcal {T}}_q$ , regarding both as being on the same node set but populated with different blocks.",
"In our actual analysis we do not consider the derivation tree for $\\hat{S}_q$ , which includes all global occurrences in $S_q$ , but just those derivation trees for $\\acute{S}_q$ and $\\grave{S}_q$ , which are restricted to global symbols making their first and last appearance in $S_q$ , respectively.",
"Define $\\acute{\\mathcal {T}}[x_q] = \\mathcal {T}(\\acute{S}_q)$ and $\\grave{\\mathcal {T}}[x_q] = \\mathcal {T}(\\grave{S}_q)$ to be any derivation trees of $\\acute{S}_q$ and $\\grave{S}_q$ that are defined on the same node set as $\\check{\\mathcal {T}}_q$ .",
"Recall that $x_q$ is the $q$ th leaf of $\\hat{\\mathcal {T}}$ .",
"One can think of the $\\acute{\\mathcal {T}}$ and $\\grave{\\mathcal {T}}$ derivation trees as filling in the gaps between wing nodes and quills.",
"Suppose $v$ were a leaf in some derivation tree $\\mathcal {T}$ whose block $\\mathcal {B}(v)$ contains a symbol $a$ .",
"By definition $\\operatorname{qu}_{|a}(v)$ is a child of $\\operatorname{wi}_{|a}(v)$ in $\\mathcal {T}_{|a}$ .",
"If $v$ were a dove (or hawk) in $\\mathcal {T}_{|a}$ then $\\operatorname{qu}_{|a}(v)$ would be identified with a leaf of $\\acute{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)]$ (or a leaf of $\\grave{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)]$ ) whose block contains $a$ .",
"However, within $\\acute{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)]$ (or $\\grave{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)]$ ), $\\operatorname{qu}_{|a}(v)$ could be a dove or hawk, feather or non-feather, wingtip or non-wingtip.",
"The new concept needed to tightly bound order-5 DS sequences is that of a double-feather.",
"See Figure REF .",
"Definition 7.1 Let $\\lbrace \\mathcal {T}\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}[u],\\grave{\\mathcal {T}}[u]\\rbrace _{u\\in \\mathcal {T}}$ be a derivation tree ensemble.",
"Let $v$ be a dove leaf in $\\mathcal {T}$ for which $a\\in \\mathcal {B}(v)$ , and let $\\acute{\\mathcal {T}}= \\acute{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)]$ .",
"We call $v$ a double-feather in $\\mathcal {T}_{|a}$ if it is a feather (that is, it is the rightmost descendant of $\\operatorname{qu}_{|a}(v)$ in $\\mathcal {T}_{|a}$ ) and $\\operatorname{qu}_{|a}(v)$ is either a dove feather or hawk wingtip in $\\acute{\\mathcal {T}}_{|a}$ .",
"The definition of double-feather is symmetric when $v$ is a hawk, that is, we substitute $\\grave{\\mathcal {T}}$ for $\\acute{\\mathcal {T}}$ and swap the roles of left and right, dove and hawk.",
"Figure: Left: a dove feather vv in 𝒯 |a \\mathcal {T}_{|a}.",
"The quill and wingnode of vv are indicated.",
"Right: the derivation tree 𝒯 ´ |a \\acute{\\mathcal {T}}_{|a} where 𝒯 ´=𝒯 ´[wi |a (v)]\\acute{\\mathcal {T}}=\\acute{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)].Within 𝒯 ´ |a \\acute{\\mathcal {T}}_{|a} the leaf v ' =qu |a (v)v^{\\prime }=\\operatorname{qu}_{|a}(v) has its own wing node wi |a ' (v ' )\\operatorname{wi}_{|a}^{\\prime }(v^{\\prime }), quill qu |a ' (v ' )\\operatorname{qu}_{|a}^{\\prime }(v^{\\prime }),and so on.",
"By virtue of v ' v^{\\prime } being a dove feather in 𝒯 ´ |a \\acute{\\mathcal {T}}_{|a}, vv is a double-feather in 𝒯 |a \\mathcal {T}_{|a}.As with the term feather, double-feather is used to refer to leaf nodes in some derivation tree $\\mathcal {T}_{|a}$ and the corresponding occurrences of $a$ in the underlying sequence $S$ .",
"Lemma REF is a more refined version of Lemma REF .",
"Lemma 7.2 Let $\\lbrace \\mathcal {T}\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}[u], \\grave{\\mathcal {T}}[u]\\rbrace _{u\\in \\mathcal {T}}$ be a derivation tree ensemble for some sequence $S$ .",
"Suppose that $v\\in \\mathcal {T}$ is a leaf and $a,b$ are symbols in a block $\\mathcal {B}(v)$ of $S$ .",
"If the following three criteria are satisfied then $a$ and $b$ are nested in $\\mathcal {B}(v)$ .",
"$v$ is not a wingtip in either $\\mathcal {T}_{|a}$ or $\\mathcal {T}_{|b}$ .",
"$v$ is not a double-feather in either $\\mathcal {T}_{|a}$ or $\\mathcal {T}_{|b}$ .",
"$v$ is a dove in both $\\mathcal {T}_{|a}$ and $\\mathcal {T}_{|b}$ or a hawk in both $\\mathcal {T}_{|a}$ and $\\mathcal {T}_{|b}$ .",
"Proof: We assume the claim is false, that $a$ and $b$ are interleaved in $\\mathcal {B}(v)$ .",
"Without loss of generality, we can assume the following additional criteria.",
"$\\operatorname{cr}_{|b}$ is equal to or strictly ancestral to $\\operatorname{cr}_{|a}$ .",
"$v$ is a dove in both $\\mathcal {T}_{|a}$ and $\\mathcal {T}_{|b}$ .",
"$\\mathcal {T}$ is the smallest derivation tree for which Criteria (REF –REF ) hold and where $a$ and $b$ are interleaved in $\\mathcal {B}(v)$ .",
"By Criterion (REF ) the leftmost descendent of $\\operatorname{wi}_{|a}(v)$ in $\\mathcal {T}_{|a}$ is $\\operatorname{wt}_{|a}(v)$ .",
"Let $u$ be its rightmost descendant.",
"Criteria (REF ,REF ) imply that $v,\\operatorname{wt}_{|a}(v)$ , and $u$ are distinct nodes.",
"Criterion (REF ) states that $v$ is distinct from $\\operatorname{wt}_{|a}(v)$ .",
"Consider the derivation tree $\\acute{\\mathcal {T}}= \\acute{\\mathcal {T}}[\\operatorname{wi}_{|a}(v)]$ .",
"The node $u$ is the rightmost descendant (in $\\mathcal {T}_{|a}$ ) of the hawk wingtip of $\\acute{\\mathcal {T}}_{|a}$ .",
"It is therefore a double-feather, and distinct from $v$ , by Criterion (REF ).",
"Partition the sequence outside of $\\mathcal {B}(v)$ into four intervals, namely $I_1$ : everything preceding the $a$ in $\\mathcal {B}(\\operatorname{wt}_{|a}(v))$ , $I_2$ : everything from the end of $I_1$ to $\\mathcal {B}(v)$ , $I_3$ : everything from $\\mathcal {B}(v)$ to the $a$ in $\\mathcal {B}(u)$ , and $I_4$ : everything following $I_3$ .",
"If $a$ and $b$ are not nested in $\\mathcal {B}(v)$ then all remaining $b$ s lie exclusively in $I_1$ and $I_3$ or exclusively in $I_2$ and $I_4$ .",
"We claim $I_1$ and $I_3$ contain no occurrences of $b$ .",
"If the contrary were true, that all occurrences of $b$ outside $\\mathcal {B}(v)$ were in $I_1$ and $I_3$ , then $I_3$ would contain $\\operatorname{rwt}_{|b}$ , which is distinct from $v$ according to Criterion (REF ).",
"By Criterion (REF ) $\\operatorname{cr}_{|b}$ is ancestral to $\\operatorname{cr}_{|a}$ , which is strictly ancestral to $\\operatorname{wi}_{|a}(v)$ , which is ancestral to both $v$ and $\\operatorname{rwt}_{|b}$ .",
"This implies that $v$ and $\\operatorname{rwt}_{|b}$ descend from the same child of $\\operatorname{cr}_{|b}$ , namely, $\\operatorname{rhe}_{|b}$ , which violates Criterion (REF ).",
"Thus, $\\operatorname{wt}_{|b}(v)$ is the dove wingtip in $\\mathcal {T}_{|b}$ and lies in interval $I_2$ .",
"Define $v^{\\prime } = \\operatorname{qu}_{|b}(v)$ to be $v$ 's quill in $\\mathcal {T}_{|b}$ .",
"It is not necessarily the case that $v^{\\prime }$ is distinct from $v$ , though we can claim that $v^{\\prime }$ is a strict descendant of $\\operatorname{wi}_{|a}(v)$ and $v=\\operatorname{fe}_{|b}(v)$ is a feather.",
"By definition quills are not wing nodes, so $v^{\\prime }$ cannot be ancestral to $\\operatorname{wt}_{|b}(v)$ .",
"However, if $v^{\\prime }$ were ancestral to $\\operatorname{wi}_{|a}(v)$ it would be ancestral to $\\operatorname{wt}_{|b}(v)$ as well, a contradiction.",
"If $v$ were distinct from $\\operatorname{fe}_{|b}(v)$ , that is, if $v$ were not the rightmost descendant of $v^{\\prime }$ in $\\mathcal {T}_{|b}$ , then $\\operatorname{fe}_{|b}(v)$ must, by Inference (REF ), lie in interval $I_4$ .",
"Since $\\overline{\\operatorname{wt}}_{|a}(v)$ is not a descendant of $\\operatorname{wi}_{|a}(v)$ it must lie to the right of $\\operatorname{fe}_{|b}(v)$ in $I_4$ .",
"However, this arrangement of nodes (namely $\\operatorname{wt}_{|a}(v),\\operatorname{wt}_{|b}(v),v,\\operatorname{fe}_{|b}(v),\\overline{\\operatorname{wt}}_{|a}(v)$ , where $\\operatorname{wt}_{|a}(v)$ and $\\operatorname{wt}_{|b}(v)$ may be equal) shows that $a$ and $b$ are nested in $\\mathcal {B}(v)$ , hence $v=\\operatorname{fe}_{|b}(v)$ is a feather.",
"From Criterion (REF ), Inference (REF ), and Lemma REF we shall infer that $v=v^{\\prime }$ .",
"Suppose $v\\ne v^{\\prime }$ .",
"Lemma REF implies that $a\\in \\mathcal {B}(v^{\\prime })$ , as witnessed by the nodes $v,v^{\\prime },\\operatorname{cr}_{|a},\\operatorname{cr}_{|b}$ on a leaf-to-root path.",
"This means that somewhere in the inductive construction of $\\mathcal {T}$ we encountered a derivation tree $\\mathcal {T}^0$ containing both $\\operatorname{cr}_{|a}$ and $\\operatorname{cr}_{|b}$ , whose leaves are at the level of $v^{\\prime }$ .",
"However, $\\mathcal {T}^0$ and $v^{\\prime }$ satisfy the conditions of the lemma, namely $a,b\\in \\mathcal {B}(v^{\\prime })$ and $v^{\\prime }$ is neither a wingtip nor double-feather nor hawk in both $\\mathcal {T}^0_{|a}$ and $\\mathcal {T}^0_{|b}$ .",
"Since $\\mathcal {T}^0$ is smaller than $\\mathcal {T}$ , Criterion (REF ) implies that $a$ and $b$ are nested in $\\mathcal {B}(v^{\\prime })$ with respect to $\\mathcal {T}^0$ , which then implies that they are nested in $\\mathcal {B}(v)$ with respect to $\\mathcal {T}$ as well.",
"This contradicts the hypothesis that $a$ and $b$ are interleaved in $\\mathcal {B}(v)$ .",
"Figure: Here v ' =qu |b (v)v^{\\prime }=\\operatorname{qu}_{|b}(v) is vv's quill in 𝒯 |b \\mathcal {T}_{|b}, and therefore a leaf of 𝒯 ´=𝒯 ´[wi |b (v)]\\acute{\\mathcal {T}}= \\acute{\\mathcal {T}}[\\operatorname{wi}_{|b}(v)].The tree 𝒯 ´ |b \\acute{\\mathcal {T}}_{|b} is rooted at the crown cr |b ' \\operatorname{cr}_{|b}^{\\prime }.Looking within 𝒯 ´ |b \\acute{\\mathcal {T}}_{|b}, v ' v^{\\prime } has its own quill qu |b ' (v ' )\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime }) and feather fe |b ' (v ' )\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime }).Since 𝒯 ´\\acute{\\mathcal {T}} is topologically identical to a corresponding subtree of 𝒯\\mathcal {T}, we can talk sensibly aboutnodes in one tree being ancestral to nodes in the other.",
"For example, it is deduced thatcr |b ' \\operatorname{cr}_{|b}^{\\prime } (in 𝒯 ´\\acute{\\mathcal {T}}) is ancestral to cr |a \\operatorname{cr}_{|a} (in 𝒯\\mathcal {T})and that wi |a (v)\\operatorname{wi}_{|a}(v) (in 𝒯\\mathcal {T}) is a strict ancestor of qu |b ' (v ' )\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime }) (in 𝒯 ´\\acute{\\mathcal {T}}).By definition $v^{\\prime }$ is the child of $\\operatorname{wi}_{|b}(v)$ in $\\mathcal {T}_{|b}$ but we have yet to deduce where $\\operatorname{wi}_{|b}(v)$ is relative to other nodes.",
"$\\operatorname{wi}_{|b}(v)$ is ancestral to $\\operatorname{cr}_{|a}$ .",
"Criterion (REF ) and Inference (REF ) imply that some $b$ appears in interval $I_4$ .",
"All such $b$ s must appear after $\\overline{\\operatorname{wt}}_{|a}(v)$ for otherwise $a$ and $b$ would be nested in $\\mathcal {B}(v)$ .",
"By Criterion (REF ) $v$ cannot be the rightmost descendant of $\\operatorname{wi}_{|b}(v)$ , which is a double-feather.",
"Thus, the rightmost descendant of $\\operatorname{wi}_{|b}(v)$ appears after the $a$ in $\\overline{\\operatorname{wt}}_{|a}(v)$ , implying that $\\operatorname{wi}_{|b}(v)$ is ancestral to $\\operatorname{cr}_{|a}$ .",
"Consider the derivation tree $\\acute{\\mathcal {T}}= \\acute{\\mathcal {T}}[\\operatorname{wi}_{|b}(v)]$ , the leaf-level of which coincides with the leaf-level of $\\mathcal {T}$ since $\\operatorname{qu}_{|b}(v) = v^{\\prime } = v$ .",
"Let $\\operatorname{cr}_{|b}^{\\prime }$ be $b$ 's crown in $\\acute{\\mathcal {T}}_{|b}$ , and let $\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ and $\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime })$ be the quill and feather of $v^{\\prime }$ in $\\acute{\\mathcal {T}}_{|b}$ .Observe that $\\operatorname{qu}^{\\prime }_{|b}(v^{\\prime })$ lies strictly between $v^{\\prime }=\\operatorname{qu}_{|b}(v)$ and $\\operatorname{wi}_{|b}(v)$ in $\\mathcal {T}$ , so its block contains $b$ only with respect to $\\acute{\\mathcal {T}}$ , not $\\mathcal {T}$ .",
"In contrast, $b$ appears in $\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime })$ 's block in both $\\mathcal {T}$ and $\\acute{\\mathcal {T}}$ .",
"We deduce the following variants of Criteria (REF ,REF ) and Inference (REF ).",
"$\\operatorname{cr}_{|b}^{\\prime }$ is ancestral to $\\operatorname{cr}_{|a}$ .",
"$v^{\\prime }$ is a dove in $\\acute{\\mathcal {T}}_{|b}$ .",
"$\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ is a strict descendant of $\\operatorname{wi}_{|a}(v)$ and strict ancestor of $u$ .",
"Furthermore, $\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime })$ lies between $u$ and $\\overline{\\operatorname{wt}}_{|a}(v)$ .",
"Inference (REF ) is just a restatement of Inference (REF ) since all $b$ s descending from $\\operatorname{wi}_{|b}(v)$ are also descendants of $\\operatorname{cr}_{|b}^{\\prime }$ .The assertion that $\\operatorname{cr}_{|b}^{\\prime }$ is ancestral to $\\operatorname{cr}_{|a}$ is only well defined if $\\acute{\\mathcal {T}}[\\operatorname{wi}_{|b}(v)]$ can be regarded as a subtree of $\\mathcal {T}$ but populated with different blocks.",
"This is why we do not permit corresponding derivation trees to have different structure.",
"Inference (REF ) follows from the fact that $\\operatorname{wt}_{|b}(v)$ is also the dove wingtip in $\\acute{\\mathcal {T}}_{|b}$ and that the least common ancestor of $\\operatorname{wt}_{|b}(v)$ and $v$ is a descendant of $\\operatorname{wi}_{|a}(v)$ and therefore a strict descendant of $\\operatorname{cr}_{|b}^{\\prime }$ .",
"We now turn to Inference (REF ).",
"By Inference (REF ) the quill $\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ must be an ancestor of $v^{\\prime }$ but not $\\operatorname{wt}_{|b}(v)$ , so it must be a strict descendant of $\\operatorname{wi}_{|a}(v)$ .",
"If $v^{\\prime }$ were the rightmost descendant of $\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ in $\\acute{\\mathcal {T}}_{|b}$ , then $v^{\\prime }=\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime })$ and $v$ would be a double-feather in $\\mathcal {T}$ , contrary to Criterion (REF ).",
"Since $I_3$ is $b$ -free, it must be that $\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ is a strict descendant of $\\operatorname{wi}_{|a}(v)$ , a strict ancestor of $u$ , and that $\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime })$ lies between $u$ and $\\overline{\\operatorname{wt}}_{|a}(v)$ .",
"The blocks $\\operatorname{wt}_{|a}(v),\\operatorname{wt}_{|b}(v),v,\\operatorname{fe}_{|b}^{\\prime }(v^{\\prime }),$ and $\\overline{\\operatorname{wt}}_{|a}(v)$ certify that $a$ and $b$ are nested in $\\mathcal {B}(v)$ , a contradicting the hypothesis that they are not.",
"$\\Box $ Remark 7.3 Whereas Lemma REF implicitly partitioned occurrences of global symbols into four categories: dove wingtips, hawk wingtips, feathers, and all remaining non-feathers, Lemma REF further distinguishes dove non-feathers and hawk non-feathers.",
"The reason for this is rather technical.",
"If Criterion (REF ) were dropped and $v$ were a dove in $\\mathcal {T}_{|a}$ and a hawk in $\\mathcal {T}_{|b}$ , we could deduce that $\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ is a strict descendant of $\\operatorname{wi}_{|a}(v)$ and strict ancestor of $\\operatorname{wt}_{|a}(v)$ .",
"However, we could not deduce that $a\\in \\mathcal {B}(\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime }))$ (that is, $\\operatorname{qu}_{|b}^{\\prime }(v^{\\prime })$ 's block in $\\mathcal {T}$ ) since Lemma REF only applies to symbols and blocks that exist in the same derivation tree.",
"Note that $\\acute{\\mathcal {T}}[\\operatorname{wi}_{|b}(v)]$ can be regarded as being superimposed on the subtree of $\\mathcal {T}$ rooted at $\\operatorname{wi}_{|b}(v)$ , but they are not identical derivation trees.",
"As in Section it is useful to define and reason about optimal derivation trees.",
"For technical reasons it is convenient to only permit uniform block partitions with widths that are powers of two.",
"Definition 7.4 (Permissible block partitions) Let $S$ be a blocked sequence and $m=\\llbracket S \\rrbracket $ be its block count.",
"A block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ is permissible if $m_q = 2^r$ for all $q < \\hat{m}$ , where $\\hat{m}= \\lceil m/2^r \\rceil $ and $r\\ge 1$ is an integer.",
"A derivation tree $\\mathcal {T}(S)$ is permissible if it was defined using only permissible block partitions.",
"Note that if $\\mathcal {T}_0$ and $\\mathcal {T}_1$ are two permissible derivation trees for some sequence with $m$ blocks, they must have exactly the same structure, though their nodes may be populated with different blocks.",
"In particular, both are binary trees with height $\\lceil \\log m \\rceil $ , where one-child nodes may only exist on the rightmost root-to-leaf path if $m$ is not a power of 2."
],
[
"Recurrences for Fifth-Order Sequences",
"Lemma REF provides us with new criteria for nestedness.",
"In order to write a new recurrence for $\\lambda _{5}$ we need to know how many double-feathers an order-5 DS sequence can have, which depends on the number of dove and hawk feathers in an order-4 DS sequence.",
"Definition 7.5 (Optimal derivation trees) When $S$ is an order-4 DS sequence let $\\mathcal {T}^*(S)$ denote the permissible derivation tree that minimizes the number of feathers of a given type (dove or hawk).",
"When $S$ is an order-5 DS sequence let $\\mathcal {E}^*(S) = \\lbrace \\mathcal {T}^*(S)\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}^*[u],\\grave{\\mathcal {T}}^*[u]\\rbrace _{u\\in \\mathcal {T}^*(S)}$ denote the ensemble of permissible derivation trees that minimize the number of double-feathers in $S$ .",
"Define $\\Phi ^{\\prime }(n,m)$ to be the maximum number of feathers of one type (dove or hawk) in an order-4 DS sequence $S$ with respect to $\\mathcal {T}^*(S)$ , where $\\Vert S\\Vert =n$ and $\\llbracket S \\rrbracket =m$ .",
"Define $\\Phi ^{\\prime \\prime }(n,m)$ to be the maximum number of double-feathers (of both types) in an order-5 DS sequence with respect to the ensemble $\\mathcal {E}^*(S)$ .",
"Recurrence 7.6 Let $m$ and $n$ be the block count and alphabet size.",
"For any permissible block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ and any alphabet partition $\\lbrace \\hat{n}\\rbrace \\cup \\lbrace \\check{n}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , we have $\\Phi ^{\\prime }(n,m) &\\;=\\; \\Phi ^{\\prime \\prime }(n,m) = 0 & \\mbox{when $m\\le 2$}\\\\\\Phi ^{\\prime }(n,m) &\\;\\le \\; \\sum _{q=1}^{\\hat{m}} \\Phi ^{\\prime }(\\check{n}_q,m_q) \\, + \\, \\Phi ^{\\prime }(\\hat{n},\\hat{m})\\, + \\, \\lambda _{3}(\\hat{n},m) - \\hat{n}\\\\\\Phi ^{\\prime \\prime }(n,m) &\\;\\le \\; \\sum _{q=1}^{\\hat{m}} \\Phi ^{\\prime \\prime }(\\check{n}_q,m_q) \\, + \\, \\Phi ^{\\prime \\prime }(\\hat{n},\\hat{m})\\, + \\, 2(\\Phi ^{\\prime }(\\hat{n},m) + \\hat{n}) & \\mbox{\\raisebox {7 mm}[0mm][0mm]{}}$ Proof: Consider an order-4 DS sequence $S$ .",
"Let $\\hat{\\mathcal {T}}^*$ and $\\lbrace \\check{\\mathcal {T}}_q^*\\rbrace _q$ be the optimal derivation trees of the global sequence $\\hat{S}^{\\prime }$ and local sequences $\\lbrace \\check{S}_q\\rbrace $ , and let $\\mathcal {T}$ be their composition.",
"The number of dove feathers of local symbols is at most $\\sum _{q} \\Phi ^{\\prime }(\\check{n}_q,m_q)$ .",
"Every global dove feather in $\\hat{S}$ is either (i) the rightmost child of a dove feather in $\\hat{S}^{\\prime }$ , or (ii) a child of a left wingtip in $\\acute{S}^{\\prime }$ , excluding the leftmost such child, which is also a left wingtip in $\\hat{S}$ .",
"Category (i) is counted by $\\Phi ^{\\prime }(\\hat{n},\\hat{m})$ and Category (ii) is counted by $\\lambda _{3}(\\hat{n},m)-\\hat{n}$ since $\\acute{S}$ (the children of dove wingtips) is an order-3 DS sequence over an $\\hat{n}$ -letter alphabet.",
"A symmetric analysis applies to hawk feathers, reversing the roles of left and right.",
"The analysis of $\\Phi ^{\\prime \\prime }(n,m)$ when $S$ is an order-5 sequence is similar.",
"After the block partition is selected, construct the optimal derivation tree ensemble $\\mathcal {E}^*(\\hat{S}^{\\prime }) = \\lbrace \\hat{\\mathcal {T}}^*\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}^*[u],\\grave{\\mathcal {T}}^*[u]\\rbrace _{u\\in \\hat{\\mathcal {T}}^*}$ for $\\hat{S}^{\\prime }$ and, separately, the optimal derivation tree ensemble $\\mathcal {E}_q^*(\\check{S}_q) = \\lbrace \\check{\\mathcal {T}}_q^*\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}^*[u],\\grave{\\mathcal {T}}^*[u]\\rbrace _{u\\in \\check{\\mathcal {T}}^*_q}$ for each $\\check{S}_q$ separately.",
"These ensembles are composed to form $\\mathcal {E}(S) = \\lbrace \\mathcal {T}\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}^*[u],\\grave{\\mathcal {T}}^*[u]\\rbrace _{u\\in \\mathcal {T}}$ in the obvious way.",
"As usual, $\\mathcal {T}= \\mathcal {T}(S)$ is the composition of $\\hat{\\mathcal {T}}^*$ and the $\\lbrace \\check{\\mathcal {T}}^*_q\\rbrace $ .",
"The only nodes $u\\in \\mathcal {T}$ whose derivation trees $\\acute{\\mathcal {T}}^*[u]$ and $\\grave{\\mathcal {T}}^*[u]$ are not well defined (that is, they are not already included in $\\mathcal {E}^*(\\hat{S}^{\\prime })$ or the $\\lbrace \\mathcal {E}^*(\\check{S}_q)\\rbrace $ ) are those at the leaf level of $\\hat{\\mathcal {T}}^*$ in $\\mathcal {T}$ .",
"Define $\\acute{\\mathcal {T}}^*[x_q]$ and $\\grave{\\mathcal {T}}^*[x_q]$ to be the optimal derivation trees for $\\acute{S}_q$ and $\\grave{S}_q$ , respectively.",
"Now let us argue that the recurrence correctly bounds the number of double-feathers in $S$ with respect to $\\mathcal {E}(S)$ .",
"The summation $\\sum _q \\Phi ^{\\prime \\prime }(\\check{n}_q,m_q)$ counts double-feathers of local symbols.",
"Every occurrence of a global double-feather in $S$ is either (i) the rightmost child of a dove double-feather in $\\hat{S}^{\\prime }$ or the leftmost child of a hawk double-feather in $\\hat{S}^{\\prime }$ (with respect to $\\mathcal {E}^*(\\hat{S}^{\\prime })$ ), (ii) a dove feather in $\\acute{S}_q$ or hawk feather in $\\grave{S}_q$ , for some $q$ , with respect to $\\acute{\\mathcal {T}}^*[x_q]$ or $\\grave{\\mathcal {T}}^*[x_q]$ , or (iii) a hawk wingtip in $\\acute{S}$ or a dove wingtip in $\\grave{S}$ .",
"Category (i) is counted by $\\Phi ^{\\prime \\prime }(\\hat{n},\\hat{m})$ , Category (ii) by $\\sum _q \\Big [ \\Phi ^{\\prime }(\\acute{n}_q,m_q) + \\Phi ^{\\prime }(\\grave{n}_q,m_q) \\Big ] \\le 2\\cdot \\Phi ^{\\prime }(\\hat{n},m)$ , and Category (iii) by $2\\hat{n}$ .",
"$\\Box $ Recurrence 7.7 Let $m$ and $n$ be the block count and alphabet size parameters.",
"For any permissible block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ and alphabet partition $\\lbrace \\hat{n}\\rbrace \\cup \\lbrace \\check{n}_q\\rbrace _{1\\le q\\le \\hat{m}}$ , $\\lambda _{5}(n,m) &\\;\\le \\; \\sum _{q=1}^{\\hat{m}} \\lambda _{5}(\\check{n}_q,m_q)\\,+\\, 2\\cdot \\lambda _{4}(\\hat{n},m) \\,+\\, \\lambda _{3}(\\Phi ^{\\prime \\prime }(\\hat{n},\\hat{m}),m) \\,+\\,\\lambda _{2}(\\lambda _{5}(\\hat{n},\\hat{m}),2m-1)$ Proof: We adopt the usual notation for an order-5 sequence $S$ with the following additions and modifications.",
"Let $\\tilde{S}^{\\prime },\\dot{S}^{\\prime },$ and $\\ddot{S}^{\\prime }$ be the subsequences of $\\hat{S}^{\\prime }$ consisting of, respectively, double-feathers, non-double-feather doves, and non-double-feather hawks, all of which exclude wingtips, and let $\\tilde{S},\\dot{S},$ and $\\ddot{S}$ be the subsequences of $\\hat{S}$ made up of their children.",
"Recall that $\\acute{S}$ and $\\grave{S}$ are the children of dove and hawk wingtips in $\\hat{S}^{\\prime }$ .",
"The contribution of local symbols to $|S|$ is bounded by $\\sum _{q} \\lambda _{5}(\\check{n}_q,m_q)$ .",
"The global sequence $\\hat{S}$ is the union of five subsequences $\\acute{S}, \\grave{S}, \\tilde{S}, \\dot{S},$ and $\\ddot{S}$ .",
"Both $\\acute{S}$ and $\\grave{S}$ are order-4 sequences, so $|\\acute{S}|+|\\grave{S}| \\le 2\\cdot \\lambda _{4}(\\hat{n},m)$ .",
"Since double-feathers are all middle occurrences in $\\hat{S}^{\\prime }$ , $\\tilde{S}$ is obtained by substituting for each block in $\\tilde{S}^{\\prime }$ an order-3 DS sequence, so $|\\tilde{S}| \\le \\lambda _{3}(|\\tilde{S}^{\\prime }|,m) \\le \\lambda _{3}(\\Phi ^{\\prime \\prime }(\\hat{n},\\hat{m}),m)$ .",
"According to Lemma REF , the blocks of $\\dot{S}^{\\prime }$ consist of mutually nested symbols.",
"The argument from Recurrence REF shows that $\\dot{S}$ is obtained by substituting an order-2 sequence for each block in $\\dot{S}^{\\prime }$ .",
"The same is true for $\\ddot{S}$ as well, so $|\\dot{S}| + |\\ddot{S}| &\\le \\lambda _{2}(|\\dot{S}^{\\prime }|,m) + \\lambda _{2}(|\\ddot{S}^{\\prime }|,m)\\\\&\\le \\lambda _{2}(|\\dot{S}^{\\prime }| + |\\ddot{S}^{\\prime }|,2m-1) & \\mbox{\\lbrace superadditivity of $\\lambda _{2}$\\rbrace }\\\\&< \\lambda _{2}(|\\hat{S}^{\\prime }|,2m-1)\\\\&\\le \\lambda _{2}(\\lambda _{5}(\\hat{n},\\hat{m}),2m-1)\\\\$ The last inequalities follow from fact that $\\dot{S}^{\\prime }$ and $\\ddot{S}^{\\prime }$ are disjoint subsequences of $\\hat{S}^{\\prime }$ , which is an order-5 DS sequence.",
"$\\Box $ Recurrences REF and REF allow us to find closed-form bounds on the number of feathers and double-feathers, and on the length of order-5 DS sequences.",
"Refer to Appendix for proof of Lemma REF .",
"Lemma 7.8 Let $n$ and $m$ be the alphabet size and block count.",
"After a parameter $i\\ge 1$ is chosen let $j\\ge 1$ be minimum such that $m \\le a_{i,j}^c$ , where $c=3$ is fixed.",
"We have the following upper bounds on $\\Phi ^{\\prime }(n,m), \\Phi ^{\\prime \\prime }(n,m),$ and $\\lambda _{5}(n,m)$ .",
"$\\Phi ^{\\prime }(n,m) &\\le \\nu _i^{\\prime }{\\left( n + (cj)^2(m-1) \\right)} & \\mbox{where $\\nu _i^{\\prime } = 3{i+1\\atopwithdelims ()2} + 3$}\\\\\\Phi ^{\\prime \\prime }(n,m) &\\le \\nu _i^{\\prime \\prime }{\\left( n + (cj)^3(m-1) \\right)} & \\mbox{where $\\nu _i^{\\prime \\prime } = 6{i+2\\atopwithdelims ()3} + 8i$}\\\\\\lambda _{5}(n,m) &\\le \\mu _{5,i}{\\left( n + (cj)^3(m-1) \\right)} & \\mbox{where $\\mu _{5,i} = i2^{i+7}$}\\\\$"
],
[
"Blocked versus 2-Sparse Order-5 Sequences",
"Lemma REF states that for any $i$ , $\\lambda _{5}(n,m) < \\mu _{5,i}(n + (3j_i)^3m)$ where $j_i$ is minimum such that $m\\le a_{i,j_i}^3$ .",
"Choose $\\iota \\ge 1$ be minimum such that $(3j_\\iota )^3 \\le \\max \\lbrace n̑{m}, (3\\cdot 3)^3\\rbrace $ .",
"One can show that $\\iota = \\alpha (n,m) + O(1)$ , implying that $\\lambda _{5}(n,m) = O((n+m)\\mu _{5,\\iota }) = O((n+m)\\alpha (n,m)2^{\\alpha (n,m)})$ , matching the construction from Section .",
"According to Lemma REF (REF ,REF ) $\\lambda _{5}(n) = O(\\alpha (\\alpha (n)))\\cdot \\lambda _{5}(n,3n-1)$ .",
"In this Section we present a more efficient reduction from 2-sparse, order-5 DS sequences to blocked order-5 sequences, thereby removing the extra $\\alpha (\\alpha (n))$ factor.",
"Theorem 7.9 $\\lambda _{5}(n) = O(n\\alpha (n)2^{\\alpha (n)})$ and $\\lambda _{5}(n,m) = m+O(n\\alpha (n,m)2^{\\alpha (n,m)})$ .",
"Proof: The second bound is asymptotically the same as $O((n+m)\\alpha (n,m)2^{\\alpha (n,m)})$ if $m=O(n)$ .",
"If not, we remove up to $m-1$ repeated symbols at block boundaries, yielding a 2-sparse, order-5 DS sequence.",
"Our remaining task is therefore to prove that $\\lambda _{5}(n) = O(n\\alpha (n)2^{\\alpha (n)})$ .",
"Let $S$ be a 2-sparse, order-5 DS sequence with $\\Vert S\\Vert =n$ .",
"Greedily partition $S$ into maximal order-3 DS sequences $S_1S_2\\cdots S_{m}$ .",
"According to Sharir's argument [73], $m\\le 2n-1$ .",
"See the proof of Lemma REF (REF ) in Appendix .",
"As usual, let $\\check{S},\\hat{S}\\prec S$ be the subsequences of local and global symbols, and let $\\hat{S}^{\\prime }$ be derived by contracting each interval to a single block.",
"The number of global symbols is $\\hat{n}= \\Vert \\hat{S}\\Vert $ .",
"In contrast to the situations we considered earlier, $\\check{S}$ and $\\hat{S}$ are neither 2-sparse nor partitioned into blocks.",
"Let $\\mathcal {E}^*(\\hat{S}^{\\prime }) = \\lbrace \\mathcal {T}^*(\\hat{S}^{\\prime })\\rbrace \\cup \\lbrace \\acute{\\mathcal {T}}^*[u],\\grave{\\mathcal {T}}^*[u]\\rbrace _{u\\in \\mathcal {T}^*(\\hat{S}^{\\prime })}$ be the optimal derivation tree ensemble for $\\hat{S}^{\\prime }$ .",
"This ensemble categorizes all occurrences in $\\hat{S}^{\\prime }$ as (i) double-feathers, (ii) non-double-feather doves, (iii) non-double-feather hawks, (iv) dove wingtips, or (v) hawk wingtips.",
"Furthermore, occurrences in $\\hat{S}$ inherit the category of their corresponding occurrence in $\\hat{S}^{\\prime }$ .",
"Let $\\tilde{S}^{\\prime },\\dot{S}^{\\prime },$ and $\\ddot{S}^{\\prime }$ be the subsequences of $\\hat{S}^{\\prime }$ in categories (i–iii) and let $\\tilde{S},\\dot{S},\\ddot{S},\\acute{S},$ and $\\grave{S}$ be the subsequences of $\\hat{S}$ in categories (i–v), none of which are necessarily 2-sparse.",
"Define $\\tilde{S}^*,\\dot{S}^*,\\ddot{S}^*,\\acute{S}^*,$ and $\\grave{S}^*$ to be their maximal length 2-sparse subsequences, and define $\\check{S}^*$ to be the maximal length 2-sparse subsequence of $\\check{S}$ .",
"Lemma REF (REF ) and the arguments from Recurrence REF imply that $|\\tilde{S}^*| + |\\dot{S}^*| + |\\ddot{S}^*| + |\\acute{S}^*| + |\\grave{S}^*|&\\le \\lambda _{3}(\\varphi ,2\\varphi -1) + 2\\cdot \\lambda _{5}(\\hat{n},2n-1) + 4\\cdot \\lambda _{4}(\\hat{n},2\\hat{n}-1)\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{7 mm}\\mbox{where }\\; \\varphi &= \\Phi ^{\\prime \\prime }(\\hat{n},2n-1)\\nonumber $ The sequence $\\tilde{S}^*$ is obtained by substituting for each block in $\\tilde{S}^{\\prime }$ a 2-sparse, order-3 DS sequence.",
"Since $|\\tilde{S}^*| \\le \\varphi $ , by the superadditivity of $\\lambda _{3}$ we have $|\\tilde{S}^*| \\le \\lambda _{3}(\\varphi )$ , which is at most $\\lambda _{3}(\\varphi ,2\\varphi -1)$ by Lemma REF (REF ).",
"By the same reasoning, $|\\acute{S}^*|$ and $|\\grave{S}^*|$ are each at most $\\lambda _{4}(\\hat{n}) \\le 2\\cdot \\lambda _{4}(\\hat{n},2\\hat{n}-1)$ and $|\\dot{S}^*| + |\\ddot{S}^*| \\le \\lambda _{2}(|\\dot{S}^{\\prime }|+ |\\ddot{S}^{\\prime }|) < \\lambda _{2}(|\\hat{S}^{\\prime }|) < 2\\cdot \\lambda _{5}(\\hat{n},2n-1)$ .",
"We can also conclude that $|\\check{S}^*| \\le \\lambda _{3}(n-\\hat{n}) \\le \\lambda _{3}(n-\\hat{n},2(n-\\hat{n})-1)$ .",
"In bounding various sequences above, the second argument of $\\lambda _{s}$ and $\\Phi ^{\\prime \\prime }$ is never more than $2\\cdot \\varphi $ .",
"Choose $\\iota $ to be minimal such that $2\\cdot \\varphi \\le a_{\\iota ,3}^3$ , so $j=3$ will be constant whenever we invoke Lemmas REF and REF with $s\\le 5$ and $c=3$ .",
"It is straightforward to show that $\\iota = \\alpha (n) + O(1)$ .",
"Observe that $S$ can be constructed by shuffling its six non-2-sparse constituent subsequences $\\tilde{S}$ , $\\dot{S}$ , $\\ddot{S}$ , $\\acute{S}$ , $\\grave{S}$ , $\\check{S}$ in some fashion that restores 2-sparseness.",
"In other words, there is a 1-1 map between positions in $S$ and positions in its six constituents, and a surjective map $\\psi $ from positions in $S$ to positions in its 2-sparse constituents $\\tilde{S}^*$ , $\\dot{S}^*$ , $\\ddot{S}^*$ , $\\acute{S}^*$ , $\\grave{S}^*$ , $\\check{S}^*$ .",
"Partition $S$ into intervals $T_1T_2\\cdots T_{\\lceil |S|/h \\rceil }$ , each with length $h = \\lceil {2} \\rceil = O(1)$ .",
"The image of $\\psi $ on two consecutive intervals $T_{p-1}$ and $T_{p}$ (where $p<\\lceil |S|/h \\rceil $ ) cannot be identical, for otherwise $T_{p-1} T_p$ would be a 2-sparse, order-5 DS sequence with length $2h > \\lambda _{5}(6)$ over a 6-letter alphabet, a contradiction.",
"Therefore, $|S| &\\le \\makebox{[}0mm][l]{h\\cdot (|\\check{S}^*|+|\\tilde{S}^*| + |\\dot{S}^*| + |\\ddot{S}^*| + |\\acute{S}^*| + |\\grave{S}^*|)}\\\\&= \\makebox{[}0mm][l]{h \\cdot n\\cdot O(\\mu _{3,\\iota } + \\mu _{3,\\iota }\\nu _{\\iota }^{\\prime \\prime } + \\mu _{5,\\iota } + \\mu _{4,\\iota })}\\\\&= O(n\\iota 2^\\iota ) & \\mbox{\\lbrace Since $\\mu _{3,\\iota } = O(\\iota ),\\; \\nu _{\\iota }^{\\prime \\prime } = O(\\iota ^3),\\; \\mu _{4,\\iota }=O(2^{\\iota })$, and$\\mu _{5,\\iota } = O(\\iota 2^{\\iota })$\\rbrace }\\\\&= O(n\\alpha (n)2^{\\alpha (n)}).$ $\\Box $"
],
[
"Discussion and Open Problems",
"Davenport-Schinzel sequences have been applied almost exclusively to problems in combinatorial and computational geometry, with only a smattering of applications in other areas.",
"For example, see [72], [9], [65], [66].",
"One explanation for this, which is undoubtedly true, is that there is a natural fit between geometric objects and their characterizations in terms of forbidden substructures.E.g., in general position two lines do not share two points, three spheres do not share three points, degree-$d$ polynomials do not have $d+1$ zeros, and so on.",
"An equally compelling explanation, in our opinion, is that DS sequences are simply underpublicized, and that the broader algorithms community is not used to analyzing algorithms and data structures with forbidden substructure arguments.",
"We are optimistic that with increased awareness of DS sequences and their generalizations (e.g., forbidden 0-1 matrices) the forbidden substructure method [66] will become a standard tool in every algorithms researcher's toolbox.",
"Our bounds on Davenport-Schinzel sequences are sharp for every order $s$ , leaving little room for improvement.That is, they cannot be expressed more tightly using a generic inverse-Ackermann function $\\alpha (n)$ .",
"See Remark REF .",
"However, there are many open problems on the geometric realizability of DS sequences and on various generalizations of DS sequences.",
"The most significant realizability result is due to Wiernik and Sharir [86], who proved that the lower envelope of $n$ line segments (that is, $n$ linear functions, each defined over a different interval) has complexity $\\Theta (\\lambda _{3}(n)) = \\Theta (n\\alpha (n))$ .",
"It is an open question whether this result can be generalized to degree-$s$ polynomials or polynomial segments.",
"In particular, it may be that the lower envelope of any set of $n$ degree-$s$ polynomials has complexity $O(n)$ , where $s$ only influences the leading constant.",
"Although our results do not address problems of geometric realizability, we suspect that modeling lower envelopes by derivation trees (rather than just sequences) will open up a new line of attack on these fundamental realizability problems.",
"There are several challenging open problems in the realm of generalized Davenport-Schinzel sequences, the foremost one being to characterize the set of all linear forbidden subsequences: those $\\sigma $ for which $\\operatorname{Ex}(\\sigma ,n)=O(n)$ [52], [68].",
"Linear forbidden subsequences and minimally nonlinear ones were exhibited by Adamec, Klazar, and Valtr [2], Klazar and Valtr [53], and Pettie [70], [68], [67], [69].",
"It is also an open problem to characterize minimally non-linear forbidden 0-1 matrices [36].",
"Though far from being solved, there has been significant progress on this problem in the last decade [60], [81], [35], [38], [67], [68]."
],
[
"Proof of Lemma ",
"Recall the four parts of Lemma REF .",
"Restatement of Lemma REF Let $\\gamma _s(n) : \\mathbb {N}\\rightarrow \\mathbb {N}$ be a non-decreasing function such that $\\lambda _{s}(n) \\le \\gamma _s(n)\\cdot n$ .",
"(Trivial) For $s\\ge 1$ , $\\lambda _{s}(n,m) \\le m-1 + \\lambda _{s}(n)$ .",
"(Sharir [73]) For $s\\ge 3$ , $\\lambda _{s}(n) \\le \\gamma _{s-2}(n)\\cdot \\lambda _{s}(n,2n-1)$ .",
"(This generalizes Hart and Sharir's proof [41] for $s=3$ .)",
"(Sharir [73]) For $s\\ge 2$ , $\\lambda _{s}(n) \\le \\gamma _{s-1}(n)\\cdot \\lambda _{s}(n,n)$ .",
"(New) For $s\\ge 3$ , $\\lambda _{s}(n) = \\gamma _{s-2}(\\gamma _s(n))\\cdot \\lambda _{s}(n,3n-1)$ .",
"Proof: Removing at most $m-1$ repeated symbols at block boundaries makes any sequence 2-sparse, which implies Part (REF ).",
"For Parts (REF ) and (REF ), consider the following method for greedily partitioning a 2-sparse, order-$s$ DS sequence $S$ with $\\Vert S\\Vert =n$ .",
"Write $S$ as $S_1S_2\\cdots S_m$ , where $S_1$ is the longest order-$(s-2)$ prefix of $S$ , $S_2$ is the longest order-$(s-2)$ prefix of the remainder of the sequence, and so on.",
"Each $S_q$ contains the first or last occurrence of some symbol, which implies $m\\le 2n-1$ since $S_1$ must contain the first occurrence of at least two symbols.",
"To see this, consider the symbol $b$ which caused the termination of $S_q$ , that is, $S_q$ has order $s-2$ but $S_qb$ contains an alternating subsequence $\\sigma _s = aba\\cdots ab$ or $ba\\cdots ab$ with length $s$ ; whether it starts with $a$ depends on the parity of $s$ .",
"If $S_q$ contained neither the first nor last occurrence of both $a$ and $b$ , $S$ would contain an alternating subsequence $\\sigma _{s+2}$ of length $s+2$ , a contradiction.",
"Obtain $S^{\\prime }$ from $S$ replacing each $S_q$ with a block containing exactly one occurrence of each symbol in $\\Sigma (S_q)$ .",
"Thus, $|S| = \\sum _{q=1}^m |S_q|&\\le \\sum _{q=1}^m \\gamma _{s-2}(\\Vert S_q\\Vert )\\cdot \\Vert S_q\\Vert & \\mbox{\\lbrace $S_q$ has order $s-2$, defn.~of $\\gamma _{s-2}$\\rbrace }\\\\&\\le \\gamma _{s-2}(n)\\cdot \\sum _{q=1}^m \\Vert S_q\\Vert & \\mbox{\\lbrace $\\gamma _{s-2}$ is non-decreasing\\rbrace }\\\\&= \\gamma _{s-2}(n) \\cdot |S^{\\prime }| \\le \\gamma _{s-2}(n)\\cdot \\lambda _{s}(n,m) & \\mbox{\\lbrace $S^{\\prime }\\prec S$ has order $s$\\rbrace }$ which proves Part (REF ).",
"Part (REF ) is proved in the same way except that we partition $S$ into order-$(s-1)$ DS sequences.",
"In this case each $S_q$ must contain the last occurrence of some symbol, so $m\\le n$ .",
"We turn now to Part (REF ).",
"Partition $S$ into order-$(s-2)$ sequences $S_1S_2\\cdots S_m$ as follows.",
"After $S_1,\\cdots ,S_{q-1}$ have been selected, let $S_{q}$ be the longest prefix of the remaining sequence that (i) has order $s-2$ and (ii) has length at most $\\gamma _s(n)$ .",
"The number of such sequences that were terminated due to (i) is at most $2n-1$ , by the same argument from Part (REF ).",
"The number terminated due to (ii) is at most $n$ since $|S| \\le \\gamma _s(n)\\cdot n$ , so $m \\le 3n-1$ .",
"Obtain an $m$ -block sequence $S^{\\prime }$ in the usual way, by replacing each $S_q$ with a block containing its alphabet.",
"Thus, $|S| = \\sum _{q=1}^m |S_q|&\\le \\sum _{q=1}^m \\gamma _{s-2}(\\Vert S_q\\Vert )\\cdot \\Vert S_q\\Vert & \\mbox{\\lbrace $S_q$ has order $s-2$, defn.~of $\\gamma _{s-2}$\\rbrace }\\\\&\\le \\gamma _{s-2}(\\gamma _s(n)) \\cdot \\sum _{q=1}^m \\Vert S_q\\Vert & \\mbox{\\lbrace $\\gamma _{s}$ is non-decreasing,$\\Vert S_q\\Vert \\le |S_q|\\le \\gamma _s(n)$\\rbrace }\\\\&= \\gamma _{s-2}(\\gamma _s(n)) \\cdot \\makebox{[}0mm][l]{|S^{\\prime }| \\le \\gamma _{s-2}(\\gamma _s(n))\\cdot \\lambda _{s}(n,m)} & \\mbox{\\lbrace $S^{\\prime }\\prec S$ has order $s$\\rbrace }$ $\\Box $ Note that while Part (REF ) is stronger than Part (REF ), it requires an upper bound on $\\gamma _s(n)$ to be applied, which is obtained by invoking Part (REF ).",
"In the end it does not matter precisely what $\\gamma _s(n)$ is.",
"Once $\\gamma _s(n)$ is known to be some primitive recursive function of $\\alpha (n)$ , it follows that $\\gamma _{s-2}(\\gamma _s(n)) = \\gamma _{s-2}(\\alpha (n)) + O(1)$ ."
],
[
"Proof of Lemma ",
"Recall our definition of Ackermann's function: $a_{1,j} = 2^j$ , $a_{i,1} = 2$ , and $a_{i,j} = w\\cdot a_{i-1,w}$ where $w=a_{i,j-1}$ .",
"Our task in this section is to prove the omnibus Lemma REF in several stages.",
"Restatement of Lemma REF Let $s\\ge 1$ be the order parameter, $c\\ge s-2$ be a constant, and $i\\ge 1$ be an arbitrary integer.",
"The following upper bounds on $\\lambda _{s}$ and $\\Phi _{s}$ hold for all $s\\ge 1$ and all odd $s\\ge 5$ , respectively.",
"Define $j$ to be maximum such that $m\\le a_{i,j}^c$ .",
"$\\lambda _{1}(n,m) &= n + m-1 & \\mbox{$s=1$}\\\\\\lambda _{2}(n,m) &= 2n + m-2 & \\mbox{$s=2$}\\\\\\lambda _{3}(n,m) &\\le (2i+2)n + (3i-2)cj(m-1) & \\mbox{$s=3$}\\\\\\lambda _{s}(n,m) &\\le \\mu _{s,i}{\\left( n + (cj)^{s-2}(m-1) \\right)} & \\mbox{all $s\\ge 4$} \\\\\\Phi _{s}(n,m) &\\le \\nu _{s,i}{\\left( n + (cj)^{s-2}(m-1) \\right)} & \\mbox{odd $s\\ge 5$}\\multicolumn{2}{l}{\\text{The values $\\lbrace \\mu _{s,i},\\nu _{s,i}\\rbrace $ are defined as follows, where $t=\\lfloor \\frac{s-2}{2} \\rfloor $.",
"}}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{5 mm}\\mu _{s,i} &= \\left\\lbrace \\begin{array}{l}2^{{i+t+3}\\atopwithdelims ()t} - 3(2(i+t+1))^t \\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm}\\mbox{$\\frac{3}{2}$}(2(i+t+1))^{t+1}2^{{i+t+3}\\atopwithdelims ()t}\\end{array}\\right.&\\begin{array}{r}\\mbox{even $s\\ge 4$\\hspace*{-5.12128pt}}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{6 mm}\\mbox{odd $s\\ge 5$\\hspace*{-5.12128pt}}\\end{array}\\\\\\rule [- 0 mm]{0mm}{0 mm}\\rule {0mm}{8 mm}\\nu _{s,i} &= 4\\cdot 2^{{i+t+3}\\atopwithdelims ()t} &\\begin{array}{r}\\mbox{odd $s\\ge 5$\\hspace*{-5.12128pt}}\\end{array}$"
],
[
"Overview.",
"The proof is by induction on $(s,i,j)$ with respect to any fixed $c\\ge s-2$ .",
"In Section REF we confirm that Lemma REF holds when $i=1$ .",
"In Section REF we discuss the role that Ackermann's function plays in selecting block partitions for Recurrences REF , REF , and REF .",
"In Section REF we confirm Lemma REF at $s=3$ .",
"In Section REF we identify sufficient lower bounds on the elements of $\\lbrace \\mu _{s,i},\\nu _{s,i}\\rbrace _{s\\ge 2,i\\ge 1}$ , then, in Section REF , prove that the particular ensemble $\\lbrace \\mu _{s,i},\\nu _{s,i}\\rbrace _{s\\ge 2,i\\ge 1}$ proposed in Lemma REF does, in fact, satisfy these lower bounds."
],
[
"Base Cases",
"Lemma B.1 Let $n,m,$ and $s\\ge 2$ be the alphabet size, block count, and order parameters.",
"Given $i\\ge 1$ , let $j^{\\prime }$ be minimum such that $m\\le a_{i,j^{\\prime }}$ .",
"Whether $i=1$ and $j^{\\prime }\\ge 1$ or $j^{\\prime }=1$ and $i>1$ , we have $\\Phi _{s}(n,m) &\\;\\le \\; \\lambda _{s}(n,m) \\;\\le \\; 2^{s-1}n + j^{\\prime s-2}(m-1).$ Proof: First note that $\\Phi _{s}(n,m) \\le \\lambda _{s}(n,m) - 2n$ holds trivially since, in the worst case, every occurrence in the sequence is a feather, except for the first and last occurrence of each letter.",
"At $s=2$ the claim follows directly from Lemma REF .",
"At $s\\ge 3,j^{\\prime }=1$ , the claim is trivial since there are only $a_{i,1}=2$ blocks and $\\lambda _{s}(n,2)=2n$ .",
"In the general case we have $s\\ge 3$ and $j^{\\prime }>1$ .",
"Let $S$ be an order-$s$ , $m$ -block sequence over an $n$ -letter alphabet, where $m\\le a_{1,j^{\\prime }}=2^{j^{\\prime }}$ .",
"Let $S=S_1S_2$ be the partition of $S$ using a uniform block partition with width $a_{1,j^{\\prime }-1} = 2^{j^{\\prime }-1}$ , so $\\llbracket S_1 \\rrbracket = a_{1,j^{\\prime }-1}$ and $\\llbracket S_2 \\rrbracket = m-a_{1,j^{\\prime }-1} \\le a_{1,j^{\\prime }-1}$ .",
"Note that $\\hat{S}^{\\prime }=\\beta _1\\beta _2$ consists of two blocks, where each $\\beta _q$ is some permutation of the global alphabet $\\hat{\\Sigma }$ .",
"Since there are no middle occurrences in $\\hat{S}^{\\prime }$ or $S$ we can apply a simplified version of Recurrence REF .",
"${\\lambda _{s}(n,m)}\\\\&\\le \\sum _{q=1,2} \\lambda _{s}(\\check{n}_q, \\llbracket S_q \\rrbracket ) \\;+\\; \\lambda _{s-1}(\\hat{n}, \\llbracket S_1 \\rrbracket ) \\;+\\; \\lambda _{s-1}(\\hat{n}, \\llbracket S_2 \\rrbracket ) & \\mbox{\\lbrace local, first, and last\\rbrace }\\\\&\\le 2^{s-1}(n-\\hat{n}) + (j^{\\prime }-1)^{s-2}(m-2) \\;+\\; 2(2^{s-2}\\hat{n})+ (j^{\\prime }-1)^{s-3}(m-2) & \\mbox{\\lbrace inductive~hypothesis\\rbrace }\\\\&< 2^{s-1}n \\,+\\, j^{\\prime s-2}(m-1)\\hspace*{14.22636pt} $ The last inequality follows from the fact that when $s\\ge 3$ , $(j^{\\prime }-1)^{s-2} + (j^{\\prime }-1)^{s-3} \\le j^{\\prime s-2}$ .",
"This concludes the induction.",
"$\\Box $ If we introduce the `$c$ ' parameter and define $j$ to be minimum such that $m\\le a_{i,j}^c$ , Lemma REF implies that $\\lambda _{s}(n,m) \\le 2^{s-1}n + (cj)^{s-2}(m-1)$ since $j^{\\prime } \\le cj$ .",
"Note that by definition of Ackermann's function, $a_{1,j}^c = (2^j)^c = a_{1,cj}$ and $a_{i,1}^c = 2^c = a_{1,c}$ .",
"Lemma REF implies the claims of Lemma REF at $i=1$ .",
"When $s=3$ , $2i+2 = 4 = 2^{s-1}$ and $3i-2 = 1$ .",
"When $s\\ge 4$ is even, $\\mu _{s,1} = 2^{t+4\\atopwithdelims ()t} - 3(2(t+2))^t \\ge 2^{2t+1} = 2^{s-1}$ .",
"When $s\\ge 5$ is odd, $\\mu _{s,1} = \\mbox{$\\frac{3}{2}$}(2(t+2))^{t+1}2^{{t+4}\\atopwithdelims ()t} \\ge 2^{2t+2} = 2^{s-1}$ and $\\nu _{s,1} = 4\\cdot 2^{t + 4\\atopwithdelims ()t} \\ge 2^{2t+2} = 2^{s-1}$ .",
"The bounds above also imply that Lemma REF holds at $j=1$ and $i>1$ since $a_{i,1}^c = a_{1,1}^c$ and both $\\mu _{s,i}$ and $\\nu _{s,i}$ are increasing in $i$ ."
],
[
"Block Partitions and Inductive Hypotheses",
"When analyzing order-$s$ DS sequences we express the block count $m$ and partition size $\\hat{m}$ in terms of constant powers of Ackermann's function $\\lbrace a_{i,j}^c\\rbrace $ , where the constant $c\\ge s-2$ is fixed.",
"Recall that once $i$ is selected, $j$ is minimal such that $m \\le a_{i,j}^c$ .",
"The base cases $i=1$ and $j=1$ have been handled so we can assume both are at least 2.",
"Let $w=a_{i,j-1}$ .",
"We always choose a uniform block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ with width $w^c$ , that is, $m_q = w^c$ for all $q<\\hat{m}= \\lceil m/w^c \\rceil $ and the leftover $m_{\\hat{m}}$ may be smaller.",
"When invoking the inductive hypothesis (Lemma REF ) on the $\\hat{m}$ -block sequence $\\hat{S}^{\\prime }$ we use parameter $i-1$ .",
"In all other invocations of the inductive hypothesis we use parameter $i$ .",
"When applied to any $m_q$ -block sequences the `$j$ ' parameter is decremented since $m_q \\le w^c = a_{i,j-1}^c$ .",
"When applied to a $\\hat{m}$ -block sequence the `$j$ ' parameter is $w$ since $\\hat{m}= \\lceil \\frac{m}{w^c} \\rceil \\le {\\left( \\frac{a_{i,j}}{w} \\right)}^c = a_{i-1,w}^c.$ Furthermore, in such an invocation the dependence on $\\hat{m}$ will always be at most linear in $m$ since $(cw)^{s-2}(\\hat{m}-1) \\le (cw)^{s-2}(\\lceil m̑{w^c} \\rceil -1) \\le c^{s-2}(m-1)$ .",
"This is the reason we require the lower bound $c\\ge s-2$ .",
"If one is more familiar with the slowly growing row-inverses of Ackermann's function, it may be helpful to remember that $cj = \\log m - O(1)$ when $i=1$ and that $j = \\log ^{[i-1]}(m) - O(1)$ when $i>1$ , the effect of the $c$ parameter being negligible since $a_{i,j}$ and $a_{i,j}^c$ are essentially identical relative to any sufficiently slowly growing function.Recall that $\\log ^{[i-1]}(m)$ is short for $\\log ^{\\star \\cdots \\star }(m)$ with $i-1$ $\\star $ s. Thus, the bounds of Lemma REF could be rephrased as $\\lambda _{s}(n,m) \\le \\mu _{s,i}{\\left( n + O{\\left( m(\\log ^{[i-1]}(m))^{s-2} \\right)} \\right)}$ .",
"Since $\\mu _{s,i}$ is increasing in $i$ , the best bounds are obtained by choosing $i$ to be minimal such that $\\log ^{[i-1]}(m) = n/m + O(1)$ ."
],
[
"Order $s=3$",
"Lemma B.2 (Order $s=3$ ) Let $n$ and $m$ be the alphabet size and block count of an order-3 DS sequence $S$ .",
"For any $i, c\\ge 1$ , define $j$ to be minimum such that $m \\le a_{i,j}^c$ .",
"Then $\\lambda _{3}$ is bounded by $\\lambda _{3}(n,m) \\le (2i+2)n + (3i-2)cj(m-1)$ Proof: The base cases $i=1$ and $j=1$ have been handled already.",
"Let $i,j>1$ and $w=a_{i,j-1}$ .",
"We invoke Recurrence REF with the uniform block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ , where $\\hat{m}=\\lceil m/w^c \\rceil $ .",
"(See Section REF .)",
"$\\lambda _{3}(n,m) &\\le \\makebox{[}0mm][l]{\\sum _{q=1}^{\\hat{m}} \\lambda _{3}(\\check{n}_q,m_q) \\;+\\; 2\\cdot \\lambda _{2}(\\hat{n},m) \\;+\\; \\lambda _{1}(\\lambda _{3}(\\hat{n},\\hat{m}) - 2\\hat{n}, m)}\\\\&\\le \\makebox{[}0mm][l]{(2i+2)(n-\\hat{n}) \\,+\\, (3i-2)c(j-1)(m-\\hat{m})} & \\mbox{\\lbrace ind.~hyp.",
": local symbols\\rbrace }\\\\&\\hspace*{14.22636pt} \\;+\\; 4\\hat{n}\\,+\\, 2(m-1) \\hspace*{142.26378pt} & \\mbox{\\lbrace global first and last occurrences\\rbrace }\\\\&\\hspace*{14.22636pt} \\;+\\; \\makebox{[}0mm][l]{(2i-2)\\hat{n}\\;+\\; (3(i-1)-2)cw(\\hat{m}-1) \\;+\\; (m-1)} & \\mbox{\\lbrace global middle occurrences\\rbrace }\\\\&\\le \\makebox{[}0mm][l]{(2i+2)n \\;+\\; (3i-2)cj(m-1)}\\\\&\\hspace*{14.22636pt} \\;+\\; \\makebox{[}0mm][l]{\\Big [ - (2i+2) + 4 + (2i - 2) \\Big ]\\hat{n}\\;+\\; \\Big [ - c(3i-2) + (3i-5) + 3 \\Big ](m-1)}\\\\&\\le \\makebox{[}0mm][l]{(2i+2)n \\;+\\; (3i-2)cj(m-1)}$ The last inequality holds since $c\\ge s-2=1$ .",
"$\\Box $ At $s=2$ and $s=3$ the terms involving $n$ and $m$ have different leading constants, namely 2 and 1 when $s=2$ and $2i+2$ and $3i-2$ when $s=3$ .",
"To provide some uniformity in the analyses below we will use the inequalities $\\lambda _{2}(n,m) \\le \\mu _{2,i}(n+m-1)$ and $\\lambda _{3}(n,m) \\le \\mu _{3,i}(n+(cj)(m-1))$ when invoking the inductive hypothesis at $i\\ge 2$ and $s\\in \\lbrace 2,3\\rbrace $ , where $\\mu _{2,i}=2$ and $\\mu _{3,i}=3i$ by definition.",
"Note that when $i\\ge 2$ , $\\mu _{3,i} = 3i \\ge \\max \\lbrace 2i+2,3i-2\\rbrace $ ."
],
[
"Lower Bounds on $\\mu _{s,i}$ and {{formula:4f4daa22-ee15-473e-85d3-53843de7c579}}",
"Call an ensemble of values $\\lbrace \\mu _{s^{\\prime },i^{\\prime }}, \\nu _{s^{\\prime },i^{\\prime }}\\rbrace _{(s^{\\prime },i^{\\prime }) \\le (s,i)}$ happy if $\\lambda _{s^{\\prime }}(n,m) \\le \\mu _{s^{\\prime },i^{\\prime }}(n+(cj)^{s^{\\prime }-2}(m-1))$ and $\\Phi _{s^{\\prime }}(n,m) \\le \\nu _{s^{\\prime },i^{\\prime }}(n+ (cj)^{s^{\\prime }-2}(m-1))$ , where $c$ and $j$ are defined as usual.",
"(In the subscript `$\\le $ ' represents lexicographic ordering on tuples.)",
"In Lemma REF we determine lower bounds on $\\mu _{s,i}$ and $\\nu _{s,i}$ in a happy ensemble.",
"In Section REF we prove that the specific ensemble proposed in Lemma REF is, in fact, happy.",
"Lemma B.3 Let $s\\ge 4$ and $i\\ge 2$ .",
"Define $n,m,c,$ and $j$ as usual.",
"If $\\lbrace \\mu _{s^{\\prime },i^{\\prime }},\\nu _{s^{\\prime },i^{\\prime }}\\rbrace _{(s^{\\prime },i^{\\prime }) \\le (s,i-1)}$ is happy then $\\lbrace \\mu _{s^{\\prime },i^{\\prime }},\\nu _{s^{\\prime },i^{\\prime }}\\rbrace _{(s^{\\prime },i^{\\prime }) \\le (s,i)}$ is as well, so long as $\\mu _{s,i} &\\;\\ge \\; 2\\mu _{s-1,i} \\;+\\; \\mu _{s-2,i}\\mu _{s,i-1} & \\mbox{even $s$}\\\\\\mu _{s,i} &\\;\\ge \\; 2\\mu _{s-1,i} \\;+\\; \\mu _{s-2,i}\\nu _{s,i-1} \\;+\\; \\mu _{s-3,i}\\mu _{s,i-1} & \\mbox{odd $s$}\\\\\\nu _{s,i} &\\;\\ge \\; \\nu _{s,i-1} \\;+\\; 2\\mu _{s-1,i} & \\mbox{odd $s$}$ Proof: When $s\\ge 4$ is even, Recurrence REF implies that $\\lambda _{s}(n,m) &\\le \\sum _{q=1}^{\\hat{m}} \\lambda _{s}(\\check{n}_q,m_q) \\;+\\; 2\\cdot \\lambda _{s-1}(\\hat{n},m) \\;+\\; \\lambda _{s-2}(\\lambda _{s}(\\hat{n},\\hat{m}), m)\\nonumber \\\\&\\le \\mu _{s,i}{\\left( (n-\\hat{n}) \\;+\\; (c(j-1))^{s-2}(m-\\hat{m}) \\right)} \\hspace*{56.9055pt} \\mbox{\\lbrace happiness of the ensemble\\rbrace }\\nonumber \\\\&\\hspace*{14.22636pt} +\\; 2\\mu _{s-1,i}{\\left( \\hat{n}\\;+\\; (cj)^{s-3}(m-1) \\right)}\\nonumber \\\\&\\hspace*{14.22636pt} +\\; \\mu _{s-2,i}\\bigg (\\mu _{s,i-1}\\Big (\\hat{n}\\;+\\; (cw)^{s-2}(\\hat{m}-1)\\Big ) \\;+\\; (cj)^{s-4}(m-1)\\bigg )\\nonumber \\\\&\\le \\mu _{s,i}{\\left( n \\;+\\; (cj)^{s-2}(m-1) \\right)}\\nonumber \\\\&\\hspace*{14.22636pt} +\\; \\Big [ -\\mu _{s,i} + 2\\mu _{s-1,i} + \\mu _{s-2,i}\\mu _{s,i-1} \\Big ]\\cdot \\hat{n}\\\\&\\hspace*{14.22636pt} +\\; \\Big [ - \\mu _{s,i}c^{s-2}j^{s-3} + 2\\mu _{s-1,i}(cj)^{s-3} + \\mu _{s-2,i}\\mu _{s,i-1}c^{s-2} + \\mu _{s-2,i}(cj)^{s-4} \\Big ]\\cdot (m-1)\\\\&\\le \\mu _{s,i}{\\left( n \\;+\\; (cj)^{s-2}(m-1) \\right)}$ Inequality () will be satisfied whenever (REF ) and () are non-positive, that is, when $\\mu _{s,i} &\\;\\ge \\; 2\\mu _{s-1,i} \\;+\\; \\mu _{s-2,i}\\mu _{s,i-1} \\\\\\mu _{s,i} &\\;\\ge \\; {s-2} \\;+\\; {2^{s-3}} \\;+\\; {2(s-2)^2} $ Inequality () was obtained by dividing () through by $c^{s-2}j^{s-3}$ and noting that $c\\ge s-2\\ge 2$ and $j\\ge 2$ .",
"Note that Inequality () is weaker than Inequality (REF ) since $\\mu _{s,i} > \\mu _{s-1,i} > \\mu _{s-2,i}$ , so it suffices to consider only the former.",
"When $s\\ge 5$ is odd, Recurrence REF implies that $\\lambda _{s}(n,m) &\\le \\sum _{q=1}^{\\hat{m}} \\lambda _{s}(\\check{n}_q,m_q) + 2\\cdot \\lambda _{s-1}(\\hat{n},m) + \\lambda _{s-2}(\\Phi _{s}(\\hat{n},\\hat{m}),m) + \\lambda _{s-3}(\\lambda _{s}(\\hat{n},\\hat{m}),m)\\nonumber \\\\&\\le \\mu _{s,i}{\\left( (n-\\hat{n}) \\;+\\; (c(j-1))^{s-2}(m-\\hat{m}) \\right)} \\hspace*{56.9055pt} \\mbox{\\lbrace happiness of the ensemble\\rbrace }\\nonumber \\\\ &\\hspace*{14.22636pt} +\\; 2\\mu _{s-1,i}{\\left( \\hat{n}\\;+\\; (cj)^{s-3}(m-1) \\right)}\\nonumber \\\\ &\\hspace*{14.22636pt} +\\; \\mu _{s-2,i}\\bigg (\\nu _{s,i-1}{\\left( \\hat{n}\\;+\\; (cw)^{s-2}(\\hat{m}-1) \\right)} \\;+\\; (cj)^{s-4}(m-1)\\bigg )\\nonumber \\\\ &\\hspace*{14.22636pt} +\\; \\mu _{s-3,i}\\bigg (\\mu _{s,i-1}{\\left( \\hat{n}\\;+\\; (cw)^{s-2}(\\hat{m}-1) \\right)} \\;+\\; (cj)^{s-5}(m-1)\\bigg )\\nonumber \\\\&\\le \\mu _{s,i}{\\left( n \\;+\\; (cj)^{s-2}(m-1) \\right)}\\nonumber \\\\&\\hspace*{14.22636pt} +\\; \\Big [ -\\mu _{s,i} \\;+\\; 2\\mu _{s-1,i} \\;+\\; \\mu _{s-2,i}\\nu _{s,i-1} \\;+\\; \\mu _{s-3,i}\\mu _{s,i-1} \\Big ]\\cdot \\hat{n}\\\\&\\hspace*{14.22636pt} +\\; \\Big [ -\\mu _{s,i}c^{s-2}j^{s-3} \\;+\\; 2\\mu _{s-1,i}(cj)^{s-3} \\;+\\; \\mu _{s-2,i}\\nu _{s,i-1}c^{s-2}\\nonumber \\\\&\\hspace*{36.98866pt} +\\; \\mu _{s-2,i}(cj)^{s-4} \\;+\\; \\mu _{s-3,i}\\mu _{s,i-1}c^{s-2} \\;+\\; \\mu _{s-3,i}(cj)^{s-5} \\Big ]\\cdot (m-1) \\\\&\\le \\mu _{s,i}{\\left( n \\;+\\; (cj)^{s-2}(m-1) \\right)}$ Inequality () will be satisfied whenever (REF ) and () are non-positive, that is, when $\\mu _{s,i} &\\;\\ge \\; 2\\mu _{s-1,i} \\;+\\; \\mu _{s-2,i}\\nu _{s,i-1} \\;+\\; \\mu _{s-3,i}\\mu _{s,i-1} \\\\\\mu _{s,i} &\\;\\ge \\; {s-2} \\;+\\; {2^{s-3}} \\;+\\; {2(s-2)^2} \\;+\\; {2^{s-3}}\\;+\\; {4(s-2)^3} $ The denominators of Inequality () follow by dividing () through by $c^{s-2}j^{s-3}$ and noting that $c\\ge s-2\\ge 3$ and $j\\ge 2$ .",
"Inequality () is weaker than Inequality (REF ) since $\\mu _{s-1,i} > \\mu _{s-2,i} > \\mu _{s-3,i}$ so it suffices to consider only Inequality (REF ).",
"Using similar calculations, one derives from Recurrence REF the claimed lower bound on $\\nu _{s,i}$ .",
"$\\nu _{s,i} &\\;\\ge \\; \\nu _{s,i-1} \\;+\\; 2\\mu _{s-1,i}$ $\\Box $"
],
[
"The Happiness of the Ensemble",
"From this point on we argue the happiness of the specific ensemble $\\lbrace \\mu _{s,i},\\nu _{s,i}\\rbrace $ stated in Lemma REF .",
"We can say an individual value $\\mu _{s,i}$ or $\\nu _{s,i}$ is happy if it satisfies the appropriate lower bound inequality, either (), (), or (REF ).",
"Lemma B.4 The ensemble $\\lbrace \\mu _{s,i}, \\nu _{s,i}\\rbrace $ defined in Lemma REF is happy.",
"Proof: All $\\nu _{s,i}$ are happy since $\\nu _{s,i-1} + 2\\mu _{s-1,i}&< 4\\cdot 2^{{i+t+2}\\atopwithdelims ()t} + 2\\cdot 2^{{i+t+3}\\atopwithdelims ()t} & \\mbox{\\lbrace by definition\\rbrace }\\\\&\\le 4 \\cdot 2^{{i+t+3}\\atopwithdelims ()t} \\;=\\; \\nu _{s,i} & \\mbox{\\lbrace $t\\ge 1,\\; 4\\cdot 2^{i+t+2\\atopwithdelims ()t} \\le 2\\cdot 2^{i+t+3\\atopwithdelims ()t}$\\rbrace }\\\\$ When $s=4$ and $t=\\lfloor \\frac{s-2}{2} \\rfloor = 1$ the expression for $\\mu _{4,i}$ simplifies to $2^{i+4} - 6(i+2)$ .",
"The happiness of $\\mu _{4,i}$ follows easily, as seen below.",
"$2\\mu _{3,i} + \\mu _{2,i}\\mu _{4,i-1} &= 2(3i) \\;+\\; 2\\cdot \\left(2^{i+3} - 6(i+1)\\right) & \\mbox{\\lbrace by definition\\rbrace }\\\\ &= 2^{i+4} \\;+\\; 6i \\,-\\, 12(i+1)\\\\&= 2^{i+4} \\,-\\, 6(i+2) = \\mu _{4,i} \\\\$ When $s=5$ and $t=\\lfloor \\frac{s-2}{2} \\rfloor =1$ , the expression for $\\mu _{5,i}$ simplifies to $\\mbox{$\\frac{3}{2}$} (2(i+2))^2 2^{i+4}$ , which lets us quickly certify the happiness of $\\mu _{5,i}$ .",
"$2\\mu _{4,i} + \\mu _{3,i}\\nu _{5,i-1} + \\mu _{2,i}\\mu _{5,i-1}&= 2{\\left( 2^{i+4} - 6(i+2) \\right)}\\;+\\; 3i \\cdot 4\\cdot 2^{i+3}\\;+\\; 2\\cdot \\mbox{$\\frac{3}{2}$}(2(i+1))^2 2^{i+3}\\\\&\\le \\Big ( 2 \\;+\\; 6i \\;+\\; \\mbox{$\\frac{3}{2}$}(2(i+1))^2 \\Big ) 2^{i+4}\\\\&\\le \\mbox{$\\frac{3}{2}$} (2(i+2))^2 2^{i+4} \\;=\\; \\mu _{5,i}$ We now turn to the happiness of $\\mu _{s,i}$ for even $s\\ge 6$ .",
"Note that when we invoke the definition of $\\mu _{s-1,i}$ and $\\mu _{s-2,i}$ their “$t$ ” parameter is $t-1 = \\lfloor \\mbox{$\\frac{(s-1)-2}{2}$} \\rfloor = \\lfloor \\mbox{$\\frac{(s-2)-2}{2}$} \\rfloor $ .",
"${2\\mu _{s-1,i} \\,+\\, \\mu _{s-2,i}\\mu _{s,i-1}}\\\\&\\le 2\\cdot \\left[\\mbox{$\\frac{3}{2}$}(2(i+t))^t2^{{i+t+2}\\atopwithdelims ()t-1}\\right] \\,+\\, \\left[2^{{i+t+2}\\atopwithdelims ()t-1} - 3(2(i+t))^{t-1}\\right]\\cdot \\left[2^{{i+t+2}\\atopwithdelims ()t} - 3(2(i+t))^t\\right]\\\\&= \\left[3(2(i+t))^t2^{{i+t+2}\\atopwithdelims ()t-1}\\right]\\,+\\, 2^{{i+t+2}\\atopwithdelims ()t-1} \\left[2^{{i+t+2}\\atopwithdelims ()t} - 3(2(i+t))^t\\right]\\,-\\, 3(2(i+t))^{t-1} \\left[2^{{i+t+2}\\atopwithdelims ()t} - 3(2(i+t))^t\\right]\\\\&= 2^{{i+t+2}\\atopwithdelims ()t-1} 2^{{i+t+2}\\atopwithdelims ()t} \\,-\\, 3(2(i+t))^{t-1} \\left[2^{{i+t+2}\\atopwithdelims ()t} - 3(2(i+t))^t\\right]\\\\&\\le 2^{{i+t+3}\\atopwithdelims ()t} - 3(2(i+t+1))^t \\, =\\, \\mu _{s,i} $ In other words, $\\mu _{s,i}$ satisfies Inequality (REF ) when $s\\ge 6$ is even.",
"It also satisfies Inequality (REF ) at odd $s\\ge 7$ , which can be seen as follows.",
"Note that the “$t$ ” parameter for $s-1$ it $t$ , whereas it is $t-1$ for $s-2$ and $s-3$ .",
"${2\\mu _{s-1,i} \\:+\\: \\mu _{s-2,i}\\nu _{s,i-1} \\:+\\: \\mu _{s-3,i}\\mu _{s,i-1} }\\\\&\\le 2\\cdot 2^{{i+t+3}\\atopwithdelims ()t} \\:+\\: \\mbox{$\\frac{3}{2}$}(2(i+t))^t2^{{i+t+2}\\atopwithdelims ()t-1} \\cdot 4\\cdot 2^{{i+t+2}\\atopwithdelims ()t}\\:+\\: 2^{{i+t+2}\\atopwithdelims ()t-1} \\cdot \\mbox{$\\frac{3}{2}$}(2(i+t))^{t+1}2^{{i+t+2}\\atopwithdelims ()t}\\\\&\\le \\Big [ 2 \\:+\\: \\mbox{$\\frac{3}{2}$}4\\cdot (2(i+t))^t \\:+\\: \\mbox{$\\frac{3}{2}$}(2(i+t))^{t+1} \\Big ] \\cdot 2^{{i+t+3}\\atopwithdelims ()t}\\\\&\\le \\Big [ 2\\:+\\: \\mbox{$\\frac{3}{2}$}(2(i+t))^t\\cdot 2(i+t+2) \\Big ] \\cdot 2^{{i+t+3}\\atopwithdelims ()t}\\\\&\\le \\mbox{$\\frac{3}{2}$}(2(i+t+1))^{t+1}\\cdot 2^{{i+t+3}\\atopwithdelims ()t} = \\mu _{s,i}$ We have shown that $\\lbrace \\mu _{s,i}\\rbrace $ and $\\lbrace \\nu _{s,i}\\rbrace $ are happy over the full range of parameters.",
"This concludes the proof of Lemma REF .",
"$\\Box $"
],
[
"Proof of Lemma ",
"The proof of Lemma REF closely mimics that of Lemma REF .",
"Restatement of Lemma REF Let $n$ and $m$ be the alphabet size and block count.",
"After a parameter $i\\ge 1$ is chosen let $j\\ge 1$ be minimum such that $m \\le a_{i,j}^c$ , where $c=3$ is fixed.",
"We have the following upper bounds on $\\Phi ^{\\prime }(n,m), \\Phi ^{\\prime \\prime }(n,m),$ and $\\lambda _{5}(n,m)$ .",
"$\\Phi ^{\\prime }(n,m) &\\le \\nu _i^{\\prime }{\\left( n + (cj)^2(m-1) \\right)} & \\mbox{where $\\nu _i^{\\prime } = 3{i+1\\atopwithdelims ()2} + 3$}\\\\\\Phi ^{\\prime \\prime }(n,m) &\\le \\nu _i^{\\prime \\prime }{\\left( n + (cj)^3(m-1) \\right)} & \\mbox{where $\\nu _i^{\\prime \\prime } = 6{i+2\\atopwithdelims ()3} + 8i$}\\\\\\lambda _{5}(n,m) &\\le \\mu _{5,i}{\\left( n + (cj)^3(m-1) \\right)} & \\mbox{where $\\mu _{5,i} = i2^{i+7}$}\\\\$ Proof: We use the following upper bounds on order-3 and order-4 DS sequences from Lemma REF .",
"$\\lambda _{3}(n,m) &\\le \\makebox{[}0mm][l]{\\mu _{3,i}[n + (cj)(m-1)]}\\hspace*{71.13188pt} & \\mbox{where $\\mu _{3,i} = 3i+1 \\ge \\max \\lbrace 2i+2,3i-2\\rbrace $}\\\\\\lambda _{4}(n,m) &\\le \\makebox{[}0mm][l]{\\mu _{4,i}[n + (cj)^2(m-1)]} & \\mbox{where $\\mu _{4,i} = 2^{i+4} - 6(i+2)$}\\multicolumn{2}{l}{\\text{and, when $i=1$,}}\\\\\\lambda _{4}(n,m) &\\le \\makebox{[}0mm][l]{2^3 n + (cj)^2(m-1)} \\\\\\lambda _{5}(n,m) &\\le \\makebox{[}0mm][l]{2^4 n + (cj)^3(m-1)} & \\mbox{See Lemma~\\ref {lem:i=1}.",
"}$"
],
[
"Base Cases.",
"In the worst case every occurrence in an order-4 sequence is a dove (or hawk) feather, except for the first and last occurrence of each symbol, which are wingtips.",
"This implies that $\\Phi ^{\\prime }(n,m) &\\le \\lambda _{4}(n,m) - 2n\\\\&\\le (2^3 - 2)n + (3j)^2(m-1) & \\lbrace \\mbox{by Lemma~\\ref {lem:i=1}}\\rbrace \\\\&\\le \\nu _1^{\\prime }{\\left( n + (3j)^2(m-1) \\right)} & \\lbrace \\mbox{since $\\nu _1^{\\prime } = 6$}\\rbrace $ The same argument implies that $\\Phi ^{\\prime \\prime }(n,m) \\le \\lambda _{5}(n,m) - 2n \\le (2^4-2)n+(3j)^3(m-1)$ , which is at most $\\nu _1^{\\prime \\prime }(n + (3j)^3(m-1))$ since $\\nu _1^{\\prime \\prime } = 14$ .",
"This confirms the claimed bounds when $i=1$ .",
"It also holds when $i>1$ and $j=1$ since $a_{i,1}^3 = a_{1,1}^3$ and all of $\\nu _i^{\\prime },\\nu _i^{\\prime \\prime },$ and $\\mu _{5,i}$ are increasing in $i$ ."
],
[
"Inductive Cases.",
"We can assume that $i,j>1$ .",
"As in the proof of Lemma REF we always apply Recurrence REF with a uniform block partition $\\lbrace m_q\\rbrace _{1\\le q\\le \\hat{m}}$ with width $w^c$ , where $w=a_{i,j-1}$ , $\\hat{m}=\\lceil m/w^c \\rceil $ , and $c=3$ is fixed.",
"According to Recurrence REF and the inductive hypothesis we have: $\\Phi ^{\\prime }(n,m) &\\le \\sum _{q=1}^{\\hat{m}} \\Phi ^{\\prime }(\\check{n}_q,m_q) \\, + \\, \\Phi ^{\\prime }(\\hat{n},\\hat{m})\\, + \\, \\lambda _{3}(\\hat{n},m) - \\hat{n}\\nonumber \\\\&\\le \\nu _i^{\\prime }\\Big [ (n-\\hat{n}) + (c(j-1))^2(m-\\hat{m}) \\Big ]\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\nu _{i-1}^{\\prime }\\Big [ \\hat{n}+ (cw)^2(\\hat{m}-1) \\Big ]\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\mu _{3,i}\\Big [ \\hat{n}+ (cj)(m-1) \\Big ] \\;-\\; \\hat{n}& \\mbox{\\lbrace inductive hypothesis\\rbrace }\\nonumber \\\\&\\le \\nu _i^{\\prime }\\left[ n + (cj)^2(m-1) \\right]\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\Big [ -\\nu _i^{\\prime } + \\nu _{i-1}^{\\prime } + \\mu _{3,i} - 1 \\Big ]\\cdot \\hat{n}\\nonumber \\\\& \\hspace*{28.45274pt}+\\; \\Big [ - (c^2j)\\nu _i^{\\prime } + (c^2/w)\\nu _{i-1}^{\\prime } + (cj)\\mu _{3,i} \\Big ]\\cdot (m-1)\\\\&\\le \\nu _i^{\\prime }\\left[ n + (cj)^2(m-1) \\right]\\\\\\multicolumn{2}{l}{\\text{Inequality (\\ref {ineq:nu_i^{\\prime }-1}) follows from the factthat $(c(j-1))^2 \\le (cj)^2 - cj$ and that $(cw)^2(\\hat{m}-1) = (cw)^2(\\lceil m̑{w^c} \\rceil -1) \\le (c^2/w) (m-1)$.Inequality (\\ref {ineq:nu_i^{\\prime }-2}) will follow so long as $\\nu _i^{\\prime }$ satisfies the following.",
"}}\\\\\\nu _i^{\\prime } &\\;\\ge \\; \\nu _{i-1}^{\\prime } + \\mu _{3,i} - 1 \\;=\\; \\nu _{i-1}^{\\prime } + 3i$ One may confirm that $\\nu _i^{\\prime } = 3{i+1 \\atopwithdelims ()2} + 3$ satisfies ().",
"In a similar fashion we can obtain a lower bound on $\\nu _i^{\\prime \\prime }$ as follows.",
"$\\Phi ^{\\prime \\prime }(n,m) &\\le \\sum _{q=1}^{\\hat{m}} \\Phi ^{\\prime \\prime }(\\check{n}_q,m_q) \\, + \\, \\Phi ^{\\prime \\prime }(\\hat{n},\\hat{m})\\, + \\, 2(\\Phi ^{\\prime }(\\hat{n},m) + \\hat{n})\\nonumber \\\\&\\le \\nu _i^{\\prime \\prime }\\Big [ (n-\\hat{n}) + (c(j-1))^3(m-\\hat{m}) \\Big ]\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\nu _{i-1}^{\\prime \\prime }\\Big [ \\hat{n}+ (cw)^3(\\hat{m}-1) \\Big ]\\nonumber \\\\&\\hspace*{28.45274pt}+\\; 2\\nu _i^{\\prime }\\Big [ \\hat{n}+ (cj)^2(m-1) \\Big ]+ 2\\hat{n}& \\mbox{\\lbrace inductive hypothesis\\rbrace }\\nonumber \\\\&\\le \\nu _i^{\\prime \\prime }\\left[ n + (cj)^3(m-1) \\right]\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\Big [ -\\nu _i^{\\prime \\prime } + \\nu _{i-1}^{\\prime \\prime } + 2\\nu _i^{\\prime } + 2 \\Big ]\\cdot \\hat{n}\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\Big [ -(c^3j^2) \\nu _i^{\\prime \\prime } + c^3\\nu _{i-1}^{\\prime \\prime } + (cj)^2\\cdot 2\\nu _i^{\\prime } \\Big ]\\cdot (m-1)\\\\&\\le \\nu _i^{\\prime \\prime }\\left[ n + (cj)^3(m-1) \\right] \\multicolumn{2}{l}{\\text{Inequality (\\ref {ineq:nu_i^{\\prime \\prime }-1}) follows since $(c(j-1))^3 \\le (cj)^3 - (cj)^2$and $(cw)^3(\\hat{m}-1) \\le c^3(m-1)$.Inequality (\\ref {ineq:nu_i^{\\prime \\prime }-2}) will follow from (\\ref {ineq:nu_i^{\\prime \\prime }-1}) if $\\nu _i^{\\prime \\prime }$ satisfies}}\\\\\\nu _i^{\\prime \\prime } &\\;\\ge \\; \\nu _{i-1}^{\\prime \\prime } + 2\\nu _i^{\\prime } + 2 \\;=\\; \\nu _{i-1}^{\\prime \\prime } + 6{i+1\\atopwithdelims ()2} + 8$ Again, one may confirm that $\\nu _i^{\\prime \\prime } = 6{i+2\\atopwithdelims ()3} + 8i$ satisfies ().",
"We are now ready to calculate a lower bound constraint on $\\mu _{5,i}$ .",
"By Recurrence REF and the inductive hypothesis we have $\\lambda _{5}(n,m) &\\le \\makebox{[}0mm][l]{\\sum _{q=1}^{\\hat{m}} \\lambda _{5}(\\check{n}_q,m_q)\\,+\\, 2\\cdot \\lambda _{4}(\\hat{n},m) \\,+\\, \\lambda _{3}(\\Phi ^{\\prime \\prime }(\\hat{n},\\hat{m}),m) \\,+\\,\\lambda _{2}(\\lambda _{5}(\\hat{n},\\hat{m}),2m-1)}\\nonumber \\\\&\\le \\makebox{[}0mm][l]{\\mu _{5,i}\\Big [ (n-\\hat{n}) + (c(j-1))^3(m-\\hat{m}) \\Big ]+ 2\\mu _{4,i}\\Big [ \\hat{n}+ (cj)^2(m-1) \\Big ]} \\nonumber \\\\&\\hspace*{28.45274pt}\\makebox{[}0mm][l]{+\\; \\mu _{3,i}\\Big [ \\nu _{i-1}^{\\prime \\prime }\\Big [ \\hat{n}+ (cw)^3(\\hat{m}-1) \\Big ] + (cj)(m-1) \\Big ]}\\nonumber \\\\&\\hspace*{28.45274pt}\\makebox{[}0mm][l]{+\\; 2\\mu _{5,i-1}\\Big [ \\hat{n}+ (cw)^3(\\hat{m}-1) \\Big ] + 2(m-1)} & \\mbox{\\lbrace inductive hypothesis\\rbrace }\\nonumber \\\\&\\le \\makebox{[}0mm][l]{\\mu _{5,i}[n + (cj)^3(m-1)]}\\nonumber \\\\&\\hspace*{28.45274pt}\\makebox{[}0mm][l]{+\\; \\Big [ -\\mu _{5,i} + 2\\mu _{4,i} + \\mu _{3,i}\\nu _{i-1}^{\\prime \\prime } + 2\\mu _{5,i-1} \\Big ]\\cdot \\hat{n}}\\nonumber \\\\&\\hspace*{28.45274pt}+\\; \\Big [ - (c^3j^2)\\mu _{5,i} + 2(cj)^2 \\mu _{4,i}+ c^3 \\mu _{3,i}\\nu _{i-1}^{\\prime \\prime } \\nonumber \\\\&\\hspace*{42.67912pt} + (cj)\\mu _{3,i} + 2(c^3)\\mu _{5,i-1} + 2 \\Big ]\\cdot (m-1)\\nonumber \\\\&\\le \\mu _{5,i}[n + (cj)^3(m-1)]\\multicolumn{2}{l}{\\text{Inequality (\\ref {ineq:mu5}) will hold so long as $\\mu _{5,i}$ satisfies}}\\\\\\mu _{5,i} &\\ge 2\\mu _{5,i-1} + 2\\mu _{4,i} + \\mu _{3,i}\\nu _{i-1}^{\\prime \\prime }\\nonumber \\\\&= 2\\mu _{5,i-1} + \\left[ 2{\\left( 2^{i+4} - 6(i+2) \\right)} + {\\left( 3i+1 \\right)}{\\left( 6{i+1\\atopwithdelims ()3} + 8(i-1) \\right)} \\right]\\nonumber \\\\&= 2\\mu _{5,i-1} + \\Big [ 2^{i+5} + {\\left( 3i+1 \\right)}{\\left( i+1 \\right)}{\\left( i \\right)}{\\left( i-1 \\right)} - 4(i+8) \\Big ]$ The bracketed term is less than $2^{i+7}$ , so it suffices to show that $\\mu _{5,i} \\ge 2\\mu _{5,i-1} + 2^{i+7}$ , when $i>1$ .",
"One may confirm that $\\mu _{5,i} = i\\cdot 2^{i+7}$ satisfies ().",
"$\\Box $"
]
] | 1204.1086 |
[
[
"Directly Imaging Tidally Powered Migrating Jupiters"
],
[
"Abstract Upcoming direct-imaging experiments may detect a new class of long-period, highly luminous, tidally powered extrasolar gas giants.",
"Even though they are hosted by ~ Gyr-\"old\" main-sequence stars, they can be as \"hot\" as young Jupiters at ~100 Myr, the prime targets of direct-imaging surveys.",
"They are on years-long orbits and presently migrating to \"feed\" the \"hot Jupiters.\"",
"They are expected from \"high-e\" migration mechanisms, in which Jupiters are excited to highly eccentric orbits and then shrink semi-major axis by a factor of ~10-100 due to tidal dissipation at close periastron passages.",
"The dissipated orbital energy is converted to heat, and if it is deposited deep enough into the atmosphere, the planet likely radiates steadily at luminosity L ~ 100-1000 L_Jup(2 x 10-7-2 x 10-6 L_Sun) during a typical ~ Gyr migration timescale.",
"Their large orbital separations and expected high planet-to-star flux ratios in IR make them potentially accessible to high-contrast imaging instruments on 10 m class telescopes.",
"~10 such planets are expected to exist around FGK dwarfs within ~50 pc.",
"Long-period radial velocity planets are viable candidates, and the highly eccentric planet HD 20782b at maximum angular separation ~0.",
"''08 is a promising candidate.",
"Directly imaging these tidally powered Jupiters would enable a direct test of high-e migration mechanisms.",
"Once detected, the luminosity would provide a direct measurement of the migration rate, and together with mass (and possibly radius) estimate, they would serve as a laboratory to study planetary spectral formation and tidal physics."
],
[
"Introduction",
"Jupiter and Saturn analogs orbiting other Sun-like main-sequence stars have evaded direct detection.",
"With an effective temperature of $\\sim 124 {\\rm K}$ , an extrasolar analogue of Jupiter is fainter than a Sun-like host by $\\gtrsim 10^{8}$ in near-IR [13], beyond the reach of instrument capabilities at present and in the near future.",
"While “hot Jupiters” (Jovian planets at $\\lesssim 0.1 {{\\,\\rm AU}}$ ) have high temperatures and thus large planet/star flux ratios, they are too close to their hosts ($\\ll 0.1^{\\prime \\prime }$ ) to be spatially resolved by 10m-class telescopes.",
"To date, direct-imaging surveys have focused on searching for long-period massive gas giants around nearby young stars with ages $\\lesssim 100 {\\,\\rm Myr}$ , a strategy that has led to a number of discoveries (e.g., [25], [16]).",
"In these systems, the planets are still “hot” – cooling down from presumably high temperatures at birth, which significantly enhances their flux ratios with host stars.",
"In this Letter, we discuss the possibility of directly imaging a third class of “hot” gas giants (besides close-in hot Jupiters and young Jupiters around young stars), consisting of a population of long-period, very luminous, tidally-powered planets undergoing orbital migration."
],
[
"High-eccentricity Migration due to Tidal Dissipation",
"It is commonly believed that progenitors of hot Jupiters are formed with semi-major axis $a$ beyond the “snow line” at a few ${\\,\\rm AU}$ and then migrate inward to their current locations by shrinking $a$ by factor of $\\sim 10 - 100$ [19].",
"One class of proposed migration mechanisms involve exciting long-period Jupiters to highly eccentric orbits, due to gravitational interactions with stellar or planetary perturbers, enabling them to lose orbital energy at successive close periastron passages through tidal interactions with their host stars.",
"Such “high-$e$ ” mechanisms include Kozai-Lidov Cycles plus Tidal Friction (KCTF) ([36], [9]), planet-planet scatter [29], “secular chaos” [37] etc.",
"Recently high-$e$ mechanisms have gained observational support.",
"A significant fraction of hot Jupiters are found to be on misaligned orbits with respect to their host stars' spin axes [35], [34], which is a natural consequence of such mechanisms [9].",
"One general expectation from all high-$e$ mechanisms is that there should exist a steady-state migrating population of long-period, highly eccentric gas giants “feeding” the hot Jupiters [31].",
"This results from the continuous generation of hot-Jupiter progenitors due to constant formation of stars and their planetary systems over the age of the Galaxy.",
"The orbital angular momentum is approximately conserved during tidal dissipation, so the actively migrating Jupiters have $a(1-e^2) \\equiv a_{\\rm F}$ , where $a_{\\rm F} 0.1 {\\,\\rm AU}$ is the semi-major axis of their final circularized orbit.",
"According to [31], the frequency of this population is likely an increasing function of their period (and the eccentricity), extending to that of their “source” (possibly at $\\gtrsim 5 {\\,\\rm AU}$ ).",
"A possible archetype of the migrating population is HD 80606b [28], [27], which is a $4 M_{\\,\\rm Jup}$ planet at semi-major axis $a=0.45{\\,\\rm AU}$ and $e=0.93$ ($a_{\\rm F}=0.06 {\\,\\rm AU}$ ), accompanied by an solar-mass companion at $\\sim 1200 {\\,\\rm AU}$ .",
"Regardless of the specifics of the high-$e$ mechanisms, a gas giant that migrates from semi-major axis $a^{\\prime }$ to $a$ over a time $\\Delta {t}_{\\rm m}$ loses orbital energy due to tidal dissipation, which is converted to heat and radiated away.",
"This leads to an averaged tidal luminosity, $L_m &=& \\frac{GM_*M_p}{2\\Delta {t}_m}\\big ({\\frac{1}{a} - \\frac{1}{a^{\\prime }}}\\big )\\cr &\\sim & 8\\times 10^{26} {\\,\\rm {erg\\,s^{-1}}}\\big (\\frac{M_*}{M_\\odot }\\big )\\big (\\frac{M_p}{3 M_{{\\,\\rm Jup}}}\\big )\\big (\\frac{\\Delta {a_m^{-1}}}{1 \\rm AU^{-1}}\\big )\\big (\\frac{\\Delta {t}_{\\rm m}}{1 {\\,\\rm Gyr}}\\big )^{-1},$ where $M_*$ is the mass of the host star, $M_p$ is the mass of the planet, $\\Delta {a_m^{-1}} = 1/a - 1/a^{\\prime }$ .",
"For comparison, $L_{\\,\\rm Jup}= 8.6\\times 10^{24}{\\,\\rm erg}\\,s^{-1}$ is the luminosity of our Jupiter.",
"[9] presents a possible migration path for HD 80606b due to Kozai-Lidov Cycles plus Tidal Friction (KCTF), and in their simulation, over $\\sim 0.1 {\\,\\rm Gyr}$ , the “migration rate” $\\Delta {a_m^{-1}}/\\Delta {t}_{\\rm m}$ at $0.5{\\,\\rm AU}$ , $1{\\,\\rm AU}$ , and $2{\\,\\rm AU}$ is $\\sim 12$ , 5, and $1.4 {\\,\\rm AU}^{-1}{\\,\\rm Gyr}^{-1}$ , corresponding to $L_m \\sim 1.3\\times 10^{28}, 5.3\\times 10^{27},1.5\\times 10^{27}{\\,\\rm {erg\\,s^{-1}}}$ , respectively, which span $\\sim 1500 - 180 \\,L_{{\\,\\rm Jup}}$ (see also [36]).",
"If tidal dissipation occurs in a deep enough layer of the planet atmosphere, the thermal relaxation time $t_{\\rm th}$ can be much longer than the orbital time scale ($P \\sim \\rm yr$ ), the planet would constantly radiate at its tidal luminosity.",
"The physics of tidal dissipation in giant planets is an unsolved theoretical problem, and there is no reliable method to calculate where the tidal dissipation happens in the planet atmosphere.",
"The upper limit of $t_{\\rm th}$ is the Kevin-Helmholtz time scale $t_{\\rm KH} \\propto M_p^2/R_p/L_p$ , which is about $0.1 {\\,\\rm Gyr}$ for a Jupiter with luminosity $L_p$ at hundreds of $L_{{\\,\\rm Jup}}$ .",
"For the Jovian planets of interest, $M_p^2/R_p$ is at most factor of 100 smaller than that for the Jupiter, and at this extreme, for $L_p = 100\\,L_{{\\,\\rm Jup}}$ , $T_{\\rm KH}$ is still $\\sim {\\rm Myr}$ , much longer than the orbital time scale.",
"The tidal forcing acts on the planet body as a whole, so it is reasonable to expect that significant tidal dissipation operates at a depth that is a considerable fraction of the planet radius and then $t_{\\rm th}$ is approximately $t_{\\rm KH}$ , which is many orders of magnitude larger than the orbital time.",
"We make the reasonable assumption $t_{\\rm th} > P$ throughout the paper.",
"The luminosities of migrating Jupiters can be tidally enhanced by $\\sim 2-3$ orders of magnitude, comparable to young Jupiters at $\\sim 100 {\\,\\rm Myr}$ [6], [4], [24], [32], which are the prime targets for ongoing and planned direct-imaging surveys [15], [20].",
"These tidally powered Jupiters could be located at $a \\sim {\\rm several} {\\,\\rm AU}$ , and their maximum separations at apastron are further enhanced by high eccentricity $e \\sim 1$ by a factor $(1+e)\\simeq 2$ , making them promising targets for direct-imaging detections.",
"Tidally powered Jupiters are not necessarily limited to those actively migrating at high eccentricity.",
"For example, in the specific cases of KCTF, while the planet visits the highly eccentric phase that enables tidal dissipation at periastron passages in each Kozai-Lidov cycle, it typically spends much longer (factor of $\\sim 10$ more, see [9]) time oscillating at low-eccentricity orbits.",
"The oscillation amplitude in $e$ is generally larger when the planets are at longer period where relativistic precession is weaker.",
"If the thermal time scale $t_{\\rm th}$ is longer than the Kozai-Lidov time scale ($t_{\\rm Kozai} \\sim 0.02 {\\,\\rm Gyr}$ for HD 80606b while migrating at $\\sim 5{\\,\\rm AU}$ ), the planet radiates approximately at the averaged tidal luminosity $L_m$ during the whole Kozai-Lidov cycle.",
"Therefore, the tidally-powered Jupiters include not only those at high $e$ (i.e., small pericenter) but also a factor of $\\sim 10$ more experiencing Kozai-Lidov oscillations at lower $e$ .",
"Figure: The planet-to-star contrast ratios for expectedtidally-powered Jupiter population as a function of angular separationsas compared to those reached or expected for current (dotted-dash lines) and upcoming (solid lines) high-contrast imaging instrumentsfor 10m-class telescopes as well as future instruments for 30m-class telescopes (dash lines).",
"The expected HH-band contrast ratios at a range of migration rate of 1-10 AU -1 Gyr -1 1-10 {\\,\\rm AU}^{-1}{\\,\\rm Gyr}^{-1} for1,31, 3 and 10 M J M_J planets are shown in dark green, blueand yellow shaded regions, respectively.",
"They are calculated usingthe models in as discussed in the text.The maximum angular separation (at apocenter) for asuper-eccentric with a=2.7 AU a = 2.7{\\,\\rm AU} (snow line for a solar-mass host)at 20pc is shown as vertical black line with shading, indicatinga plausible upper boundary in orbital separations for these Jupiters.",
"Upcoming instrument such as Subaru/SCExAO may have substantial sensitivity at small enough angular separations to probe the tidally powered Jupiter population, and high-contrast imaging instruments on future 30-m class telescopes (such as TMT, GMT and ELT) are expected to have excellent capabilities to detect such planets.The sources for the contrast curves are listed below.",
"(1) Gemini/NICI HH-band: (2) Subaru/SCExAO with extreme AO, 1- hr 1-{\\,\\rm hr} integration: http://www.naoj.org/Projects/newdev/intmtg/20100426/files/SCExAO_overview_2010-04-26.pdf (3) Gemini/GPI – http://planetimager.org/pages/gpi_tech_contrast.html (4) VLT SPHERE and ELT EPICS in JJ-band: (5) TMT Planet Formation Instrument at 1.65μm1.65 {\\,\\rm \\mu m}: https://e-reports-ext.llnl.gov/pdf/333450.pdf (6) GMT HRCAM at 1.6μm1.6 {\\,\\rm \\mu m} with>1 hr >1 {\\,\\rm hr} exposure: http://www.physics.berkeley.edu/research/genzel/Physics250.5/literature-talks/ELT/GMT_Science_Case.pdf"
],
[
"Direct Imaging Observations",
"The achievable sensitivity of high-contrast imaging instruments degrades substantially within the so-called inner working angle, which is often quoted to be $\\sim 2-4$ times the diffraction limit, $\\theta _{\\rm diff} \\sim \\lambda /D \\sim 0.02(\\lambda /1 {\\,\\rm \\mu m})(D/10 {\\rm m})^{-1}$ , where $\\lambda $ is the observed wavelength and $D$ is telescope aperture.",
"Several high-contrast imaging instruments are and will be commissioned in the near future on a number of 8-10m telescopes, such as Gemini GPI [22], VLT SPHERE [5], and Subaru SCExAO [26], LBTI [10].",
"The best contrast goals of these instruments are $10^{-7} - 10^{-8}$ in near-IR beyond the inner working angle [22], [5].",
"See Fig.",
"1 for their expected contrast ratio curves in near IR.",
"At a migration rate of $\\Delta {a_m^{-1}}/\\Delta {t}_{\\rm m} = 1 - 10{\\,\\rm AU}^{-1}{\\,\\rm Gyr}^{-1}$ , tidal luminosities for $\\sim 3 M_{{\\,\\rm Jup}}$ planets are $\\sim 2\\times 10^{-7} - 2\\times 10^{-6} L_\\odot $ ($\\sim 10^2$ - $10^3 L_{{\\,\\rm Jup}}$ ), corresponding to blackbody effective temperatures $T_{\\rm eff} \\sim 390K - 690K$ for a planet radius $1 R_{Jup}$ .",
"The peaks of the black body radiation at these temperatures are at $\\sim 9-5 {\\,\\rm \\mu m}$ .",
"The black body contrast ratios at $3.78 {\\,\\rm \\mu m}$ ($L^{\\prime }$ -band) of the planet to a Sun-like star ($T_{\\rm eff} \\sim 5800 K$ ) are $\\sim 5.0\\times 10^{-7} - 3.8\\times 10^{-5}$ .",
"At bands further away from the peak (e.g., $\\sim 1 {\\,\\rm \\mu m}$ ), black body radiation is exponentially suppressed, resulting in very low flux ratios.",
"However, the near-IR spectral energy distribution is unlikely to be well described by black body emission.",
"For example, Jupiter has a spectral window allowing for probing deeper, warmer layers of the atmosphere that leads to orders of magnitude larger flux than that of a $125 \\rm K$ blackbody in near-IR [13].",
"We note that according to [6]'s model, at effective temperatures $400K$ and $600K$ , with surface gravity $10^4 {\\rm cm\\,s^{-2}}$ , the contrast ratios in [$J$ , $H$ , $K$ , $L^{\\prime }$ ]-band with a Sun-like star are approximately $[3.6\\times 10^{-7}, 4\\times 10^{-7}, 3.9\\times 10^{-8}, 4\\times 10^{-6}]$ , $[3.5\\times 10^{-6}, 4.8\\times 10^{-6}, 1.4\\times 10^{-6}, 2.2\\times 10^{-5}]$ , respectively.",
"Other models [4], [32] generally predict similar contrast ratios in $L^{\\prime }$ -band, but the predictions can vary by order of magnitude in $J$ -band and $H$ -band, possibly reflecting theoretical uncertainties in such calculations such as the treatment of clouds and planet luminosity evolution.",
"[31] estimate that the frequency of long-period (hundreds of days), highly-eccentric Jupiters is about $10\\%$ that of hot Jupiters, whose occurrence rate is estimated to be $\\sim 1\\%$ around FGK dwarfs [23].",
"Therefore, $\\sim 0.1\\%$ of solar-type stars may host this population.",
"There are $\\sim 10^4$ FGK dwarfs within $\\sim 50 {\\,\\rm pc}$ of the Sun, which amounts to $\\sim 10$ potentially tidally-powered Jupiters.",
"The closest super-eccentric Jupiter host is then at $\\sim 20{\\,\\rm pc}$ , and suppose it starts migration near the snow line at $\\sim 2.7{\\,\\rm AU}$ , its maximum angular separation is $\\sim 0.27$ (achieved at apastron for $e \\sim 1$ ), which is marked as a shaded black line in Fig.1.",
"In Fig.1, the expected ranges of $H$ -band contrast ratio assuming migration rate varying from $1 {\\,\\rm AU}^{-1}{\\,\\rm Gyr}^{-1}$ to $10 {\\,\\rm AU}^{-1}{\\,\\rm Gyr}^{-1}$ for planets with $1, 3, 10 M_J$ are shown in shaded regions in green, blue and yellow, respectively.",
"It is challenging for present and most of the upcoming high-contrast imaging instruments such as Gemini/NICI, Gemini/GPI and VLT/SPHERE to detect the population of tidally-powered Jupiters due to their relatively large inner working angles ($\\gtrsim 0.2 - 0.3 $ ).",
"The most promising instrument is SCExAO at Subaru.",
"The extreme AO system of SCExAO is expected to allow for significant sensitivity inside $0.1$ , which may potentially probe the tidal Jupiters as close as $a \\sim 0.5 {\\,\\rm AU}$ at $20 {\\,\\rm pc}$ .",
"One strategy to identify these tidally-powered Jupiters is to follow up long-period, highly eccentric planets with known radial velocity (RV) orbits, which are likely to be actively migrating.",
"For $e \\sim 1$ , one obtains maximum projected separation very close to apastron at $r_{\\perp , \\rm max} \\approx 2a\\sqrt{\\sin ^2\\omega \\cos ^2 i + \\cos ^2\\omega }$ , where $\\omega $ is the argument of periastron.",
"There is one RV planet, with $a\\sim 1{\\,\\rm AU}$ and $a_{\\rm F}\\lesssim 0.1{\\,\\rm AU}$ , known as HD 20782b [11], [33] at $36 {\\,\\rm pc}$ , and the best-fit parameters are $M \\sin i=1.9 M_{{\\,\\rm Jup}}$ , $a = 1.38{\\,\\rm AU}$ , $e = 0.97$ , $\\omega = 148^o$ (note that the eccentricity needs to confirmed as the periastron passage was not sufficiently probed).",
"It reaches maximum angular separation at $(0.28\\cos ^2 i + 0.72) 0.076$ at apocenter, which corresponds to $\\lesssim 2.9 \\lambda /D$ at $J$ -band, $\\lesssim 2.2 \\lambda /D$ at $H$ -band, $\\lesssim 1.7 \\lambda /D$ at $K$ -band and $\\sim \\lambda /D$ at $L^{\\prime }$ -band for a 10m telescope.",
"The target is difficult to image for most of the current and upcoming high-contrast imaging instruments but it is likely to be accessible by SCExAO at Subaru given its small expected inner working angle allowed by its extreme AO system (see Fig.1).",
"The long baseline of LBTI (22.8m) may be advantageous in terms of spatial resolution, but it is challenging given the inner working angle achievable for the current available instrument.",
"It may be accessible when the LBTI high contrast imaging system is refined (Private Communication, Philip Hinz 2012).",
"As previously discussed, if KCTF takes place, many long-period, low-$e$ Jupiters can be tidally-powered by radiating heat accumulated from past high-$e$ visits, and their occurrence rate can be $\\sim 10$ larger than that of the actively-migrating, high-$e$ population.",
"A possible observation strategy would be to directly image all RV planets with sufficiently large maximum angular separations and probe this population (as well as those that show a linear trend, which indicates the probable presence of long-period planets).",
"Planetary systems with known perturbers from RV residuals or imaging may receive preference since the planets in these system have a higher probability of Kozai-Lidov oscillation.",
"Advanced AO/coronagraphs and better post processing techniques (e.g., [14]) may help to increase the contrast and decrease the inner working angle.",
"It may also be possible to remove the speckle noise more efficiently and improve the detection sensitivity with information from the known orbital phase from RV, which will further boost the detectable contrast ratio and obtain smaller inner working angle.",
"A large fraction of nearby main-sequence stars have not been monitored by RV sufficiently long to find long-period Jupiters.",
"The all-sky astrometric mission Gaia is expected to be sensitive to planets more massive than Jupiter between $\\sim 0.5 - 3 {\\,\\rm AU}$ for all solar-type hosts within $50 {\\,\\rm pc}$ [17], so it will make a nearly complete census of tidally-powered Jupiters in the solar neighborhood and provide an excellent sample for direct-imaging follow-ups."
],
[
"Discussion and Future Prospect",
"If the tidally-powered Jupiters are directly imaged, their luminosities provide a direct measure of planet migration rate due to tidal dissipation and thus constrain high-$e$ migration mechanisms.Note however thermal tidal power generated at pericenter passages could be responsible for tidally powering long-period Jupiters [1], [2], [3].",
"In that case, the source of energy is from star light rather than orbit.",
"Combined with RV orbits, one can break the inclination degeneracy in RV to obtain the de-projected true mass and full orbital solution, making them an excellent laboratory to study planetary dynamics.",
"With measured mass and luminosity, one may probe the spectral formation of gas giant atmosphere as well as the physics of tidal dissipation.",
"It is interesting to note that, even though similar high-$e$ population may exist for binary stars [8], it is much more difficult to measure tidal luminosity directly due to nuclear burning.",
"As discussed in [31] and [7], transit surveys such as Kepler are ideal to find the eccentric migrating Jupiters due to their enhanced transit probability.",
"Space-based transit surveys that target bright stars may potentially provide an excellent sample of tidally-powered planets hosted by nearby stars that are suitable for direct-imaging study.",
"For transiting planets, high-precision IR light curves with secondary eclipse could in principle directly measure the tidal luminosity for these planets as well (see [18]).",
"Ground-based high-contrast imaging instruments are experiencing rapid development over the last few years, and the tidally powered Jupiters may turn out to be the most luminous planets to image for nearby solar-type main-sequence stars.",
"New instruments such as SCExAO should already be able to probe this population.",
"Future telescopes such as TMT, GMT and ELT can conduct a thorough survey for this population around nearby stars due to their smaller diffraction limits (see Fig.",
"1).",
"We thank Andy Gould, Cullen Blake, Jose Prieto, Scott Tremaine, Matias Zaldarriaga, Dave Spiegel, Rashid Sunyaev, Sasha Hinkley, Adam Burrows, and Phil Hinz for discussions.",
"Work by SD was performed under contract with the California Institute of Technology (Caltech) funded by NASA through the Sagan Fellowship Program.",
"BK is supported by NASA through the Einstein Postdoctoral Fellowship awarded by Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060.",
"AS acknowledges support from a John N. Bahcall Fellowship awarded at the Institute for Advanced Study, Princeton."
]
] | 1204.1056 |
[
[
"Stimulated generation of superluminal light pulses via four-wave mixing"
],
[
"Abstract We report on the four-wave mixing of superluminal pulses, in which both the injected and generated pulses involved in the process propagate with negative group velocities.",
"Generated pulses with negative group velocities of up to $v_{g}=-1/880c$ are demonstrated, corresponding to the generated pulse's peak exiting the 1.7\\,cm long medium $\\approx50$\\,ns earlier than if it had propagated at the speed of light in vacuum, $c$.",
"We also show that in some cases the seeded pulse may propagate with a group velocity larger than $c$, and that the generated conjugate pulse peak may exit the medium even earlier than the amplified seed pulse peak.",
"We can control the group velocities of the two pulses by changing the seed detuning and the input seed power."
],
[
"1 Stimulated generation of superluminal light pulses via four-wave mixing Ryan T. GlasserR.",
"Glasser and U. Vogl contributed equally to this work.",
"Ulrich Vogl[] Paul D. Lett National Institute of Standards and Technology and Joint Quantum Institute, NIST and University of Maryland, Gaithersburg, MD 20899 USA We report on the four-wave mixing of superluminal pulses, in which both the injected and generated pulses involved in the process propagate with negative group velocities.",
"Generated pulses with negative group velocities of up to $v_{g}=-\\frac{1}{880}c$ are demonstrated, corresponding to the generated pulse's peak exiting the 1.7 cm long medium $\\approx 50$ ns earlier than if it had propagated at the speed of light in vacuum, $c$ .",
"We also show that in some cases the seeded pulse may propagate with a group velocity larger than $c$ , and that the generated conjugate pulse peak may exit the medium even earlier than the amplified seed pulse peak.",
"We can control the group velocities of the two pulses by changing the seed detuning and the input seed power.",
"42.65.-k,42.65.Hw,42.65.Ky,42.50.Nn There has been substantial recent interest in modifying the group velocity of optical pulses, resulting in slow, stopped, and superluminal light [1].",
"Experiments have resulted in extremely slow light with very small group velocities [2], [3], [4], [5], [6], [7], [8], while others exhibit the ability to store and retrieve optical pulses [9], [10], [11], [12].",
"It is also possible to generate a dispersion relation that results in negative group velocities [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25].",
"In such cases when the group velocity of light in a material is negative, the exiting pulse's peak can appear to exit the medium before the peak of the input pulse enters.",
"The peak that exits the medium does not correspond point-to-point to the peak of the input pulse.",
"Rather, the phenomena of re-phasing allows for different frequency components of the input pulse to propagate at different velocities, which results in the apparent pulse peak advancement seen in fast light experiments [26], [27].",
"We report the stimulated generation of light pulses that propagate with a group velocity faster than the speed of light in vacuum, via four-wave mixing (4WM) in hot rubidium vapor.",
"The 4WM process employed here involves injecting one weak beam into the medium and pumping with a beam at a different frequency, as seen in Fig.",
"1.",
"A beam at a third frequency is generated via the process, as photons from the pump beam are converted into photons in the injected seed and generated conjugate modes.",
"The amplified seed pulse is shown to have a negative group velocity due to the 4WM dispersion, and stimulates the generation of the conjugate pulse that may appear to propagate even faster, as seen in Fig.",
"2.",
"The anomalous dispersion results from asymmetric gain and absorption lines at the seed and generated conjugate pulse frequencies.",
"We show that it is possible to manipulate the group velocities of the two modes relative to one another, to some extent, by detuning the seed pulse or varying the input seed intensity.",
"The scheme could be applicable to a variety of optical communications scenarios, where the correction of pulse jitter by advancing or delaying pulses may be necessary.",
"The present results will allow us to investigate the effects of superluminal group velocities on quantum entanglement and squeezed light, both of which may be produced via the 4WM process [7], [28].",
"The first experiments to produce fast light used absorption lines [29], [14], [13], which exhibit strong negative dispersion at the center of the line, but the dispersion is accompanied by significant attenuation of the input pulses.",
"More recently, schemes using gain doublets have been shown to produce fast light ([1], [15], [17] and references therein).",
"All previous experiments involving fast light, to our knowledge, involve injecting a pulse into the fast light medium, and observing that it appears to exit the medium faster than a reference pulse traveling at the speed of light in vacuum.",
"In our experiment, we show not only that the injected pulse propagates with a negative group velocity, but also that a second pulse at another frequency and in a separate spatial mode is generated by the 4WM process, which may appear to propagate faster than a vacuum-traversing reference pulse as well.",
"In contrast to some previous 4WM experiments [7], the group velocities of the seed and conjugate modes under the fast light conditions in the present experiment exhibit little coupling due to the large differential absorption between the two modes.",
"The gain line for the generated conjugate frequency sits inside the edge of an absorption line, resulting in conjugate pulses with a significantly weaker amplitude than the amplified seed pulses which are far off-resonance.",
"Previous theoretical investigations involving four-level atomic systems to generate fast light have been performed in which the probe field is co-propagating with the coupling field [24].",
"Additional theoretical work has shown that the double-lambda scheme and electromagnetically induced transparency may result in slow and fast light for two input probe beams [30].",
"Experimental work has been done in which Raman gain of an input probe results in slow light, while pump depletion results in anomalous dispersion at the pump frequency (heterodyne detection was used to resolve the different frequency components) [23].",
"The present experiment differs from these theoretical and experimental investigations in that we generate a conjugate pulse in a separate spatial mode that may propagate superluminally, in addition to the seed pulse.",
"This also has the benefit of allowing for direct detection of both pulses, rather than having to use heterodyne detection to resolve the separate frequency modes.",
"For the generation of a strong anomalous dispersion we use the 4WM process shown schematically in Fig.",
"1.",
"The gain line that the input seed pulse experiences results in negative dispersion at the gain line edges, giving rise to superluminal propagation of the injected pulse.",
"The generated conjugate pulse experiences a broad region of negative dispersion due to the gain and absorption line profile resulting from this 4WM process, which is asymmetric (see Fig.",
"1) and exhibits absorption on the high frequency side of the gain line [7], [28].",
"Here the seed and conjugate pulses sample independent dispersion features approximately 6 GHz apart, in contrast to the gain-doublet scheme that is sometimes used to create a single dispersion feature [15].",
"Figure: Experimental setup and double-lambda scheme.",
"The pump laser is detuned approximately 400 MHz to the blue of the Rb D1 line.",
"Generated conjugate pulses are created inside the cell, and exit at the same angle as the seed pulses.",
"The generated conjugate is on the wing of an absorption line, while the injected seed is blue detuned ≈\\approx 6 GHz relative to the conjugate.",
"The double-lambda scheme is shown, and the two-photon detuning δ\\delta is indicated.",
"The lineshapes of the gain lines at the seed and conjugate frequencies are also shown.The 4WM process involves the annihilation of two pump photons, and the creation of a single probe and conjugate photon, as is evident when examining the simplified phenomenological interaction Hamiltonian, $\\hat{H}_{I}=\\chi \\hat{a}^{\\dag 2}\\hat{b}\\hat{c}+\\chi ^{*}\\hat{a}^{2}\\hat{b}^{\\dag }\\hat{c}^{\\dag }$ [31], [32].",
"Here, the modes $\\hat{a}$ , $\\hat{b}$ and $\\hat{c}$ correspond to the pump, injected seed and generated conjugate, respectively, and $\\chi $ is the effective interaction strength that depends on the third-order susceptibility and the length of the interaction.",
"Combined with energy conservation, this results in the constraint on frequencies such that $2\\omega _{pump}=\\omega _{s}+\\omega _{c}$ .",
"The generated conjugate pulses in the present experiment experience both absorption and gain, whereas the seed pulses experience only gain, as seen in Fig.",
"1.",
"Figure: Traces of the amplified seed and generated conjugate pulses exhibiting negative group velocities, for two different seed pulse detunings.",
"The blue, green dashed and red dot-dashed curves correspond to the generated, amplified seed and reference pulses, respectively.",
"The pulse amplitudes have been normalized relative to the reference pulses, and are averaged 512 times.",
"The amplified seed and generated pulses on the left are scaled by a factor of 0.36×\\times and 0.32×\\times relative to the reference pulses.",
"On the right figure, the seed pulse is scaled by a factor of 0.12×\\times and the generated pulse is scaled by 0.78×\\times relative to the reference.The 4WM process is pumped with a strong ($\\approx $ 220 mW) continuous-wave linearly polarized laser detuned $\\approx \\,$ 400 MHz to the blue of the Rb D1 line $\\left|5S_{1/2},F=2\\right\\rangle \\rightarrow \\left|5P_{1/2}\\right\\rangle $ , at $\\lambda \\,\\approx \\,795$ nm (see Fig.",
"1).",
"Weak input seed pulses with peak powers of $\\approx \\,$ 5$\\,\\mu $ W with a full-width at half-maximum (FWHM) of 200 ns and a frequency bandwidth of $\\approx 5$ MHz, are injected at an angle of $\\approx 1^{\\circ }$ relative to the pump.",
"The seed pulses are orthogonally polarized and detuned $\\approx 3$ GHz to the blue of the pump beam.",
"The injected seed pulse frequency is varied by changing the operating frequency of a double-passed 1.5 GHz acousto-optic modulator, which is used to generate the seed from part of the pump beam.",
"This allows us to precisely control the relative frequency detuning between the pump and seed beams, though the absolute position of the pump beam frequency varies slightly in each experimental run.",
"The pump and probe are focused into the center of the cell with focal spot sizes of $\\approx 800\\,\\mu m\\times 1000\\,\\mu $ m and $\\approx 700\\,\\mu m\\times 700\\,\\mu $ m, respectively.",
"A conjugate pulse is created which propagates at the same angle as the seed relative to the pump, but in the opposite azimuthal direction.",
"The $^{85}$ Rb cell is 1.7 cm long and is held at a temperature of $\\approx 116^{\\circ }$ C. The seed and generated beams are spatially filtered with irises after the cell in order to select only the central spots of the 4WM beams and filter out residual pump light.",
"Approximately $2/3$ of the pulse power is detected.",
"The pulses are then detected with a high-gain avalanche photodiode operating in the linear mode and fed directly to an oscilloscope and averaged 512 times.",
"Reference pulses are obtained by measuring the seed pulses when the pump beam is blocked, and serve as a measure for the vacuum propagation speed of the pulses for each experimental run.",
"The reference pulses were also measured with the Rb cell removed, to ensure that they propagate at the same speed with and without the cell present (when the pump is blocked) to within our experimental time resolution.",
"Most previous experiments involving fast light use absorption lines, single gain lines, or double gain lines that are Lorentzian in shape to generate the desired dispersion [1], [13], [15], [17].",
"In the present experiment, the generated conjugate experiences an asymmetric gain line with an absorption dip on one of the wings, while the seed experiences only a gain line, as seen in Fig.",
"1 [7].",
"Modeling the line shape for the generated conjugate mode as a Lorentzian for the gain and a second Lorentzian for the absorption results in: k()=cn0+2 12(gg(-g)+ig+aa(-a)+ia) Here $n_{0}$ is the background index, and $\\alpha _{g}$ and $\\alpha _{a}$ are the coefficients of the gain and absorption components, respectively, with $\\alpha _{g}<0$ resulting in gain.",
"The operating frequency and the center frequencies of the gain and absorption lines are denoted by $\\omega $ , $\\omega _{g}$ , and $\\omega _{a}$ .",
"The linewidths of the gain and absorption lines are $\\gamma _{g}$ and $\\gamma _{a}$ , respectively.",
"Due to the large absorption present at the generated pulse frequency, we use the maximum gain of approximately $G\\,=\\,e^{-\\alpha _{g}L}=20$ measured at the seed frequency to determine the gain coefficient in the presence of absorption.",
"The modeled gain line is then fit to the measured gain line, which has a FWHM of $\\approx $ 20 MHz, as well as an absorption dip at the wing.",
"The parameters in the model are $\\alpha _{g}=-175$ m$^{-1}$ , $\\gamma _{g}=20$ MHz, $\\alpha _{a}=95$ m$^{-1}$ and $\\gamma _{a}=23$ MHz.",
"The group velocity $v_{g}$ is defined by k1=dkd=1vg= 3 n0c-gg2(-g+ig)2-aa2(-a+ia)2, where $k_{1}$ is the first term in the power series expansion of $k(\\omega )$ about the central frequency.",
"The resulting expected time delays and advancements from this model are shown in Fig.",
"3, with the inset showing a comparison between the measured and modeled conjugate lineshape.",
"Both the magnitude and detuning dependence of the pulse advancement/delay fit the experiment remarkably well.",
"The seed gain line is modeled as a single gain lineshape that has an asymmetric lineshape as measured (modeled as the sum of several Lorentzians to account for the asymmetry).",
"We speculate that the modeled relative advancements for the seed pulses do not fit the measured data very well due to the steep dispersion of the seed gain line.",
"This results in the seed pulse's bandwidth exceeding some regions of “linear\" dispersion, making higher order terms in the power series expansion of $k(\\omega )$ contribute more significantly.",
"Figure: Generated conjugate and amplified seed pulse peak advancement versus relative seed detuning.",
"The black triangles correspond to the amplified seed pulse peak advancement when the pump is present.",
"The blue circles correspond to the generated conjugate pulse peak advancement.",
"Below seed detunings of ≈-20\\approx -20 MHz, the generated conjugate is too weak to be measured.",
"The input seed peak power is approximately 5μ5\\,\\mu W. The blue curve is the generated pulse's expected advancement derived from the double-Lorentzian gain and absorption line model corresponding to the solid curve in the inset.",
"The inset shows the frequency dependent total absorption coefficient α=\\alpha =Im(n(ω))(n(\\omega )), with n(ω)=ck(ω) ωn(\\omega )=\\frac{ck(\\omega )}{\\omega } (plotted is -α\\alpha , so that gain corresponds to positive values).",
"The dotted curve in the inset is the measured conjugate lineshape.",
"The black curve is the expected advancement for the modeled seed gain line.Experimentally the group velocity is determined by the measured arrival time of the pulse peak relative to a reference pulse propagating at $c$ .",
"The next order term is the lowest-order term that contributes to pulse reshaping, as it describes dispersion in the group velocity.",
"The superluminal generated pulses in this experiment exhibit some reshaping relative to the reference pulses.",
"The amount of pulse reshaping is sensitive to various experimental parameters such as beam waists, cell temperature and input pulse width.",
"In the present experiment we operated with parameters that allow for minimal reshaping, but consequently sub-optimal pulse advancement.",
"When the strong pump beam is present, it is possible for both the injected seed and generated conjugate pulses to exhibit negative group velocities, depending on the pump and injected seed detunings.",
"A negative group velocity can be understood by considering its relation to a pulse's arrival time delay, $\\Delta T=\\frac{L}{v_{g}}-\\frac{L}{c}$ , after propagating some distance $L$ .",
"When the group velocity is negative, so is the time delay, which corresponds to the pulse peak exiting the medium at a time $-\\Delta T$ sooner than if a similar pulse had traversed the same distance in vacuum.",
"With a cell length of 1.7 cm, we obtain a maximum pulse peak advancement of 50 ns for the generated pulse, which corresponds to a group velocity of $v_{g}=-\\frac{1}{880}c$ , and a relative pulse peak advancement of $25\\%$ .",
"This is the first observation of stimulated generation of superluminal pulses, in particular via the 4WM interaction with an injected seed.",
"The peak of the stimulated conjugate pulse exits the medium significantly before the peak of the seed pulse enters.",
"In vacuum, the seed pulse would traverse the Rb cell length in $\\approx $ 0.057 ns, but the generated conjugate's peak exits $\\approx $ 50 ns earlier when optimized for minimal pulse distortion.",
"Allowing for more severe distortion, we have obtained a maximum conjugate pulse peak relative advancement of 90 ns.",
"In Figs.",
"2b and 3 we show cases where the amplified seed pulse exhibits a negative group velocity and the generated conjugate pulse's group velocity may be tuned such that the pulse peak exits the medium prior to the exit of the peak of the seed pulse (all uncertainties shown in the figures are one standard deviation, combined statistical and systematic uncertainties).",
"As previously mentioned, the absolute frequency of the pump beam varies somewhat during each experimental run.",
"This has the effect of slightly changing the position of the gain lines as well their shape.",
"The relative pulse peak advancement as a function of the two-photon detuning maintains the same overall shape as in Fig.",
"3 despite this.",
"One consequence is that there may be slight shifts in the relative advancements between the seed and conjugate pulses.",
"This results in the conjugate pulse in Fig.",
"2b being slightly more advanced than the seed pulse when the two-photon detuning is set to 17 MHz, a variance from what is indicated in Fig. 3.",
"The generated pulse is somewhat reshaped relative to the seed pulse, but the pulse peak and leading edge at half-maximum are significantly advanced relative to the reference pulse.",
"At a detuning set for the maximum advancement of the generated pulse relative to the injected pulse ($\\delta \\approx $ 23 MHz), the generated pulse has approximately $20\\%$ of the peak amplitude of the reference (input) pulse.",
"Additionally, under the same conditions, the generated conjugate pulse is shown to exit the superluminal medium with a pulse peak advancement of 8$\\%$ relative to the superluminal seeded pulse, and 25$\\%$ relative to a 200 ns FWHM reference pulse traveling at $c$ .",
"We note that this advancement is achieved for a peak gain of $G\\,\\approx $ 20, which is a relatively large advancement considering it has been theoretically shown that for systems consisting of an absorption line in the center of a broad gain background of $G\\,=\\,e^{32}$ the max relative advancement is only $2\\sqrt{2}$ [33].",
"Figure: Generated conjugate and amplified seed pulse peak advancements versus normalized input seed power at a relative detuning of δ≈18\\delta \\approx 18MHz.",
"The black triangles correspond to the amplified seed pulse and the blue dots correspond to the generated conjugate pulse.",
"The curves are fits to the data as a function of the logarithm of the input seed power.The measured pulse peak advancement for both the amplified seed pulse and the generated conjugate pulse as a function of the seed pulse detuning is shown in Fig.",
"3.",
"For a variety of detunings the generated pulse closely resembles the input seed pulse shape.",
"We have some flexibility in controlling the relative group velocities of the amplified seed and generated conjugate pulses, in particular we see cases when both pulses are superluminal, with the generated pulse peak exiting the medium first.",
"At the largest detunings the generated pulse exhibits significant reshaping relative to the seed.",
"The generated pulse and amplified seed pulse advancements versus injected seed power are shown in Fig.",
"4.",
"This situation is unusual in that the intensity of the beam influences its own group velocity.",
"This is somewhat analogous to a Kerr medium in which the intensity-dependent index of refraction alters the phase velocity.",
"To account for the nonlinear, intensity-dependent part of the refractive index, typically the nonlinear Schrödinger equation is used [34].",
"We speculate that in addition to this, group velocity dispersion and higher order terms, as well as some coupling between the seed and generated beams all contribute to the nontrivial intensity dependence of the relative pulse peak advancement.",
"Despite this complexity, the pulse advancement is well approximated by a natural logarithm dependence on the input intensity, as seen by the fits in Fig. 4.",
"The generated conjugate pulse advancement is also controllable by varying the input seed power, resulting in a tunable advancement.",
"We have shown that a superluminal seed pulse can stimulate the creation of an additional superluminal conjugate pulse by the 4WM interaction.",
"This generated pulse can propagate even faster than the superluminally-propagating seed pulse, with relatively small distortion.",
"The pulse peak advancements of the two pulses are tunable by changing the input seed detuning and power.",
"This could have applications in optical communication schemes in which pulse jitter may be compensated for by advancing or delaying pulses accordingly.",
"In an on-off keying optical communications system, if the jitter is small relative to the pulse width, it may be beneficial to advance a pulse rather than delay the entire pulse train.",
"The benefits of applying this type of jitter correction would have to be weighed against the disadvantages, which include added noise and some degree of pulse reshaping.",
"Additionally, due to the multi-spatial-mode nature of 4WM in atomic vapors, the present results suggest that the superluminal propagation of images may be possible in future experiments.",
"The high level of squeezing obtainable via 4WM, and the fact that fast light may also be obtained at a relatively low gain (and added noise), suggests that one may be able to use the quantum correlations between the twin beams to further investigate the details of superluminal light pulse propagation in such media.",
"In particular, one can hope to use the quantum correlations to examine experimentally which part of the input pulse is causally linked to the peak of the output pulse.",
"This work was supported by the Air Force Office of Scientific Research.",
"This research was performed while Ryan Glasser held a National Research Council Research Associateship Award at NIST.",
"Ulrich Vogl would like to thank the Alexander von Humboldt Foundation."
]
] | 1204.0810 |
[
[
"Colored-noise thermostats \\`a la carte"
],
[
"Abstract Recently, we have shown how a colored-noise Langevin equation can be used in the context of molecular dynamics as a tool to obtain dynamical trajectories whose properties are tailored to display desired sampling features.",
"In the present paper, after having reviewed some analytical results for the stochastic differential equations forming the basis of our approach, we describe in detail the implementation of the generalized Langevin equation thermostat and the fitting procedure used to obtain optimal parameters.",
"We discuss in detail the simulation of nuclear quantum effects, and demonstrate that, by carefully choosing parameters, one can successfully model strongly anharmonic solids such as neon.",
"For the reader's convenience, a library of thermostat parameters and some demonstrative code can be downloaded from an on-line repository."
],
[
"Markovian and non-Markovian formulations",
"The Langevin equation for a particle with position $q$ and momentum $p$ , subject to a potential $V(q)$ , can be written as $\\begin{split}\\dot{q}=&p\\\\\\dot{p}=&-V^{\\prime }(q) -a_{pp} p + b_{pp}\\xi (t).\\end{split}$ where $\\xi (t)$ represent an uncorrelated, Gaussian-distributed random force with unitary variance and zero mean [$\\left<\\xi \\right>=0$ , $\\left<\\xi (t)\\xi (0)\\right>=\\delta (t)$ ].",
"Here and what follows we use mass-scaled coordinates.",
"Furthermore, for consistency, the friction coefficient (usually denoted by $\\gamma $ ) is here given the symbol $a_{pp}$ , while $b_{pp}$ is the intensity of the random force.",
"In this notation, the fluctuation-dissipation theorem (FDT) reads $b_{pp}^2= 2 a_{pp} k_B T$ .",
"If this relation holds, the dynamics generated by Eq.",
"(REF ) will sample the canonical ensemble at temperature $T$[18], [4].",
"As explained in the Introduction, in order to bypass the complexity of dealing with a non-Markovian formulation directly, we supplement the system with $n$ additional degrees of freedom $\\mathbf {s}=\\left\\lbrace s_i\\right\\rbrace $ which are linearly coupled to the physical momentum and between themselves.",
"The resulting SDE can be cast into the compact form $\\begin{split}\\dot{q}=&p\\\\\\!\\left(\\!",
"\\begin{array}{c}\\dot{p}\\\\ \\dot{\\mathbf {s}} \\end{array}\\!\\right)\\!=&\\left(\\!\\begin{array}{c}-V^{\\prime }(q)\\\\ \\mathbf {0}\\end{array}\\!\\!\\right)\\!-\\!\\left(\\!\\begin{array}{cc}a_{pp} & \\mathbf {a}_p^T \\\\\\bar{\\mathbf {a}}_p & \\mathbf {A}\\end{array}\\!\\right)\\!\\left(\\!\\begin{array}{c}p\\\\ \\mathbf {s}\\end{array}\\!\\right)\\!+\\!\\left(\\!\\begin{array}{cc}b_{pp} & \\mathbf {b}_p^T \\\\\\bar{\\mathbf {b}}_p & \\mathbf {B}\\end{array}\\!\\right)\\!\\left(\\!\\begin{array}{c}{2}{*}{\\xi }\\\\ \\\\\\end{array}\\!\\right),\\end{split}$ Here, $\\xi $ is a vector of $n+1$ uncorrelated Gaussian random numbers, with $\\left<\\xi _i\\left(t\\right)\\xi _j\\left(0\\right)\\right>=\\delta _{ij}\\delta \\left(t\\right)$ .",
"Clearly, Eq.",
"(REF ) is recovered when $n=0$ .",
"For an harmonic potential $V(q)=\\frac{1}{2}\\omega ^2 q^2$ , Eqs.",
"(REF ) are linear, and an Ornstein-Uhlenbeck process is recovered whose time propagation can be evaluated analytically.",
"In the non-linear case one can use the Trotter-decomposition to split the dynamics into a linear part, which evolves the $(p,\\mathbf {s})$ momenta, and a non-linear part, which evolves the Hamilton equations[19].",
"This is facilitated by the fact that the dynamics of $(p,\\mathbf {s})$ alone is linear, and its exact finite-time propagator can be analytically evaluated (see Subsection REF ).",
"Here and in the rest of the paper, we adopt the same notation introduced in Ref.",
"[9] to distinguish between matrices acting on the full state vector $\\mathbf {x}=\\left(q,p,\\mathbf {s}\\right)^T$ or on parts of it, as illustrated below: $\\begin{array}{ccccc}& q & p & \\mathbf {s} & \\rule {0pt}{12pt}\\\\ \\cline {2-4}\\multicolumn{1}{c|}{q} & m_{qq} & m_{qp} & \\multicolumn{1}{c|}{\\mathbf {m}_q^T} & \\rule {0pt}{12pt}\\\\\\cline {3-4}\\multicolumn{1}{c|}{p} & \\multicolumn{1}{c|}{\\bar{m}_{qp}} & m_{pp} & \\multicolumn{1}{c|}{\\mathbf {m}_p^T} & \\rule {0pt}{12pt}\\\\\\cline {4-4}\\multicolumn{1}{c|}{\\mathbf {s}} & \\multicolumn{1}{c|}{\\bar{\\mathbf {m}}_q} & \\multicolumn{1}{c|}{\\bar{\\mathbf {m}}_p} & \\multicolumn{1}{c|}{\\mathbf {M}} & \\rule {0pt}{12pt}\\\\\\cline {2-4}\\end{array}\\hspace{-8.0pt}\\begin{array}{cc}\\rule {0pt}{12pt}\\\\ \\rule {0pt}{12pt}\\\\\\left.\\rule {0pt}{12pt}\\right\\rbrace \\!\\mathbf {M}_p \\\\\\end{array}\\hspace{-8.0pt}\\begin{array}{cc}\\rule {0pt}{12pt}\\\\\\left.\\rule {0pt}{20pt}\\right\\rbrace \\!\\mathbf {M}_{qp}\\end{array}$ The Markovian dynamical equations (REF ) are equivalent to a non-Markovian process for the physical variables only.",
"This is best seen by first considering the evolution of the $(p,\\mathbf {s})$ variables in the free-particle analogue of Eqs.",
"(REF ).",
"The additional degrees of freedom $\\mathbf {s}$ can be integrated away, and one is left with (see Ref.",
"[4] and Appendix ) $\\dot{p}=-\\int _{-\\infty }^t K(t-s) p(s)\\mathrm {d} s +\\zeta (t)$ where the memory kernel $K(t)$ is related to the elements of $\\mathbf {A}_p$ by $K(t)=2a_{pp} \\delta (t)-\\mathbf {a}_p^T e^{-\\left|t\\right|\\mathbf {A}}\\bar{\\mathbf {a}}_p.",
"$ Based on the fact that the the free-particle dynamics of $(p,\\mathbf {s})$ is an OU process, one also finds than the relationship between the static covariance matrix $\\mathbf {C}_p=\\left< \\left(p,\\mathbf {s}\\right)^T \\left(p,\\mathbf {s}\\right)\\right>$ , the drift matrix $\\mathbf {A}_p$ and the diffusion matrix $\\mathbf {B}_p$ is given by:Note the remarkable formal analogy between Eq.",
"(REF ) and the equations for the orthogonality constraints in Car-Parrinello dynamics, see e.g.",
"Ref.",
"[41] $\\mathbf {A}_p\\mathbf {C}_p+\\mathbf {C}_p\\mathbf {A}_p^T=\\mathbf {B}_p\\mathbf {B}_p^T.$ In Appendix we show that setting $\\mathbf {C}_p=k_B T$ is sufficient to satisfy the FDT.",
"In this case, Eq.",
"(REF ) fixes $\\mathbf {B}_p$ once $\\mathbf {A}_p$ is given.",
"FDT also implies that the colored-noise autocorrelation function $H(t)=\\left<\\zeta (t)\\zeta (0)\\right>$ is equal to $k_BTK(t)$ , whereas the more complex relation between $K(t)$ and $H(t)$ , valid in the general case, is reported in Eq.",
"(REF ).",
"Since there is no explicit coupling between the position $q$ and the additional momenta $\\mathbf {s}$ , one can check that exactly the same dimensional reduction can be performed in the case of an arbitrary potential coupling $p$ and $q$ , and that Eqs.",
"(REF ) correspond to the non-Markovian process $\\begin{split}\\dot{q}&=p\\\\\\dot{p}&=-\\frac{\\partial V}{\\partial q}-\\int _{-\\infty }^t K(t-s) p(s)\\mathrm {d} s +\\zeta (t).\\end{split}$ In the memory kernel (REF ), $\\mathbf {A}$ can be chosen to be a general real matrix, and can have complex eigenvalues, provided they have a positive real part.",
"This results in a $K(t)$ that is a linear combination of exponentially damped oscillations.",
"Therefore, a vast class of non-Markovian dynamics can be represented by Markovian equations such as (REF )."
],
[
"Exact solution in the harmonic limit",
"The thermostats typically used in MD simulations have a few parameters, that are chosen by trial and error.",
"A thermostat based on Eqs.",
"(REF ) depends on a much larger number of parameters, and hence the fitting procedure is more complex.",
"It is therefore important to find ways to compute a priori analytical estimates so as to guide the tuning of the thermostat.",
"To this end, we examine the harmonic oscillator, which is commonly used to model physical and chemical systems.",
"By choosing $V(q)=\\frac{1}{2}\\omega ^2 q^2$ the force term in (REF ) becomes linear, and the dynamics of $\\mathbf {x}=(q,p,\\mathbf {s})^T$ is the OU process $\\dot{\\mathbf {x}}=-\\mathbf {A}_{qp}\\mathbf {x}+\\mathbf {B}_{qp}\\xi $ .",
"In Eqs.",
"(REF ) the $\\mathbf {s}$ degrees of freedom are coupled to the momentum only.",
"Therefore, most of the additional entries in $\\mathbf {A}_{qp}$ and $\\mathbf {B}_{qp}$ are zero, and the equations for $\\mathbf {x}$ read $\\!\\left(\\!\\begin{array}{c}\\dot{q} \\\\\\dot{p}\\\\ \\dot{\\mathbf {s}}\\end{array}\\!\\right)\\!=-\\!\\left(\\!\\begin{array}{ccc}0 & -1 & \\mathbf {0} \\\\\\omega ^2 & a_{pp} & \\mathbf {a}_p^T \\\\\\mathbf {0} & \\bar{\\mathbf {a}}_p & \\mathbf {A}\\end{array}\\right)\\!\\left(\\!\\begin{array}{c}q\\\\ p\\\\ \\mathbf {s}\\end{array}\\!\\right)\\!+\\!\\left(\\!\\begin{array}{ccc}0 & 0& \\mathbf {0}\\\\0 & \\multicolumn{2}{c}{{2}{*}{\\mathbf {B}_p}}\\\\\\mathbf {0} & & \\\\\\end{array}\\!\\right)\\!\\!\\left(\\!\\begin{array}{c}0\\\\{2}{*}{\\xi } \\\\ \\\\\\end{array}\\!\\right)\\!.$ The exact finite-time propagator for Eqs.",
"(REF ) can be computed, and so it is possible to obtain any ensemble average or time-correlation function analytically.",
"Of course, one is most interested in the expectation values of the physical variables $q$ and $p$ .",
"In particular, one can obtain the fluctuations $\\left<q^2\\right>$ and $\\left<p^2\\right>$ and correlation functions of the form $\\left<q^2(t)q^2(0)\\right>$ , which can be used to measure the coupling between the thermostat and the system.",
"The resulting expressions are simple to evaluate but lengthy, and we refer the reader to Appendix for their explicit form.",
"One can envisage, using the estimates computed for an oscillator of frequency $\\omega $ , to predict and hence optimize the response of a normal mode of a similar frequency in the system being studied.",
"Furthermore, thanks to the properties of Eq.",
"(REF ), one does not need to perform a normal-modes analysis to turn this idea into a practical method.",
"Consider indeed a perfect harmonic crystal, and apply an independent instance of the GLE thermostat to the three cartesian coordinates of each atom.",
"It is easy to see that, since Eq.",
"(REF ) is linear, and contains Gaussian noise, the thermostatted equations of motion are invariant under any orthogonal transformation of the coordinates.",
"Therefore, the resulting dynamics can be described on the basis of the normal modes just as in ordinary Hamiltonian lattice dynamics.",
"As a consequence, each phonon will respond independently as a 1-D oscillator with its own characteristic frequency.",
"Thus, to tune the GLE thermostat, one only needs the analytical results in the one-dimensional case, evaluated as a function of $\\omega $ .",
"The parameters can then be optimized for a number of different purposes, based solely on minimal information on the vibrational spectrum of the system under investigation, without any knowledge of the phonons eigenmodes.",
"The invariance properties of the GLE thermostat lead to additional advantages.",
"For instance, we can contrast its behavior with that of Nosé-Hoover (NH) chains, based on equations which are quadratic in $p$ (see Appendix ).",
"As a consequence of the nonlinearity, the efficiency of an NH chains thermostat for a multidimensional oscillator depends on the orientation of the eigenmodes relative to the cartesian axes, an artefact which is absent in our case.",
"Having set the background, we now turn to the description of the various applications of Eqs.",
"(REF )."
],
[
"Efficient canonical sampling",
"We first discuss the design of a GLE which can optimally sample phase space.",
"In this case, the target stationary distribution is the canonical ensemble, so the equations of motion need to satisfy the detailed-balance condition.",
"Still, there is a great deal of freedom available in the choice of the autocorrelation kernel or, equivalently, in the choice of $\\mathbf {A}_p$ and $\\mathbf {B}_p$ matrices.",
"These free parameters can be used to optimize the sampling efficiency.",
"To this end, we must first define an appropriate merit function.",
"Standard choices are the autocorrelation times of the potential and total energy ($V$ and $H$ respectively): $\\begin{split}\\tau _V=&\\frac{1}{\\left<V^2\\right>}\\int _0^\\infty \\left<(V(t)-\\left<V\\right>)(V(0)-\\left<V\\right>)\\right> \\mathrm {d}t \\\\\\tau _H=&\\frac{1}{\\left<H^2\\right>}\\int _0^\\infty \\left<(H(t)-\\left<H\\right>)(H(0)-\\left<H\\right>)\\right> \\mathrm {d}t. \\\\\\end{split} $ In the harmonic case, these can be readily computed in terms of correlation times of $q^2$ and $p^2$ (see Appendix ), and will depend on $\\mathbf {A}_p$ and the oscillator's frequency $\\omega $ .",
"For example, one easily finds that in the white-noise limit, with no additional degrees of freedom as in Eq.",
"(REF ), $\\tau _H\\left(\\omega \\right)=\\frac{1}{a_{pp}}+\\frac{a_{pp}}{4\\omega ^2}, \\quad \\tau _V\\left(\\omega \\right)=\\frac{1}{2a_{pp}}+\\frac{a_{pp}}{2\\omega ^2}, \\quad .",
"$ Both response times are constant in the high-frequency limit, and increase quadratically in the low-frequency extreme of the spectrum.",
"For a given frequency one can choose $a_{pp}$ so as to minimize the correlation time - thus enhancing sampling.",
"It should be noted that Eqs.",
"(REF ) contain a “trivial” dependence on $\\omega $ , as one expects that sampling a normal mode would require at least a time of the order of its vibrational period.",
"One can thus define a renormalized $\\kappa (\\omega )=\\left[\\tau (\\omega ) \\omega \\right]^{-1}$ as a measure of the efficiency of the coupling.",
"In the white-noise case, $\\kappa =1$ for the optimally-coupled frequency ($\\omega _H=a_{pp}/2$ and $\\omega _V=a_{pp}$ , respectively), and decreases linearly for lower and higher values of $\\omega $ .",
"While this result in itself provides a guide to choose a good value of the friction coefficient in conventional (white-noise) Langevin dynamics, we can enhance the value of $\\kappa (\\omega )$ over a broader frequency range, by using a colored-noise SDE.",
"If we want to obtain canonical sampling, the FDT has to hold, so that $\\mathbf {C}_p=k_B T$ .",
"We therefore consider the entries of $\\mathbf {A}_p$ as the only independent parameters, since $\\mathbf {B}_p$ is then determined by Eq.",
"(REF ).",
"In practice, we set up a fitting procedure, in which we choose a set of frequencies $\\omega _i$ , distributed over a broad range $\\left(\\omega _{min},\\omega _{max}\\right)$ .",
"For an initial guess for the thermostat matrix $\\mathbf {A}_p$ we compute $\\kappa (\\omega )$ for each of these frequencies.",
"We then vary $\\mathbf {A}_p$ , so as to optimize $\\min _i \\kappa (\\omega _i)$ , and aim at a sampling efficiency on the range $\\left(\\omega _{min},\\omega _{max}\\right)$ which is as high and frequency-independent as possible.",
"We will discuss this fitting procedure in more detail in Section .",
"Figure: Sampling efficiency as estimated from Eq.",
"()for an harmonic oscillator, plotted as a function of the frequency ω\\omega .",
"The κ(ω)\\kappa (\\omega )curve for a white-noise Langevin thermostat optimized for ω=1\\omega =1 [black, dotted lines, Eq.",
"()]is contrasted with those for a set of optimized GLE thermostats.The panels, from bottom to top, contain the results fitted respectively over a frequency range spanningtwo, four and six orders of magnitudes around ω=1\\omega =1.",
"Dark, continuous lines correspond tomatrices with n=4n=4, and dashed, lighter lines to n=2n=2.The GLE curves correspond to the sets of parameters kv_4-2, kv_2-2,kv_4-4, kv_2-2, kv_4-6, kv_2-6,which can be downloaded from an on-line repository.In Figure REF we compare the optimized $\\kappa (\\omega )$ for different frequency ranges and number of additional degrees of freedom.",
"We find empirically that $\\kappa (\\omega )=1$ is the best result which can be attained, and that nearly-optimal efficiency can be reached over a very broad range of frequencies.",
"This constant efficiency decreases slightly as the fitted range is extended, regardless of the number $n$ of $s_i$ employed.",
"For a given frequency range, however, increasing $n$ has the effect of making the response flatter.",
"Clearly this scheme will work optimally in harmonic or quasi-harmonic systems, and anharmonicity will introduce deviations from the predicted behavior.",
"In the extreme case of diffusive systems such as liquids, one has to ask the question of how much diffusion will be affected by the thermostat, especially since in an overdamped LE equation the diffusive modes are considerably slowed down (see e.g. Ref. [20]).",
"To estimate the impact of the thermostat on the diffusion, we define the free-particle diffusion coefficient $D^*$ as that calculated switching off the physical forces.",
"Its value when a GLE thermostat is used is $\\begin{split}\\frac{m D^*}{k_B T}=&\\frac{1}{\\left<p^2\\right>} \\int _0^\\infty \\left<p(t)p(0)\\right>\\mathrm {d}t=\\\\=&\\left[\\mathbf {A}_p^{-1}\\right]_{pp}=\\left(a_{pp} -\\mathbf {a}_p^T\\mathbf {A}^{-1}\\bar{\\mathbf {a}}_p\\right)^{-1}.\\end{split}$ where we have assumed the FDT to hold.",
"In practical cases, if an estimate of the unthermostated (intrinsic) diffusion coefficient $D$ is available, one should choose the matrix $\\mathbf {A}_p$ in such a way that $D^*\\gg D$ , so that the thermostat will not behave as an additional bottleneck for diffusion.",
"Equation (REF ) has the interesting consequence that $D^*$ can be enhanced either by reducing the overall strength of the noise, as in white-noise LE, but also by carefully balancing the terms in the denominator of Eq.",
"(REF ).",
"We have found empirically that for an $\\mathbf {A}_p$ matrix fitted to harmonic modes over the frequency range $\\left(\\omega _{min},\\omega _{max}\\right)$ , the diffusion coefficient computed by (REF ) is $D^*\\approx k_BT/\\omega _{min}$ .",
"This latter expression gives a useful recipe for choosing the minimal frequency to be considered when fitting a GLE thermostat for a system whose diffusion coefficient can be roughly estimated."
],
[
"Frequency-dependent thermostatting",
"The ability to control the strength of the thermostat-system coupling as a function of the frequency – demonstrated above – points quite naturally at more sophisticated applications.",
"For instance, one can apply two thermostats with distinct target temperatures and different efficiencies $\\kappa (\\omega )$ (see Figure REF ).",
"Obviously, such a simulation is not an equilibrium one, since energy is systematically injected in some modes and removed from others, but leads to a steady state that has useful properties.",
"Indeed, the normal modes will couple differently to the two thermostat, so that the effective temperature of each mode can be controlled as a function of $\\omega $ .",
"This two-thermostats example is just an instance of a broader class of stochastic processes, for whom the FDT is violated.",
"In general, we can relax the assumption that $\\mathbf {C}_p=k_B T$ , and for a given drift matrix we can choose a $\\mathbf {B}_p$ which is suitable to our purpose.",
"Figure: A cartoon representing a two-thermostat setup, which we take as the simplestexample of a stochastic process violating the fluctuation-dissipation theorem.If the relaxation time versus frequency curves for the two thermostats are different,a steady-state will be reached in which normal modes corresponding to different frequencieswill equilibrate at different effective temperatures.Returning to the harmonic oscillator case, one can solve exactly the dynamics for a given choice of $\\mathbf {A}_p$ , $\\mathbf {B}_p$ and frequency $\\omega $ .",
"The resulting dynamics is performed in the $n+2$ -dimensional space defined by the variables $(q,p,\\mathbf {s})$ , according to Eq.",
"(REF ).",
"For a compact notation, we used the full matrices $\\mathbf {A}_{qp}$ and $\\mathbf {B}_{qp}$ .",
"The full $\\mathbf {C}_{qp}(\\omega )$ , which defines the stationary distribution in the steady state, can be computed solving an equation analogous to (REF ): $\\mathbf {A}_{qp}\\mathbf {C}_{qp}+\\mathbf {C}_{qp}\\mathbf {A}_{qp}^T=\\mathbf {B}_{qp}\\mathbf {B}_{qp}^T$ One can tune the free parameters ($\\mathbf {A}_p$ and $\\mathbf {B}_p$ ) so as to make the $c_{qq}(\\omega )$ and $c_{pp}(\\omega )$ elements of the extended covariance matrix as close as possible to the desired target functions $\\left<q^2\\right>(\\omega )$ and $\\left<p^2\\right>(\\omega )$ .",
"In a previous paper [9] we applied this method to obtain $\\left<q^2\\right>(\\omega )$ and $\\left<p^2\\right>(\\omega )$ in agreement with the values appropriate for a quantum harmonic oscillator, and obtained a good approximation to the quantum-corrected structural properties in quasi-harmonic systems.",
"Many other applications can be envisaged, which take advantage of frequency-dependent thermostatting.",
"For instance, one could use this technique in accelerated sampling methods[21], [22], [23], which work by artificially heating the low-frequency modes, whilst keeping the other modes at the correct temperature."
],
[
"Implementation",
"The implementation of a GLE thermostat in molecular-dynamics simulations is straightforward.",
"Here, we consider the case of a velocity-Verlet integrator, which updates positions and momenta by a time step $\\Delta t$ , according to the scheme: $\\begin{split}p\\leftarrow & p + V^{\\prime }(q) \\Delta t /2\\\\q\\leftarrow & q + p \\Delta t\\\\p\\leftarrow & p + V^{\\prime }(q) \\Delta t /2.\\\\\\end{split}$ Eqs.",
"(REF ) can be obtained using Trotter splitting in a Liouville operator formalism[24].",
"In the same spirit one can introduce our GLE thermostat by performing two free-particle steps by $\\Delta t/2$ on the $(p,\\mathbf {s})$ variables[19]: $\\begin{split}\\left(p,\\mathbf {s}\\right)&\\leftarrow \\mathcal {P}\\left[\\left(p,\\mathbf {s}\\right),\\Delta t/2\\right]\\\\p&\\leftarrow p + V^{\\prime }(q) \\Delta t /2\\\\q&\\leftarrow q + p \\Delta t\\\\p&\\leftarrow p + V^{\\prime }(q) \\Delta t /2.\\\\\\left(p,\\mathbf {s}\\right)&\\leftarrow \\mathcal {P}\\left[\\left(p,\\mathbf {s}\\right),\\Delta t/2\\right]\\\\\\end{split}$ At variance with thermostats based on second-order equations of motion such as Nosé-Hoover, where a multiple time-step approach is required to obtain accurate trajectories[25], [26], this free-particle step can be performed without introducing additional sampling errors.",
"The exact finite-time propagator for $(p,\\mathbf {s})$ reads: $\\mathcal {P}\\left[\\left(p,\\mathbf {s}\\right),\\Delta t\\right]^T=\\mathbf {T}(\\Delta t)\\left(p,\\mathbf {s}\\right)^T+\\mathbf {S}(\\Delta t)\\xi ^T$ where $\\xi $ is a vector of $n+1$ uncorrelated Gaussian numbers, and the matrices $\\mathbf {T}$ and $\\mathbf {S}$ can be computed once, at the beginning of the simulation and for all degrees of freedom[10], [27].",
"The relations between $\\mathbf {T}$ , $\\mathbf {S}$ , $\\mathbf {A}_p$ , $\\mathbf {C}_p$ and $\\Delta t$ read $\\mathbf {T}=e^{-\\Delta t \\mathbf {A}_p},\\mathbf {S}\\mathbf {S}^T=\\mathbf {C}_p-e^{-\\Delta t \\mathbf {A}_p}\\mathbf {C}_p e^{-\\Delta t \\mathbf {A}_p^T}.$ It is worth pointing out that when FDT holds, the canonical distribution is invariant under the action of (REF ), whatever the size of the time-step.",
"A useful consequence of this property is that, in the rare cases where applying (REF ) introduces a significant overhead over the force calculation, the thermostat can be applied every $m$ steps of dynamics, using a stride of $m\\;\\Delta t$ .",
"This will change the trajectory, but does not affect the accuracy of sampling.",
"The velocity-Verlet algorithm (REF ) introduces finite-$\\Delta t$ errors, whose effect needs to be monitored.",
"In microcanonical simulations, this is routinely done by checking conservation of the total energy $H$ .",
"Following the work of Bussi et al.",
"[28] we introduce a conserved quantity $\\tilde{H}$ , which can be used to the same purpose: $\\tilde{H}=H-\\sum _i \\Delta K_i $ where $\\Delta K_i$ is the change in kinetic energy due to the action of the thermostat at the $i$ -th time-step, and the sum is extended over the past trajectory.",
"In cases where the FDT holds, such as that described in Section REF , the drift of the effective energy quantitatively measures the violation of detailed balance induced by the velocity-Verlet step, similarly to Refs.",
"[28], [19].",
"In the cases where the FDT does not hold, such as the frequency dependent thermostating described in Section REF , the conservation of this quantity just measures the accuracy of the integration, similarly to Refs.",
"[29], [30]."
],
[
"Fitting of colored-noise parameters",
"A key feature of our approach resides in our ability to optimize the performance of the thermostat based on analytical estimates, making the method effectively parameterless.",
"Such optimization, however, is not trivial, even if computationally inexpensive.",
"The relationship between $\\mathbf {A}_p$ , $\\mathbf {B}_p$ and the correlation properties of the resulting trajectory is highly nonlinear.",
"Furthermore, we found empirically that many local minima exist which greatly hinder the optimization process.",
"With these difficulties in mind, we provide a downloadable library of fitted parameters[17] which can be adapted to most of the foreseeble applications, according to the prescriptions given in Section REF .",
"Details about the fitting procedure are given in the following three subsections."
],
[
"Parameterization of GLE matrices",
"A number of constraints must be enforced on the drift and diffusion matrices in order to guarantee that the resulting SDE is well-behaved.",
"It is therefore important to find a representation of the matrices such that during fitting these conditions are automatically enforced, and that the parameters space is efficiently explored.",
"A first condition, required to yield a memory kernel with exponential decay, is that all the eigenvalues of $\\mathbf {A}_p$ must have positive real part.",
"A second requirement is that the kernel $K(\\omega )$ is positive for all real $\\omega $ .",
"This ensures that the stochastic process will be consistent with the second law of thermodynamics[31].",
"Finding the general conditions for $\\mathbf {A}_p$ to satisfy this second constraint is not simple.",
"However, we can state that a sufficient condition for $K(\\omega )>0$ is that $\\mathbf {A}_p+\\mathbf {A}_p^T$ is positive definite.",
"For simplicity we shall assume such a positivity condition to hold, since we found empirically that this modest loss of generality does not significantly affect the accuracy or the flexibility of the fit.",
"Moreover, in the case of canonical sampling, $\\mathbf {A}_p+\\mathbf {A}_p^T>0$ is also required in order to obtain a real diffusion matrix, since $\\mathbf {B}_p\\mathbf {B}_p^T=k_BT\\left(\\mathbf {A}_p+\\mathbf {A}_p^T\\right)$ according to Eq.",
"(REF ).",
"One would like to find a convenient parameterization, which enforces automatically these constraints.",
"This is best done by writing $\\mathbf {A}_p=\\mathbf {A}_p^{(S)}+\\mathbf {A}_p^{(A)}$ , the sum of a symmetric and antisymmetric part.",
"Since any orthogonal transform of the $\\mathbf {s}$ degrees of freedom would not change the dynamics (see Appendix ), one can assume without loss of generality that the $\\mathbf {A}^{(S)}$ block in $\\mathbf {A}_p^{(S)}$ is diagonal (see Eq.",
"(REF ) for the naming convention).",
"Since in the general case the antisymmetric $\\mathbf {A}_p^{(A)}$ does not commute with $\\mathbf {A}_p^{(S)}$ , we will assume it to be full, while $\\mathbf {A}_p^{(S)}$ can be written in the form $\\mathbf {A}_p^{(S)}=\\!\\left(\\!\\begin{array}{ccccc}a & a_1 & a_2 & \\cdots & a_n \\\\a_1 & \\alpha _1 & 0 & \\cdots & 0 \\\\a_2 & 0 & \\alpha _2 & \\ddots & 0 \\\\\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\a_n & 0 & 0 &\\cdots & \\alpha _n\\end{array}\\!\\right)\\!.$ In order to enforce the positive-definiteness, one use an analytical Cholesky decomposition $\\mathbf {A}_p^{(S)}=\\mathbf {Q}_p\\mathbf {Q}_p^T$ , with $\\mathbf {Q}_p$ $\\mathbf {Q}_p=\\!\\left(\\!\\begin{array}{ccccc}q & q_1 & q_2 & \\cdots & q_n \\\\0 & d_1 & 0 & \\cdots & 0 \\\\0 & 0 & d_2 & \\ddots & 0 \\\\\\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\0 & 0 & 0 &\\cdots &d_n\\end{array}\\!\\right)\\!$ and $\\alpha _i=d_i^2$ , $a_i=d_i q_i$ , and $a=q^2+\\sum _i q_i^2$ .",
"Such a parameterization guarantees that $\\mathbf {A}_p$ will generate a dynamics with a stationary probability distribution, and requires $2n+1$ parameters for the symmetric part (the elements of $\\mathbf {Q}_p$ , Eq.",
"(REF )), and $n(n+1)/2$ for the antisymmetric part $\\mathbf {A}_p^{(A)}$ .",
"If we want the equilibrium distribution to be the canonical, one we must enforce the FDT, and $\\mathbf {B}_p\\mathbf {B}_p^T$ is uniquely determined.",
"If we aim at a generalized formulation, which allows for frequency-dependent thermalization, there are no constraints on the choice of $\\mathbf {B}_p$ other than the fact that both $\\mathbf {B}_p\\mathbf {B}_p^T$ and the covariance $\\mathbf {C}_p$ must be positive-definite.",
"Clearly, a real, lower-triangular $\\mathbf {B}_p$ is the most general parameterization of a positive-definite $\\mathbf {B}_p\\mathbf {B}_p^T$ , and amounts at introducing $(n+1)(n+2)/2$ extra parameters.",
"Together with the assumption that $\\mathbf {A}_p^{(S)}>0$ , the condition $\\mathbf {B}_p\\mathbf {B}_p^T>0$ is sufficient to ensure that the unique symmetric $\\mathbf {C}_p$ which satisfies (REF ) is also positive-definite."
],
[
"Fitting for canonical sampling",
"Armed with such a robust and fairly general parameterization, one only needs to define a merit function to be optimized.",
"Again, we first consider the simpler case of canonical sampling.",
"Here, we want to obtain a flat response over a wide, physically-relevant frequency range $\\left(\\omega _{min},\\omega _{max}\\right)$ .",
"We have chosen the form $\\chi _1=\\left[\\sum _i \\left|\\log \\kappa (\\omega _i)\\right|^m\\right]^{1/m},$ where $\\omega _i$ s are equally spaced on a logarithmic scale over the fitted range.",
"If a large value of $m$ is chosen, the $\\omega _i$ which yields the lowest efficiency is weighted more, and a flat response curve is obtained.",
"We found empirically that values of $m$ larger than $~10$ lead to a proliferation of local minima, and hinder efficient optimization.",
"To resolve this, one can use the optimal parameters for $m=2$ as input for further refinement at larger $m$ , until convergence is achieved.",
"Figure: Thermostatting efficiency, as estimated from Eq.",
"(),for a colored-noise thermostat optimized for Car-Parrinello dynamics.",
"Sampling efficiencyis optimized for ω∈(10 -3 ,1)\\omega \\in (10^{-3},1), and an abrupt drop in efficiency is enforcedfor ω∈(1,10)\\omega \\in (1,10), using the penalty function ()in the fitting.",
"The continuous (dark red) curve corresponds to k=9k=9, the dashed (orange)curve to k=6k=6 and the dotted (light orange) curve to k=3k=3.",
"The κ(ω)\\kappa (\\omega ) curvefor a white-noise thermostat centered on the optimized range is also reported forreference (dotted black curve).The three curves correspond to the parameters set cp-9_4-3, cp-6_4-3 andcp-3_4-3.This procedure can be modified so as to provide an efficient thermostat which can be used in Car-Parrinello-like dynamics.",
"In this case, the GLE has to act as a low-pass filter in which only the low ionic frequencies are affected, and fast electronic modes are not perturbed.",
"To obtain this effect, we compute (REF ) only for the $\\omega _i$ 's which are smaller than a cutoff frequency $\\omega _{CP}$ , and we introduce an additional term $\\chi _2=\\left[\\sum _{\\omega _i>\\omega _{CP}}\\left|\\max \\left[\\log \\kappa (\\omega _i)- k (\\omega _{CP}-\\omega _i),0\\right]\\right|^m\\right]^{1/m}.$ $\\chi _2$ enforces a steep decrease of $\\kappa (\\omega )$ above $\\omega _{CP}$ , with a slope $k$ on a logarithmic scale.",
"Values of $k$ as large as 9 can be used, which guarantee an abrupt drop in thermalization efficiency for the fast modes (see Figure REF )."
],
[
"Non-thermal noise and quantum thermostat",
"We now discuss the case in which the thermostat is permitted to violate FDT, in order to achieve frequency-dependent equilibration.",
"For these applications, one must also fit the fluctuations $c_{pp}(\\omega )$ and $c_{qq}(\\omega )$ to some target function $\\tilde{c}_{pp}$ and $\\tilde{c}_{qq}$ .",
"We shall not treat the general case, but rather investigate the example of the quantum thermostat (Ref. [9]).",
"The procedure followed provides a clear guide for future extensions to different applications.",
"In order to reproduce quantum ions effects, one must selectively heat high-frequency phonons, for which zero-point energy effects are important, without affecting the low-frequency modes which behave classically.",
"The required frequency dependence of the variance for this case is that of a quantum oscillator, i.e.",
"$\\tilde{c}_{pp}(\\omega )=\\omega ^2 \\tilde{c}_{qq}(\\omega )=\\frac{\\hbar \\omega }{2} \\coth \\frac{\\hbar \\omega }{2 k_B T}$ The $\\omega \\rightarrow 0$ , classical limit can be proved to correspond to two conditions on the elements of the free-particle covariance matrix $\\mathbf {C}_p$ ; namely, $c_{pp}=k_B T$ and $\\mathbf {a}_p^T\\mathbf {A}^{-1}\\mathbf {c}_p=0$ .",
"One could enforce such constraints exactly, by considering the entries of $\\mathbf {C}_p$ as independent fitting parameters, and obtaining the diffusion matrix from Eq.",
"(REF ).",
"We found however that this choice makes it difficult to obtain a positive-definite $\\mathbf {B}_p\\mathbf {B}_p^T$ , and that the fitting becomes more complex and inefficient.",
"As an alternative, we decided to enforce the low-frequency limit with an appropriate penalty function, $\\chi _3=(c_{pp}/k_B T-1)^2+\\left(\\mathbf {a}_p^T\\mathbf {A}^{-1}\\mathbf {c}_p/k_B T\\right)^2,$ to be optimized together with the sampling efficiency (REF ) and a term which measures how well the finite-frequency fluctuations were fitted: $\\chi _4=\\left[\\sum _i \\left|\\log \\frac{c_{qq}(\\omega _i)}{\\tilde{c}_{qq}(\\omega _i)} \\right|^m+\\left|\\frac{\\log c_{pp}(\\omega _i)}{\\tilde{c}_{pp}(\\omega _i)} \\right|^m \\right]^{1/m}$ Since the low-frequency limit is already enforced by (REF ), we compute (REF ) on a set of points equally spaced between the maximum frequency $\\omega _{max}$ and one half of the onset frequency for quantum effects $\\omega _q=k_BT/\\hbar $ ."
],
[
"Transferability of fitted parameters",
"The scheme described in the previous Sections allowed us to obtain matrices suitable for all the applications discussed in previous works.",
"Furthermore, it provides a starting point for obtaining matrices which one might deem useful for novel applications.",
"However, the reader is advised that the fitting is still far from being a black-box procedure.",
"It is thus necessary to experiment with a combination of different initial parameters and minimization schemes.",
"We found the downhill simplex methods[32] to be particularly effective, but resorted to simulated annealing when the optimization got stuck in a local minimum.",
"There is a great deal of arbitrariness in the choice of the terms (REF -REF ), and in their weighted combination $\\chi =\\sum w_i \\chi _i$ .",
"To make the procedure even more delicate, we observe that in high-$n$ cases the parameters tend to collapse into “degenerate” minima, where the full dimensionality of the search space is not exploited.",
"This phenomenon can be successfully circumvented by enforcing an even spacing of the eigenvalues of $\\mathbf {A}$ over the frequency range of interest, and slowly releasing this restraint during the later stages of optimization.",
"However, the problems mentioned above have no major practical consequences, as the computation of analytical estimates is inexpensive and one can afford a great deal of trial-and-error during the optimization.",
"Moreover, fitted parameters can be reused, since the optimized parameters can be easily transferred to similar problems because of the scaling properties of the dynamics (REF ).",
"In fact, one can see that if the drift and covariance matrices $(\\mathbf {A}_p,\\mathbf {C}_p)$ lead to the efficiency curves $\\kappa (\\omega )$ and fluctuations $c_{pp}(\\omega )$ , the scaled matrices $(\\alpha \\, \\mathbf {A}_p,\\beta \\, \\mathbf {C}_p)$ will yield $\\kappa (\\alpha ^{-1}\\omega )$ , and the fluctuations $\\beta c_{pp}(\\alpha ^{-1}\\omega )$ .",
"This means that if $\\mathbf {A}_p$ is optimized for sampling over the range $\\left(\\omega _{min},\\omega _{max}\\right)$ , $\\alpha \\, \\mathbf {A}_p$ will be optimal over $\\left(\\alpha \\,\\omega _{min},\\alpha \\,\\omega _{max}\\right)$ .",
"We also remark that if $(\\mathbf {A}_p,\\mathbf {C}_p)$ are fitted to the quantum harmonic oscillator fluctuations at temperature $T$ , $(\\alpha \\, \\mathbf {A}_p,\\alpha \\, \\mathbf {C}_p)$ will be suitable for temperature $\\alpha \\,T$ .",
"Care must be taken in this case to ensure that the scaled frequency range still encompasses the whole vibrational spectrum of the system being studied."
],
[
"Understanding the quantum thermostat",
"As discussed in Ref.",
"[9], one must pay a great deal of attention when using a “quantum thermostat”, because energy is transferred between modes of different frequency, as a consequence of the anharmonic coupling.",
"This is reminiscent of zero-point energy (ZPE) leakage which plagues semiclassical approaches to the computation of nuclear quantum effects[33], [34].",
"In the cases we explored so far, empirical evidence suggests that quasi-harmonic solids, can be treated with good accuracy down to temperatures as low as 10% of the Debye temperature $\\Theta _D$ .",
"Clearly, the ultimate test to assess of the accuracy of the method is a comparison with path-integral calculations, to be performed on a similar but computationally cheaper model, such as a smaller-size box or a simpler force field.",
"One would like however to obtain some qualitative measure of the quality of the fit, and to gauge the transferability of a given set of parameters.",
"To this end, we first state a couple of empirical rules, and then validate them on two fairly different real systems.",
"A first observation is that it is useless to push the fitting of the fluctuations $c_{pp}(\\omega )$ and $c_{qq}(\\omega )$ to very high accuracy, if this comes at the expense of the coupling efficiency.",
"In fact, we would be trading a small, controlled fitting error with a possibly larger, uncontrollable and system-dependent error stemming from anharmonicity.",
"Secondly, we observed that in order to contrast more effectively the flow of energy between different phonons, one should try to reduce the correlation time of the kinetic energy $\\tau _K$ , rather than focus solely on the terms (REF ), which are better suited to measure sampling efficiency.",
"In fact, a low $\\tau _K(\\omega )$ corresponds to a slightly overdamped regime, where sampling efficiency is sub-optimal, but ZPE is enforced more tightly.",
"Figure: (a): ω\\omega -dependence of the kinetic energy correlation time τ k (ω)\\tau _k(\\omega ) (light, dotted line) andof the ratio of the fitted fluctuations c pp (ω)c_{pp}(\\omega ) (dashed line) and of ω 2 c qq (ω)\\omega ^2 c_{qq}(\\omega ) (full line)with the exact, quantum-mechanical target function.",
"(b): normal-mode-projected kinetic temperature for a few, selected phonons.",
"The dashed line is thevalue expected from the fitting c pp (ω)c_{pp}(\\omega ), while the full line is the exact, quantum-mechanicalexpectation value for a harmonic oscillator.Calculations have been performed with the parameters qt-20_6_BAD.Figure: (a): ω\\omega -dependence of the kinetic energy correlation time τ K (ω)\\tau _K(\\omega ) (light, dotted line) andof the ratio of the fitted fluctuations c pp (ω)c_{pp}(\\omega ) (dashed line) and of ω 2 c qq (ω)\\omega ^2 c_{qq}(\\omega ) (full line)with the exact, quantum-mechanical target function.",
"(b): normal-mode-projected kinetic temperature for a few, selected phonons.",
"The dashed line is thevalue expected from the fitting c pp (ω)c_{pp}(\\omega ), while the full line is the exact, quantum-mechanicalexpectation value for a harmonic oscillator.Calculations have been performed with the parameters qt-20_6.To demonstrate these concepts in a real system, we performed some calculations with a Tersoff model of diamond at a temperature $T=200$ K. At this low temperature, slightly below $0.1 \\Theta _D$ , quantum effects are very strong, and we therefore expect to have problems maintaining the large difference in temperature between the stiff and soft phonons.",
"Using a very harmonic system such as diamond is particularly useful, since one can monitor directly the efficiency of the thermostat by projecting the atomic velocities on a selection of normal modes.",
"Hence, a projected kinetic temperature $T^{\\prime }(\\omega )$ can be computed, and its value checked against the predictions in the harmonic limit, in the same spirit as in Ref.[9].",
"In Figure REF we report the results with a matrix fitted taking into account only the terms (REF ) and (REF ).",
"Even in a harmonic system such as diamond there are major errors due to ZPE leakage from the high-frequency to the low-frequency modes, which the thermostat compensates only partially.",
"These poor results should be compared with those of Figure REF .",
"Here, we have also introduced in the fit a term analogous to (REF ) to reduce the value of $\\tau _K(\\omega )$ .",
"The projected kinetic temperature now agrees almost perfectly with the analytical predictions $c_{pp}(\\omega )$ for most of the modes.",
"The only ones displaying significant deviations are the faster ones, for whom the value of $\\tau _K(\\omega )$ is slightly larger.",
"The $c_{pp}(\\omega )$ curve deviates by nearly 10% from the exact, quantum-mechanical expectation value.",
"However, thanks to the more efficient coupling, the errors due to anharmonicities are better compensated, and in actuality, the overall error is much smaller than for the parameters presented in Figure REF .",
"Figure: Radial distribution function as computed from fully-converged path-integral calculations(black, dotted line), and from a quantum-thermostat MD trajectory for a Lennard-Jones modelof solid neon at T=20T=20 K. Distances are in reduced units.",
"Full line corresponds tothe parameters set qt-20_6 (cfr.",
"Figure ), andlighter, dashed line to the set qt-20_6_BAD (cfr.",
"Figure ).To test whether these prescriptions work for less harmonic problems, we now turn to a completely different system; namely, the structural properties of solid neon at 20 K. At variance with diamond, quantum-ions effects are less pronounced, but the system is close to its melting temperature, and it is significantly anharmonic.",
"As shown in Figure REF , the agreement between our results and those of accurate path-integral calculations[35] is almost perfect if the parameters of Figure REF are used.",
"As expected, large errors are present if qt-20_6_BAD is used.",
"Further improvements on the fitting strategy, and the application to strongly anharmonic systems is currently being investigated, and will be the subject of further work."
],
[
"Conclusions",
"In this paper we have discussed in detail the use of colored-noise dynamics, based on Ornstein-Uhlenbeck processes, as a tool for performing molecular dynamics.",
"Applications range from enhanced sampling, which we demonstrate in the harmonic limit and which will be applied to real systems in forthcoming publications, to thermostats for adiabatically separated problems and frequency-dependent thermalization.",
"Our idea exploits the linear nature of the OU stochastic differential equations, which allows one to use the one-dimensional harmonic oscillator as a simple but physically-motivated reference model.",
"On the basis of the analytical prediction obtained in that case, we describe a recipe for fitting the thermostat parameters so as to obtain the desired response properties in real systems.",
"The procedure is not simple, and we are considering different approaches to make it more robust and effective.",
"Fortunately however, fitted matrices can be easily transferred from one system to another.",
"With this in mind we have provided an extensive library of optimized parameters[17], which makes fitting unnecessary for most applications.",
"We also comment on practical issues concerning the implementation of the generalized-Langevin thermostat in a molecular-dynamics program and its use in applications.",
"In particular, we discuss in detail how one can use colored noise to model nuclear quantum effects[9].",
"We provide some empirical rules to guide the fitting in this difficult case, and we demonstrate that a normal-mode analysis in a quasi-harmonic system is a valuable tool for assessing the quality of a set of parameters.",
"We believe that further investigation will find many other applications for colored-noise in molecular-dynamics, and in computer simulations of molecular systems in general.",
"As an example, we are currently investigating using a zero-temperature, optimal-sampling GLE thermostat in order to perform structural optimization.",
"On similar lines, and taking inspiration from “quantum annealing”[36], [37], one can envisage using frequency-dependent thermalization to improve the performance of simulated annealing."
],
[
"Acknowledgements",
"The authors would like to thank David Manolopoulos for important suggestions and fruitful discussion, Marcella Iannuzzi for having implemented the GLE thermostat in CP2K[38], Alessandro Curioni for implementation in CPMD[39], Grigorios Pavliotis and Michela Ottobre for discussion and references on stochastic processes.",
"We also acknowledge Stefano Angioletti-Uberti, Paolo Elvati, Hagai Eshet, Kuntal Hazra, Rustam Khaliullin and Tom Markland for discussion and for having preliminarily tested the thermostat in real applications, providing valuable feedback.",
"We are especially in debt with Gareth Tribello, who contibuted to testing and greatly helped us improving the manuscript."
],
[
"Memory kernels for the non-Markovian formulation",
"The connection between the Markovian (REF ) and non-Markovian (REF ) formulations of the colored-noise Langevin equation can be understood using techniques similar to those adopted in Mori-Zwanzig theory[4], [11].",
"Let us first consider a very general, multidimensional OU process, where we single out some degrees of freedom ($\\mathbf {y}$ ) that we wish to integrate out, leaving only the variables marked as $\\mathbf {x}$ .",
"$\\!\\left(\\!\\begin{array}{c}\\dot{\\mathbf {x}} \\\\\\hline \\dot{\\mathbf {y}} \\\\\\end{array}\\!\\right)\\!=-\\!\\left(\\!\\begin{array}{c|c}\\mathbf {A}_{xx} & \\mathbf {A}_{xy} \\\\\\hline \\mathbf {A}_{yx} & \\mathbf {A}_{yy} \\\\\\end{array}\\!\\right)\\!\\!\\left(\\!\\begin{array}{c}\\mathbf {x} \\\\\\hline \\mathbf {y} \\\\\\end{array}\\!\\right)\\!+\\!\\left(\\!\\begin{array}{c}\\quad \\mathbf {B}_{x\\xi } \\quad \\quad \\\\\\hline \\quad \\mathbf {B}_{y\\xi } \\quad \\quad \\\\\\end{array}\\!\\right)\\!\\!\\left(\\!\\begin{array}{c}{2}{*}{\\xi }\\\\\\\\\\end{array}\\!\\right)\\!$ Assuming that the dynamics has finite memory, one can safely take $\\mathbf {y}(-\\infty )=0$ , and the ansatz $\\mathbf {y}(t)=\\int _{-\\infty }^t\\!\\!\\!",
"e^{-(t-t^{\\prime })\\mathbf {A}_{yy}} \\left[-\\mathbf {A}_{yx}\\mathbf {x}(t^{\\prime })+\\mathbf {B}_{y\\xi } \\xi (t^{\\prime })\\right] \\mathrm {d}t^{\\prime }.$ Substituting into (REF ), one sees that $\\mathbf {y}$ can be eliminated from the dynamics of $\\mathbf {x}$ , and arrives at $\\begin{split}\\dot{\\mathbf {x}}(t)=&-\\int _{-\\infty }^t \\mathbf {K}(t-t^{\\prime }) \\mathbf {x}(t^{\\prime })\\mathrm {d}t^{\\prime }+\\zeta (t)\\\\\\mathbf {K}(t)=&2\\mathbf {A}_{xx}\\delta (t)-\\mathbf {A}_{xy} e^{-t\\mathbf {A}_{yy}}\\mathbf {A}_{yx}\\quad \\left(t\\ge 0\\right)\\\\\\zeta (t)=&\\mathbf {B}_{x\\xi } \\xi (t) -\\int _{-\\infty }^t \\mathbf {A}_{xy} e^{-(t-t^{\\prime })\\mathbf {A}_{yy}} \\mathbf {B}_{y\\xi } \\xi (t^{\\prime }).\\end{split}$ One can see that (REF ) are invariant under any orthogonal transformation of the $\\mathbf {y}$ dynamical variables, meaning that such a transformation leaves the dynamics of the $\\mathbf {x}$ 's unchanged.",
"The colored noise is better described in terms of its time-correlation function, $\\mathbf {H}(t)=\\left<\\zeta (t)\\zeta (0)^T\\right>$ .",
"Let us first introduce the symmetric matrix $\\mathbf {D}=\\mathbf {B}\\mathbf {B}^T$ , whose parts we shall label using the same scheme used for $\\mathbf {A}$ in Eq.",
"(REF ).",
"We shall also need $\\mathbf {Z}_{yy}=\\int _0^\\infty e^{-\\mathbf {A}_{yy}t}\\mathbf {D}_{yy} e^{-\\mathbf {A}_{yy}^T t}\\mathrm {d}t$ .",
"With these definitions in mind, one finds $\\mathbf {H}(t)=\\delta (t)\\mathbf {D}_{xx}+\\mathbf {A}_{xy} e^{-t\\mathbf {A}_{yy}}\\left[\\mathbf {Z}_{yy}\\mathbf {A}_{xy}^T-\\mathbf {D}_{yx}\\right] \\quad \\left(t\\ge 0\\right).$ Note that the value of $\\mathbf {H}(t)$ for $t<0$ is determined by the constraint $\\mathbf {H}(-t)=\\mathbf {H}(t)^T$ ; the value of $\\mathbf {K}(t)$ instead, is irrelevant for negative times: we will assume $\\mathbf {K}(-t)=\\mathbf {K}(t)^T$ to hold, since this will simplify some algebra below.",
"Let's now switch to the case of the free-particle counterpart of Eqs.",
"(REF ), which is relevant to the memory functions entering Eqs.",
"(REF ).",
"Here, we want to integrate away all the $\\mathbf {s}$ degrees of freedom, retaining only the momentum $p$ .",
"Hence, we can transform Eqs.",
"(REF ) and (REF ) to the less cumbersome form $\\begin{split}K(t)=&2a_{pp} \\delta (t)-\\mathbf {a}_p^T e^{-\\left|t\\right|\\mathbf {A}}\\bar{\\mathbf {a}}_p\\\\H(t)=& d_{pp} \\delta (t)-\\mathbf {a}_p^T e^{-\\left|t\\right|\\mathbf {A}}\\left[\\mathbf {Z}\\mathbf {a}_p-\\mathbf {d}_p\\right]\\end{split}$ This compact notation hides certain relevant property of the memory kernels, which are more apparent when the kernels are written in their Fourier representation.",
"If $\\mathbf {D}_p=\\mathbf {B}_p\\mathbf {B}_p^T$ is transformed according to Eq.",
"(REF ).",
"$K(\\omega )$ and $H(\\omega )$ read $\\begin{split}K(\\omega )=& 2a_{pp}-2 \\mathbf {a}_p^T \\frac{\\mathbf {A}}{\\mathbf {A}^2+\\omega ^2}\\bar{\\mathbf {a}}_p \\\\H(\\omega )=& K(\\omega ) \\left(c_{pp}- \\mathbf {a}_p^T \\frac{\\mathbf {A}}{\\mathbf {A}^2+\\omega ^2}\\mathbf {c}_p\\right) +\\\\&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+2\\omega ^2 \\left(\\mathbf {a}_p^T \\frac{1}{\\mathbf {A}^2+\\omega ^2}\\mathbf {c}_p \\right)\\!\\!",
"\\left(1+\\mathbf {a}_p^T \\frac{1}{\\mathbf {A}^2+\\omega ^2} \\bar{\\mathbf {a}}_p \\right).\\end{split}$ It is seen that the memory functions (hence the dynamical trajectory) are independent of the value of $\\mathbf {C}$ , the covariance of the fictitious degrees of freedom.",
"Moreover, a sufficient condition for the FDT to hold is readily found.",
"By setting $c_{pp}=k_BT$ and $\\mathbf {c}_p=0$ , one obtains $H(\\omega )=k_B T K(\\omega )$ , which is precisely the FDT for a non-Markovian Langevin equation.",
"Since the value of $\\mathbf {C}$ is irrelevant we can take $\\mathbf {C}_p=k_B T$ , which simplifies the algebra and leads to numerically-stable trajectories."
],
[
"Covariance matrix and correlation times for the harmonic oscillator",
"Given $\\mathbf {A}$ and $\\mathbf {C}$ matrices (the drift term and the static covariance for a generic OU process), one can find the diffusion matrix $\\mathbf {B}$ by an expression analogous to Eq.",
"(REF ).",
"The same relation can be used to obtain the elements of $\\mathbf {C}$ given the drift and diffusion matrices, by solving the linear system.",
"However, the covariance matrix can be computed more efficiently by finding the eigendecomposition of $\\mathbf {A}=\\mathbf {O}\\;\\mathrm {diag}(\\alpha _i)\\;\\mathbf {O}^{-1}$ , and computing $C_{ij}=\\sum _{kl} \\frac{O_{ik}\\left[\\mathbf {O}^{-1}\\mathbf {B}\\mathbf {B}^T {\\mathbf {O}^{-1}}^T\\right]_{kl}O_{jl}}{\\alpha _k+\\alpha _l}.",
"$ Now, let $\\mathbf {x}$ be the vector describing the trajectory of the OU process.",
"In order to compute $\\tau _H$ or $\\tau _V$ (Eq.",
"(REF )) one needs time correlation functions of the form $\\left<x_i(t)x_j(t) x_k(0)x_l(0)\\right>$ .",
"The corresponding, non-normalized integrals $\\tau _{ijkl}=\\int _0^\\infty \\left[ \\left<x_i(t)x_j(t) x_k(0)x_l(0)\\right> -\\left<x_i x_j\\right>\\left<x_k x_l\\right>\\right] \\mathrm {d}t$ can be computed in terms of the tensorial quantity $X_{ijkl}= \\sum _{mn}\\frac{O_{im}\\left[\\mathbf {O}^{-1}\\mathbf {C}\\right]_{ml}O_{jn}\\left[\\mathbf {O}^{-1}\\mathbf {C}\\right]_{nk}}{\\alpha _m+\\alpha _n}$ as $\\tau _{ijkl}=\\frac{1}{4}\\left(X_{ijkl}+X_{ijlk}+X_{klij}+X_{lkij}\\right)$ .",
"For example – if we consider the full OU process in the harmonic case – one computes $\\tau _H=\\frac{\\omega ^4\\tau _{qqqq}+2\\omega ^2\\tau _{qqpp}+\\tau _{pppp}}{\\omega ^4 c_{qq}^2 +2 \\omega ^2 c_{qp}^2+c_{pp}^2},\\quad \\tau _V=\\frac{\\tau _{qqqq}}{c_{qq}^2}$ where we use an obvious notation for the indices in $\\tau _{ijkl}$ ."
],
[
"A comparison with Nosé-Hoover Chains ",
"The most widespread techniques for canonical sampling in MD are probably white-noise Langevin and Nosé-Hoover chains (NHC).",
"White-noise Langevin can be considered as a limit case of the thermostatting method we describe in this work, but NHC is based on a redically different philosophy.",
"It is therefore worth performing a brief comparison between the latter and the GLE thermostat.",
"In the “massive” version of the NH thermostat[13], [14], each component of the physical momentum is coupled to an additional degree of freedom with a fictitious mass $Q$ , by means of a second-order equation of motion.",
"The resulting dynamics ensures that the physically-relevant degrees of freedom will sample the correct, constant-temperature ensemble, with the advantage of having deterministic equations of motion, and a well-defined conserved quantity.",
"However, in the harmonic case, trajectories are poorly ergodic.",
"This problem can be addressed by coupling the fictitious momentum to a second bath variable with a similar equation of motion.",
"By repeating this process further a “Nosé-Hoover chain” can be formed, which ensures that the dynamics is sufficiently chaotic to achieve efficient sampling[15], [40].",
"The drawback of this approach is that the thermostat equations are second-order in momenta.",
"It is therefore difficult to obtain analytical predictions for the properties of the dynamics, and the integration of the additional degrees of freedom must be performed with a multiple time-step approach, which makes the thermostat more expensive.",
"To examine the performances of NHC and GLE, one could envisage comparing the sampling efficiency as defined by the correlation times (REF ).",
"Obtaining such estimates is not straightforward, not only because the the harmonic case cannot be treated analytically, but also because in the multidimensional case the properties of the trajectory will not be invariant under an orthogonal transformation of coordinates, as discussed in Section .",
"The simplest model we can conceive for comparing NHC and GLE is therefore a two-dimensional harmonic oscillator, with different vibrational frequencies on the two normal modes and adjustable relative orientations of the eigenvectors with respect to the thermostatted coordinates.",
"Figure: Correlation time for the potential energy of a 2-D harmonic oscillator, as a function ofthe angle between the eigenmodes and the cartesian axes.",
"τ V \\tau _V is computed for different values ofthe condition number ω max /ω min \\omega _{max}/\\omega _{min}, from bottom to top 10, 31.6 and 100.Thin lines serve as an aid for the eye, connecting the results obtainedin the three cases using a massive NH chains thermostat, with four additional degrees of freedom andQ=k B T/ω max 2 Q=k_BT/\\omega _{max}^2.",
"Error bars are also shown for individual data points.Thick lines correspond to the (constant) result predicted for a GLE thermostat,using respectively the thermostat parameters kv_2-1, centered on 0.32ω max 0.32 \\omega _{max},kv_4-2, centered on 0.18ω max 0.18 \\omega _{max}, andkv_4-2 centered on 0.1ω max 0.1 \\omega _{max}.",
"The values obtained inactual GLE simulations agree with the predictions within the statistical errorbar,and are not reported.The resulting $\\tau _V$ is reported in Figure REF : in the highly anisotropic cases, the efficiency of the NH chains depends dramatically on the orientation of the axes, while for well-conditioned problems is almost constant.",
"The linear stochastic thermostat, on the other hand, has a predictable response, which is completely independent on orthogonal transforms of the coordinates.",
"In the one-dimensional case – or when eigenvectors are perfectly aligned with axes – NH chains are very efficient for all modes with frequency $\\omega <\\sqrt{\\frac{k_BT}{Q}}$ .",
"One should however consider that, in the absence of an exact propagator, choosing a small $Q$ implies that integration of the trajectory for the chains will become more expensive.",
"Obviously, such a simple toy model does not give quantitative information on the behavior in real-life cases, where modes of different frequencies coexist with anharmonicity and diffusive behavior.",
"However, it demonstrates that the colored-noise Langevin thermostat performs almost as well as the axis-aligned NH chains.",
"Furthermore, unlike the NHC, there are no unpredictable failures for anisotropic potentials."
]
] | 1204.0822 |
[
[
"The Anomalous Behavior of Solid $^{4}$He in Porous Vycor Glass"
],
[
"Abstract The low temperature properties of solid $^4$He contained in porous Vycor glass have been investigated utilizing a two-mode compound torsional oscillator.",
"At low temperatures, we find period shift signals for the solid similar to those reported by Kim and Chan \\cite{ref1}, which were taken at the time as evidence for a supersolid helium phase.",
"The supersolid is expected to have properties analogous to those of a conventional superfluid, where the superfluid behavior is independent of frequency and the ratio of the superfluid signals observed at two different mode periods will depend only on the ratio of the sensitivities of the mode periods to mass-loading.",
"In the case of helium studies in Vycor, one can compare the period shift signals seen for a conventional superfluid film with signals obtained for a supersolid within the same Vycor sample.",
"We find, contrary to our own expectations, that the signals observed for the solid display a marked period dependence not seen in the case of the superfluid film.",
"This surprising result suggests that the low temperature response of solid $^4$He in a Vycor is more complex than previously assumed and cannot be thought of as a simple superfluid."
],
[
"The Anomalous Behavior of Solid $^{4}$ He in Porous Vycor Glass Xiao Mi John D. Reppy [email protected] Laboratory of Atomic and Solid State Physics and the Cornell Center for Materials Research, Cornell University, Ithaca, New York 14853-2501 The low temperature properties of solid $^4$ He contained in porous Vycor glass have been investigated utilizing a two-mode compound torsional oscillator.",
"At low temperatures, we find period shift signals for the solid similar to those reported by Kim and Chan [1], which were taken at the time as evidence for a supersolid helium phase.",
"The supersolid is expected to have properties analogous to those of a conventional superfluid, where the superfluid behavior is independent of frequency and the ratio of the superfluid signals observed at two different mode periods will depend only on the ratio of the sensitivities of the mode periods to mass-loading.",
"In the case of helium studies in Vycor, one can compare the period shift signals seen for a conventional superfluid film with signals obtained for a supersolid within the same Vycor sample.",
"We find, contrary to our own expectations, that the signals observed for the solid display a marked period dependence not seen in the case of the superfluid film.",
"This surprising result suggests that the low temperature response of solid $^4$ He in a Vycor is more complex than previously assumed and cannot be thought of as a simple superfluid.",
"67.80.Bd, 66.30.Ma Figure: Cross section of the compound vycor oscillatorIn 2004, Kim and Chan (KC) [1], employing a torsional oscillator (TO) containing a sample of porous Vycor glass impregnated with solid $^4$ He, reported the first clear evidence for a possible $^4$ He supersolid phase.",
"In their measurement they found an anomalous decrease in the period of the oscillator as the temperature was lowered below about 200 mK.",
"This period-shift signal was interpreted as evidence for a superfluid-like decoupling of a fraction of the moment of inertia of the solid $^4$ He from the TO.",
"Such a mass decoupling or non-classical rotational inertia (NCRI) signal is a characteristic of the supersolid state [2].",
"At the same time, they also reported a sensitivity of the period-shift signals to the level of $^3$ He impurities and a “critical velocity” effect where the signal size was substantially reduced by increases in the drive level.",
"In a second publication [3], KC extended their measurements to bulk solid and found an essentially identical supersolid phenomenology to that observed in solid $^4$ He in Vycor including the sensitivity to drive amplitude and $^3$ He impurity level.",
"Given the four orders of magnitude difference in the confining dimension of the Vycor pores as compared to the millimeter scale geometry for the bulk experiments, this similarity between the Vycor and bulk $^4$ He results is remarkable.",
"A second important development was the observation by Day and Beamish (DB) [4] of an anomalous increase in the elastic shear modulus of the solid $^4$ He extending over the same temperature range as the supersolid phenomenon and also accompanied by a sensitivity to $^3$ He impurity level and dependence on the drive level similar to that seen in the KC supersolid experiments.",
"This similarity suggests a close association between the elastic anomaly and the supersolid phenomenon.",
"In the case of bulk samples in relatively large containers, the acceleration of the solid during oscillation through the elastic interaction between the solid and the TO might lead to observable effects.",
"However, in the case of Vycor it is difficult to believe that the shear modulus could play a significant role given the rigidity of the Vycor glass and the small dimensions of pores confining the solid $^4$ He.",
"A recent torsional oscillator (TO) study [5] of bulk samples of solid $^4$ He utilizing a multiple frequency oscillator showed that period shift signals observed with this oscillator, although similar in form to classic supersolid period shift signals, are in fact attributable in large part to the temperature dependence of the shear modulus.",
"Measurements at two different oscillator periods allow a separation of the observed period shift signals into two components: first a period-independent contribution, such as that expected for the supersolid or a conventional superfluid, and second a period-dependent contribution due to dynamic effects.",
"The period shift signal arising from the shear modulus anomaly is expected to be negligible for solid $^4$ He in Vycor, so a compound oscillator study of this system should provide definitive test of the nature of the signals observed by KC.",
"The compound oscillator we have constructed for the current series of Vycor measurements is shown in Figure REF .",
"Our oscillator design is similar to that of Aoki, Graves and Kojima [6] who pioneered the use of the compound oscillator in supersolid studies of the bulk solid.",
"The oscillator is largely constructed from high strength aluminum alloy and consists of two moments of inertia and two nearly equal torsion rods, 0.5715 cm in outer diameter and 0.127 cm in inner diameter.",
"The entire oscillator structure is rigidly attached to a massive block of Cu which is thermally anchored to a dilution refrigerator.",
"The torsion constant for the upper torsion rod is $k_1 = 1.41 \\times 10^9$ dyncm and $ k_2 = 1.65 \\times 10^9$ dyncm for the lower rod.",
"The upper moment of inertia, $ I_1 = 20.95$ gcm$^2$ , contains a sample of porous Vycor glass sealed with epoxy in an aluminum cylinder with an outer diameter of 1.778 cm.",
"The Vycor sample is slightly elliptical with a mean diameter of 1.43 cm and length 2.96 cm.",
"The lower moment of inertia, including the magnesium electrodes used for exciting and detecting the two rotational modes of the oscillator, is $I_2 = 14.55$ gcm$^2$ .",
"The oscillator structure has two rotational modes, an upper mode (designated by $+$ ) where the moments of inertia execute a counter rotational (anti-phase) motion and a lower mode (designated by $-$ ) where the two moments of inertia rotate together in-phase.",
"The low temperature oscillator periods are $P_+ = 0.389$ ms and $P_- = 1.112$ ms for the high and low modes, respectively.",
"The periods of oscillation for the modes are given by $P_\\pm = 2\\pi \\lbrace [\\frac{k_1(I_1 + I_2) + k_2I_1}{2I_1I_2}][1 \\pm \\sqrt{1 - \\frac{4k_1k_2I_1I_2}{(k_1(I_1 + I_2) + k_2I_1)^2}}]\\rbrace ^{-\\frac{1}{2}}$ .",
"The electrodes mounted on the oscillator are connected to a low-impedance 220 V source.",
"Since we employ a current amplifier to detect the motion of the electrodes, the recorded signal amplitude is proportional to the angular velocity, $\\dot{\\theta }_2$ .",
"The relation between the angular velocity, $\\dot{\\theta }_1$ , of the Vycor sample and that of the electrodes, obtained from the compound oscillator equations of motion is $\\dot{\\theta }_1 = [(\\frac{k_2}{I_1})(\\frac{P_\\pm }{2\\pi })-\\frac{I_2}{I_1}] \\dot{\\theta }_2$ .",
"For the high ($+$ ) mode, $\\dot{\\theta }_1 = -0.381\\dot{\\theta }_2$ , and for the low ($-$ ) mode, $\\dot{\\theta }_1 = 1.848\\dot{\\theta }_2$ .",
"Thus the angular velocity of the sample is a factor of 4.5 larger for the low mode as compared to the high mode for a given detected signal amplitude.",
"Commercial $^4$ He gas, with a nominal 0.3 ppm $^3$ He impurity level, is used for the samples that are formed by the blocked-capillary method.",
"Typically, the cell is loaded at a pressure of 70 bar ($\\sim $ 1000 psi) at a temperature of 3 K. Upon cooling, the fill line to the cell freezes before the helium in the Vycor.",
"After the freezing of the $^4$ He within the Vycor, the sample pressure is approximately 40 bar.",
"It is our usual practice to excite both modes simultaneously and employ separate lock-in amplifiers to record the individual response for each mode.",
"A feed-back system is used to control the drive voltage and maintain the signal amplitudes at a constant level, thus minimizing any non-linear amplitude-dependent effect.",
"The two modes have different sensitivities for changes in the moment of inertia of the $^4$ He sample.",
"These sensitivities were experimentally determined from the period shift data, $\\delta P_\\pm $ , observed while condensing measured volumes of $^4$ He in the Vycor sample.",
"The shifts, following filling of the Vycor with solid, are $\\delta P_+=156.4$ ns for the high mode and $\\delta P_-=1426.6$ ns for the low mode.",
"The contribution of the solid $^4$ He, $I_{\\text{He}}$ , to the torsion bob moment of inertia, $I_1$ , can also be obtained from the filling data and is found to be approximately $8.1 \\times 10^{-2}$ gcm$^2$ .",
"The mass-loading sensitivities are $m_+ = 1.931 \\times 10^3$ ns/(gcm$^2$ ) for the high mode and $m_- = 17.612 \\times 10^3$ ns/(gcm$^2$ ) for the low mode.",
"The sensitivity ratio is $m_+/m_-=$ 0.11.",
"Figure: The upper panel shows the period shift data for a superfluid film adsorbed in the Vycor sample.",
"The lower panel shows comparable data for the case of 4 ^4He frozen within the Vycor.",
"Note the difference in the vertical axis scales.For the superfluid film data shown in Figure REF , the period difference between the mode period at the transition temperature of 750 mK and the period data at lower temperatures, $\\Delta P_{\\text{Film}} (T) = P_{\\text{Film}} (750 \\text{ mK}) - P_{\\text{Film}} (T)$ , is proportional to the superfluid mass of the adsorbed film.",
"The striking aspect of the film data is the large ratio between the magnitudes of the high and low mode signals.",
"However, this is just what should be expected for a superfluid on the basis of the mass loading calibration.",
"The lower panel shows comparable data for the case of solid $^4$ He within the Vycor sample.",
"Here the situation is quite different and there is only factor of approximately 2 between the magnitudes of the signals for the two modes.",
"Both the superfluid film data and the solid data have been corrected for the empty cell temperature-dependent backgrounds.",
"Although both modes for the solid show the same anomalous drop in the oscillator period as that reported by KC for solid $^4$ He in Vycor, the ratio of the magnitudes of the signals for the two modes is much less than what would be expected from the mass loading calibration.",
"To emphasize this point we treat the period change, $\\Delta P_{\\text{Sol}} = P_{\\text{Sol}} (150 \\text{ mK}) - P_{\\text{Sol}}(20 \\text{ mK})$ , between 150 mK and 20 mK as a standard NCRI signal.",
"These changes in period are 1.354 ns and 0.916 ns for the low and high mode respectively.",
"We can calculate a supersolid fraction or NCRIF in the usual way by dividing these period shifts by the corresponding period shift values obtained when filling the cell with solid.",
"The NCRIF value for the low mode is then 0.095 %, while for the high mode value it is 0.586 %.",
"For a genuine superfluid signal these values should be equal and independent of the TO frequency.",
"Thus we are forced to consider other mechanisms for the observed period shifts for the solid beyond the usual supersolid scenario.",
"Figure: The increase with velocity in the mode periods at the temperature of 20 mK are plotted as a function of the maximum rim velocity.In addition to the temperature dependence of the solid signals, we have also investigated their sensitivities to increasing velocity.",
"We find, in agreement with KC, that as the angular velocity of the torsion bob is increased, the low temperature values of the mode periods increase, thus decreasing the overall magnitude of the solid period shift signals.",
"In Figure REF we plot the values, obtained at 20 mK, for the increase in mode periods with increasing total angular velocity as functions of the maximum rim velocity.",
"This data set was obtained while driving both modes of the oscillator.",
"In contrast to what is often seen in bulk $^4$ He measurements of this type, there is no evidence in these measurements of a subcritical velocity region where the period shift signal is independent of the rim velocity.",
"In our data, the initial change in period signal appears to be linearly proportional to the velocity starting from our lowest velocity.",
"Figure: In this figure the period shift data for the two modes displayed in Figures and are plotted against each other.",
"In addition, data obtained while incrementally filling the cell are also shown.",
"The period shift values for the film and mass loading data sets have been divided by the factors of 2 and 40 respectively.An interesting way to display these data as well as the period shift data for the temperature dependent solid signals is to plot the period shifts for the two modes against each other.",
"This representation of the data is shown in Figure REF .",
"Also included in this figure are film data from the transition temperature down to 300 mK and period shift data obtained while filling the cell with $^4$ He.",
"In contrast to the superfluid film data, which agree quite well with the mass-loading calibration as indicated by the straight line in the figure, the solid data follow a much steeper trend.",
"The velocity and temperature sweep data show a very similar behavior.",
"This similarity gives support to the hypothesis [7] of the Davis group at Cornell that velocity and temperature play complementary roles in controlling the low temperature dynamics of solid $^4$ He.",
"The solid signals show an unanticipated sensitivity to the frequency of the measurement and bear a resemblance to the period shift signals due to elastic modulus effects seen in earlier bulk solid experiments [5].",
"There are two obvious ways in which the temperature dependent shear modulus might influence this experiment.",
"The first would be the influence of the temperature dependent shear modulus on the dynamic response of the torsion bob itself, and second, the effect due to the temperature dependence of the shear modulus of the solid $^4$ He in the torsion rods of the compound oscillator.",
"An explanation based on possible shear modulus changes of the solid within the Vycor seems unlikely, however, given the rather large value of the Vycor shear modulus, on the order of $7 \\times 10^9$ Pa, as compared to the value of $1.5 \\times 10^7$ Pa for the solid $^4$ He.",
"It is known, however, from acoustic measurements [8], [9] that the elastic modulus of a Vycor sample containing $^4$ He increases by about 0.3 % upon freezing of $^4$ He.",
"An estimate based on a shear modulus change of this magnitude leads to period shifts of less than 0.1 ns, much smaller than the observed signals.",
"In the case of period shifts due to the temperature dependent changes of the shear modulus of the solid within the torsion rods, the period shifts are somewhat larger when assuming a 100 % change in the shear modulus, on the order of 0.19 ns for the high mode and 0.55 ns for the low mode.",
"These period shifts are larger than the shifts due to shear modulus effect on the dynamics of the TO, but are still relatively small compared with the observed period shift signals for the solid.",
"When corrected for the mass loading sensitivity factors, these period shifts yield a frequency independent contribution to the effective moment of inertia, $\\Delta I_{\\text{eff}} = 1.035 \\times 10^{-5}$ gcm$^2$ , for an equivalent NCRIF of about $1.26 \\times 10^{-4}$ .",
"Figure: This figure examines the relationship between the effective moment of inertia, I eff I_{\\text{eff}}, and the square of the mode frequency.",
"The zero frequency intercept represents a superfluid-like (i.e.",
"frequency independent) contribution to the total signal.Although our estimates of the possible period shifts due to the elastic properties of the $^4$ He are too small to account for the observed period shifts, the strong frequency dependence of both the velocity and temperature sweep signals suggests that there may be some other, as yet unknown, dynamic process coming into play.",
"In Figure REF we analyze the data in terms of a simple model consisting of a dynamic contribution to the effective moment of inertia for each mode proportional to the frequency squared and a frequency independent term corresponding to a possible NCRI contribution.",
"In this figure the effective moment of inertia values derived from the temperature dependent period shift signals at 150 mK are plotted against the square of the mode frequency.",
"In this model the zero frequency intercept of the line, $\\Delta I_{\\text{eff}} (f^2) = (2.23 + 9.04f^2) \\times 10^{-5}$ gcm$^2$ , determined by our two data points representing the frequency independent or NCRI contribution to the moment of inertia.",
"For the data shown the intercept is small but positive and would correspond to a NCRIF of about $2.72 \\times 10^{-4} \\pm 3.2 \\times 10^{-4}$ .",
"The error estimate was made assuming a resolution of $\\pm $ 0.05 ns in the period shift data.",
"One would then conclude from these data that there is no evidence for a significant contribution from a supersolid term to the Vycor period shift data.",
"This conclusion must be treated with caution since we do not know the origin of the frequency dependence of the data or even, as has been assumed in the foregoing analysis, that this dependence follows the square of the frequency.",
"Clearly measurements at least three different frequencies are mandated.",
"In conclusion, we have examined the behavior of a Vycor sample containing solid $^4$ He mounted on a two-frequency compound TO.",
"Period shift signals similar to those reported by KC are observed.",
"The response at the two different frequencies, however, demonstrates a pronounced dependence on frequency not expected for a true superfluid.",
"The temperature dependent shear modulus of the solid does not appear to lead to large enough period shifts in the oscillator to explain the observations.",
"Thus we are left with an as yet unexplained phenomenon for solid $^4$ He in the porous Vycor glass.",
"This problem will require further experimental and theoretical effort to achieve a solution.",
"Xiao Mi wishes to thank the Hunter R. Rawlings III Cornell Presidential Research Scholarship for its continuing support.",
"The authors acknowledge useful and encouraging discussions with J.R. Beamish, M.H.W.",
"Chan, H. Kojima and Erich J. Mueller.",
"This work was supported by the National Science Foundation through Grant DMR-060586 and CCMR Grant DMR-0520404.",
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]
] | 1204.0749 |
[
[
"A Complexity Preserving Transformation from Jinja Bytecode to Rewrite\n Systems"
],
[
"Abstract We revisit known transformations from Jinja bytecode to rewrite systems from the viewpoint of runtime complexity.",
"Suitably generalising the constructions proposed in the literature, we define an alternative representation of Jinja bytecode (JBC) executions as \"computation graphs\" from which we obtain a novel representation of JBC executions as \"constrained rewrite systems\".",
"We prove non-termination and complexity preservation of the transformation.",
"We restrict to well-formed JBC programs that only make use of non-recursive methods and expect tree-shaped objects as input.",
"Our approach allows for simplified correctness proofs and provides a framework for the combination of the computation graph method with standard techniques from static program analysis like for example \"reachability analysis\"."
],
[
"Introduction",
"In recent years research on complexity of rewrite systems has matured and a number of noteworthy results could be established.",
"We give a quantitative assessment based on the annual competition of complexity analysers within TERMCOMP.http://termcomp.uibk.ac.at/.",
"With respect to last year's run of TERMCOMP, we see a success rate of 38 % in the category Runtime Complexity – Innermost Rewriting.",
"Note that the corresponding testbed is not restricted to polynomial runtime complexity in any way.",
"With respect to a qualitative assessment we want to mention the very recent efforts to apply methods from linear algebra and automata theory to complexity [19]; recent efforts on adaption of the dependency pair method to complexity [11], [12], [22], [13] and the ongoing quest to incorporate compositionality [31], [2].",
"(See [20] for an overview in methods of complexity analysis of term rewrite systems.)",
"In this paper we are concerned with the applicability of these results to automated runtime complexity analysis of imperative programs, in particular of Jinja bytecode (JBC) programs.",
"Jinja is a Java-like language that exhibits the core features of Java [29].",
"Its semantics is clearly defined and machine checked in the theorem prover Isabelle/HOL [15].",
"We establish a complexity preserving transformation from JBC programs $$ to constrained term rewrite systems $$ , that is, the runtime complexity function with respect to $$ is bounded by the runtime complexity function with respect to $$ (Theorem REF ).",
"As a simple corollary to this result we obtain that the proposed transformation is non-termination preserving (Corollary REF ).",
"In our analysis we restrict to well-formed JBC programs that only make use of non-recursive methods.",
"The proposed transformation encompasses two stages.",
"The first stage provides a finite representation of all execution paths of $$ through a graph, dubbed computation graph (Theorem REF ).",
"The nodes of the computation graph are abstractions of JVM states and the graph is formed by symbolic execution essentially employing widening akin to those used in abstract interpretations [7].",
"We develop a new graph-based representation of abstractions of JVM states (Definition REF ).",
"Furthermore we show that finiteness of the computation graph can always be guaranteed (Lemma REF ).",
"In the second stage, we encode the (finite) computation graph as constrained term rewrite system (cTRS for short).",
"CTRSs form a special type of rewrite systems that allow the formulation of conditions $C$ over a theory $$ , such that a rule can only be used if the condition $C$ is satisfied in $$ .",
"Constraints are used to express relations on program variables.",
"We emphasise, that the proposed transformation is not directly automatable, but its implementation asks for a combination with an external shape analysis as presented for example in [27], [25], [33].",
"This allows the mating of the proposed term-based abstraction technique with more standard concepts from static program analysis.",
"In principle, the established transformation allows for the use of rewriting-based runtime complexity analysis for the resource analysis of JBC programs.",
"However, currently existing methods for complexity analysis do not (yet) extend to cTRSs; this is subject to future work.",
"Our work was inspired by Panitz and Schmidt-Schauß original observation that term-based abstraction can provide powerful termination analysis [24].",
"Furthermore, we got inspiration from the ongoing quest to establish non-termination preserving transformations from JBC programs to integer term rewrite system [23], [6], [4].",
"The approach has been implemented in http://aprove.informatik.rwth-aachen.de/.",
"and has shown significant power in comparison to dedicated complexity and termination tools for JBC programs [28], [1].",
"Comparing our work with earlier results reported for the termination graph method [23], [6] we see that a similar transformation from graphs to rewrite systems is employed.",
"On the other hand in Otto et al.",
"[23] (and follow-up work) sharing is dealt with explicitly, while in our context sharing is always allowed if not stated otherwise.",
"Furthermore Otto et al.",
"rely on heuristics to obtain a finite termination graph, while we can prove finiteness of computation graphs.",
"Termination behaviour and complexity of JBC programs is studied by Albert et al.",
"in [1].",
"The approach employs program transformations to constrained logic programs and has been successfully implemented in the http://costa.ls.fi.upm.es/.",
"tool; it often allows precise bounds on the resource usage and is not restricted to runtime complexity.",
"A theoretical limitation of the work is the focus on a path-length analysis of the heap, which does not provide the same detail as the term based abstraction presented here.",
"Zuleger et al.",
"[32] employs size-change abstraction to analyse the runtime complexity of C programs automatically.",
"In connection with pathwise analysis and contextualisation size-change abstraction yields a powerful analysis.",
"The approach has been implemented in the tool .",
"Our approach extends the use of transition systems by cTRSs, which theoretically form a strict extension.",
"Furthermore, as our methods are rooted in rewriting we are not limited to the powers of invariant generation tools.",
"Very recently Hofmann and Rodrigues proposed in [14] an automated resource analysis based on Tarjan's amortised cost analysis [30] for object-oriented programs.",
"The method is implemented in the prototype http://raja.tcs.ifi.lmu.de.."
],
[
"Structure",
"This paper is structured as follows.",
"In Sections and we fix some basic notions to be used in the sequel.",
"In particular, we give an overview over the Jinja programming language.",
"Our notion of abstract states is presented in Section , while computation graphs are proposed in Section .",
"Section introduces cTRSs and presents the transformation from computation graphs to rewrite systems.",
"In Section we briefly mention crucial design choices for our prototype implementation.",
"Finally, in Section we conclude.",
"Let $f$ be a mapping from $A$ to $B$ , denoted $f:A \\rightarrow B$ , then $(f) = \\lbrace x \\mid f(x) \\in B \\rbrace $ and $(f) = \\lbrace f(x) \\mid x \\in A\\rbrace $ .",
"Let $a \\in (f)$ .",
"We define: $f\\lbrace a \\mapsto v\\rbrace (x) {\\left\\lbrace \\begin{array}{ll}v & \\text{if $x = a$}\\\\f(x) & \\text{otherwise} \\end{array}\\right.",
"}$ We compare partial functions with Kleene equality: Two partial functions $f \\colon \\rightarrow $ and $g \\colon \\rightarrow $ are equal, denoted $f g$ , if for all $n \\in $ either $f(n)$ and $g(n)$ are defined and $f(n) = g(n)$ or $f(n)$ and $g(n)$ are not defined.",
"We usually use square brackets to denote a list.",
"Further, ($$ ) denotes the cons operator, and ($$ ) is used to denote the concatenation of two lists.",
"Definition 2.1 A directed graph $G=({G},{G},{G})$ over the set $$ of labels is a structure such that ${G}$ is a finite set, the nodes or vertices, ${G} \\colon {G} \\rightarrow {G}^{\\ast }$ is a mapping that associates a node $u$ with an (ordered) sequence of nodes, called the successors of $u$ .",
"Note that the sequence of successors of $u$ may be empty: ${G}(u) = []$ .",
"Finally ${G} \\colon {G} \\rightarrow $ is a mapping that associates each node $u$ with its label ${G}(u)$ .",
"Let $u$ , $v$ be nodes in $G$ such that $v \\in {G}(u)$ , then there is an edge from $u$ to $v$ in $G$ ; the edge from $u$ to $v$ is denoted as $u \\rightarrow v$ .",
"Definition 2.2 A structure $G=({G},{G},{G},{G})$ is called directed graph with edge labels if $({G},{G},{G})$ is a directed graph over the set $$ and ${G} \\colon {G} \\times {G} \\rightarrow $ is a mapping that associates each edge $e$ with its label ${G}(e)$ .",
"Edges in $G$ are denoted as $u {\\ell } v$ , where ${G}(u \\rightarrow v) = l$ and $u,v \\in {G}$ .",
"We often write $u \\rightarrow v$ if the label is either not important or is clear from context.",
"If not mentioned otherwise, in the following a graph is a directed graph with edge labels.",
"Usually nodes in a graph are denoted by $u,v, \\dots $ possibly followed by subscripts.",
"We drop the reference to the graph $G$ from ${G}$ , ${G}$ , and ${G}$ , ie., we write $G = (,,)$ if no confusion can arise from this.",
"Further, we also write $u \\in G$ instead of $u \\in $ .",
"Let $G=(,,)$ be a graph and let $u \\in G$ .",
"Consider $(u) = [u_1,\\dots ,u_{k}]$ .",
"We call $u_i$ ($1 \\leqslant i \\leqslant k$ ) the $i$ -th successor of $u$ (denoted as $u {i} u_i$ ).",
"If $u {i} v$ for some $i$ , then we simply write $u v$ .",
"A node $v$ is called reachable from $u$ if $u v$ , where $$ denotes the reflexive and transitive closure of $$ .",
"We write $$ for $\\circ $ .",
"A graph $G$ is acyclic if $u v$ implies $u \\ne v$ .",
"We write ${G}{u}$ for the subgraph of $G$ reachable from $u$ ."
],
[
"Jinja Bytecode",
"In this section, we give an overview over the Jinja programming language [15].",
"In particular we inspect the internal state of the Jinja Virtual Machine (JVM).",
"We expect the reader to be familiar with the Java programming language.",
"Definition 3.1 A Jinja value can be a Boolean of type $$ , an (unbounded) integer of type $$ , the dummy value $$ of type $$ , the null reference $$ of type $$ , or a reference (or address).",
"We usually refer to (non-null) references as addresses.",
"The dummy value $$ is used for the evaluation of assignments (see [15]) and also used in the JVM to allocate uninitialised local variables.",
"The actual type of addresses is not important and we usually identify the type of an address with the type of the object bounded to the address.",
"Example 3.1 Figure REF depicts a program defining a List class with the append method.",
"Deviating from the notation employed by Klein and Nipkow in [15], we present Jinja code in a Java-like syntax.",
"Figure: The append program.In preparation for the sequent sections, we reflect the structure and properties of JBC programs and the JVM.",
"Definition 3.2 A JBC program $$ consists of a set of class declarations.",
"Each class is identified by a class name and further consists of the name of its direct superclass, field declarations and method declarations.",
"The superclass declaration is non-empty, except for a dedicated class termed Object.",
"Moreover, the subclass hierarchy of $$ is tree-shaped.",
"A field declaration is a pair of field name and field type.",
"A method declaration consists of the method name, a list of parameter types, the result type and the method body.",
"A method body is a triple of $(mxs \\times mxl \\times instructionlist)$ , where $mxs$ and $mxl$ are natural numbers denoting the maximum size of the operand stack and the number of local variables, not including the $this$ reference and the parameters of the method, while $instructionlist$ gives a sequence of bytecode instructions.",
"The $this$ reference can be conceived as a hidden parameter and references the object that invokes the method.",
"The set of Jinja bytecode instructions is adapted for our needs and listed in Figure REF .",
"We employ following conventions: Let $n$ denote a natural number, $i$ an integer, $v$ a Jinja value, $cn$ a class name, and $mn$ a method name.",
"Figure: The Jinja bytecode instruction set.Definition 3.3 A (JVM) state is a pair consisting of the heap and a list of frames.",
"Let $$ denote the strict subclass relation and $$ its reflexive closure.",
"A heap is a mapping from addresses to objects, where an object is a pair $(,)$ such that: $$ denotes the class name, and $$ denotes the fieldtable, ie., a mapping from $(^{\\prime },)$ to values, where $$ is a field name and $^{\\prime }$ is a (not necessarily proper) superclass of $$ , ie., $^{\\prime }$ .",
"A frame represents the environment of a method and is a quintuple $(,,,,)$ , such that: $$ denotes the operation stack, ie., an array of values, $$ denotes the registers, ie., an array of values, $$ denotes the class name, $$ denotes the method name, and $$ is the program counter.",
"Let $$ ($$ ) denote the operation stack (registers) of a given frame.",
"Typically the structure of $$ is as follows: the 0th register holds the this-pointer, followed by the parameters and the local variables of the method.",
"Uninitialised registers are preallocated with the dummy value $$ .",
"We denote the entries of $$ ($$ ), by $(i)$ ($(i)$ ) for $i\\in $ and write $()$ ($()$ ) for the set of indices of the array $$ ($$ ).",
"The collection of all stack (register) indices of a state is denoted $$ ($$ ).",
"Often there is no need to separate between the local variables of a Jinja program and the registers in a JBC program.",
"Hence we use registers and local variables interchangeably.",
"Observe that the domain of the fieldtable for a given object of class $$ contains all fields declared for $$ together with all fields declared for superclasses of $$ .",
"Clearly the domain of the fieldtable is equal for any instance of class $$ .",
"Figure REF illustrates the one-step execution of the IAdd bytecode instruction.",
"We have extended the original set of instructions by some standard operations on values, taking ideas from Jinja with Threads into account [17], [18].",
"The semantics of all employed JBC instructions can be found in the Appendix.",
"Figure: The bytecode instruction.Example 3.2 Consider the append program from Example REF .",
"Figure REF depicts the corresponding bytecode program, resulting from the compilation rules in [15].",
"In the following we name the registers 0,1, and 2 as this, ys, and cur, respectively.",
"Figure: The bytecode for the List program.Definition 3.4 We extend the subclass relation to a partial order on types, denoted $$ .",
"The types of $$ consists of $\\lbrace , , , \\rbrace $ together with all classes $cn$ defined in $$ .",
"We use $(v)$ to denote the type of value $v$ and $()$ to denote the collection of types in $$ .",
"Recall that we usually identify the type of an address with the type of the object bound to the address.",
"Let $t, t^{\\prime }, cn, cn^{\\prime }$ be types in $$ .",
"Then $t t^{\\prime }$ holds if $t = t^{\\prime }$ or $t = $ , $t = $ and $t^{\\prime } = cn$ , $t = cn$ , $t^{\\prime } = cn^{\\prime }$ and $cn cn^{\\prime }$ .",
"The least common superclass is the least upper bound for a set of classes $CN \\subseteq ()$ and is always defined.",
"The bytecode verifier established in [15] ensures following properties: All bytecode instructions are provided with arguments of the expected type.",
"No instruction tries to get a value from the empty stack, nor puts more elements on the stack or access more registers than specified in the method.",
"The program counter is always within the code array of the method.",
"All registers except from the register storing $this$ must be first written to before accessed.",
"Furthermore the verifier ensures that for states with equal program counter the size of the stack is of equal length.",
"Moreover, the list of registers is of fixed length.",
"The compiler presented in [15] transforms a well-formed Jinja program into a well-formed JBC program.",
"A JBC program that passes the bytecode verification is again called well-formed.",
"While the set of instruction used here are a (slight) extension of the minimalistic set considered in [15], this notion of well-formedness is still applicable, as all considered extensions are present in Jinja with Threads [17], [18].",
"In the following we consider Jinja programs and JBC programs to be well-formed.",
"To ease readability we do not consider exception handling, that is, an exception yields immediate termination of the program.",
"This is not a restriction of our analysis, as it could be easily integrated, but complicates matters without gaining additional insight.",
"While Definition REF provides a succinct presentation of the state, it is more natural to conceive the heap (and conclusively a state) as a graph.",
"We omit the technical definition here but provide the general idea: Let $s = (,)$ be a state.",
"We define the state graph of $s$ as $= ({},{},{},{})$ .",
"For all non-address values of $s$ we define an unique implicit reference.",
"The idea is that sharing is only induced via references but not implicit references.",
"The nodes of $S$ consists of all stack (register) indices, the references in $$ and the implicit references of $s$ .",
"The successors of a node indicate the values bound to stack (register) indices and the fields of instances in $$ , and is an implicit reference if a non-address value is bound and a reference otherwise.",
"The label of a node is either a stack (register) index, the type of an instance $(u)$ or a non-address value.",
"The label of an edge indicates the fields $(cn,id)$ for instances $(u)$ , and is empty otherwise.",
"In presenting state graphs, we indicate references, but do not depict implicit references.",
"Furthermore, we use representative names for stack (register) indices.",
"Example 3.3 Recall the ${append}$ program of example REF .",
"Suppose $this$ is initially a list of length one, and $ys$ is $$ .",
"Figure REF depicts the state graph after the assignment $cur = this$ .",
"Figure: State graph.Let $P$ be a program and let $s$ and $t$ be states.",
"Then we denote by ${s}{t}$ the one-step transition relation of the JVM.",
"If there exists a (normal) evaluation of $s$ to $t$ , we write ${s}{t}$ .",
"Let $$ denote the set of states.",
"The complete lattice $() ((),\\subseteq ,\\cup ,\\cap ,\\varnothing ,)$ denotes the concrete computation domain.",
"The size of a state is defined on a per-reference basis, which unravels sharing.",
"We explicitly add 1 to the overall construction.",
"This does not affect the results but allows a more convenient relation to the size of its term representation we present later.",
"Definition 3.5 Let $$ be a state and let $$ be its state graph.",
"Let $u,v$ be nodes in $$ and $u [S] v$ denote a simple path $P$ in $$ from $u$ to $v$ .",
"Note that $P$ does not contain cycles.",
"Then the size of a stack or register index $u$ , denoted as ${u}$ , is defined as follows: ${u} \\sum _{\\raisebox {-2mm}{u [S] v}} {{}(v)} $ where ${l}$ is $(l)$ if $l \\in $ , otherwise 1, for $l \\in {}$ .",
"Here, $(z)$ denotes the absolute value of the integer $z$ .",
"Then the size of $$ is the sum of all sizes of stack or register indices in $$ plus 1.",
"In the following we use ${s}$ to denote the size of a state $s$ .",
"We define the runtime of a JVM for a given normal evaluation ${s}{t}$ as the number of single-step executions in the course of the evaluation from $s$ to $t$ .",
"Definition 3.6 Let $$ denote the set of JVM states of $$ , and $\\mathcal {S} \\subseteq $ .",
"We define the runtime complexity with respect to $$ as follows: $(n) \\max \\lbrace m |\\text{${}{t}$ holds such that the runtime is $m$,$\\in \\mathcal {S}$ and ${} \\leqslant n$}\\rbrace $ Note that we adopt a (standard) unit cost model for system calls."
],
[
"Abstract States",
"In this section, we introduce abstract states as generalisations of JVM states.",
"The intuition being that abstract states represent sets of states in the JVM.",
"The idea of abstracting JVM states in this way is due to Otto et al. [23].",
"However, our presentation crucially differs from [23] (and also from follow-up work in the literature) as we employ an implicit representation of sharing that makes use of graph morphisms, rather than the explicit sharing information proposed in [23], [6], [5], [4].",
"Furthermore, abstract states as defined below are a straightforward generalisation of JVM states as defined in [15].",
"This circumvents an additional transformation step as presented in [6].",
"Definition 4.1 We extend Jinja expressions by countable many abstract variables $X_1,X_2,X_3,\\dots $ , denoted by $x$ , $y$ , $z$ , ...An abstract variable may either abstract an object, an integer or a Boolean value.",
"In denoting abstract variables typically the name is of less importance than the type, that is we denote an abstract variable for an object of class $$ , simply as $$ , while abstract integer or Boolean variables are denoted as $$ , and $$ , respectively.",
"The (strict) subclass relation ($$ ) $$ is extended in the natural way to abstract variables for classes.",
"For brevity we sometimes refer to an abstract variable of integer or Boolean type, as abstract integer or abstract Boolean, respectively.",
"Definition 4.2 An abstract value is either a Jinja value (cf.",
"Definition REF ), or an abstract Boolean or integer.",
"In turn a Jinja value is also called a concrete value.",
"Note that, as in the JVM, only (abstract) objects can be shared.",
"In particular abstract variables for objects are only referenced via the heap.",
"The next definition abstracts the heap of a JVM through the use of abstract variables and values.",
"Definition 4.3 An abstract heap is a mapping from addresses to abstract objects, where an abstract object is either a pair $(,)$ or an abstract variable.",
"Abstract frames are defined like frames of the JVM, but registers and operand stack of an abstract frame store abstract values.",
"We define (partial) projection functions $$ and $$ as follows: $(obj) & {\\left\\lbrace \\begin{array}{ll}& \\text{if $obj$ is an object and $obj = (,)$} \\\\& \\text{if $obj$ is an abstract variable of type $$}\\end{array}\\right.",
"}\\\\(obj) & {\\left\\lbrace \\begin{array}{ll}& \\text{if $obj$ is an object and $obj = (,)$} \\\\\\text{undefined} & \\text{otherwise}\\end{array}\\right.",
"}$ Furthermore, we define annotations of addresses in an abstract state $s$ , denoted as $$ .",
"Formally, annotations are pairs ${p}{q}$ of addresses, where $p,q \\in $ and $p$ is not $q$ .",
"Definition 4.4 An abstract state $s=(,,)$ is either a triple consisting of an abstract heap $$ , a list of abstract frames $$ , and a set of annotations $$ , the maximal abstract state, denoted as $\\top $ , or the minimal abstract state, denoted as $\\bottom $ .",
"If $s=(,,)$ , we demand that all addresses in $$ are reachable from local variables or stack entries in the list of frames $$ .",
"The set of abstract states is collected in the set $$ .",
"When depicting (abstract) states, we replace stack and register indices by intuitive names, denoted in roman font.",
"Furthermore, we make use of the following conventions: we use an italic font (and lower-case) to describe abstract variables and a sans serif (and upper-case) to depict class names.",
"Example 4.1 Consider the List program from Example REF together with the well-formed JBC program depicted in Figure REF .",
"Consider the state $A$ depicted below: state](A) $\\begin{array}[ht]{l|l}{04} & \\epsilon \\mid \\text{this} = o_1, \\text{ys} = o_2, \\text{cur} = o_1 \\\\& o_1 = {List} (\\text{List.val} = int, \\text{List.next} = o_3 ) \\\\A & o_2 = list, o_3 = list\\end{array}$ ; The operation stack in $A$ is empty.",
"The registers this and cur contain the same address $o_1$ and ys is mapped to $o_2$ .",
"In the heap $o_1$ is mapped to an object of type ${List}$ whose value is abstracted to $int$ and whose next element is referenced by $o_3$ .",
"It is not difficult to see that $A$ forms an abstraction of any JVM state obtained at instruction 04 in the List program (if this initially references a non-empty list) before any iteration of the while-loop.",
"Furthermore, consider the following state $B$ : state](B) $\\begin{array}[ht]{l|l}{04} & \\epsilon \\mid \\text{this} = o_1, \\text{ys} = o_2, \\text{cur} = o_3 \\\\& o_1 = {List}(\\text{List.val} = int, \\text{List.next} = o_3) \\\\& o_2 = list, o_4 = list \\\\B & o_3 = {List}(\\text{List.val} = int, \\text{List.next} = o_4 )\\end{array}$ ; Again it is not difficult to see that $B$ abstracts any JVM state obtained if exactly one iteration of the loop has been performed.",
"Due to the presence of abstract variables, abstract states can represent sets of states as the variables can be suitably instantiated.",
"The annotation ${p}{q} \\in $ will be used to disallow aliasing of addresses in JVM states represented by the abstract state.",
"Different JVM states can be abstracted to a single abstract state.",
"To make this precise, we will augment $$ with a partial order $$ , the instance relation (see Definition REF ).",
"We will extend the partial order $(,)$ to a complete lattice $(,,,,\\bottom ,\\top )$ and show a Galois insertion between $()$ and $$ .",
"Definition 4.5 We define a preorder on abstract values, which are not references, and abstract objects.",
"We extend $(v)$ (cf.",
"Definition REF ) to abstract values the intended way, ie., $(int) = int, (bool) = bool$ and $(cn) = cn$ for an integer variable $int$ , a Boolean variable $bool$ , and class variable $cn$ .",
"Then the preorder $$ is defined as follows: We have $v w$ , if either $v = w$ , or $(v) (w)$ and $w$ is an abstract variable.",
"We write $w v$ , if $v w$ .",
"Let ${}$ , ${}$ denote the maximum size of the operand stack and the number of variables respectively.",
"We make use of the following abbreviation: $w {m} v$ if either $w v$ or $v,w$ are references and we have $v = m(w)$ , where $m$ denotes a mapping on references.",
"Definition 4.6 Let $s = (,,)$ be a state in $\\setminus \\lbrace \\top ,\\bottom \\rbrace $ with $= [_1,\\dots ,_k]$ and $_i = (_i,_i,_i,_i,_i)$ , and let $t=(^{\\prime },^{\\prime },^{\\prime })$ be a state with $^{\\prime } = [^{\\prime }_1,\\dots ,^{\\prime }_k]$ and $^{\\prime }_i = (^{\\prime }_i,^{\\prime }_i,^{\\prime }_i,^{\\prime }_i,^{\\prime }_i)$ .",
"Then $s$ is an abstraction of $t$ (denoted as $s t$ ) if the following conditions hold: for all $1 \\leqslant i \\leqslant k$ : $_i = ^{\\prime }_i$ , $_i = ^{\\prime }_i$ , and $_i = ^{\\prime }_i$ , for all $1 \\leqslant i \\leqslant k$ : $(_i) = (^{\\prime }_i)$ and $(_i) = (^{\\prime }_i)$ , and there exists a mapping $m \\colon () \\rightarrow (^{\\prime })$ such that for all $1 \\leqslant i \\leqslant k$ , $1 \\leqslant j \\leqslant {_i}$ : ${_i(j) {m} ^{\\prime }_i(j)}$ , for all $1 \\leqslant i \\leqslant k$ , $1 \\leqslant j \\leqslant {_i}$ : $_i(j) {m} ^{\\prime }_i(j)$ , for all $a \\in ()$ : $(a) heap^{\\prime }(m(a))$ , for all $a \\in ()$ , such that $((a))$ is defined and for all $1 \\leqslant i \\leqslant \\ell $ : $f(_i,id_i) {m} f^{\\prime }(^{\\prime }_i,id_i)$ ,where $f ((a))$ with $(f) = \\lbrace (_1,id_1),\\dots ,(_\\ell ,id_\\ell )\\rbrace $ , and $f^{\\prime } (^{\\prime }(m(a)))$ with $(f^{\\prime }) = \\lbrace (_1,id_1),\\dots ,(_\\ell ,id_\\ell )\\rbrace $ .",
"finally, we have $^{\\prime } \\supseteq m^\\ast ()$ .",
"Here, $m^\\ast $ denotes the lifting of the mapping $m$ to sets: $m(\\lbrace isunshared_1,\\dots ,_k\\rbrace ) = \\lbrace m(_1),\\dots ,m(_k)\\rbrace $ .",
"Furthermore for all $s \\in $ : $s \\top $ and $\\bottom s$ .",
"Example 4.2 Consider the states $A$ and $B$ described in Example REF .",
"For the state $S$ depicted below we obtain that $A S$ and $B S$ , ie., $S$ forms an abstraction of both states.",
"state](S) $\\begin{array}[ht]{l|l}{04} & \\epsilon \\mid \\text{this} = o_1, \\text{ys} = o_2, \\text{cur} = o_4 \\\\& o_1 = {List}(\\text{List.val} = int, \\text{List.next} = o_3 ) \\\\& o_2 = list, o_3 = list, o_5 = list\\\\S & o_4 = {List}(\\text{List.val} = int, \\text{List.next} = o_5)\\end{array}$ ; The definition of state graphs naturally extends to abstract states, when incorporating $_{}$ and considering abstract values.",
"Furthermore, we use $\\top $ to denote the state graph of $\\top \\in $ and the empty graph to denote $\\bot \\in $ .",
"Example 4.3 Consider the states $A$ , $B$ , and $S$ presented in Examples REF and REF .",
"The state graph of $A$ and $B$ are given in Figure REF and Figure REF , respectively.",
"The state graph of the abstraction $S$ is depicted in Figure REF .",
"Figure: Abstract State BBWe introduce state homomorphisms that allow an alternative, but equivalent definition of the instance relation $$ .",
"Definition 4.7 Let $S$ and $T$ be state graphs of states $s$ and $t$ , respectively such that $S, T \\ne \\varnothing $ .",
"A state homomorphism from $S$ to $T$ (denoted $m \\colon S \\rightarrow T$ ) is a function $m \\colon {S} \\rightarrow {T}$ such that for all $u \\in S$ and $u \\in \\cup $ , ${S}(u) = {T}(m(u))$ , for all $u \\in S \\setminus (\\cup )$ , ${S}(u) {T}(m(u))$ , for all $u \\in S$ : if $u [S]{i} v$ , then $m(u) [T]{i} m(v)$ and for all ${u {\\ell } v} \\in S$ and ${m(u) {\\ell ^{\\prime }} m(v)} \\in T$ , $\\ell = \\ell ^{\\prime }$ .",
"If no confusion can arise we refer to a state homomorphism simply as morphism.",
"It is easy to see that the composition $m_1 \\circ m_2$ of two morphisms $m_1$ , $m_2$ is again a morphism.",
"We say that two states $s, t \\in $ are isomorphic if there exists a morphism from $s$ to $t$ and vice versa.",
"Suppose the abstract states $s$ and $t$ are isomorphic.",
"Then they differ only in their abstract variables and can be transformed into each other through a renaming of variables.",
"Thus the set of JVM states represented by $s$ and $t$ is equal; we call $s$ and $t$ equivalent (denoted $s t$ ).",
"Let $, t \\in $ and let $$ and $T$ denote their state graphs.",
"Then $s t$ if one of the following alternatives holds: (i) $S = \\top $ , (ii) $T$ is empty, or (iii) $S,T \\ne \\varnothing $ and there exists a state morphism $m$ from $S$ to $T$ ; $s=(,,)$ , $t=(^{\\prime },^{\\prime },^{\\prime })$ and the program counters, the class and method names of all frames in $s$ and $t$ coincide; $^{\\prime } \\supseteq m^\\ast ()$ .",
"Lemma 4.1 Let $, t \\in $ .",
"Then $s t$ iff $s t$ .",
"Straightforward.",
"Due to Lemma REF and the composability of morphism it follows that the instance relation $$ is transitive.",
"Hence the relation $$ is a preorder.",
"Furthermore $$ can be lifted to a partial order, if we consider the factorisation of the set of abstract states with respect to the equivalence relation $$ .",
"In order to express this fact notationally, we identify isomorphic states and replace $$ by $=$ .",
"Conclusively $(,)$ is a partial order.",
"We are left to provide a least upper bound definition of the join of abstract states.",
"Definition 4.8 Let $s$ and $s^{\\prime }$ be states such that there exists an abstraction $t$ of $s$ and $s^{\\prime }$ .",
"We call $t$ the join of $s$ and $s^{\\prime }$ , denoted as $s s^{\\prime }$ , if $t$ is a least upper bound of $\\lbrace s,s^{\\prime }\\rbrace $ with respect to the preorder $$ .",
"The limit cases are handled as usual.",
"If the program locations of $s$ and $s^{\\prime }$ differ, then $s s^{\\prime } = \\top $ .",
"Otherwise, we can identify invariants to construct an upper bound $t \\ne \\top $ and prove well-definedness of $s s^{\\prime }$ .",
"Let $=({},{},{},{},_{})$ and $^{\\prime }=({^{\\prime }},{^{\\prime }},{^{\\prime }},{^{\\prime }},_{^{\\prime }})$ be the two state graphs of state $s$ and $s^{\\prime }$ , respectively.",
"Furthermore, let $t$ be an abstraction of $s$ and $s^{\\prime }$ , and let $T=({T},{T},{T},{T},_{T})$ be its state graph.",
"By definition we have the following properties: Let $$ ($$ ) collect the stack (register) indices of state $s$ .",
"As $s t$ , $$ ($$ ) coincides with the set of stack (register) indices of $t$ .",
"Similarly for $s^{\\prime }$ and thus ${T} \\supseteq {\\cup }$ .",
"For any node $u \\in T$ there exist uniquely defined nodes $v \\in {}$ , $w \\in {^{\\prime }}$ such that ${}(v) {T}(u)$ , ${^{\\prime }}(w) {T}(u)$ .",
"We say the nodes $v$ and $w$ correspond to $u$ .",
"For any node $u \\in T$ and any successor $u^{\\prime }$ of $u$ in $T$ there exists a successor $v^{\\prime }$ ($w^{\\prime }$ ) in $$ ($^{\\prime }$ ) of the corresponding node $v$ ($w$ ) in $$ ($^{\\prime }$ ).",
"Furthermore $v^{\\prime }$ and $w^{\\prime }$ correspond to $u^{\\prime }$ .",
"For any edge ${u {\\ell } u^{\\prime }} \\in T$ such that $v$ ($w$ ) corresponds to $u$ in $$ ($^{\\prime }$ ) there is an edge ${v {k} v^{\\prime }} \\in $ and an edge ${w {k^{\\prime }} w^{\\prime }} \\in ^{\\prime }$ such that $\\ell = k = k^{\\prime }$ .",
"For any annotation ${u \\ne u^{\\prime }} \\in _{T}$ there exists ${v \\ne v^{\\prime }}$ in $_{}$ and ${w \\ne w^{\\prime }}$ in $_{^{\\prime }}$ , where $v$ ($v^{\\prime }$ ) and $w$ ($w^{\\prime }$ ) correspond to $u$ ($u^{\\prime }$ ).",
"In order to construct an abstraction $t$ of $s$ and $s^{\\prime }$ we use the above properties as invariants and define its state graph $T$ by iterated extension.",
"We define $T^0$ by setting ${T^0} \\cup $ .",
"Due to Property REF these nodes exist in $$ and $^{\\prime }$ as well.",
"The labels of stack or register indices trivially coincide in $$ and $^{\\prime }$ , cf.",
"Definition REF .",
"Thus we set ${T^0}$ accordingly.",
"Furthermore we set ${T^0} = {T^0} = _{T^0} \\varnothing $ .",
"Then $T^0$ satisfies Properties REF –REF .",
"Suppose state graph $T^n$ has already been defined such that the Properties REF – REF are fulfilled.",
"In order to update $T^n$ , let $u \\in {T^n}$ such that $v$ and $w$ correspond to $u$ .",
"Suppose ${v {k} v^{\\prime }} \\in $ and ${w {k} w^{\\prime }} \\in ^{\\prime }$ such that there is no node $u^{\\prime }$ in $T^n$ where $v^{\\prime }$ and $w^{\\prime }$ correspond to $u^{\\prime }$ .",
"Let $u^{\\prime }$ denote a node fresh to $T^n$ .",
"We define ${T^{n+1}} {T^{n}} \\cup \\lbrace u^{\\prime }\\rbrace $ and establish Property REF by setting ${T^{n+1}}(u^{\\prime })$ such that ${}(v^{\\prime }) {T^{n+1}}(u^{\\prime })$ and ${^{\\prime }}{w^{\\prime }} {T^{n+1}}(u^{\\prime })$ where ${T^{n+1}}(u^{\\prime })$ is as concrete as possible.",
"If we succeed, we fix that $v^{\\prime }$ and $w^{\\prime }$ correspond to $u^{\\prime }$ .",
"It remains to update $_{T^{n+1}}$ suitably such that Property REF is fulfilled.",
"If this also succeeds Properties REF – REF are fulfilled for $T^{n+1}$ .",
"On the other hand, if no further update is possible we set $T T^n$ .",
"By construction $T$ is an abstraction of $S$ and $S^{\\prime }$ and indeed represents $s s^{\\prime }$ .",
"Example 4.4 Consider the states $A$ , $B$ , and $S$ described in Example REF .",
"In Figure REF an abstraction of $A$ and $B$ is given.",
"In particular, abstraction $S$ results of the construction defined above, ie., $S = A B$ .",
"Figure: Abstraction SSA sequence of states $(s_i)_{i \\geqslant 0}$ forms an ascending sequence, if $i < j$ implies $s_i s_j$ .",
"An ascending sequence $(s_i)_{i \\geqslant 0}$ eventually stablises, if there exits $i_0 \\in $ such that for all $i \\geqslant i_0$ : $s_i = s_{i_0}$ .",
"The next lemma shows that any ascending sequence eventually stabilises.",
"Lemma 4.2 The partial order $(,)$ satisfies the ascending chain condition, that is, any ascending chain eventually stabilises.",
"In order to derive a contradiction we assume the existence of an ascending sequence $(s_i)_{i \\geqslant 0}$ that never stabilises.",
"By definition for all $i \\geqslant 0$ : ${s_i} \\geqslant {s_{i+1}}$ .",
"By assumption there exists $i \\in $ such that for all $j > i$ : ${s_i} = {s_j}$ and $s_{i} s_{j}$ .",
"The only possibility for two different states $s_{i}, s_j$ of equal size that $s_i s_j$ holds, is that addresses shared in $s_i$ become unshared in $s_j$ .",
"Clearly this is only possible for a finite amount of cases.",
"Contradiction.",
"Lemma REF in conjunction with the fact that $(,)$ has a least element $\\bottom $ and binary least upper bounds implies that $(,,,,\\bottom ,\\top )$ is a complete lattice.",
"In particular any set of states $\\mathcal {S}$ has a least upper bound, denoted as ${\\mathcal {S}}$ .",
"The meet operation ${}$ can be expressed by ${}$ , yet in practice we do not need it."
],
[
"Correctness",
"In the remainder of the paper we fix to a concrete JBC program $$ .",
"Above, we already restricted our attention to well-formed JBC programs $$ using the expressions and instructions defined in Section .",
"For the proposed static analysis of these programs we additionally restrict to non-recursive methods.",
"Note that the states in $$ can in principle express recursive methods, but for recursive methods, we cannot use the below proposed construction to obtain finite computation graphs, as the graphs defined in Definition REF cannot handle unbounded list of frames.",
"In the following we use superscript $\\natural $ , if we want to distinguish between concrete and abstract states, or between operations on concrete and abstract states.",
"Let $s=(,) \\in $ , we define a mapping $\\beta \\colon \\rightarrow $ , that injects JVM states into $$ .",
"For that let $() = \\lbrace p_1,\\dots ,p_n\\rbrace $ and define $$ such that all $p_i \\ne p_j \\in $ for all different $i,j$ .",
"Definition 4.9 We define the abstraction function $\\alpha \\colon () \\rightarrow $ and the concretisation function $\\gamma \\colon \\rightarrow ()$ as follows: $\\alpha (\\mathcal {S}) &{\\lbrace \\beta (s) \\mid s \\in \\mathcal {S}\\rbrace } \\\\\\gamma () &\\lbrace s \\in \\mid \\beta (s) \\rbrace $ We set $\\alpha (s) \\alpha (\\lbrace s\\rbrace )$ .",
"It is easy to see that $$ contains redundant states: Consider abstract states $,\\in $ .",
"Let $= (, , )$ , $p,q \\in ()$ and $p \\ne q \\in $ .",
"Let $$ be defined like $$ but $p \\ne q \\notin $ .",
"Now suppose that the types of $p$ and $q$ are not related with respect to the subclass order.",
"Then $$ and $\\gamma () = \\gamma ()$ .",
"To form a Galois insertion between $()$ and $$ , we introduce a reduction operator that adds annotations for non-aliasing addresses.",
"Definition 4.10 Let $= (, , )$ be an abstract state.",
"We define the reduction operator $\\varsigma \\colon \\rightarrow $ as follows: $\\varsigma () (, , ^{\\prime })$ where $^{\\prime } \\lbrace p \\ne q \\mid p, q \\in (heap)\\rbrace \\setminus \\lbrace p \\ne q \\mid s \\in \\gamma (), m: \\rightarrow \\beta (s), m(p) = m(q) \\rbrace $ .",
"Then $\\varsigma () $ and $\\gamma (\\varsigma ()) = \\gamma ()$ .",
"In practice, we compute the reduction by a unification argument of $p$ and $q$ in $$ : We try to construct a new state ${} $ , where $r = m(p) = m(q)$ .",
"Let $$ and $$ be the state graphs of $$ and $$ .",
"Suppose $u, v, w$ represent $r, p, q$ in $$ and ${}$ .",
"We can use a similar reasoning we used for the join construction, but now require ${}(u) {}(v)$ and ${}(u) {}(w)$ if $v$ and $w$ correspond to $u$ .",
"If the construction succeeds, we can easily find a concrete state from ${}$ such that $m(p) = m(q)$ .",
"The construction does not succeed if, for example, successors of corresponding nodes have different concrete values; then we add $p \\ne q$ .",
"Lemma 4.3 The maps $\\alpha $ and $\\gamma $ define a Galois insertion between the complete lattices $()$ and $\\varsigma ^\\ast ()$ , where $\\varsigma ^\\ast $ denotes the set extension of $\\varsigma $ .",
"It suffices to prove that $\\gamma $ is injective, ie., for all $, \\in \\varsigma ^\\ast ()$ if $\\ne $ then $\\gamma () \\ne \\gamma ()$ .",
"Suppose $\\ne $ but $\\gamma () = \\gamma ()$ .",
"It is a simple consequence of our morphism definition that $\\gamma () \\ne \\gamma ()$ , if the state graphs of $$ and $$ differ.",
"Hence, $$ can only be different from $$ if the annotations of $$ and $$ differ.",
"However, by assumption they are equal.",
"Contradiction.",
"It follows that the reduction operator defined in Definition REF , indeed returns the greatest lower bound that represents the same element in the concrete domain as required.",
"In the following we identify the $\\varsigma ^\\ast ()$ with $$ .",
"In order to prove that the abstract domain $$ correctly approximates the concrete domain $()$ we need to define a suitable notion of abstract computation on abstract states.",
"Recall that Figure REF presents the single-step execution of the $$ instruction on the JVM.",
"Based on these instructions, and actually mimicking them quite closely, we define how abstract states are evaluated symbolically.",
"This is straightforward in most cases, with the exception of $$ and $$ instructions.",
"With respect to the former, we suppose a preliminary analysis on different heap shape properties.",
"In particular our analysis requires may-share, may-reachable, and maybe-cyclic analyses as given, see for example [27], [25], [33].",
"Definition 4.11 Let $$ be an abstract state and $p,q$ be addresses in the heap of $s$ .",
"We use $S$ to denote the state graph of $\\beta (s)$ for some concrete state $s$ .",
"We say that: $p$ and $q$ may-alias, if $m(p) = m(q)$ for some $s \\in \\gamma ()$ and morphism $m \\colon \\rightarrow \\beta (s)$ ; $p$ may-reaches $q$ , if $m(p) [S] m(q)$ for some $s \\in \\gamma ()$ and morphism $m \\colon \\rightarrow \\beta (s)$ ; $p$ is maybe-cyclic, if $m(p) [S] m(p)$ for some $s \\in \\gamma ()$ and morphism $m \\colon \\rightarrow \\beta (s)$ ; $p$ is acyclic, if $p$ is not maybe-cyclic.",
"Note that our representation does not provide a precise approximation of these properties, as abstract variables generally also present cyclic instances.",
"In Figure REF we have worked out the cases for the instructions $$ , $$ , $$ , $$ , $$ and $$ .",
"We follow the notation used in Figure REF above.",
"The other cases are left to the reader.",
"In addition to symbolic evaluations, we define refinement steps on abstract states $$ if the information given in $$ is not concrete enough to execute a given instruction.",
"It will be a consequence of our definitions that for any refinement $_i$ of $$ , we have $_i $ .",
"In the following assume $= (,,)$ .",
"Some comments: The symbolic instruction $~n$ loads the value of the $n$ th register onto the stack.",
"The only difference to $~n$ is that the value may be an integer or Boolean variable.",
"For the $$ instruction, we introduce a new abstract integer $i_3$ and the side-condition $i_1 + i_2 = i_3$ , if either $i_1$ or $i_2$ is an integer variable.",
"The $$ splits into different cases, depending on the status of the compared values.",
"We adapt the instruction to abstract values as follows: Let $val_1$ and $val_2$ be addresses.",
"If the addresses of $val_1$ and $val_2$ are the same then the test evaluates to true.",
"Otherwise, we have to check if $val_1$ and $val_2$ may alias and perform a unsharing refinement (cf.",
"Definition REF ) if necessary.",
"In the latter case the test returns false.",
"Wlog.",
"let $val_1$ be an address and $val_2$ be $$ .",
"If $(val_1)=obj$ and $(obj) = $ , we perform a instance refinement according to Definition REF on $val_1$ and re-consider the condition.",
"If $val_1$ and $val_2$ are concrete non-address Jinja values, then the test $(val_1 = val_2)$ can be directly executed and the symbolic execution equals the instruction on the JVM.",
"If $val_1$ and $val_2$ are abstract Boolean or integer variables, then we introduce a new Boolean variable $b_3$ and the side condition $(val_1 = val_2) \\equiv b_3$ .",
"Figure REF only shows the latter case.",
"$~cn$ allocates a new instance of type $cn$ in the heap and pushes the corresponding address onto the stack.",
"All fields of the fresh created instance are instantiated with the default value.",
"That is, 0 for integer typed fields, $$ for Boolean typed fields, and $$ otherwise.",
"If the top element of the stack is a concrete value, $$ can be executed directly.",
"Otherwise we perform a Boolean refinement, replacing the variable with values $$ and $$ .",
"Recall that a class variable $cn$ represents $$ as well as instances of $cn$ and its subtypes.",
"Hence, $~~^{\\prime }$ may require an instance refinement (cf.",
"Definition REF ).",
"Let $v$ be a value and $p$ be an address such that $(p) = (^{\\prime \\prime },)$ .",
"Due to abstraction there may exist addresses $q \\in (heap)$ different from $p$ that alias with $p$ .",
"Hence they are affected by the field update.",
"We introduce unsharing refinements (cf.",
"Definition REF ) for all $q$ , where $p \\ne q \\notin iu$ .",
"Definition 4.12 Let $=(,,)$ be a state and let $p$ be an address such that $(p) = cn^{\\prime }$ .",
"Let $cn \\in subclasses(cn^{\\prime })$ .",
"Furthermore, suppose $(_1,id_1),\\dots ,(_n,id_n)$ denote fields of $$ (together with the defining classes).",
"We perform the following class instance steps, where the second takes care of the case, where address $p$ is replaced by $$ .",
"${(\\lbrace p \\mapsto (,ftable_1)\\rbrace ,,)}{(,,)}\\hspace{42.67912pt}{(_2,_2,)}{(,,)}$ Here $ftable_1((_i,id_i)) v_i$ such that the type of the abstract variable $v_i$ is defined in correspondence to the type of field $(_i, id_i)$ , eg., a fresh $int$ variable for integer fields.",
"On the other hand we set $_2$ ($_2$ ) equal to $$ ($$ ), but $p \\notin (_2)$ and all occurrences of $p$ are replaced by $$ .",
"Definition 4.13 Let $=(,,)$ and let $p$ and $q$ denote different addresses in $$ such that $p \\ne q \\notin $ .",
"We perform the following unsharing steps: The first case forces these addresses to be distinct.",
"The second case substitutes all occurrences of $q$ with $p$ .",
"${(,,\\cup \\lbrace {p}{q} \\rbrace )}{(,,)}\\hspace{42.67912pt}{(^{\\prime },^{\\prime },)}{(,,)}$ where $^{\\prime }$ ($^{\\prime }$ ) is equal to $$ ($$ ) with all occurrences of $q$ replaced by $p$ .",
"Figure: Symbolic evaluations of Jinja bytecode instructionsExample 4.5 In Figure REF we present an example detailing the need for the given definition of class instantiation.",
"Here class B overrides method m inherited from class A.",
"We only know the static type of the parameter when analysing method call(A a).",
"Method call(A a) accepts any instances of class A or any instances of a subclass of A as parameter.",
"In particular any instance of class B.",
"Due to the overridden method call(A a) does not terminate for instances of class B.",
"Figure: All subclasses need to be considered.Let $, {^{\\prime }}$ and $$ be abstract states such that ${^{\\prime }}$ is obtained by zero or multiple refinement steps from $$ .",
"Furthermore, suppose $$ is obtained from ${^{\\prime }}$ due to a symbolic evaluation.",
"Then we say $$ is obtained form $$ by an abstract computation.",
"To prove correctness of an symbolic evaluation step, we have to show that $f^\\ast (\\gamma ()) \\subseteq \\gamma (())$ .",
"Hence, it is enough to show that for all $s \\in \\gamma ()$ and ${s}{t}{}$ it follows that $t \\in \\gamma ()$ , where $$ is obtained from a symbolic evaluation step, ie., $= ()$ .",
"Similarly, to prove correctness of the refinement steps it is enough to show that for all $s \\in \\gamma ()$ there exists a state $_i$ obtained by a state refinement of $$ such that $s \\in \\gamma (_i)$ .",
"Correctness of an abstract computation step follows from the correctness of refinement and symbolic evaluation steps.",
"Lemma 4.4 Let $\\in $ .",
"Suppose $_1, \\ldots , _n$ is obtained by a state refinement from $$ .",
"Then $_i$ for all $_i$ .",
"Furthermore, $s \\in \\gamma ()$ implies that there exists an abstract state $_i$ such that $s \\in \\gamma (_i)$ .",
"The claim follows easily by the definition of Boolean and class variables, and the fact that two addresses in the heap of $$ either alias or not.",
"Lemma 4.5 Let $, \\in $ such that $$ is obtained by a symbolic evaluation from $$ .",
"Suppose $s \\in \\gamma ()$ and ${s}{t}{}$ .",
"Then $t \\in \\gamma ()$ .",
"The proof is straightforward in most cases; we only treat some informative ones.",
"Let $= (, , )$ and $s = (, )$ .",
"By assumption the domain of $$ and $$ coincide.",
"Consider $~$ .",
"By assumption $(n) {m} loc(n)$ .",
"In the abstract computation step $(n)$ is loaded on to the top of the stack.",
"Obviously $_i(n) {m^{\\prime }} _i(n)$ , where $_i$ represents the top of the stack.",
"Then $t \\in \\gamma ()$ .",
"Consider $$ .",
"Let $i_2,i_1$ denote the first two stack elements of $$ .",
"Wlog.",
"suppose that $i_1$ is abstract.",
"By definition of the symbolic evaluation of $$ we perform the step by introducing a new abstract integer $i_3$ and adding the constraint $i_3 = i_1 + i_2$ .",
"Then $t \\in \\gamma ()$ , since $i_3 {} z$ for all numbers $z$ .",
"Consider $~i$ .",
"Wlog.",
"let $$ be the top element of the stack of $$ .",
"Executing the symbolic step yields a state $$ , which is an abstraction of $t$ by assumption on $s$ and $$ .",
"Then $t \\in \\gamma ()$ .",
"Consider $~~$ on address $p$ .",
"By assumption the instruction can be symbolically evaluated and $p$ does not alias with some address $q \\in ()$ different from $p$ .",
"The only interesting case to consider is when $(q)$ is a class variable and there exists $s \\in \\gamma ()$ such that $m(q) [S] r [S] m(p)$ , where $r \\in ()$ .",
"Then $m(q)$ reaches $m(p)$ via $r$ and is affected by the update instruction.",
"This does not matter, since $(q)$ is also a class variable in $$ , thus also representing the affected instance.",
"Then $t \\in \\gamma ()$ .",
"Consider $$ .",
"By assumption the instruction can be symbolically executed.",
"That is the necessary refinement steps are already performed.",
"Then $t \\in \\gamma ()$ follows directly.",
"The next theorem is an immediate result of the lemma.",
"Theorem 4.1 Let $s$ and $t$ be JVM states, such that ${s}{t}$ .",
"Suppose $s \\in \\gamma ()$ for some state $$ .",
"Then there exists an abstract computation of $$ from $$ such that $t \\in \\gamma ()$ .",
"Theorem REF formally proves the correctness of the proposed abstract domain with respect to the operational semantics for Jinja, established by Klein and Nipkow [15].",
"In order to exploit this abstract domain we require a finite representation of the abstract domain $$ induced by $$ .",
"For that we propose in the next section computation graphs as finite representations of all relevant states in $$ , abstracting JVM states in $$ ."
],
[
"Computation Graphs",
"In this section, we define computation graphs as finite representations of the abstract domain $$ with respect to $$ .",
"Definition 5.1 A computation graph $=({},{})$ is a directed graph with edge labels, where ${} \\subset $ and ${\\ell } \\in {}$ if either $$ is obtained from $$ by an abstract computation or $$ is an instance of $$ .",
"Furthermore, if there exists a constraint $C$ in the symbolic evaluation, then $\\ell C$ .",
"For all other cases $\\ell \\varnothing $ .",
"We say that $$ is the computation graph of program $$ if for all initial states $$ of $$ there exists an abstract state $ \\in $ such that $\\in \\gamma ()$ .",
"We obtain a finite representation of loops, if we suitably exploit the fact that any subset of $$ has a least upper bound.",
"The intution is best conveyed by an example.",
"Example 5.1 Consider the List program from Example REF together with the well-formed JBC program depicted in Figure REF .",
"Figure REF illustrates the computation graph of append.",
"For the sake of readability we omit the $val$ field of the list, the unsharing annotations and some intermediate nodes.",
"Figure: The (incomplete) computation graph of append.Consider the initial node $I$ .",
"It is easy to see that $I$ is an abstraction of all concrete initial states, when $this$ is not $$ .",
"We assume that $this$ is acyclic and initially do not share with $ys$ .",
"Nodes $A$ , $B$ and $S$ correspond to the situation described in Example REF and Example REF .",
"That is, node $A$ is obtained after assigning $cur$ to $this$ before any iteration of the loop, node $B$ is obtained after exactly one iteration of the loop and node $S = \\lbrace A,B\\rbrace $ .",
"Intermediate iterations are normally removed.",
"This is indicated by a dashed border for $B$ .",
"After pushing the reference of $cur.next$ and $$ onto the operand stack, we reach node $C$ .",
"At ${pc} = 7$ we want to compare the reference of $cur.next$ with $$ .",
"But, $cur.next$ is not concrete.",
"Therefore, a class instance refinement is performed, yielding nodes $C_1$ and $C_2$ .",
"First, we consider that $cur.next$ is not $$ , but references an arbitrary instance, as illustrated in node $C_1$ .",
"The step from $C_1$ to $D$ is trivial.",
"Let $id$ denote the identity function and $m = id({S})$ .",
"Then $m\\lbrace o_4 \\mapsto o_5, o_5 \\mapsto o_6\\rbrace $ is a morphism from $S$ to $D$ .",
"Therefore, $D$ is an instance of $S$ .",
"Second, we consider the case when $cur.next$ is $$ , as depicted in node $C_2$ .",
"Node $E$ is obtained from $C_2$ after loading registers $cur$ and $ys$ onto the stack.",
"At program counter 19 a $$ instruction is performed.",
"Therefore we perform a refinement according to Definition REF .",
"We obtain nodes $E_1, E_2$ and $E_3$ .",
"In $E_1$ , $this$ and $cur$ point to the same reference, in $E_2$ $this.next$ and $cur$ point to the same reference, and in $E_3$ the abstracted part from $cur$ is distinct from $this$ , yet $this$ and $cur$ shares.",
"Nodes $F_1, F_2$ and $F_3$ are obtained after performing the $$ instruction.",
"To concretise the employed strategy, note that whenever we are about to finish a loop, we attempt to use an instance refinement to the state starting this loop.",
"If this fails, for example in an attempted step from $B$ to $A$ in Example REF , we widen the corresponding state.",
"Here we collect all states that need to be abstracted and join them to obtain an abstraction.",
"Complementing the proposed strategy, we restrict the applications of refinements, such that refinement steps are only performed if no other steps are applicable.",
"We say that this strategy is an eager strategy.",
"The next lemma shows that if an eager strategy is followed we are guaranteed to obtain a finite computation graph.",
"Lemma 5.1 Let $$ be the computation graph of a program $P$ such that in the construction of $$ an eager strategy is applied.",
"Then $$ is finite.",
"We argue indirectly.",
"Suppose the computation graph $$ of $P$ is infinite.",
"This is only possible if there exists an initial state $$ of $$ that is non-terminating, which implies that starting from $$ we reach a loop in $$ that is called infinitely often.",
"As $$ is infinite this implies that the widening operation for this loop gives rise to an infinite sequence of states $(_j)_{j \\geqslant 0}$ such that $_j _{j+1}$ for all $j$ .",
"However, this is impossible as any ascending chain of abstract states eventually stabilises, cf.",
"Lemma REF .",
"Let $$ be a computation graph.",
"We write ${}{}$ to indicate that state $$ is directly reachable in $G$ from $$ .",
"Sometimes we want to distinguish whether $$ is obtained by a refinement (denoted as ${}{}$ ) or by a symbolic evaluation (denoted as ${}{}$ ), or whether $$ is an instance of $$ (denoted as ${}{}$ ).",
"If $$ is reachable from $$ in $$ we write ${}{}$ .",
"If $\\ne $ this is denoted by ${}{}$ .",
"Lemma 5.2 Let $s, t \\in $ such that ${s}{t}$ .",
"Let $G$ denote the computation graph of $P$ , and $, \\in G$ .",
"Suppose $s \\in \\gamma ()$ , then there exists $$ such that $t \\in \\gamma ()$ and ${} [\\text{ins}] \\cdot [\\text{ref}] \\cdot [\\text{eva}] $ .",
"By construction of $G$ we have to consider two cases: Suppose $$ is obtained by an abstract computation from $$ .",
"We employ Lemma REF to conclude that $t \\in \\gamma ()$ .",
"Then $[\\text{ref}] \\cdot [\\text{eva}] $ .",
"Next, suppose $$ is obtained by an abstract computation from ${^{\\prime }}$ , where ${^{\\prime }}$ .",
"Hence, we also have $s \\in \\gamma ({^{\\prime }})$ .",
"We employ Lemma REF to conclude that $t \\in \\gamma ()$ .",
"Then $[\\text{ins}] \\cdot [\\text{ref}] \\cdot [\\text{eva}] $ .",
"Since $G$ is finite we conclude that ${} [\\text{ins}] \\cdot [\\text{ref}] \\cdot [\\text{eva}] $ has finitely many instance and refinement steps, only depending on $G$ .",
"We arrive at the main result of this section.",
"Theorem 5.1 Let $, t \\in $ and suppose ${}{t}$ , where the runtime of the execution is $m$ .",
"Let $$ denote the computation graph of $$ obtained from some initial state $$ such that $\\in \\gamma ()$ .",
"Then there exists an abstraction $\\in G$ and a path ${} [G] {}$ of length $m^{\\prime }$ such that $m \\leqslant m^{\\prime } \\leqslant K \\cdot m$ .",
"Here constant $K \\in $ only depends on $$ .",
"By induction on $m$ (employing Lemma REF ), we conclude the existence of state $$ such that ${}{}$ .",
"Hence, the first part of the theorem follows.",
"Furthermore by Lemma REF there exists $m^{\\prime }$ such that $m \\leqslant m^{\\prime } \\leqslant K \\cdot m$ ."
],
[
"Constrained Rewrite Systems",
"Let $$ be the computation graph for program $$ with initial state $$ ; $$ is kept fixed for the remainder of the section.",
"In the following we describe the translation from $$ into a constrained term rewrite system (cTRS for short).",
"Our definition is a variation of cTRSs as for example defined by Falke and Kapur [8], [9] or Sakata et al. [26].",
"Recently, Kop and Nishida introduced a very general formalism of term rewrite systems with constraints, termed logical constrained term rewrite systems (LCTRSs) [16].",
"The proposed notion of cTRSs is not directly interchangeable with LCTRSs, yet the rewrite system resulting from the transformation could also be formalised as LCTRS.",
"The here proposed transformation is inspired by [23].",
"Otto et al.",
"transform termination graphs into integer term rewrite systems (ITRSs for short) [10].",
"Let $$ be a (not necessarily finite) sorted signature, let $^{\\prime }$ denote a countably infinite set of sorted variables.",
"Furthermore let $$ denote a theory over $$ .",
"Quantifier-free formulas over $$ are called constraints.",
"Suppose $$ is a sorted signature that extends $$ and let $\\supseteq ^{\\prime }$ denote an extension of the variables in $^{\\prime }$ .",
"Let $(,)$ denote the set of (sorted) terms over the signature $$ and $$ .",
"Note that the sorted signature is necessary to distinguish between theory variables that are to be interpreted over the theory $$ and term variables whose interpretation is free.",
"A constrained rewrite rule, denoted as ${l}{r}{C}$ , is a triple consisting of terms $l$ and $r$ , together with a constraint $C$ .",
"We assert that $l \\notin $ , but do not require that $(l) \\supseteq (r) \\cup (C)$ , where $(t)$ ($(C)$ ) denotes the variables occurring in the term $t$ (constraint $C$ ).",
"A constrained term rewrite system (cTRS) is a finite set of constrained rewrite rules.",
"Let $$ denote a cTRS.",
"A context $D$ is a term with exactly one occurrence of a hole $$ , and $D[t]$ denotes the term obtained by replacing the hole $$ in $D$ by the term $t$ .",
"A substitution $\\sigma $ is a function that maps variables to terms, and $t\\sigma $ denotes the homomorphic extension of this function to terms.",
"We define the rewrite relation $$ as follows.",
"For terms $s$ and $t$ , $s t$ holds, if there exists a context $D$ , a substitution $\\sigma $ and a constrained rule ${l}{r}{C} \\in $ such that $s D[l\\sigma ]$ and $t=D[r\\sigma ]$ with $C\\sigma $ .",
"Here $$ denotes unification modulo $$ .",
"For extra variables $x$ , possibly occurring in $t$ , we demand that $\\sigma (x)$ is in normal-form.",
"We often drop the reference to the cTRS $$ , if no confusion can arise from this.",
"A function symbol in $$ is called defined if $f$ occurs as the root symbol of $l$ , where ${l}{r}{C} \\in $ .",
"Function symbols in $\\setminus $ that are not defined, are called constructor symbols, and the symbols in $$ are called theory symbols.",
"A cTRS $$ is called terminating, if the relation $$ is well-founded.",
"For a terminating cTRS $$ , we define its runtime complexity, denoted as $$ .",
"We adapt the runtime complexity with respect to a standard TRS suitable for cTRS $$ .",
"(See [11] for the standard definition.)",
"The derivation height of a term $t$ (with respect to $$ ) is defined as the maximal length of a derivation (with respect to $$ ) starting in $t$ .",
"The derivation height of $t$ is denoted as $(t)$ .",
"Note that $$ is not necessarily finitely branching for finite cTRSs, as fresh variables on the right-hand side of a rule can occur.",
"Definition 6.1 We define the runtime complexity (with respect to $$ ) as follows: $(n) \\max \\lbrace (t) \\mid \\text{$t$ is basic and${t} \\leqslant n$}\\rbrace $ where a term $t = f(t_1,\\dots ,t_k)$ is called basic if $f$ is defined, and the terms $t_i$ are only built over constructor, theory symbols, and variables.",
"We fix the size measure ${\\cdot }$ below.",
"In the following we are only interested in cTRS over a specific theory $$ , namely Presburger arithmetic, that is, we have $C$ , if all ground instances of the constraint $C$ are valid in Presburger arithmetic.",
"Recall, that Presburger arithmetic is decidable.",
"If $C$ , then $C$ is valid.",
"On the other hand, if there exists a substitution $\\sigma $ , such that $C\\sigma $ , then $C$ is satisfiable.",
"To represent the basic operations in the Jinja bytecode instruction set (cf.",
"Figure REF ) we collect the following connectives and truth constants in $$ : $\\wedge $ , $\\vee $ , $\\lnot $ , $$ , and $$ , together with the following relations and operations: $=$ , $\\ne $ , $\\geqslant $ , $+$ , $-$ .",
"Furthermore, we add infinitely many constants to represent integers.",
"We often write $l \\rightarrow r$ instead of ${l}{r}{}$ .",
"As expected $$ makes use of two sorts: ${bool}$ and ${int}$ .",
"We suppose that all abstract variables $X_1,X_2,\\dots $ are present in the set of variables $$ , where abstract integer (Boolean) variables are assigned sort ${int}$ (${bool}$ ) and all other variables are assigned sort ${univ}$ .",
"The remaining elements of the signature $$ will be defined in the course of this section.",
"As the signature of these function symbols is easily read off from the translation given below, in the following the sort information is left implicit, to simplify the presentation.",
"The size of a term $t$ , denoted as ${t}$ is defined as follows: ${t} {\\left\\lbrace \\begin{array}{ll}1 & \\text{if $t$ is a variable}\\\\(t) & \\text{if $t$ is an integer}\\\\1 + \\sum _{i=1}^n {t_i} & \\text{if $t=f(t_1,\\dots ,t_n)$ and $f$ is notan integer}\\end{array}\\right.",
"}$ In the next definition, we show how a state becomes representable as term over $$ .",
"Definition 6.2 Let $=(,,)$ be a state and let the index sets $$ and $$ be defined as above.",
"Suppose $v$ is a value.",
"Then the value $v$ is translated as follows: $(v) & {\\left\\lbrace \\begin{array}{ll}& \\text{if $v \\in \\lbrace ,\\rbrace $}\\\\v & \\text{if $v$ is a non-address value, except $$ or$$}\\\\(v) & \\text{if $v$ is an address}\\end{array}\\right.",
"}\\\\\\multicolumn{2}{l}{\\text{Let $a$ be an address.",
"Then $a$ is translated as follows:}}\\\\(a) &{\\left\\lbrace \\begin{array}{ll}x &\\begin{minipage}[t]{40ex}if a is maybe-cyclic andx is a fresh variable\\end{minipage}\\\\[3mm]x &\\begin{minipage}[t]{40ex}if (a) denotes an abstract variable~x\\end{minipage}\\\\((v_1), \\ldots , (v_n)) & \\text{if $(a) = (, )$}\\end{array}\\right.",
"}$ Here we suppose in the last case that $() = \\lbrace (_1,id_1),\\ldots , (_n,id_n)\\rbrace $ and for all $1 \\leqslant i \\leqslant n$ : $((_i,id_i) = v_i$ .",
"Finally, to translate the state $s$ into a term, it suffices to translate the values of the registers and the operand stacks of all frames in the list $$ .",
"Let $(, i,j) \\in $ such that $_i(j)$ denotes the $j^{\\text{th}}$ value in the operation stack of the $i^{\\text{th}}$ frame in $$ .",
"Similarly for $(, i^{\\prime },j^{\\prime }) \\in $ .",
"Then we set $(s) [(_1(1)),\\dots ,(_k({_k}))),(_1(1)),\\dots ,(_k({_k}))] $ where the list $[\\dots ]$ , is formalised by an auxiliary binary symbol $$ and the constant $$ .",
"Example 6.1 Consider the simplified presentation of state $C$ in Figure REF .",
"Then $(C)$ yields following term: $(C) = [list5, ,{List}(list3), list2, {List}({List}(list5))] $ Note that we can omit the information of the defining classes of the fields, since this is already captured in the symbolic evaluation.",
"Furthermore, observe that our term representation can only fully represent acyclic data.",
"In this sense, the term representation of a state $s$ is less general, than its graph-based representation.",
"However, we still obtain the following lemma.",
"Lemma 6.1 Let $$ and $$ be abstract states.",
"If $$ , then there exists a substitution $\\sigma $ such that $() = ()\\sigma $ .",
"Let $$ and $$ be the state graphs of $$ and $$ , respectively.",
"By assumption there exists a morphism $m \\colon \\rightarrow $ .",
"The lemma is a direct consequence of the following observations: Consider the terms $()$ and $()$ .",
"By definition these terms encode the standard term representations of the graphs $$ and $$ .",
"Let $u$ and $v$ be nodes in $$ and $$ such that $m(u) = v$ .",
"The label of $u$ (in $$ ) can only be distinct from the label of $v$ (in $$ ), if ${}(u)$ is an abstract variable or $$ .",
"In the former case $({}(u))$ is again a variable and the latter case implies that ${}(v) = $ .",
"Thus in both cases, $({}(u))$ matches $({}(v))$ .",
"By correctness of our abstraction, we have $m(u)$ is maybe-cyclic, if $v$ is maybe-cyclic.",
"In this case $({}(u))$ and $({}(v))$ are fresh variables.",
"Hence, $({}(u))$ matches $({}(v))$ .",
"The next lemma relates the size of a state to its term representation and vice versa.",
"Lemma 6.2 Let $s=(,)$ be a state such that $$ does not admit cyclic data structures.",
"Then ${(\\beta (s))} = {s}$ .",
"As a consequence of Definition REF and the above proposed variant of the term complexity we see that ${(\\beta (s))} = {s}$ for all states $s$ .",
"Lemma 6.3 Let $s=(,)$ be a state such that $$ may contain cyclic data structures.",
"Then ${(\\beta (s))} \\leqslant {s}$ and therefore ${(\\beta (s))} \\in ({s})$ .",
"Follows from the previous lemma and the fact that addresses bounded to cyclic data structures are replaced by fresh variables.",
"Let $$ be a computation graph.",
"For any state $$ in $$ we introduce a new function symbol ${}$ .",
"Suppose $() = [_1,\\dots ,_n]$ .",
"To ease presentation we write ${}(())$ instead of ${}(_1,\\dots ,_n)$ .",
"Definition 6.3 Let $$ be a finite computation graph and $= (, , )$ and $$ be states in $$ .",
"We define the constrained rule corresponding to the edge $(,)$ , denoted by $(,)$ , as follows: $(,) ={\\left\\lbrace \\begin{array}{ll}{}(()) \\rightarrow {}(()) & \\text{if $$}\\\\{}(()) \\rightarrow {}(()) & \\text{if $$ is a state refinement of $$} \\\\{{}(())}{{}(())}{(C)} & \\text{the edge is labelled by $C$} \\\\{}(()) \\rightarrow {}(^\\ast ()) &\\begin{minipage}[t]{30ex} corresponds to a on address p, (q) is variable cn, and q may-reach p\\end{minipage}\\\\[2mm]{}(()) \\rightarrow {t}(()) &\\text{otherwise} \\end{array}\\right.",
"}$ Here $(C)$ denotes the standard extension of the mapping $$ to labels of edges and $^\\ast $ is defined as $$ but employs fresh variables for any reference $q$ that may-reach the object that is updated.",
"The cTRS obtained from $$ consists of rules $(,)$ for all edges $\\rightarrow \\in $ .",
"Example 6.2 Figure REF illustrates the cTRS obtained from the computation graph of Example REF .",
"We use following conventions: ${L}$ denotes the list constructor symbol and $l$ followed by a number a list variable.",
"In the last rule $l4$ is fresh on the right-hand side.",
"This is because we update $cur$ and have a side-effect on $this$ that is not directly observable in the abstraction.",
"Figure: The cTRS of append.In the following we show that the rewrite relation of the obtained cTRS safely approximates the concrete semantics of the concrete domain.",
"We first argue informally: By Lemma REF there exists a path ${} [\\text{ins}] \\cdot [\\text{ref}] \\cdot [\\text{eva}] $ in $G$ for ${s}{t}$ such that $s \\in \\gamma ()$ and $t \\in \\gamma ()$ .",
"Together with Lemma REF we have to show that ${}((\\beta (s))) {}((\\beta (t)))$ .",
"We do this by inspecting the rules obtained from the transformation.",
"We will see that instance steps and refinement steps do not modify the term instance.",
"In case of evaluation steps the effect is either directly observable in the abstract state, as it happens for $$ for example, or indirectly by requiring that the substitution is conform with the constraint.",
"In the case of the $$ instructions we have to find a suitable substitution for fresh variables to accommodate possible side-effects.",
"Lemma 6.4 Let $$ and $$ be states in $$ connected by an edge ${\\ell } $ from $$ to $$ .",
"Suppose $s \\in $ with $s \\in \\gamma ()$ .",
"Suppose further that if the constraint $\\ell $ labelling the edge is non-empty, then $s$ satisfies $\\ell $ .",
"Moreover, if ${\\ell } $ follows due to a refinement step, then $s$ is consistent with the chosen refinement.",
"Then there exists $t \\in \\gamma ()$ such that ${}((s^{\\prime })) [(,)] {}((t^{\\prime }))$ with $s^{\\prime } = \\beta (s)$ , $t^{\\prime } = \\beta (t)$ .",
"The proof proceeds by case analysis on the edge ${\\ell } $ in $$ , where we only need to consider the following four cases.",
"The argument for the omitted fifth case is very similar to the third case.",
"Case ${\\ell } $ , as $$ ; $\\ell = \\varnothing $ .",
"By assumption $s^{\\prime } $ .",
"Hence, $s \\in \\gamma ()$ by transitivity of the instance relation.",
"By Lemma REF there exists a substitution $\\sigma $ such that $(s^{\\prime }) = ()\\sigma $ .",
"In sum, we obtain: ${}((s^{\\prime })) = {}(())\\sigma [(,)]{}(())\\sigma = {}((t^{\\prime })) $ where we set $t^{\\prime } s^{\\prime }$ .",
"Case ${\\ell } $ , as $$ is a refinement of $$ ; $\\ell = \\varnothing $ .",
"By assumption $s^{\\prime } $ and $s$ is concrete.",
"Hence, $s^{\\prime } $ by definition of $$ .",
"Again by Lemma REF there exists a substitution $\\sigma $ , such that $(s^{\\prime }) = ()\\sigma $ .",
"In sum, we obtain: ${}((s^{\\prime })) = {}(())\\sigma [(,)]{}(())\\sigma = {}((t^{\\prime })) $ where we again set $t^{\\prime } s^{\\prime }$ .",
"Case ${\\ell } $ , as $$ is the result of the symbolic evaluation of $$ and $\\ell = C \\ne \\varnothing $ .",
"By assumption $s$ satisfies the constraint $C$ .",
"More precisely, there exists a substitution $\\sigma $ such that $(s^{\\prime }) = ()\\sigma $ and $(C)\\sigma $ .",
"We obtain: ${}((s^{\\prime })) = {}(())\\sigma [(,)]{}(())\\sigma $ Let $t$ be defined such that ${s}{t}$ .",
"By Lemma REF we obtain $t^{\\prime } $ and by inspection of the proof of Lemma REF we observe that $(t^{\\prime }) = ()\\sigma $ .",
"In sum, ${}((s^{\\prime })) [(,)] {}((t^{\\prime }))$ .",
"Case ${\\ell } $ , as $$ is the result of a $$ instruction on $p$ and there exists an address $q$ in $$ that may-reaches $p$ .",
"By assumption $s^{\\prime } $ and thus $(s^{\\prime }) = ()\\sigma $ for some substitution $\\sigma $ .",
"Let $t$ be defined such that ${s}{t}$ .",
"Due to Lemma REF , we have $t^{\\prime } $ and thus there exists a substitution $\\tau $ such that $(t^{\\prime }) = ^\\ast ()\\tau $ .",
"Consider the rule ${}(()) \\rightarrow {}(^\\ast ())$ .",
"By definition address $q$ points in $$ to an abstract variable $x$ such that $x$ occurs in $()$ and $()$ .",
"Furthermore, $x$ is replaced by an extra variable $x^{\\prime }$ in $^\\ast ()$ .",
"Wlog., we assume that $x^{\\prime }$ is the only extra variable in $^\\ast ()$ .",
"Let $m$ be a morphism such that $m \\colon \\rightarrow s^{\\prime }$ and $m(q) [] m(p)$ .",
"By definition of $$ , $m(p)$ and $m(q)$ exist in $t^{\\prime }$ and only the part of the heap reachable from these addresses can differ in $s^{\\prime }$ and $t^{\\prime }$ .",
"In order to show the admissibility of the rewrite step ${}((s^{\\prime })) \\rightarrow {}((t^{\\prime }))$ we define a substitution $\\rho $ such that $()\\rho = (s^{\\prime })$ and $^\\ast ()\\rho = (t^{\\prime })$ .",
"We set: $\\rho (y) {\\left\\lbrace \\begin{array}{ll}\\tau (x) & \\text{if $y = x^{\\prime }$}\\\\\\sigma (y) & \\text{otherwise}\\end{array}\\right.",
"}$ Then $()\\rho = (s^{\\prime })$ by definition as $x^{\\prime } \\notin ()$ .",
"On the other hand $^\\ast ()\\rho = (t^{\\prime })$ follows as the definition of $\\rho $ forces the correct instantiation of $x^{\\prime }$ and Lemma REF in conjunction with Lemma REF implies that $\\sigma $ and $\\tau $ coincide on the portion of the heap that is not changed by the field update.",
"The next lemma emphasises that any execution step is represented by finitely many but at least one rewrite steps in $$ .",
"Lemma 6.5 Let $\\in $ and $s \\in $ such that $s \\in \\gamma ()$ .",
"Then ${s}{t}$ implies that there exists a state $\\in $ such that $t \\in \\gamma ()$ and ${}((\\beta (s))) {{} \\leqslant K} {}((\\beta (t)))$ .",
"Here $K$ depends only on $$ and ${{} \\leqslant K}$ denotes at least one and at most $K$ many rewrite steps in $$ .",
"The lemma follows from the proof of Lemma REF and Lemma REF .",
"We arrive at the main result of this thesis.",
"Theorem 6.1 Let $s, t \\in $ .",
"Suppose ${s}{t}$ , where $s$ is reachable in $$ from some initial state $$ .",
"Set $s^{\\prime } = \\beta (s)$ , $t^{\\prime } = \\beta (t)$ .",
"Then there exists $, \\in $ and a derivation ${}((s^{\\prime })) {}((t^{\\prime }))$ such that $s \\in \\gamma ()$ and $t \\in \\gamma ()$ .",
"Furthermore, for all $n$ : $(n) \\in ((n))$ .",
"The existence of $$ follows from the correctness of abstract computation together with the construction of the computation graph.",
"Let $m$ denote the runtime of the execution ${s}{t}$ .",
"Then by induction on $m$ in conjunction with Lemma REF we obtain the existence of a state $$ such that $t^{\\prime } $ and a derivation: ${}((s^{\\prime })) {{} \\leqslant K \\cdot m} {}((t^{\\prime }))$ Here the constant $K$ depends only on $$ .",
"In particular we have ${s}((s^{\\prime })) {t}((t^{\\prime }))$ from which we conclude the first part of the theorem.",
"To conclude the second part, let $n$ be arbitrary and suppose $m$ denotes the runtime of the execution ${}{t}$ , where ${} \\leqslant n$ .",
"We set $^{\\prime } = \\beta ()$ .",
"As $$ is the computation graph of $$ we obtain $^{\\prime } $ .",
"From Lemma REF it follows that ${(\\beta ())} \\leqslant {}$ .",
"Specialising (REF ) to $$ and $^{\\prime }$ yields ${}((^{\\prime })) {{} \\leqslant K \\cdot m} {}((t^{\\prime }))$ .",
"Thus we obtain $({i}) = m \\leqslant K \\cdot m \\leqslant ({{(\\beta ())}}) \\leqslant ({i})$ It is tempting to think that the precise bound on the number of rewrite steps presented in Lemma REF should translate to a linear simulation between JVM executions and rewrite derivation.",
"Unfortunately this is not the case as the transformation is not termination preserving.",
"For this consider Figure REF .",
"Figure: The inits program.Here the outer loop cuts away the last cell until the initial list consists only of one cell whereas the inner loop is used to iterate through the list.",
"It is easy to see that the main function terminates if the argument is an acyclic list.",
"Since variables $ys$ and $cur$ share during iteration, the proposed transformation introduces a fresh variable for the next field of the initial argument $ys$ when performing the $$ instruction.",
"Termination of the resulting rewrite system can not be shown any more.",
"However non-termination preservation follows as an easy corollary of Theorem REF .",
"Corollary 6.1 The computation graph method, that is the transformation from a given JBC program $$ to a cTRS $$ is non-termination preserving.",
"Suppose there exists an infinite run in $$ , but $$ is terminating.",
"Let $$ be some initial state $$ of $$ .",
"By Theorem REF there exists a state $t$ such that ${}{t}$ and ${}((^{\\prime })) {}((t^{\\prime }))$ , where $\\in \\gamma ()$ , $^{\\prime } = \\beta ()$ , $t \\in \\gamma ()$ , and $t^{\\prime } = \\beta (t)$ .",
"Furthermore, as $$ is terminating we can assume ${}((t^{\\prime }))$ is in normalform.",
"However, as $t^{\\prime }$ is non-terminating, there exists a successor, thus Lemma REF implies that ${}((t^{\\prime }))$ cannot be in normalform.",
"Contradiction."
],
[
"Implementation",
"A prototype, termed , of the proposed method has been implemented in the Haskell programming language.",
"We use [27], [25], [33] to provide acyclicity and reachability facts.",
"Example 7.1 Figure REF depicts a slightly modified version of the motivating example from [23].",
"The program ${flatten}$ collects all integers from a list of trees storing integers.",
"The complexity tool $$ is able to show that the rewrite system resulting from our proposed transformation has linear runtime complexity.",
"Figure: The flatten program.Currently $$ only provides limited support for cTRSs.",
"A meaningful experimental evaluation will be provided in the future."
],
[
"Conclusion and Future Work",
"In this paper we define a representation of JBC executions as computation graphs from which we obtain a representation of JBC executions as constrained rewrite systems.",
"We precise the widening of abstract states so that the representation of JBC executions is provably finite.",
"Furthermore, we show that the resulting transformation is complexity preserving.",
"As emphasised above our approach does not directly give rise to an automatable complexity-preserving transformation, but for that requires an extension by annotation or a dedicated shape analysis [21].",
"However our main result applies to any computable approximation of the transformation and in particular it shows complexity preservation of the transformation proposed by Otto et al. [23].",
"Moreover, it allows for an easy incorporation of the existing wealth of results on shape analysis present in the literature and thus improves upon the modularity of the proposed transformational approach.",
"Future work will be dedicated towards new methods for complexity analysis of cTRSs."
],
[
"Semantics of Jinja Bytecode Instructions",
" We use ${BOp}$ together with $\\otimes = \\lbrace +, - , \\vee , \\wedge , \\geqslant , ==, \\ne \\rbrace $ to define instructions $$ , $$ , $$ , $$ , $$ , $$ and $$ .",
"${{BOp}} \\quad & {{}{}{{v_2 \\otimes v_1 }{}{}{}{+1}}}{{}{}{{v_1 v_2 }{}{}{}{}}}\\\\[]{} \\quad & {{}{}{{\\lnot b }{}{}{}{+1}}}{{}{}{{ b }{}{}{}{}}}\\\\[]{~i} \\quad & {{}{}{{}{}{}{}{+i}}}{{}{}{{}{}{}{}{}}}\\\\[]& {{}{}{{}{}{}{}{+1}}}{{}{}{{}{}{}{}{}}}\\\\[]{~i} \\quad & {{}{}{{}{}{}{}{+i}}}{{}{}{{}{}{}{}{}}}$ $~^{\\prime }$ creates a new instance $obj$ of class $cn^{\\prime }$ .",
"The fields of $obj$ are instantiated with the default values, ie., 0 for $$ , ${false}$ for $$ and $$ otherwise.",
"Instance $obj$ is mapped to by a fresh address $a$ in $heap$ .",
"$~fn~cn^{\\prime }$ access field $(cn^{\\prime }, fn)$ of $((a))$ .",
"$~fn~cn^{\\prime }$ updates field $(cn^{\\prime }, fn)$ in $(cn^{\\prime \\prime }, ftable) = (a)$ with value $v$ .",
"$~cn^{\\prime }$ fails if $cn^{\\prime } cn$ does not hold.",
"$$ and $$ fail if $a$ is $$ .",
"${~^{\\prime }} \\quad & {{}{\\lbrace \\mapsto obj\\rbrace }{{}{}{}{}{+ 1} }}{{}{}{}}\\\\[]{~~^{\\prime }} \\quad & {{\\varnothing }{}{{(^{\\prime }, ) }{}{}{}{+ 1} }}{{\\varnothing }{}{{}{}{}{}{} }}\\\\[]{~~^{\\prime }} \\quad & {{\\varnothing }{\\lbrace \\mapsto (^{\\prime \\prime }, ^{\\prime })\\rbrace }{{}{}{}{}{+ 1} }}{{\\varnothing }{}{{}{}{}{}{} }}\\\\[]{~^{\\prime }} \\quad & {{\\varnothing }{}{{cn }{}{}{}{+ 1} }}{{\\varnothing }{}{{cn }{}{}{}{} }}$ $~mn^{\\prime }~n$ inspects the type of $(a)$ , and performs a bottom-up search (with respect to the subclass hierarchy) for the first method declaration $mn^{\\prime }$ .",
"The new frame is $^{\\prime } = (\\epsilon , , cn^{\\prime }, mn^{\\prime },0)$ , where $loc$ consists of the $this$ reference (address $a$ ), parameters $p_0\\dots p_{n-1}$ and $mxl$ registers instantiated with $$ ($mxl$ is defined in the method declaration), and $cn^{\\prime }$ denotes the class where $mn^{\\prime }$ is declared.",
"The program terminates if $$ is executed and $frms$ consists of a single frame.",
"Otherwise, the top frame is dropped and the next frame updated; $frm^{\\prime }$ drops the parameters and the reference and pushes the return value $v$ onto the stack.",
"${~^{\\prime }~n} \\quad & {{\\varnothing }{}{^{\\prime } (p_{n-1} \\dots p_{0} a ,,,,) }}{{\\varnothing }{}{(p_{n-1} \\dots p_{0} a ,,,,) }}\\\\[]{} \\quad & {{\\varnothing }{}{[]}}{{\\varnothing }{}{[]}}\\qquad {{\\varnothing }{}{^{\\prime } }}{{\\varnothing }{}{{}{}{}{}{} }}$"
]
] | 1204.1568 |
[
[
"A large scale structure traced by [OII] emitters hosting a distant\n cluster at z=1.62"
],
[
"Abstract We present a panoramic narrow-band imaging survey of [OII] emitters in and around the ClG J0218.3-0510 cluster at z=1.62 with Suprime-Cam on Subaru telescope.",
"352 [OII] emitters were identified on the basis of narrow-band excesses and photometric redshifts.",
"We discovered a huge filamentary structure with some clumps traced by [OII] emitters and found that the ClG J0218.3-0510 cluster is embedded in an even larger super-structure than the one reported previously.",
"31 [OII] emitters were spectroscopically confirmed with the detection of H-alpha and/or [OIII] emission lines by FMOS observations.",
"In the high density regions such as cluster core and clumps, star-forming [OII] emitters show a high overdensity by a factor of more than 10 compared to the field region.",
"Although the star formation activity is very high even in the cluster core, some massive quiescent galaxies also exits at the same time.",
"Furthermore, the properties of the individual [OII] emitters, such as star formation rates, stellar masses and specific star formation rates, do not show a significant dependence on the local density, either.",
"Such lack of environmental dependence is consistent with our earlier result by Hayashi et al.",
"(2011) on a z=1.5 cluster and its surrounding region.",
"The fact that the star-forming activity of galaxies in the cluster core is as high as that in the field at z~1.6 may suggest that the star-forming galaxies are probably just in a transition phase from a starburst mode to a quiescent mode, and are thus showing comparable level of star formation rates to those in lower density environments.",
"We may be witnessing the start of the reversal of the local SFR--density relation due to the \"biased\" galaxy formation and evolution in high density regions at high this redshift, beyond which massive galaxies would be forming vigorously in a more biased way in proto-cluster cores."
],
[
"Introduction",
"It is widely known that the formation and evolution of galaxies strongly depend on their surrounding environments.",
"Galaxy clusters are most massive object and the densest structures in the universe.",
"As such, clusters and their surrounding regions serve as ideal sites for studying the roles of galaxy environment on galaxy formation and evolution.",
"In the local universe, it is known that the star formation has already ceased long time ago in high density regions such as clusters and most of the galaxies therein are only passively evolving since then, while galaxies in the general field are still actively forming stars even at the present day (e.g., [36], [16], [24]).",
"Such environmental variation has been well recognized as a sharp correlation between galaxy morphology and local number density of galaxies (e.g., [9], [44], [58], [10]).",
"In clusters at $z\\sim 1$ , [39] found that the specific star formation rate (SSFR) of star-forming galaxies is independent of environment at fixed stellar mass but the fraction of star-forming galaxies is decreased in high density environment.",
"This suggests that the environmental-quenching timescale would be rapid.",
"Recent observations are discovering very distant clusters at $z\\sim 1.5-2.0$ [5], [12], [42], [54], [19], [15].",
"It is crucial to observe clusters at such frontier redshifts so as to directly witness and understand the early evolution of galaxies and the physical processes which are responsible for the strong environmental dependence of galaxy properties at later times.",
"To survey star-forming galaxies with a narrow-band filter, that captures nebular emission lines such as H$\\alpha $ and [O ii] , is very efficient and effective for studying cluster galaxies because we can construct a large sample of star-forming galaxies without any expensive spectroscopic observation.",
"The correlation between the star formation activity, traced by a nebular emission, and the environment has already been investigated by many authors rather intensively.",
"For field environment and groups at $z=$ 0.84, [51] have constructed a large sample of H$\\alpha $ emitters with a narrow-band filter as a part of the High-z Emission Line Survey (HiZELS), and found that the fraction of star-forming galaxies falls sharply as a function of local number density of photo-$z$ sample from about 40 percent in the field to almost zero in rich groups.",
"For clusters at $z \\sim $ 0.5–1.5, our previous studies on H$\\alpha $ emitters in CL0939+4713 (Abell 851) at $z=0.41$ and in RX J1716.4+6708 at $z=0.81$ show that the fraction of emitters decreases towards cluster central region, and the star formation activity peaks in the intermediate density regions such as groups and filaments at the cluster outskirts [31], [32].",
"On the other hand, the studies in higher redshift clusters ($z\\sim $ 1.5) have produced two different results.",
"While there are no H$\\alpha $ emitters within a radius of 200 kpc from the centre of the most massive cluster XMMU J2235.3$-$ 2557 at $z=1.39$ [2], the other cluster XMMXCS 2215.9$-$ 1738 at $z=1.46$ [18] shows much higher star formation activity as traced by [O ii] emitters even in the cluster core.",
"One possible interpretation would be that, in a more massive system galaxy evolution is accelerated and star formation is truncated at an earlier epoch.",
"However, there is an caveat that some of the [O ii] emitters in the cluster core may be contaminated by AGN.",
"To obtain more general picture on the star formation history in clusters, we need to study many clusters at each epoch.",
"In fact, we are currently conducting the MAHALO-Subaru project (MApping HAlpha and Lines of Oxygen with Subaru: Kodama et al.",
"in prep), which is a large and systematic narrow-band survey of star forming galaxies in many clusters at $z\\sim $ 1.5–3.0 and in the field.",
"This paper reports one of the initial results of the project.",
"This survey will provide us with the dependence of star formation history on environment.",
"We comment here, however, that if we only target clusters that have already formed and matured at each epoch, the results may be biased towards apparently weaker evolution.",
"This difficult problem should be tackled by targeting lower-mass systems as well as the rich clusters.",
"By tracing the surrounding large scale structures around rich clusters, however, we can automatically include many smaller mass groups and/or filaments embedded in the structures.",
"This is another good advantage of panoramic studies of distant clusters like ours.",
"In this paper, we report a narrow-band survey of [O ii] emitters in and around the spectroscopically confirmed distant cluster ClG J0218.3-0510 (IRC 0218-A) at $z=1.62$ [42], [54] and is in Subaru/$XMM-Newton$ Deep Survey Field (SXDS; [13]).",
"The coordinates of this cluster is $\\alpha =2^h18^m21.3^s, \\delta =-5^\\circ 1027$ (J2000), derived from the centroid of the sources selected by $Spitzer$ IRAC colour in this overdensity.",
"We assume cosmological parameters of H$_0$ = 70 km s$^{-1}$ Mpc$^{-1}$ , $\\Omega _\\mathrm {M}$ = 0.3, and $\\Omega _\\Lambda $ = 0.7, and adopt AB magnitudes throughout this paper.",
"At $z=1.62$ , 1corresponds to 8.47 kpc in a physical distance and to 22.20 kpc in a comoving distance.",
"To identify line emitters, we used a narrow-band filter (NB973) and a medium-band filter ($z_R$ ).",
"The NB973 filter ($\\lambda _\\mathrm {c}=9755$ Å, $\\Delta \\lambda _\\mathrm {FWHM}=202$ Å) can detect an [O ii] line emission at $1.590\\le z\\le 1.644$ and $z_R$ filter ($\\lambda _\\mathrm {c}=9860~$ Å, $\\Delta \\lambda _\\mathrm {FWHM}=590$ Å) can measure a continuum flux at same wavelength range as the NB973 filter.",
"These filters enable us to sample most of the star-forming galaxies associated to the cluster at $z=1.62$ as shown in Figure REF .",
"The imaging data with NB973 filter was obtained in an area between the SXDS-C and SXDS-S by [40].",
"The seeing size in the combined image is 1.0 and the 5$\\sigma $ limiting magnitude is 25.4 in a 2.0 diameter aperture.",
"The details of the observations and the data reduction are described in [40].",
"For the $z_R$ data, we conducted a new observation with Subaru Prime Focus Camera (Suprime-Cam; [38]) on the Subaru telescope in October 2010.",
"The weather was good during our observation, and the sky conditions were photometric with the seeing of 0.5 - 0.7.",
"Data reduction was carried out in a normal manner with SDFRED [60], [41].",
"The frames for the total of 327 minutes are integrated in the final image.",
"The field coverage is 830 arcmin$^2$ , corresponds to $1.4\\times 10^5$ Mpc$^3$ in the survey volume.",
"The 5$\\sigma $ limiting magnitude is 25.3 in a 2.0 diameter aperture.",
"Point spread function (PSF) of the $z_R$ image was degraded to 1.0 to match the quality of the NB973 image."
],
[
"Photometric catalogue",
"In the SXDS field, many multi-wavelength data are already available.",
"In this work, we used a total of 12 multi-band data including the NB973 and $z_R$ -band ones.",
"Deep optical broadband images ($B,V,R,i^{\\prime },z^{\\prime }$ ) were taken by the Suprime-Cam [13].",
"For near-infrared wavelength, the UKIDSS Ultra Deep Survey (UKIDSS-UDS;[34]) has been conducted in this region, and we used the DR8 data ($J,H,K$ ).",
"The photometric system and calibration are described by [20] and [21], respectively.",
"From 10-band (NB973,$B,V,R,i^{\\prime },z^{\\prime },z_\\mathrm {R},J,H,K$ ) images, we made a photometric catalogue using SExtractor [3].",
"PSFs of all the images were matched to 1.0and the positions were matched with respect to the NB973 image.",
"A source detection was performed on the NB973 image.",
"The extraction criteria were at least 9 pixels with fluxes above 2$\\sigma $ level, where 1$\\sigma $ is the sky noise of the image.",
"The sources fainter than 5$\\sigma $ limiting magnitude in NB973 were rejected.",
"Photometries in all the images were carried out using the double image mode.",
"An aperture magnitude within a diameter of 2was used to derive a colour and Kron magnitude was used to calculate a total magnitude [33].",
"For mid-infrared wavelength, we used IRAC catalogue (only 3.6 $\\mu $ m and 4.5 $\\mu $ m) for a $Spitzer$ Public legacy survey of the UKIDSS-UDS (SpUDS; PI: J. Dunlop).",
"From our catalogue, we identified sources that have an association in the IRAC catalogue within a 1.0 arcsec radius.",
"Moreover, not only the NB973-detected catalogue but also the K-detected one was created above 5$\\sigma $ in the same manner when we select passively evolving galaxies later in section 3.3.",
"The NB973-detected and the K-detected catalogues include 39229 objects and 40786 objects, respectively, of which 33242 objects are in common."
],
[
"Spectroscopy",
"Near-infrared spectroscopy of our photometrically identified [O ii] emitter candidates was conducted with Fiber Multi Object Spectrograph (FMOS; [26]) on the Subaru telescope in January 2012.",
"Observations were made with low-resolution (LR) mode in IRS1 and high resolution (HR) mode at J-long and H-long bands in IRS2.",
"Low-resolution, J-long and H-long modes can cover the wavelength ranges of 1.0–1.8 $\\mu $ m, 1.12–1.35 $\\mu $ m and 1.59–1.80 $\\mu $ m, respectively.",
"Cross-beam switching mode was adopted and the total integration times were 375 minutes in LR, 60 minutes in J-long and 300 minutes in the H-long mode.",
"The seeing was typically 1 in R-band during observations.",
"Reduction was carried out with the FMOS pipeline FIBRE-pac [22].",
"The flux calibration was done using some F,G or K-type stars from the Two Micron All Sky Survey (2MASS; [47]), observed simultaneously.",
"The flux loss was estimated to roughly 50% for a point source and this might be caused by various factors (e.g.",
"weather condition, instrument focus, and position error in the catalogue).",
"In this paper, only spectroscopic redshifts, derived from fitting emission lines, are used in order to estimate the accuracy of the photometric redshift, and the level of completeness and contamination.",
"Other analyses such as line fluxes, line widths and the line ratios will be presented in detail in our forthcoming paper.",
"Also, we used the catalogues of spectroscopic redshifts from [46] and [48].",
"In order to investigate galaxy evolution, we need to sample both actively star-forming galaxies and passively evolving galaxies at the same epoch.",
"As noted in section , the former can be selected by the presence of nebular emission lines such as [O ii] .",
"The latter population can be selected by utilizing the photometric redshift and the colour-colour diagram.",
"Using the multi-band data in section 2, a total of 352 [O ii] emitters at $z\\simeq 1.62$ have been identified over a 830 arcmin$^2$ area.",
"Also, we have constructed a photo-$z$ selected sample with M$_\\mathrm {star}>10^{10}$ M$_\\odot $ at $1.56 \\le z_\\mathrm {phot} \\le 1.68$ , which includes 132 [O ii] emitters and 259 quiescent galaxies at $z\\simeq 1.6$ ."
],
[
"[O ",
"We can identify emission line galaxies on the basis of the excesses of NB973 fluxes over $z_R$ fluxes.",
"NB973-detected catalogue created in section 2.2 is used.",
"Figure REF shows the $z_R$ $-$ NB973 colour-magnitude diagram for all the objects satisfying $\\mathrm {NB973} > 5\\sigma $ .",
"Note that we made a correction of +0.11 in the $z_R$ $-$ NB973 colour because the effective wavelengths of two filters are not exactly same.",
"For faint objects ($<2\\sigma $ ) in $z_R$ -band, $2\\sigma $ value was used to calculate colour.",
"If an strong emission line comes into the $z_R$ band but not on the NB973 filter (e.g., at $z=1.58$ or $z=1.66$ ), such object may be detected as false-absorbers.",
"Also, because the effective wavelength of NB973 filter is slightly shorter than that of the $z_R$ filter, very red objects may have negative colours in $z_R$ - NB973.",
"[6] have defined the significance of the excess in the narrow-band by the parameter $\\Sigma $ with taking account of the fact that the fainter the narrow-band flux is, the larger the photometric error is.",
"$\\Sigma $ =2.5 was adopted as our first criteria to select NB emitters [49].",
"Since the standard deviation of $z_R$ $-$ NB973 is 0.03 in the range of 18$<$ NB973$<$ 21, the colour excess of $z_R$ $-$ NB973 $> 0.1$ , that is corresponds to 3$\\sigma $ for the bright objects, was also adopted.",
"These criteria correspond to the limiting line flux of $1.8\\times 10^{-17}$ erg s$^{-1}$ cm$^{-2}$ , that is 4-5 M$_\\odot $ yr$^{-1}$ in dust-uncorrected star formation rate (SFR) using the calibration of [25], and to the equivalent width of about 30 Å in the observed frame.",
"Based on first criteria, 969 objects are identified as NB973 emitters.",
"Figure: z R z_R - NB973 colour-magnitude diagram to select [O ii] emitter candidates at z=1.62.",
"Gray dots show all objects with NB 973>5σ\\mathrm {NB973} > 5\\sigma .",
"Black and red filled circles are NB973 emitters and [O ii] emitters which satisfy the criteria in section 3.1.The NB973 filter can pick out not only [O ii] lines at $z=1.62$ but some other lines at different redshifts: Ly $\\alpha $ at $z=7.02$ , H$\\beta $ at $z=1.01$ , [O iii] at $z=0.95$ and H$\\alpha $ line at $z=0.49$ .",
"To discriminate between [O ii] emitters and those other line emitters, we utilize photometric redshifts for those objects that have satisfied our criteria.",
"Photometric redshifts were determined with the EAZY code [4].",
"11 band data ($B, V, R, i^{\\prime }, z^{\\prime }, z_R, J, H, K, 3.6 \\mu \\mathrm {m}, 4.5 \\mu \\mathrm {m}$ ) are used for SED fitting.",
"Capturing Balmer/4000Å break, $z_R$ -band should improve the accuracy of photometric redshift for galaxies at $z\\sim $ 1.4–2.0.",
"Because $z_R$ -band can be substantially affected by the presence of our target emission line itself, the emission line flux and the emission line subtracted continuum flux density are estimated from the $z_R$ -band and NB973-band fluxes as follows; $F_\\mathrm {line}=\\Delta _\\mathrm {NB973} \\frac{f_\\mathrm {NB973}-f_{z_R}}{1-\\Delta _\\mathrm {NB973} /\\Delta _{z_R}},$ $f_\\mathrm {continuum}=\\frac{f_{z_R}-f_\\mathrm {NB973}(\\Delta _\\mathrm {NB973}/\\Delta _{z_R})}{1-\\Delta _\\mathrm {NB973}/\\Delta _{z_R}},$ where $F_\\mathrm {line}$ is the emission line flux, $f$ denotes a flux density, and $\\Delta $ indicates a FWHM of a filter.",
"The emission line subtracted flux density in $z_R$ -band is used for SED fitting.",
"Figure REF is the distribution of the photometric redshift for NB973 emitters.",
"We can clearly recognize three peaks in the photometric redshift distribution (i.e., $z\\sim $ 0.5, 1, and 1.6).",
"These redshift peaks neatly correspond to H$\\alpha $ , [O iii] or H$\\beta $ , and [O ii] respectively, assuring the detection of line emitters as expected.",
"This also indicates that our first criterion of $\\Sigma $ =2.5 and $z_R$ $-$ NB973 $> 0.1$ is effective in identifying secure line emitters.",
"Out of NB973 emitters, we down selected 352 [O ii] emitters that fall between $1.4< z_{\\mathrm {phot}}<1.9$ , as second criterion, to distinguish between our target [O ii] emitters at $z=1.62$ and other unwanted line emitters at other redshifts.",
"Figure: The distribution of photometric redshift for NB973 emitters.",
"Red dash lines indicate the expected redshift of emission lines that can fall onto the NB973 filter (Hα\\alpha at z∼0.49z\\sim 0.49, [O iii] at z∼0.95z\\sim 0.95, Hβ\\beta at z∼1.01z\\sim 1.01 and [O ii] at z∼1.62z\\sim 1.62).",
"Blue lines indicate the photometric redshift ranges (1.4<z phot <1.91.4< z_{\\mathrm {phot}}< 1.9) to define [O ii] emitters.While most of the [O ii] emitters are likely to be star-forming galaxies, there is a possibility that some of the [O ii] emission lines are originated from central active galactic nuclei (AGN) rather than star forming regions.",
"In particular, some authors reported that “red” [O ii] emitters tend to be contaminated by AGN activities [61], [35], [17], [53].",
"Alternatively, red emitters are dusty starburst galaxies which are reddened by a large amount of dust extinction.",
"In any case, these red emitters are interesting objects which may be key populations to understanding galaxy evolution in dense environment.",
"However, we can not disentangle these two possibilities at this stage without spectroscopy.",
"With spectroscopy, we should be able to distinguish between AGNs and dusty starbursts by measuring emission line ratios of [O iii]/H$\\beta $ and [N ii]/H$\\alpha $ and plot them on the BPT diagram [1], [23].",
"Here, we defined red emitters with the $z - J$ versus $J - K$ colour-colour diagram (hereafter $zJK$ diagram) and treated only the [O ii] emitters that are outside the quiescent zone defined in section 3.3 as star-forming galaxies.",
"Also, our criteria of [O ii] emitters select three X-ray detected objects [57], of which two are located at different redshifts.",
"Therefore, the remaining one is not used in our analysis due to some uncertainty.",
"Eventually, we have identified totally 340 star-forming [O ii] emitters and 12 red emitters associated to the cluster at $z=1.62$ ."
],
[
"Spectroscopic follow-up for [O ",
"At $z=1.62$ , [O iii] and H$\\alpha $ emission lines shift to 1.31 $\\mu $ m and 1.72 $\\mu $ m, respectively.",
"From the NB973 emitter sample with F([O ii])$>3.0\\times 10^{-17}$ erg s$^{-1}$ cm $^{-2}$ identified in the previous section, 46 (IRS1, LR mode) and 39 objects (IRS2, HR mode) were observed with FMOS.",
"H$\\alpha $ emission lines are detected for 4 objects in LR mode and for 24 in H-long.",
"[O iii] emission lines are detected for 7 objects in LR mode and for 4 in J-long.",
"The sensitivity of the LR mode is not good enough to detect emission lines due to the large flux loss.",
"If only the H-long mode is considered, the detection rate of emission lines is about 60 percent in total and 80 percent for bright [O ii] emitters with F([O ii])$>7.0\\times 10^{-17}$ erg s$^{-1}$ cm $^{-2}$ .",
"These value are really good since 20-25 percent of the wavelength coverage is masked in the FMOS spectrograph due to OH-airglow lines.",
"Figure REF shows some examples of H$\\alpha $ spectra of our [O ii] emitters.",
"Spectroscopic redshifts are derived by gaussian fitting with free parameters of redshift, line width, and flux densities of H$\\alpha $ and [N ii]$\\lambda 6584$ or [O iii]$\\lambda 4959,\\lambda 5007$ .",
"As a result, 31 spectroscopic redshifts are obtained in total from the FMOS data, since there are 8 objects whose H$\\alpha $ and [O iii] lines are both detected.",
"There are a total of 42 spectroscopically confirmed galaxies with emission lines, including 11 spectroscopic redshifts with a emission line in the literatures.",
"40 out of them were selected by the criteria in the previous section.",
"On the other hand, there were no objects with a different redshift in our [O ii] emitter sample.",
"Figure: The spectroscopic redshift versus the photometric redshift for galaxies with 0.8<z spec <1.80.8<z_{\\mathrm {spec}}<1.8.",
"Red line is z phot =z spec z_\\mathrm {phot}=z_\\mathrm {spec}.",
"Blue dashed lines show the criterion of the photo-zz selected sample (1.56≤z phot ≤1.681.56\\le z_{\\mathrm {phot}}\\le 1.68)."
],
[
"Quiescent galaxies",
"We also need to select passively evolving galaxies located at $z$ =1.62 to quantify the averaged star formation activity.",
"First, photometric redshifts were measured for all the objects in the K-band detected catalogue created in section 2.",
"X-ray detected objects were rejected because the accuracy of photometric redshift is so bad that could increase contaminations.",
"In Figure REF , photometric redshifts are plotted against spectroscopic redshifts for confirmed galaxies.",
"The standard deviation of ($z_\\mathrm {spec}-z_\\mathrm {phot}$ ) is 0.05 at $1.590\\le z_\\mathrm {spec}\\le 1.644$ .",
"Our photometric redshifts are fairly good for them.",
"We selected only the objects that fall within a narrow redshift interval of $1.56\\le z_{\\mathrm {phot}}\\le 1.68$ since the K-band selected galaxies can include contaminations at contiguous redshift unlike the NB973 emitters.",
"Such a stringent criterion of photometric redshift range is reasonable because our photometric redshifts are good enough to recover most of the spectroscopically confirmed cluster members.",
"In fact the photometric redshift distribution of the [O ii] emitters shows a narrow concentration at $z\\sim 1.6$ as shown in Figure REF .",
"Figure: The z-Jz - J versus J-KJ - K colour-colour (zJKzJK) diagram of the photo-zz selected galaxies (black dots).",
"Blue circles show the [O ii] emitter samples.",
"Red dashed and green dashed lines shows a model track of a passive galaxy and a star-forming galaxy, respectively .The squares indicate the model colour at z=0,z=1.6z=0, z=1.6 and z=2.5z=2.5.",
"The arrow shows the dust reddening vector of E(B-V)=0.2 .",
"The solid lines show the adopted criteria to separate between quiescent galaxies and star-forming galaxies at z=1.6z=1.6.We have constructed a mass-limited sample with the stellar mass M$_\\mathrm {star}>10^{10}$ M$_\\odot $ on the basis of photometric redshifts.",
"Stellar masses are estimated from total K-band magnitudes, $K^{\\mathrm {total}}$ , and $z-K$ colours, using the ratio between mass-to-luminosity ratio in K-band and z-K colour of the population synthesis bulge-disk composite models [28] after it is scaled to the Salpeter IMF for consistency.",
"At $z=1.6$ , we used the following equations: $\\mathrm {log}(M_\\mathrm {star}/10^{11})=-0.4(K^{\\mathrm {total}}-21.64)$ $\\Delta \\mathrm {log}M_\\mathrm {star}=0.1-1.12\\exp [-0.82(z^{\\prime }-K)].$ The amount of contamination and that of completeness of the sample are estimated based on the spectroscopically confirmed galaxies.",
"There are 32 galaxies with M$_\\mathrm {star}>10^{10}$ M$_\\odot $ which are located within $1.590\\le z_\\mathrm {spec}\\le 1.644$ .",
"Since 6 spectroscopically confirmed cluster member galaxies do not fall within the photometric redshift range of $1.56\\le z_\\mathrm {phot}\\le 1.68$ , the completeness can be estimated to be 81%.",
"We include them in our sample.",
"Also, because our sample include three objects at different spectroscopic redshift, from the cluster redshift, the contamination can be estimated to be 10%.",
"However, in order to accurately estimate them, we need to make more spectroscopic follow-up observations.",
"132 [O ii] emitters with M$_\\mathrm {star}>10^{10}$ M$_\\odot $ satisfy our photometric redshift selection ($1.56\\le z_\\mathrm {phot}\\le 1.68$ ).",
"After excluding these [O ii] emitters from our photo-$z$ selected sample, we constructed a non-emitter sample of 1174 galaxies.",
"Figure: (Left) The 2-D distributions of 352 [O ii] emitters.",
"Blue and red filled circles show star-forming and red [O ii] emitters, respectively.",
"Squares indicate the spectroscopically confirmed objects with 1.590≤z spec <1.6201.590\\le z_\\mathrm {spec}<1.620 (black), 1.620≤z spec ≤1.630\\le z_\\mathrm {spec}\\le 1.630 (green) and 1.630<z spec ≤1.6441.630<z_\\mathrm {spec}\\le 1.644 (red).",
"Contours denote the local number density of [O ii] emitters.",
"Outside of dash lines is the field region defined in section 4.1.",
"A gray filled circle is a masked region near a bright star.",
"(Right) Close-up view of the cluster core (r<2<2).",
"Dashed line circle shows the radius of r==1 Mpc.",
"Note that our NB973 filter can prove about ±47\\pm 47 Mpc (comving) in the redshift direction.It is known that cluster galaxies exhibit a conspicuous “red sequence” on the colour-magnitude diagram which are mainly composed of passively evolving galaxies.",
"The red sequence is well recognized even in distant clusters out to $z\\sim 1$ [27] and it persists at the bright end even in proto-clusters at $z\\sim 2$ [30].",
"The red sequence consists of not only passive galaxies but also dusty star-forming galaxies, and we use a $zJK$ diagram to separate them in the same manner as in [59].",
"Figure REF shows the $zJK$ diagram for the photo-$z$ selected galaxies.",
"Using the model colour tracks of [27] and the dust reddening vector [7], we defined the quiescent galaxies at $z=1.6$ using the three criteria: $z - J > 1.3, J - K < 1.6$ and $z - J > 0.64 (J - K)+0.8$ .",
"In total, 259 quiescent galaxies have been selected from photo-$z$ selected sample.",
"On the other hand, there are a number of blue galaxies (915) in photo-$z$ selected sample.",
"It is likely that either their [O ii] emissions are too weak to be detected as the narrow-band excess or they are outside of the redshift range of the NB973 filter ($z<1.590$ or $z>1.644$ ).",
"Since the contamination of the latter objects are probably large, we do not use those blue galaxies in our analyses.",
"Three quiescent galaxies have been spectroscopically confirmed and a spectral fit of the brightest member yields an age of 1.8 Gyr with a mass of $3.8\\times 10^{11}$ M$_\\odot $ in Salpeter IMF [54].",
"Because most of galaxies with spectroscopic redshift are line emitters, a follow-up spectroscopy are needed to confirm whether other quiescent galaxies are truly cluster members.",
"We thus constructed a combined sample of the photo-$z$ selected and mass-limited galaxies at $1.56\\le z_\\mathrm {phot}\\le 1.68$ , composed of 132 [O ii] emitters and 259 quiescent galaxies.",
"Table 1 gives a summary of our galaxy samples used in this paper."
],
[
"Spatial distribution",
"With the panoramic imaging of the $z=1.62$ cluster with Suprime-Cam, we have revealed a gigantic structure surrounding the cluster traced by [O ii] emitters for the first time, as shown in Figure REF .",
"The cluster appears to be embedded in a huge filament extending from North to East/South.",
"In particular, there are two dense regions to the East/South and to the South of the cluster, respectively.",
"Also a filament of relatively dense region extends to the North.",
"They all appear to be associated to the cluster core and constitute a huge structure of about 20 Mpc in comoving scale.",
"Many star-forming [O ii] emitters are located within the projected radius of 1 Mpc in comoving scale from the cluster centre.",
"This is in stark contrast to the nearby Universe where galaxies in dense environment tend to be in-active red galaxies and star forming galaxies are preferentially located in lower density environments.",
"We evaluated the overdenisity of the [O ii] emitters in the cluster core (r$<$ 1 Mpc).",
"The average number density of the [O ii] emitters is 0.24 Mpc$^{-2}$ (352/1472) for the entire region of the survey ($\\sim $ 1472 Mpc$^2$ ).",
"This value is roughly comparable to the number density (0.35 Mpc$^{-2}$ ) expected from the [O ii] luminosity function at $z=1.47$ in the general field [37] because the criterion of the equivalent width in our selection is more stringent than the one used in the previous studies.",
"On the other hand, since 13 [O ii] emitters exist in the cluster core ($\\sim $ 3.142 Mpc$^2$ ), the number density is 4.1 Mpc$^{-2}$ which is about 17 times larger than that in the entire region.",
"The quiescent galaxies show the overdensity by factor of 9.",
"This clearly indicates that the star formation activity has not been ceased yet even in the core of the $z=1.62$ cluster, and rather the integrated star formation activity in the unit volume is actually higher in the cluster core than in the lower density regions.",
"Note that our narrow-band filter is probing $\\sim $ 94 Mpc (comving) in the redshift direction, and the 2-D overdensity may not be free from the projection effect.",
"In order to robustly measure the overdensity, the spectroscopic redshifts would be required.",
"In fact, it is found from the FMOS spectroscopy that the [O ii] emitters in the East and the South clumps are located at somewhat different redshifts from the cluster centre (Figure REF ).",
"This difference corresponds to 1000–2000 km s$^{-1}$ in the line of sight velocity or to 15–30 Mpc in the comoving distance.",
"It is not clear at this stage whether these clumps are gravitationally bound, physically associated systems to the cluster.",
"More intensive spectroscopic follow-up observations are needed to address this issue."
],
[
"Density dependence",
"It is widely recognized that the star formation activity of a galaxy is strongly related to the local density of galaxies surrounding it.",
"In the local universe, the star forming activity decreases with increasing local galaxy density [16], [43].",
"This relation holds out to at least $z\\sim 0.8$ (e.g., [29], [55]).",
"At $z\\sim 1$ , it is suggested that SFR-density relation may, in part, be inverted relative to the local relation [11], [8].",
"However, these studies do not include the very dense environment such as rich cluster cores.",
"In the ClG J2018-0510 cluster at $z=1.62$ , [56] reported that the relative fraction of star-forming galaxies increases with increasing local density based on the SFRs derived from the 24 $\\mu $ m fluxes and the SED-fitting.",
"Note that they used sample based on only photometric redshift.",
"On the other hand, [45] claimed that the star formation-density relation holds out to at least $z\\sim 1.8$ although they similarly selected sample on the basis of the photometric redshift.",
"In the case of photo-z selected sample, any results must be interpreted with caution due to a large contamination.",
"Therefore, the reversal of SFR-density relation at $z>1$ is very controversial at this stage.",
"Table: The numbers of our photo-zz selected samples at 1.56≤z phot ≤1.681.56\\le z_\\mathrm {phot}\\le 1.68.Figure: Relative fractions of [O ii] emitters (blue circles) and quiescent galaxies (red circles) plotted as a function of local density.",
"The local density is calculated from the combined samples of [O ii] emitters and quiescent galaxies.",
"Dashed line shows the number density in the field region (∼1.5\\sim 1.5 Mpc -2 ^{-2}).",
"In our analysis, we did not use the area below this density.",
"Vertical and horizontal error bars indicate the Poissonian errors and the sizes of each bin, respectively.Our panoramic narrow-band survey of the [O ii] emitters enables us to investigate the environmental dependence over a much larger area hence covering a much wider range in environment.",
"Moreover, due to the great advantage of narrow-band imaging survey combined with photometric redshifts, our [O ii] emitters are more robust members of star-forming galaxies associated to the cluster even without spectroscopy because we just need to separate out [O ii] emitters at $z\\sim 1.6$ among a few other possibilities of different redshifts (corresponding to other lines) based on photometric redshifts.",
"In order to discuss the environmental effect, we define the local density within the circled area of a radius to the 5th nearest neighbour object ($\\Sigma _{\\mathrm {5th}}$ ), by using the combined sample of $z\\sim 1.6$ galaxies (section 3.3).",
"And we calculated the fraction of each sample among the combined sample, i.e., $f=N_{\\mathrm {each~sample}}/N_{\\mathrm {combined~sample}}$ at each bin of the local density.",
"However, due to the photometric redshift selection, contaminations would dominate the number density in the low density regions.",
"Therefore, we do not use sample in the field region and concentrated on the region within the dashed rectangle in Figure REF , and trust only the range of local number density that is more than the field density ($\\Sigma _{\\mathrm {5th,combined}}>1.5$ Mpc$^{-2}$ ).",
"The quiescent galaxies sample inevitably contains some foreground or background objects because the photometric redshift range that we adopted ($\\Delta z_{\\mathrm {phot}}=0.12$ ) is larger than that of the [O ii] emitters ($\\Delta z_{\\mathrm {NB}}=0.054$ ).",
"We tried to correct for this effect by multiplying $\\Delta z_{\\mathrm {NB}}/\\Delta z_{\\mathrm {phot}}$ to the number of quiescent galaxies, assuming conservatively that their redshift distribution is uniform.",
"As shown in Figure REF , the fraction of star-forming galaxies in the combined sample do not shows a significant dependence on the local density at $z\\sim 1.6$ .",
"It is consistent with a constant value ($\\sim $ 60%) across different environments within errors.",
"Star-forming activity in the cluster core is very high in this cluster, and the well established star formation–density relation in the local Universe no longer exists.",
"Although a lot of massive, quiescent galaxies do exist in the high density environment such as the cluster core, the star formation activity has not been ceased yet.",
"Rather the integrated star formation rate per unit volume is actually much higher in the cluster core due to its high over-density of the emitters.",
"Figure: SFRs (top), stellar masses (middle) and specific SFRs (bottom) for individual [O ii] emitters, plotted against the local density.",
"The local density is calculated from the combined samples of [O ii] emitters and quiescent galaxies.",
"Red circles show the median value in each density bin.In order to quantitatively compare the properties of the individual [O ii] emitters as a function of environment, we estimate stellar mass, SFR and specific SFR (SSFR=SFR/M$_\\mathrm {star}$ ) for the [O ii] emitters.",
"[O ii] emission line is widely used as a good SFR indicator but strongly depends on metallicity and dust-extinction.",
"[O ii] -SFR relation are recently being calibrated by using lower redshift samples [14], [50].",
"We used the [O ii] -SFR calibration given by [50], which is described as a function of stellar mass.",
"The results are shown in Figure REF as a function of local density.",
"We do not find any significant environmental dependence in all of the individual quantities of the [O ii] emitters.",
"In fact, the median values of all the three quantities are almost constant with local density.",
"[52] claimed that the averaged star formation activity is rapidly declined from $z$ =2.5 to $z$ =0.8 in high density regions such as clusters while it is only gradually declined in the low density regions.",
"At some point at high redshifts, we may expect that the averaged star formation rate of galaxies in cluster cores should exceed that in the general field probably because galaxy formation processes (such as gas cooling and mergers) are accelerated in dense environments.",
"The fact that we observe just a comparable level of star formation activity in the [O ii] emitters irrespective of environment may suggest that, at $z=1.62$ , the [O ii] emitters in the high density regions are just in the transition phase from a star-bursting mode to a quiescent mode as a result of environmental effect."
],
[
"Summary",
"We have conducted a narrow-band survey of [O ii] emitters in and around the ClG J0218.3–0510 cluster at $z=1.6$ , using Suprime-Cam on Subaru.",
"The observation with $z_\\mathrm {R}$ filter was newly carried out to measure the continuum at the same wavelength as the narrow-band filter (NB973).",
"We combined these data with multi broad-band data ($B,V,R,i^{\\prime },z^{\\prime },J,H,K,3.6~\\mu $ m,$4.5~\\mu $ m) to create the NB973-detected and K-detected photometric catalogues.",
"The photometric redshifts were determined with 11-bands including $z_\\mathrm {R}$ -band.",
"By comparing the spectroscopic members and our [O ii] emitter sample, we estimated that the accuracy of our photometric redshifts is $\\sigma _z=0.05$ at $z=1.6$ .",
"Our main results with these samples are summarized as follows.",
"On the basis of narrow-band excesses and photometric redshifts, our survey provides a sample of 352 [O ii] emitters over a 830 arcmin$^2$ area.",
"Our very recent FMOS near-infrared spectroscopic observations have confirmed 31 [O ii] emitters at $z\\sim 1.6$ by the presence of H$\\alpha $ or [O iii] lines at the expected wavelengths.",
"The [O ii] emitters constitute a large scale structure at $z=1.62$ in which the ClG J0218.3–0510 cluster is embedded.",
"Also, we find that many star-forming [O ii] emitters are located even in the cluster core (r$<$ 1 Mpc in the comoving scale) and in the surrounding clumps, and show a high overdensity by a factor of 17 compared to the entire region.",
"This suggests that the integrated star formation activity per unit volume is activated in such regions.",
"The galaxies with $1.56 \\le z_{\\mathrm {phot}}\\le 1.68$ show a clear bimodal distribution in the $zJK$ diagram, namely, quiescent and star-forming galaxies.",
"We selected total of 259 quiescent galaxies on the basis of the population synthesis model and the extinction law by dust.",
"We calculated the local number density of galaxies in the photo-$z$ selected and mass-limited sample to examine the environmental dependence of the star-formation activity.",
"We obtain a large fraction of [O ii] emitters even in the cluster core, showing that the star forming activity in the cluster core is elevated substantially compared to the local clusters where there are little star-forming galaxies in their cores.",
"There is no longer a environmental dependence in the relative fraction of [O ii] emitters in the combined sample, and the well known SFR-density relation in the present-day Universe no longer exists within errors.",
"Furthermore, the properties of the individual [O ii] emitters, such as star formation rates, stellar masses and specific star formation rates, do not depend on the local density, either.",
"These results suggest that the [O ii] emitters in the high density regions are just in the transition phase from a star-bursting mode to a quiescent mode due to some environmental effects and the star formation rates in these systems may be rapidly declining.",
"Note that the results presented in this paper for the environmental effect of galaxies in and around a $z$ =1.62 cluster may not necessarily apply to all the clusters at the same epoch of the Universe, and the results may well depend on the degree of matureness of clusters even at the same redshift.",
"For this reason, we require a systematic studies of distant clusters and general fields to construct a more general picture, and the MAHALO-Subaru project (Kodama et al.",
"in prep) will provide us more comprehensive views of galaxy evolution at its most active phase in the Universe."
],
[
"Acknowledgments",
"This paper is based on data collected at Subaru Telescope, which is operated by the National Astronomical Observatory of Japan.",
"We thank the Subaru telescope staff for their help in the observation.",
"We thank the anonymous referee who gave us many useful comments, which improved the paper.",
"T.K.",
"acknowledges the financial support in part by a Grant-in-Aid for the Scientific Research (No.",
"21340045) by the Japanese Ministry of Education, Culture, Sports, Science and Technology.",
"Y.K.",
"acknowledge the support from the Japan Society for the Promotion of Science (JSPS) through JSPS research fellowships for young scientists."
]
] | 1204.1165 |
[
[
"Characterization of radially symmetric finite time blowup in\n multidimensional aggregation equations,"
],
[
"Abstract This paper studies the transport of a mass $\\mu$ in $\\real^d, d \\geq 2,$ by a flow field $v= -\\nabla K*\\mu$.",
"We focus on kernels $K=|x|^\\alpha/ \\alpha$ for $2-d\\leq \\alpha<2$ for which the smooth densities are known to develop singularities in finite time.",
"For this range This paper studies the transport of a mass $\\mu$ in $\\real^d, d \\geq 2,$ by a flow field $v= -\\nabla K*\\mu$.",
"We focus on kernels $K=|x|^\\alpha/ \\alpha$ for $2-d\\leq \\alpha<2$ for which the smooth densities are known to develop singularities in finite time.",
"For this range we prove the existence for all time of radially symmetric measure solutions that are monotone decreasing as a function of the radius, thus allowing for continuation of the solution past the blowup time.",
"The monotone constraint on the data is consistent with the typical blowup profiles observed in recent numerical studies of these singularities.",
"We prove monotonicity is preserved for all time, even after blowup, in contrast to the case $\\alpha >2$ where radially symmetric solutions are known to lose monotonicity.",
"In the case of the Newtonian potential ($\\alpha=2-d$), under the assumption of radial symmetry the equation can be transformed into the inviscid Burgers equation on a half line.",
"This enables us to prove preservation of monotonicity using the classical theory of conservation laws.",
"In the case $2 -d < \\alpha < 2$ and at the critical exponent $p$ we exhibit initial data in $L^p$ for which the solution immediately develops a Dirac mass singularity.",
"This extends recent work on the local ill-posedness of solutions at the critical exponent."
],
[
"Introduction",
"This manuscript considers the problem of dynamic nonlocal aggregation equations of the form $\\frac{\\partial \\rho }{\\partial t} - \\text{div}(\\rho \\nabla K*\\rho ) = 0$ in $\\mathbb {R}^d$ for $d \\ge 2$ .",
"This problem has been a very active area of research in the literature [7], [8], [9], [10], [12], [15], [16], [17], [18], [19], [21], [22], [23], [24], [29], [32], [39], [33], [41], [42], [44], [46], [48], [50], [51], [52], [53], [62], [63], [69], [70], [71].",
"These models arise in a number of applications including aggregation in materials science [41], [42], [64], [65], cooperative control [39], granular flow [25], [26], [71], biological swarming models [62], [69], [70], evolution of vortex densities in superconductors [36], [3], [2], [35], [58] and bacterial chemotaxis [21], [46], [16], [17].",
"A body of recent work has focused on the problem of finite time singularities and local vs global well-posedness in multiple space dimension for both the inviscid case (REF ) [8], [9], [10], [12], [18], [13], [24], [33], [44], [48] and the cases with various kinds of diffusion [5], [16], [13], [50], [51].",
"The highly studied Keller-Segel problem typically has a Newtonian potential and linear diffusion.",
"For the pure transport problem (REF ), of particular interest is the transition from smooth solutions to weak and measure solutions with mass concentration.",
"This paper presents a general framework for radially symmetric solutions that blowup in finite time in which the initial data decreases monotonically from the origin.",
"This paper differentiates itself from the previous work in that it considers continuation of the solution as a measure past the initial singularity for the range of $1>\\alpha >2-d$ .",
"Prior work on measure solutions considers the case $\\alpha \\ge 1$ in general dimension [24] and the Newtonian case with a defect measure in two dimensions [65].",
"The monotone constraint is typical of the local structure of the solution near the blowup time as shown in numerical simulations.",
"This paper presents rigorous theory for solutions with such structure, showing that the nonlocal evolution preserves this global structure.",
"Much work has been done on related problems in fluid dynamics to understand measure solutions of active scalar equations [30], [31], [60], [65], [77].",
"The results of this paper may provide further insight for those problems.",
"In the case of power-law kernels e.g.",
"$K(x) = |x|^\\alpha / \\alpha $ , it is already known that the critical power is $\\alpha =2$ .",
"For $\\alpha <2$ finite time singularities always arise and for $\\alpha \\ge 2 $ solutions stay smooth for all time if the initial data is smooth [9], [10], [24].",
"In the case of finite time blowup, i.e.",
"$\\alpha <2$ , it has been observed numerically [44], [45], [43] that, starting with a smooth radially symmetric initial data which is monotone decreasing as a function of the radius, the solution evolves so that the monotonicity is preserved and at some finite time develops a power-law singularity at the origin.",
"The power is sufficiently singular that, based on the results in this paper, the solution produces an instantaneous mass concentration after the initial singularity.",
"To make this rigorous we must develop a general theory for such singular solutions.",
"This paper addresses this particular class of measure solutions, namely those with a radially symmetric decreasing profile and possibly a Dirac mass at the origin.",
"We prove that this structure is maintained for all time when $\\alpha <2$ .",
"We conjecture that this class of radially symmetric decreasing solutions including a Dirac mass at the origin describes well the local behavior at the blow up time of general non radially symmetric solutions.",
"We also note that for $\\alpha >2$ it has been observed in numerical simulations [45], [43] that monotone decreasing structures are not preserved: indeed there is an attracting solution of the form of a collapsing delta-ring which causes mass to collect on the ring during the collapse thereby destroying any initial monotone property of the solution [45], [43].",
"We note that our work fits nicely between the general measure theory in [24] (for $\\alpha \\ge 1$ ) and previous works which consider more regular classes of weak solutions including $L^\\infty $ [8], [9] and $L^p$ [12].",
"For $L^\\infty $ and $L^p$ solutions we typically have only local well-posedness whereas the measure solutions have global well-posedness.",
"Our work extends the global existence results in [24] to the case of more singular kernels with power $\\alpha \\ge 2-d$ , for the special case of monotone decreasing radially symmetric measure solutions.",
"This includes that of the Newtonian potential, which is discussed separately in the next paragraph.",
"Uniqueness of solutions for $2-d<\\alpha <1$ is still an open problem.",
"In two space dimensions, when $K(x)= \\log (|x|)$ (i.e.",
"$K$ is the Newtonian potential), the aggregation equation arises as a model for the evolution of vortex densities in superconductors [36], [66], [67], [55], [3], [2], [56], [35], [58], and also in models for adhesion dynamics [64], [65].",
"In these models singularities are known to appear in finite time, and the question of interest is how to continue the solution after the initial formation of singularities.",
"Since these singularities are expected to be Dirac masses one has to consider measure solutions.",
"Unfortunately, due to the very singular behavior of the Newtonian potential at the origin, most of the results to date concern the existence of measure solutions which contain an error term (a defect measure) compared to the original equation [35], [64], [2].",
"Also uniqueness is lacking in these works.",
"In this paper we consider the Newtonian case in all dimensions and show that for general radially symmetric data there is no need to consider a defect measure because the symmetry allows the problem to be reduced to a form of the inviscid Burgers equation on the half line, for which many things are known.",
"In particular, the case of radially symmetric monotone decreasing densities maps to classical Lipschitz solutions of the inviscid Burgers equation, without shocks, allowing us to prove such solutions exist and are unique for all time.",
"For the non-monotone case, shocks can form, corresponding to mass concentrations along spherical shells, and their evolution is not immediately well-defined, due to a jump in the velocity field at the shell.",
"However, one can use the classical weak solution theory for Burgers equation to define a jump condition through a weak form of the evolution equation, or through some other convention.",
"If we use the classical Burgers shock solution then the solution is unique since it automatically satisfies the Lax entropy condition.",
"The more singular case of signed measures can also be studied in this framework, in which case one must consider rarefaction solutions as well as shocks.",
"Going back to the case $K(x)=\\left|x\\right|^\\alpha /\\alpha $ , $2-d<\\alpha <2$ , recent computational results [44] show that the initial finite time blowup from radially symmetric data has a simple self-similar form in which the power-laws of the similarity solution have anomalous scaling but the shape of the similarity solution has a simple monotonically decreasing structure with powerlaw tail.",
"The power in the tail determines the degree of singularity of the solution at the initial blowup time - we observe that at the initial blowup time the solution leaves $L^\\infty $ but remains in some $L^p$ spaces and does not concentrate mass.",
"This result prompted a careful study of the well-posedness of the equation in $L^p$ spaces [12].",
"In that paper it was proved that for a given interaction kernel, there exists a critical $L^p$ space such that the problem is locally well-posed for $p>p_c$ .",
"Moreover it was proved in [12] that the power $p_c$ is sharp for $K=|x|$ , i.e.",
"the problem is locally ill-posed for $p<p_c$ .",
"Some of these results, in particular the critical $L^{p}$ space, have been extended to general power-law kernels in [33].",
"In the present work we examine the mechanism by which initial data in the critical space $L^{p_c}$ leave instantaneously this space.",
"Taking advantage of our existence theory for radially symmetric decreasing measure solutions when $2-d<\\alpha <2$ , we exhibit a large class of radially symmetric decreasing initial data in $L^{p_c}$ for which a Dirac mass forms instantaneously in the solution.",
"This is a natural extension of the results in [12] and [33].",
"This paper is organized as follows: below we review the mathematical notation and basic functional analysis used in this paper.",
"Section develops a general existence theory for radially symmetric solutions with the monotonicity constraint.",
"Section proves instantaneous mass concentration for the critical $L^p$ spaces.",
"Section considers the case of the Newtonian potential, for which we can show that radial symmetry results in a transformation of the nonlocal problem to the inviscid Burgers equation on a half line.",
"Section summarizes the results and discusses some open problems.",
"In the appendix we derive some background theory of ordinary differential equations needed for the proofs in this paper and not derived in standard references (although the arguments are similar to standard methods).",
"The aggregation equation, for smooth solutions, in Eulerian coordinates, is $&\\frac{\\partial \\rho }{\\partial t} + \\text{div} (\\rho v)=0 \\\\&v(x,t)=-\\nabla K*\\rho .$ For very singular kernels, and correspondingly singular solutions - in general measure solutions - it makes sense to reformulate the problem in Lagrangian coordinates and work mainly in this framework to develop the theory.",
"It is easy to see, in the case of strong solutions, that the above formulation is equivalent to $\\rho (t)= \\sigma ^t \\# \\rho _{init} ,\\\\ \\text{ $\\sigma $ is the flow map associated to the field $v=-(\\nabla K * \\rho (t))(x)$} .$ In other words, the mass $\\rho $ is transported by characteristics $\\sigma $ that satisfy the ordinary differential equation $\\frac{d}{dt}{\\sigma (x,t)} = v(\\sigma (x,t), t) \\quad , \\quad \\sigma (x,0)=x.$ The map $\\sigma ^t: \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ is defined by $\\sigma ^t(x)=\\sigma (x,t)$ and $\\sigma ^t \\# \\rho _{init}$ stands for the push forward of the measure $\\rho _{init}$ by the map $\\sigma ^t$ (see below for a precise definition of the push forward).",
"We work with formulation (REF -) to prove existence of solutions, rather than (REF -).",
"We refer to this as the Lagrangian formulation of the problem.",
"Note, in particular, that the flux $\\rho v$ in (REF ) may be difficult to define, for the product of a measure $\\rho $ and a velocity field that blows up precisely at the point where $\\rho $ concentrates mass.",
"This phenomena can occur, for example, in the case of power law potentials potentials $K(x)=\\left|x\\right|^\\alpha / \\alpha $ , $\\alpha <1$ , for which existence theory was not known prior to this work.",
"Since we are working with a purely transport problem it is very natural to work in a Lagrangian framework.",
"The radial symmetry combined with monotonicity provides a focusing effect in which the only mass concentration occurs precisely at the origin, providing a natural way to keep track of mass transport in this problem.",
"We now introduce some technical notation and corresponding well-known functional analytic results.",
"${\\cal M}(\\mathbb {R}^d)$ stands for the space of Borel non-negative measure on $\\mathbb {R}^d$ which have finite mass.",
"${\\cal M}_R(\\mathbb {R}^d)$ is the set of $\\mu \\in {\\cal M}(\\mathbb {R}^d)$ which are radially symmetric.",
"${\\cal M}_{RD}(\\mathbb {R}^d)$ is the set of $\\mu \\in {\\cal M}(\\mathbb {R}^d)$ which are radially symmetric and decreasing.",
"To be more precise, $\\mu $ belongs to ${\\cal M}_{RD}(\\mathbb {R}^d)$ if and only if it can be written $\\mu =m \\delta + g$ , where $m\\in [0,+\\infty )$ , $\\delta $ is the Dirac delta measure at the origin and $g$ is an $L^1$ function which is nonnegative, radially symmetric and monotone decreasing as a function of the radius.",
"${\\cal P}(\\mathbb {R}^d)$ , ${\\cal P}_R(\\mathbb {R}^d)$ and ${\\cal P}_{RD}(\\mathbb {R}^d)$ are the subset of ${\\cal M}(\\mathbb {R}^d)$ , ${\\cal M}_R(\\mathbb {R}^d)$ and ${\\cal M}_{RD}(\\mathbb {R}^d)$ respectively which are made of measure of mass 1.",
"${\\cal P}_2(\\mathbb {R}^d)\\subset {\\cal P}(\\mathbb {R}^d)$ , is the subspace of probability measure of finite second moment, i.e.",
"$\\int _{\\mathbb {R}^d} |x|^2 d\\mu (x) <\\infty $ .",
"We say that a sequence $(\\mu _n) \\subset {\\cal P}(\\mathbb {R}^d)$ converges narrowly to $\\mu \\in {\\cal P}(\\mathbb {R}^d)$ , denoted by $\\mu _n \\rightharpoonup \\mu $ , if $\\lim _{n\\rightarrow \\infty } \\int _{\\mathbb {R}^d} f(x) d\\mu _n(x) = \\int _{\\mathbb {R}^d} f(x) d\\mu (x)$ for every $f \\in C_b^0(\\mathbb {R}^d)$ , the space of continuous and bounded real function defined on $\\mathbb {R}^d$ .",
"$C_{w}([0,+\\infty ),{\\cal P}(\\mathbb {R}^d))$ is the set of functions $\\mu :[0,+\\infty )\\rightarrow {\\cal P}(\\mathbb {R}^d)$ which are narrowly continuous, i.e.",
"$\\mu (t+h) \\rightharpoonup \\mu (t)$ as $h\\rightarrow 0$ $\\forall t\\ge 0$ .",
"For $\\mu $ and $\\nu $ in ${\\cal P}_2(\\mathbb {R}^d)$ , $W_2(\\mu ,\\nu )$ stands for the Wasserstein distance with quadratic cost between $\\mu $ and $\\nu $ (See [65] for the definition and properties of the Wasserstein distance $W_2(\\mu ,\\nu )$ ).",
"Recall that ${\\cal P}_2(\\mathbb {R}^d)$ , endowed with the metric $W_2$ is a complete metric space.",
"Furthermore, $\\lim _{n\\rightarrow \\infty } W_2(\\mu _n,\\mu ) = 0 \\;\\; \\Rightarrow \\;\\; \\mu _n\\rightharpoonup \\mu \\quad \\text{ as } n\\rightarrow \\infty .$ $C([0,+\\infty ),{\\cal P}_2(\\mathbb {R}^d))$ is the set of functions from $[0,+\\infty )$ to ${\\cal P}_2(\\mathbb {R}^d)$ which are continuous with respect to $W_2$ .",
"Note that $C([0,+\\infty ),{\\cal P}_2(\\mathbb {R}^d)) \\subset C_w([0,+\\infty ),{\\cal P}(\\mathbb {R}^d)).$ The space $C([0,+\\infty ),{\\cal P}_2(\\mathbb {R}^d))$ is endowed with the distance $\\mathcal {W}_2(\\mu ,\\nu ) = \\sup _{t\\ge 0} W_2(\\mu (t),\\nu (t)).$ If $T:\\mathbb {R}^d\\rightarrow \\mathbb {R}^d$ is a Borel map, and if $\\mu \\in {\\cal M}(\\mathbb {R}^d)$ , we denote by $T\\# \\mu $ the push forward of $\\mu $ through $T$ , defined by $T\\#\\mu (B) = \\mu (T^{-1} (B))$ , $\\forall B\\in \\mathcal {B}(\\mathbb {R}^d)$ .",
"More generally we have $\\int _{\\mathbb {R}^d} f(T(x)) d \\mu (x)= \\int _{\\mathbb {R}^d} f(x) d (T \\# \\mu )(x)$ for every bounded Borel function $f : \\mathbb {R}^d \\rightarrow \\mathbb {R}$ .",
"Both $B(0,R)$ and $B_R$ will be used to denote the open ball of radius $R$ , $\\lbrace x\\in \\mathbb {R}^d: \\left|x\\right| < R \\rbrace $ .",
"$A_\\epsilon $ , $\\epsilon <1$ , denotes the annulus $\\lbrace x\\in \\mathbb {R}^d: \\epsilon < |x| < 1\\rbrace $ .",
"All the probability measures in this paper are compactly supported and therefore belong to the space ${\\cal P}_2(\\mathbb {R}^n)$ on which the Wasserstein distance with quadratic cost is defined."
],
[
" Lagrangian solutions versus distributional solutions",
"Let us focus on power law kernels $K=|x|^\\alpha / \\alpha $ for simplicity.",
"If $v$ is bounded on compact sets, which is only true for $\\alpha \\ge 1$ , it is then standard (see [1] or [12] for example) to prove that if $\\rho $ and $\\sigma $ satisfy (REF ) and (), then $\\rho $ is a distributional solution of the aggregation equation, i.e.",
"$\\int _0^{+\\infty } \\int _{\\mathbb {R}^d} \\Big ( \\; \\frac{d\\xi }{dt}(x,t) +\\nabla \\xi (x,t) \\cdot v_t (x) \\; \\Big ) \\; d\\rho _t(x) \\; dt =0 ,\\\\v_t(x)=v(x,t)= -(\\nabla K * \\rho (t))(x) $ for all $\\xi \\in C_0^\\infty (\\mathbb {R}^d\\times (0,+\\infty ))$ .",
"On the other hand, if $2-d<\\alpha < 1$ , then the velocity field $x \\mapsto v_t(x)$ is not bounded and it is not clear how to give a sense to (REF ).",
"Hence we use the Lagrangian formulation of the problem throughout most of this paper.",
"In the special case of the Newtonian potential, in Section we transform using mass variables to Burgers equation for which it again makes sense to use a distributional form of the problem albeit in a different coordinate system.",
"This section is devoted to the proof of the following theorem: Let $\\nabla K(x)=x {\\left|x\\right|^{\\alpha -2}}$ , $\\alpha \\in (2-d,2)$ .",
"Given $\\rho _{init} \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ with compact support, there exists $\\rho \\in C([0,+\\infty ),{\\cal P}_{RD}(\\mathbb {R}^d))$ and a continuous map $\\sigma : [0,+\\infty ) \\times \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ satisfying $\\rho (t)= \\sigma ^t \\# \\rho _{init} ,\\\\ \\text{$\\sigma $ is the flow map associated to the field }v(x,t)={\\left\\lbrace \\begin{array}{ll}-(\\nabla K * \\rho (t))(x),& x\\ne 0\\\\0, & x=0.\\end{array}\\right.",
"}$ Remark 1 In () we could have let $v(x,t)=-(\\nabla K * \\rho (t))(x)$ for all $x$ with the understanding $\\nabla K(0)=0$ and the convolution is taken in the sense of principle value.",
"Theorem is interesting for two reasons: first it provides global existence of radially symmetric decreasing measure solutions with potential more singular than the one considered previously.",
"In [24] global existence and uniqueness of measure solutions is proven for $\\alpha \\ge 1$ ; here we restrict our attention to radially symmetric decreasing solution but we obtain global existence for $2-d<\\alpha <2$ .",
"Secondly this theorem shows that radially symmetric decreasing profiles are preserved for all time when $2-d<\\alpha <2$ .",
"Monotonicity is also preserved for the Newtonian case however in this case the problem localizes and the simpler proof is carried out in Section ."
],
[
"Formula for the convolution in radial coordinates and properties of the kernel",
"In this section we recall some known results about radially symmetric solutions of the aggregation equation and we prove additional results needed in the following subsections.",
"Let $\\mu \\in {\\cal M}_R(\\mathbb {R}^d)$ .",
"We define $\\hat{\\mu } \\in {\\cal M}([0,+\\infty ))$ to be the Borel measure on $[0,+\\infty )$ which satisfies $\\hat{\\mu }(I)= \\mu (\\lbrace x\\in \\mathbb {R}^d: |x|\\in I\\rbrace )$ for all $I\\in {\\mathcal {B}}([0,+\\infty ))$ .",
"Remark 2 It is straightforward to check that if a sequence $\\mu _n \\in {\\cal P}_{R}(\\mathbb {R}^d)$ converges narrowly to $\\mu \\in {\\cal P}_{R}(\\mathbb {R}^d)$ then $ \\lim _{n\\rightarrow \\infty } \\int _{[0,+\\infty )} f(r) d\\hat{\\mu }_n(r) = \\int _{[0,+\\infty )} f(r) d\\hat{\\mu }(r)$ for every $f \\in C_b^0([0,+\\infty ))$ , the space of continuous and bounded real function defined on $[0,+\\infty )$ .",
"Let $\\mu ,\\nu \\in {\\cal P}_R(\\mathbb {R}^d)$ .",
"We say that $\\mu $ is more concentrated than $\\nu $ , and we write $\\mu \\succ \\nu $ , if ${\\hat{\\mu }}=T\\#{\\hat{\\nu }}$ for some Borel map $T: [0,+\\infty ) \\mapsto [0,+\\infty )$ satisfying $T(r)\\le r$ for all $r \\in [0,+\\infty )$ .",
"Remark 3 Suppose that $\\mu , \\nu \\in {\\cal P}_{R}(\\mathbb {R}^d)$ with $\\nu =m \\delta + f$ for some $m \\ge 0$ and $f \\in L^1(\\mathbb {R}^d)$ .",
"It can then be proven that $ \\mu \\succ \\nu \\quad \\Longleftrightarrow \\quad \\mu \\left( \\overline{B (0,r)}\\right) \\ge \\nu \\left(\\overline{B(0,r)} \\right) \\quad \\forall r>0.$ This equivalence is actually true for any $\\mu ,\\nu \\in {\\cal P}_{R}(\\mathbb {R}^d)$ if one uses transport plans in Definition REF rather than only transport maps (see [73] for a definition of transport plans).",
"The proof is a consequence of the fact that, given any two probability measures on $[0,+\\infty )$ , an optimal transport plan with respect to the quadratic cost can be explicitly constructed in term of the cumulative distributions of these two probability measures [73].",
"It can then be checked that if the cumulative distribution of $\\hat{\\mu }$ is greater than the cumulative distribution of $\\hat{\\nu }$ , this optimal transport plan takes elements of mass from $\\hat{\\nu }$ and move them toward the origin.",
"It can also be checked that if $\\hat{\\nu }=m \\delta + \\hat{f}$ for some $m \\ge 0$ and $\\hat{f} \\in L^1([0,+\\infty ))$ , then this optimal transport plan is induced by a transport map.",
"Since the equivalence (REF ) is not used in this paper, we omit the proof.",
"We note that other authors (see for example [72]) use the right hand side of (REF ) to define the notion of concentration.",
"For $\\alpha \\in (2-d,2)$ define the function $\\phi : [0,+\\infty ) \\rightarrow \\mathbb {R}$ by $\\phi (r)= \\frac{1}{\\omega _{d-1}}\\int _{S^{d-1}} \\frac{e_1-r y}{\\left|e_1-r y\\right|^{2-\\alpha }} \\cdot e_1 \\; d \\sigma (y),$ where $S^{d-1} =\\lbrace x\\in \\mathbb {R}^d: \\left|x\\right|=1\\rbrace $ is the unit sphere and $\\omega _{d-1}$ its surface measure.",
"The following lemma was proven in [12] for the case $\\alpha =1$ and in [33] for general $\\alpha $ .",
"Let $\\nabla K(x)=x {\\left|x\\right|^{\\alpha -2}}$ , $\\alpha \\in (2-d,2)$ .",
"Let $\\mu \\in {\\cal M}_{R}(\\mathbb {R}^d)$ .",
"Then for any $x\\ne 0$ , we have $ (\\nabla K * \\mu ) (x)= \\left|x\\right|^{\\alpha -1} \\int _0^{+\\infty } \\phi \\left(\\frac{r}{\\left|x\\right|}\\right) d \\hat{\\mu }(r) \\; \\frac{x}{\\left|x\\right|}.$ Moreover, $\\phi $ is continuous, strictly positive, non-increasing on $[0,+\\infty )$ , and $\\phi (0)=1, \\qquad \\lim _{r \\rightarrow \\infty } \\phi (r) r^{2-\\alpha }=\\frac{d+\\alpha -2}{d}.$ Note in particular that $\\phi \\in C_0^b([0,+\\infty ))$ which, in view of (REF ), will be convenient in order to pass to the limit in expressions such as (REF ).",
"The positivity, monotonicity and boundedness of $\\phi $ have three important consequences that can be directly read from (REF ).",
"Let $\\mu , \\nu \\in {\\cal M}_{R}(\\mathbb {R}^d)$ .",
"Since $\\phi $ is strictly positive, we have $ \\mu \\ge \\nu \\quad \\Longrightarrow \\quad \\left|\\nabla K * \\mu \\right| \\ge \\left|\\nabla K * \\nu \\right| .$ Let $\\mu , \\nu \\in {\\cal P}_R(\\mathbb {R}^d)$ .",
"Since $\\phi $ is non-increasing, we have $ \\mu \\succ \\nu \\quad \\Longrightarrow \\quad \\left|\\nabla K * \\mu \\right| \\ge \\left|\\nabla K * \\nu \\right| .$ Let $\\mu \\in {\\cal P}_R(\\mathbb {R}^d)$ .",
"Since $0 < \\phi \\le 1$ we have $ \\left|\\nabla K * \\mu \\right| \\le \\left|x\\right|^{\\alpha -1} .$ In the next Lemma we prove that that $\\phi $ is $C^1$ for $\\alpha > 3-d$ , quasi-Lipschitz continuous for $\\alpha =3-d$ and Hölder continuous for $2-d <\\alpha < 3-d$ .",
"The lack of smoothness for $\\alpha \\le 3-d$ is due to a singularity in the derivative $r=1$ .",
"Below we prove sharp estimates on the regularity of $\\phi $ ; later we will only use the fact that $\\phi $ is Hölder continuous in the range of $\\alpha $ considered.",
"If $\\alpha \\in (3-d,2)$ then $\\phi \\in C^1(0,+\\infty )$ and $\\phi ^{\\prime }$ is bounded on $(0,+\\infty )$ .",
"If $\\alpha =3-d$ , then there exists a constant $C_1>0$ such that $ \\left|\\phi (r_1)-\\phi (r_2)\\right| \\le C_1 \\left|r_1-r_2\\right|(1- \\log \\left|r_1-r_2\\right| )$ for all $r_1, r_2$ satisfying $\\left|r_1-r_2\\right|<1/2$ .",
"If $\\alpha \\in (2-d, 3-d)$ , then there exists a constant $C_2>0$ such that $ \\left|\\phi (r_1)-\\phi (r_2)\\right| \\le C_2 \\left|r_1-r_2\\right|^{\\alpha -(2-d)}$ for all $r_1, r_2$ satisfying $\\left|r_1-r_2\\right|<1/2$ .",
"In [33] it was proven that $\\phi $ is differentiable on $[0,1)\\cup (1+\\infty )$ and that, for $r \\ne 1$ : $ \\phi ^{\\prime } (r)= -C_{\\alpha ,d} \\int _0^\\pi \\frac{r (\\sin \\theta )^d}{A(r,\\theta )^{4-\\alpha }}d \\theta \\\\\\text{where } A(r, \\theta ) = (1 + r^2 -2 r \\cos \\theta )^{1/2} \\\\\\text{and } C_{\\alpha ,d}=\\frac{\\omega _{d-2}(2- \\alpha )(d+\\alpha -2)}{\\omega _{d-1} (d-1)}>0.$ Note first that for fixed $r$ the function $\\theta \\mapsto A(r, \\theta )$ reaches its minimum at $\\theta =0$ .",
"So $A(r,\\theta ) \\ge \\left|r-1\\right|$ and one can easily see from (REF ) that for all the $\\alpha $ considered, $\\phi ^{\\prime }(r)$ is bounded on $[0,1/2] \\cup [3/2,+\\infty )$ .",
"It is therefore enough to prove the statements of the Lemma only on the interval $(1/2,3/2)$ .",
"We first prove (i).",
"Note that for fixed $\\theta $ the function $r \\mapsto A(r, \\theta )$ reaches its minimum at $r=\\cos \\theta $ and therefore $A(r,\\theta ) \\ge \\left|\\sin \\theta \\right| $ .",
"As a concequence we have the following estimate for the integrand in (REF ): $\\frac{r (\\sin \\theta )^d}{A(r,\\theta )^{4-\\alpha }} \\le \\frac{3/2}{(\\sin \\theta )^{4-\\alpha -d}} \\qquad \\text{ for all $r \\in \\left(\\frac{1}{2},\\frac{3}{2}\\right)$ and $\\theta \\in [0,\\pi ]$.",
"}$ Since the right hand side of (REF ) is integrable if $\\alpha >3-d$ we obtain (i) by the dominated convergence theorem.",
"We now turn to the proof of (ii) and (iii).",
"We first derive the estimates $\\left|\\phi ^{\\prime }(1+h)\\right| \\le \\frac{C}{\\left|h\\right|^{3-\\alpha -d}} & \\qquad \\text{ if } \\alpha \\in (2-d,3-d)\\\\\\left|\\phi ^{\\prime }(1+h)\\right| \\le - C \\log \\left|h\\right| & \\qquad \\text{ if } \\alpha = 3-d $ for all $h \\in (-1/2,1/2)$ , $h \\ne 0$ .",
"The constant $C>0$ depends on $\\alpha $ but not on $h$ .",
"We will prove (REF ) only for $h \\in (0,1/2)$ .",
"The proof for $h \\in (-1/2,0)$ is precisely the same.",
"Write $\\left|\\phi ^{\\prime } (1+h)\\right|= C_{\\alpha ,d} \\left( \\int _h^{\\pi -h}\\frac{ (1+h) (\\sin \\theta )^d}{A(1+h,\\theta )^{4-\\alpha }} d \\theta +\\int _{[0,h] \\cup [\\pi -h, \\pi ] }\\frac{ (1+h) (\\sin \\theta )^d}{A(1+h,\\theta )^{4-\\alpha }} d \\theta \\right)$ and let $(I)$ be the first integral and $(II)$ the second one.",
"Using the fact that $A(r,\\theta ) \\ge \\sin \\theta $ we find that $(I) \\le \\frac{3}{2} \\int _h^{\\pi -h}\\frac{ 1}{(\\sin \\theta )^{4-\\alpha -d}} d \\theta \\le 3 \\int _h^{\\pi /2}\\frac{ 1}{(\\sin \\theta )^{4-\\alpha -d}} d \\theta \\\\\\le 3 \\int _h^{\\pi /2}\\frac{ 1}{( \\frac{2}{\\pi } \\theta )^{4-\\alpha -d}} d \\theta \\le \\frac{C}{h^{3-\\alpha -d}}$ where we have used the symmetry of $\\sin \\theta $ around $\\theta =\\pi /2$ and the fact that $\\sin \\theta \\ge (2/ \\pi ) \\theta $ on the interval $[0, \\pi /2]$ .",
"To estimate $(II)$ we use the fact that $A(1+h, \\theta ) \\ge h$ : $(II) \\le \\frac{3/2}{h^{4-\\alpha }} \\int _{[0,h] \\cup [\\pi -h, \\pi ] } (\\sin \\theta )^d d\\theta \\le \\frac{3}{h^{4-\\alpha }} \\int _0^h (\\sin \\theta )^d d\\theta \\\\\\le \\frac{3}{h^{4-\\alpha }} \\int _0^h \\theta ^d d\\theta \\le \\frac{C}{h^{3-\\alpha -d}}.$ This concludes the proof of (REF ).",
"The proof of () is similar.",
"Let $\\omega (r)= {\\left\\lbrace \\begin{array}{ll}\\frac{C}{\\alpha -(2-d)}r^{\\alpha -(2-d)} & \\text{ if }\\alpha \\in (2-d,3-d)\\\\C r (1-\\log r) & \\text{ if }\\alpha =3-d\\end{array}\\right.",
"}$ and note that $\\omega $ is the antiderivative of the right hand side of (REF ) and ().",
"Note also that $\\omega $ is a nonnegative, increasing, concave function on $[0,1]$ which is equal to 0 at $r=0$ .",
"To conclude the proof of (ii) and (iii) we need to show that $\\omega $ is the modulus of continuity of $\\phi $ on the interval $(1/2,3/2)$ , that is $ \\left|\\phi (r_1) -\\phi (r_2)\\right| \\le \\omega (\\left|r_1-r_2\\right|) \\qquad \\text{for all $r_1,r_2 \\in \\left(\\frac{1}{2},\\frac{3}{2}\\right) $}.$ Since $\\phi ^{\\prime }$ is negative, from (REF ) and () we have that $0\\le -\\phi ^{\\prime }(1+h) \\le \\omega ^{\\prime }(h).$ for $h>0$ .",
"Integrating this inequality on $[h_1,h_2]$ and using the fact that $\\omega (h_2)-\\omega (h_1)\\le \\omega (h_2-h_1)-\\omega (0)= \\omega (h_2-h_1)$ due to the concavity of $\\omega $ , we obtain that $ 0 \\le \\phi (1+h_1)-\\phi (1+h_2) \\le \\omega (h_2-h_1) \\qquad \\text{ for all } 0 \\le h_1 < h_2 < 1/2.$ This prove that (REF ) holds for all $r_1, r_2 \\in [1,3/2)$ .",
"A similar proof leads to the same result on the interval $(1/2,1]$ .",
"To obtain the result on the full interval $(1/2,3/2)$ , let $h_1,h_2 \\in (0,1/2)$ and write $0 \\le \\phi (1-h_1)-\\phi (1+h_2)= \\phi (1-h_1)-\\phi (1)+\\phi (1)-\\phi (1+h_2) \\\\\\le \\omega (h_1)+ \\omega (h_2) \\le 2 \\omega (h_1+h_2).$ To obtain the last inequality we have used the fact that $\\omega $ is increasing."
],
[
"Regularity of the velocity field",
"We now study the regularity of a the velocity field associated with a radially symmetric decreasing measure solution of the aggregation equation.",
"Recall that $A_\\epsilon := \\lbrace x\\in \\mathbb {R}^d: \\epsilon < |x| < 1\\rbrace .$ Obviously $A_0= B(0,1)\\backslash \\lbrace 0\\rbrace $ .",
"Let $\\rho \\in C_w([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d))$ and assume that $\\text{supp}(\\rho (t)) \\subset B(0,1)$ for all $t\\ge 0$ .",
"Let $\\nabla K(x)=x {\\left|x\\right|^{\\alpha -2}}$ , $\\alpha \\in (2-d,2)$ .",
"Then the velocity field $v(x,t)$ defined by $v(x,t)={\\left\\lbrace \\begin{array}{ll}-(\\nabla K * \\rho (t))(x),& x\\ne 0\\\\0, & x=0\\end{array}\\right.",
"}$ satisfies: $v(x,t)$ is continuous on $A_0 \\times [0,+\\infty )$ for all $t \\ge 0$ .",
"For every $t \\ge 0$ , the function $x\\mapsto v(x,t)$ is continuously differentiable on $A_0$ .",
"Given $\\epsilon >0$ there exists $C>0$ such that $\\left|\\nabla v(x,t)\\right|< C$ for all $(x,t) \\in A_\\epsilon \\times [0,+\\infty )$ .",
"Given $\\epsilon >0$ and $\\eta >0$ , there exists $\\delta >0$ such that $ \\left|\\nabla v(y,t)-\\nabla v(x,t)\\right| \\le \\eta $ for all $x,y \\in A_\\epsilon $ satifying $\\left|x-y\\right|< \\delta $ and for all $t \\ge 0$ .",
"The notation $\\nabla v(x,t)$ stands for the derivative of $v$ with respect to $x$ .",
"In order to prove this proposition, we will need the following lemma and its corollary.",
"Suppose $g\\in L^1(\\mathbb {R}^d)$ is nonnegative, radially symmetric decreasing, and supported in $B(0,1)$ .",
"Suppose also that $\\Phi \\in C(\\mathbb {R}^d \\backslash \\lbrace 0\\rbrace ) \\cap L^1_{loc}(\\mathbb {R}^d)$ .",
"Then $\\Phi *g \\in C(\\mathbb {R}^d \\backslash \\lbrace 0\\rbrace ) \\cap L^1_{loc}(\\mathbb {R}^d)$ and $\\Vert \\Phi *g\\Vert _{L^\\infty (A_{\\epsilon })} \\le \\Vert g\\Vert _{L^1(B_{\\epsilon /2})} \\Big \\lbrace \\sup _{\\epsilon /2<|y|<2}|\\Phi (y)|+ \\frac{\\Vert \\Phi \\Vert _{L^1(B_2)}}{|B_{\\epsilon /2}|}\\Big \\rbrace .$ Since $\\Phi $ belongs to $L^1_{loc}$ , by Young's inequality $\\Phi *g$ is also in $L^1_{loc}$ .",
"We now prove estimate (REF ).",
"Fix $x$ in $A_\\epsilon $ .",
"$\\left|(\\Phi *g)(x)\\right| &\\le \\int _{B_{\\epsilon /2}} \\left|\\Phi (x-y)\\right| g(y) dy + \\int _{A_{\\epsilon /2} } \\left|\\Phi (x-y)\\right| g(y) dy\\\\& \\le \\Big ( \\sup _{\\epsilon /2<|y|<1+\\epsilon /2}|\\Phi (y)|\\Big ) \\; \\Vert g\\Vert _{L^1(B_{\\epsilon /2})}\\\\ & \\hspace{142.26378pt}+ \\Big ( \\sup _{y \\in A_{\\epsilon /2}} g(y) \\Big ) \\Vert \\Phi \\Vert _{L^1(B_2)} .$ Since $g$ is radially symmetric decreasing, we have that $g(z) \\ge \\sup _{y \\in A_{\\epsilon /2}} g(y)$ for almost every $z\\in B_{\\epsilon /2}$ .",
"Therefore we obtain $\\Vert g\\Vert _{L^1(B_{\\epsilon /2})} \\ge |B_{\\epsilon /2} | \\;\\;\\sup _{y \\in A_{\\epsilon /2}} g(y)$ which, combined with (), leads to the desired estimate.",
"We now prove that $\\Phi *g$ is continuous.",
"Reasoning as above we obtain that if $x\\in A_\\epsilon $ then $ \\left|(\\Phi *g)(x+h)-(\\Phi *g)(x)\\right| \\le \\Vert g\\Vert _{L^1(B_{\\epsilon /2})} \\\\ \\Big \\lbrace \\sup _{\\epsilon /2<|y|<2}|\\Phi (y+h)-\\Phi (y)|+ \\frac{\\Vert \\Phi (\\cdot +h ) - \\Phi (\\cdot ) \\Vert _{L^1(B_2)}}{|B_{\\epsilon /2}|}\\Big \\rbrace .$ We conclude that $\\Phi * g$ is continuous using the fact that $\\Phi $ is uniformly continuous on compact sets which do not contain the origin and the continuity of the translation $h \\rightarrow \\Phi (\\cdot +h)$ from $\\mathbb {R}^d$ to $L^1(B_2)$ .",
"Recall that ${\\cal P}_{RD}(\\mathbb {R}^d)$ is the space of probability measure $\\mu $ which can be written $\\mu = m \\delta + g$ for some $m \\ge 0$ and for some nonnegative, radially symmetric decreasing function $g\\in L^1(\\mathbb {R}^d)$ .",
"From the previous Lemma we directly obtain the following corollary: Let $K: \\mathbb {R}^d \\rightarrow \\mathbb {R}$ be a potential such that $K_{x_i x_j}\\in C(\\mathbb {R}^d \\backslash \\lbrace 0\\rbrace ) \\cap L^1_{loc}(\\mathbb {R}^d)$ .",
"Then the family of functions $\\lbrace K_{x_i x_j}*\\mu : {\\mu \\in {\\cal P}_{RD}(\\mathbb {R}^d)}\\rbrace $ is uniformly bounded and equicontinuous on every annulus $A_\\epsilon $ , $\\epsilon >0$ .",
"To be more precise we have: $\\Vert K_{x_i x_j}*\\mu \\Vert _{L^\\infty (A_{\\epsilon })} \\le \\sup _{\\epsilon /2<|y|<2}|K_{x_i x_j}(y)|+ \\frac{\\Vert K_{x_i x_j}\\Vert _{L^1(B_2)}}{|B_{\\epsilon /2}|}\\qquad \\text{ for all $\\mu \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ }$ And also: given $\\epsilon >0$ and $\\eta >0$ , there exists $\\delta >0$ such that $ \\left|(K_{x_i x_j}*\\mu ) (x) - (K_{x_i x_j}*\\mu ) (y)\\right| \\le \\eta $ for all $x,y \\in A_\\epsilon $ satisfying $\\left|x-y\\right|< \\delta $ and for all $\\mu \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ .",
"Since $\\mu \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ it can be written $\\mu =m \\delta +g$ where $m \\in [0,1]$ and $g$ satisfies the hypothesis of Lemma REF .",
"So $K_{x_i x_j}*\\mu = m K_{x_i x_j} +K_{x_i x_j}*g$ and it is easy to conclude using Lemma REF with $\\Phi =K_{x_ix_j}$ .",
"The second statement is a consequence of (REF ).",
"We now prove Proposition REF : Proof of Proposition REF .",
"Let us first check that $v(x,t)$ is continuous on $A_0 \\times [0,+\\infty )$ .",
"Continuity with respect to time comes from the fact that $t \\mapsto \\rho (t)$ is narrowly continuous together with the fact that the kernel $\\phi $ appearing in formula (REF ) belongs to $C^0_b([0,+\\infty ))$ .",
"Continuity with respect to space comes from the Hölder continuity of $\\phi $ .",
"To prove (P1), (P2) and (P3), note first that if $K(x)=\\left|x\\right|^{\\alpha }$ , $\\alpha >2-\\alpha $ , then $K_{x_i x_j}\\in C(\\mathbb {R}^d \\backslash \\lbrace 0\\rbrace ) \\cap L^1_{loc}(\\mathbb {R}^d)$ , and then use Corollary REF .",
"$\\Box $ $\\mathcal {V}$ is the space of velocity fields $v: B(0,1) \\times [0,+\\infty ) \\rightarrow \\mathbb {R}^d$ which are radially symmetric and pointing inward (i.e.",
"$v(0,t)=0$ and $v(x,t)=- \\lambda (\\left|x\\right|,t) {x}$ , $|x|>0$ , for some nonnegative function $\\lambda : (0,1) \\times [0,+\\infty ) \\rightarrow \\mathbb {R}$ ) and which satisfies (P0)–(P3).",
"Obviously a velocity field defined as in the statement of Proposition REF belongs to $\\mathcal {V}$ (the fact that it points inward comes from formula (REF ) together with the positivity of the kernel $\\phi $ ).",
"We now investigate properties of flow maps generated by velocity fields in $\\mathcal {V}$ .",
"Suppose $v \\in \\mathcal {V}$ .",
"Then there exists a unique continuous function $\\sigma : B(0,1) \\times [0,+\\infty ) \\mapsto B(0,1)$ satisfying $\\sigma (x,t)=x + \\int _{0}^t v(\\sigma (x,s),s) ds \\qquad \\text{ for all }(x,t) \\in B(0,1) \\times [0,+\\infty ).$ Moreover if the point $(x_0,t_0) \\in B(0,1) \\times [0,+\\infty )$ is such that $\\sigma (x_0,t_0) \\ne 0$ , then the mapping $ x \\mapsto \\sigma (x,t_0)$ is continuously differentiable at $x_0$ and we have $\\text{det }\\nabla \\sigma (x_0,t_0) = \\text{exp} \\big ( {\\int _{0}^{t_0} (\\text{div }v)( \\sigma (x_0,s),s) ds} \\big ) >0 .$ In particular, if $R(t)$ is such that $(\\sigma ^t)^{-1}(\\lbrace 0\\rbrace )= \\overline{B(0,R(t))}$ , then $\\sigma ^t$ is a diffeomorphism from $B(0,1) \\backslash \\overline{B(0,R(t))}$ to $\\mathbb {R}^d \\backslash \\lbrace 0\\rbrace $ .",
"The global existence and forward uniqueness of solution of the ODE $\\dot{x}=v(x,t)$ simply come from the fact that $v$ is Lipschitz continuous with respect to space away from the origin (because of (P2)) together with the fact that $v$ is pointing inward.",
"Indeed, solutions can be continued as long as they are in $B(0,1)\\backslash \\lbrace 0\\rbrace $ and since $v$ is pointing inward, the only way for a solution to escape this domain is to reach the origin.",
"When a solution reaches the origin, it stays there forever in accordance to the fact that $v(0,t)=0$ for all $t \\ge 0$ .",
"The differentiability of $\\sigma ^t$ on $B(0,1) \\backslash \\overline{B(0,R(t))}$ and formula (REF ) are more delicate.",
"In classical ODE textbooks, such results are obtained under the assumption that $v(x,t)$ is continuously differentiable in both space and time.",
"In our case $v$ is continuously differentiable in space but only continuous in time.",
"However, by revisiting classical proofs, one can easily check that assumption (P3) is enough to obtained differentiability of the flow map as well as formula (REF ).",
"This is done in section REF of the appendix.",
"Since the flow map $\\sigma ^t$ generated by a velocity field in $\\mathcal {V}$ is a diffeomorphism from $B(0,1) \\backslash \\overline{B(0,R(t))}$ to $B(0,1) \\backslash \\lbrace 0\\rbrace $ , we can use the change of variable formula to express the push forward of a measure in ${\\cal P}_{RD}(\\mathbb {R}^d)$ by $\\sigma ^t$ : Suppose $v \\in \\mathcal {V}$ and let $\\sigma $ be the associated flow map provided by Proposition REF .",
"Let $\\mu =m_0 \\delta + g_0 \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ and assume that $\\text{supp} \\mu \\in B(0,1)$ .",
"Then $\\sigma ^t \\# \\mu = m(t) \\delta + g(t)$ where $m(t) \\in \\mathbb {R}^+$ and $g(t) \\in L^1(\\mathbb {R}^d)$ satisfy : $m(t)=m_0+\\int _ {(\\sigma ^t)^{-1}(\\lbrace 0\\rbrace ) }g_0(x) \\; dx \\\\g(x,t)= \\left(\\frac{g_{0}}{\\text{det} \\, \\nabla \\sigma ^t} \\circ (\\sigma ^t)^{-1}\\right)(x) \\quad \\text{ for }x \\ne 0.$"
],
[
"Radially symmetric decreasing profiles are preserved",
"In this subsection we give a heuristic argument (which will be made rigorous in the next subsection) explaining why radially symmetric decreasing profiles are preserved by the aggregation equation when $2-d<\\alpha \\le 2$ .",
"We recall here that it is observed numerically that when $\\alpha >2$ , radially symmetric decreasing profiles are not preserved.",
"A key ingredient in our argument is the known fact that the convolution of two radially symmetric decreasing functions is still radially symmetric decreasing (see [54] for example).",
"For completeness we give a quick proof of this fact: Suppose $g\\in L^1(\\mathbb {R}^d)$ with compact support and $f\\in L^1_{loc}(\\mathbb {R}^d)$ .",
"If $f$ and $g$ are nonnegative radially symmetric decreasing functions, then $f*g$ is also a nonnegative radially symmetric decreasing function.",
"The mononicity of $g$ allows us to use a “layer cake\" decomposition of g, namely $g(x) =\\int _0^\\infty \\chi _{B(0,\\tilde{r}(g))} (x) dg$ where $\\tilde{r}(g)$ denotes the inverse function of $g(r)$ and $\\chi _{B(0,s)}$ denotes the characteristic function of the ball of radius $s$ .",
"Thus $f*g(x) = \\int _0^\\infty f*\\chi _{B(0,\\tilde{r}(g))} (x) dg$ and we note that the integrand of (REF ) is monotone decreasing because the characteristic function of a ball convolved with a nonnegative $L^1_{loc}$ monotone decreasing function is itself monotone decreasing.",
"By integrating a monotone integrand with respect to g we obtain the monotonicity result for $f*g$ .",
"We now present the heuristic argument.",
"Let us assume that $u(x,t)$ is a smooth solution of the aggregation equation.",
"Clearly we have: $ \\frac{\\partial {u}}{{\\partial t}} + \\nabla u \\cdot {v}= (\\Delta K * u) u.$ Suppose that $\\Delta K$ is locally integrable, nonnegative and radially symmetric decreasing.",
"When $\\nabla K(x)=x |x|^{\\alpha -2}$ , these hold true if and only if $2-d<\\alpha \\le 2$ .",
"We then use Lemma REF to see that if for some $t_0\\ge 0$ , $u(\\cdot ,t_0)$ is radially symmetric decreasing then the right hand side of (REF ) is also radially symmetric decreasing at $t_0$ .",
"This indicates that the rate of change along the characteristic is greater the closer we are to the origin.",
"Therefore the solution is expected to remain radially symmetric decreasing for $t>t_0$ .",
"For the special case of the Newtonian potential, $\\Delta K*u=u$ and monotonicity is similarly preserved - this is discussed in more detail in Section ."
],
[
"Proof of Theorem ",
"The proof is inspired by the work in [28], where global existence of measure solutions for some kinetics model was obtained by using a fixed point iteration in the space of probability measures endowed with the Wasserstein distance.",
"Let $\\nabla K(x)=x {\\left|x\\right|^{\\alpha -2}}$ , $\\alpha \\in (2-d,2)$ , and let $\\rho _{init} \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ with $\\text{supp}(\\rho _{init}) \\subset B(0,1)$ .",
"Define: $&\\rho _0(t)=\\rho _{init} \\qquad \\qquad \\qquad \\forall t \\in [0,+\\infty ) \\\\&v_0(x,t)={\\left\\lbrace \\begin{array}{ll}-(\\nabla K * \\rho _0(t))(x),& \\text{ if } x\\ne 0\\\\0, & \\text{ if } x=0\\end{array}\\right.",
"}\\qquad \\forall t \\in [0,+\\infty ) \\\\&\\sigma _0^t : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d := \\text{flow map associated with } v_0&$ and for $n \\ge 1$ define recursively $&\\rho _n(t)= \\sigma ^t_{n-1} \\# \\rho _{init} \\qquad \\qquad \\forall t \\in [0,+\\infty ) \\\\&v_n(x,t)={\\left\\lbrace \\begin{array}{ll}-(\\nabla K * \\rho _n(t))(x),& \\text{ if } x\\ne 0\\\\0, & \\text{ if } x=0\\end{array}\\right.",
"}\\qquad \\forall t \\in [0,+\\infty ) \\\\&\\sigma _n^t : \\mathbb {R}^d \\rightarrow \\mathbb {R}^d := \\text{flow map associated with } v_n.$ For all $n\\ge 0$ , $\\rho _n \\in C([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d)) $ and $\\text{supp}(\\rho _n(t)) \\subset B(0,1)$ for all $t\\ge 0$ .",
"Given $\\epsilon >0$ , there exists $L_\\epsilon > 0$ such that $\\left|v_n(x,t)-v_n(y,t)\\right| \\le L_\\epsilon \\left|x-y\\right|$ for all $x,y \\in A_\\epsilon $ , for all $t \\ge 0$ , and for all $n \\ge 0$ .",
"There exists a constant $\\theta \\in (0,1]$ depending only on $\\alpha $ such that the following holds: Given $\\epsilon >0$ , there exists $C_\\epsilon >0$ and $\\delta >0$ such that $\\left|t-s\\right| < \\delta $ implies $\\left|v_n(x,t)-v_n(x,s)\\right| \\le C_\\epsilon \\left|s-t\\right|^{\\theta }$ for all $x \\in A_\\epsilon $ and for all $n \\ge 0$ .",
"$\\rho _{n+1}(t) \\succ \\rho _n(t)$ for all $n\\ge 0$ and all $t\\in [0,+\\infty )$ .",
"This implies $|v_{n+1}(x,t)| \\ge |v_n(x,t)|$ for all $(x,t) \\in \\mathbb {R}^d \\times [0,+\\infty )$ and for all $n\\ge 0$ .",
"Before we prove this proposition let us explain how it will be used in the proof of Theorem .",
"Because of statements (ii), (iii), (iv) and the bound $\\left|v_n(x,t)\\right|\\le \\left|x\\right|^{\\alpha -1}$ (see (REF )), we can use the Arzela-Ascoli theorem to conclude that the $v_n$ 's converge uniformly on $A_\\epsilon \\times [0,+\\infty )$ to some function $v$ which is Lipschitz continuous in space and Hölder continuous in time, with same constants $L_\\epsilon $ and $C_\\epsilon $ .",
"Since $\\epsilon $ can be chosen as small as we want, $v(x,t)$ is well define on $B(0,1)\\backslash \\lbrace 0\\rbrace \\times [0,+\\infty )$ .",
"Let $v(0,t)=0$ so that $v$ is now well defined on $B(0,1) \\times [0,+\\infty )$ .",
"This velocity field $v(x,t)$ generates a flow map $\\sigma ^t: B(0,1) \\rightarrow B(0,1)$ and from this flow map we can construct $\\rho (t)=\\sigma ^t\\# \\rho _{init}$ .",
"In Proposition REF it will be shown that $\\sigma _n$ converges uniformly to $\\sigma $ on $B(0,1) \\times [0,\\infty )$ .",
"This implies in particular that for a given $t$ , $\\rho _n(t)$ converges narrowly to $\\rho (t)$ .",
"The narrow convergence preserves the monotonicity (see Proposition REF of the Appendix), and therefore $\\rho (t)$ is radially symmetric decreasing.",
"In order to prove that the radially symmetric decreasing function $\\rho (t)$ and the flow map $\\sigma ^t$ obtained by the above limiting process satisfy (REF ) and () we just need to show that $v(x,t)=-(\\nabla K *\\rho (t))(x)$ for $x\\ne 0$ , and this fact will follow easily from passing to the limit in the relationship $v_n(x,t)=-(\\nabla K *\\rho _n(t))(x)$ for $x \\ne 0$ .",
"Proof of Proposition REF .",
"Let us first prove (i).",
"The initial iterate $\\rho _0(t) \\equiv \\rho _{init}$ obviously belongs to $C([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d))$ with $\\text{supp}(\\rho _0(t)) \\subset B(0,1)$ for all $t\\ge 0$ .",
"Assume that $\\rho _n \\in C([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d))$ with $\\text{supp}(\\rho _n(t)) \\subset B(0,1)$ for all $t\\ge 0$ .",
"From Proposition REF $v_n \\in \\mathcal {V}$ and from Corollary REF $ &\\rho _{n+1}(t) = m_{n+1}(t) \\delta + g_{n+1}(t) \\\\&m_{n+1}(t)=m_0+\\int _ {(\\sigma _n^t)^{-1}(\\lbrace 0\\rbrace ) }g_0(x) \\; dx \\\\& g_{n+1}(x,t)= \\left(\\frac{g_{0}}{\\text{det} \\, \\nabla \\sigma _n^t} \\circ (\\sigma _n^t)^{-1}\\right)(x) \\quad \\text{ for }x \\ne 0.$ Here $m_0$ and $g_0$ are such that $\\rho _{init}=m_0 \\delta + g_0$ .",
"Also from Proposition REF we know that $\\text{det} \\nabla \\sigma _n^t$ satisfies $ \\text{det} \\nabla \\sigma _n^t(x) = \\text{exp} {\\int _0^t (\\text{div } {v_n}) (\\sigma _n^s(x),s) ds }$ for all $(x,t)$ such that $\\sigma _n^t(x) \\ne 0$ .",
"Since we have assumed that $\\rho _n(t)$ is in ${\\cal P}_{RD}(\\mathbb {R}^d)$ with compact support, and since for $\\alpha \\in (2-d,2)$ $\\Delta K$ is nonnegative, radially symmetric decreasing and locally integrable, we know from Lemma REF that the function $x \\mapsto - \\text{div} v_n(x,t) = [\\Delta K * \\rho _n(t)](x)$ is nonnegative, radially symmetric and decreasing.",
"Since $|x|\\le |y|$ implies $|\\sigma _n^s(x)| \\le |\\sigma _n^s(y)|$ one can easily see from (REF ) that $x\\mapsto \\frac{1}{\\text{det} \\nabla \\sigma _n^t(x)} \\text{ is nonnegative, radially symmetric and decreasing.",
"}$ Then we easily see from () that, since $g_0$ is radially symmetric and decreasing, so is $x \\mapsto g_{n+1}(x,t)$ .",
"Let us now remark that the estimate $\\left|v_n(x,t)\\right|\\le \\left|x\\right|^{\\alpha -1}$ together with Lemma REF of the Appendix lead to the following: if $\\alpha \\in (2-d,1)$ then $ \\left|\\sigma _n^t(x)-\\sigma _n^s(x)\\right| \\le C_\\alpha \\left|t-s\\right|^{\\frac{1}{2-\\alpha }}$ for all $x\\in B(0,1)$ and for all $t,s \\ge 0$ .",
"Here $C_\\alpha := (2-\\alpha )^{\\frac{1}{2-\\alpha }}$ .",
"If $\\alpha \\in [1,2)$ then $v(x,t)\\le 1$ on $B(0,1) \\times [0,+\\infty )$ and therefore we get $ \\left|\\sigma _n^t(x)-\\sigma _n^s(x)\\right| \\le \\left|t-s\\right|$ for all $x\\in B(0,1)$ and for all $t,s \\ge 0$ .",
"Using Lemma REF from the Appendix, together with (REF ) we obtain that, if $\\alpha \\in (2-d,1)$ then $W_2(\\rho _{n+1}(t),\\rho _{n+1}(s))\\le \\Vert \\sigma _n^t-\\sigma _n^s\\Vert _{L^\\infty (B(0,1))}\\le C_\\alpha |t-s|^{\\frac{1}{2-\\alpha }}.$ This prove that $t \\mapsto \\rho _{n+1}(t)$ is Hölder continuous with respect to $W_2$ when $\\alpha \\in (2-d,1)$ .",
"If $\\alpha \\in [1,2)$ , we obtain from (REF ) that $t \\mapsto \\rho _{n+1}(t)$ is Lipschitz continuous with respect to $W_2$ .",
"Statement (ii) is a direct consequence of Corollary REF .",
"We now prove (iii).",
"Suppose $2-d< \\alpha <1$ .",
"Recall from Lemma REF that $\\phi $ is $\\gamma $ -Hölder continuous for some $\\gamma \\in (0,1]$ .",
"Choose $\\delta $ such that $\\left|t-s\\right|< \\delta $ implies $C_\\alpha \\left|t-s\\right|^{\\frac{1}{2-\\alpha }} \\le \\epsilon /2$ .",
"Using Lemma REF and estimate (REF ) we obtain that $\\left|t-s\\right|< \\delta $ and $x \\in A_\\epsilon $ implies that $\\left|v_n(x,t)-v_n(x,s)\\right| &= \\left|x\\right|^{\\alpha -1} \\int _0^{+\\infty } \\left| \\phi \\left(\\frac{\\sigma ^t_{n-1}(r)}{\\left|x\\right|}\\right)-\\phi \\left(\\frac{\\sigma ^s_{n-1}(r)}{\\left|x\\right|}\\right)\\right| d \\hat{\\rho }_{init}(r) \\\\& \\le \\left|x\\right|^{\\alpha -1-\\gamma } \\int _0^{+\\infty } c\\left| \\sigma ^t_{n-1}(r)-\\sigma ^s_{n-1}(r)\\right|^\\gamma d \\hat{\\rho }_{init}(r) \\\\& \\le c \\;C_\\alpha \\left|x\\right|^{\\alpha -1-\\gamma } \\left|t-s\\right|^{\\frac{\\gamma }{2-\\alpha }} .$ The first equality is a simple consequence of formula (REF ), the fact that $\\rho _n(t)= \\sigma ^t_{n-1}\\# \\rho _{init}$ and the definition of the push forward.",
"Note that $C_\\alpha \\left|t-s\\right|^{\\frac{1}{2-\\alpha }} \\le \\epsilon /2$ and (REF ) imply that $\\left| \\frac{\\sigma ^t_{n-1}(r)}{\\left|x\\right|}-\\frac{\\sigma ^s_{n-1}(r)}{\\left|x\\right|}\\right| \\le 1/2$ for $x \\in A_\\epsilon $ .",
"This allowed us to use Lemma REF in order to go from (REF ) to ().",
"The case $\\alpha \\in [1,2)$ is dealt with similarly.",
"We finally prove (iv).",
"Obviously $\\rho _1(t) \\succ \\rho _0(t)\\equiv \\rho _{init}$ for all $t\\ge 0$ .",
"Assume that for a given $n$ , $\\rho _n(t) \\succ \\rho _{n-1}(t)$ for all $t \\ge 0$ .",
"Then (REF ) implies $|v_n(x,t)|\\ge |v_{n-1}(x,t)|$ .",
"Lemma REF , which is proven in the next section, implies then that $\\rho _{n+1}(t) \\succ \\rho _{n}(t)$ for all $t \\ge 0$ .",
"$\\Box $ As already mentioned, (ii) (iii) and (iv) imply that the sequence $\\lbrace v_n\\rbrace $ converges uniformly on $A_\\epsilon \\times [0,+\\infty )$ to some function $v$ (which is Lipschitz continuous in space away from the origin).",
"Setting $v(0,t)=0$ we obtain a velocity field well defined on $B(0,1) \\times [0,+\\infty )$ .",
"This velocity field $v(x,t)$ generates a flow map $\\sigma ^t: B(0,1) \\rightarrow B(0,1)$ .",
"$\\sigma _n(x,t)$ converges uniformly to $\\sigma (x,t)$ on $B(0,1) \\times [0,+\\infty )$ .",
"Let $\\epsilon >0$ be fixed.",
"From formula (REF ) it is clear that $\\left|v_0\\right|$ is strictly positive away from the origin.",
"Since $\\left|v_{n+1}\\right| \\ge \\left|v_n\\right|$ we have that $\\left|v\\right|$ is also strictly positive away from the origin.",
"Therefore there exists a time $T_\\epsilon >0$ such that $\\sigma ^{T_\\epsilon }(B(0,1)) \\subset B(0,\\epsilon )$ .",
"Choose $N$ so that $n \\ge N$ implies $\\left\\Vert v-v_n\\right\\Vert _{L^\\infty (A_\\epsilon \\times [0,T_\\epsilon ])} \\le \\epsilon / (T_\\epsilon e^{L_\\epsilon T_\\epsilon })$ .",
"Case 1: Assume first that $(x,t) \\in B(0,1) \\times [0,+\\infty )$ is such that $\\left|\\sigma ^t(x)\\right| \\ge \\epsilon $ .",
"Note that such a $t$ is necessarily smaller than $T_\\epsilon $ .",
"For all $\\tau \\le t$ and for all $n \\ge 0$ we have $\\left|\\sigma ^\\tau (x)-\\sigma ^\\tau _n(x)\\right| &\\le \\int _0^\\tau \\left|v(\\sigma ^s(x),s)-v_n(\\sigma _n^s(x),s)\\right| ds\\\\& \\le \\int _0^\\tau \\left|v(\\sigma ^s(x),s)-v_n(\\sigma ^s(x),s)\\right| + \\left|v_n(\\sigma ^s(x),s)-v_n(\\sigma _n^s(x),s)\\right| ds\\\\& \\le \\tau \\left\\Vert v-v_n\\right\\Vert _{L^\\infty (A_\\epsilon \\times [0,\\tau ])} + L_\\epsilon \\int _0^\\tau \\left|\\sigma ^s(x) -\\sigma _n^s(x)\\right| ds.$ We have use the fact that $\\left|\\sigma ^t(x)\\right| \\ge \\epsilon $ implies that $\\left|\\sigma ^s(x)\\right| \\ge \\epsilon $ for all $s \\le \\tau \\le t$ .",
"We have also use the fact that, since $\\left|v\\right| \\ge \\left|v_n\\right|$ , $\\left|\\sigma _n^s(x)\\right| \\ge \\left|\\sigma ^s(x)\\right| \\ge \\epsilon $ for all $s \\le \\tau \\le t$ and for all $n\\ge 0$ .",
"Using Gronwall's lemma and the fact that $t \\le T_\\epsilon $ we obtain that for $n\\ge N$ : $ \\left|\\sigma ^t(x)-\\sigma ^t_n(x)\\right| \\; \\le \\;T_\\epsilon \\left\\Vert v-v_n\\right\\Vert _{L^\\infty ( A_\\epsilon \\times [0,T_\\epsilon ])} \\; e^{L_\\epsilon T_\\epsilon } \\le \\epsilon .$ Case 2: Assume that $(x,t) \\in B(0,1) \\times [0,+\\infty )$ is such that $|x| < \\epsilon $ .",
"Since the velocity fields $v$ and $v_n$ are focussing we clearly have that $\\left|\\sigma ^t(x)-\\sigma ^t_n(x)\\right| < 2 \\epsilon $ for all $n$ .",
"Case 3: Assume finally that $(x,t) \\in B(0,1) \\times [0,+\\infty )$ is such that $\\left|\\sigma ^t(x)\\right| < \\epsilon $ and $|x| \\ge \\epsilon $ .",
"Since $\\tau \\mapsto \\sigma ^{\\tau }(x)$ is continuous there exists a time $s \\in [0, t]$ such that $|\\sigma ^s(x)|= \\epsilon $ .",
"So from case 1 we get $\\left|\\sigma ^s(x)-\\sigma ^s_n(x)\\right|\\le \\epsilon $ for $n \\ge N$ .",
"Since $|\\sigma ^s(x)|= \\epsilon $ we have that $\\left|\\sigma ^s_n(x)\\right|\\le 2\\epsilon $ for $n \\ge N$ .",
"Since $s\\le t$ we have $\\left|\\sigma ^t_n(x)\\right| \\le \\left|\\sigma ^s_n(x)\\right|\\le 2\\epsilon $ for $n \\ge N$ .",
"Therefore $\\left|\\sigma ^t(x)-\\sigma ^t_n(x)\\right| < 3 \\epsilon $ for all $n \\ge N$ .",
"We are now ready to prove the Theorem : Proof of Theorem .",
"Define $\\rho (t):=\\sigma ^t \\# \\rho _{init}$ .",
"Recall that from Lemma REF of the Appendix $\\mathcal {W}_2(\\rho ,\\rho _n):= \\sup _{t\\in [0,\\infty )} W_2(\\rho (t),\\rho _n(t)) \\le \\left\\Vert \\sigma -\\sigma _n\\right\\Vert _{L^\\infty ( B_1 \\times [0,+\\infty ))}.$ So from Proposition REF we get that $\\mathcal {W}_2(\\rho _n-\\rho )\\rightarrow 0$ .",
"This implies in particular that for every $t \\in [0,+\\infty )$ , $\\rho _n(t)$ converges narrowly to $\\rho (t)$ .",
"Since narrow convergence preserves the monotonicity (Lemma REF of the Appendix), we know that $\\rho (t)$ is radially symmetric decreasing.",
"We are now going to prove that $\\rho $ and $\\sigma $ satisfy (REF ) and ().",
"Since $\\rho $ is defined by $\\rho (t)=\\sigma ^t \\# \\rho _{init}$ where $\\sigma ^t: \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ is the flow map associated to the velocity field $v(x,t)$ , we just need to prove that $v(x,t)= -(\\nabla K * \\rho (t))(x) $ for $x \\ne 0$ .",
"This is obtain by passing to the limit in the relation $v_n(x,t)= -(\\nabla K * \\rho _n(t))(x)$ for $x \\ne 0$ .",
"Indeed $v_n$ converges pointwise to $v$ in $A_0 \\times [0,+\\infty )$ .",
"And since for fixed $t$ , $\\rho _n(t)$ converges narrowly to $\\rho (t)$ , we obtain from (REF ) that $\\nabla K * \\rho _n$ converges pointwise to $\\nabla K * \\rho $ in $A_0 \\times [0,+\\infty )$ .",
"$\\Box $"
],
[
"Instantaneous mass concentration",
"The recent work of [12], [33] concerns local well-posedness of the problem with initial data in $L^p$ .",
"One can prove a sharp condition on $p$ for local well-posedness by considering a family of initial data that behave as a powerlaw near the origin.",
"Such initial conditions satisfy the monotonicity assumptions considered in this paper.",
"In this section, using existence results from the prior section and a bootstrap argument, we prove results about the behavior of these solutions as measure solutions that concentrate mass.",
"Such results are not discussed in the prior literature for the singular power law potential $K(x)=|x|^\\alpha $ , $\\alpha <1$ .",
"More specifically in [12] it was proven that the aggregation equation with potential $\\nabla K(x)= x\\left|x\\right|^{\\alpha -2}$ , $2-d< \\alpha <2$ , is locally well posed in any $L^p$ -space with $p>\\frac{d}{d + \\alpha -2}$ .",
"Note that given $\\beta \\in (\\frac{d+\\alpha -2}{d},1)$ the function $h(x)={\\left\\lbrace \\begin{array}{ll}\\frac{c}{|x|^{{d + \\alpha -2}}}\\; \\frac{1}{(-\\log \\left|x\\right|)^{\\beta }}& \\text{ if } \\left|x\\right| \\le 1\\\\0 & \\text{ otherwise}\\end{array}\\right.",
"}$ belongs to the critical space $L^{\\frac{d}{d + \\alpha -2}}(\\mathbb {R}^d)$ but does not belong to any $L^p$ space with $p>\\frac{d}{d + \\alpha -2}$ .",
"In [33] it was proved that if the initial data is exactly equal to $h(x)$ then a solution of the aggregation equation instantaneously leaves the space $L^{\\frac{d}{d + \\alpha -2}}$ .",
"In this section we go a little further and show that the solution not only leaves $L^{{{d} \\over {d+ \\alpha -2}}}$ but also instantaneously concentrates some point mass at the origin.",
"Our results make use of the existence theory from the previous section.",
"Also compared to the work in [12] and [33], our argument here is local in essence and holds for any radially symmetric decreasing initial data which is locally more singular than $h(x)$ at the origin.",
"The main theorem of the section is the following: Let $\\nabla K(x)=x \\left|x\\right|^{\\alpha -2}$ , $2-d< \\alpha <2$ .",
"Suppose $\\rho _{init} \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ is compactly supported and absolutely continuous with respect to the Lebesgue measure.",
"Suppose that there exists $c>0$ , $r_0>0$ and $\\beta \\in (\\frac{d+\\alpha -2}{d},1)$ such that the density $u_{init}$ of $\\rho _{init}$ satisfies $u_{init}(x) \\ge \\frac{c}{|x|^{d + \\alpha -2}}\\; \\frac{1}{(-\\log \\left|x\\right|)^{\\beta }} \\quad \\text{ for all } |x|<r_0.$ Suppose finally that $\\rho \\in C([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d))$ satisfies the Lagrangian formulation (REF )-() of the aggregation equation.",
"Then $\\rho (t)(\\lbrace 0\\rbrace )>0$ for all $t>0$ ."
],
[
"Comparison principles",
"In this subsection we derive a few comparison principles which will be necessary in order to make the arguments local.",
"Suppose $v_1,v_2 \\in \\mathcal {V}$ and $\\left|v_1\\right| \\ge \\left|v_2\\right|$ .",
"Then $\\sigma _1^t \\# \\mu \\succ \\sigma _2^t \\# \\mu \\qquad \\text{ for all } \\mu \\in {\\cal P}_R(\\mathbb {R}^d) \\text{ and } t \\ge 0$ where $\\sigma _1$ and $\\sigma _2$ are the flow maps associated to $v_1$ and $v_2$ respectively.",
"Since $v_2 \\in \\mathcal {V}$ the flow map $\\sigma _2^t$ is invertible away from the origin.",
"Define $\\Lambda ^t_2(x)=(\\sigma _2^t)^{-1}(x)$ if $x \\ne 0$ and $\\Lambda _2^t(0)=0$ .",
"One can then easily check that $(\\sigma _1^t \\circ \\Lambda _2^t) \\# (\\sigma _2^t\\# \\mu )= \\sigma _1^t\\# \\mu $ Moreover since $\\left|v_1\\right| \\ge \\left|v_2\\right|$ we have that $\\left|(\\sigma _1^t \\circ \\Lambda _2^t)(x)\\right| \\le \\left|x\\right|$ , which concludes the proof.",
"Suppose $v \\in \\mathcal {V}$ .",
"Suppose also that $\\mu , \\nu \\in {\\cal P}_R(\\mathbb {R}^d)$ and $\\mu \\succ \\nu $ .",
"Then $\\sigma ^t \\# \\mu \\succ \\sigma ^t \\# \\nu \\qquad \\text{ for all } t \\ge 0$ where $\\sigma $ is the flow maps associated to $v$ .",
"Since $\\mu \\succ \\nu $ there is a map $P$ satisfying $\\left|P(x)\\right| \\le \\left|x\\right|$ such that $\\mu =P \\#\\nu $ .",
"As in the previous lemma, define $\\Lambda ^t(x)=(\\sigma ^t)^{-1}(x)$ if $x \\ne 0$ and $\\Lambda ^t(0)=0$ .",
"One can then easily check that $(\\sigma ^t \\circ P \\circ \\Lambda ^t) \\# (\\sigma ^t \\# \\mu )=\\sigma ^t \\# \\nu $ and $\\left|(\\sigma ^t \\circ P \\circ \\Lambda ^t)(x)\\right| \\le \\left|x\\right|$ which conclude the proof.",
"The following definition will be needed in order to compare two measures of different mass.",
"Suppose $\\rho \\in {\\cal P}_R(\\mathbb {R}^d)$ and $\\mu \\in {\\cal M}_R(\\mathbb {R}^d)$ , with $\\mu (\\mathbb {R}^d) \\le 1$ .",
"We write $\\rho \\triangleright \\mu $ if there exists a measure $\\nu \\in {\\cal P}_R(\\mathbb {R}^d)$ such that $\\rho \\succ \\nu \\quad \\text{and} \\quad \\nu (A) \\ge \\mu (A) \\;\\; \\forall A \\in \\mathcal {B}(\\mathbb {R}^d).$ In view of (REF ) and (REF ) it is clear that: $ \\rho \\triangleright \\mu \\quad \\Longrightarrow \\quad \\left|\\nabla K * \\rho \\right| \\ge \\left|\\nabla K * \\mu \\right|$ The following Lemma will be useful in order to make localized comparisons.",
"Suppose $v_1, v_2 \\in \\mathcal {V}$ and $\\left|v_1\\right| \\ge \\left|v_2\\right|$ in $B(0,2R) \\times [0,+\\infty )$ .",
"Suppose also that $\\rho \\in {\\cal P}_R(\\mathbb {R}^d)$ , $\\mu \\in {\\cal M}_R(\\mathbb {R}^d)$ and $\\rho \\triangleright \\mu $ .",
"Then $\\sigma ^t_1 \\# \\rho \\;\\; \\triangleright \\; \\; \\sigma ^t_2 \\# (\\mu \\, \\chi _{B(0,R)}) \\qquad \\text{ for all } t \\ge 0, $ where $\\sigma _1$ and $\\sigma _2$ are the flow maps associated to $v_1$ and $v_2$ respectively, and $\\chi _{B(0,R)}$ is the indicator function of the set $B(0,R)$ .",
"Since $\\rho \\triangleright \\mu $ there exists a probability measure $\\nu $ such that $\\rho \\succ \\nu \\ge \\mu $ .",
"Let $\\xi (x)$ be a smooth radially symmetric function which satisfies $\\xi (x)=1$ if $\\left|x\\right| \\le R$ , $ \\xi (x)=0$ if $\\left|x\\right| \\ge 2R$ and $\\chi (x) \\le 1$ for all $x \\in \\mathbb {R}^d$ .",
"The velocity field $v_3(x,t):=v_2(x,t) \\xi (x)$ is still in $\\mathcal {V}$ .",
"Moreover we have $\\left|v_3\\right| \\le \\left|v_1\\right|$ for all $x \\in \\mathbb {R}^d$ and $t \\ge 0$ .",
"We can therefore use the two previous Lemmas to obtain that $\\sigma _1^t \\# \\rho \\succ \\sigma _3^t\\# \\rho \\succ \\sigma _3^t \\#\\nu \\ge \\sigma _3^t\\# \\mu \\ge \\sigma ^t_3 \\# (\\mu \\, \\chi _{B[0,R]})$ The last two inequalities are a simple consequence of the definition of the push-forward together with the fact that $\\nu \\ge \\mu \\ge \\mu \\, \\chi _{B[0,R]}$ .",
"Finally, note that since $v_3=v_2$ on $B(0,R) \\times [0,+\\infty )$ , then $\\sigma ^t_3 \\# (\\mu \\, \\chi _{B[0,R]})=\\sigma ^t_2 \\# (\\mu \\, \\chi _{B[0,R]})$ ."
],
[
"Proof of Theorem ",
"Fix $\\alpha \\in (2-d, 2)$ and define the functions $&f_{\\epsilon ,r_0}(x) = \\frac{1}{\\left|x\\right|^{d+\\alpha -2 +\\epsilon }} \\;\\; \\chi _{B(0,r_0)}(x) \\qquad \\text{for }\\epsilon \\in (0,1)\\\\&g_{r_0}(x) = \\frac{1}{\\left|x\\right|^{d + \\alpha -2}} \\;\\; \\chi _{B(0,r_0)}(x) \\\\&h_{\\beta , r_0}(x)= \\frac{1}{\\left|x\\right|^{d + \\alpha -2}}\\frac{1}{(- \\ln \\left|x\\right|)^{\\beta }} \\;\\;\\chi _{B(0,r_0)}(x) \\qquad \\text{for }\\beta \\in (\\frac{d+\\alpha -2}{d},1).$ Note that at the origin $f_{\\epsilon ,r_0}$ is more singular than $g_{r_0}$ which itself is more singular than $h_{\\beta , r_0}$ .",
"In [12] it was proved that if $\\alpha =1$ and the initial data is exactly equal to $Cf_{\\epsilon ,r_0}(x)$ ($C$ is a normalizing constant) then a Dirac delta function appears instantaneously in the solution.",
"The proof relied on the fact that solutions of the ODE $\\dot{x}=-(\\nabla K * f_{\\epsilon ,r_0})(x)$ reach the origin in finite time.",
"However this strategy does not work with $g_{r_0}$ and $f_{\\epsilon ,r_0}$ , because solutions of $\\dot{x}=-(\\nabla K * g_{r_0})(x)$ and $\\dot{x}=-(\\nabla K * h_{\\beta ,r_0})(x)$ do not reach the origin in finite time.",
"For that reason we will use a bootstrap argument to prove that a delta function appears instantaneously when the initial data is equal to or more singular than $h_{\\beta ,r_0} \\in L^{\\frac{d}{d + \\alpha -2}}(\\mathbb {R}^d)$ .",
"Roughly speaking, we will show that the velocity field $-\\nabla K * h_{\\beta ,r_0}$ instantaneously deforms $h_{\\beta ,r_0}$ into a function more singular than $g_{r_0}$ , then we will show that the velocity field $-\\nabla K * g_{r_0}$ instantaneously deforms $g_{r_0}$ into a function more singular than $f_{\\epsilon ,r_0}$ , and finally we will use the argument from [12] to show that the velocity field $-\\nabla K * f_{\\epsilon ,r_0}$ deforms $f_{\\epsilon ,r_0}$ in such a way that a delta function appears instantly.",
"The following definition is consistent with Definition REF : Given a radially symmetric, non-negative function $u\\in L^1(\\mathbb {R}^d)$ , we define $\\hat{u} \\in L^1((0,+\\infty ))$ to be the unique function satisfying $ \\int _{r_1}^{r_2} \\hat{u}(r) dr=\\int _{r_1<\\left|x\\right|<r_2} u(x) dx \\qquad \\text{for all } r_1,r_2 \\ge 0.",
"$ In other words, $\\hat{u}(r)=u(r) \\omega _d r^{d-1}.$ With this notation we have: $ &\\hat{f}_{\\epsilon ,r_0}(r)= \\omega _d\\;\\; \\frac{1}{r^{\\alpha -1 + \\epsilon }}\\;\\; \\chi _{[0,r_0]}(r) \\\\ &\\hat{g}_{r_0}(r) = \\omega _d \\;\\; \\frac{1}{r^{\\alpha -1}} \\chi _{[0,r_0]}(r) \\\\&\\hat{h}_{\\beta ,r_0}(r)= \\omega _d \\;\\;\\frac{1}{r^{\\alpha -1}} \\frac{1}{(- \\ln r)^{\\beta }} \\;\\; \\chi _{[0,r_0]}(r).$ We remind the reader that by Lagrangian solution we mean a function $\\rho (x,t)$ that satisfies the Lagrangian formulation (REF )-() of the aggregation equation.",
"Let $\\rho \\in C([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d))$ be a Lagrangian solution of the aggregation equation with compactly supported initial data $\\rho _{init}$ and potential $K$ satisfying $\\nabla K(x)=x \\left|x\\right|^{\\alpha -2}$ , $2-d < \\alpha <2$ .",
"Let $v(x,t)= -(\\nabla K * \\rho (t))(x)$ be the associated velocity field.",
"If $\\rho _{init} \\triangleright c f_{\\epsilon ,r_0} $ for some $c,r_0>0$ and $\\epsilon \\in (0,1)$ , then there exist $R,C>0$ such that $\\left|v(x,t)\\right| \\ge C \\left|x\\right|^{1-\\epsilon } \\qquad \\text{for all } (x,t)\\in B(0,R)\\times [0+\\infty ).$ If $\\rho _{init} \\triangleright c\\; g_{r_0} $ for some $c,r_0>0$ , then there exist $R,C>0$ such that $\\left|v(x,t)\\right| \\ge C \\; \\left|x\\right| (-\\ln \\left|x\\right| ) \\qquad \\text{for all } (x,t)\\in B(0,R)\\times [0+\\infty ).$ If $\\rho _{init} \\triangleright c h_{\\beta ,r_0}$ for some $c,r_0>0$ and $\\beta \\in (\\frac{d + \\alpha -2}{d},1)$ , then there exist $R,C>0$ such that $\\left|v(x,t)\\right| \\ge C \\; \\left|x\\right| (-\\ln \\left|x\\right| )^{1-\\beta } \\qquad \\text{for all } (x,t)\\in B(0,R)\\times [0+\\infty ).$ Let us prove (i).",
"On one hand, from (REF ) we see that $\\left|v(x,0)\\right| \\ge c \\left|(\\nabla K* f_{\\epsilon ,r_0})(x)\\right|$ for all $x\\in \\mathbb {R}^d$ .",
"On the other hand, since the velocity field is always pointing inward (this is due to the positivity of $\\phi $ ), we have that $\\rho (t) \\succ \\rho (0)$ for all $t \\ge 0$ , and therefore from (REF ) we get that $\\left|v(x,t)\\right| \\ge \\left|v(x,0)\\right|$ for all $x\\in \\mathbb {R}^d$ and $t\\ge 0$ .",
"So we only need to show that $\\left|(\\nabla K* f_{\\epsilon ,r_0})(x)\\right| \\ge C \\left|x\\right|^{1-\\epsilon }$ in some neighborhood of the origin, and this estimate follows easily from Lemma REF .",
"Indeed, by Lemma REF we have for $|x|\\le r_0$ $\\left|\\nabla K * f_{\\epsilon , r_0} (x)\\right| &= \\omega _d\\int _0^{|x|} \\phi \\left(\\frac{r}{\\left|x\\right|}\\right) \\left(\\frac{\\left|x\\right|}{r}\\right)^{\\alpha -1} r^{- \\epsilon } dr \\\\ & \\qquad \\qquad \\qquad + \\omega _d |x| \\int _{|x|}^{r_0}\\phi \\left(\\frac{r}{\\left|x\\right|}\\right) \\left(\\frac{r}{\\left|x\\right|}\\right)^{2-\\alpha } r^{-1-\\epsilon } dr \\\\&\\ge \\omega _d C_1\\int _0^{|x|}r^{- \\epsilon } dr + \\omega _d C_2 |x| \\int _{|x|}^{r_0}r^{-1-\\epsilon } dr \\\\& \\ge \\omega _d C_1 \\frac{|x|^{1-\\epsilon }}{1-\\epsilon }$ where $C_1= \\inf _{[0,1]}\\phi =\\phi (1)$ and $C_2= \\inf _{(1,+\\infty )} \\phi (r)r^{2-\\alpha }>0$ .",
"Let us now prove (ii).",
"Reasoning as above we see that it is enough to show that $\\left|(\\nabla K* g_{r_0})(x)\\right| \\ge C \\; \\left|x\\right| (-\\ln \\left|x\\right| ) $ in some neighborhood of the origin.",
"Then the argument is similar.",
"From (REF ) with $\\epsilon =0$ we get $\\left|\\nabla K * g_{r_0} (x)\\right| \\ge \\omega _d C_2 \\left|x\\right| \\ln \\left(\\frac{r_0}{\\left|x\\right|}\\right)$ which yields to the desired estimate.",
"To prove (iii) it is enough to show $\\left|(\\nabla K* h_{\\beta ,r_0})(x)\\right|\\ge C \\; \\left|x\\right| (-\\ln \\left|x\\right| )^{1-\\beta }$ in some neighborhood of the origin, and the argument is similar.",
"In this case we have $|\\nabla K * u_0(x)| \\ge \\omega _d C_2|x| \\int _{|x|}^{r_0} \\frac{1}{|\\log r|^{\\beta }}{\\frac{dr}{r}},$ which yields to the desired estimate.",
"This last estimate was derived independently in [33].",
"The ODE's $\\dot{r}=-C r^{1-\\epsilon }, \\quad \\dot{r}=-C r (-\\ln r ) \\quad \\text{ and } \\quad \\dot{r}=-C r (-\\ln r )^{1-\\beta }$ suggested by the previous proposition have explicit solutions and their flow maps are respectively: $&\\sigma _1^t(r) = {\\sigma _1}(r,t)={\\left\\lbrace \\begin{array}{ll}(r^\\epsilon -\\epsilon Ct)^{1/\\epsilon }& \\text{ if } r>(\\epsilon Ct)^{1/\\epsilon }\\\\0 &\\text{ if } r\\le (\\epsilon Ct)^{1/\\epsilon }\\end{array}\\right.}",
"\\\\&\\sigma _2^t(r) = {\\sigma _2}(r,t)=r^{e^{Ct}} \\\\&\\sigma _3^t(r) = {\\sigma _3}(r,t)=e^{-\\left(C\\beta t +\\left(\\ln {\\frac{1}{r}}\\right)^\\beta \\right)^{1/\\beta }} $ Solutions of the first ODE reach the origin in finite time but solutions of the other two ODE's only approach the origin as $t \\rightarrow \\infty $ .",
"Corresponding to the flow maps $\\sigma _i: [0,+\\infty ) \\times [0,+\\infty ) \\rightarrow [0,+\\infty )$ there are flow maps $S_i: \\mathbb {R}^d \\times [0,+\\infty ) \\rightarrow \\mathbb {R}^d$ defined by $S_i(x,t)=\\sigma _i(\\left|x\\right|,t)\\frac{x}{\\left|x\\right|}$ .",
"The $S_i$ are the flow maps associated to the velocity fields $w_1(x)=-C \\left|x\\right|^{1-\\epsilon } \\frac{x}{\\left|x\\right|}$ , $w_2(x)=-C \\left|x\\right| (-\\ln \\left|x\\right| ) \\frac{x}{\\left|x\\right|}$ , and $w_3(x)=-C \\left|x\\right| (-\\ln \\left|x\\right| )^{1-\\beta }\\frac{x}{\\left|x\\right|}$ .",
"Let $u\\in L^1(\\mathbb {R}^d)$ be a radially symmetric, non-negative function.",
"It is clear from (REF ) that $S^t_1 \\# u$ has a point mass at the origin if $u$ has non-zero mass in $B(0,(\\epsilon Ct)^{1/\\epsilon })$ .",
"On the other hand, because $S^t_2$ and $S^t_3$ are smooth invertible maps, $S^t_2 \\# u$ and $S^t_3 \\# u$ are continuous with respect to the Lebesgue measure, and by the change of variable formula, we have $ (S_i^t \\# u)\\hat{}\\;(r) =(\\sigma _i^t \\# \\hat{u})(r) = \\hat{u}(\\tau _i^t(r)) \\;\\; \\frac{\\partial \\tau _i^t (r)}{\\partial r} \\qquad i=2,3 \\\\\\text{ where }\\tau _i^t(r)= (\\sigma _i^t)^{-1}(r)$ [Bootstrap] Let $\\rho \\in C([0,+\\infty ), {\\cal P}_{RD}(\\mathbb {R}^d))$ be a Lagrangian solution of the aggregation equation with compactly supported initial data $\\rho _{init}$ and potential $K$ satisfying $\\nabla K(x)=x \\left|x\\right|^{\\alpha -2}$ , $2-d < \\alpha <2$ .",
"If $\\rho _{init} \\triangleright c f_{\\epsilon , r_0} $ for some $c,r_0>0$ and $\\epsilon \\in (0,1)$ , then $\\rho (t)(\\lbrace 0\\rbrace )>0$ for all $t >0$ .",
"If $\\rho _{init} \\triangleright c\\; g_{r_0} $ for some $c,r_0>0$ , then for any $t>0$ there exist constants $c_1,r_1>0$ and $\\epsilon \\in (0,1)$ , such that $\\rho (t) \\triangleright c_1\\, f_{\\epsilon ,r_1} $ .",
"If $\\rho _{init} \\triangleright c \\;h_{\\beta ,r_0}$ , for some $c,r_0>0$ and $\\beta \\in (\\frac{d +\\alpha -2}{d},1)$ , then for any $t>0$ there exists constants $c_1,r_1>0$ such that $\\rho (t) \\triangleright c_1\\; g_{r_1} $ .",
"Let us prove (i).",
"Let $S_1^t(x)=S_1(x,t)$ be the flow map generated by the velocity field $w_1(x)=-C \\left|x\\right|^{1-\\epsilon } \\frac{x}{\\left|x\\right|}$ suggested by Proposition REF .",
"From Lemma REF and Proposition REF we then obtain that $ \\rho (t) \\triangleright S_1^t \\# c f_{\\epsilon , r_1} $ for $r_1$ small enough and for all $t \\ge 0$ .",
"Let us fix a $t>0$ .",
"Since $f_{\\epsilon , r_1}$ has non-zero mass in $B(0,(\\epsilon Ct)^{1/\\epsilon })$ , it is clear from (REF ) that the measure $S_1^t \\# c f_{\\epsilon , r_1}$ has a point mass at the origin.",
"Then by (REF ) we conclude that $\\rho (t)$ also has a point mass at the origin.",
"Let us now prove (ii).",
"Again Lemma REF and Proposition REF imply that $\\rho (t) \\triangleright S_2^t \\# c g_{ r_1}$ for $r_1$ small enough.",
"As already mentioned $S_2^t \\# c g_{ r_1}$ is continuous with respect to Lebesgue measure.",
"We are going to show that given any $t>0$ , $S_2^t \\# g_{ r_1} \\ge c_2 f_{r_2,\\epsilon }$ for some constant $c_2,r_2>0$ and $\\epsilon \\in (0,1)$ which will conclude the proof of (ii).",
"Let $\\tau _2^t(r)= (\\sigma _2^t)^{-1}(r)=r^{e^{-ct}}$ where $\\sigma _2^t(r)$ is defined by ().",
"Using the change of variable formula, we get that $(\\sigma _2 \\# \\hat{g}_{r_1})(r)&=\\hat{g}_{r_1}(\\tau _2^t(r)) \\;\\; \\frac{\\partial \\tau _2^t}{\\partial r} (r)\\\\&= \\frac{\\omega _d}{\\left(r^{e^{-ct}}\\right)^{\\alpha -1}} e^{-ct} r^{e^{-ct}-1} \\quad \\text{for $r$ small enough}\\\\&= \\frac{\\omega _d}{r^{\\alpha -1 + (2-\\alpha )(1-e^{-ct})}} e^{-ct}$ Since $2-\\alpha >0$ it is clear that $(\\sigma _2 \\# \\hat{g}_{r_1})(r) \\ge c_2 \\hat{f}_{r_2,\\epsilon }(r)$ for $r$ small enough.",
"Let us now prove (iii).",
"Once more Lemma REF and Proposition REF imply that $\\rho (t) \\; \\triangleright \\; S_3^t \\# c h_{\\beta , r_1}$ for $r_1$ small enough.",
"Let us fix $t>0$ and show that $S_3^t \\# c h_{\\beta , r_1} \\ge c_2 g_{r_2}$ for $r_2$ small enough.",
"In view of () it is enough to prove that $ \\lim _{r \\rightarrow 0} r^{\\alpha -1} \\left(\\sigma _3^t \\# \\hat{h}_{\\beta , r_1}(r)\\right)>0.$ Let $\\tau _3^t(r)=(\\sigma _3^t)^{-1}(r)$ and note that $ \\ln \\frac{1}{\\tau _3^t(r)}= \\left(-c \\beta t+\\left(\\ln \\frac{1}{r}\\right)^\\beta \\right)^{1/\\beta }$ From now on we drop the lower subscript.",
"From the change of variable formula we have $\\sigma ^t \\# \\hat{h}(r)&= \\hat{h}(\\tau ^t(r)) \\; \\frac{\\partial \\tau ^t}{\\partial r} (r)\\\\&= \\frac{\\tau ^t(r)^{1-\\alpha }}{\\left(\\ln \\frac{1}{\\tau ^t(r)}\\right)^\\beta } \\frac{\\partial \\tau ^t}{\\partial r} (r) \\\\&= \\frac{\\tau ^t(r)^{2-\\alpha }}{-c \\beta t+\\left(\\ln \\frac{1}{r}\\right)^\\beta } \\frac{ \\frac{\\partial \\tau ^t}{\\partial r} (r)}{\\tau ^t(r)} $ where we have used (REF ) to go from (REF ) to ().",
"Then note that using (REF ) again we get $\\frac{ \\frac{\\partial \\tau ^t}{\\partial r} (r)}{\\tau ^t(r)}=- \\frac{\\partial }{\\partial r} \\ln \\left(\\frac{1}{\\tau ^t(r)}\\right)=\\left(-c \\beta t + \\left(\\ln \\frac{1}{r}\\right)^\\beta \\right)^{\\frac{1}{\\beta }-1} \\left(\\ln \\frac{1}{r}\\right)^{\\beta -1} \\frac{1}{r}$ which combined with () gives $r^{\\alpha -1} \\left(\\sigma ^t\\# \\hat{h}(r)\\right)&= \\left(\\frac{\\tau ^t(r)}{r}\\right)^{2-\\alpha } \\left(-c \\beta t + \\left(\\ln \\frac{1}{r}\\right)^\\beta \\right)^{\\frac{1}{\\beta }-2} \\left(\\ln \\frac{1}{r}\\right)^{\\beta -1}\\\\&= \\left(\\frac{\\tau ^t(r)}{r}\\right)^{2-\\alpha } \\left(1- \\frac{c \\beta t}{ \\left(\\ln \\frac{1}{r}\\right)^\\beta }\\right)^{\\frac{1}{\\beta }-2} \\left(\\ln \\frac{1}{r}\\right)^{-\\beta }\\\\& \\ge \\frac{1}{2}\\left(\\frac{\\tau ^t(r)}{r}\\right)^{2-\\alpha } \\left(\\ln \\frac{1}{r}\\right)^{-\\beta } \\quad \\text{for $r$ small enough} $ Using (REF ) and doing a Taylor expansion we find that $\\ln \\left(\\frac{\\tau ^t(r)}{r}\\right)&=\\ln \\left(\\frac{1}{r}\\right) - \\ln \\left(\\frac{1}{r}\\right) \\left(1- \\frac{c \\beta t}{ \\left(\\ln \\frac{1}{r}\\right)^\\beta }\\right)\\\\&= ct \\left(\\ln \\frac{1}{r}\\right)^{1-\\beta } \\left( 1 + o\\left(\\frac{c \\beta t}{ \\left(\\ln \\frac{1}{r}\\right)^\\beta }\\right)\\right) \\\\& \\ge \\frac{1}{2}ct \\left(\\ln \\frac{1}{r}\\right)^{1-\\beta } \\quad \\text{for $r$ small enough} $ Combining (REF ) and (REF ) we get $\\ln \\left(r^{\\alpha -1} \\left(\\sigma ^t \\# \\hat{h}(r)\\right)\\right) \\ge \\ln (1/2) + \\frac{1}{2}(2-\\alpha )ct \\left(\\ln \\frac{1}{r}\\right)^{1-\\beta } - \\beta \\ln \\ln \\frac{1}{r}$ for $r$ small enough.",
"Since $2-\\alpha >0$ it is clear that $\\lim _{r \\rightarrow \\infty }\\ln \\left(r^{\\alpha -1} \\left(\\sigma ^t \\# \\hat{h}(r)\\right)\\right)=+\\infty $ which implies (REF ).",
"We now prove Theorem : Proof of Theorem .",
"If $\\rho _{init} \\triangleright c \\;h_{\\beta ,r_0}$ for some for some $c,r_0>0$ and $\\beta \\in (\\frac{d + \\alpha -2}{d},1)$ , we can apply the previous proposition to get that for any $t_1>0$ , $\\rho (t_1) \\triangleright c\\; g_{r_0}$ for some different constants $c,r_0>0$ .",
"Applying the proposition again, we get that for any $t_2>t_1$ , $\\rho (t_2) \\triangleright c\\; f_{\\epsilon ,r_0}$ for some other constants $c,r_0>0$ and for some $\\epsilon \\in (0,1)$ .",
"Applying the proposition one last time we get that for any $t_3>t_2$ , $\\rho (t_3)(\\lbrace 0\\rbrace )>0$ .",
"Since $t_1<t_2< t_3$ can be chosen arbitrarily small, this conclude the proof.",
"$\\Box $"
],
[
"Newtonian potential case",
"An even more singular case is that of the Newtonian potential, $\\alpha = 2-d$ in general, with $K(x) = \\log |x|$ in the special case of 2D.",
"Without loss of generality we use the normalization for $K$ that yields $\\Delta (K*\\rho )= \\rho $ , i.e.",
"the fundamental solution of the Poisson equation.",
"This simple fact localizes the dynamics as compared to the nonlocal case studied in previous sections.",
"In Eulerian coordinates, for smooth densities, we have $\\rho _t + v\\cdot \\nabla \\rho = \\rho ^2.",
"$ Recall that for radially symmetric problems, the Laplace operator is $\\Delta f = \\frac{1}{r^{d-1}} \\frac{\\partial }{\\partial r}\\bigl ( r^{d-1} \\frac{\\partial f}{\\partial r}\\bigr ).$ Likewise we have the following formulae for the gradient and divergence operators: $\\nabla f = \\frac{\\partial f}{\\partial r} \\vec{r},$ where $\\vec{r}$ is the unit outward pointing radial vector and $div\\ v = \\frac{1}{r^{d-1}} \\frac{\\partial }{\\partial r} r^{d-1} v.$ Using the latter formula we can rewrite $v$ above in terms of $\\rho $ simply by inverting $div v = -\\rho $ : $v(r) = -\\frac{1}{r^{d-1}} \\int _0^r s^{d-1} \\rho (s) ds : = - \\frac{m(r)}{r^{d-1}},$ where $m(r)$ is proportional to the mass contained inside a ball of radius $r$ .",
"Thus it makes sense to rewrite the evolution equation (REF ) in mass coordinates - in general regardless of the kernel it is $m_t + vm_r = 0.$ However this greatly simplifies in the special Newtonian case.",
"Formula (REF ) gives $m_t - \\frac{m m_r}{r^{d-1}} = 0.",
"$ By changing variables to $z$ coordinates, where $z=\\frac{r^d}{d}$ , we have the inviscid Burgers equation, $m_t -m m_z=0.$ The transformation to equation (REF ) is well-known; the transformation to the $z$ variable appeared in [14] in the context of a viscous version of our problem arising in astrophysics.",
"Here we use the classical conservation law theory for the inviscid (purely transport) problem to prove that monotonicity is preserved by the flow in all dimensions.",
"In one dimension it is known that $K=|x|$ can be transformed to the inviscid Burgers problem see e.g.",
"[18].",
"The connection to Burgers equation allows us to prove quite a lot about radially symmetric solutions of the aggregation equation with Newtonian potential, by directly connecting to the classical theory of conservation laws.",
"We consider three cases: (a) monotone decreasing radial densities for which we have a unique forward time solution; (b) general radial densities for which we have existence of solutions but uniqueness requires the specification of a jump condition (akin to choosing a particular entropy-flux pair for the definition of distribution solution, and; (c) the case of radially symmetric signed measures for which one requires an additional entropy condition in order to have a unique solution.",
"All of these cases can be distinguished by the known properties of the inviscid Burgers equation [4], [49], [37]."
],
[
"Case 1: $\\rho \\in {\\cal P}_{RD} (\\mathbb {R}^d)$ - existence of unique classical solutions",
"In the case of radially symmetric monotone decreasing probability measures, we have unique classical solutions by virtue of the fact that the corresponding flow field $v$ is Lipschitz for $r>0$ .",
"Monotonicity is preserved by virtue of the localization of the equations as described above.",
"More specifically, we have the heuristic that $\\rho $ satisfies $\\rho _t = \\rho ^2$ along characteristics so the initial ordering of the density is preserved provided that the characteristics remain well ordered and are well defined.",
"We can prove this to be the case by going to the mass coordinate formulation above.",
"The condition that the characteristics remain well defined is akin to proving that shocks will not form from any initial data satisfying the monotonicity condition.",
"If a shock forms - which we define as a singularity in $m_z$ in the mass equation (REF ), the first time of formation will occur at $t_{shock} = 1/ \\sup _z \\lbrace m_{init}^{\\prime }(z)\\rbrace $ .",
"So we need the characteristic to reach the origin before this time occurs.",
"Denote by $z_s$ the location at time zero of this characteristic.",
"Then our condition on the shock occurring after the characteristic crosses zero is $\\frac{z_s}{m_{init}(z_s)} < \\frac{1}{m_{init}^{\\prime }(z_s)}\\iff m_{init}(z_s)>z_s m_{init}^{\\prime }(z_s)$ since both $m_{init}$ and $m_{init}^{\\prime }$ are nonnegative.",
"Using the definition of the mass $m$ and converting back to regular radial coordinates, the above is equivalent to the following condition on the density $\\rho $ : $ \\int _{B(0,R)} \\rho _{init}(R) dx \\le \\int _{B(0,R)} \\rho _{init} (x) dx$ for all $R$ , which is true for monotone decreasing initial data $\\rho _{init}$ .",
"The special case of the equals sign in (REF ) corresponds to the shock happening right when the characteristic reaches the origin.",
"There are exact solutions that satisfy this - corresponding to a density that is the characteristic function of a collapsing ball.",
"The corresponding solution in $(m,z)$ coordinates is the well-known Burgers solution of the form $-z/(1-t)$ that forms a shock in finite time in which all of the characteristics on an interval collapse at the origin simultaneously.",
"This example is the most singular case of the general class of solutions considered in this subsection.",
"Since the only shocks that form occur at the origin, which is a boundary of the domain, this results in a global-in-time classical solution of (REF ) for any initial condition $m_{init}(z)$ arising from a probability density $\\rho _{init} \\in {\\cal P}_{RD}(\\mathbb {R}^d)$ .",
"The classical solution of the inviscid Burgers equation easily gives us a unique solution of the Lagrangian formulation of the problem as well.",
"We state these results below: Given compactly supported initial data $\\rho _{init}\\in {\\cal P}_{RD} (\\mathbb {R}^d)$ , define $m_{init} = \\int _0^r s^{d-1} d\\rho $ .",
"Then there exists a unique classical solution to equation (REF ) on the half space $(x,t) \\in (0,\\infty )\\times [0,\\infty )$ and a corresponding unique solution of the Lagrangian mapping formulation of the density transport problem.",
"The solution retains its monotonicity property for all time.",
"As we show below the situation is much more complicated for general radially solutions without the monotonicity condition.",
"Some observations can be immediately made using classical results from conservation laws.",
"Moreover these results connect directly to related problems in fluid dynamics such as vortex sheet solutions of the 2D Euler equations.",
"The next two subsections provide a discussion of these observations."
],
[
"Case 2: $\\rho \\in {\\cal P}_{R} (\\mathbb {R}^d)$ - existence of unique solutions with jump condition",
"In the case of general radially symmetric probability densities, we no longer have classical solutions.",
"Let us consider the simplest example of data that violates the monotonicity condition - that of a uniform delta-concentration on the boundary of the ball of radius $R_*$ .",
"Following the mass coordinates, we see that this example has a jump discontinuity in $m(z)$ at $z^*= (R^*)^d/d$ .",
"Since $m$ is the characteristic speed, this results in a jump in the velocity across the delta-ring.",
"One way to define the solution is to consider a distribution solution of (REF ) in which case the speed of the shock (velocity of the delta-ring) is defined, in $z$ coordinates as $s_{z-shock} = (m_1+m_2)/2$ , i.e.",
"the Rankine-Hugoniot condition associated with equation (REF ).",
"As is well-known for scalar conservation laws, we could transform equation (REF ) by multiplying by any function of $m$ , $(F(m))_t -(G(m))_z=0, \\quad F^{\\prime }(m) = f(m), \\quad G^{\\prime }(m) = m f(m)$ for some function $f$ , yieldling a different jump condition in the weak-distribution form of (REF ), $s_{z-shock} = \\frac{[F(m)]}{[G(m)]}, $ where $[\\phantom{F}]$ denotes the jump across the shock.",
"By virtue of well-known results for scalar conservation laws, we obtain families of weak solutions for the general radially symmetric problem.",
"For a given formulation of the form (REF ) there exists a unique distribution solution.",
"Uniqueness for inviscid Burgers often requires an additional entropy condition.",
"In the case of formulation (REF ), the entropy condition is automatically satisfied by the monotonicity of $m$ , which is guaranteed for any radial probability density, not necessarily monotone.",
"The full entropy condition would only be required in the case of non-monotone $m$ such as would arise in the case of a signed measure $\\rho $ .",
"It would be interesting to know whether there is an optimal choice of shock speeds for these under-determined problems.",
"For example one might also consider an optimal transport framework in which the best choice of shock speed would be one in which the interaction energy is most quickly dissipated.",
"For the aggregation problem this would result in the fastest speed possible for the delta-ring which would satisfy a Lagrangian formulation of the problem but perhaps not a classical distribution solution in Eulerian variables - even in the $m-z$ framework described above.",
"We note that the entropy solution discussed above, in which the speed of the shock is chosen to be the average of the speeds on either side, is a natural generalization of the choice conventionally made for 2D vortex sheets, in which $v=\\nabla ^{\\perp } K*\\rho $ rather than $v=\\nabla K*\\rho $ , and $\\rho $ is the vorticity.",
"For that problem one ascribes a velocity to the sheet that is the arithmetic average of the speeds on either side [57].",
"The frozen time calculation can be made by analogy to the incompressible flow problem, however the ensuing dynamics is quite different.",
"For the vortex sheet problem the flow is tangential to the sheet so the issue of shocks does not arise.",
"For the aggregation problem the flow is normal to the sheet and affects the solution on either side of it, because the speed of the shock determines the rate at which characteristics on either side of the discontinuity are absorbed and the rate at which information is lost in the discontinuity.",
"To summarize, if we define a solution as satisfying an equation of the form (REF ) in the sense of distributions, then we expect a unique solution, however, in the case of jump discontinuities in $m$ , the shock speed will depend on the choice of entropy-flux pair as discussed above.",
"Moreover we believe even more general examples may exist that could satisfy an optimality condition associated with dissipation of the interaction energy.",
"We finally briefly mention the case of signed measures below."
],
[
"Case 3: signed measures",
"The case of signed measures introduces yet another source of nonuniqueness of solutions, which we briefly discuss.",
"A signed measure corresponds to a non-montone (but $L^\\infty $ ) solution of the inviscid Burgers problem.",
"This general formulation introduces the need for something like an entropy condition to achieve unique distribution solutions.",
"For example, in the case of a negative delta-ring measure, we have a decreasing jump in $m$ which introduces the possibility of a rarefaction solution going forward in time.",
"In the classical weak solution formulation of Burgers equation, the entropy condition would select the rarefaction as the unique forward-time solution.",
"Nevertheless there exist other solutions, such as the outward-moving shock, that are bonafide distribution solutions, albeit ones that violate Lax's entropy condition whereby the speed of the shock should be faster than the characteristic speed ahead of it, and slower than the characteristic speed behind it."
],
[
"Conclusions",
"We have considered existence of radially symmetric, monotone decreasing solutions to the aggregation equation in the case of more singular potentials $|x|^\\alpha /\\alpha $ for $\\alpha $ in the range $2-d\\le \\alpha <1$ .",
"We remind the reader that the problem with $\\alpha \\ge 1$ is known to be globally well-posed for measure data including the case without radial symmetry and monotonicity [24].",
"For $2-d\\le \\alpha <2$ we find that monotonicity is preserved, a feature that is not true for $\\alpha > 2$ .",
"Our results provide a rigorous framework for monotonicity behavior observed in numerical simulations of finite time blowup [44], [45] for radially symmetric data.",
"The results also provide an understanding of the continuation of the solution after blowup.",
"That understanding includes the result that one obtains instantaneous mass concentration for certain classes of $L^1$ initial data including those observed as the asymptotic form of the blowup profile in numerical simulations [44], [45].",
"The special case of the Newtonian potential results in a localization of the problem, reducing to a form of the inviscid Burgers equation on the half line.",
"In particular for radially symmetric decreasing data, there is a unique classical solution of the Burgers problem for all time, resulting in a unique solution of the original density problem.",
"This solution also retains its monotonicity.",
"In contrast to the Newtonian potential, for the case $1>\\alpha >2-d$ the ensuing velocity field is at best Hölder continuous in time and our results are less precise.",
"For example, uniqueness of solutions is still an open problem in this range, as is existence in the case of non-monotone, radially symmetric data.",
"The existence problem is complicated by the fact that the velocity field is at best Hölder continuous which makes it difficult to get convergence estimates for the flow map - something we use to prove existence of solutions in the case of monotone data.",
"It is somewhat ironic that the more singular case of the Newtonian potential can be more easily solved - the velocity field is more singular, with a jump discontinuity.",
"However the localization of the dynamics results in a better understanding of the problem.",
"For the general nonlocal problem in the range $2-d< \\alpha <2$ , the monotonicity assumption allows for smoother estimates on the velocity field, namely Lipschitz estimates, which allow us to prove convergence of approximations and hence existence of a Lagrangian solution.",
"In addition to the above problems for data with symmetry, the general problem of measure solutions with non-radially symmetric data is wide open.",
"Some insights can be gained from recent work on special families of weak solutions.",
"In the case of the Newtonian potential there exists a class of `patch solutions' that are the time-dependent characteristic functions of of a domain in $\\mathbb {R}^d$ .",
"These solutions have recently been observed [11] to converge in finite time to a measure supported on a set of codimension one.",
"Other works considers the analogue of vortex sheets for general aggregation equations with potentials that include both attraction and repulsion [75], [68], [47]."
],
[
"Acknowledgments",
"We thank the referees for many helpful comments."
],
[
"Some general ODE results",
"In standard ODE textbooks such as [27], it is proven that the flow map associated to a velocity field which is continuously differentiable in both space and time is itself differentiable.",
"In our case of interest the velocity field is continuously differentiable in space but only continuous in time.",
"We show here that the hypothesis of continuous differentiability in time can be replaced by a weaker assumption that holds true in our case.",
"Only very minor modifications are needed compared to standard proofs found in ODE textbooks such as [27].",
"We will refer to [27] and we will indicate the necessary modifications to be made in the proof there.",
"Let $\\Omega \\subset \\mathbb {R}^d$ and $J \\subset (-\\infty ,+\\infty ) $ be two open sets.",
"Suppose the function $v: \\Omega \\times J \\rightarrow \\mathbb {R}^d$ satisfies the following: $v$ is continuous on $\\Omega \\times J$ .",
"For every $t \\in J$ , the function $x\\mapsto v(x,t)$ is continuously differentiable on $\\Omega $ .",
"Given compact sets $\\bar{\\Omega }_1 \\subset \\subset \\Omega $ and $\\bar{J}_1 \\subset \\subset J$ , there exists $C>0$ such that $\\left|\\nabla v(x,t)\\right| \\le C$ for all $(x,t) \\in \\bar{\\Omega }_1 \\times \\bar{J}_1$ .",
"Given compact sets $\\bar{\\Omega }_1 \\subset \\subset \\Omega $ and $\\bar{J}_1 \\subset \\subset J$ , and given $\\epsilon >0$ , there exists $\\delta >0$ such that $ \\left|\\nabla v(y,t)-\\nabla v(x,t)\\right| \\le \\epsilon $ for all $x,y \\in \\bar{\\Omega }_1$ satifying $\\left|x-y\\right|< \\delta $ and for all $t \\in \\bar{J}_1$ .",
"In all the above $\\nabla v(x,t)$ always stands for the derivative of $v$ with respect to $x$ .",
"Under the above hypothesis, given $(x_0,t_0)\\in \\Omega \\times J$ there exists open sets $\\Omega _0 \\subset \\Omega $ and $J_0 \\subset J$ such that $(x_0,t_0) \\in \\Omega _0 \\times J_0$ and a unique continuous function $\\sigma : \\Omega _0 \\times J_0 \\mapsto \\mathbb {R}^d$ such that $\\sigma (x,t)=x + \\int _{t_0}^t v(\\sigma (x,s),s) ds.$ Moreover, given $t \\in J_0$ , the mapping $x \\mapsto \\sigma (x,t)$ is continuously differentiable on $\\Omega _0$ and we have $\\nabla \\sigma (x,t) = Id+ \\int _{t_0}^t \\nabla v( \\sigma (x,s),s) \\nabla \\sigma (x,s) ds.$ The proof of Theorem 1.184, page 120 in [27] (or Theorem 1.261, page 138 of the online version of [27]) can be carried out with very minor modifications.",
"Let us just mentioned where and how hypothesis (H3) is needed.",
"In the proof of [27] the space $X$ and $Y$ are defined by $X= C(b(t_0, \\delta ) \\times B(x_0,\\nu /2), \\bar{B}(x_0,\\nu ))$ and $Y= C_b(b(t_0, \\delta ) \\times B(x_0,\\nu /2), L(\\mathbb {R}^d, \\mathbb {R}^d))$ , where $b(t_0, \\delta )$ and $B(x_0,\\nu /2)$ denotes balls of radius $\\delta $ and $\\nu /2$ and $L(\\mathbb {R}^d, \\mathbb {R}^d)$ denotes the set of linear transformations on $\\mathbb {R}^n$ .",
"Both $X$ and $Y$ are endowed with the sup norm.",
"$C_b$ stands for continuous and bounded.",
"The mapping $\\Psi : X \\times Y \\rightarrow Y$ is defined by $\\Psi (\\phi ,\\Phi )(x,t)= Id+ \\int _{t_0}^t \\nabla v( \\phi (x,s),s) \\Phi (x,s) ds.$ In order to use the fiber contraction principle from [27], we must verify that $\\Psi $ is continuous.",
"From (H2) we easily obtain $\\left\\Vert \\Psi (\\phi ,\\Phi _1)-\\Psi (\\phi ,\\Phi _2)\\right\\Vert \\le K \\delta \\left\\Vert \\Phi _1-\\Phi _2\\right\\Vert $ where $K=\\sup _{ B(x_0,\\nu /2) \\times b(t_0, \\delta )} \\left|\\nabla v\\right|$ .",
"Hypothesis (H3) is needed in order to obtain continuity of $\\Psi $ with respect to its first argument.",
"To see this write $\\Psi (\\phi _1,\\Phi )(x,t)-\\Psi (\\phi _2,\\Phi )(x,t) = \\int _{t_0}^t \\left( \\nabla v( \\phi _1(x,s),s)-\\nabla v_2( \\phi (x,s),s) \\right) \\Phi (x,s) ds.$ Using (H3) we see that $\\left\\Vert \\Psi (\\phi _1,\\Phi )-\\Psi (\\phi _2,\\Phi )\\right\\Vert $ can be made as small as we want by choosing $\\phi _1$ and $\\phi _2$ close enough with respect to the sup-norm.",
"Remark 4 Since $\\nabla v$ is not assumed to be continuous with respect to time, the function $t \\mapsto \\nabla \\sigma (x,t)$ is not necessarily continuously differentiable.",
"However it is absolutely continuous as can been seen from (REF ).",
"Therefore, given $x \\in \\Omega _0$ , the function $Y(t)= \\nabla \\sigma (x,t)$ is differentiable for almost every $t \\in J_0$ and the differential equation $Y^{\\prime }(t)= \\nabla v( \\sigma (x,t),t)\\; Y(t)$ holds almost everywhere in $J_0$ .",
"Then we can use Liouville Theorem (which is stated below) to deduce that $\\frac{d}{dt}\\text{det }\\nabla \\sigma (x,t) = (\\text{div }v)( \\sigma (x,t),t) \\; \\text{det } \\nabla \\sigma (x,t)$ also holds almost everywhere in $J_0$ .",
"This of course implies that $\\text{det }\\nabla \\sigma (x,t) = \\text{exp} \\big ( {\\int _{t_0}^t (\\text{div }v)( \\sigma (x,s),s) ds} \\big ),$ which is the formula needed in our case (see (REF )).",
"[Liouville] Let $A$ be $d \\times d$ matrix and let Y(t) be a $d \\times d$ time dependent matrix which is differentiable at $t=t_0$ and satisfies $Y^{\\prime }(t_0)= A \\; Y(t_0)$ .",
"Then the function $\\Lambda (t) = \\text{ det } Y(t) $ is differentiable at $t_0$ and satisfies $\\Lambda ^{\\prime } (t_0) = (\\text{Tr } A) \\; \\Lambda (t_0).$ See for example Hartman [40]."
],
[
"Some general Lemmas",
"A proof of the following Lemma can be found in [28].",
"Let $T,S: \\mathbb {R}^d \\rightarrow \\mathbb {R}^d$ be two Borel maps.",
"Also take $\\rho \\in {\\cal P}_2(\\mathbb {R}^d)$ .",
"Then $W_2(S\\# \\rho , T\\# \\rho ) \\le \\left\\Vert S-T\\right\\Vert _{L^\\infty (\\text{supp} \\rho )}.$ Suppose that $\\rho \\in {\\cal P}(\\mathbb {R}^d)$ has compact support and suppose that ${\\cal P}_{RD}(\\mathbb {R}^d) \\ni \\rho _n$ converges narrowly to $\\rho $ .",
"Then $\\rho $ also belongs to ${\\cal P}_{RD}(\\mathbb {R}^d)$ .",
"Let $R$ be a rotation of $\\mathbb {R}^d$ and let $f \\in C(\\mathbb {R}^d).$ Then $f \\circ R \\in C(\\mathbb {R}^d)$ and $\\int f \\circ R d\\rho _n = \\int f d\\rho _n$ .",
"Taking limits we see that $\\int f\\circ R d\\rho = \\int f d\\rho $ so that $\\rho \\in {\\cal M}_R(\\mathbb {R}^d).$ To prove $\\rho $ is decreasing fix $0 < r_1 < r_2$ and take disjoint small rings $A_j = \\lbrace r_j - \\eta _j \\le |x| \\le r_j + \\delta _j\\rbrace , j = 1,2$ having the same volume.",
"We may assume $\\rho (\\partial A_j) =0$ .",
"Then there exist continuous functions $f_1 \\ge \\chi _{A_1}$ and $f_2 \\le \\chi _{A_2}$ with disjoint supports such that $|\\rho (A_j) - \\int f_j d\\rho | < \\epsilon .$ By hypothesis we have $\\inf f_1 d\\rho _n \\ge \\int f_2 d\\rho _n$ , so that $\\rho (A_2) + 2\\epsilon \\le \\rho (A_1).$ Then shrinking $\\epsilon ,$ $A_1$ and $A_2$ shows that $\\rho \\in {\\cal P}_{RD}(\\mathbb {R}^d).$ Let $\\alpha > 2-d$ and suppose $y: [0,+\\infty ) \\rightarrow [0,+\\infty )$ is an absolutely continuous function satisfying $- y(t)^{\\alpha -1} \\le y^{\\prime }(t) \\le 0$ for almost every $t \\in [0,+\\infty )$ for which $y(t)>0$ .",
"If $\\alpha \\le 1$ then $y(t)$ is Hölder continuous.",
"To be more precise: $- ( (2-\\alpha )(t-s))^\\frac{1}{2-\\alpha } \\le y(t)-y(s) \\le 0$ for all $0\\le s \\le t$ .",
"If $\\alpha >1$ then $y(t)$ is Lipschitz continuous.",
"To be more precise: $-y(0)^{\\alpha -1} (t-s) \\le y(t)-y(s) \\le 0$ for all $0\\le s \\le t$ .",
"The case $\\alpha >1$ is trivial since the inequality $- y(t)^{\\alpha -1} \\le y^{\\prime }(t) \\le 0$ together with the non-negativity of $y$ implies $- y(0)^{\\alpha -1} \\le y^{\\prime }(t) \\le 0$ .",
"We now prove the Lemma for $2-d<\\alpha \\le 1$ .",
"For almost every $t\\ge 0$ for which $y(t)>0$ we have $- (2-\\alpha ) \\le \\frac{d}{dt}\\left( y(t)^{2-\\alpha }\\right) \\le 0$ .",
"It is then clear that $ -(2-\\alpha )(t-s) \\le y(t)^{2-\\alpha }-y(s)^{2-\\alpha } \\le 0$ for all $0\\le s \\le t$ .",
"But because of the convexity of the function $r \\mapsto r^{2-\\alpha }$ we have that $(y(s)-y(t))^{2-\\alpha } \\le y(s)^{2-\\alpha }-y(t)^{2-\\alpha }$ , which gives the result."
]
] | 1204.1095 |
[
[
"A compilation of metallic systems that show a quantum ferromagnetic\n transition"
],
[
"Abstract We provide a compilation of metallic systems in which a low-temperature ferromagnetic or similar transition is observed.",
"Our objective is to demonstrate the universal first-order nature of such transitions in clean systems in two or three spatial dimensions.",
"Please contact the authors with information about omissions, corrections, or any other information."
],
[
"A compilation of metallic systems that show a quantum ferromagnetic transition D. Belitz$^{1,2}$ and T. R. Kirkpatrick$^3$ $^{1}$ Department of Physics and Institute of Theoretical Science, University of Oregon, Eugene, OR 97403, USA $^{2}$ Materials Science Institute, University of Oregon, Eugene, OR 97403, USA $^{3}$ Institute for Physical Science and Technology,and Department of Physics, University of Maryland, College Park, MD 20742, USA We provide a compilation of metallic systems in which a low-temperature ferromagnetic or similar transition is observed.",
"Our objective is to demonstrate the universal first-order nature of such transitions in clean systems in two or three spatial dimensions.",
"Please contact the authors with information about omissions, corrections, or any other information.",
"Quantum phase transitions are phenomena of great interest.",
"[1], [2] Perhaps the most obvious quantum phase transition, and the first one considered historically, is the transition from a paramagnetic metal to a ferromagnetic metal at zero temperature ($T=0$ ) as a function of some non-thermal control parameter.",
"Stoner's theory of itinerant ferromagnetism[3] describes both the thermal transition and the static properties of the quantum transition in a mean-field approximation.",
"It predicts a second-order or continuous transition with standard Landau or mean-field static critical exponents.",
"For the thermal or classical transition this constitutes an approximation for spatial dimensions $d\\le 4$ .",
"In the physical dimensions $d=3$ or lower, fluctuations of the magnetization order parameter, which are neglected in Stoner theory, lead to deviations from the mean-field critical behavior that require the renormalization group (RG) for a theoretical understanding.",
"[4] In a seminal paper, Hertz[1] derived a Landau-Ginzburg-Wilson (LGW) functional for the ferromagnetic transition from a model of itinerant electrons that interact via a point-like potential in the particle-hole spin-triplet channel.",
"Hertz analyzed this dynamical LGW functional by means of RG methods.",
"He concluded that, at $T=0$ , Stoner theory is exact as far as the static critical behavior is concerned, i.e., the transition is of second order with mean-field static critical exponents, and a dynamical critical exponent $z=3$ , for all $d>1$ .",
"This is because the coupling of the statics to the dynamics makes the system effectively behave as if it were in a higher spatial dimension, given by $D = d+z$ .",
"It became clear in the late 1990s that this theoretical picture is not correct.",
"It was shown that particle-hole excitations about the Fermi surface, which exists in all metals in dimensions $d>1$ , couple to the magnetization and invalidate Hertz's conclusions.",
"[5], [14] As a result, the quantum ferromagnetic transition was predicted to be generically of first order in clean metallic ferromagnets in $d>1$ .",
"Physically, the mechanism that drives the transition first order is very similar to the fluctuation-induced first-order transition that was predicted earlier for the classical transition in superconductors and smectic liquid crystals,[6] and to the spontaneous mass-generation mechanism known as the Coleman-Weinberg mechanism in particle physics.",
"[7] In all of these cases a generic soft mode (the photon in the cases of superconductors and scalar electrodynamics; the nematic Goldstone mode in the case of liquid crystals) that is distinct from the order parameter fluctuations couples to the latter and qualitatively changes the nature of the phase transition.",
"For the quantum ferromagnetic transition in metals, the resulting prediction is the generic phase diagram shown in Fig.",
"REF .",
"Figure: Generic phase diagram of a metallic ferromagnet in the space spannedby temperature (TT), magnetic field (HH), and the control parameter (tt).",
"Shownare the ferromagnetic (FM) and paramagnetic (PM) phases,lines of second-order transitions, surfaces of first-order transitions(“tricritical wings”), the tricritical point (TCP), and the two quantumcritical points (QCP).At zero temperature ($T=0$ ), there is a first-order transition triggered by a non-thermal control parameter $t$ .",
"A nonzero temperature gives the generic particle-hole excitations a mass, and as a result the mechanism driving the first-order transition becomes weaker with increasing temperature.",
"This leads to a tricritical point (TCP) at some temperature $T_{\\text{tc}} > 0$ , and for $T > T_{\\text{tc}}$ the transition is generally of second order with classical critical exponents.",
"Upon the application of an external field conjugate to the order parameter, i.e., a homogeneous magnetic field $H$ in the case of a ferromagnet, surfaces of first-order transitions called tricritical wings emerge from the tricritical point.",
"This is true for any classical phase diagram that contains a tricritical point,[8] and it holds for quantum ferromagnets as well.",
"[9] These tricritical wings are bounded by lines of second-order transitions and end in a pair of quantum critical points (QCPs) in the $T=0$ plane.",
"The critical behavior at these QCPs can be determined exactly and is a slight modification of the critical behavior predicted by Hertz for the quantum phase transition at $H=0$ that is pre-empted by the first-order transition.",
"[9] This general picture is theoretically predicted to apply to all transitions from a metallic paramagnetic phase to a metallic ferromagnetic one in dimensions $d>1$ , irrespective of whether the magnetism is caused by the conduction electrons (“itinerant ferromagnets”) of by electrons in a different band, and irrespective of the isotropy or lack thereof of the magnetization.",
"That is, it applies to easy-axis (Ising) and easy-plane (XY) magnets as well as to isotropic (Heisenberg) magnets.",
"It also applies to ferrimagnets and canted ferromagnets, and more generally to any metallic system that has a nonvanishing homogeneous magnetization,[10] but the most extensive experimental information is available for ferromagnets.",
"There are only two ways to avoid these conclusions: (1) In 1-d or quasi-1-d systems there is no Fermi surface, and hence no particle-hole excitations, and the soft-mode mechanism is not operative.",
"(2) In the presence of sufficiently strong quenched disorder the nature of the particle-hole excitations changes from ballistic to diffusive, and the nature of the coupling to the magnetization changes as well.",
"This can lead to a second-order quantum phase transition with non-mean-field critical exponents that still can be determined exactly.",
"[11], [12], [13], [14] Disorder may also have stronger effects, leading to Griffiths phases and smeared transitions, see Ref.",
"Vojta2010.",
"Experimentally, the picture summarized above is confirmed with remarkable uniformity.",
"To the authors's knowledge, all metallic ferromagnets that do not fall into one of the two exceptional classes mentioned above, show a first-order transition if the transition temperature is sufficiently low, or can be driven sufficiently low by a non-thermal control parameter, such as pressure, or composition.",
"This is especially remarkable if compared with the case of classical liquid crystals, where the observed transition is usually of second order, and only recently have examples of weakly first-order transitions been found.",
"[16] The reason why the theory is so much more successful in the quantum case is not entirely understood, but it is likely related to the fact that order-parameter fluctuations, which can invalidate the fluctuation-induced first-order mechanism, are strongly suppressed in the quantum case for the same reasons that lead to a mean-field critical behavior in Hertz's theory.",
"[10] The purpose of this informal communication is to demonstrate this remarkable agreement between theory and experiment by compiling a list of metallic systems in which a quantum ferromagnetic transition has been observed.",
"The systems are listed roughly in order of completeness of the experimental information available.",
"All but three of the systems listed display a confirmed or suspected first-order transition.",
"The three expections are, URu$_{2-x}$ Re$_x$ Si$_2$ , which is strongly disordered, YbNi$_4$ P$_2$ , which is quasi-1-d, and Ni$_x$ Pd$_{1-x}$ , where the lowest transition temperature achieved is 7K, which may be above the tricritical point, if one exists.",
"All other examples are consistent with the phase diagram shown in Fig.",
"REF .",
"In some cases (e.g., UCoGe) the transition is first order at the highest, or only, temperature observed, so the tricritical point is not accessible.",
"In all cases where a tricritical point is accessible and the behavior in a magnetic field has been studied, tricritical wings have been observed.",
"One of the best studied materials, MnSi, is actually a helimagnet,[17] but the helical wavelength ($\\approx 200Å$ ) is so long compared to the atomic length scale that the system is well approximated as a ferromagnet.",
"[18] We conclude with a few general remarks.",
"First, there also are cases of transitions from a metallic ferromagnet to some insulating phase.",
"Examples include, FeSi$_{1-x}$ Ge$_x$ ,[19] and RE$_{0.55}$ Sr$_{0.45}$ Mn$_3$ , with RE a rare earth or a combination of rare earths.",
"[20] In these cases the theoretical situation is more complicated, and we do not include them in our discussion.",
"Second, in any given material a first-order transition may occur for reasons other than the coupling to particle-hole excitations.",
"This is likely the case in systems that have a tricritical point at a relatively high temperature, such as various manganites, see, e.g., Ref.",
"Kimetal2002.",
"Finally, we mention that some of the materials listed in the table have gotten a lot of attention for properties other than, although possibly related to, the ferromagnetic transition.",
"Examples are the coexistence of superconductivity and ferromagnetism observed in UGe$_2$ ,[22] URhGe,[23] and UCoGe,[24] or the non-Fermi-liquid phase and the A-phase in MnSi.",
"[25], [26] As a result, there is a large body of literature on some of these materials; we quote only papers that are directly relevant to properties reflected by the entries in the table.",
"We gratefully acknowledge discussions and correspondence with Greg Stewart, Jeff Lynn, Nick Butch, and Christoph Geibel.",
"This work was supported by the National Science Foundation under Grant Nos.",
"DMR-09-29966, and DMR-09-01907.",
"Table: Systems with low-TT ferromagnetic transitions and their properties.",
"T c =T_{\\text{c}} =Curie temperature, T tc =T_{\\text{tc}} = tricritical temperature.",
"ρ 0 =\\rho _0 =residual resistivity.",
"FM = ferromagnet, SC = superconductor.",
"N/A = not applicable; n.a.",
"= not available."
]
] | 1204.0873 |
[
[
"Differential equations for multi-loop integrals and two-dimensional\n kinematics"
],
[
"Abstract In this paper we consider multi-loop integrals appearing in MHV scattering amplitudes of planar N=4 SYM.",
"Through particular differential operators which reduce the loop order by one, we present explicit equations for the two-loop eight-point finite diagrams which relate them to massive hexagons.",
"After the reduction to two-dimensional kinematics, we solve them using symbol technology.",
"The terms invisible to the symbols are found through boundary conditions coming from double soft limits.",
"These equations are valid at all-loop order for double pentaladders and allow to solve iteratively loop integrals given lower-loop information.",
"Comments are made about multi-leg and multi-loop integrals which can appear in this special kinematics.",
"The main motivation of this investigation is to get a deeper understanding of these tools in this configuration, as well as for their application in general four-dimensional kinematics and to less supersymmetric theories."
],
[
"Introduction",
"Scattering amplitudes in planar $\\mathcal {N}= 4$ super Yang-Mills (SYM) have received a lot of attention in recent years, mostly due to the high amount of symmetry of this theory.",
"This characteristic leads to many simplifications which give hope to exactly solve the theory.",
"It was indeed discovered that the theory possesses a hidden dual conformal symmetry [1]-[6], not visible at the Lagrangian level, which is related to the duality between maximally helicity-violating (MHV) amplitudes and light-like Wilson loops [2], [3], [4], [6]-[10].",
"This was revealed at weak coupling by considering the amplitudes in a dual coordinate space, defined through the change of variables $p_i^{\\alpha \\dot{\\alpha }} = x_i^{\\alpha \\dot{\\alpha }}-x_{i+1}^{\\alpha \\dot{\\alpha }}$ .",
"This symmetry extends to the supersymmetric level [11], [12] and, together with the standard superconformal symmetry, forms a Yangian structure [13].",
"Yangian symmetry constrains and completely fixes the amplitude structure at tree level, when collinear anomalies are also taken into account, but at loop level it is broken by infrared divergences.",
"This was made more explicit by considering the Grassmannian [14], [15], which gives all possible invariants [16], [17] under the free Yangian generators.",
"At loop level the all-loop planar integrand manifests the full Yangian symmetry [21] but at the level of the actual amplitudes, the invariance is broken when integrating over contours.",
"Therefore at loop level, even though much progress has been done, finding analytic results for amplitudes seems to be very complicated.",
"The multi-parameter nature of the loop integrals increases so enormously their complexity that only very recently an analytical expression for the six-point two-loop remainder function was proposed [26].",
"Efficient methods to evaluate the integrals appearing in the amplitudes are therefore relevant and can turn out to be very important even for less supersymmetric theories.",
"In this respect, in [18] it was presented a new type of second-order differential equations for on-shell loop integrals.",
"The important feature of these equations is their iterative structure, as they reduce the loop degree by one.",
"In a series of recent papers, the action of differential operators has been proven to be very successful in finding the six-dimensional hexagon integrals [19], [20], which are related to four-dimensional MHV amplitudes.",
"Moreover, all-loop differential equations for the BDS-subtracted amplitude have been proposed [33], which give the action of superconformal and dual superconformal generators.",
"Even more recently [34], Mellin space representation of amplitudes has been used to derive differential equations for dual conformal integrals.",
"Meanwhile, a very powerful mathematical tool has been applied to amplitude results, the symbol [27] of pure functions.",
"It allowed a remarkable simplification [28] of the six-point two-loop remainder function found in [26], as the complicated identities between polylogarithms are trivialized.",
"The integrals for scattering amplitudes in $\\mathcal {N}=4$ SYM exhibit indeed an iterated structure and all known perturbative results can be expressed in terms of generalized polylogarithms, multi-dimensional iterated integrals of uniform transcendentality.",
"Symbols have been efficiently used in the evaluation of the two-loop $n$ -point remainder function [46] and to make Ansätze on the three-loop six-point remainder function [32] and the two-loop six-point ratio function [29].",
"The symbol loses nevertheless some information about the function (for some recent development using Hopf algebra of multiple polylogarithms see [35]; this method encodes also the $\\zeta $ value information).",
"It is indeed invisible to terms which, to a given transcendentality degree, are lower-degree functions multiplied by constants of the appropriate degree.",
"Therefore other tools must be thought to reconstruct the complete result.",
"We will show the usefulness for this scope of boundary conditions given by soft limits.",
"Further important achievements have been reached considering a special kinematics, where the external momenta live in two dimensions.",
"This kinematics has been used at strong coupling, when considering Wilson loops with light-like contours contained in a two-dimensional subspace of Minkowski space-time [36].",
"Differential equations in this regime have been proposed in [42] for two-loop eight-point MHV amplitudes.",
"The analitycal expressions for two-loop Wilson loops with arbitrary number of points [30] and three-loop eight-point Wilson loops [31] have been evaluated.",
"At least in the one-loop and two-loop cases only logarithms enter in the Wilson loop expressions [36], [25], [30].",
"In this kinematics it has also been possible to demonstrate the Yangian invariance of Wilson loops at one-loop [37].",
"The main motivation for this paper is to combine these new tools in the search for efficient and powerful methods to find analytic results for multi-loop multi-leg diagrams appearing in scattering amplitudes.",
"We will focus our attention on MHV scattering amplitudes in the basis given in [21], [22].",
"Starting with the two-loop eight-point finite diagrams, we will present the second-order differential operators, in momentum twistor variables [23], which lower the loop degree by one relating them to hexagon integrals.",
"Then we will truncate the differential equations to the restricted two-dimensional kinematical regime.",
"This configuration simplifies the results obtained and allows to analitically solve the equations, when boundary conditions are taken into account.",
"The same equations can be used for higher loops, extending the results of [42].",
"Through these results, beyond the explicit analytic expressions of the diagrams, we also want to give further contribution to the investigation of these methods.",
"First of all, it is becoming more and more challenging to understand which letters can enter in a symbol subject to physical constraints, such as e.g.",
"the operator product expansion [38]-[41].",
"In this respect, more data are needed to possibly constraining them.",
"Another important goal is the analysis of boundary conditions, coming for instance from soft limits as we will discuss.",
"The differential equations and the symbols leave indeed some ambiguities about the function, which must be fixed demanding additional constraints.",
"The differential equations discussed here, as already stressed, are valid for an infinite series of loop integrals which are part of higher-loop MHV amplitudes.",
"Behind them, other topologies appear, as well as in NMHV amplitudes [22], where also different tensor numerators show up.",
"New differential operator are therefore necessary and further investigation needed.",
"The paper is organized as follows.",
"The first part is an introduction on the subject, in order to review the main ingredients which will be used throughout the paper.",
"In particular we recall the dual conformal integrands appearing in one-loop and two-loop MHV amplitudes and how specific differential operators act on their master topology.",
"We also review the two-dimensional kinematics.",
"We then discuss the boundary conditions that can be used to fix the lower transcendental-degree functions.",
"Then in section REF , an explicit example is presented, appearing in the two-loop eight-point MHV amplitude.",
"We describe the action of a particular operator in momentum twistor space and then reduce it to the two-dimensional kinematics in order to find an explicit result.",
"In section REF we summarize the results for the other finite diagrams appearing in the two-loop eight-point case.",
"We then discuss the generalization of such tools for the cases of two-loop $n$ -point integrals and $L$ -loop double pentaladder integrals.",
"In the appendices we review the notion of symbols and give technical details."
],
[
"Background material and setup",
"We recall here some basics about MHV scattering amplitudes in momentum twistor space, differential operators for on-shell integrals and two-dimensional kinematics.",
"Later we will discuss boundary conditions which are helpful in evaluating diagrams from differential equations and symbols.",
"As we already pointed out in the introduction, many developments have been reached in studying MHV amplitudes.",
"By using a BCFW-generalized recursive formula for the all-loop integrand, it has been possible to reconstruct the integrand of multi-loop amplitudes [21], [22].",
"In the case of one- and two-loop MHV amplitudes, a single master topology contributes, namely the tensor-pentagon diagram.",
"The one-loop case is depicted in (REF ), which is \"formally\" dual conformal invariant, where \"formally\" means that IR divergences can appear when the integrals are performed.",
"The dashed line represents the tensor numerator and its explicit expression will be discussed later on.",
"The box-integrals are \"boundary terms\" of this topology, which are retrieved by taking soft limits [18].",
"$A_{\\mathrm {MHV}}^{\\mathrm {1-loop}} = \\sum _{j<k}\\qquad {\\parbox [c]{30mm}{\\includegraphics [height = 4cm] {basis1.eps}}}$ The same topology appears at two-loops (REF ), where the tensor pentagon is now a subintegral and penta-boxes, as well as double-boxes are again boundary terms: $A_{\\mathrm {MHV}}^{\\mathrm {2-loop}} = \\frac{1}{2} \\sum _{i<j<k<l}\\qquad {\\parbox [c]{30mm}{\\includegraphics [height = 4cm] {basis2.eps}}}~$ There are different spaces where amplitudes can be defined in and in the following we want to review them briefly, in order to show the relation of momentum twistor variables to the dual space.",
"In the spinor helicity formalism, the massless momenta are written as $p_i^{\\alpha \\dot{\\alpha }} = \\lambda _i^{\\alpha }\\tilde{\\lambda }_i^{\\dot{\\alpha }}$ , where $\\lambda _i^{\\alpha }$ and $\\tilde{\\lambda }_i^{\\dot{\\alpha }}$ are two-component commuting spinors and define the on-shell space.",
"The dual coordinates $x_i$ are related to the momenta through $p_i^{\\alpha \\dot{\\alpha }} = x_i^{\\alpha \\dot{\\alpha }}-x_{i+1}^{\\alpha \\dot{\\alpha }} \\, ,$ and describe a light-like closed polygon.",
"The momentum twistor variables [23] are the twistors associated to the dual coordinates $x_i$ 's and are defined as follows: $Z_i^A = (\\lambda _i^\\alpha , \\mu _i^{\\dot{\\alpha }}), \\qquad \\mu _i^{\\dot{\\alpha }} = x_i^{\\alpha \\dot{\\alpha }} \\lambda _{i \\alpha } = x_{i+1}^{\\alpha \\dot{\\alpha }} \\lambda _{i \\alpha }\\,$ where the second relation is the so-called incidence relation and the index $A$ is in the fundamental of $sl(4)$ .",
"The most powerful property of momentum twistors is that they are free variables, since momentum conservation and light-likeness constraint are already encoded in the definition.",
"The dual distances are therefore expressed in the following form $x_{ij}^2 = \\frac{(i-1\\,\\, i\\,\\, j-1\\,\\, j)}{\\langle i-1 \\, i \\rangle \\langle j-1 \\, j\\rangle }\\,$ where the four-bracket $(ijkl) = \\epsilon _{ABCD} Z_i^A Z_j^B Z_k^C Z_l^D$ is dual conformal invariant, while $\\langle i \\, j \\rangle = \\epsilon _{\\alpha \\beta }\\lambda _i^{\\alpha }\\lambda _j^{\\beta }$ is invariant under Lorentz and (dual) translations only.",
"To a point $x_i$ corresponds therefore the line $(Z_{i-1} Z_i)$ , while two light-like separated dual points are mapped to intersecting lines in twistor space.",
"Let us now consider a particular diagram, present in (REF ), in momentum twistor space, namely a nine-point one-loop tensor pentagon, see Fig.",
"(REF ) where dual points are indicated.",
"We choose this specific integral to show differential operators at work in an explicit example which will be useful later on.",
"Its expression is $F_9 ^{(1)} = \\int \\frac{\\mathrm {d}^4 Z_{AB}}{i \\pi ^2} \\frac{N~ (AB37)}{(AB23)(AB34)(AB67)(AB78)(AB91)} \\equiv \\int \\frac{\\mathrm {d}^4 Z_{AB}}{i \\pi ^2} I_9^{(1)}$ where $N$ is the normalization factor $N = (1234) (6789)$ and the tensor numerator $(AB37)$ is represented pictorially with a dashed line.",
"All these characteristics assure that this is a pure integral with unit leading singularity [21], [22].",
"The integration is performed over the space of lines in momentum twistor space $d^4 Z_{AB} \\sim \\langle AB \\rangle ^4 d^4x_0 \\,.$ Figure: One-loop tensor pentagon integral F 9 (1) F_{9}^{(1)}.",
"Dual points i≡x i i\\equiv x_i's are showed.We want now to act on this integral with one of the operators considered in [18] and whose iterative structure allows to relate $L$ -loop integrals to lower-loop ones.",
"The twistor derivative is defined to be $\\mathcal {O}_{ij} = Z_i \\cdot \\frac{\\partial }{\\partial Z_j} \\, ,$ which on four-brackets simply gives $\\mathcal {O}_{ij} (j l k m) = (i l k m)$ .",
"Considering the explicit action of the operator e.g.",
"$\\mathcal {O}_{24} \\mathcal {O}_{86} \\equiv Z_2 \\cdot \\frac{\\partial }{\\partial Z_4} Z_8 \\cdot \\frac{\\partial }{\\partial Z_6} $ on the integrand $I_9^{(1)}$ we find $\\mathcal {O}_{24} \\mathcal {O}_{86} I_9^{(1)} = \\frac{N~ (AB37)}{(AB34)^2 (AB67)^2 (AB91)} \\, ,$ where the normalization factor $N$ has not been touched by the particular operator.",
"Integration using Feynman parametrization gives $\\mathcal {O}_{24} \\mathcal {O}_{86} F_9 ^{(1)} \\propto \\frac{N~ (9137)}{(9134) (9167) (3467)} .$ This substitution can be used whenever the pentagon $F_9 ^{(1)}$ is a subintegral and pictorially it can be depicted as $\\mathcal {O}_{24} \\mathcal {O}_{86}\\qquad {\\parbox [c]{30mm}{\\includegraphics [height = 3cm] {massivepent.eps}}}\\qquad \\propto \\qquad {\\parbox [c]{30mm}{\\includegraphics [height = 3cm] {reduction.eps}}}$ This method is very powerful since the second-order differential operator $\\mathcal {O}_{24} \\mathcal {O}_{86}$ reduces the loop order by one: in the next section we will apply this and other twistor derivatives of the same kind to derive differential equations.",
"Specifically, we will restrict our analysis to finite integrals; IR divergent diagrams need separate consideration [18] and the introduction of e.g.",
"the AdS mass regulator of [44].",
"Since these integrals are dual conformal, they have a restricted variable dependence.",
"Indeed, they can be expressed in terms of the $(3n-15)$ conformal invariant cross-ratios only (in case of three-point functions the integrals are rational functions), $u_{ij} = \\frac{x_{i,j+1}^2 x_{i+1,j}^2}{x_{i,j}^2 x_{i+1,j+1}^2} = \\frac{(i-1\\,\\, i\\,\\, j\\,\\, j+1) (i\\,\\, i+1\\,\\, j-1\\,\\, j)}{(i-1\\,\\, i\\,\\, j-1\\,\\, j) (i\\,\\, i+1\\,\\, j\\,\\, j+1)} \\, ,$ where (REF ) is used to translate from dual coordinate space to momentum twistor variables.",
"One can therefore rewrite the twistor operators in terms of derivatives acting on cross-ratios, by using the chain rule.",
"We want to end this section by reviewing the two-dimensional kinematics, where the momenta of the external particles lie in a two-dimensional plane.",
"The momentum twistor variables are restricted and preserve two commuting copies of $sl(2)$ inside $sl(4)$ .",
"We choose (see appendix ) $Z_{2i+1}=\\left( \\begin{array}{c} 0 \\\\ 0 \\\\ Z_{2i+1}^3 \\\\ Z_{2i+1}^4 \\\\ \\end{array} \\right),~~~Z_{2i}=\\left( \\begin{array}{c} Z_{2i}^1 \\\\ Z_{2i}^2 \\\\ 0 \\\\ 0 \\\\ \\end{array} \\right) \\, .$ Four-brackets with an odd number of odd or even twistors vanish and many cross-ratios become trivial, in particular $u_{i,i + \\mathrm {odd}} = 1$ .",
"Specifically the number of independent cross-ratios is reduced to $(n-6)$ out of $(3n-15)$ of the original four-dimensional space."
],
[
"Boundary conditions from soft limits ",
"In this section we explain boundary conditions deriving from soft limits, which will be necessary to fix the ambiguities in symbol technology for lower-transcendental functions.",
"We first consider the eight-point tensor pentagon $\\tilde{\\Psi }^{(1)} (u,v,w)$ of Fig.",
"REF which depends on the following three cross-ratios: $u = \\frac{x^2_{28} x^2_{36}}{x^2_{26} x^2_{38}} \\, , \\qquad v = \\frac{x^2_{68} x^2_{25}}{x^2_{26} x^2_{58}} \\, , \\qquad w = \\frac{x^2_{35} x^2_{26}}{x^2_{36} x^2_{25}} \\, ,$ and whose expression is known [18]: $\\tilde{\\Psi }^{(1)} (u,v,w) = \\log u \\log v + {\\rm Li} _2(1-u) + {\\rm Li} _2(1-v) + {\\rm Li} _2(1-w) - {\\rm Li} _2(1-u w) - {\\rm Li} _2(1-v w) \\, .$ A simple soft limit consists on taking $p_4 \\rightarrow 0 $ ($w\\rightarrow 0$ ) to relate the eight-point pentagon integral with the seven-point one.",
"Another limit, which will be useful later on, is $p_5 \\rightarrow 0$ .",
"In dual coordinates this means $x_5 = x_6$ and the cross-ratios take the values $(v \\rightarrow 1, w \\rightarrow 1)$ , with $u$ arbitrary.",
"On the function the limit becomes $\\lim _{\\tau \\rightarrow 0} \\tilde{\\Psi }^{(1)} (u, 1- \\tau \\xi , 1- \\tau ) = 0 \\, ,$ where the ratio $\\frac{1-v}{1-w} = \\xi $ can be arbitrary.",
"This limit, as well as the previous one, can be used also for diagrams where this pentagon is a subintegral.",
"For instance, the seven-point pentagon case can be used in the six-point double-pentagon diagram $\\Omega ^{(2)}(v_1, v_2, v_3)$ , defined in [18], to obtain other boundary conditions.",
"If we now consider the integral $F_b^{(2)}$ , see Fig.",
"REF , appearing in the two-loop eight-point MHV scattering amplitude, we can take the double soft limit $p_5, p_6 \\rightarrow 0$ .",
"Since we are most interested in the two-dimensional kinematics, we can use the fixing of appendix to find that the relevant cross-ratios are $u_1 = \\frac{x^2_{27} x^2_{36}}{x^2_{26} x^2_{37}} = \\frac{x}{1+x} \\, , \\qquad u_2 = \\frac{x^2_{38} x^2_{47}}{x^2_{37} x^2_{48}} = \\frac{y}{1+y} \\, .$ The double soft limit under consideration implies that $(u_1 \\rightarrow 1 \\, , \\, u_2 \\rightarrow 0) \\, \\equiv \\, (x \\rightarrow \\infty \\, ,\\, y \\rightarrow 0) \\, .$ Since the product $(x \\, y)$ can be arbitrary, calling $x=\\frac{1}{\\tau }$ and $y = \\tau \\xi $ we expect $\\lim _{\\tau \\rightarrow 0} F_{b}^{(2)}\\left(\\frac{1}{\\tau }, \\tau \\xi \\right) = 0 \\, .$ Due to the symmetry of the diagram, the limit $(x \\leftrightarrow y)$ is also valid.",
"The same mechanisms work for the case $ F_{c}^{(2)}$ , which vanish when $(p_5, p_6 \\rightarrow 0)$ , so that $(x \\rightarrow 0 \\, , \\, y \\rightarrow \\infty )$ , giving a boundary condition similar to (REF ).",
"Considering instead the double soft limit $(p_6, p_7 \\rightarrow 0)$ , this diagram can be related to $\\Omega ^{(2)}(1,1,1)$ .",
"Appropriate double soft limits give therefore powerful constraints for diagrams appearing in scattering amplitudes and, as we will show, allow to reconstruct the complete result for an integral from its symbol."
],
[
"Loop amplitudes and differential equations",
"Let us start by considering the finite diagrams which appear in the two-loop eight-point MHV amplitudes and given by the compact expression (REF ).",
"Specifically, there are three topologies contributing to the amplitude, which are collected in Fig.",
"REF .",
"As already mentioned, they are composed by tensor-pentagon diagrams.",
"Some differential equations for these diagrams have been already found in [42]; here we want to extend those results in such a way to be possible to use them for higher-loop higher-point diagrams.",
"We reviewed in the previous section that the number of independent cross-ratios is $(3n-15)$ ; since the integrals do not depend on some dual points, the respective cross-ratios are not present and their number is actually reduced diagram by diagram.",
"The goal is to find differential operators for each of these diagrams such that the loop-level is lowered by one: the operators will be therefore valid for the $L$ -loop double pentaladder diagrams.",
"We will show that the iterative structure relates them to one-loop hexagon integrals."
],
[
"Double-pentagon integral $F^{(2)}_{c}$",
"We will present the $F^{(2)}_{c}$ case in detail here, and resume the results for the others in the next subsection.",
"Since this diagram is independent of the dual point $x_7$ , the following cross-ratios will be present only: $u_{1}&:=& u_{14} \\,, \\qquad u_{2} := u_{25} \\,, \\qquad u_{3} := u_{58} \\, , \\nonumber \\\\u_{4}&:=& u_{83} \\,, \\qquad u_{5} := u_{15} \\,,\\qquad u_{6} := u_{48} \\, ,$ where they are defined as in (REF ).",
"Going to two-dimensional kinematics by using the fixing (REF ), they become $u_{1} = u_{2} = u_{3} = u_{4} = 1 \\,,\\qquad u_{5} = \\frac{1}{1+y} \\,,\\qquad u_{6} = \\frac{1}{1+x} \\,.$ Since there are two legs attached to one of the gluing points, we can use the following equation [18], $\\mathcal {O}_{24} \\mathcal {O}_{46} \\frac{1}{N_{\\rm p}} F^{(1)}_{7} = \\frac{(7135)}{(7134)(7156)(3456)} \\, ,$ with $ F^{(1)}_{7} $ being the seven-point pentagon integral, i.e.",
"$F_9 ^{(1)}$ with the right-most corner massless.",
"Note that this is slightly different from the operator taken in account in the previous section, as the second twistor derivative does not commute with the normalization factor $N_{\\rm p}$ .",
"We therefore consider the differential equation $\\mathcal {O}_{24} \\mathcal {O}_{46} \\frac{1}{N_{2c}} F_{c}^{(2)} = \\frac{1}{(3456)} \\frac{1}{N_{1c}} F_c^{(1)}\\,,$ where $N_{2c} = -(1234)\\left[(3781)(5624)+(7812)(3456)\\right]$ and $N_{1c} = (1234) (6781)$ are the respective normalizations (see appendix ).",
"$F_c^{(1)}$ is the one-loop hexagon integral with two massive corners as shown in Fig.",
"REF (c).",
"Therefore the second-order differential operators has related a two-loop double-pentagon diagram with a simpler one-loop hexagon one.",
"By using the chain rule we obtain $\\mathcal {O}_{46} = -\\frac{(8145)}{(8156)} [u_5 u_2 (1-u_2)\\partial _2 + (1-u_3)\\partial _3 + u_5 (1-u_5)\\partial _5] \\,.$ The expression for the operator $\\mathcal {O}_{24}$ involves four-brackets that are related to the cross-ratios via quadratic equations.",
"We do not need to present it here explicitly since the coefficients of the derivatives $\\partial _1,\\partial _3 $ and $\\partial _4$ vanish for two-dimensional kinematics, such that both $\\mathcal {O}_{46}$ and $\\mathcal {O}_{24}$ are compatible with (REF ).",
"Implementing the choice of momentum twistors as in (REF ), the differential equation becomes $y \\partial _{y} (1 + y) \\partial _{y} F_{c}^{(2)}(x,y) = F_{c}^{(1)}(x,y) \\, .$ The one-loop integral $F_{c}^{(1)}(x,y)$ is calculated from its Feynman parameterization, $ F_{c}^{(1)}(x,y) &=& \\frac{y}{(x - y)} \\left[ -\\log (y) \\log (1+x) + \\log (1+y) \\log (1+ x) \\right] \\nonumber \\\\&+& \\frac{x}{(x - y)} \\Bigg [ \\log (y) \\log (1+y) + \\log (1+y) \\log (x) -\\log (1+x) \\log (1+y) \\nonumber \\\\&& \\qquad \\qquad -\\log (x) \\log (1+x) + 2 \\, {\\rm Li} _{2}(-y) -2 \\, {\\rm Li} _{2}(-x) \\Bigg ] \\,.$ As discussed before, the boundary conditions for this case are given by the double soft limit $p_5,p_6 \\rightarrow 0$ so that in the two-dimensional regime we have $F_{c}^{(2)}(x \\rightarrow \\infty , y \\rightarrow 0) = 0 $ and the symmetric $F_{c}^{(2)}(x \\rightarrow 0, y \\rightarrow \\infty ) = 0$ .",
"Now we are ready to use symbol technology and solve (REF ).",
"As first step, we evaluate the symbol of $F^{(1)}_c$ , $\\mathcal {S}\\left(F^{(1)}_c\\right)&=&\\frac{y}{x-y}(- [1+x,y]+ [1+x,1+y]- [y,1+x]+ [1+y,1+x]) \\nonumber \\\\&+& \\frac{x}{x-y} \\left(- [x,1+x]+ [x,1+y]+ [1+x,x]- [1+x,1+y]+ [y,1+y] \\right.",
"\\nonumber \\\\&& \\left.",
"\\qquad \\qquad + [1+y,x]- [1+y,1+x]- [1+y,y]\\right)$ where every square bracket is composed by two letters, being the symbol of a transcendentality-two function (we use the notation of appendix ).",
"The next step is to integrate the symbol using the differential equation and then add terms given by the integrability condition (REF ), which assures that the symbol actually corresponds to a function.",
"To the result, we can still add any single-variable symbol $\\mathcal {S}(g(x))$ which is invisible for the differential equation and is fixed by imposing the boundary conditions on the symbol.",
"We finally find that $\\mathcal {S}(F^{(2)}_c)&=& [x,1+x,x,1+x]- [x,1+x,x-y,1+x]+ [x,1+x,x-y,1+y]- [x,1+x,y,1+y] \\nonumber \\\\&+& [x,1+x,1+y,1+x]- [x,y,1+x,1+y]- [x,y,1+y,1+x]+ [x,1+y,1+x,1+y] \\nonumber \\\\&+& [x,1+y,x-y,1+x]- [x,1+y,x-y,1+y]+ [x,1+y,y,1+y]- [1+x,x,x,1+x] \\nonumber \\\\&+& [1+x,x,x-y,1+x]- [1+x,x,x-y,1+y]+ [1+x,x,y,1+y]- [1+x,x,1+y,1+x] \\nonumber \\\\&+& [1+x,y,x,1+x]- [1+x,y,x-y,1+x]+ [1+x,y,x-y,1+y]+ [1+x,y,1+y,1+x] \\nonumber \\\\&-& [1+x,1+y,x,1+x]- [1+x,1+y,y,1+y] + (x \\leftrightarrow y) \\, .$ This symbol corresponds to the one found in [42] when appropriate differential operator and change of variables are taken into account.",
"One can now express the symbol in term of functions.",
"To this extent, we have considered the letters $\\lbrace x, 1+x, y, 1+y, x-y\\rbrace $ and their combinations as arguments of degree four classical harmonic polylogarithms.",
"Nevertheless, since the constraint (REF ) is not satisfied by $\\mathcal {S}(F^{(2)}_c)$ , the expression cannot be written in terms of classical polylogarithms only.",
"Therefore additional functions must be added, and we find that in case $(c)$ only the following contribute $\\mathcal {W}_1 &=& \\int _0^x \\frac{H_{011}(-z) - H_{011}(-y)}{z-y} dz \\, ,\\\\\\mathcal {W}_2 &=& \\int _0^y \\frac{H_{011}(-z) - H_{011}(-x)}{z-x} dz$ and appear in the final expression of $F^{(2)}_c$ .",
"Imposing the validity of the differential equation and the boundary conditions implies that the terms to be subtracted are $\\mathcal {B}_c = \\frac{\\pi ^4}{60} + 2 \\log (1+x) \\zeta _3 +\\frac{\\pi ^2}{6} \\left(\\log x \\log (1+x) - \\frac{1}{2} \\log (1+x)^2+ {\\rm Li} _2(-x)\\right) + (x \\rightarrow y)$ which are indeed lower-degree functions multiplied by constants of the right degree.",
"The final expression for the two-loop integral is $F^{(2)}_c(x,y) = -2 \\mathcal {W}_1 -2 \\mathcal {W}_2 -\\mathcal {B}_c + {\\rm classical \\, polylogs} \\, .$ The classical-polylogarithm part is lengthy and explicitly showed in appendix .",
"It deserves further investigation to be simplified.",
"Figure: Double pentaladder integrals which obey the same iterative differential equations.To conclude this section, we want to remark again that the differential equations appearing in this and in the other cases are valid at all-loop order $\\mathcal {O}_{24} \\mathcal {O}_{46} \\frac{1}{N_{Lc}} F_{c}^{(L)} = \\frac{1}{(3456)} \\frac{1}{N_{(L-1)c}} F_c^{(L-1)}\\, ,$ where with $ F_{c}^{(L)}$ we mean the double pentaladder integrals of Fig.",
"REF , where the pentagon subintegrals correspond case by case to the ones at two loops.",
"which are present in loop amplitudes.",
"This iterative structure, relating higher-loop orders to one-loop hexagons, is very powerful since it allows to obtain analytical results from one-loop information."
],
[
"Double-pentagon integrals $F^{(2)}_{a}$ and {{formula:ff4cd5a3-ba01-47ee-b543-b5bb57934baa}}",
"We now consider the other two topologies appearing in the two-loop eight-point MHV amplitude.",
"The case $F_{a}^{(2)}$ is the simplest, as it turns out to depend on one cross-ratio only when projected to the two-dimensional subspace.",
"Considering the massive pentagon, we have in four dimensions $\\mathcal {O}_{24} \\mathcal {O}_{86} \\frac{1}{(3782)} F_{a}^{(2)} = -\\frac{1}{(3467)} F_{a}^{(1)}\\,,$ where $F_{a}^{(1)}$ is the one-loop hexagon integral with one massive and one massless corner as in Fig.",
"REF (a).",
"The reduction to two-dimensional kinematics leaves only one independent cross-ratio and reduces the differential equation to $y \\partial _y (1+y) \\partial _y F_{a}^{(2)}(y) = F_{a}^{(1)}(y) \\, ,$ where the boundary condition is $F_{a}^{(2)}(y\\rightarrow \\infty )=0$ .",
"The inhomogeneous term is known [22] and in two dimensions becomes $F^{(1)}_{a}(y ) &=& - \\log \\frac{1}{1+y} \\log (\\frac{y}{1+y}) - 2 {\\rm Li} _2 (\\frac{1}{1+y}) \\, .$ Actually in this particular case it would be easily possible to directly write the solution in terms of harmonic polylogarithms.",
"If we nevertheless make use of symbology, starting from the symbol for $F^{(1)}_{a}(y )$ , ${\\mathcal {S}}(F_{a}^{(1)}) = - \\left[\\frac{1}{(1+y)} , \\frac{y}{(1+y)} \\right]+ \\left[\\frac{y}{(1+y)} , \\frac{1}{(1+y)}\\right]\\, ,$ and given the dependence of the integrals from one cross-ratio only, we can directly write the result coming from the differential equation at $L$ -th loop order ${\\cal S}(F_{a}^{(L)}) &=& [{\\cal S}(F_{a}^{(1)}) \\underbrace{, \\frac{1}{y} , \\frac{1}{(1+y)} , \\ldots , \\frac{1}{y} , \\frac{1}{(1+y)}}_{2(L-1) - {\\rm fold}\\;{\\rm tensor}\\;{\\rm product}} ] \\,.$ In terms of functions, the two-loop expression for the case $(a)$ is easily written as $F_{a}^{(2)} &=& -\\text{Li}^2_2(-y)-2 \\text{Li}_4(-y)-2 \\text{Li}_4\\left(\\frac{1}{y+1}\\right)-2 \\text{Li}_4\\left(\\frac{y}{y+1}\\right) \\nonumber \\\\&& -\\frac{1}{3} \\pi ^2 \\left(\\text{Li}_2(-y)-\\frac{1}{2} \\log ^2(y+1)+\\log (y) \\log (y+1)\\right) \\nonumber \\\\&& -\\text{Li}_2(-y) \\log (y+1) \\log (y)+2 \\text{Li}_3(-y) \\log (y)+2 \\text{Li}_3\\left(\\frac{y}{y+1}\\right) \\log (y) \\nonumber \\\\&& -\\text{Li}_3(-y) \\log (y+1)-2 \\zeta (3) \\log (y+1)-\\frac{1}{4} \\log ^4(y)-\\frac{1}{12} \\log ^4(y+1) \\nonumber \\\\&& -\\frac{1}{12} \\log ^4\\left(\\frac{y+1}{y}\\right)+\\frac{2}{3} \\log (y+1) \\log ^3(y)-\\frac{1}{3} \\log ^3\\left(\\frac{y+1}{y}\\right) \\log (y)\\nonumber \\\\&& -\\frac{1}{2} \\log ^2(y+1) \\log ^2(y)-\\frac{2 \\pi ^4}{45} \\, .$ The case $F_{b}^{(2)}$ is more similar to case (c) and the independence on the dual points $x_{1}$ and $x_{5}$ reduces the number of cross-ratios.",
"Let us consider again the differential operators adapted to the massive corners as in Section , i.e $\\mathcal {O}_{24} \\mathcal {O}_{75} \\frac{1}{(3672)} F_{b}^{(2)} = -\\frac{1}{(3456)} F_{b}^{(1)}\\,.$ After utilization of the chain rule and the verification of the compatibility between the differential operator and the two-dimensional kinematics, we can write: $x \\partial _{x} y \\partial _{y} F_{b}^{(2)}(x,y) = F_{b}^{(1)}(x,y) \\,,$ where the hexagon diagram is given by [22] $F_{b}^{(1)}(x,y) &=&\\frac{x }{(y - x)} \\left[2 \\, {\\rm Li} _2(-x) + 2 \\zeta _2 - \\log x \\log y +\\log ( x y ) \\log (1 + x)\\right] + (y \\leftrightarrow x) \\, .$ As discussed in section REF , the double soft limit $p_5, p_6 \\rightarrow 0 $ fixes the boundary conditions to $F_{b}^{(2)}(x\\rightarrow 0,y\\rightarrow \\infty ) = 0 $ and the symmetric one $F_{b}^{(2)}(x\\rightarrow \\infty ,y\\rightarrow 0)=0$ .",
"Using the differential equation and the integrability condition as before, we arrive at $\\mathcal {S}(F_b^{(2)}) &=& [x,x,y,y]- [x,x,1+y,y]+ [x,1+x,x-y,x]- [x,1+x,x-y,y] \\nonumber \\\\&-& [x,1+x,y,x]+ [x,1+x,1+y,y]+ [x,y,x,y]- [x,y,1+x,x] \\nonumber \\\\&+& [x,y,y,x]- [x,y,1+y,y]- [x,1+y,x,y]+ [x,1+y,1+x,x] \\nonumber \\\\&-& [x,1+y,x-y,x]+ [x,1+y,x-y,y]- [1+x,x,x-y,x]+ [1+x,x,x-y,y] \\nonumber \\\\&+& [1+x,x,y,x]- [1+x,x,y,y]+ [1+x,y,x-y,x]- [1+x,y,x-y,y] \\nonumber \\\\&-& [1+x,y,y,x]+ [1+x,y,1+y,y] + (x\\leftrightarrow y) \\, .$ From the failure of the constraint (REF ), as in case $(c)$ , the final result will contain functions different from classical polylogarithms.",
"For this diagram two of such terms are necessary: $\\mathcal {W}_3 &=& \\int _0^x \\frac{H_{001}(-z) - H_{001}(-y)}{z-y} dz \\\\ \\,\\mathcal {W}_4 &=& \\int _0^y \\frac{H_{001}(-z) - H_{001}(-x)}{z-x} dz \\,$ and the boundary conditions imply the subtraction of the following terms $\\mathcal {B}_b = \\frac{\\pi ^4}{45}+\\frac{\\pi ^2}{12} \\log ^2 x -\\frac{\\pi ^2 }{4} \\log ^2(1+x)+4 \\zeta _3 \\log x + 2 \\zeta _3 \\log (1+x) +\\frac{\\pi ^2}{3} \\log x \\log y + (x \\rightarrow y) \\, ,$ The final result will therefore be $F_b^{(2)}(x,y) = - 2 \\mathcal {W}_3 - 2 \\mathcal {W}_4 -\\mathcal {B}_b + {\\rm classical \\, polylogs}$ where the classical-polylogarithm terms are presented in appendix .",
"As for case $(c)$ , we can extend these differential equations to the ladder diagrams depicted in Fig.",
"REF ."
],
[
"Two-loop result",
"We combine now the previous results found for the finite diagrams appearing in the two-dimensional two-loop MHV scattering amplitude.",
"To do so, we must consider the sum over cyclic permutations of each diagram.",
"This is taken into account by evaluating for the three different cases: $F_{a}^{(2)} &=& F_{a}^{(2)}(x) + F_{a}^{(2)}(y) + F_{a}^{(2)}\\left(\\frac{1}{x}\\right) + F_{a}^{(2)}\\left(\\frac{1}{y}\\right) \\\\F_{b,c}^{(2)} &=& F_{b,c}^{(2)}(x,y) + F_{b,c}^{(2)}\\left(y,\\frac{1}{x}\\right) + F_{b,c}^{(2)}\\left(\\frac{1}{x},\\frac{1}{y}\\right) + F_{b,c}^{(2)}\\left(\\frac{1}{y},x\\right)$ and considering twice the diagrams $(a)$ and $(c)$ .",
"The symbol is then given by $\\mathcal {S}(F^{(2)}) &=& [x,x,y,y]-2 [x,1+x,x,y]+4 [x,1+x,x,1+y]-2 [x,1+x,y,x] \\nonumber \\\\&+& 4 [x,1+x,1+y,x]+ [x,y,x,y]-2 [x,y,1+x,x] + [x,y,y,x] \\nonumber \\\\&-& 2 [x,y,1+y,y]+4 [x,1+y,1+x,x]+2 [x,1+y,y,y] + 2 [1+x,x,x,y] \\nonumber \\\\&-& 4 [1+x,x,x,1+y]+2 [1+x,x,y,x]-4 [1+x,x,1+y,x]+2 [1+x,y,x,x] \\nonumber \\\\&+& 4 [1+x,y,1+y,y]-4 [1+x,1+y,x,x]-4 [1+x,1+y,y,y]+ [y,x,x,y] \\nonumber \\\\&-& 2 [y,x,1+x,x]+ [y,x,y,x]-2 [y,x,1+y,y]+2 [y,1+x,x,x] \\nonumber \\\\&+& 4 [y,1+x,1+y,y]+ [y,y,x,x]-2 [y,1+y,x,y]+4 [y,1+y,1+x,y] \\nonumber \\\\&-& 2 [y,1+y,y,x]+4 [y,1+y,y,1+x]+4 [1+y,x,1+x,x]+2 [1+y,x,y,y] \\nonumber \\\\&-& 4 [1+y,1+x,x,x]-4 [1+y,1+x,y,y]+2 [1+y,y,x,y]-4 [1+y,y,1+x,y] \\nonumber \\\\&+& 2 [1+y,y,y,x]-4 [1+y,y,y,1+x] \\, .$ We want to remark here that all the terms with the $(x-y)$ letter have disappeared.",
"This was indeed noticed already through the differential equations of [42].",
"The expression for the highest-degree terms of the final function drastically simplifies: $F^{(2)}_{\\rm highest} &=& 2 \\text{Li}_2(-x) \\log (x) \\log (y)-4 \\text{Li}_2(-y) \\log (1+x) \\log (y)+2 \\text{Li}_2(-y) \\log (x) \\log (y) \\nonumber \\\\&-& 6 \\text{Li}_3(-x) \\log (y)-4 \\text{Li}_2(-x) \\log (x) \\log (1+y)+12 \\text{Li}_3(-x) \\log (1+y) \\nonumber \\\\&+& 12 \\text{Li}_3(-y) \\log (1+x)-6 \\text{Li}_3(-y) \\log (x)+\\frac{1}{4} \\log ^2(x) \\log ^2(y) \\, .$ We have checked that taking the $x$ and $y$ derivatives of $F^{(2)}_{\\rm highest} $ , we find the same function of [42].",
"One important point that we want to stress here is that the functions which were not expressible in terms of classical polylogarithms have now disappeared: if we indeed use the relation (REF ) for the symbol of $F^{(2)}$ we correctly find that it vanishes."
],
[
"Two-loop $n$ -point and {{formula:bb4ff4db-9331-4046-91f0-51a4b2d27cae}} -loop ladder integrals in two dimensions",
"We want to consider now particular diagrams appearing in scattering amplitudes with high number of legs and at higher loop, to see which topologies survive to the two-dimensional reduction.",
"We start by looking at double-pentagon integrals with arbitrary number of legs in four dimensions.",
"To this extent, we use the Feynman parametrization, $\\frac{1}{A_1^{\\lambda _1}... A_n^{\\lambda _n}} = \\frac{\\Gamma (\\lambda _1+...+\\lambda _n)}{\\Gamma (\\lambda _1)...\\Gamma (\\lambda _n)} \\int _0^{\\infty } d\\alpha _1...d\\alpha _n \\delta (\\sum \\alpha _i -1) \\frac{\\alpha _1^{\\lambda _1-1}...\\alpha _n^{\\lambda _n-1}}{(\\alpha _1 A_1 +...+\\alpha _n A_n)^{\\lambda _1+...+\\lambda _n}}$ to rewrite the general formula for double pentagons $\\mathcal {F}^{(2)}_n = N \\int \\frac{d^4Z_{AB}}{i \\pi ^2} \\frac{d^4Z_{CD}}{i \\pi ^2} \\frac{(A B i j)(C D k l)}{(AB \\,a-1\\,a)...(ABCD)...(CD\\,b-1\\,b)} \\, ,$ where $i, j, k$ and $l$ denote the legs of the tensor numerators in the pentagons, in terms of the Feynman parameters $\\alpha _i$ and $\\beta _i$ $\\mathcal {F}^{(2)}_n \\propto \\int _0^{\\infty } \\Pi _i d\\alpha _i d\\beta _i \\delta (\\sum \\alpha _i -1) \\delta (\\sum \\beta _i -1) \\frac{f(\\alpha _i, \\beta _i) }{\\mathcal {M}} \\left(6 \\frac{(i j Y) (k l Y)}{(Y Y)^4}-\\frac{(i j k l) }{(Y Y)^3}\\right) \\, .$ In going from (REF ) to (REF ) we have performed several steps.",
"At first, one has to Feynman parametrize one of the two loops, say the $(AB)$ one, and then integrate over the $d^4 Z_{AB}$ following the techniques of [47].",
"After the integration of a specific Feynman parameter, the same steps for the second loop $(CD)$ leads to (REF ).",
"For what we want to argue, we do not need to specify the function $f(\\alpha , \\beta )$ , neither the $\\mathcal {M}$ denominator, which is just a combination of four-brackets of the type $(Z_{a-1} \\, Z_{a} \\,Z_{b-1} \\, Z_b )$ .",
"$Y$ instead is a particular bitwistor: it is indeed a linear combination of bitwistors of the form $(Z_{m-1}, Z_m)$ with coefficients the Feynman parameters.",
"This particular form is given by the parametrization, which combines indeed the propagators in one denominator.",
"The interesting term for what we are going to discuss, is the combination in the bracket $6 \\frac{(i j Y) (k l Y)}{(Y Y)^4}-\\frac{(i j k l) }{(Y Y)^3} \\, .$ The reduction to two dimensions makes indeed vanish some or even all the terms in (REF ) in particular configurations.",
"Given the special form of $Y$ , the first term will survive only when $(i j)$ and $(k l)$ $ \\sim ({{\\rm even ~odd}})$ .",
"Concerning the second term, many more combinations will survive for $(i j k l)$ , provided that two of them are odd and two even.",
"Requiring the number of legs to be even, as we are in two dimensions, the only topologies which survive are depicted in Fig.",
"REF , where with two (three) legs we mean an even (odd) number of them.",
"Figure: The double pentaladder diagrams which simplify in two dimensions.We want now to comment on the reduction to two dimensions for the double pentaladder integrals of Fig.",
"REF , where any number of legs is inserted between the pentagons and the boxes.",
"When written in the Feynman parametrization they will contain again terms of the type $(1 3 i i+3)$ and $(1 3 Y)$ , where $Y$ is a bitwistor formed by some linear combination of $(Z_{m-1}, Z_m)$ as before.",
"Therefore, due to the particular structure of these diagrams, it actually happens that they are subject to strong simplifications when projected to two dimensions.",
"Moreover, using the differential operators we have worked out and in particular the ones similar to the case $(b)$ , it can be showed that at three loop they are reduced to two-loop diagrams which vanish in two dimensions, given the comments above (see Fig.",
"REF for an example).",
"This is a very powerful constraint for the function, as it fixes the inhomogeneous term of the differential equation to be zero.",
"$\\mathcal {D}^{(2)}~{\\parbox [c]{40mm}{\\includegraphics [height = 3cm] {3loop.eps}}}\\qquad \\propto \\,{\\parbox [c]{30mm}{\\includegraphics [height = 3cm] {vanish.eps}}}\\qquad = 0 \\, ({\\rm two-dim})$ Therefore, given the differential operators and the two-dimensional kinematics, the analytic results for the topologies present in $L$ -loop $n$ -point scattering amplitudes strongly simplify."
],
[
"Acknowledgments",
"I would like to thank J. Drummond and J. Henn for collaboration at different stages of this project.",
"This work was supported by the Volkswagen-Foundation."
],
[
"Review on symbology",
"In this appendix we want to recall the notion of symbol and its properties.",
"A transcendental function of degree $k$ is defined iteratively as the linear combination with rational coefficients of $k$ iterated integrals: $f_k &=& \\int _a^b d\\log R_1 \\otimes ... \\otimes d\\log R_k \\\\&=& \\int _a^b d\\log R_k(t) \\left[\\int _0^t d\\log R_1 \\otimes ... \\otimes d\\log R_{k-1}\\right] \\, ,$ where $R_i$ are rational functions.",
"For instance the trilogarithm $ {\\rm Li} _3(z)$ is given by $ {\\rm Li} _3(z) = - \\int _0^z d\\log t_1 \\int _0^{t_1} d\\log t_2 \\int _0^{t_2} d\\log (1-t_3) \\, .$ The symbol of the pure function $f_k$ is defined as $\\mathcal {S}(f_k) = R_1 \\otimes ... \\otimes R_{k} \\equiv \\left[R_1,..., R_k\\right] \\, ,$ where for sake of simplicity throughout the paper we use a square-bracket notation.",
"For instance the symbol of $ {\\rm Li} _3(z)$ is $\\mathcal {S}( {\\rm Li} _3(z)) = - [1-t, t, t] \\, .$ The symbol of a function makes clear the locations of the discontinuities of the function.",
"It satisfies, besides the shuffle algebra, many properties which are useful in simplifying expressions.",
"For instance, for given rational functions $A, B$ and $ \\frac{a_i a_j}{a_k}$ we have $\\left[A, \\frac{a_i a_j}{a_k}, B\\right] = \\left[A, a_i, B\\right] + \\left[A, a_j, B\\right]-\\left[A, a_k, B\\right] \\, ,$ while if $c$ is a constant different from zero, the symbol is invisible to it: $\\left[A, c~ a_i, B\\right] = \\left[A, a_i, B\\right] \\, .$ The derivative of a symbol is instead performed acting on its last entry: $\\frac{\\partial }{\\partial x} [a_1,...,a_n] = [a_1,...,a_{n-1}] \\times \\frac{\\partial }{\\partial x} \\log a_n \\, .$ The symbol loses some information about the function.",
"Besides missing the knowledge of which logarithmic branch the integrand of an iterated integral is on, at a given transcendental degree it does not detect functions which are transcendental constants times pure functions of lower degree.",
"Let us consider for instance the following identity $ {\\rm Li} _2(-z) + {\\rm Li} _2\\left(-\\frac{1}{z}\\right) = -\\frac{1}{2} \\log ^2(z)-\\frac{\\pi ^2}{6} \\, ,$ and compute the symbol of the left-hand side $\\mathcal {S}( {\\rm Li} _2(-z)) + \\mathcal {S}\\left( {\\rm Li} _2\\left(-\\frac{1}{z}\\right)\\right) = - [z,z] \\, .$ Recalling the symbol of the $n$ -th power of the logarithm $\\mathcal {S}\\left(\\log ^n(z)\\right) = n!",
"\\, \\underbrace{[z,..., z]}_{n-times} \\, ,$ we see that the $\\frac{\\pi ^2}{6}$ term is missed.",
"Due to the fact that $d^2 f_k = 0$ , the letters of its symbol $[R_1,...,R_i, R_{i+1},...R_k]$ are not all independent, but constrained by the integrability condition $d\\log R_i \\wedge d\\log R_{i+1} [R_1,...,R_{i-1}, R_{i+2},...R_k] = 0 \\, .$ This means that, if to a pure function always corresponds a symbol, the other way round is not always true.",
"For instance $[x, y]$ is not the symbol of a function, since it does not satisfy (REF ), but $\\left([x, y] + [y, x]\\right)$ is.",
"There is a further constraint that we want to mention here, as it is used in the main text.",
"For degree $k \\leqslant 4$ , Goncharov conjectured that symbols obeying $\\mathcal {S}^{(k)}_{abcd} - \\mathcal {S}^{(k)}_{bacd} - \\mathcal {S}^{(k)}_{abdc} + \\mathcal {S}^{(k)}_{badc} - \\left(a \\leftrightarrow c, b\\leftrightarrow d\\right) = 0$ can be obtained from a function involving classical polylogarithms only."
],
[
"Explicit expression for the classical HPLs in $F_b^{(2)}$",
"In the following, we present the expression given by the symbol we have found for $F_b^{(2)}$ when written in terms of HPLs.",
"We recall that, in order to get the complete answer, two functions which are not classical HPLs and lower-degree functions multiplied by constants must be added.",
"$F_{b,class}^{(2)}&=& \\frac{\\log ^4(x)}{12}-\\frac{1}{6} \\log (y) \\log ^3(x)+\\frac{1}{4} \\log ^2(x+1) \\log ^2(x)+\\frac{1}{4} \\log ^2(y) \\log ^2(x)-\\frac{1}{4} \\log ^2(y+1) \\log ^2(x) \\nonumber \\\\&+&\\frac{1}{2} \\text{Li}_2(-x) \\log ^2(x)-\\frac{1}{2} \\text{Li}_2\\left(1-\\frac{x+1}{y+1}\\right) \\log ^2(x) +\\frac{1}{12} \\log ^3(x+1) \\log (x)+\\frac{1}{6} \\log ^3(y) \\log (x) \\nonumber \\\\&+&\\frac{1}{12} \\log ^3(y+1) \\log (x)-\\frac{1}{4} \\log (x+1) \\log ^2(y+1) \\log (x)+\\frac{1}{4} \\log ^2(x+1) \\log (y+1) \\log (x) \\nonumber \\\\&-&\\log (x+1) \\text{Li}_2(-x) \\log (x)+\\log (y) \\text{Li}_2(-x) \\log (x)-\\frac{1}{2} \\log (y+1) \\text{Li}_2(-x) \\log (x) \\nonumber \\\\&+&\\frac{1}{2} \\log (x+1) \\text{Li}_2\\left(1-\\frac{x}{y}\\right) \\log (x)-\\frac{1}{2} \\log (y+1) \\text{Li}_2\\left(1-\\frac{x}{y}\\right) \\log (x)-\\frac{1}{2} \\log (x+1) \\text{Li}_2(-y) \\log (x) \\nonumber \\\\&+&\\log (y) \\text{Li}_2(-y) \\log (x)+\\text{Li}_3(-x) \\log (x)-3 \\text{Li}_3(-y) \\log (x)-\\frac{5}{2} H_{1,0,1}(-x) \\log (x) \\nonumber \\\\&+&\\frac{1}{2} H_{1,0,1}\\left(\\frac{x-y}{x}\\right) \\log (x)-\\frac{1}{2} H_{1,0,1}(-y) \\log (x)-\\frac{1}{2} H_{1,0,1}\\left(\\frac{y-x}{(x+1) y}\\right) \\log (x) \\nonumber \\\\&+&\\frac{1}{2} H_{1,0,1}\\left(1-\\frac{x+1}{y+1}\\right) \\log (x)-\\frac{1}{24} \\log ^4(x+1)-\\frac{\\log ^4(y)}{12}-\\frac{1}{24} \\log ^4(y+1)+\\frac{1}{4} \\log (y) \\log ^3(y+1) \\nonumber \\\\&-&\\frac{1}{4} \\log ^2(x+1) \\log ^2(y)+\\frac{1}{4} \\log ^2(y) \\log ^2(y+1)-\\frac{1}{4} \\log (x+1) \\log (y) \\log ^2(y+1)-\\frac{3}{2} \\text{Li}^2_2(-x){} \\nonumber \\\\&-&\\frac{3}{2} \\text{Li}^2_2(-y){}-\\frac{1}{12} \\log ^3(x+1) \\log (y)+\\frac{1}{4} \\log ^2(x+1) \\log (y) \\log (y+1)+\\frac{1}{2} \\log ^2(x+1) \\text{Li}_2(-x) \\nonumber \\\\&-&\\frac{1}{2} \\log (y) \\log (y+1) \\text{Li}_2(-x)+\\frac{1}{2} \\log (x+1) \\log (y) \\text{Li}_2\\left(1-\\frac{x}{y}\\right)-\\frac{1}{2} \\log (y) \\log (y+1) \\text{Li}_2\\left(1-\\frac{x}{y}\\right) \\nonumber \\\\&+&\\frac{1}{2} \\log ^2(y) \\text{Li}_2(-y)+\\frac{1}{2} \\log ^2(y+1) \\text{Li}_2(-y)-\\frac{1}{2} \\log (x+1) \\log (y) \\text{Li}_2(-y)-\\log (y) \\log (y+1) \\text{Li}_2(-y) \\nonumber \\\\&-&\\text{Li}_2(-x) \\text{Li}_2(-y)+\\frac{1}{2} \\log ^2(y) \\text{Li}_2\\left(1-\\frac{x+1}{y+1}\\right)-2 \\log (x+1) \\text{Li}_3(-x)-3 \\log (y) \\text{Li}_3(-x) \\nonumber \\\\&+&\\log (y) \\text{Li}_3(-y)-2 \\log (y+1) \\text{Li}_3(-y)+\\log (x+1) H_{1,0,1}(-x)-\\frac{1}{2} \\log (y) H_{1,0,1}(-x) \\nonumber \\\\&+&\\frac{1}{2} \\log (y) H_{1,0,1}\\left(\\frac{x-y}{x}\\right)-\\frac{5}{2} \\log (y) H_{1,0,1}(-y)+\\log (y+1) H_{1,0,1}(-y)-\\frac{1}{2} \\log (y) H_{1,0,1}\\left(\\frac{y-x}{(x+1) y}\\right) \\nonumber \\\\&+&\\frac{1}{2} \\log (y) H_{1,0,1}\\left(1-\\frac{x+1}{y+1}\\right)+3 H_{0,1,0,1}\\left(\\frac{1}{x+1}\\right)+3 H_{0,1,0,1}\\left(\\frac{1}{y+1}\\right)+2 H_{1,1,0,1}(-x) \\nonumber \\\\&+&2 H_{1,1,0,1}\\left(\\frac{1}{x+1}\\right)+2 H_{1,1,0,1}(-y)+2 H_{1,1,0,1}\\left(\\frac{1}{y+1}\\right)$"
],
[
"Explicit expression for the classical HPLs in $F_c^{(2)}$",
" $&+&\\log (y) H_{1,0,1}(-x)-\\frac{1}{2} \\log (y+1) H_{1,0,1}(-x)+\\frac{1}{2} \\log (y+1) H_{1,0,1}\\left(\\frac{x-y}{x}\\right)+\\log (x) H_{1,0,1}(-y) \\nonumber \\\\&+&\\frac{5}{6} \\log (y+1) H_{1,0,1}(-y)+\\frac{1}{2} \\log (y+1) H_{1,0,1}\\left(\\frac{y-x}{(x+1) y}\\right)-\\log (x) H_{1,0,1}\\left(1-\\frac{x+1}{y+1}\\right) \\nonumber \\\\&+& \\frac{4}{3} H_{1,1,0,1}(-y) + H_{0,1,0,1}\\left(\\frac{1}{y+1}\\right) +\\frac{4}{3} H_{1,1,0,1}(-x) +H_{0,1,0,1}\\left(\\frac{1}{x+1}\\right)$"
],
[
"Special choice of twistors for two-dimensional kinematics",
"We implement the two-dimensional kinematics by making this special choice for the momentum twistor variables: $Z_1=\\left( \\begin{array}{c} 0 \\\\ 0 \\\\ i \\sqrt{2} \\\\ - i \\sqrt{2} \\\\ \\end{array} \\right),~~~Z_2=\\left( \\begin{array}{c} -i \\sqrt{2} \\\\ \\frac{i}{\\sqrt{2}} \\\\ 0 \\\\ 0 \\\\ \\end{array} \\right),~~~Z_3=\\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ \\frac{i}{\\sqrt{2}} \\\\ \\end{array} \\right),~~~Z_4=\\left( \\begin{array}{c} i \\sqrt{2} y \\\\ \\frac{i (1-y)}{\\sqrt{2}} \\\\ 0 \\\\ 0 \\\\ \\end{array} \\right) \\nonumber \\\\Z_5=\\left( \\begin{array}{c} 0 \\\\ 0 \\\\ i \\sqrt{2} x \\\\ \\frac{i (1-x)}{\\sqrt{2}} \\\\ \\end{array} \\right),~~~Z_6=\\left( \\begin{array}{c} 0 \\\\ \\frac{i}{\\sqrt{2}} \\\\ 0 \\\\ 0 \\\\ \\end{array} \\right),~~~Z_7=\\left( \\begin{array}{c} 0 \\\\ 0 \\\\ - i \\sqrt{2} \\\\ \\frac{i}{\\sqrt{2}} \\\\ \\end{array} \\right),~~~Z_8=\\left( \\begin{array}{c} i \\sqrt{2} \\\\ - i \\sqrt{2} \\\\ 0 \\\\ 0 \\\\ \\end{array} \\right)$"
],
[
"Definitions of $F_{a,b,c}^{(1)}$ and {{formula:6fbbbb44-db5a-4f8c-a025-b6a8b1e828e2}} diagrams",
"In this appendix we give the definition of two-loop double-pentagon and one-loop hexagon integrals, with the respective normalizations, which are used throughout the paper:"
],
[
"Integrals of type (c)",
"where $N_{2c} = -(1234)\\left[(3781)(5624)+(7812)(3456)\\right]$ $F_{c}^{(1)} = \\int \\frac{d^4Z_{AB}}{i \\pi ^2} \\frac{(1234)(6781) (AB28) (AB35) }{(AB12) (AB23) (AB34) (AB56) (AB78)(AB81)}$"
]
] | 1204.1031 |
[
[
"Minimality of planes in normed spaces"
],
[
"Abstract We prove that a region in a two-dimensional affine subspace of a normed space $V$ has the least 2-dimensional Hausdorff measure among all compact surfaces with the same boundary.",
"Furthermore, the 2-dimensional Hausdorff area density admits a convex extension to $\\Lambda^2 V$.",
"The proof is based on a (probably) new inequality for the Euclidean area of a convex centrally-symmetric polygon."
],
[
"Introduction",
"The main purpose of this paper is to prove the following seemingly elementary fact.",
"Consider a bounded region in a two-dimensional affine plane in a normed vector space.",
"Then the region has the least possible two-dimensional Hausdorff measure among all compact two-dimensional surfaces with the same boundary.",
"Even though the problem sounds almost silly, it stood open for over 50 years and still remains open in dimensions greater than 2.",
"We prove our result by showing the convexity of the density function for the Busemann–Hausdorff surface area in normed spaces.",
"This contrasts with a result of Busemann, Ewald and Shephard [7] who demonstrated that the density of the Holmes–Thompson (symplectic) surface area may fail to be convex.",
"We first embed the problem in a more general set-up, borrowing some preliminaries from [3].",
"Here we consider various notions of $k$ -dimensional surface areas, although the rest of the paper is devoted to the case $k=2$ and the Busemann–Hausdorff definition of area in a normed space.",
"A common way to introduce a (translation-invariant) $k$ -dimensional surface area in $\\mathbb {R}^n$ is to define its density in every $k$ -dimensional linear subspace.",
"Namely for a continuous function $A\\colon G(n,k)\\rightarrow \\mathbb {R}_+$ , where $G(n,k)=G_k(\\mathbb {R}^n)$ is the Grassmannian manifold of $k$ -dimensional linear subspaces of $\\mathbb {R}^n$ , one defines the associated surface area functional $Area_A$ by the obvious formula $Area_A(S)=\\int _S A(T_xS)\\,dm(x),$ where $S$ is a smooth (more generally, Lipschitz) surface, $m$ is the $k$ -dimensional Euclidean surface area, and the tangent spaces $T_xS$ are regarded as points in $G(n,k)$ .",
"If $\\mathbb {R}^n$ is equipped with a norm $\\Vert \\cdot \\Vert $ , then every subspace $P\\in G(n,k)$ becomes a $k$ -dimensional normed space.",
"In the Busemann–Hausdorff definition of area, the density $A(P)$ is defined so that the area of the norm's unit ball (in $P$ ) equals the standard constant $\\varepsilon _k$ depending only on $k$ , namely $\\varepsilon _k$ is the Euclidean volume of the Euclidean unit ball in $\\mathbb {R}^k$ .",
"The resulting area density $A^{bh}\\colon G(n,k)\\rightarrow \\mathbb {R}_+$ has the form $A^{bh}(P)= \\frac{\\varepsilon _k}{m_k(P \\cap B)}$ where $B\\subset \\mathbb {R}^n$ is the unit ball of the norm $\\Vert \\cdot \\Vert $ and $m_k$ is the $k$ -dimensional Euclidean area.",
"The corresponding surface area functional has a clear geometric meaning: for embedded surfaces, it is just the $k$ -dimensional Hausdorff measure of the surface as a subset of the normed space.",
"Remark 1.1 Another commonly used notion is the Holmes–Thompson area whose density $A^{ht}$ is given by $A^{ht}(P)= \\frac{1}{\\varepsilon _k} m_k((P \\cap B)^*),$ where $(P \\cap B)^*$ is the polar body to $P\\cap B$ with respect to the Euclidean structure in $P$ .",
"Note that $(P\\cap B)^*$ is the orthogonal projection of $B^*$ to $P$ where $B^*$ is polar to $B$ in $\\mathbb {R}^n$ .",
"These two definitions of surface area come from Finsler geometry.",
"A smooth immersed surface in a normed space naturally acquires the induced structure of a Finsler manifold exactly the same way as a surface in Euclidean space gets a Riemannian structure.",
"The surface areas in question can be regarded as Finsler volumes in the induced Finsler metric and belong to its intrinsic geometry.",
"In Riemannian geometry, Riemannian volume bears two main meanings.",
"Geometrically, it is the Hausdorff measure.",
"Dynamically, it is the projection of the Liouville measure from the unit tangent bundle.",
"In Finsler geometry, there is no notion of volume which would enjoy the two properties.",
"The Busemann–Hausdorff and Holmes–Thompson definitions inherit one of the properties of Riemannian volume each.",
"The Busemann–Hausdorff volume of a $k$ -dimensional Finsler manifold $M$ equals the $k$ -dimensional Hausdorff measure of the Finsler metric (see [4]).",
"The Holmes–Thompson volume of $M$ (see [10]) is the (normalised by a suitable constant) symplectic volume of the bundle of the unit balls in $T^*M$ .",
"The two Finsler volumes can be expressed by the following coordinate formulas (for $\\Omega \\subset M$ identified with a subset in $\\mathbb {R}^k$ , that is in one chart).",
"For the Busemann–Hausdorff volume one gets $\\operatorname{vol}^{bh}(\\Omega ) = {\\varepsilon _k} \\int _\\Omega m_k(B_x)^{-1} \\,dm(x),$ and the Holmes–Thompson volume is given by $\\operatorname{vol}^{ht}(\\Omega ) = \\frac{1}{\\varepsilon _k} \\int _\\Omega m(B^*_x) \\,dm(x),$ where $m_k$ is the coordinate Lebesgue measure, $B_x$ is the unit ball of the Finsler norm at $x$ : $B_x=\\lbrace v \\in T_xM: \\Phi (v) \\le 1\\rbrace $ .",
"Here we identify $T_xM$ , $T^*_xM$ and $\\mathbb {R}^k$ since $M=\\mathbb {R}^k$ .",
"The normalizing coefficient $\\varepsilon _k$ makes the volume definitions agree with the Riemannian one for Riemannian manifolds.",
"In the above definition of area densities as functions on $G(n,k)$ , the Euclidean structure of $\\mathbb {R}^n$ is irrelevant.",
"Here is an affine-invariant definition.",
"Let $V$ be an $n$ -dimensional vector space, $\\Lambda ^k V$ the $k$ th exterior power of $V$ , and $GC_k(V)\\subset \\Lambda ^k V$ the $k$ -dimensional Grassmannian cone, that is the set of all simple $k$ -vectors.",
"(A $k$ -vector $\\sigma \\in \\Lambda ^k V$ is called simple if it is decomposable: $\\sigma =v_1\\wedge \\dots \\wedge v_k$ for some $v_1,\\dots ,v_k\\in V$ .)",
"A (translation invariant) $k$ -dimensional density in $V$ is a continuous function $A\\colon GC_k(V)\\rightarrow \\mathbb {R}_+$ which is symmetric and positively homogeneous, that is $A(\\lambda \\sigma )=|\\lambda | A(\\sigma )$ for all $\\lambda \\in \\mathbb {R}$ , $\\sigma \\in GC_k(V)$ .",
"For a Lipschitz surface $S\\colon M\\rightarrow V$ (parametrized by a smooth $k$ -dimensional manifold $M$ ), the integral $\\int _S A$ is defined in an obvious way.",
"We refer to this integral as the $k$ -dimensional surface area associated with $A$ and denote it by $Area_A(S)$ .",
"In the case of the Busemann–Hausdorff area in a normed space $(V,\\Vert \\cdot \\Vert )$ , the density $A^{bh}\\colon GC_k(V)\\rightarrow \\mathbb {R}_+$ is defined as follows.",
"For a simple $k$ -vector $\\sigma =v_1\\wedge \\dots \\wedge v_k$ , the value $A^{bh}(\\sigma )$ equals the $k$ -dimensional Hausdorff measure (with respect to the metric defined by $\\Vert \\cdot \\Vert $ ) of the parallelotope spanned by the vectors $v_1,\\dots ,v_k$ .",
"It can be expressed by the formula $A^{bh}(v_1\\wedge \\dots \\wedge v_k) = \\frac{\\varepsilon _k}{m_k(L^{-1}(B))},$ where $B$ is the unit ball of $\\Vert \\cdot \\Vert $ and $L\\colon \\mathbb {R}^k\\rightarrow V$ is a linear map given by $L(e_i)=v_i$ for the standard basis $(e_1,\\dots ,e_k)$ of $\\mathbb {R}^k$ .",
"We abuse notation and write $A^{bh}(S)$ instead of $Area_{A^{bh}}(S)$ for the Busemann–Hausdorff area of a surface $S$ .",
"For a Lipschitz chain $S=\\sum a_i S_i$ , $S_i\\colon \\Delta \\rightarrow \\mathbb {R}^n$ , where each $\\Delta $ is a standard simplex, we define $Area_A(S)=\\sum |a_i| Area_A(S_i)$ .",
"The coefficients $a_i$ can be taken from $\\mathbb {Z}$ , $\\mathbb {R}$ , or $\\mathbb {Z}_2:=\\mathbb {Z}/2\\mathbb {Z}$ .",
"In the case of $\\mathbb {Z}_2$ , the absolute values $|a_i|$ are defined as follows: $|a_i|=0$ if $a_i=0$ and $|a_i|=1$ otherwise.",
"Note that a two-dimensional chain over $\\mathbb {Z}$ or $\\mathbb {Z}_2$ can be parameterized by a manifold (which is oriented in the case of $\\mathbb {Z}$ ).",
"A density $A\\colon GC_k(V) \\rightarrow \\mathbb {R}$ is said to be convex if it can be extended to a convex function on the vector space $\\Lambda ^k V$ of all $k$ -vectors.",
"The functional $Area_A$ is said to be semi-elliptic over $\\mathbb {R}$ , $\\mathbb {Z}$ , or $\\mathbb {Z}_2$ (see [1]) if, whenever the boundary $\\partial S$ of a chain $S$ over the respective ring is equal to the boundary of a $k$ -disc $D$ embedded into an affine $k$ -plane, one has $Area_A(S)\\ge Area_A(D)$ .",
"It is rather obvious that convexity of $A$ implies semi-ellipticity of $Area_A$ over $\\mathbb {R}$ and $\\mathbb {Z}$ .",
"The converse is true over $\\mathbb {R}$ but in general may fail over $\\mathbb {Z}$ (see [3]).",
"For the Busemann-Hausdorff surface area density Busemann [5] proved that it is convex in co-dimension one, that is if $\\dim V=k+1$ , and left the general case as a conjecture (see e.g.",
"[8], [6] or [11]).",
"The main result of this paper is a proof of this conjecture for the 2-dimensional Busemann-Hausdorff surface area: Theorem 1 In every finite-dimensional normed space $V$ , the two-dimensional Busemann–Hausdorff area density admits a convex extension to $\\Lambda ^2 V$ .",
"Hence the area is semi-elliptic over $\\mathbb {Z}$ , that is, planar discs minimize the area among orientable surfaces with the same boundary.",
"Furthermore, an easy analysis of the proof shows that it also works in the non-orientable case: Theorem 2 In every finite-dimensional normed space $V$ , the two-dimensional Busemann–Hausdorff area density is semi-elliptic over $\\mathbb {Z}_2$ .",
"That is, every two-dimensional affine disc in $V$ minimizes the Busemann–Hausdorff area among all compact Lipschitz surfaces with the same boundary.",
"Remark 1.2 Since we do not assume that the norm of $V$ is strictly convex, the area functional is not elliptic in general.",
"However it is easy to show that the theorems imply that the Busemann–Hausdorff area is elliptic if the norm is strictly (quadratically) convex.",
"Remark 1.3 For the Holmes–Thompson surface area, it had been noticed by Busemann, Ewald and Shephard [7] that the density fails to be convex already for the two-dimensional surface area for a certain norm on $\\mathbb {R}^4$ .",
"Hence it is not elliptic over $\\mathbb {R}$ ([3], see also [2] for explicit examples.)",
"It turns out however that discs in affine 2-planes minimize the Homes–Thompson area among all surfaces (with the same boundary) parametrized by topological discs ([2]).",
"The problem of whether the Holmes–Thomson area is semi-elliptic over $\\mathbb {Z}$ (that is, for competing surfaces that may have handles) and beyond dimension 2 remains widely open and intriguing.",
"The rest of the paper is organized as follows.",
"Sections and are devoted to proofs of Theorems REF and REF respectively.",
"The proof of the main Theorem REF goes via constructing calibrating forms with constant coefficients, and the construction of the forms is based on a certain inequality for the Euclidean area of a convex centrally-symmetric polygon.",
"As it was pointed out by the anonymous referees, the key Proposition REF is connected to investigations related to areas of random triangles carried out by Blaschke and others in response to Sylvester's Four-Point Problem.",
"Namely, the proposition actually describes the probability measure supported on a centrally-symmetric convex curve so as to maximize the expectation of the area of a triangle formed by the center and two random points on the curve.",
"In Section we discuss prospectives and limitations of our methods for feasible generalizations to higher dimensions."
],
[
"Proof of Theorem ",
"Definition 2.1 Let $V$ be a finite-dimensional vector space, $A\\colon GC_k(V)\\rightarrow \\mathbb {R}_+$ a $k$ -dimensional density, and $P\\subset V$ a $k$ -dimensional linear subspace.",
"A calibrator (or a calibrating form) for $P$ with respect to $A$ is an exterior $k$ -form $\\omega \\in \\Lambda ^k V^*$ such that for every simple $k$ -vector $\\sigma \\in GC_k(V)$ , one has $|\\omega (\\sigma )|\\le A(\\sigma )$ , and this inequality turns into equality if $\\sigma \\in \\Lambda ^k P$ .",
"One easily sees that $A$ admits a convex extension to $\\Lambda ^k V$ if and only if every 2-plane $P$ admits a calibrator.",
"(Indeed, an exterior $k$ -form is a linear function on $\\Lambda ^kV$ , and a calibrator is a linear support for $A$ .)",
"Hence our plan is to give an explicit construction of such calibrators for $k=2$ and $A=A^{bh}$ .",
"Let $V$ be a finite-dimensional normed space and $B$ its unit ball.",
"By means of approximation, it suffices to prove Theorem REF in the case when $B$ is a polyhedron.",
"Fix a two-dimensional linear subspace $P\\subset V$ .",
"Our goal is to construct a calibrator for $P$ with respect to the Busemann–Hausdorff area density.",
"Consider the intersection $B\\cap P$ .",
"It is a centrally symmetric polygon $a_1a_2\\dots a_{2n}$ (whose center is the origin 0 of $V$ ).",
"For each $i=1,\\dots ,n$ , let $F_i\\colon V\\rightarrow \\mathbb {R}$ be a supporting linear function to $B$ such that $F_i=1$ on the segment $[a_ia_{i+1}]$ .",
"For each $i=1,\\dots ,n$ , let $p_i = \\frac{2A(\\triangle 0a_ia_{i+1})}{A(B\\cap P)}$ where $A$ is an (arbitrarily normalized) area form on $P$ and $\\triangle 0a_ia_{i+1}$ is the triangle with vertices $0,a_i,a_{i+1}$ .",
"Note that $\\sum p_i=1$ .",
"Define a 2-form $\\omega \\in \\Lambda ^2 V^*$ by $\\omega = \\pi \\cdot \\!\\!\\!",
"\\sum _{1\\le i<j\\le n} p_ip_j\\, F_i\\wedge F_j .$ We are going to prove that $\\omega $ is a desired calibrator for $P$ .",
"Consider a simple 2-vector $\\sigma =v_1\\wedge v_2$ where $v_1,v_2\\in V$ are linearly independent vectors.",
"We need to prove that $|\\omega (v_1\\wedge v_2)| \\le A^{bh}(v_1\\wedge v_2)$ with equality in the case when $v_1,v_2\\in P$ .",
"Identify the plane $(v_1,v_2)$ with $\\mathbb {R}^2$ by means of the linear embedding $I\\colon \\mathbb {R}^2\\rightarrow V$ that takes the standard basis of $\\mathbb {R}^2$ to $v_1$ and $v_2$ .",
"Then $|\\omega (v_1\\wedge v_2)| = |I^*\\omega | =\\pi \\cdot \\biggl | \\sum _{1\\le i<j\\le n} p_ip_j\\, f_i\\wedge f_j \\biggr |$ where $f_i=I^*F_i=F_i\\circ I$ and the norms of 2-forms are taken with respect to the Euclidean structure of $\\mathbb {R}^2$ .",
"The fact that $F_i$ is a supporting function for $B$ implies that $f_i\\le 1$ on the set $K=I^{-1}(B)$ which corresponds to the unit ball of the norm restricted to the plane $(v_1,v_2)$ .",
"By the definition of the Busemann–Hausdorff area, $A^{bh}(v_1\\wedge v_2) = \\frac{\\pi }{A(K)}$ where $A$ is the Euclidean area.",
"Thus the problem reduces to the following statement from convex geometry on the plane.",
"Proposition 2.2 Let $K\\subset \\mathbb {R}^2$ be a symmetric convex polygon, $f_1,\\dots ,f_n\\colon \\mathbb {R}^2\\rightarrow \\mathbb {R}$ are linear functions such that $f_i|_K\\le 1$ for all $i$ , and $p_1,\\dots ,p_n$ are nonnegative real numbers such that $\\sum p_i=1$ .",
"Then $\\biggl | \\sum _{1\\le i<j\\le n} p_ip_j\\, f_i\\wedge f_j \\biggr |\\le \\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j|\\le \\frac{1}{A(K)} .$ In addition, if $K$ is a convex $2n$ -gon $a_1a_2\\dots a_{2n}$ , $f_i$ are supporting functions of $K$ corresponding to its sides (that is, such that $f_i=1$ on $[a_ia_{i+1}]$ ), and $p_i = 2A(\\triangle 0a_ia_{i+1})/A(K)$ , then the above inequalities turn into equalities.",
"The proof of Proposition REF occupies the rest of this section.",
"It consists of several elementary lemmas.",
"Lemma 2.3 Let $K=a_1a_2\\dots a_{2n}$ be a symmetric $2n$ -gon in $\\mathbb {R}^2$ .",
"Let $v_i = \\overrightarrow{a_ia_{i+1}}$ for $i=1,\\dots ,n$ .",
"Then $A(K) = \\sum _{1\\le i<j\\le n} |v_i\\wedge v_j|= \\biggl | \\sum _{1\\le i<j\\le n} v_i\\wedge v_j \\biggr | .$ The second identity follows from the fact that all pairs $(v_i,v_j)$ , ${1\\le i<j\\le n}$ , are of the same orientation.",
"To prove the first one, observe that $A(K) = 2 A (a_1a_2\\dots a_{n+1}) = 2\\sum _{j=2}^{n} A(\\triangle a_1a_ja_{j+1})$ since $K$ is symmetric.",
"Further, $A(\\triangle a_1a_ia_{i+1}) = \\frac{1}{2} |\\overrightarrow{a_1a_j}\\wedge \\overrightarrow{a_ja_{j+1}}|= \\frac{1}{2} \\sum _{i=1}^j |v_i\\wedge v_j|$ since $\\overrightarrow{a_1a_j}=v_1+v_2+\\dots +v_{j-1}$ and all pairs $(v_i,v_j)$ , $i<j$ , are of the same orientation.",
"Plugging the second identity into the first one yields the result.",
"The following lemma takes care of the equality case in Proposition REF .",
"Lemma 2.4 Let $K=a_1a_2\\dots a_{2n}$ be a symmetric $2n$ -gon in $\\mathbb {R}^2$ .",
"For each $i=1,\\dots ,n$ , let $v_i=\\overrightarrow{a_ia_{i+1}}$ , $f_i\\colon \\mathbb {R}^2\\rightarrow \\mathbb {R}$ the linear function such that $f_i=1$ on $[a_ia_{i+1}]$ and $p_i = 2A(\\triangle 0a_ia_{i+1})/A(K)$ .",
"Then $p_ip_j |f_i\\wedge f_j| = \\frac{1}{A(K)^2} |v_i\\wedge v_j| .$ for all $i,j$ , and therefore $\\biggl |\\sum _{1\\le i<j\\le n} p_ip_j\\, f_i\\wedge f_j \\biggr |= \\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j|= \\frac{1}{A(K)} .$ Denote $S_i=2A(\\triangle 0a_ia_{i+1})=|a_i\\wedge a_{i+1}|$ , then $p_i=S_i/A(K)$ .",
"The oriented area form of $\\mathbb {R}^2$ determines a linear isometry $J\\colon (\\mathbb {R}^2)^*\\rightarrow \\mathbb {R}^2$ in the standard way, and one easily sees that $J(S_if_i) =\\pm v_i$ (depending on the orientation).",
"Hence $S_iS_j |f_i\\wedge f_j | = |(S_if_i)\\wedge (S_jf_j) | = |v_i\\wedge v_j |$ and therefore $p_ip_j |f_i\\wedge f_j| = \\frac{1}{A(K)^2} S_iS_j |f_i\\wedge f_j| = \\frac{1}{A(K)^2} |v_i\\wedge v_j| .$ To prove the second assertion, observe that all pairs $(f_i,f_j)$ , $1\\le i<j\\le n$ , are of the same orientation, hence $\\biggl |\\sum _{1\\le i<j\\le n} p_ip_j\\, f_i\\wedge f_j \\biggr | = \\sum _{1\\le i<j\\le n} p_ip_j |f_i\\wedge f_j |= \\frac{1}{A(K)^2} \\sum _{1\\le i<j\\le n} |v_i\\wedge v_j | = \\frac{1}{A(K)}$ where the last identity follows from Lemma REF .",
"It remains to prove the inequality part of Proposition REF .",
"The next lemma covers the principal case when the $f_i$ 's are supporting functions of the sides.",
"Lemma 2.5 Let $K=a_1a_2\\dots a_{2n}$ be a symmetric $2n$ -gon in $\\mathbb {R}^2$ .",
"For each $i=1,\\dots ,n$ , let $f_i\\colon \\mathbb {R}^2\\rightarrow \\mathbb {R}$ be linear function such that $f_i=1$ on $[a_ia_{i+1}|$ .",
"Let $p_1,\\dots ,p_n$ be nonnegative real numbers such that $\\sum p_i=1$ .",
"Then $\\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j| \\le \\frac{1}{A(K)} .$ Denote $v_i=\\overrightarrow{a_ia_{i+1}}$ , $q_i = 2A(\\triangle 0a_ia_{i+1})/A(K)$ and $\\lambda _i=p_i/q_i$ .",
"By the first part of Lemma REF , $q_iq_j |f_i\\wedge f_j| = \\frac{1}{A(K)^2} |v_i\\wedge v_j| .$ Let $v^{\\prime }_i=\\lambda _i v_i$ for $i=1,\\dots ,n$ .",
"Consider a symmetric $2n$ -gon $K^{\\prime }=a^{\\prime }_1\\dots a^{\\prime }_{2n}$ such that $\\overrightarrow{a^{\\prime }_ia^{\\prime }_{i+1}}=v^{\\prime }_i$ .",
"Then $\\begin{aligned}\\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j |&= \\sum _{1\\le i<j\\le n} \\lambda _i\\lambda _j q_iq_j\\, |f_i\\wedge f_j | \\\\&= \\frac{1}{A(K)^2} \\sum _{1\\le i<j\\le n} \\lambda _i\\lambda _j \\, |v_i\\wedge v_j | \\\\&= \\frac{1}{A(K)^2} \\sum _{1\\le i<j\\le n} |v^{\\prime }_i\\wedge v^{\\prime }_j |= \\frac{A(K^{\\prime })}{A(K)^2}\\end{aligned}$ where the last identity follows from Lemma REF .",
"Therefore $\\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j | = \\frac{A(K^{\\prime })}{A(K)^2} .$ Denote $\\ell _i=|v_i|$ and $\\ell ^{\\prime }_i=|v^{\\prime }_i|=\\lambda _i|v_i|$ .",
"Let $h_i$ denote the distance from the origin to the line containing the side $[a_ia_{i+1}]$ .",
"Then $2A(\\triangle 0a_ia_{i+1})=h_i\\ell _i$ , hence $q_i = h_i\\ell _i/A(K)$ .",
"Therefore $1 = \\sum p_i = \\sum \\lambda _iq_i = \\frac{1}{A(K)}\\sum \\lambda _i h_i\\ell _i= \\frac{1}{A(K)}\\sum h_i\\ell ^{\\prime }_i .$ The last sum is the two-dimensional mixed volume $V(K,K^{\\prime })$ , thus $V(K,K^{\\prime })=A(K)$ .",
"By the Minkowski inequality (which is a special case of the Alexandrov–Fenchel inequality), we have $V(K,K^{\\prime })^2\\ge A(K)A(K^{\\prime })$ .",
"Therefore $A(K^{\\prime })\\le A(K)$ .",
"Substituting this inequality into (REF ) yields the assertion of the lemma.",
"To complete the proof of Proposition REF it remains to prove the inequality (REF ) in a slightly more general setting, namely when $K$ is a symmetric polygon (not necessarily with $2n$ sides) and $f_1,\\dots ,f_n$ are arbitrary linear functions such that $f_i|_K\\le 1$ .",
"The condition $f_i|_K\\le 1$ means that $f_i$ belongs to the polar polygon $K^*\\subset (\\mathbb {R}^2)^*$ .",
"Consider the left-hand side of (REF ) as a function in one variable $f_i$ with others staying fixed.",
"This function is convex (since it is a sum of the absolute values of linear functions), therefore it attains its maximum on $K^*$ at a vertex of $K^*$ .",
"The vertices of $K^*$ are supporting linear functions to $K$ at its sides.",
"So it suffices to consider the case when each $f_i$ equals 1 on one of the sides of $K$ .",
"If two of the functions $f_i$ and $f_j$ coincide (without loss of generality, $f_1=f_n$ ), one reduces the problem to a smaller number of functions as follows: drop $f_n$ from the list of functions and replace $p_1,p_2,\\dots ,p_n$ by $p_1+p_n,p_2,\\dots ,p_{n-1}$ .",
"Also note that changing sign of one of the functions $f_i$ does not change the left-hand side of (REF ).",
"Thus it suffices to consider the case when $\\pm f_1,\\dots ,\\pm f_n$ are all distinct.",
"If $n=1$ , the left-hand side of (REF ) is zero so the inequality is trivial.",
"If $n>1$ , applying Lemma REF to the polygon $K^{\\prime } = \\bigcap _{i=1}^n \\lbrace x:|f_i(x)|\\le 1\\rbrace $ yields that $\\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j| \\le \\frac{1}{A(K^{\\prime })} \\le \\frac{1}{A(K)}$ since $K\\subset K^{\\prime }$ .",
"This completes the proof of Proposition REF and hence the proof of Theorem REF ."
],
[
"Proof of Theorem ",
"Let $V$ be a finite-dimensional normed space and $B$ its unit ball.",
"Let $M$ be a compact two-dimensional smooth manifold with $\\partial M\\simeq S^1$ and $S\\colon M\\rightarrow V$ a Lipschitz map such that $S|_{\\partial M}$ parametrizes the boundary of a 2-disc $D$ lying in a two-dimensional linear subspace $P\\subset V$ .",
"Our goal is to prove that $A^{bh}(S)\\ge A^{bh}(D)$ .",
"By means of approximation, we may assume that $B$ is a polyhedron.",
"The intersection $B\\cap P$ is a symmetric polygon $a_1a_2\\dots a_{2n}$ .",
"Define linear functions $F_i\\colon V\\rightarrow \\mathbb {R}$ and coefficients $p_i$ ($i=1,\\dots ,n$ ) as in the proof of Theorem REF .",
"Define a function $\\alpha \\colon GC_2(V)\\rightarrow \\mathbb {R}_+$ by $\\alpha (\\sigma ) = \\pi \\cdot \\!\\!\\!",
"\\sum _{1\\le i<j\\le n} p_ip_j\\, |(F_i\\wedge F_j)\\cdot \\sigma |$ for all $\\sigma \\in GC_2(V)$ .",
"Similarly to the proof of Theorem REF , Proposition REF implies that $\\alpha (\\sigma )\\le A^{bh}(\\sigma )$ for all $\\sigma \\in GC_2(V)$ , and this inequality turns into equality if $\\sigma \\in \\Lambda ^2(P)$ .",
"Indeed, let $\\sigma = v_1\\wedge v_2$ where $v_1,v_2\\in V$ are linearly independent vectors and consider the linear embedding $I\\colon \\mathbb {R}^2\\rightarrow V$ that takes the standard basis of $\\mathbb {R}^2$ to $v_1$ and $v_2$ .",
"Let $K=I^{-1}(B)$ , then $A^{bh}(\\sigma ) = \\frac{\\pi }{A(K)}$ where $A$ is the Euclidean area.",
"On the other hand $\\alpha (\\sigma ) =\\pi \\cdot \\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j|$ where $f_i=F_i\\circ I$ .",
"Recall that $F_i|_B\\le 1$ , hence $f_i|_K\\le 1$ for all $i$ .",
"By Proposition REF , we have $\\sum _{1\\le i<j\\le n} p_ip_j\\, |f_i\\wedge f_j|\\le \\frac{1}{A(K)}$ and (REF ) follows.",
"In the case when $v_1,v_2\\in P$ , the equality case of Proposition REF and the construction of $f_i$ and $p_i$ yields the equality in (REF ).",
"It remains to show that (REF ) implies the inequality $A^{bh}(S)\\ge A^{bh}(D)$ .",
"For each pair $i,j$ , $1\\le i<j\\le n$ , define a linear map $F_{ij}\\colon V\\rightarrow \\mathbb {R}^2$ by $F_{ij}(v) = (F_i(v),F_j(v))\\in \\mathbb {R}^2, \\qquad v\\in V,$ and consider the map $F_{ij}\\circ S\\colon M\\rightarrow \\mathbb {R}^2$ .",
"The Euclidean area $A(F_{ij}\\circ S)$ of this map is given by $A(F_{ij}\\circ S) = \\int _S |F_i\\wedge F_j| .$ (The term $|F_i\\wedge F_j|$ here is a two-dimensional density in $V$ given by $\\sigma \\mapsto |(F_i\\wedge F_j)\\cdot \\sigma |$ , $\\sigma \\in GC_2(V)$ .",
"Recall that the integration of a density over a surface, orientable or not, is well-defined.)",
"Therefore $\\int _S \\alpha = \\int _S \\sum _{1\\le i<j\\le n} p_ip_j|F_i\\wedge F_j|= \\sum _{1\\le i<j\\le n} p_ip_j A(F_{ij}\\circ S) .$ Similarly, for the planar disc $D\\subset P$ we have $\\int _D \\alpha = \\int _D \\sum _{1\\le i<j\\le n} p_ip_j|F_i\\wedge F_j|= \\sum _{1\\le i<j\\le n} p_ip_j A(F_{ij}(D)) .$ Observe that $F_{ij}(D)\\subset F_{ij}\\circ S(M)$ because $S|_{\\partial M}$ is a degree 1 map from $\\partial M$ to $\\partial D$ .",
"Therefore $A(F_{ij}\\circ S) \\ge A(F_{ij}(D))$ for all $i,j$ , and hence the above identities imply that $\\int _S \\alpha \\ge \\int _D \\alpha $ .",
"By (REF ), we have $A^{bh}(S)=\\int _S A^{bh} \\ge \\int _S \\alpha $ , and $A^{bh}(D) = \\int _D A^{bh} \\ge \\int _D \\alpha $ by the equality case of (REF ).",
"Thus $A^{bh}(S)\\ge \\int _S \\alpha \\ge \\int _D \\alpha =A^{bh}(D)$ and Theorem REF follows."
],
[
"Remarks on the higher-dimensional case",
"Although we cannot generalize Theorem REF to surfaces of dimension $k>2$ at the moment, some of the arguments from Section apply in this case as well.",
"Moreover, the convexity of a $k$ -dimensional Busemann–Hausdorff surface area density is equivalent to a $k$ -dimensional analogue of Proposition REF .",
"In hope that this approach will be useful, we formulate this equivalence as the following proposition.",
"Proposition 4.1 For every positive integer $k$ , the following two assertions are equivalent.",
"(i) In every finite-dimensional normed space $V$ , the $k$ -dimensional Busemann–Hausdorff area density admits a convex extension to $\\Lambda ^k V$ .",
"(ii) For every $n\\ge k$ and every central symmetric convex polyhedron $K\\subset \\mathbb {R}^k$ with $n$ pairs of opposite faces $F_1,F_1^{\\prime },\\dots ,F_n,F_n^{\\prime }$ there exist a collection $\\mu _{i_1i_2\\dots i_k}$ , $1\\le i_1<\\dots <i_k\\le n$ , of real coefficients such that the following holds.",
"For every convex polyhedron $K^{\\prime }\\in \\mathbb {R}^k$ and every collection of linear functions $f_1,\\dots ,f_n\\colon \\mathbb {R}^k\\rightarrow \\mathbb {R}$ such that $f_i|_{K^{\\prime }}\\le 1$ for all $i$ , one has $\\left| \\sum \\mu _{i_1i_2\\dots i_k} f_{i_1}\\wedge \\dots \\wedge f_{i_k}\\right| \\le \\frac{1}{\\operatorname{vol}(K^{\\prime })},$ and this inequality turns into equality if $K^{\\prime }=K$ and $f_i$ 's are supporting linear functions corresponding to faces $F_i$ 's (i.e., $f_i|_{F_i}=1$ ).",
"Remark 4.2 In the two-dimensional case, we just defined $\\mu _{ij}=p_ip_j$ where $p_i$ is the portion of the area of $K$ spanned by the $i$ th pair of its faces.",
"This construction is not unique: for many polygons $K$ , other choice of $\\mu _{ij}$ works as well.",
"In higher dimensions, we have no idea how a suitable collection of coefficients (depending on a polyhedron) could be defined, and the most straightforward generalization of the two-dimensional construction does not work.",
"The implication (ii)$\\Rightarrow $ (i) is similar to the deduction of Theorem REF from Proposition REF in Section .",
"To see that (i) implies (ii), consider a polyhedron $K$ as in (ii) and equip $\\mathbb {R}^k$ with a norm $\\Vert \\cdot \\Vert $ whose unit ball is $K$ .",
"Let $f_1^K,\\dots ,f_n^K\\colon \\mathbb {R}^k\\rightarrow \\mathbb {R}$ be the linear functions corresponding to the faces of $K$ , then $K = \\lbrace x\\in \\mathbb {R}^k : |f_i^K(x)|\\le 1, \\ i=1,\\dots ,n \\rbrace .$ Hence the linear map $f^K\\colon \\mathbb {R}^k\\rightarrow \\mathbb {R}^n$ given by $f^K(x) = (f^K_1(x),\\dots ,f^K_n(x)), \\qquad x\\in \\mathbb {R}^k,$ is an isometric embedding of the normed space $(\\mathbb {R}^k,\\Vert \\cdot \\Vert )$ into $\\mathbb {R}^n_\\infty =(\\mathbb {R}^n,\\Vert \\cdot \\Vert _\\infty )$ where the norm $\\Vert \\cdot \\Vert _\\infty $ on $\\mathbb {R}^n$ is defined by $\\Vert x\\Vert _\\infty = \\max _{1\\le i\\le n} |x_i|.$ Assuming (i), the $k$ -dimensional Busemann–Hausdorff area density in $\\mathbb {R}^n_\\infty $ admits a convex extension, and therefore there exists a calibrating form $\\omega \\in \\Lambda ^k(\\mathbb {R}^n)^*$ for the linear subspace $f^K(\\mathbb {R}^k)$ .",
"The fact that $\\omega $ is a calibrator means that for every linear map $f\\colon \\mathbb {R}^k\\rightarrow \\mathbb {R}^n$ one has $|f^*\\omega | \\le \\varepsilon _k \\operatorname{vol}(f^{-1}(B))^{-1}$ with the equality for $f=f^K$ , where $\\varepsilon _k$ is the volume of the Euclidean unit ball in $\\mathbb {R}^k$ and $B=[-1,1]^n$ is the unit ball of $\\mathbb {R}^n_\\infty $ .",
"Let $\\mu _{i_1i_2\\dots i_k}$ , $1\\le i_1<\\dots <i_k\\le n$ , be the coefficients of $\\varepsilon _k^{-1}\\omega $ , that is, $\\omega = \\varepsilon _k \\sum \\mu _{i_1i_2\\dots i_k} dx_{i_1}\\wedge dx_{i_2}\\wedge \\dots \\wedge dx_{i_k} .$ Given a polyhedron $K^{\\prime }\\subset \\mathbb {R}^k$ and linear functions $f_i$ as in (ii), consider $f=(f_1,\\dots ,f_n)\\colon \\mathbb {R}^k\\rightarrow \\mathbb {R}^n$ .",
"Then $f^*\\omega = \\varepsilon _k \\sum \\mu _{i_1i_2\\dots i_k} f_{i_1}\\wedge \\dots \\wedge f_{i_k}$ and $\\operatorname{vol}(f^{-1}(B))^{-1}\\le \\operatorname{vol}(K^{\\prime })^{-1}$ since $f^{-1}(B) = \\lbrace x\\in \\mathbb {R}^k : |f_i(x)| \\le 1, i=1,\\dots ,n\\rbrace \\supset K^{\\prime }$ Thus (REF ) implies (REF ), and the equality in (REF ) for $f=f^K$ yields the equality in (REF ) for $K^{\\prime }=K$ and $f_i=f_i^K$ ."
]
] | 1204.1543 |
[
[
"Mass of highly magnetized white dwarfs exceeding the Chandrasekhar\n limit: An analytical view"
],
[
"Abstract In recent years a number of white dwarfs has been observed with very high surface magnetic fields.",
"We can expect that the magnetic field in the core of these stars would be much higher (~ 10^{14} G).",
"In this paper, we analytically study the effect of high magnetic field on relativistic cold electron, and hence its effect on the stability and the mass-radius relation of a magnetic white dwarf.",
"In strong magnetic fields, the equation of state of the Fermi gas is modified and Landau quantization comes into play.",
"For relatively very high magnetic fields (with respect to the energy density of matter) the number of Landau levels is restricted to one or two.",
"We analyse the equation of states for magnetized electron degenerate gas analytically and attempt to understand the conditions in which transitions from the zero-th Landau level to first Landau level occur.",
"We also find the effect of the strong magnetic field on the star collapsing to a white dwarf, and the mass-radius relation of the resulting star.",
"We obtain an interesting theoretical result that it is possible to have white dwarfs with mass more than the mass set by Chandrasekhar limit."
],
[
"Introduction",
"The origin of high magnetic fields in compact stars is explained by the fossil field hypothesis, as proposed by Ginzburg [1] and Woltjer [2].",
"The magnetic flux $\\phi _b \\sim 4\\pi BR^2 $ of a star is conserved during its evolution, thus a degenerate collapsed star is expected to have a very high magnetic field if the original star had a magnetic field, say, $B \\sim 10^8 ~G $ [3] [4].",
"Neutron stars were of the main interest in this category, but in recent years some white dwarfs have been found to have quite high surface magnetic fields with field strength varying from $10^6 ~G$ to $10^9~ G $ [5], [6], [7], [8].",
"It is highly intuitive that the fields near the core of the white dwarf would be much higher.",
"Ostriker and Hartwick [9] constructed models of white dwarf with central magnetic field $10^{12}~ G$ but a much smaller field in surface.",
"The maximum limit of the field strength is set by the virial theorem $2T+W+3U+{\\cal M}=0 , $ where $T$ is the total kinetic energy, $W$ the gravitational potential energy, $U$ the total internal energy and $\\cal M$ the magnetic energy.",
"Since $T$ and $U$ are both positive definite, the maximum magnetic energy is always less than the total gravitational energy in equilibrium.",
"For a star of mass $M$ and radius $R$ this gives $B_{max} \\sim 2\\times 10^8\\left(\\frac{M}{M_\\odot }\\right)\\left(\\frac{R}{R_\\odot }\\right)^{-2} ~G, $ where $M_\\odot $ and $R_\\odot $ respectively denote solar mass and radius.",
"For a white dwarf this sets the limit of $B \\sim 10^{12} ~G $ at the center but relatively lower fields outside.",
"The mass-radius relation for a non-magnetic relativistic white dwarf as determined by Chandrasekhar [10] sets its maximum mass to $1.44~M_{\\odot }$ such that the electron degeneracy pressure is adequate for counterbalancing the gravitational collapse.",
"This mass limit is strengthened by higher central densities [3].",
"The effect of weak magnetic field $B\\le 10^{13}~G$ on white dwarfs was studied by Suh et al.",
"[11] by applying Euler-MacLaurin expansion on the equation of state for a degenerate electron gas in a magnetic field [12].",
"They found that both the mass and radius of the white dwarf increase in presence of magnetic fields.",
"In this report we analyse the case of relativistic white dwarfs with magnetic fields $B~>~10^{13}~G$ .",
"In order to have a white dwarf of such magnetic field, the original solar type star should have a magnetic field $\\sim 10^9~G$ , from the flux freezing theorem.",
"Existence of such stars is not ruled out [3].",
"First we analyse the effect of such strong magnetic field on the equation of state of degenerate electron gas at zero temperature and analyse some of its properties, subsequently we consider considerable higher magnetic field for the same and then study explicitly the mass-radius relation for the case of just one Landau level (produced by high magnetic field).",
"The results of higher Landau level occupancy (hence lower magnetic field) are addressed quantitatively from the equation of state.",
"In doing so, we find for central densities $\\rho \\sim 10^{10} ~g/cc$ the maximum mass of the white dwarf can be much greater than the Chandrasekhar limit $M \\sim 1.44M_{\\odot }$ .",
"The energy eigenstates of the free electrons in the magnetic field are quantized to what is known as Landau orbitals.",
"The electrons motion perpendicular to the magnetic field is no longer independent but is quantized.",
"This can be seen by solving the time-independent Schrödinger equation of a particle in a magnetic field $B$ directed along $z-$ axis given by [13] $\\hat{H}\\Psi =\\left[\\frac{(p_{x}+eBy/c)^{2}}{2m_e}+\\frac{p_{y}^{2}}{2m_e}+\\frac{p_{z}^{2}}{2m_e}\\right]\\Psi -\\frac{\\mu \\sigma }{s}B\\Psi =E\\Psi , $ where $p_x,p_y,p_z$ are the components of linear momentum, $y$ is the arbitrary position on the $y-$ axis, $m_e$ and $e$ are the mass and charge of the electron respectively, $c$ the speed of light, $\\mu $ the magnetic moment, $s$ the magnitude of spin, $\\sigma =\\pm \\frac{1}{2}$ , $E$ the energy eigenvalue.",
"We obtain the classical quantization of energy levels of the system given by $E_{\\nu }=\\nu \\hbar \\omega _{H}+\\frac{p_{z}^{2}}{2m_e}, $ where $\\omega _{H}=\\frac{eB}{m_ec}$ , is the critical cyclotron frequencies at which quantization occurs, $\\hbar $ the Planck's constant, and $\\nu =(l+\\frac{1}{2}+\\sigma )$ , gives the Landau levels where $l$ is the principal quantum number for the electron.",
"Relativistically we obtain energy by solving Dirac equation in a magnetic field, hence the energy above is modified to $E_{\\nu }=\\sqrt{m_e^{2}c^{4}+p_{z}^{2}c^{2}+2\\nu e\\hbar Bc}.",
"$ Thus electrons with up ($\\uparrow $ ) spin and down ($\\downarrow $ ) spin have different energies.",
"We see that the ground state ($\\nu =0$ ) has degeneracy 1, while the Landau levels from $\\nu =1$ have degeneracy 2.",
"The spin $\\downarrow $ electrons can occupy Landau levels $\\nu =0, 1, 2, 3...$ , while the spin $\\uparrow $ ones take Landau levels $\\nu =1, 2, 3...$ [14].",
"Note that the motion of the electrons in the $x-y$ plane is coupled.",
"Therefore the density of states of the electron will change, as the motion is restricted and quantized in the plane.",
"As the motion in the $x-y$ plane is quantized, the phase space occupied by the electrons will change.",
"The number of states per unit volume in the interval $\\triangle p_z$ for a given Landau level $\\nu $ is $g_\\nu \\left(\\frac{eB}{h^2 c}\\right)\\triangle p_z$ .",
"The modified density of state is then $2\\frac{dp_{x}dp_{y}dp_{z}}{h^{3}}\\Rightarrow 2\\frac{eB}{h^{2}c}{\\sum _\\nu }g_{\\nu 0}dp_{z}(\\nu ), $ where $g_{\\nu 0}=(2-\\delta _{0,\\nu })$ , is the degeneracy in each Landau level, and $dp_{z}(\\nu )$ is the small element of component of momentum in the $z-$ direction in the $\\nu -$ th Landau level.",
"Thus each Landau level has its own distribution of states.",
"The separation between two Landau levels depends on the strength of the magnetic field.",
"For high magnetic fields the separation of the Landau levels is large, hence electrons with low energy (non-relativistic) can only occupy the ground state.",
"As the strength of the magnetic field decreases, the separation between the levels decreases.",
"Hence it becomes energetically favourable for the electrons to jump to higher levels, thus the number of occupied Landau level increases.",
"Similarly in the case of relativistic electrons, if the magnetic field is low, the separation of the Landau levels is comparable to the rest mass energy of the electrons, and hence the electrons can freely move between the Landau levels, thus it will behave as continuum.",
"But again in presence of very high magnetic fields, the electrons, although relativistic, cannot jump to higher levels.",
"The electrons can become relativistic in two different cases: (1) when the density is high enough such that the mean Fermi energy of the electron exceeds rest mass energy of the electron, (2) when the term in the energy associated with the cyclotron frequency of the electron exceeds the rest mass of the electron."
],
[
"Equation of state",
"We now analyse the equation of state for the relativistic electrons in magnetic field, as was obtained earlier [12].",
"We define the Fermi momentum, analogous to that in the non-magnetic case, as $p_{F}^{2}=p_{z}^{2}+\\frac{2 \\nu eB\\hbar }{c}$ .",
"Therefore for $\\nu $ -th Landau level the $z-$ component of the momentum is given by $p_{z}(\\nu )=\\sqrt{p_{F}^{2}-\\frac{2\\nu eB\\hbar }{c}}$ .",
"We introduce a convenient dimensionless parameter $x_{F}=\\frac{p_{F}}{m_e c}$ , as relativity parameter, and $B_{c}=\\frac{m_e^{2}c^{3}}{e\\hbar }=4.14\\times 10^{13}~G$ , as the critical magnetic field giving rise to the significant effect due to the Landau quantization.",
"We further define $\\gamma =\\frac{B}{B_{c}}$ , and $\\lambda =\\frac{\\hbar }{m_ec}$ , the Compton wavelength of the electron.",
"Using these definitions we can write $p_{z}(\\nu )$ as $\\frac{p_z}{m_ec}=x_{z}(\\nu )=x(\\nu )=\\sqrt{x_{F}^{2}-2\\nu \\gamma }, $ where $x(\\nu )$ denotes the relativity parameter of the $z-$ component of momentum for $\\nu $ -th Landau level.",
"The number density is then given by $n_{e}=\\frac{2\\gamma }{(2\\pi )^{2}\\lambda ^{3}}\\sum _{\\nu =0}^{\\nu _{max}}g_{\\nu 0}dx(\\nu )=\\frac{2\\gamma }{(2\\pi )^{2}\\lambda ^{3}}\\sum _{\\nu =0}^{\\nu _{max}}g_{\\nu 0}x(\\nu ) , $ where $\\nu _{max}$ is the maximum number of Landau level that can be filled which is determined by the condition that $x_{z}(\\nu )$ is real.",
"Hence $\\nu \\le \\nu _{max}= {\\rm Integer}(\\frac{x_{F}^{2}}{2\\gamma })= {\\rm Integer}(\\frac{\\epsilon _F^2-1}{2\\gamma }), $ where $\\epsilon _F=\\frac{E_\\nu }{m_ec^{2}}$ which is the dimensionless chemical potential and from equation (REF ) $\\epsilon _F^{2}=1+x_F^{2}$ .",
"This is related to matter density by $\\rho =\\mu _em_nn_e=n_e\\Theta , $ where $\\mu _e$ is the mean molecular weight given by $A/Z$ , $Z$ the atomic number, $A$ the mass number, and $m_n$ the mass of the neutron.",
"Note that $\\Theta =\\mu _em_n$ which has the dimension of mass and denoting the effective mass per electron.",
"Its value depends on the constituents of the white dwarf.",
"In our case value of $\\mu _e$ is taken to be 2, and hence $\\Theta =2m_n$ .",
"The electron energy density at zero temperature is given by [12] $E=\\frac{2\\gamma }{(2\\pi )^{2}\\lambda ^{3}}\\sum _{\\nu =0}^{\\nu _{max}}g_{\\nu 0}^{x_F}_0 E_\\nu dx(\\nu ), $ and hence $E=\\sum _{\\nu =0}^{\\nu _{max}}\\frac{\\gamma g_{\\nu 0}m_ec^{2}}{(2\\pi )^{2}\\lambda ^{3}}\\left[x(\\nu )\\epsilon _{F}+(1+2\\nu \\gamma )\\ln [\\frac{x(\\nu )+\\epsilon _{F}}{\\sqrt{1+2\\nu \\gamma }}]\\right].$ Then the pressure of the Fermi gas at zero temperature can be found from the relation $P=\\sum \\epsilon _{F}x(\\nu )-E, $ which gives the relation $P=\\sum _{\\nu =0}^{\\nu _{max}}\\frac{\\gamma g_{\\nu 0}m_ec^{2}}{(2\\pi )^{2}\\lambda ^{3}}\\left[x(\\nu )\\epsilon _{F}-(1+2\\nu \\gamma )\\ln [\\frac{x(\\nu )+\\epsilon _{F}}{\\sqrt{1+2\\nu \\gamma }}]\\right].",
"$"
],
[
"one level and the two level systems and equation of states",
"First we consider the one level system in which only the ground level is occupied.",
"This is possible at a very high magnetic field.",
"Expanding equation (REF ) to first term (for $\\nu =0$ ) and with equation (REF ) we obtain $\\rho =\\frac{\\gamma \\mu _e m_n}{2\\pi ^2\\lambda ^3}x(0)=\\frac{\\gamma \\mu _e m_n}{2\\pi ^2\\lambda ^3}x_F, $ from which we further obtain $x_F=\\frac{2\\pi ^2\\lambda ^3}{\\gamma \\mu _e m_n}\\rho =~\\frac{\\rho }{K}, $ where $K=\\mu _e m_n \\frac{2 \\gamma }{(2\\pi )^2\\lambda ^3}$ .",
"Here we obtain $x_F$ in terms of density and hence the relation of Fermi energy to the density of state.",
"As we increase the density of the matter, we in turn increase the Fermi energy associated with it, hence electrons acquire probability to jump to higher Landau levels and therefore $\\nu $ can not be set 0.",
"From equation (REF ) we find that once the Fermi energy and $\\nu $ are fixed, the magnetic field and the maximum density are restricted.",
"The equation of state (REF ) then reduces to $P=\\frac{m_ec^{2}}{2K\\Theta }\\left(\\rho \\sqrt{K^2+\\rho ^2}-K^2\\ln (\\frac{\\rho +\\sqrt{K^2+\\rho ^2}}{K})\\right) $ for $\\nu =0$ .",
"For the two level system (when $\\nu =0,1$ ) $n_{e}=\\frac{\\gamma }{2\\pi ^{2}\\lambda ^{3}}\\sum _{\\nu =0}^{1}\\left(\\left(2-\\delta _{0,\\nu }\\right)\\sqrt{x_{F}^{2}-2\\gamma \\nu }\\right)=\\frac{\\gamma }{2\\pi ^{2}\\lambda ^{3}}\\left(x_{F}+2\\sqrt{x_{F}^{2}-2\\gamma }\\right) $ and the pressure then reduces to $P=\\frac{\\gamma m_ec^{2}}{(2\\pi )^{2}\\lambda ^{3}}\\left(x_{F}\\epsilon _{F}+2\\epsilon _{F}\\sqrt{x_{F}^{2}-2\\gamma }-\\ln (x_{F}+\\epsilon _{F})-2(1+2\\gamma )\\ln (\\frac{\\sqrt{x_{F}^{2}-2\\gamma }+\\epsilon _{F}}{\\sqrt{1+2\\gamma }})\\right).",
"$ The density can be written in terms of $x_F$ as $\\rho =K\\left(x_{F}+2\\sqrt{x_{F}^{2}-2\\gamma }\\right).",
"$ Hence, $x_{F}=\\frac{-\\rho +2\\sqrt{6K^{2}\\gamma +\\rho ^{2}}}{3K}, $ which is always positive definite.",
"The other solution is neglected as infeasible solution as density cannot be negative for real particles.",
"Substituting this $x_F$ in equation (REF ) we obtain $\\nonumber P=\\frac{m_e c^2 }{6 \\Theta } \\left(\\epsilon _F \\left(6 K x_1-\\rho +2 \\sqrt{6 K^2 \\gamma +\\rho ^2}\\right)-6 (K+2 K \\gamma ) \\ln \\left(\\frac{x_1+\\epsilon _F }{\\sqrt{1+2 \\gamma }}\\right)\\right.",
"\\\\\\left.",
"-3 K \\ln \\left(\\epsilon _F -\\frac{\\rho -2 \\sqrt{6 K^2 \\gamma +\\rho ^2}}{3 K}\\right)\\right),$ where $x_1=\\sqrt{-2\\gamma +\\frac{\\left(\\rho -2\\sqrt{6K^{2}\\gamma +\\rho ^{2}}\\right)^{2}}{9K^{2}}} $ and $\\epsilon _{F}=\\sqrt{1+\\frac{\\left(\\rho -2\\sqrt{6K^{2}\\gamma +\\rho ^{2}}\\right)^{2}}{9K^{2}}}.",
"$ We must be careful that the two level system is only valid above a density given by equation (REF ).",
"The net equation of state is represented by two functions given by: $&&{\\rm Equation~~(16) }~~{\\rm for}~~\\rho < \\rho _t~~(\\nu =0) \\\\\\nonumber &&{\\rm Equation~~(21)}~~~{\\rm for}~~\\rho > \\rho _t~~(\\nu =1).$ where $\\rho _t$ is the density at which transition occurs between the Landau levels.",
"A diagrammatic representation of the equations (REF ) and (REF ) is given in Fig REF .",
"Figure: Equations of state for one level (solid line) and two level (dashed line) systems as represented by equations ()and () for ϵ F =20\\epsilon _F=20.",
"The kink in dashed curve at ρ t \\rho _t is found from equation ().Here the values of $K$ will depend on the limiting magnetic field taken.",
"As for one level system the limiting magnetic field is much higher than that of two level system, the values of $K$ will be in general different between one level and two level systems.",
"Hence, the steepness of the equation of state representing ground Landau level will be different between the systems restricted to one level and that restricted to higher levels.",
"The higher level systems can be computed in a similar algebraic way.",
"It must be noted that the value of $\\gamma $ obtained from equation (REF ) corresponds to the lower limit for the given value of $\\epsilon _F$ .",
"For example, for say maximum Fermi energy $E_{Fmax}=20m_ec^2$ , if $\\gamma =199.5$ it corresponds to one level system, but for any other values greater than $199.5$ will not alter the condition for the same level system."
],
[
"Discussion of equation of states",
"The electrons in the system become relativistic when the factor $\\epsilon _F$ becomes more than one.",
"For relatively low magnetic fields, the spacing between Landau levels becomes small and $\\nu _{max}$ becomes large.",
"Thus relativistic electrons can freely jump between the states as said earlier.",
"In this case the levels can be treated as continuum and the summation over states can be replaced by an integral and the resulting equation gives back the normal non-magnetic relativistic equation of state $P=K^{\\prime }\\rho ^\\Gamma $ with $\\Gamma =4/3$ , just the case which was analysed by Chandrasekhar [10].",
"Note that $\\Gamma =4/3$ corresponds to the equation of state of relativistic but non-magnetized electrons giving rise to the Chandrasekhar mass limit, which we recover in a particular range of density even in the magnetized system.",
"In just the opposite case, which we are interested in, we consider just one/two Landau level(s), i.e.",
"the ground/first excited state, is filled in.",
"From equation (REF ) we fix $\\nu _{max}$ to a value less than 1 (2) such that only the ground level (and first level) is (are) occupied.",
"The electrons in the ground level would all be in the spin $\\downarrow $ state.",
"From equation (REF ) we can see that once we fix $\\nu _{max}$ and $x_F$ , then the minimum magnetic field is automatically fixed.",
"If there are finite Landau levels, then the ground level would be filled first.",
"If the density is increased further, then the ground state will saturate (it cannot contain more electrons) and there will be a phase transition like phenomena, in which electrons would start filling the first excited Landau level.",
"This process will continue to higher Landau level transitions.",
"During these phase transition like phenomena, the $E-\\rho $ relation shows a sharp discontinuity.",
"Thus $\\frac{dE}{d \\rho }=\\infty $ .",
"An analytic condition can be found out for the critical densities at which this transition occurs by differentiating equation (REF ) with respect to $\\rho $ .",
"In $\\frac{dE}{d\\rho }$ for two states, we note there is only one unique denominator which is not positive definite, hence can be set to zero which makes $\\frac{dE}{d\\rho }=\\infty $ .",
"Thus this gives us the condition for the critical density.",
"Solving the equation $2\\gamma =x_F^2 $ and equation (REF ) simultaneously we obtain $ \\rho _t=\\sqrt{2\\gamma }K. $ This gives the critical density below which the electrons cannot occupy the first Landau level ($\\nu =1$ ) and would all be in the ground level ($\\nu =0$ ) and hence it would be inappropriate to use the two level equation of state.",
"This result can also be estimated by using $\\sqrt{2\\gamma }\\le x_F$ which gives on combining with $x_F=\\frac{\\rho }{K}$ the same formula as equation (REF ).",
"Similarly, the transition from first level to second level would then be given by the equation $\\sqrt{2\\times 2 \\gamma } \\le \\frac{-\\rho +2\\sqrt{6 K^2\\gamma +\\rho ^2}}{3 K}.",
"$ The critical density for the transition from the first to second levels is given by $\\rho _t = 2(1+\\sqrt{2})\\sqrt{\\gamma }K. $ The higher phase change also occurs under the same principle.",
"Therefore we can write the formula of critical density for the transition from $(\\nu -1)$ -th level to $\\nu $ -th level for $1 \\le \\nu \\le 2$ as $\\rho _t(\\nu )=(\\sqrt{2\\nu }+2\\sqrt{2(\\nu -1)})\\sqrt{\\gamma }K, $ where it must be remembered that $K$ is a function of $\\gamma $ ."
],
[
"Determining mass and radius of the white dwarf analytically",
"For a magnetized star in hydrostatic equilibrium, we require to solve the condition for equilibrium given by [3] $\\frac{1}{r^{2}}\\frac{d}{dr}(\\frac{r^{2}}{\\rho }\\frac{dP}{dr})=-4\\pi G\\rho (r), $ where $P$ is the electron degeneracy pressure, varying with the radial coordinate of the star $r$ , playing the role to balance the gravitational force, $G$ the Newton's gravitation constant.",
"Now we choose the equation of state as a polytropic relation $P=K^{\\prime } \\rho ^ \\Gamma , $ where $K^{\\prime }$ is a dimensional constant, $\\rho =\\rho _c \\theta ^n , $ and $r=a\\xi , $ where $\\rho _c$ is the central density of the white dwarf and $a$ is a constant defined by $a=\\left[\\frac{(n+1)K^{^{\\prime }}}{4\\pi G}\\right]^{\\frac{1}{2}}\\rho _{c}^{\\frac{(1-n)}{2n}}.",
"$ With the use of equations (REF ), (REF ), (REF ) and (REF ), equation (REF ) reduces to famous Lane-Emden equation $\\frac{1}{\\xi ^{2}}\\frac{d}{d\\xi }(\\xi ^{2}\\frac{d\\theta }{d\\xi })=-\\theta ^{n}, $ where $\\Gamma =1+\\frac{1}{n}$ .",
"The Lane-Emden equation can be solved for a given $n$ using boundary conditions $\\theta (\\xi =0)=1 $ and $\\left(\\frac{d\\theta }{d\\xi }\\right)_{\\xi =0}=0.",
"$ For $n<5$ , the value of $\\theta $ falls to zero at a finite $\\xi $ say $\\xi _0$ , which basically defines the surface of the star where pressure goes to zero.",
"The physical radius is given by $R=a \\xi _0 .",
"$ Here we notice that $n\\ge -1$ so that $a$ (see equation (REF )) must be real and so does $R$ .",
"Since the equation of state in presence of magnetic field cannot be written in a simple polytropic form as in equation (REF ), we fit equation (REF ) for various density ranges.",
"The actual equation of state is reconstructed by using multiple fits in various density range.",
"The values for $K^{\\prime }$ and $\\Gamma $ in various ranges of density, which are found from the fitting function, also carry information about the magnetic field of the system.",
"The idea behind the fit is to be able to solve the Lame-Emden equation piecewise ranges of density and to obtain an idea of the mass-radius relation from it easily.",
"The mass of the star is then given by $M=\\int 4\\pi \\rho r^{2}dr = 4\\pi \\rho _c a^3\\int ^{\\xi _0}_0 \\xi ^2 \\theta ^n d\\xi .",
"$ Now we consider only one level system explicitly, hence fit the one level equation of state and obtain different values of $K^{\\prime }$ and $\\Gamma $ .",
"We see from Fig.",
"REF that the equation of state representing the one level system for $\\epsilon _F=20$ is given by $P=2\\times 10^8\\left(\\rho \\sqrt{34\\times 10^{16} +\\rho ^2}-34 \\times 10^{16} \\ln \\left((\\rho +\\sqrt{34\\times 10^{16}+\\rho ^2})~16\\times 10^{-10}\\right)\\right) , $ which is obtained from equation (REF ).",
"This curve is fitted with two functions $\\nonumber &&P(\\rho )= 6.98\\times 10^{-1} \\rho ^{2.92}~~~{\\rm for}~~~\\rho \\le \\: 1.4\\times 10^9,\\\\\\nonumber &&P(\\rho )= 1.98\\times 10^{8} \\rho ^2~~~~~~~{\\rm for}~~~\\rho \\ge \\: 1.4\\times 10^9,$ shown by Fig.",
"REF (b).",
"Figure: Equations of state for one Landau level system, when (a) ϵ F =2\\epsilon _F=2, (b) ϵ F =20\\epsilon _F=20.Solid and broken lines respectively indicate original and fitting results, both overlap each other perfectly.Here we consider a uniform magnetic field.",
"In very low densities where we consider the pressure goes to zero, the equation of state defined above cannot be used as it becomes non-linear in nature.",
"We have extended the fitting curve until this region arises.",
"However, this region is a very small part of the total density region, hence would not affect our results effectively.",
"Similarly, the fitting of the one level equation of state for $\\epsilon _F=2$ is given by Fig.",
"REF (a).",
"Then we integrate equation (REF ) in two ranges of density to obtain $M=4\\pi \\rho _c \\left( a(n_1)^3\\int ^{\\xi _i}_0 \\xi ^2 \\theta ^n d\\xi + a(n_2)^3(\\int ^{\\xi _0}_{0} \\xi ^2 \\theta ^n d\\xi - \\int ^{\\xi _i}_0 \\xi ^2 \\theta ^n d\\xi )\\right), $ which gives $M=4 \\pi \\rho _c \\left({a(n_1)}^3|{\\xi _i}^2\\theta ^{^{\\prime }}(\\xi _i)|+{a(n_2)}^3(|{\\xi _0}^2\\theta ^{^{\\prime }}(\\xi _0)|-|{\\xi _i}^2\\theta ^{^{\\prime }}(\\xi _i)|)\\right) , $ where $\\xi _i$ is the radius corresponding to the intermediate density where the $\\Gamma $ changes its value drastically in the equation of state.",
"We then substitute this intermediate density ($\\rho (\\xi _i)$ ) in equation (REF ) with a particular value of central density and corresponding polytropic exponent and find the corresponding $\\theta $ .",
"From the solution to Lane-Emden equation we find the corresponding values of $\\xi $ and ${\\xi }^2|\\theta ^{^{\\prime }}(\\xi )|$ .",
"The radius in this case will be $R=a(n_1) \\xi _i +a(n_2) ( \\xi _0- \\xi _i) , $ where the values of $a(n_1)$ and $ a(n_2)$ are obtained from equation (REF ) by substituting the corresponding values of $n$ and $K^{^{\\prime }}$ ."
],
[
"Mass-Radius relation",
"We now discuss the mass-radius relation for the highly magnetized star.",
"We again consider to study the extreme case having only one Landau level.",
"For the higher level occupancy, as shown by the equation of state in Fig.",
"REF by the dashed curve for a two level system, the magnetic effect will be softened and we will discuss the results qualitatively.",
"From equation (REF ) we obtain mass as a function of central density and equation (REF ) gives the radius of the star as a function of central density, for one level systems.",
"The values of coefficients are calculated from the values of $K^{\\prime }$ and $\\Gamma $ from the fitting curves in various zones and $\\gamma $ is determined from the Fermi energy.",
"We can eliminate the central density from equations (REF ) and (REF ) and obtain mass as a function of radius.",
"In order to express the results in convenient units we write the mass in the units of solar mass and radius in the units of $10^8 ~cm$ .",
"The values of $K^{\\prime }$ and $\\Gamma $ obtained from the fitting are used in equations (REF ) and (REF ) which gives for $\\epsilon _F$ =20 $M/M_{\\odot }= \\: 3.81291\\times 10^{-11} \\rho _c \\, + 2.86571 \\times 10^{-23} \\rho _c^{2.38} $ and $R/10^8=\\: 0.05839 \\,+ 3.196\\times 10^{-5} \\rho _c ^{0.46}.",
"$ Now eliminating $\\rho _c$ from above equations we obtain the mass-radius relation.",
"Figure REF shows the mass-radius relation for $\\epsilon _F=2,20$ .",
"The extreme right point of the curves (maximum radius) denotes the typical maximum central density of white dwarfs for the respective values of $\\epsilon _F$ .",
"Thus the mass corresponding to that radius gives the maximum mass possible for the white dwarf.",
"Figure: Mass-Radius relation for magnetic white dwarfs for one Landau level systems, when (a)ϵ F =2\\epsilon _F=2, (b) ϵ F =20\\epsilon _F=20.",
"MM is expressed in the units of M ⊙ M_\\odot and RR in the units of 10 8 cm10^8~cm.The maximummass is determined from the maximum value of ρ c \\rho _c for one level system as determined from equation ().For ϵ F =2\\epsilon _F=2 , the corresponding ρ c \\rho _c is 7.4×10 6 gm/cc7.4 \\times 10^6 gm/cc and for ϵ F =20\\epsilon _F=20,the corresponding ρ c \\rho _c is 1.2×10 10 gm/cc1.2 \\times 10^{10} gm/cc.We know from the non-magnetic case that a stiffer equation of state can balance gravity more effectively leading to a higher value of radius (non-relativistic case).",
"The same argument can be applied here for $\\Gamma =2.92$ (lower density and hence outer region of the star), owing to the fact that the stiffness arises from the effect of magnetic field on degeneracy pressure.",
"Hence from the mass-radius relation, mass would turn out to be higher than the usual one at a particular radius.",
"Here we must be careful that even though the resulting analytical form of equation apparently does not have any restriction on the value of the maximum mass or radius, the maximum mass is restricted by the typical values of central density for white dwarf [3] and the maximum Fermi energy of the star determined by equation (REF ).",
"Figure REF however shows that the magnetic white dwarfs have typically higher mass and higher radius.",
"In general for typical densities of white dwarf, the mass of the white dwarf can exceed the Chandrasekhar mass limit.",
"For $\\epsilon _F=2$ , however, we see from the mass-radius relation that the typical radius turns out to be higher than that of $\\epsilon _F=20$ , but the Chandrasekhar mass limit is not exceeded.",
"In the equation of state for one level, $\\Gamma $ decreases from $2.92$ in the low density region to the value 2 in the high density region.",
"Quantitatively for a two level system, we can see from the equation of state that after the transition from the ground level to the first level, the slope of the curve changes drastically to very low values and then rises again.",
"At this transition point, the pressure is nearly independent of density.",
"This is physically improbable and the corresponding mass-radius relation would give an unstable branch.",
"If a star starts out in this density range, it cannot form a compact star and will go into a runaway process."
],
[
"Discussion and summary",
"In this work we have calculated the equation of state of a degenerate electron gas in high magnetic field at zero temperature analytically.",
"For simplicity, and keeping it analytically tractable, we have discussed the extreme cases when there are only one and two Landau levels for the system and studied the mass-radius relation for the case of one Landau level.",
"Our pure analytical approach helps in understanding the underlying physics in great details.",
"In the forthcoming work, we will show that the detailed numerical unrestricted solutions indeed match with our analytical results [15].",
"To make the problem theoretically tractable easily, we have chosen constant magnetic field throughout.",
"However, this does not matter for the present purpose due to the following reason.",
"Although the matter density and the magnetic field both vary from center (highest) to surface (lowest), by the time the density falls to about half the value of the central density, the mass generally crosses the Chandrasekhar mass limit.",
"Thus, although we have considered a constant magnetic field, the field strength plays its major role and brings in new results in the high density regions of the star only.",
"Hence it appears equivalent to the central magnetic field of the star.",
"Now following previous work [16], [14], one can adopt an inhomogeneous magnetic field profile in any compact star which is nearly constant throughout most of the star and then gradually falls off close to the surface (see Figure 5(b) in [14]).",
"Thus the choice of an inhomogeneous magnetic field would not affect our main finding.",
"A detailed description for the same will be given in a follow up paper based on detailed numerical analyses [15].",
"Note that the size of a white dwarf is very large so that the effect of general relativity is insignificant there.",
"In addition, the central density of the star is almost an order of magnitude larger than the energy density arised due to the corresponding magnetic field.",
"As the effect of magnetic field arises in the high density regime only when the matter density exceeds the energy density arised due to the magnetic field, even the magnetic field does not contribute to the gravitational field leaving the system Newtonian.",
"It is found that the ground level of the system is occupied first and then the higher levels start getting filled in.",
"This transition appears as a kink in the pressure-density plot, what we have addressed analytically.",
"For one level system that we have studied explicitly for the mass-radius relation (our numerical solutions addressed in a separate paper [15] would show the mass-radius relations for multi level systems), it is known to have no kink, for two level systems one kink, and so on for higher level systems, what we have studied analytically.",
"We have derived a general expression for the positions of the kinks in the equation of state in terms of the magnetic field and other constants and the relation among them.",
"It has been found that they bear a simple ratio, thus if we know exactly the value of density for a kink, then we can find out the density for other kinks.",
"One of the cases considered here is for the Fermi energy $20m_ec^2$ and the system is of a one level.",
"The minimum magnetic field required for this system is found to be $8.7 \\times 10^{15}~G$ .",
"The corresponding mass of the resulting white dwarf has been found to be larger than that predicted by Chandrasekhar and is about $2.3M_{\\odot }$ when the radius is $8\\times 10^7~cm$ .",
"We have also analysed analytically how is the mass-radius relation dependent on the Fermi energy and the magnetic field.",
"We end by addressing the possible reason for not observing such a high field yet in a white dwarf.",
"This could be due to the magnetic screening effects on the surface of the star.",
"If the white dwarf is an accreting one, then the current in accreting plasma depositing on the surface of the white dwarf could create an induced magnetic moment of sign opposite to that of the original magnetic dipole, thus reducing the surface magnetic field of the white dwarf unaffecting the central field.",
"In addition, the surface field could be several orders of magnitude smaller than the field in the central region which brings in the main results.",
"Hence by estimating the surface field one should not interpret the rest.",
"Acknowledgments: This work was partly supported by an ISRO grant ISRO/RES/2/367/10-11 and a KVPY grant of DST.",
"We thank Upasana Das for discussion and cross checking several calculations."
]
] | 1204.1463 |
[
[
"Deterministic Vector Freak Waves"
],
[
"Abstract We construct and discuss a semi-rational, multi-parametric vector solution of coupled nonlinear Schr\\\"odinger equations (Manakov system).",
"This family of solutions includes known vector Peregrine solutions, bright-dark-rogue solutions, and novel vector unusual freak waves.",
"The vector freak (or rogue) waves could be of great interest in a variety of complex systems, from optics to Bose-Einstein condensates and finance."
],
[
"Deterministic Vector Freak Waves Fabio Baronio$^1$ , Antonio Degasperis$^2$ , Matteo Conforti$^1$ , and Stefan Wabnitz$^1$ $^1$ CNISM, Dipartimento di Ingegneria dell'Informazione, Università di Brescia, Via Branze 38, Brescia 25123, Italy.",
"$^2$ INFN, “Sapienza” Università di Roma, P.le A. Moro 2, 00185 Roma, Italy We construct and discuss a semi-rational, multi-parametric vector solution of coupled nonlinear Schrödinger equations (Manakov system).",
"This family of solutions includes known vector Peregrine solutions, bright-dark-rogue solutions, and novel vector unusual freak waves.",
"The vector freak (or rogue) waves could be of great interest in a variety of complex systems, from optics to Bose-Einstein condensates and finance.",
"05.45.Yv, 02.30.Ik, 42.65.Tg Introduction.",
"Extreme wave events, also referred to as freak or rogue waves, are mostly known as an oceanic phenomenon responsible for a large number of maritime disasters.",
"These waves, which have height and steepness much greater than expected from the sea average state [1], have recently become a topic of intense research.",
"Freak waves appear both in deep ocean and in shallow waters [2].",
"In contrast to tsunamis and storms associated with typhoons that can be predicted hours (sometimes days) in advance, the particular danger of oceanic rogue waves is that they suddenly appear from nowhere only seconds before they hit a ship.",
"The grim reality, however, is that although the existence of freak waves has now been confirmed by multiple observations, uncertainty remains on their fundamental origins.",
"This hinders systematic approaches to study their characteristics, including the predictability of their appearance [3].",
"In fact, research on rogue waves is in an emerging state [1], [3], [4].",
"These waves not only appear in oceans [5] but also in the atmosphere [7], in optics [8], [9], in plasmas [11], in superfluids, in Bose–-Einstein condensates [12] and also as capillary waves [13].",
"The common features and differences among freak wave manifestations in their different contexts is a subject of intense discussion [2].",
"New studies of rogue waves in any of these disciplines enrich their concept and lead to progress towards a comprehensive understanding of a phenomenon which still remains largely unexplored.",
"A formal mathematical description of a rogue wave is provided by the so-called Peregrine soliton [14].",
"This solitary wave is a solution of the one-dimensional nonlinear Schrödinger equation (NLSE) with the property of being localized in both the transverse and the longitudinal coordinate: thus it describes a unique wave event.",
"This solution is also unique in a mathematical sense, as it is written in terms of rational functions of coordinates, in contrast to most of other known solutions of the NLSE.",
"Recent experiments have provided a path to generating Peregrine solitons in optical fibers with standard telecommunication equipment [15].",
"The further experimental observation of Peregrine solitons in a water tank [16] indicates that they can also describe rogue waves in oceans.",
"The Peregrine soliton is not the only fully localized waveform [17].",
"In fact, there is an infinite hierarchy of rational solutions of the NLSE which enjoy the same property [18], [19], [20], [21].",
"In a variety of complex systems such as Bose–Einstein condensates [22], optical fibers [23], and financial systems [24], [25], several variables rather than a single wave amplitude need to be considered.",
"For instance, in the financial world it is necessary to couple cash to the value of other assets such as shares, bonds, options, etc., as well as to consider all correlations between these variables.",
"The resulting systems of equations may thus describe extreme waves with higher accuracy than the single NLSE model.",
"Approaches to rogue wave phenomena involving multiple coupled waves are the coupled Gross-Pitaevskii (GP) equations [26] and the Manakov system (or vector NLSE) [27].",
"Indeed, vector rogue waves of GP equations and the Manakov system have been recently presented [22], [25], [28].",
"In this Letter, we construct and discuss a novel semi-rational, multi-parametric vector solution of the Manakov system.",
"For special parameter values this solution reproduces known vector rogue waves (such as the vector Peregrine soliton [22] and bright-dark-rogue waves [28]).",
"Our treatment below goes as follows.",
"We give the essential Darboux dressing transformation to construct freak solutions of the Manakov system.",
"We present the multi-parametric, semi-rational deterministic freak waves.",
"Moreover, we discuss their experimental feasibility in nonlinear optics.",
"Darboux dressing technique.",
"Waves are assumed to be modeled by the dimensionless vector coupled nonlinear Schrödinger equations (VNLSE) or Manakov system: $\\left\\lbrace \\begin{array} {lll}iu^{(1)}_t+ u^{(1)}_{xx} +2 ( |u^{(1)}|^2 + |u^{(2)}|^2 ) u^{(1)} & = & 0 \\\\iu^{(2)}_t+ u^{(2)}_{xx} +2 ( |u^{(1)}|^2 + |u^{(2)}|^2 ) u^{(2)} & = & 0,\\end{array} \\right.$ where $u^{(1)}(x,t),\\,u^{(2)}(x,t)$ represent the wave envelopes and $t,x$ are the longitudinal and transverse coordinates, respectively.",
"Each subscripted variable in Eqs.",
"(REF ) stands for partial differentiation.",
"It should be pointed out that the meaning of the dependent variables $u^{(1)}(x,t),u^{(2)}(x,t)$ , and of the coordinates $t,x$ depends on the particular applicative context (f.i.",
"plasma physics, nonlinear optics, finance).",
"Note also that Eqs.",
"(REF ) refer to the self-focusing (or anomalous dispersion) regime.",
"Eqs.",
"(REF ) are integrable: the associated Lax pair is $\\Psi _x=(ik\\sigma +Q)\\Psi \\;,\\;\\Psi _t=[2ik^2\\sigma +2kQ +i\\sigma (Q^2-Q_x)]\\Psi ,$ where $\\Psi =\\Psi (x,t,k)$ is a $ 3\\times 3$ matrix solution, $k$ is the complex spectral variable, the matrix $\\sigma =\\text{diag}\\lbrace 1\\,,\\,-1\\,,\\,-1\\rbrace $ is constant and diagonal, and $Q=Q(x,t)$ is the $ 3\\times 3$ matrix $Q=\\left( \\begin{array} {ccc}0 & -u^{(1)*} & -u^{(2)*} \\\\ u^{(1)} & 0 & 0 \\\\ u^{(2)} & 0 & 0 \\end{array} \\right).$ The starting point here is the construction of the solution representing one soliton wave nonlinearly superimposed to the following plane wave background solution of Eqs.",
"(REF ) $\\left( \\begin{array} {l}u^{(1)}_0(x,t) \\\\ u^{(2)}_0(x,t) \\end{array} \\right) = e^{2i\\omega t} \\left( \\begin{array} {l}a_{1} \\\\ a_{2} \\end{array} \\right),$ where $a_1$ and $a_2$ are arbitrary parameters which, with no loss of generality, are taken real.",
"Moreover the frequency $\\omega $ reads as $\\omega =a^2$ where, from now on, we set $a=\\sqrt{a_1^2+a_2^2}$ .",
"The Darboux method to obtain such one–soliton solution $u^{(1)}(x,t),\\,u^{(2)}(x,t)$ is well known, therefore we omit detailed computations, limiting ourselves to list the essential few steps.",
"Our results have been obtained by following the general formulation and construction as presented in [29] (the interested reader may find additional references quoted there).",
"The chain of calculations ends up with the following general formula [29] $\\left( \\begin{array} {l}u^{(1)} \\\\ u^{(2)} \\end{array} \\right) = e^{2i\\omega t} \\left( \\begin{array} {l}a_{1} \\\\ a_{2} \\end{array} \\right) + \\frac{2i(\\beta ^*-\\beta ) \\zeta ^*}{|\\zeta |^2+z^{\\dagger }z} \\left( \\begin{array} {r}z^{(1)} \\\\ z^{(2)} \\end{array} \\right).$ The constant parameter $\\beta $ is complex (with non vanishing imaginary part), while the functions $\\zeta (x,t)\\,,\\,z^{(1)}(x,t)\\,,\\,z^{(2)}(x,t)$ are the components of a generic 3–dimensional vector solution $Z(x,t)$ of the Lax pair of equations (REF ), which corresponds to the spectral parameter $k=\\beta $ and to the background solution (REF ).",
"Thus, if $Z_0$ is an arbitrary complex 3–dimensional vector, $Z(x,t)$ reads as $Z= \\left( \\begin{array} {c}\\zeta \\\\ z^{(1)} \\\\ z^{(2)} \\end{array} \\right)=\\left( \\begin{array} {ccc}1 & 0 & 0 \\\\ 0 & e^{2i \\omega t} & 0 \\\\ 0 & 0 & e^{2i\\omega t} \\end{array} \\right)\\exp (i\\Lambda x - i \\Omega t) Z_0,$ where $\\Lambda $ and $\\Omega $ are the constant matrices $\\Lambda =\\left( \\begin{array} {ccc}\\beta & ia_{1} & ia_{2} \\\\ -ia_{1} & -\\beta & 0 \\\\ -ia_{2} & 0 & -\\beta \\end{array} \\right), \\; \\;\\Omega = -\\Lambda ^2 -2\\beta \\Lambda +\\beta ^2 + 2a^2.$ Formula (REF ) shows that, if the matrix $\\Lambda $ (and therefore $\\Omega $ ) possesses three linearly independent eigenvectors, then the vector $Z(x,t)$ is a linear combination of exponential functions of $x$ and $t$ .",
"Therefore the solution (REF ) cannot be rational or semi–rational.",
"Since only the exponential function of a nilpotent matrix is polynomial, one has to find those particular values of $\\beta $ (see (REF )) such that the matrix $\\Lambda $ (and therefore $\\Omega $ ) is similar to a Jordan matrix.",
"Indeed this happens if and only if $\\beta =\\pm ia$ .",
"By taking f.i.",
"$\\beta =ia$ , in this way we arrive to the following semi–rational solution of the VNLSE Eqs.",
"(REF ) $\\left( \\begin{array} {l}u^{(1)}(x,t) \\\\ u^{(2)}(x,t) \\end{array} \\right) = e^{2i\\omega t} \\left[ \\frac{L}{B} \\left( \\begin{array} {l}a_{1} \\\\ a_{2} \\end{array} \\right) + \\frac{M}{B} \\left( \\begin{array} {r}a_{2} \\\\ - a_{1} \\end{array} \\right) \\right],$ with the following notation: $ L = \\frac{3}{2} -8\\omega ^2 t^2 -2 a^2 x^2 +8i\\omega t + |f|^2 e^{2a x}\\;, $ $ M= 4f (a x -2i\\omega t -\\frac{1}{2} ) e^{(a x + i \\omega t)}\\;, $ and $ B= \\frac{1}{2} +8\\omega ^2 t^2 + 2 a^2 x^2 + |f|^2 e^{2a x} $ , where $ f $ is a complex arbitrary constant.",
"It should be remarked that the dressing construction of the vector rogue wave (REF ) has introduced as arbitrary parameters the three complex components of the vector $Z_0$ , see (REF ) and (REF ).",
"However only the complex parameter $f$ is left essential out of these components, since the other parameters enter as trivial translations of the coordinates $x\\,,\\,t$ , which have been set to zero for simplicity.",
"The two other real parameters $a_1, a_2$ originate instead from the naked solution, namely from the background plane wave (REF ).",
"We note also that the dependence of $L, M$ and $B$ (see (REF )) on $x,t$ is both polynomial and exponential only through the dimensionless variables $ax$ and $\\omega t=a^2 t$ .",
"Moreover the vector solution (REF ) turns out to be a combination of the two constant orthogonal vectors $(a_1\\,,\\,a_2)^T$ and $(a_2\\,,\\,-a_1)^T$ .",
"Vector semi–rational rogue waves.",
"Setting $f=0$ implies $M=0$ : in this particular case the expression (REF ) yields the trivial vector generalization of the rational Peregrine solution [14], [22].",
"In this case $u^{(1)}(x,t)$ is merely proportional to $u^{(2)}(x,t)$ .",
"For future reference we note that the amplitude $|u^{(j)}(x,t)|$ is picked at $x=0$ with the maximum value $3 |a_j|$ at $t=0$ .",
"Figure: Deterministic vector freak waves envelope distributions |u (1) (x,t)||u^{(1)}(x,t)|and |u (2) (x,t)||u^{(2)}(x,t)| of ().",
"Here, a 1 =1,a 2 =0.1,f=0a_1=1, a_2=0.1, f=0.If instead $f\\ne 0$ , the Peregrine bump coexists and interacts with a pulse which propagates with non constant speed, and, depending on the value of $|f|$ , may have different looks.",
"In order to better describe this behavior we note that the ratios $L(x,t)/B(x,t)$ and $M(x,t)/B(x,t)$ which appear in (REF ) describe asymptotically as $t\\rightarrow \\pm \\infty $ a dark and a bright pulse, respectively.",
"Thus each wave component $u^{(j)}(x,t)$ is generically a mixture of a dark and a bright pulse.",
"Leaving aside the detailed analytic description of the solution (REF ) at intermediate times, we limit our present analysis to the large time behavior.",
"The pulse motion, for each individual dark and bright contribution, asymptotically reads as $x=\\xi (t) \\rightarrow x_0 +\\frac{1}{a} \\ln (\\omega |t|) + O\\left(\\frac{\\ln ^2(|t|)}{t^2}\\right),\\,\\,t\\rightarrow \\pm \\infty ,$ where $x_0=(1/a) \\ln (2\\sqrt{2}/|f|)$ .",
"This implies that this pulse goes to infinity where it “stops” since its velocity slowly vanishes as $d\\xi /dt \\rightarrow 1/(at)$ .",
"The shape of the dark and bright contributions at large times as a function of the parameter $\\chi $ which measures the displacement from the peak position, takes the expected form as $t\\rightarrow \\pm \\infty $ $\\frac{L}{B} \\rightarrow \\tanh (\\chi ) \\,,\\;\\frac{M}{B} \\rightarrow -i\\sqrt{2}\\left(\\frac{f t}{|f t|}\\right) \\frac{e^{i\\omega t}}{\\cosh (\\chi )} \\;.$ The superposition of the dark and bright contributions in each of the two wave components $|u^{(j)}|$ may give rise to complicated breather–like pulses.",
"These results are well represented in Figs.",
"REF -REF .",
"The single contributions of the dark shape $L/B$ and bright shape $M/B$ are better displayed when f.i.",
"$a_2=0$ .",
"In this case typical distributions $|u^{(1)}(x,t)|, |u^{(2)}(x,t)|$ are displayed in Figs.",
"REF , REF .",
"Figure REF shows a vector dark–bright soliton together with a single Peregrine soliton.",
"Decreasing the value of $|f|$ , Peregrine and dark–bright solitons separate.",
"By increasing $|f|$ , Peregrine and dark–bright solitons merge and the Peregrine bump cannot be identified while the resulting dark–bright pulse apprears as a boomeron-type soliton (see Fig.",
"REF ), i.e.",
"a soliton solution with a time–dependent velocity [30], [31].",
"Note that the solution (REF ) includes as a special case the bright-dark-rogue wave solution that was reported in [28].",
"Figure: As in Fig.",
", with f=0.1,a 1 =1,a 2 =0f=0.1, a_1=1, a_2=0.Figure: As in Fig.",
", with f=10,a 1 =1,a 2 =0f=10, a_1=1, a_2=0.Finally our formula (REF ), if all parameters $f, a_1, a_2$ are non vanishing, describes the dynamics of a breather–like wave resulting from the interference between the dark and bright contributions.",
"Distributions $|u^{(1)}(x,t)|, |u^{(2)}(x,t)|$ which are typical of this general case are displayed in Fig.",
"REF .",
"Again, by decreasing $|f|$ Peregrine and breather solitons separate while Peregrine and breather solitons merge, with boomeronic behavior, if instead $|f|$ increases.",
"Figure: As in Fig.",
", with f=0.1i,a 1 =1.2,a 2 =1.2f=0.1i, a_1=1.2, a_2=1.2.These results provide evidence of an attractive interaction between the dark–bright wave and the Peregrine rogue wave.",
"The observed behavior may also be interpreted as a mechanism of generation of one rogue wave out of a slowly moving boomeronic soliton.",
"Let us discuss the experimental conditions for the observation of the vector, semi-rational freak solitons.",
"Nonlinear optics is a fertile ground to develop the knowledge of the phenomenon of vector freak or rogue waves.",
"Figure: Numerical transmission of two 50 ps spaced dark-bright solitonsin optical fibers, y-polarized dark waves (E Y E_Y), andx-polarized bright envelopes (E X E_X).As first scenery, consider the propagation of arbitrarily polarized optical pulses in a weakly dispersive and nonlinear dielectric.",
"In fact, Eqs.",
"(REF ) apply to a Kerr medium with the electrostrictive mechanism of nonlinearity [34], as well as to randomly birefringent fiber optic transmission links [35], [36].",
"Indeed, the use of the polarization degree of freedom for doubling the capacity of long-distance fiber-based transmission systems has been currently widely adopted by means of the technique of polarization multiplexing.",
"To be specific, we consider the transmission at the 40 Gbit/s rate of a train of dark-bright solitons, dark in one polarization, and bright in the orthogonal polarization.",
"Fig.",
"REF shows that a Peregrine soliton is generated at 800 km, and it attracts a dark-bright soliton.",
"In this example, we numerically integrated Eqs.",
"(REF ) for properly rescaled wave envelope amplitudes $E_Y$ , $E_X$ and rescaled coordinates, with initial conditions two dark-bright solitons plus a small noise seed.",
"We used a fiber nonlinear coefficient of $1.3$ km$^{-1}$ W$^{-1}$ , the anomalous average fiber dispersion of $0.1$ ps$^2$ /km, and a dark-bright full width at half maximum of $8.25$ ps ($33\\%$ of the 25 ps bit period); the peak power of the two polarizations is equal to 3 mW and 6 mW, respectively.",
"As second scenery, we may consider incoherently coupled photorefractive spatial waves in strontium barium niobate (SBN).",
"Modulation instability and the existence of unstable dark-bright pairs (first steps in demonstrating vector Peregrine waves and dark-bright-Peregrine dynamics) have been already demonstrated in SBN [32].",
"Set-ups proposed in Ref.",
"[33], [32] can be exploited to observe and characterize spatial vector rogue waves in SBN.",
"Conclusions.",
"Here we have analytically constructed and discussed, a multi-parametric vector freak solution of the vector NLSE.",
"This family of exact solutions includes known vector Peregrine (rational) solutions, and novel freak solutions which feature both exponential and rational dependence on coordinates.",
"Because of the universality of the vector NLSE model (REF ), our solutions contribute to a better control and understanding of rogue wave phenomena in a variety of complex dynamics, ranging from optical communications to Bose-Einstein condensates and financial systems.",
"Acknowledgement.",
"The present research was supported in Brescia by the Italian Ministry of University and Research (MIUR) (Project Nb.2009P3K72Z)."
]
] | 1204.1449 |
[
[
"Online multipath convolutional coding for real-time transmission"
],
[
"Abstract Most of multipath multimedia streaming proposals use Forward Error Correction (FEC) approach to protect from packet losses.",
"However, FEC does not sustain well burst of losses even when packets from a given FEC block are spread over multiple paths.",
"In this article, we propose an online multipath convolutional coding for real-time multipath streaming based on an on-the-fly coding scheme called Tetrys.",
"We evaluate the benefits brought out by this coding scheme inside an existing FEC multipath load splitting proposal known as Encoded Multipath Streaming (EMS).",
"We demonstrate that Tetrys consistently outperforms FEC in both uniform and burst losses with EMS scheme.",
"We also propose a modification of the standard EMS algorithm that greatly improves the performance in terms of packet recovery.",
"Finally, we analyze different spreading policies of the Tetrys redundancy traffic between available paths and observe that the longer propagation delay path should be preferably used to carry repair packets."
],
[
"Introduction",
"Multipath streaming has gained much attention recently thanks to overlay networks and multiple access technologies (e.g., Wi-Fi, Cellular) available by default in handheld devices.",
"The benefits of multipath overlay routing and multipath streaming are presented in [1], [2] (e.g., reduction in correlation between consecutive packet losses, throughput gain, ability to react to congestion variation in different parts of the network).",
"Another interesting property of multipath has been illustrated in [3].",
"Fashandi et al.",
"[3] showed that the loss rate after packet recovery decays exponentially with the number of paths.",
"However, the challenging task in multipath streaming is to split the data flow among available paths to achieve better perceived video quality.",
"As a potential solution, in [4], Jurca et al.",
"proposed a load splitting scheme based on an end-to-end (E2E) distortion model for single layer video streaming.",
"Later in [5], they proposed a similar E2E distortion model for scalable video streaming as an objective function and used optimization algorithms to minimize the distortion.",
"One of the most achieved algorithms is Encoded Multipath Streaming (EMS) framework proposed by Chow et al.",
"[6].",
"In their proposal, the receiver observes the loss rate on each path, calculates the overall loss rate after packet recovery and sends the load splitting vector to the sender.",
"However, all these proposals ([3]-[7]) use Forward Error Correction (FEC) to protect the video from losses.",
"The main problem is this block code scheme requires to dynamically adapt its initial parameters and as a result, complex probing and network feedback analysis.",
"Recently, a novel erasure coding approach that prevents such complex configuration has been proposed [8], [9], [10].",
"In this paper, we propose to use an on-the-fly erasure coding scheme called Tetrys [10] to real-time multipath streaming and in particular, inside the EMS framework [6].",
"The rationale of using this framework is because EMS obtains better performance in terms of computation compared to [4], [5] [11].",
"We show that enabling Tetrys instead of FEC inside EMS greatly improves the overall performance in terms of packet delivery ratio in both uniform and burst losses.",
"We also study the decoupling between load allocation and redundancy traffic with Tetrys and propose several other measurements with different propagation delay not tackled in [6].",
"The results show that sending Tetrys repair packets on paths with longer propagation delay increases the packet delivery ratio before the E2E delay constraint in real-time transmission limited to hundreds milliseconds.",
"Furthermore, we improve the EMS scheme to better follow the network dynamics and to reduce the loss rate after packet recovery.",
"The rest of this article is organized as follows.",
"Section introduces briefly the EMS scheme.",
"Section presents the basic principle of Tetrys and the decoupling between load allocation and Tetrys redundancy traffic.",
"Section shows the results obtained from Tetrys compared to FEC with different settings and the benefits of decoupling between load allocation and Tetrys redundancy traffic.",
"Section presents the modified EMS algorithm and results.",
"We conclude and provide future work in section ."
],
[
"EMS principle",
"Fig.",
"REF shows an overview of the EMS scheme.",
"The FEC encoder in EMS sender takes the live stream and encodes with FEC parameters specified by k source packets and n-k repair packets.",
"The encoded stream is then splitted among available paths with different characteristics (e.g., propagation delay, loss rate, available bandwidth) thanks to the packet scheduler.",
"The EMS receiver stores all received packets and checks whether it can decode all lost packets in a FEC block specified by FEC(k,n).",
"In the context of live streaming, any packets arrived or recovered after the deadline are discarded.",
"The EMS scheme is detailed in [6].",
"Thus, we introduce the most important part of EMS, the Online Load Splitting (OLS).",
"At bootstrap, EMS sender equally splits the load between available paths so that the receiver measures the loss rate in each path.",
"At each period defined by OLS Adapt Window (in second), the EMS receiver executes the OLS algorithm as depicted in the pseudo code .",
"The information loss rate indicates the percentage of data that can not be recovered.",
"After performing the OLS, EMS receiver sends a feedback containing the load splitting vector and the FEC parameters.",
"The packet scheduler of EMS sender follows the load vector upon reception of the feedback.",
"Figure: EMS overview Online Load Splitting (OLS) [1] Compute the asymptotic optimal solution and split the load accordingly Sort the paths in the increasing order of loss rate Pick the first path in the list Increase the load on the chosen path by pre-defined $\\Delta _L$ (3% by default) Decrease the load on each of remaining paths by a fraction of $\\delta $ , proportional to their respective loss rates measured information loss rate increases Remove the chosen path from the list Revert to the previous load splitting the path list is empty goto Step 2 It is noted that the load splitting vector only decides the amount of traffic that each path should carry.",
"This inspires our study of decoupling between the load splitting vector and redundancy traffic with Tetrys (will be described in Section REF )."
],
[
"Tetrys multipath",
"We introduce in this section an on-the-fly erasure coding scheme called Tetrys coupled with EMS scheme for real-time multipath streaming.",
"Then, we present the rationale of decoupling between load allocation and Tetrys redundancy traffic."
],
[
"Basic principle of Tetrys",
"Tetrys uses an elastic encoding window buffer $B_{EW}$ which includes all the source packets sent without acknowledgment.",
"For every $k$ source packets, Tetrys sender sends a repair packet $R_{(i..j)}$ which is built as a linear combination of all packets currently in $B_{EW}$ from packet indexed i to j.",
"The receiver is expected to periodically acknowledge the received or decoded packets.",
"Upon reception of acknowledgment, the sender removes the acknowledged packets out of its $B_{EW}$ .",
"Generally, the receiver can decode lost packets as soon as the number of repair packets received is equal to the number of lost packets.",
"By this principle, Tetrys is tolerant to burst losses in neither source, repair nor acknowledgment packets as long as the redundancy ratio exceeds the packet loss rate (PLR).",
"Furthermore, the lost packets are recovered within a delay that does not depend on the Round Trip Time (RTT).",
"This property is very important for real-time applications.",
"Let us show in Fig.",
"REF a simple Tetrys data exchange with $k$ = 2 which implies that a repair packet is sent for every two sent source packets (or redundancy ratio of 33.3%).",
"The packet $P_2$ is lost during the data exchange.",
"However, the reception of repair packet $R_{(1,2)}$ allows to rebuild $P_2$ .",
"The acknowledgment for packets $P_1$ and $P_2$ from the receiver is lost.",
"This loss does not interrupt the transmission, the sender simply continues to compute the repair packets from $P_1$ .",
"Later, the lost packets $P_3$ , $P_4$ are rebuilt thanks to $R_{(1..6)}$ and $R_{(1..8)}$ .",
"The reception of second acknowledgment packet allows the sender to remove the acknowledged packets and build the repair packets from $P_9$ .",
"The reader is referred to [10] for further details.",
"Figure: A simple data exchange with Tetrys (k=2)"
],
[
"Decoupling load allocation and redundancy traffic with Tetrys",
"In [12], Kurant showed that the propagation time differences between paths reach several tens of milliseconds by measurements.",
"In case of FEC, the last packet (source or repair) in a FEC block arrived at the receiver must be sooner than the end-to-end (E2E) delay requirements normally specified by the path with longest propagation delay.",
"If the arrival date of last FEC packet exceeds the deadline due to long block size or queuing delay in the network, the sender should reduce the block size.",
"The size reduction makes FEC less tolerant to burst losses (see later in Fig.",
"REF ).",
"Thus, we believe that FEC repair packets can be sent to any available paths without changing the result with well dimensioning block size.",
"On the other hand, the arrival time of Tetrys repair packets is rather important since they are used to recover all previous lost packets if possible.",
"If a repair packet built from sent source packets without acknowledgment is transmitted to the path with short propagation delay, it is likely that the repair packet arrives sooner than the source packets sent on longer paths.",
"This means that the arrival of repair packet can not be used to recovered the previous lost packets at its arrival even though the source packets sent on longer paths arrive successfully.",
"This reduces the effectiveness of Tetrys repair packets in real-time streaming.",
"Based on this observation and the independence between load splitting vector and packet scheduler (see section ), we propose to decouple the load allocation on each path specified by the load splitting scheme and the way Tetrys repair packets are sent.",
"This implies that Tetrys repair packets are preferably sent to the path with longer propagation delay while keeping the same load allocation.",
"Table REF show different strategies to send Tetrys repair packets.",
"For instance, with “Tetrys long” strategy, the Tetrys repair packets are first sent to the path with longest propagation delay.",
"If the load on longest path is fulfilled, the Tetrys repair packets are sent to the path with second longest propagation delay and so on.",
"While Tetrys repair packets are sent to the available path according to the packet scheduler in ”Tetrys” strategy.",
"Table: Different strategies of sending Tetrys repair packets"
],
[
"Simulations and results",
"We use ns-2 [13] to evaluate Tetrys and FEC using EMS scheme.",
"The number of paths is specified in each simulation.",
"These paths can be built thanks to multiple physical interfaces or overlay network.",
"The path establishment is out of scope of this article.",
"We assume that the available bandwidth exceeds the application rate.",
"The one-way E2E delay constraint is set to 150ms based on ITU-T/G.114 [14] which is recommended for highly interactive applications.",
"One of the main characteristics of Tetrys is to be fully reliable whatever the burst size [10].",
"Indeed, all lost packets are recovered if the redundancy ratio exceeds the PLR.",
"However, we consider the packets as lost at the application level if their arrival or recovery date exceeds the deadline.",
"The information loss rate indicates the percentage of lost data that can not be recovered or be recovered after the deadline of 150ms.",
"To simulate the burst losses, we use a Gilbert-Elliot model in [10] which is specified by an average PLR and an average length of consecutive lost packets (or shortly mean burst size).",
"In each simulation, the streaming server sends a Constant Bit Rate (CBR) traffic at 1900 kb/s with packet size of 210 bytes.",
"The frequency of Tetrys acknowledgment packet is set to 10ms.",
"The feedback frequency does not change the result since it only affects the buffer sizes."
],
[
"Comparison between FEC and Tetrys with the same EMS scheme",
"In this simulation, there are two paths between a sender and a receiver.",
"The propagation delay on each path is set to 50ms.",
"The streaming lasts 4 hours and the OLS Adapt Window is set to 60s.",
"The redundancy ratio is set to 10% which is equivalent to FEC(45,50).",
"The PLR on path 1 is set to 3% and the PLR on path 2 varies from 0% to 5%.",
"Fig.",
"REF shows that Tetrys consistently outperforms FEC(45,50) in both uniform and mean burst size of 2 and 3 packets.",
"More specifically, Tetrys reduces up to more than 1% information loss rate in case of mean burst size of 3 packets.",
"With the video coding standard H.264/AVC, the Peak Signal to Noise Ratio (PSNR) with Tetrys can gain up to several dBs [15].",
"It is noted that the result with FEC(45,50) in Fig.",
"REF is similar to Fig.",
"14 in [6].",
"In fact, when the PLR on path 2 is less than 3%, the EMS scheme tends to assign more load on path 2, thus the information loss rate proportionally increases with the PLR on path 2.",
"When the PLR on path 2 is greater than 3%, the EMS scheme switches to assign more load on path 1.",
"This results in a rather flat in information loss rate at PLR on path 2 greater than 3%.",
"EMS scheme comes with FEC redundancy and FEC block size adaptations (see Fig.",
"REF ).",
"On the other hand, the configuration with Tetrys is simpler than FEC since Tetrys does not need to scale the block size.",
"With the same redundancy ratio, Tetrys achieves smaller information loss rate.",
"Thus, with the redundancy adaptation so that the loss requirement less than a threshold (normally 1% for video), Tetrys requires less redundancy than FEC.",
"In fact, in Fig.",
"REF , the information loss rate of Tetrys is much less than 1% at mean burst size of 3 packets at redundancy ratio of 10% while the redundancy for FEC must be greater than 10% to lower its information loss rate to less than 1%.",
"Fig.",
"REF shows the information loss rate of Tetrys and FEC(45,50) at PLR of 3% on both paths and mean burst size of 3 packets.",
"Since Tetrys is fully reliable, the lost packets are due to missed deadline.",
"Thus, the information loss rate of Tetrys is reduced with the relaxation of the delay requirement.",
"This implies that the gain of Tetrys against FEC is increased if the delay constraint is relaxed.",
"It is noted that we use EMS as load splitting scheme to demonstrate the better performance of Tetrys against FEC, we believe that Tetrys still outperform FEC in any load splitting scheme.",
"Figure: Tetrys vs FEC(45,50) at PLR of 3% on path 1 with uniform losses and burst losses with mean size of 2 and 3 packets, redundancy ratio is 10%Figure: Information loss rate of Tetrys and FEC(45,50) as a function of delay requirement with PLR of 3% on both paths"
],
[
"Propagation delay differences",
"In [6], the authors did not perform the tests where the available paths have different propagation delays.",
"We compare Tetrys and FEC in case of 3 paths with different settings to REF .",
"The PLR on each path is 14%, 10% and 12%, respectively.",
"The redundancy ratio is set to 25% which is equivalent to FEC(15,20), FEC(24,32), FEC(30,40) and FEC(45,60).",
"The simulation duration is 1000s with OLS Adapt Window of 1 second.",
"Kurant showed in [12] that the propagation delay differences between paths reach several tens of milliseconds.",
"Thus, we vary the propagation delay on each path from 50 to 80ms so that no path has the same delay to the others and the maximum delay difference between paths is 30ms.",
"This results in 24 simulations.",
"First, we compare different strategies of sending Tetrys repair packet (see Table REF ).",
"Fig.",
"REF shows the difference in information loss rate of “Tetrys long” against “Tetrys short” and “Tetrys” strategies for the mean burst size of 2 packets.",
"The positive value means that the information loss rate of “Tetrys long” is less than the compared strategy (“Tetrys long” is better) and vice versa.",
"It is clear that “Tetrys long” strategy outperforms other strategies in most cases.",
"Table REF shows the mean information loss rate and standard deviation of 24 simulations in case of uniform and mean burst size of 2 and 3 packets.",
"“Tetrys long” strategy shows better results in all cases.",
"At uniform losses and mean burst size of 2, 3 packets, “Tetrys long” gains 50%, 24% and 6%, respectively, against the best strategy among “Tetrys short” and “Tetrys”.",
"These results confirm our analyis in REF .",
"Thus, we consider Tetrys as “Tetrys long” strategy from now on.",
"Figure: “Tetrys long” performance against “Tetrys short” and “Tetrys“Table: Mean information loss rate and standard deviation of different strategies of sending Tetrys repair packets at uniform losses and burst losses with mean size of 2 and 3 packetsWe then compare Tetrys with different FEC settings (FEC(15,20), FEC(24,32), FEC(30,40), FEC(45,60)).",
"Fig.",
"REF shows the information loss rate of different FEC settings.",
"The larger FEC block size makes FEC more tolerant to burst losses but leads to more delay to recover the lost packets.",
"Fig.",
"REF shows the comparison between Tetrys and FEC(45,60), the best FEC among 4 settings, at mean burst size of 3 packets.",
"Tetrys outperforms FEC(45,60) regardless the propagation delay on each path.",
"Table REF shows the results of different FEC settings and Tetrys at both uniform losses and burst losses with mean size of 2 and 3 packets.",
"We can see that Tetrys has a significant gain in information loss rate compared to FEC.",
"Specifically, Tetrys has an average gain of 75% in information loss rate against the best FEC at mean burst size of 3 packets.",
"Figure: Comparison between different FEC settings at mean burst size of 2 packets.",
"PLR on each path is 14%, 10%, 12%.",
"Redundancy ratio is 25%Figure: FEC(45,60) vs Tetrys at mean burst size of 3 packets.",
"PLR on each path is 14%, 10%, 12%.",
"Redundancy ratio is 25%Table: Mean and standard deviation information loss rate with different FEC settings and Tetrys"
],
[
"EMS algorithm improvement",
"The original OLS algorithm (see pseudo code ) shows very good results.",
"However, it does not adapt well to the network dynamics.",
"In fact, assuming that the OLS is increasing the load on path 1, the loss rate on path 2 reduces significantly and is lower than path 1.",
"This might lead to the better information loss rate, the original OLS scheme is still in a repeat-until loop and continues to increase the load on path 1.",
"This make the OLS scheme goes farther from the new optimal load splitting, while it is better to stop increasing the load on path 1 and to increase the load on path 2.",
"Thus, we propose to add a pre-defined threshold of loss rate $\\theta $ .",
"At each period, the scheme compares the loss rate on each path with the one in previous period, if the absolute difference is greater than $\\theta $ , the OLS scheme quits the repeat-until loop and re-sorts the paths.",
"The improved OLS scheme is depicted in the pseudo code .",
"Modified OLS [1] Compute the asymptotic optimal solution and split the load accordingly Sort the paths in the increasing order of loss rate Pick the first path in the list the absolute difference of loss rate on one path exceeds the pre-defined threshold $\\theta $ goto Step 2 Increase the load on the chosen path by pre-defined $\\Delta _L$ Decrease the load on each of remaining paths by a fraction of $\\delta $ , proportional to their respective loss rates measured information loss rate increases Remove the chosen path from the list Revert to the previous load splitting the path list is empty goto Step 2 With the same settings as in REF , we re-run the simulations with a pre-defined threshold $\\theta $ = 5%.",
"The information loss rate of both FEC(45,60) and Tetrys with threshold is lower than the one without threshold in case of mean burst size of 2 packets (Fig.",
"REF and REF ).",
"Table REF shows the improvement in information loss rate with modified EMS scheme in both uniform and burst losses.",
"At mean burst size of 2 packets, FEC(45,60) and Tetrys with threshold has an average gain of 30% and 65%, respectively, compared to the case without threshold.",
"While FEC(45,60) and Tetrys with modified EMS scheme have an average gain of 21% and 49%, respectively in comparison to the original EMS scheme at mean burst size of 3 packets.",
"These simulations show that Tetrys achieves much lower information loss rate with modified OLS algorithm although it has a very small information loss rate using the original one.",
"Figure: FEC(45,60) without threshold and with threshold θ\\theta = 5% at mean burst size of 2 packetsFigure: Tetrys without threshold and with threshold θ\\theta = 5% at mean burst size of 2 packetsTable: Mean and standard deviation information loss rate with and without threshold"
],
[
"Conclusions and future work",
"In this paper, we introduced an on-the-fly coding scheme named Tetrys to real-time multipath streaming.",
"With the same load splitting scheme, the EMS scheme presented in [6], we have shown that Tetrys consistently has significant reduction in information loss rate compared to the FEC approach in both uniform and burst losses.",
"We showed that the decoupling between load allocation and Tetrys redundancy traffic improves the performance in terms of loss rate after packet recovery.",
"The Tetrys repair packets are preferably sent to the path with longer propagation delay shows best performance.",
"Furthermore, we showed that the EMS scheme can be improved to provide better results.",
"By introducing a threshold parameter, the modified EMS scheme adapts well to the network dynamics and showed a significant reduction in information loss rate compared to the original one.",
"For future work, we plan to analyze the multipath streaming in more realistic contexts (e.g., 2 paths with Wi-Fi and 3G/LTE) and to validate the results with video data."
],
[
"Acknowledgments",
"This work was supported by the French ANR grant ANR-VERS-019-02 (ARSSO project)."
]
] | 1204.1428 |
[
[
"Branching structure for the transient random walk on a strip in a random\n environment"
],
[
"Abstract An intrinsic branching structure within the transient random walk on a strip in a random environment is revealed.",
"As applications, which enables us to express the hitting time explicitly, and specifies the density of the absolutely continuous invariant measure for the \"environments viewed from the particle\"."
],
[
"Introduction",
"Let $d\\ge 1$ be any integer and denote ${D}=\\lbrace 1,2,\\cdots ,d\\rbrace $ , we consider random walks in a random environment on the strip $S= \\mathbb {Z} \\times \\lbrace 1,2,\\cdots ,d\\rbrace $ .",
"This model was introduced by Bolthausen and Goldsheid ([1], 2000), where the conditions for recurrent and transient has been obtained.",
"After that, Goldsheid ([4], 2008) considered the hitting time of the walk by the method of “enlarged random environments\"; Bolthausen and Goldsheid ([2], 2008) obtained the $(\\log t)^2$ asymptotic behaviour and Roitershtein ([10], 2008) proved a strong law of large numbers and an annealed central limit theorem for the walk in a suitable environment situation; etc..",
"The aim of this paper is to reveal the intrinsic branching structure within the transient random walk on a strip in a random environment, which enables us to express the hitting time explicitly.",
"Roitershtein (Theorem 2.3, [10], 2008) figured out the stationary distribution for the Markov chain of “environments viewed from the particle\" which is equivalent to the original distribution.",
"To specify the density of the absolutely continuous invariant measure is another application of our branching structure.",
"And as a by product, the rate of the LLN can be obtained.",
"For the nearest random walk in random environment (RWRE, for short) on the line, as we known, the branching structure is a powerful tool in the proof of the famous result about “stable law\" by Kesten et al ([8], 1975), and is also used by Ganterta and Shi in ([5]).",
"The branching structure for the one dimensional RWRE with bounded jumps has been considered by Key ([9], 1987), Hong & Wang ([6], 2009) and Hong & Zhang ([7], 2010).",
"We adapt the description of the model as that of [1].",
"Let $(P_n,Q_n,R_n), -\\infty < n < \\infty $ , be a strictly stationary ergodic sequence of triples of $m\\times m$ matrices with non-negative elements such that for all $n$ the sum $P_n + Q_n + R_n$ is a stochastic matrix, i.e., $(P_{n}+Q_{n}+R_{n})\\mathbf {1}=\\mathbf {1}$ , where $\\mathbf {1}$ is a column vector whose components are all equal to 1.",
"We write the components of $P_n$ as $P_n(i, j ), 1\\le i, j \\le m $ , and similarly for $Q_n$ and $R_n$ .",
"Let $(\\Omega ,{F},P,\\theta )$ be the corresponding dynamical system with $\\Omega $ denoting the space of all sequences $\\omega := (\\omega _n) = ((P_n,Q_n,R_n))$ of triples described above, ${F}$ being the corresponding natural $\\sigma $ -algebra, $P$ denoting the probability measure on $(\\Omega ,{F})$ , and the shift operator on $\\Omega $ defined by $\\theta $ : $(\\theta \\omega )_{n}=\\omega _{n+1},~n\\in \\mathbb {Z}$ .",
"The random walk on the strip $S=\\mathbb {Z} \\times {D}:= \\mathbb {Z} \\times \\lbrace 1,2£¬\\cdots , d\\rbrace $ is denoted by $X=\\lbrace X_{n},n \\in \\mathbb {Z}\\rbrace $ , $X_{n}=(\\xi _{n},Y_{n}),~~\\xi _{n}\\in \\mathbb {Z},~Y_{n}\\in {D}.$ $\\xi _{n}$ is the $\\mathbb {Z}$ -coordinate of the walk and $Y_{n}$ takes values in ${D}:=\\lbrace 1,2£¬\\cdots , d\\rbrace $ .",
"For describing the initial distribution, we introduce $\\mathcal {M}_{d}$ , $\\mathcal {M}_{d}=\\Big \\lbrace (\\mu _{\\omega })_{\\omega \\in \\Omega }: \\mu _{w} \\mbox{ is a probability measure vector on} \\ {D}=\\lbrace 1,2,\\cdots , d\\rbrace \\Big \\rbrace .\\nonumber $ Given a environment $\\omega \\in \\Omega $ and an $\\mu =(\\mu _{\\omega }) \\in \\mathcal {M}_{d}$ , one can define the random walk $X_{n}$ on the strip $S= \\mathbb {Z} \\times {D}$ to be a time-homogeneous Markov chain taking values in $ \\mathbb {Z} \\times \\lbrace 1,2,\\cdots ,d\\rbrace $ , which is determined by its transition probabilities $\\mathfrak {Q}_{\\omega }(z,z_{1})$ : $\\mathfrak {Q}(z,z_{1})=\\left\\lbrace \\begin{array}{r@{\\quad \\quad }l}P_{n}(i,j) & \\mbox{if}~ \\ z=(n,i),z_{1}=(n+1,j),\\\\ R_{n}(i,j) & \\mbox{if}~ \\ z=(n,i),z_{1}=(n,j),\\\\ Q_{n}(i,j) & \\mbox{if}~ \\ z=(n,i),z_{1}=(n-1,j),\\\\ 0 & \\mbox{otherwise},\\end{array}\\right.$ and initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=z_{0})=\\mu _{\\omega }(z_{0})\\quad \\mbox{for any}\\quad z_{0}\\in {D}.$ This defines for any starting point $x_{0}=(0,y_{0})\\in S$ and for any $\\omega \\in (\\Omega ,{F},P)$ , the quenched law $P_{\\omega }^{\\mu }$ for the Markov chain by $P_{\\omega }^{\\mu }(X_{0}=x_{0},X_{1}=x_{1},\\cdots ,X_{n}=x_{n}):=\\mu _{\\omega }(y_{0}) \\mathfrak {Q}_{\\omega }(x,x_{1}) \\mathfrak {Q}_{\\omega }(x_{1},x_{2}) \\cdots \\mathfrak {Q}_{\\omega }(x_{n-1},x_{n}).$ Then we define a annealed law $\\mathbb {P}^{\\mu }=P\\bigotimes P_{\\omega }^{\\mu }$ on $(\\Omega \\times (\\mathbb {Z} \\times {D})^{\\mathbb {N}},{F}\\times {G})$ by $\\mathbb {P}^{\\mu }(F\\times G)=\\int _{F}P_{\\omega }^{\\mu }(G)P(d\\omega )\\quad F\\in {F},~G \\in {G},$ and the expectation with respect to $\\mathbb {P}^{\\mu }$ defined by $\\mathbb {E}^{\\mu }$ .",
"Statements involving $P_{\\omega }^{\\mu }$ and $\\mathbb {P}^{\\mu }$ are called quenched and annealed, respectively.",
"Notations and assumption.",
"Throughout the paper we use the notation $\\mathbf {0}=(0,0,\\cdots ,0)\\in \\mathbb {R}^{d}$ , $\\mathbf {1}=(1,1,\\cdots ,1)\\in \\mathbb {R}^{d}$ , and denote $\\mathbf {e_{i}}=(0,\\cdots ,1,\\cdots ,0),~(i=1,2,\\cdots ,d)$ as the canonical basis of $\\mathbb {R}^{d}$ .",
"For the vector $\\mathbf {x}=(x_{j})$ and matrix $A=(a(i,j))$ , define $\\Vert \\mathbf {x}\\Vert :=\\max _{j}|x_{j}| \\quad \\mbox{and}\\quad \\Vert A\\Vert :=\\max _{i}\\sum _{j}|a(i,j)|.$ We say that $A$ is strictly positive (denoted by $A>0$ ) if all its components satisfy $a(i,j)>0$ , and $A$ is non-negative (which is denoted by $A\\ge 0$ ) if all $a(i,j)$ are negative.",
"If a $d\\times d$ real matrix $A$ is non-negative, $\\Vert A\\Vert :=\\Vert A\\mathbf {1}\\Vert $ .",
"Finally, we use the notation $\\mathbf {I}_{A}$ for the indicator function of the set $A$ .",
"For the random walk $X_{n}=(\\xi _{n},Y_{n})$ , we often use the expressions like $\\lim _{n\\rightarrow \\infty }X_{n}=+\\infty $ which simply means $\\xi _{n}$ tends to $+\\infty $ as $n\\rightarrow \\infty $ .",
"The hitting time $T_{n}$ is defined as the the first time when the walk reaches layer $n$ , $L_{n}:=\\lbrace (n,j),1\\le j\\le m\\rbrace $ starting from a point $z\\in L_{0}:=\\lbrace (0,j),1\\le j\\le m\\rbrace $ .",
"Let $T_o=0$ , and for $n\\ge 1$ , $T_{n}:=\\inf \\lbrace t:~X(t)\\in L_{n}\\rbrace \\quad \\mbox{and} \\quad \\tau _{n}:=T_{n}-T_{n-1},$ with the usual convention that the infimum over an empty set is $\\infty $ and $\\infty -\\infty =\\infty $ .",
"The following Condition C is from Bolthausen and Goldshied [1].",
"Condition C. C1 The dynamical system $(\\Omega ,{F},\\mathbb {P},\\mathcal {T})$ is ergodic.",
"C2 $\\mathbb {E}\\log (1-\\Vert R_{n}+P_{n}\\Vert )^{-1}<\\infty \\quad \\mbox{and}\\quad \\mathbb {E}\\log (1-\\Vert R_{n}+Q_{n}\\Vert )^{-1}<\\infty .$ C3 For all $j\\in \\lbrace 1,2,\\cdots , m\\rbrace $ and all $n$ , $\\sum _{i=1}^{m}Q_{n}(i,j) > 0,\\quad \\sum _{i=1}^{m}P_{n}(i,j) > 0 \\quad \\mathbb {P}\\mbox{-almost surely}.$ C4 With positive $\\mathbb {P}$ -probability, the layer 0 is in one communication class.",
"Known results.",
"Let us first review some known results about the random walk in a random environment on the strip.",
"1.recurrence and transience.",
"If Condition C is satisfied, Theorem 1 in [1] proved $\\zeta _{n},~n\\in \\mathbb {Z}$ of $m\\times m$ matrices is the unique sequence of stochastic matrices which satisfies the following system of equations: $\\zeta _{n}=(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}P_{n},\\quad P-a.s.~n\\in \\mathbb {Z},$ and the enlarged sequence $(P_n,Q_n,R_n, \\zeta _{n}), -\\infty < n < \\infty ,$ is stationary and ergodic.",
"Let $A_{n}:=(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}Q_{n}\\quad \\mbox{and}\\quad u_{n}:=(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}\\mathbf {1}$ and $\\lambda ^{+}:=\\lim _{n\\rightarrow \\infty }\\frac{1}{n}\\log \\parallel A_{n}A_{n-1}\\cdots A_{1} \\parallel ,$ Theorem 2 in [1] gave the criterion of recurrent and transient behavior for $X_{n}=(\\xi _{n},Y_{n})$ .",
"One of the cases is $\\lim _{t\\rightarrow \\infty } \\xi (t)= \\infty , ~~\\mathbb {P}-a.e.\\quad \\mbox{if and only if} \\quad \\lambda ^{+}< 0.$ 2.exit probability.",
"Let $\\eta _{n}(i,j)$ be the probability of a random walk starting in $(n,i)$ reaches the layer $n+1$ at point $(n,j)$ finally, we usually called it the exiting probability.",
"If the random walk is transient to the right, we have $\\eta _{n}=\\zeta _{n},~P-a.e.$ (see [4], (1.15) ).",
"And if Condition C is satisfied, $\\zeta _{n}>0$ for $P-a.s.~\\omega $ .",
"We only concentrate on random walks which are transient to the right in our paper.",
"3.stationary sequence of probability vectors $y_n$ .",
"If Condition C is satisfied then following limit exists for $P-a.s.$ $\\omega $ (Lemma 1, [4]): $\\mathbf {y}_{n}:=\\lim _{a\\rightarrow -\\infty }\\mathbf {u}_{a}\\zeta _{a}(\\omega )\\zeta _{a+1}(\\omega )\\cdots \\zeta _{n}(\\omega ).$ where $\\mathbf {u}_{a}$ is any sequence of row-vectors with non-negative components $u_{a}(i)$ , and $\\sum _{i=1}^{d}u_{a}(i)=1$ .",
"Note that the sequence $\\lbrace \\mathbf {y}_{n}\\rbrace $ is the unique solution of $\\mathbf {y}_{n}=\\mathbf {y}_{n-1}\\zeta _{n}$ in the class of probability vectors and it has the property $y_n > 0$ , which is a probability measure on ${D}=\\lbrace 1,2,\\cdots , d\\rbrace $ whose support is the whole ${D}$ .",
"It is clear that vectors $\\mathbf {y}_n := \\mathbf {y}(\\omega _{\\le n}) $ form a stationary sequence."
],
[
"Statement of main results.",
"We assume the walk $X_{n}=(\\xi _{n},Y_{n})$ starts from layer 0, the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i), ~P-a.s. \\omega ,$ for any $i\\in {D}$ with $\\mu _{\\omega }\\in \\mathcal {M}_{d}$ .",
"In what follows, suppose Condition C is satisfied and $\\lambda ^{+}< 0$ , i.e., we concentrate on random walks $X_{n}=(\\xi _{n},Y_{n})$ transient to the right $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ .",
"In this case, suppose $T_{0}=0$ and we have $T_{k}<\\infty ,~\\mathbb {P}-a.s.$ for any positive integer $k\\ge 1$ .",
"The aim of this paper is to calculate the hitting time $T_{1}=\\inf \\lbrace i:~\\xi (i)=1\\rbrace $ accurately in terms of the intrinsic branching structure within the walk.",
"For $n\\le 1$ , define $\\mathbf {U}_{n}=(U_{n}^{1},U_{n}^{2},\\cdots ,U_{n}^{d})$ , where $U_{n}^{i}~(1\\le i\\le d)$ is the number of steps from layer $n$ to layer $n-1$ at the site $(n-1,i)$ before time $T_{1}$ .",
"$\\mathbf {Z}_{n}=(Z_{n}^{1},Z_{n}^{2},\\cdots ,Z_{n}^{d})$ , where $Z_{n}^{i}~(1\\le i\\le d)$ is the number of steps from layer $n$ to the same layer at the site $(n,i)$ before time $T_{1}$ .",
"And $|\\mathbf {U}_{n}|=\\sum _{i=1}^{d}U_{n}^{i}=\\mathbf {U}_{n}\\mathbf {1}\\quad \\mbox{and} \\quad |\\mathbf {Z}_{n}|=\\sum _{i=1}^{d}Z_{n}^{i}=\\mathbf {Z}_{n}\\mathbf {1}.$ All steps before $T_1$ can be recorded by $\\mathbf {U}_{n}$ and $\\mathbf {Z}_{n}$ .",
"Since $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , if the random walk takes a step to the left from any layer $n~(n\\le 0)$ , it must come back finally from layer $n-1$ to layer $n$ , so $T_{1}= 1+\\sum _{n\\le 0}(2|\\mathbf {U}_{n}|+|\\mathbf {Z}_{n}|),$ and the following theorem tells us that $\\lbrace |\\mathbf {U}_{n}|, |\\mathbf {Z}_{n}|, ~n\\le 1 \\rbrace $ is an inhomogeneous branching process with immigration.",
"The exit probability $\\eta _{n}$ plays an important role, when $X_{n}\\rightarrow +\\infty $ , $\\eta _{n}=\\zeta _{n},~P-a.e.$ , (see [4], (1.15) ), which is given by (REF ).",
"Theorem 1.1 Assume Condition C is satisfied and $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i), ~P-a.s.. $ Then (1) for $P-a.s.~\\omega $ , $\\lbrace |\\mathbf {U}_{n}|,~n\\le 1\\rbrace $ and $\\lbrace |\\mathbf {Z}_{n}|,~n\\in \\mathbb {Z}\\rbrace $ are inhomogeneous branching processes with immigration.",
"The offspring distribution is given by for $n\\le 0$ $P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m \\Big | \\mathbf {U}_{n+1}=\\mathbf {e}_{i}\\Big )=\\mathbf {e}_{i} [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{m}(I-R_{n})^{-1}P_{n}\\mathbf {1},$ $P_{\\omega }^{\\mu }\\Big (|\\mathbf {Z}_{n}|=K \\Big | \\mathbf {U}_{n+1}=\\mathbf {e}_{i}\\Big )=\\mathbf {e}_{i}[(I-Q_{n}\\zeta _{n-1})^{-1} R_{n}]^{K} (I-Q_{n}\\zeta _{n-1})^{-1} P_{n} \\mathbf {1},$ with immigration $P_{\\omega }^{\\mu }\\Big (\\mathbf {U}_{1}=\\mathbf {e}_{i} \\Big )=\\mu _{\\omega }(i), \\ \\ \\ \\ i\\in {D},$ where $\\zeta _{n}=\\eta _{n}$ (see [4], (1.15) ) is exit probability, which is given by (REF ).",
"(2) The first hitting time $T_1$ is given by $T_{1}= 1+\\sum _{n\\le 0}(2|\\mathbf {U}_{n}|+|\\mathbf {Z}_{n}|).$ $\\Box $ Remark (1) In Theorem REF , we restrict ourselves only to the trajectory of the walk $X_t$ for $t\\in [0, T_1]$ , and all the steps have been counted in $\\lbrace |\\mathbf {U}_{n}|, |\\mathbf {Z}_{n}|, ~n\\le 1 \\rbrace $ which formulate a branching structure as (REF ) and (REF ) with immigration (REF ).",
"After that, the trajectory of the walk $X_t$ follows the same structure.",
"For example, the trajectory of the walk $X_t$ for $t\\in [T_1, T_2]$ , all the steps have been counted in $\\lbrace |\\mathbf {U}_{n}|, |\\mathbf {Z}_{n}|, ~n\\le 2 \\rbrace $ which formulate a branching structure as (REF ) and (REF ) with immigration $P_{\\omega }^{\\mu }\\Big (\\mathbf {U}_{2}=\\mathbf {e}_{i} \\Big )=Y_{T_1}(i)$ , and so on.",
"(2) Note that it is “unsymmetrical\" in the branching structure (REF ) and (REF ) between the “father \" and “children\".",
"It can be explained as that we focus on the number of the “children\" but the individual of the “father \" (determine the probability).",
"$\\Box $ As an immediate application of the branching structure, we can calculate the mean of the hitting time explicitly.",
"Theorem 1.2 Assume Condition C is satisfied and $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , and the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i), ~P-a.s.. $ Then $&E{T_1}=&E({\\overrightarrow{\\mu _\\omega }}(u_{0}+A_{0}u_{-1}+ \\cdots +A_{0}A_{-1}\\cdots A_{-k}u_{-k-1}+\\cdots ))\\nonumber ,$ where $A_n, u_n$ is given in (REF ).",
"$\\Box $ Another application of the branching structure is to specify the density of the absolutely continuous invariant measure for the “environments viewed from the particle\".",
"Let us review the process discussed in Section 4 of $\\cite {[Roi08]}$ .",
"Let $\\overline{\\omega _{n}}=\\theta ^{\\xi _{n}}w$ , for $n\\ge 0$ , and consider the process $Z_{n}:=(\\overline{\\omega _{n}},Y_{n})$ , defined in $(\\Omega \\times {D},{F}\\otimes {H})$ , where ${H}$ as the set of all subsets of ${D}$ , and the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i)=\\mathbf {y}_{-1} (i) $ given by (REF ).",
"$(Z_{n})_{n\\ge 0}$ is a Markov chain under $\\mathbb {P}^{\\mu }$ with transition kernel $K(\\omega ,i;B,j)=P_{0}(i,j)I_{B}(\\theta w)+R_{0}(i,j)I_{B}(w)+Q_{0}(i,j)I_{B}(\\theta ^{-1} w).$ Usually, $Z_{n}=(\\overline{\\omega _{n}},Y_{n})$ be called as auxiliary Markov chain.",
"Let $v_{p}=\\frac{1}{\\mathbb {E}T_{1}}$ , whenever $\\mathbb {E}T_{1} < \\infty $ .",
"For $B \\in {F}$ , $i\\in {D}$ , define a probability measure $Q$ on $(\\Omega \\times {D},{F}\\otimes {H})$ : $Q(B,i):=v_{p}\\mathbb {E}\\left(\\sum _{n=0}^{T_{1}-1}I_{B}(\\overline{\\omega _{n}})I_{Y_{n}}(i)\\right).$ $Q(\\cdot )$ is a invariant measure under the Markov kernel $K$ (Proposition 4.1, $\\cite {[Roi08]}$ ).",
"Define a probability measure $\\overline{Q}(\\cdot )$ on $(\\Omega ,{F})$ by setting $\\overline{Q}(B):=Q(B,{D}),\\quad B \\in {F}.$ and let $~Q_{i}(B):=Q(B,i)~\\mbox{for}~B \\in {F}$ .",
"Then both $Q_{i}(\\cdot )$ and $\\overline{Q}(\\cdot )$ are absolutely continuous with regard to $P$ (Proposition 4.1, $\\cite {[Roi08]}$ ), but where only the up bound of the density have been proved.",
"The branching structure enable us to specify the density completely in the following theorem.",
"Theorem 1.3 Assume Condition C is satisfied and $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i)=\\mathbf {y}_{-1} (i) $ , for ${P}-a.s.~\\omega $ , and assume in addition that $v_{p}>0$ .",
"Then $Q_{i}(\\cdot )$ is absolutely continuous with regard to $P$ , and so is $\\overline{Q}(\\cdot )$ .",
"The density is given by $\\frac{dQ_{i}}{dP}=\\Lambda _{\\omega }^{(i)},$ where $\\Lambda _{\\omega }^{(i)}=v_{p} [\\mathbf {\\mu }_{\\omega }\\left(\\widetilde{u}_{0}+\\zeta _{0}A_{1}\\widetilde{u}_{0}+ \\zeta _{0}\\zeta _{1}A_{2}A_{1}\\widetilde{u}_{0}+\\cdots \\right)](i).$ and $\\frac{d\\overline{Q}}{dP}=\\Lambda _{\\omega },$ where $\\Lambda _{\\omega }= v_{p} [\\mu _{\\omega }\\left(\\widetilde{u}_{0}+\\zeta _{0}A_{1}\\widetilde{u}_{0}+ \\zeta _{0}\\zeta _{1}A_{2}A_{1}\\widetilde{u}_{0}+\\cdots \\right)]\\mathbf {1},$ where $\\widetilde{u}_{n}:=(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}$ .",
"$\\Box $ Remark (1) The first part of the Theorem $\\ref {thm3}$ is obtained in Proposition 4.1 of $\\cite {[Roi08]}$ .",
"We will focus on the “density\" part only.",
"(2) As a by product, we can prove the LLN from two different method as the situation for the nearest RWRE on the line ($\\cite {[Zei04]}$ ).",
"On the one hand, If $\\overrightarrow{\\mu _\\omega }=\\mathbf {y}_{-1} $ , then $\\lbrace \\tau _i: i\\in N\\rbrace $ in (REF ) is a stationary and ergodic sequence variables (Lemma 3.2, $\\cite {[Roi08]}$ ), so the LLN can be obtained from the hitting time decomposition; on the other hand, with the “density\" in hand, it is easy to obtain the LLN again from the point of view “environments viewed from the particle\".",
"We omit the details of the proof.",
"Corollary 1.4 Assume Condition C is satisfied and $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i)=\\mathbf {y}_{-1} (i) $ , for ${P}-a.s.~\\omega $ , and assume in addition that $v_{p}>0$ .",
"Then $\\mathbb {P}-a.s.$ , $&&\\lim _{n\\rightarrow \\infty } \\frac{\\xi (n)}{n}=\\frac{1}{E\\Big (\\mathbf {y}_{-1}(u_{0}+A_{0}u_{-1}+ \\cdots +A_{0}A_{-1}\\cdots A_{-k}u_{-k-1}+\\cdots )\\Big )}.$ $\\Box $"
],
[
"Intrinsic branching structure—",
"Assume Condition C is satisfied and $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i), ~P-a.s.. $ Note that $T_{k}<\\infty ,~\\mathbb {P}-a.s.$ for any positive integer $k\\ge 1$ .",
"We will analyze the trajectory of the walk, and restrict to the first excursion between lay 0 to lay 1, i.e., the path of $X_k$ for $k\\in [0,T_1]$ .",
"Define for $n\\le 0$ , $\\alpha _{n,0}&=&\\min \\lbrace k\\le T_{1}:~X_{k}\\in L_{n} \\rbrace ,\\nonumber \\\\\\beta _{n,0}&=&\\min \\lbrace \\alpha _{n,0}< k \\le T_{1}:~X_{k-1}\\in L_{n},~X_{k}\\in L_{n-1}\\rbrace .\\nonumber $ And for $b\\ge 1$ , $\\alpha _{n,b}&=&\\min \\lbrace \\beta _{n,b-1}< k\\le T_{1}:~X_{k}\\in L_{n} \\rbrace ,\\nonumber \\\\\\beta _{n,b}&=&\\min \\lbrace \\alpha _{n,b}< k \\le T_{1}:~X_{k-1}\\in L_{n},~X_{k}\\in L_{n-1}\\rbrace .\\nonumber $ (with the usual convention that the minimum over an empty set is $+\\infty $ ).",
"We refer to the time interval $[\\beta _{n,b-1},~\\alpha _{n,b}]$ as the $b$ -th excursion from $n-1$ layer to $n$ layer.",
"For any $b\\ge 0$ , any $n\\le 0$ , and $i\\in \\lbrace 1,2,\\cdots ,d\\rbrace $ , define $U_{n,b}^{i}&:=&\\sharp \\lbrace k\\ge 0:~X_{k-1}\\in L_{n},~X_{k}=(n-1,i),~\\beta _{n+1,b}<k<\\alpha _{n+1,b+1}\\rbrace ,\\\\Z_{n,b}^{i}&:=&\\sharp \\lbrace k\\ge 0:~X_{k-1}\\in L_{n},~X_{k}=(n,i),~\\beta _{n+1,b}<k<\\alpha _{n+1,b+1}\\rbrace .$ Note that $U_{n,b}^{i}$ is the number of steps from layer $n$ to $(n-1,i)$ during the $b+1$ -th excursion from layer $n$ to layer $n+1$ , whereas $Z_{n,b}^{i}$ is the number of steps from layer $n$ to $(n,i)$ during the same excursion.",
"Define for $n\\le 0$ and $i\\in \\lbrace 1,2,\\cdots ,d\\rbrace $ , $U_{n}^{i}:=\\sum _{b\\ge 0} U_{n,b}^{i}$ , then $U_{n}^{i}$ is the number of steps from layer $n$ to $(n-1,i)$ before time $T_{1}$ .",
"Similarly define $Z_{n}^{i}:=\\sum _{b\\ge 0}Z_{n,b}^{i}$ .",
"$\\mathbf {U}_{n}=(U_{n}^{1},U_{n}^{2},\\cdots ,U_{n}^{d})$ , and $|\\mathbf {U}_{n}|=\\sum _{i=1}^{d}U_{n}^{i}=\\mathbf {U}_{n}\\mathbf {1}$ ; $\\mathbf {Z}_{n}=(Z_{n}^{1},Z_{n}^{2},\\cdots ,Z_{n}^{d})$ , and $|\\mathbf {Z}_{n}|=\\sum _{i=1}^{d}Z_{n}^{i}=\\mathbf {Z}_{n}\\mathbf {1}$ which have been defined in (REF ).",
"By Markov property, we obtain $&& P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m, |\\mathbf {Z}_{n}|=K \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )\\nonumber \\\\&=& \\mathbf {e}_{i} \\sum \\nolimits _{k_{0}+k_{1}+\\cdots +k_{m}=K} R_{n}^{k_{0}} Q_{n}\\zeta _{n-1} R_{n}^{k_{1}} \\cdots Q_{n}\\zeta _{n-1} R_{n}^{k_{m}} P_{n} \\mathbf {1}.$ where $\\zeta _{n}=\\eta _{n}$ is the exiting probability matrix (see ($\\ref {ep}$ )).",
"In (REF ), the path of an excursion is considered: the particle start from layer $n$ (given by $ \\mathbf {U}_{n+1}=\\mathbf {e_{i}}$ ), moves at layer $n$ by $|\\mathbf {Z}_{n}|=K$ steps (each step with probability $R_{n} $ ) and $|\\mathbf {U}_{n}|=m$ steps from layer $n$ to layer $n-1$ ( but in the trajectory point, each “down\" step with probability $Q_{n}$ must connect with a path “from layer $n-1$ finally goes back to layer $n$ \" with probability $\\zeta _{n-1}$ ), the last step of the excursion is from layer $n$ to layer $n+1$ with probability $P_{n}$ .",
"The idea of (REF ) is that we only care the number of the “children\", which lead to the “unsymmetrical\".",
"Note that only the “$ \\mathbf {U}$ \" type particles produce “children\".",
"With a similar consideration, the branching mechanism can also be expressed as $&& P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m, |\\mathbf {Z}_{n}|=K \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )\\nonumber \\\\&=& \\mathbf {e}_{i} \\sum \\nolimits _{m_{0}+m_{1}+\\cdots +m_{K}=m} Q_{n}\\zeta _{n-1}^{m_{0}} R_{n} Q_{n}\\zeta _{n-1}^{m_{1}} \\cdots R_{n} Q_{n}\\zeta _{n-1}^{m_{K}} P_{n} \\mathbf {1}.$ In what follows, we will derive the marginal distribution of $|\\mathbf {U}_{n}|$ and $|\\mathbf {Z}_{n}|$ respectively.",
"Let's discuss the marginal distribution of $|\\mathbf {U}_{n}|$ first, summarize over $K$ in (REF ), $&&P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m \\Big | \\mathbf {U}_{n+1}= \\mathbf {e_{i}}\\Big )\\nonumber \\\\&=&\\sum _{K=0}^{+\\infty }P_{w}^{\\mu }\\Big (|\\mathbf {U}_{n}|=m, |\\mathbf {Z}_{n}|=K \\Big | \\mathbf {U}_{n+1}= \\mathbf {e_{i}}\\Big )\\nonumber \\\\&=&\\mathbf {e_{i}} \\Big [\\sum _{K=0}^{+\\infty } \\sum \\nolimits _{k_{0}+k_{1}+\\cdots +k_{m}=K} R_{n}^{k_{0}} Q_{n}\\zeta _{n-1} R_{n}^{k_{1}} \\cdots Q_{n}\\zeta _{n-1} R_{n}^{k_{m}} P_{n} \\mathbf {1} \\Big ].$ It's not hard to see $&&\\sum _{K=0}^{+\\infty } \\sum \\nolimits _{k_{0}+k_{1}+\\cdots +k_{m}=K} R_{n}^{k_{0}} Q_{n}\\zeta _{n-1} R_{n}^{k_{1}} \\cdots Q_{n}\\zeta _{n-1} R_{n}^{k_{m}}\\nonumber \\\\&=&(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}(I-R_{n})^{-1} \\cdots Q_{n}\\zeta _{n-1}(I-R_{n})^{-1} \\nonumber \\\\&=&[(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{m}(I-R_{n})^{-1}.$ Taking together (REF ) and (REF ), derives the marginal distribution of $|\\mathbf {U}_{n}|$ , $P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )=\\mathbf {e_{i}} [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{m} (I-R_{n})^{-1}P_{n} \\mathbf {1}.$ For the marginal distribution of $|\\mathbf {Z}_{n}|$ , summarize over $m$ in (REF ), we have $&& P_{\\omega }^{\\mu }\\Big (|\\mathbf {Z}_{n}|=K \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )\\nonumber \\\\&=&\\mathbf {e_{i}} \\sum _{m=0}^{+\\infty }\\sum \\nolimits _{m_{0}+m_{1}+\\cdots +m_{K}=m} Q_{n}\\zeta _{n-1}^{m_{0}} R_{n} Q_{n}\\zeta _{n-1}^{m_{1}} \\cdots R_{n} Q_{n}\\zeta _{n-1}^{m_{K}} P_{n} \\mathbf {1}\\nonumber \\\\&=&\\mathbf {e_{i}} (I-Q_{n}\\zeta _{n-1})^{-1} R_{n} (I-Q_{n}\\zeta _{n-1})^{-1} \\cdots R_{n}(I-Q_{n}\\zeta _{n-1})^{-1} P_{n} \\mathbf {1}\\nonumber \\\\&=&\\mathbf {e_{i}} [(I-Q_{n}\\zeta _{n-1})^{-1} R_{n}]^{K} (I-Q_{n}\\zeta _{n-1})^{-1} P_{n} \\mathbf {1}.$ Complete the proof of part (1) of Theorem (REF ); and part (2) is immediate.",
"$\\Box $ Remark.",
"From the marginal distribution, we also can test of the validity of the branching structure.",
"In fact, $\\sum _{m=0}^{+\\infty } P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m \\Big | \\mathbf {U}_{n+1}=\\mathbf {e}_{i}\\Big )& = & \\mathbf {e_{i}}[ \\sum _{m=0}^{+\\infty } [(I-R_{n}) Q_{n}\\zeta _{n-1}]^{m} ](I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}}[ I-(I-R_{n}) Q_{n}\\zeta _{n-1}]^{-1} ](I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}}[ (I-R_{n}) - Q_{n}\\zeta _{n-1}]^{-1} ]P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}}\\zeta _{n} \\mathbf {1}=1.$"
],
[
"The random walk $X_{n}=(\\xi _{n},Y_{n})$ starts from layer 0 with the initial distribution $\\mathbf {\\mu }_{\\omega }$ .",
"With the branching structure in hand, we can calculate the mean of the first hitting time $T_1$ .",
"We discuss it by four steps as follows.",
"Step 1.",
"$E_{\\omega }^{\\mu }\\Big (| \\mathbf {U}_{n}| \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )$ and $E_{\\omega }^{\\mu }\\Big (| \\mathbf {Z}_{n}| \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )$ .",
"From (REF ) of Theorem REF , $E_{\\omega }^{\\mu }\\Big (| \\mathbf {U}_{n}| \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )&=& \\sum _{m=0}^{+\\infty } m P_{\\omega }^{\\mu }\\Big (|\\mathbf {U}_{n}|=m \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )\\nonumber \\\\&=&\\mathbf {e_{i}} \\sum _{m=1}^{+\\infty } m [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{m} (I-R_{n})^{-1}P_{n} \\mathbf {1}.$ To process the calculation, we need the following lemma 2.1 For matrix $B$ , $I-B$ is non-degenerate, then $\\sum _{m=1}^{+\\infty } m B^{m}=B(I-B)^{-2}$ .",
"Proof.",
"$\\sum _{m=1}^{+\\infty } m B^{m}=(B+2B^{2}+3B^{3}+\\cdots )=B(I+2B+3B^{2}+\\cdots ),$ and $\\begin{split}(I-B)^{-2}=((I-B)^{-1})^{2}=(\\sum _{m=1}^{+\\infty }B^{n})^{2}=(\\sum _{m=1}^{+\\infty }B^{n})(\\sum _{m=1}^{+\\infty }B^{n})= (I+2B+3B^{2}+4B^{3}\\cdots ).\\end{split}$ Thus $\\sum _{m=1}^{+\\infty } m B^{m}=B(I-B)^{-2}$ .",
"$\\Box $ Let $B=(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}$ , (REF ) can be continued as $E_{\\omega }^{\\mu }\\Big (| \\mathbf {U}_{n}| \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big ) &=&\\mathbf {e_{i}} (I-R_{n})^{-1} Q_{n}\\zeta _{n-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-2} (I-R_{n})^{-1}P_{n} \\mathbf {1} \\nonumber \\\\&=&\\mathbf {e_{i}} (I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}Q_{n}\\zeta _{n-1}\\zeta _{n} \\mathbf {1} \\\\&=&\\mathbf {e_{i}} A_{n} \\mathbf {1}.$ The second equality (REF ) need a series calculations about the matrix which we leave it as Appendix, where $A_n$ is given in (REF ).",
"Similarly, $E_{\\omega }^{\\mu }\\Big (| \\mathbf {Z}_{n}| \\Big | \\mathbf {U}_{n+1}=\\mathbf {e_{i}}\\Big )&=&\\sum _{K=0}^{+\\infty } \\mathbf {e_{i}}K [(I-Q_{n}\\zeta _{n-1})^{-1}R_{n} ]^{K} (I-Q_{n}\\zeta _{n-1})^{-1}P_{n} \\mathbf {1}\\nonumber \\\\&=&\\mathbf {e_{i}} (I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}R_{n}\\zeta _{n} \\mathbf {1}\\nonumber \\\\&=&\\mathbf {e_{i}} (I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}R_{n} \\mathbf {1}.$ As a result, we have $&&E_{\\omega }^{\\mu }\\Big (| \\mathbf {U}_{n}| \\Big | \\mathbf {U}_{n+1}\\Big )=\\mathbf {U}_{n+1}A_{n} \\mathbf {1},\\\\&&E_{\\omega }\\Big (|\\mathbf {Z}_{n}| \\Big | \\mathbf {U}_{n+1}\\Big )=\\mathbf {U}_{n+1}(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}R_{n} \\mathbf {1}.$ Step 2.",
"Steps visited on layer $n$ .",
"For any $n\\le 0$ , define $N_{n}^{i}=\\sharp \\lbrace k \\in [0,T_{1}):~X_{k}=(n,i)\\rbrace .$ Note that $N_{n}^{i}$ is the number of steps visited $(n,i)$ before time $T_{1}$ .",
"Let $\\mathbf {N}_{n}=(N_{n}^{1},N_{n}^{2},\\cdots ,N_{n}^{d})$ and $|\\mathbf {N}_{n}|=\\sum _{i=1}^{d}N_{n}^{i}=\\mathbf {N}_{n}\\mathbf {1}$ .",
"Define a vector valued random variable $\\mathbf {U^{\\prime }}_{n}$ where $\\mathbf {U^{\\prime }}$$_{n}^{i},~1 \\le i \\le d$ is the number of steps from layer $n-1$ to $(n,i)$ .",
"Then $|\\mathbf {N}_{n}|=|\\mathbf {U^{\\prime }}_{n}|+|\\mathbf {Z}_{n}|+|\\mathbf {U}_{n+1}|,\\quad \\mathbb {P}-a.s..$ For another perspective, $T_{1}=\\sum _{n\\le 0}(|\\mathbf {N}_{n}|), ~ \\mathbb {P}-a.s..$ Since $X_{n}\\rightarrow +\\infty ,~\\mathbb {P}-a.s.$ , if the random walk takes a step to the left from any layer $n~(n\\le 0)$ to layer $n-1$ , it must come back finally from layer $n-1$ to layer $n$ , therefore $|\\mathbf {U}_{n}|=|\\mathbf {U^{\\prime }}_{n}|,\\quad \\mathbb {P}-a.s..$ Together with (REF ), we have $&& E_{\\omega }^{\\mu }(| \\mathbf {N}_{n}|)=E_{\\omega }^{\\mu }(|\\mathbf {U}_{n}|+|\\mathbf {Z}_{n}|+|\\mathbf {U}_{n+1}|)\\nonumber \\\\&=&E_{\\omega }^{\\mu }\\Big [E_{\\omega }^{\\mu }(|\\mathbf {U}_{n}|\\Big | \\mathbf {U}_{n+1})+E_{\\omega }^{\\mu }(|\\mathbf {Z}_{n}|\\Big | \\mathbf {U}_{n+1})+E_{\\omega }^{\\mu }(|\\mathbf {U}_{n+1}| \\Big | \\mathbf {U}_{n+1})\\Big ]\\nonumber .$ By using (REF ), one can calculate the quenched expectation of $| \\mathbf {N}_{n}|$ as $&& E_{\\omega }^{\\mu }(| \\mathbf {N}_{n}|)=E_{\\omega }^{\\mu }[\\mathbf {U}_{n+1}A_{n} \\mathbf {1}+ \\mathbf {U}_{n+1}(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}R_{n} \\mathbf {1}+\\mathbf {U}_{n+1}\\mathbf {1}]\\nonumber \\\\&=&E_{\\omega }^{\\mu }[\\mathbf {U}_{n+1}(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}(Q_{n}\\zeta _{n-1}+R_{n}+I-Q_{n}\\zeta _{n-1}-R_{n})\\mathbf {1}]\\nonumber \\\\&=&E_{\\omega }^{\\mu }(\\mathbf {U}_{n+1})(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}\\mathbf {1}.$ Step 3.",
"The next object is to discuss $E_{\\omega }^{\\mu }(\\mathbf {U}_{n+1})$ .",
"Define a probability matrix $B_{m}$ , where $B_{m}(i,j)$ is the probability of a particle starting from $(n+1,i)$ , takes more than $m$ steps to the layer $n$ , and the $m$ -th step located at $(n,j)$ .",
"$B_{m}(i,j)$ can be expressed by our branching structure $B_{m}(i,j)&=& \\mathbf {e_{i}}\\Big [\\sum _{\\overline{K}=0}^{+\\infty } \\sum \\nolimits _{k_{0}+k_{1}+\\cdots +k_{m-1}=\\overline{K}} R_{n+1}^{k_{0}} Q_{n+1}\\zeta _{n} R_{n+1}^{k_{1}}Q_{n+1}\\zeta _{n} R_{n+1}^{k_{2}} \\cdots Q_{n+1}\\zeta _{n} R_{n+1}^{k_{m-1}} Q_{n+1}\\Big ]\\mathbf {e_{j}}\\nonumber \\\\&=& \\mathbf {e_{i}} [(I- R_{n+1}) Q_{n+1}\\zeta _{n}]^{m-1} (I- R_{n+1})^{-1} Q_{n+1}\\mathbf {e_{j}}.$ Let $\\widetilde{P}_{i,j}^{m}:={B}_{m}(i,j) - {B}_{m+1}(i,j),$ be the probability of a particle starts from $(n+1,i)$ , the $m$ -th step takes to the left and located at $(n,j)$ .",
"We have $E_{\\omega }^{\\mu }({U}_{n+1}^{j} \\Big | \\mathbf {U}_{n+2}=\\mathbf {e_{i}})&= & \\sum _{m=1}^{+\\infty } m \\widetilde{P}_{i,j}^{m}= \\mathbf {e_{i}}\\sum _{m=1}^{+\\infty } m({B}_{m}-{B}_{m+1})\\mathbf {e_{j}}\\nonumber \\\\&= & \\mathbf {e_{i}}\\sum _{m=1}^{+\\infty }B_{m}\\mathbf {e_{j}}.$ Combine with (REF ), $E_{\\omega }^{\\mu }(\\mathbf {U}_{n+1} | \\mathbf {U}_{n+2})&=&\\mathbf {U}_{n+2} \\sum _{m=1}^{+\\infty } {B}_{m} \\nonumber \\\\&=& \\mathbf {U}_{n+2} \\sum _{m=1}^{+\\infty } [(I- R_{n+1}) Q_{n+1}\\zeta _{n}]^{m-1} (I- R_{n+1})^{-1} Q_{n+1} \\nonumber \\\\&= & \\mathbf {U}_{n+2} (I- Q_{n+1}\\zeta _{n}-R_{n+1})^{-1} Q_{n+1} \\nonumber \\\\&=& \\mathbf {U}_{n+2}A_{n+1}.$ Then $E_{\\omega }^{\\mu }(\\mathbf {U}_{n+1})= E_{\\omega }^{\\mu }[E_{\\omega }^{\\mu }(\\mathbf {U}_{n+1}| \\mathbf {U}_{n+2})]= E_{\\omega }^{\\mu }(\\mathbf {U}_{n+2})A_{n+1},$ where $A_n$ is given in (REF ).",
"By recursive argument, we obtain $E_{\\omega }^{\\mu }(\\mathbf {U}_{n+1})& = & E_{\\omega }^{\\mu }(\\mathbf {U}_{n+3})A_{n+2}A_{n+1}\\nonumber \\\\& = & \\cdots \\nonumber \\\\& = & E_{\\omega }^{\\mu }(\\mathbf {U}_{1})A_{0}A_{-1}A_{-2} \\cdots A_{n+2}A_{n+1}.$ Step 4.Calculate $\\mathbb {E}(T_{1})$.",
"It follows from (REF ) and (REF ) that, $\\mathbb {E}(T_{1})&=&E(E_{\\omega }(T_{1}))= E\\left(\\sum _{n\\le 0}E_{\\omega }(|\\mathbf {N}_{n}|)\\right)\\nonumber \\\\&=& E\\left(\\sum _{n\\le 0}E_{\\omega }(\\mathbf {U}_{n+1})(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}\\mathbf {1}\\right)\\nonumber \\\\&=&E\\left(\\sum _{n\\le 0}E_{\\omega }(\\mathbf {U}_{1})A_{0}A_{-1}A_{-2} \\cdots A_{n+2}A_{n+1}(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}\\mathbf {1}\\right)\\nonumber \\\\&=&E(E_{\\omega }^{\\mu }(\\mathbf {U}_{1})(u_{0}+A_{0}u_{-1}+ \\cdots +A_{0}A_{-1}\\cdots A_{-k}u_{-k-1}+\\cdots ))\\nonumber \\\\&=&E(\\overrightarrow{\\mu _\\omega }(u_{0}+A_{0}u_{-1}+ \\cdots +A_{0}A_{-1}\\cdots A_{-k}u_{-k-1}+\\cdots )).$ where $A_n, u_n$ is given in (REF ).",
"$\\Box $"
],
[
"Density of the absolutely continuous invariant measure–",
"Let us review the process discussed in Section 4 of $\\cite {[Roi08]}$ .",
"From the point of view “environments viewed from the particle\", let $\\overline{\\omega _{n}}=\\theta ^{\\xi _{n}}w$ , for $n\\ge 0$ , and consider the process $Z_{n}:=(\\overline{\\omega _{n}},Y_{n})$ , defined in $(\\Omega \\times {D},{F}\\otimes {H})$ , where ${H}$ as the set of all subsets of ${D}$ , and the initial distribution $P_{\\omega }^{\\mu }(\\xi _{0}=0,~Y_{0}=i)=\\mu _{\\omega }(i)=\\mathbf {y}_{-1} (i) $ given by (REF ).",
"$(Z_{n})_{n\\ge 0}$ is a Markov chain under $\\mathbb {P}^{\\mu }$ with transition kernel $K(\\omega ,i;B,j)=P_{0}(i,j)I_{B}(\\theta w)+R_{0}(i,j)I_{B}(w)+Q_{0}(i,j)I_{B}(\\theta ^{-1} w).$ Let $v_{p}=\\frac{1}{\\mathbb {E}T_{1}}$ , whenever $\\mathbb {E}T_{1} < \\infty $ .",
"For $B \\in {F}$ , $i\\in {D}$ , define a probability measure $Q$ on $(\\Omega \\times {D},{F}\\otimes {H})$ : $Q(B,i):=v_{p}\\mathbb {E}\\left(\\sum _{n=0}^{T_{1}-1}I_{B}(\\overline{\\omega _{n}})I_{Y_{n}}(i)\\right)=v_{p}\\sum _{j\\in {D}}E_{p}\\left(\\mu _{\\omega }(j)E_{\\omega }^{j}\\left(\\sum _{n=0}^{T_{1}-1}I_{B}(\\theta ^{\\xi _{n}}\\omega )I_{Y_{n}}(i)\\right)\\right)\\nonumber $ $Q(\\cdot )$ is a invariant measure under the Markov kernel $K$ (Proposition 4.1, $\\cite {[Roi08]}$ ).",
"Define a probability measure $\\overline{Q}(\\cdot )$ on $(\\Omega ,{F})$ by setting $\\overline{Q}(B):=Q(B,{D}),\\quad B \\in {F}.$ and let $~Q_{i}(B):=Q(B,i)~\\mbox{for}~B \\in {F}$ .",
"Then both $Q_{i}(\\cdot )$ and $\\overline{Q}(\\cdot )$ are absolutely continuous with regard to $P$ (Proposition 4.1, $\\cite {[Roi08]}$ ), but where only the up bound of the density have been proved.",
"For $m\\le 0,~i\\in {D}$ , define $N_{m}^{i}$ as in (REF ): $N_{m}^{i}:=\\left\\lbrace \\sharp n \\in [0,T_{1}):~\\xi _{n}=m,~Y_{n}=i \\right\\rbrace .\\nonumber $ Note that for any bounded measurable function $f:~\\Omega \\rightarrow \\mathbb {R}$ and $\\forall ~ i \\in {D}$ , $\\int _{\\Omega }f(\\omega )Q(d\\omega ,i)&=&v_{p}\\sum _{n=0}^{+\\infty }\\mathbb {E}^{\\mu }(f(\\overline{w_{n}});~Y_{n}=i,~T_{1}>n)\\nonumber \\\\&=& v_{p}\\sum _{m\\le 0}\\mathbb {E}^{\\mu }(f(\\theta ^{m} \\omega )N_{m}^{i})\\nonumber \\\\&=& v_{p}E_{p}\\left(\\sum _{k\\in {D}}\\mu _{\\omega }(k) \\sum _{m\\le 0}\\left(f(\\theta ^{m} \\omega ) E_{\\omega }^{k}N_{m}^{i}\\right)\\right)\\nonumber \\\\&=&v_{p}E_{p}\\left(f(\\omega ) \\sum _{k\\in {D}} \\sum _{m\\le 0} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}N_{m}^{i}\\right).$ Therefore, $Q_{i}$ is absolutely continuous with respect to $P$ , and also $\\overline{Q}$ is absolutely continuous with respect to $P$ .",
"And the density is $&&\\Lambda _{\\omega }^{(i)}:=\\frac{dQ_{i}}{d P}=v_{p} \\sum _{k\\in {D}} \\sum _{m\\le 0} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}(N_{m}^{i}) \\\\&&\\Lambda _{\\omega }:=\\frac{d\\overline{Q}}{d P}= v_{p} \\sum _{k\\in {D}} \\sum _{m\\le 0} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}(\\mathbf {N}_{m}\\mathbf {1}).$ We intend to spesity the density $\\Lambda _{\\omega }^{(i)}$ and $\\Lambda _{\\omega }$ by branching structure.",
"Note that $\\mu _{\\omega }(i)=\\mathbf {y}_{-1} (i) $ given by (REF ), and $\\zeta _{-n}=\\eta _{-n}$ is the exit probability, $\\mu _{\\theta \\omega }&=& \\lim _{n\\rightarrow \\infty }\\mathbf {e}_{i}\\zeta _{-n}(\\theta \\omega )\\cdots \\zeta _{-2}(\\theta \\omega )\\zeta _{-1}(\\theta \\omega )\\nonumber \\\\&=& \\lim _{n\\rightarrow \\infty }\\mathbf {e}_{i}\\zeta _{-n+1}(\\omega )\\cdots \\zeta _{-1}(\\omega )\\zeta _{0}(\\omega )\\nonumber \\\\&=& \\mu _{\\omega }\\zeta _{0}(\\omega )\\nonumber .$ Thus $\\sum _{k\\in {D}} \\mu _{\\theta \\omega }(k) E_{\\theta \\omega }^{k}(N_{-1}^{i})=\\sum _{k\\in {D}} \\mu _{\\theta \\omega }(k) E_{\\omega }^{k}(N_{0}^{i})=\\sum _{k\\in {D}} \\mu _{\\omega }\\zeta _{0}(k) E_{\\omega }^{k}(N_{0}^{i}).\\nonumber $ Similarly, for $m\\le 0$ and $ i\\in {D}$ $\\sum _{k\\in {D}} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}(N_{m}^{i})=\\sum _{k\\in {D}} \\mu _{\\omega }\\zeta _{0}\\zeta _{1} \\cdots \\zeta _{-m-1}(k) E_{\\omega }^{k}(N_{0}^{i}).$ The following lemma is closely related to branching structure.",
"lemma 2.2 For $n < 0$ , $E_{\\omega }(\\mathbf {N}_{n})=\\mu _{\\omega }A_{0}A_{-1}\\cdots A_{n+1}\\widetilde{u}_{n}.$ Proof.",
"Due to the definition of $\\mathbf {N}_{n}$ , $\\mathbf {U}_{n}^{\\prime }$ , $\\mathbf {Z}_{n}$ and $\\mathbf {U}_{n+1}$ , $\\mathbf {N}_{n}=\\mathbf {U}_{n}^{\\prime }+\\mathbf {Z}_{n}+\\mathbf {U}_{n+1}.\\nonumber $ Recall the branching structure and by similarly argument as in the proof of Theorem REF , we obtain that $E_{\\omega }(\\mathbf {U}_{n}^{\\prime })&=& \\mathbf {U}_{n+1} \\sum _{m=1}^{+\\infty }\\left((I-R_{n})^{-1}Q_{n}\\zeta _{n-1}\\right)^{m-1}(I-R_{n})^{-1}Q_{n}\\zeta _{n-1}\\nonumber \\\\&=& \\mathbf {U}_{n+1} \\left((I-Q_{n}\\zeta _{n-1}-R_{n}\\right)^{-1}Q_{n}\\zeta _{n-1}= \\mathbf {U}_{n+1} A_{n}\\zeta _{n-1}\\nonumber ,$ and $E_{\\omega }(\\mathbf {Z}_{n})&=& \\mathbf {U}_{n+1} \\sum _{K=0}^{+\\infty }\\left((I-Q_{n}\\zeta _{n-1})^{-1}R_{n}\\right)^{K}(I-Q_{n}\\zeta _{n-1})^{-1}R_{n}\\nonumber \\\\&=& \\mathbf {U}_{n+1} \\left(I-Q_{n}\\zeta _{n-1}-R_{n}\\right)^{-1}R_{n}\\nonumber .$ Thus $E_{\\omega }(\\mathbf {N}_{n}\\mid \\mathbf {U}_{n+1})&=& E_{\\omega }(\\mathbf {N}_{n}\\mid \\mathbf {U}_{n+1})\\nonumber \\\\&=& E_{\\omega }(\\mathbf {U}_{n}^{\\prime }\\mid \\mathbf {U}_{n+1})+E_{\\omega }(\\mathbf {Z}_{n}\\mid \\mathbf {U}_{n+1})+E_{\\omega }(\\mathbf {U}_{n+1}\\mid \\mathbf {U}_{n+1})\\nonumber \\\\&=& E_{\\omega }(\\mathbf {U}_{n+1}\\left[(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}Q_{n}\\zeta _{n-1}+(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}R_{n}+I \\right])\\nonumber \\\\&=& E_{\\omega }(\\mathbf {U}_{n+1})(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}\\nonumber .$ Together with the fact $E_{\\omega }(\\mathbf {U}_{n+1})=\\mathbf {U}_{n+2}A_{n+1},\\nonumber $ we have $E_{\\omega }(\\mathbf {N}_{n})&=&\\mu _{\\omega }A_{0}A_{-1}\\cdots A_{n+1}(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}\\nonumber \\\\&=& \\mu _{\\omega }A_{0}A_{-1}\\cdots A_{n+1}\\widetilde{u}_{n}\\nonumber .$ Then Lemma REF follows.$\\Box $ It follows from equation (REF ) and lemma REF that $\\sum _{k\\in {D}} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}(N_{m}^{i})&=& \\sum _{k\\in {D}} \\mu _{\\omega }\\zeta _{0}\\zeta _{1} \\cdots \\zeta _{-m-1}(k) E_{\\omega }^{k}(N_{0}^{i})\\nonumber \\\\&=& \\mu _{\\omega }\\zeta _{0}\\zeta _{1} \\cdots \\zeta _{-m-1}A_{-m}A_{-m-1}\\cdots A_{2}A_{1}\\widetilde{u}_{0}(i).$ Thus $\\Lambda _{\\omega }^{(i)}=\\frac{dQ_{i}}{dP}&=& v_{p} \\sum _{k\\in {D}} \\sum _{m\\le 0} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}(N_{m}^{i})\\nonumber \\\\&=& v_{p} \\sum _{m\\le 0}[ \\mu _{\\omega }\\zeta _{0}\\zeta _{1} \\cdots \\zeta _{-m-1}A_{-m}A_{-m-1}\\cdots A_{2}A_{1}\\widetilde{u}_{0}](i)\\nonumber \\\\&=& v_{p}[ \\mu _{\\omega }\\left(\\widetilde{u}_{0}+\\zeta _{0}A_{1}\\widetilde{u}_{0}+ \\zeta _{0}\\zeta _{1}A_{2}A_{1}\\widetilde{u}_{0}+\\cdots \\right)](i)\\nonumber .$ and similarly $\\frac{d\\overline{Q}}{dP}=\\Lambda _{\\omega }&=& v_{p} \\sum _{k\\in {D}} \\sum _{m\\le 0} \\mu _{\\theta ^{-m}\\omega }(k) E_{\\theta ^{-m}\\omega }^{k}(\\mathbf {N}_{m}\\mathbf {1})\\nonumber \\\\&=& v_{p} \\sum _{m\\le 0} [\\mu _{\\omega }\\zeta _{0}\\zeta _{1} \\cdots \\zeta _{-m-1}A_{-m}A_{-m-1}\\cdots A_{2}A_{1}\\widetilde{u}_{0}]\\mathbf {1} \\nonumber \\\\&=& v_{p} [\\mu _{\\omega }\\left(\\widetilde{u}_{0}+\\zeta _{0}A_{1}\\widetilde{u}_{0}+ \\zeta _{0}\\zeta _{1}A_{2}A_{1}\\widetilde{u}_{0}+\\cdots \\right)]\\mathbf {1}\\nonumber .$ $\\Box $"
],
[
"Appendix",
"The following calculation is needed in (REF ), it is a details calculations on the matrix.",
"$& & E_{w}^{\\mu }(| \\mathbf {U}_{n}| \\mid \\mathbf {U}_{n+1}=\\mathbf {e_{i}})\\\\& = & \\sum _{m=0}^{+\\infty } m P_{w}^{\\mu }(|\\mathbf {U}_{n}|=m \\mid \\mathbf {U}_{n+1}=\\mathbf {e_{i}})\\\\& = & \\sum _{m=0}^{+\\infty } \\mathbf {e_{i}}m [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{m} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\sum _{m=1}^{+\\infty } m [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{m} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} (I-R_{n})^{-1} Q_{n}\\zeta _{n-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-2} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} (I-R_{n})^{-1} Q_{n}\\zeta _{n-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1}\\rbrace ^{-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}][(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1}\\rbrace ^{-1}[I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1}-I\\rbrace ^{-1} [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1} (I-R_{n})^{-1}P_{n}\\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}][[(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1}-I]\\rbrace ^{-1} (I-R_{n})^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace (I-R_{n})[I-(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}][[(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1}-I]\\rbrace ^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [(I-R_{n}) -Q_{n}\\zeta _{n-1}][[(I-R_{n})^{-1} Q_{n}\\zeta _{n-1}]^{-1}-I]\\rbrace ^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [(I-R_{n}) -Q_{n}\\zeta _{n-1}][( Q_{n}\\zeta _{n-1})^{-1}(I-R_{n})-I]\\rbrace ^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} \\lbrace [(I-R_{n}) -Q_{n}\\zeta _{n-1}][( Q_{n}\\zeta _{n-1})^{-1}](I-R_{n}-Q_{n}\\zeta _{n-1})\\rbrace ^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} [(I-Q_{n}\\zeta _{n-1}-R_{n})( Q_{n}\\zeta _{n-1})^{-1}(I-Q_{n}\\zeta _{n-1}-R_{n})]^{-1}P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} (I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}Q_{n}\\zeta _{n-1}(I-Q_{n}\\zeta _{n-1}-R_{n})^{-1} P_{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} (I-Q_{n}\\zeta _{n-1}-R_{n})^{-1}Q_{n}\\zeta _{n-1}\\zeta _{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} A_{n} \\zeta _{n-1}\\zeta _{n} \\mathbf {1}\\\\& = & \\mathbf {e_{i}} A_{n} \\mathbf {1} .$ Acknowledgements: The authors would like to thank Hongyan Sun, Huaming Wang, Lin Zhang and Zhou Ke for the stimulating discussions."
]
] | 1204.1104 |
[
[
"An Attractor for Natural Supersymmetry"
],
[
"Abstract We propose an attractor mechanism which generates the \"more minimal\" supersymmetric standard model from a broad class of supersymmetry breaking boundary conditions.",
"The hierarchies in the fermion masses and mixings are produced by the same dynamics and a natural weak scale results from gaugino mediation.",
"These features arise from augmenting the standard model with a new SU(3) gauge group under which only the third generation quarks are charged.",
"The theory flows to a strongly interacting fixed point which induces a negative anomalous dimension for the third generation quarks and a positive anomalous dimension for the Higgs.",
"As a result, a split-family natural spectrum and the flavor hierarchies are dynamically generated."
],
[
"Introduction and Summary",
"The stability of the electroweak scale and the hierarchical structure of the fermion masses and mixing angles are two of the central mysteries of the Standard Model (SM).",
"It is possible that these puzzles are explained by the same underlying mechanism.",
"One approach is to supersymmetrize the SM and augment it with a new strongly interacting gauge theory.",
"Supersymmetry tames the quadratically divergent contributions to the Higgs mass while the strong dynamics can yield a parametric suppression of the first and second generation Yukawa couplings.",
"Various realizations of this possibility have been proposed so far, including single sector models [1], models of superconformal flavor [2], [3], warped extra-dimensional realizations [4], and theories based on deconstruction [5].For some other models which connect the supersymmetry breaking spectrum and flavor, see [6].",
"For some of these constructions, the dynamics which gives rise to the flavor textures also produces an inverted squark hierarchy, where the lightest SM fermions have the heaviest sfermion partners.",
"This provides a microscopic realization of the “more minimal” supersymmetric SM of [7], which was motivated by considerations of naturalness and flavor constraints.",
"The phenomenology of these models has been studied thoroughly in e.g. [8].",
"Furthermore, the recent LHC bounds on first and second generation squark masses [9] together with attempts to minimize fine tuning have reinvigorated interest in the phenomenology and collider signatures of such “natural supersymmetry” spectra [10], [11].",
"In this work, we will present a new model to explain the flavor hierarchies which simultaneously yields the natural supersymmetry spectrum and radiative electroweak symmetry breaking (REWSB).",
"This will be accomplished by adding a new strongly coupled conformal sector to the minimal supersymmetric standard model (MSSM).",
"We will show that starting from rather generic supersymmetry breaking boundary conditions (with some assumptions on certain approximate global symmetries), the infrared theory after escaping from the conformal regime is the more minimal supersymmetric SM.",
"The MSSM fields are weakly coupled both in the UV and in the IR.",
"The conformal dynamics will generate order one negative anomalous dimensions for the third generation fields once the theory becomes strongly coupled.This differs from previously studied constructions, which relied on large positive anomalous dimensions for the first two SM generations.",
"This can result from compositeness or localization in the IR region of a Randall-Sundrum throat.",
"Negative anomalous dimensions are only possible if the third generation is charged under this new gauge group — the unitarity bound on dimensions only applies to gauge invariant operators.",
"The third generation Yukawa couplings are marginal operators in the conformal field theory (CFT).",
"These marginal Yukawa couplings will induce a large positive anomalous dimension for the Higgs field.",
"Hence, the remaining Yukawas become irrelevant deformations.",
"It will be shown that this structure can lead to viable flavor hierarchies.",
"Additionally, the strong dynamics will suppress soft masses for the third generation squarks and Higgs fields.",
"Below the exit scale, these will be regenerated by gaugino mediation [13].",
"The model acts as an attractor for the more minimal supersymmetry spectrum and REWSB.",
"The goal of this work is to analyze the simplest realization of this mechanism and its main dynamical consequences.",
"The gauge group is $SU(3)_\\text{CFT} \\times SU(3)_X \\times SU(2)_W \\times U(1)_Y$ where $SU(3)_\\mathrm {CFT}$ will flow to a strongly coupled fixed point, $SU(3)_X$ is weakly coupled, and $SU(2)_W\\times U(1)_Y$ are as in the MSSM.",
"The third generation quark superfields transform under $SU(3)_\\text{CFT}$ .",
"The first and second generations transform under $SU(3)_X$ .",
"The $SU(3)$ groups are connected by bifundamental `link' fields.",
"With this matter content, $SU(3)_\\text{CFT}$ is in the conformal window [14].",
"The link fields eventually acquire a nonzero expectation value causing an exit from the conformal regime; this also breaks $SU(3)_\\text{CFT} \\times SU(3)_X \\rightarrow SU(3)_C$ , giving rise the visible color interactions.The super top color model of [15] utilizes a similar group structure and matter content.",
"However, unlike models of top color (see [16] for a review), the mechanism studied in this work does not utilize top condensation to break electroweak symmetry.",
"This structure is summarized in Fig.",
"REF .",
"Figure: The model presented here is given by an SU(3) X ×SU(3) CFT SU(3)_X \\times SU(3)_\\mathrm {CFT} quiver gauge theory.",
"The node SU(3) CFT SU(3)_\\mathrm {CFT} flows to an interacting fixed point and provides the necessary dynamics for generating flavor and an attractor mechanism for natural supersymmetry.",
"SU(3) X SU(3)_X is IR free.",
"The bifundamental link fields Σ\\Sigma and Σ ¯\\overline{\\Sigma } break the group to the diagonal visible SU(3) C SU(3)_C, providing an exit from the conformal regime.The rest of this paper is organized as follows.",
"§ describes the basic mechanism and its implications for the spectrum of soft masses and the flavor hierarchies.",
"In § we discuss the low energy phenomenology, including some general remarks about the spectrum and the Higgs sector.",
"We also provide some concrete example spectra.",
"Our conclusions and future directions are presented in §."
],
[
"The Model",
"We begin by describing the model of Fig.",
"REF .",
"For simplicity we will ignore the leptons, which do not affect our discussion other than to ensure anomaly cancellation.",
"The matter content and charge assignments are given in Table REF .",
"The third generation quarks are charged under $SU(3)_\\text{CFT}$ , while the first two generations are charged under $SU(3)_X$ .",
"The bi-fundamental link fields are denoted by $\\Sigma $ and $\\overline{\\Sigma }$ .",
"The field $A$ is an adjoint plus a singlet of $SU(3)_X$ .",
"The superpotential will be chosen so that the $F$ -term for $A$ forces $\\langle \\Sigma \\rangle = \\langle \\overline{\\Sigma } \\rangle \\ne 0$ .",
"This will cause an escape from the conformal regime while also giving masses to the bi-fundamentals.",
"The superpotential contains the following relevant terms: $W\\supset Q_3\\, H_u\\, \\overline{u}_3 + Q_3\\, H_d \\, \\overline{d}_3 + \\Sigma \\,A\\, \\overline{\\Sigma } + W_{\\leavevmode {U(1)}}\\,.$ Contractions over gauge indices are implicit.",
"$W_{\\leavevmode {U(1)}}$ will be instrumental in breaking some of the Abelian symmetries which can spoil the desired low energy spectrum.",
"We will discuss this term in detail below.",
"Table: The particle content and charge assignments for the MSSM quark and CFT sectors.",
"The subscripts denote generation assignments, and the leptons are charged as in the MSSM.",
"The visible color gauge group is a diagonal subgroup of SU(3) CFT ×SU(3) X SU(3)_\\mathrm {CFT} \\times SU(3)_X.With this matter content, $SU(3)_\\text{CFT}$ has five flavors and flows to a strongly interacting superconformal fixed point in the IR.",
"The crossover scale below which this theory becomes strong is denoted by $\\Lambda _\\text{CFT}$ .",
"The remaining gauge groups are IR free and act as spectators to this strong dynamics.",
"A crucial property of the model is that the third generation Yukawa couplings appear as relevant interactions in the CFT.",
"The Higgs fields will then also be part of the CFT — they will receive a positive anomalous dimension.",
"These couplings, as well as the rest of the interactions in Eq.",
"(REF ), will naturally flow to order one values below $\\Lambda _\\text{CFT}$ .",
"In contrast, the remaining Yukawas will arise as irrelevant deformations, resulting in a flavor hierarchy between the third and first two generations.",
"If we do not add extra fields, this matter content spoils gauge coupling unification.",
"However, there are no issues with Landau poles up to the GUT scale, and one could imagine UV completing the model using full $SU(5)$ representations.",
"We will come back to this point briefly in §, while here we continue to focus on this minimal realization.",
"The energy scales in our model are as follows, see Fig.",
"REF .",
"At the messenger scale $M$ , soft supersymmetry breaking operators are generated.",
"The supersymmetry breaking mechanism and mediation can be arbitrary, up to certain assumptions on global symmetries which we describe below.",
"The scale $M$ could be above or below $\\Lambda _\\text{CFT}$ , but the physical soft masses should be smaller than $\\Lambda _\\text{CFT}$ so that the superconformal dynamics dominate.",
"At a scale $v < \\Lambda _\\text{CFT}$ , we exit the CFT regime.",
"This is done supersymmetrically by adding $W \\supset - v^2\\, \\mathrm {Tr}A\\,$ to Eq.",
"(REF ).",
"This new scale can be generated dynamically as explained in [17].",
"The link fields acquire an expectation value $\\langle \\Sigma \\overline{\\Sigma } \\rangle =v^2$ , which breaks $SU(3)_\\mathrm {CFT}\\times SU(3)_X \\rightarrow SU(3)_C$ .",
"The visible gauge coupling becomes $\\frac{1}{g_C^2}=\\frac{1}{g_X^2}+\\frac{1}{g_\\text{CFT}^2}\\,,$ which is dominated by $g_X^2\\simeq g_C^2$ .",
"We assume that the exit from the conformal regime happens quickly, such that at energy scales $E < v$ a perturbative description is valid.",
"As we show below, the weak scale $m_W < v$ is radiatively generated.",
"We note that, in contrast with composite models, here the MSSM fields are weakly coupled both in the UV (above $\\Lambda _\\text{CFT}$ ) and in the IR (below the exit scale).",
"Figure: The relevant scales for our model."
],
[
"An Attractor for Natural Supersymmetry",
"In this section, we will analyze the conformal regime and how it affects the soft masses.",
"For more details, see [18] and the references therein.",
"We will first neglect the effects from the weakly interacting gauge groups and the first two generations.",
"This corresponds to setting $g_\\mathrm {SM} \\rightarrow 0$ and ignoring mixings from Yukawa couplings.",
"We will then show that such effects amount to small finite corrections.",
"Our main dynamical assumption is that the fixed point is stable, which means that small perturbations of the couplings away from their fixed point value are irrelevant.",
"Equivalently, the matrix $\\partial \\beta _i/\\partial y_j$ must be positive definite, where $y_i$ are the couplings of the theory and $\\beta _i$ are the corresponding beta functions.",
"With this assumption, all the physical couplings flow to their fixed point values, and their higher $\\theta $ components flow to zero.",
"This can be seen by promoting the couplings to background superfields.",
"One implication is that all soft supersymmetry breaking terms associated with relevant couplings are highly suppressed by the conformal dynamics.For a model which uses this mechanism to suppress the Higgs soft mass, see [19].",
"To understand the consequences for our model, consider a relevant superpotential interaction $W \\supset \\lambda \\, \\prod _i \\,\\Phi _i^{n_i},$ for superfields $\\Phi _i$ and positive integers $n_i$ .",
"The physical coupling is $\\lambda _\\mathrm {phys} = \\lambda \\, \\prod _i \\left(Z_{\\Phi _i}^{n_i}\\right)^{-1/2}$ where the $Z_{\\Phi _i}$ are the wave function renormalizations for the superfields $\\Phi _i$ and encode the soft masses as their $\\theta ^4$ components.",
"As $\\lambda _\\mathrm {phys}$ flows to its fixed point value, its $\\theta ^4$ component flows to zero.",
"Equivalently, this implies that the combination of soft masses $\\sum _i n_i \\,\\widetilde{m}_i^2$ flows to zero at the fixed point, where $\\widetilde{m}_i$ is the soft mass for $\\Phi _i$ .",
"Since the $\\theta ^2$ component also flows to zero, the same conclusion holds for the $a$ -terms.",
"Similarly, promoting the gauge coupling to a superfield implies that the CFT gaugino mass and $\\sum _r \\text{dim}(r)\\, T_r\\, \\widetilde{m}_r^2$ are also suppressed by the CFT dynamics.",
"Here the field $\\Phi _r$ has index $T_r$ under the gauge group, e.g.",
"$T(\\Box )=1/2$ , and $\\text{dim}(r)$ is the dimension of $\\Phi _r$ for a fixed gauge index.",
"As we mentioned above, we assume that the CFT is IR attractive, which means that the eigenvalues $\\lambda _i$ of the matrix $\\partial \\beta _i/ g_j$ are positive and, generically at a strongly coupled fixed point, order one.",
"The previous soft parameters are then suppressed by a power-law $(\\frac{\\mu }{\\Lambda _\\text{CFT}})^{\\lambda _i}$ , where $\\mu $ is the RG scale.",
"This effect can be seen explicitly in weakly coupled examples such as the Wilson-Fisher fixed point.",
"Below we will take into account the small contributions from the perturbative SM couplings.",
"On the other hand, due to the non-renormalization of conserved currents, combinations of soft masses proportional to conserved $U(1)$ symmetries, $\\sum _i \\text{dim}(i) \\,q_i\\, \\widetilde{m}_i^2$ are not renormalized by the strong dynamics.",
"Here $q_i$ denotes the $U(1)$ charge.",
"This effect will be an important constraint on the viability of obtaining the more minimal supersymmetric SM.",
"Our goal is to use the conformal dynamics to suppress the soft masses for the third generation squarks and Higgs fields, which is essentially the idea of conformal sequestering [20].",
"We must explicitly break some of the non-anomalous global $U(1)$ symmetries.",
"Otherwise, Eq.",
"(REF ) shows that they would lead to unsuppressed tachyonic soft masses.",
"We accomplish this with the term $W_{\\leavevmode {U(1)}}$ in the superpotential of Eq.",
"(REF ).",
"As a concrete example, let us investigate a specific choice: $W_{\\leavevmode {U(1)}}= (Q_3 \\,\\overline{u}_3)(Q_3\\, \\overline{d}_3)\\,.$ In this theory, the superconformal $R$ -charges are uniquely determined in terms of symmetries and anomaly cancellation.",
"The anomaly-free Abelian symmetries are given in Table REF .",
"Table: The global anomaly free U(1)U(1) symmetries for the model given by Eq.",
"() with the U(1)U(1) breaking superpotential in Eq. ().",
"The charge assignments for the gauged symmetries are given in Table .One can verify self-consistently that all terms in the superpotential in Eq.",
"(REF ) have $R$ -charge 2 at the fixed point.",
"The remaining first and second generation fields decouple from the strong dynamics; they are neutral under the non-$R$ symmetries and are (approximately) free fields.",
"It is useful to explain in more detail how the flow to this fixed point proceeds, starting from the UV free theory.",
"In the UV, the cubic terms in the superpotential are classically marginal, but the quartic symmetry breaking term is classically irrelevant.",
"In terms of the canonical UV fields, $W_{\\leavevmode {U(1)}}= \\frac{1}{M_*}(Q_3 \\,\\overline{u}_3)(Q_3\\, \\overline{d}_3)$ , where $M_*$ is some large mass scale.",
"First, consider the limit $M_* \\rightarrow \\infty $ .",
"The resulting theory is Supersymmetric QCD with extra singlets and cubic superpotential deformations (SSQCD).",
"Below the strong coupling scale for this model $\\Lambda _\\text{SSQCD}$ , the theory flows to a superconformal fixed point.",
"This CFT can be studied using $a$ -maximization [21]; we find that the superconformal $R$ -charges of $Q_3$ and $(\\bar{u}_3, \\bar{d}_3)$ are $2/3 \\times \\sqrt{2/7}$ .",
"The cubic interactions are then relevant perturbations of the free fixed point, driving the theory to the nontrivial SSQCD fixed point.",
"Next, we can add the quartic superpotential, taking $M_*/\\Lambda _\\text{SSQCD}$ large but finite.",
"The theory first flows close to the SSQCD fixed point which we just described.",
"According to the previous $R$ -charges, in this regime the quartic operator $(Q_3 \\,\\overline{u}_3)(Q_3\\, \\overline{d}_3)$ is relevant.",
"So for any nonzero value of $M_*/\\Lambda _\\text{SSQCD}$ it will drive the theory away from the SSQCD fixed point.",
"The crossover scale $\\Lambda _\\text{CFT}$ at which such effects become important is of order $\\Lambda _\\text{CFT}^{3-4\\Delta } \\sim \\Lambda _\\text{SSQCD}^{4-4\\Delta }/M_*$ .",
"Below this scale the fixed point value of the quartic coupling is order one irrespective of the initial $M_*$ , and we recover the CFT with the $R$ -charges given in Table REF .",
"We note that for larger $M_*$ it takes longer to flow to this fixed point; however, since the fixed point values are always order one, the $U(1)$ symmetry breaking term will suppress the unwanted soft masses as long as $\\widetilde{m}< \\Lambda _\\text{CFT}$ .This analysis does not conflict with the results of [12] since $(Q_3 \\,\\overline{u}_3)(Q_3\\, \\overline{d}_3)$ is not a chiral primary operator at the fixed point with superconformal $R$ -charges given in Table REF .",
"We thank Dan Green for discussions on this point.",
"Having analyzed the renormalization group (RG) evolution towards our fixed point, let us return to the behavior of the soft parameters.",
"The only combinations of soft masses which are not suppressed as the fixed point is approached are $&&\\widetilde{m}_\\Sigma ^2 - \\widetilde{m}_{\\overline{\\Sigma }}^2 \\nonumber \\\\&&2 \\,\\widetilde{m}_{Q_3}^2 - \\widetilde{m}_{\\overline{u}_3}^2 -\\widetilde{m}_{\\overline{d}_3}^2 \\nonumber \\\\&& \\widetilde{m}_{H_u}^2 -\\widetilde{m}_{H_d}^2 + \\widetilde{m}_{\\overline{d}_3}^2- m_{\\overline{u}_3}^2\\,.$ This implies that for arbitrary UV boundary conditions, the model does not fully sequester soft masses.",
"However, if the supersymmetry breaking mechanism preserves approximate charge conjugation and custodial symmetries, then the contributions from Eq.",
"(REF ) are negligible at the messenger scale and are not generated by the strong dynamics.",
"This is the case in minimal gauge mediation [22], where at the messenger scale the first difference in Eq.",
"(REF ) vanishes identically, while the linear combinations in the second and third lines are much smaller than each of their respective terms.",
"These combinations can also be suppressed by going beyond minimal gauge mediation or in gravity mediation by imposing discrete symmetries.",
"As we noted before, this analysis neglects effects from the weakly interacting sector of the theory.",
"The first two generations and the SM gauginos continuously feed supersymmetry breaking contributions to the CFT fields, giving rise to “driving terms” in the beta functions for the CFT superfield couplings.",
"However, these supersymmetry breaking effects are much smaller than the soft masses of the first two generation sfermions and gauginos, since the CFT couples to such fields only through irrelevant interactions.",
"Specifically, they are suppressed by loop factors and by SM gauge couplings or Yukawa interactions.",
"These corrections will be taken into account in §.",
"Hence, under the assumption that the supersymmetry breaking mechanism (approximately) respects the above symmetries, the strong conformal dynamics fully suppresses the soft masses of the third generation quarks and Higgs fields, up to small corrections from the weakly coupled sector.",
"It would be interesting to modify the model to accomplish a complete sequestering without having to assume these symmetries, e.g.",
"by adding new flavors and turning on different deformations.",
"Some of these possibilities will be discussed briefly in §.",
"Finally, at the scale $v$ we exit the conformal regime.",
"This happens in an approximately supersymmetric way and does not lead to appreciable finite corrections for the soft parameters.",
"Therefore, the theory at energies below $v$ is the weakly coupled MSSM with the soft masses for the third generation fields and Higgses suppressed with respect to the first two generations and the gauginos.",
"The third generation squark masses are then regenerated by gaugino mediation [13], which in turn can drive the up-type Higgs soft mass squared negative.",
"Starting from generic supersymmetry breaking mechanisms, the CFT acts an an attractor for realizing the natural supersymmetry spectrum of the more minimal supersymmetric SM.",
"The resulting phenomenology will be studied in §."
],
[
"Generating the Flavor Hierarchies",
"Starting from the pioneering work of Nelson and Strassler [2], it has been understood how CFT dynamics can generate the flavor hierarchies at low energies from arbitrary order one Yukawas in the UV.",
"We will explain how this works in the context of our construction.",
"The model presented here differs from previous models of compositeness/superconformal flavor since the third generation superfields have negative anomalous dimensions.",
"Above the dynamical scale $\\Lambda _\\text{CFT}$ , the renormalizable Yukawa couplings are $W \\supset Y^u_{ij} \\,Q_i\\, H_u\\, \\overline{u}_j+Y^d_{ij} \\,Q_i\\, H_d \\,\\overline{d}_j+Y^u_{33} \\,Q_3 \\,H_u \\,\\overline{u}_3+Y^d_{33} \\,Q_3\\, H_d\\, \\overline{d}_3$ where $i,j=1,2$ and all the coefficients are taken to be order one.",
"Renormalizable mixing terms between the third generation and the first two are forbidden by gauge invariance.",
"They will be generated by irrelevant operators as we explain below.",
"The third generation Yukawas are relevant in the CFT regime.",
"Below $\\Lambda _\\mathrm {CFT}$ they flow to order one fixed values.",
"In contrast, the first two generation Yukawas become irrelevant because the Higgs fields acquire positive anomalous dimension.",
"For energies, $v<E<\\Lambda _\\text{CFT}$ we findThe anomalous dimension is defined as $\\Delta =1+\\gamma /2$ .",
"For chiral primary operators it is related to the superconformal $R$ -charge by $\\gamma =3R-2$ .",
"$Y_{ij}^u(E) =\\left(\\frac{E}{\\Lambda _\\mathrm {CFT}} \\right)^\\frac{\\gamma _{Q_i}+\\gamma _{u_j}+\\gamma _{H_u}}{2} Y_{ij}^u(\\Lambda _\\mathrm {CFT})\\,,$ and a similar expression for $Y^d$ .",
"Defining the ratio between the exit scale and dynamical scale $\\epsilon \\equiv \\frac{v}{\\Lambda _\\text{CFT}}\\,,$ the suppression in the first two generation Yukawas at the exit scale becomes $Y_{ij}(v) = \\epsilon ^{\\frac{\\gamma _H}{2}} Y_{ij}(\\Lambda _\\mathrm {CFT}) \\ll Y_{33}(v)\\,.$ We have neglected the perturbative anomalous dimensions for the first two generations.",
"This dynamically generates a hierarchy between the first/second and the third generation Yukawa couplings.",
"Next we consider the off-diagonal Yukawa interactions between the third and first/second generations.",
"The lowest dimension operators allowed by gauge invariance are of the form $W\\supset \\frac{1}{\\Lambda _*} \\overline{\\Sigma } \\,Q_3\\, H_u\\, \\overline{u}_{1,2}+\\frac{1}{\\Lambda _*} Q_{1,2}\\, H_u\\, \\Sigma \\, \\overline{u}_3 + \\ldots \\,,$ where $\\Lambda _*$ is the scale at which these operators are generated.",
"These lead to off-diagonal Yukawas after setting $\\langle \\Sigma \\overline{\\Sigma } \\rangle =v^2$ at the exit scale.",
"The RG evolution between $\\Lambda _\\text{CFT}$ and $v$ yields $Y^u_{i3}(v) = \\frac{v}{\\Lambda _*} \\epsilon ^{\\frac{\\gamma _{H_u}+\\gamma _{Q_3}+ \\gamma _{\\overline{\\Sigma }}}{2}}\\;,\\;Y^u_{3i}(v) = \\frac{v}{\\Lambda _*} \\epsilon ^{\\frac{\\gamma _{H_u}+\\gamma _{u_3}+ \\gamma _{\\Sigma }}{2}}\\,.$ Note that the theory near the UV free fixed point contains two types of classically irrelevant operators: the $U(1)$ symmetry breaking term Eq.",
"(REF ) and the interactions Eq.",
"(REF ).",
"However, their IR fate is very different.",
"As we showed before, the interaction $W_{\\leavevmode {U(1)}}$ becomes relevant in the IR, driving the theory to a strongly coupled fixed point (where it becomes order one), while Eq.",
"(REF ) is irrelevant along the whole flow toward the fixed point.",
"Such irrelevant perturbations do not modify the RG flow or the suppression of soft parameters.",
"They become marginal only after the exit of the conformal regime.",
"Combining these results, we obtain the following the Yukawa textures at $v$ : $Y^u \\sim \\left( \\begin{array}{ccc}\\epsilon ^{\\frac{\\gamma _{H_u}}{2}} & \\epsilon ^{\\frac{\\gamma _{H_u}}{2}} & \\xi _Q \\,\\epsilon ^{\\frac{\\gamma _{H_u}}{2}}\\\\\\epsilon ^{\\frac{\\gamma _{H_u}}{2}} & \\epsilon ^{\\frac{\\gamma _{H_u}}{2}} & \\xi _Q\\,\\epsilon ^{\\frac{\\gamma _{H_u}}{2}}\\\\\\xi _u \\,\\epsilon ^{\\frac{\\gamma _{H_u}}{2}}& \\xi _u \\,\\epsilon ^{\\frac{\\gamma _{H_u}}{2}}& 1\\end{array} \\right),$ with $\\epsilon \\ll 1$ defined in Eq.",
"(REF ) and $\\xi _Q \\equiv \\frac{v}{\\Lambda _*}\\,\\epsilon ^{\\frac{\\gamma _{\\overline{\\Sigma }}+\\gamma _{Q_3}}{2}}\\;,\\;\\xi _u \\equiv \\frac{v}{\\Lambda _*}\\,\\epsilon ^{\\frac{\\gamma _{\\Sigma }+\\gamma _{\\bar{u}_3}}{2}}\\,.$ A similar expression holds for $Y^d$ .",
"Choosing $\\Lambda _*$ below the dynamical scale of the CFT and requiring negative $\\gamma _{\\overline{\\Sigma }}+\\gamma _{Q_3}$ and $\\gamma _{\\overline{\\Sigma }}+\\gamma _{\\bar{u}_3}$ (as is the case in our model) gives $\\xi _{Q, u} \\gtrsim 1$ .",
"For the model with the superpotential given in Eq.",
"(REF ), the anomalous dimensions which determine the Yukawa couplings are $\\gamma _{H_u}= \\gamma _{H_d}=1$ and $\\gamma _{Q_3}+ \\gamma _{\\overline{\\Sigma }}=\\gamma _{\\bar{u}_3}+ \\gamma _{\\Sigma }=-3/2$ .",
"The flavor hierarchies between the third and second generations can be generated when $\\frac{v}{\\Lambda _\\text{CFT}} \\sim 10^{-4}\\;,\\;\\frac{\\Lambda _*}{\\Lambda _\\text{CFT}} \\sim 10^{-1} - 10^{-2}\\,.$ This model does not explain why the first generation Yukawa is smaller than the second generation one.",
"However, this additional small Yukawa could arise by an accidental degeneracy of Eq.",
"(REF ), or by approximate flavor symmetries as in [23].",
"We have checked that by scanning over order one coefficients, we can reproduce the quark spectrum and the CKM matrix to a good approximation.",
"This ends the general analysis of our mechanism.",
"The rest of the work is devoted to a study of its phenomenological consequences."
],
[
"Low Energy Phenomenology",
"Having explained the main features of our mechanism, we will now analyze the properties of the spectrum and Higgs sector and the parameter ranges which lead to a realistic low energy phenomenology."
],
[
"General Properties of the Spectrum",
"In this section, we discuss the features of the low energy spectrum in models which use the dynamics described in §.",
"Supersymmetry breaking is communicated to the MSSM at the messenger scale $M$ , where the operators $c^2_{f}\\int d^4 \\theta \\frac{X^\\dag X}{M^2} \\Phi ^\\dag _\\mathrm {SM} \\Phi _\\mathrm {SM}\\;,\\;c_{g} \\int d^2 \\theta \\,\\frac{X}{M} \\mathcal {W}_\\alpha \\mathcal {W}^\\alpha \\;,\\;\\ldots $ are generated, where $X$ is a supersymmetry breaking spurion with $\\langle X \\rangle \\supset \\theta ^2 F$ , $\\Phi _\\mathrm {SM}$ is an MSSM matter superfield, $\\mathcal {W}_\\alpha $ is the field strength for an MSSM gauge group, and the factors of $c$ are model dependent coefficients.",
"These terms give sfermion and gaugino masses which are determined by the $F$ -term of $X$ .",
"Supersymmetry breaking is external to the dynamics described in §, and we do not constrain the soft UV boundary values, up to the assumptions on approximate symmetries required to suppress the differences given in Eq.",
"(REF ).",
"Typically in concrete models of supersymmetry breaking, the sfermion masses at the messenger scale are comparable for the three generations.",
"On the other hand, sfermion and gaugino masses need not arise at the same order in $F/M$ .",
"This happens in many known cases.",
"For instance, an approximate $R$ -symmetry or gaugino screening (which occurs for a wide class of gauge mediated models [24]) can lead to subleading gaugino masses.",
"We will assume that gauginos are around the TeV scale.",
"In principle the sfermions can be much heavier at the messenger scale, but we do require that $\\widetilde{m}_{f} \\sim c_{f}\\frac{F}{M} \\ll \\Lambda _\\text{CFT}$ so that the conformal dynamics will be relatively unperturbed.",
"Generic sfermion masses will lead to flavor changing neutral currents (FCNCs).",
"In our setup, flavor problems can be somewhat alleviated by having heavy enough sfermions, while also imposing some degree of degeneracy between the first two generations.This is satisfied automatically if the mediation mechanism is flavor-blind.",
"In this case, $\\widetilde{m}_{f_{1,2}} \\gtrsim \\mathcal {\\mathcal {O}}(10 \\mbox{ TeV})$ avoids dangerous FCNCs.",
"On the other hand, there is a limit on how heavy the first two generations can be so that the third generation sfermion masses do not become tachyonic [25].",
"To account for this constraint, we include the dominant 2-loop contributions from the heavy states in the analysis of §REF .",
"It would also be interesting to study models where the CFT dynamics alleviates such tachyonic contributions, allowing a more complete decoupling of the first two generation sfermions.",
"Once we enter the conformal regime, the soft masses for the third generation sfermions and Higgs fields are renormalized by the strong dynamics as described in §, while the first two generation sfermions and gauginos are not appreciably modified.",
"When evaluating the running of the soft parameters in the conformal regime we must consider that gauginos and first/second generation fields are continuously feeding supersymmetry breaking contributions into the third generation and Higgs fields [18].",
"For most of the viable parameter space, the dominant contribution comes from the gauginos, leading to finite contributions $\\widetilde{m}^2_\\text{CFT} \\sim \\frac{g_X^2}{16\\, \\pi ^2} |M_3|^2$ where $M_3$ is the MSSM gluino mass and $g_X$ is the gauge coupling of the weakly interacting $SU(3)_X$ .",
"After escaping the conformal regime, we find a soft spectrum with $\\widetilde{m}_{1,2}\\sim \\mathcal {O}\\mbox{(few TeV)}$ , $M_3 \\sim {\\mathcal {O}}(1 \\mbox{ TeV})$ , and small masses for the third generation squarks and Higgs fields.",
"The masses for the light fields are then predominantly regenerated by gaugino mediation [13] between $v$ and the electroweak scale.",
"The gauginos drive the stop mass to positive values.",
"For $v\\gtrsim 50 \\mbox{ TeV}$ , this makes the up-type Higgs tachyonic and triggers electroweak symmetry breaking.",
"The RG evolution will be studied explicitly below."
],
[
"Comments on the Higgs Sector",
"Next we discuss the interplay between the MSSM Higgs sector and our model.",
"First we consider the supersymmetric Higgs mass $\\mu $ and the bi-linear supersymmetry breaking Higgs mass $b_\\mu $ .",
"Our model contains a solution to the $\\mu $ problem via the irrelevant interaction $W \\supset \\frac{1}{\\Lambda _{\\mu }} \\Sigma \\, \\overline{\\Sigma }\\, H_u\\, H_d\\,.$ (The tree level $\\mu $ term $W \\supset H_u H_d$ can be forbidden by symmetries.)",
"The operator in Eq.",
"(REF ) can be generated by the same mechanism which produces the off-diagonal Yukawas in Eq.",
"(REF ).",
"This is another interesting connection between flavor textures and the Higgs sector.",
"Assuming this occurs, $\\Lambda _{\\mu } \\sim \\Lambda _*$ and no new scale is needed.One can also imagine a different discrete symmetry such that $\\Sigma ^3 H_u H_d$ is the lowest dimension operator which could generate an effective $\\mu $ term.",
"Taking into account the CFT suppression, the $\\mu $ term at the exit scale becomes $\\mu =\\left( \\frac{v}{\\Lambda _*}\\, \\epsilon ^{\\gamma _{H}+\\gamma _\\Sigma }\\right)\\,v\\,.$ In this approach, $b_\\mu $ is zero at $v$ and is generated radiatively as we run down to the weak scale.",
"In the leading log approximation, $b_\\mu \\simeq -\\frac{1}{16 \\,\\pi ^2}\\,\\mu \\left(6\\,g_W^2 M_2 + \\frac{6}{5}g_Y^2 M_1\\right)\\log \\left(\\frac{v}{m_W}\\right),$ where $M_1$ is the bino mass and $M_2$ is the wino mass.",
"This solution to $\\mu $ and $b_\\mu $ can lead to REWSB.",
"For our model, $\\gamma _{H}+\\gamma _\\Sigma =0$ .",
"Requiring $\\mu \\sim 100$ GeV and using the approximate values in Eq.",
"(REF ), $v \\sim 100\\;\\text{TeV},\\;\\Lambda _* \\sim 10^4 - 10^5\\;\\text{TeV},\\;\\mbox{and }\\Lambda _\\text{CFT} \\sim 10^6\\;\\text{TeV}.$ While this is an attractive solution to the $\\mu $ problem, when coupling our mechanism to a specific supersymmetry breaking model, there could be additional dynamics which explains $\\mu /b_\\mu $ .",
"In this case, it is not necessary to introduce Eq.",
"(REF ), and the scales Eq.",
"(REF ) could be different.",
"We now discuss the physical Higgs mass.",
"Below the exit scale, the gluino mass will drive the stop mass positive, which in turn contributes negatively to $\\widetilde{m}_{H_u}^2$ .",
"As long as the bino and wino masses are not too large, this will trigger electroweak symmetry breaking.",
"Models with unified gauginos provide an example of successful REWSB.",
"The down-type Higgs soft mass will be generated though a combination of competing effects from the sbottom and the heavy first/second generations (which drive it negative), and the bino and wino (which drive it positive).",
"Since the mechanism described in this work yields light stops and negligible $a$ -terms, there is tension with a physical Higgs mass of order 125 GeV, as currently hinted at by the LHC [26].",
"Thus, a realistic model must include an additional source to raise the physical Higgs mass.",
"In the simplest version of our construction, an NMSSM type extension does not solve this problem because the CFT makes the interaction $W \\supset S\\, H_u\\, H_d$ (with $S$ the extra singlet in the NMSSM) irrelevant.",
"This leads to a negligible increase in the physical Higgs mass.",
"One option beyond singlet extensions would be to add “non-decoupling $D$ -terms” [27] below the exit scale.",
"While we do not attempt to embed this or other mechanisms into our model, we see no fundamental obstruction.",
"The validity of our conclusions require that this additional module does not lead to appreciable shifts for any of the soft masses."
],
[
"An Example Spectrum",
"In order to perform a concrete analysis, we will work in the context of a model with unified gaugino masses.",
"We will also assume that the mediation of supersymmetry breaking respects custodial symmetry and a “charge conjugation\" symmetry between $Q$ and $\\overline{u},\\,\\overline{d}$ , i.e., $\\widetilde{m}_{Q_3}^2 = \\widetilde{m}_{\\overline{u}_3}^2 = \\widetilde{m}_{\\overline{d}_3}^2$ .",
"For example, both of these assumptions are well approximated by models of minimal gauge mediation.",
"The following analysis demonstrates in a concrete setup the viability of the mechanism for splitting the third generation from the first and second.",
"The techniques presented here can be applied to a wide class of supersymmetry breaking scenarios.",
"Given this framework, the spectrum is determined by choosing a gluino mass and solving the RG equations with the boundary condition at the scale $v$ which the third generation and Higgs soft masses are given byEq.",
"(REF ).",
"While there is an incalculable order one coefficient, we find that such effects are small in the regime of interest.",
"If we also assume the solution to the $\\mu $ problem proposed in §REF , the exit scale is fixed at $v \\sim \\mathcal {O}(100)\\; \\text{TeV}$ .",
"The model is then very predictive: all we need to specify are the messenger scale, gaugino and first/second generation masses.",
"As an example, we find the viable spectrum presented in Table REF , with first/second generation sfermion masses chosen to be 5 TeV.",
"We have assumed an additional contribution to the Higgs quartic from a coupling $g_\\mathrm {new}$ so that $m_Z^2 = \\frac{g_Z^2}{2} \\left(\\langle H_u \\rangle ^2 +\\langle H_d \\rangle ^2\\right) \\ \\quad \\longrightarrow \\quad \\Xi ^2 \\equiv \\frac{g_Z^2+g_\\mathrm {new}^2}{2}\\left(\\langle H_u \\rangle ^2 +\\langle H_d \\rangle ^2\\right).$ in all tree-level MSSM expressions for electroweak symmetry breaking and the Higgs sector.",
"In our numerical analysis below, we will take $\\Xi \\simeq 150 \\mbox{ GeV}$ .",
"As discussed in §REF , this could in principle arise from a non-decoupling $D$ -term — we are agnostic about its source and find that this leads to a small change for all the parameters except for the physical value of the CP even Higgs masses.",
"This yields a Higgs mass of 105 GeV at tree-level which (given the stop masses in Table REF ) will lead to a mass consistent with 125 GeV once loop corrections are taken into account.",
"This point is also consistent with the relevant experimental bounds considered in §REF below.",
"This demonstrates the viability of our mechanism.",
"Table: An example set of consistent parameters with the solution to the μ\\mu problem given in Eq. ().",
"We have assumed gaugino mass unification and to good approximation m ˜ Q i 2 =m ˜ u i 2 =m ˜ d i 2 =m ˜ i 2 \\widetilde{m}_{Q_i}^2=\\widetilde{m}_{u_i}^2=\\widetilde{m}_{d_i}^2=\\widetilde{m}_i^2 at low energies.",
"We find that the tree-level value of the Higgs mass is ≃105GeV \\simeq 105\\mbox{ GeV} which is consistent with 125 GeV when loop corrections are taken into account."
],
[
"Exploring the Parameter Space",
"In this subsection we will briefly explore the possible range of predictions for the soft mass spectrum.",
"In order to do this we will relax the relationship between the $\\mu $ term and $v$ given in Eq.",
"(REF ).",
"Noting that in our concrete model the coupling $W \\supset H_u H_d$ is exactly marginal at the fixed point, one can in principle generate $\\mu $ and $b_{\\mu }$ using an unrelated mechanism at scales above $\\Lambda _\\mathrm {CFT}$ .",
"We can thus take $v$ and $\\tan \\beta $ as free parameters and explore the resultant phenomenology.",
"In Figure REF we have plotted the low energy values of $\\widetilde{m}_{Q_3}^2\\simeq \\widetilde{m}_{u_3}^2 \\simeq \\widetilde{m}_{d_3}^2$ [black, solid] , $m_A\\, (\\mbox{with }\\tan \\beta = 2)$ [red, dashed], and $m_A\\, (\\mbox{with }\\tan \\beta = 10)$ [orange, dotted-dashed] for two choices of $v$ as a function of the gluino mass.",
"The mass of the $A$ is the only parameter with a strong dependence on $\\tan \\beta $ .",
"As in §REF , we assume that supersymmetry breaking respects $\\widetilde{m}_{Q_3}^2 = \\widetilde{m}_{\\overline{u}_3}^2 = \\widetilde{m}_{\\overline{d}_3}^2$ to a good approximation.",
"In order to generate this plot, we use the RG equations for the MSSM to flow from $v$ to the weak scale including the leading 2-loop contributions from the first and second generation sparticles which we fix at 5 TeV.",
"It is this choice which causes the third generation squarks to become tachyonic for small gluino masses.",
"This is the excluded region plotted in opaque grey in Figure REF .",
"The opaque blue region is excluded due to a lack of REWSB (these conditions are unchanged from the MSSM).",
"The light translucent green region is excluded due to the LEP bound on the $A$ massThis exclusion is highly dependent on $\\tan \\beta $ .",
"Furthermore, one could imagine a model where $H_d$ is not a part of the CFT.",
"It would have a large mass and the model would generically be in the decoupling limit.",
"[28].",
"This constraint is cut-off by the kinematic reach of LEP for the process $e^+ e^- \\rightarrow h\\, A$ .",
"For $m_h = 125\\mbox{ GeV}$ $(115\\mbox{ GeV})$ , this implies that $m_A \\gtrsim 90\\mbox{ GeV}$ ($m_A \\gtrsim 100\\mbox{ GeV}$ ).",
"As a conservative estimate, we impose $m_A > 100 \\mbox{ GeV}$ in Fig.",
"REF .",
"We have not included the bino, wino, and first/second generation soft masses in Fig.",
"REF since they are unaffected by our mechanism up to small effects due to off-diagonal Yukawa couplings and 2-loop diagrams.",
"Figure: Low energy spectrum for a model with unified gaugino masses, for v=10 3 TeVv = 10^3\\mbox{ TeV} (left) and v=10 6 TeVv = 10^6\\mbox{ TeV} (right).",
"The curves represent m ˜ Q 3 ≃m ˜ u 3 ≃m ˜ d 3 \\widetilde{m}_{Q_3}\\simeq \\widetilde{m}_{u_3} \\simeq \\widetilde{m}_{d_3} [black, solid], m A (withtanβ=2)m_A\\, (\\mbox{with }\\tan \\beta = 2) [red, dashed], m A (withtanβ=10)m_A\\, (\\mbox{with }\\tan \\beta = 10) [orange, dotted-dashed].",
"(Only m A m_A has a strong dependence on tanβ\\tan \\beta .)",
"The first/second generation squark masses are at 5 TeV.",
"The opaque grey region is excluded due to tachyonic third generation squarks.",
"The opaque blue region is excluded by requiring radiative electroweak symmetry breaking.",
"The light translucent green region is excluded due to the LEP bound on the AA mass.",
"Both of these regions are plotted for the tanβ=2\\tan \\beta = 2 case.",
"We fix Ξ=150GeV\\Xi = 150 \\mbox{ GeV} and for simplicity do not attempt the model dependent task of reproducing the Higgs mass for all points in this plot.The bounds for the $\\tan \\beta = 10$ case are $M_3 > 0.85\\, (0.92) \\mbox{ TeV}$ to avoid having tachyonic squarks, $M_3 > 1.2\\, (1.1) \\mbox{ TeV}$ for REWSB and $M_3 > 2.1 \\,(1.9) \\mbox{ TeV}$ for the $A$ mass, given $v = 10^3 \\,\\left(10^6\\right) \\mbox{ TeV}$ .",
"Note that the LHC also places strong bounds on $m_A$ from searches for di-tau resonances [29].",
"In fact, the LHC excludes the range $120 \\mbox{ GeV}\\lesssim m_A \\lesssim 220 \\mbox{ GeV}$ for $\\tan \\beta = 10$ in the context of the MSSM (with $\\Xi = m_Z$ ).",
"We do not show these constraints in Fig.",
"REF since the excluded regions are for the $\\tan \\beta = 2$ example.",
"Recall that achieving a Higgs mass of 125 GeV requires physics beyond the simple model proposed here.",
"Hence, we will only make a few comments about the mass of the Higgs.",
"First, we note that $A$ is light in the region of parameter space with the lightest squarks and gluino, which can have a non-trivial impact on the mass and couplings of the $h$ .",
"In the pure MSSM, this manifests as a dependence on both $\\tan \\beta $ and the Higgs mixing angle $\\alpha $ for the Higgs couplings (for a review, see [30]).",
"More generally, the dependence of the Higgs couplings on $m_A$ is model dependent.",
"It would be interesting to develop a realistic model for the Higgs sector based on our general mechanism, where this and related questions could be addressed in detail.",
"In generating Fig.",
"REF , we took $\\Xi = 150\\mbox{ GeV}$ , see Eq.",
"(REF ); we find only mild sensitivity to the choice of $\\Xi $ .",
"When $m_A < \\Xi $ , $m_h \\simeq m_A \\cos (2\\,\\beta )$ , independent of $\\Xi $ .",
"For the choice $\\tan \\beta = 10$ , the one-loop corrections from the stops are approximately of the right size to generate a Higgs mass of 125 GeV in the allowed window $100 \\mbox{ GeV} \\lesssim m_A \\lesssim 120 \\mbox{ GeV}$ .",
"For larger values of $m_A$ , the tree-level contribution to the Higgs boson mass would be set by $\\Xi $ , which could be carefully chosen to reproduce the desired result.",
"Alternatively, one could attempt to alter the Higgs quartic with a different mechanism than the one captured by our parameter $\\Xi $ .",
"Since we have a splitting between the first/second and third generation squarks, we must worry about FCNC effects induced by rotating the Yukawa matrices of Eq.",
"(REF ) to the physical basis.",
"To leading order in $\\epsilon $ , the relevant 1-3 and 2-3 mixing is given by $\\delta _{i3} \\sim \\epsilon ^{\\gamma _{H}/2}\\xi $ , where $\\delta _{ij} \\equiv \\widetilde{m}^2_{ij}/\\mathrm {max}(\\widetilde{m}^2_{ii},\\,\\widetilde{m}^2_{jj})$ .",
"Assuming some degree of degeneracy between the first and second generations, negligible $a$ -terms, and an absence of CP violating phases (as in minimal gauge mediation), the strongest flavor bound is from $\\left(\\delta ^d_{13}\\right)_{\\mathrm {LL}=\\mathrm {RR}} \\lesssim 5\\times 10^{-3}$ [31].",
"There are also potential constraints from $b\\rightarrow s\\,\\gamma $ and $B_s\\rightarrow \\mu ^+\\mu ^-$ which are sensitive to model dependent choices, such as details of the chargino sector.",
"Overall, we find no impediment to accommodating these constraints in our model.",
"Figure: We plot contours of the “fine tuning\" parameter Δ -1 \\Delta ^{-1} in the M 3 M_3 versus vv plane.",
"We make the same assumptions as in Fig.",
"with first/second generation squark masses at 5 TeV and tanβ=2\\tan \\beta = 2.",
"The solid grey region is excluded due to tachyonic third generation squarks, the solid blue region is excluded due to a lack of REWSB and the green translucent region is excluded due to the LEP bound on the AA mass.Finally, let us briefly discuss the contributions to fine tuning which result from our mechanism.",
"The problem of naturalness is related to the question of curvature in the symmetry breaking direction — it is a one-dimensional problem for a Higgs field $H$ as in the standard model with a potential $V = m_H^2 |H|^2 + \\lambda |H|^4.$ When $\\langle H \\rangle \\ne 0$ , the physical Higgs mass squared $m_h^2 = - 2\\,m_H^2$ .",
"One simple measure of fine tuning, advocated in [32], [11], is then $\\Delta ^{-1} \\equiv -2 \\frac{\\delta m_H^2}{m_h^2} = -2 \\frac{\\widetilde{m}_{H_u}^2}{m_h^2},$ where in the last equality we are interested in the contribution to the Higgs soft mass in our model.",
"In Fig.",
"REF we have plotted contours of $\\Delta ^{-1}$ from our dynamics in the $M_3$ versus $v$ plane.",
"The most important assumption from the point of view of fine tuning is the gaugino mass spectrum.",
"We have also plotted the region which is excluded due to tachyonic third generation squarks in solid grey, a lack of REWSB in solid blue, and the LEP bound on the $A$ mass for $\\tan \\beta = 2$ in translucent green.",
"We see that there is an allowed region with $\\Delta ^{-1}\\simeq \\mathcal {O}(10\\%)$ where $v\\simeq 10^2 \\mbox{ TeV}$ and $M_3 \\gtrsim 2.5 \\mbox{ TeV}$ .",
"We note that in a complete model which addresses the physical Higgs boson mass, there may be additional sources of fine tuning."
],
[
"Conclusions and Future Directions",
"In this work we have presented a mechanism which acts as an attractor for the more minimal supersymmetric standard model and radiative electroweak symmetry breaking, while also generating the hierarchical structure of the quark Yukawa matrix.",
"We have presented the simplest realization, which is accomplished by adding a new $SU(3)$ gauge group under which the third generation quarks are charged.",
"The model flows to a strongly interacting fixed point where these quarks acquire order one negative anomalous dimensions, while the Higgs gets a positive anomalous dimension.",
"The mechanism applies to generic supersymmetry breaking scenarios, as long as appropriate symmetries ensure that the combinations of masses in Eq.",
"(REF ) are small.",
"It also leads to a simple solution of the $\\mu $ problem.",
"For concreteness we analyzed the low energy phenomenology starting from unified gaugino masses, finding a natural supersymmetry spectrum with split families.",
"It would be interesting to build a fully realistic model based on this mechanism.",
"The main points which need to be addressed are unification (which has been explicitly broken here by the extra matter charged under $SU(3)$ ) and the generation of a realistic physical Higgs mass.",
"This motivates extending our approach by having two copies of the full SM gauge group, instead of just the $SU(3)$ group.",
"One of the nodes will then become strongly coupled, leading to the properties analyzed here.",
"In this context, the NMSSM can naturally become part of the strong dynamics, and unification is in principle possible.",
"The mechanism itself can also be improved in different directions.",
"Here we had to assume that certain approximate symmetries of the supersymmetry breaking sector were forbidding the combinations of soft masses given in Eq.",
"(REF ).",
"In particular, combinations proportional to $U(1)_Y$ can not be screened.",
"This can be avoided if $U(1)_Y$ is embedded into a larger gauge group for the duration of the conformal regime.",
"One possibility would be to weakly gauge the custodial $SU(2)$ .",
"In this case, the only combinations which are not sequestered are $\\widetilde{m}_\\Sigma ^2-\\widetilde{m}_{\\overline{\\Sigma }}^2$ and $\\widetilde{m}_{Q_3}^2-\\widetilde{m}_{\\overline{Q}_3}^2$ , where $\\overline{Q} = (\\overline{u},\\,\\overline{d})$ — both of these combinations can be suppressed by imposing a discrete symmetry.",
"This can lead to a stronger attractor mechanism.",
"For models which realize this stronger attractor, there is a novel possibility of decoupling the first/second generation squarks beyond the bound of [25].",
"If $\\widetilde{m}_{1,2} \\gg v$ , at scales below $\\widetilde{m}_{1,2}$ there will be a quadratically divergent contribution to the stop masses at 2-loops and the Higgs mass at 1-loop (which is proportional to small Yukawa couplings).",
"If it is possible to construct a CFT which would be strong enough to suppress these quadratic divergences, the contribution to the mass from these effects will be schematically given by $y^2/(16\\, \\pi ^2)\\, v^2$ for the Higgs soft mass squared and $g_C^4/(16\\,\\pi ^2)^2\\, v^2$ for the stop soft mass squared.",
"For $v\\simeq 50\\mbox{ TeV}$ , these contributions are small enough to not destabilize our mechanism.",
"Hence, the flavor problem could be completely decoupled in these models.",
"It may also be possible to find a microscopic realization where the CFT and the sector which breaks supersymmetry are part of the same dynamics.",
"In this setup, the exit scale $v$ would be related to the scale of supersymmetry breaking.",
"This may be done at the level of the superpotential, or by destabilizing some of the flat directions of the CFT.",
"Exploring a concrete supersymmetry breaking sector which minimizes the mass differences in Eq.",
"(REF ) would also be an interesting avenue for future work.",
"If nature cares about naturalness, it is plausible that the dynamics between the weak scale and Planck scale could be highly non-trivial.",
"We have demonstrated that coupling the supersymmetric standard model to a new strongly coupled conformal sector can give rise to the flavor hierarchies and the more minimal spectrum."
],
[
"Acknowledgements",
"We thank J. Wacker for collaboration at early stages of this work.",
"We also thank I. Bah, S. El Hedri, T. Gherghetta, D. Green, H. Haber, S. Kachru, J. March-Russell, and M. Peskin for useful conversations and N. Craig for detailed comments on the manuscript.",
"This work was supported by the US Department of Energy under contract number DE-AC02-76SF00515.",
"tocsectionBibliography"
]
] | 1204.1337 |
[
[
"Free energy fluctuations for directed polymers in random media in 1+1\n dimension"
],
[
"Abstract We consider two models for directed polymers in space-time independent random media (the O'Connell-Yor semi-discrete directed polymer and the continuum directed random polymer) at positive temperature and prove their KPZ universality via asymptotic analysis of exact Fredholm determinant formulas for the Laplace transform of their partition functions.",
"In particular, we show that for large time tau, the probability distributions for the free energy fluctuations, when rescaled by tau^{1/3}, converges to the GUE Tracy-Widom distribution.",
"We also consider the effect of boundary perturbations to the quenched random media on the limiting free energy statistics.",
"For the semi-discrete directed polymer, when the drifts of a finite number of the Brownian motions forming the quenched random media are critically tuned, the statistics are instead governed by the limiting Baik-Ben Arous-Peche distributions from spiked random matrix theory.",
"For the continuum polymer, the boundary perturbations correspond to choosing the initial data for the stochastic heat equation from a particular class, and likewise for its logarithm -- the Kardar-Parisi-Zhang equation.",
"The Laplace transform formula we prove can be inverted to give the one-point probability distribution of the solution to these stochastic PDEs for the class of initial data."
],
[
"Introduction and main results",
"The main results of this paper are contained in Sections REF , REF and REF .",
"The below section introduces the study of directed polymers and motivates our main results within that field."
],
[
"Directed polymers in random media.",
"This article studies the effect of quenched disorder (homogeneous and inhomogeneous) on a class of path measures introduced first by Huse and Henley [37] which are commonly called directed polymers in a random media (DPRM).",
"Such polymers are directed in what is often referred to as a time direction, and then are free to configure themselves in the remaining $d$ spatial dimensions.",
"The probability $dP_{\\beta ,Q}(\\pi (\\cdot ))$ of a given configuration $\\pi (\\cdot )$ of the polymer is then given relative to an underlying reference path measure $dP_0$ by a Radon-Nikodym derivative.",
"This derivative is written as a Boltzmann weight involving a Hamiltonian $H_{Q}$ which assigns an energy to the pathActually, the energy is $-H_{Q}(\\pi (\\cdot ))$ but we keep the same convention as in the models analyzed.",
": $dP_{\\beta ,Q}(\\pi (\\cdot )) = Z_{\\beta ,Q}^{-1} e^{\\beta H_{Q}(\\pi (\\cdot ))} dP_0(\\pi (\\cdot )).$ In the above equation $\\beta $ is the inverse temperature which balances the entropy of the underlying reference path measure with the energy of the Hamiltonian.",
"The subscript $Q$ stands for quenched which means that this $H_{Q}(\\pi (\\cdot ))$ is a function of some disorder $\\omega $ which we think of as an element of a probability space.",
"Hence, with respect to this probability space, $H_{Q}$ is a random function.",
"The normalization constant $Z_{\\beta ,Q}$ , given by $Z_{\\beta ,Q}= \\int e^{\\beta H_{Q}(\\pi (\\cdot ))} dP_0(\\pi (\\cdot )),$ is the quenched partition function, and is a function of $\\omega $ as well.",
"The measure $dP_{\\beta ,Q}$ is a quenched polymer measure since it is also a function of $\\omega $ .",
"We denote averages with respect to the disorder $\\omega $ by $\\mathbb {E}$ , so that $\\mathbb {E}(Z_{\\beta ,Q})$ is the average of the quenched partition function.",
"We denote by $\\mathbb {P}$ the probability measure for the disorder $\\omega $ and denote the variance with respect to the disorder as $\\operatornamewithlimits{Var}{\\cdot }$ .",
"From now on we will assume $dP_0$ is the path measure of a random walk with either a free end point, or a specified pinned endpoint.",
"The latter case is called a point-to-point polymer.",
"We will focus mainly on point-to-point polymers herein.",
"At infinite temperature, $\\beta =0$ , and under standard hypotheses on $dP_0$ (i.e., i.i.d.",
"finite variance increments) the measure $dP_{\\beta ,Q}(\\pi (\\cdot ))$ rescales diffusively to that of a Brownian motion and thus the polymer is purely maximizing entropy.",
"At zero temperature, $\\beta =\\infty $ , the polymer measure concentrates on the path (or paths) $\\pi $ which maximize the polymer energy $H_Q(\\pi )$ .",
"A well studied challenge is to understand the effect of quenched disorder at positive $\\beta $ on the behavior of a $dP_{\\beta ,Q}$ -typical path of the free energy $F_{Q,\\beta }:=\\beta ^{-1} \\ln (Z_{\\beta ,Q})$ .",
"A rough description of the behaviour is given by the transversal fluctuation exponent $\\xi $ and the longitudinal fluctuation exponent $\\chi $ .",
"There are many different ways these exponents have been defined, and it is not at all obvious that they exist for a typical polymer model – though it is believed that they do.",
"As $n$ goes to infinity, the first exponent describes the fluctuations of the endpoint of the path $\\pi $ : typically $|\\pi (n)| \\approx n^{\\xi }$ .",
"The second exponent describes the fluctuations of the free energy: $\\operatornamewithlimits{Var}{F_{\\beta ,Q}} \\approx n^{2\\chi }$ .",
"Assuming the existence of these exponents, in order to have a better understanding of the system it is of essential interest to understand the statistics for the properly scaled location of the endpoint and fluctuations of the free energy.",
"From now on we will focus entirely on Hamiltonians which take the form of a path integral through a space-time independent noise field (the quenched disorder).",
"In the discrete setting of $dP_0$ given as a simple symmetric random walk (SSRW) of length $n$ , we consider a homogeneous noise field given by i.i.d.",
"random variables $w_{t,x}$ .",
"The quenched Hamiltonian is then defined as $H_{Q}(\\pi (\\cdot )) = \\sum _{t=0}^{n} w_{t,\\pi (t)}$ .",
"The first rigorous mathematical work on directed polymers was by Imbrie and Spencer [40] in 1988 where they proved that in dimensions $d\\ge 3$ and with small enough $\\beta $ , the polymer is diffusive ($\\xi =1/2$ ).",
"Bolthausen [15] strengthened the result to a central limit theorem for the endpoint of the walk.",
"This means that in $d\\ge 3$ entropy dominates at high enough temperature, i.e., the polymer behaves as if there were no noise.",
"Bolthausen's work relied on the now fundamental observation that renormalized partition function (for $dP_0$ a SSRW of length $n$ ) $W_n=Z_{\\beta ,Q} /\\mathbb {E}(Z_{\\beta ,Q})$ is a martingale.",
"By a zero-one law, the limit $W_\\infty =\\lim _{n\\rightarrow \\infty } W_n$ is either almost surely 0 or almost surely positive.",
"Since at $\\beta =0$ the effect of the randomness $\\omega $ vanishes and one has $W_\\infty =1$ , one refers to the case $W_{\\infty }>0$ as weak disorder, while the term strong disorder is used when $W_{\\infty }=0$ .",
"There is a critical value $\\beta _c$ such that weak disorder holds for $\\beta <\\beta _c$ and strong for $\\beta >\\beta _c$ .",
"It is known that $\\beta _c=0$ for $d\\in \\lbrace 1,2\\rbrace $ [21] and $0<\\beta _c\\le \\infty $ for $d\\ge 3$ .",
"In $d\\ge 3$ and weak disorder the walk converges to a Brownian motion [22].",
"On the other hand, the behavior of the polymer paths in strong disorder is different since there exist (random) points at which the path $\\pi $ has a uniform (in $n$ ) positive probability (under $dP_{\\beta ,Q}$ ) of ending (see [21], [20])."
],
[
"The KPZ universality conjecture for $d=1$ .",
"In this paper we focus on the $d=1$ case.",
"The universality conjecture says that the scaling exponents and limiting fluctuation statistics for the free energy exist and are universal in the sense that they do not depend of the details of the model, provided some generic requirements are satisfied (e.g.",
"the variables $w$ have only local correlations, and enough non-trivial finite moments, see also the review [23]).",
"The models we consider below are predicted to belong to the so-called the Kardar-Parisi-Zhang (KPZ) universality class [44] for which $\\xi =2/3$ and $\\chi =1/3$ [34], [68].",
"In particular, since the transition from strong to weak disorder happens at $\\beta =0$ , the universality conjecture predicts that for any $\\beta >0$ the scaling exponents $\\xi ,\\chi $ and the fluctuation statistics equal the corresponding zero temperature ones ($\\beta =\\infty $ ).",
"The first step to this conjecture is to identify $\\xi ,\\chi $ and describe the scaling limit for polymers and their free energy.",
"This can be done by studying any model believed to be in the universality class.",
"The second step is to show universality, i.e., show that the results are independent of the chosen model.",
"In this paper we show such universality for any $\\beta > 0$ for two particular polymer models – the semi-discrete and continuum directed random polymers.",
"Consider now $Z_{\\beta ,Q}^{\\rm pp}(\\tau ,x)$ to be the point-to-point partition function of polymers ending at $x$ at time $\\tau $ .",
"Denote by $F_{\\beta ,Q}^{\\rm pp}:=\\frac{1}{\\beta }\\ln (Z_{\\beta ,Q}^{\\rm pp})$ the free energy and by $f_{\\beta ,Q}^{\\rm pp}$ the free energy limit shape (law of large numbers of $F_{\\beta ,Q}^{\\rm pp}$ along different velocities).",
"Then, a stronger version of the KPZ universality conjecture is to claim that over all models, there exists a unique limitMore precisely, the uniqueness should be up to a space-time scaling by parameters which can be calculated from the microscopic properties of the polymer such as $dP_0$ , $\\beta $ and the disorder distribution (see, for example, [60], [46]).",
"$\\lim _{\\varepsilon \\rightarrow 0} \\varepsilon ^{\\chi } \\left(F_{\\beta ,Q}^{\\rm pp}(\\varepsilon ^{-1}\\tau ,\\varepsilon ^{-\\xi }x)-\\varepsilon ^{-1}f_{\\beta ,Q}^{\\rm pp}(\\tau ,\\varepsilon ^{1-\\xi }x)\\right).$ One issue is that, since we are in the strong disorder regime, the quenched free energy differs from the annealed one (easier to compute) and there is no general way of determining it.",
"The conjectural space-time limit is described in [26] where it is called the KPZ renormalization fixed point.",
"Information about this fixed point has generally come from studying exactly solvable models at zero temperature, $\\beta =\\infty $ , such as last passage percolation, TASEP or PNG (see the review [23]).",
"For these models, one indeed finds $\\xi =2/3$ , $\\chi =1/3$ , and although the existence of the full space-time limit under these scalings is not known, for fixed time $\\tau >0$ the limiting spatial process is the Airy$_2$ process [53], [42].",
"This extends the results of [9], [41] that show that the limiting one-point distribution (fixed $(\\tau ,x)$ ) is governed by the GUE Tracy-Widom distribution [62].",
"Some of the zero temperature models can be reinterpreted as growth models of interfaces belonging to the KPZ universality class (where the free energy plays the role of a height function).",
"Thus, the universality conjecture says that the height function (properly rescaled) converges to the same limit as the polymer free energy.",
"These other models require initial data (i.e., height function profile at time zero) and the description of the KPZ fixed point takes the limit of this initial data into consideration.",
"The scalings $\\xi $ and $\\chi $ do not change, but the limiting statistics reflect the initial data (see the reviews [29], [30] and recent experiments [61]).",
"For random polymers the change in statistics occurs for instance if one looks at point-to-line problem instead.",
"We will look at a different way to change the statistics by considering point-to-point polymers with boundary perturbations.",
"The role of initial data for growth processes and particle systems can be mimicked in the case of polymers by introducing inhomogeneity into the quenched disorder so as to encourage the path measure to spend more time on the boundary of the underlying path space.",
"For instance consider a polymer with $dP_0$ given by a SSRW (either free or point-to-point) in a quenched disorder formed by i.i.d.",
"centered weights $w_{t,x}$ , for $x<t$ ; and i.i.d.",
"boundary weights $w_{t,t}$ with positive mean, for $t\\ge 0$ .",
"As the mean of the boundary weights $w_{t,t}$ increases, the paths which stay along the boundary for a long time tend to have a larger Boltzmann weight, and above a critical threshold, the energy-entropy competition is dominated by the boundary attraction so that the path spends a macroscopic proportion of its time pinned along the boundary.",
"Note that once a path leaves the boundary it can not return.",
"This leads to Gaussian scalings and statistics for the free energy and hence takes us beyond the basin of attraction for the KPZ renormalization fixed point.",
"However, in a scaling window of perturbation strength around the critical value, the boundary energy is strong enough to modify the free energy statistics and polymer measure, without modifying the scaling exponents.",
"Modulating the strength within the window one sees a transition from sub to super critical behaviors (in terms of exponents and statistics).",
"Under the scaling (REF ) these critically perturbed polymers should converge to the KPZ fixed point with a certain type of initial data, which results in different statistics than in the homogeneous disorder case.",
"Information about these statistics originated from the analysis of a few solvable zero temperature, $\\beta =\\infty $ , models [8], [18], [10], [17], [38], [12].",
"In this work we consider two polymer models with boundary perturbations at positive temperature: the O'Connell-Yor semi-discrete directed polymer and the continuum directed random polymer (CDRP).",
"Each of these two models are themselves scaling limits of discrete polymers with general weight distributions when the inverse temperature $\\beta $ scales to 0 simultaneously with the other parameters scaling to infinity.",
"This type of scaling is called intermediate disorder scaling as it moves between weak and strong disorder (see [2], [4], [3], [31], [32]).",
"This distinguishes these two models as being somewhat universal in their own rights, and hence provides increased motivation to prove the KPZ universality conjecture in their cases.",
"The first contribution of this work is to rigorously establish the KPZ universality conjecture (for one-point scalings and statistics) for these two positive temperature polymer models with boundary perturbations.",
"In particular, we determine the critical perturbation strength and show that the phase diagram of one-point scalings and statistics for these polymers match that of the solvable zero temperature models.",
"These results significantly expand upon the previous proved results for positive temperature polymer models [5], [25], [16], [57], [58].",
"Our results rely on an algebraic framework for the exact solvability of these polymers which comes from Macdonald processes and their limits and degenerations [16].",
"One output of that work is an exact formula for the Laplace transform of the partition function for the semi-discrete polymer with arbitrary boundary perturbation strength.",
"From this, [16] showed KPZ universality for the one-point scalings and statistics for the unperturbed semi-discrete polymer at sufficiently low temperatures.",
"The second contribution of this work is to develop a variant on the formulas of [16] for the semi-discrete polymer which are well-adapted for rigorous asymptotic analysis at all temperatures (and for boundary perturbations).",
"This requires modifications at a fairly high level of the hierarchy of models arising from Macdonald processes.",
"The third contribution is the discovery and proof of exact formulas for the Laplace transform of the partition function for the CDRP with boundary perturbations.",
"The semi-discrete polymer has an intermediate disorder scaling limit in which it converges to the CDRP [31], [32], and these formulas are found via rigorous asymptotics of the aforementioned variant on the formulas of [16].",
"These formulas display a striking similarity (and a clear limit transition) to those of the KPZ fixed point with initial data corresponding to the boundary perturbations [8].",
"The CDRP free energy solves the Kardar-Parisi-Zhang (KPZ) stochastic PDE and the boundary perturbation corresponds to particular class of half Brownian-like initial data.",
"The discovered formulas also provide a new class of statistics for this equation, which is believed to model the fluctuations of randomly growing interfaces.",
"Let us now explain in detail our results."
],
[
"O'Connell-Yor semi-discrete polymer free energy",
"The first result is about a semi-discrete directed polymer model introduced by O'Connell and Yor [52].",
"Definition 1.1 An up/right path in $\\mathbb {R}\\times \\mathbb {Z}$ is an increasing path which either proceeds to the right or jumps up by one unit.",
"For each sequence $0<s_1<\\cdots <s_{N-1}<\\tau $ we can associate an up/right path $\\phi $ from $(0,1)$ to $(\\tau ,N)$ which jumps between the points $(s_i,i)$ and $(s_{i},i+1)$ , for $i=1,\\ldots , N-1$ , and is continuous otherwise.",
"Fix a real vector $a=(a_1,\\ldots , a_N)$ and let $B(s) = (B_1(s),\\ldots , B_N(s))$ for $s\\ge 0$ be independent standard Brownian motions such that $B_i$ has drift $a_i$ .",
"Let $\\mathbb {P}$ and $\\mathbb {E}$ denote the probability measure and expectation with respect to the disorder $B(\\cdot )$ .",
"Define the energy of a path $\\phi $ to be $E(\\phi ) = B_1(s_1)+\\left(B_2(s_2)-B_2(s_1)\\right)+ \\cdots + \\left(B_N(\\tau ) - B_{N}(s_{N-1})\\right).$ Then the O'Connell-Yor semi-discrete directed polymer partition function $\\mathbf {Z}^{N}(\\tau )$ is given by $\\mathbf {Z}^{N}(\\tau ) = \\int e^{E(\\phi )} d\\phi ,$ where the integral is with respect to Lebesgue measure on the Euclidean set of all up/right paths $\\phi $ (i.e., the simplex of jumping times $0<s_1<\\cdots <s_{N-1}<\\tau $ ).",
"The quenched free energy $\\mathbf {F}^{N}(\\tau )$ is defined as $\\mathbf {F}^{N}(\\tau ) = \\ln \\mathbf {Z}^{N}(\\tau ),$ whereas the annealed free energy is given by $\\mathbf {F}_{\\rm an}^N(\\tau ):=\\ln \\mathbb {E}\\mathbf {Z}^{N}(\\tau )$ .",
"This model is related to the discrete polymer discussed in the introduction as follows: By rotating an angle of $\\pi /4$ the discrete polymer becomes a measure on up/right lattice paths starting at $(1,1)$ with weights in $(\\mathbb {Z}_{> 0})^2$ .",
"Consider the point-to-point partition function for paths going from $(1,1)$ to $(\\tau \\varepsilon ^{-1},N)$ and rescale the lattice weights like $\\varepsilon ^{1/2} w_{i,j}$ .",
"Then as $\\varepsilon \\rightarrow 0$ , due to the invariance principle the up/right lattice paths become like $\\phi $ of the above definition, and the discrete path integral energy becomes $E(\\phi )$ .",
"We focus on scaling the semi-discrete polymer as $\\tau =\\kappa N$ for some $\\kappa >0$ .",
"Due to Brownian scaling, the parameter $\\kappa $ can be changed to 1 by replacing $E(\\phi )$ by $\\kappa ^{1/2} E(\\phi )$ in the exponential of (REF ).",
"In this way, $\\kappa $ corresponds to an inverse temperature parameter $\\beta = \\kappa ^{1/2}$ .",
"It is easy to calculate the annealed free energy exactly as $\\mathbf {F}^{N}_{\\rm an}(\\tau ) = \\frac{\\tau }{2} + (N-1)\\ln \\tau - \\ln (N-1)!$ which after setting $\\tau =\\kappa N$ implies that the annealed free energy density converges to $\\textbf {f}_{\\rm an}(\\kappa ):=\\lim _{N\\rightarrow \\infty } N^{-1} \\mathbf {F}_{\\rm an}^N(\\kappa N) =\\frac{\\kappa }{2} + \\ln \\kappa +1.$ Let us briefly relate what the law of large numbers is for the quenched free energy of the unperturbed polymer.",
"Definition 1.2 The Digamma function is given by $\\Psi (z) = \\frac{d}{dz}\\ln \\Gamma (z)$ .",
"For $\\theta \\in \\mathbb {R}_+$ , define $\\kappa _\\theta :=\\Psi ^{\\prime }(\\theta ),\\quad f_\\theta :=\\theta \\Psi ^{\\prime }(\\theta )-\\Psi (\\theta ),\\quad c_\\theta :=(-\\Psi ^{\\prime \\prime }(\\theta )/2)^{1/3},$ or, equivalently, for $\\kappa \\in \\mathbb {R}_+$ , define $\\theta ^\\kappa :=(\\Psi ^{\\prime })^{-1}(\\kappa )\\in \\mathbb {R}_+,\\quad f^\\kappa :=\\inf _{t>0} (\\kappa t - \\Psi (t))\\equiv f_{\\theta ^\\kappa },\\quad c^\\kappa :=c_{\\theta ^\\kappa }.$ The following law of large numbers for $\\mathbf {F}^{N}(\\kappa N)$ was conjectured in [52] and proved in [48].",
"Consider the semi-discrete directed polymer with drift vector $a=(0,\\ldots ,0)$.",
"Then for all $\\kappa >0$ , $\\lim _{N\\rightarrow \\infty } N^{-1} \\mathbf {F}^{N}(\\kappa N) = f^{\\kappa }, \\qquad \\textrm {a.s.}$ It follows that for all $\\kappa >0$ , $\\textbf {f}_{\\rm an}(\\kappa )=\\frac{\\kappa }{2} + \\ln \\kappa +1>f^{\\kappa }$ , and the quenched and annealed free energy density converge in the $\\kappa \\rightarrow 0$ limit.",
"This is in agreement with strong disorder.",
"Below we analyze the large $N$ asymptotics of the fluctuations of $\\mathbf {F}^{N}(\\kappa N)$ when centered by $N f^\\kappa $ .",
"Theorem 1.3 Consider the semi-discrete directed polymer with drift vector $a=(a_1,\\ldots , a_m,0,\\ldots ,0)$ where $m\\le N$ is fixed and the $m$ non-zero real numbers $a_1,a_2,\\ldots , a_m$ may depend on $N$ .",
"We can consider without loss of generality that $a_1\\ge a_2\\ge \\ldots \\ge a_m$ as the free energy is invariant under permuting these parameters.",
"Then for all $\\kappa >0$ , we have the following characterization of the limiting behavior of the free energy $\\mathbf {F}^{N}(\\kappa N)$ as $N\\rightarrow \\infty $ .",
"(a) If $\\limsup _{N\\rightarrow \\infty }N^{1/3} (a_1(N)-\\theta ^\\kappa )=-\\infty $ , then $\\lim _{N\\rightarrow \\infty } \\mathbb {P}\\left( \\frac{ \\mathbf {F}^{N}(\\kappa N) - N f^{\\kappa }}{c^\\kappa N^{1/3}}\\le r\\right) = F_{{\\rm GUE}}(r),$ where $F_{{\\rm GUE}}$ is the GUE Tracy-Widom distribution [62].",
"(b) If $\\lim _{N\\rightarrow \\infty } N^{1/3}(a_i(N) -\\theta ^\\kappa ) = b_i\\in \\mathbb {R}\\cup \\lbrace -\\infty \\rbrace $ for $i=1,\\ldots ,m$ , then $\\lim _{N\\rightarrow \\infty } \\mathbb {P}\\left( \\frac{ \\mathbf {F}^{N}(\\kappa N) - N f^{\\kappa }}{c^\\kappa N^{1/3}}\\le r\\right) = F_{{\\rm BBP},b}(r),$ where $F_{{\\rm BBP},b}$ is the Baik-Ben Arous-Péché [8] distribution from spiked random matrix theory, with $b=(b_1,\\ldots ,b_m)$ .",
"The definitions of $F_{{\\rm GUE}}(r)$ and $F_{{\\rm BBP},b}$ are provided below in Definition REF .",
"The fact that the result is independent of the ordering of $a_1,\\ldots ,a_N$ is apparent from the formulas, see e.g.",
"Theorem REF below.",
"In Section we reduce the proof of this result to a claim on certain asymptotics of the Fredholm determinant formula presented in Section REF .",
"We provide a formal critical point derivation of these asymptotics in Section and a rigorous proof later in Section .",
"Remark 1.4 If $\\lim _{N\\rightarrow \\infty } N^{1/3}(a_i(N) -\\theta ^\\kappa )=\\infty $ then the boundary perturbation overwhelms the free energy fluctuations and the scalings and statistics become Gaussian in nature.",
"This super critical regime is proved for zero temperature polymers [8], [55].",
"We do not include a proof of this regime here, but it should be readily accessible from the exact formulas via asymptotic analysis.",
"Remark 1.5 In the unperturbed case, i.e., $a=(0,\\ldots ,0)$, a tight upper bound on the exponent for the free energy fluctuation scalings was determined in [58].",
"In [16] the full one-point scaling limit was proved for $\\kappa >\\kappa ^*$ with $\\kappa ^*$ a large (enough) constant forced by some technical consideration in the asymptotic analysis.",
"In this article we do away with those technical issues which allows us to rigorously extend the asymptotics to all positive $\\kappa $ , as well as to $\\kappa $ tending to zero simultaneously with $N$ , as we soon will consider.",
"Theorem REF is expected by universality, because the same results hold in the zero-temperature limit ($\\beta =\\infty $ ) and the phase transition is expected to be at $\\beta =0$ .",
"More precisely, the limit of the free energy (divided by $\\beta $ ) as $\\beta $ goes to infinity and $(N,\\tau )$ is fixed, is described by the ground-state maximization problem $\\begin{aligned}M^N(\\tau ) :=& \\lim _{\\beta \\rightarrow \\infty } \\frac{1}{\\beta } \\ln \\int e^{\\beta E(\\phi )} d\\phi \\\\=& \\max _{0<s_1<\\cdots <s_{N-1}<\\tau } B_1(s_1)+\\left(B_2(s_2)-B_2(s_1)\\right)+ \\cdots + \\left(B_N(\\tau ) - B_{N}(s_{N-1})\\right).\\end{aligned}$ In the unperturbed case, $a_i\\equiv 0$ , $M^N(1)$ is distributed as the largest eigenvalue of an $N\\times N$ GUE random matrix [11], see also Theorem 1.1 of [51].",
"In fact, as a process of $\\tau $ , $M^N(\\tau )$ has the law of the largest eigenvalue of the standard Dyson Brownian motion on Hermitian matrices (the lower eigenvalues are also connected to certain generalizations of the free energy [51]).",
"It follows from the original work of Tracy and Widom [62] and also [33], [50] that $\\lim _{N \\rightarrow \\infty } \\mathbb {P}\\left( \\frac{ M^N(N) - 2N}{N^{1/3}}\\le r\\right) = F_{{\\rm GUE}}(r).$ For general drift parameters, $M^N(\\tau )$ is related to Dyson Brownian motion with drifts and the distribution of $M^N(\\tau )$ then coincides with the largest eigenvalue of a spiked GUE matrix, for which the analog of Theorem REF was proved by Péché [55].",
"The first such results were for spiked LUE matrices in the work of Baik-Ben Arous-Péché [8].",
"We now record the definitions of these limiting distributions in terms of Fredholm determinants.",
"Note that there are many equivalent ways to rewrite these formulas (cf.",
"[7]) and we use the most convenient for our purposes.",
"Definition 1.6 The GUE Tracy-Widom distribution [62] is defined as $F_{{\\rm GUE}}(r) =\\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )},$ where $K_{\\rm Ai}$ is the Airy kernel, that has integral representations $K_{\\rm Ai}(\\eta ,\\eta ^{\\prime })=\\frac{1}{(2\\pi I)^2} \\int _{e^{-2\\pi I/3}\\infty }^{e^{2\\pi I/3}\\infty } dw \\int _{e^{-\\pi I/3}\\infty }^{e^{\\pi I/3}\\infty } dz\\frac{1}{z-w}\\frac{e^{z^3/3-z\\eta }}{e^{w^3/3-w\\eta ^{\\prime }}}=\\int _{\\mathbb {R}_+}d\\lambda \\operatorname{Ai}(\\eta +\\lambda )\\operatorname{Ai}(\\eta ^{\\prime }+\\lambda ),$ where in the first representation the contours $z$ and $w$ must not intersect.",
"The BBP distribution from spiked random matrix theory [8] is defined as $F_{{\\rm BBP},b}(r) =\\det (\\mathbb {1}-K_{{\\rm BBP},b})_{L^2(r,\\infty )},$ where $b=(b_1\\ge b_2\\ge \\ldots \\ge b_m)\\in \\mathbb {R}^m$ and $K_{{\\rm BBP},b}$ is given by $K_{{\\rm BBP},b}(\\eta ,\\eta ^{\\prime })=\\frac{1}{(2\\pi I)^2} \\int _{e^{-2\\pi I/3}\\infty }^{e^{2\\pi I/3}\\infty } dw \\int _{e^{-\\pi I/3}\\infty }^{e^{\\pi I/3}\\infty } dz\\frac{1}{z-w}\\frac{e^{z^3/3-z\\eta }}{e^{w^3/3-w\\eta ^{\\prime }}}\\prod _{k=1}^m\\frac{z-b_k}{w-b_k},$ where the path for $w$ passes on the right of $b_1,\\ldots ,b_m$ and does not intersect with the path $z$ , see Figure REF .",
"It is convenient to extend this definition to allow for $b_i=-\\infty $ for all $i=\\ell +1,\\ldots ,m$ .",
"Calling $\\tilde{b}=(b_1,\\ldots ,b_{\\ell })$ the finite values of $b$ , we then set $K_{{\\rm BBP},b}=K_{{\\rm BBP},\\tilde{b}}$ and likewise define $F_{{\\rm BBP},b}(r)= F_{{\\rm BBP},\\tilde{b}}(r)$ .",
"Notice that if $m=0$ then $K_{{\\rm BBP},b}=K_{\\rm Ai}$ .",
"For representations of this kernel in terms of Airy functions see [8].",
"Figure: Integration contours for ww (dashed) and zz (solid) of the kernel K BBP ,b K_{{\\rm BBP},b}.",
"The black dots are b 1 ,...,b m b_1,\\ldots ,b_m."
],
[
"Continuum directed random polymer free energy",
"Just as Brownian motion serves as a paradigm for and universal scaling limit of random walks, the continuum directed random polymer (CDRP) serves a similar role for $1+1$ dimensional directed random polymers [19], [2].",
"It is proved that the CDRP is the limit of (general weight distribution) discrete [4] or semi-discrete polymers [31] under intermediate disorder scaling in which the inverse temperature (or noise strength) is scaled to zero as the system size grows (so as to converge to space-time white noise).",
"One such result is quoted as Theorem REF .",
"The directed polymers considered earlier are measures on random walk trajectories reweighted by a Boltzmann weight with Hamiltonian given in terms of a path integral through a space-time independent noise field along the walk's trajectory.",
"In the continuum, random walks are replaced by Brownian motions and the space-time noise field becomes space-time Gaussian white-noise.",
"It is possible to define a path measure in the continuum [3].",
"Here we will only focus on the continuum limit of the polymer partition function (REF ), not the path measure.",
"We define the CDRP partition function with respect to general boundary perturbations.",
"These perturbations are the limit of critically tuned boundary perturbations of the discrete polymers.",
"Definition 1.7 The partition function for the continuum directed random polymer with boundary perturbation $\\ln \\mathcal {Z}_0(X)$ is given by the solution to the stochastic heat equation with multiplicative Gaussian space-time white noise and $\\mathcal {Z}_0(X)$ initial data: $\\partial _T \\mathcal {Z} = \\tfrac{1}{2}\\partial _X^2 \\mathcal {Z} +\\mathcal {Z}\\dot{{W}}, \\qquad \\mathcal {Z}(0,X)=\\mathcal {Z}_0(X).$ The initial data $\\mathcal {Z}_0(X)$ may be random but is assumed to be independent of the Gaussian space-time white noise $\\dot{{W}}$ and is assumed to be almost surely a sigma-finite positive measure.",
"Observe that even if $\\mathcal {Z}_0(X)$ is zero in some regions, the stochastic PDE makes sense and hence the partition function is well-defined.",
"The stochastic heat equation (REF ) is really short-hand for its integrated (mild) form $\\mathcal {Z}(T,X) = \\int _{-\\infty }^{\\infty } p(T,X-Y) \\mathcal {Z}_0(Y) dY + \\int _{0}^{T} \\int _{-\\infty }^\\infty p(T-S,X-Y)\\mathcal {Z}(S,Y) \\dot{{W}}(dS,dY)$ where (formally) Gaussian space-time white noise has covariance $\\mathbb {E}\\left[\\dot{{W}}(T,X)\\dot{{W}}(S,Y)\\right] = \\delta (T-S)\\delta (Y-X),$ and $p(T,X) = (2\\pi T)^{-1/2} \\exp (-X^2/2T)$ is the standard heat kernel.",
"A detailed description of this stochastic PDE and the class of initial data for which it is well-posed can be found in [5], [14], or the review [54].",
"As long as $\\mathcal {Z}_0$ is an almost surely sigma-finite positive measure, it follows from work of Müller [49] that, almost surely, $\\mathcal {Z}(T,X)$ is positive for all $T>0$ and $X\\in \\mathbb {R}$ .",
"Hence we can take its logarithm.",
"Definition 1.8 For $\\mathcal {Z}_0$ an almost surely sigma-finite positive measure define the free energy for the continuum directed random polymer with boundary perturbation $\\ln \\mathcal {Z}_0(X)$ as $\\mathcal {F}(T,X) = \\ln \\mathcal {Z}(T,X).$ The random space-time function $\\mathcal {F}$ is also the Hopf-Cole solution to the Kardar-Parisi-Zhang equation with initial data $\\mathcal {F}_0(X)=\\ln \\mathcal {Z}_0(X)$ [5], [14].",
"The solution to the stochastic heat equation is interpreted as a polymer partition function due to a version of the Feynman-Kac representation (see [13] or the review [23]): $\\mathcal {Z}(T,X) = \\mathbb {E}_{B(T)=X}\\left[\\mathcal {Z}_0(B(0)):\\,\\exp \\,: \\left\\lbrace \\int _0^{T} \\dot{{W}}(t,B(t))dt\\right\\rbrace \\right]$ where the expectation $\\mathbb {E}$ is taken over the law of a Brownian motion $B$ which is run backwards from time $T$ position $X$ .",
"The $:\\,\\exp \\,:$ is the Wick exponential which accounts for the fact that one can not naïvely integrate white noise along a Brownian motion (various equivalent ways exist to define this exponential [13], [5], [23]).",
"By time reversal we may consider this as the partition function for Brownian bridges which can depart at time 0 from any location $B(0)\\in \\mathbb {R}$ with an energetic cost of $\\ln \\left(p(T,X-B(0))\\mathcal {Z}_0(B(0))\\right)$ and then must end at $X$ at time $T$ .",
"This departure energy cost is the limit of the boundary perturbations for the discrete and semi-discrete polymers.",
"Let us introduce the class of boundary perturbations which arise in the limit of the semi-discrete directed polymer partition function with the first few drift parameters $a_1,\\ldots , a_m$ tuned as a function of $N$ in a critical way (and all other drifts zero).",
"Definition 1.9 The continuum directed random polymer partition function with $m$ -spiked boundary perturbation corresponds to choosing initial data for (REF ) as follows: Fix $m\\ge 1$ and a real vector $b=(b_1,\\ldots , b_m)$ ; then $\\mathcal {Z}_0(X) = \\mathbf {Z}^m(X)\\mathbf {1}_{X\\ge 0}$ where $\\mathbf {Z}^m(X)$ is defined as in (REF ) with drift vector $b$ , and where $\\mathbf {1}_{X\\ge 0}$ is the indicator function for $X\\ge 0$ .",
"When $m=0$ we will define 0-spiked initial data as corresponding to $\\mathcal {Z}_0(X)=\\mathbf {1}_{X=0}$ , where $\\mathbf {1}_{X=0}$ is the indicator function that $X=0$ .",
"Theorem 1.10 Fix $m\\ge 0$ and a real vector $b=(b_1,\\ldots , b_m)$ .",
"Consider the free energy of the continuum directed random polymer with $m$ -spiked boundary perturbation with drift vector $b$ (Definition REF ).",
"(a) If $m=0$ , then for any $T>0$ and any $S$ with positive real part, $\\mathbb {E}\\left[e^{-S \\exp \\left(\\mathcal {F}(T,0) + T/4!\\right)}\\right] = \\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}.$ (b) If $m\\ge 1$ , then for any $T>0$ and any $S$ with positive real part, $\\mathbb {E}\\left[e^{-S \\exp \\left(\\mathcal {F}(T,0) + T/4!\\right)}\\right] = \\det (\\mathbb {1}-K_{{\\rm CDRP},b})_{L^2(\\mathbb {R}_+)}.$ The kernel in the above theorem are given now.",
"Definition 1.11 Fix $m\\ge 0$ and a real vector $b=(b_1,\\ldots , b_m)$ .",
"The integral kernel $K_{{\\rm CDRP},b}(\\eta ,\\eta ^{\\prime })=\\frac{1}{(2\\pi I)^2}\\int _{\\mathcal {C}_w} dw \\int _{\\mathcal {C}_z} dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-z\\eta ^{\\prime }}}{e^{w^3/3-w\\eta }} \\prod _{k=1}^m\\frac{\\Gamma (\\sigma w-b_k) }{\\Gamma (\\sigma z-b_k)},$ where $\\sigma =(2/T)^{1/3}$ .",
"When $m=0$ the product of Gamma function ratios is replaced by 1 and the resulting kernel is denoted $K_{{\\rm CDRP}}$ .",
"The $w$ contour $\\mathcal {C}_w$ is from $-\\frac{1}{4\\sigma }-I\\infty $ to $-\\frac{1}{4\\sigma }+I\\infty $ and crosses the real axis on the right of $b_1/\\sigma ,\\ldots ,b_m/\\sigma $ .",
"The $z$ contour $\\mathcal {C}_z$ is taken as $\\mathcal {C}_w$ shifted to the right by $\\frac{1}{2\\sigma }$ (see Figure REF for an illustration).",
"Just as in Definition REF there exist integral representations for these kernels involving Airy functions.",
"In particular, $K_{{\\rm CDRP}}(\\eta ,\\eta ^{\\prime })=\\int _{\\mathbb {R}} dt \\frac{S}{S+e^{-t/\\sigma }} \\operatorname{Ai}(t+\\eta )\\operatorname{Ai}(t+\\eta ^{\\prime }).$ Similar formulas exist for $K_{{\\rm CDRP},b}$ involving Gamma deformed Airy functions [25], [39].",
"Figure: Integration contours for ww (dashed) and zz (solid) of the kernel K CDRP ,b K_{{\\rm CDRP},b}.",
"The black dots are b 1 /σ,...,b m /σb_1/\\sigma ,\\ldots ,b_m/\\sigma .",
"The path 𝒞 z \\mathcal {C}_z equals 1 2σ+𝒞 w \\frac{1}{2\\sigma }+\\mathcal {C}_w, so as to avoid the zeros of the sine function.Remark 1.12 To recover the case of $(m-1)$ -spiked from the $m$ -spiked boundary perturbation case, one needs to take $b_m \\rightarrow -\\infty $ and simultaneously replace $S$ by $-b_m S$ .",
"In Section , Theorem REF we explain how the semi-discrete polymer partition function limits to that of the CDRP.",
"This reduces the proof of the above theorem to a claim on the asymptotics of the Fredholm determinant formula presented in Section REF .",
"We provide a formal critical point derivation of these asymptotics in Section and a rigorous proof later in Section .",
"Remark 1.13 The above Laplace transform can be inverted via a contour integral in $S$ so as to give the probability distribution for the free energy.",
"Note that the branch cut in $S^{(z-w)\\sigma }$ in the integrand in (REF ) should be taken as the negative real axis.",
"For $m=0$ the free energy probability distribution was discovered simultaneously and independently in both [5], [56] and rigorously proved in [5] via Tracy and Widom's ASEP formulas [63], [65], [64], [67]; the above Laplace transform formula was soon after (non-rigorously) derived from the replica trick approach in [19], [27].",
"For $m=1$ the free energy probability distribution was discovered and rigorously proved in [25] also via ASEP [66]; the above Laplace transform formula was later (non-rigorously) derived from the replica trick approach in [39].",
"The general $m\\ge 2$ result above is, to our knowledge, new and it is not clear how one would derive or prove it from ASEP.",
"There are other ways to write the kernel as well as the Fredholm determinant in the theorem, as can be seen in the above mentioned citations.",
"Remark 1.14 It is not necessary to focus just on the free energy at $(T,0)$ .",
"When $m=0$, for $T$ fixed, $\\mathcal {F}(T,X) - \\ln p(T,X)$ is a stationary process in $X$ [5] due to the fact that space-time white noise is statistically invariant under affine shifts.",
"For $m\\ge 1$ , $\\mathcal {F}(T,X)$ is no longer stationary, however, a calculation given in Section REF shows that if we let $\\mathcal {\\tilde{Z}}_0(X)\\stackrel{(d)}{=}\\mathbf {Z}^m(X)\\mathbf {1}_{X\\ge 0}$ for a shifted drift vector $b=(b_1+X/T,\\ldots , b_m+X/T)$ , then $\\mathcal {Z}(T,X) = e^{-\\frac{X^2}{2T}} \\mathcal {\\tilde{Z}}(T,0),$ where $ \\mathcal {\\tilde{Z}}(T,X)$ solves the stochastic heat equation with initial data $\\mathcal {\\tilde{Z}}_0(X)$ .",
"A corollary of the above theorem is the large $T$ asymptotics of the free energy fluctuations for the CDRP with $m$ -spiked boundary perturbation.",
"Corollary 1.15 Fix $m\\ge 0$ and a real vector $b=(b_1,\\ldots , b_m)$ .",
"Consider the free energy of the continuum directed random polymer with $m$ -spiked boundary perturbation with drift vector $\\sigma b$ , with $\\sigma =(2/T)^{1/3}$ (Definition REF ).",
"(a) If $m=0$ , then for any $r\\in \\mathbb {R}$ , $\\lim _{T\\rightarrow \\infty } \\mathbb {P}\\left(\\frac{ \\mathcal {F}(T,0) +T/4!",
"}{(T/2)^{1/3}} \\le r \\right) = F_{{\\rm GUE}}(r),$ where $F_{{\\rm GUE}}$ is the GUE Tracy-Widom distribution [62] (see Definition REF ).",
"(b) If $m\\ge 1$ , then for any $r\\in \\mathbb {R}$ , $\\lim _{T\\rightarrow \\infty } \\mathbb {P}\\left(\\frac{ \\mathcal {F}(T,0) +T/4!",
"}{ (T/2)^{1/3}} \\le r \\right) = F_{{\\rm BBP},b}(r),$ where $F_{{\\rm BBP};b}$ is the Baik-Ben Arous-Péché [8] distribution from spiked random matrix theory (see Definition REF ).",
"For $m=0,1$ the above corollary is proved in [5] and [25] (respectively).",
"Given the new $m\\ge 2$ formulas, it is straightforward to prove the full corollary as is done in Section REF ."
],
[
"Fredholm determinant formula for semi-discrete polymer free energy",
"Theorems REF and REF are proved via asymptotic analysis of a Fredholm determinant formula for the Laplace transform of the O'Connell-Yor semi-discrete directed polymer partition function which we now give as Theorem REF .",
"The formula is written in terms of a Fredholm determinant.",
"One of the surprising aspects of the known exactly solvable positive temperature directed random polymers is that the Laplace transform of their partition functions are given by Fredholm determinants.",
"Here we write the Laplace transform as a double-exponential transform of the free energy.",
"Definition 1.16 For $\\alpha \\in \\mathbb {R}$ and $\\varphi \\in (0,\\pi /4)$ , we define a contour $\\mathcal {C}_{\\alpha ,\\varphi }$ that surrounds the portion of the real axis with values less than $\\alpha $ by $\\mathcal {C}_{\\alpha ,\\varphi }=\\lbrace \\alpha +e^{I(\\pi +\\varphi )}y\\rbrace _{y\\in \\mathbb {R}_{+}}\\cup \\lbrace \\alpha +e^{I(\\pi -\\varphi )}y\\rbrace _{y\\in \\mathbb {R}_{+}}$ .",
"The contour is oriented so as to have increasing imaginary part.",
"For every $v\\in \\mathcal {C}_{\\alpha ,\\varphi }$ we choose $R=-\\operatorname{Re}(v)+\\alpha +1$ , $d>0$ , and define a contour $\\mathcal {D}_{v}$ as follows: $\\mathcal {D}_{v}$ goes by straight lines from $R-I\\infty $ , to $R-Id$ , to $1/2-Id$ , to $1/2+Id$ , to $R+Id$ , to $R+I\\infty $ .",
"The parameter $d$ is taken small enough so that $v+\\mathcal {D}_{v}$ do not intersect $\\mathcal {C}_{\\alpha ,\\varphi }$ .",
"We also call $\\mathcal {D}_{v,\\vert }$ the portion of $\\mathcal {D}_{v}$ with real part $R$ and $\\mathcal {D}_{v,\\sqsubset }$ the remaining part.",
"See Figure REF for an illustration.",
"Figure: Left: The contour 𝒞 α,ϕ \\mathcal {C}_{\\alpha ,\\varphi } (dashed), where the black dots are a 1 ,...,a N a_1,\\ldots ,a_N and α>max{a 1 ,...,a N }\\alpha >\\max \\lbrace a_1,\\ldots ,a_N\\rbrace .",
"The contour v+𝒟 v v+\\mathcal {D}_{v} is the solid line.",
"Right: The contour 𝒟 v \\mathcal {D}_{v}, where the thick part is 𝒟 v,⊏ \\mathcal {D}_{v,\\sqsubset } and the thin part is 𝒟 v,| \\mathcal {D}_{v,\\vert }.",
"The grey dots are at {1,2,...}\\lbrace 1,2,\\ldots \\rbrace Theorem 1.17 For fixed $N\\ge 1$ , $\\tau \\ge 0$ , a drift vector $a=(a_1,\\ldots , a_N)$ , and $\\alpha >\\max \\lbrace a_1,\\ldots ,a_N\\rbrace $, the Laplace transform of the partition function for the O'Connell-Yor semi-discrete directed polymer with drift $a$ is given by the Fredholm determinant formula $\\mathbb {E}\\left[ e^{-u \\mathbf {Z}^{N}(\\tau )}\\right] = \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}$ where $\\mathcal {C}_{\\alpha ,\\varphi }$ is given in Definition REF for any $\\varphi \\in (0,\\pi /4)$ .",
"The operator $K_u$ is defined in terms of its integral kernel $K_u(v,v^{\\prime }) = \\frac{1}{2\\pi I}\\int _{\\mathcal {D}_{v}}ds \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}.$ Theorem 5.2.10 of [16] gives a similar formula for the Laplace transform of the semi-discrete directed polymer partition function.",
"The difference between the two formulas is the contours involved – both the contour of the $L^2$ space and the contour involved in defining the kernel.",
"The contours of the above formula are unbounded.",
"This somewhat technical modification is important since it enables us to perform rigorous steepest descent analysis as necessary to prove Theorems REF and REF .",
"In [16], the corresponding contours were bounded, thus limiting the asymptotic analysis of the semi-discrete polymer to a low temperature regime.",
"In order to prove this modified formula we modify the choice of contours very early in the proof of Theorem 5.2.10 of [16].",
"In Section we provide a proof of the above formula (see in particular the end of Section REF ) with the more detailed and technical pieces of the proof delayed until Section .",
"The occurrence of unbounded contours introduce some new considerations in proving this theorem.",
"It is not presently clear how to derive the above theorem directly from Theorem 5.2.10 of [16]."
],
[
"Acknowledgements",
"We wish to thank the American Institute of Mathematics, since this work originated at the workshop on: The Kardar-Parisi-Zhang equation and universality class.",
"In particular, we appreciate early discussions with Gérard Ben Arous, Tomohiro Sasamoto and Thomas Weiss.",
"We have also benefited from correspondence with Jeremy Quastel, Gregorio Moreno Flores, and Daniel Remenik on their work.",
"We area also grateful to Bálint Vető for careful reading of part of the manuscript.",
"AB was partially supported by the NSF grant DMS-1056390.",
"IC was partially supported by the NSF grant DMS-1208998; as well as the Clay Mathematics Institute through a Clay Research Fellowship and Microsoft Research through the Schramm Memorial Fellowship.",
"PF was supported by the German Research Foundation via the SFB611–A12 project."
],
[
"Free energy fluctuations for the O'Connell-Yor semi-discrete polymer",
"In this section we reduce the proof of Theorems REF to a statement about the asymptotics of a Fredholm determinant (Theorem REF below).",
"We then provide a formal critical point derivation of the asymptotics, delaying the rigorous proof until Section .",
"The starting point for the proof of Theorem REF is the Fredholm determinant formula given in Theorem REF for $\\mathbb {E}[e^{-u\\mathbf {Z}^{N}(t)}]$ .",
"We rely on the fact that under the scalings we consider, the Laplace transform of the partition function converges to the asymptotic probability distribution of the free energy (a similar approach is used in the proof of Corollary REF ).",
"Towards this aim, define a sequence of functions $\\lbrace \\Theta _N\\rbrace _{N\\ge 1}$ by $\\Theta _N(x) = \\exp \\left(-\\exp \\left(c^\\kappa N^{1/3}x\\right)\\right)$ , where $c^\\kappa $ is given in Definition REF .",
"Recall also that we are scaling $\\tau =\\kappa N$ for $\\kappa >0$ fixed.",
"Assume that the drift vector $a=(a_1,\\ldots ,a_N)$ is as specified in the statement of Theorem REF .",
"Set $u=u(N,r,\\kappa )=e^{-Nf^\\kappa - r c^\\kappa N^{1/3}}$ where $f^\\kappa $ is as in Definition REF and observe that $\\mathbb {E}\\left[e^{-u\\mathbf {Z}^{N}(\\kappa N)}\\right] = \\mathbb {E}\\left[\\Theta _N\\left(\\frac{\\mathbf {F}^{N}(\\kappa N) - N f^{\\kappa }}{c^\\kappa N^{1/3}}-r\\right)\\right].$ The $N\\rightarrow \\infty $ asymptotics of the left-hand side of (REF ) can be computed from taking asymptotics of the Fredholm determinant formula given in Theorem REF which states that $\\mathbb {E}\\left[e^{-u\\mathbf {Z}^{N}(\\kappa N)}\\right] = \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}$ for $\\alpha >\\max \\lbrace a_1,\\ldots ,a_N\\rbrace $ and $\\varphi \\in (0,\\pi /4)$ .",
"This asymptotic result is stated as Theorem REF below and proved in Section .",
"After explaining how it implies Theorem REF we provide a formal critical point derivation of the asymptotics.",
"Theorem 2.1 Consider a vector $a=(a_1,\\ldots , a_m,0,\\ldots ,0)$ where $m\\le N$ is fixed and the $m$ non-zero real numbers $a_1,a_2,\\ldots , a_m$ may depend on $N$ .",
"We can consider without loss of generality that $a_1\\ge a_2\\ge \\ldots \\ge a_m> 0$.",
"Then for all $\\kappa >0$ , (a) For the unperturbed case, $m=0$ , $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}=\\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}=F_{{\\rm GUE}}(r),$ where $F_{{\\rm GUE}}$ is the GUE Tracy-Widom distribution [62].",
"(b) If $\\lim _{N\\rightarrow \\infty } N^{1/3}(a_i(N) -\\theta ^\\kappa ) = b_i\\in \\mathbb {R}\\cup \\lbrace -\\infty \\rbrace $ for $i=1,\\ldots ,m$ , then $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}=\\det (\\mathbb {1}-K_{{\\rm BBP},b})_{L^2(r,\\infty )}=F_{{\\rm BBP},b}(r),$ where $F_{{\\rm BBP},b}$ is the Baik-Ben Arous-Péché [8] distribution from spiked random matrix theory.",
"We remind from Definition REF that when $b_i=-\\infty $ , $1\\le i \\le m$ , then $F_{{\\rm BBP},b}=F_{\\rm GUE}$ .",
"The above result implies the the right-hand side of (REF ) has a limit $p(r)$ that is a continuous probability distribution function.",
"Here $p(r)$ is the limiting distribution function in cases (a) or (b) of Theorem REF .",
"The functions $\\Theta _N(x-r)$ approximate $\\mathbf {1}(x\\le r)$ in the sense necessary to apply Lemma REF , and hence $p(r)$ also describes the limiting probability distribution of the free energy: $\\lim _{N\\rightarrow \\infty } \\mathbb {P}\\left( \\frac{ \\mathbf {F}^{N}(\\kappa N) - N f^{\\kappa }}{c^\\kappa N^{1/3}}\\le r\\right) = p(r).$ This implies Theorem REF ."
],
[
"Formal critical point asymptotics for Theorem ",
"We provide a formal analysis of the asymptotics of the Fredholm determinant $\\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}$ .",
"In particular, we only focus on the limit of the kernel $K_u$ , and even in that pursuit, we only consider the pointwise limit of the kernel in (REF ).",
"We also disregard issues respecting the choice of contours.",
"All of these issues are considered in the rigorous proof contained in Section .",
"The first set of manipulations to $K_u$ that we make are to rewrite $\\Gamma (-s)\\Gamma (1+s) = -\\pi / \\sin (\\pi s)$ , and to factor out the ratios of Gamma functions involving those $a_i$ for $1\\le i \\le m$ .",
"With a change of variable $\\tilde{z}=s+v$ , we obtain $K_{u}(v,v^{\\prime }) = \\frac{-1}{2\\pi I}\\int d\\tilde{z}\\, \\frac{\\pi }{\\sin (\\pi (\\tilde{z}-v))}\\frac{\\exp \\left(N G(v)+ r c^\\kappa N^{1/3} v\\right)}{\\exp \\left(N G(\\tilde{z})+ r c^\\kappa N^{1/3} \\tilde{z}\\right)} \\frac{1}{\\tilde{z}-v^{\\prime }} \\prod _{k=1}^{m}\\frac{\\Gamma (v-a_k) \\Gamma (\\tilde{z})}{\\Gamma (\\tilde{z}-a_k)\\Gamma (v)}\\, ,$ where $G(z) = \\ln \\Gamma (z) - \\kappa \\frac{z^2}{2} + f^\\kappa z.$ The problem is now prime for steepest descent analysis of the integral defining the kernel above.",
"The idea of steepest descent is to find critical points for the function in the exponential, and then to deform contours so as to go close to the critical point.",
"The contours should be engineered so that away from the critical point, the real part of the function $G$ in the exponential decays and hence as $N$ gets large, has negligible contribution.",
"This then justifies localizing and rescaling the integration around the critical point.",
"The order of the first non-zero derivative (here third order) determines the rescaling in $N$ (here $N^{1/3}$ ) which in turn corresponds with the scale of the fluctuations in the problem we are solving.",
"It is exactly this third order nature that accounts for the emergence of Airy functions and hence the Tracy Widom (GUE) distribution as well as the BBP transition distributions.",
"The critical point equation for $G$ is given by $G^{\\prime }(z)=0$ with $G^{\\prime }(z) = \\Psi (z) - \\kappa z + f^\\kappa .$ The Digamma function $\\Psi (z)=\\frac{d}{dz}\\ln (\\Gamma (z))$ is given in Definition REF .",
"Also given in that definition is $\\theta ^\\kappa \\in \\mathbb {R}_+$ which is the critical point, i.e., $G^{\\prime }(\\theta ^\\kappa ) = 0$ .",
"At the critical point $G^{\\prime \\prime }(\\theta ^\\kappa )=0$ and $G^{(3)}(\\theta ^\\kappa )=\\Psi ^{\\prime \\prime }(\\theta ^\\kappa )=-2 (c^\\kappa )^3$ so that Taylor expansion at the critical point gives (up to higher order terms) $G(v) \\simeq G(\\theta ^\\kappa )- \\frac{(c^\\kappa )^3}{3} (v-\\theta ^\\kappa )^3,\\quad G(\\tilde{z}) \\simeq G(\\theta ^\\kappa )- \\frac{(c^\\kappa )^3}{3} (\\tilde{z}-\\theta ^\\kappa )^3.$ This cubic behavior suggests rescaling around $\\theta ^\\kappa $ by the change of variables $w = c^\\kappa N^{1/3}(v-\\theta ^\\kappa ), \\qquad w^{\\prime } = c^\\kappa N^{1/3}(v^{\\prime }-\\theta ^\\kappa ), \\qquad z = c^\\kappa N^{1/3}(\\tilde{z}-\\theta ^\\kappa ).$ Under the above change of variables we find that as $N\\rightarrow \\infty $ , $\\frac{\\exp \\left(N G(v)+ r c^\\kappa N^{1/3} v\\right)}{\\exp \\left(N G(\\tilde{z})+ r c^\\kappa N^{1/3} \\tilde{z}\\right)}\\rightarrow \\frac{\\exp \\left(z^3/3-r z\\right)}{\\exp \\left(w^3/3-r w\\right)}\\,.$ Note that since the $v,v^{\\prime }$ variables were scaled, there is a Jacobian factor of $1/(c^\\kappa N^{1/3})$ introduced into the kernel.",
"Grouping this with the reciprocal sine function we see that as $N\\rightarrow \\infty $ , $\\frac{1}{c^\\kappa N^{1/3}}\\frac{\\pi }{\\sin (\\pi (v-\\tilde{z}))} \\rightarrow \\frac{1}{w-z}, \\qquad \\frac{d\\tilde{z}}{\\tilde{z}-v^{\\prime }} \\rightarrow \\frac{dz}{z-w^{\\prime }}\\,.$ It remains to study the ratio of Gamma functions.",
"This is where the subcriticality (a) versus criticality (b) becomes important.",
"First notice that the factor $\\prod _{k=1}^{m}\\Gamma (\\tilde{z})/\\Gamma (v)\\rightarrow 1$ .",
"The fact that the critical value for the $a_i$ 's is $\\theta ^\\kappa $ coincides with the centering of the change of variables is not an accident as we now explain.",
"After the above change of variables $\\frac{\\Gamma (v-a_k)}{\\Gamma (\\tilde{z}-a_k)} = \\frac{\\Gamma (\\theta ^\\kappa -a_k+ w/(c^\\kappa N^{1/3}))}{\\Gamma (\\theta ^\\kappa -a_k+z/(c^\\kappa N^{1/3}))}\\,.$ As long as $\\limsup _{N\\rightarrow \\infty }N^{1/3} (a_k(N)-\\theta ^\\kappa )=-\\infty $ , as $N\\rightarrow \\infty $ , the numerator and denominator both converge to $\\Gamma (\\theta ^\\kappa -a_k)$ and hence their ratio is 1.",
"Thus for the subcritical case (a) the limiting kernel is given by $\\frac{1}{2\\pi I}\\int dz\\, \\frac{1}{(w-z)(z-w^{\\prime })}\\frac{\\exp \\left(z^3/3-r z\\right)}{\\exp \\left(w^3/3-r w\\right)}\\,.$ In the critical case, $\\limsup _{N\\rightarrow \\infty }N^{1/3} (a_k(N)-\\theta ^\\kappa ) = b_k$ for $1\\le k\\le m$ .",
"This means that after the change of variables $\\frac{\\Gamma (v-a_k)}{\\Gamma (\\tilde{z}-a_k)} = \\frac{\\Gamma ((w-b_k)/(c^\\kappa N^{1/3}))}{\\Gamma ((z-b_k)/(c^\\kappa N^{1/3}))}\\simeq \\frac{z-b_k}{w-b_k}$ for large $N$ , since $\\Gamma (z)\\simeq 1/z$ near 0.",
"Therefore, in the critical case (b) the limiting kernel is given by $\\widetilde{K}_{{\\rm BBP},b}(w,w^{\\prime })=\\frac{1}{2\\pi I}\\int dz\\, \\frac{1}{(w-z)(z-w^{\\prime })}\\frac{\\exp \\left(z^3/3-r z\\right)}{\\exp \\left(w^3/3-r w\\right)}\\prod _{k=1}^{m}\\frac{z-b_k}{w-b_k}\\,.$ The variables $w,w^{\\prime }$ are integrated along a contour $\\mathcal {C}$ from $e^{-2\\pi I/3}\\infty $ to $e^{2\\pi I/3}\\infty $ , passing on the right of $b_1,\\ldots ,b_m$ .",
"The variable $z$ is integrated along a contour which goes from $e^{-\\pi I/3}\\infty $ to $e^{\\pi I/3}\\infty $ without crossing $\\mathcal {C}$ .",
"The limiting Fredholm determinants we have derived can be rewritten as Fredholm determinants on $L^2(r,\\infty )$ as shown in Section REF , and hence one sees their equivalence to those given in Definition REF .",
"This completes the formal critical point derivation of Theorem REF .",
"Let us note, that in order to make this formal manipulations into a proof it is necessary (among other things) to be careful about the contours.",
"First one has to find a steep descent pathFor an integral $I=\\int _\\gamma dz\\, e^{t f(z)}$ , we say that $\\gamma $ is a steep descent path if (1) $\\operatorname{Re}(f(z))$ reaches the maximum at some $z_0\\in \\gamma $ : $\\operatorname{Re}(f(z))< \\operatorname{Re}(f(z_0))$ for $z\\in \\gamma \\setminus \\lbrace z_0\\rbrace $ , and (2) $\\operatorname{Re}(f(z))$ is monotone along $\\gamma $ except at its maximum point $z_0$ and, if $\\gamma $ is closed, at a point $z_1$ where the minimum of $\\operatorname{Re}(f)$ is reached.",
"for $G(v)$ , which might not be obvious due to the Digamma function (it turns out a useful representation for the real part of the Digamma function is as an infinite sum, see Section REF ).",
"Secondly, one would like to find a steep descent path for $-G(\\tilde{z})$ , but because the path for $\\tilde{z}$ has to include all the poles at $v+1,v+2,\\ldots $ , such path does not exists.",
"The way out we used was to find a steep descent path for $-G(\\tilde{z})$ and then add the contributions of the poles at the $v+1,\\ldots ,v+\\ell $ which lie on the left of the path (see Figures REF and REF ).",
"Finally, one needs to get estimates so that not only the kernel converges, but also the Fredholm determinant.",
"Further technical details are presented in the proof, see Section ."
],
[
"Laplace transform of the CDRP partition function",
"In this section we reduce the proof of Theorem REF to a statement about the asymptotics of a Fredholm determinant (Theorem REF below).",
"We then provide a formal critical point derivation of the asymptotics, delaying the rigorous proof until Section .",
"We also include two brief calculations delayed from the introduction.",
"The CDRP occurs as limits of discrete and semi-discrete polymers under what has been called intermediate disorder scaling.",
"This means that the inverse temperature should be scaled to zero in a critical way as the system size scales up.",
"For the discrete directed polymer it was observe independently by Calabrese, Le Doussal and Rosso [19] and by Alberts, Khanin and Quastel [2] that if one scaled the inverse temperature to zero at the right rate while diffusively scaling time and space, then the discrete polymer partition function converges to CDRP partition function.",
"Using convergence of discrete to continuum chaos series, [4] provide a proof of this result that is universal with respect to the underlying i.i.d.",
"random variables which form the random media (subject to certain moment conditions).",
"Concerning the semi-discrete directed polymer, [31] prove convergence of the partition function to that of the CDRP.",
"The results of [31] deal with zero drift vector, and in [32] the convergence is extended to deal with a finite number of non-zero drifts, critically tuned so as to result in an boundary perturbation for the CDRP partition function.",
"Theorem 3.1 ([31], [32]) Fix $T>0$ , $X\\in \\mathbb {R}$ , $m\\ge 0$ , and a real vector $b=(b_1,\\ldots ,b_m)$ .",
"Set $\\kappa =\\sqrt{T/N}$ from which $\\tau =\\kappa N =\\sqrt{TN}$ , and $\\theta =\\theta ^\\kappa \\simeq \\sqrt{N/T}+\\frac{1}{2}$ .",
"For each $N\\ge m$ define a drift vector $a=(a_1,\\ldots ,a_m,0,\\ldots ,0)$ where $a_k:=\\theta +b_k$ for $1\\le k\\le m$ .",
"Consider the O'Connell-Yor semi-discrete polymer partition function $\\mathbf {Z}^{N}(\\tau )$ with drift vector $a$ .",
"For $\\sqrt{T N}+X>0$ , define its rescaling as $\\mathcal {Z}^N(T,X)=\\frac{\\mathbf {Z}^{N}(\\sqrt{T N}+X)}{C(N,T,X)}$ with scaling constant $C(N,T,X) = \\exp \\left(N+\\tfrac{1}{2} N\\ln (T/N)+\\tfrac{1}{2}(\\sqrt{T N}+X) + X\\sqrt{N/T}\\right) \\exp \\left(-\\tfrac{1}{2} m\\ln (T/N)\\right).$ Then as $N\\rightarrow \\infty $ , $\\mathcal {Z}^N(T,X)$ converges in distribution to $\\mathcal {Z}(T,X)$ which is the partition function for the continuum directed random polymer with $m$ -spiked boundary perturbation corresponding to drift vector $b$ .",
"Remark 3.2 For $S$ with positive real part, due to the almost sure positivity of $\\mathcal {Z}(T,X)$ , the above weak convergence also implies convergence of Laplace transforms: $\\mathbb {E}\\left[e^{-S \\mathcal {Z}(T,X)}\\right]=\\lim _{N\\rightarrow \\infty }\\mathbb {E}\\left[e^{-S \\mathcal {Z}^N(T,X)}\\right]$ Let us focus on the semi-discrete polymer with drift vector as specified in the above theorem, and $X=0$ .",
"Then, rewrite the CDRP partition function in terms of the free energy, the previous remark implies $\\mathbb {E}\\left[e^{-S \\exp \\left(\\mathcal {F}(T,0) + T/4!\\right)}\\right] = \\lim _{N\\rightarrow \\infty } \\mathbb {E}\\left[e^{-u \\mathbf {Z}^{N}(\\sqrt{TN})}\\right] =\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}$ where $u=S e^{-N-\\frac{1}{2} N \\ln (T/N)-\\frac{1}{2} \\sqrt{TN}+T/4!}",
"e^{\\frac{1}{2} m \\ln (T/N)},$ $\\alpha >\\max \\lbrace a_1,\\ldots ,a_N\\rbrace $ and $\\varphi \\in (0,\\pi /4)$ .",
"This reduces the proof of Theorem REF to the following.",
"Theorem 3.3 Fix $S$ with positive real part, $T>0$ , $m\\ge 0$ , and a real vector $b=(b_1,\\ldots ,b_m)$.",
"Set $\\kappa =\\sqrt{T/N}$ from which $\\tau =\\kappa N =\\sqrt{TN}$ , and $\\theta =\\theta ^\\kappa \\simeq \\sqrt{N/T}+\\frac{1}{2}$ .",
"For each $N\\ge m$ define a drift vector $a=(a_1,\\ldots ,a_m,0,\\ldots ,0)$ where $a_k:=\\theta +b_k$ for $1\\le k\\le m$ , and define $u$ by (REF ).",
"Fix any $\\alpha >\\max \\lbrace a_1,\\ldots ,a_N\\rbrace $ and $\\varphi \\in (0,\\pi /4)$ .",
"Then, recalling the kernel $K_u$ from (REF ), it holds that: (a) In the unperturbed case, $m=0$ , $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ with $K_{{\\rm CDRP}}$ given in Definition REF .",
"(b) For the perturbed case, $m\\ge 1$ , $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}=\\det (\\mathbb {1}-K_{{\\rm CDRP},b})_{L^2(\\mathbb {R}_+)}$ with $K_{{\\rm CDRP},b}$ given in Definition REF .",
"This theorem is proved in Section .",
"However, we now include a formal critical point derivation of the asymptotics."
],
[
"Formal critical point aysmptotics for Theorem ",
"We provide a formal analysis of the asymptotics of the Fredholm determinant $\\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}$ under the prescribed scalings.",
"In particular, we only focus on the limit of the kernel $K_u$ , and even in that pursuit, we only consider the pointwise limit of the kernel in (REF ).",
"We also disregard issues respecting the choice of contours.",
"All of these issues are considered in the rigorous proof contained in Section .",
"Set $\\kappa =\\sqrt{T/N}$ and note that $u= (S/\\theta ^m) e^{-N f^\\kappa +\\mathcal {O}(N^{-1/2})}$ (recall Definition REF ) so that the result is the same if we just set $u=(S/\\theta ^m) e^{-N f^\\kappa }$ .",
"It is convenient to do the change of variable $v\\rightarrow \\theta + \\sigma w$ and $\\tilde{z}\\rightarrow \\theta + \\sigma z$ .",
"In these new variables, the kernel is (up to the approximations) given by $K_\\theta (w,w^{\\prime })=\\frac{-1}{2\\pi I}\\int dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\theta +\\sigma w)-N G(\\theta + \\sigma z)}}{z-w^{\\prime }}\\prod _{k=1}^m\\frac{\\Gamma (\\sigma w-b_k) \\Gamma (\\theta + \\sigma z)\\theta ^{\\sigma w}}{\\Gamma (\\sigma z-b_k)\\Gamma (\\theta + \\sigma w)\\theta ^{\\sigma z}},$ where $G$ is given by (REF ).",
"$G$ has a double critical point at $\\theta $ , as $G^{\\prime }(\\theta )=G^{\\prime \\prime }(\\theta )=0$ .",
"Therefore, using the large $N$ Taylor expansion of $G$ one sees that $N G(\\theta +\\sigma w) \\simeq N G(\\theta ) - \\frac{1}{3} w^3.$ Also, as $\\theta $ is going to infinity with $N$ , it is immediate that $\\Gamma (\\theta + \\sigma z)\\theta ^{\\sigma w}/\\Gamma (\\theta + \\sigma w)\\theta ^{\\sigma z}\\rightarrow 1$ .",
"Thus, as $N\\rightarrow \\infty $ , the kernel $K_{\\theta }$ goes to $\\widetilde{K}_{{\\rm CDRP},b}(w,w^{\\prime })=\\frac{-1}{2\\pi I}\\int dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}\\prod _{k=1}^m\\frac{\\Gamma (\\sigma w-b_k)}{\\Gamma (\\sigma z-b_k)}.$ The variables $w,w^{\\prime }$ are on the contour $\\mathcal {C}_w$ and $z$ on $\\mathcal {C}_z$ of Figure REF .",
"The Fredholm determinant with this kernel can be rewritten as a Fredholm determinant on $L^2(\\mathbb {R}_+)$ as shown in Section REF and one checks that this corresponds with $\\det (\\mathbb {1}-K_{{\\rm CDRP},b})_{L^2(\\mathbb {R}_+)}$ as desired.",
"This completes the brief and formal critical point derivation of Theorem REF .",
"The technical challenges in order to produce a rigorous proof are similar to those explained at the end of Section .",
"There is a new difficulty with regards to contours which arises in this case, however.",
"The variables $w,w^{\\prime },z$ arose from the change of variables of $v,v^{\\prime },\\tilde{z}$ .",
"Besides a shift by $\\theta $ (which goes to infinity when $N$ ), the variables are just scaled by $\\sigma $ , which is of order 1 (here $T$ is fixed).",
"This means that all of the singularities of $1/\\sin (\\pi (\\tilde{z}-v))$ remain in the $N\\rightarrow \\infty $ scaling.",
"This necessitates significant care in the choice of contours along which to take asymptotics.",
"The complete proof of these asymptotics is given in Section ."
],
[
"Affine shifting of the perturbed CDRP partition function",
"The following calculation shows how the claim (for $m\\ge 1$ ) in Remark REF is derived.",
"As explained in Section REF , we can write $\\mathcal {Z}(T,X) = \\int _{-\\infty }^{\\infty } p(T,Y-X) \\mathcal {Z}_0(Y) \\mathbb {E}_{\\begin{array}{c}B(0)=Y\\\\ B(T)=X\\end{array}}\\left[:\\,\\exp \\,: \\left\\lbrace \\int _0^{T} \\dot{{W}}(t,B(t))dt\\right\\rbrace \\right] dY,$ where $\\mathbb {E}_{\\begin{array}{c}B(0)=Y\\\\ B(T)=X\\end{array}}$ denotes the expectation over a Brownian bridge $B$ starting at $B(0)=Y$ and ending at $B(T)=X$ .",
"Let $\\tilde{B}(s)=B(s)-s X/T$ , then observe $\\mathcal {Z}(T,X) = \\int _{-\\infty }^{\\infty } \\frac{p(T,Y)}{p(T,Y)} p(T,Y-X) \\mathcal {Z}_0(Y) \\mathbb {E}_{\\begin{array}{c}\\tilde{B}(0)=Y\\\\ \\tilde{B}(T)=0\\end{array}}\\left[:\\,\\exp \\,: \\left\\lbrace \\int _0^{T} \\dot{{W}}(t,\\tilde{B}(t))dt\\right\\rbrace \\right] dY,$ where the space-time white noise here is the affine shift of the previous one (but, in any case, equal in law).",
"We have also inserted a factor of 1 so that we can now rewrite $\\mathcal {Z}(T,X) = e^{-\\frac{X^2}{2T}} \\int _{-\\infty }^{\\infty } p(T,Y)\\mathcal {Z}_0(Y)e^{XY/T} \\mathbb {E}_{\\begin{array}{c}\\tilde{B}(0)=Y\\\\ \\tilde{B}(T)=0\\end{array}}\\left[:\\,\\exp \\,: \\left\\lbrace \\int _0^{T} \\dot{{W}}(t,\\tilde{B}(t))dt\\right\\rbrace \\right] dY.$ If $\\mathcal {Z}_0(Y) = \\mathbf {Z}^m(Y)\\mathbf {1}_{Y\\ge 0}$ for drift vector $b=(b_1,\\ldots , b_m)$ then $\\mathcal {\\tilde{Z}}_0(Y) = \\mathcal {Z}_0(Y) e^{XY/T}= \\mathbf {Z}^m(Y)\\mathbf {1}_{Y\\ge 0}$ for drift vector $b=(b_1+X/T,\\ldots , b_m+X/T)$ .",
"Letting $\\mathcal {\\tilde{Z}}$ correspond to the solution to the stochastic heat equation with $\\mathcal {\\tilde{Z}}_0(Y)$ initial data, we find that $\\mathcal {Z}(T,X) = e^{-\\frac{X^2}{2T}} \\mathcal {\\tilde{Z}}(T,0).$ Thus, after a parabolic shift, and an addition of drift into the boundary perturbation, the distribution of the general $X$ free energy can also be determined from Theorem REF as well."
],
[
"Proof of Corollary ",
"Recall $\\sigma = (2/T)^{1/3}$ .",
"Let us focus on $m\\ge 1$ and note that $m=0$ is proved identically.",
"Define a sequence of function $\\lbrace \\Theta _T\\rbrace _{T\\ge 0}$ by $\\Theta _T(x) = e^{-e^{x/\\sigma }}$ .",
"Then if we set $S=e^{-r/\\sigma }$ $\\mathbb {E}\\left[e^{-S \\exp \\left(\\mathcal {F}(T,0) + T/4!\\right)}\\right]=\\mathbb {E}\\left[\\Theta _{T}\\left(\\frac{ \\mathcal {F}(T,0) +T/4!",
"}{ \\sigma ^{-1}}-r\\right)\\right].$ Let us first calculating the $T\\rightarrow \\infty $ limit of the Fredholm determinant expression for the left-hand side of the above equality.",
"It is easy to see that in this limit $K_{{\\rm CDRP},\\sigma b}(\\eta ,\\eta ^{\\prime }) \\rightarrow K_{{\\rm BBP},b}(\\eta +r,\\eta ^{\\prime }+r).$ This is because $T\\rightarrow \\infty $ corresponds to $\\sigma \\rightarrow 0$ and thus $\\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\rightarrow \\frac{e^{-r(z-w)}}{z-w},\\quad \\frac{\\Gamma (\\sigma (w-b_k)) }{\\Gamma (\\sigma (z-b_k))} \\rightarrow \\frac{z-b_k}{w-b_k}.$ One readily identifies the resulting expression with that of Definition REF .",
"The tail bounds necessary to justify this are not hard (see [5], [25] for instance).",
"Going back to (REF ), this implies that $\\lim _{T\\rightarrow \\infty } \\mathbb {E}\\left[\\Theta _{T}\\left(\\frac{ \\mathcal {F}(T,0) +T/4!",
"}{ \\sigma ^{-1}}-r\\right)\\right] = F_{{\\rm BBP},b}(r).$ Since (away from $x=r$ ) $\\Theta _{T}(x-r)$ is converging to $\\mathbf {1}_{x\\le r}$ (and in particular due to Lemma REF ), $\\lim _{T\\rightarrow \\infty } \\mathbb {E}\\left[\\Theta _{T}\\left(\\frac{ \\mathcal {F}(T,0) +T/4!",
"}{ \\sigma ^{-1}}-r\\right)\\right]= \\lim _{T\\rightarrow \\infty } \\mathbb {P}\\left(\\frac{ \\mathcal {F}(T,0) +T/4!",
"}{\\sigma ^{-1}} \\le r \\right)$ and hence the corollary is proved."
],
[
"Proof of Theorem ",
"The proof of Theorem REF follows closely the proof of Theorem 5.2.10 in [16].",
"The major difference is that the formula resulting from Theorem 5.2.10 had bounded contours which were unsuitable for the full scope of asymptotic analysis necessary to prove Theorems REF and REF .",
"This somewhat minor modification to the final formula requires us to modify the proof quite early on and in fact the unboundedness of contours results in a fair number of new technical steps in the proof.",
"We produce in this section the complete proof.",
"Results used along the way which are stated and proved in [16] are not reproved.",
"Some of the more technical points in the proof are delayed until Section .",
"To prove Theorem REF we use the theory of Macdonald processes as developed in [16].",
"As we explain below in Section REF , due to O'Connell's work [51] on a continuum version of tropical RSK correspondence, the partition function $\\mathbf {Z}^N(t)$ arises as a marginal of the Whittaker process (or measure).",
"Macdonald processes sit above Whittaker processes due to the hierarchy of symmetric functions.",
"Under suitable scaling, Macdonald processes converge weakly to Whittaker processes (and hence a suitable marginal converges to $\\mathbf {Z}^N(t)$ ).",
"Due to the Macdonald difference operators and the Macdonald version of the Cauchy identity it is possible to compute simple formulas for expectations of a large class of observables of Macdonald processes.",
"In particular it is possible to compute $q$ -moments for the marginal random variable which converges to $\\mathbf {Z}^N(t)$ .",
"A $q$ -Laplace transform can be rigorously computed by taking an appropriate generating function of the $q$ -moments, and switching the expectations and summation is rigorously justified.",
"For the $q$ -Laplace transform we find a nice Fredholm determinant.",
"Taking the degeneration of Macdonald processes to Whittaker processes, the $q$ -Laplace transform becomes the usual Laplace transform, and due to the weak convergence, we recover the desired formula for the Laplace transform of $\\mathbf {Z}^N(t)$ .",
"We do not provide an introduction to the theory of Macdonald symmetric functions here.",
"Instead we refer readers to Section 2.1 of [16] for all of the relevant details, or to Chapter VI of [47].",
"We also do not define Macdonald processes in their full generality but content ourselves with studying a certain set of marginals of the processes which are called Macdonald measures, and a certain Plancherel specialization."
],
[
"O'Connell's Whittaker measure and its relation to polymers",
"In order to state the pre-asymptotic Laplace transform formula and set up its derivation, it is useful to introduce a few concepts.",
"Initially they may seem a little out of place, but due to O'Connell's work [51] (some of which is recorded in Theorem REF ) the connection to the semi-discrete polymer becomes clear.",
"This connection is analogous to the relation between last passage percolation and the Schur process (see e.g. [43]).",
"The class-one $\\mathfrak {gl}_{N}$ -Whittaker functions are basic objects of representation theory and integrable systems [45], [28].",
"One of their properties is that they are eigenfunctions for the quantum $\\mathfrak {gl}_{N}$ -Toda chain.",
"As showed by Givental [36], they can also be defined via the following integral representation $\\psi _{\\lambda }(x_{N,1},\\ldots ,x_{N,N})=\\int _{\\mathbb {R}^{N(N-1)/2}} e^{\\mathcal {F}_\\lambda (x)} \\prod _{k=1}^{N-1}\\prod _{i=1}^k dx_{k,i},$ where $\\lambda =(\\lambda _1,\\ldots ,\\lambda _N)$ and $\\mathcal {F}_{\\lambda }(x)=I\\sum _{k=1}^{N} \\lambda _k\\left(\\sum _{i=1}^k x_{k,i}-\\sum _{i=1}^{k-1} x_{k-1,i}\\right)-\\sum _{k=1}^{N-1}\\sum _{i=1}^k \\left(e^{x_{k,i}-x_{k+1,i}}+e^{x_{k+1,i+1}-x_{k,i}}\\right).$ For any $\\tau >0$ set $\\theta _{\\tau }(x_1,\\ldots ,x_N)=\\int _{\\mathbb {R}^N} \\psi _{\\nu }(x_1,\\ldots ,x_{N}) e^{-\\tau \\sum _{j=1}^N\\nu _j^2/2} m_N(\\nu )\\prod _{j=1}^{N} d\\nu _j$ with the Skylanin measure $m_{N}(\\nu )=\\frac{1}{(2\\pi )^{N} (N)!",
"}\\prod _{j\\ne k} \\frac{1}{\\Gamma (I\\nu _k-I\\nu _j)}\\,.$ Note that our definition of Whittaker functions differs by factors of $I$ from those considered by O'Connell [51].",
"Definition 4.1 For $N\\ge 1$ , define the Whittaker measure with respect to a vector $a=(a_1,\\ldots , a_N)\\in \\mathbb {R}^N$ via the density function (with respect to Lebesgue) given by $\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}\\big (\\lbrace T_{i}\\rbrace _{1\\le i\\le N}\\big )=e^{-\\tau \\sum _{j=1}^N a_j^2/2}\\psi _{\\iota a}(T_{1},\\dots ,T_{N})\\,{\\theta _{\\tau }(T_{1},\\dots ,T_{N})}.$ The fact that this measure integrates to one follows from analytic continuation of the orthogonality relation for Whittaker functions (see [16] Proposition 4.1.17).",
"We write expectations with respect to Whittaker measures as $\\langle \\cdot \\rangle _{\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}}$ .",
"Let us remark that in [51], [16] the Whittaker measure is a level $N$ marginal of a measure on triangular arrays called the Whittaker process, which is also defined via Whittaker functions, but whose precise definition will not be important for us in this work.",
"The connection between the Whittaker measure and the semi-discrete polymer is best explained by introducing an extension to the polymer partition function and free energy.",
"Definition 4.2 The hierarchy of partition functions $\\mathbf {Z}^{N}_{n}(\\tau )$ for $0\\le n\\le N$ is defined so that $\\mathbf {Z}^{N}_{0}(\\tau )=1$ and for $n\\ge 1$ , $\\mathbf {Z}^{N}_{n}(\\tau ) = \\int _{D^N_{n}(\\tau )} e^{\\sum _{i=1}^{n} E(\\phi _i)} d\\phi _1 \\cdots d\\phi _n,$ where the integral is with respect to the Lebesgue measure on the Euclidean set $D^N_n(\\tau )$ of all $n$ -tuples of non-intersecting (disjoint) up/right paths with initial points $(0,1),\\ldots , (0,n)$ and endpoints $(\\tau ,N-n+1),\\ldots , (\\tau ,N)$ .",
"For a path $\\phi _i$ which starts at $(0,i)$ , ends at $(\\tau ,N-n+i)$ and jumps between levels $j$ and $j+1$ at times $s_j$ , the energy $E(\\phi _i)$ is defined as $E(\\phi _i) = B_i(s_i)+\\left(B_{i+1}(s_{i+1})-B_{i+1}(s_i)\\right)+ \\cdots + \\left(B_N(\\tau ) - B_{N}(s_{N-1})\\right),$ with $B_1,\\ldots ,B_N$ independent Brownian motions.",
"It follows that $\\mathbf {Z}^{N}_{1}(\\tau ) = \\mathbf {Z}^{N}(\\tau )$ as defined in (REF ).",
"The hierarchy of free energies $\\mathbf {F}^{N}_{n}(\\tau )$ for $1\\le n\\le N$ is defined via $\\mathbf {F}^{N}_{n}(\\tau ) = \\ln \\left(\\frac{ \\mathbf {Z}^{N}_{n}(\\tau )}{ \\mathbf {Z}^{N}_{n-1}(\\tau )}\\right).$ It follows that $\\mathbf {F}^{N}_{1}(\\tau ) = \\mathbf {F}^{N}(\\tau )$ as defined in (REF ).",
"The following result is shown in [51] by utilizing a continuous version of the tropical RSK correspondence (see also [24] for a discrete analog) and certain Markov kernel intertwining relations.",
"Theorem 4.3 Fix $N\\ge 1$ , $\\tau \\ge 0$ and a vector of drifts $a=(a_1,\\ldots , a_N)$ , then $\\left\\lbrace \\mathbf {F}^{N}_n(\\tau )\\right\\rbrace _{1\\le n\\le N}$ is distributed according to the Whittaker measure $\\mathbf {WM}_{(-a_1,\\ldots , -a_N;\\tau )}$ of (REF ).",
"More is shown in [51] including the fact that the collection of free energies evolves as a Markov diffusion with infinitesimal generator given in terms of the quantum $\\mathfrak {gl}_N$ Toda lattice Hamiltonian; as well as the fact that the entire triangular array $\\mathbf {F}(\\tau )$ is distributed according to the Whittaker process which we briefly mentioned earlier.",
"Remark 4.4 It is useful to note the following symmetry: The transformation $T_{i}\\leftrightarrow -T_{N+1-i}$ maps $\\mathbf {WM}_{(-a_1,\\ldots , -a_N;\\tau )}$ to $\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}$ (the sign of $a_j$ 's changes).",
"This easily follows from the definition of $\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}$ .",
"The result below is similar to the formula found in Theorem 4.1.40 of [16] except that the contours have been modified.",
"This technical change requires us to go through the proof of this theorem and perform a number of estimates which are harder than in [16], due to the unboundedness of the contours.",
"Theorem 4.5 Fix $N\\ge 1$ , $\\tau >0$ and a vector $a=(a_1,\\ldots , a_N)\\in \\mathbb {R}^N$ and $\\alpha >\\max \\lbrace a_i\\rbrace $ .",
"Then for all $u\\in with positive real part\\begin{equation*}\\left\\langle e^{-u e^{-T_{N}}} \\right\\rangle _{\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}} = \\det (\\mathbb {1}+ K_{u})_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}\\end{equation*}where the operator $ Ku$ is defined in terms of its integral kernel\\begin{equation}K_{u}(v,v^{\\prime }) = \\frac{1}{2\\pi I}\\int _{\\mathcal {D}_{v}}ds \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}.\\end{equation}The paths $ C,$ and $ Dv$ are as in Definition~\\ref {CaCsdef}, where $$ is any angle in $ (0,/4)$.$ This theorem is proved over the course of the rest of this section, with the more technical aspects of the proof relegated to the later Section .",
"Before going into this, let us note that in view Theorem REF , the proof of Theorem REF is now immediate.",
"By appealing to Theorem REF and Remark REF we relate $\\mathbf {Z}^{N}(\\tau )$ to $T_{N}$ as $\\left\\langle e^{-u e^{-T_{N}}} \\right\\rangle _{\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}} = \\mathbb {E}\\left[ e^{-u \\mathbf {Z}^{N}(\\tau )}\\right].$ Applying Theorem REF gives the claimed Laplace transform and completes the proof."
],
[
"Macdonald measures",
"We now turn to the proof of Theorem REF .",
"Before giving the proof (as we do in Section REF ) we need to recall and develop a variation on the solvability framework of Macdonald measures [16].",
"The Macdonald measure is defined as (following the notation of [16] where it is introduced and studied) $\\mathbf {MM}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )(\\lambda )= \\frac{P_{\\lambda }(\\tilde{a}_1,\\ldots ,\\tilde{a}_N) Q_{\\lambda }(\\rho )}{\\Pi (\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )}\\,.$ Here $\\lambda $ is a partition of length $\\ell (\\lambda )\\le N$ (i.e., $\\lambda =(\\lambda _1\\ge \\lambda _2\\ge \\cdots \\lambda _N\\ge 0)$ ) and $P_{\\lambda }$ and $Q_\\lambda $ are Macdonald symmetric functions which are defined with respect to two additional parametersThe parameter $t$ does not correspond to time in any of the processes we have considered.",
"For time we have either used $\\tau $ in the semi-discrete polymer or $T$ in the CDRP $q,t\\in [0,1)$ .",
"The notation $P_{\\lambda }(\\tilde{a}_1,\\ldots ,\\tilde{a}_N)$ means to specialize the symmetric function to an $N$ variable symmetric polynomial, and then evaluate those $N$ variables at the $\\tilde{a}_i$ 'sWe use tildes presently since after a limit transition of Macdonald measures to Whittaker measures the $\\tilde{a}_i$ 's will become $a_i$ 's.",
"The notation $Q_{\\lambda }(\\rho )$ means to apply a specialization $\\rho $ to $Q_{\\lambda }$ .",
"A specialization of a symmetric function $f$ in the space $\\mathrm {Sym}$ of symmetric functions of an infinite number of indeterminants is an algebra homomorphism $\\rho :\\mathrm {Sym}\\rightarrow , and its application to $ f$ is written as $ f()$.",
"We will focus here on a single class of {\\it Plancherel} specializations $$ which are defined with respect to a parameter $ >0$ via the relation\\begin{equation}\\sum _{n\\ge 0} g_n(\\rho ) u^n= \\exp (\\gamma u) =: \\Pi (u;\\rho ).\\end{equation}Here $ u$ is a formal variable and $ gn=Q(n)$ is the $ (q,t)$-analog of the complete homogeneous symmetric function $ hn$.",
"The above generating function in $ u$ is denoted $ (u;)$.",
"Since $ gn$ forms a $$\\mathbb {Q}$ [q,t]$ algebraic basis of $$\\mathrm {Sym}$$, this uniquely defines the specialization $$.",
"The Plancherel specialization has the property that $ Q()0$ for all partitions $$.$ On account of the Cauchy identity for Macdonald polynomials $\\sum _{\\lambda :\\ell (\\lambda )\\le N} P_{\\lambda }(\\tilde{a}_1,\\ldots ,\\tilde{a}_N) Q_{\\lambda }(\\rho ) =: \\Pi (\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho ) = \\Pi (\\tilde{a}_1;\\rho )\\cdots \\Pi (\\tilde{a}_N;\\rho ).$ Our ability to compute expectations of observables for Macdonald measures comes from the following idea: Assume we have a linear operator $\\mathcal {D}$ on the space of functions in $N$ variables whose restriction to the space of symmetric polynomials diagonalizes in the basis of Macdonald polynomials: $\\mathcal {D} P_\\lambda =d_\\lambda P_\\lambda $ for any partition $\\lambda $ with $\\ell (\\lambda )\\le N$ .",
"Then we can apply $\\mathcal {D}$ to both sides of the identity $\\sum _{\\lambda :\\ell (\\lambda )\\le N} P_{\\lambda }(\\tilde{a}_1,\\dots ,\\tilde{a}_N) Q_{\\lambda }(\\rho )=\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho ).$ Dividing the result by $\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )$ we obtain $\\langle d_\\lambda \\rangle _{\\mathbf {MM}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}=\\frac{\\mathcal {D}\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}{\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}\\,,$ where $\\langle \\cdot \\rangle _{\\mathbf {MM}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ represents averaging $\\cdot $ over the specified Macdonald measure.",
"If we apply $\\mathcal {D}$ several times we obtain $\\langle d_\\lambda ^k \\rangle _{\\mathbf {MM}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}=\\frac{\\mathcal {D}^k \\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}{\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}\\,.$ If we have several possibilities for $\\mathcal {D}$ we can obtain formulas for averages of the observables equal to products of powers of the corresponding eigenvalues.",
"One of the remarkable features of Macdonald polynomials is that there exists a large family of such operators for which they form the eigenbasis (and this fact can be used to define the polynomials).",
"These are the Macdonald difference operators.",
"We need only consider the first of these operators $D_N^1$ which acts on functions $f$ of $N$ independent variables as $(D_N^1 f)(x_1,\\ldots , x_N) = \\sum _{i=1}^{N} f(x_1,\\ldots ,x_{i-1},qx_{i},x_{i+1},\\ldots ,x_N) \\prod _{j\\ne i} \\frac{ tx_i-x_j}{x_i-x_j}.$ This difference operator takes symmetric polynomials to symmetric polynomials and acts diagonally on Macdonald polynomials.",
"Proposition 4.6 (VI(4.15) of [47]) For any partition $\\lambda =(\\lambda _1\\ge \\lambda _2\\ge \\cdots )$ with $\\lambda _m=0$ for $m>N$ $D_N^1 P_\\lambda (x_1,\\cdots ,x_N)=\\left(q^{\\lambda _1}t^{N-1}+q^{\\lambda _2}t^{N-2}+\\cdots +q^{\\lambda _N}\\right) P_\\lambda (x_1,\\cdots ,x_N).$ Although the operator $D_N^1$ does not look particularly simple, it can be represented by contour integrals by properly encoding the shifts in terms of residues.",
"Proposition 4.7 (Proposition 2.2.13 of [16]) Fix $k\\ge 1$ .",
"Assume that $F(u_1,\\dots ,u_N)=f(u_1)\\cdots f(u_N)$ .",
"Take $x_1,\\dots ,x_N >0$ and assume that $f(u)$ holomorphic and nonzero in a complex neighborhood of an interval in $\\mathbb {R}$ that contains $\\lbrace q^ix_j\\mid i=0,\\ldots ,k,j=1\\ldots ,N\\rbrace $ .",
"Then $\\frac{\\bigl ({(D_n^1)}^k F\\bigr )(x)}{F(x)}=\\frac{(t-1)^{-k}}{(2\\pi I)^k} \\oint \\!",
"\\cdots \\!",
"\\oint \\!",
"\\!",
"\\prod _{1\\le a<b\\le k} \\!",
"\\frac{(tz_a-qz_b)(z_a-z_b)}{(z_a-qz_b)(tz_a-z_b)} \\prod _{c=1}^k\\!",
"\\left(\\prod _{m=1}^N \\frac{(tz_c-x_m)}{(z_c-x_m)}\\right)\\!\\frac{f(qz_c)}{f(z_c)}\\frac{dz_c}{z_c},$ where the $z_c$ -contour contains $\\lbrace qz_{c+1},\\ldots ,qz_k,x_1,\\ldots ,x_N\\rbrace $ and no other singularities for $c=1,\\dots ,k$.",
"Observe that $\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )=\\prod _{i=1}^N \\Pi (\\tilde{a}_i;\\rho )$ .",
"Hence, Proposition REF is suitable for evaluating the right-hand side of equation (REF ) and hence computing the associated observable of the Macdonald process."
],
[
"The emergence of a Fredholm determinant",
"Macdonald polynomials in $N$ variables with $t=0$ are also known of as $q$ -deformed $\\mathfrak {gl}_{N}$ Whittaker functions [35].",
"We now denote the Macdonald measure as $\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )$ and refer to these as $q$ -Whittaker measures.",
"The partition function for the corresponding $q$ -Whittaker measure $\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )$ when $\\rho $ is Plancherel simplifies as $\\sum _{\\lambda \\in \\mathbb {Y}(N)} P_\\lambda (\\tilde{a}_1,\\dots ,\\tilde{a}_N)Q_\\lambda (\\rho )=\\Pi (\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )=\\prod _{j=1}^N \\exp (\\gamma \\tilde{a}_j).$"
],
[
"Formulas for q-moments",
"Let us take the limit $t\\rightarrow 0$ of Proposition REF .",
"Write $\\mu _k=\\left\\langle q^{k\\lambda _N}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}.$ Then $\\mu _k= \\frac{(-1)^k q^{\\frac{k(k-1)}{2}}}{(2\\pi I)^k} \\oint \\cdots \\oint \\prod _{1\\le A<B\\le k} \\frac{z_A-z_B}{z_A-qz_B} \\frac{f(z_1)\\cdots f(z_k)}{z_1\\cdots z_k} dz_1\\cdots dz_k,$ where $f(z) = \\prod _{i=1}^{N}\\frac{\\tilde{a}_i}{\\tilde{a}_i-z}\\exp \\lbrace (q-1)\\gamma z\\rbrace $ and where the $z_j$ -contours contain $\\lbrace q z_{j+1},\\ldots , q z_k\\rbrace $ as well as $\\lbrace \\tilde{a}_1,\\ldots , \\tilde{a}_N\\rbrace $ but not 0.",
"For example when $k=2$ and all $\\tilde{a}_i\\equiv 1$ , $z_2$ can be integrated along a small contours around 1, and $z_1$ is integrated around a slightly larger contour which includes 1 and the image of $q$ times the $z_2$ contour.",
"We have encountered the point at which we diverge from [16].",
"In particular, we will now deform our contours from bounded curves to infinite contours (as justified by Cauchy's theorem).",
"This may not seem so significant presently, but in the final formula this frees us up to deform contours to steepest descent contours and hence to rigorously prove the various asymptotics desired for our limit theorems.",
"However, the unboundedness of contours introduces new complications which we must address.",
"First, however, let us define two important unbounded contours we will soon encounter.",
"Definition 4.8 For an illustration, see Figure REF .",
"For any $\\tilde{\\alpha }>0$ and $\\varphi \\in (0,\\pi /4)$ define $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }=\\lbrace \\tilde{\\alpha }+e^{-I\\varphi \\operatorname{sgn}(y)}y,y\\in \\mathbb {R}\\rbrace $ (note that it is oriented so as to have decreasing imaginary part).",
"For $w\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ for some choice of parameters, define a contour $\\widetilde{\\mathcal {D}}_{w}$ in the same way as $\\mathcal {D}_{w}$ of Definition REF but with $R$ and $d$ taken such that the following holds: For all $s\\in \\widetilde{\\mathcal {D}}_{w}$ (i) if $b=\\frac{\\pi }{4}-\\frac{\\varphi }{2}$ , then $\\arg (w(q^s-1))\\in (\\pi /2+b,3\\pi /2-b)$ ; (ii) $q^s w$ lies to the left of $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"Remark 4.9 Let us check that the contours $\\widetilde{\\mathcal {D}}_{w}$ in above definition actually exist.",
"Fix $\\varphi \\in (0,\\pi /4)$ and fix a contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ as above.",
"For $w\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ we must show that there exists $R$ and $d$ as desired to satisfy (i) and (ii).",
"The existence of these contains is clear for $|w|$ small, so let us consider when $|w|$ is sufficiently large.",
"Then the argument of $w$ is roughly $\\varphi $ (actually, $\\pm \\varphi $ but let us focus on $w$ with positive imaginary part).",
"Then, for $R>R_0$ large enough and $d<d_0$ small enough (though positive) it follows from basic geometry that the argument of $w(q^s-1)$ can be bounded in $(\\pi /2+b,3\\pi /2-b)$ .",
"In order that $q^s w$ avoid $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"it suffices that $d$ be less than a constant times $|w|^{-1}$ and that $|w|q^R$ be small (but still of order 1).",
"This implies that $R$ can be chosen roughly as $-\\ln _q|w|$ .",
"Figure: (a) ww-contour 𝒞 ˜ α ˜,ϕ \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }.",
"The black dots are a ˜ 1 ,...,a ˜ N \\tilde{a}_1,\\ldots ,\\tilde{a}_N; (b) ss-contour 𝒟 ˜ w \\widetilde{\\mathcal {D}}_{w} which depends on the value of w∈𝒞 ˜ α ˜,ϕ w\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }; (c) The contour 𝒟 ˜ w \\widetilde{\\mathcal {D}}_{w} is chosen so that q s wq^sw sits entirely to the left of 𝒞 ˜ α ˜,ϕ \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi } as is demonstrated here; (d) The image of q s q^s for s∈𝒟 ˜ w s\\in \\widetilde{\\mathcal {D}}_{w}.",
"The value of dd and RR are also chosen so that the argument of w(q s -1)w(q^s-1) is contained in (π/2+b,3π/2-b)(\\pi /2+b,3\\pi /2-b).",
"This amounts to making sure that the angle q s -1q^s-1 makes with the negative real axis is within (-b,b)(-b,b).For the moment we will only make use of the contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"Lemma 4.10 Fix $k\\ge 1$ and consider positive $\\tilde{a}_1,\\ldots , \\tilde{a}_N$ .",
"Then, for any $\\varphi \\in (0,\\pi /4)$ , $\\mu _k= \\frac{(-1)^k q^{\\frac{k(k-1)}{2}}}{(2\\pi I)^k} \\int \\cdots \\int \\prod _{1\\le A<B\\le k} \\frac{z_A-z_B}{z_A-qz_B} \\frac{f(z_1)\\cdots f(z_k)}{z_1\\cdots z_k} dz_1\\cdots dz_k,$ with $f(z) = \\exp ((q-1)\\gamma z)\\prod _{i=1}^{N}\\frac{\\tilde{a}_i}{\\tilde{a}_i-z}$ and the $z_j$ -contours given by $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha }_j,\\varphi }$ where $\\lbrace \\tilde{\\alpha }_j\\rbrace _{j=1}^{N}$ are positive real numbers such that $\\tilde{\\alpha }_j<q\\tilde{\\alpha }_{j+1}$ for all $1\\le j\\le N-1$ and $\\tilde{\\alpha }_N<\\min _i\\lbrace \\tilde{a}_i\\rbrace $.",
"The formula in (REF ) for $\\mu _k$ involves closed contours for the $z_j$ which can be chosen to be concentric circles centered on the real axis and with their left crossing point with the real axis a $\\tilde{\\alpha }_j$ (the $z_A$ contours containing $q$ times the $z_B$ contour for all $A<B$ as well as the poles at the $\\tilde{a}_i$ 's).",
"We now proceed to expand these contours to the infinite contours $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha }_j,\\varphi }$ , one at a time.",
"We start with $z_1$ .",
"We may freely deform $z_1$ to a pie slice shape with radius $r\\gg 1$ made up of the portion of $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha }_1,\\varphi }$ with distance from $\\alpha _1$ less than $r$ , and then a circular arc through the real axis: precisely the contour is $\\lbrace \\tilde{\\alpha }_1+e^{I\\varphi }t\\rbrace _{0\\le t\\le r}\\cup \\lbrace \\tilde{\\alpha }_1+e^{-I\\varphi }t\\rbrace _{0\\le t\\le r}\\cup \\lbrace \\tilde{\\alpha }+ e^{I\\sigma }r\\rbrace _{-\\varphi \\le \\sigma \\le \\varphi }$ .",
"We would like to replace this by the infinite contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha }_1,\\varphi }$ .",
"To justify this observe that for $z$ with positive real part, $f(z)$ decays exponentially with respect to the real part of $z$ .",
"Since $\\varphi \\in (0,\\pi /4)$ , as $r$ goes to infinity, the contributions of the contour integral away from the origin along both the pie slice and the limiting contour become negligible, and hence the replacement is justified.",
"This procedure can be repeated for $z_2$ and so on until the contours are as claimed in the lemma.",
"Note that as the original contours were positively oriented circles, and the expanded sections were on the left of the circles, the contours are now oriented so as to have decreasing imaginary part.",
"In order to combine the $q$ -moments $\\mu _k$ into a Fredholm determinant it is useful to have all of the contours coincide.",
"This, however, involves keeping track of the residues which result from such a deformation.",
"First, define the $q$ -Pochhammer symbol and $q$ -factorial as $(a;q)_{n} = \\prod _{i=0}^{n-1} (1-q^i a), \\qquad k_q!",
":= \\frac{(q;q)_{k}}{(1-q)^k}.$ Proposition 4.11 Fix $k\\ge 1$ and consider a meromorphic function $f(z)$ with $N$ poles $\\tilde{a}_{1},\\ldots ,\\tilde{a}_N$ with positive real part.",
"Then setting $\\mu _k=\\frac{(-1)^k q^{\\frac{k(k-1)}{2}}}{(2\\pi I)^k} \\int \\cdots \\int \\prod _{1\\le A<B\\le k} \\frac{z_A-z_B}{z_A-qz_B} \\frac{f(z_1)\\cdots f(z_k)}{z_1\\cdots z_k} dz_1\\cdots dz_k,$ we have $\\begin{aligned}\\mu _k =& k_q!",
"\\sum _{\\begin{array}{c}\\lambda \\vdash k\\\\ \\lambda =1^{m_1}2^{m_{2}}\\cdots \\end{array}} \\frac{1}{m_1!m_2!\\cdots } \\\\& \\times \\frac{(1-q)^{k}}{(2\\pi I)^{\\ell (\\lambda )}} \\int \\cdots \\int \\det \\left[\\frac{1}{w_i q^{\\lambda _i}-w_j}\\right]_{i,j=1}^{\\ell (\\lambda )} \\prod _{j=1}^{\\ell (\\lambda )} f(w_j)f(qw_j)\\cdots f(q^{\\lambda _j-1}w_j) dw_j,\\end{aligned}$ with the $z_j$ -contours given by $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha }_j,\\varphi }$ where $\\lbrace \\tilde{\\alpha }_j\\rbrace _{j=1}^{k}$ are positive real numbers such that $\\tilde{\\alpha }_j<q\\tilde{\\alpha }_{j+1}$ for all $1\\le j\\le k-1$ and $\\tilde{\\alpha }_k<\\min _i\\lbrace \\tilde{a}_i\\rbrace $ , and where the $w_j$ -contours are all the same and given by $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ for any $\\tilde{\\alpha }\\in (0,\\min _i\\lbrace \\tilde{a}_i\\rbrace )$ .",
"The notation $\\lambda \\vdash k$ above means that $\\lambda $ partitions $k$ (i.e., if $\\lambda =(\\lambda _1,\\lambda _2,\\ldots )$ then $k=\\sum \\lambda _i$ ), and the notation $\\lambda = 1^{m_1}2^{m_2}\\cdots $ means that $i$ shows up $m_i$ times in the partition $\\lambda $ .",
"Word-for-word repetition of that for Proposition 3.2.1 of [16]."
],
[
"A first Fredholm determinant",
"Before stating our first Fredholm determinant in this derivation, let us recall how such determinants are defined.",
"Fix a Hilbert space $L^2(X,\\mu )$ where $X$ is a measure space and $\\mu $ is a measure on $X$ .",
"When $X=\\mathcal {C}$ is a simple smooth contour in $, we write $ L2(C)$ where $$ is understood to be the path measure along $ C$ divided by $ 2I$.",
"When $ X$ is the product of a discrete set $ D$ and a contour $ C$, $$ is understood to be the product of the counting measure on $ D$ and the path measure along $ C$ divided by $ 2I$.",
"Let $ K$ be an {\\it integral operator} acting on $ f()L2(X,)$ by $ Kf(x) = X K(x,y)f(y) d(y)$.",
"$ K(x,y)$ is called the {\\it kernel} of $ K$.",
"One way of defining the Fredholm determinant of $ K$, for trace class operators $ K$, is via the Fredholm series\\begin{equation}\\det (\\mathbb {1}+K)_{L^2(X,\\mu )} = 1+\\sum _{n=1}^{\\infty } \\frac{1}{n!}",
"\\int _{X} \\cdots \\int _{X} \\det \\left[K(x_i,x_j)\\right]_{i,j=1}^{n} \\prod _{i=1}^{n} d\\mu (x_i).\\end{equation}In fact, if an operator $ K$ is such that the above right-hand side series is absolutely convergent, then we still write it as $ (1+K)$ as short-hand even if $ K$ is not trace class.",
"This is sufficient for our purposes as we will deal directly with the expansion.",
"We will often write only $ (1+K)L2(X)$ for $ (1+K)L2(X,)$.$ Our first Fredholm determinant formula now follows.",
"Proposition 4.12 Fix $k\\ge 1$ and consider positive $\\tilde{a}_1,\\ldots ,\\tilde{a}_N$ .",
"Then for any $\\varphi \\in (0,\\pi /4)$ and any $\\tilde{\\alpha }\\in (0,\\min _{i}\\lbrace \\tilde{a}_i\\rbrace )$ , there exists a positive constant $C\\ge 1$ such that for all $|\\zeta |<C^{-1}$ , $\\left\\langle \\frac{1}{\\left(\\zeta q^{\\lambda _N};q\\right)_{\\infty }}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )} = \\det (\\mathbb {1}+K)_{L^2(\\mathbb {Z}_{>0}\\times \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi })}$ where the kernel $K$ is given by $K(n_1,w_1;n_2,w_2) = \\frac{\\zeta ^{n_1} f(w_1)f(qw_1)\\cdots f(q^{n_1-1}w_1)}{q^{n_1} w_1 - w_2}$ with $f(w) = \\exp ((q-1)\\gamma w)\\prod _{m=1}^{N} \\frac{\\tilde{a}_m}{\\tilde{a}_m-w}.$ Recall that the contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ is given in Definition REF .",
"The left-hand-side of (REF ) is the $q$ -Laplace transform of the random variable $q^{\\lambda _N}$ (with respect to the $e_q$ exponential) — see Section 3.1.1 of [16].",
"The $q$ -Binomial theorem is analogous to Taylor expansion of the $e_q$ exponential.",
"Because $q^{\\lambda _N}<1$ it we can justify interchanging the summation (of the series expansion resulting from the $q$ -Binomial theorem) and the expectation and hence we find that the $q$ -Laplace transform can be written as a generating series of $q$ -moments.",
"Using the expression for these moments coming from Proposition REF it is an easy formal manipulation of terms in a summation to turn this generating series into the desired Fredholm determinant.",
"A little extra arguing shows that these formal manipulations are numerical equalities and proves the claimed result.",
"The details are given in Section REF ."
],
[
"A Fredholm determinant suitable for asymptotics",
"By using a Mellin-Barnes type integral representation and analytic continuation we can reduce our Fredholm determinant to that of an operator acting on a single contour.",
"The above developments all lead to the following result.",
"Theorem 4.13 Fix $\\rho $ a Plancherel (see equation ()) Macdonald nonnegative specialization and positive $\\tilde{a}_1,\\ldots , \\tilde{a}_N$ .",
"Then for all $\\zeta \\in \\mathbb {R}_{+}$ $\\left\\langle \\frac{1}{\\left(\\zeta q^{\\lambda _N};q\\right)_{\\infty }}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )} = \\det (\\mathbb {1}+\\tilde{K}_{\\zeta })_{L^2(\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi })}$ where $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ as in Definition REF with any $\\tilde{\\alpha }\\in (0,\\min _i\\lbrace \\tilde{a}_i\\rbrace )$ and any $\\varphi \\in (0,\\pi /4)$ .",
"The operator $\\tilde{K}_{\\zeta }$ is defined in terms of its integral kernel $\\tilde{K}_{\\zeta }(w,w^{\\prime }) = \\frac{1}{2\\pi I}\\int _{\\widetilde{\\mathcal {D}}_{w}} \\Gamma (-s)\\Gamma (1+s)(-\\zeta )^s g_{w,w^{\\prime }}(q^s)ds$ where $g_{w,w^{\\prime }}(q^s) = \\frac{\\exp \\big (\\gamma w(q^{s}-1)\\big )}{q^s w - w^{\\prime }} \\prod _{m=1}^{N} \\frac{(q^{s}w/\\tilde{a}_m;q)_{\\infty }}{(w/\\tilde{a}_m;q)_{\\infty }},$ and the contour $\\widetilde{\\mathcal {D}}_{w}$ is as in Definition REF .",
"The details of the proof of this result are given in Section REF ."
],
[
"Weak convergence to the Whittaker measure",
"We are now almost prepared to relate the above discussion to the Fredholm determinant in Theorem REF .",
"The connection relies on the following weak convergence of probability measures result.",
"Theorem 4.14 (Theorem 4.1.20 of [16]) Fix $N\\ge 1$, a drift vector $a=(a_1,\\ldots , a_N)\\in \\mathbb {R}^N$, and a time parameter $\\tau >0$ .",
"For positive $\\tilde{a}_1,\\dots ,\\tilde{a}_N$ and $\\gamma >0$ , consider a partition $\\lambda =(\\lambda _1,\\ldots ,\\lambda _N)$ distributed according to the $q$ -Whittaker measure (i.e., Macdonald measure at $t=0$ ) $\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )$ with Plancherel specialization $\\rho $ determined by $\\gamma $ .",
"For a small parameter $\\varepsilon >0$ let $q=e^{-\\epsilon },\\quad \\gamma =\\tau \\epsilon ^{-2}, \\quad \\tilde{a}_j=e^{-\\epsilon a_j}, \\quad \\lambda _j=\\tau \\epsilon ^{-2}-(N+1-2j)\\epsilon ^{-1}\\ln \\epsilon +T_{j}\\epsilon ^{-1}, \\quad 1\\le j\\le N.$ Consider the $\\varepsilon $ -indexed measure induced on $\\lbrace T_{j}\\rbrace _{1\\le j\\le N}$ .",
"This measure weakly converges, as $\\varepsilon \\rightarrow 0$ , to the Whittaker measure $\\mathbf {WM}_{(a_1,\\ldots , a_N,\\tau )}$ on $\\lbrace T_{j}\\rbrace _{1\\le j\\le N}$ ."
],
[
"Proof of Theorem ",
"We may now combine the above results to provide a proof of Theorem REF .",
"We follow approach taken in proving Theorem 4.1.40 of [16] but due to the unboundedness of the contours with which we are dealing, there are a number of extra and somewhat involved estimates we must make.",
"These are stated as propositions and proved in Sections REF and REF .",
"The proof splits into two parts.",
"Step 1: We prove that the left-hand side of equation (REF ) of Theorem REF converges to $\\left\\langle e^{-u e^{-T_{N}}} \\right\\rangle _{\\mathbf {WM}_{(a_1,\\ldots , a_N;\\tau )}}$ .",
"This relies on combining Theorem REF (which provides weak convergence of the $q$ -Whittaker measure to the Whittaker measure) with Lemma REF and the fact that the $q$ -Laplace transform converges to the usual Laplace transform.",
"Step 2: We prove that the Fredholm determinant expression coming from the right-hand side of Theorem REF converges to the Fredholm determinant given in the theorem we are presently proving (see ).",
"In accordance with the scalings of Theorem REF , we scale the parameters of Theorem REF as $\\begin{aligned}&q =e^{-\\varepsilon }, \\qquad \\gamma =\\tau \\varepsilon ^{-2}, \\qquad \\tilde{a}_k=e^{-\\varepsilon a_k},\\quad 1\\le k\\le N\\\\&w= q^v, \\qquad \\zeta = -\\varepsilon ^{N} e^{\\tau \\varepsilon ^{-1}} u, \\qquad \\lambda _N = \\tau \\varepsilon ^{-2} + (N-1) \\varepsilon ^{-1} \\ln \\varepsilon + T_N \\varepsilon ^{-1}.\\end{aligned}$"
],
[
"Step 1:",
"We assume throughout that $u\\in with positive real part.",
"Rewrite the left-hand side of equation (\\ref {thmlaplaceeqn}) in Theorem~\\ref {PlancherelfredThm} as\\begin{equation*}\\left\\langle \\frac{1}{\\left(\\zeta q^{\\lambda _N};q\\right)_{\\infty }}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )} = \\left\\langle e_q(x_q)\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )}\\end{equation*}where\\begin{equation*}x_q=(1-q)^{-1}\\zeta q^{\\lambda _N} =-ue^{-T_N} \\varepsilon /(1-q)\\end{equation*}and\\begin{equation*}e_q(x)=\\frac{1}{((1-q)x;q)_{\\infty }}\\end{equation*}is a $ q$-exponential.",
"Combine this with the fact that $ eq(x)ex$ uniformly on $ Re(x)<0$ to show that, considered as a function of $ TN$, $ eq(xq)e-u e-TN$ uniformly for $ TN$\\mathbb {R}$$.",
"By Theorem~\\ref {theorem26}, the measure on $ TN$ (induced from the $ q$-Whittaker measure on $ N$) converges weakly in distribution as $ 0$ to the marginal of the Whittaker measure $$\\mathbf {WM}$ (a1,...,aN;)$ on the $ TN$ coordinate.",
"Combining this weak convergence with the uniform convergence of $ eq(xq)$ and Lemma~\\ref {problemma2} gives that\\begin{equation*}\\left\\langle \\frac{1}{\\left(\\zeta q^{\\lambda _N};q\\right)_{\\infty }}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )} \\rightarrow \\left\\langle e^{-u e^{-T_{N}}} \\right\\rangle _{\\mathbf {WM}_{(a_1,\\ldots ,a_N;\\tau )}}\\end{equation*}as $ q1$, or equivalently $ 0$.$ Recall the kernel in the right-hand side of equation (REF ) in Theorem REF .",
"It can be rewritten as a Fredholm determinant of a kernel with the variables $v$ and $v^{\\prime }$ (recall $w=q^v$ ) as follows: $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta }) = \\det (\\mathbb {1}+K_u^{\\varepsilon }).$ Here the $L^2$ space with respect to which this determinant is defined is that of the contour specified in Definition REF below.",
"The kernel is denoted by $K_u^{\\varepsilon }$ to denote the dependence on $u$ (through $\\zeta = -\\varepsilon ^{N} e^{\\tau \\varepsilon ^{-1}} u$ ) and $\\varepsilon $ (through $q^{-\\varepsilon }$ ).",
"It is given by $K_u^{\\varepsilon }(v,v^{\\prime }) = \\frac{1}{2\\pi I}\\int _{\\widetilde{\\mathcal {D}}_{q^v}}h^q(s) ds,$ where $h^q(s)=\\Gamma (-s)\\Gamma (1+s)\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s \\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}} e^{\\gamma q^v(q^{s}-1)} \\prod _{m=1}^{N} \\frac{\\Gamma _q(v- a_m)}{\\Gamma _q(s+v- a_m)}$ where the new term $q^v \\ln q$ came from the Jacobian of changing $w$ to $v$ and where the $q$ -Gamma function is defined as $\\Gamma _q(x) = \\frac{(q;q)_{\\infty }}{(q^x;q)_{\\infty }} (1-q)^{1-x}.$ The contour on which this kernel $K_u^{\\varepsilon }$ acts is the image of the contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ under the map $x\\mapsto \\ln _q x$ .",
"There is a fair amount of freedom in specifying this contour, so we will fix a particular such pre-image contour.",
"Definition 4.15 Let $\\alpha = 1+\\max a_i$ then we define the contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }$ as the image of $q^{\\alpha } + e^{\\pm \\varphi I}\\mathbb {R}_{+}$ under the map $x\\mapsto \\ln _{q} x$ .",
"This contour is illustrated in Figure REF .",
"We will assume that $K_u^{\\varepsilon }$ acts on the contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }$ .",
"Note that as $\\varepsilon \\rightarrow 0$ this contour converges locally uniformly to $\\alpha + e^{(\\pi \\pm \\varphi )I}\\mathbb {R}_{+}$ as can readily be seen by Taylor expanding the map $x\\mapsto \\ln _{q} x$ .",
"It follows from the above observation that the contour on which the kernel $K_u^{\\varepsilon }$ is defined converges as $\\varepsilon \\rightarrow 0$ to the contour a $\\mathcal {C}_{\\alpha ,\\varphi }$ on which the kernel in Theorem REF is defined.",
"Let us likewise demonstrate the pointwise convergence of the integrand in the integral (REF ) defining kernel $K_u^{\\varepsilon }$ to that of the kernel $K_u$ .",
"Consider the behavior of each term as $q\\rightarrow 1$ (or equivalently as $\\varepsilon \\rightarrow 0$ as $q=e^{-\\varepsilon }$ ): $e^{-\\tau s \\varepsilon ^{-1}}\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s & \\rightarrow u^s,\\\\\\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}} &\\rightarrow \\frac{1}{v+s-v^{\\prime }}, \\\\\\frac{\\Gamma _q(v-a_m)}{\\Gamma _q(v+s-a_m)} & \\rightarrow \\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)},\\\\e^{\\tau s \\varepsilon ^{-1}} \\exp \\left(\\gamma q^v(q^{s}-1)\\right) & \\rightarrow e^{v \\tau s + \\tau s^2/2}.$ Combining these pointwise limits together gives the integrand of the kernel $K_u$ given in ().",
"However, in order to prove convergence of the determinants, or equivalently the Fredholm expansion series, one needs more than just this straightforward pointwise convergence.",
"There are four things we must do to complete Step 2 and prove convergence of the determinants.",
"In proving convergence of Fredholm determinants it is convenient to have the contour on which the operator acts be fixed with $\\varepsilon $ .",
"In Step 2a we deform $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }$ to a contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r}$ with a portion $\\mathcal {C}_{\\alpha ,\\varphi ,<r }$ (of distance $<r$ to the origin) which coincides with the limiting contour $\\mathcal {C}_{\\alpha ,\\varphi }$ .",
"Then in Step 2b we show that for any fixed $\\eta >0$ , by choosing $\\varepsilon _0$ small enough and $r_0$ large enough, for all $\\varepsilon <\\varepsilon _0$ and $r>r_0$ the determinant restricted to $L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})$ differs from the entire determinant on $L^2(\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r})$ by less than $\\eta $ .",
"Thus, at an arbitrarily small cost of $\\eta $ we can restrict to a sufficiently large radius on which the contour is independent of $\\varepsilon $ .",
"In Step 2c we show that for any $\\eta >0$ , for $\\varepsilon $ small, the Fredholm determinant of $K_u^{\\varepsilon }$ restricted to $L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})$ differs by at most $\\eta $ from the Fredholm determinant of $K_u$ restricted to the same space.",
"Finally, Step 2d shows that for $r_0$ large enough, for all $r>r_0$ the Fredholm determinant of $K_u$ restricted to $L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})$ differs from the Fredholm determinant of $K_u$ on $L^2(\\mathcal {C}_{\\alpha ,\\varphi })$ by at most $\\eta $ .",
"Summing up the steps, we deform the contour, we cut the contour to be finite, we take the $\\varepsilon \\rightarrow 0$ limit, and then we repair the contour to its final form – all at cost $3\\eta $ for $\\eta $ arbitrarily small.",
"We must define the contour to which we want to deform $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }$ , and then justify this deformation as not changing the value of the Fredholm determinant.",
"Definition 4.16 Fix $\\varphi \\in (0,\\pi /4)$ and $r>0$ .",
"For $\\alpha \\in \\mathbb {R}$ , define the finite contour $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ to be $\\lbrace \\alpha +te^{(\\pi \\pm \\varphi )I}:0\\le t\\le r\\rbrace $ .",
"The maximal imaginary part along $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ is $r\\sin (\\varphi )$ .",
"Define the infinite contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r}$ to be the union of $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ with $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,>r}$ and $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,=r}$ .",
"Here, the contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,>r}$ is the portion of the contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }$ which has imaginary part exceeding $r\\sin (\\varphi )$ in absolute value; and the contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,=r}$ is composed of the two horizontal line segments which join $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ with $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,>r}$ .",
"These contours are illustrated in Figure REF .",
"Figure: Left: The infinite contour 𝒞 α,ϕ ϵ \\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi } and the limiting contour 𝒞 α,ϕ \\mathcal {C}_{\\alpha ,\\varphi }.",
"Right: The infinite contour 𝒞 α,ϕ,r ϵ \\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r} (which we deform from 𝒞 α,ϕ ϵ \\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }).Now we justify replacing the contour $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi }$ by $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r}$ .",
"Lemma 4.17 For any $r>0$ there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ , $\\det (\\mathbb {1}+K_u^{\\varepsilon })_{L^2(\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi })} = \\det (\\mathbb {1}+K_u^{\\varepsilon })_{L^2(\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r})}.$ The two contours differ only by a finite length modification.",
"We can continuously deform between the two contours.",
"We will employ Lemma REF which says that as long as the kernel is analytic in a neighborhood of the contour as we continuously deform then the Fredholm determinant remains unchanged throughout the deformation.",
"The only things which could threaten the analyticity of the kernel are the poles coming from the left-hand side terms of () and ().",
"On account of the condition satisfied by the contour $\\widetilde{\\mathcal {D}}_{q^v}$ (see Definition REF ), it follows that these poles are avoided.",
"By choosing $\\varepsilon $ small enough, the two contours we are deforming between can be made as close as desired.",
"Taking them close enough ensures it is possible then to deform between them while avoiding poles of the kernel in $v$ or $v^{\\prime }$ – hence proving the lemma.",
"We must now show that we can, with small error, restrict our Fredholm determinant to acting on the finite, fixed contour $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ .",
"This requires us choosing $r>r_0$ for $r_0$ large enough, and also choosing $\\varepsilon <\\varepsilon _0$ for $\\varepsilon _0$ small enough.",
"Proposition 4.18 Fix $\\varphi \\in (0,\\pi /4)$ .",
"For any $\\eta >0$ there exists $r_0>0$ and $\\varepsilon _0>0$ such that for all $r>r_0$ and $\\varepsilon <\\varepsilon _0$ $\\left|\\det (\\mathbb {1}+ K_u^{\\varepsilon })_{L^2(\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r})} - \\det (\\mathbb {1}+ K_u^{\\varepsilon })_{L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})}\\right| \\le \\eta .$ The proof of this proposition is fairly technical and is given in Section REF .",
"Having restricted our attention to the finite, unchanging (with $\\varepsilon $ ) contour $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ we may now take the limit of Fredholm determinant on the restricted $L^2$ space as $\\varepsilon \\rightarrow 0$ .",
"Proposition 4.19 Fix $\\varphi \\in (0,\\pi /4)$ .",
"For any $\\eta >0$ and any $r>0$ there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ $\\left|\\det (\\mathbb {1}+ K_u^{\\varepsilon })_{L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})} - \\det (\\mathbb {1}+ K_u)_{L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})}\\right|\\le \\eta $ where $K_u(v,v^{\\prime })$ is given by the integral (REF ).",
"The proof of this proposition is also fairly technical and is given in Section REF .",
"Finally, we show that post-asymptotics we can return to the simple infinite contour $\\mathcal {C}_{\\alpha ,\\varphi }$ .",
"Proposition 4.20 Fix $\\varphi \\in (0,\\pi /4)$ .",
"For any $\\eta >0$ there exists $r_0>0$ such that for all $r>r_0$ $\\left| \\det (\\mathbb {1}+ K_u)_{L^2(\\mathcal {C}_{\\alpha ,\\varphi ,<r})} - \\det (\\mathbb {1}+ K_u)_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}\\right|\\le \\eta .$ The proof of this proposition is given in Section REF .",
"It is a fair amount more straight forward than the previous two proofs and hence is given first.",
"Having completed the four substeps we may combine Propositions REF , REF and REF to show that for any $\\eta >0$ , there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ , $\\left|\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })_{L^2(\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi })} - \\det (\\mathbb {1}+ K_u)_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })} \\right| \\le 3\\eta $ where $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ is as in the right-hand side of equation (REF ) in Theorem REF , subject to the scalings given in (REF ).",
"Since $\\eta $ is arbitrary this shows that $\\lim _{\\varepsilon \\rightarrow 0} \\det (\\mathbb {1}+\\tilde{K}_{\\zeta })_{L^2(\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi })} = \\det (\\mathbb {1}+ K_u)_{L^2(\\mathcal {C}_{\\alpha ,\\varphi })}.$ The above result completes the proof of Theorem REF modulo proving Propositions REF , REF and REF ."
],
[
"Details in the proof of Theorem ",
"As discussed in Section , to finish the proof of Theorem REF we need to show Theorem REF .",
"For $\\kappa >0$ , Definition REF associates the scaling parameters $\\theta ^\\kappa $ , $f^\\kappa $ and $c^\\kappa $ which appear in the statement of this result.",
"The variable $\\kappa $ and $\\theta $ are dual in the sense that one could instead start with some fixed $\\theta >0$ and then associated scaling parameters $\\kappa _{\\theta }$ , $f_{\\theta }$ and $c_{\\theta }$ .",
"In particular, for $\\theta =\\theta ^{\\kappa }$ one recovers $f_{\\theta }=f^\\kappa $ and $c_{\\theta }=c^\\kappa $ .",
"In the proof it is more natural to parameterize everything by $\\theta $ instead of $\\kappa $ , so we will do it.",
"First we prove the convergence to the GUE Tracy-Widom distribution without boundary perturbations, since the proof with boundary perturbations is a small modification of it."
],
[
"Proof of Theorem ",
"We first give explicit expansions for some of the functions from Definition REF .",
"Let $\\Psi (z)=\\frac{d}{dz} \\ln (\\Gamma (z))$ be the Digamma function and fix $\\theta \\in (0,\\infty )$ .",
"Then $\\Psi (z)=-\\gamma _{\\rm E}+\\sum _{n=0}^\\infty \\left(\\frac{1}{n+1}-\\frac{1}{n+z}\\right),$ where $\\gamma _{\\rm E}$ is the Euler constant.",
"Hence, $\\kappa _\\theta &=\\Psi ^{\\prime }(\\theta )=\\sum _{n=0}^\\infty \\frac{1}{(\\theta +n)^2}, \\\\f_\\theta &=\\theta \\Psi ^{\\prime }(\\theta )-\\Psi (\\theta )=\\gamma _{\\rm E} +\\sum _{n=0}^\\infty \\left(\\frac{n+2\\theta }{(n+\\theta )^2}-\\frac{1}{n+1}\\right), \\\\c_\\theta &=(-\\Psi ^{\\prime \\prime }(\\theta )/2)^{1/3}=\\bigg (\\sum _{n=0}^\\infty \\frac{1}{(n+\\theta )^3}\\bigg )^{1/3}.",
"$ Under the scaling limit $\\tau =\\kappa _\\theta N,\\quad u=e^{-N f_\\theta - r c_\\theta N^{1/3}}.$ we have to show the following: For $K_u$ as in (REF ) and a contour $\\mathcal {C}_v:=\\mathcal {C}_{0,\\varphi }$ , $\\lim _{N\\rightarrow \\infty }\\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)}= \\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}.$ To show this we start with the kernel (REF ), replace $\\Gamma (-s)\\Gamma (1+s) = -\\pi / \\sin (\\pi s)$ and then perform the change of variable $\\tilde{z}=s+w$ to obtain $K_{u}(v,v^{\\prime }) = \\frac{-1}{2\\pi I}\\int d\\tilde{z}\\, \\frac{\\pi }{\\sin (\\pi (\\tilde{z}-v))}\\frac{e^{N G(v)-N G(\\tilde{z})} e^{r N^{1/3} (v-\\tilde{z})}}{\\tilde{z}-v^{\\prime }}.$ where $G(z) = \\ln \\Gamma (z) - \\kappa \\frac{z^2}{2} + f^\\kappa z.$ We will show that the leading contribution to the Fredholm determinant comes for $v,v^{\\prime }$ in a $N^{-1/3}$ -neighborhood of $\\theta $ .",
"Now let us specify the exact choice for the contour $\\mathcal {C}_v$ as well as the contour along which $\\tilde{z}$ is integrated.",
"We chooseTheorem REF is stated for $\\varphi \\in (0,\\pi /4)$ since one uses the quadratic decay (REF ) to control the linear term in the bound (REF ).",
"For $\\varphi =\\pi /4$ one gets a linear decay instead of (REF ) whose strength depends on the parameter $\\alpha $ too, it would not strong enough general $\\alpha $ .",
"However, in our case, with $\\alpha =\\theta $ , it still works, as can be seen from the bound obtained in Proposition REF .",
"The proof could also be adapted to any other asymptotic direction $0<\\varphi <\\pi /4$ by simply modifying the path away at a distance greather than some (arbitrary but fixed with $N$ ) value $R_0$ (one can not employ any angle $\\varphi \\in (0,\\pi /4)$ right away from the critical point since some steep descent properties are then locally not satisfied).",
"$\\mathcal {C}_v:=\\lbrace \\theta -|y|+Iy, y\\in \\mathbb {R}\\rbrace .$ $\\mathcal {C}_v$ is a steep descent path (see the footnote in Section REF ) for the function $\\operatorname{Re}(G(v))$ .",
"The path for $\\tilde{z}$ is dependent on $v$ , since it has to pass to the left of, or contain the simple poles $v+1,v+2,\\ldots $ , see Figure REF (left).",
"Consider the sequence of points $S=\\lbrace \\operatorname{Re}(v)+1,\\operatorname{Re}(v)+2,\\ldots \\rbrace $ .",
"There are three possibilities: (1) If the sequence $S$ does not contain points in $[\\theta ,\\theta +3c_\\theta ^{-1} N^{-1/3}]$ , then let $\\ell \\in \\mathbb {N}_0$ be such that $\\operatorname{Re}(v)+\\ell \\in [\\theta -1,\\theta ]$ and we set $\\tilde{\\varepsilon }=c_\\theta ^{-1} N^{-1/3}$ .",
"(2) If the sequence $S$ contains a point in $[\\theta ,\\theta +2c_\\theta ^{-1} N^{-1/3}]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in [\\theta ,\\theta +2c_\\theta ^{-1} N^{-1/3}]$ and set $\\tilde{\\varepsilon }=3 c_\\theta ^{-1} N^{-1/3}$ .",
"(3) If the sequence $S$ contains a point in $(\\theta +2c_\\theta ^{-1} N^{-1/3},\\theta +3c_\\theta ^{-1} N^{-1/3}]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in (\\theta -1+2c_\\theta ^{-1} N^{-1/3},\\theta -1+3c_\\theta ^{-1} N^{-1/3}]$ and set $\\tilde{\\varepsilon }= c_\\theta ^{-1} N^{-1/3}$ .",
"With this choice, the singularity of the sine along the line $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ is not present, since the poles are at a distance at least $c_\\theta ^{-1} N^{-1/3}$ from it.",
"Then, the path for $\\tilde{z}$ is given by $\\mathcal {C}_{\\tilde{z}}:=\\lbrace \\theta +\\tilde{\\varepsilon }+Iy, y\\in \\mathbb {R}\\rbrace \\cup \\bigcup _{k=1}^\\ell B_{v+k},$ and $B_{v+k}$ denotes a small circle (radius smaller than $1/2$ ) around $v+k$ and clockwise oriented.",
"If $\\ell =0$ then the small circles are simply not present.",
"The idea behind this choice of the path $\\mathcal {C}_{\\tilde{z}}$ is that the $z$ -contour consists of a fixed line that is (almost) independent of kernel arguments, and an additional number of little circles (i.e., poles) as needed.",
"Moreover, the leading contribution of the kernel comes only from the cases where $\\ell =0$ (i.e., situation (1)) for which $\\tilde{\\varepsilon }=c_\\theta ^{-1} N^{-1/3}$ .",
"Figure: Left: Integration paths 𝒞 v \\mathcal {C}_v (dashed) and 𝒞 z ˜ \\mathcal {C}_{\\tilde{z}} (the solid line plus circles at v+1,...,v+ℓv+1,\\ldots ,v+\\ell ).",
"The small black dots are poles either of the sine or of the gamma function.",
"Right:Integration paths after the change of variables 𝒞 w \\mathcal {C}_w (dashed) and 𝒞 z \\mathcal {C}_z (the solid line plus circles at w+1,...,w+ℓw+1,\\ldots ,w+\\ell ), with p=p(w)∈{1,3}p=p(w)\\in \\lbrace 1,3\\rbrace .Also, we do the change of variable $\\lbrace v,v^{\\prime },\\tilde{z}\\rbrace =\\lbrace \\Phi (w),\\Phi (w^{\\prime }),\\Phi (z)\\rbrace \\quad \\textrm {with}\\quad \\Phi (z):=\\theta +z c_\\theta ^{-1} N^{-1/3}.$ After this change of variable, $\\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)}=\\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)}$ , the path $\\mathcal {C}_v$ becomes (see Figure REF (right)) $\\mathcal {C}_w:=\\lbrace -|y|+Iy,y\\in \\mathbb {R}\\rbrace $ and the accordingly rescaled kernel $\\begin{aligned}K_N(w,w^{\\prime })&:=c_\\theta ^{-1} N^{-1/3} K_u(\\Phi (w),\\Phi (w^{\\prime })) \\\\&=\\frac{-c_\\theta ^{-1} N^{-1/3}}{2\\pi I}\\int _{\\mathcal {C}_z:=\\Phi ^{-1}(\\mathcal {C}_{\\tilde{z}})}dz \\frac{\\pi e^{N G(\\Phi (w))-N G(\\Phi (z))}}{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})} \\frac{e^{r (w-z)}}{z-w^{\\prime }}\\end{aligned}$ where $G(w)=\\ln (\\Gamma (w))+ f_\\theta w - \\kappa _\\theta w^2/2.$ In Proposition REF we show that for any $w,w^{\\prime }\\in \\mathcal {C}_w$ , there exists a constant $C\\in (0,\\infty )$ such that $|K_N(w,w^{\\prime })|\\le C e^{-|\\operatorname{Im}(w)|}$ uniformly for all $N$ large enough.",
"Therefore, $\\left|\\det (K_N(w_i,w_j))_{1\\le i,j\\le n}\\right|\\le n^{n/2} C^n \\prod _{i=1}^n e^{-|\\operatorname{Im}(w_i)|}$ where the factor $n^{n/2}$ is Hadamard's bound.",
"From this bound, it follows that the Fredholm expansion of the determinant, $\\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)} =\\sum _{n=0}^\\infty \\frac{1}{n!}",
"\\int _{\\mathcal {C}_w} dw_1 \\cdots \\int _{\\mathcal {C}_w} dw_n \\det (K_N(w_i,w_j))_{1\\le i,j\\le n},$ is absolutely integrable and summable.",
"Thus we can by dominated convergence take the $N\\rightarrow \\infty $ limit inside the series, i.e., replace $K_N$ by its pointwise limit, $\\lim _{N\\rightarrow \\infty } K_N(w,w^{\\prime }) = \\widetilde{K}_{\\rm Ai}(w,w^{\\prime }):=\\frac{-1}{2\\pi I} \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{e^{z^3/3-w^3/3}e^{r w-r z}}{(z-w)(z-w^{\\prime })},$ derived in Proposition REF , i.e., we have shown that $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)} = \\det (\\mathbb {1}+\\widetilde{K}_{\\rm Ai})_{L^2(\\mathcal {C}_w)}.$ The last part is a standard reformulation, which we report in Lemma REF , see also [65].",
"This ends the proof of Theorem REF ."
],
[
"Pointwise convergence and bounds",
"The function $G$ satisfies $G^{\\prime }(\\theta )=G^{\\prime \\prime }(\\theta )=0,\\quad G^{(3)}(\\theta )=-2\\sum _{n=0}^\\infty \\frac{1}{(n+\\theta )^3}=-2c_\\theta ^{3},\\quad G^{(4)}(\\theta )=\\sum _{n=0}^\\infty \\frac{6}{(n+\\theta )^4},$ therefore $G$ has a double critical point at $\\theta $ .",
"For the steep descent analysis we need to analyze the function $g(x,y)=\\operatorname{Re}(G(x+Iy))$.",
"It holdsSee for example $\\texttt {http://functions.wolfram.com/06.11.19.0001.01}$ $\\begin{aligned}\\operatorname{Re}(\\ln \\Gamma (x+Iy)) & = \\sum _{n=1}^\\infty \\left(\\frac{x}{n} - \\frac{1}{2} \\ln \\left(\\frac{(x+n)^2+y^2}{n^2}\\right)\\right)- \\gamma _{\\rm E} x -\\frac{1}{2} \\ln (x^2+y^2) \\\\&=\\sum _{n=0}^\\infty \\left(\\frac{x}{n+1} - \\frac{1}{2} \\ln \\left((x+n)^2+y^2\\right)+\\ln (n)\\mathbf {1}_{n\\ge 1}\\right)- \\gamma _{\\rm E} x.\\end{aligned}$ Together with (REF ) and () we get $\\begin{aligned}g(x,y)&=\\operatorname{Re}(\\ln \\Gamma (x+Iy))+ f_\\theta x - \\frac{1}{2} \\kappa _\\theta (x^2-y^2)\\\\&=\\sum _{n=0}^\\infty \\left(\\frac{(n+2\\theta )x-(x^2-y^2)/2}{(n+\\theta )^2}-\\frac{1}{2} \\ln \\left((x+n)^2+y^2\\right)+\\ln (n)\\mathbf {1}_{n\\ge 1}\\right).\\end{aligned}$ It follows that $g_1(x,y):=\\frac{\\partial g(x,y)}{\\partial x}=\\sum _{n=0}^\\infty \\left(\\frac{n+2\\theta -x}{(n+\\theta )^2}-\\frac{x+n}{(x+n)^2+y^2}\\right)$ and $g_2(x,y):=\\frac{\\partial g(x,y)}{\\partial y} =\\sum _{n=0}^\\infty \\left(\\frac{y}{(\\theta +n)^2}-\\frac{y}{(x+n)^2+y^2}\\right).$ Proposition 5.1 Uniformly for $w,w^{\\prime }$ in a bounded set of $\\mathcal {C}_w$ , $\\lim _{N\\rightarrow \\infty } K_N(w,w^{\\prime }) = \\frac{-1}{2\\pi I} \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{e^{z^3/3-w^3/3}e^{r w-r z}}{(z-w)(z-w^{\\prime })}.$ Consider $w,w^{\\prime }$ in a bounded set of $\\mathcal {C}_w$ , i.e., the original variables $v,v^{\\prime }$ of order $N^{-1/3}$ around the critical point $\\theta $ .",
"For $N$ large enough and $w$ bounded, $\\operatorname{Re}((-w)c_\\theta ^{-1} N^{-1/3}) \\in (0,1)$, and $\\mathcal {C}_z:=\\Phi ^{-1}(\\mathcal {C}_{\\tilde{z}})=\\lbrace 1+Iy,y\\in \\mathbb {R}\\rbrace $ .",
"Using (REF ) we have the expansion $\\begin{aligned}N G(\\Phi (w)) &= N G(\\theta )-\\frac{1}{3} w^3 + \\mathcal {O}(w^4 N^{-1/3})\\\\-N G(\\Phi (z)) &= -N G(\\theta )+\\frac{1}{3} z^3 -\\mu _\\theta z^4 N^{-1/3}+\\mathcal {O}(z^5 N^{-2/3})\\end{aligned}$ with $\\mu _\\theta =G^{(4)}(\\theta ) c_\\theta ^{-4}/24>0$ and $\\frac{\\pi }{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})}=\\frac{c_\\theta N^{1/3}}{z-w}(1+\\mathcal {O}((z-w)^2 N^{-1/3})).$ It is also easy to control the $w^{\\prime }$ -dependence because $|z-w^{\\prime }|\\ge 1$ .",
"Now we divide the integral over $z$ into (a) $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ and (b) $|\\operatorname{Im}(z)|\\le \\delta N^{1/3}$ for some $\\delta >0$ which can be taken as small as desired (but independent of $N$ ).",
"(a) Contribution of the integration over $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ .",
"We need to estimate $\\left|\\frac{-c_\\theta ^{-1} N^{-1/3}}{2\\pi I}\\int _{1+Iy,\\\\|y|> \\delta N^{1/3}}dz \\frac{\\pi e^{N G(\\Phi (w))-N G(\\Phi (z))}}{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})} \\frac{e^{r (w-z)}}{z-w^{\\prime }}\\right|.$ From (REF ), (REF ), and the fact that $w$ is in a bounded neighborhood of 0, we have $(\\ref {PFeq33})\\le \\mathcal {O}(1) \\int _{|y|\\ge \\delta N^{1/3}}dy\\, e^{N \\operatorname{Re}(G(\\Phi (0))-G(\\Phi (1+Iy)))}.$ Setting $\\tilde{\\varepsilon }=c_\\theta ^{-1} N^{-1/3}$ and doing the change of variable $\\tilde{y}=y c_\\theta ^{-1} N^{-1/3}$ we obtain $(\\ref {PFeq34})\\le \\mathcal {O}(N^{1/3}) \\int _{\\delta /c_\\theta }^\\infty d\\tilde{y}\\, e^{N (g(\\theta +\\tilde{\\varepsilon },\\tilde{y})-g(\\theta ,0))}$ The function $g(x,y):=\\operatorname{Re}(G(x+Iy))$ is given in (REF ).",
"Finally, in Lemma REF we show that the path $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ is steep descent for the function $-G(\\tilde{z})$ with derivative of $-\\operatorname{Re}(G(\\tilde{z}))$ going to $-\\infty $ linearly in $\\operatorname{Im}(\\tilde{z})$ .",
"It then follows that (REF ) is of order $N^{1/3} e^{Ng(\\theta ,0)-N g(\\theta +\\tilde{\\varepsilon },0)}e^{-c_1(\\delta ) N}$ for some positive constant $c_1(\\delta )\\sim \\delta ^4$ for small $\\delta $ .",
"But $Ng(\\theta ,0)-N g(\\theta +\\tilde{\\varepsilon },0)= \\frac{1}{3}+\\mathcal {O}(N^{-1/3}).$ Thus the contribution of the integration over $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ is $\\mathcal {O}(e^{-c_2(\\delta ) N})$ for some positive constant $c_2(\\delta )\\sim \\delta ^4$ for small $\\delta $ .",
"(b) Contribution of the integration over $|\\operatorname{Im}(z)|\\le \\delta N^{1/3}$ .",
"We need to determine the asymptotics of $\\frac{-c_\\theta ^{-1} N^{-1/3}}{2\\pi I}\\int _{1+Iy,\\\\|y|\\le \\delta N^{1/3}}dz \\frac{\\pi e^{N G(\\Phi (w))-N G(\\Phi (z))}}{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})} \\frac{e^{r (w-z)}}{z-w^{\\prime }}.$ Using the expansion (REF ) and (REF ) we get $(\\ref {PFeq37})=\\frac{-1}{2\\pi I} \\int _{1-I\\delta N^{1/3}}^{1+I\\delta N^{1/3}}dz \\frac{e^{-\\mu _\\theta z^4 N^{-1/3}}}{(z-w)(z-w^{\\prime })} \\frac{e^{z^3/3-rz}}{e^{w^3/3-rw}}(1+\\mathcal {O}((z-w)^2 N^{-1/3}))e^{\\mathcal {O}(w^4 N^{-1/3}; z^5 N^{-2/3})}.$ Denoting $z=1+Iy$ we have $\\operatorname{Re}(z^3/3)= -3y^2+1,\\quad \\operatorname{Re}(z^4)=y^4-6 y^2+1.$ The convergence of the integral is controlled by $e^{-\\mu _\\theta y^4 N^{-1/3}-3 y^2}$ .",
"One employs the bound $|e^{x}-1|\\le |x| e^{|x|}$ with $x=\\mathcal {O}(w^4 N^{-1/3}; z^5 N^{-2/3})$ to control the error terms.",
"Altogether they are only of order $\\mathcal {O}(N^{-1/3})$ , i.e., we have obtained $(\\ref {PFeq32})=\\mathcal {O}(N^{-1/3})+\\frac{-1}{2\\pi I}\\int _{1-I\\delta N^{1/3}}^{1+I\\delta N^{1/3}}dz \\frac{e^{-\\mu _\\theta z^4 N^{-1/3}}}{(z-w)(z-w^{\\prime })} \\frac{e^{z^3/3-rz}}{e^{w^3/3-rw}}$ Finally, we deform the integration contour to the following one: from $\\delta N^{1/3} (1-I)$ to $\\delta N^{1/3} (1+I)$.",
"The error term is again of order $e^{-c_1(\\delta ) N}$ .",
"However, with the new contour, using again $|e^{x}-1|\\le |x| e^{|x|}$ but with $x=-\\mu _\\theta z^4 N^{-1/3}$ one sees that the eliminating the quartic power in $z$ amounts in an error of order $\\mathcal {O}(N^{-1/3})$ .",
"The last step is to replace $\\delta $ by $\\infty $ in the integration boundaries.",
"This leads to an extra error $\\mathcal {O}(e^{-c_3(\\delta ) N})$ with some positive constant $c_3(\\delta )\\sim \\delta ^3$ for small $\\delta $ .",
"To summarize, we first choose $\\delta $ small enough so that all the $c_k(\\delta )>0$ .",
"Then for all $N$ large enough we have shown that the contribution of the integration over $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ is of order $\\mathcal {O}(e^{-c_1(\\delta ) N})$ and the integration over $|\\operatorname{Im}(z)|\\le \\delta N^{1/3}$ is given by $\\mathcal {O}(N^{-1/3})+\\frac{-1}{2\\pi I}\\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{1}{(z-w)(z-w^{\\prime })} \\frac{e^{z^3/3-rz}}{e^{w^3/3-rw}}.$ Taking the $N\\rightarrow \\infty $ limit we obtain the result.",
"Proposition 5.2 For any $w,w^{\\prime }\\in \\mathcal {C}_w$ , there exists a constant $C\\in (0,\\infty )$ such that $|K_N(w,w^{\\prime })|\\le C e^{-|\\operatorname{Im}(w)|}$ uniformly for all $N$ large enough.",
"Since the $z$ -contour can be chosen such that $|z-w^{\\prime }|\\ge 1/2$ , we can estimate the absolute value of the factor $(z-w^{\\prime })^{-1}$ by 2 and discard it from further considerations.",
"For $w$ in a bounded set of $\\mathcal {C}_w$ , the statement is a consequence of the computations in the proof of Proposition REF .",
"Thus, it is enough to consider $w=-|y|+Iy$ for $y\\ge L$ , for $L$ which will be chosen large enough (but independent of $N$ ).",
"In the original variables $v,v^{\\prime }$ , this means that we need to consider $v=\\theta -|y|+Iy$ for $y\\ge L c_\\theta ^{-1} N^{-1/3}$ .",
"Let $v=\\Phi (w)$ , $v^{\\prime }=\\Phi (w^{\\prime })$ , then the kernel $K_N$ is given by $K_N(w,w^{\\prime })=\\frac{e^{N (G(v)-G(\\theta ))+r(v-\\theta ) c_\\theta N^{1/3}}}{c_\\theta N^{1/3}\\, 2\\pi I}\\int _{\\mathcal {C}_{\\tilde{z}}}d\\tilde{z} \\frac{\\pi e^{N G(\\theta )-N G(\\tilde{z})} e^{r(\\theta -\\tilde{z}) c_\\theta N^{1/3}}}{\\sin (\\pi (\\tilde{z}-v))(\\tilde{z}-v^{\\prime })}.$ We divide the bound dividing in two contributions: (a) integration over $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ , with $\\tilde{\\varepsilon }=p c_\\theta ^{-1} N^{-1/3}$ (with $p\\in \\lbrace 1,3\\rbrace $ depending on the value of $v$ , see the proof of Theorem REF (a) above), and (b) integration over the circles $B(v+k)$ , $k=1,\\ldots ,\\ell (v)$ .",
"(a) Integration over $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ .",
"The relevant dependence on $v$ is in the prefactor $e^{N (G(v)-G(\\theta ))+r(v-\\theta ) c_\\theta N^{1/3}}$ and in the sine.",
"The dependence of $\\tilde{\\varepsilon }$ on $v$ is marginal, as the needed bounds can be made for any $\\tilde{\\varepsilon }$ small enough.",
"The estimates as in the proof of Proposition REF imply that this contribution is bounded by $C e^{N (\\operatorname{Re}(G(v))-G(\\theta ))+r(\\operatorname{Re}(v)-\\theta ) c_\\theta N^{1/3}} = C e^{N (g(\\theta -y,y)-g(\\theta ,0))-r y c_\\theta N^{1/3}},$ where we used the parametrization $v=\\theta -|y|+Iy$ and, by symmetry, considered only $y>0$ .",
"In Lemma REF we show that $g(\\theta -y,y)$ is strictly decreasing as $y$ increases and for large $y$ the derivative goes to $-\\infty $ (logarithmically).",
"Thus, for any fixed $\\delta >0$ , there exists a constant $c_1>0$ such that for all $y\\ge \\delta $ , $\\partial _y g(\\theta -y,y)\\le -c_1$ .",
"In Lemma REF we also show that for small $y$ , $g(\\theta -y,y)-g(\\theta ,0)=-\\tfrac{2}{3} c_\\theta ^3 y^3+\\mathcal {O}(y^4)$ .",
"Therefore, we can choose $\\delta >0$ small enough such that: for $L c_\\theta ^{-1} N^{-1/3}\\le y\\le \\delta $ , $g(\\theta -y,y)\\le g(\\theta ,0) -\\tfrac{1}{3} c_\\theta ^3 y^3$ , for $y>\\delta /2$ , $\\partial _y g(\\theta -y,y)\\le -2 c_1$ .",
"It follows $g(\\theta -y,y)\\le g(\\theta ,0)-c_1 y$ for all $y>\\delta $ .",
"Replacing $y=\\operatorname{Im}(v)=\\operatorname{Im}(w)/(c_\\theta N^{1/3})$ we get the bounds: for $L\\le \\operatorname{Im}(w) \\le \\delta c_\\theta N^{1/3}$ , $C e^{- \\operatorname{Im}(w)^3/3-r \\operatorname{Im}(w)}\\le C e^{-\\operatorname{Im}(w)^3/6}\\le 3 C e^{-\\operatorname{Im}(w)}$ for $L$ large enough (depending on $r$ only).",
"for $\\operatorname{Im}(w)\\ge \\delta c_\\theta N^{1/3}$ , $C e^{-\\operatorname{Im}(w) (N^{2/3} c_1/c_\\theta +r)}\\le C e^{-\\operatorname{Im}(w)}$ for $N$ large enough.",
"(b) Integration over the circles $B(v+k)$ , $k=1,\\ldots ,\\ell (v)$ .",
"This happens only if $y+3 c_\\theta ^{-1} N^{-1/3} \\ge 1$, where $y=\\operatorname{Im}(v)=\\operatorname{Im}(w)/(c_\\theta N^{1/3})$ .",
"The contribution of the integration over $B_{v+k}$ (up to a $\\pm $ sign depending on $k$ ) is $\\frac{e^{N G(v)-N G(v+k)} e^{-rk c_\\theta N^{1/3}}}{(v+k-v^{\\prime })}.$ We have $|v+k-v^{\\prime }|\\ge 1/\\sqrt{2}$ , thus the contribution from the pole at $v+k$ is bounded by $2 e^{N (g(\\theta -v,v)-g(\\theta -v+k,v))}e^{|r| k c_\\theta N^{1/3}}.$ Define the function $h(v,k):=g(\\theta -v,v)-g(\\theta -v+k,v)$ .",
"In Lemma REF we show that $h(v,k)$ is strictly decreasing as a function of $k$ , for $k\\in [0,y+\\tilde{\\varepsilon }]$ (we have a positive $\\delta $ instead of $\\tilde{\\varepsilon }$ , but for $N$ large enough $\\tilde{\\varepsilon }<\\delta $ ).",
"Also, $k\\le \\ell (v) = \\lfloor y+\\tilde{\\varepsilon }\\rfloor $ , so that the contribution of the poles at $v+1,\\ldots ,v+\\ell (v)$ is bounded by $2 \\ell (v) e^{N h(v,1) +|r| \\ell (v) c_\\theta N^{1/3}}.$ We consider separately the cases (1) $y\\le \\theta $ (i.e., $\\operatorname{Re}(v)\\ge 0$ ) and (2) $y>0$ (i.e., $\\operatorname{Re}(v)<0$ ).",
"For $y\\le \\theta $ , from the bound on $\\partial _k h(v,k)$ , see Lemma REF , we get $h(v,1)\\le -y c_\\theta ^3/4$ .",
"For $y>\\theta $ , we know that $h(v,1)<0$ for all $y$ and when $y\\rightarrow \\infty $ , $\\partial _k h(v,k)|_{k=0}\\simeq -y \\kappa _\\theta $ .",
"Since the function $h(v,1)$ is continuous in $y$ , there exists a positive constant $c>0$ such that $h(v,1)\\le -c y$ for all $y>\\theta $ .",
"Thus, with $c^{\\prime }=\\min \\lbrace c,c_\\theta ^3/4\\rbrace $ we get $(\\ref {PFeq45})\\le e^{-\\operatorname{Im}(w) N^{2/3}c^{\\prime }/c_\\theta + \\mathcal {O}(1)} \\mathcal {O}(\\operatorname{Im}(w) N^{-1/3})\\le C e^{-\\operatorname{Im}(w)}$ for $N$ large enough.",
"This ends the proof of the Proposition.",
"Finally let us collect the lemmas on the steep descent properties used in the propositions above.",
"Lemma 5.3 The function $g(\\theta -y,y)$ is strictly decreasing for $y>0$ .",
"For $y\\rightarrow \\infty $ it holds $\\partial _y g(\\theta -y,y)\\sim -\\ln (y)$ .",
"For $y\\searrow 0$ we have $g(\\theta -y,y)=g(\\theta ,0)-\\tfrac{2}{3} c_\\theta ^3 y^3+\\mathcal {O}(y^4)$ .",
"Using (REF ) and (REF ) we have $\\begin{aligned}&\\frac{\\partial g(\\theta -y,y)}{\\partial y} = g_2(\\theta -y,y)-g_1(\\theta -y,y)\\\\&=-\\sum _{n=0}^\\infty \\left(\\frac{1}{\\theta +n}-\\frac{n+\\theta -2y}{(n+\\theta -y)^2+y^2}\\right)=-\\sum _{n=0}^\\infty \\frac{2 y^2}{(\\theta +n)((\\theta +n-y)^2+y^2)}\\end{aligned}$ which is 0 for $y=0$ and strictly negative for $y>0$ .",
"The asymptotics for large $y$ are obtained by writing (REF ) as $\\frac{I}{2}\\Psi (-y+\\theta +Iy)-\\frac{I}{2}\\Psi (-y+\\theta -Iy)-\\frac{1}{2}\\Psi (-y+\\theta -Iy)-\\frac{1}{2}\\Psi (-y+\\theta +Iy)+\\Psi (\\theta )$ and using the large-$z$ expansion $\\Psi (z)=\\ln (z)-\\frac{1}{2z}+\\mathcal {O}(z^{-2}).$ Taylor expansion gives the small $y$ estimate.",
"Lemma 5.4 For any $x\\ge \\theta $ , the function $g(x,y)$ is strictly increasing for $y>0$ .",
"For $y\\rightarrow \\infty $ it holds $\\partial _y g(x,y)\\sim \\kappa _{\\theta } y$ .",
"From (REF ) we have $\\frac{\\partial g(x,y)}{\\partial y} =\\sum _{n=0}^\\infty \\left(\\frac{y}{(\\theta +n)^2}-\\frac{y}{(x+n)^2+y^2}\\right),$ which is 0 for $y=0$ and for $y>0$ is strictly positive.",
"For large $y$ , the second term goes to zero, leading to the estimate.",
"Lemma 5.5 Let $y>0$ be fixed.",
"The function $h(y,k):=g(\\theta -y,y)-g(\\theta -y+k,y)$ satisfies $h(y,0)=0$ , $h(y,k)$ is strictly decreasing for $k\\in [0,y]$ .",
"For any $\\delta \\in (0,\\theta )$ , $y\\ge \\delta $ , $h(y,k)$ is strictly decreasing in $k\\in [0,y+\\delta /2]$ .",
"For $y\\rightarrow \\infty $ , $\\partial _k h(y,k)|_{k=0}\\sim -y \\kappa _\\theta $ .",
"For $y\\le \\theta $ , $\\partial _k h(y,k)\\le -\\frac{k y c_\\theta ^3}{2}$ for $k\\in [0,y]$ .",
"From (REF ) we have $\\begin{aligned}\\frac{\\partial h(y,k)}{\\partial k}&=-g_1(\\theta -y+k,y)=-\\sum _{n=0}^\\infty \\left(\\frac{\\theta +n+y-k}{(\\theta +n)^2}-\\frac{\\theta +n-y+k}{(\\theta +n-y+k)^2+y^2}\\right)\\\\&=-\\sum _{n=0}^\\infty \\frac{(\\theta +n)(y^2-(k-y)^2)+(y-k)^3+y^2(y-k)}{(\\theta +n)^2((\\theta +n-y+k)^2+y^2)}\\end{aligned}$ which strictly negative for $k\\in [0,y]$ .",
"The second statement follows from $\\begin{aligned}& (\\theta +n)(y^2-(k-y)^2)+(y-k)^3+y^2(y-k) \\\\&\\quad \\ge \\theta (y^2-(k-y)^2)+(y-k)^3+y^2(y-k)\\\\&\\quad \\quad \\ge \\theta (\\delta ^2/2-\\delta ^2/4)+\\theta y^2/2-\\delta ^3/8-\\delta y^2/2\\ge \\delta ^8/8.\\end{aligned}$ To get the asymptotics of the derivative for large $y$ , we can rewrite $\\frac{\\partial h(y,k)}{\\partial k}\\bigg |_{k=0}=\\Psi (\\theta )-\\Psi ^{\\prime }(\\theta ) y-\\frac{1}{2} \\Psi (-y+\\theta -Iy)-\\frac{1}{2} \\Psi (-y+\\theta +Iy)$ and use (REF ) and $\\Psi ^{\\prime }(\\theta )=\\kappa _\\theta $ .",
"Moreover, for $k\\in [0,y]$ and $y\\le \\theta $ we have the bound $\\begin{aligned}\\frac{\\partial h(y,k)}{\\partial k}&\\le -\\sum _{n=0}^\\infty \\frac{(y^2-(k-y)^2)}{(\\theta +n)((\\theta +n-y+k)^2+y^2)}\\\\&\\le -k y\\sum _{n=0}^\\infty \\frac{1}{(\\theta +n)((\\theta +n-y+k)^2+y^2)}\\le -k y\\sum _{n=0}^\\infty \\frac{1}{2(\\theta +n)^3}=-\\frac{k y c_\\theta ^3}{2}.\\end{aligned}$"
],
[
"Proof of Theorem ",
"Now we turn to the proof of the Theorem with boundary perturbations.",
"Note that due to the ordering of the $a_i$ 's, $b_1\\ge b_2>\\cdots $ .",
"Call $\\bar{b}=b_1$ .",
"The scaling of the $a_i$ 's implies that the contours $\\mathcal {C}_w$ and $\\mathcal {C}_z$ can be chosen as before except for a modification in a $N^{-1/3}$ -neighborhood of the critical point, since they have to pass on the right of $\\theta +\\bar{b} c_{\\theta }^{-1} N^{-1/3}$ (see Figure REF ).",
"Figure: Perturbation of the integration paths, compare with Figure (right).",
"The white dots on the right are the values of b 1 ,...,b m b_1,\\ldots ,b_m.Let us denote $P(w,z,a):=\\frac{\\Gamma (\\Phi (w)-a)}{\\Gamma (\\Phi (w))}\\frac{\\Gamma (\\Phi (z))}{\\Gamma (\\Phi (z)-a)}.$ Then, the only difference with respect to the kernel (REF ) is that in the $N\\rightarrow \\infty $ limit there might remains a factor coming from $\\prod _{k=1}^m P(w,z,a_k)$ .",
"Using $\\frac{\\Gamma (z+a)}{\\Gamma (z+b)}\\sim z^{a-b}(1+\\mathcal {O}(1/z))$ (see (6.1.47) of [1]), for any $w,z$ on $\\mathcal {C}_w,\\mathcal {C}_z$ we have the bound $|P(w,z,a_k)|\\le C e^{c|a_k| (|\\operatorname{Im}(w)|+|\\operatorname{Im}(z)|) N^{-1/3}}$ for some constants $C,c$ .",
"The local modification of the paths has no influence on any of the bounds for large $w$ and $z$ , so that the proof of pointwise convergence and of the bounds are minor modifications of Proposition REF and Proposition REF .",
"It remains to determine the pointwise limits of $P(w,z,a_k)$ as $N\\rightarrow \\infty $ .",
"Case 1: If $\\limsup _{N\\rightarrow \\infty } (a_k(N)-\\theta ) N^{1/3} = -\\infty $ , then $\\lim _{N\\rightarrow \\infty }P(w,z,a_k)=1.$ Case 2: If $\\limsup _{N\\rightarrow \\infty } (a_k(N)-\\theta ) N^{1/3} = b_k$ , then $\\begin{aligned}&\\lim _{N\\rightarrow \\infty } \\frac{\\Gamma (\\Phi (w)-\\Phi (b_k))}{\\Gamma (\\Phi (w))}\\frac{\\Gamma (\\Phi (z))}{\\Gamma (\\Phi (z)-\\Phi (b_k))}\\\\=& \\lim _{N\\rightarrow \\infty } \\frac{\\Gamma ((w-b_k) c_{\\theta }^{-1} N^{-1/3})}{\\Gamma (\\theta +w c_{\\theta }^{-1} N^{-1/3})}\\frac{\\Gamma (\\theta +z c_{\\theta }^{-1} N^{-1/3})}{\\Gamma ((z-b_k)c_{\\theta }^{-1} N^{-1/3})}= \\frac{z-b_k}{w-b_k}\\end{aligned}$ because $\\Gamma (z)=z^{-1}-\\gamma _E+\\mathcal {O}(z)$ as $z\\rightarrow 0$ .",
"Therefore, one obtains $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)} = \\det (\\mathbb {1}+\\widetilde{K}_{{\\rm BBP},b})_{L^2(\\mathcal {C}_w)}$ where $\\widetilde{K}_{{\\rm BBP},b}(s,s^{\\prime })=\\frac{-1}{2\\pi I} \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{e^{z^3/3-w^3/3}e^{r w-r z}}{(z-w)(z-w^{\\prime })}\\prod _{k=1}^m \\frac{z-b_k}{w-b_k}.$ The reformulation of Lemma REF complete the proof."
],
[
"Details in the proof of Theorem ",
"As discussed at the beginning of Section , in the proof we parameterize using the position of the critical point $\\theta $ instead of $\\kappa $ .",
"Let us set $(0,\\infty )$ and consider the scaling limit $\\theta :=\\sqrt{N/,\\quad \\tau =\\kappa _\\theta N,\\quad u=S e^{-N f_\\theta }.",
"}One has the following large-\\theta expansion of (\\ref {PFeqKappa}) and (\\ref {PFeqF}) gives\\begin{equation*}\\begin{aligned}\\kappa _\\theta &=\\frac{1}{\\theta }+\\frac{1}{2 \\theta ^2}+\\frac{1}{6\\theta ^3}+\\mathcal {O}(\\theta ^{-5}),\\\\f_\\theta &=1-\\ln (\\theta )+\\frac{1}{\\theta }+\\frac{1}{4\\theta ^2}+\\mathcal {O}(\\theta ^{-4}).\\end{aligned}\\end{equation*}Thus,\\begin{equation*}\\begin{aligned}\\tau &=\\kappa _\\theta N=\\sqrt{N̰}+\\tfrac{1}{2} \\mathcal {O}(N^{-1/2}),\\\\u&=S e^{-N f_\\theta }=S e^{-N-\\frac{1}{2} N \\ln (N)+\\sqrt{N̰}+\\frac{1}{4} \\mathcal {O}(N^{-1})}.\\end{aligned}\\end{equation*}Equivalently, we can set \\tau =\\sqrt{T N}, then \\theta =\\sqrt{N/T}+\\frac{1}{2}-\\frac{1}{12}\\sqrt{T/N}+\\mathcal {O}(N^{-3/2}), so that\\begin{equation*}\\begin{aligned}=T-T^{3/2} /N^{1/2}+\\tfrac{11}{12}T^2/N+\\mathcal {O}(N^{-3/2}),\\\\u&=S e^{-N-\\frac{1}{2} N\\ln (T/N)-\\tfrac{1}{2} \\sqrt{T N}+T/4!+\\mathcal {O}(N^{-1})}.\\end{aligned}\\end{equation*}$ As shown in Section , what it remains is to prove Theorem REF .",
"We first prove the statement for the unperturbed case, and then we will show how the generalization is obtained."
],
[
"Proof of Theorem ",
"We have to determine is $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)}$ .",
"Consider the case of a drift vector $b=0$ .",
"The path $\\mathcal {C}_v$ is chosen as $\\begin{aligned}\\mathcal {C}_v&=\\lbrace \\theta -1/4+Ir, |r|\\le r^*\\rbrace \\cup \\lbrace \\theta e^{It}, t^*\\le |t| \\le \\pi /2\\rbrace \\cup \\lbrace \\theta -|y|+Iy,|y|\\ge \\theta \\rbrace ,\\\\\\end{aligned}$ where $r^*=\\sqrt{\\theta /2-1/16}$ , $t^*=\\arcsin (\\sqrt{1/2\\theta -1/16\\theta ^2})$ .",
"The path $\\mathcal {C}_{\\tilde{z}}$ is set as $\\mathcal {C}_{\\tilde{z}}=\\lbrace \\theta +p/4+I\\tilde{y}, \\tilde{y}\\in \\mathbb {R}\\rbrace \\cup \\bigcup _{k=1}^{\\ell } B_{v+k},$ where $B_{z}$ is a small circle around $z$ clockwise oriented and $p\\in \\lbrace 1,2\\rbrace $ depending on the value of $v$ , see Figure REF .",
"More precisely, for given $v$ , we consider the sequence of points $S=\\lbrace \\operatorname{Re}(v)+1,\\operatorname{Re}(v)+2,\\ldots \\rbrace $ and we choose $p=p(v)$ and $\\ell =\\ell (v)$ as follows: (1) If the sequence $S$ does not contain points in $[\\theta ,\\theta +1/2]$ , then let $\\ell \\in \\mathbb {N}_0$ be such that $\\operatorname{Re}(v)+\\ell \\in [\\theta -1,\\theta ]$ and we set $p=1$ .",
"(2) If the sequence $S$ contains a point in $[\\theta ,\\theta +3/8]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in [\\theta ,\\theta +3/8]$ and set $p=2$ .",
"(3) If the sequence $S$ contains a point in $[\\theta +3/8,\\theta +1/2]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in [\\theta -5/8,\\theta -1/2]$ and set $p=1$ .",
"With this choice, the singularity of the sine along the line $\\theta +p/4+I\\mathbb {R}$ is not present, since the poles are at a distance at least $1/8$ from it.",
"Also, the leading contribution of the kernel will come from situation (1) with $\\ell =0$ and $p=1$ .",
"Figure: Left: Integration paths 𝒞 v \\mathcal {C}_v (dashed) and 𝒞 z ˜ \\mathcal {C}_{\\tilde{z}} (the solid line plus circles at v+1,...,v+ℓv+1,\\ldots ,v+\\ell ), where θ + =θ+p/4\\theta _+=\\theta +p/4 and θ - =θ-1/4\\theta _-=\\theta -1/4, the small black dots are poles either of the sine or of the gamma function.",
"Right:Integration paths after the change of variables 𝒞 w \\mathcal {C}_w (dashed) and 𝒞 z \\mathcal {C}_z (the solid line plus circles at w+1,...,w+ℓw+1,\\ldots ,w+\\ell ), with p=p(w)∈{1,2}p=p(w)\\in \\lbrace 1,2\\rbrace We denote $\\sigma :=(2/^{1/3}$ and we do the change of variable $\\lbrace v,v^{\\prime },\\tilde{z}\\rbrace =\\lbrace \\Phi (w),\\Phi (w^{\\prime }),\\Phi (z)\\rbrace \\quad \\textrm {with}\\quad \\Phi (z):=\\theta +z\\sigma $ and $K_\\theta (w,w^{\\prime }):=\\sigma K_u(\\Phi (w),\\Phi (w^{\\prime }))=\\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }}.$ After this change of variable, the paths $\\mathcal {C}_w=\\Phi ^{-1}(\\mathcal {C}_v)$ and $\\mathcal {C}_z=\\Phi ^{-1}(\\mathcal {C}_{\\tilde{z}})$ are given by $\\begin{aligned}\\mathcal {C}_w=&\\lbrace -1/(4\\sigma )+Ir/\\sigma , |r|\\le r^*\\rbrace \\cup \\lbrace (e^{It}-1)\\theta /\\sigma , t^*\\le |t|\\le \\pi /2\\rbrace \\cup \\lbrace -|y|+Iy,|y|\\ge \\theta /\\sigma \\rbrace ,\\\\\\mathcal {C}_z=&\\lbrace p/(4\\sigma )+Iy, y\\in \\mathbb {R}\\rbrace \\cup \\bigcup _{k=1}^{\\ell } B_{w+k/\\sigma },\\end{aligned}$ where $r^*=\\sqrt{\\theta /2-1/16}$ , $t^*=\\arcsin (\\sqrt{1/2\\theta -1/16\\theta ^2})$ , and $B_{z}$ is a small circle around $z$ clockwise oriented.",
"After this change of variable, we have $\\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)} = \\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)}.$ Thus, we need to prove that $\\lim _{N\\rightarrow \\infty }\\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ with $K_{{\\rm CDRP}}$ given in Definition REF .",
"The proof is very similar to the second part of the proof of Theorem REF (a), where this time the convergence of the kernel is in Proposition REF and the exponential bound in Proposition REF .",
"We then obtain $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)} = \\det (\\mathbb {1}+\\widetilde{K}_{{\\rm CDRP}})_{L^2(\\mathcal {C}_w)}$ with $\\widetilde{K}_{{\\rm CDRP}}$ given in (REF ).",
"Lemma REF shows that the limiting Fredholm determinant is equivalent to $\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ and thus completes the proof of Theorem REF (a)."
],
[
"Pointwise convergence and bounds",
"The leading contribution of the Fredholm determinant and in the kernel comes from $w,w^{\\prime },z$ of order 1 away from $\\theta \\sim \\mathcal {O}(\\sqrt{N})$ .",
"The scale for steep descent analysis is $N\\theta $ instead of $N$ as in the case of the convergence to the GUE Tracy-Widom distribution function.",
"So, the function whose real part has to be controlled is this time $\\widetilde{G}(Z):=\\frac{G(\\theta +\\theta Z)}{\\theta },$ that satisfies $\\begin{aligned}\\widetilde{G}^{(3)}(0)&=-1+\\mathcal {O}(\\theta ^{-1}),\\\\\\widetilde{G}^{(4)}(0)&=2+\\mathcal {O}(\\theta ^{-1}),\\\\\\widetilde{G}^{(n)}(0)&=\\mathcal {O}(1),\\quad n\\ge 3,\\\\G^{(n)}(\\theta )&=\\theta ^{-n+1}\\widetilde{G}^{(n)}(0).\\end{aligned}$ For asymptotic analysis we need to control the real part of $\\widetilde{G}$ , which we denote $\\widetilde{g}(X,Y):=\\operatorname{Re}(\\widetilde{G}(X+IY))=\\frac{g(\\theta +\\theta X,\\theta Y)}{\\theta }.$ In Lemmas REF , REF , and REF we will analyze the steep descent properties for $\\widetilde{G}$ (those are analogs of Lemmas REF -REF ), that we use to prove Proposition REF and Proposition REF below.",
"Proposition 6.1 Uniformly for $w,w^{\\prime }$ in a bounded set of $\\mathcal {C}_w$ , $\\lim _{N\\rightarrow \\infty } K_\\theta (w,w^{\\prime }) = \\widetilde{K}_{{\\rm CDRP}}(w,w^{\\prime })$ where $\\widetilde{K}_{{\\rm CDRP}}(w,w^{\\prime })= \\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}.$ First remark that the only dependence on $N$ in the kernel (REF ) is in the factor $\\exp \\left[N \\left(G(\\Phi (w))-G(\\Phi (z))\\right)\\right]=\\exp \\left[N\\theta \\left(\\widetilde{G}(w\\sigma /\\theta )-\\widetilde{G}(z\\sigma /\\theta )\\right)\\right].$ Let $w,w^{\\prime }$ be in a bounded set of $\\mathcal {C}_w$ around the origin.",
"For $N$ large enough and $w$ bounded in $\\mathcal {C}_w$ , $\\operatorname{Re}(w\\sigma +1)>1/2$ and $\\operatorname{Re}((z-w)\\sigma )\\in (0,1)$ so that we have $\\ell =0$ and $p=1$ , i.e., in this case $\\mathcal {C}_z=\\lbrace \\frac{1}{4\\sigma }+Iy,y\\in \\mathbb {R}\\rbrace $ .",
"We have $\\begin{aligned}N G(\\Phi (w))=N G(\\theta +w\\sigma )&=N G(\\theta )+\\frac{N}{6} G^{(3)}(\\theta )\\sigma ^3 w^3+ \\mathcal {O}(N w^4/\\theta ^3)\\\\&=N G(\\theta )-\\frac{N w^3 \\sigma ^3}{6\\theta ^2}+\\mathcal {O}(N w^4/\\theta ^3,N w^3/\\theta ^3)\\\\&=N G(\\theta )-\\frac{w^3}{3}+\\mathcal {O}(w^4/\\theta )\\end{aligned}$ where the $\\theta $ -dependence in the error term follows from $G^{(4)}(\\theta )=\\mathcal {O}(\\theta ^{-3})$ and then we used the expansion (REF ) for $G^{(3)}(\\theta )$ .",
"We divide the integral over $z$ into two parts: (a) $|\\operatorname{Im}(z)|>\\theta ^{1/3}$ and (b) $|\\operatorname{Im}(z)|\\le \\theta ^{1/3}$.",
"(a) Contribution of the integration over $|\\operatorname{Im}(z)|>\\theta ^{1/3}$ .",
"For $w,w^{\\prime }$ on $\\mathcal {C}_w$ of order 1 and $z\\in \\mathcal {C}_z$ , $|z-w^{\\prime }|\\ge \\mathcal {O}(1)$ , $|\\sin (\\pi (z-w)\\sigma )^{-1}|=\\mathcal {O}(1)$ .",
"So, $|K_\\theta (w,w^{\\prime })|\\le \\mathcal {O}(\\theta )\\int _{\\theta ^{-2/3}}^\\infty dY \\exp \\left[N\\theta \\left(\\widetilde{g}(0,0)- \\widetilde{g}((4\\sigma \\theta )^{-1},Y)\\right)\\right].$ From Lemma REF we have that $-\\widetilde{g}((4\\sigma \\theta )^{-1},Y)$ is strictly decreasing with derivative going to $-\\infty $ as $Y$ goes to infinity.",
"Then, the integral over $Y$ is bounded and of leading order $\\exp \\left[N\\theta \\left(\\widetilde{g}(0,0)- \\widetilde{g}((4\\sigma \\theta )^{-1},\\theta ^{-2/3})\\right)\\right].$ The estimates for small $Y$ of Lemma REF with $X=(4\\sigma \\theta )^{-1}$ and $Y=\\theta ^{-2/3}$ lead then to $(\\ref {PFeq74})\\le \\exp \\left[N\\theta \\left(\\widetilde{g}(0,0)- \\widetilde{g}((4\\sigma \\theta )^{-1},0)-\\frac{1}{12\\theta ^{8/3}}+\\mathcal {O}(\\theta ^{-11/3})\\right)\\right]=\\mathcal {O}(1) \\exp \\left(-\\frac{{5/6} N^{1/6}}{12}\\right)$ where we also used the fact that $\\widetilde{g}((4\\sigma \\theta )^{-1},0)=\\widetilde{g}(0,0)+\\mathcal {O}(\\theta ^{-3})$ .",
"Thus, the contribution in the kernel from the integration over $|\\operatorname{Im}(z)|\\ge \\theta ^{1/3}$ is of order $e^{-c N^{1/6}}$ for some positive constant $c>0$ .",
"(b) Contribution of the integration over $|\\operatorname{Im}(z)|\\le \\theta ^{1/3}$ .",
"We need to estimate $\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/3}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }}.$ Unlike the scaling where we have proven the convergence to the GUE Tracy-Widom distribution, in this case the sine function survives in the limiting expression and we do not have to employ the quartic term in the estimates (since it was used only to control the error term of the sine).",
"First we verify that the convergence is controlled by the third order term.",
"For this purpose, we set $z=Iy +1/(4\\sigma )$ .",
"Then, using (REF ) we obtain (as in (REF )) $-N G(\\theta +\\sigma z) =-N G(\\theta )+\\frac{z^3}{3}+\\mathcal {O}(z^4/\\theta ).$ The real part of the cubic term is given by $\\operatorname{Re}\\left(\\frac{z^3}{3}\\right)=-\\frac{y^2}{4\\sigma ^2}+\\frac{1}{192\\sigma ^3}.$ In our situation we have $|y|\\le \\theta ^{1/3}$ , therefore $-\\frac{y^2}{4\\sigma ^2}$ dominates $\\mathcal {O}(z^4/\\theta )$ for large $\\theta $ (since $y^2=\\mathcal {O}(\\theta ^{2/3})$ ).",
"We have $(\\ref {PFeq2.20})=\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/3}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3+\\mathcal {O}(w^4/\\theta ;z^4/\\theta )}}{z-w^{\\prime }}.$ We divide the integration in (b.1) $\\theta ^{1/6}\\le |y|\\le \\theta ^{1/3}$ and (b.2) $|y|\\le \\theta ^{1/6}$ .",
"Since the quadratic term in $y$ from (REF ) dominates the others, the contribution of (b.1) is only of order $\\mathcal {O}(e^{-c_1 \\theta ^{1/3}})=\\mathcal {O}(e^{-c_2 N^{1/6}})$ for some constants $c_1,c_2>0$ .",
"The contribution from (b.2) is given by $\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/6}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3+\\mathcal {O}(w^4/\\theta ;z^4/\\theta )}}{z-w^{\\prime }}.$ For $|y|\\le \\theta ^{1/6}$ , $\\mathcal {O}(z^4/\\theta )=\\mathcal {O}(\\theta ^{-1/3})$ .",
"Using $|e^x-1|\\le |x| e^{|x|}$ for $x=\\mathcal {O}(z^4/\\theta )$ and then for $x=\\mathcal {O}(w^4/\\theta )$ we can delete the error term by making an error of order $\\mathcal {O}(\\theta ^{-1/3})=\\mathcal {O}(N^{-1/6})$ .",
"Thus, $(\\ref {PFeq2.24})=\\mathcal {O}(N^{-1/6})+\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/6}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}.$ Finally, extending the last integral to $\\frac{1}{4\\sigma }+I\\mathbb {R}$ we make an error of order $\\mathcal {O}(e^{-c_3 \\theta ^{1/3}})$ for some constant $c_3>0$ .",
"Putting all the above estimates together we obtain that, for $w,w^{\\prime }\\in \\mathcal {C}_w$ in a bounded set around 0, $K_\\theta (w,w^{\\prime })=\\mathcal {O}(N^{-1/6})+\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}.$ Proposition 6.2 For any $w,w^{\\prime }$ in $\\mathcal {C}_w$ , uniformly for all $N$ large enough, $|K_\\theta (w,w^{\\prime })|\\le C e^{-|\\operatorname{Im}(w)|}$ for some constant $C$ .",
"First recall the expression of the kernel, $\\begin{aligned}K_\\theta (w,w^{\\prime })&=\\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }}\\\\&=S^{-w\\sigma }e^{N G(\\Phi (w))-N G(\\theta )} \\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{z\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{N G(\\theta )-N G(\\Phi (z))}}{z-w^{\\prime }}.\\end{aligned}$ As in the proof of Proposition REF , the dependence on $w^{\\prime }$ is marginal because (a) we can choose the integration variable $z$ such that $|z-w^{\\prime }|\\ge 1/(4\\sigma )$ and (b) we will get the bound through evaluating the absolute value of the integrand of (REF ).",
"Case 1: $w\\in \\lbrace -1/(4\\sigma )+Iy, |y|\\le r^*/\\sigma \\rbrace $ with $r^*=\\sqrt{\\theta /2-1/16}$ .",
"In this case, the integration path for $z$ is $1/(4\\sigma )+I\\mathbb {R}$ and no extra contributions from poles of the sine are present.",
"The factor $1/\\sin (\\pi (z-w)\\sigma )$ is uniformly bounded from above.",
"Doing the change of variable $z=\\frac{1}{4\\sigma }+I\\frac{Y\\, \\theta }{\\sigma }$ we get $|K_\\theta (w,w^{\\prime })|\\le \\mathcal {O}(1) e^{N \\operatorname{Re}(G(\\theta +w\\sigma ))-N G(\\theta )}\\int _{\\mathbb {R}}dY e^{N\\theta \\left(\\widetilde{g}(0,0)-\\widetilde{g}(\\tilde{\\varepsilon },Y)\\right)} \\theta $ with $\\tilde{\\varepsilon }=1/(4\\theta )$ .",
"The estimates as in the proof of Proposition REF on the integral over $Y$ yield $(\\ref {PFeq2.30})\\le \\mathcal {O}(1)\\times e^{N \\operatorname{Re}(G(\\theta +w\\sigma ))-N G(\\theta )} =\\mathcal {O}(1)\\times e^{N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)}.$ Since $|w\\sigma /\\theta |\\le \\mathcal {O}(\\theta ^{-1/2})$ is small, we can use Taylor expansion and with (REF ) we obtain $N\\theta \\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0) = -\\frac{1}{3} w^3(1+\\mathcal {O}(\\theta ^{-1}))+\\frac{\\sigma }{6\\theta }w^4(1+\\mathcal {O}(\\theta ^{-1})),$ substituting $w=-1/(4\\sigma )+Iy$ and taking the real part we get $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0) = -\\frac{1}{4\\sigma } y^2+\\frac{\\sigma }{6\\theta }y^4+\\mathcal {O}(1)+\\mathcal {O}(y^3/\\theta ,y^4/\\theta ^2).$ Now, for $|y|\\le \\sqrt{\\theta /2}/\\sigma $ , $\\frac{\\sigma }{6\\theta }y^4\\le \\frac{1}{12\\sigma } y^2$ and the quadratic term dominates $\\mathcal {O}(y^3/\\theta ,y^4/\\theta ^2)$ for large $\\theta $ .",
"Therefore, for all $\\theta $ large enough, we have $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0) \\le -\\frac{1}{8\\sigma } y^2+\\mathcal {O}(1).$ Consequently, $|K_\\theta (w,w^{\\prime })|\\le \\mathcal {O}(1) e^{-\\frac{1}{8\\sigma } |\\operatorname{Im}(w)|^2}\\le C e^{-|\\operatorname{Im}(w)|}$ for some finite constant $C$ .",
"Case 2: $w\\in \\lbrace (e^{It}-1)\\theta /\\sigma , t^*\\le |t|\\le \\pi /2\\rbrace \\cup \\lbrace -|y|+Iy,|y|\\ge \\theta /\\sigma \\rbrace $.",
"We divide the estimation of the bound by dividing into the contributions from (a) integration over $\\frac{p}{4\\sigma }+I\\mathbb {R}$ with $p\\in \\lbrace 1,2\\rbrace $ depending on $w$ (see the definitions after (REF )) and (b) integration over the circles $B_{w+k/\\sigma }$ , $k=1,\\ldots ,\\ell $ .",
"Case 2(a).",
"First notice that the estimate (REF ) of Case 1 still holds with the minor difference that $\\tilde{\\varepsilon }=p/(4\\theta )$ where $p\\in \\lbrace 1,2\\rbrace $ depending on the value of $w$ .",
"Then, also (REF ) still holds, so that we need only to estimate $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)$ .",
"For $w\\in \\lbrace (e^{It}-1)\\theta /\\sigma , t^*\\le |t|\\le \\pi /2\\rbrace $ , in Lemma REF we show that $\\widetilde{g}(\\cos (t)-1,\\sin (t))-\\widetilde{g}(0,0)\\le -\\sin (t)^4/16$ .",
"Replacing $\\operatorname{Im}(w)=\\sin (t) \\theta /\\sigma $ and using $|\\operatorname{Im}(w)|\\ge \\sqrt{\\theta /2-1/16}$ we obtain $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)\\le -c_1 |\\operatorname{Im}(w)|^4/\\theta \\le -c_2 |\\operatorname{Im}(w)| \\sqrt{\\theta } \\le -|\\operatorname{Im}(w)|$ for all $\\theta $ large enough, where $c_1,c_2$ are some (explicit) constants.",
"This is the desired bound.",
"For $w\\in \\lbrace -|y|+Iy,|y|\\ge \\theta /\\sigma \\rbrace $ , from Lemma REF it follows that there exists a constant $c_3>0$ such that $\\partial _Y\\widetilde{g}(-Y,Y)\\le -c_3$ and from Lemma REF we know that $\\widetilde{g}(-1,1)-\\widetilde{g}(0,0)\\le -1/16$ .",
"Thus, for $c_4=\\min \\lbrace \\sigma /16,c_3\\rbrace $ it holds $\\widetilde{g}(-1,1)-\\widetilde{g}(0,0)\\le -c_4 Y$ for all $|Y|\\ge 1/\\sigma $ .",
"This means that $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)\\le -c_4 N \\theta |\\operatorname{Im}(w)|/\\theta \\le -|\\operatorname{Im}(w)|$ for $N$ large enough, giving us the needed bound.",
"Case 2(b).",
"It remains to check that the extra contributions of the poles of the sine also tend to zero exponentially in $|\\operatorname{Im}(w)|$ .",
"The contribution of the integration over $B_{w+k/\\sigma }$ is (up to a $\\pm $ sign depending on $k$ ) given by $\\frac{S^k e^{N G(\\Phi (w))-N G(\\Phi (w+k/\\sigma ))}}{w+k/\\sigma -w^{\\prime }}.$ Let us set $\\widetilde{h}(Y,k):=\\widetilde{g}(-Y,Y)-\\widetilde{g}(-Y+k,Y)$ .",
"From Lemma REF it follows that the largest contribution comes from the integration over $B_{w+1/\\sigma }$ .",
"We have at most $\\mathcal {O}(|\\operatorname{Im}(w)|)$ poles and also $|w+k/\\sigma -w^{\\prime }|\\ge \\mathcal {O}(1/\\theta )$ (the worst case is at the junction between the arc of circle and the straight lines).",
"Thus, the contribution of all the poles is bounded by $\\mathcal {O}(\\theta |\\operatorname{Im}(w)|) S^{|\\operatorname{Im}(w)|} e^{N G(\\Phi (w))-N G(\\Phi (w+1/\\sigma ))} =\\mathcal {O}(\\theta |\\operatorname{Im}(w)|) S^{|\\operatorname{Im}(w)|} e^{N\\theta \\widetilde{h}(Y,1)},$ where $Y=|\\operatorname{Im}(w)|\\sigma /\\theta $ .",
"We consider separately the cases $1/\\sqrt{2\\theta }\\le Y\\le 1$ and $Y>1$ : (1) For $1/\\sqrt{2\\theta }\\le Y\\le 1$ , the bound on $\\partial _k \\widetilde{h}(Y,k)$ leads to $\\widetilde{h}(Y,1)\\le -Y/8$ .",
"(2) For $Y>1$ we know that $\\widetilde{h}(Y,1)<0$ and $\\partial _k \\widetilde{h}(Y,k)|_{k=0}\\simeq -Y$ as $Y\\rightarrow \\infty $ .",
"By the continuity of $\\widetilde{h}(Y,1)$ in $Y$ , there exists a constant $c_5>0$ such that $\\widetilde{h}(Y,1)\\le -c_5 Y$ for all $Y\\ge 1$ .",
"Therefore, with $c_6=\\min \\lbrace c_5,1/8\\rbrace $ and inserting $Y=|\\operatorname{Im}(w)|\\sigma /\\theta $ we have $(\\ref {PFeq2.40})\\le \\mathcal {O}(\\theta |\\operatorname{Im}(w)|) S^{|\\operatorname{Im}(w)|} e^{-N c_5\\sigma |\\operatorname{Im}(w)|} \\le \\mathcal {O}(1) e^{-|\\operatorname{Im}(w)|}$ for $N$ large enough.",
"We have shown that also the contributions of the poles have the desired bound.",
"Lemma 6.3 The function $\\widetilde{g}(\\cos (t)-1,\\sin (t))$ is zero at $t=0$ and strictly decreasing for $t\\in (0,\\pi /2]$ .",
"For $t\\in [0,\\pi /2]$ and $\\theta $ large enough, $\\partial _t \\widetilde{g}(\\cos (t)-1,\\sin (t)) \\le -\\sin (t)(1-\\cos (t))/2$ so that $\\widetilde{g}(\\cos (t)-1,\\sin (t))- \\widetilde{g}(0,0)\\le - \\sin (t)^4/16.$ We have $\\widetilde{g}(\\cos (t)-1,\\sin (t))=\\theta ^{-1}g(\\theta \\cos (t),\\theta \\sin (t))$ , thus $\\begin{aligned}\\frac{\\partial \\widetilde{g}(\\cos (t)-1,\\sin (t))}{\\partial t}&=\\cos (t) g_2(\\theta \\cos (t),\\theta \\sin (t))-\\sin (t) g_1(\\theta \\cos (t),\\theta \\sin (t))\\\\&=-\\sum _{n=0}^\\infty \\frac{2\\theta ^2\\sin (t)(1-\\cos (t))(2n\\cos (t)+\\theta )}{(2n\\theta \\cos (t)+n^2+\\theta ^2)(n+\\theta )^2}\\end{aligned}$ is strictly negative for all $t\\in (0,\\pi /2]$ , which shows the first result.",
"We can further bound $\\begin{aligned}(\\ref {PFeq2.34})&\\le -\\sin (t)(1-\\cos (t)) \\sum _{n=0}^\\infty \\frac{2\\theta ^3}{(2n\\theta \\cos (t)+n^2+\\theta ^2)(n+\\theta )^2}\\\\&\\le -\\sin (t)(1-\\cos (t)) \\sum _{n=0}^\\infty \\frac{2\\theta ^3}{(n+\\theta )^4}=-\\sin (t)(1-\\cos (t))(\\tfrac{2}{3}+\\mathcal {O}(\\theta ^{-1})).\\end{aligned}$ Thus, for large enough $\\theta $ , the derivative is bounded by $-\\sin (t)(1-\\cos (t))/2$ , $t\\in [0,\\pi /2]$ .",
"Integrating over $[0,t]$ gives $\\widetilde{g}(\\cos (t)-1,\\sin (t))- \\widetilde{g}(0,0)\\le -(1-\\cos (t))^2/4\\le -\\sin (t)^4/16$ for $t\\in [0,\\pi /2]$ .",
"Lemma 6.4 For any $X\\ge 0$ , the function $\\widetilde{g}(X,Y)$ is strictly increasing for $Y>0$ , with $\\partial _Y \\widetilde{g}(X,Y)\\ge \\partial _Y \\widetilde{g}(0,Y)$ .",
"For $Y\\searrow 0$ , $\\partial _Y \\widetilde{g}(0,Y)= Y^3/3+\\mathcal {O}(Y^3/\\theta ;Y^5)$ , so that $\\widetilde{g}(X,Y)\\ge \\widetilde{g}(X,0)+Y^4/12+\\mathcal {O}(Y^4/\\theta ;Y^6).$ For $Y\\rightarrow \\infty $ it holds $\\partial _Y \\widetilde{g}(X,Y)\\sim Y$ .",
"We have $\\frac{\\partial \\widetilde{g}(X,Y)}{\\partial Y} =g_2(\\theta +\\theta X,\\theta Y)=\\sum _{n=0}^\\infty \\left(\\frac{\\theta Y}{(\\theta +n)^2}-\\frac{\\theta Y}{(\\theta +\\theta X+n)^2+\\theta ^2 Y^2}\\right),$ which is 0 for $Y=0$ and for $Y>0$ is strictly positive.",
"The inequality $\\partial _Y \\widetilde{g}(X,Y)\\ge \\partial _Y \\widetilde{g}(0,Y)=\\sum _{n=0}^\\infty \\frac{\\theta ^3 Y^3}{(\\theta +n)^2((\\theta +n)^2+\\theta ^2 Y^2)},$ whose expansion for small $Y$ and large $\\theta $ is given by $Y^3/3+\\mathcal {O}(Y^3/\\theta ;Y^5)$ .",
"For large $Y$ , the second term becomes irrelevant with respect to the first, so that $\\partial _Y \\widetilde{g}(X,Y)\\sim \\theta \\kappa _\\theta Y= Y(1+\\mathcal {O}(\\theta ^{-1}))$.",
"Lemma 6.5 The function $\\widetilde{g}(-Y,Y)$ is strictly decreasing for $Y>0$ .",
"For $Y\\rightarrow \\infty $ it holds $\\partial _Y \\widetilde{g}(-Y,Y)\\sim -\\ln (Y)$ .",
"For $Y\\searrow 0$ we have $\\widetilde{g}(-Y,Y)=\\widetilde{g}(0,0)-\\tfrac{1}{3} Y^3+\\mathcal {O}(Y^3/\\theta ;Y^4)$ .",
"As in the proof of Lemma REF , we use (REF ) and (REF ) to obtain $\\begin{aligned}\\frac{\\partial \\widetilde{g}(-Y,Y)}{\\partial Y} &= g_2(\\theta -\\theta Y,\\theta Y)-g_1(\\theta -\\theta Y,\\theta Y)=-\\sum _{n=0}^\\infty \\frac{2 Y^2\\theta ^2}{(\\theta +n)((\\theta +n-\\theta Y)^2+\\theta ^2 Y^2)}.\\end{aligned}$ which is 0 for $Y=0$ and strictly negative for $Y>0$ .",
"We can rewrite the sum with the variable $\\eta =n/\\theta $ .",
"Then, for large (but still fixed) $\\theta $ the sum over $\\eta \\in \\lbrace 0,1,2,\\ldots \\rbrace /\\theta $ is very close to the integral over $\\eta \\in [0,\\infty )$ .",
"From this one deduces that for large $Y$ , $\\partial _Y \\widetilde{g}(-Y,Y)\\sim -\\ln (Y)$ .",
"The asymptotics for $Y\\searrow 0$ is obtained by writing the Taylor series of $\\widetilde{G}(Z)$ around $Z=1$ and taking the real part of it.",
"Lemma 6.6 Let $Y>0$ be fixed.",
"The function $\\widetilde{h}(Y,k):=\\widetilde{g}(-Y,Y)-\\widetilde{g}(-Y+k,Y)$ satisfies $\\widetilde{h}(Y,0)=0$ , $\\widetilde{h}(Y,k)$ is strictly decreasing for $k\\in [0,Y]$ .",
"For any $\\delta \\in (0,1)$ , $Y\\ge \\delta $ , $\\widetilde{h}(Y,k)$ is strictly decreasing in $k\\in [0,Y+\\delta /2]$ .",
"For $Y\\rightarrow \\infty $ , $\\partial _k \\widetilde{h}(Y,k)|_{k=0}\\sim -Y$ .",
"For $Y\\le 1$ , $\\partial _k \\widetilde{h}(Y,k)\\le - kY/4$ for $k\\in [0,Y]$ .",
"The first statements follows directly from Lemma REF .",
"The asymptotics for large $Y$ can be obtained by approximating the sums in $\\partial _k \\widetilde{h}$ by integrals.",
"The bound for $Y\\le 1$ follows from (REF ) with $k\\rightarrow \\theta k$ , $y\\rightarrow \\theta Y$ , and $\\sum _{n=0}^\\infty (\\theta +n)^{-3}\\ge 1/(2\\theta ^2).$"
],
[
"Proof of Theorem ",
"Now we consider the perturbed case, where $a_k:=\\theta +b_k,\\quad k=1,\\ldots ,m.$ Then, the change of variable as in (REF ) leads to the kernel $K_\\theta (w,w^{\\prime }):=\\sigma K_u(\\Phi (w),\\Phi (w^{\\prime }))=\\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }} \\prod _{k=1}^m P(w,z,b_k)$ where the perturbation term is $P(w,z,b_k)=\\frac{\\Gamma (\\sigma w-b_k) \\Gamma (\\Phi (z)) \\theta ^{\\sigma w}}{\\Gamma (\\sigma z-b_k)\\Gamma (\\Phi (w))\\theta ^{\\sigma z}}.$ The difference from Theorem REF (a) is that now (as it was the case for Theorem REF (b)), the paths $\\mathcal {C}_z$ and $\\mathcal {C}_w$ have to be locally modified around the critical point, $\\theta $ , so that they remains on the right of all the $b_1/\\sigma ,\\ldots ,b_m/\\sigma $ , see Figure REF for an illustration.",
"Figure: Perturbation of the integration paths, compare with Figure (right).",
"The white dots on the right are the values of b 1 /σ,...,b m /σb_1/\\sigma ,\\ldots ,b_m/\\sigma .We just have to show that $\\lim _{N\\rightarrow \\infty }\\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP},b})_{L^2(\\mathbb {R}_+)}$ with $K_{{\\rm CDRP},b}$ as in (REF ).",
"The proof is a minor modification of the one of Theorem REF (b).",
"The local modification of the paths have no influence on the bounds for large $z$ and/or for large $w$ .",
"This is because $N G(\\theta +b_k)-N G(\\theta ) = \\mathcal {O}(1)$ and the path for $z$ is the same away from a distance $\\mathcal {O}(1)$ from the origin.",
"What remains to be clarified is the limit kernel.",
"We can choose the path $\\mathcal {C}_w$ to be as before with a small perturbation (e.g.",
"a circle) around 0 so that it passes on the right of the all the $b_k$ 's.",
"Then, we modify the path $\\mathcal {C}_z$ in the same way, i.e., by doing the same small perturbation but shifted to the right by $1/2\\sigma $ (see Figure REF too).",
"This ensures that we do not get extra poles from the sine.",
"Finally, for the pointwise convergence of the kernel the new term remaining is $\\lim _{N\\rightarrow \\infty }P(w,z,b_k)=\\frac{\\Gamma (\\sigma w-b_k)}{\\Gamma (\\sigma z-b_k)}.$ Indeed, since $\\Phi (z)=\\theta +\\mathcal {O}(1)$ and $\\Phi (w)=\\theta +\\mathcal {O}(1)$ , then $\\lim _{N\\rightarrow \\infty }\\frac{\\Gamma (\\Phi (z)) \\theta ^{\\sigma w}}{\\Gamma (\\Phi (w))\\theta ^{\\sigma z}}=1$ .",
"Finally one reformulate the Fredholm determinant into one on $L^2(\\mathbb {R}_+)$ in the same way as the unperturbed case of Lemma REF .",
"The only small difference is that we the first step requires $\\operatorname{Re}(z-w^{\\prime })>0$ , which holds only for $b_k<\\frac{1}{4}$ , for all $k$ .",
"Under this condition the rewriting holds.",
"By looking at the final expressions one verifies that both sides are analytic in the parameters $b_1,\\ldots ,b_m$ .",
"Thus we have equality by analytic continuation.",
"This ends the proof of Theorem REF (b)."
],
[
"Proof of Proposition ",
"This closely follows the proof of [16] Proposition 3.2.8 and Corollary 3.2.10.",
"However, in that case the contour playing the role of $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ is bounded whereas it is unbounded presently.",
"As such, some additional estimates must be made, so we include the entire proof here.",
"First observe that we may combine the $q$ -moments $\\mu _{k}=\\langle q^{k\\lambda _N}\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ (see definitions in Section REF ) into a generating function $G_{q}(\\zeta )=\\sum _{k\\ge 0} \\frac{(\\zeta /(1-q))^k}{k_q!}",
"\\left\\langle q^{k\\lambda _N}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ where $k_q!= (q;q)_n/(1-q)^n$ and $(a;q)_k=(1-a)\\cdots (1-aq^{k-1})$ (when $k=\\infty $ the product is infinite, though convergent since $|q|<1$ ).",
"The convergence of the series defining $G_{q}(\\zeta )$ follows from the fact that $q^{k\\lambda _N}\\le 1$ and $\\frac{(\\zeta /(1-q))^k}{k_q!}",
"= \\frac{\\zeta ^k}{(1-q)\\cdots (1-q^k)},$ which shows geometric decay for large enough $k$ .",
"This justifies writing $G_{q}(\\zeta )=\\left\\langle \\sum _{k\\ge 0} \\frac{(\\zeta /(1-q))^k}{k_q!}",
"q^{k\\lambda _N}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )} = \\left\\langle \\frac{1}{\\left(\\zeta q^{\\lambda _N};q\\right)_{\\infty }}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ where the second equality follows from the $q$ -Binomial theorem [6].",
"It now suffices to show that $G_{q}(\\zeta ) =\\det (\\mathbb {1}+K)$ as in the statement of the proposition.",
"From now on, all contour integrals are along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"Observe that we can rewrite the summation in the definition of $\\mu _k$ so that $\\mu _k \\frac{\\zeta ^k}{k_q!",
"}= \\sum _{L\\ge 0}\\sum _{\\begin{array}{c}m_1,m_2,\\ldots \\\\ \\sum m_i =L \\\\ \\sum i m_i = k\\end{array}} \\frac{1}{(m_1+m_2+\\cdots )!",
"}\\cdot \\frac{(m_1+m_2+\\cdots )!}{m_1!",
"m_2!",
"\\cdots }\\int \\cdots \\int I_L(\\lambda ;w;\\zeta ) \\prod _{j=1}^{L} \\frac{dw_j}{2\\pi I},$ where $w=(w_1,\\ldots , w_L)$ , $\\lambda =(\\lambda _1,\\ldots , \\lambda _L)$ and is specified by $\\lambda =1^{m_1}2^{m_2}\\cdots $ , and where the integrand is $I_L(\\lambda ;w;\\zeta ) = \\det \\left[\\frac{1}{w_i q^{\\lambda _i}-w_j}\\right]_{i,j=1}^{L} \\prod _{j=1}^{L}(1-q)^{\\lambda _j}\\zeta ^{\\lambda _j} f(w_j)f(qw_j)\\cdots f(q^{\\lambda _j-1}w_j).$ The term $\\tfrac{(m_1+m_2+\\cdots )!}{m_1!",
"m_2!",
"\\cdots }$ is a multinomial coefficient and can be removed by replacing the inner summation by $\\sum _{n_1,\\ldots ,n_L\\in \\mathcal {L}_{k,m_1,m_2,\\ldots }} \\int \\cdots \\int I_L(n;w;\\zeta )\\frac{dw_j}{2\\pi I},$ with $n=(n_1,\\ldots ,n_L)$ and where $\\mathcal {L}_{k,m_1,m_2,\\ldots } = \\lbrace n_1,\\ldots ,n_L\\ge 1: \\sum _i n_i = k \\textrm { and for each } j\\ge 1, m_j \\textrm { of the } n_i \\textrm { equal } j\\rbrace .$ This gives $\\mu _k \\frac{\\zeta ^k}{k_q!}",
"= \\sum _{L\\ge 0}\\frac{1}{L!}",
"\\sum _{\\begin{array}{c}n_1,\\ldots ,n_L\\ge 1\\\\ \\sum n_i=k\\end{array}} \\int \\cdots \\int I_L(n;w;\\zeta )\\frac{dw_j}{2\\pi I}.$ Now we may sum over $k$ which removes the requirement that $\\sum _i n_i = k$ .",
"This yields that the left-hand side of equation (REF ) can be expressed as $\\sum _{L\\ge 0} \\frac{1}{L!}",
"\\sum _{n_1,\\ldots ,n_L\\ge 1} \\int \\cdots \\int \\det \\left[\\frac{1}{q^{n_i}w_i-w_j}\\right]_{i,j=1}^{L} \\prod _{j=1}^{L} (1-q)^{n_j}\\zeta ^{n_j} f(w_j)f(qw_j)\\cdots f(q^{n_j-1}w_j) \\frac{dw_j}{2\\pi I} .$ This is the definition of the Fredholm determinant expansion $\\det (\\mathbb {1}+K)$ , as desired.",
"As these were purely formal manipulations, at this point to complete the proof we must justify the rearrangements in the above argument.",
"In order to do this, we will show that the double summation of (REF ) is absolutely convergent.",
"This is the point at which the unboundedness of the $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ contour introduces a slight divergence from the analogous proof of [16] Proposition 3.2.8 where the contour was bounded and of finite length.",
"Basically, the absolute convergence follows from the exponential decay of the function $f$ as the real part of $w$ increases to positive infinity, combined with Hadamard's inequality.",
"Let us bound the absolute value of the integrand in (REF ).",
"Note that by assumption $q^{n_i}w_i/w_j-1$ is bounded from 0 uniformly as $w_i$ , $w_j$ , and $n_i$ vary.",
"Thus, it follows that for some finite constant $B_1$ , $\\left| \\det \\left[\\frac{1}{q^{n_i}w_i-w_j}\\right]_{i,j=1}^{L}\\right| \\le B_1^{L} L^{L/2}.$ Since the function $f(w)$ is bounded as $w$ varies and has exponential decay with respect to the real part of $w$ , we can replace $\\left|f(w_j)f(qw_j)\\cdots f(q^{n_j-1}w_j)\\right| \\le (B_2)^{n_j}e^{-c\\operatorname{Re}(w_j)}$ for constants $c>0$ and $B_2<\\infty $ .",
"Thus we find that $|(\\ref {fredexpabove})|\\le \\sum _{L\\ge 0} \\frac{1}{L!}",
"B_1^L L^{L/2} \\left(\\sum _{n\\ge 1} (B_2 (1-q) \\zeta )^n \\int \\frac{|dw|}{2\\pi } e^{-c \\operatorname{Re}(w)}\\right)^L.$ Since $w$ is being integrated along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , the integral over $w$ is bounded by some constant $B_3<\\infty $ .",
"Finally, for $|\\zeta |$ small enough the geometric series converges and it is bounded by a constant $B_4$ .",
"Therefore $(\\ref {eq7.12})\\le \\sum _{L\\ge 0} \\frac{(B_1 B_3 B_4)^{L} L^{L/2}}{L!",
"}<\\infty .$ Thus we have shown that the double summation in (REF ) is absolutely convergent, completing the proof of Proposition REF ."
],
[
"Proof of Theorem ",
"This theorem and its proof are adapted from [16] Theorem 3.2.11.",
"However, in that theorem, the $w$ -contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , was of finite length and the $s$ -contour $\\widetilde{\\mathcal {D}}_{w}$ was just a vertical line.",
"The need for slightly more involved contours comes from the unboundedness of the $w$ -contour and the necessity that $\\tilde{K}_{\\zeta }(w,w^{\\prime })$ goes to zero sufficiently fast as $|w|$ grows along the $w$ -contour.",
"The starting point for this proof is Proposition REF .",
"There are, however, two issues we must deal with.",
"First, the operator in the proposition acts on a different $L^2$ space; second, the equality is only proved for $|\\zeta |<C^{-1}$ for some constant $C>1$ .",
"We split the proof into three steps.",
"Step 1: We present a general lemma which provides an integral representation for an infinite sum.",
"Step 2: Assuming $\\zeta \\in \\lbrace \\zeta :|\\zeta |<C^{-1}, \\zeta \\notin \\mathbb {R}_{+}\\rbrace $ we derive equation (REF ).",
"Step 3: A direct inspection of the left-hand side of that equation shows that for all $\\zeta \\ne q^{-M}$ for $M\\ge 0$ the expression is well-defined and analytic.",
"The right-hand side expression can be analytically extended to all $\\zeta \\notin \\mathbb {R}_{+}$ and thus by uniqueness of the analytic continuation, we have a valid formula on all of $\\mathbb {R}_{+}$ ."
],
[
"Step 1:",
"The purpose of the next lemma is to change that $L^2$ space we are considering and to replace the summation in Proposition REF by a contour integral.",
"Lemma 7.1 For all functions $g$ which satisfy the conditions below, we have the identity that for $\\zeta \\in \\lbrace \\zeta :|\\zeta |<1, \\zeta \\notin \\mathbb {R}_{+}\\rbrace $ $\\sum _{n=1}^{\\infty } g(q^n) (\\zeta )^n = \\frac{1}{2\\pi I} \\int _{C_{1,2,\\ldots }} \\Gamma (-s)\\Gamma (1+s)(-\\zeta )^s g(q^s) ds,$ where the infinite contour $C_{1,2,\\ldots }$ is a negatively oriented contour which encloses $1,2,\\ldots $ and no singularities of $g(q^s)$ , and $z^s$ is defined with respect to a branch cut along $z\\in \\mathbb {R}_-$ .",
"For the above equality to be valid the left-hand-side must converge, and the right-hand-side integral must be able to be approximated by integrals over a sequence of contours $C_{k}$ which enclose the singularities at $1,2,\\ldots , k$ and which partly coincide with $C_{1,2,\\ldots }$ in such a way that the integral along the symmetric difference of the contours $C_{1,2,\\ldots }$ and $C_{k}$ goes to zero as $k$ goes to infinity.",
"The identity follows from $\\underset{{s=k}}{\\mathrm {Res}}\\Gamma (-s)\\Gamma (1+s) = (-1)^{k+1}$ .",
"For this step let us assume that $\\zeta \\in \\lbrace \\zeta :|\\zeta |<C^{-1}, \\zeta \\notin \\mathbb {R}_{+}\\rbrace $ .",
"We may rewrite equation (REF ) as $K(n_1,w_1;n_2,w_2) = \\zeta ^{n_1} g_{w_1,w_2}(q^{n_1})$ where $g$ is given in equation (REF ).",
"Writing out the $M^{th}$ term in the Fredholm expansion we have $\\frac{1}{M!}",
"\\sum _{\\sigma \\in S_M} \\operatorname{sgn}(\\sigma )\\prod _{j=1}^{M} \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }} \\frac{dw_j}{2\\pi I} \\sum _{n_j=1}^{\\infty } \\zeta ^{n_j} g_{w_j,w_{\\sigma (j)}}(q^{n_j}).$ In order to apply Lemma REF we need to define the sequence of contours $C_{k}$ (in fact we need only specify the contours for $k$ large).",
"Let $C_{k}$ be composed of the union of two parts – the first part is the portion of the contour $\\widetilde{\\mathcal {D}}_{w}$ which lies within the ball of radius $k+1/2$ centered at the origin; the second part is the arc of the boundary of that ball which causes the union to be a closed contour which encloses $\\lbrace 1,2,\\ldots , k\\rbrace $ and no other integers.",
"The contours $C_{k}$ are oriented positively and illustrated in the left-hand-side of Figure REF .",
"The infinite contour $C_{1,2,\\ldots }$ is chosen to be $\\widetilde{\\mathcal {D}}_{w}$ oriented as in the statement of the theorem (decreasing imaginary part).",
"By the definition of the contours $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and $\\widetilde{\\mathcal {D}}_{w}$ we are assured that the contours $C_k$ do not contain any poles beyond those of the Gamma function $\\Gamma (-s)$ .",
"This is due to the fact that the contours have been chosen such that as $s$ varies, $q^sw$ stays entirely to the left of $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and hence does not touch $w^{\\prime }$ .",
"Figure: Left: The contour C k C_{k} composed of the union of two parts – the first part is the portion of the contour 𝒟 ˜ w \\widetilde{\\mathcal {D}}_{w} which lies within the ball of radius k+1/2k+1/2 centered at the origin; the second part is the arc of that ball which causes the union to be a closed contour which encloses {1,2,...,k}\\lbrace 1,2,\\ldots , k\\rbrace and no other integers.",
"Right: The symmetric difference between C k C_k and 𝒟 ˜ w \\widetilde{\\mathcal {D}}_{w} is given by two parts: a semi-circle arc which we call C k arc C^{arc}_k and a portion of R+IℝR+I\\mathbb {R} with magnitude exceeding k+1/2k+1/2 which we call C k seg C^{seg}_k.In order to apply the above lemma we must also estimate the integral along the symmetric difference.",
"Identify the part of the symmetric difference given by the circular arc as $C^{arc}_k$ and the part given by the portion of $R+I\\mathbb {R}$ with magnitude exceeding $k+1/2$ as $C^{seg}_k$ (see the right-hand-side of Figure REF ).",
"First observe that for $w_1,w_2$ fixed, $g_{w_1,w_2}(q^s)$ stays uniformly bounded as $s$ varies along these contours.",
"Consider next $(-\\zeta )^s$ .",
"If $-\\zeta = r e^{i\\sigma }$ for $\\sigma \\in (-\\pi ,\\pi )$ and $r>0$ we have $(-\\zeta )^s =r^s e^{Is\\sigma }$ .",
"Writing $s=x+Iy$ we have $|(-\\zeta )^s| = r^{x}e^{-y\\sigma }$ .",
"Note that our assumption on $\\zeta $ corresponds to $r<1$ and $\\sigma \\in (-\\pi ,\\pi )$ .",
"Concerning the product of Gamma functions, recall Euler's Gamma reflection formula $\\Gamma (-s)\\Gamma (1+s) = \\frac{\\pi }{\\sin (-\\pi s)}.$ One readily confirms that for all $s$ : $\\operatorname{dist}(s,\\mathbb {Z})>c$ for some $c>0$ fixed, $\\left| \\frac{\\pi }{\\sin (-\\pi s)} \\right| \\le \\frac{c^{\\prime }}{e^{\\pi \\operatorname{Im}(s)}}$ for a fixed constant $c^{\\prime }>0$ which depends on $c$ .",
"Therefore, along the $C^{seg}_k$ contour where $s=R+Iy$ , $|(-\\zeta )^s\\Gamma (-s)\\Gamma (1+s)|\\sim r^R e^{-y\\sigma }e^{-\\pi |y|},$ and since $\\sigma \\in (-\\pi ,\\pi )$ is fixed, this product decays exponentially in $|y|$ and the integral goes to zero as $k$ goes to infinity.",
"Along the $C^{arc}_k$ contour, the product of Gamma functions still behaves like $c^{\\prime }e^{-\\pi |y|}$ for some fixed $c^{\\prime }>0$ .",
"Thus along this contour (again using the notation $s=x+Iy$ ) $|(-\\zeta )^s\\Gamma (-s)\\Gamma (1+s)| \\sim e^{-y\\sigma }r^x e^{-\\pi |y|}.$ Since $r<1$ and $-(\\pi +\\sigma )<0$ these terms behave like $e^{-c^{\\prime \\prime }(x+|y|)}$ ($c^{\\prime \\prime }>0$ fixed) along the circular arc.",
"Clearly, as $k$ goes to infinity, the integrand decays exponentially in $k$ (versus the linear growth of the length of the contour) and the conditions of the lemma are met.",
"Applying Lemma REF we find that the $M^{th}$ term in the Fredholm expansion can be written as $\\frac{1}{M!}",
"\\sum _{\\sigma \\in S_M} \\operatorname{sgn}(\\sigma )\\prod _{j=1}^{M} \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }} \\frac{dw_j}{2\\pi I}\\int _{\\widetilde{\\mathcal {D}}_{w_j}}\\frac{ds_j}{2\\pi I} \\,\\Gamma (-s)\\Gamma (1+s)(-\\zeta )^{s} g_{w_j,w_{\\sigma (j)}}(q^{s}).$ Therefore the determinant can be written as $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ as desired.",
"At this point we now make critical use of the choice for the contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ on which $w$ varies, since proving analyticity in $\\zeta $ of the Fredholm determinant requires the decay properties of the kernel with respect to $w$ varying along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"In order to analytically extend our formula we must prove two facts.",
"First, that the left-hand side of equation (REF ) is analytic for all $\\zeta \\notin \\mathbb {R}_{+}$ ; and second, that the right-hand side determinant is defined (i.e., its expansion is convergent) and analytic for all $\\zeta \\notin \\mathbb {R}_{+}$ .",
"Expand the left-hand side of equation (REF ) as $\\sum _{n=0}^{\\infty } \\frac{ \\mathbb {P}(\\lambda _N = n) }{(1-\\zeta q^n)(1-\\zeta q^{n+1})\\cdots },$ where $\\mathbb {P}=\\mathbb {P}_{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )}$ .",
"Observe that for any $\\zeta \\notin \\lbrace q^{-M}\\rbrace _{M=0,1,\\ldots }$ , within a neighborhood of $\\zeta $ the infinite products are uniformly convergent and bounded away from zero.",
"As a result the series is uniformly convergent in a neighborhood of any such $\\zeta $ which implies that its limit is analytic, as desired.",
"Turning to the Fredholm determinant, we must show that the series denoted by $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ is an analytic function of $\\zeta $ away from $\\mathbb {R}_{+}$ .",
"For this we will appeal to the fact that limits of uniformly absolutely convergent series of analytic functions are themselves analytic.",
"Recall that $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta }) = 1 + \\sum _{n=1}^{\\infty } \\frac{1}{n!}",
"\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{dw_1}{2\\pi I} \\cdots \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }} \\frac{dw_n}{2\\pi I} \\det (\\tilde{K}_{\\zeta }(w_i,w_j))_{i,j=1}^n.$ It is clear from the definition of $\\tilde{K}_{\\zeta }$ that $\\det (\\tilde{K}_{\\zeta }(w_i,w_j))_{i,j=1}^{n}$ is analytic in $\\zeta $ away from $\\mathbb {R}_{+}$ .",
"Thus any partial sum of the above series is analytic in the same domain.",
"What remains is to show that the series is uniformly absolutely convergent on any fixed neighborhood of $\\zeta $ not including $\\mathbb {R}_{+}$ .",
"Towards this end consider the $n^{th}$ term in the Fredholm expansion: $\\begin{aligned}F_n(\\zeta ) &=& \\frac{1}{n!}",
"\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{dw_1}{2\\pi I}\\cdots \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{dw_n}{2\\pi I} \\int _{\\widetilde{\\mathcal {D}}_{w_1}} \\frac{ds_1}{2\\pi I} \\cdots \\int _{\\widetilde{\\mathcal {D}}_{w_n}} \\frac{ds_n}{2\\pi I} \\det \\left(\\frac{1}{q^{s_i}w_i -w_j}\\right)_{i,j=1}^{n}\\\\&&\\times \\prod _{j=1}^{n} \\left(\\Gamma (-s_j)\\Gamma (1+s_j) (-\\zeta )^{s_j} \\exp \\big (\\gamma w_j(q^{s_j}-1)\\big )\\prod _{m=1}^{N}\\frac{(q^{s_j}w_j/\\tilde{a}_m;q)_{\\infty }}{(w_j/\\tilde{a}_m;q)_{\\infty }} \\right).\\end{aligned}$ We wish to bound the absolute value of this.",
"We may pull the absolute values inside the integration.",
"Now observe the following bounds which hold uniformly over all $w_j\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and then all $s_j\\in \\widetilde{\\mathcal {D}}_{w_j}$ .",
"For the first bound note that for all $z:|z|\\le 1$ , there exists a constant $c_q$ such that $|(z;q)_{\\infty }|<c_q$ .",
"From that it follows that for any $|z|>1$ , $|(z;q)_{\\infty }|\\le c_q|(z;q)_{k}|$ where $k$ is such that $|zq^k|\\le 1$ .",
"This $k$ is approximately $-\\ln (|z|)/\\ln (q)$ and hence bounded by $k\\le c_q^{\\prime } \\ln |z|$ for some other constant $c_q^{\\prime }$ .",
"Finally $|(z;q)_{k}|\\le c_q^{\\prime \\prime } |z|^k \\le c_q^{\\prime \\prime } |z|^{c_q^{\\prime }\\ln |z|}$ .",
"From this and the fact that $|q^{s_j}|<1$ (recall that $\\operatorname{Re}(s_j)>0$ along $\\widetilde{\\mathcal {D}}_{w_j}$ ), it follows that for $|w_j|\\ge 1$ along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ we can bound $\\left| \\prod _{m=1}^{N} \\frac{(q^{s_j}w_j/\\tilde{a}_m;q)_{\\infty }}{(w_j/\\tilde{a}_m;q)_{\\infty }}\\right| \\le c_1 |w_{j}|^{N c_q^{\\prime }\\ln (|w_j|/\\tilde{a})}$ for some constant $c_1$ and $\\tilde{a}=\\min _i\\lbrace \\tilde{a}_i\\rbrace $ .",
"For $|w_j|\\le 1$ along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , we can bound the above left-hand side by a constant and since $|w_j|$ is bounded from below along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , it follows that the above bound (REF ) holds for all $w\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , possibly with a modified value of $c_1$ .",
"By Hadamard's inequality and the conditions we have imposed on $\\widetilde{\\mathcal {D}}_{w_j}$ we get the crude bound $\\left| \\det \\left(\\frac{1}{q^{s_i}w_i -w_j}\\right)_{i,j=1}^{n} \\right| \\le c_2^n n^{n/2}.$ for some fixed constant $c_2>0$ .",
"Finally note that by the conditions we imposed in choosing the contours, for $w_j$ on $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and $s_j$ on $\\widetilde{\\mathcal {D}}_{w_j}$ , we have $\\operatorname{Re}\\left(w_j(q^s-1)\\right)\\le -c_{\\varphi } |w_j|$ where $c_{\\varphi }>0$ is a constant depending on $\\varphi \\in (0,\\pi /4)$ .",
"From this it follows that $\\left|\\exp \\big (\\gamma w_j(q^{s_j}-1)\\big )\\right| \\le \\exp \\big (-\\gamma c_{\\varphi }|w_j|\\big ).$ Taking the absolute value of (REF ) and bringing the absolute value all the way inside the integrals, we find that after plugging in the results of (REF ), (REF ) and (REF )For a complex contour $C$ and a function $f:C\\rightarrow \\mathbb {R}$ we write $\\int _C |dz|f(z)$ for the integral of $f$ along $C$ with respect to the arc length $|dz|$ .",
"$|F_n(\\zeta )| \\le \\frac{(c_1 c_2)^n n^{n/2}}{n!}",
"\\left(\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{|dw|}{2\\pi }\\int _{\\widetilde{\\mathcal {D}}_{w}}\\frac{|ds|}{2\\pi } \\left|\\Gamma (-s)\\Gamma (1+s) (-\\zeta )^{s}\\right| |w|^{N c_q^{\\prime }\\ln (|w|/\\tilde{a})} \\exp (-\\gamma c_{\\varphi }|w|)\\right)^n.$ We integrate this in the $s$ variables first.",
"For $\\zeta \\notin \\mathbb {R}_{+}$ we would like to bound $\\int _{\\widetilde{\\mathcal {D}}_{w}} \\frac{|ds|}{2\\pi } |\\Gamma (-s)\\Gamma (1+s) (-\\zeta )^{s}|$ for a neighborhood of $\\zeta $ which does not touch $\\mathbb {R}_{+}$ .",
"We divide the contour of integral into two regions and bound the integral along each region: (1) The portion of the contour from $R-Id$ to $1/2-Id$ and then vertical to $1/2$ and its reflection through the real axis; (2) The portion of the contour which is infinite from $R-I\\infty $ to $R-Id$ and then from $R+Id$ to $R+I\\infty $ .",
"Recall that by Remark REF we may assume that up to constants $d\\approx |w|^{-1}$ and $R\\approx \\ln |w|$ for $|w|$ large enough.",
"Case (1): By standard bounds $|\\Gamma (-s)\\Gamma (1+s)|\\le 1/d \\approx |w|$ (since $1/\\sin (x)\\approx 1/x$ near $x=0$ ).",
"Calling $r$ the maximal modulus over the neighborhood of $|\\zeta |$ in question, it follows that since the $s$ integral in Case (1) has length less than $2R$ (when $d<1/2$ ), the first part of the integral is bounded by a constant times $|w|\\ln |w| r^{c_3\\ln |w|}$ with a constant $c_3>0$ .",
"Case (2): The product of Gamma functions decays exponentially in $s$ and so the integral is estimated by $r^{R}$ which, by Remark REF is like $r^{c_3\\ln |w|}$ .",
"Summing up the above two cases we have that for $|w|$ large, $\\int _{\\widetilde{\\mathcal {D}}_{w}} \\frac{|ds|}{2\\pi } |\\Gamma (-s)\\Gamma (1+s) (-\\zeta )^{s}| \\le c_4 r^{c_3\\ln |w|} |w| \\ln |w|.$ This estimate can be plugged in to the right-hand side of (REF ) to reduce the bound to just an integral in the $w_j$ .",
"This integral factors and thus we have $\\begin{aligned}|F_n(\\zeta )| &\\le & \\frac{(c_1 c_2)^n n^{n/2}}{n!}",
"\\left(\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{|dw|}{2\\pi } |w|^{N c_q^{\\prime }\\ln (|w|/\\tilde{a})} c_4 r^{c_3\\ln |w|}|w|\\ln |w|\\exp (-\\gamma c_{\\varphi }|w|)\\right)^n.\\end{aligned}$ The integral in $|w|$ is clearly convergent due to the exponential decay (which easily overwhelms the growth of $|w|^{Nc_q^{\\prime }\\ln |w|}$ as well as the other terms).",
"Thus the right-hand side above is bounded by $c_5^n n^{n/2}/n!$ for some constant $c_5$ .",
"Thus $F_n(\\zeta )$ is absolutely convergent, uniformly over any fixed neighborhood of a $\\zeta \\notin \\mathbb {R}_{+}$ .",
"This implies that $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ is analytic in $\\zeta \\notin \\mathbb {R}_{+}$ and hence completes the proof of Step 3 and hence the proof of the theorem."
],
[
"Proof of Proposition ",
"By virtue of Lemma REF it suffices to show that for some $c,C>0$ , $|K_u(v,v^{\\prime })|\\le Ce^{-c|v|}.$ Before proving this let us recall from Definition REF the contours with which we are dealing.",
"The variable $v$ lies on $\\mathcal {C}_{\\alpha ,\\varphi }$ and hence can be written as $v=\\alpha -\\kappa \\cos (\\varphi ) \\pm I\\kappa \\sin (\\varphi )$ , for $\\kappa \\in \\mathbb {R}_+$ , where the $\\pm $ represents the two rays of the contour.",
"The $s$ variables lies on $\\mathcal {D}_{v}$ which depends on $v$ and has two parts: The portion (which we have denoted $\\mathcal {D}_{v,\\sqsubset }$ ) with real part bounded between $1/2$ and $R$ and imaginary part $\\pm d$ for $d$ sufficiently small, and the vertical portion (which we have denoted $\\mathcal {D}_{v,\\vert }$ ) with real part $R$ .",
"The condition on $R$ implies that $R\\ge -\\operatorname{Re}(v)+\\alpha +1$ and for our purposes we will assume $R=-\\operatorname{Re}(v)+\\alpha +1$ .",
"Let us denote by $h(s)$ the integrand in (), through which $K_u(v,v^{\\prime })$ is defined.",
"We split the proof into three steps.",
"Step 1: We show the integral of $h(s)$ over $s\\in \\mathcal {D}_{v,\\sqsubset }$ is bounded by an expression with exponential decay in $|v|$ .",
"Step 2: We show the integral of $h(s)$ over $s\\in \\mathcal {D}_{v,\\vert }$ is bounded by an expression with exponential decay in $|v|$ .",
"Step 3: We show that the integral of $h(s)$ over the entire contour $s\\in \\mathcal {D}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"The combination of these three steps imply the inequality (REF ) and hence complete the proof."
],
[
"Step 1:",
"We the various terms in $h(s)$ separately and develop bounds for each.",
"Let us write $s=x+Iy$ and note that along the contour $\\mathcal {D}_{v,\\sqsubset }$ , $y\\in [-d,d]$ for $d$ small, and $x\\in [1/2,R]$ .",
"Let us start with $e^{v\\tau s + \\tau s^2/2}$ .",
"The norm of the above expression is bounded by the exponential of the real part of the exponent.",
"For $s$ along $\\mathcal {D}_{v,\\sqsubset }$ $\\operatorname{Re}(v s + s^2/2) = x\\operatorname{Re}(v)+\\frac{x^2}{2}-y\\operatorname{Im}(v)-\\frac{y^2}{2} .$ We take $R=-\\operatorname{Re}(v)+\\alpha +1$ , $d$ sufficiently small, and the bound $\\operatorname{Re}(v)\\le \\tilde{c}^{\\prime }-c^{\\prime } |v|$ for some constants $c^{\\prime },\\tilde{c}^{\\prime }$ (depending on $\\varphi $ ), to deduce $\\operatorname{Re}(v s + s^2/2) \\le \\tilde{c} -c |v| x$ for some constants $c,\\tilde{c}>0$ , from which $|e^{v\\tau s + \\tau s^2/2}| \\le C e^{-c\\tau |v|x}.$ Let us now turn to the other terms in $h(s)$ .",
"We have $|u^s|\\le e^{x\\ln |u| - y\\arg (u)}$ and we may also bound $\\left|\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)}\\right| \\le C, \\qquad \\left|\\frac{1}{v+s-v^{\\prime }}\\right|\\le C,\\qquad |\\Gamma (-s)\\Gamma (1+s)|\\le C,$ for some constants $C>0$ (which may be different in each case).",
"The first bound comes from the functional equation for the Gamma function, and the last from the fact that $s$ is bounded away from $\\mathbb {Z}$ .",
"Combining these together shows that for $|v|$ large, the portion of the integral of $h(s)$ for $s$ in $\\mathcal {D}_{v,\\sqsubset }$ is bounded by (recall $s=x+Iy$ ) $\\int _{\\mathcal {D}_{v,\\sqsubset }}|ds| C^{\\prime } e^{-c\\tau |v|x + x\\ln |u| -\\arg (u)y} \\le C e^{-c|v|}$ for some constants $c,C>0$ .",
"Since for $|y|$ in a bounded set, everything starting from the integration path is clearly bounded, the bound holds.",
"As above, we consider the various terms in $h(s)$ separately and develop bounds for each.",
"Let us write $s=R+Iy$ and note that $s\\in \\mathcal {D}_{v,\\vert }$ corresponds to $y$ varying over all $|y|\\ge d$ .",
"As in Step 1, the most important bound will be that of $e^{v\\tau s + \\tau s^2/2}$ .",
"Observe that $\\operatorname{Re}(v s + s^2/2) = \\operatorname{Re}(v)R - \\operatorname{Im}(v) y + \\frac{R^2}{2} -\\frac{y^2}{2} = -\\frac{(y+\\operatorname{Im}(v))^2}{2} + \\frac{\\operatorname{Im}(v)^2}{2} +\\frac{R^2}{2} +\\operatorname{Re}(v)R.$ Observe that because $\\varphi \\in (0,\\pi /4)$ and $R=-\\operatorname{Re}(v)+\\alpha +1$ , $\\frac{\\operatorname{Im}(v)^2}{2} +\\frac{R^2}{2} +\\operatorname{Re}(v)R \\le \\tilde{c} -c |v|^2$ for some constants $c,\\tilde{c}>0$ .",
"Thus $\\operatorname{Re}(v s + s^2/2) \\le -\\frac{(y+\\operatorname{Im}(v))^2}{2} +\\tilde{c}-c |v|^2.$ Let us now turn to the other terms in $h(s)$ .",
"We bound $|u^s|\\le e^{R\\ln |u| - y\\arg (u)}.$ By standard bounds for the large imaginary part behavior we can show $\\left|\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)}\\right| \\le C e^{\\frac{\\pi }{2} |y|}$ for some constant $C>0$ sufficiently large.",
"Also, $|1/(v+s-v^{\\prime })|\\le C$ for a fixed constant.",
"Finally, the term $|\\Gamma (-s)\\Gamma (1+s)|\\le Ce^{-\\pi |y|},$ for some constant $C>0$ .",
"Combining these together shows that the integral of $h(s)$ over $s$ in $\\mathcal {D}_{v,\\vert }$ is bounded by a constant time $\\int _{\\mathbb {R}} \\exp \\left(-\\tau \\frac{(y+\\operatorname{Im}(v))^2}{2} -\\tau c |v|^2 + R\\ln |u| -y\\arg (u) -\\pi |y| + N\\frac{\\pi }{2}|y|\\right) dy.$ We can factor out the terms above which do not depend on $y$ , giving $\\exp \\left(-\\tau c |v|^2 + R\\ln |u| \\right) \\int _{\\mathbb {R}} \\exp \\left(-\\tau \\frac{(y+\\operatorname{Im}(v))^2}{2} -y \\arg (u) + N\\frac{\\pi }{2}|y|\\right) dy.$ Notice that the prefactors on $y$ and $|y|$ in the integrand's exponential are fixed constants.",
"We can therefore use the following bound that for $a$ fixed and $b\\in \\mathbb {R}$ , there exists a constant $C$ such that $\\int _{\\mathbb {R}} e^{-\\beta (y+b)^2 + a|y|}dy \\le C e^{|ab|},\\quad \\beta >0.$ Using this we find that we can upper-bound (REF ) by $\\exp \\left(-\\tau c |v|^2 + R\\ln |u|+c^{\\prime }|v|\\right).$ For $|v|$ large enough the Gaussian decay in the above bound dominates, and hence integral of $h(s)$ over $s$ in $\\mathcal {D}_{v,\\vert }$ is bounded by $C e^{-c|v|}$ for some constants $c,C>0$ .",
"Since $v^{\\prime }$ only comes in to the term $1/(v+s-v^{\\prime })$ in the integrand, it is clear that the above arguments imply that the integral of $h(s)$ over the entire contour $s\\in \\mathcal {D}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"This completes the third step and hence completes the proof of Proposition REF ."
],
[
"Proof of Proposition ",
"Fix $\\eta ,r>0$ .",
"We are presently considering the Fredholm determinant of the kernels $K_u^{\\varepsilon }$ and $K_u$ restricted to the fixed finite contour $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ .",
"By Lemma REF we need only show convergence as $\\varepsilon \\rightarrow 0$ of the kernel $K_u^{\\varepsilon }(v,v^{\\prime })$ to $K_u(v,v^{\\prime })$ , uniformly in $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ .",
"This is achieved via the following lemma.",
"Lemma 7.2 For all $\\eta ^{\\prime }>0$ there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ and for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$, $\\left|K_u^{\\varepsilon }(v,v^{\\prime })-K_u(v,v^{\\prime })\\right|\\le \\eta ^{\\prime }.$ The kernels $K_u^{\\varepsilon }$ and $K_u$ are both defined via integrals over $s$ .",
"The contour on which $s$ is integrated can be fixed for ($\\varepsilon <\\varepsilon _0$ ) to equal $\\mathcal {D}_{v}$ , which is the $s$ contour used to define $K_u$ .",
"The fact that the $s$ contours are the same for $K_u^{\\varepsilon }$ and $K_u$ is convenient.",
"The proof of this lemma will follow from three claims.",
"The first deals with the uniformity of convergence of the integrand defining $K_u^{\\varepsilon }$ to the integrand defining $K_u$ for $s$ restricted to any fixed compact set.",
"Before stating this claim, let us define some notation.",
"Definition 7.3 Let $\\mathcal {D}_{v,>M}= \\lbrace s\\in \\mathcal {D}_{v}: |s|\\ge M\\rbrace $ be the portion of $\\mathcal {D}_{v}$ of magnitude greater than $M$ and similarly let $\\mathcal {D}_{v,<M}= \\lbrace s\\in \\mathcal {D}_{v}: |s|<M\\rbrace $ .",
"Let us assume $M$ is large enough so that $\\mathcal {D}_{v,>M}$ is the union of two vertical rays with fixed real part $R=-\\operatorname{Re}(v)+\\alpha +1$ .",
"Assuming this, we will write $s=R+Iy$ .",
"Then for $y_M=(M^2- (-\\operatorname{Re}(v)+\\alpha +1)^2)^{1/2}$ , the contour $\\mathcal {D}_{v,>M}=\\lbrace R+Iy: |y|\\ge y_M\\rbrace $ .",
"Claim 7.4 For all $\\eta ^{\\prime \\prime }>0$ and $M>0$ there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ , for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , and for all $s\\in \\mathcal {D}_{v,<M}$ , $\\left| h^q(s) - \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}\\right|\\le \\eta ^{\\prime \\prime },$ where $h^q$ is given in (REF ).",
"This is a strengthened version of the pointwise convergence in (REF ) through ().",
"It follows from the uniform convergence of the $\\Gamma _q$ function to the $\\Gamma $ function on compact regions away from the poles, as well as standard Taylor series estimates.",
"The choice of contours is such that the pole arising from $1/(v+s-v^{\\prime })$ is uniformly avoided in the limiting procedure as well.",
"It remains to show that for $M$ large enough, the integrals defining $K_u^{\\varepsilon }(v,v^{\\prime })$ and $K_u(v,v^{\\prime })$ restricted to $s$ in $\\mathcal {D}_{v,>M}$ , have negligible contribution to the kernel, uniformly over $v,v^{\\prime }$ and $\\varepsilon $ .",
"This must be done separately for each of the kernels and hence requires two claims.",
"Claim 7.5 For all $\\eta ^{\\prime }>0$ there exists $M_0>0$ and $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ , for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , and for all $M>M_0$ , $\\left|\\int _{\\mathcal {D}_{v,>M}} ds h^q(s)\\right|\\le \\eta ^{\\prime }.$ We will use the notation introduced in Definition REF and assume $M_0$ is large enough so that $\\mathcal {D}_{v,>M}$ is only comprised of two vertical rays.",
"Let us first consider the behavior of the left-hand side of ().",
"The magnitude of this term is bounded by the exponential of $\\operatorname{Re}(\\tau \\varepsilon ^{-1} s + \\varepsilon ^{-2} \\tau q^v (q^s-1)).$ This equation is periodic in $y$ (recall $s=R+Iy$ ) with a fundamental domain $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$ .",
"For $\\varepsilon ^{-1}\\pi >|y|>y_0$ for some $y_0$ which can be chosen uniformly in $v$ and $\\varepsilon $ , the following inequality holds $\\operatorname{Re}(\\tau \\varepsilon ^{-1} s + \\varepsilon ^{-2} \\tau q^v (q^s-1)) \\le -\\tau y^2/6$ This can is proved by careful Taylor series estimation and the inequality that for $x\\in [-\\pi ,\\pi ]$ , $\\cos (x)-1\\le -x^2/6$ .",
"This provides Gaussian decay in the fundamental domain of $y$ .",
"Turning to the ratio of $q$ -Gamma functions in (), observe that away from its poles, the denominator $\\left|\\frac{1}{\\Gamma _q(s+v-a_m)}\\right|\\le c e^{c^{\\prime } f^{\\varepsilon }(s)}$ where $c,c^{\\prime }$ are positive constants independent of $\\varepsilon $ and $v$ (as it varies in its compact contour) and $f^{\\varepsilon }(s) = {\\rm dist}(\\operatorname{Im}(s),2\\pi \\varepsilon ^{-1} \\mathbb {Z})$ .",
"This establishes a periodic bound on this denominator, which grows at most exponentially in the fundamental domain.",
"The numerator $\\Gamma _q(v-a_m)$ in () is bounded uniformly by a constant.",
"This is because the $v$ contour was chosen to avoid the poles of the Gamma function, and the convergence of the $q$ -Gamma function to the Gamma function is uniform on compact sets away from those poles.",
"Finally, the magnitude of (REF ) corresponds to $|u^s|$ and behaves like $e^{-R\\ln {|u|} + y\\arg (u)}$ .",
"Thus, we have established the following inequality which is uniform in $v,v^{\\prime }$ and $\\varepsilon $ as $y$ varies: $\\left|\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s \\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}} e^{\\gamma q^v(q^{s}-1)} \\prod _{m=1}^{N} \\frac{\\Gamma _q(v-\\ln _q(\\tilde{a}_m))}{\\Gamma _q(s+v-\\ln _q(\\tilde{a}_m))} \\right| \\le c^{\\prime \\prime }\\, e^{-f^{\\varepsilon }(s)^2/6+c^{\\prime }N|f^{\\varepsilon }(s)|}$ for some constant $c^{\\prime \\prime }>0$ .",
"Notice that this inequality is periodic with respect to the fundamental domain for $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$.",
"The last term to consider is $\\Gamma (-s)\\Gamma (1+s)$ which is not periodic in $y$ and decays like $e^{-\\pi |y|}$ for $y\\in \\mathbb {R}$ .",
"Since $\\mathcal {D}_{v,>M}$ is only comprised of two vertical rays we must control the integral of $h^q(s)$ for $s=R+Iy$ and $|y|>y_M$ .",
"By making sure $M$ is large enough, we can use the periodic bound (REF ) to show that the integral over $y_M<|y|<\\varepsilon ^{-1} \\pi $ is less than $\\eta $ (with the desired uniformity in $v,v^{\\prime }$ and $\\varepsilon $ .",
"For the integral over $|y|>\\varepsilon ^{-1}\\pi $ , we can use the above exponential decay of $\\Gamma (-s)\\Gamma (1+s)$ .",
"On shifts by $2\\pi \\varepsilon ^{-1}\\mathbb {Z}$ of the fundamental domain, the exponential decay of $\\Gamma (-s)\\Gamma (1+s)$ can be compared to the boundedness of the other terms (which is certainly true considering the bounds we established above).",
"The integral of each shift can be bounded by a term in a convergent geometric series.",
"Taking $\\varepsilon _0$ small then implies that the sum can be bounded by $\\eta ^{\\prime }$ as well.",
"Since $\\eta ^{\\prime }$ was arbitrary the proof is complete.",
"Claim 7.6 For all $\\eta ^{\\prime }>0$ there exists $M_0>0$ such that for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , and for all $M>M_0$ , $\\left|\\int _{\\mathcal {D}_{v,>M}} ds \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}\\right|\\le \\eta ^{\\prime }.$ The desired decay here comes easily from the behavior of $vs+s^2/2$ as $s$ varies along $\\mathcal {D}_{v,>M}$ .",
"As before, assume that $M_0$ is large enough so that this contour is only comprised of two vertical rays and set $s=R+ Iy$ for $y\\in \\mathbb {R}$ for $|y|>y_M$ .",
"As in the proof of Proposition REF given in Section REF one shows that $|e^{v\\tau s+\\tau s^2/2}|\\le C e^{-cy^2}$ uniformly over $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<R}$ , and for all $M>M_0$ .",
"This behavior should be compared to that of the other terms: $|\\Gamma (-s)\\Gamma (1+s)|\\approx e^{-\\pi |y|}$ ; $|u^s|= e^{-R\\ln {|u|} + y\\arg (u)}$ ; $\\left|\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)}\\right|\\le C e^{|y| \\pi /2}$ ; and $|1/(v+s+v^{\\prime })|\\le C$ as well.",
"Combining these observations we see that the integral decays in $|y|$ at worst like $C e^{-cy^2+c^{\\prime } |y|}$ .",
"Thus, by choosing $M$ large enough so that $y_M\\gg 1$ we can be assured that the integral over $|y|>y_M$ is as small as desired, proving the above claim.",
"Let us now combine the above three claims to finish the proof of the Proposition REF .",
"Choose $\\eta ^{\\prime }=\\eta /3$ and fix $M_0$ and $\\varepsilon _0^{\\prime }$ as specified by the second and third of the above claims.",
"Fix some $M>M_0$ and let $L$ equal the length of the finite contour $\\mathcal {D}_{v,<M}$ .",
"Set $\\eta ^{\\prime \\prime }=\\frac{\\eta ^{\\prime }}{3L}$ and apply Claim REF .",
"This yields an $\\varepsilon _0$ (which we can assume is less than $\\varepsilon _0^{\\prime }$ ) so that (REF ) holds.",
"This implies that for $\\varepsilon <\\varepsilon _0$ , and for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , $\\left| \\int _{\\mathcal {D}_{v,<M}} h^q(s) ds - \\int _{\\mathcal {D}_{v,<M}} \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}ds \\right|\\le \\eta ^{\\prime }/3.$ From the triangle inequality and the three factors of $\\eta ^{\\prime }/3$ we arrive at the claimed result of (REF ) and thus complete the proof of the lemma and hence also Proposition REF ."
],
[
"Proof of Proposition ",
"The proof of this proposition is essentially a finite $\\varepsilon $ (recall $q=e^{-\\varepsilon }$ ) perturbation of the proof of Proposition REF given in Section REF .",
"The estimates presently are a little more involved since the functions involved are $q$ -deformations of classic functions.",
"However, by careful Taylor approximation with remainder estimates, all estimates can be carefully shown.",
"By virtue of Lemma REF it suffices to show that for some $c,C>0$ , $|K_u^{\\varepsilon }(v,v^{\\prime })|\\le C e^{-c|v|}.$ Before proving this let us recall from Definition REF the contours with which we are dealing.",
"The variable $v$ lies on $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r}$ for $\\varphi \\in (0,\\pi /4)$ .",
"The $s$ variables lies on $\\widetilde{\\mathcal {D}}_{v}$ which depends on $v$ and has two parts: The portion (which we have denoted $\\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ ) with real part bounded between $1/2$ and $R$ and imaginary part $\\pm d$ for $d$ sufficiently small, and the vertical portion (which we have denoted $\\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ ) with real part $R$ .",
"The condition on $R$ is that $R\\ge -\\operatorname{Re}(v)+\\alpha +1$ and for our purposes we can take that to be an equality.",
"Let us recall that the integrand in (REF ), through which $K_u^{\\varepsilon }(v,v^{\\prime })$ is defined, is denoted by $h^q(s)$ .",
"We split the proof into three steps.",
"Step 1: We show the integral of $h^q(s)$ over $s\\in \\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ is bounded for all $\\varepsilon <\\varepsilon _0$ by an expression with exponential decay in $|v|$ .",
"Step 2: We show the integral of $h^q(s)$ over $s\\in \\widetilde{\\mathcal {D}}_{v,\\vert }$ is bounded for all $\\varepsilon <\\varepsilon _0$ by an expression with exponential decay in $|v|$ .",
"Step 3: We show that for all $\\varepsilon <\\varepsilon _0$ , the integral of $h^q(s)$ over the entire contour $\\widetilde{\\mathcal {D}}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"The combination of these three steps imply the inequality (REF ) and hence complete the proof."
],
[
"Step 1:",
"We consider the various terms in $h^q(s)$ separately (in particular we consider the left-hand sides of (REF ) through ()) and develop bounds for each which are valid uniformly for $\\varepsilon <\\varepsilon _0$ and $\\varepsilon <0$ small enough.",
"Let us write $s=x+Iy$ and note that along the contour $\\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ , $y\\in [-d,d]$ for $d$ small, and $x\\in [1/2,R]$ .",
"Let us start with the left-hand side of () which can be rewritten as $\\exp \\left(\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1))\\right).$ The norm of the above expression is bounded by the exponential of the real part of the exponent.",
"For $\\varphi \\in (0,\\pi /4)$ one shows (as a perturbation of the analogous estimate in Step 1 of the Proof of Proposition REF ) via Taylor expansion with remainder estimates that $\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1))\\le \\tilde{c}- \\tau c|v| x,$ for some constants $c,\\tilde{c}$ .",
"The above bound implies $\\left|\\exp \\left(\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1))\\right)\\right|\\le C e^{-c|v|x}.$ Let us now turn to the other terms in $h^q(s)$ .",
"We bound the left-hand side of (REF ) as $\\left|e^{-\\tau s \\varepsilon ^{-1}}\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s\\right| \\le C |u^s| \\le C e^{x\\ln |u| - y\\arg (u)}.$ We may also bound the left-hand sides of () and (), as well as the remaining product of Gamma functions by constants: $\\left|\\frac{\\Gamma _q(v-\\ln _q(\\tilde{a}_m))}{\\Gamma _q(s+v-\\ln _q(\\tilde{a}_m))} \\right|\\le C,\\qquad \\left|\\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}}\\right|\\le C, \\qquad |\\Gamma (-s)\\Gamma (1+s)|\\le C,$ for some constants $C>0$ (which may be different in each case).",
"The first bound comes from the functional equation for the $q$ -Gamma function, and the last from the fact that $s$ is bounded away from $\\mathbb {Z}$ .",
"Combining these together shows that for $|v|$ large, $\\left|\\int _{\\widetilde{\\mathcal {D}}_{v,\\sqsubset }}h^q(s) ds\\right| \\le \\int _{\\widetilde{\\mathcal {D}}_{v,\\sqsubset }} C e^{-\\tau c|v| \\operatorname{Re}(s) + x\\ln |u|-y \\arg (u)} |ds| \\le C^{\\prime } e^{-c^{\\prime }|v|}$ for some constants $c^{\\prime },C^{\\prime }>0$ , while for bounded $|v|$ the integral is just bounded as well.",
"As above, we consider the various terms in $h^q(s)$ separately and develop bounds for each.",
"Let us write $s=R+Iy$ and note that $s\\in \\widetilde{\\mathcal {D}}_{v,\\vert }$ corresponds to $y$ varying over all $|y|\\ge d$ .",
"Three of the terms we consider (corresponding to the left-hand sides of (), () and ()) are periodic functions in $y$ with fundamental domain $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$ .",
"We will first develop bounds on these three terms in this fundamental domain, and then turn to the non-periodic terms.",
"We start by controlling the behavior of the left-hand side of () as $y$ varies in its fundamental domain.",
"For each $\\varphi <\\pi /4$ there exists a sufficiently small (yet positive) constant $c^{\\prime }$ such that as $y$ varies in its fundamental domain $\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1)) \\le c^{\\prime } \\tau \\operatorname{Re}(vs+s^2/2).$ On account of this, we can use the bound (REF ) from the proof of Proposition REF .",
"This implies that $\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1)) \\le c^{\\prime } \\tau \\left(-\\frac{(y+\\operatorname{Im}(v))^2}{2} -c |v|^2\\right).$ Let us now turn to the other $y$ -periodic terms in $h^q(s)$ .",
"By bounds for the large imaginary part behavior of the $q$ -Gamma function we can show $\\left|\\frac{\\Gamma _q(v-\\ln _q(\\tilde{a}_m))}{\\Gamma _q(s+v-\\ln _q(\\tilde{a}_m))} \\right| \\le C e^{c f^{\\varepsilon }(s+v)}$ for some constants $c,C>0$ where $f^{\\varepsilon }(s) = {\\rm dist}(\\operatorname{Im}(s),2\\pi \\varepsilon ^{-1} \\mathbb {Z})$ .",
"Note that as opposed to (REF ) when $|v|$ was bounded, in the above inequality we write $f^{\\varepsilon }(s+v)$ in the exponential on the right-hand side.",
"This is because we are presently considering unbounded ranges for $v$ .",
"Also, we can bound $\\left|\\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}}\\right|\\le C$ for some constant $C>0$ .",
"The parts of $h^q(s)$ which are not periodic in $y$ can easily be bounded.",
"We bound the left-hand side of (REF ) as in Step 1 by $\\left|e^{-\\tau s \\varepsilon ^{-1}}\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s\\right| \\le C |u^s| \\le C e^{x\\ln |u| - y\\arg (u)}.$ Finally, the term $|\\Gamma (-s)\\Gamma (1+s)|\\le Ce^{-\\pi |y|},$ for some constant $C>0$ .",
"We may now combine the estimates above.",
"The idea is to first prove that the integral on the fundamental domain $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$ is exponentially small in $|v|$ .",
"Then, by using the decay of the two non-periodic terms above, we can get a similar bound for the integral as $y$ varies over all of $\\mathbb {R}$ .",
"For $j\\in \\mathbb {Z}$ , define the $j$ shifted fundamental domain as $D_j=j\\varepsilon ^{-1}2\\pi + [-\\varepsilon ^{-1}\\pi ,\\varepsilon ^{-1}\\pi ]$ .",
"Let $I_j:= \\int _{D_j} h^q(R+Iy) dy$ and observe that combining all of the bounds developed above, we have that $|I_j|\\le C \\int _{-\\varepsilon ^{-1}\\pi }^{\\varepsilon ^{-1}\\pi } F_1(y) F_2(y) dy,$ where $\\begin{aligned}F_1(y) &= \\exp \\left(c^{\\prime } \\tau \\left(-\\frac{(y+\\operatorname{Im}(v))^2}{2} -c |v|^2\\right) +c^{\\prime \\prime } f^{\\varepsilon }(s+v) +x\\ln |u|\\right),\\\\F_2(y) &= \\exp \\left(- (y+j\\varepsilon ^{-1}2\\pi )\\arg (u) -\\pi |y+j\\varepsilon ^{-1}2\\pi | \\right).\\end{aligned}$ The term $F_1(y)$ is from the periodic bounds while $F_2(y)$ from the non-periodic terms (hence explaining the $j\\varepsilon ^{-1}2\\pi $ shift in $y$ ).",
"By assumption on $u$ , we have $-\\arg (u)-\\pi =\\delta \\le c$ for some $\\delta $ .",
"Therefore $F_2(y) \\le C e^{-c\\varepsilon ^{-1} |j|}$ form some constants $c,C>0$ .",
"Thus $|I_j|\\le C e^{-c\\varepsilon ^{-1} |j|} \\int _{-\\varepsilon ^{-1}\\pi }^{\\varepsilon ^{-1}\\pi } F_1(y)dy.$ Just as in the end of Step 2 in the proof of Proposition REF we can estimate the integral $\\int _{-\\varepsilon ^{-1}\\pi }^{\\varepsilon ^{-1}\\pi } F_1(y)dy \\le \\hat{C} e^{-\\hat{c}|v|}$ for some constants $\\hat{C},\\hat{c}>0$ .",
"This implies $|I_j|\\le \\hat{C} C e^{-c\\varepsilon ^{-1} |j|} e^{-\\hat{c}|v|}.$ Finally, observe that $\\left|\\int _{\\widetilde{\\mathcal {D}}_{v,\\vert }} h^{q}(s) ds\\right| \\le \\sum _{j\\in \\mathbb {Z}} |I_j| \\le \\hat{C} C e^{-\\hat{c}|v|} \\sum _{j\\in \\mathbb {Z}}e^{-c\\varepsilon ^{-1} |j|} \\le C^{\\prime } e^{-\\hat{c}|v|}$ where $C^{\\prime }$ is independent of $\\varepsilon $ as long as $\\varepsilon <\\varepsilon _0$ for some fixed $\\varepsilon _0$ .",
"This is the bound desired to complete this step.",
"Since $v^{\\prime }$ only comes in to the term $\\frac{q^v \\ln q}{q^{s+v}-q^{v^{\\prime }}}$ in the integrand, it is clear that the above arguments imply that the integral of $h^q(s)$ over the entire contour $s\\in \\widetilde{\\mathcal {D}}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"This completes the third step and hence completes the proof of Proposition REF ."
],
[
"Two probability lemmas",
"Lemma 8.1 (Lemma 4.1.38 of [16]) Consider a sequence of functions $\\lbrace f_n\\rbrace _{n\\ge 1}$ mapping $\\mathbb {R}\\rightarrow [0,1]$ such that for each $n$ , $f_n(x)$ is strictly decreasing in $x$ with a limit of 1 at $x=-\\infty $ and 0 at $x=\\infty $ , and for each $\\delta >0$ , on $\\mathbb {R}\\setminus [-\\delta ,\\delta ]$ , $f_n$ converges uniformly to $\\mathbf {1}_{x\\le 0}$ .",
"Consider a sequence of random variables $X_n$ such that for each $r\\in \\mathbb {R}$ , $\\mathbb {E}[f_n(X_n-r)] \\rightarrow p(r)$ and assume that $p(r)$ is a continuous probability distribution function.",
"Then $X_n$ converges weakly in distribution to a random variable $X$ which is distributed according to $\\mathbb {P}(X\\le r) = p(r)$.",
"Lemma 8.2 (Lemma 4.1.39 of [16]) Consider a sequence of functions $\\lbrace f_n\\rbrace _{n\\ge 1}$ mapping $\\mathbb {R}\\rightarrow [0,1]$ such that for each $n$ , $f_n(x)$ is strictly decreasing in $x$ with a limit of 1 at $x=-\\infty $ and 0 at $x=\\infty $ , and $f_n$ converges uniformly on $\\mathbb {R}$ to $f$ .",
"Consider a sequence of random variables $X_n$ converging weakly in distribution to $X$ .",
"Then $\\mathbb {E}[f_n(X_n)] \\rightarrow \\mathbb {E}[f(X)].$"
],
[
"Some properties of Fredholm determinants",
"We give some important properties for Fredholm determinants.",
"For a more complete treatment of this theory see, for example, [59].",
"Lemma 8.3 (Proposition 1 of [65]) Suppose $t\\rightarrow \\Gamma _t$ is a deformation of closed curves and a kernel $L(\\eta ,\\eta ^{\\prime })$ is analytic in a neighborhood of $\\Gamma _t\\times \\Gamma _t\\subset 2$ for each $t$ .",
"Then the Fredholm determinant of $L$ acting on $\\Gamma _t$ is independent of $t$ .",
"Lemma 8.4 Consider the Fredholm determinant $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ on an infinite complex contour $\\Gamma $ and an integral operator $K$ on $\\Gamma $ .",
"Parameterize $\\Gamma $ by arc length with some fixed point corresponding to $\\Gamma (0)$ .",
"Assume that $|K(v,v^{\\prime })|\\le C$ for some constant $C$ and for all $v,v^{\\prime }\\in \\Gamma $ and that either of the following two exponential decay conditions holds: There exists constants $c,C>0$ such that $|K(\\Gamma (s),\\Gamma (s^{\\prime }))|\\le Ce^{-c|s|},$ Then the Fredholm series defining $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ is well-defined.",
"Moreover, for any $\\eta >0$ there exists an $r_0>0$ such that for all $r>r_0$ $|\\det (\\mathbb {1}+K)_{L^2(\\Gamma )} - \\det (\\mathbb {1}+K)_{L^2(\\Gamma _r)}|\\le \\eta $ where $\\Gamma _r=\\lbrace \\Gamma (s):|s|\\le r\\rbrace $ .",
"The Fredholm series expansion () is given by $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )} = \\sum _{n\\ge 0}\\frac{1}{n!}",
"\\int _{\\Gamma }ds_1\\cdots \\int _{\\Gamma }ds_n \\det (K(\\Gamma (s_i),\\Gamma (s_j)))_{i,j=1}^n$ is well-defined since by using Hadamard's boundHadamard's bound: the determinant of a $n\\times n$ matrix with entries of absolute value not exceeding 1 is bounded by $n^{n/2}$ .",
"one gets that $\\left|\\det (K(\\Gamma (s_i),\\Gamma (s_j)))_{i,j=1}^n\\right|\\le n^{n/2} C^n \\prod _{j=1}^n e^{-c|s_j|}$ which is absolutely integrable / summable.",
"To show is $\\det (\\mathbb {1}+K)_{L^2(\\Gamma _r)}\\rightarrow \\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ as $r\\rightarrow \\infty $ .",
"From (REF ) one immediately gets that $\\det (\\mathbb {1}+K)_{L^2(\\Gamma _r)} = \\det (\\mathbb {1}+P_r K)_{L^2(\\Gamma )}.$ where $P_r$ is the projection onto $\\Gamma _r$ .",
"The kernel $(P_r K)(s_i,s_j)$ converges pointwise to $K(s_i,s_j)$ and (REF ) provides a in $r$ uniform, integrable / summable bound for $\\det (K(\\Gamma (s_i),\\Gamma (s_j)))_{i,j=1}^n$ .",
"Therefore, by dominated convergence as $r\\rightarrow \\infty $ the two Fredholm determinant converge.",
"Lemma 8.5 Consider a finite length complex contour $\\Gamma $ and a sequence of integral operators $K^{\\varepsilon }$ on $\\Gamma $ , as well as an addition integral operator $K$ also on $\\Gamma $ .",
"Assume that for all $\\eta >0$ there exists $\\varepsilon _0$ such that for all $\\varepsilon <\\varepsilon _0$ and all $z,z^{\\prime }\\in \\Gamma $ , $|K^{\\varepsilon }(z,z^{\\prime }) - K(z,z^{\\prime })|\\le \\eta $ and that there is some constant $C$ such that $|K(z,z^{\\prime })|\\le C$ for all $z,z^{\\prime }\\in \\Gamma $ .",
"Then $\\lim _{\\varepsilon \\rightarrow 0} \\det (\\mathbb {1}+K^{\\varepsilon })_{L^2(\\Gamma )} = \\det (\\mathbb {1}+K)_{L^2(\\Gamma )}.$ As in Lemma REF one writes the Fredholm series.",
"Since $\\Gamma $ is finite, the Fredholm determinants $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ is well-defined because $|K(z,z^{\\prime })|\\le C$ (use Hadamard's bound).",
"By assumption, $K^\\varepsilon $ converges pointwise to $K$ and we have the uniform bound $|K^\\varepsilon (z,z^{\\prime })|\\le C+\\eta $.",
"This ensures that $\\det (\\mathbb {1}+K^{\\varepsilon })_{L^2(\\Gamma )}$ is well-defined and that we can take the limit inside the Fredholm series, providing our result."
],
[
"Reformulation of Fredholm determinants",
"Lemma 8.6 Let $\\widetilde{K}_{\\rm Ai}$ as in (REF ), $\\mathcal {C}_w$ as in (REF ), and $K_{\\rm Ai}$ the Airy kernel.",
"Then it holds $\\det (\\mathbb {1}+\\widetilde{K}_{\\rm Ai})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}.$ The integration path in (REF ) can be chosen to have $\\operatorname{Re}(z)>0$ and since $\\operatorname{Re}(w)<0$ for $w\\in \\mathcal {C}_w$ , we can use $\\frac{1}{z-w}=\\int _{\\mathbb {R}_+}d\\lambda e^{-\\lambda (z-w)}$ to write $\\widetilde{K}_{\\rm Ai}(w,w^{\\prime })=-(A B)(w,w^{\\prime })$ with $A:L^2(\\mathcal {C}_w)\\rightarrow L^2(\\mathbb {R}_+)$ and $B:L^2(\\mathbb {R}_+)\\rightarrow L^2(\\mathcal {C}_w)$ have kernels $A(w,\\lambda )=e^{-w^3/3+w(r+\\lambda )}, \\quad B(\\lambda ,w^{\\prime })=\\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty } \\frac{dz}{2\\pi I} \\frac{e^{z^3/3-z(r+\\lambda )}}{z-w^{\\prime }}.$ We also have $\\begin{aligned}(BA)(\\eta ,\\eta ^{\\prime })&=\\frac{1}{2\\pi I}\\int _{\\mathcal {C}_w} dw B(\\eta ,w) A(w,\\eta ^{\\prime })\\\\&=\\frac{1}{(2\\pi I)^2} \\int _{e^{-3\\pi I/4}\\infty }^{e^{3\\pi I/4}\\infty } dw \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty } dz\\frac{1}{z-w}\\frac{e^{z^3/3-z(r+\\eta )}}{e^{w^3/3-w(r+\\eta ^{\\prime })}}= K_{\\rm Ai}(\\eta +r,\\eta ^{\\prime }+r).\\end{aligned}$ Then, since $\\det (\\mathbb {1}-AB)_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-BA)_{L^2(\\mathbb {R}_+)}=\\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}$ we get the claimed result.",
"The first equality is a general result which applies as long as $AB$ and $BA$ are both trace-class operators [59].",
"Lemma 8.7 Let $\\widetilde{K}_{\\rm BBP}$ as in (REF ), $\\mathcal {C}_w$ as in Theorem REF (b), and $K_{\\rm BBP}$ as in (REF ).",
"Then it holds $\\det (\\mathbb {1}+\\widetilde{K}_{\\rm BBP})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{\\rm BBP})_{L^2(r,\\infty )}.$ The proof is as the one of Lemma REF , except that in $A(w,\\lambda )$ is multiplied by $\\prod _{k=1}^m\\frac{1}{w-b_k}$ and $B(\\lambda ,w^{\\prime })$ by $\\prod _{k=1}^m (z-b_k)$ .",
"Lemma 8.8 Let $\\widetilde{K}_{{\\rm CDRP}}$ as in (REF ), $\\mathcal {C}_w$ as in (REF ), and $K_{{\\rm CDRP}}$ the CDRP kernel given in (REF ).",
"Then it holds $\\det (\\mathbb {1}+\\widetilde{K}_{{\\rm CDRP}})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}.$ Using $\\frac{1}{z-w^{\\prime }}=\\int _{\\mathbb {R}_+}d\\eta e^{-\\eta (z-w^{\\prime })}$ we get $\\widetilde{K}_{{\\rm CDRP}}(w,w^{\\prime })=\\int _{\\mathbb {R}_+} d\\eta A(w,\\eta ) B(\\eta ,w^{\\prime })$ with $B(\\eta ,w^{\\prime })=e^{\\eta w^{\\prime }}$ and $A(w,\\eta )=\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} e^{z^3/3-w^3/3-\\eta z}.$ Thus $\\det (\\mathbb {1}+\\widetilde{K}_{{\\rm CDRP}})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ where $K_{{\\rm CDRP}}=-BA$ , namely $\\begin{aligned}K_{{\\rm CDRP}}(\\eta ,\\eta ^{\\prime })&=-\\frac{1}{2\\pi I}\\int _{-\\frac{1}{4\\sigma }+I\\mathbb {R}}dw B(\\eta ,w) A(w,\\eta ^{\\prime })\\\\&=\\frac{1}{(2\\pi I)^2}\\int _{-\\frac{1}{4\\sigma }+I\\mathbb {R}} dw \\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}} dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-z\\eta ^{\\prime }}}{e^{w^3/3-w\\eta }}.\\end{aligned}$ The next step uses the following identity: for $0<\\operatorname{Re}(u)<1$ it holds $\\frac{\\pi \\, S^{u}}{\\sin (\\pi u)}=\\int _{\\mathbb {R}} \\frac{S e^{u t}}{S+e^t}dt$ from which, for $0<\\operatorname{Re}(u)<1/\\sigma $ it holds $\\frac{\\pi \\, \\sigma S^{\\sigma u}}{\\sin (\\pi \\sigma u)}=\\int _{\\mathbb {R}} \\frac{S e^{-u t}}{S+e^{-t/\\sigma }}dt.$ We can use wit $u=z-w$ and obtain $\\begin{aligned}K_{{\\rm CDRP}}(\\eta ,\\eta ^{\\prime })&=\\int _{\\mathbb {R}}dt \\frac{S}{S+e^{-t/\\sigma }} \\bigg (\\frac{1}{2\\pi I}\\int _{-\\frac{1}{4\\sigma }+I\\mathbb {R}} dw e^{-w^3/3+w(\\eta +t)}\\bigg )\\bigg (\\frac{1}{2\\pi I} \\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}} dz e^{z^3/3-z(\\eta ^{\\prime }+t)}\\bigg )\\\\&=\\int _{\\mathbb {R}}dt \\frac{S}{S+e^{-t/\\sigma }} \\operatorname{Ai}(\\eta +t)\\operatorname{Ai}(\\eta ^{\\prime }+t),\\end{aligned}$ the expression of (REF )."
],
[
"Details in the proof of Theorem ",
"As discussed in Section , to finish the proof of Theorem REF we need to show Theorem REF .",
"For $\\kappa >0$ , Definition REF associates the scaling parameters $\\theta ^\\kappa $ , $f^\\kappa $ and $c^\\kappa $ which appear in the statement of this result.",
"The variable $\\kappa $ and $\\theta $ are dual in the sense that one could instead start with some fixed $\\theta >0$ and then associated scaling parameters $\\kappa _{\\theta }$ , $f_{\\theta }$ and $c_{\\theta }$ .",
"In particular, for $\\theta =\\theta ^{\\kappa }$ one recovers $f_{\\theta }=f^\\kappa $ and $c_{\\theta }=c^\\kappa $ .",
"In the proof it is more natural to parameterize everything by $\\theta $ instead of $\\kappa $ , so we will do it.",
"First we prove the convergence to the GUE Tracy-Widom distribution without boundary perturbations, since the proof with boundary perturbations is a small modification of it."
],
[
"Proof of Theorem ",
"We first give explicit expansions for some of the functions from Definition REF .",
"Let $\\Psi (z)=\\frac{d}{dz} \\ln (\\Gamma (z))$ be the Digamma function and fix $\\theta \\in (0,\\infty )$ .",
"Then $\\Psi (z)=-\\gamma _{\\rm E}+\\sum _{n=0}^\\infty \\left(\\frac{1}{n+1}-\\frac{1}{n+z}\\right),$ where $\\gamma _{\\rm E}$ is the Euler constant.",
"Hence, $\\kappa _\\theta &=\\Psi ^{\\prime }(\\theta )=\\sum _{n=0}^\\infty \\frac{1}{(\\theta +n)^2}, \\\\f_\\theta &=\\theta \\Psi ^{\\prime }(\\theta )-\\Psi (\\theta )=\\gamma _{\\rm E} +\\sum _{n=0}^\\infty \\left(\\frac{n+2\\theta }{(n+\\theta )^2}-\\frac{1}{n+1}\\right), \\\\c_\\theta &=(-\\Psi ^{\\prime \\prime }(\\theta )/2)^{1/3}=\\bigg (\\sum _{n=0}^\\infty \\frac{1}{(n+\\theta )^3}\\bigg )^{1/3}.",
"$ Under the scaling limit $\\tau =\\kappa _\\theta N,\\quad u=e^{-N f_\\theta - r c_\\theta N^{1/3}}.$ we have to show the following: For $K_u$ as in (REF ) and a contour $\\mathcal {C}_v:=\\mathcal {C}_{0,\\varphi }$ , $\\lim _{N\\rightarrow \\infty }\\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)}= \\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}.$ To show this we start with the kernel (REF ), replace $\\Gamma (-s)\\Gamma (1+s) = -\\pi / \\sin (\\pi s)$ and then perform the change of variable $\\tilde{z}=s+w$ to obtain $K_{u}(v,v^{\\prime }) = \\frac{-1}{2\\pi I}\\int d\\tilde{z}\\, \\frac{\\pi }{\\sin (\\pi (\\tilde{z}-v))}\\frac{e^{N G(v)-N G(\\tilde{z})} e^{r N^{1/3} (v-\\tilde{z})}}{\\tilde{z}-v^{\\prime }}.$ where $G(z) = \\ln \\Gamma (z) - \\kappa \\frac{z^2}{2} + f^\\kappa z.$ We will show that the leading contribution to the Fredholm determinant comes for $v,v^{\\prime }$ in a $N^{-1/3}$ -neighborhood of $\\theta $ .",
"Now let us specify the exact choice for the contour $\\mathcal {C}_v$ as well as the contour along which $\\tilde{z}$ is integrated.",
"We chooseTheorem REF is stated for $\\varphi \\in (0,\\pi /4)$ since one uses the quadratic decay (REF ) to control the linear term in the bound (REF ).",
"For $\\varphi =\\pi /4$ one gets a linear decay instead of (REF ) whose strength depends on the parameter $\\alpha $ too, it would not strong enough general $\\alpha $ .",
"However, in our case, with $\\alpha =\\theta $ , it still works, as can be seen from the bound obtained in Proposition REF .",
"The proof could also be adapted to any other asymptotic direction $0<\\varphi <\\pi /4$ by simply modifying the path away at a distance greather than some (arbitrary but fixed with $N$ ) value $R_0$ (one can not employ any angle $\\varphi \\in (0,\\pi /4)$ right away from the critical point since some steep descent properties are then locally not satisfied).",
"$\\mathcal {C}_v:=\\lbrace \\theta -|y|+Iy, y\\in \\mathbb {R}\\rbrace .$ $\\mathcal {C}_v$ is a steep descent path (see the footnote in Section REF ) for the function $\\operatorname{Re}(G(v))$ .",
"The path for $\\tilde{z}$ is dependent on $v$ , since it has to pass to the left of, or contain the simple poles $v+1,v+2,\\ldots $ , see Figure REF (left).",
"Consider the sequence of points $S=\\lbrace \\operatorname{Re}(v)+1,\\operatorname{Re}(v)+2,\\ldots \\rbrace $ .",
"There are three possibilities: (1) If the sequence $S$ does not contain points in $[\\theta ,\\theta +3c_\\theta ^{-1} N^{-1/3}]$ , then let $\\ell \\in \\mathbb {N}_0$ be such that $\\operatorname{Re}(v)+\\ell \\in [\\theta -1,\\theta ]$ and we set $\\tilde{\\varepsilon }=c_\\theta ^{-1} N^{-1/3}$ .",
"(2) If the sequence $S$ contains a point in $[\\theta ,\\theta +2c_\\theta ^{-1} N^{-1/3}]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in [\\theta ,\\theta +2c_\\theta ^{-1} N^{-1/3}]$ and set $\\tilde{\\varepsilon }=3 c_\\theta ^{-1} N^{-1/3}$ .",
"(3) If the sequence $S$ contains a point in $(\\theta +2c_\\theta ^{-1} N^{-1/3},\\theta +3c_\\theta ^{-1} N^{-1/3}]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in (\\theta -1+2c_\\theta ^{-1} N^{-1/3},\\theta -1+3c_\\theta ^{-1} N^{-1/3}]$ and set $\\tilde{\\varepsilon }= c_\\theta ^{-1} N^{-1/3}$ .",
"With this choice, the singularity of the sine along the line $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ is not present, since the poles are at a distance at least $c_\\theta ^{-1} N^{-1/3}$ from it.",
"Then, the path for $\\tilde{z}$ is given by $\\mathcal {C}_{\\tilde{z}}:=\\lbrace \\theta +\\tilde{\\varepsilon }+Iy, y\\in \\mathbb {R}\\rbrace \\cup \\bigcup _{k=1}^\\ell B_{v+k},$ and $B_{v+k}$ denotes a small circle (radius smaller than $1/2$ ) around $v+k$ and clockwise oriented.",
"If $\\ell =0$ then the small circles are simply not present.",
"The idea behind this choice of the path $\\mathcal {C}_{\\tilde{z}}$ is that the $z$ -contour consists of a fixed line that is (almost) independent of kernel arguments, and an additional number of little circles (i.e., poles) as needed.",
"Moreover, the leading contribution of the kernel comes only from the cases where $\\ell =0$ (i.e., situation (1)) for which $\\tilde{\\varepsilon }=c_\\theta ^{-1} N^{-1/3}$ .",
"Figure: Left: Integration paths 𝒞 v \\mathcal {C}_v (dashed) and 𝒞 z ˜ \\mathcal {C}_{\\tilde{z}} (the solid line plus circles at v+1,...,v+ℓv+1,\\ldots ,v+\\ell ).",
"The small black dots are poles either of the sine or of the gamma function.",
"Right:Integration paths after the change of variables 𝒞 w \\mathcal {C}_w (dashed) and 𝒞 z \\mathcal {C}_z (the solid line plus circles at w+1,...,w+ℓw+1,\\ldots ,w+\\ell ), with p=p(w)∈{1,3}p=p(w)\\in \\lbrace 1,3\\rbrace .Also, we do the change of variable $\\lbrace v,v^{\\prime },\\tilde{z}\\rbrace =\\lbrace \\Phi (w),\\Phi (w^{\\prime }),\\Phi (z)\\rbrace \\quad \\textrm {with}\\quad \\Phi (z):=\\theta +z c_\\theta ^{-1} N^{-1/3}.$ After this change of variable, $\\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)}=\\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)}$ , the path $\\mathcal {C}_v$ becomes (see Figure REF (right)) $\\mathcal {C}_w:=\\lbrace -|y|+Iy,y\\in \\mathbb {R}\\rbrace $ and the accordingly rescaled kernel $\\begin{aligned}K_N(w,w^{\\prime })&:=c_\\theta ^{-1} N^{-1/3} K_u(\\Phi (w),\\Phi (w^{\\prime })) \\\\&=\\frac{-c_\\theta ^{-1} N^{-1/3}}{2\\pi I}\\int _{\\mathcal {C}_z:=\\Phi ^{-1}(\\mathcal {C}_{\\tilde{z}})}dz \\frac{\\pi e^{N G(\\Phi (w))-N G(\\Phi (z))}}{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})} \\frac{e^{r (w-z)}}{z-w^{\\prime }}\\end{aligned}$ where $G(w)=\\ln (\\Gamma (w))+ f_\\theta w - \\kappa _\\theta w^2/2.$ In Proposition REF we show that for any $w,w^{\\prime }\\in \\mathcal {C}_w$ , there exists a constant $C\\in (0,\\infty )$ such that $|K_N(w,w^{\\prime })|\\le C e^{-|\\operatorname{Im}(w)|}$ uniformly for all $N$ large enough.",
"Therefore, $\\left|\\det (K_N(w_i,w_j))_{1\\le i,j\\le n}\\right|\\le n^{n/2} C^n \\prod _{i=1}^n e^{-|\\operatorname{Im}(w_i)|}$ where the factor $n^{n/2}$ is Hadamard's bound.",
"From this bound, it follows that the Fredholm expansion of the determinant, $\\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)} =\\sum _{n=0}^\\infty \\frac{1}{n!}",
"\\int _{\\mathcal {C}_w} dw_1 \\cdots \\int _{\\mathcal {C}_w} dw_n \\det (K_N(w_i,w_j))_{1\\le i,j\\le n},$ is absolutely integrable and summable.",
"Thus we can by dominated convergence take the $N\\rightarrow \\infty $ limit inside the series, i.e., replace $K_N$ by its pointwise limit, $\\lim _{N\\rightarrow \\infty } K_N(w,w^{\\prime }) = \\widetilde{K}_{\\rm Ai}(w,w^{\\prime }):=\\frac{-1}{2\\pi I} \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{e^{z^3/3-w^3/3}e^{r w-r z}}{(z-w)(z-w^{\\prime })},$ derived in Proposition REF , i.e., we have shown that $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)} = \\det (\\mathbb {1}+\\widetilde{K}_{\\rm Ai})_{L^2(\\mathcal {C}_w)}.$ The last part is a standard reformulation, which we report in Lemma REF , see also [65].",
"This ends the proof of Theorem REF ."
],
[
"Pointwise convergence and bounds",
"The function $G$ satisfies $G^{\\prime }(\\theta )=G^{\\prime \\prime }(\\theta )=0,\\quad G^{(3)}(\\theta )=-2\\sum _{n=0}^\\infty \\frac{1}{(n+\\theta )^3}=-2c_\\theta ^{3},\\quad G^{(4)}(\\theta )=\\sum _{n=0}^\\infty \\frac{6}{(n+\\theta )^4},$ therefore $G$ has a double critical point at $\\theta $ .",
"For the steep descent analysis we need to analyze the function $g(x,y)=\\operatorname{Re}(G(x+Iy))$.",
"It holdsSee for example $\\texttt {http://functions.wolfram.com/06.11.19.0001.01}$ $\\begin{aligned}\\operatorname{Re}(\\ln \\Gamma (x+Iy)) & = \\sum _{n=1}^\\infty \\left(\\frac{x}{n} - \\frac{1}{2} \\ln \\left(\\frac{(x+n)^2+y^2}{n^2}\\right)\\right)- \\gamma _{\\rm E} x -\\frac{1}{2} \\ln (x^2+y^2) \\\\&=\\sum _{n=0}^\\infty \\left(\\frac{x}{n+1} - \\frac{1}{2} \\ln \\left((x+n)^2+y^2\\right)+\\ln (n)\\mathbf {1}_{n\\ge 1}\\right)- \\gamma _{\\rm E} x.\\end{aligned}$ Together with (REF ) and () we get $\\begin{aligned}g(x,y)&=\\operatorname{Re}(\\ln \\Gamma (x+Iy))+ f_\\theta x - \\frac{1}{2} \\kappa _\\theta (x^2-y^2)\\\\&=\\sum _{n=0}^\\infty \\left(\\frac{(n+2\\theta )x-(x^2-y^2)/2}{(n+\\theta )^2}-\\frac{1}{2} \\ln \\left((x+n)^2+y^2\\right)+\\ln (n)\\mathbf {1}_{n\\ge 1}\\right).\\end{aligned}$ It follows that $g_1(x,y):=\\frac{\\partial g(x,y)}{\\partial x}=\\sum _{n=0}^\\infty \\left(\\frac{n+2\\theta -x}{(n+\\theta )^2}-\\frac{x+n}{(x+n)^2+y^2}\\right)$ and $g_2(x,y):=\\frac{\\partial g(x,y)}{\\partial y} =\\sum _{n=0}^\\infty \\left(\\frac{y}{(\\theta +n)^2}-\\frac{y}{(x+n)^2+y^2}\\right).$ Proposition 5.1 Uniformly for $w,w^{\\prime }$ in a bounded set of $\\mathcal {C}_w$ , $\\lim _{N\\rightarrow \\infty } K_N(w,w^{\\prime }) = \\frac{-1}{2\\pi I} \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{e^{z^3/3-w^3/3}e^{r w-r z}}{(z-w)(z-w^{\\prime })}.$ Consider $w,w^{\\prime }$ in a bounded set of $\\mathcal {C}_w$ , i.e., the original variables $v,v^{\\prime }$ of order $N^{-1/3}$ around the critical point $\\theta $ .",
"For $N$ large enough and $w$ bounded, $\\operatorname{Re}((-w)c_\\theta ^{-1} N^{-1/3}) \\in (0,1)$, and $\\mathcal {C}_z:=\\Phi ^{-1}(\\mathcal {C}_{\\tilde{z}})=\\lbrace 1+Iy,y\\in \\mathbb {R}\\rbrace $ .",
"Using (REF ) we have the expansion $\\begin{aligned}N G(\\Phi (w)) &= N G(\\theta )-\\frac{1}{3} w^3 + \\mathcal {O}(w^4 N^{-1/3})\\\\-N G(\\Phi (z)) &= -N G(\\theta )+\\frac{1}{3} z^3 -\\mu _\\theta z^4 N^{-1/3}+\\mathcal {O}(z^5 N^{-2/3})\\end{aligned}$ with $\\mu _\\theta =G^{(4)}(\\theta ) c_\\theta ^{-4}/24>0$ and $\\frac{\\pi }{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})}=\\frac{c_\\theta N^{1/3}}{z-w}(1+\\mathcal {O}((z-w)^2 N^{-1/3})).$ It is also easy to control the $w^{\\prime }$ -dependence because $|z-w^{\\prime }|\\ge 1$ .",
"Now we divide the integral over $z$ into (a) $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ and (b) $|\\operatorname{Im}(z)|\\le \\delta N^{1/3}$ for some $\\delta >0$ which can be taken as small as desired (but independent of $N$ ).",
"(a) Contribution of the integration over $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ .",
"We need to estimate $\\left|\\frac{-c_\\theta ^{-1} N^{-1/3}}{2\\pi I}\\int _{1+Iy,\\\\|y|> \\delta N^{1/3}}dz \\frac{\\pi e^{N G(\\Phi (w))-N G(\\Phi (z))}}{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})} \\frac{e^{r (w-z)}}{z-w^{\\prime }}\\right|.$ From (REF ), (REF ), and the fact that $w$ is in a bounded neighborhood of 0, we have $(\\ref {PFeq33})\\le \\mathcal {O}(1) \\int _{|y|\\ge \\delta N^{1/3}}dy\\, e^{N \\operatorname{Re}(G(\\Phi (0))-G(\\Phi (1+Iy)))}.$ Setting $\\tilde{\\varepsilon }=c_\\theta ^{-1} N^{-1/3}$ and doing the change of variable $\\tilde{y}=y c_\\theta ^{-1} N^{-1/3}$ we obtain $(\\ref {PFeq34})\\le \\mathcal {O}(N^{1/3}) \\int _{\\delta /c_\\theta }^\\infty d\\tilde{y}\\, e^{N (g(\\theta +\\tilde{\\varepsilon },\\tilde{y})-g(\\theta ,0))}$ The function $g(x,y):=\\operatorname{Re}(G(x+Iy))$ is given in (REF ).",
"Finally, in Lemma REF we show that the path $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ is steep descent for the function $-G(\\tilde{z})$ with derivative of $-\\operatorname{Re}(G(\\tilde{z}))$ going to $-\\infty $ linearly in $\\operatorname{Im}(\\tilde{z})$ .",
"It then follows that (REF ) is of order $N^{1/3} e^{Ng(\\theta ,0)-N g(\\theta +\\tilde{\\varepsilon },0)}e^{-c_1(\\delta ) N}$ for some positive constant $c_1(\\delta )\\sim \\delta ^4$ for small $\\delta $ .",
"But $Ng(\\theta ,0)-N g(\\theta +\\tilde{\\varepsilon },0)= \\frac{1}{3}+\\mathcal {O}(N^{-1/3}).$ Thus the contribution of the integration over $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ is $\\mathcal {O}(e^{-c_2(\\delta ) N})$ for some positive constant $c_2(\\delta )\\sim \\delta ^4$ for small $\\delta $ .",
"(b) Contribution of the integration over $|\\operatorname{Im}(z)|\\le \\delta N^{1/3}$ .",
"We need to determine the asymptotics of $\\frac{-c_\\theta ^{-1} N^{-1/3}}{2\\pi I}\\int _{1+Iy,\\\\|y|\\le \\delta N^{1/3}}dz \\frac{\\pi e^{N G(\\Phi (w))-N G(\\Phi (z))}}{\\sin (\\pi (z-w)c_\\theta ^{-1} N^{-1/3})} \\frac{e^{r (w-z)}}{z-w^{\\prime }}.$ Using the expansion (REF ) and (REF ) we get $(\\ref {PFeq37})=\\frac{-1}{2\\pi I} \\int _{1-I\\delta N^{1/3}}^{1+I\\delta N^{1/3}}dz \\frac{e^{-\\mu _\\theta z^4 N^{-1/3}}}{(z-w)(z-w^{\\prime })} \\frac{e^{z^3/3-rz}}{e^{w^3/3-rw}}(1+\\mathcal {O}((z-w)^2 N^{-1/3}))e^{\\mathcal {O}(w^4 N^{-1/3}; z^5 N^{-2/3})}.$ Denoting $z=1+Iy$ we have $\\operatorname{Re}(z^3/3)= -3y^2+1,\\quad \\operatorname{Re}(z^4)=y^4-6 y^2+1.$ The convergence of the integral is controlled by $e^{-\\mu _\\theta y^4 N^{-1/3}-3 y^2}$ .",
"One employs the bound $|e^{x}-1|\\le |x| e^{|x|}$ with $x=\\mathcal {O}(w^4 N^{-1/3}; z^5 N^{-2/3})$ to control the error terms.",
"Altogether they are only of order $\\mathcal {O}(N^{-1/3})$ , i.e., we have obtained $(\\ref {PFeq32})=\\mathcal {O}(N^{-1/3})+\\frac{-1}{2\\pi I}\\int _{1-I\\delta N^{1/3}}^{1+I\\delta N^{1/3}}dz \\frac{e^{-\\mu _\\theta z^4 N^{-1/3}}}{(z-w)(z-w^{\\prime })} \\frac{e^{z^3/3-rz}}{e^{w^3/3-rw}}$ Finally, we deform the integration contour to the following one: from $\\delta N^{1/3} (1-I)$ to $\\delta N^{1/3} (1+I)$.",
"The error term is again of order $e^{-c_1(\\delta ) N}$ .",
"However, with the new contour, using again $|e^{x}-1|\\le |x| e^{|x|}$ but with $x=-\\mu _\\theta z^4 N^{-1/3}$ one sees that the eliminating the quartic power in $z$ amounts in an error of order $\\mathcal {O}(N^{-1/3})$ .",
"The last step is to replace $\\delta $ by $\\infty $ in the integration boundaries.",
"This leads to an extra error $\\mathcal {O}(e^{-c_3(\\delta ) N})$ with some positive constant $c_3(\\delta )\\sim \\delta ^3$ for small $\\delta $ .",
"To summarize, we first choose $\\delta $ small enough so that all the $c_k(\\delta )>0$ .",
"Then for all $N$ large enough we have shown that the contribution of the integration over $|\\operatorname{Im}(z)|> \\delta N^{1/3}$ is of order $\\mathcal {O}(e^{-c_1(\\delta ) N})$ and the integration over $|\\operatorname{Im}(z)|\\le \\delta N^{1/3}$ is given by $\\mathcal {O}(N^{-1/3})+\\frac{-1}{2\\pi I}\\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{1}{(z-w)(z-w^{\\prime })} \\frac{e^{z^3/3-rz}}{e^{w^3/3-rw}}.$ Taking the $N\\rightarrow \\infty $ limit we obtain the result.",
"Proposition 5.2 For any $w,w^{\\prime }\\in \\mathcal {C}_w$ , there exists a constant $C\\in (0,\\infty )$ such that $|K_N(w,w^{\\prime })|\\le C e^{-|\\operatorname{Im}(w)|}$ uniformly for all $N$ large enough.",
"Since the $z$ -contour can be chosen such that $|z-w^{\\prime }|\\ge 1/2$ , we can estimate the absolute value of the factor $(z-w^{\\prime })^{-1}$ by 2 and discard it from further considerations.",
"For $w$ in a bounded set of $\\mathcal {C}_w$ , the statement is a consequence of the computations in the proof of Proposition REF .",
"Thus, it is enough to consider $w=-|y|+Iy$ for $y\\ge L$ , for $L$ which will be chosen large enough (but independent of $N$ ).",
"In the original variables $v,v^{\\prime }$ , this means that we need to consider $v=\\theta -|y|+Iy$ for $y\\ge L c_\\theta ^{-1} N^{-1/3}$ .",
"Let $v=\\Phi (w)$ , $v^{\\prime }=\\Phi (w^{\\prime })$ , then the kernel $K_N$ is given by $K_N(w,w^{\\prime })=\\frac{e^{N (G(v)-G(\\theta ))+r(v-\\theta ) c_\\theta N^{1/3}}}{c_\\theta N^{1/3}\\, 2\\pi I}\\int _{\\mathcal {C}_{\\tilde{z}}}d\\tilde{z} \\frac{\\pi e^{N G(\\theta )-N G(\\tilde{z})} e^{r(\\theta -\\tilde{z}) c_\\theta N^{1/3}}}{\\sin (\\pi (\\tilde{z}-v))(\\tilde{z}-v^{\\prime })}.$ We divide the bound dividing in two contributions: (a) integration over $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ , with $\\tilde{\\varepsilon }=p c_\\theta ^{-1} N^{-1/3}$ (with $p\\in \\lbrace 1,3\\rbrace $ depending on the value of $v$ , see the proof of Theorem REF (a) above), and (b) integration over the circles $B(v+k)$ , $k=1,\\ldots ,\\ell (v)$ .",
"(a) Integration over $\\theta +\\tilde{\\varepsilon }+I\\mathbb {R}$ .",
"The relevant dependence on $v$ is in the prefactor $e^{N (G(v)-G(\\theta ))+r(v-\\theta ) c_\\theta N^{1/3}}$ and in the sine.",
"The dependence of $\\tilde{\\varepsilon }$ on $v$ is marginal, as the needed bounds can be made for any $\\tilde{\\varepsilon }$ small enough.",
"The estimates as in the proof of Proposition REF imply that this contribution is bounded by $C e^{N (\\operatorname{Re}(G(v))-G(\\theta ))+r(\\operatorname{Re}(v)-\\theta ) c_\\theta N^{1/3}} = C e^{N (g(\\theta -y,y)-g(\\theta ,0))-r y c_\\theta N^{1/3}},$ where we used the parametrization $v=\\theta -|y|+Iy$ and, by symmetry, considered only $y>0$ .",
"In Lemma REF we show that $g(\\theta -y,y)$ is strictly decreasing as $y$ increases and for large $y$ the derivative goes to $-\\infty $ (logarithmically).",
"Thus, for any fixed $\\delta >0$ , there exists a constant $c_1>0$ such that for all $y\\ge \\delta $ , $\\partial _y g(\\theta -y,y)\\le -c_1$ .",
"In Lemma REF we also show that for small $y$ , $g(\\theta -y,y)-g(\\theta ,0)=-\\tfrac{2}{3} c_\\theta ^3 y^3+\\mathcal {O}(y^4)$ .",
"Therefore, we can choose $\\delta >0$ small enough such that: for $L c_\\theta ^{-1} N^{-1/3}\\le y\\le \\delta $ , $g(\\theta -y,y)\\le g(\\theta ,0) -\\tfrac{1}{3} c_\\theta ^3 y^3$ , for $y>\\delta /2$ , $\\partial _y g(\\theta -y,y)\\le -2 c_1$ .",
"It follows $g(\\theta -y,y)\\le g(\\theta ,0)-c_1 y$ for all $y>\\delta $ .",
"Replacing $y=\\operatorname{Im}(v)=\\operatorname{Im}(w)/(c_\\theta N^{1/3})$ we get the bounds: for $L\\le \\operatorname{Im}(w) \\le \\delta c_\\theta N^{1/3}$ , $C e^{- \\operatorname{Im}(w)^3/3-r \\operatorname{Im}(w)}\\le C e^{-\\operatorname{Im}(w)^3/6}\\le 3 C e^{-\\operatorname{Im}(w)}$ for $L$ large enough (depending on $r$ only).",
"for $\\operatorname{Im}(w)\\ge \\delta c_\\theta N^{1/3}$ , $C e^{-\\operatorname{Im}(w) (N^{2/3} c_1/c_\\theta +r)}\\le C e^{-\\operatorname{Im}(w)}$ for $N$ large enough.",
"(b) Integration over the circles $B(v+k)$ , $k=1,\\ldots ,\\ell (v)$ .",
"This happens only if $y+3 c_\\theta ^{-1} N^{-1/3} \\ge 1$, where $y=\\operatorname{Im}(v)=\\operatorname{Im}(w)/(c_\\theta N^{1/3})$ .",
"The contribution of the integration over $B_{v+k}$ (up to a $\\pm $ sign depending on $k$ ) is $\\frac{e^{N G(v)-N G(v+k)} e^{-rk c_\\theta N^{1/3}}}{(v+k-v^{\\prime })}.$ We have $|v+k-v^{\\prime }|\\ge 1/\\sqrt{2}$ , thus the contribution from the pole at $v+k$ is bounded by $2 e^{N (g(\\theta -v,v)-g(\\theta -v+k,v))}e^{|r| k c_\\theta N^{1/3}}.$ Define the function $h(v,k):=g(\\theta -v,v)-g(\\theta -v+k,v)$ .",
"In Lemma REF we show that $h(v,k)$ is strictly decreasing as a function of $k$ , for $k\\in [0,y+\\tilde{\\varepsilon }]$ (we have a positive $\\delta $ instead of $\\tilde{\\varepsilon }$ , but for $N$ large enough $\\tilde{\\varepsilon }<\\delta $ ).",
"Also, $k\\le \\ell (v) = \\lfloor y+\\tilde{\\varepsilon }\\rfloor $ , so that the contribution of the poles at $v+1,\\ldots ,v+\\ell (v)$ is bounded by $2 \\ell (v) e^{N h(v,1) +|r| \\ell (v) c_\\theta N^{1/3}}.$ We consider separately the cases (1) $y\\le \\theta $ (i.e., $\\operatorname{Re}(v)\\ge 0$ ) and (2) $y>0$ (i.e., $\\operatorname{Re}(v)<0$ ).",
"For $y\\le \\theta $ , from the bound on $\\partial _k h(v,k)$ , see Lemma REF , we get $h(v,1)\\le -y c_\\theta ^3/4$ .",
"For $y>\\theta $ , we know that $h(v,1)<0$ for all $y$ and when $y\\rightarrow \\infty $ , $\\partial _k h(v,k)|_{k=0}\\simeq -y \\kappa _\\theta $ .",
"Since the function $h(v,1)$ is continuous in $y$ , there exists a positive constant $c>0$ such that $h(v,1)\\le -c y$ for all $y>\\theta $ .",
"Thus, with $c^{\\prime }=\\min \\lbrace c,c_\\theta ^3/4\\rbrace $ we get $(\\ref {PFeq45})\\le e^{-\\operatorname{Im}(w) N^{2/3}c^{\\prime }/c_\\theta + \\mathcal {O}(1)} \\mathcal {O}(\\operatorname{Im}(w) N^{-1/3})\\le C e^{-\\operatorname{Im}(w)}$ for $N$ large enough.",
"This ends the proof of the Proposition.",
"Finally let us collect the lemmas on the steep descent properties used in the propositions above.",
"Lemma 5.3 The function $g(\\theta -y,y)$ is strictly decreasing for $y>0$ .",
"For $y\\rightarrow \\infty $ it holds $\\partial _y g(\\theta -y,y)\\sim -\\ln (y)$ .",
"For $y\\searrow 0$ we have $g(\\theta -y,y)=g(\\theta ,0)-\\tfrac{2}{3} c_\\theta ^3 y^3+\\mathcal {O}(y^4)$ .",
"Using (REF ) and (REF ) we have $\\begin{aligned}&\\frac{\\partial g(\\theta -y,y)}{\\partial y} = g_2(\\theta -y,y)-g_1(\\theta -y,y)\\\\&=-\\sum _{n=0}^\\infty \\left(\\frac{1}{\\theta +n}-\\frac{n+\\theta -2y}{(n+\\theta -y)^2+y^2}\\right)=-\\sum _{n=0}^\\infty \\frac{2 y^2}{(\\theta +n)((\\theta +n-y)^2+y^2)}\\end{aligned}$ which is 0 for $y=0$ and strictly negative for $y>0$ .",
"The asymptotics for large $y$ are obtained by writing (REF ) as $\\frac{I}{2}\\Psi (-y+\\theta +Iy)-\\frac{I}{2}\\Psi (-y+\\theta -Iy)-\\frac{1}{2}\\Psi (-y+\\theta -Iy)-\\frac{1}{2}\\Psi (-y+\\theta +Iy)+\\Psi (\\theta )$ and using the large-$z$ expansion $\\Psi (z)=\\ln (z)-\\frac{1}{2z}+\\mathcal {O}(z^{-2}).$ Taylor expansion gives the small $y$ estimate.",
"Lemma 5.4 For any $x\\ge \\theta $ , the function $g(x,y)$ is strictly increasing for $y>0$ .",
"For $y\\rightarrow \\infty $ it holds $\\partial _y g(x,y)\\sim \\kappa _{\\theta } y$ .",
"From (REF ) we have $\\frac{\\partial g(x,y)}{\\partial y} =\\sum _{n=0}^\\infty \\left(\\frac{y}{(\\theta +n)^2}-\\frac{y}{(x+n)^2+y^2}\\right),$ which is 0 for $y=0$ and for $y>0$ is strictly positive.",
"For large $y$ , the second term goes to zero, leading to the estimate.",
"Lemma 5.5 Let $y>0$ be fixed.",
"The function $h(y,k):=g(\\theta -y,y)-g(\\theta -y+k,y)$ satisfies $h(y,0)=0$ , $h(y,k)$ is strictly decreasing for $k\\in [0,y]$ .",
"For any $\\delta \\in (0,\\theta )$ , $y\\ge \\delta $ , $h(y,k)$ is strictly decreasing in $k\\in [0,y+\\delta /2]$ .",
"For $y\\rightarrow \\infty $ , $\\partial _k h(y,k)|_{k=0}\\sim -y \\kappa _\\theta $ .",
"For $y\\le \\theta $ , $\\partial _k h(y,k)\\le -\\frac{k y c_\\theta ^3}{2}$ for $k\\in [0,y]$ .",
"From (REF ) we have $\\begin{aligned}\\frac{\\partial h(y,k)}{\\partial k}&=-g_1(\\theta -y+k,y)=-\\sum _{n=0}^\\infty \\left(\\frac{\\theta +n+y-k}{(\\theta +n)^2}-\\frac{\\theta +n-y+k}{(\\theta +n-y+k)^2+y^2}\\right)\\\\&=-\\sum _{n=0}^\\infty \\frac{(\\theta +n)(y^2-(k-y)^2)+(y-k)^3+y^2(y-k)}{(\\theta +n)^2((\\theta +n-y+k)^2+y^2)}\\end{aligned}$ which strictly negative for $k\\in [0,y]$ .",
"The second statement follows from $\\begin{aligned}& (\\theta +n)(y^2-(k-y)^2)+(y-k)^3+y^2(y-k) \\\\&\\quad \\ge \\theta (y^2-(k-y)^2)+(y-k)^3+y^2(y-k)\\\\&\\quad \\quad \\ge \\theta (\\delta ^2/2-\\delta ^2/4)+\\theta y^2/2-\\delta ^3/8-\\delta y^2/2\\ge \\delta ^8/8.\\end{aligned}$ To get the asymptotics of the derivative for large $y$ , we can rewrite $\\frac{\\partial h(y,k)}{\\partial k}\\bigg |_{k=0}=\\Psi (\\theta )-\\Psi ^{\\prime }(\\theta ) y-\\frac{1}{2} \\Psi (-y+\\theta -Iy)-\\frac{1}{2} \\Psi (-y+\\theta +Iy)$ and use (REF ) and $\\Psi ^{\\prime }(\\theta )=\\kappa _\\theta $ .",
"Moreover, for $k\\in [0,y]$ and $y\\le \\theta $ we have the bound $\\begin{aligned}\\frac{\\partial h(y,k)}{\\partial k}&\\le -\\sum _{n=0}^\\infty \\frac{(y^2-(k-y)^2)}{(\\theta +n)((\\theta +n-y+k)^2+y^2)}\\\\&\\le -k y\\sum _{n=0}^\\infty \\frac{1}{(\\theta +n)((\\theta +n-y+k)^2+y^2)}\\le -k y\\sum _{n=0}^\\infty \\frac{1}{2(\\theta +n)^3}=-\\frac{k y c_\\theta ^3}{2}.\\end{aligned}$"
],
[
"Proof of Theorem ",
"Now we turn to the proof of the Theorem with boundary perturbations.",
"Note that due to the ordering of the $a_i$ 's, $b_1\\ge b_2>\\cdots $ .",
"Call $\\bar{b}=b_1$ .",
"The scaling of the $a_i$ 's implies that the contours $\\mathcal {C}_w$ and $\\mathcal {C}_z$ can be chosen as before except for a modification in a $N^{-1/3}$ -neighborhood of the critical point, since they have to pass on the right of $\\theta +\\bar{b} c_{\\theta }^{-1} N^{-1/3}$ (see Figure REF ).",
"Figure: Perturbation of the integration paths, compare with Figure (right).",
"The white dots on the right are the values of b 1 ,...,b m b_1,\\ldots ,b_m.Let us denote $P(w,z,a):=\\frac{\\Gamma (\\Phi (w)-a)}{\\Gamma (\\Phi (w))}\\frac{\\Gamma (\\Phi (z))}{\\Gamma (\\Phi (z)-a)}.$ Then, the only difference with respect to the kernel (REF ) is that in the $N\\rightarrow \\infty $ limit there might remains a factor coming from $\\prod _{k=1}^m P(w,z,a_k)$ .",
"Using $\\frac{\\Gamma (z+a)}{\\Gamma (z+b)}\\sim z^{a-b}(1+\\mathcal {O}(1/z))$ (see (6.1.47) of [1]), for any $w,z$ on $\\mathcal {C}_w,\\mathcal {C}_z$ we have the bound $|P(w,z,a_k)|\\le C e^{c|a_k| (|\\operatorname{Im}(w)|+|\\operatorname{Im}(z)|) N^{-1/3}}$ for some constants $C,c$ .",
"The local modification of the paths has no influence on any of the bounds for large $w$ and $z$ , so that the proof of pointwise convergence and of the bounds are minor modifications of Proposition REF and Proposition REF .",
"It remains to determine the pointwise limits of $P(w,z,a_k)$ as $N\\rightarrow \\infty $ .",
"Case 1: If $\\limsup _{N\\rightarrow \\infty } (a_k(N)-\\theta ) N^{1/3} = -\\infty $ , then $\\lim _{N\\rightarrow \\infty }P(w,z,a_k)=1.$ Case 2: If $\\limsup _{N\\rightarrow \\infty } (a_k(N)-\\theta ) N^{1/3} = b_k$ , then $\\begin{aligned}&\\lim _{N\\rightarrow \\infty } \\frac{\\Gamma (\\Phi (w)-\\Phi (b_k))}{\\Gamma (\\Phi (w))}\\frac{\\Gamma (\\Phi (z))}{\\Gamma (\\Phi (z)-\\Phi (b_k))}\\\\=& \\lim _{N\\rightarrow \\infty } \\frac{\\Gamma ((w-b_k) c_{\\theta }^{-1} N^{-1/3})}{\\Gamma (\\theta +w c_{\\theta }^{-1} N^{-1/3})}\\frac{\\Gamma (\\theta +z c_{\\theta }^{-1} N^{-1/3})}{\\Gamma ((z-b_k)c_{\\theta }^{-1} N^{-1/3})}= \\frac{z-b_k}{w-b_k}\\end{aligned}$ because $\\Gamma (z)=z^{-1}-\\gamma _E+\\mathcal {O}(z)$ as $z\\rightarrow 0$ .",
"Therefore, one obtains $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_N)_{L^2(\\mathcal {C}_w)} = \\det (\\mathbb {1}+\\widetilde{K}_{{\\rm BBP},b})_{L^2(\\mathcal {C}_w)}$ where $\\widetilde{K}_{{\\rm BBP},b}(s,s^{\\prime })=\\frac{-1}{2\\pi I} \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty }dz \\frac{e^{z^3/3-w^3/3}e^{r w-r z}}{(z-w)(z-w^{\\prime })}\\prod _{k=1}^m \\frac{z-b_k}{w-b_k}.$ The reformulation of Lemma REF complete the proof."
],
[
"Details in the proof of Theorem ",
"As discussed at the beginning of Section , in the proof we parameterize using the position of the critical point $\\theta $ instead of $\\kappa $ .",
"Let us set $(0,\\infty )$ and consider the scaling limit $\\theta :=\\sqrt{N/,\\quad \\tau =\\kappa _\\theta N,\\quad u=S e^{-N f_\\theta }.",
"}One has the following large-\\theta expansion of (\\ref {PFeqKappa}) and (\\ref {PFeqF}) gives\\begin{equation*}\\begin{aligned}\\kappa _\\theta &=\\frac{1}{\\theta }+\\frac{1}{2 \\theta ^2}+\\frac{1}{6\\theta ^3}+\\mathcal {O}(\\theta ^{-5}),\\\\f_\\theta &=1-\\ln (\\theta )+\\frac{1}{\\theta }+\\frac{1}{4\\theta ^2}+\\mathcal {O}(\\theta ^{-4}).\\end{aligned}\\end{equation*}Thus,\\begin{equation*}\\begin{aligned}\\tau &=\\kappa _\\theta N=\\sqrt{N̰}+\\tfrac{1}{2} \\mathcal {O}(N^{-1/2}),\\\\u&=S e^{-N f_\\theta }=S e^{-N-\\frac{1}{2} N \\ln (N)+\\sqrt{N̰}+\\frac{1}{4} \\mathcal {O}(N^{-1})}.\\end{aligned}\\end{equation*}Equivalently, we can set \\tau =\\sqrt{T N}, then \\theta =\\sqrt{N/T}+\\frac{1}{2}-\\frac{1}{12}\\sqrt{T/N}+\\mathcal {O}(N^{-3/2}), so that\\begin{equation*}\\begin{aligned}=T-T^{3/2} /N^{1/2}+\\tfrac{11}{12}T^2/N+\\mathcal {O}(N^{-3/2}),\\\\u&=S e^{-N-\\frac{1}{2} N\\ln (T/N)-\\tfrac{1}{2} \\sqrt{T N}+T/4!+\\mathcal {O}(N^{-1})}.\\end{aligned}\\end{equation*}$ As shown in Section , what it remains is to prove Theorem REF .",
"We first prove the statement for the unperturbed case, and then we will show how the generalization is obtained."
],
[
"Proof of Theorem ",
"We have to determine is $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)}$ .",
"Consider the case of a drift vector $b=0$ .",
"The path $\\mathcal {C}_v$ is chosen as $\\begin{aligned}\\mathcal {C}_v&=\\lbrace \\theta -1/4+Ir, |r|\\le r^*\\rbrace \\cup \\lbrace \\theta e^{It}, t^*\\le |t| \\le \\pi /2\\rbrace \\cup \\lbrace \\theta -|y|+Iy,|y|\\ge \\theta \\rbrace ,\\\\\\end{aligned}$ where $r^*=\\sqrt{\\theta /2-1/16}$ , $t^*=\\arcsin (\\sqrt{1/2\\theta -1/16\\theta ^2})$ .",
"The path $\\mathcal {C}_{\\tilde{z}}$ is set as $\\mathcal {C}_{\\tilde{z}}=\\lbrace \\theta +p/4+I\\tilde{y}, \\tilde{y}\\in \\mathbb {R}\\rbrace \\cup \\bigcup _{k=1}^{\\ell } B_{v+k},$ where $B_{z}$ is a small circle around $z$ clockwise oriented and $p\\in \\lbrace 1,2\\rbrace $ depending on the value of $v$ , see Figure REF .",
"More precisely, for given $v$ , we consider the sequence of points $S=\\lbrace \\operatorname{Re}(v)+1,\\operatorname{Re}(v)+2,\\ldots \\rbrace $ and we choose $p=p(v)$ and $\\ell =\\ell (v)$ as follows: (1) If the sequence $S$ does not contain points in $[\\theta ,\\theta +1/2]$ , then let $\\ell \\in \\mathbb {N}_0$ be such that $\\operatorname{Re}(v)+\\ell \\in [\\theta -1,\\theta ]$ and we set $p=1$ .",
"(2) If the sequence $S$ contains a point in $[\\theta ,\\theta +3/8]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in [\\theta ,\\theta +3/8]$ and set $p=2$ .",
"(3) If the sequence $S$ contains a point in $[\\theta +3/8,\\theta +1/2]$ , then let $\\ell \\in \\mathbb {N}$ such that $\\operatorname{Re}(v)+\\ell \\in [\\theta -5/8,\\theta -1/2]$ and set $p=1$ .",
"With this choice, the singularity of the sine along the line $\\theta +p/4+I\\mathbb {R}$ is not present, since the poles are at a distance at least $1/8$ from it.",
"Also, the leading contribution of the kernel will come from situation (1) with $\\ell =0$ and $p=1$ .",
"Figure: Left: Integration paths 𝒞 v \\mathcal {C}_v (dashed) and 𝒞 z ˜ \\mathcal {C}_{\\tilde{z}} (the solid line plus circles at v+1,...,v+ℓv+1,\\ldots ,v+\\ell ), where θ + =θ+p/4\\theta _+=\\theta +p/4 and θ - =θ-1/4\\theta _-=\\theta -1/4, the small black dots are poles either of the sine or of the gamma function.",
"Right:Integration paths after the change of variables 𝒞 w \\mathcal {C}_w (dashed) and 𝒞 z \\mathcal {C}_z (the solid line plus circles at w+1,...,w+ℓw+1,\\ldots ,w+\\ell ), with p=p(w)∈{1,2}p=p(w)\\in \\lbrace 1,2\\rbrace We denote $\\sigma :=(2/^{1/3}$ and we do the change of variable $\\lbrace v,v^{\\prime },\\tilde{z}\\rbrace =\\lbrace \\Phi (w),\\Phi (w^{\\prime }),\\Phi (z)\\rbrace \\quad \\textrm {with}\\quad \\Phi (z):=\\theta +z\\sigma $ and $K_\\theta (w,w^{\\prime }):=\\sigma K_u(\\Phi (w),\\Phi (w^{\\prime }))=\\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }}.$ After this change of variable, the paths $\\mathcal {C}_w=\\Phi ^{-1}(\\mathcal {C}_v)$ and $\\mathcal {C}_z=\\Phi ^{-1}(\\mathcal {C}_{\\tilde{z}})$ are given by $\\begin{aligned}\\mathcal {C}_w=&\\lbrace -1/(4\\sigma )+Ir/\\sigma , |r|\\le r^*\\rbrace \\cup \\lbrace (e^{It}-1)\\theta /\\sigma , t^*\\le |t|\\le \\pi /2\\rbrace \\cup \\lbrace -|y|+Iy,|y|\\ge \\theta /\\sigma \\rbrace ,\\\\\\mathcal {C}_z=&\\lbrace p/(4\\sigma )+Iy, y\\in \\mathbb {R}\\rbrace \\cup \\bigcup _{k=1}^{\\ell } B_{w+k/\\sigma },\\end{aligned}$ where $r^*=\\sqrt{\\theta /2-1/16}$ , $t^*=\\arcsin (\\sqrt{1/2\\theta -1/16\\theta ^2})$ , and $B_{z}$ is a small circle around $z$ clockwise oriented.",
"After this change of variable, we have $\\det (\\mathbb {1}+K_u)_{L^2(\\mathcal {C}_v)} = \\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)}.$ Thus, we need to prove that $\\lim _{N\\rightarrow \\infty }\\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ with $K_{{\\rm CDRP}}$ given in Definition REF .",
"The proof is very similar to the second part of the proof of Theorem REF (a), where this time the convergence of the kernel is in Proposition REF and the exponential bound in Proposition REF .",
"We then obtain $\\lim _{N\\rightarrow \\infty } \\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)} = \\det (\\mathbb {1}+\\widetilde{K}_{{\\rm CDRP}})_{L^2(\\mathcal {C}_w)}$ with $\\widetilde{K}_{{\\rm CDRP}}$ given in (REF ).",
"Lemma REF shows that the limiting Fredholm determinant is equivalent to $\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ and thus completes the proof of Theorem REF (a)."
],
[
"Pointwise convergence and bounds",
"The leading contribution of the Fredholm determinant and in the kernel comes from $w,w^{\\prime },z$ of order 1 away from $\\theta \\sim \\mathcal {O}(\\sqrt{N})$ .",
"The scale for steep descent analysis is $N\\theta $ instead of $N$ as in the case of the convergence to the GUE Tracy-Widom distribution function.",
"So, the function whose real part has to be controlled is this time $\\widetilde{G}(Z):=\\frac{G(\\theta +\\theta Z)}{\\theta },$ that satisfies $\\begin{aligned}\\widetilde{G}^{(3)}(0)&=-1+\\mathcal {O}(\\theta ^{-1}),\\\\\\widetilde{G}^{(4)}(0)&=2+\\mathcal {O}(\\theta ^{-1}),\\\\\\widetilde{G}^{(n)}(0)&=\\mathcal {O}(1),\\quad n\\ge 3,\\\\G^{(n)}(\\theta )&=\\theta ^{-n+1}\\widetilde{G}^{(n)}(0).\\end{aligned}$ For asymptotic analysis we need to control the real part of $\\widetilde{G}$ , which we denote $\\widetilde{g}(X,Y):=\\operatorname{Re}(\\widetilde{G}(X+IY))=\\frac{g(\\theta +\\theta X,\\theta Y)}{\\theta }.$ In Lemmas REF , REF , and REF we will analyze the steep descent properties for $\\widetilde{G}$ (those are analogs of Lemmas REF -REF ), that we use to prove Proposition REF and Proposition REF below.",
"Proposition 6.1 Uniformly for $w,w^{\\prime }$ in a bounded set of $\\mathcal {C}_w$ , $\\lim _{N\\rightarrow \\infty } K_\\theta (w,w^{\\prime }) = \\widetilde{K}_{{\\rm CDRP}}(w,w^{\\prime })$ where $\\widetilde{K}_{{\\rm CDRP}}(w,w^{\\prime })= \\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}.$ First remark that the only dependence on $N$ in the kernel (REF ) is in the factor $\\exp \\left[N \\left(G(\\Phi (w))-G(\\Phi (z))\\right)\\right]=\\exp \\left[N\\theta \\left(\\widetilde{G}(w\\sigma /\\theta )-\\widetilde{G}(z\\sigma /\\theta )\\right)\\right].$ Let $w,w^{\\prime }$ be in a bounded set of $\\mathcal {C}_w$ around the origin.",
"For $N$ large enough and $w$ bounded in $\\mathcal {C}_w$ , $\\operatorname{Re}(w\\sigma +1)>1/2$ and $\\operatorname{Re}((z-w)\\sigma )\\in (0,1)$ so that we have $\\ell =0$ and $p=1$ , i.e., in this case $\\mathcal {C}_z=\\lbrace \\frac{1}{4\\sigma }+Iy,y\\in \\mathbb {R}\\rbrace $ .",
"We have $\\begin{aligned}N G(\\Phi (w))=N G(\\theta +w\\sigma )&=N G(\\theta )+\\frac{N}{6} G^{(3)}(\\theta )\\sigma ^3 w^3+ \\mathcal {O}(N w^4/\\theta ^3)\\\\&=N G(\\theta )-\\frac{N w^3 \\sigma ^3}{6\\theta ^2}+\\mathcal {O}(N w^4/\\theta ^3,N w^3/\\theta ^3)\\\\&=N G(\\theta )-\\frac{w^3}{3}+\\mathcal {O}(w^4/\\theta )\\end{aligned}$ where the $\\theta $ -dependence in the error term follows from $G^{(4)}(\\theta )=\\mathcal {O}(\\theta ^{-3})$ and then we used the expansion (REF ) for $G^{(3)}(\\theta )$ .",
"We divide the integral over $z$ into two parts: (a) $|\\operatorname{Im}(z)|>\\theta ^{1/3}$ and (b) $|\\operatorname{Im}(z)|\\le \\theta ^{1/3}$.",
"(a) Contribution of the integration over $|\\operatorname{Im}(z)|>\\theta ^{1/3}$ .",
"For $w,w^{\\prime }$ on $\\mathcal {C}_w$ of order 1 and $z\\in \\mathcal {C}_z$ , $|z-w^{\\prime }|\\ge \\mathcal {O}(1)$ , $|\\sin (\\pi (z-w)\\sigma )^{-1}|=\\mathcal {O}(1)$ .",
"So, $|K_\\theta (w,w^{\\prime })|\\le \\mathcal {O}(\\theta )\\int _{\\theta ^{-2/3}}^\\infty dY \\exp \\left[N\\theta \\left(\\widetilde{g}(0,0)- \\widetilde{g}((4\\sigma \\theta )^{-1},Y)\\right)\\right].$ From Lemma REF we have that $-\\widetilde{g}((4\\sigma \\theta )^{-1},Y)$ is strictly decreasing with derivative going to $-\\infty $ as $Y$ goes to infinity.",
"Then, the integral over $Y$ is bounded and of leading order $\\exp \\left[N\\theta \\left(\\widetilde{g}(0,0)- \\widetilde{g}((4\\sigma \\theta )^{-1},\\theta ^{-2/3})\\right)\\right].$ The estimates for small $Y$ of Lemma REF with $X=(4\\sigma \\theta )^{-1}$ and $Y=\\theta ^{-2/3}$ lead then to $(\\ref {PFeq74})\\le \\exp \\left[N\\theta \\left(\\widetilde{g}(0,0)- \\widetilde{g}((4\\sigma \\theta )^{-1},0)-\\frac{1}{12\\theta ^{8/3}}+\\mathcal {O}(\\theta ^{-11/3})\\right)\\right]=\\mathcal {O}(1) \\exp \\left(-\\frac{{5/6} N^{1/6}}{12}\\right)$ where we also used the fact that $\\widetilde{g}((4\\sigma \\theta )^{-1},0)=\\widetilde{g}(0,0)+\\mathcal {O}(\\theta ^{-3})$ .",
"Thus, the contribution in the kernel from the integration over $|\\operatorname{Im}(z)|\\ge \\theta ^{1/3}$ is of order $e^{-c N^{1/6}}$ for some positive constant $c>0$ .",
"(b) Contribution of the integration over $|\\operatorname{Im}(z)|\\le \\theta ^{1/3}$ .",
"We need to estimate $\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/3}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }}.$ Unlike the scaling where we have proven the convergence to the GUE Tracy-Widom distribution, in this case the sine function survives in the limiting expression and we do not have to employ the quartic term in the estimates (since it was used only to control the error term of the sine).",
"First we verify that the convergence is controlled by the third order term.",
"For this purpose, we set $z=Iy +1/(4\\sigma )$ .",
"Then, using (REF ) we obtain (as in (REF )) $-N G(\\theta +\\sigma z) =-N G(\\theta )+\\frac{z^3}{3}+\\mathcal {O}(z^4/\\theta ).$ The real part of the cubic term is given by $\\operatorname{Re}\\left(\\frac{z^3}{3}\\right)=-\\frac{y^2}{4\\sigma ^2}+\\frac{1}{192\\sigma ^3}.$ In our situation we have $|y|\\le \\theta ^{1/3}$ , therefore $-\\frac{y^2}{4\\sigma ^2}$ dominates $\\mathcal {O}(z^4/\\theta )$ for large $\\theta $ (since $y^2=\\mathcal {O}(\\theta ^{2/3})$ ).",
"We have $(\\ref {PFeq2.20})=\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/3}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3+\\mathcal {O}(w^4/\\theta ;z^4/\\theta )}}{z-w^{\\prime }}.$ We divide the integration in (b.1) $\\theta ^{1/6}\\le |y|\\le \\theta ^{1/3}$ and (b.2) $|y|\\le \\theta ^{1/6}$ .",
"Since the quadratic term in $y$ from (REF ) dominates the others, the contribution of (b.1) is only of order $\\mathcal {O}(e^{-c_1 \\theta ^{1/3}})=\\mathcal {O}(e^{-c_2 N^{1/6}})$ for some constants $c_1,c_2>0$ .",
"The contribution from (b.2) is given by $\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/6}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3+\\mathcal {O}(w^4/\\theta ;z^4/\\theta )}}{z-w^{\\prime }}.$ For $|y|\\le \\theta ^{1/6}$ , $\\mathcal {O}(z^4/\\theta )=\\mathcal {O}(\\theta ^{-1/3})$ .",
"Using $|e^x-1|\\le |x| e^{|x|}$ for $x=\\mathcal {O}(z^4/\\theta )$ and then for $x=\\mathcal {O}(w^4/\\theta )$ we can delete the error term by making an error of order $\\mathcal {O}(\\theta ^{-1/3})=\\mathcal {O}(N^{-1/6})$ .",
"Thus, $(\\ref {PFeq2.24})=\\mathcal {O}(N^{-1/6})+\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+Iy,|y|\\le \\theta ^{1/6}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}.$ Finally, extending the last integral to $\\frac{1}{4\\sigma }+I\\mathbb {R}$ we make an error of order $\\mathcal {O}(e^{-c_3 \\theta ^{1/3}})$ for some constant $c_3>0$ .",
"Putting all the above estimates together we obtain that, for $w,w^{\\prime }\\in \\mathcal {C}_w$ in a bounded set around 0, $K_\\theta (w,w^{\\prime })=\\mathcal {O}(N^{-1/6})+\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-w^3/3}}{z-w^{\\prime }}.$ Proposition 6.2 For any $w,w^{\\prime }$ in $\\mathcal {C}_w$ , uniformly for all $N$ large enough, $|K_\\theta (w,w^{\\prime })|\\le C e^{-|\\operatorname{Im}(w)|}$ for some constant $C$ .",
"First recall the expression of the kernel, $\\begin{aligned}K_\\theta (w,w^{\\prime })&=\\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }}\\\\&=S^{-w\\sigma }e^{N G(\\Phi (w))-N G(\\theta )} \\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{z\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{N G(\\theta )-N G(\\Phi (z))}}{z-w^{\\prime }}.\\end{aligned}$ As in the proof of Proposition REF , the dependence on $w^{\\prime }$ is marginal because (a) we can choose the integration variable $z$ such that $|z-w^{\\prime }|\\ge 1/(4\\sigma )$ and (b) we will get the bound through evaluating the absolute value of the integrand of (REF ).",
"Case 1: $w\\in \\lbrace -1/(4\\sigma )+Iy, |y|\\le r^*/\\sigma \\rbrace $ with $r^*=\\sqrt{\\theta /2-1/16}$ .",
"In this case, the integration path for $z$ is $1/(4\\sigma )+I\\mathbb {R}$ and no extra contributions from poles of the sine are present.",
"The factor $1/\\sin (\\pi (z-w)\\sigma )$ is uniformly bounded from above.",
"Doing the change of variable $z=\\frac{1}{4\\sigma }+I\\frac{Y\\, \\theta }{\\sigma }$ we get $|K_\\theta (w,w^{\\prime })|\\le \\mathcal {O}(1) e^{N \\operatorname{Re}(G(\\theta +w\\sigma ))-N G(\\theta )}\\int _{\\mathbb {R}}dY e^{N\\theta \\left(\\widetilde{g}(0,0)-\\widetilde{g}(\\tilde{\\varepsilon },Y)\\right)} \\theta $ with $\\tilde{\\varepsilon }=1/(4\\theta )$ .",
"The estimates as in the proof of Proposition REF on the integral over $Y$ yield $(\\ref {PFeq2.30})\\le \\mathcal {O}(1)\\times e^{N \\operatorname{Re}(G(\\theta +w\\sigma ))-N G(\\theta )} =\\mathcal {O}(1)\\times e^{N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)}.$ Since $|w\\sigma /\\theta |\\le \\mathcal {O}(\\theta ^{-1/2})$ is small, we can use Taylor expansion and with (REF ) we obtain $N\\theta \\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0) = -\\frac{1}{3} w^3(1+\\mathcal {O}(\\theta ^{-1}))+\\frac{\\sigma }{6\\theta }w^4(1+\\mathcal {O}(\\theta ^{-1})),$ substituting $w=-1/(4\\sigma )+Iy$ and taking the real part we get $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0) = -\\frac{1}{4\\sigma } y^2+\\frac{\\sigma }{6\\theta }y^4+\\mathcal {O}(1)+\\mathcal {O}(y^3/\\theta ,y^4/\\theta ^2).$ Now, for $|y|\\le \\sqrt{\\theta /2}/\\sigma $ , $\\frac{\\sigma }{6\\theta }y^4\\le \\frac{1}{12\\sigma } y^2$ and the quadratic term dominates $\\mathcal {O}(y^3/\\theta ,y^4/\\theta ^2)$ for large $\\theta $ .",
"Therefore, for all $\\theta $ large enough, we have $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0) \\le -\\frac{1}{8\\sigma } y^2+\\mathcal {O}(1).$ Consequently, $|K_\\theta (w,w^{\\prime })|\\le \\mathcal {O}(1) e^{-\\frac{1}{8\\sigma } |\\operatorname{Im}(w)|^2}\\le C e^{-|\\operatorname{Im}(w)|}$ for some finite constant $C$ .",
"Case 2: $w\\in \\lbrace (e^{It}-1)\\theta /\\sigma , t^*\\le |t|\\le \\pi /2\\rbrace \\cup \\lbrace -|y|+Iy,|y|\\ge \\theta /\\sigma \\rbrace $.",
"We divide the estimation of the bound by dividing into the contributions from (a) integration over $\\frac{p}{4\\sigma }+I\\mathbb {R}$ with $p\\in \\lbrace 1,2\\rbrace $ depending on $w$ (see the definitions after (REF )) and (b) integration over the circles $B_{w+k/\\sigma }$ , $k=1,\\ldots ,\\ell $ .",
"Case 2(a).",
"First notice that the estimate (REF ) of Case 1 still holds with the minor difference that $\\tilde{\\varepsilon }=p/(4\\theta )$ where $p\\in \\lbrace 1,2\\rbrace $ depending on the value of $w$ .",
"Then, also (REF ) still holds, so that we need only to estimate $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)$ .",
"For $w\\in \\lbrace (e^{It}-1)\\theta /\\sigma , t^*\\le |t|\\le \\pi /2\\rbrace $ , in Lemma REF we show that $\\widetilde{g}(\\cos (t)-1,\\sin (t))-\\widetilde{g}(0,0)\\le -\\sin (t)^4/16$ .",
"Replacing $\\operatorname{Im}(w)=\\sin (t) \\theta /\\sigma $ and using $|\\operatorname{Im}(w)|\\ge \\sqrt{\\theta /2-1/16}$ we obtain $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)\\le -c_1 |\\operatorname{Im}(w)|^4/\\theta \\le -c_2 |\\operatorname{Im}(w)| \\sqrt{\\theta } \\le -|\\operatorname{Im}(w)|$ for all $\\theta $ large enough, where $c_1,c_2$ are some (explicit) constants.",
"This is the desired bound.",
"For $w\\in \\lbrace -|y|+Iy,|y|\\ge \\theta /\\sigma \\rbrace $ , from Lemma REF it follows that there exists a constant $c_3>0$ such that $\\partial _Y\\widetilde{g}(-Y,Y)\\le -c_3$ and from Lemma REF we know that $\\widetilde{g}(-1,1)-\\widetilde{g}(0,0)\\le -1/16$ .",
"Thus, for $c_4=\\min \\lbrace \\sigma /16,c_3\\rbrace $ it holds $\\widetilde{g}(-1,1)-\\widetilde{g}(0,0)\\le -c_4 Y$ for all $|Y|\\ge 1/\\sigma $ .",
"This means that $N\\theta \\operatorname{Re}(\\widetilde{G}(w\\sigma /\\theta ))-N\\theta \\widetilde{G}(0)\\le -c_4 N \\theta |\\operatorname{Im}(w)|/\\theta \\le -|\\operatorname{Im}(w)|$ for $N$ large enough, giving us the needed bound.",
"Case 2(b).",
"It remains to check that the extra contributions of the poles of the sine also tend to zero exponentially in $|\\operatorname{Im}(w)|$ .",
"The contribution of the integration over $B_{w+k/\\sigma }$ is (up to a $\\pm $ sign depending on $k$ ) given by $\\frac{S^k e^{N G(\\Phi (w))-N G(\\Phi (w+k/\\sigma ))}}{w+k/\\sigma -w^{\\prime }}.$ Let us set $\\widetilde{h}(Y,k):=\\widetilde{g}(-Y,Y)-\\widetilde{g}(-Y+k,Y)$ .",
"From Lemma REF it follows that the largest contribution comes from the integration over $B_{w+1/\\sigma }$ .",
"We have at most $\\mathcal {O}(|\\operatorname{Im}(w)|)$ poles and also $|w+k/\\sigma -w^{\\prime }|\\ge \\mathcal {O}(1/\\theta )$ (the worst case is at the junction between the arc of circle and the straight lines).",
"Thus, the contribution of all the poles is bounded by $\\mathcal {O}(\\theta |\\operatorname{Im}(w)|) S^{|\\operatorname{Im}(w)|} e^{N G(\\Phi (w))-N G(\\Phi (w+1/\\sigma ))} =\\mathcal {O}(\\theta |\\operatorname{Im}(w)|) S^{|\\operatorname{Im}(w)|} e^{N\\theta \\widetilde{h}(Y,1)},$ where $Y=|\\operatorname{Im}(w)|\\sigma /\\theta $ .",
"We consider separately the cases $1/\\sqrt{2\\theta }\\le Y\\le 1$ and $Y>1$ : (1) For $1/\\sqrt{2\\theta }\\le Y\\le 1$ , the bound on $\\partial _k \\widetilde{h}(Y,k)$ leads to $\\widetilde{h}(Y,1)\\le -Y/8$ .",
"(2) For $Y>1$ we know that $\\widetilde{h}(Y,1)<0$ and $\\partial _k \\widetilde{h}(Y,k)|_{k=0}\\simeq -Y$ as $Y\\rightarrow \\infty $ .",
"By the continuity of $\\widetilde{h}(Y,1)$ in $Y$ , there exists a constant $c_5>0$ such that $\\widetilde{h}(Y,1)\\le -c_5 Y$ for all $Y\\ge 1$ .",
"Therefore, with $c_6=\\min \\lbrace c_5,1/8\\rbrace $ and inserting $Y=|\\operatorname{Im}(w)|\\sigma /\\theta $ we have $(\\ref {PFeq2.40})\\le \\mathcal {O}(\\theta |\\operatorname{Im}(w)|) S^{|\\operatorname{Im}(w)|} e^{-N c_5\\sigma |\\operatorname{Im}(w)|} \\le \\mathcal {O}(1) e^{-|\\operatorname{Im}(w)|}$ for $N$ large enough.",
"We have shown that also the contributions of the poles have the desired bound.",
"Lemma 6.3 The function $\\widetilde{g}(\\cos (t)-1,\\sin (t))$ is zero at $t=0$ and strictly decreasing for $t\\in (0,\\pi /2]$ .",
"For $t\\in [0,\\pi /2]$ and $\\theta $ large enough, $\\partial _t \\widetilde{g}(\\cos (t)-1,\\sin (t)) \\le -\\sin (t)(1-\\cos (t))/2$ so that $\\widetilde{g}(\\cos (t)-1,\\sin (t))- \\widetilde{g}(0,0)\\le - \\sin (t)^4/16.$ We have $\\widetilde{g}(\\cos (t)-1,\\sin (t))=\\theta ^{-1}g(\\theta \\cos (t),\\theta \\sin (t))$ , thus $\\begin{aligned}\\frac{\\partial \\widetilde{g}(\\cos (t)-1,\\sin (t))}{\\partial t}&=\\cos (t) g_2(\\theta \\cos (t),\\theta \\sin (t))-\\sin (t) g_1(\\theta \\cos (t),\\theta \\sin (t))\\\\&=-\\sum _{n=0}^\\infty \\frac{2\\theta ^2\\sin (t)(1-\\cos (t))(2n\\cos (t)+\\theta )}{(2n\\theta \\cos (t)+n^2+\\theta ^2)(n+\\theta )^2}\\end{aligned}$ is strictly negative for all $t\\in (0,\\pi /2]$ , which shows the first result.",
"We can further bound $\\begin{aligned}(\\ref {PFeq2.34})&\\le -\\sin (t)(1-\\cos (t)) \\sum _{n=0}^\\infty \\frac{2\\theta ^3}{(2n\\theta \\cos (t)+n^2+\\theta ^2)(n+\\theta )^2}\\\\&\\le -\\sin (t)(1-\\cos (t)) \\sum _{n=0}^\\infty \\frac{2\\theta ^3}{(n+\\theta )^4}=-\\sin (t)(1-\\cos (t))(\\tfrac{2}{3}+\\mathcal {O}(\\theta ^{-1})).\\end{aligned}$ Thus, for large enough $\\theta $ , the derivative is bounded by $-\\sin (t)(1-\\cos (t))/2$ , $t\\in [0,\\pi /2]$ .",
"Integrating over $[0,t]$ gives $\\widetilde{g}(\\cos (t)-1,\\sin (t))- \\widetilde{g}(0,0)\\le -(1-\\cos (t))^2/4\\le -\\sin (t)^4/16$ for $t\\in [0,\\pi /2]$ .",
"Lemma 6.4 For any $X\\ge 0$ , the function $\\widetilde{g}(X,Y)$ is strictly increasing for $Y>0$ , with $\\partial _Y \\widetilde{g}(X,Y)\\ge \\partial _Y \\widetilde{g}(0,Y)$ .",
"For $Y\\searrow 0$ , $\\partial _Y \\widetilde{g}(0,Y)= Y^3/3+\\mathcal {O}(Y^3/\\theta ;Y^5)$ , so that $\\widetilde{g}(X,Y)\\ge \\widetilde{g}(X,0)+Y^4/12+\\mathcal {O}(Y^4/\\theta ;Y^6).$ For $Y\\rightarrow \\infty $ it holds $\\partial _Y \\widetilde{g}(X,Y)\\sim Y$ .",
"We have $\\frac{\\partial \\widetilde{g}(X,Y)}{\\partial Y} =g_2(\\theta +\\theta X,\\theta Y)=\\sum _{n=0}^\\infty \\left(\\frac{\\theta Y}{(\\theta +n)^2}-\\frac{\\theta Y}{(\\theta +\\theta X+n)^2+\\theta ^2 Y^2}\\right),$ which is 0 for $Y=0$ and for $Y>0$ is strictly positive.",
"The inequality $\\partial _Y \\widetilde{g}(X,Y)\\ge \\partial _Y \\widetilde{g}(0,Y)=\\sum _{n=0}^\\infty \\frac{\\theta ^3 Y^3}{(\\theta +n)^2((\\theta +n)^2+\\theta ^2 Y^2)},$ whose expansion for small $Y$ and large $\\theta $ is given by $Y^3/3+\\mathcal {O}(Y^3/\\theta ;Y^5)$ .",
"For large $Y$ , the second term becomes irrelevant with respect to the first, so that $\\partial _Y \\widetilde{g}(X,Y)\\sim \\theta \\kappa _\\theta Y= Y(1+\\mathcal {O}(\\theta ^{-1}))$.",
"Lemma 6.5 The function $\\widetilde{g}(-Y,Y)$ is strictly decreasing for $Y>0$ .",
"For $Y\\rightarrow \\infty $ it holds $\\partial _Y \\widetilde{g}(-Y,Y)\\sim -\\ln (Y)$ .",
"For $Y\\searrow 0$ we have $\\widetilde{g}(-Y,Y)=\\widetilde{g}(0,0)-\\tfrac{1}{3} Y^3+\\mathcal {O}(Y^3/\\theta ;Y^4)$ .",
"As in the proof of Lemma REF , we use (REF ) and (REF ) to obtain $\\begin{aligned}\\frac{\\partial \\widetilde{g}(-Y,Y)}{\\partial Y} &= g_2(\\theta -\\theta Y,\\theta Y)-g_1(\\theta -\\theta Y,\\theta Y)=-\\sum _{n=0}^\\infty \\frac{2 Y^2\\theta ^2}{(\\theta +n)((\\theta +n-\\theta Y)^2+\\theta ^2 Y^2)}.\\end{aligned}$ which is 0 for $Y=0$ and strictly negative for $Y>0$ .",
"We can rewrite the sum with the variable $\\eta =n/\\theta $ .",
"Then, for large (but still fixed) $\\theta $ the sum over $\\eta \\in \\lbrace 0,1,2,\\ldots \\rbrace /\\theta $ is very close to the integral over $\\eta \\in [0,\\infty )$ .",
"From this one deduces that for large $Y$ , $\\partial _Y \\widetilde{g}(-Y,Y)\\sim -\\ln (Y)$ .",
"The asymptotics for $Y\\searrow 0$ is obtained by writing the Taylor series of $\\widetilde{G}(Z)$ around $Z=1$ and taking the real part of it.",
"Lemma 6.6 Let $Y>0$ be fixed.",
"The function $\\widetilde{h}(Y,k):=\\widetilde{g}(-Y,Y)-\\widetilde{g}(-Y+k,Y)$ satisfies $\\widetilde{h}(Y,0)=0$ , $\\widetilde{h}(Y,k)$ is strictly decreasing for $k\\in [0,Y]$ .",
"For any $\\delta \\in (0,1)$ , $Y\\ge \\delta $ , $\\widetilde{h}(Y,k)$ is strictly decreasing in $k\\in [0,Y+\\delta /2]$ .",
"For $Y\\rightarrow \\infty $ , $\\partial _k \\widetilde{h}(Y,k)|_{k=0}\\sim -Y$ .",
"For $Y\\le 1$ , $\\partial _k \\widetilde{h}(Y,k)\\le - kY/4$ for $k\\in [0,Y]$ .",
"The first statements follows directly from Lemma REF .",
"The asymptotics for large $Y$ can be obtained by approximating the sums in $\\partial _k \\widetilde{h}$ by integrals.",
"The bound for $Y\\le 1$ follows from (REF ) with $k\\rightarrow \\theta k$ , $y\\rightarrow \\theta Y$ , and $\\sum _{n=0}^\\infty (\\theta +n)^{-3}\\ge 1/(2\\theta ^2).$"
],
[
"Proof of Theorem ",
"Now we consider the perturbed case, where $a_k:=\\theta +b_k,\\quad k=1,\\ldots ,m.$ Then, the change of variable as in (REF ) leads to the kernel $K_\\theta (w,w^{\\prime }):=\\sigma K_u(\\Phi (w),\\Phi (w^{\\prime }))=\\frac{-1}{2\\pi I}\\int _{\\mathcal {C}_z}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )}\\frac{e^{N G(\\Phi (w))-N G(\\Phi (z))}}{z-w^{\\prime }} \\prod _{k=1}^m P(w,z,b_k)$ where the perturbation term is $P(w,z,b_k)=\\frac{\\Gamma (\\sigma w-b_k) \\Gamma (\\Phi (z)) \\theta ^{\\sigma w}}{\\Gamma (\\sigma z-b_k)\\Gamma (\\Phi (w))\\theta ^{\\sigma z}}.$ The difference from Theorem REF (a) is that now (as it was the case for Theorem REF (b)), the paths $\\mathcal {C}_z$ and $\\mathcal {C}_w$ have to be locally modified around the critical point, $\\theta $ , so that they remains on the right of all the $b_1/\\sigma ,\\ldots ,b_m/\\sigma $ , see Figure REF for an illustration.",
"Figure: Perturbation of the integration paths, compare with Figure (right).",
"The white dots on the right are the values of b 1 /σ,...,b m /σb_1/\\sigma ,\\ldots ,b_m/\\sigma .We just have to show that $\\lim _{N\\rightarrow \\infty }\\det (\\mathbb {1}+K_\\theta )_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP},b})_{L^2(\\mathbb {R}_+)}$ with $K_{{\\rm CDRP},b}$ as in (REF ).",
"The proof is a minor modification of the one of Theorem REF (b).",
"The local modification of the paths have no influence on the bounds for large $z$ and/or for large $w$ .",
"This is because $N G(\\theta +b_k)-N G(\\theta ) = \\mathcal {O}(1)$ and the path for $z$ is the same away from a distance $\\mathcal {O}(1)$ from the origin.",
"What remains to be clarified is the limit kernel.",
"We can choose the path $\\mathcal {C}_w$ to be as before with a small perturbation (e.g.",
"a circle) around 0 so that it passes on the right of the all the $b_k$ 's.",
"Then, we modify the path $\\mathcal {C}_z$ in the same way, i.e., by doing the same small perturbation but shifted to the right by $1/2\\sigma $ (see Figure REF too).",
"This ensures that we do not get extra poles from the sine.",
"Finally, for the pointwise convergence of the kernel the new term remaining is $\\lim _{N\\rightarrow \\infty }P(w,z,b_k)=\\frac{\\Gamma (\\sigma w-b_k)}{\\Gamma (\\sigma z-b_k)}.$ Indeed, since $\\Phi (z)=\\theta +\\mathcal {O}(1)$ and $\\Phi (w)=\\theta +\\mathcal {O}(1)$ , then $\\lim _{N\\rightarrow \\infty }\\frac{\\Gamma (\\Phi (z)) \\theta ^{\\sigma w}}{\\Gamma (\\Phi (w))\\theta ^{\\sigma z}}=1$ .",
"Finally one reformulate the Fredholm determinant into one on $L^2(\\mathbb {R}_+)$ in the same way as the unperturbed case of Lemma REF .",
"The only small difference is that we the first step requires $\\operatorname{Re}(z-w^{\\prime })>0$ , which holds only for $b_k<\\frac{1}{4}$ , for all $k$ .",
"Under this condition the rewriting holds.",
"By looking at the final expressions one verifies that both sides are analytic in the parameters $b_1,\\ldots ,b_m$ .",
"Thus we have equality by analytic continuation.",
"This ends the proof of Theorem REF (b)."
],
[
"Proof of Proposition ",
"This closely follows the proof of [16] Proposition 3.2.8 and Corollary 3.2.10.",
"However, in that case the contour playing the role of $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ is bounded whereas it is unbounded presently.",
"As such, some additional estimates must be made, so we include the entire proof here.",
"First observe that we may combine the $q$ -moments $\\mu _{k}=\\langle q^{k\\lambda _N}\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ (see definitions in Section REF ) into a generating function $G_{q}(\\zeta )=\\sum _{k\\ge 0} \\frac{(\\zeta /(1-q))^k}{k_q!}",
"\\left\\langle q^{k\\lambda _N}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ where $k_q!= (q;q)_n/(1-q)^n$ and $(a;q)_k=(1-a)\\cdots (1-aq^{k-1})$ (when $k=\\infty $ the product is infinite, though convergent since $|q|<1$ ).",
"The convergence of the series defining $G_{q}(\\zeta )$ follows from the fact that $q^{k\\lambda _N}\\le 1$ and $\\frac{(\\zeta /(1-q))^k}{k_q!}",
"= \\frac{\\zeta ^k}{(1-q)\\cdots (1-q^k)},$ which shows geometric decay for large enough $k$ .",
"This justifies writing $G_{q}(\\zeta )=\\left\\langle \\sum _{k\\ge 0} \\frac{(\\zeta /(1-q))^k}{k_q!}",
"q^{k\\lambda _N}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )} = \\left\\langle \\frac{1}{\\left(\\zeta q^{\\lambda _N};q\\right)_{\\infty }}\\right\\rangle _{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\dots ,\\tilde{a}_N;\\rho )}$ where the second equality follows from the $q$ -Binomial theorem [6].",
"It now suffices to show that $G_{q}(\\zeta ) =\\det (\\mathbb {1}+K)$ as in the statement of the proposition.",
"From now on, all contour integrals are along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"Observe that we can rewrite the summation in the definition of $\\mu _k$ so that $\\mu _k \\frac{\\zeta ^k}{k_q!",
"}= \\sum _{L\\ge 0}\\sum _{\\begin{array}{c}m_1,m_2,\\ldots \\\\ \\sum m_i =L \\\\ \\sum i m_i = k\\end{array}} \\frac{1}{(m_1+m_2+\\cdots )!",
"}\\cdot \\frac{(m_1+m_2+\\cdots )!}{m_1!",
"m_2!",
"\\cdots }\\int \\cdots \\int I_L(\\lambda ;w;\\zeta ) \\prod _{j=1}^{L} \\frac{dw_j}{2\\pi I},$ where $w=(w_1,\\ldots , w_L)$ , $\\lambda =(\\lambda _1,\\ldots , \\lambda _L)$ and is specified by $\\lambda =1^{m_1}2^{m_2}\\cdots $ , and where the integrand is $I_L(\\lambda ;w;\\zeta ) = \\det \\left[\\frac{1}{w_i q^{\\lambda _i}-w_j}\\right]_{i,j=1}^{L} \\prod _{j=1}^{L}(1-q)^{\\lambda _j}\\zeta ^{\\lambda _j} f(w_j)f(qw_j)\\cdots f(q^{\\lambda _j-1}w_j).$ The term $\\tfrac{(m_1+m_2+\\cdots )!}{m_1!",
"m_2!",
"\\cdots }$ is a multinomial coefficient and can be removed by replacing the inner summation by $\\sum _{n_1,\\ldots ,n_L\\in \\mathcal {L}_{k,m_1,m_2,\\ldots }} \\int \\cdots \\int I_L(n;w;\\zeta )\\frac{dw_j}{2\\pi I},$ with $n=(n_1,\\ldots ,n_L)$ and where $\\mathcal {L}_{k,m_1,m_2,\\ldots } = \\lbrace n_1,\\ldots ,n_L\\ge 1: \\sum _i n_i = k \\textrm { and for each } j\\ge 1, m_j \\textrm { of the } n_i \\textrm { equal } j\\rbrace .$ This gives $\\mu _k \\frac{\\zeta ^k}{k_q!}",
"= \\sum _{L\\ge 0}\\frac{1}{L!}",
"\\sum _{\\begin{array}{c}n_1,\\ldots ,n_L\\ge 1\\\\ \\sum n_i=k\\end{array}} \\int \\cdots \\int I_L(n;w;\\zeta )\\frac{dw_j}{2\\pi I}.$ Now we may sum over $k$ which removes the requirement that $\\sum _i n_i = k$ .",
"This yields that the left-hand side of equation (REF ) can be expressed as $\\sum _{L\\ge 0} \\frac{1}{L!}",
"\\sum _{n_1,\\ldots ,n_L\\ge 1} \\int \\cdots \\int \\det \\left[\\frac{1}{q^{n_i}w_i-w_j}\\right]_{i,j=1}^{L} \\prod _{j=1}^{L} (1-q)^{n_j}\\zeta ^{n_j} f(w_j)f(qw_j)\\cdots f(q^{n_j-1}w_j) \\frac{dw_j}{2\\pi I} .$ This is the definition of the Fredholm determinant expansion $\\det (\\mathbb {1}+K)$ , as desired.",
"As these were purely formal manipulations, at this point to complete the proof we must justify the rearrangements in the above argument.",
"In order to do this, we will show that the double summation of (REF ) is absolutely convergent.",
"This is the point at which the unboundedness of the $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ contour introduces a slight divergence from the analogous proof of [16] Proposition 3.2.8 where the contour was bounded and of finite length.",
"Basically, the absolute convergence follows from the exponential decay of the function $f$ as the real part of $w$ increases to positive infinity, combined with Hadamard's inequality.",
"Let us bound the absolute value of the integrand in (REF ).",
"Note that by assumption $q^{n_i}w_i/w_j-1$ is bounded from 0 uniformly as $w_i$ , $w_j$ , and $n_i$ vary.",
"Thus, it follows that for some finite constant $B_1$ , $\\left| \\det \\left[\\frac{1}{q^{n_i}w_i-w_j}\\right]_{i,j=1}^{L}\\right| \\le B_1^{L} L^{L/2}.$ Since the function $f(w)$ is bounded as $w$ varies and has exponential decay with respect to the real part of $w$ , we can replace $\\left|f(w_j)f(qw_j)\\cdots f(q^{n_j-1}w_j)\\right| \\le (B_2)^{n_j}e^{-c\\operatorname{Re}(w_j)}$ for constants $c>0$ and $B_2<\\infty $ .",
"Thus we find that $|(\\ref {fredexpabove})|\\le \\sum _{L\\ge 0} \\frac{1}{L!}",
"B_1^L L^{L/2} \\left(\\sum _{n\\ge 1} (B_2 (1-q) \\zeta )^n \\int \\frac{|dw|}{2\\pi } e^{-c \\operatorname{Re}(w)}\\right)^L.$ Since $w$ is being integrated along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , the integral over $w$ is bounded by some constant $B_3<\\infty $ .",
"Finally, for $|\\zeta |$ small enough the geometric series converges and it is bounded by a constant $B_4$ .",
"Therefore $(\\ref {eq7.12})\\le \\sum _{L\\ge 0} \\frac{(B_1 B_3 B_4)^{L} L^{L/2}}{L!",
"}<\\infty .$ Thus we have shown that the double summation in (REF ) is absolutely convergent, completing the proof of Proposition REF ."
],
[
"Proof of Theorem ",
"This theorem and its proof are adapted from [16] Theorem 3.2.11.",
"However, in that theorem, the $w$ -contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , was of finite length and the $s$ -contour $\\widetilde{\\mathcal {D}}_{w}$ was just a vertical line.",
"The need for slightly more involved contours comes from the unboundedness of the $w$ -contour and the necessity that $\\tilde{K}_{\\zeta }(w,w^{\\prime })$ goes to zero sufficiently fast as $|w|$ grows along the $w$ -contour.",
"The starting point for this proof is Proposition REF .",
"There are, however, two issues we must deal with.",
"First, the operator in the proposition acts on a different $L^2$ space; second, the equality is only proved for $|\\zeta |<C^{-1}$ for some constant $C>1$ .",
"We split the proof into three steps.",
"Step 1: We present a general lemma which provides an integral representation for an infinite sum.",
"Step 2: Assuming $\\zeta \\in \\lbrace \\zeta :|\\zeta |<C^{-1}, \\zeta \\notin \\mathbb {R}_{+}\\rbrace $ we derive equation (REF ).",
"Step 3: A direct inspection of the left-hand side of that equation shows that for all $\\zeta \\ne q^{-M}$ for $M\\ge 0$ the expression is well-defined and analytic.",
"The right-hand side expression can be analytically extended to all $\\zeta \\notin \\mathbb {R}_{+}$ and thus by uniqueness of the analytic continuation, we have a valid formula on all of $\\mathbb {R}_{+}$ ."
],
[
"Step 1:",
"The purpose of the next lemma is to change that $L^2$ space we are considering and to replace the summation in Proposition REF by a contour integral.",
"Lemma 7.1 For all functions $g$ which satisfy the conditions below, we have the identity that for $\\zeta \\in \\lbrace \\zeta :|\\zeta |<1, \\zeta \\notin \\mathbb {R}_{+}\\rbrace $ $\\sum _{n=1}^{\\infty } g(q^n) (\\zeta )^n = \\frac{1}{2\\pi I} \\int _{C_{1,2,\\ldots }} \\Gamma (-s)\\Gamma (1+s)(-\\zeta )^s g(q^s) ds,$ where the infinite contour $C_{1,2,\\ldots }$ is a negatively oriented contour which encloses $1,2,\\ldots $ and no singularities of $g(q^s)$ , and $z^s$ is defined with respect to a branch cut along $z\\in \\mathbb {R}_-$ .",
"For the above equality to be valid the left-hand-side must converge, and the right-hand-side integral must be able to be approximated by integrals over a sequence of contours $C_{k}$ which enclose the singularities at $1,2,\\ldots , k$ and which partly coincide with $C_{1,2,\\ldots }$ in such a way that the integral along the symmetric difference of the contours $C_{1,2,\\ldots }$ and $C_{k}$ goes to zero as $k$ goes to infinity.",
"The identity follows from $\\underset{{s=k}}{\\mathrm {Res}}\\Gamma (-s)\\Gamma (1+s) = (-1)^{k+1}$ .",
"For this step let us assume that $\\zeta \\in \\lbrace \\zeta :|\\zeta |<C^{-1}, \\zeta \\notin \\mathbb {R}_{+}\\rbrace $ .",
"We may rewrite equation (REF ) as $K(n_1,w_1;n_2,w_2) = \\zeta ^{n_1} g_{w_1,w_2}(q^{n_1})$ where $g$ is given in equation (REF ).",
"Writing out the $M^{th}$ term in the Fredholm expansion we have $\\frac{1}{M!}",
"\\sum _{\\sigma \\in S_M} \\operatorname{sgn}(\\sigma )\\prod _{j=1}^{M} \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }} \\frac{dw_j}{2\\pi I} \\sum _{n_j=1}^{\\infty } \\zeta ^{n_j} g_{w_j,w_{\\sigma (j)}}(q^{n_j}).$ In order to apply Lemma REF we need to define the sequence of contours $C_{k}$ (in fact we need only specify the contours for $k$ large).",
"Let $C_{k}$ be composed of the union of two parts – the first part is the portion of the contour $\\widetilde{\\mathcal {D}}_{w}$ which lies within the ball of radius $k+1/2$ centered at the origin; the second part is the arc of the boundary of that ball which causes the union to be a closed contour which encloses $\\lbrace 1,2,\\ldots , k\\rbrace $ and no other integers.",
"The contours $C_{k}$ are oriented positively and illustrated in the left-hand-side of Figure REF .",
"The infinite contour $C_{1,2,\\ldots }$ is chosen to be $\\widetilde{\\mathcal {D}}_{w}$ oriented as in the statement of the theorem (decreasing imaginary part).",
"By the definition of the contours $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and $\\widetilde{\\mathcal {D}}_{w}$ we are assured that the contours $C_k$ do not contain any poles beyond those of the Gamma function $\\Gamma (-s)$ .",
"This is due to the fact that the contours have been chosen such that as $s$ varies, $q^sw$ stays entirely to the left of $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and hence does not touch $w^{\\prime }$ .",
"Figure: Left: The contour C k C_{k} composed of the union of two parts – the first part is the portion of the contour 𝒟 ˜ w \\widetilde{\\mathcal {D}}_{w} which lies within the ball of radius k+1/2k+1/2 centered at the origin; the second part is the arc of that ball which causes the union to be a closed contour which encloses {1,2,...,k}\\lbrace 1,2,\\ldots , k\\rbrace and no other integers.",
"Right: The symmetric difference between C k C_k and 𝒟 ˜ w \\widetilde{\\mathcal {D}}_{w} is given by two parts: a semi-circle arc which we call C k arc C^{arc}_k and a portion of R+IℝR+I\\mathbb {R} with magnitude exceeding k+1/2k+1/2 which we call C k seg C^{seg}_k.In order to apply the above lemma we must also estimate the integral along the symmetric difference.",
"Identify the part of the symmetric difference given by the circular arc as $C^{arc}_k$ and the part given by the portion of $R+I\\mathbb {R}$ with magnitude exceeding $k+1/2$ as $C^{seg}_k$ (see the right-hand-side of Figure REF ).",
"First observe that for $w_1,w_2$ fixed, $g_{w_1,w_2}(q^s)$ stays uniformly bounded as $s$ varies along these contours.",
"Consider next $(-\\zeta )^s$ .",
"If $-\\zeta = r e^{i\\sigma }$ for $\\sigma \\in (-\\pi ,\\pi )$ and $r>0$ we have $(-\\zeta )^s =r^s e^{Is\\sigma }$ .",
"Writing $s=x+Iy$ we have $|(-\\zeta )^s| = r^{x}e^{-y\\sigma }$ .",
"Note that our assumption on $\\zeta $ corresponds to $r<1$ and $\\sigma \\in (-\\pi ,\\pi )$ .",
"Concerning the product of Gamma functions, recall Euler's Gamma reflection formula $\\Gamma (-s)\\Gamma (1+s) = \\frac{\\pi }{\\sin (-\\pi s)}.$ One readily confirms that for all $s$ : $\\operatorname{dist}(s,\\mathbb {Z})>c$ for some $c>0$ fixed, $\\left| \\frac{\\pi }{\\sin (-\\pi s)} \\right| \\le \\frac{c^{\\prime }}{e^{\\pi \\operatorname{Im}(s)}}$ for a fixed constant $c^{\\prime }>0$ which depends on $c$ .",
"Therefore, along the $C^{seg}_k$ contour where $s=R+Iy$ , $|(-\\zeta )^s\\Gamma (-s)\\Gamma (1+s)|\\sim r^R e^{-y\\sigma }e^{-\\pi |y|},$ and since $\\sigma \\in (-\\pi ,\\pi )$ is fixed, this product decays exponentially in $|y|$ and the integral goes to zero as $k$ goes to infinity.",
"Along the $C^{arc}_k$ contour, the product of Gamma functions still behaves like $c^{\\prime }e^{-\\pi |y|}$ for some fixed $c^{\\prime }>0$ .",
"Thus along this contour (again using the notation $s=x+Iy$ ) $|(-\\zeta )^s\\Gamma (-s)\\Gamma (1+s)| \\sim e^{-y\\sigma }r^x e^{-\\pi |y|}.$ Since $r<1$ and $-(\\pi +\\sigma )<0$ these terms behave like $e^{-c^{\\prime \\prime }(x+|y|)}$ ($c^{\\prime \\prime }>0$ fixed) along the circular arc.",
"Clearly, as $k$ goes to infinity, the integrand decays exponentially in $k$ (versus the linear growth of the length of the contour) and the conditions of the lemma are met.",
"Applying Lemma REF we find that the $M^{th}$ term in the Fredholm expansion can be written as $\\frac{1}{M!}",
"\\sum _{\\sigma \\in S_M} \\operatorname{sgn}(\\sigma )\\prod _{j=1}^{M} \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }} \\frac{dw_j}{2\\pi I}\\int _{\\widetilde{\\mathcal {D}}_{w_j}}\\frac{ds_j}{2\\pi I} \\,\\Gamma (-s)\\Gamma (1+s)(-\\zeta )^{s} g_{w_j,w_{\\sigma (j)}}(q^{s}).$ Therefore the determinant can be written as $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ as desired.",
"At this point we now make critical use of the choice for the contour $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ on which $w$ varies, since proving analyticity in $\\zeta $ of the Fredholm determinant requires the decay properties of the kernel with respect to $w$ varying along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ .",
"In order to analytically extend our formula we must prove two facts.",
"First, that the left-hand side of equation (REF ) is analytic for all $\\zeta \\notin \\mathbb {R}_{+}$ ; and second, that the right-hand side determinant is defined (i.e., its expansion is convergent) and analytic for all $\\zeta \\notin \\mathbb {R}_{+}$ .",
"Expand the left-hand side of equation (REF ) as $\\sum _{n=0}^{\\infty } \\frac{ \\mathbb {P}(\\lambda _N = n) }{(1-\\zeta q^n)(1-\\zeta q^{n+1})\\cdots },$ where $\\mathbb {P}=\\mathbb {P}_{\\mathbf {MM}_{t=0}(\\tilde{a}_1,\\ldots ,\\tilde{a}_N;\\rho )}$ .",
"Observe that for any $\\zeta \\notin \\lbrace q^{-M}\\rbrace _{M=0,1,\\ldots }$ , within a neighborhood of $\\zeta $ the infinite products are uniformly convergent and bounded away from zero.",
"As a result the series is uniformly convergent in a neighborhood of any such $\\zeta $ which implies that its limit is analytic, as desired.",
"Turning to the Fredholm determinant, we must show that the series denoted by $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ is an analytic function of $\\zeta $ away from $\\mathbb {R}_{+}$ .",
"For this we will appeal to the fact that limits of uniformly absolutely convergent series of analytic functions are themselves analytic.",
"Recall that $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta }) = 1 + \\sum _{n=1}^{\\infty } \\frac{1}{n!}",
"\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{dw_1}{2\\pi I} \\cdots \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }} \\frac{dw_n}{2\\pi I} \\det (\\tilde{K}_{\\zeta }(w_i,w_j))_{i,j=1}^n.$ It is clear from the definition of $\\tilde{K}_{\\zeta }$ that $\\det (\\tilde{K}_{\\zeta }(w_i,w_j))_{i,j=1}^{n}$ is analytic in $\\zeta $ away from $\\mathbb {R}_{+}$ .",
"Thus any partial sum of the above series is analytic in the same domain.",
"What remains is to show that the series is uniformly absolutely convergent on any fixed neighborhood of $\\zeta $ not including $\\mathbb {R}_{+}$ .",
"Towards this end consider the $n^{th}$ term in the Fredholm expansion: $\\begin{aligned}F_n(\\zeta ) &=& \\frac{1}{n!}",
"\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{dw_1}{2\\pi I}\\cdots \\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{dw_n}{2\\pi I} \\int _{\\widetilde{\\mathcal {D}}_{w_1}} \\frac{ds_1}{2\\pi I} \\cdots \\int _{\\widetilde{\\mathcal {D}}_{w_n}} \\frac{ds_n}{2\\pi I} \\det \\left(\\frac{1}{q^{s_i}w_i -w_j}\\right)_{i,j=1}^{n}\\\\&&\\times \\prod _{j=1}^{n} \\left(\\Gamma (-s_j)\\Gamma (1+s_j) (-\\zeta )^{s_j} \\exp \\big (\\gamma w_j(q^{s_j}-1)\\big )\\prod _{m=1}^{N}\\frac{(q^{s_j}w_j/\\tilde{a}_m;q)_{\\infty }}{(w_j/\\tilde{a}_m;q)_{\\infty }} \\right).\\end{aligned}$ We wish to bound the absolute value of this.",
"We may pull the absolute values inside the integration.",
"Now observe the following bounds which hold uniformly over all $w_j\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and then all $s_j\\in \\widetilde{\\mathcal {D}}_{w_j}$ .",
"For the first bound note that for all $z:|z|\\le 1$ , there exists a constant $c_q$ such that $|(z;q)_{\\infty }|<c_q$ .",
"From that it follows that for any $|z|>1$ , $|(z;q)_{\\infty }|\\le c_q|(z;q)_{k}|$ where $k$ is such that $|zq^k|\\le 1$ .",
"This $k$ is approximately $-\\ln (|z|)/\\ln (q)$ and hence bounded by $k\\le c_q^{\\prime } \\ln |z|$ for some other constant $c_q^{\\prime }$ .",
"Finally $|(z;q)_{k}|\\le c_q^{\\prime \\prime } |z|^k \\le c_q^{\\prime \\prime } |z|^{c_q^{\\prime }\\ln |z|}$ .",
"From this and the fact that $|q^{s_j}|<1$ (recall that $\\operatorname{Re}(s_j)>0$ along $\\widetilde{\\mathcal {D}}_{w_j}$ ), it follows that for $|w_j|\\ge 1$ along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ we can bound $\\left| \\prod _{m=1}^{N} \\frac{(q^{s_j}w_j/\\tilde{a}_m;q)_{\\infty }}{(w_j/\\tilde{a}_m;q)_{\\infty }}\\right| \\le c_1 |w_{j}|^{N c_q^{\\prime }\\ln (|w_j|/\\tilde{a})}$ for some constant $c_1$ and $\\tilde{a}=\\min _i\\lbrace \\tilde{a}_i\\rbrace $ .",
"For $|w_j|\\le 1$ along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , we can bound the above left-hand side by a constant and since $|w_j|$ is bounded from below along $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , it follows that the above bound (REF ) holds for all $w\\in \\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ , possibly with a modified value of $c_1$ .",
"By Hadamard's inequality and the conditions we have imposed on $\\widetilde{\\mathcal {D}}_{w_j}$ we get the crude bound $\\left| \\det \\left(\\frac{1}{q^{s_i}w_i -w_j}\\right)_{i,j=1}^{n} \\right| \\le c_2^n n^{n/2}.$ for some fixed constant $c_2>0$ .",
"Finally note that by the conditions we imposed in choosing the contours, for $w_j$ on $\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }$ and $s_j$ on $\\widetilde{\\mathcal {D}}_{w_j}$ , we have $\\operatorname{Re}\\left(w_j(q^s-1)\\right)\\le -c_{\\varphi } |w_j|$ where $c_{\\varphi }>0$ is a constant depending on $\\varphi \\in (0,\\pi /4)$ .",
"From this it follows that $\\left|\\exp \\big (\\gamma w_j(q^{s_j}-1)\\big )\\right| \\le \\exp \\big (-\\gamma c_{\\varphi }|w_j|\\big ).$ Taking the absolute value of (REF ) and bringing the absolute value all the way inside the integrals, we find that after plugging in the results of (REF ), (REF ) and (REF )For a complex contour $C$ and a function $f:C\\rightarrow \\mathbb {R}$ we write $\\int _C |dz|f(z)$ for the integral of $f$ along $C$ with respect to the arc length $|dz|$ .",
"$|F_n(\\zeta )| \\le \\frac{(c_1 c_2)^n n^{n/2}}{n!}",
"\\left(\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{|dw|}{2\\pi }\\int _{\\widetilde{\\mathcal {D}}_{w}}\\frac{|ds|}{2\\pi } \\left|\\Gamma (-s)\\Gamma (1+s) (-\\zeta )^{s}\\right| |w|^{N c_q^{\\prime }\\ln (|w|/\\tilde{a})} \\exp (-\\gamma c_{\\varphi }|w|)\\right)^n.$ We integrate this in the $s$ variables first.",
"For $\\zeta \\notin \\mathbb {R}_{+}$ we would like to bound $\\int _{\\widetilde{\\mathcal {D}}_{w}} \\frac{|ds|}{2\\pi } |\\Gamma (-s)\\Gamma (1+s) (-\\zeta )^{s}|$ for a neighborhood of $\\zeta $ which does not touch $\\mathbb {R}_{+}$ .",
"We divide the contour of integral into two regions and bound the integral along each region: (1) The portion of the contour from $R-Id$ to $1/2-Id$ and then vertical to $1/2$ and its reflection through the real axis; (2) The portion of the contour which is infinite from $R-I\\infty $ to $R-Id$ and then from $R+Id$ to $R+I\\infty $ .",
"Recall that by Remark REF we may assume that up to constants $d\\approx |w|^{-1}$ and $R\\approx \\ln |w|$ for $|w|$ large enough.",
"Case (1): By standard bounds $|\\Gamma (-s)\\Gamma (1+s)|\\le 1/d \\approx |w|$ (since $1/\\sin (x)\\approx 1/x$ near $x=0$ ).",
"Calling $r$ the maximal modulus over the neighborhood of $|\\zeta |$ in question, it follows that since the $s$ integral in Case (1) has length less than $2R$ (when $d<1/2$ ), the first part of the integral is bounded by a constant times $|w|\\ln |w| r^{c_3\\ln |w|}$ with a constant $c_3>0$ .",
"Case (2): The product of Gamma functions decays exponentially in $s$ and so the integral is estimated by $r^{R}$ which, by Remark REF is like $r^{c_3\\ln |w|}$ .",
"Summing up the above two cases we have that for $|w|$ large, $\\int _{\\widetilde{\\mathcal {D}}_{w}} \\frac{|ds|}{2\\pi } |\\Gamma (-s)\\Gamma (1+s) (-\\zeta )^{s}| \\le c_4 r^{c_3\\ln |w|} |w| \\ln |w|.$ This estimate can be plugged in to the right-hand side of (REF ) to reduce the bound to just an integral in the $w_j$ .",
"This integral factors and thus we have $\\begin{aligned}|F_n(\\zeta )| &\\le & \\frac{(c_1 c_2)^n n^{n/2}}{n!}",
"\\left(\\int _{\\widetilde{\\mathcal {C}}_{\\tilde{\\alpha },\\varphi }}\\frac{|dw|}{2\\pi } |w|^{N c_q^{\\prime }\\ln (|w|/\\tilde{a})} c_4 r^{c_3\\ln |w|}|w|\\ln |w|\\exp (-\\gamma c_{\\varphi }|w|)\\right)^n.\\end{aligned}$ The integral in $|w|$ is clearly convergent due to the exponential decay (which easily overwhelms the growth of $|w|^{Nc_q^{\\prime }\\ln |w|}$ as well as the other terms).",
"Thus the right-hand side above is bounded by $c_5^n n^{n/2}/n!$ for some constant $c_5$ .",
"Thus $F_n(\\zeta )$ is absolutely convergent, uniformly over any fixed neighborhood of a $\\zeta \\notin \\mathbb {R}_{+}$ .",
"This implies that $\\det (\\mathbb {1}+\\tilde{K}_{\\zeta })$ is analytic in $\\zeta \\notin \\mathbb {R}_{+}$ and hence completes the proof of Step 3 and hence the proof of the theorem."
],
[
"Proof of Proposition ",
"By virtue of Lemma REF it suffices to show that for some $c,C>0$ , $|K_u(v,v^{\\prime })|\\le Ce^{-c|v|}.$ Before proving this let us recall from Definition REF the contours with which we are dealing.",
"The variable $v$ lies on $\\mathcal {C}_{\\alpha ,\\varphi }$ and hence can be written as $v=\\alpha -\\kappa \\cos (\\varphi ) \\pm I\\kappa \\sin (\\varphi )$ , for $\\kappa \\in \\mathbb {R}_+$ , where the $\\pm $ represents the two rays of the contour.",
"The $s$ variables lies on $\\mathcal {D}_{v}$ which depends on $v$ and has two parts: The portion (which we have denoted $\\mathcal {D}_{v,\\sqsubset }$ ) with real part bounded between $1/2$ and $R$ and imaginary part $\\pm d$ for $d$ sufficiently small, and the vertical portion (which we have denoted $\\mathcal {D}_{v,\\vert }$ ) with real part $R$ .",
"The condition on $R$ implies that $R\\ge -\\operatorname{Re}(v)+\\alpha +1$ and for our purposes we will assume $R=-\\operatorname{Re}(v)+\\alpha +1$ .",
"Let us denote by $h(s)$ the integrand in (), through which $K_u(v,v^{\\prime })$ is defined.",
"We split the proof into three steps.",
"Step 1: We show the integral of $h(s)$ over $s\\in \\mathcal {D}_{v,\\sqsubset }$ is bounded by an expression with exponential decay in $|v|$ .",
"Step 2: We show the integral of $h(s)$ over $s\\in \\mathcal {D}_{v,\\vert }$ is bounded by an expression with exponential decay in $|v|$ .",
"Step 3: We show that the integral of $h(s)$ over the entire contour $s\\in \\mathcal {D}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"The combination of these three steps imply the inequality (REF ) and hence complete the proof."
],
[
"Step 1:",
"We the various terms in $h(s)$ separately and develop bounds for each.",
"Let us write $s=x+Iy$ and note that along the contour $\\mathcal {D}_{v,\\sqsubset }$ , $y\\in [-d,d]$ for $d$ small, and $x\\in [1/2,R]$ .",
"Let us start with $e^{v\\tau s + \\tau s^2/2}$ .",
"The norm of the above expression is bounded by the exponential of the real part of the exponent.",
"For $s$ along $\\mathcal {D}_{v,\\sqsubset }$ $\\operatorname{Re}(v s + s^2/2) = x\\operatorname{Re}(v)+\\frac{x^2}{2}-y\\operatorname{Im}(v)-\\frac{y^2}{2} .$ We take $R=-\\operatorname{Re}(v)+\\alpha +1$ , $d$ sufficiently small, and the bound $\\operatorname{Re}(v)\\le \\tilde{c}^{\\prime }-c^{\\prime } |v|$ for some constants $c^{\\prime },\\tilde{c}^{\\prime }$ (depending on $\\varphi $ ), to deduce $\\operatorname{Re}(v s + s^2/2) \\le \\tilde{c} -c |v| x$ for some constants $c,\\tilde{c}>0$ , from which $|e^{v\\tau s + \\tau s^2/2}| \\le C e^{-c\\tau |v|x}.$ Let us now turn to the other terms in $h(s)$ .",
"We have $|u^s|\\le e^{x\\ln |u| - y\\arg (u)}$ and we may also bound $\\left|\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)}\\right| \\le C, \\qquad \\left|\\frac{1}{v+s-v^{\\prime }}\\right|\\le C,\\qquad |\\Gamma (-s)\\Gamma (1+s)|\\le C,$ for some constants $C>0$ (which may be different in each case).",
"The first bound comes from the functional equation for the Gamma function, and the last from the fact that $s$ is bounded away from $\\mathbb {Z}$ .",
"Combining these together shows that for $|v|$ large, the portion of the integral of $h(s)$ for $s$ in $\\mathcal {D}_{v,\\sqsubset }$ is bounded by (recall $s=x+Iy$ ) $\\int _{\\mathcal {D}_{v,\\sqsubset }}|ds| C^{\\prime } e^{-c\\tau |v|x + x\\ln |u| -\\arg (u)y} \\le C e^{-c|v|}$ for some constants $c,C>0$ .",
"Since for $|y|$ in a bounded set, everything starting from the integration path is clearly bounded, the bound holds.",
"As above, we consider the various terms in $h(s)$ separately and develop bounds for each.",
"Let us write $s=R+Iy$ and note that $s\\in \\mathcal {D}_{v,\\vert }$ corresponds to $y$ varying over all $|y|\\ge d$ .",
"As in Step 1, the most important bound will be that of $e^{v\\tau s + \\tau s^2/2}$ .",
"Observe that $\\operatorname{Re}(v s + s^2/2) = \\operatorname{Re}(v)R - \\operatorname{Im}(v) y + \\frac{R^2}{2} -\\frac{y^2}{2} = -\\frac{(y+\\operatorname{Im}(v))^2}{2} + \\frac{\\operatorname{Im}(v)^2}{2} +\\frac{R^2}{2} +\\operatorname{Re}(v)R.$ Observe that because $\\varphi \\in (0,\\pi /4)$ and $R=-\\operatorname{Re}(v)+\\alpha +1$ , $\\frac{\\operatorname{Im}(v)^2}{2} +\\frac{R^2}{2} +\\operatorname{Re}(v)R \\le \\tilde{c} -c |v|^2$ for some constants $c,\\tilde{c}>0$ .",
"Thus $\\operatorname{Re}(v s + s^2/2) \\le -\\frac{(y+\\operatorname{Im}(v))^2}{2} +\\tilde{c}-c |v|^2.$ Let us now turn to the other terms in $h(s)$ .",
"We bound $|u^s|\\le e^{R\\ln |u| - y\\arg (u)}.$ By standard bounds for the large imaginary part behavior we can show $\\left|\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)}\\right| \\le C e^{\\frac{\\pi }{2} |y|}$ for some constant $C>0$ sufficiently large.",
"Also, $|1/(v+s-v^{\\prime })|\\le C$ for a fixed constant.",
"Finally, the term $|\\Gamma (-s)\\Gamma (1+s)|\\le Ce^{-\\pi |y|},$ for some constant $C>0$ .",
"Combining these together shows that the integral of $h(s)$ over $s$ in $\\mathcal {D}_{v,\\vert }$ is bounded by a constant time $\\int _{\\mathbb {R}} \\exp \\left(-\\tau \\frac{(y+\\operatorname{Im}(v))^2}{2} -\\tau c |v|^2 + R\\ln |u| -y\\arg (u) -\\pi |y| + N\\frac{\\pi }{2}|y|\\right) dy.$ We can factor out the terms above which do not depend on $y$ , giving $\\exp \\left(-\\tau c |v|^2 + R\\ln |u| \\right) \\int _{\\mathbb {R}} \\exp \\left(-\\tau \\frac{(y+\\operatorname{Im}(v))^2}{2} -y \\arg (u) + N\\frac{\\pi }{2}|y|\\right) dy.$ Notice that the prefactors on $y$ and $|y|$ in the integrand's exponential are fixed constants.",
"We can therefore use the following bound that for $a$ fixed and $b\\in \\mathbb {R}$ , there exists a constant $C$ such that $\\int _{\\mathbb {R}} e^{-\\beta (y+b)^2 + a|y|}dy \\le C e^{|ab|},\\quad \\beta >0.$ Using this we find that we can upper-bound (REF ) by $\\exp \\left(-\\tau c |v|^2 + R\\ln |u|+c^{\\prime }|v|\\right).$ For $|v|$ large enough the Gaussian decay in the above bound dominates, and hence integral of $h(s)$ over $s$ in $\\mathcal {D}_{v,\\vert }$ is bounded by $C e^{-c|v|}$ for some constants $c,C>0$ .",
"Since $v^{\\prime }$ only comes in to the term $1/(v+s-v^{\\prime })$ in the integrand, it is clear that the above arguments imply that the integral of $h(s)$ over the entire contour $s\\in \\mathcal {D}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"This completes the third step and hence completes the proof of Proposition REF ."
],
[
"Proof of Proposition ",
"Fix $\\eta ,r>0$ .",
"We are presently considering the Fredholm determinant of the kernels $K_u^{\\varepsilon }$ and $K_u$ restricted to the fixed finite contour $\\mathcal {C}_{\\alpha ,\\varphi ,<r}$ .",
"By Lemma REF we need only show convergence as $\\varepsilon \\rightarrow 0$ of the kernel $K_u^{\\varepsilon }(v,v^{\\prime })$ to $K_u(v,v^{\\prime })$ , uniformly in $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ .",
"This is achieved via the following lemma.",
"Lemma 7.2 For all $\\eta ^{\\prime }>0$ there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ and for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$, $\\left|K_u^{\\varepsilon }(v,v^{\\prime })-K_u(v,v^{\\prime })\\right|\\le \\eta ^{\\prime }.$ The kernels $K_u^{\\varepsilon }$ and $K_u$ are both defined via integrals over $s$ .",
"The contour on which $s$ is integrated can be fixed for ($\\varepsilon <\\varepsilon _0$ ) to equal $\\mathcal {D}_{v}$ , which is the $s$ contour used to define $K_u$ .",
"The fact that the $s$ contours are the same for $K_u^{\\varepsilon }$ and $K_u$ is convenient.",
"The proof of this lemma will follow from three claims.",
"The first deals with the uniformity of convergence of the integrand defining $K_u^{\\varepsilon }$ to the integrand defining $K_u$ for $s$ restricted to any fixed compact set.",
"Before stating this claim, let us define some notation.",
"Definition 7.3 Let $\\mathcal {D}_{v,>M}= \\lbrace s\\in \\mathcal {D}_{v}: |s|\\ge M\\rbrace $ be the portion of $\\mathcal {D}_{v}$ of magnitude greater than $M$ and similarly let $\\mathcal {D}_{v,<M}= \\lbrace s\\in \\mathcal {D}_{v}: |s|<M\\rbrace $ .",
"Let us assume $M$ is large enough so that $\\mathcal {D}_{v,>M}$ is the union of two vertical rays with fixed real part $R=-\\operatorname{Re}(v)+\\alpha +1$ .",
"Assuming this, we will write $s=R+Iy$ .",
"Then for $y_M=(M^2- (-\\operatorname{Re}(v)+\\alpha +1)^2)^{1/2}$ , the contour $\\mathcal {D}_{v,>M}=\\lbrace R+Iy: |y|\\ge y_M\\rbrace $ .",
"Claim 7.4 For all $\\eta ^{\\prime \\prime }>0$ and $M>0$ there exists $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ , for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , and for all $s\\in \\mathcal {D}_{v,<M}$ , $\\left| h^q(s) - \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}\\right|\\le \\eta ^{\\prime \\prime },$ where $h^q$ is given in (REF ).",
"This is a strengthened version of the pointwise convergence in (REF ) through ().",
"It follows from the uniform convergence of the $\\Gamma _q$ function to the $\\Gamma $ function on compact regions away from the poles, as well as standard Taylor series estimates.",
"The choice of contours is such that the pole arising from $1/(v+s-v^{\\prime })$ is uniformly avoided in the limiting procedure as well.",
"It remains to show that for $M$ large enough, the integrals defining $K_u^{\\varepsilon }(v,v^{\\prime })$ and $K_u(v,v^{\\prime })$ restricted to $s$ in $\\mathcal {D}_{v,>M}$ , have negligible contribution to the kernel, uniformly over $v,v^{\\prime }$ and $\\varepsilon $ .",
"This must be done separately for each of the kernels and hence requires two claims.",
"Claim 7.5 For all $\\eta ^{\\prime }>0$ there exists $M_0>0$ and $\\varepsilon _0>0$ such that for all $\\varepsilon <\\varepsilon _0$ , for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , and for all $M>M_0$ , $\\left|\\int _{\\mathcal {D}_{v,>M}} ds h^q(s)\\right|\\le \\eta ^{\\prime }.$ We will use the notation introduced in Definition REF and assume $M_0$ is large enough so that $\\mathcal {D}_{v,>M}$ is only comprised of two vertical rays.",
"Let us first consider the behavior of the left-hand side of ().",
"The magnitude of this term is bounded by the exponential of $\\operatorname{Re}(\\tau \\varepsilon ^{-1} s + \\varepsilon ^{-2} \\tau q^v (q^s-1)).$ This equation is periodic in $y$ (recall $s=R+Iy$ ) with a fundamental domain $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$ .",
"For $\\varepsilon ^{-1}\\pi >|y|>y_0$ for some $y_0$ which can be chosen uniformly in $v$ and $\\varepsilon $ , the following inequality holds $\\operatorname{Re}(\\tau \\varepsilon ^{-1} s + \\varepsilon ^{-2} \\tau q^v (q^s-1)) \\le -\\tau y^2/6$ This can is proved by careful Taylor series estimation and the inequality that for $x\\in [-\\pi ,\\pi ]$ , $\\cos (x)-1\\le -x^2/6$ .",
"This provides Gaussian decay in the fundamental domain of $y$ .",
"Turning to the ratio of $q$ -Gamma functions in (), observe that away from its poles, the denominator $\\left|\\frac{1}{\\Gamma _q(s+v-a_m)}\\right|\\le c e^{c^{\\prime } f^{\\varepsilon }(s)}$ where $c,c^{\\prime }$ are positive constants independent of $\\varepsilon $ and $v$ (as it varies in its compact contour) and $f^{\\varepsilon }(s) = {\\rm dist}(\\operatorname{Im}(s),2\\pi \\varepsilon ^{-1} \\mathbb {Z})$ .",
"This establishes a periodic bound on this denominator, which grows at most exponentially in the fundamental domain.",
"The numerator $\\Gamma _q(v-a_m)$ in () is bounded uniformly by a constant.",
"This is because the $v$ contour was chosen to avoid the poles of the Gamma function, and the convergence of the $q$ -Gamma function to the Gamma function is uniform on compact sets away from those poles.",
"Finally, the magnitude of (REF ) corresponds to $|u^s|$ and behaves like $e^{-R\\ln {|u|} + y\\arg (u)}$ .",
"Thus, we have established the following inequality which is uniform in $v,v^{\\prime }$ and $\\varepsilon $ as $y$ varies: $\\left|\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s \\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}} e^{\\gamma q^v(q^{s}-1)} \\prod _{m=1}^{N} \\frac{\\Gamma _q(v-\\ln _q(\\tilde{a}_m))}{\\Gamma _q(s+v-\\ln _q(\\tilde{a}_m))} \\right| \\le c^{\\prime \\prime }\\, e^{-f^{\\varepsilon }(s)^2/6+c^{\\prime }N|f^{\\varepsilon }(s)|}$ for some constant $c^{\\prime \\prime }>0$ .",
"Notice that this inequality is periodic with respect to the fundamental domain for $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$.",
"The last term to consider is $\\Gamma (-s)\\Gamma (1+s)$ which is not periodic in $y$ and decays like $e^{-\\pi |y|}$ for $y\\in \\mathbb {R}$ .",
"Since $\\mathcal {D}_{v,>M}$ is only comprised of two vertical rays we must control the integral of $h^q(s)$ for $s=R+Iy$ and $|y|>y_M$ .",
"By making sure $M$ is large enough, we can use the periodic bound (REF ) to show that the integral over $y_M<|y|<\\varepsilon ^{-1} \\pi $ is less than $\\eta $ (with the desired uniformity in $v,v^{\\prime }$ and $\\varepsilon $ .",
"For the integral over $|y|>\\varepsilon ^{-1}\\pi $ , we can use the above exponential decay of $\\Gamma (-s)\\Gamma (1+s)$ .",
"On shifts by $2\\pi \\varepsilon ^{-1}\\mathbb {Z}$ of the fundamental domain, the exponential decay of $\\Gamma (-s)\\Gamma (1+s)$ can be compared to the boundedness of the other terms (which is certainly true considering the bounds we established above).",
"The integral of each shift can be bounded by a term in a convergent geometric series.",
"Taking $\\varepsilon _0$ small then implies that the sum can be bounded by $\\eta ^{\\prime }$ as well.",
"Since $\\eta ^{\\prime }$ was arbitrary the proof is complete.",
"Claim 7.6 For all $\\eta ^{\\prime }>0$ there exists $M_0>0$ such that for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , and for all $M>M_0$ , $\\left|\\int _{\\mathcal {D}_{v,>M}} ds \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}\\right|\\le \\eta ^{\\prime }.$ The desired decay here comes easily from the behavior of $vs+s^2/2$ as $s$ varies along $\\mathcal {D}_{v,>M}$ .",
"As before, assume that $M_0$ is large enough so that this contour is only comprised of two vertical rays and set $s=R+ Iy$ for $y\\in \\mathbb {R}$ for $|y|>y_M$ .",
"As in the proof of Proposition REF given in Section REF one shows that $|e^{v\\tau s+\\tau s^2/2}|\\le C e^{-cy^2}$ uniformly over $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<R}$ , and for all $M>M_0$ .",
"This behavior should be compared to that of the other terms: $|\\Gamma (-s)\\Gamma (1+s)|\\approx e^{-\\pi |y|}$ ; $|u^s|= e^{-R\\ln {|u|} + y\\arg (u)}$ ; $\\left|\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)}\\right|\\le C e^{|y| \\pi /2}$ ; and $|1/(v+s+v^{\\prime })|\\le C$ as well.",
"Combining these observations we see that the integral decays in $|y|$ at worst like $C e^{-cy^2+c^{\\prime } |y|}$ .",
"Thus, by choosing $M$ large enough so that $y_M\\gg 1$ we can be assured that the integral over $|y|>y_M$ is as small as desired, proving the above claim.",
"Let us now combine the above three claims to finish the proof of the Proposition REF .",
"Choose $\\eta ^{\\prime }=\\eta /3$ and fix $M_0$ and $\\varepsilon _0^{\\prime }$ as specified by the second and third of the above claims.",
"Fix some $M>M_0$ and let $L$ equal the length of the finite contour $\\mathcal {D}_{v,<M}$ .",
"Set $\\eta ^{\\prime \\prime }=\\frac{\\eta ^{\\prime }}{3L}$ and apply Claim REF .",
"This yields an $\\varepsilon _0$ (which we can assume is less than $\\varepsilon _0^{\\prime }$ ) so that (REF ) holds.",
"This implies that for $\\varepsilon <\\varepsilon _0$ , and for all $v,v^{\\prime }\\in \\mathcal {C}_{\\alpha ,\\varphi ,<r}$ , $\\left| \\int _{\\mathcal {D}_{v,<M}} h^q(s) ds - \\int _{\\mathcal {D}_{v,<M}} \\Gamma (-s)\\Gamma (1+s) \\prod _{m=1}^{N}\\frac{\\Gamma (v-a_m)}{\\Gamma (s+v-a_m)} \\frac{ u^s e^{v\\tau s+\\tau s^2/2}}{v+s-v^{\\prime }}ds \\right|\\le \\eta ^{\\prime }/3.$ From the triangle inequality and the three factors of $\\eta ^{\\prime }/3$ we arrive at the claimed result of (REF ) and thus complete the proof of the lemma and hence also Proposition REF ."
],
[
"Proof of Proposition ",
"The proof of this proposition is essentially a finite $\\varepsilon $ (recall $q=e^{-\\varepsilon }$ ) perturbation of the proof of Proposition REF given in Section REF .",
"The estimates presently are a little more involved since the functions involved are $q$ -deformations of classic functions.",
"However, by careful Taylor approximation with remainder estimates, all estimates can be carefully shown.",
"By virtue of Lemma REF it suffices to show that for some $c,C>0$ , $|K_u^{\\varepsilon }(v,v^{\\prime })|\\le C e^{-c|v|}.$ Before proving this let us recall from Definition REF the contours with which we are dealing.",
"The variable $v$ lies on $\\mathcal {C}^{\\epsilon }_{\\alpha ,\\varphi ,r}$ for $\\varphi \\in (0,\\pi /4)$ .",
"The $s$ variables lies on $\\widetilde{\\mathcal {D}}_{v}$ which depends on $v$ and has two parts: The portion (which we have denoted $\\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ ) with real part bounded between $1/2$ and $R$ and imaginary part $\\pm d$ for $d$ sufficiently small, and the vertical portion (which we have denoted $\\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ ) with real part $R$ .",
"The condition on $R$ is that $R\\ge -\\operatorname{Re}(v)+\\alpha +1$ and for our purposes we can take that to be an equality.",
"Let us recall that the integrand in (REF ), through which $K_u^{\\varepsilon }(v,v^{\\prime })$ is defined, is denoted by $h^q(s)$ .",
"We split the proof into three steps.",
"Step 1: We show the integral of $h^q(s)$ over $s\\in \\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ is bounded for all $\\varepsilon <\\varepsilon _0$ by an expression with exponential decay in $|v|$ .",
"Step 2: We show the integral of $h^q(s)$ over $s\\in \\widetilde{\\mathcal {D}}_{v,\\vert }$ is bounded for all $\\varepsilon <\\varepsilon _0$ by an expression with exponential decay in $|v|$ .",
"Step 3: We show that for all $\\varepsilon <\\varepsilon _0$ , the integral of $h^q(s)$ over the entire contour $\\widetilde{\\mathcal {D}}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"The combination of these three steps imply the inequality (REF ) and hence complete the proof."
],
[
"Step 1:",
"We consider the various terms in $h^q(s)$ separately (in particular we consider the left-hand sides of (REF ) through ()) and develop bounds for each which are valid uniformly for $\\varepsilon <\\varepsilon _0$ and $\\varepsilon <0$ small enough.",
"Let us write $s=x+Iy$ and note that along the contour $\\widetilde{\\mathcal {D}}_{v,\\sqsubset }$ , $y\\in [-d,d]$ for $d$ small, and $x\\in [1/2,R]$ .",
"Let us start with the left-hand side of () which can be rewritten as $\\exp \\left(\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1))\\right).$ The norm of the above expression is bounded by the exponential of the real part of the exponent.",
"For $\\varphi \\in (0,\\pi /4)$ one shows (as a perturbation of the analogous estimate in Step 1 of the Proof of Proposition REF ) via Taylor expansion with remainder estimates that $\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1))\\le \\tilde{c}- \\tau c|v| x,$ for some constants $c,\\tilde{c}$ .",
"The above bound implies $\\left|\\exp \\left(\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1))\\right)\\right|\\le C e^{-c|v|x}.$ Let us now turn to the other terms in $h^q(s)$ .",
"We bound the left-hand side of (REF ) as $\\left|e^{-\\tau s \\varepsilon ^{-1}}\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s\\right| \\le C |u^s| \\le C e^{x\\ln |u| - y\\arg (u)}.$ We may also bound the left-hand sides of () and (), as well as the remaining product of Gamma functions by constants: $\\left|\\frac{\\Gamma _q(v-\\ln _q(\\tilde{a}_m))}{\\Gamma _q(s+v-\\ln _q(\\tilde{a}_m))} \\right|\\le C,\\qquad \\left|\\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}}\\right|\\le C, \\qquad |\\Gamma (-s)\\Gamma (1+s)|\\le C,$ for some constants $C>0$ (which may be different in each case).",
"The first bound comes from the functional equation for the $q$ -Gamma function, and the last from the fact that $s$ is bounded away from $\\mathbb {Z}$ .",
"Combining these together shows that for $|v|$ large, $\\left|\\int _{\\widetilde{\\mathcal {D}}_{v,\\sqsubset }}h^q(s) ds\\right| \\le \\int _{\\widetilde{\\mathcal {D}}_{v,\\sqsubset }} C e^{-\\tau c|v| \\operatorname{Re}(s) + x\\ln |u|-y \\arg (u)} |ds| \\le C^{\\prime } e^{-c^{\\prime }|v|}$ for some constants $c^{\\prime },C^{\\prime }>0$ , while for bounded $|v|$ the integral is just bounded as well.",
"As above, we consider the various terms in $h^q(s)$ separately and develop bounds for each.",
"Let us write $s=R+Iy$ and note that $s\\in \\widetilde{\\mathcal {D}}_{v,\\vert }$ corresponds to $y$ varying over all $|y|\\ge d$ .",
"Three of the terms we consider (corresponding to the left-hand sides of (), () and ()) are periodic functions in $y$ with fundamental domain $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$ .",
"We will first develop bounds on these three terms in this fundamental domain, and then turn to the non-periodic terms.",
"We start by controlling the behavior of the left-hand side of () as $y$ varies in its fundamental domain.",
"For each $\\varphi <\\pi /4$ there exists a sufficiently small (yet positive) constant $c^{\\prime }$ such that as $y$ varies in its fundamental domain $\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1)) \\le c^{\\prime } \\tau \\operatorname{Re}(vs+s^2/2).$ On account of this, we can use the bound (REF ) from the proof of Proposition REF .",
"This implies that $\\tau \\operatorname{Re}(\\varepsilon ^{-1}s + \\varepsilon ^{-2} q^v(q^s-1)) \\le c^{\\prime } \\tau \\left(-\\frac{(y+\\operatorname{Im}(v))^2}{2} -c |v|^2\\right).$ Let us now turn to the other $y$ -periodic terms in $h^q(s)$ .",
"By bounds for the large imaginary part behavior of the $q$ -Gamma function we can show $\\left|\\frac{\\Gamma _q(v-\\ln _q(\\tilde{a}_m))}{\\Gamma _q(s+v-\\ln _q(\\tilde{a}_m))} \\right| \\le C e^{c f^{\\varepsilon }(s+v)}$ for some constants $c,C>0$ where $f^{\\varepsilon }(s) = {\\rm dist}(\\operatorname{Im}(s),2\\pi \\varepsilon ^{-1} \\mathbb {Z})$ .",
"Note that as opposed to (REF ) when $|v|$ was bounded, in the above inequality we write $f^{\\varepsilon }(s+v)$ in the exponential on the right-hand side.",
"This is because we are presently considering unbounded ranges for $v$ .",
"Also, we can bound $\\left|\\frac{q^v \\ln q}{q^{s+v} - q^{v^{\\prime }}}\\right|\\le C$ for some constant $C>0$ .",
"The parts of $h^q(s)$ which are not periodic in $y$ can easily be bounded.",
"We bound the left-hand side of (REF ) as in Step 1 by $\\left|e^{-\\tau s \\varepsilon ^{-1}}\\left(\\frac{-\\zeta }{(1-q)^N}\\right)^s\\right| \\le C |u^s| \\le C e^{x\\ln |u| - y\\arg (u)}.$ Finally, the term $|\\Gamma (-s)\\Gamma (1+s)|\\le Ce^{-\\pi |y|},$ for some constant $C>0$ .",
"We may now combine the estimates above.",
"The idea is to first prove that the integral on the fundamental domain $y\\in [-\\pi \\varepsilon ^{-1},\\pi \\varepsilon ^{-1}]$ is exponentially small in $|v|$ .",
"Then, by using the decay of the two non-periodic terms above, we can get a similar bound for the integral as $y$ varies over all of $\\mathbb {R}$ .",
"For $j\\in \\mathbb {Z}$ , define the $j$ shifted fundamental domain as $D_j=j\\varepsilon ^{-1}2\\pi + [-\\varepsilon ^{-1}\\pi ,\\varepsilon ^{-1}\\pi ]$ .",
"Let $I_j:= \\int _{D_j} h^q(R+Iy) dy$ and observe that combining all of the bounds developed above, we have that $|I_j|\\le C \\int _{-\\varepsilon ^{-1}\\pi }^{\\varepsilon ^{-1}\\pi } F_1(y) F_2(y) dy,$ where $\\begin{aligned}F_1(y) &= \\exp \\left(c^{\\prime } \\tau \\left(-\\frac{(y+\\operatorname{Im}(v))^2}{2} -c |v|^2\\right) +c^{\\prime \\prime } f^{\\varepsilon }(s+v) +x\\ln |u|\\right),\\\\F_2(y) &= \\exp \\left(- (y+j\\varepsilon ^{-1}2\\pi )\\arg (u) -\\pi |y+j\\varepsilon ^{-1}2\\pi | \\right).\\end{aligned}$ The term $F_1(y)$ is from the periodic bounds while $F_2(y)$ from the non-periodic terms (hence explaining the $j\\varepsilon ^{-1}2\\pi $ shift in $y$ ).",
"By assumption on $u$ , we have $-\\arg (u)-\\pi =\\delta \\le c$ for some $\\delta $ .",
"Therefore $F_2(y) \\le C e^{-c\\varepsilon ^{-1} |j|}$ form some constants $c,C>0$ .",
"Thus $|I_j|\\le C e^{-c\\varepsilon ^{-1} |j|} \\int _{-\\varepsilon ^{-1}\\pi }^{\\varepsilon ^{-1}\\pi } F_1(y)dy.$ Just as in the end of Step 2 in the proof of Proposition REF we can estimate the integral $\\int _{-\\varepsilon ^{-1}\\pi }^{\\varepsilon ^{-1}\\pi } F_1(y)dy \\le \\hat{C} e^{-\\hat{c}|v|}$ for some constants $\\hat{C},\\hat{c}>0$ .",
"This implies $|I_j|\\le \\hat{C} C e^{-c\\varepsilon ^{-1} |j|} e^{-\\hat{c}|v|}.$ Finally, observe that $\\left|\\int _{\\widetilde{\\mathcal {D}}_{v,\\vert }} h^{q}(s) ds\\right| \\le \\sum _{j\\in \\mathbb {Z}} |I_j| \\le \\hat{C} C e^{-\\hat{c}|v|} \\sum _{j\\in \\mathbb {Z}}e^{-c\\varepsilon ^{-1} |j|} \\le C^{\\prime } e^{-\\hat{c}|v|}$ where $C^{\\prime }$ is independent of $\\varepsilon $ as long as $\\varepsilon <\\varepsilon _0$ for some fixed $\\varepsilon _0$ .",
"This is the bound desired to complete this step.",
"Since $v^{\\prime }$ only comes in to the term $\\frac{q^v \\ln q}{q^{s+v}-q^{v^{\\prime }}}$ in the integrand, it is clear that the above arguments imply that the integral of $h^q(s)$ over the entire contour $s\\in \\widetilde{\\mathcal {D}}_{v}$ is bounded by a fixed constant, independent of $v$ or $v^{\\prime }$ .",
"This completes the third step and hence completes the proof of Proposition REF ."
],
[
"Two probability lemmas",
"Lemma 8.1 (Lemma 4.1.38 of [16]) Consider a sequence of functions $\\lbrace f_n\\rbrace _{n\\ge 1}$ mapping $\\mathbb {R}\\rightarrow [0,1]$ such that for each $n$ , $f_n(x)$ is strictly decreasing in $x$ with a limit of 1 at $x=-\\infty $ and 0 at $x=\\infty $ , and for each $\\delta >0$ , on $\\mathbb {R}\\setminus [-\\delta ,\\delta ]$ , $f_n$ converges uniformly to $\\mathbf {1}_{x\\le 0}$ .",
"Consider a sequence of random variables $X_n$ such that for each $r\\in \\mathbb {R}$ , $\\mathbb {E}[f_n(X_n-r)] \\rightarrow p(r)$ and assume that $p(r)$ is a continuous probability distribution function.",
"Then $X_n$ converges weakly in distribution to a random variable $X$ which is distributed according to $\\mathbb {P}(X\\le r) = p(r)$.",
"Lemma 8.2 (Lemma 4.1.39 of [16]) Consider a sequence of functions $\\lbrace f_n\\rbrace _{n\\ge 1}$ mapping $\\mathbb {R}\\rightarrow [0,1]$ such that for each $n$ , $f_n(x)$ is strictly decreasing in $x$ with a limit of 1 at $x=-\\infty $ and 0 at $x=\\infty $ , and $f_n$ converges uniformly on $\\mathbb {R}$ to $f$ .",
"Consider a sequence of random variables $X_n$ converging weakly in distribution to $X$ .",
"Then $\\mathbb {E}[f_n(X_n)] \\rightarrow \\mathbb {E}[f(X)].$"
],
[
"Some properties of Fredholm determinants",
"We give some important properties for Fredholm determinants.",
"For a more complete treatment of this theory see, for example, [59].",
"Lemma 8.3 (Proposition 1 of [65]) Suppose $t\\rightarrow \\Gamma _t$ is a deformation of closed curves and a kernel $L(\\eta ,\\eta ^{\\prime })$ is analytic in a neighborhood of $\\Gamma _t\\times \\Gamma _t\\subset 2$ for each $t$ .",
"Then the Fredholm determinant of $L$ acting on $\\Gamma _t$ is independent of $t$ .",
"Lemma 8.4 Consider the Fredholm determinant $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ on an infinite complex contour $\\Gamma $ and an integral operator $K$ on $\\Gamma $ .",
"Parameterize $\\Gamma $ by arc length with some fixed point corresponding to $\\Gamma (0)$ .",
"Assume that $|K(v,v^{\\prime })|\\le C$ for some constant $C$ and for all $v,v^{\\prime }\\in \\Gamma $ and that either of the following two exponential decay conditions holds: There exists constants $c,C>0$ such that $|K(\\Gamma (s),\\Gamma (s^{\\prime }))|\\le Ce^{-c|s|},$ Then the Fredholm series defining $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ is well-defined.",
"Moreover, for any $\\eta >0$ there exists an $r_0>0$ such that for all $r>r_0$ $|\\det (\\mathbb {1}+K)_{L^2(\\Gamma )} - \\det (\\mathbb {1}+K)_{L^2(\\Gamma _r)}|\\le \\eta $ where $\\Gamma _r=\\lbrace \\Gamma (s):|s|\\le r\\rbrace $ .",
"The Fredholm series expansion () is given by $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )} = \\sum _{n\\ge 0}\\frac{1}{n!}",
"\\int _{\\Gamma }ds_1\\cdots \\int _{\\Gamma }ds_n \\det (K(\\Gamma (s_i),\\Gamma (s_j)))_{i,j=1}^n$ is well-defined since by using Hadamard's boundHadamard's bound: the determinant of a $n\\times n$ matrix with entries of absolute value not exceeding 1 is bounded by $n^{n/2}$ .",
"one gets that $\\left|\\det (K(\\Gamma (s_i),\\Gamma (s_j)))_{i,j=1}^n\\right|\\le n^{n/2} C^n \\prod _{j=1}^n e^{-c|s_j|}$ which is absolutely integrable / summable.",
"To show is $\\det (\\mathbb {1}+K)_{L^2(\\Gamma _r)}\\rightarrow \\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ as $r\\rightarrow \\infty $ .",
"From (REF ) one immediately gets that $\\det (\\mathbb {1}+K)_{L^2(\\Gamma _r)} = \\det (\\mathbb {1}+P_r K)_{L^2(\\Gamma )}.$ where $P_r$ is the projection onto $\\Gamma _r$ .",
"The kernel $(P_r K)(s_i,s_j)$ converges pointwise to $K(s_i,s_j)$ and (REF ) provides a in $r$ uniform, integrable / summable bound for $\\det (K(\\Gamma (s_i),\\Gamma (s_j)))_{i,j=1}^n$ .",
"Therefore, by dominated convergence as $r\\rightarrow \\infty $ the two Fredholm determinant converge.",
"Lemma 8.5 Consider a finite length complex contour $\\Gamma $ and a sequence of integral operators $K^{\\varepsilon }$ on $\\Gamma $ , as well as an addition integral operator $K$ also on $\\Gamma $ .",
"Assume that for all $\\eta >0$ there exists $\\varepsilon _0$ such that for all $\\varepsilon <\\varepsilon _0$ and all $z,z^{\\prime }\\in \\Gamma $ , $|K^{\\varepsilon }(z,z^{\\prime }) - K(z,z^{\\prime })|\\le \\eta $ and that there is some constant $C$ such that $|K(z,z^{\\prime })|\\le C$ for all $z,z^{\\prime }\\in \\Gamma $ .",
"Then $\\lim _{\\varepsilon \\rightarrow 0} \\det (\\mathbb {1}+K^{\\varepsilon })_{L^2(\\Gamma )} = \\det (\\mathbb {1}+K)_{L^2(\\Gamma )}.$ As in Lemma REF one writes the Fredholm series.",
"Since $\\Gamma $ is finite, the Fredholm determinants $\\det (\\mathbb {1}+K)_{L^2(\\Gamma )}$ is well-defined because $|K(z,z^{\\prime })|\\le C$ (use Hadamard's bound).",
"By assumption, $K^\\varepsilon $ converges pointwise to $K$ and we have the uniform bound $|K^\\varepsilon (z,z^{\\prime })|\\le C+\\eta $.",
"This ensures that $\\det (\\mathbb {1}+K^{\\varepsilon })_{L^2(\\Gamma )}$ is well-defined and that we can take the limit inside the Fredholm series, providing our result."
],
[
"Reformulation of Fredholm determinants",
"Lemma 8.6 Let $\\widetilde{K}_{\\rm Ai}$ as in (REF ), $\\mathcal {C}_w$ as in (REF ), and $K_{\\rm Ai}$ the Airy kernel.",
"Then it holds $\\det (\\mathbb {1}+\\widetilde{K}_{\\rm Ai})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}.$ The integration path in (REF ) can be chosen to have $\\operatorname{Re}(z)>0$ and since $\\operatorname{Re}(w)<0$ for $w\\in \\mathcal {C}_w$ , we can use $\\frac{1}{z-w}=\\int _{\\mathbb {R}_+}d\\lambda e^{-\\lambda (z-w)}$ to write $\\widetilde{K}_{\\rm Ai}(w,w^{\\prime })=-(A B)(w,w^{\\prime })$ with $A:L^2(\\mathcal {C}_w)\\rightarrow L^2(\\mathbb {R}_+)$ and $B:L^2(\\mathbb {R}_+)\\rightarrow L^2(\\mathcal {C}_w)$ have kernels $A(w,\\lambda )=e^{-w^3/3+w(r+\\lambda )}, \\quad B(\\lambda ,w^{\\prime })=\\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty } \\frac{dz}{2\\pi I} \\frac{e^{z^3/3-z(r+\\lambda )}}{z-w^{\\prime }}.$ We also have $\\begin{aligned}(BA)(\\eta ,\\eta ^{\\prime })&=\\frac{1}{2\\pi I}\\int _{\\mathcal {C}_w} dw B(\\eta ,w) A(w,\\eta ^{\\prime })\\\\&=\\frac{1}{(2\\pi I)^2} \\int _{e^{-3\\pi I/4}\\infty }^{e^{3\\pi I/4}\\infty } dw \\int _{e^{-\\pi I/4}\\infty }^{e^{\\pi I/4}\\infty } dz\\frac{1}{z-w}\\frac{e^{z^3/3-z(r+\\eta )}}{e^{w^3/3-w(r+\\eta ^{\\prime })}}= K_{\\rm Ai}(\\eta +r,\\eta ^{\\prime }+r).\\end{aligned}$ Then, since $\\det (\\mathbb {1}-AB)_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-BA)_{L^2(\\mathbb {R}_+)}=\\det (\\mathbb {1}-K_{\\rm Ai})_{L^2(r,\\infty )}$ we get the claimed result.",
"The first equality is a general result which applies as long as $AB$ and $BA$ are both trace-class operators [59].",
"Lemma 8.7 Let $\\widetilde{K}_{\\rm BBP}$ as in (REF ), $\\mathcal {C}_w$ as in Theorem REF (b), and $K_{\\rm BBP}$ as in (REF ).",
"Then it holds $\\det (\\mathbb {1}+\\widetilde{K}_{\\rm BBP})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{\\rm BBP})_{L^2(r,\\infty )}.$ The proof is as the one of Lemma REF , except that in $A(w,\\lambda )$ is multiplied by $\\prod _{k=1}^m\\frac{1}{w-b_k}$ and $B(\\lambda ,w^{\\prime })$ by $\\prod _{k=1}^m (z-b_k)$ .",
"Lemma 8.8 Let $\\widetilde{K}_{{\\rm CDRP}}$ as in (REF ), $\\mathcal {C}_w$ as in (REF ), and $K_{{\\rm CDRP}}$ the CDRP kernel given in (REF ).",
"Then it holds $\\det (\\mathbb {1}+\\widetilde{K}_{{\\rm CDRP}})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}.$ Using $\\frac{1}{z-w^{\\prime }}=\\int _{\\mathbb {R}_+}d\\eta e^{-\\eta (z-w^{\\prime })}$ we get $\\widetilde{K}_{{\\rm CDRP}}(w,w^{\\prime })=\\int _{\\mathbb {R}_+} d\\eta A(w,\\eta ) B(\\eta ,w^{\\prime })$ with $B(\\eta ,w^{\\prime })=e^{\\eta w^{\\prime }}$ and $A(w,\\eta )=\\frac{-1}{2\\pi I}\\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}}dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} e^{z^3/3-w^3/3-\\eta z}.$ Thus $\\det (\\mathbb {1}+\\widetilde{K}_{{\\rm CDRP}})_{L^2(\\mathcal {C}_w)}=\\det (\\mathbb {1}-K_{{\\rm CDRP}})_{L^2(\\mathbb {R}_+)}$ where $K_{{\\rm CDRP}}=-BA$ , namely $\\begin{aligned}K_{{\\rm CDRP}}(\\eta ,\\eta ^{\\prime })&=-\\frac{1}{2\\pi I}\\int _{-\\frac{1}{4\\sigma }+I\\mathbb {R}}dw B(\\eta ,w) A(w,\\eta ^{\\prime })\\\\&=\\frac{1}{(2\\pi I)^2}\\int _{-\\frac{1}{4\\sigma }+I\\mathbb {R}} dw \\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}} dz \\frac{\\sigma \\pi S^{(z-w)\\sigma }}{\\sin (\\pi (z-w)\\sigma )} \\frac{e^{z^3/3-z\\eta ^{\\prime }}}{e^{w^3/3-w\\eta }}.\\end{aligned}$ The next step uses the following identity: for $0<\\operatorname{Re}(u)<1$ it holds $\\frac{\\pi \\, S^{u}}{\\sin (\\pi u)}=\\int _{\\mathbb {R}} \\frac{S e^{u t}}{S+e^t}dt$ from which, for $0<\\operatorname{Re}(u)<1/\\sigma $ it holds $\\frac{\\pi \\, \\sigma S^{\\sigma u}}{\\sin (\\pi \\sigma u)}=\\int _{\\mathbb {R}} \\frac{S e^{-u t}}{S+e^{-t/\\sigma }}dt.$ We can use wit $u=z-w$ and obtain $\\begin{aligned}K_{{\\rm CDRP}}(\\eta ,\\eta ^{\\prime })&=\\int _{\\mathbb {R}}dt \\frac{S}{S+e^{-t/\\sigma }} \\bigg (\\frac{1}{2\\pi I}\\int _{-\\frac{1}{4\\sigma }+I\\mathbb {R}} dw e^{-w^3/3+w(\\eta +t)}\\bigg )\\bigg (\\frac{1}{2\\pi I} \\int _{\\frac{1}{4\\sigma }+I\\mathbb {R}} dz e^{z^3/3-z(\\eta ^{\\prime }+t)}\\bigg )\\\\&=\\int _{\\mathbb {R}}dt \\frac{S}{S+e^{-t/\\sigma }} \\operatorname{Ai}(\\eta +t)\\operatorname{Ai}(\\eta ^{\\prime }+t),\\end{aligned}$ the expression of (REF )."
]
] | 1204.1024 |
[
[
"The stellar-subhalo mass relation of satellite galaxies"
],
[
"Abstract We extend the abundance matching technique (AMT) to infer the satellite-subhalo and central-halo mass relations (MRs) of galaxies, as well as the corresponding satellite conditional mass functions (CMFs).",
"We use the observed galaxy stellar mass function (GSMF) decomposed into centrals and satellites and the LCDM halo/subhalo mass functions as inputs.",
"We explore the effects of defining the subhalo mass at the time of accretion (m_acc) vs. at the time of observation (m_obs).",
"We test the standard assumption that centrals and satellites follow the same MRs, showing that this assumption leads to predictions in disagreement with observations, specially for m_obs.",
"Instead, when the satellite-subhalo MRs are constrained following our AMT, they are always different from the central-halo MR: the smaller the stellar mass (Ms), the less massive is the subhalo of satellites as compared to the halo of centrals of the same Ms. On average, for Ms<2x10^11Msol, the dark mass of satellites decreased by 60-65% with respect to their masses at accretion time.",
"The resulting MRs for both definitions of subhalo mass yield satellite CMFs in agreement with observations.",
"Also, when these MRs are used in a HOD model, the predicted correlation functions agree with observations.",
"We show that the use of m_obs leads to less uncertain MRs than m_acc, and discuss implications of the obtained satellite-subhalo MR. For example, we show that the tension between abundance and dynamics of MW satellites in LCDM gives if the slope of the GSMF faint-end slope upturns to -1.6."
],
[
"Introduction",
"In recent years the abundance matching technique (AMT) has emerged as a simple yet powerful statistical approach for connecting galaxies to halos without requiring knowledge of the underlying physics [45], [24], [12], [36], [4], [11], [15], [5].",
"Briefly, the AMT assumes a one-to-one monotonic relationship between stellar and halo masses which can be constrained by matching the cumulative observed galaxy stellar mass function (GSMF) to the theoretical halo plus subhalo cumulative mass function.",
"Interestingly enough, this simple approach successfully reproduces the observed spatial clustering of galaxies [12], [30].",
"The AMT allows to probe the average galaxy stellar–halo mass relation, $M_{\\rm *}$($M_{\\rm h}$) (hereafter SHMR), delivering very useful information for constraining models of galaxy evolution [22], [17], [2].",
"The above has motivated several authors to use the AMT extensively.",
"For example, with the advent of large galaxy surveys at different redshifts, the AMT has been applied for constraining the evolution of the average SHMR [12], [15], [11], [30], [5].",
"As a natural extension, these studies have been combined with predicted average halo mass aggregation histories in order to infer average galaxy $M_{\\rm *}$ growth histories as a function of mass [11], [16].",
"By including observational information on the gas content of galaxies, the AMT has been also used to constrain the baryon mass to $M_{\\rm h}$ relation of galaxies [4], [35].",
"Finally, variants of the AMT, where instead of mass functions, circular velocity functions or functions of any other galaxy/halo global property are employed, have been explored, too [12], [7], [44].",
"The AMT has been commonly applied to the total (central plus satellite galaxies) GSMF matched against the total (distinct plus satellite) halo population.",
"This approach has been criticized, because quite different average SHMRs are obtained for different proposed forms of the satellite stellar-subhalo mass relation (SSMR, $m_{\\rm *}$($m_{\\rm sub}$)) and the central SHMR ($M_{\\rm *}$($M_{\\rm h}$)In order to make the distinction explicit, we shall use upper-case letters for the central galaxy and the distinct halo masses and lower-case letters for the satellite galaxy and subhalo masses.",
"; [33]).",
"A common (questionable) assumption is that the SSMR is identical to the central SHMR.",
"Under this assumption, it is also common to define subhalo mass at the time of accretion ($m_{\\rm sub}^{\\rm acc}$) rather than at the time of observation ($m_{\\rm sub}^{\\rm obs}$), when subhalos have lost a significant fraction of mass due to tidal stripping.",
"The use of $m_{\\rm sub}^{\\rm acc}$ has been justified because this way is avoided the question of subhalo mass loss, and regarding the satellite $m_{\\rm *}$, it is expected that it remains almost constant since its infall into the host halo.",
"The projected two-point correlation function of galaxies is reproduced under these assumptions [12], [30].",
"It should also be said that while the (local) SHMR for central galaxies has been determined [27], [29], the stellar–subhalo mass relation for satellites/subhalos, SSMR, has not been yet discussed in detail in the literature.",
"In view of the above mentioned, some important questions arise.",
"Why does using $m_{\\rm sub}^{\\rm acc}$ instead of $m_{\\rm sub}^{\\rm obs}$ lead to the correct clustering of galaxies?",
"Does the $m_{\\rm sub}^{\\rm acc}$–$M_{\\rm *}$ relation reproduce the observed satellite GSMF, the conditional stellar mass function, and spatial clustering of galaxies at the same time?",
"Even more fundamentally, if is not assumed that the SSMR is identical to the central SHMR, then, what follows for the SSMR, either using $m_{\\rm sub}^{\\rm obs}$ or $m_{\\rm sub}^{\\rm acc}$?",
"Does it deviate from the central SHMR?",
"In this paper we extend the common AMT to constrain both the central SHMR and the SSMR separately, as well as the average (total) SHMR.",
"By construction, this formalism also allows to predict the mean satellite conditional mass function (CSMF), i.e., the probability that satellites of a given stellar mass reside in distinct host halos of a given mass.",
"We will (i) test whether the SSMR and the central SHMR follow the same shape; (ii) discuss the consequences of defining the subhalo mass at accretion time vs. at observed (present) time; and (iii) check the self-consistency of our predicted present-day central SHMR and SSMR by comparison with the observed satellite CSMF and the spatial clustering of galaxies.",
"This paper is laid out as follows.",
"In Section 2 we present the AMT, focusing on the details of our extended abundance matching.",
"In Section 3 we present the predicted stellar-halo mass relations (§§3.1) and satellite $\\mbox{CSMF}$ s (§§3.2) for cases when the SSMR is assumed equal to the central SHMR, and when both mass relations are independently constrained.",
"In §§3.3, a Halo Occupation Distribution (HOD) model is used to explore whether the predicted central SHMR's and SSMR's are consistent with the observed spatial clustering of galaxies.",
"Section 4 is devoted to our conclusions and a discussion of the results and their implications.",
"All our calculations are based on a flat $\\Lambda $ CDM cosmology with $\\Omega _\\Lambda =0.73$ , $h=0.7$ , and $\\sigma _8=0.84$ ."
],
[
"The Abundance Matching Technique",
"In this section we describe the technique of matching abundances between central galaxies and halos and satellite galaxies and subhalos, separetely, which we present here as an extension to the standard AMT."
],
[
"Modeling the central & satellite $\\mbox{GSMF}$ s",
"To model the central GSMF, let $\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}$ denotes the probability distribution function that a distinct halo of mass $M_{\\rm h}$ hosts a central galaxy of stellar mass $\\mbox{$M_{\\rm *}$}$ .",
"Then the number density of central galaxies with stellar masses between $\\mbox{$M_{\\rm *}$}$ and $\\mbox{$M_{\\rm *}$}+ d\\mbox{$M_{\\rm *}$}$ is given by $\\phi _{\\rm cen}(\\mbox{$M_{\\rm *}$})d\\mbox{$M_{\\rm *}$}=d\\mbox{$M_{\\rm *}$}\\int ^{\\infty }_{0}\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$})d\\mbox{$M_{\\rm h}$}.$ For the population of satellite galaxies in individual subhalos, let $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ be the probability distribution function that a subhalo $m_{\\rm sub}$Whenever we use $m_{\\rm sub}$ we refer to subhalo mass generically.",
"In practice, that can either be the mass at accretion time, $m_{\\rm sub}^{\\rm acc}$, or at observation (present-day) time, $m_{\\rm sub}^{\\rm obs}$.",
"hosts a satellite galaxy of stellar mass $\\mbox{$m_{\\rm *}$}$ .",
"Thus the average satellite CSMF (the number of satellite galaxies of stellar mass between $\\mbox{$m_{\\rm *}$}$ and $\\mbox{$m_{\\rm *}$}+ d\\mbox{$m_{\\rm *}$}$ that reside in distinct host halos of mass $M_{\\rm h}$, e.g., [52]) is $\\mbox{$\\Phi _s(\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}d\\mbox{$m_{\\rm *}$}=d\\mbox{$m_{\\rm *}$}\\int ^{\\infty }_{0}\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}\\mbox{$\\Phi _{\\rm sub}(\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})$}d\\mbox{$m_{\\rm sub}$},$ where $\\Phi _{\\rm sub}(\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})$ is the subhalo conditional mass function [9].",
"A natural link between the satellite GSMF, $\\phi _{\\rm sat}$, and the distinct halo mass function (HMF, $\\phi _{\\rm h}$) arises once the satellite CSMF is given: $\\mbox{$\\phi _{\\rm sat}$}(\\mbox{$m_{\\rm *}$})d\\mbox{$m_{\\rm *}$}=d\\mbox{$m_{\\rm *}$}\\int ^{\\infty }_{0}\\mbox{$\\Phi _s(\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$}) d\\mbox{$M_{\\rm h}$}.$ Inserting equation (REF ) into equation (REF ) and rearranging terms, the satellite GSMF can be rewritten in terms of $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$: $\\mbox{$\\phi _{\\rm sat}$}(\\mbox{$m_{\\rm *}$})d\\mbox{$m_{\\rm *}$}=d\\mbox{$m_{\\rm *}$}\\int ^{\\infty }_{0}\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}\\mbox{$\\phi _{\\rm sub}$}(\\mbox{$m_{\\rm sub}$})d\\mbox{$m_{\\rm sub}$},$ where the subhalo mass function (subHMF) is given by $\\mbox{$\\phi _{\\rm sub}$}(\\mbox{$m_{\\rm sub}$})d\\mbox{$m_{\\rm sub}$}=d\\mbox{$m_{\\rm sub}$}\\int ^{\\infty }_{0}\\mbox{$\\Phi _{\\rm sub}(\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})$}\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$})d\\mbox{$M_{\\rm h}$}.$ Equations (REF ) and (REF ) describe the abundance matching in its differential form for the central-halo and satellite-subhalo populations, respectively.",
"The distribution probability $P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$ is defined by the mean $M_{\\rm *}$($M_{\\rm h}$) relation and a scatter around it of $\\sigma _c$ , while the distribution probability $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$, assumed to be independent of host halo mass, is defined by the mean $m_{\\rm *}$($m_{\\rm sub}$) relation and a scatter around it of $\\sigma _s$ .",
"Observe that once $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ is given, the satellite CSMF is a prediction according to equation (REF ).",
"Here, $P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$ and $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ are modeled as lognormal distributions with a width (scatter around the stellar mass) assumed to be constant and the same for both centrals and satellites, $\\sigma _c=\\sigma _s=0.173$ dex.",
"Such a value was inferred for central galaxies from the analysis of general large group catalogs (YMB09) and it is supported by recent studies on the kinematics of satellite galaxies [29].",
"Regarding the intrinsic scatter of the satellite-subhalo relation, it has not been discussed in detail in the literature.",
"While the exploration of this scatter is beyond the scope of the present paper, our conclusions will not depend critically on the assumed value for it or even if it is allowed to depend on host halo mass.",
"We will further discuss this question in §4.2.",
"Both $m_{\\rm *}$($m_{\\rm sub}$) and $M_{\\rm *}$($M_{\\rm h}$) are parametrized by the same modified two-power-law form proposed in [5].",
"This five-parameters function is quite general and, in the context of the AMT, has been shown to reproduce the main features of a Schechter-like GSMF."
],
[
"The relation to standard abundance matching",
"In the standard AMT the cumulative halo+subhalo mass function and the total observed cumulative GSMF are matched to determine the mass relation between halos and galaxies, which is assumed to be monotonic.",
"In this context, no intrinsic scatter in the stellar mass at a given halo is assumed.",
"In our approach, where the galaxy and halo populations are separated into centrals/satellites and distinct halo/subhalos, the latter entails that the probability distribution functions of centrals and satellites take the particular forms: $P_{\\rm cen}(\\mathcal {M}|\\mbox{$M_{\\rm h}$})=\\delta (\\mathcal {M}-\\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$}))$ and $P_{\\rm sat}(\\mathcal {M}|\\mbox{$m_{\\rm sub}$})=\\delta (\\mathcal {M}-\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}$}))$ , where $M_{\\rm *}$($M_{\\rm h}$) and $m_{\\rm *}$($m_{\\rm sub}$) are the mean central-halo and satellite-subhalo mass relations, and $\\delta $ is for the $\\delta $ -Dirac function.",
"The above \"no scatter” probability distribution function for centrals applied in Eq.",
"($\\ref {Pcen}$ ) would lead us to conclude that the cumulative central GSMF, $n_{\\rm cen}(>\\mbox{$M_{\\rm *}$})$ , should match the cumulative distinct halo mass function, $n_{\\rm h}(>\\mbox{$M_{\\rm h}$}(\\mbox{$M_{\\rm *}$}))$ .",
"The same reasoning applies for satellites/subhalos.",
"Therefore, we arrive to the standard AMT formulation: $\\mbox{$n_g$}(>\\mbox{$M_{\\rm *}$})=n_{h}(>M_{\\rm h})+n_{\\rm sub}(>M_{\\rm h}),$ where $\\mbox{$n_g$}(>\\mbox{$M_{\\rm *}$})\\equiv n_{\\rm cen}(>\\mbox{$M_{\\rm *}$})+n_{\\rm sat}(>\\mbox{$M_{\\rm *}$})$ is the total GSMF.",
"Since the abundance matching can be applied to centrals/halos and satellites/subhalos separately, let analyze now only the latter.",
"Under the assumption that the $m_{\\rm *}$($m_{\\rm sub}$) relation is independent of the host halo mass, it is clear that using either the abundance matching of all satellites and all subhalos, $n_{\\rm sat}(>\\mbox{$m_{\\rm *}$})=n_{\\rm sub}(>\\mbox{$m_{\\rm sub}$})$ , or the matching of their corresponding mean occupational numbers, one may find exactly the same $m_{\\rm *}$($m_{\\rm sub}$) relation.",
"In this sense, we state that matching abundances is equivalent to matching occupational numbers: $\\mbox{$\\langle N_s(>\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})\\rangle $}=\\mbox{$\\langle N_{\\rm sub}(>\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})\\rangle $}& \\\\\\Longleftrightarrow \\nonumber n_{\\rm sat}(>\\mbox{$m_{\\rm *}$})=n_{\\rm sub}(>\\mbox{$m_{\\rm sub}$}).$ In the case that the probability distribution function $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ includes scatter around the mean SSMR, as we consider here, the above conclusion remains the same whilst $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ is assumed to be independent on halo mass.",
"In general, the inclusion of constant scatter in the galaxy-halo mass relations is not a conceptual problem for the AMT, but it slightly modifies the shape of the mass relations at the high mass end [5].",
"Finally, note that if $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ depends on $M_{\\rm h}$, then $\\mbox{$\\phi _{\\rm sat}$}$ may not be directly related to $\\mbox{$\\phi _{\\rm sub}$}$ (see Eq.",
"4) and using either the matching of satellites and subhalo abundances or the matching of their corresponding occupational numbers would not lead to find exactly the same $m_{\\rm *}$($m_{\\rm sub}$)."
],
[
"Inputs for matching abundances",
"The inputs required for the procedure described above are the subhCMF, the distinct HMF, and the observed satellite and central $\\mbox{GSMF}$ s. For the subhCMF, we use the results obtained in [9] based on the analysis of the Millennium-II Simulation.",
"This is a re-simulation with the same resolution of a smaller volume of the Millennium Simulation.",
"It consists of $2160^3$ particles, each of mass $m=6.885\\times 10^6h^{-1}\\mbox{M$_\\odot $}$ in a periodic cube of length $L=100h^{-1}$ Mpc.",
"Observe that this mass particle resolution is around four orders of magnitude below the subhalo masses required ($\\sim 10^{10}h^{-1}\\mbox{M$_\\odot $}$ ) to match the lower stellar mass limit in the YMB09 GSMF.",
"The fitting formula for the cumulative subhCMF reported in Boylan-Kolchin10 at the $[10^{12},10^{12.5}]h^{-1}\\mbox{M$_\\odot $}$ mass interval is: $\\mbox{$\\langle N_{\\rm sub}(>\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})\\rangle $}=\\mu _0\\left(\\frac{\\mu }{\\mu _1}\\right)^a\\exp \\left[-\\left(\\frac{\\mu }{\\mu _{\\rm cut}}\\right)^b\\right],$ where $\\mu =\\mbox{$m_{\\rm sub}$}/\\mbox{$M_{\\rm h}$}$ .",
"For $\\mbox{$m_{\\rm sub}$}$$=\\mbox{$m_{\\rm sub}^{\\rm acc}$}$ , $(\\mu _0,\\mu _1,\\mu _{\\rm cut},a,b,)=(1,0.038,0.225,-0.935,0.75)$ , while for $\\mbox{$m_{\\rm sub}$}=\\mbox{$m_{\\rm sub}^{\\rm obs}$}$ , $=(1.15^{(\\log M_{\\rm h}-12.25)},0.01,0.096,-0.935,1.29)$ .",
"According to Boylan-Kolchin10, the shape of the $m_{\\rm sub}^{\\rm obs}$ subhCMF remains the same for other halo masses but its normalization, $\\mu _0$ , systematically increases with $M_{\\rm h}$, roughly by 15% per dex in $M_{\\rm h}$.",
"Such a behavior has been reported in an analysis of the Millennium simulations by [18].",
"we introduce the dependence $\\mu _0=1.15^{(\\log M_{\\rm h}-12.25)}$ , where $\\mu _0=1$ at $\\log \\mbox{$M_{\\rm h}$}=12.25$ .",
"In the case of the $m_{\\rm sub}^{\\rm acc}$ subhCMF, the normalization factor is nearly independent on $M_{\\rm h}$, i.e., $\\mu _0=1$ (Boylan-Kolchin10; see also [21]).",
"The subhCMF is given by $\\Phi _{\\rm sub}=dN_{\\rm sub}/d\\mbox{$m_{\\rm sub}$}$ .",
"In order to construct the $m_{\\rm sub}^{\\rm acc}$ subhCMF, Boylan-Kolchin10 traced each subhalo back in time so that they were able to find the point at which its bound mass reached a maximum, i.e., the time the halo became a subhalo.",
"The latter guarantees that we are working with the surviving population of accreted halos and no further assumptions on the merging process are necessary.",
"The difference between the Millennium-II simulation cosmology and ours leads to differences in the resulting abundances of subhalos of roughly a few percent in the amplitude of the subhalo mass function (Boylan-Kolchin10).",
"This is also supported by previous works that explored the impact of changing cosmological parameters on the subhalo occupational statistics (e.g., [58]).",
"Additionally, to be consistent with the same cosmology for which the subhalo subhCMF was inferred, we repeated all the analysis to be showed below but using the WMAP1 cosmology.",
"We find that all our results are practically the same.",
"For the distinct HMF, we will use the formula given by [38].",
"This formula provides a reasonable fit to the the virial massThe mass enclosed within the radius at which, according to the spherical collapse model, the overdensity of a sphere is $\\Delta _{\\rm vir}$ times larger than the matter critical density of the used cosmological model; for the cosmology assumed here, $\\Delta _{\\rm vir}$ ($z=0$ )$=97$ .",
"function at $z\\sim 0$ measured in large cosmological N-body simulations [23], [14].",
"For our purpose, the decomposition of the GSMF and the $\\mbox{CSMF}$ s into centrals and satellites galaxies is necessary.",
"Using a large general group catalog [53] based on the data from the SDSS, YMB09 constructed and studied the decomposition of the GSMF and the $\\mbox{CSMF}$ s into centrals and satellites galaxies.",
"In that paper, a central galaxy was defined as the most massive galaxy in a group and the remaining galaxies as satellites.",
"For the mass completeness limit in the GSMF, they adopted the value as function of redshift proposed in [46].",
"They have also taken into account incompleteness in the group mass by considering an empirical halo-mass completeness limit (for details we refer the reader to YMB09).",
"Where necessary, halo masses are converted to match our definition of virial mass and stellar masses are converted to the [10] IMF.",
"In particular, YMB09 defined halo masses as being 180 times the background density, according to [20] these halos are $\\sim 11\\%$ larger than our definition of virial mass."
],
[
"Procedure and uncertainties",
"We constrain the parameters of the functions proposed to describe the central SHMR and SSMR by means of Eqs.",
"(REF ) and (REF ), and by using the Powell's directions set method in multi-dimensions for the minimization [34].",
"Note that in our analysis the reported statistical errors in the GSMFs, as well as the intrinsic scatter in the mass relations are taken into account.",
"However, we will not analyze rigorously here the effects of uncertainties on the mass relations as well as their errors.",
"Instead, we remit the reader to previous works [30], [5], [35].",
"[5] studied in detail the uncertainties and effects on the average SHMR due to different sources of error like those in the observed GSMFs, including stellar mass estimates; in the halo mass functions; in the uncertainty of the cosmological parameters; and in linking galaxies to halos, including the intrinsic scatter in this connection.",
"These authors have found that the largest uncertainty by far in the SHMR is due to systematic shifts in the stellar estimates.",
"The second important source of uncertainty is due to the intrinsic scatter, that we take into account in our analysis.",
"Other statistical and sample variance errors have negligible effects, at least for local galaxies.",
"According to the [5] study, the statistical and systematical uncertainties account for 1$\\sigma $ errors in the SHMR of approximately 0.25 dex at all masses, which is almost totally due to the uncertainty in stellar mass estimates.",
"We have explored here also the effects of the subhalo CMF uncertainty on the SSMR.",
"By using the 25% per dex in $M_{\\rm h}$ variation reported by [20] (instead of 15%), we find that the SSMR shifts in $m_{\\rm *}$ by only $\\approx 0.04$ dex.",
"Figure: Satellite GSMF\\mbox{GSMF}s calculated under the assumptionthat P sat (m * |m sub )=P cen (M * |M h )\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}=\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$} and for the cases m sub acc m_{\\rm sub}^{\\rm acc} (long-dashed line)and m sub obs m_{\\rm sub}^{\\rm obs} (short-dashed line) were used for the subhalo massdefinition.",
"Filled circles and squares with error bars show theYMB09 central and satelliteGSMF\\mbox{GSMF}s, respectively.",
"The solid line is for the case whenP sat (m * |m sub )P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$}) was determined using the YMB09 satellite GSMF as a constraint,i.e., is the best model fit to this function."
],
[
"The satellite GSMF and the SSMR",
"By means of the procedure described in Section 2, we calculate first the satellite GSMF (Fig.",
"REF ) when the SSMR and the central SHMR are assumed to be the same, i.e., $\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}$})=\\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$})$ .",
"This is equivalent to assume that $\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}=\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}$ if the intrinsic scatter of both relations is the same.",
"We obtain the central SHMR by matching abundances of YMB09 central galaxies to distinct halos.",
"This relation and the subhalo mass function obtained from the theoretical subhCMF (eq.",
"REF ), are used to infer the satellite GSMF (eq.",
"REF ).",
"The satellite GSMF is presented for the two cases of subhalo mass definition: $\\phi _{\\rm sat,acc}$ when $m_{\\rm sub}^{\\rm acc}$ is used (long-dashed line), and $\\phi _{\\rm sat,obs}$ when $m_{\\rm sub}^{\\rm obs}$ is used (short-dashed line).",
"The observational results of YMB09 are plotted as well.",
"Under the assumption that $\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}$})=\\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$})$ , the predicted number density of satellites at masses below the knee is underestimated on the average by a factor of $\\sim 2$ when using $m_{\\rm sub}^{\\rm acc}$, and $\\sim 5$ when using $m_{\\rm sub}^{\\rm obs}$.",
"Note that the former is closer to the YMB09 data.",
"The reason is simply because the normalization of the $m_{\\rm sub}^{\\rm acc}$ subHMF is higher and closer to the distinct HMF than the normalization of the $m_{\\rm sub}^{\\rm obs}$ subHMF.",
"Therefore, satellites of equal $m_{\\rm *}$ are expected to have a higher number density when using the accreted-time ($m_{\\rm sub}^{\\rm acc}$) subHMF compared to using the observed-time ($m_{\\rm sub}^{\\rm obs}$; present-day) subHMF.",
"However, neither $m_{\\rm sub}^{\\rm acc}$ nor $m_{\\rm sub}^{\\rm obs}$ are able to reproduce the observed satellite GSMF, and the discrepancy is due to the basic assumption of a common stellar mass–(sub)halo mass relation for centrals and satellites.",
"In the case that $\\mbox{$m_{\\rm sub}$}=\\mbox{$m_{\\rm sub}^{\\rm acc}$}$ , this is equivalent to assume that the SSMR is independent of redshift.",
"But in fact this cannot be the case since the satellite mass $m_{\\rm *}$ hardly will remain the same since it was accreted to the present epoch.",
"On the other hand, when using $\\mbox{$m_{\\rm sub}$}=\\mbox{$m_{\\rm sub}^{\\rm obs}$}$ , that the SSMR is equal to the central SHMR implies that both have evolved, on average, identically.",
"This cannot be the case because it is evident that the population of subhalos evolved differently to distinct halos, mainly by losing mass due to tidal striping [24], [47].",
"The next step in our analysis is to allow the SSMR and central SHMR to be different, i.e., $\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}$})\\ne \\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$})$ .",
"In this case, $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ is determined by means of Eq.",
"(REF ) using the YMB09 satellite GSMF as a constraint.",
"For the subhCMF, we again use both definitions of subhalo mass, $m_{\\rm sub}^{\\rm acc}$ and $m_{\\rm sub}^{\\rm obs}$.",
"For illustrative purpose, we present the resulting satellite GSMF for the case when $m_{\\rm sub}^{\\rm obs}$ was used (solid line in Fig REF ; an almost identical GSMF is obtained when $m_{\\rm sub}^{\\rm acc}$ was used).",
"As shown in Fig.",
"REF , the SSMRs obtained by using $m_{\\rm sub}^{\\rm acc}$ (long-dashed line) and $m_{\\rm sub}^{\\rm obs}$ (solid line) are quite different.",
"The central SHMR (dot-dashed line) is the same for both cases.",
"The error bar in the left panel shows a 1$\\sigma $ uncertainty of 0.25 dex in the normalization of the mass relations.",
"This is roughly the uncertainty estimated by [5] taking into account all the systematical and statistical sources of errors (see §§2.4).",
"When using the accretion-time subhalo mass, $m_{\\rm sub}^{\\rm acc}$, we note that the resulting SSMR at $\\log (\\mbox{$m_{\\rm *}$}/\\mbox{M$_\\odot $})<11$ systematically lies above the central SHMR, with differences in the stellar-mass axis (halo-mass axis) of $\\sim 0.5$ dex (0.2 dex) at the smallest masses.",
"For $\\log (\\mbox{$M_{\\rm *}$}/\\mbox{M$_\\odot $})>11$ this trend is inverted, but the differences between central and satellites are very small.",
"However, the relation obtained this way should be taken with caution.",
"By construction, each $m_{\\rm sub}^{\\rm acc}$ is itself a cumulative distribution of all the objects accreted in a time interval $\\Delta z$ .",
"Therefore such a SSMR entails that all accreted objects of mass $m_{\\rm sub}^{\\rm acc}$ would evolve, on average, to host the same $m_{\\rm *}$ despite having been accreted at different times.",
"We discuss this in §§4.1.",
"When using the observation-time (present-day) subhalo mass, $m_{\\rm sub}^{\\rm obs}$, the SSMR (solid blue line) and central SHMR are very different, though they show the same trend as when using $m_{\\rm sub}^{\\rm acc}$.",
"For example, on average, a satellite with $\\log (\\mbox{$M_{\\rm *}$}/\\mbox{M$_\\odot $})=10$ resides in a subhalo a factor of $\\sim 4$ less massive than the halo of a central galaxy with the same stellar mass.",
"Notice that $\\mbox{$m_{\\rm sub}^{\\rm obs}$}(\\mbox{$m_{\\rm *}$})<\\mbox{$m_{\\rm sub}^{\\rm acc}$}(\\mbox{$m_{\\rm *}$})$ and that the difference increases the lower the mass is.",
"This is consistent with the picture that most massive subhalos, on average, fell into larger halos just very recently and they have not had time to lose significant amounts of mass due to tidal striping, in contrast to the lowest mass subhalos.",
"This also suggests that the SSMR for both definitions of subhalo mass should tend to the central SHMR at the high-mass end, but this is not the case as seen in Fig.",
"REF where small differences remain.",
"The possible reasons are that, firstly, the intrinsic scatter around the stellar–(sub)halo mass relations is actually lower for the former than for the latter (here we assumed it to be the same for satellites and centrals, see §4.2).",
"Secondly, that the YMB09 satellite GSMF may underestimate the true satellite mass function at large masses [39].",
"Fiber collisions could introduce some systematic error that may affect the YMB09 group catalog.",
"To study the impact of this possible systematic error, YMB09 divided their group catalog into two sample: one that uses galaxies with known redshifts, and another that includes galaxies that lack redshifts due to fiber collisions.",
"When compared the corresponding satellite $\\mbox{CSMF}$ s from both samples (see their Fig.",
"6), they found that the sample for which the correction for fiber collisions has been taking into account, has a higher amplitude of the $\\mbox{CSMF}$ s than when this correction has not been applied, specially in low mass halos.",
"However, the difference is very marginal and well within the error bars.",
"We conclude that fiber collisions in the YMB09 group catalog are not a serious source of systematics able to affect our conclusions.",
"Regarding completeness and contamination of their group catalog [53], 80% have a completeness greater than 0.6, while 85% have a contamination lower than 0.5.",
"In terms of purity, their halo-based group finder is consistent with the ideal situation.",
"Finally, we note that the mass relation usually obtained by matching abundances between the total GSMF and the halo plus subhalo mass function, in the light of the decomposition into centrals and satellites, could be interpreted as a density–weighted average SHMR: $\\langle \\mbox{$M_{\\rm *}$}(M)\\rangle _{\\phi }=\\frac{\\mbox{$\\phi _{\\rm sub}$}(M)}{\\phi _{\\rm DM}(M)}\\mbox{$m_{\\rm *}$}(M)+\\frac{\\mbox{$\\phi _{\\rm h}$}(M)}{\\phi _{\\rm DM}(M)}\\mbox{$M_{\\rm *}$}(M),$ where $\\phi _{\\rm DM}(M)=\\mbox{$\\phi _{\\rm sub}$}(M)+\\mbox{$\\phi _{\\rm h}$}(M)$ , $\\mbox{$m_{\\rm *}$}(M)$ is the mean SSMR and $\\mbox{$M_{\\rm *}$}(M)$ is the mean central SHMR.",
"This relation is plotted in Fig.",
"REF with short-dashed-dot and short-dashed lines when using $m_{\\rm sub}^{\\rm acc}$ and $m_{\\rm sub}^{\\rm obs}$, respectively.",
"Since most galaxies in the YMB09 catalog are centrals, the central $\\mbox{SHMR}$ is very close to the density-weighted average SHMR.",
"For comparison, we plotted the [5] average mass relation (filled circles), which is in excellent agreement with our density-weighted average SHMR when using the accreted-time subhalo mass, $m_{\\rm sub}^{\\rm acc}$.",
"Observe that differences between the satellite and the average (total) mass relations are small when $m_{\\rm sub}^{\\rm acc}$ is used, while differences become dramatic when $m_{\\rm sub}^{\\rm obs}$ is used.",
"The above explains why under the assumption that $\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}^{\\rm acc}$}) = \\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$})=\\langle \\mbox{$M_{\\rm *}$}(M)\\rangle _{\\phi }$ , the resulting satellite GSMF are closer to observations.",
"On the other hand, since the $m_{\\rm sub}^{\\rm acc}$ subHMF has a higher normalization than the $m_{\\rm sub}^{\\rm obs}$ subHMF, the above shows that when assuming $\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}^{\\rm obs}$}) =\\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$})=\\langle \\mbox{$M_{\\rm *}$}(M)\\rangle _{\\phi }$ , we should expect that the resulting satellite GSMF is significantly below the observed satellite GSMF.",
"In the right panel of Fig.",
"REF , we plot some observational inferences of halo and subhalo masses as a function of stellar mass.",
"The inferred $\\langle \\mbox{$M_{\\rm h}$}\\rangle (\\mbox{$M_{\\rm *}$})$ of central galaxies from staked weak-lensing studies using the SDSS [27] are shown as filled circles with error bars.",
"[27] reported the data actually for blue and red galaxies separately.",
"We estimated the average mass relation for central galaxies as: $\\langle \\mbox{$M_{\\rm h}$}\\rangle (\\mbox{$M_{\\rm *}$})=f_b(\\mbox{$M_{\\rm *}$})\\langle \\mbox{$M_{\\rm h}$}\\rangle _b(\\mbox{$M_{\\rm *}$})+f_r(\\mbox{$M_{\\rm *}$})\\langle \\mbox{$M_{\\rm h}$}\\rangle _r(\\mbox{$M_{\\rm *}$})$ , where $f_b(\\mbox{$M_{\\rm *}$})$ and $f_r(\\mbox{$M_{\\rm *}$})$ are the blue and red galaxy fractions in the sample, and $\\langle \\mbox{$M_{\\rm h}$}\\rangle _b$ and $\\langle \\mbox{$M_{\\rm h}$}\\rangle _r$ are the corresponding blue and red mass relations.",
"The inferred $\\langle \\log (\\mbox{$M_{\\rm *}$})\\rangle (\\mbox{$M_{\\rm h}$})$ for central galaxies from staked kinematics of satellites [29] are plotted as the dashed area indicating the 68% of confidence.",
"Our inferred central SHMR (dotted-dashed curve) is consistent with the weak-lensing inferences at all masses, and with the satellite kinematics inferences at masses $\\mbox{$M_{\\rm *}$}$ > $$ 1011$ \\mbox{M$_\\odot $};for smaller masses, our halo masses are a factor up to $ 2$smaller than the satellite kinematics inferences.",
"In fact, it wasalready noted that using the kinematics of satellite galaxies yields halo masses aroundlow mass galaxies that are systematically larger than most othermethods, specially for red central galaxies \\cite {More11,Skibba+2011,Rodriguez+2011}.$ Regarding satellites, unfortunately, there are not direct inferences of their subhalo masses.",
"Some model-dependent estimates based on dynamical observations of Milky-Way (MW) satellites were presented in the literature.",
"For example, using the line-of-sight velocity dispersions measured for the brightest spheroidal dwarf galaxies, [41] and [42] determined their masses within their tidal radii.",
"These dynamical masses, plotted in Fig.",
"REF (filled squares and triangles, respectively), are expected to be of the order of $m_{\\rm sub}^{\\rm obs}$.",
"We also plot an estimate of the mass at the tidal radius for the Large-Magallanic Cloud [48].",
"The SSMRs constrained here do not extend to the small masses of MW satellites but we plot their extrapolations to these masses (dashed curves).",
"The gray dotted-dashed curve will be discussed in §§4.3"
],
[
"The satellite CSMF",
"From the approach described in Section 2, another statistical quantity that deserves to be subject of study is the satellite CSMF (Eq.",
"REF ).",
"We calculate the mean halo–density–weighted CSMF at the $[M_{\\rm h_1},M_{\\rm h_2}]$ bin as: $\\langle \\Phi _s\\rangle =\\frac{\\int _{M_{\\rm h_1}}^{M_{\\rm h_2}}\\mbox{$\\Phi _s(\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$}) d\\mbox{$M_{\\rm h}$}}{\\int _{M_{\\rm h_1}}^{M_{\\rm h_2}}\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$}) d\\mbox{$M_{\\rm h}$}}.$ This quantitiy has been inferred from observations by YMB09, again using their SDSS galaxy catalog (filled circles with error bars in Fig.",
"REF ).",
"First, we consider again the case assuming $\\mbox{$m_{\\rm *}$}(\\mbox{$m_{\\rm sub}$})=\\mbox{$M_{\\rm *}$}(\\mbox{$M_{\\rm h}$})$ .",
"When $m_{\\rm sub}$ is defined at the observation time, the resulting $\\mbox{CSMF}$ s are lower than the YMB09 $\\mbox{CSMF}$ s by a factor of $\\sim 5$ in the power-law regime (roughly the same factor by which $\\phi _{\\rm sat,obs}$ is lower than the YMB09 observed satellite GSMF).",
"Similarly, when $m_{\\rm sub}$ is defined at the accretion time, the predicted $\\mbox{CSMF}$ s in the power-law regime are below the YMB09 $\\mbox{CSMF}$ s by nearly the same factor, $\\sim 2$ , that $\\phi _{\\rm sat,acc}$ lies below the satellite GSMF.",
"The normalization of the $\\mbox{CSMF}$ increases faster with $M_{\\rm h}$ when $m_{\\rm sub}^{\\rm obs}$ is used instead of $m_{\\rm sub}^{\\rm acc}$.",
"This is because we allow the $m_{\\rm sub}^{\\rm obs}$ subhCMF normalization to vary with host halo mass, while the $m_{\\rm sub}^{\\rm acc}$ subhCMF normalization is independent of host halo mass.",
"The black continuous ($m_{\\rm sub}^{\\rm obs}$) and blue long-dashed ($m_{\\rm sub}^{\\rm acc}$) lines in Fig.",
"REF (almost indistinguishable one from other) are the predictions when $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ has been constrained by means of the observed satellite GSMF.",
"The agreement of the predicted satellite $\\mbox{CSMF}$ 's, for both $m_{\\rm sub}^{\\rm obs}$ and $m_{\\rm sub}^{\\rm acc}$ with the YMB09 $\\mbox{CSMF}$ 's is now remarkable at all halo mass bins for low/intermedium stellar masses.",
"Although, as above, the normalization of the $\\mbox{CSMF}$ 's increases faster when $\\mbox{$m_{\\rm sub}$}=\\mbox{$m_{\\rm sub}^{\\rm obs}$}$ than when $\\mbox{$m_{\\rm sub}$}=\\mbox{$m_{\\rm sub}^{\\rm acc}$}$ , the differences between both cases at any mass are less than 0.05 dex, within the error bars of the observational data.",
"Despite the overall agreement, for halo mass bins lower than $\\sim 10^{13}$ M$_\\odot $, the number of massive satellite galaxies is overestimated, specially at the lowest $M_{\\rm h}$ bins.",
"A possible reason for this is the assumption that the scatter in $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ is constant while in reality it could depend on $M_{\\rm h}$ as well as on $m_{\\rm sub}$.",
"However, the probability of finding massive satellite galaxies in halos less massive than $\\sim 10^{13}$ M$_\\odot $ is low and they do not contribute significantly to the mean total density of satellite galaxies.",
"Therefore, this assumption does not change our conclusions, see also §4.",
"Our analysis shows that assuming $\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}=\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}$ the resulting satellite $\\mbox{CSMF}$ s are not consistent with observations.",
"Instead, when $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ is independently constrained using the observed satellite GSMF, there is a clear agreement, no matter what definition of $m_{\\rm sub}$ was employed for the subhCMF.",
"Figure: Density–weighted average satellite CSMF\\mbox{CSMF} in eight halo mass intervals.Red short-dashed and green short-dashed-dot lines are for thecases when the central SHMR and the SSMR were assumed to beequal and m sub obs m_{\\rm sub}^{\\rm obs} and m sub acc m_{\\rm sub}^{\\rm acc} were used, respectively.",
"The black solid andblue long-dashed lines are again for m sub obs m_{\\rm sub}^{\\rm obs} and m sub acc m_{\\rm sub}^{\\rm acc}, respectively,but in the case the central and satellite mass functions were independentlyconstrained by means of our extended AMT (they overlap most of time).Filled circles with error bars show the CSMF\\mbox{CSMF}s inferred from observations byYMB09.",
"Note that their halo masses were converted to match our virial definition."
],
[
"Abundance matching and clustering",
"It has been noted in the literature that the average (total) SHMR obtained with the standard AMT is consistent with the observed spatial clustering of galaxies [12], [30].",
"We will test now whether this is the case for the mass relations of central and satellite galaxies obtained here with our extended AMT.",
"We will compute the galaxy projected correlation function by means of a HOD model for each of the mass relation obtained in §3.1.",
"A HOD model is a statistical tool mainly used to describe the clustering of galaxies (e.g., [6], [13], [50], [56], [55], [25], [26], [54], and more references therein).",
"In contrast to the AMT, which is a quasi empirical tool, a HOD employs modeling motivated by results of cosmological $N$ -body [24] and hydrodynamical [59] simulations.",
"In short, a HOD model describes the probability that a halo of mass $M_{\\rm h}$ hosts a number of $N$ galaxies with stellar masses greater than $M_{\\rm *}$.",
"Once the occupational numbers are defined, the two-point correlation function can be computed assuming that the total mean number of galaxy pairs is the contribution of all pairs coming from galaxies in the same halo (one-halo term) and pairs from different halos (two-halo term).",
"For a detailed description for the HOD model we employ here, see Appendix A.",
"First, consider the case when $\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}=\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}$ .",
"The short-dashed curves in Fig.",
"REF show the projected correlation functions in five stellar mass bins for the case the $m_{\\rm sub}^{\\rm obs}$ subhCMF was used.",
"The [54] galaxy projected correlation functions from the DR7 SDSS are plotted as filled circles with error bars.",
"The resulting correlation functions are clearly below observations, mainly in the one-halo term.",
"This is because using $m_{\\rm sub}^{\\rm obs}$ underestimates the satellite GSMF and CSMF, resulting in a strong deficit of satellite galaxies.",
"Observe that if $N_s\\sim 0$ , then $N\\sim N_c$ and therefore, $b_g(\\mbox{$M_{\\rm *}$})\\sim \\langle b(\\mbox{$M_{\\rm h}$}) \\rangle $ where $\\langle b(\\mbox{$M_{\\rm h}$}) \\rangle $ is the mean weighted halo bias function, see Eq.",
"REF .",
"When using the subhCMF for $m_{\\rm sub}^{\\rm acc}$ instead of $m_{\\rm sub}^{\\rm obs}$ (dot-short-dashed curves), the agreement with the observed correlation functions is better, though at scales where the one-halo term dominates, the predictions are still below observations.",
"This is, again, because the satellite GSMF and CSMF are underestimated in this case.",
"We remark that using the (average or total) SHMR obtained with the standard AMT in the HOD model by matching the total GSMF to the total halo+subhalo mass function (in the case of $m_{\\rm sub}^{\\rm acc}$), leads to excellent agreement with the observed correlation functions, a result that is well known.",
"However in this case the SSMR is not constrained, instead it is implicitly assumed to be equal to the central SHMR (for $m_{\\rm sub}^{\\rm acc}$).",
"With our extended AMT, we can explicitly separate both mass relations.",
"When they are assumed to be equal and the central SHMR is constrained with the central GSMF, then we obtain the predictions already shown, in particular the correlation functions.",
"The fact that the predicted correlation functions, when $m_{\\rm sub}^{\\rm acc}$ is used, are close to those predicted in the standard AMT (and to the observed ones) is because the central and average SHMR are indeed close, as we discussed in §§3.1, see Fig.",
"REF .",
"Thus, under the assumption that $\\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$}=\\mbox{$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}$ , the observed clustering of galaxies is better reproduced when the subhalo mass in abundance matching is defined as $m_{\\rm sub}^{\\rm acc}$ rather than $m_{\\rm sub}^{\\rm obs}$.",
"Nevertheless, even in the former case, the agreement with observations is only marginal.",
"We now turn the analysis to the cases where the SSMR is not assumed to be equal to the central SHMR.",
"The black solid and blue long-dashed lines in Fig.",
"REF show the predicted correlation functions in the cases where either $m_{\\rm sub}$ or $m_{\\rm sub}^{\\rm acc}$ were used.",
"Both cases lead to very similar results and agree very well with observations.",
"Therefore, the HOD model combined with the central and satellite mass relations independently constrained with the extended AMT, is able to reproduce the observed correlation functions, no matter if $m_{\\rm sub}^{\\rm obs}$ or $m_{\\rm sub}^{\\rm acc}$ are used.",
"This successful prediction is a consequence of the good agreement obtained between our predicted satellite $\\mbox{CSMF}$ s and those inferred from observations (§§3.2 and Fig.",
"3).",
"Figure: Projected two–point correlation functions of galaxies in fivestellar mass intervals obtained with the HOD model using differentgalaxy-halo mass relations obtained with our AMT.",
"Gray short-dashedand green short-dashed-dot lines are for thecases when the central SHMR and the SSMR were assumed to beequal and m sub obs m_{\\rm sub}^{\\rm obs} and m sub acc m_{\\rm sub}^{\\rm acc} were used, respectively.",
"The black solid andblue long-dashed lines are again for m sub obs m_{\\rm sub}^{\\rm obs} and m sub acc m_{\\rm sub}^{\\rm acc}, respectively,but in the case the central and satellite mass functions were independentlyconstrained (they overlap most of time).",
"The observed projected correlation functionsreported in Yang et al.",
"(2011) are shown by filled circles with error bars."
],
[
"Summary and Discussion",
"In this paper, we extend the AMT in order to constrain both the central stellar–halo and the satellite–subhalo mass relations separately, using as an input (i) the distinct halo and subhalo mass functions, and (ii) the observed central and satellite $\\mbox{GSMF}$ s. Our formalism, by construction, predicts also the satellite $\\mbox{CSMF}$ s as a function of host halo mass, and when applied to a HOD model, allows to predict the spatial correlation functions.",
"We present results for the cases when the SSMR is assumed to be equal to the central SHMR, $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$=$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$, and when both mass relations are constrained independently (i.e., it is not assumed that $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$=$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$).",
"All our analysis is carried out for subhalo masses defined at accretion time, $m_{\\rm sub}^{\\rm acc}$, and at the observed time (present day), $m_{\\rm sub}^{\\rm obs}$.",
"The main results and conclusions are as follows: Assuming that the mass relation between satellites and subhalos is identical to the mass relation between centrals and distinct halos (including their intrinsic scatters), $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$=$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$, the predicted satellite GSMF, $\\mbox{CSMF}$ s and projected two–point correlation functions lie below those obtained from observations for both definitions of $m_{\\rm sub}$, though the disagreements are small when $m_{\\rm sub}$=$m_{\\rm sub}^{\\rm acc}$ (Figs.",
"REF , REF , REF ).",
"We conclude that assuming $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$=$P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$ leads to predictions in disagreement with observations, specially when $m_{\\rm sub}^{\\rm obs}$ is used.",
"When the SSMR is no longer assumed to be equal to the central SHMR and instead is constrained by means of the observed satellite GSMF, the predicted satellite $\\mbox{CSMF}$ s and projected correlation functions agree in general with observations, both for $m_{\\rm sub}^{\\rm obs}$ and $m_{\\rm sub}^{\\rm acc}$.",
"However, for halo masses lower than $\\sim 10^{13}$ M$_\\odot $, the number of very massive (rare) satellites is over-predicted.",
"The resulting $m_{\\rm sub}$–$m_{\\rm *}$ relations when using either $\\mbox{$m_{\\rm sub}^{\\rm obs}$}$ or $\\mbox{$m_{\\rm sub}^{\\rm acc}$}$ are quite different from each other, and in each case are different from the central SHMR (Fig.",
"REF ).",
"For a given stellar mass, the satellite subhalo mass is smaller than central halo mass, and the mass difference is increasing the lower the mass is.",
"These differences are dramatic when $m_{\\rm sub}^{\\rm obs}$ is used.",
"Our density-weighted average (centrals + satellites) $\\mbox{SHMR}$ s are close to the central SHMR when either $m_{\\rm sub}^{\\rm obs}$ or $m_{\\rm sub}^{\\rm acc}$ is used (central galaxies dominate in the YMB09 catalog).",
"Such an average SHMR coincides conceptually with the one inferred from matching the total (centrals+ satellites) cumulative GSMF and the halo + subhalo cumulative mass function (standard AMT)."
],
[
"On the inference of the SSMR and its implications for\nthe average mass relation",
"The conclusions listed above can be well understood by examining the basic ideas behind the extended AMT, as we show in Section 2.2.",
"Essentially, matching abundances of satellite galaxies to subhalos is equivalent to matching their corresponding occupational numbers, that is: $\\mbox{$\\langle N_s(>\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})\\rangle $}=\\mbox{$\\langle N_{\\rm sub}(>\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})\\rangle $}.$ The opposite is also true: matching their corresponding occupational numbers is equivalent to matching their abundances.",
"This is an important result because it shows that once $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ (and $P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$) is properly constrained, we will obtain the correct conditional mass functions and consequently the correct spatial clustering for galaxies.",
"The above means that there is a unique $m_{\\rm sub}$($m_{\\rm *}$) relationship for each definition of $m_{\\rm sub}$, which depends solely on $\\langle N_{\\rm sub}(>\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})\\rangle $.",
"Because of this uniqueness, it follows that the $m_{\\rm sub}^{\\rm obs}$($m_{\\rm *}$) and $m_{\\rm sub}^{\\rm acc}$($m_{\\rm *}$) relations should be different, and any incorrect assumption on each one of these relations will lead to inconsistencies in the conditional mass functions and spatial clustering of galaxies, as for example those that we have found here when $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}$})$ was assumed to be equal to $P_{\\rm cen}(\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$.",
"Under this assumption, when $m_{\\rm sub}^{\\rm acc}$ was used, the inconsistencies were actually small.",
"This is because in this case the \"incorrect\" assumption for the SSMR is actually not too far from the \"correct\" result obtained when $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}^{\\rm obs}$})$ is independently constrained (compare dot-dashed and solid green curves in Fig.",
"REF ), contrary to what happens when $m_{\\rm sub}^{\\rm obs}$ is used.",
"It is important to remark that in the standard AMT, only the average SHMR is constrained (using the total GSMF), leaving unconstrained the SSMR, something that on its own introduces a large uncertainty in the average SHMR [33].",
"We have shown that such average SHMR is conceptually equal to the density–weighted average mass relation obtained here from the observationally constrained central SHMR and SSMR.",
"Therefore, our resulting average mass relation is expected to be less uncertain than previous determinations.",
"On the other hand, this average mass relation is expected to be close to the central SHMR because most of the galaxies in the used observational catalog are centrals.",
"We conclude that in order to properly infer the SSMR and the central SHMR at the same time, and this way reduce the uncertainty in the average SHMR, more observational constraints than the total GSMF are necessary.",
"The most obvious and direct is the GSMF decomposed into central and satellite galaxies, something that was provided by YMB09.",
"However, observe that, according to eq.",
"(REF ), the satellite CSMFs or the clustering of galaxies, modulo the observational errors, provide observational constraints that lead to similar inferences of the SSMR, because of the uniqueness of this relation for a given well defined $\\langle N_{\\rm sub}(>\\mbox{$m_{\\rm sub}$}|\\mbox{$M_{\\rm h}$})\\rangle $ (see above).",
"Finally, we note that obtaining the SSMR for the subhalo mass defined at the accretion time introduces uncertainties due to our ignorance about evolutionary processes of the stellar mass since accretion .",
"This does not happen when the SSMR is obtained for both the satellite and subhalo masses defined at the same epoch, for instance the present time.",
"When matching abundances for the $m_{\\rm sub}^{\\rm acc}$ case, the fact that (1) $m_{\\rm sub}^{\\rm acc}$ is itself a cumulative distribution of all objects accreted over a period of time, and that (2) $m_{\\rm *}$ may have changed between accretion and observation, are not taken into account.",
"In other words, it is implicitly assumed that the satellite stellar mass stops evolving soon after accretion.",
"In reality the situation is actually quite complex in the sense that, depending on the accretion time and the orbit of the satellites, the evolution of their stellar masses is diverse, with some of them early quenched and others actively evolving, perhaps in some cases as the central ones of the same mass [49].",
"This diversity introduces an intrinsic uncertainty on the results.",
"Such an uncertainty might be accounted for the probability distribution functions: $P(\\mbox{$m_{\\rm *}$}|m_{*,\\rm acc},z)$ , which gives the probability that a satellite accreted at epoch $z$ evolves, on average, to the observed satellite $m_{\\rm *}$, and $P(m_{*,\\rm acc}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$ , which gives the probability that a subhalo $\\mbox{$m_{\\rm sub}^{\\rm acc}$}$ hosts a galaxy of mass $m_{*,\\rm acc}$ at the time of accretion.",
"Now, the satellite CSMF (Eq.",
"REF ) can be written as [28]: $\\mbox{$\\Phi _s(\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}=\\int \\int \\int P(\\mbox{$m_{\\rm *}$}|m_{*,\\rm acc},z)P(m_{*,\\rm acc}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)\\times & \\nonumber \\\\\\Phi (\\mbox{$m_{\\rm sub}^{\\rm acc}$}|\\mbox{$M_{\\rm h}$},z)dm_{*,\\rm acc} d\\mbox{$m_{\\rm sub}^{\\rm acc}$}dz.& \\nonumber \\\\$ Note that in our analysis in §§3.1, we implicitly assume that the stellar mass of satellite galaxies does not change once they become satellites, i.e.",
"$P(\\mbox{$m_{\\rm *}$}|m_{*,\\rm acc},z)=\\delta (\\mbox{$m_{\\rm *}$}-m_{*,\\rm acc},z)$ , and that $P(m_{*,\\rm acc}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$ is independent of redshift.",
"Thus, the application of the AMT to infer the satellite CSMF and the $m_{\\rm *}$–$m_{\\rm sub}$ relation for subhalo mass defined at its accretion time formally requires more observational constraints at higher redshifts.",
"This is a problem already faced by previous authors [54].",
"The above is not the only way to formally write the satellite CSMF; it can be written in a way that instead of implying knowledge of the change of $m_{\\rm *}$ from accretion to observation, implies just knowledge on the change of the subhCMF between these two epochs.",
"Let us consider the distribution function, $P_{\\rm acc}(\\mbox{$m_{\\rm sub}^{\\rm obs}$}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$, giving the probability that halos accreted at epoch $z$ evolve, on average, to the observed (present-day) subhalos $m_{\\rm sub}^{\\rm obs}$, and the probability distribution function of these subhalos of hosting a galaxy of mass $m_{\\rm *}$, $P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}^{\\rm obs}$})$.",
"In this case, the satellite CSMF (Eq.",
"REF ) is written as $\\mbox{$\\Phi _s(\\mbox{$m_{\\rm *}$}|\\mbox{$M_{\\rm h}$})$}=\\int \\int \\int \\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}^{\\rm obs}$})$}\\mbox{$P_{\\rm acc}(\\mbox{$m_{\\rm sub}^{\\rm obs}$}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$}\\times & \\nonumber \\\\\\Phi (\\mbox{$m_{\\rm sub}^{\\rm acc}$}|\\mbox{$M_{\\rm h}$},z)d\\mbox{$m_{\\rm sub}^{\\rm obs}$}d\\mbox{$m_{\\rm sub}^{\\rm acc}$}dz,& \\nonumber \\\\$ and therefore the satellite GSMF, Eq.",
"REF , is given by $\\mbox{$\\phi _{\\rm sat}$}(\\mbox{$m_{\\rm *}$})=\\int \\int \\int \\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}^{\\rm obs}$})$}\\mbox{$P_{\\rm acc}(\\mbox{$m_{\\rm sub}^{\\rm obs}$}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$}\\times & \\nonumber \\\\\\mbox{$\\phi _{\\rm sub}$}(\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)d\\mbox{$m_{\\rm sub}^{\\rm obs}$}d\\mbox{$m_{\\rm sub}^{\\rm acc}$}dz.& \\nonumber \\\\$ Since the $m_{\\rm sub}^{\\rm acc}$ subHMF would evolve into the $m_{\\rm sub}^{\\rm obs}$ subHMF, we write $\\mbox{$\\phi _{\\rm sub}$}(\\mbox{$m_{\\rm sub}^{\\rm obs}$})=\\int \\int \\mbox{$P_{\\rm acc}(\\mbox{$m_{\\rm sub}^{\\rm obs}$}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$}\\mbox{$\\phi _{\\rm sub}$}(\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)d\\mbox{$m_{\\rm sub}^{\\rm acc}$}dz.$ This last equation is the abundance matching of accreted subhalos to present-day subhalos.",
"Therefore, $\\mbox{$\\phi _{\\rm sat}$}(\\mbox{$m_{\\rm *}$})=\\int \\mbox{$P_{\\rm sat}(\\mbox{$m_{\\rm *}$}|\\mbox{$m_{\\rm sub}^{\\rm obs}$})$}\\mbox{$\\phi _{\\rm sub}$}(\\mbox{$m_{\\rm sub}^{\\rm obs}$})d\\mbox{$m_{\\rm sub}^{\\rm obs}$}.$ This equation is nothing but abundance matching satellite galaxies to subhalos at the time they are observed.",
"Hence, the reason that the satellite GSMF matches the subHMF in a more direct way for subhalo masses defined at the observation time (eq.",
"REF ) than at the accretion time (eq.",
"REF ), is that in the latter case the unknown $P(\\mbox{$m_{\\rm *}$}|m_{*,\\rm acc},z)$ and $P(m_{*,\\rm acc}|\\mbox{$m_{\\rm sub}^{\\rm acc}$},z)$ \"evolutionary\" functions have to be introduced.",
"However, we acknowledge that for the former case, our ignorance on the scatter around the SSMR is a also potential source of uncertainty.",
"All our calculations are under the assumption that this scatter is the same as the scatter of the central SHMR.",
"In any case, even if these scatters are different, note that including scatter affects the stellar-to-(sub)halo mass relation only at its high-mass end, where on average satellites are expected to be accreted recently, hence their SSMR and scatter are yet similar to those of centrals/halos."
],
[
"On the intrinsic scatter in the SSMR ",
"A possible source of systematic errors in our analysis is the assumption that the intrinsic scatter around the SSMR, $\\sigma _s$ is constant and equal to the scatter around the central SHMR.",
"To probe the impact of this assumption we repeated all our analysis but this time assuming $\\sigma _s=0$ .",
"When comparing the results using $\\sigma _s=0$ to those obtained based on $\\sigma _s=0.173$ dex, we find that they are consistent with each other, and therefore with the satellite CSMFs and with the galaxy spatial clustering measured from the YMB09 catalog.",
"In more detail, we find that the resulting $\\mbox{CSMF}$ 's reproduce observations for $\\sigma _s=0$ slightly better than for $\\sigma _s=0.173$ dex, especially at the massive end.",
"This is because when the intrinsic scatter is not taken into account ($\\sigma _s=0$ ), the shape of the SSMRs steepens at the massive end (see also [5]).",
"Consequently, for a given $m_{\\rm *}$, the subhalo mass is larger, and the abundances of larger (sub)halos is lower in general than those of smaller halos.",
"Therefore, the number density of satellites at the massive end is lower.",
"However, the projected correlation functions remain almost the same because the probability of finding a massive satellite galaxy in host halos less massive than $\\log \\mbox{$M_{\\rm h}$}\\sim 13$ is very low.",
"They do not contribute significantly to the mean total density of galaxies.",
"Although better models are needed in order to give a realistic form for $\\sigma _s$ , our main conclusions seem to be robust to variations in the adopted value for $\\sigma _s$ ."
],
[
"Implications for satellite/subhalo evolution",
"The local SSMR obtained for both definitions of the subhalo mass, $m_{\\rm sub}^{\\rm acc}$ and $m_{\\rm sub}^{\\rm obs}$, are such that at halos masses smaller than $2-10\\times 10^{13}$ M$_\\odot $ and at a given galaxy stellar mass, the corresponding subhalo mass is smaller on average than the halo mass of centrals (Fig.",
"REF ).",
"This difference increases the smaller the mass is, and much more for the subhalo mass defined at the observed time (present-day).",
"In the case of $m_{\\rm sub}^{\\rm acc}$, the differences might be because the halo mass at the epoch it became a subhalo (accretion time) is smaller than its present-day counterpart at a given stellar mass and/or because the satellite stellar mass increased faster than the central one for a given halo mass.",
"In fact, it is difficult to make any inference in this case because the abundance matching is between local galaxies and (sub)halos at different past epochs.",
"In any case, the fact that the inferred mass relations for satellites and centrals when $m_{\\rm sub}^{\\rm acc}$ is used are not too different, suggests that the central galaxy–distinct halo mass relation does not change too much with time, at least since the epoch at which most of the subhalos were accreted.",
"When $m_{\\rm sub}^{\\rm obs}$ is used, both abundances of satellites and subhalos are matched at the same epoch, the observation (present-day) time.",
"In this case the strong difference between the satellite and central mass relations can be interpreted mainly as the result of subhalo mass loss due to tidal stripping.",
"Besides, the smaller the subhalo, the larger is the mass loss on average.",
"Probably, the different evolution in stellar mass between central and satellite galaxies could also play a role for the differences but not as significant a role the one related to halo and subhalo mass evolution.",
"From Fig.",
"REF one sees that for a given $m_{\\rm *}$, the $m_{\\rm sub}^{\\rm obs}$–to–$m_{\\rm sub}^{\\rm acc}$ ratio is 0.35–0.40 for the smallest masses up to $\\mbox{$m_{\\rm *}$}\\sim 2\\times 10^{11}$ M$_\\odot $.",
"At larger masses, this ratio rapidly tends to 1.",
"Therefore, the subhalos of satellites galaxies less massive than $\\mbox{$m_{\\rm *}$}\\sim 2\\times 10^{11}$ M$_\\odot $ have lost, on average (for all host halo massesThe dependence of the satellite subhalo mass loss on host halo mass will be explored elsewhere.",
"), 65–60% of their masses since they were accreted.",
"It should be noted that this is a rough approximation and the evolution of the stellar mass since the satellite was accreted should be taken into account, see §4.1.",
"This above result shows us that the galaxy-(sub)halo connection for satellite galaxies is far from direct; present-day satellites of masses $\\mbox{$m_{\\rm *}$}\\sim 7\\times 10^8$ M$_\\odot $ and larger have formed in subhalos that at the time they were accreted onto galaxy sized halos were on average a factor 2.5–3 larger than at the present epoch.",
"This has severe implications for studies aimed to constrain the $\\Lambda $ -CDM scenario at the level of subhalo/satellite distributions.",
"For example, it has been discussed that seeding the subhalos in simulations of MW-like halos by using an extrapolation to low masses of the stellar–halo mass relation obtained by means of the AMT, predicts a MW dwarf spheroidal (dSph) luminosity function in agreement with the observed one.",
"However, the circular velocities at the maximum (or the masses at the infall) of the subhalos associated to the dSphs result significantly larger than inferences from observed kinematics [8].",
"In the right panel of Fig.",
"REF we have plotted the extrapolation to low masses of our SSMRs, both for subhalo masses defined at the present day (red line) and at the infall time (blue line).",
"The observational points in the panel are for MW satellites, which subhalo masses were estimated at their truncation radii.",
"Thus, if we assume that these masses are roughly equal to the present-day subhalo masses in the $\\Lambda $ -CDM simulations, then the simulated subhalo masses, $m_{\\rm sub}^{\\rm obs}$, are up to $\\approx 10-30$ times larger than those associated to dSphs.",
"If the comparison is done with the extrapolation of the average (or central) SHMR, then the differences increases by a factor of $\\sim 3$ more [8].",
"Our extrapolated results show that the discrepancy in subhalo mass between MW bright dSphs and $\\Lambda $ -CDM simulations is smaller than previously reported but is still significant.",
"Note that for the extrapolation, we have used the same slope of the YMB09 satellite GSMF at the low mass end, $\\alpha =-1.25$ (Fig.",
"REF ).",
"If this slope steepens for smaller masses, for example to a value of $\\alpha =-1.6$ , then our extended AMT predict the $m_{\\rm sub}^{\\rm obs}$ SSMR plotted as the gray dotted-dashed curve in Fig.",
"REF , which is already consistent with the dynamical estimates.",
"The GSMF at low masses may be significantly incomplete because of missing low-surface brightness galaxies.",
"By taking into account the bivariate distribution of stellar mass versus surface brightness, [4] have found evidence for an upturn in the faint-end GSMF slope ($\\alpha \\approx -1.6$ ) for a subsample of field SDSS galaxies.",
"More recently, using the GAMA survey, a slope of $\\alpha \\approx -1.47$ has been reported [3].",
"Steep faint-end slopes have been also found at higher redshifts.",
"For instance, using the COSMOS field, [15] have measured slopes of $\\alpha \\sim -1.7$ at all redshifts $z\\le 1$ There are also pieces of evidence that the faint-end slope of the GSMF (or luminosity function) changes with the environment: in clusters of galaxies it steepens significantly [4].",
"The cluster GSMF is actually related to the satellite GSMF, through the satellite CSMF.",
"We conclude that using a correct AMT for connecting satellite galaxies to their present-day subhalos and assuming a steep faint-end slope in the satellite GSMF ($\\alpha \\sim -1.6$ ), the predicted SSMR for the $\\Lambda $ -CDM cosmogony would be consistent with the dynamics of MW satellites.",
"We thank Alexie Leathaud, Ramin Skibba, Surhud More, Xiaohu Yang, and the anonymous Referee for useful comments, and suggestions.",
"A. R-P acknowledges a graduate student fellowship provided by CONACyT.",
"N. D. and V. A. acknowledge to CONACyT grant 128556 (Ciencia Básica).",
"V.A acknowledges PAPIIT-UNAM grant IN114509.",
"We are grateful to X. Yang for providing us in electronic form their data for the $\\mbox{CSMF}$ s."
],
[
"The spatial clustering of galaxies\nin the HOD model",
"Here we review the main ideas used to infer the spatial clustering of galaxies based on a HOD model.",
"We assume that the most massive galaxy in terms of stellar mass within a halo of mass $M_{\\rm h}$ is its central galaxy.",
"Consequently the remaining galaxies are all satellites.",
"We let them follow the mass density profile of the host halo.",
"We denote the cumulative number of central and satellite galaxies with stellar masses greater than $M_{\\rm *}$ as $N_c$ and $N_s$ , respectively.",
"The two point correlation function is decomposed into two terms, $1+\\mbox{$\\xi _{\\rm gg}(r)$}=[1+\\mbox{$\\xi _{\\rm gg}^{\\rm 1h}(r)$}]+[1+\\mbox{$\\xi _{\\rm gg}^{\\rm 2h}(r)$}],$ with $1+\\mbox{$\\xi _{\\rm gg}^{\\rm 1h}(r)$}$ describing galaxy pairs within the same halo (the one-halo term), and $1+\\mbox{$\\xi _{\\rm gg}^{\\rm 2h}(r)$}$ describing the correlation between galaxies occupying different halos (the two-halo term).",
"We compute the one-halo term as $1+\\mbox{$\\xi _{\\rm gg}^{\\rm 1h}(r)$}=\\frac{1}{2\\pi r^2n_g^2}\\int _{0}^{\\infty }\\frac{\\langle N(N-1)\\rangle }{2}\\lambda (r)\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$}) d\\mbox{$M_{\\rm h}$},$ for pairs separated by a distance $r\\pm dr/2$ .",
"Here $\\langle N(N-1)\\rangle /2$ is the total mean number of galaxy pairs within halos $M_{\\rm h}$ following a pair distribution $\\lambda (r)dr$ , and a mean number density $n_g$($M_{\\rm *}$).",
"The contribution to the total mean number of galaxy pairs from central-satellite pairs and satellite-satellite pairs is $\\frac{\\langle N(N-1)\\rangle }{2}\\lambda (r)dr=\\langle N_c\\rangle \\mbox{$\\langle N_s\\rangle $}\\lambda _{c,s}(r)dr& & \\nonumber \\\\+\\frac{\\langle N_s(N_s-1)\\rangle }{2}\\lambda _{s,s}(r)dr.$ As commonly assumed in HOD models, the number of central-satellite pairs follow the normalized mass density halo profile, taken to be of [32] shape.",
"The number of satellite-satellite pairs is then related to the normalized density profile convolved with itself.",
"Halo profiles are defined in terms of the total halo mass and the concentration parameter.",
"We use the relation between concentration parameter $c_{\\rm NFW}$ and halo mass obtained by [31] from fits to $N$ -body simulations.",
"Based on results of high-resolution $N$ -body [24] and hydrodynamic simulations of galaxy formation [59], we model the second moment of satellite galaxies, $\\langle N_s(N_s-1)\\rangle $ , as a Poisson distribution with mean $\\mbox{$\\langle N_s\\rangle $}^2=\\langle N_s(N_s-1)\\rangle $ .",
"We compute the two-halo term as $\\mbox{$\\xi _{\\rm gg}^{\\rm 2h}(r)$}=b_g^2\\zeta ^2(r)\\xi _m(r),$ where $\\xi _m(r)$ is the non-linear matter correlation function [40], $\\zeta (r)$ is the scale dependence of dark matter halo bias [43], and $b_g= \\frac{1}{\\mbox{$n_g$}}\\int _{0}^{\\infty }b(\\mbox{$M_{\\rm h}$})\\mbox{$\\langle N(>\\mbox{$M_{\\rm *}$}|\\mbox{$M_{\\rm h}$})\\rangle $}\\mbox{$\\phi _{\\rm h}$}(\\mbox{$M_{\\rm h}$}) d\\mbox{$M_{\\rm h}$},$ is the galaxy bias with $b(\\mbox{$M_{\\rm h}$})$ the halo bias function [37].",
"Once we have calculated $\\xi _{\\rm gg}(r)$, we relate it to the projected correlation function, $w_{\\rm p}(r_{\\rm p})$, integration over the line of sight, $\\mbox{$w_{\\rm p}(r_{\\rm p})$}=2\\int _{0}^{\\infty }\\xi _{\\rm gg}(\\sqrt{r_{\\rm p}^2+x^2})dx.$"
]
] | 1204.0804 |
[
[
"Generalized index theorem for topological superconductors with\n Yang-Mills-Higgs couplings"
],
[
"Abstract We investigate an index theorem for a Bogoliubov-de Gennes Hamiltonian (BdGH) describing a topological superconductor with Yang-Mills-Higgs couplings in arbitrary dimensions.",
"We find that the index of the BdGH is determined solely by the asymptotic behavior of the Higgs fields and is independent of the gauge fields.",
"It can be nonvanishing if the dimensionality of the order parameter space is equal to the spatial dimensions.",
"In the presence of point defects there appear localized zero energy states at the defects.",
"Consistency of the index with the existence of zero energy bound states is examined explicitly in a vortex background in two dimensions and in a monopole background in three dimensions."
],
[
"Introduction",
"Yang-Mills-Higgs systems admit topologically nontrivial field configurations such as vortices and magnetic monopoles [1], [2].",
"They are considered to be responsible for various nonperturbative effects in particle physics.",
"Recently topological insulators and superconductors have attracted much interest as a new phase of materials in condensed matter physics.",
"Topological insulators are band insulators whose ground states are characterized by topological numbers [3].",
"The method of topologically classifying insulating ground states has been generalized to superconducting states described by the mean-field Bogoliubov-de Gennes Hamiltonian (BdGH).",
"This provides a unified framework of the classification of non-interacting fermion systems with respect to time reversal and particle-hole symmetries [4], [5], [6], [7].",
"The topological classification was further extended to systems with topological defects [8], which enables us to classify zero modes along line defects and zero energy bound states localized at point defects from the point of view of the topological universality class.",
"For a superconductor with time-reversal symmetry (class BDI), there exists a unitary operator that anticommutes with the BdGH.",
"It defines an extended chiral symmetry.",
"Generically, chiral symmetry ensures the topological stability of the zero energy states, since these states are controlled by the index theorem [9], [10], [11] for the BdGH, implying the robustness under continuous deformations of the order parameters.",
"Remarkably, the extended chiral symmetry can be defined in arbitrary dimensions.",
"This is a sharp contrast to the usual chiral invariance in massless Dirac theories, which only exists in even dimensions.",
"The extended chiral invariance also incorporates internal spin symmetry.",
"By gauging the internal spin symmetry we can reformulate the system as a Yang-Mills-Higgs system, where the space varying gap parameters can be regarded as Higgs fields.",
"In this paper we investigate the index theorem of the Dirac-type BdGH corresponding to the Yang-Mills-Higgs systems in arbitrary dimensions with special emphasis on the extended chiral invariance.",
"Previously, zero energy bound states in two and three dimensions were investigated by Jackiw and Rebbi [12], and Jackiw and Rossi [13].",
"The index theoretical approaches of their models were explored by Callias [9] and Weinberg [10].",
"In particular, the Weinberg index theorem turned out to be quite useful for much more complicated systems such as non-Abelian vortices in a color superconductor [14].",
"We show that for more generic models with Yang-Mills-Higgs couplings in arbitrary dimensions the Weinberg index theorem is applicable, and the index of the BdGH can be nonvanishing, implying that there appear topological zero energy bound states localized at the point defect if the dimensions of order parameter space is equal to the spatial dimensions.",
"We also see that the index is solely determined by the behaviors of the Higgs fields at the spatial infinities.",
"This is expected in odd dimensions [9], since we have no topological invariant defined by the gauge field.",
"In even dimensions, however, we have two topological invariants, one defined by the Higgs fields and one by the Yang-Mills fields.",
"The expression of the topological invariants seems to contain both of them.",
"One might consider that the number of zero modes would be affected by including the Yang-Mills fields.",
"We show that this is not the case.",
"These two topological invariants are so combined that a unique index is obtained.",
"This was noted for vortex background in two dimensions [10].",
"We show this happens in arbitrary dimensions.",
"This paper is organized as follows.",
"In Sect.",
"we introduce BdGH in arbitrary dimensions and their generalized chiral symmetry.",
"We then define the generalized index for the BdGH.",
"A computation of the topological index is given in Sect.",
".",
"The chiral current at the spatial infinities is investigated in Sect.",
".",
"General properties of the chiral current and the relation with the topological index in even dimensions are given in Sect.",
".",
"Zero modes for vortex and monopole configurations in two and three dimensions are investigated in Sect.",
".",
"Finally, Sect.",
".",
"is devoted to summary and discussions."
],
[
"Extended chiral symmetry of\nBdGH",
"We begin with a $d$ dimensional fermion system coupled to an O($d$ ) Higgs field $\\phi _a$ $(a=1,\\cdots , d)$ .",
"We assume that the system is described by the following Hamiltonian $H_0&=&-i\\gamma ^j\\partial _j+\\phi (x), \\qquad (\\phi (x)=\\Gamma ^a\\phi _a(x))$ where $\\gamma ^j$ ($j=1,\\cdots ,d$ ) and $\\gamma ^{d+a}=\\Gamma ^a$ form a set of $2d$ dimensional $\\gamma $ matrices satisfying $\\lbrace \\gamma ^\\mu ,\\gamma ^\\nu \\rbrace =\\delta ^{\\mu \\nu }$ .",
"We introduce the $2d$ dimensional chiral matrix by $\\gamma _{2d+1}=(-i)^d\\gamma ^1\\cdots \\gamma ^{2d}=(-i)^d\\gamma ^1\\cdots \\gamma ^d\\Gamma ^1\\cdots \\Gamma ^d$ .",
"It anti-commutes with both $\\gamma ^j$ and $\\Gamma ^a$ .",
"The system then possesses the chiral symmetry in the sense that $\\lbrace \\gamma _{2d+1},H_0\\rbrace &=&0.$ Note that the chiral symmetry can be defined in any dimensions.",
"This is contrasted with the usual chiral symmetry which is only defined in even dimensions.",
"The Hamiltonian (REF ) concerns itself with $2d$ dimensions, $d$ spatial and $d$ internal.",
"The chiral invariance is known to be violated in the presence of the chemical potential.",
"In the present work we are concerned with the chiral symmetric case, where the index of the Hamiltonian is well-defined.",
"In addition to the chiral invariance we can define particle-hole symmetry.",
"Let us introduce the charge conjugation matrix $C$ by $C(\\gamma ^j)^\\ast C^{-1}=\\gamma ^{j}, \\qquad C(\\Gamma ^a)^\\ast C^{-1}=-\\Gamma ^a.$ It is always possible to find $C$ for a given set of $\\gamma ^\\mu $ .",
"Then $H_0$ satisfies ${\\cal C}H_0{\\cal C}^{-1}&=&-H_0,$ where ${\\cal C}=CK$ denotes complex conjugation $K$ followed by the multiplication by $C$ .",
"To each particle state $\\psi $ of an energy $E>0$ we can define a state $\\psi _c$ with energy $-E$ by $\\psi _c(x)={\\cal C}\\psi (x)=C\\psi ^\\ast (x).$ Eq.",
"(REF ) ensures the particle-hole symmetry of the spectrum of $H_0$ .",
"Before going into detailed analysis of the index of the Hamiltonian we introduce spin($d$ ) gauge field $A_j(x)&=&\\frac{1}{2}\\Sigma ^{ab}A_{abj}(x),$ where $\\Sigma ^{ab}\\equiv [\\Gamma ^a,\\Gamma ^b]/4$ are the spin($d$ ) generators and $A_{abj}$ are the components of the gauge field satisfying $A_{abj}^\\ast =A_{abj}=-A_{baj}$ .",
"The gauge symmetric generalization of $H_0$ is then given by $H&=&-i\\gamma ^jD_j+\\phi (x),$ where $D_j=\\partial _j+A_j$ is the covariant derivative.",
"For the Higgs field it is given $D_j\\phi =\\Gamma ^aD_j\\phi _a, \\quad D_j\\phi _a=\\partial _j\\phi _a+A_{abj}\\phi _b.$ The field strength is also defined as usual $F_{ij}&=&\\frac{1}{2}\\Sigma ^{ab}F_{abij}, \\quad F_{abij}=\\partial _iA_{abj}-\\partial _jA_{abi}+A_{aci}A_{cbj}-A_{acj}A_{cbi}.$ Since the spin($d$ ) generators commute with $\\gamma _{2d+1}$ and ${\\cal C}$ , the chiral and particle-hole symmetries remain intact by the generalization.",
"Unlike Nielsen-Olsen vortex and 't Hooft-Polyakov monopole, which will be treated in Sect.",
", we consider the Yang-Mills and Higgs fields as independent classical background.",
"For the present we only assume that the gauge potential approaches to pure gauge and the Higgs field satisfies $\\phi ^2\\rightarrow |\\phi _0|^2$ at the spatial infinities, where $|\\phi _0|^2$ is a nonvanishing constant.",
"This gives rise to a finite energy gap.",
"Due to the particle-hole symmetry mentioned above, any eigenstate of nonvanishing energy $E$ is necessarily paired with an eigenstate of energy $-E$ .",
"These states are also related with the chiral transformation.",
"The zero modes of $H$ , however, can be unpaired.",
"They can be made self-conjugate under the particle-hole symmetry and can be regarded as Majorana states.",
"For a general Yang-Mills-Higgs background it is not possible to find an explicit form of zero mode wave function.",
"In the chirally symmetric case, however, we can investigate the existence or nonexistence of zero modes by computing the index of $H$ defined by $\\mathrm {index}H&=&n_+-n_-,$ where $n_\\pm $ are the numbers of positive and negative $\\gamma _{2d+1}$ chirality zero energy states.",
"The index of $H$ is known to be a topological invariant, i.e., it is invariant under continuous deformations of the gauge and Higgs fields.",
"The index theorem relates the index with a topological invariant of these fields [9], [10], [11].",
"To establish the index theorem we rewrite Eq.",
"(REF ) as ${\\rm index}H&=&\\lim _{m\\rightarrow 0}{\\rm Tr}\\gamma _{2d+1}\\frac{m^2}{H^2+m^2}\\nonumber \\\\&=&\\lim _{m\\rightarrow 0}\\int d^dx\\lim _{y\\rightarrow x}{\\rm tr}\\gamma _{2d+1}\\frac{m^2}{H^2+m^2}\\delta ^d(x-y),$ where $\\mathrm {Tr}$ stands for the integration over the spatial coordinates as well as the trace on the $\\gamma $ matrices.",
"As is well-known, index of the Dirac operator is related with the chiral anomaly of the axial current divergence.",
"To show this we introduce an axial current $J^i(x)$ by $J^i(x)&=&\\lim _{m\\rightarrow 0\\atop M\\rightarrow \\infty }\\lim _{y\\rightarrow x}{\\rm tr}\\gamma _{2d+1}\\gamma ^i\\left(\\frac{1}{iH+m}-\\frac{1}{iH+M}\\right)\\delta ^d(x-y),$ where $M$ is a Pauli-Villars mass to regularize the current at short distances.",
"It is straightforward to show that the axial current divergence can be cast into the form $\\partial _iJ^i(x)&=&-2\\lim _{m\\rightarrow 0\\atop M\\rightarrow \\infty }\\lim _{y\\rightarrow x}{\\rm tr}\\gamma _{2d+1}\\left(\\frac{m^2}{H^2+m^2}-\\frac{M^2}{H^2+M^2}\\right)\\delta ^d(x-y).$ At this stage we can take the two limits, $m\\rightarrow 0$ and $M\\rightarrow \\infty $ , separately.",
"Eq.",
"(REF ) then can be written as $\\mathrm {index}H&=&-\\frac{1}{2}\\int _{S_\\infty ^{d-1}} dS_iJ^i(x)+c_d,$ where $c_d$ is the topological index defined by $c_d&=&\\lim _{M\\rightarrow \\infty }\\mathrm {Tr}\\gamma _{2d+1}\\frac{M^2}{H^2+M^2}$ and $S_\\infty ^{d-1}$ denotes the infinities of the $d$ dimensional euclidean space.",
"In the case of index theorem on compact manifolds without boundaries the contribution from the chiral current on the rhs of (REF ) vanishes and the index coincides with $c_d$ .",
"Since we are working with the Hamiltonian (REF ) defined on a euclidean space, the surface term can be nontrivial [9], [10], [11].",
"We expect a nonvanishing contribution to $\\mathrm {index}H$ if $J^i(x)$ is of order $|x|^{-d+1}$ for $|x|\\rightarrow \\infty $ .",
"To compute the integral on the rhs of Eq.",
"(REF ) we only need the leading behavior of the chiral current at spatial infinities.",
"This will be done in Sect.",
"."
],
[
"Topological index",
"We have shown that the index can be written as a sum of the topological index (REF ) and the surface integral of the chiral current.",
"The evaluation of the functional trace on the rhs of (REF ) is similar to that of chiral anomalies.",
"In this section we compute $c_d$ .",
"Eq.",
"(REF ) can be explicitly written as $c_d&=&\\lim _{M\\rightarrow \\infty }\\int d^dx\\lim _{y\\rightarrow x}{\\rm tr}\\gamma _{2d+1}\\frac{M^2}{H^2+M^2}\\delta ^d(x-y) \\nonumber \\\\&=&\\lim _{M\\rightarrow \\infty }\\int d^dx\\int \\frac{d^dk}{(2\\pi )^d}e^{-ikx}{\\rm tr}\\gamma _{2d+1}\\frac{M^2}{H^2+M^2}e^{ikx}.$ Using $e^{-ikx}He^{ikx}&=&-i\\gamma ^i(D_j+ik_j)+\\phi , \\\\e^{-ikx}H^2e^{ikx}&=&-(D_j+ik_j)^2-\\frac{1}{2}\\gamma ^i\\gamma ^jF_{ij}-i\\gamma ^jD_j\\phi +\\phi ^2,$ we can compute the rhs of Eq.",
"(REF ) as $c_d&=&\\lim _{M\\rightarrow \\infty }\\int d^dx\\int \\frac{d^dk}{(2\\pi )^d}{\\rm tr}\\gamma _{2d+1}\\frac{M^2}{-(D_j+ik_j)^2-\\frac{1}{2}\\gamma ^i\\gamma ^jF_{ij}-i\\gamma ^jD_j\\phi +\\phi ^2+M^2} \\nonumber \\\\&=&\\lim _{M\\rightarrow \\infty }M^2\\int d^dx\\int \\frac{d^dk}{(2\\pi )^d}\\sum _{n=0}^\\infty \\mathrm {tr}\\Biggl [\\gamma _{2d+1}\\frac{1}{-(D_i+ik_i)^2+\\phi ^2+M^2} \\nonumber \\\\&&\\times \\left(\\left(i\\gamma ^iD_i\\phi +\\frac{1}{2}\\gamma ^i\\gamma ^jF_{ij}\\right)\\frac{1}{-(D_i+ik_i)^2+\\phi ^2+M^2}\\right)^n\\Biggr ].$ Due to the presence of $\\gamma _{2d+1}$ the terms with $n<d/2$ vanish under the trace, whereas the terms with $n>d/2$ do not contribute to the sum in the limit $M^2\\rightarrow \\infty $ as can be easily seen by the scaling argument of the momentum variables $k\\rightarrow Mk$ .",
"This immediately gives $c_d=0$ in odd dimensions.",
"For $d=2N$ even we obtain $c_d&=&\\int d^dx\\int \\frac{d^{2N}k}{(2\\pi )^d}\\frac{1}{(k^2+1)^{N+1}}\\mathrm {tr}\\gamma _{2d+1}\\left(\\frac{1}{2}\\gamma ^i\\gamma ^jF_{ij}\\right)^N \\nonumber \\\\&=&\\frac{(-1)^N}{(2\\pi )^NN!",
"}\\int d^dx\\epsilon ^{i_1\\cdots i_d}\\mathrm {tr}_\\eta F_{i_1i_2}\\cdots F_{i_{d-1}i_d},$ where $\\epsilon ^{i_1\\cdots i_d}$ is the Levi-Civita symbol in $d$ dimensions.",
"We have also introduced $\\mathrm {tr}_\\eta $ by $\\mathrm {tr}_\\eta (\\cdots )&=& \\mathrm {tr}(\\eta \\cdots ),$ where $\\eta $ is defined by $\\eta &=&\\frac{(-1)^{d(d-1)/2}}{2^d}\\Gamma ^1\\cdots \\Gamma ^d.$ The overall normalization is chosen to give $\\mathrm {tr}_\\eta \\Gamma ^{a_1}\\cdots \\Gamma ^{a_d}=\\epsilon ^{a_1\\cdots a_d}.$ It satisfies anti-cyclic property $\\mathrm {tr}_\\eta \\Gamma \\Gamma ^{\\prime }=-\\mathrm {tr}_\\eta \\Gamma ^{\\prime }\\Gamma \\quad \\hbox{for}\\quad \\lbrace \\eta ,\\Gamma \\rbrace =0.$ Later we need to compute traces involving $\\phi $ .",
"In even dimensions $\\eta $ anti-commutes with $\\phi $ , whereas it commutes in odd dimensions.",
"We have seen that only the gauge field contributes to the topological index and the Higgs field is irrelevant to the computation of $c_d$ .",
"The result (REF ) coincides with computation of index of a Dirac operator without Higgs fields."
],
[
"Computation of chiral current",
"The chiral current defined by (REF ) is written by the regularized fermion propagator.",
"It is in general a nonlocal quantity of the background fields.",
"To find its contribution to the index of the BdGH we only need the asymptotic behaviors at spatial infinities, where the chiral current turns out to become local.",
"In this section we evaluate the asymptotic form of the chiral current.",
"We first rewrite the current (REF ) as $J^i(x)&=&-i\\lim _{m\\rightarrow 0\\atop M\\rightarrow \\infty }\\int \\frac{d^dk}{(2\\pi )^d}e^{-ikx}{\\rm tr}\\gamma _{2d+1}\\gamma ^iH\\left(\\frac{1}{H^2+m^2}-\\frac{1}{H^2+M^2}\\right)e^{ikx}.$ The computation of $J^i(x)$ is similar to the one presented in Sect.",
".",
"We obtain $J^i(x)&=&-i\\lim _{m\\rightarrow 0\\atop M\\rightarrow \\infty }\\int \\frac{d^dk}{(2\\pi )^d}\\sum _{n=0}^\\infty \\Biggl \\lbrace \\frac{1}{(k^2+|\\phi _0|^2+m^2)^{n+1}}-(m^2\\rightarrow M^2)\\Biggr \\rbrace \\nonumber \\\\&&\\times \\mathrm {tr}\\gamma _{2d+1}\\gamma ^i(\\gamma ^jk_j-i\\gamma ^jD_j+\\phi )\\left(i\\gamma ^kD_k\\phi +\\frac{1}{2}\\gamma ^k\\gamma ^lF_{kl}+\\Delta \\right)^n,$ where $\\Delta $ is given by $\\Delta &=&2ik_iD_i+D_i^2-|\\phi |^2+|\\phi _0|^2.$ At the spatial infinities $\\phi ^2$ approaches to a constant $|\\phi _0|^2$ .",
"We further assume $\\phi ^2-|\\phi _0|^2, ~\\partial _j\\phi , ~ A_j&\\sim &\\mathrm {O}(|x|^{-1})\\quad \\hbox{for}\\quad |x|\\rightarrow \\infty .$ To find $\\mathrm {index}H$ it is only necessary to know the leading $\\mathrm {O}(|x|^{-d+1})$ terms of the current for $|x|\\rightarrow \\infty $ .",
"It is easy to convince oneself that the terms of the rhs of Eq.",
"(REF ) vanish for $n<(d-1)/2$ because of the trace with $\\gamma _{2n+1}$ , whereas the terms with $n\\ge d$ can be ignored since they decay faster than $|x|^{-d+1}$ for $|x|\\rightarrow \\infty $ .",
"Eq.",
"(REF ) then can be written as $J^i(x)&=&-i\\lim _{m\\rightarrow 0\\atop M\\rightarrow \\infty }\\int \\frac{d^dk}{(2\\pi )^d}\\sum _{n\\ge (d-1)/2}^{d-1}\\Biggl \\lbrace \\frac{1}{(k^2+|\\phi _0|^2+m^2)^{n+1}}-(m^2\\rightarrow M^2)\\Biggr \\rbrace \\nonumber \\\\&&\\times \\mathrm {tr}\\gamma _{2d+1}\\gamma ^i\\phi \\left(i\\gamma ^iD_i\\phi +\\frac{1}{2}\\gamma ^i\\gamma ^jF_{ij}\\right)^n+\\mathrm {O}(|x|^{-d}).$ The $k$ integral can be done for $n\\ge (d-1)/2$ as $\\int \\frac{d^dk}{(2\\pi )^d}\\frac{1}{(k^2+\\mu ^2)^{n+1}}&=&\\frac{\\Gamma (n+1-d/2)}{(4\\pi )^{d/2}n!",
"}\\frac{1}{\\mu ^{2(n+1)-d}}.$ Now the limits $m\\rightarrow 0$ and $M\\rightarrow \\infty $ can be taken safely.",
"Keeping only the nonvanishing contributions, we obtain $J^i(x)&=&-i\\sum _{l=0}^{\\left[\\frac{d-1}{2}\\right]}\\frac{\\Gamma (d/2-l)}{(4\\pi )^{d/2}(d-l-1)!",
"}\\mathrm {tr}\\gamma _{2d+1}\\gamma ^i\\hat{\\phi }\\mathrm {Symm}\\left[\\left(i\\gamma ^jD_j\\hat{\\phi }\\right)^{d-2l-1}\\left(\\frac{1}{2}\\gamma ^j\\gamma ^kF_{jk}\\right)^l\\right]+\\mathrm {O}(|x|^{-d}), \\nonumber \\\\$ where $\\hat{\\phi }=\\phi /|\\phi _0|$ and $\\mathrm {Symm}$ denotes symmetrized product defined by $\\mathrm {Symm}A^nB^m=\\frac{(n+m)!}{n!m!",
"}\\left.\\frac{\\partial ^{n+m}}{\\partial s^n\\partial t^m}\\exp [sA+tB]\\right|_{s=t=0}.$ Since $\\gamma ^jD_j\\hat{\\phi }$ and $\\gamma ^j\\gamma ^kF_{jk}$ are effectively commutative in the trace of Eq.",
"(REF ), we can simplify the symmetrized product further.",
"We thus arrive at the asymptotic form of the chiral current $J^i(x)&=&\\frac{(-1)^{d-1}}{(4\\pi )^{d/2}}\\sum _{k=0}^{\\left[\\frac{d-1}{2}\\right]}\\frac{2^{d-k}\\Gamma (d/2-k)}{k!(d-2k-1)!",
"}\\epsilon ^{ii_2\\cdots i_d}\\mathrm {tr}_\\eta \\hat{\\phi }D_{i_2}\\hat{\\phi }\\cdots D_{i_{d-2k}}\\hat{\\phi }F_{i_{d-2k+1}i_{d-2k+2}}\\cdots F_{i_{d-1}i_d}, \\nonumber \\\\$ where nonleading contributions are suppressed.",
"The asymptotic chiral current is local with respect to the background fields.",
"Unlike the topological index $c_d$ it exists in odd as well as even dimensions.",
"It is easy to obtain explicit forms of $J^i(x)$ in low dimensions.",
"In two dimensions the chiral current and topological index are given by $J^i(x)&=&\\frac{1}{\\pi }\\epsilon ^{ij}\\epsilon ^{ab}\\hat{\\phi }_a\\partial _j\\hat{\\phi }_b-\\frac{1}{2\\pi }\\epsilon ^{ij}\\epsilon ^{ab}A_{abj}, \\\\c_2&=&-\\frac{1}{4\\pi }\\int d^2x\\epsilon ^{ij}\\epsilon ^{ab}\\partial _iA_{abj}.$ These lead to the index $\\mathrm {index}H&=&\\frac{1}{2\\pi }\\int _{S_\\infty ^1}dS_i\\epsilon ^{ij}\\epsilon ^{ab}\\hat{\\phi }_a\\partial _j\\hat{\\phi }_b.$ We see that the topological index is canceled by the gauge field dependent term of the chiral current [10].",
"Similar thing also happens in three dimensions.",
"The topological index vanish identically in odd dimensions and the gauge field dependent terms of the chiral current can be cast into a total derivative term as $J^i(x)&=&-\\frac{1}{4\\pi }\\epsilon ^{ijk}\\epsilon ^{abc}\\left(\\hat{\\phi }_aD_j\\hat{\\phi }_bD_k\\hat{\\phi }_c+\\frac{1}{2}\\hat{\\phi }_aF_{jkbc}\\right) \\nonumber \\\\&=&-\\frac{1}{4\\pi }\\epsilon ^{ijk}\\epsilon ^{abc}\\hat{\\phi }_a\\partial _j\\hat{\\phi }_b\\partial _k\\hat{\\phi }_c-\\frac{1}{4\\pi }\\epsilon ^{ijk}\\epsilon ^{abc}\\partial _j(A_{abk}\\hat{\\phi }_c).$ We thus obtain the index $\\mathrm {index}H&=&\\frac{1}{8\\pi }\\int _{S_\\infty ^2} dS_i\\epsilon ^{ijk}\\epsilon ^{abc}\\hat{\\phi }_a\\partial _j\\hat{\\phi }_b\\partial _k\\hat{\\phi }_c.$ Again the gauge field dependence disappears and the index is only determined by the asymptotic behavior of the Higgs field at the infinities [9].",
"In the next section we show that this holds true in arbitrary dimensions."
],
[
"Differential geometric approach",
"To establish the gauge field independence of $\\mathrm {index}H$ it is convenient to introduce Lie algebra valued differential forms [16], the gauge potential 1-form and the field strength 2-form, as $&&A=\\frac{1}{4}\\Gamma ^a\\Gamma ^b\\mathrm {d}x^iA_{abi}, \\\\&&F=\\frac{1}{2}\\mathrm {d}x^i\\mathrm {d}x^jF_{ij}=\\frac{1}{8}\\mathrm {d}x^i\\mathrm {d}x^j\\Gamma ^a\\Gamma ^bF_{abij}=\\mathrm {d}A+A^2.$ Note that $F$ satisfies Bianchi identity $DF=dF+AF-FA=0.$ The exterior covariant derivative of $\\hat{\\phi }$ is given by $D\\hat{\\phi }=\\mathrm {d}\\hat{\\phi }+[A,\\hat{\\phi }].$ Taking $D$ once more we obtain $D^2\\hat{\\phi }&=&[F,\\hat{\\phi }].$ Since the generators of spin($d$ ) commutes with $\\eta $ defined by (REF ), so do $A$ and $F$ in arbitrary dimensions.",
"The topological index (REF ) in $d=2N$ dimensions can be written in terms of $F$ as $c_d&=&\\frac{(-1)^N}{\\pi ^NN!",
"}\\int \\mathrm {tr}_\\eta F^N,$ where the integral is taken over the entire $d$ dimensional space.",
"If we introduce Chern-Simons form $\\omega _{d-1}^0$ by $\\mathrm {tr}_\\eta F^N&=&\\mathrm {d}\\omega _{d-1}^0,$ the topological index (REF ) can be converted to the surface integral $c_d&=&\\frac{(-1)^N}{\\pi ^NN!",
"}\\int _{S_\\infty ^{d-1}}\\omega ^0_{d-1},$ where $S_\\infty ^{d-1}$ denotes the $(d-1)$ -sphere at the spatial infinities.",
"The Chern-Simons form $\\omega ^0_{d-1}$ can be written as $\\omega ^0_{d-1}&=&N\\int _0^1dt \\mathrm {tr}_\\eta A(t\\mathrm {d}A+t^2A^2)^{N-1}.$ For $d=2,4,6$ the Chern-Simons forms are explicitly given by $&&\\omega ^0_1=\\mathrm {tr}_\\eta A, \\nonumber \\\\&&\\omega ^0_3=\\mathrm {tr}_\\eta \\left(A\\mathrm {d}A+\\frac{2}{3}A^3\\right),\\nonumber \\\\&&\\omega ^0_5=\\mathrm {tr}_\\eta \\left(A(\\mathrm {d}A)^2+\\frac{3}{2}A^3\\mathrm {d}A+\\frac{3}{5}A^5\\right).$ We now turn to the chiral current (REF ).",
"The first integral in the rhs of Eq.",
"(REF ) can be written as $\\int _{S_\\infty ^{d-1}}dS_iJ^i&=&\\int _{S_\\infty ^{d-1}}{}^\\ast J,$ where ${}^\\ast J$ is the Hodge dual of the chiral current 1-form $J=J_i(x)\\mathrm {d}x^i$ and is given by ${}^\\ast J&=&\\frac{1}{(d-1)!",
"}\\epsilon _{i_1i_2\\cdots i_d}J^{i_1}(x)\\mathrm {d}x^{i_2}\\cdots \\mathrm {d}x^{i_d}.$ For the current Eq.",
"(REF ), the dual can be written as ${}^\\ast J&=&\\sum _{k=0}^{\\left[\\frac{d-1}{2}\\right]}C_{d,k}\\mathrm {tr}_\\eta \\hat{\\phi }(D\\hat{\\phi })^{d-2k-1}F^k,$ where $C_{d,k}$ is defined by $C_{d,k}&=&-(-1)^{\\left[\\frac{d+1}{2}\\right]}\\frac{\\Gamma (d/2-k)}{\\pi ^{d/2}k!(d-2k-1)!",
"}.$ Taking the exterior derivative of the current and using Eq.",
"(REF ) and (REF ), we get $\\mathrm {d}{}^\\ast J&=&\\sum _{k=0}^{\\left[\\frac{d-1}{2}\\right]}C_{d,k}\\mathrm {tr}_\\eta ((D\\hat{\\phi })^{d-2k}+\\hat{\\phi }[F,\\hat{\\phi }](D\\hat{\\phi })^{d-2k-2}-\\hat{\\phi }D\\hat{\\phi }[F,\\hat{\\phi }](D\\hat{\\phi })^{d-2k-3} \\nonumber \\\\&&+\\cdots +(-1)^{d-2k-2}\\hat{\\phi }(D\\hat{\\phi })^{d-2k-2}[F,\\hat{\\phi }])F^k \\nonumber \\\\&=&\\sum _{k=0}^{\\left[\\frac{d-1}{2}\\right]}C_{d,k}\\mathrm {tr}_\\eta (D\\hat{\\phi })^{d-2k}F^k-\\sum _{k=0}^{\\left[\\frac{d-1}{2}\\right]}(d-2k-1)C_{d,k}\\mathrm {tr}_\\eta [F,\\hat{\\phi }]\\hat{\\phi }(D\\hat{\\phi })^{d-2k-2}F^k.",
"\\nonumber \\\\$ In deriving this use has been made of the fact that $[F,\\hat{\\phi }]$ effectively anti-commutes with $\\hat{\\phi }$ and $D\\hat{\\phi }$ in the trace.",
"We can simplify the trace in the second summand on the rhs of Eq.",
"(REF ) as $\\mathrm {tr}_\\eta [F,\\hat{\\phi }]\\hat{\\phi }(D\\hat{\\phi })^{d-2k-2}F^k&=&-\\frac{1}{k+1}\\mathrm {tr}_\\eta \\hat{\\phi }(D\\hat{\\phi })^{d-2k-2}([F,\\hat{\\phi }]F^k+F[F,\\hat{\\phi }]F^{k-1}\\cdots +F^k[F,\\hat{\\phi }]) \\nonumber \\\\&=&-\\frac{1}{k+1}\\mathrm {tr}_\\eta \\hat{\\phi }(D\\hat{\\phi })^{d-2k-2}[F^{k+1},\\hat{\\phi }] \\nonumber \\\\&=&(-1)^d\\frac{2}{k+1}\\mathrm {tr}_\\eta (D\\hat{\\phi })^{d-2k-2}F^{k+1}.$ Inserting this into Eq.",
"(REF ), we obtain $\\mathrm {d}{}^\\ast J&=&-\\left.\\frac{2(d-2k+1)}{k}C_{d,k-1}\\mathrm {tr}_\\eta (D\\hat{\\phi })^{d-2k}F^k\\right|_{k=\\left[\\frac{d-1}{2}\\right]+1},$ where use has been made of $\\mathrm {tr}_\\eta (D\\hat{\\phi })^d=0$ and the relation $C_{d,k}=\\frac{2(d-2k+1)}{k}C_{d,k-1} , \\qquad \\left(k=1,2,\\cdots ,\\left[\\frac{d-1}{2}\\right]\\right) .$ The former can be verified by noting $\\hat{\\phi }^2=1$ .",
"In odd dimensions the coefficient on the rhs of (REF ) vanishes.",
"We therefore obtain $\\mathrm {d}{}^\\ast J&=&0.$ This implies that the current can be written as $-\\frac{1}{2}{}^\\ast J&=&\\frac{(-1)^\\frac{d+1}{2}}{2^d\\pi ^{\\frac{d-1}{2}}(\\frac{d-1}{2})!",
"}\\mathrm {tr}_\\eta \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^{d-1}+\\mathrm {d}\\Omega _{d-2}(\\hat{\\phi },A),$ where $\\Omega _{d-2}$ is a $(d-2)$ -form.",
"In even ($d=2N$ ) dimensions Eq.",
"(REF ) can be written as $\\mathrm {d}{}^\\ast J&=&2\\frac{(-1)^N}{\\pi ^NN!",
"}\\mathrm {tr}_\\eta F^N.$ This is reminiscent of chiral anomaly.",
"It also implies that ${}^\\ast J$ can be written as $-\\frac{1}{2}{}^\\ast J&=&(-1)^\\frac{d}{2}\\frac{\\left(\\frac{d}{2}\\right)!",
"}{\\pi ^\\frac{d}{2}d!",
"}\\mathrm {tr}_\\eta \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^{d-1}+\\mathrm {d}\\Omega _{d-2}(\\hat{\\phi },A)-\\frac{(-1)^\\frac{d}{2}}{\\pi ^\\frac{d}{2}\\left(\\frac{d}{2}\\right)!",
"}\\omega _{d-1}^0.$ Interestingly enough, there appears the the Chern-Simons form.",
"It is straightforward to check Eq.",
"(REF ) and (REF ) in low dimensions.",
"We give explicit expressions for $d=2,~3,~4,~5$ : $d=2: &&{}^\\ast J=\\frac{1}{\\pi }\\mathrm {tr}_\\eta \\hat{\\phi }\\mathrm {d}\\hat{\\phi }-\\frac{1}{\\pi }\\mathrm {tr}_\\eta A, \\nonumber \\\\d=3: &&{}^\\ast J=-\\frac{1}{4\\pi }\\mathrm {tr}_\\eta \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^2-\\frac{1}{\\pi }\\mathrm {d}\\mathrm {tr}_\\eta \\hat{\\phi }A, \\nonumber \\\\d=4: &&{}^\\ast J=-\\frac{1}{6\\pi ^2}\\mathrm {tr}_\\eta \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^3+\\frac{1}{\\pi ^2}\\mathrm {d}\\mathrm {tr}_\\eta \\left(\\hat{\\phi }\\mathrm {d}\\hat{\\phi }A+\\frac{1}{2}\\hat{\\phi }A\\hat{\\phi }A\\right)+\\frac{1}{\\pi ^2}\\mathrm {tr}_\\eta \\left(A\\mathrm {d}A+\\frac{2}{3}A^3\\right), \\nonumber \\\\d=5: && \\nonumber \\\\&&\\hspace{-56.9055pt}{}^\\ast J=\\frac{1}{32\\pi ^2}\\mathrm {tr}_\\eta \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^4+\\frac{1}{4\\pi ^2}\\mathrm {d}\\mathrm {tr}_\\eta \\Biggl \\lbrace \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^2A+\\hat{\\phi }\\mathrm {d}\\hat{\\phi }A\\hat{\\phi }A+\\frac{1}{3}(\\hat{\\phi }A)^3+\\hat{\\phi }(A\\mathrm {d}A+\\mathrm {d}AA+A^3)\\Biggr \\rbrace .",
"\\nonumber \\\\$ We now turn to $\\mathrm {index}H$ .",
"We see from Eq.",
"(REF ) or (REF ) that $\\Omega _{d-2}$ does not contribute to the index and the topological index in even dimensions is canceled by the Chern-Simons form.",
"We thus obtain $\\mathrm {index}H&=&(-1)^{\\left[\\frac{d+1}{2}\\right]}\\frac{\\Gamma (d/2+1)}{\\pi ^{d/2}d!",
"}\\int _{S_\\infty ^{d-1}} \\mathrm {tr}_\\eta \\hat{\\phi }(\\mathrm {d}\\hat{\\phi })^{d-1}.$ The gauge fields disappear completely and the index is determined only by the behaviors of the Higgs fields at the spatial infinities.",
"This generalizes the observation of Ref.",
"[10] for the two dimensional model of Jackiw and Rossi [13].",
"In the next section we will introduce Yukawa coupling constant $g$ by substituting $\\phi $ by $g\\phi $ in Eq.",
"(REF ).",
"In even dimensions the index (REF ) is not affected by this, whereas an extra overall factor $\\mathrm {sgn}(g)$ appears on the rhs of Eq.",
"(REF ) in odd dimensions."
],
[
"Topological configurations in two and three dimensions",
"So far we have considered the Higgs and gauge fields as independent background fields.",
"In some model field theories topological objects can be realized dynamically as a solution to the field equations.",
"More specifically we assume the gauge-Higgs system in $d$ spatial dimensions with a static energy ${\\cal H}&=&\\int d^dx\\left(\\frac{1}{8e^2}(F_{abij})^2+\\frac{1}{2}(D_i\\phi _a)^2+\\frac{\\lambda _0}{8}(|\\phi _0|^2-|\\phi |^2)^2\\right),$ where $e$ is the gauge coupling constant.",
"It becomes stationary for the fields satisfying $D_iD_i\\hat{\\phi }_a+\\lambda (1-|\\hat{\\phi }|^2)\\hat{\\phi }_a=0,\\qquad D_iF_{abij}+\\kappa \\hat{\\phi }_a\\overleftrightarrow{D}_j\\hat{\\phi }_b=0,$ where $\\kappa =e^2|\\phi _0|^2$ and $\\lambda =\\lambda _0|\\phi _0|^2/2$ .",
"These nonlinear field equations give rise to topological objects, Nielsen-Olesen vortex in two dimensions [1] and 't Hooft-Polyakov monopole in three dimensions [2].",
"We consider the BdG equation with these topological configurations and see more closely how the index relation is fulfilled."
],
[
"Vortex in Maxwell-Higgs system",
"In two dimensions Eq.",
"(REF ) can describe vortices.",
"Ansatz for a vortex with unit vorticity is given by $\\hat{\\phi }_a(x)=h(r)\\frac{x^a}{r}, \\qquad A_{abi}(x)=-\\epsilon _{ab}\\epsilon _{ij}(1-k(r))\\frac{x^j}{r^2},$ where $h(r)$ and $k(r)$ are assumed to satisfy the following boundary conditions $h(0)=0, \\quad k(0)=1, \\quad h(\\infty )=1, \\quad k(\\infty )=0.$ Eqs.",
"(REF ) give differential equations for $h(r)$ and $k(r)$ as $h^{\\prime \\prime }+\\frac{h^{\\prime }}{r}=\\frac{k^2h}{r^2}-\\lambda (1-h^2)h, \\qquad k^{\\prime \\prime }-\\frac{k^{\\prime }}{r}=\\kappa h^2k.$ We see that $1-h$ and $k$ decrease exponentially for large $r$ .",
"We now turn to $\\mathrm {index}H$ for the vortex background.",
"The covariant derivative $D_i\\hat{\\phi }_a$ is given by $D_i\\hat{\\phi }_a&=&h^{\\prime }\\frac{x^ix^a}{r^2}+\\epsilon _{ij}\\epsilon _{ab}hk\\frac{x^jx^b}{r^3}.$ It decays exponentially at $r\\rightarrow \\infty $ , so does the chiral current $\\ast J$ .",
"We see that $\\mathrm {index}H$ coincides with the topological index $c_2$ since the chiral current has no contribution to the index.",
"It is easy to compute $c_2$ .",
"The field strength can be found as $F_{abij}&=&-\\epsilon _{ab}\\epsilon _{ij}\\frac{k^{\\prime }}{r} .$ Eq.",
"(REF ) then immediately gives $\\mathrm {index}H=c_2=-1$ .",
"This might be felt contradictive with the result of Sect.",
"that $\\mathrm {index}H$ is determined completely by the asymptotic behavior of the Higgs field.",
"It is of course not the case.",
"The gauge field is related with the Higgs field by the field equations.",
"In particular the chiral current vanishes exponentially as $r\\rightarrow \\infty $ .",
"Therefore, the contribution from the Higgs current, the first term on the rhs of (REF ), cancels that from the gauge current, the second term of the same equation, which in turn exactly cancels the topological index of the gauge field strength (REF ).",
"This implies that the topological invariant (REF ) coincides with the topological index.",
"The nonvanishing index suggests that the Hamiltonian given by Eq.",
"(REF ) has one negative chirality zero mode.",
"For the vortex background we can find zero mode of $H$ explicitly.",
"To see this we employ the following set of $\\gamma $ matrices $\\gamma ^j=\\sigma ^j\\otimes \\sigma ^1, \\qquad \\Gamma ^1=\\sigma ^3\\otimes \\sigma ^1, \\qquad \\Gamma ^2=1\\otimes \\sigma ^2.",
"\\qquad (j=1,2)$ The $\\mathrm {spin}(2)$ generator and chiral matrix is given by $\\Sigma _3=\\Sigma ^{12}=\\frac{i}{2}\\sigma ^3\\otimes \\sigma ^3, \\qquad \\gamma _5=(-i)^2\\gamma ^1\\gamma ^2\\Gamma ^1\\Gamma ^2=1\\otimes \\sigma ^3.$ Zero mode wave function can be chosen to be chiral.",
"We write it in chiral spinors $\\psi _\\pm $ as $\\psi _\\pm (x)&=&\\left(\\begin{matrix}u_\\pm (x) \\\\ v_\\pm (x)\\end{matrix}\\right)\\otimes \\chi _\\pm ,$ where $\\chi _\\pm $ are two component eigenspinors with $\\sigma ^3\\chi _\\pm =\\pm \\chi _\\pm $ .",
"Each component of the chiral zero mode satisfies in polar coordinates $\\left\\lbrace -ie^{i\\theta }\\left(\\partial _r+\\frac{i}{r}\\partial _\\theta \\right)\\pm i\\frac{1-k}{r}e^{i\\theta }\\right\\rbrace u_\\pm -g|\\phi _0|he^{\\mp i\\theta }v_\\pm &=&0, \\nonumber \\\\g|\\phi _0|he^{\\pm i\\theta }u_\\pm +\\left\\lbrace -ie^{-i\\theta }\\left(\\partial _r-\\frac{i}{r}\\partial _\\theta \\right)\\pm i\\frac{1-k}{r}e^{-i\\theta }\\right\\rbrace v_\\pm &=&0,$ where we have introduced Yukawa coupling constant $g$ by replacing $\\phi $ with $g\\phi $ in Eq.",
"(REF ).",
"The index (REF ) is not affected by this change as mentioned in Sect.",
".",
"For the negative chirality zero mode we can assume that $u_-$ and $v_-$ are independent of $\\theta $ .",
"It is easy to check that a normalizable solution is given by $u_-=-i \\mathrm {sgn}(g)v_-=C_0\\exp \\left[-\\int _0^r\\left(|g\\phi _0|h(r^{\\prime })+\\frac{1-k(r^{\\prime })}{r^{\\prime }}\\right)dr^{\\prime }\\right],$ where $C_0$ is a normalization constant.",
"In Figure REF we give a plot for the profile of $u_-(r)$ together with $h(r)$ and $k(r)$ .",
"The zero mode wave function is localized around the vortex core.",
"Figure: Profiles of h(r)h(r), k(r)k(r) and u - (r)u_-(r) with λ=κ\\lambda =\\kappa and g=e/2g=e/2.As for the positive chirality zero mode, we can separate the angle variable by assuming $u_+(x)=f_m(r)e^{im\\theta }, \\qquad v_+(x)=ig_m(r)e^{i(m+2)\\theta },$ where $m$ is an integer.",
"$f_m$ and $g_m$ satisfy $&&\\left(\\frac{d}{dr}-\\frac{m}{r}-\\frac{1-k}{r}\\right)f_m+g|\\phi _0|hg_m=0, \\nonumber \\\\&&\\left(\\frac{d}{dr}+\\frac{m+2}{r}-\\frac{1-k}{r}\\right)g_m+g|\\phi _0|hf_m=0.$ These lead to the behaviors $f_m\\sim r^m$ and $g_m\\sim r^{-m-2}$ as $r\\rightarrow 0$ .",
"We see that only one of the two independent solution is regular at the origin.",
"Such a regular solution, however, contains exponentially growing component $\\sim e^{|\\phi _0|r}$ as $r\\rightarrow \\infty $ .",
"We thus conclude that there is no positive chirality zero mode."
],
[
"'t Hooft-Polyakov monopole",
"Next we consider a Yang-Mills-Higgs system with $\\mathrm {spin}(3)$ gauge symmetry in three spatial dimensions.",
"The ansatz for the monopole of unit magnetic charge is given by $\\hat{\\phi }_a(x)=h(r)\\frac{x^a}{r}, \\qquad A_{abi}(x)=-(1-k(r))\\frac{\\delta _{ia}x^b-\\delta _{ib}x^a}{r^2}.$ Eqs.",
"(REF ) are satisfied if $h(r)$ and $k(r)$ obey the following differential equations $h^{\\prime \\prime }+\\frac{2}{r}h^{\\prime }=\\frac{2}{r^2}k^2h-\\lambda (1-h^2)h, \\qquad k^{\\prime \\prime }=\\kappa h^2k-\\frac{1}{r^2}(1-k^2)k.$ The boundary conditions for $h(r)$ and $k(r)$ take the same form as Eq.",
"(REF ) for the vortex.",
"From Eqs.",
"(REF ) we see the asymptotic behavior $k \\sim e^{-\\sqrt{\\kappa }r}$ and $1-h\\sim e^{-\\sqrt{2\\lambda } r}$ for sufficiently large $r$ .",
"No analytic solution is not known for Eqs.",
"(REF ).",
"See Ref.",
"[15] for a recent high precision numerical study.",
"It is straightforward to evaluate $\\mathrm {index}H$ .",
"In odd dimensions only chiral current Eq.",
"(REF ) contributes to the index.",
"Note that the covariant derivatives $D_i\\hat{\\phi }$ decays exponentially as $r\\rightarrow \\infty $ and the field strength approaches $F_{abij}\\rightarrow \\frac{\\delta _{ia}\\delta _{jb}}{r^2}-\\frac{\\delta _{ia}x^jx^b-\\delta _{ib}x^jx^a}{r^4}-(i\\leftrightarrow j).$ Keeping terms that survive at $r\\rightarrow \\infty $ , we obtain $J^i\\approx -\\frac{1}{8\\pi }\\mathrm {sgn}(g)\\epsilon ^{ijk}\\epsilon ^{abc}\\hat{\\phi }_aF_{jkbc}=-\\mathrm {sgn}(g)\\frac{x^i}{2\\pi r^3},$ where the overall factor $\\mathrm {sgn}(g)$ comes from the introduction of the Yukawa coupling constant.",
"In odd dimensions the index depends on the sign of $g$ .",
"This immediately gives $\\mathrm {index}H=\\mathrm {sgn}(g)$ .",
"As in the vortex case, it is also possible to obtain the same result by computing the topological invariant (REF ).",
"The index obtaind above implies the existence of a zero mode of chirality $\\mathrm {sgn}(g)$ .",
"For the monopole background Eq.",
"(REF ) it is also possible to find the wave function for the zero mode.",
"We employ the following representation of the $\\gamma $ matrices $\\gamma ^i=\\sigma ^i\\otimes 1\\otimes \\sigma ^1, \\qquad \\Gamma ^a=1\\otimes \\sigma ^a\\otimes \\sigma ^2.",
"\\qquad (i,a=1,2,3)$ The $\\mathrm {spin}(3)$ generators $\\Sigma _a=\\frac{1}{2}\\epsilon _{abc}\\Sigma ^{ab}$ and the chiral matrix $\\gamma _7$ are given by $\\Sigma _a=\\frac{i}{2}1\\otimes \\sigma ^a\\otimes 1, \\qquad \\gamma _7=1\\otimes 1\\otimes \\sigma ^3.$ Let us denote the zero mode wave function by chiral components as $\\psi &=&\\left(\\begin{matrix}\\psi _+ \\\\ \\psi _-\\end{matrix}\\right).$ Then $\\psi _\\pm $ must satisfy $\\left(-i\\sigma ^j\\otimes 1\\partial _j+\\frac{1}{2}\\sigma ^j\\otimes \\sigma ^aA_{aj}\\pm ig1\\otimes \\sigma ^a\\phi _a\\right)\\psi _\\pm &=&0.$ where $A_{ai}$ is defined by $A_{ai}=\\frac{1}{2}\\epsilon _{abc}A_{bci}$ .",
"These can be cast into $2\\times 2$ matrix equations by noting $(A\\otimes B\\psi _\\pm )_{\\alpha \\beta }=A_{\\alpha \\gamma }B_{\\beta \\delta }(\\psi _\\pm )_{\\gamma \\delta }=(A\\psi _\\pm B^T)_{\\alpha \\beta }$ .",
"With this notation Eq.",
"(REF ) for the monopole background Eq.",
"(REF ) can be expressed as $-i\\sigma ^j\\partial _j\\Psi _\\pm -\\epsilon _{jab}(1-k)\\frac{x^b}{2r^2}\\sigma ^j\\Psi _\\pm \\sigma ^a\\mp ig|\\phi _0|h \\frac{x^a}{r}\\Psi _\\pm \\sigma ^a&=&0.$ where $\\Psi _\\pm $ are defined by $\\Psi _\\pm =i\\psi _\\pm \\sigma ^2$ .",
"These have spherically symmetric solutions $(\\Psi _\\pm (x))_{\\alpha \\beta }&=&F_\\pm (r)\\delta _{\\alpha \\beta },$ where $F_\\pm $ satisfy $F_\\pm ^{\\prime }(r)&=&-\\left(\\pm g|\\phi _0|h(r)+\\frac{1-k(r)}{r}\\right)F_\\pm (r).$ For $g>0$ the negative chiral component $F_-$ must vanish, otherwise it grows exponentially as $r\\rightarrow \\infty $ .",
"We thus arrive at the normalizable positive chiral zero mode $&&\\psi =\\left(\\begin{matrix}iF_+(r)\\sigma ^2 \\\\ 0\\end{matrix}\\right),$ where $F_+$ is given by $F_+&=&C_0\\exp \\left[-\\int _0^r\\left(g|\\phi _0|h(r^{\\prime })+\\frac{1-k(r^{\\prime })}{r^{\\prime }}\\right)dr^{\\prime }\\right].$ Again $C_0$ is a normalization constant.",
"The zero mode is localized around the monopole and the wave function decays exponentially for $r\\rightarrow \\infty $ .",
"In Figure REF we give a plot of $F_+(r)$ together with $h(r)$ and $k(r)$ .",
"The case of $g<0$ can be analyzed similarly.",
"We obtain one normalizable zero mode with negative chirality.",
"This is consistent with the index theorem.",
"Figure: Profiles of h(r)h(r), k(r)k(r) and F + (r)F_+(r) for the monopolebackground with λ=κ/2\\lambda =\\kappa /2 and g=e/2g=e/2."
],
[
"Summary and Discussion",
"We have evaluated the index of BdGH of a gauged topological insulator or Yang-Mills-Higgs system in arbitrary dimensions by regarding the Higgs and Yang-Mills fields as external backgrounds, which can be set up independently.",
"The index can be expressed as a surface integral of a gauge invariant chiral current plus topological index of the Yang-Mills fields.",
"In odd dimensions the topological index vanishes identically and the gauge field dependent terms of the chiral current can be gathered into a total derivative at spatial infinities, giving no contribution to the index.",
"In even dimensions the gauge field dependent terms of the chiral current can be converted into a total derivative plus the Chern-Simons form, which exactly cancels the topological index.",
"We have thus shown that the index of the BdGH is determined solely by the asymptotic behavior of the Higgs fields whatever topological charge the Yang-Mills field carries.",
"If the behavior of the Yang-Mills and Higgs fields are governed by some effective Hamiltonian, $D_i\\phi _a$ and $F_{ij}$ must decay faster than $|x|^{-d/2}$ for $|x|\\rightarrow \\infty $ to ensure the finiteness of the Hamiltonian.",
"In such systems nonvanishing topological invariant can be obtained only in spatial dimensions less than four.",
"In two dimensions the index of the BdGH is saturated by the topological index.",
"In three dimensions only the $\\phi F$ term of the chiral current contributes to the index.",
"This apparently contradicts to the general conclusion that the Yang-Mills field does not contribute to the index.",
"It is , however, possible to have expressions for the index only in terms of $\\phi $ by noting that the gauge fields are related with the Higgs fields by $D_j\\hat{\\phi }_a=0$ at the spatial infinities.",
"We have considered the case of the BdGH containing $d$ order parameters from the beginning.",
"It is possible to consider other systems with less or more order parameters.",
"The evaluation of index of the corresponding BdGH is straightforward.",
"It is rather obvious from our explicit calculations that one cannot obtain nontrivial index unless the number of the order parameters coincides with the spatial dimensions.",
"Our result is consistent with the the topological classification by the Chern number computed from the Berry connection of the Bloch wave functions."
],
[
"Acknowledgements",
"This work is supported in part by the Grant-in-Aid for Scientific Research (No.",
"21540378) from the Japan Society for the Promotion of Science (JSPS) and by the “Topological Quantum Phenomena” Grant-in Aid for Scientific Research on Innovative Areas (No.",
"23103502) from the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT)."
]
] | 1204.0988 |
[
[
"Riemann compatible tensors"
],
[
"Abstract Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists.",
"Several properties are shown to remain valid in this broader setting.",
"Riemann compatibility is equivalent to the Bianchi identity of the new \"Codazzi deviation tensor\" with a geometric significance.",
"Examples are given of manifolds with Riemann-compatible tensors, in particular those generated by geodesic mapping.",
"Compatibility is extended to generalized curvature tensors with an application to Weyl's tensor and general relativity."
],
[
"Introduction",
"The Riemann tensor $R_{ijk}{}^m$ and its contractions, $R_{kl}=R_{kml}{}^m$ and $R=g^{kl}R_{kl}$ , are the fundamental tensors to describe the local structure of a Riemannian manifold $(M_n,g)$ of dimension $n$ .",
"In a remarkable theorem [10], [3] Derdzinski and Shen showed that the existence of a non trivial Codazzi tensor poses strong constraints on the structure of the Riemann tensor.",
"Because of their geometric relevance, Codazzi tensors have been studied by several authors, as Berger and Ebin [1], Bourguignon [4], Derdzinski [8], [9], Derdzinski and Shen [10], Ferus [11], Simon [29]; a compendium of results is found in Besse's book [3].",
"Recently, we showed [22] that the Codazzi differential condition $\\nabla _i b_{jk}-\\nabla _j b_{ik}=0$ is sufficient for the theorem to hold, and can be replaced by the more general notion of Riemann-compatibility, which is instead algebraic: Definition 1.1 A symmetric tensor $b_{ij}$ is Riemann compatible ($R$ -compatible) if: $b_{im} R_{jkl}{}^m+b_{jm} R_{kil}{}^m+b_{km} R_{ijl}{}^m=0.$ With this requirement, we proved the following extension of Derdzinski-Shen's theorem: Theorem 1.2 [22] Suppose that a symmetric $R$ -compatible tensor $b_{ij}$ exists.",
"Then, if $X$ , $Y$ and $Z$ are three eigenvectors of the matrix $b_r{}^s$ at a point of the manifold, with eigenvalues $\\lambda $ , $\\mu $ and $\\nu $ , it is $R_{ijkl}\\,X^i \\, Y^j\\, Z^k \\, =0 $ provided that both $\\lambda $ and $\\mu $ are different from $\\nu $ .",
"The concept of compatibility allows for a further extension of the theorem, where the Riemann tensor $R$ is replaced by a generalized curvature tensor $K$ , and $b$ is required to be $K$ -compatible [22].",
"This paper studies the properties of Riemann compatibility, and its implications on the geometry of the manifold.",
"In section 2 $R$ -compatibility is shown to be equivalent to the Bianchi identity of a new tensor, the Codazzi deviation.",
"In section 3 the irreducible components of the covariant derivative of a symmetric tensor are classified in a simple manner, based on the decomposition into traceless terms.",
"This is of guidance in the study of different structures suited for $R$ -compatibility.",
"The general properties of Riemann compatibility are presented in section 4.",
"In section 5 several properties of manifolds in presence of a Riemann compatible tensor that were obtained by Derdzinsky-Shen and Bourguignon for manifolds with a Codazzi tensor, are recovered.",
"In particular, it is shown that $R$ -compatibility implies pureness, a property of the Riemann tensor introduced by Maillot that implies the vanishing of Pontryagin forms.",
"Manifolds that display $R$ -compatible tensors are presented in section 6; interesting examples are generated by geodesic mappings, that induce metric tensors that are $R$ -compatible.",
"Finally, in section 7, $K$ -tensors and $K$ -compatibility are presented, with applications to the standard curvature tensors.",
"In the end, an application to general relativity is mentioned, that will be discussed fully elsewhere."
],
[
"The Codazzi deviation tensor and R-compatibility",
"Since Codazzi tensors are Riemann compatible, for a non Codazzi differentiable simmetric tensor field $b$ it is useful to define its deviation from the Codazzi condition.",
"This tensor solves an unexpected relation that generalizes Lovelock's identity for the Riemann tensor, and shows that Riemann compatibility is a condition for closedness of certain 2-forms.",
"Definition 2.1 The Codazzi deviation of a symmetric tensor $b_{kl}$ is $C_{jkl}=:\\nabla _j b_{kl}-\\nabla _k b_{jl}$ Simple properties are: $C_{jkl}=-C_{kjl}$ and $C_{jkl}+C_{klj}+C_{ljk} =0$ .",
"The following identity holds in general, and relates the Bianchi differential combination for $C$ to the Riemann compatibility of $b$ : Proposition 2.2 $\\nabla _iC_{jkl}+\\nabla _jC_{kil}+\\nabla _kC_{ijl} =b_{im} R_{jkl}{}^m+b_{jm} R_{kil}{}^m+b_{km} R_{ijl}{}^m$ $&\\nabla _iC_{jkl}+\\nabla _jC_{kil}+\\nabla _kC_{ijl}=[\\nabla _i,\\nabla _j]b_{kl}+[\\nabla _k,\\nabla _i]b_{jl}+[\\nabla _j,\\nabla _k]b_{il}\\\\&= b_{ml}(R_{ijk}{}^m+R_{kij}{}^m+R_{jki}{}^m)+b_{im} R_{jkl}{}^m+b_{jm} R_{kil}{}^m+b_{km} R_{ijl}{}^m$ the first term vanishes by the first Bianchi identity.",
"Remark 2.3 The identity holds true if $b_{ij}$ is replaced by $b^{\\prime }_{ij}=b_{ij}+\\chi a_{ij}$ , where $a_{ij}$ is a Codazzi tensor and $\\chi $ a scalar field.",
"Then: $C^{\\prime }_{jkl}=C_{jkl} -(a_{kl}\\nabla _j -a_{jl} \\nabla _k)\\chi $ .",
"The deviation tensor is associated to the 2-form $C_l=\\frac{1}{2}C_{jkl}dx^j\\wedge dx^k$ .",
"The closedness condition $0=DC_l =\\frac{1}{2}\\nabla _iC_{jkl}dx^i\\wedge dx^j\\wedge dx^k$ ($D$ is the exterior covariant derivative) is the second Bianchi identity for the Codazzi deviation: $\\nabla _iC_{jkl}+\\nabla _jC_{kil}+\\nabla _kC_{ijl} =0 $ .",
"This gives a geometric picture of Riemann compatibility: Theorem 2.4 $b_{ij}$ is Riemann compatible if and only if $DC_l =0 $ .",
"Remark 2.5 The Codazzi deviation of the Ricci tensor is, by the contracted second Bianchi identity: $ C_{jkl} =: \\nabla _jR_{kl}-\\nabla _k R_{jl} = -\\nabla _m R_{jkl}{}^m $ .",
"For the Ricci tensor the identity (REF ) identifies with Lovelock's identity [18] for the Riemann tensor: $&&\\nabla _i\\nabla _m R_{jkl}{}^m +\\nabla _j\\nabla _m R_{kil}{}^m +\\nabla _k\\nabla _mR_{ijl}{}^m \\\\&&\\qquad = - R_{im} R_{jkl}{}^m - R_{jm} R_{kil}{}^m - R_{km} R_{ijl}{}^m .\\nonumber $ A Veblen-like identity holds, that corresponds to (REF ) (For $b_{ij}=R_{ij}$ it specializes to Veblen's identity for the divergence of the Riemann tensor [20]): Proposition 2.6 $\\nabla _i C_{jlk} + \\nabla _j C_{kil} +\\nabla _k C_{lji} + \\nabla _l C_{ikj}\\\\= b_{im} R_{jlk}{}^ m+ b_{jm} R_{kil}{}^m + b_{km} R_{lji}{}^m + b_{lm} R_{ikj}{}^m\\nonumber $ Write four equations (REF ) with cycled indices $i,j,k,l$ and sum up.",
"Then simplify by means of the first Bianchi identity for the Riemann tensor and the cyclic identity $C_{jkl} + C_{klj} + C_{ljk} = 0$ ."
],
[
"Irreducible components for $\\nabla _j b_{kl}$ and {{formula:ba045b3c-9306-4f4a-9f70-d2427e4a6aaa}} -compatibility",
"We begin with a simple procedure to classify the $O(n)$ invariant components of the tensor $\\nabla _j b_{kl}$ .",
"They will guide us in the study of $R$ -compatibility.",
"If $b$ is the Ricci tensor, this simple construction reproduces the seven equations linear in $\\nabla _iR_{jk}$ , invariant for the $O(n)$ group, that are enumerated and discussed in Besse's treatise “Einstein Manifolds” [3].",
"For a simmetric tensor $b_{kl}$ with $\\nabla _jb_{kl}\\ne 0$ , the tensor $\\nabla _jb_{kl}$ can be decomposed into $O(n)$ invariant terms, where $B^0_{jkl}$ is traceless ($B^0_{jk}{}^j=B^0_{kj}{}^j=0$ ) [14], [17]: $\\nabla _jb_{kl}= B^0_{jkl}+ A_j g_{kl}+B_kg_{jl}+B_l g_{jk}\\\\A_j=\\frac{(n+1)\\nabla _j b^m{}_m-2\\nabla _mb^m{}_j}{n^2+n-2},\\qquad B_j=-\\frac{\\nabla _j b^m{}_m-n\\nabla _mb^m{}_j}{n^2+n-2}$ The traceless tensor can then be written as a sum of orthogonal components [18]: $B^0_{jkl}= \\frac{1}{3}\\left[B^0_{jkl}+B^0_{klj}+B^0_{ljk}\\right]+ \\frac{1}{3}\\left[B^0_{jkl}-B^0_{kjl}\\right]+\\frac{1}{3}\\left[B^0_{jlk}-B^0_{ljk}\\right]$ The orthogonal subspaces classify the $O(n)$ invariant equations that are linear in $\\nabla _jb_{kl}$ .",
"The trivial subspace: $\\nabla _jb_{kl}=0$ .",
"The subspace $\\mathcal {I}$ (we follow Gray's notation, [13]) where $B^0_{jkl}=0$ : $ \\nabla _j b_{kl}= A_j g_{kl}+B_kg_{jl}+B_l g_{jk}.",
"$ The complement $\\mathcal {I}^\\perp $ is characterized by $A_j,B_j=0$ i.e.",
"$\\nabla _j b_{kl}$ is traceless.",
"This gives two invariant equations: $\\nabla _jb^j{}_l=0$ , and $\\nabla _jb^m{}_m=0$ .",
"Since $\\nabla _jb_{kl}= B^0_{jkl}$ , the structure of $B^0$ specifies two orthogonal subspaces $\\mathcal {I}^\\perp =\\mathcal {A}\\oplus \\mathcal {B}$ .",
"In $\\mathcal {A}$ : $ \\nabla _jb_{kl}+\\nabla _k b_{lj}+\\nabla _lb_{jk}=0.",
"$ In $\\mathcal {B}$ : $ \\nabla _jb_{kl}-\\nabla _k b_{jl}=0.",
"$ The subspace $\\mathcal {I}\\oplus \\mathcal {A}$ contains tensors with traceless part $\\nabla _jb_{kl}-A_j g_{kl}-B_kg_{jl}-B_l g_{jk}$ that solves the cyclic condition: $[\\nabla _jb_{kl}-\\frac{1}{n+2}(\\nabla _jb^m{}_m +2\\nabla _mb^m{}_j)g_{kl}]+cyclic =0.$ The subspace $\\mathcal {I}\\oplus \\mathcal {B}$ contains tensors with traceless part that solves the Codazzi condition: $[\\nabla _jb_{kl}-\\frac{1}{n-1}(\\nabla _jb^m{}_m -\\nabla _mb^m{}_j)g_{kl}]=[\\nabla _kb_{jl}-\\frac{1}{n-1}(\\nabla _kb^m{}_m -\\nabla _mb^m{}_k)g_{jl}] $ Accordingly, the Codazzi deviation tensor has the (unique) decomposition in irreducible components $C_{jkl}= C^0_{jkl}+\\lambda _jg_{kl}-\\lambda _k g_{jl} ,\\qquad \\lambda _j=A_j-B_j=\\frac{\\nabla _j b^m{}_m-\\nabla _mb^m{}_j}{n-1}$ where $C^0$ is traceless.",
"Eq.",
"(REF ) becomes $b_{im} R_{jkl}{}^m + b_{jm} R_{kil}{}^m + b_{km}R_{ijl}{}^m=\\nabla _iC^0_{jkl}+\\nabla _jC^0_{kil}+\\nabla _kC^0_{ijl}\\\\+g_{il}(\\nabla _j\\lambda _k-\\nabla _k\\lambda _j)+g_{jl}(\\nabla _k\\lambda _i-\\nabla _i\\lambda _k)+g_{kl}(\\nabla _i\\lambda _j-\\nabla _j\\lambda _i)\\nonumber $ There are only two orthogonal invariant cases: - $C^0_{jkl}=0$ , then $b$ is $R-$ compatible if and only if $\\lambda $ is closed.",
"If $b$ is the Ricci tensor, this requirement gives Nearly conformally symmetric $(NCS)_n$ manifolds, that were introduced by Roter [28].",
"-$\\nabla _j b^m{}_m-\\nabla _mb^m{}_j=0$ then $b$ is $R-$ compatible if and only if $C=C^0$ solves the second Bianchi dentity.",
"If $b$ is the Ricci tensor, this corresponds to $\\nabla _j R=0$ .",
"Remark 3.1 The decomposition (REF ) for the deviation of the Ricci tensor turns out to be $C_{jkl}= -\\frac{n-2}{n-3} \\nabla _m C_{jkl}{}^m+\\frac{1}{2(n-1)}\\left[g_{kl}\\nabla _j R -g_{jl} \\nabla _k R\\right]$ where $C_{jkl}{}^m$ is the conformal curvature tensor, or Weyl's tensor.",
"In this case the $\\lambda $ covector is closed."
],
[
"Riemann compatibility: general properties",
"The existence of a Riemann compatible tensor has various implications.",
"A first one is the existence of a generalized curvature tensor.",
"This leads to the generalization of Derdzinski-Shen theorem and other relations that were obtained for Codazzi tensors.",
"We need the definition, from Kobayashi and Nomizu's book [16]: Definition 4.1 A tensor $K_{ijlm}$ is a generalized curvature tensor (or, briefly, a $K$ -tensor) if it has the symmetries of the Riemann curvature tensor: a) $K_{ijkl} =-K_{jikl} =-K_{ijlk}$ , b) $K_{ijkl} =K_{klij}$ , c) $K_{ijkl} +K_{jkil} +K_{kijl} =0$ (first Bianchi identity).",
"It follows that the tensor $K_{jk}=: -K_{mjk}{}^m$ is symmetric.",
"Theorem 4.2 If $b$ is $R$ -compatible then $K_{ijkl}=:R_{ijpq}b^p{}_kb^q{}_l$ is a $K$ -tensor.",
"a) For example: $K_{ijlk}= R_{ijrs}b_l{}^r b_k{}^s = R_{ijsr} b_l{}^s b_k{}^r=- R_{ijrs}b_l{}^s b_k{}^r =-K_{ijkl}$ .",
"Property c) follows from (REF ): $K_{ijkl} +K_{jkil} +K_{kijl} =R_{ijrs}b_k{}^r b_l{}^s + R_{jkrs} b_i{}^r b_l{}^s + R_{kirs} b_j{}^r b_l{}^s= ( R_{jis}{}^r b_{kr}+ R_{kjs}{}^r b_{ir}+ R_{iks}{}^r b_{jr}) b_l{}^s=0 $ .",
"Property b) follows from c): $K_{ijkl} +K_{jkil} +K_{kijl} =0$ .",
"Sum the identity over cyclic permutations of all indices $i,j,k,l$ and use the symmetries a).",
"It is easy to see that a first Bianchi identity holds also for the last three indices: $K_{ijkl} +K_{iklj} +K_{iljk} =0$ .",
"The next result remarks the relevance of the local basis of eigenvectors of the Ricci tensor.",
"Another symmetric contraction of the Riemann tensor was introduced by Bourguignon [4]: $\\text{R̊}_{ij}=: b^{pq}R_{pijq}.$ Theorem 4.3 If $b$ is $R$ -compatible then: 1) $b_{im} R_j{}^m-b_{jm} R_i{}^m=0$ , 2) $b_{im}\\text{R̊}_j{}^m - b_{jm}\\text{R̊}_i{}^m =0$ The first identity is proven by transvecting (REF ) with $g^{kl}$ .",
"The second one is a restatement of the symmetry of the tensor $K_{ij}$ .",
"Remark 4.4 A) Identies 1 and 2 are here obtained directly from $R$ -compatibility.",
"Bourguignon [4] obtained them from Weitzenböck's formula for Codazzi tensors, and Derdzinski and Shen [10] from their theorem.",
"B) As the symmetric matrices $b_{ij}$ , $R_{ij}$ , R̊$_{ij}$ commute, they share at each point of the manifold an orthonormal set of $n$ eigenvectors.",
"C) If $b^{\\prime }$ is a symmetric tensor that commutes with a Riemann compatible $b$ , then it can be shown that R̊$^{\\prime }_{ij}=: b^{\\prime pq}R_{pijq}$ commutes with $b$ .",
"Finally, this Veblen-type identity holds: Proposition 4.5 If $b$ is $R$ -compatible, then: $b_{im}R_{jlk}{}^m +b_{jm}R_{kil}{}^m +b_{km}R_{lji}{}^m +b_{lm}R_{ikj}{}^m = 0$ Write four equations (REF ) with cycled indices $i,j,k,l$ and sum up, and use the first Bianchi identity."
],
[
"Pure Riemann tensors and Pontryagin forms",
"Riemann compatibility and nondegeneracy of the eigenvalues of $b$ imply directly that the Riemann tensor is pure and Pontryagin forms vanish.",
"We quote two results from Maillot's paper [19]: Definition 5.1 In a Riemann manifold $M_n$ , the Riemann curvature tensor is pure if at each point of the manifold there is an orthonormal basis of $n$ tangent vectors $X(1),\\ldots ,X(n)$ , $X(a)^iX(b)_i = \\delta _{ab}$ , such that the tensors $X(a)^i\\wedge X(b)^j=: X(a)^iX(b)^j-X(a)^jX(b)^i$ , $a<b$ , diagonalize it: $R_{ij}{}^{lm} X(a)^i\\wedge X(b)^j = \\lambda _{ab} X(a)^l\\wedge X(b)^m$ Theorem 5.2 If a Riemannian manifold has pure Riemann curvature tensor, then all Pontryagin forms vanish.",
"Consider the maps on tangent vectors, built with the Riemann tensor, $&\\omega _4(X_1\\ldots X_4)=R_{ija}{}^bR_{klb}{}^a(X_1^i\\wedge X_2^j)(X_3^k\\wedge X_4^l), \\\\&\\omega _8(X_1\\ldots X_8)=R_{ija}{}^bR_{klb}{}^c R_{mnc}{}^dR_{pqd}{}^a(X_1^i\\wedge X_2^j)\\cdots (X_7^p\\wedge X_8^q),\\\\&\\ldots \\ldots $ They are antisymmetric under exchange of vectors in the single pairs, and for cyclic permutation of pairs.",
"The Pontryagin forms [26] $\\Omega _{4k}$ result from total antisymmetrization of $\\omega _{4k}$ : $ \\Omega _{4k} (X_1\\ldots X_{4k})=\\sum _P (-1)^P \\omega _{4k}(X_{i_1}\\ldots X_{i_{4k}})$ where $P$ is the permutation taking $(1\\ldots 4k)$ to $(i_1\\ldots i_{4k})$ .",
"$\\Omega _{4k}=0$ if two vectors repeat, intermediate forms $\\Omega _{4k-2}$ vanish identically.",
"Pontryagin forms on generic tangent vectors are linear combinations of forms evaluated on basis vectors.",
"If the Riemann tensor is pure, all Pontryagin forms on the basis of eigenvectors of the Riemann tensor vanish.",
"For example, if $X,Y,Z,W$ are orthogonal: $\\omega _4 (XYZW)= \\lambda _{XY}\\lambda _{ZW}(X^a\\wedge Y^b)(Z_b\\wedge W_a)=0 $ and $\\Omega _4(XYZU)=0$ .",
"A consequence of the extended Derdzinski-Shen theorem REF is the following: Theorem 5.3 If a symmetric tensor field $b_{ij}$ exists, that is $R$ -compatible and has distinct eigenvalues at each point of the manifold, then the Riemann tensor is pure and all Pontryagin forms vanish.",
"At each point of the manifold the symmetric matrix $b_{ij}(x)$ is diagonalized by $n$ tangent orthonormal vectors $X(a)$ , with distinct eigenvalues.",
"Since $b$ is $R$ -compatible, theorem REF holds and, because of antisymmetry of $R$ in first two indices: $0=R_{ij}{}^{kl}X(a)^i\\wedge X(b)^jX(c)_k, \\quad a\\ne b\\ne c. $ This means that all column vectors of the matrix $V(a,b)^{kl}=R_{ij}{}^{kl}X(a)^i\\wedge X(b)^j$ are orthogonal to vectors $X(c)$ i.e.",
"they belong to the subspace spanned by $X(a)$ and $X(b)$ .",
"Because of antisymmetry in indices $k,l$ , it is necessarily $V(a,b)=\\lambda _{ab} X(a)\\wedge X(b)$ , i.e.",
"the Riemann tensor is pure.",
"This property has been checked by Petersen [27] in various examples with rotationally invariant metrics, by giving explicit orthonormal frames such that $R(e_i,e_j)e_k=0$ ."
],
[
"Two and three dimensional manifolds",
"Riemannian manifolds of dimension $n=2$ and $n=3$ are special, as the Riemann tensor is expressible in terms of the Ricci and metric tensors.",
"Therefore, Riemann-compatibility and ensuing pureness of the Riemann tensor can be established by simple means.",
"$n=2$ ) $R_{jklm}= R_{jl}g_{km}-g_{jm} R_{kj}$ .",
"Explicit evaluation proves that any symmetric tensor $b$ is Riemann compatible.",
"$n=3$ ) $R_{jklm}=g_{jl}R_{km}+g_{km}R_{jl}-g_{kl}R_{jm}-g_{jm}R_{kl}-\\frac{R}{2}(g_{jl}g_{km}-g_{jm}g_{kl})$ .",
"Then, for any symmetric tensor $b$ it is: $b_{im} R_{jkl}{}^m+b_{jm} R_{kil}{}^m+b_{km} R_{ijl}{}^m=g_{kl}(b_{jm}R_i{}^m -b_{im}R_j{}^m)\\\\+g_{il}(b_{km}R_j{}^m - b_{jm}R_k{}^m)+g_{jl}(b_{im}R_k{}^m - b_{km}R_i{}^m)$ Thus in $n=3$ the Ricci tensor is always $R$ -compatible.",
"Moreover, if $b$ commutes with the Ricci tensor, then $b$ is $R$ -compatible.",
"Since a symmetric tensor that commutes with the Ricci tensor can always be constructed, with arbitrarily chosen distinct eigenvalues, by theorem REF we conclude: Proposition 5.4 In Riemannian manifolds of dimension $n=2$ and $n=3$ the Riemann tensor is pure."
],
[
"Quasi-constant curvature spaces",
"The same conclusions can be drawn in any dimension $n$ for quasi-constant curvature spaces.",
"They were introduced by Chen and Yano [5] and are defined by a Riemann tensor with the following structure: $R_{jklm} =p[g_{mj}g_{kl}-g_{mk}g_{jl}]+q[g_{mj}t_kt_l -g_{mk}t_jt_l+g_{kl}t_mt_j-g_{jl}t_mt_k]$ $p$ and $q$ are scalar functions.",
"The first term describes constant curvature, the second one contains a vector field with $t_kt^k=1$ .",
"The following identity holds: $b_i{}^m R_{jklm} + b_j{}^m R_{kilm} + b_k{}^m R_{ijlm} =q[g_{kl}(t_jb_i{}^mt_m-t_ib_j{}^mt_m) \\\\+g_{il}(t_kb_j{}^mt_m-t_jb_k{}^mt_m)+g_{jl}(t_ib_k{}^mt_m-t_kb_i{}^mt_m)]\\nonumber $ Contraction with $g^{kl}$ gives: $ -b_i{}^m R_{jm} + b_j{}^m R_{im} = q(n-2)(t_jb_i{}^mt_m-t_ib_j{}^mt_m)$ .",
"Therefore, if $b$ commutes with the Ricci tensor and $n\\ne 2$ , the r.h.s.",
"is zero and, by (REF ), $b$ is $R$ -compatible.",
"Then the Riemann tensor is pure and all Pontryagin forms vanish."
],
[
"Structures for Riemann compatibility",
"Some differential structures are presented that yield Riemann compatibility.",
"Of particular interest are geodesic mappings, which leave the condition for $R$ -compatibility form-invariant, and generate $R$ -compatible tensors."
],
[
"Quasi Codazzi tensors",
"Let $b_{ij}$ be a symmetric tensor that solves the Codazzi condition deformed by a closed gauge field [22]: $(\\nabla _j - \\beta _j) b_{kl} = (\\nabla _k - \\beta _k) b_{jl}$ The Codazzi deviation is $C_{jkl}= \\beta _j b_{kl}- \\beta _k b_{jl}$ , and $b$ is $R$ -compatible.",
"Since $\\beta _i =\\nabla _i\\xi $ , the gauge field cancels for $b_{ij}=e^\\xi b^{\\prime }_{ij}$ , where $b^{\\prime }$ is a Codazzi tensor.",
"Of this type are Weakly $b$ -symmetric manifolds, defined by the recurrency $\\nabla _i b_{kl}= A_i b_{kl}+ B_k b_{il}+ D_l b_{ik} $ where $A$ , $B$ and $D$ are covector fields.",
"Eq.",
"(REF ) is obtained for $\\beta _i=A_i-B_i$ , and $b$ is Riemann compatible if $A-B$ is closed.",
"Examples are: Weakly Ricci-symmetric manifolds, where $b_{ij}=R_{ij}$ [20], [21], Weakly and pseudo Z-symmetric manifolds, where $b_{ij}$ is a $Z$ -tensor [21], [23].",
"Another example are manifolds with a recurrent generalized curvature tensor [20]: $\\nabla _iK_{jkl}{}^m= A_i K_{jkl}{}^m$ , then $b_{kl}=: K_{kml}{}^m\\ne 0$ has the form (REF )."
],
[
"Pseudo-$K$ symmetric manifolds",
"They are characterized by a generalized curvature tensor $K$ such that ([6], [24]) $\\nabla _iK_{jkl}{}^m = 2A_iK_{jkl}{}^m + A_j K_{ikl}{}^m + A_k K_{jil}{}^m+A_l K_{jki}{}^m +A^m K_{jkli},$ The tensor $b_{jk}=:K_{jmk}{}^m$ is symmetric.",
"It is $R-$ compatible if its Codazzi deviation $C_{ikl}= A_i b_{kl}-A_k b_{il} + 3A_m K_{ikl}{}^m $ fulfills the II Bianchi identity.",
"This is ensured by $A_m$ being concircular, i.e.",
"$\\nabla _iA_m=A_iA_m+\\gamma \\,g_{im}$ ."
],
[
"Generalized Weyl tensors",
"A Riemannian manifold is a $(NCS)_n$ [28] if the Ricci tensor satisfies $\\nabla _jR_{kl}-\\nabla _kR_{jl} = \\frac{1}{2(n-1)}[g_{kl}\\nabla _j\\, R-g_{jl}\\nabla _k\\,R ]$ .",
"As such, the Ricci tensor is the Weyl tensor, and the left hand side is its Codazzi deviation.",
"This condition, by (REF ), is equivalent to $\\nabla _mC_{jkl}{}^m=0$ .",
"This suggests a class of deviations of a symmetric tensor $b$ with $C^0_{jkl}=0$ in (REF ): $C_{jkl}=\\lambda _j g_{kl}-\\lambda _k g_{jl}$ Proposition 6.1 $b$ is $R$ -compatible if and only if $\\lambda _i$ is closed.",
"Transvect (REF ) with $g^{kl}$ and obtain: $ -b_i{}^m R_{jm} + b_j{}^m R_{im} =(n-2)(\\nabla _i\\lambda _j-\\nabla _j\\lambda _i)$ .",
"Then $b$ commutes with the Ricci tensor iff $\\lambda $ is closed and, by the previous equation, $b$ is $R$ -compatible.",
"An example is provided by spaces with $\\nabla _j b_{kl}=A_j g_{kl}+B_k g_{jl}+B_l g_{jk},$ where $C_{jkl}= \\lambda _jg_{kl}-\\lambda _kg_{jl}$ with $\\lambda _j=A_j-B_j$ .",
"Sinyukov manifolds [30] are of this sort, with $b_{ij}$ being the Ricci tensor itself."
],
[
"Geodesic mappings",
"Riemann compatible tensors arise naturally in the study of geodesic mappings, i.e.",
"mappings that preserve geodesic lines.",
"Their importance arise from the fact that Sinyukov manifolds are $(NCS)_n$ manifolds and they always admit a nontrivial geodesic mapping.",
"Geodesic mappings preserve Weyl's projective curvature tensor [30].",
"We show that they also preserve the form of the compatibility relation.",
"A map $f:\\; (M_n,g)\\rightarrow (M_n,\\overline{g})$ is geodesic if and only if Christoffel symbols are linked by $\\overline{\\Gamma }_{ij}^k = \\Gamma _{ij}^k +\\delta _i^k X_j + \\delta _j^k X_i$ where, on a Riemannian manifold, $X$ is closed ($\\nabla _iX_j= \\nabla _jX_i$ ).",
"The condition is equivalent to: $\\nabla _k \\overline{g}_{jl} = 2X_k\\overline{g}_{jl} + X_j\\overline{g}_{kl}+X_l\\overline{g}_{kj}$ which has the form (REF ).",
"The corresponding relation among Riemann tensors is $\\overline{R}_{jkl}{}^m = R_{jkl}{}^m +\\delta _j^m P_{kl} - \\delta _k^m P_{jl}$ where $P_{kl}=\\nabla _kX_l-X_kX_l$ is the deformation tensor.",
"The symmetry $P_{kl}=P_{lk}$ is ensured by closedness of $X$ .",
"Proposition 6.2 Geodesic mappings preserve $R$ -compatibility $b_{im} \\overline{R}_{jkl}{}^m+b_{jm} \\overline{R}_{kil}{}^m+b_{km} \\overline{R}_{ijl}{}^m =b_{im} R_{jkl}{}^m+b_{jm} R_{kil}{}^m+b_{km} R_{ijl}{}^m$ where $b$ is a symmetric tensor.",
"Let's show that the difference of the two sides is zero.",
"Eq.",
"(REF ) gives: $b_{im} (\\delta _j^m P_{kl} - \\delta _k^m P_{jl}) +b_{jm}(\\delta _k^m P_{il} - \\delta _i^m P_{kl})+b_{km}(\\delta _i^m P_{jl} - \\delta _j^m P_{il})$ $=b_{ij}P_{kl} - b_{ik} P_{jl} +b_{jk} P_{il} - b_{ji}P_{kl}+b_{ki} P_{jl} - b_{kj} P_{il}=0$ Since $\\overline{g}$ is trivially $\\overline{R}$ -compatible (first Bianchi identity), form invariance implies: Corollary 6.3 $\\overline{g} $ is $R$ -compatible.",
"Sinyukov [30] (see also [25], [12]) showed that a manifold admits a geodesic mapping if and only if there are a scalar field $\\varphi $ and a symmetric non singular tensor $b_{ij}$ such that: $ \\nabla _k b_{jl} = g_{kl} \\nabla _j\\varphi + g_{kj}\\nabla _l \\varphi .",
"$ The Codazzi deviation of $b$ , $C_{jkl}= g_{kl}\\nabla _j\\varphi -g_{jl}\\nabla _k\\varphi $ , has the form (REF ).",
"Therefore $b$ is $R$ -compatible."
],
[
"Generalized curvature tensors.",
"Several results that are valid for the Riemann tensor with a Riemann compatible tensor, extend to generalized curvature tensors $K_{ijkl}$ (hereafter referred to as $K$ -tensors) with a $K$ -compatible symmetric tensor $b_{jk}$ .",
"The classical curvature tensors are $K$ -tensors.",
"The compatibility with the Ricci tensor is then examined.",
"Definition 7.1 A symmetric tensor $b_{ij}$ is K-compatible if $b_{im} K_{jkl}{}^m+b_{jm} K_{kil}{}^m+b_{km} K_{ijl}{}^m=0.$ The metric tensor is always $K$ -compatible, as (REF ) then coincides with the first Bianchi identity for $K$ .",
"Proposition 7.2 If $K_{ijlm}$ is a $K$ -tensor and $b_{kl}$ is $K$ -compatible, then $\\hat{K}_{ijkl} =: K_{ijrs} b_k{}^r b_l{}^s$ is a $K$ -tensor.",
"We quote without proof the extension of Derdzinski and Shen theorem for generalized curvature tensors [22]: Theorem 7.3 Suppose that $K_{ijkl}$ is a $K$ -tensor, and a symmetric $K$ -compatible tensor $b_{ij}$ exists.",
"Then, if $X$ , $Y$ and $Z$ are three eigenvectors of the matrix $b_r{}^s$ at a point $x$ of the manifold, with eigenvalues $\\lambda $ , $\\mu $ and $\\nu $ , it is $X^i \\, Y^j\\, Z^k \\, K_{ijkl} =0$ provided that both $\\lambda $ and $\\mu $ are different from $\\nu $ .",
"Proposition 7.4 If $b$ is $K-$ compatible, and $b$ commutes with a tensor $h$ , then the symmetric tensor K̊$_{kl}=: K_{jklm}h^{jm}$ commutes with $b$ .",
"Multiply relation of $K$ compatibility for $b$ by $h^{kl}$ .",
"The last term vanishes for symmetry.",
"The remaining terms give the null commutation relation.",
"In ref.",
"[20] (prop.2.4) we proved that a generalization of Lovelock's identity (REF ) holds for certain $K$ -tensors, that include all classical curvature tensors: Proposition 7.5 Let $K_{jkl}{}^m$ be a $K$ -tensor with the property $\\nabla _m K_{jkl}{}^m = \\alpha \\,\\nabla _m R_{jkl}{}^m+\\beta \\, \\left(a_{kl}\\nabla _j - a_{jl}\\nabla _k\\right) \\varphi ,$ where $\\alpha $ , $\\beta $ are non zero constants, $\\varphi $ is a real scalar function and $a_{kl}$ is a Codazzi tensor.",
"Then: $&&\\nabla _i\\nabla _m K_{jkl}{}^m+ \\nabla _j\\nabla _m K_{kil}{}^m +\\nabla _k\\nabla _mK_{ijl}{}^m \\\\&&\\qquad =-\\alpha (R_{im} R_{jkl}{}^m+R_{jm} R_{kil}{}^m + R_{km} R_{ijl}{}^m).\\nonumber $"
],
[
"ABC curvature tensors",
"A class of $K$ -tensors with the structure (REF ) are the $ABC$ curvature tensors.",
"They are combinations of the Riemann tensor and its contractions ($A$ , $B$ , $C$ are constants unless otherwise stated): $K_{jkl}{}^m = R_{jkl}{}^m \\,+\\, A(\\delta _j{}^m R_{kl}-\\delta _k{}^m R_{jl})+ B(R_j{}^m g_{kl}-R_k{}^m g_{jl})\\\\+ C(R\\delta _j{}^m g_{kl}-R\\delta _k{}^m g_{jl})\\nonumber $ The canonical curvature tensors are of this sort: Conformal tensor $C_{ijkl}$ : $A=B=\\frac{1}{n-2}$ , $C= -\\frac{1}{(n-1)(n-2)}$ ; Conharmonic tensor $N_{ijkl}$ : $A=B=\\frac{1}{n-2}$ , $C=0$ ; Projective tensor: $P_{ijkl}$ : $A= \\frac{1}{n-1}$ , $B=C=0$ ; Concircular tensor: $\\tilde{C}_{ijkl}$ : $A-B=0$ , $C= \\frac{1}{n(n-1)}$ .",
"Proposition 7.6 Let $K_{ijkl}$ be an $ABC$ tensor ($A$ , $B$ , $C$ may be scalar functions) and $b_{ij}$ a symmetric tensor; 1) if $b$ is $R$ -compatible then $b$ is $K$ -compatible.",
"2) if $b$ is $K$ -compatible and $B\\ne \\frac{1}{n-2}$ then $b$ is $R$ -compatible.",
"The following identity holds for $ABC$ tensors and a symmetric tensor $b$ : $b_{im} K_{jkl}{}^m + b_{jm} K_{kil}{}^m + b_{km} K_{ijl}{}^m= b_{im} R_{jkl}{}^m + b_{jm} R_{kil}{}^m + b_{km} R_{ijl}{}^m \\\\+ B\\left[g_{kl} (b_{im} R_j{}^m - b_{jm} R_i{}^m ) + g_{il} (b_{jm} R_k{}^m - b_{km}R_j{}^m ) + g_{jl} (b_{km} R_i{}^m - b_{im} R_k{}^m )\\right].\\nonumber $ 1) by theorem REF , if $b$ is $R$ -compatible then it commutes with the Ricci tensor, and $K$ -compatibility follows.",
"2) if $b$ is $K$ -compatible it commutes with $K_{ij}$ .",
"Contraction with $g^{kl}$ gives: $b_{im}K_j{}^m - b_{jm} K_i{}^m = (b_{im}R_j{}^m - b_{jm} R_i{}^m)[1-B(n-2)],$ then, if $B\\ne \\frac{1}{n-2}$ , $b$ commutes with the Ricci tensor and by (REF ) it is $R$ -compatible.",
"Proposition 7.7 Let $K$ be an $ABC$ tensor with constant $A\\ne 1$ and $B$ .",
"If $\\nabla _i\\nabla _m K_{jkl}{}^m+ \\nabla _j\\nabla _m K_{kil}{}^m +\\nabla _k\\nabla _mK_{ijl}{}^m=0$ then the Ricci tensor is $K$ -compatible.",
"If $A$ and $B$ are constants, one evaluates $\\nabla _m K_{jkl}{}^m = (1-A) \\nabla _m R_{jkl}{}^m + \\frac{1}{2}(B +2C)\\left(g_{kl}\\nabla _jR - g_{jl}\\nabla _k R \\right),$ Lovelock's identity (REF ) for the Riemann tensor implies $&&\\nabla _i\\nabla _m K_{jkl}{}^m+ \\nabla _j\\nabla _m K_{kil}{}^m +\\nabla _k\\nabla _mK_{ijl}{}^m \\\\&&\\qquad =-(1-A)(R_{im} R_{jkl}{}^m+R_{jm} R_{kil}{}^m + R_{km} R_{ijl}{}^m).\\nonumber $ In the r.h.s.",
"the Riemann tensor can be replaced by tensor $K$ by (REF ) written for the Ricci tensor.",
"Sufficient conditions are: $K$ is harmonic, $K$ is recurrent (with closed recurrency parameter, see eq.",
"(3.13) in [20]).",
"Note that prop.",
"REF remains valid for the Weyl's conformal tensor, which is traceless."
],
[
"Weyl-compatibility and General Relativity",
"In general relativity, the Ricci tensor is related to the energy-momentum tensor by the Einstein equation: $R_{jl}=\\frac{1}{2}Rg_{jl}+kT_{jl}$ with curvature $R=-2kT/(n-2)$ ($T=T^k{}_k$ ).",
"The contracted II Bianchi identity gives $\\nabla _mR_{jkl}{}^m= k \\left(\\nabla _k T_{jl}-\\nabla _j T_{kl}\\right)+\\frac{1}{2} \\left( g_{jl} \\nabla _k R - g_{kl}\\nabla _j R \\right).",
"$ Let $K$ be an $ABC$ tensor, with constant $A$ , $B$ , $C$ .",
"Its divergence (REF ) can be expressed in terms of the gradients of the energy momentum tensor $T_{ij}$ .",
"In the same way Einstein's equations and (REF ) give an equation which is local in the energy momentum tensor: $&&\\nabla _i\\nabla _m K_{jkl}{}^m+ \\nabla _j\\nabla _m K_{kil}{}^m +\\nabla _k\\nabla _mK_{ijl}{}^m \\\\&&\\qquad =-(1-A)k\\left(T_{im} K_{jkl}{}^m+T_{jm} K_{kil}{}^m + T_{km} K_{ijl}{}^m\\right).\\nonumber $ The Weyl tensor $C_{jkl}{}^m$ is the traceless part of the Riemann tensor, and it is an $ABC$ tensor.",
"There are advantages in discussing General Relativity by taking the Weyl tensor as the fundamental geometrical quantity [2], [15], [7].",
"The first equation (REF ) $ \\nabla _m C_{jkl}{}^m =k \\frac{n-3}{n-2} \\left[\\nabla _k T_{jl}-\\nabla _j T_{kl} +\\frac{1}{n-1}\\left( g_{jl} \\nabla _k T - g_{kl}\\nabla _j T \\right)\\right] $ is reported in textbooks, as De Felice [7], Hawking Ellis [15], Stephani [31], and in the paper [2].",
"Instead, a further derivation yields a Bianchi-like equation for the divergence, Eq.",
"(REF ), which contains no derivatives of the sources $&&\\nabla _i\\nabla _m C_{jkl}{}^m+ \\nabla _j\\nabla _m C_{kil}{}^m +\\nabla _k\\nabla _mC_{ijl}{}^m \\\\&&\\qquad =-k\\frac{n-3}{n-2} \\left( T_{im} C_{jkl}{}^m+T_{jm} C_{kil}{}^m +T_{km} C_{ijl}{}^m \\right).\\nonumber $ It can be viewed as a condition for Weyl-compatibility for the energy momentum tensor.",
"In view of prop.REF and the previous equation, the following holds: Proposition 8.1 If $T_{ij}$ is Weyl-compatible, the symmetric tensor $\\text{C̊}_{kl}=: T^{jm}C_{jklm}$ commutes with $T_{ij}$ .",
"In 4 dimensions, given a time-like velocity field $u^i$ , Weyl's tensor is projected in longitudinal (electric) and transverse (magnetic) tensorial components [2] $ E_{kl}=u^ju^m C_{jklm}, \\qquad H_{kl}=\\frac{1}{4}u^ju^m (\\epsilon _{pqjk}C^{pq}{}_{lm} + \\epsilon _{pqjl} C^{pq}{}_{km}) $ that solve equations that resemble Maxwell's equations with source.",
"Therefore, the tensor $E_{kl}=\\text{C̊}_{kl}$ can be viewed as a generalized electric field.",
"It coincides with the standard definition if $T_{ij}=(p+\\rho )u_iu_j+pg_{ij}$ (perfect fluid).",
"The generalized magnetic field is $H_{kl} =\\frac{1}{4}T^{jm} (\\epsilon _{pqjk}C^{pq}{}_{lm} + \\epsilon _{pqjl} C^{pq}{}_{km}) $ .",
"Proposition 8.2 If $T_{kl}$ is Weyl compatible then $H_{kl}=0$ .",
"From the condition for Weyl compatibility we obtain $\\epsilon _{ijkp}[T^{im}C^{jk}{}_{lm}+T^{jm}C^{ki}{}_{lm}+T^{km}C^{ij}{}_{lm}]=0 $ .",
"The first and the second term are modified as follows: $&\\epsilon _{ijkp} T^{im}C^{jk}{}_{lm}=\\epsilon _{kijp}T^{km}C^{ij}{}_{lm}=\\epsilon _{ijkp}T^{km}C^{ij}{}_{lm}\\\\&\\epsilon _{ijkp}T^{jm}C^{ki}{}_{lm}=\\epsilon _{jkip}T^{km}C^{ij}{}_{lm}=\\epsilon _{ijkp}T^{km}C^{ij}{}_{lm}.$ Then, since the sum becomes $\\epsilon _{ijkp}T^{km}C^{ij}{}_{lm}=0$ , then the magnetic part of Weyl's tensor is zero."
]
] | 1204.1211 |
[
[
"Quantum-gravity-induced matter self-interactions in the\n asymptotic-safety scenario"
],
[
"Abstract We investigate the high-energy properties of matter theories coupled to quantum gravity.",
"Specifically, we show that quantum gravity fluctuations generically induce matter self-interactions in a scalar theory.",
"Our calculations apply within asymptotically safe quantum gravity, where our results indicate that the UV is dominated by an interacting fixed point, with non-vanishing gravitational as well as matter couplings.",
"In particular, momentum-dependent scalar self-interactions are non-zero and induce a non-vanishing momentum-independent scalar potential.",
"Furthermore we point out that terms of this type can have observable consequences in the context of scalar-field driven inflation, where they can induce potentially observable non-Gaussianities in the CMB."
],
[
"Introduction",
"In quantum gravity research, much work focuses on pure gravity, not taking into account dynamical matter degrees of freedom.",
"However it is a priori unclear that a consistent quantum theory of gravity can straightforwardly be coupled to matter without changing some of its main properties.",
"In other words, a consistent quantisation of gravity might require the inclusion of matter degrees of freedom from the start.",
"It is thus mandatory to study the complete system of gravitational and matter degrees of freedom in a quantum theory.",
"Then dynamical quantum gravity effects might alter some properties of the matter sector.",
"Most importantly, quantum gravity fluctuations will typically generate new operators for the matter fields in the effective action.",
"This effect occurs even if the matter theory that is coupled to gravity is just a free theory.",
"It is the main goal of this paper to point out that to preserve asymptotic freedom in a matter theory that is coupled to quantum gravity is highly non-trivial.",
"We will show that quantum gravity fluctuations will induce non-vanishing matter couplings, and thus not allow the matter theory to remain asymptotically free under its coupling to gravity.",
"We will exemplify this by considering a simple theory of a free scalar field coupled to gravity, where we point out that the so-called Gaußian matter fixed point of asymptotically safe quantum gravity does not exist, instead all existing fixed points have non-vanishing matter self-interactions.",
"Extensions of this work to the case of non-abelian gauge fields are discussed in the outlook.",
"We investigate this question in a framework, in which quantum gravitational degrees of freedom are carried by the metric.",
"Irrespective of the UV completion for gravity, such a description in terms of the metric holds within the effective-field theory framework [1], [2].",
"It thus holds at scales, presumably below the Planck scale, where the microscopic degrees of freedom of gravity can be integrated out and traded for effective degrees of freedom carried by the metric.",
"Thereby many UV completions for gravity can be analysed within one single framework, and observable consequences, e.g.",
"for the CMB, can be studied.",
"Among the candidates for a quantum theory of gravity there is even one where the parameterisation of quantum gravitational degrees of freedom in terms of the metric in a continuum quantum field theory holds up to arbitrarily high momentum scales, namely asymptotic safety [3].",
"It allows to construct a predictive, continuum quantum field theory of the metric within the path-integral framework.",
"The UV finiteness of observable quantities follows from the existence of an interacting fixed point in the running couplings, i.e.",
"a zero in their $\\beta $ functions.",
"In the vicinity of the fixed point, the theory becomes scale-free, thus allowing for a UV limit without any divergences, in which all dimensionless couplings approach their fixed-point values.",
"A fully non-perturbative formulation of the functional Renormalisation Group (FRG) [4], for reviews see [5], [6], [7], [8], [9], has allowed to collect a substantial amount of evidence for the existence of the fixed point [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [23], [22], [24], [25], [26], and its compatibility with Standard Model matter [29], [30], [31], [32], [33], [34], [27], [28], even allowing for a possibility to solve the triviality problem in QED and the Higgs sector [33], [35], [36], [37], for reviews see [38], [40], [39], [41], [42].",
"For further indications for the existence of the fixed point, see also [43], [44], [45], [46], [47], [48].",
"Such a scenario faces the following challenge: If the fixed point exists, the theory can only be predictive, if the fixed point has a finite number of UV-attractive directions (in the case of the non-interacting, Gaußian fixed point, these are exactly given by the couplings with positive or vanishing canonical dimension).",
"Since the fixed point is interacting, the determination of these relevant directionsThe relevant directions are determined from the critical exponents $\\theta _i$ , which denote the negative eigenvalues of the stability matrix $\\theta _i = {\\rm eig} \\left(-\\frac{\\partial \\beta _{g_i}}{\\partial _{g_j}} \\right)\\Big |_{g_{\\ast }}.$ In the vicinity of a fixed point, the solution to the linearised flow equation is given by $g_i (k)= g_{i\\,\\ast }+ \\sum _n C_n V_i^n \\left( \\frac{k}{k_0}\\right)^{-\\theta _n},$ with constants of integration $C_n$ and $V^n$ the $n$ th eigenvector of the stability matrix.",
"$k_0$ denotes a reference scale.",
"(This reflects the fact that the RG flow cannot predict a numerical value for a physical scale.",
"Thus scales such as the transition scale to the fixed-point regime have to be fixed from observations.)",
"Thus $\\theta _i>0$ implies that towards the IR, the coupling flows away from its fixed-point value, and its IR value corresponds to a free parameter of the theory, that has to be determined by experiment.",
"is technically challenging.",
"Studies within truncated RG flows without matter degrees of freedom indicate that asymptotically safe quantum gravity has at least 3 relevant couplings, and therefore more than 3 free parameters [19], [20], [22], [24].",
"Including matter into the theory, there are three distinct scenarios with regard to predictivity: Either, matter comes with the relevant couplings it already has in the Standard Model, i.e.",
"without coupling it to gravity.",
"A more exciting possibility is, that gravity might actually turn some or all of these couplings into irrelevant, and thus predictable couplings.",
"Indications for a possible realisation of this scenario in QED are discovered in [33].",
"Here we point out that gravity might actually induce further matter interactions which correspond to relevant couplings.",
"Thus a viable quantum theory of dynamical gravitational and matter degrees of freedom might actually have more relevant couplings, then the sum of relevant couplings in both matter and gravity theories taken separately.",
"We will discuss how this scenario might be realised within a truncation of the full matter and gravity effective action.",
"Further, the fact that residual interactions exist at the fixed point will of course be crucial for physical predictions obtained from this scenario.",
"In particular, the question if matter couplings have residual interactions in the far UV can open a window into the realm of quantum gravity phenomenology.",
"In fact, predictions for high-energy scattering experiments can be obtained from the effective action, see, e.g.",
"[49], [51], [50], [52], and crucially depend on the type of operators that are present in the action.",
"Here, we use this framework to address the question how different quantum gravity proposals could be distinguished by the amount of non-Gaußianity that they induce in the CMB, assuming an inflationary scenario with a single scalar field satisfying slow-roll conditions.",
"In the following we will examine the structure of the fixed point, i.e.",
"its number of relevant directions, and its non-vanishing couplings, in the context of a scalar matter theory.",
"We will show that a non-vanishing value for the Newton coupling induces non-vanishing matter self-interactions.",
"This conclusion holds in particular at the fixed point in the context of the asymptotic-safety scenario, but it is also valid within an effective field-theory framework for quantum gravity.",
"In the context of a scalar theory, asymptotic freedom is usually not expected to hold in any case; but in a broader context, our calculation exemplifies a general mechanism, that presumably also applies in the case of asymptotically free Yang-Mills theories.",
"Before we present the details of the calculation, let us explain the basic idea underlying it: The reason why a free matter theory does not remain such when coupled to gravity is simple to understand: Consider any free matter theory which is coupled to gravity minimally.",
"Then the integration measure contains a factor $\\sqrt{g}$ Note that we work in a Euclidean setting here, however our arguments carry over to a Lorentzian setting.",
"Explicit RG calculations indicate that results regarding asymptotically safe quantum gravity hold also in the Lorentzian case [25].. An expansion of $\\sqrt{g}$ around any background metric produces an arbitrary number of vertices.",
"Thus the matter theory is not a free theory any more, instead it contains interaction vertices with the gravitational field of arbitrarily high power.",
"It is then straightforward to construct loop diagrams which contain metric loops only, and an arbitrary number of external matter legs.",
"These loop diagrams generate further effective matter-gravity interactions, and most importantly, matter self-interactions.",
"The situation can be compared to QED, where no photon self-interactions are present at the classical, microscopic level, but fermionic loops induce photon-photon-interactions already in the one-loop effective action.",
"As an example, consider a simple scalar theory described by the following effective action $\\Gamma _k= \\frac{1}{2} Z_{\\phi } (k) \\int d^4x\\, \\sqrt{g}g^{\\mu \\nu }\\partial _{\\mu } \\phi \\partial _{\\nu }\\phi $ with a momentum-scale dependent wave-function renormalisation $Z_{\\phi }(k)$ .",
"The momentum scale $k$ indicates an infrared cutoff, such that all fluctuations in the path integral with momenta $p^2 > k^2$ have been integrated out.",
"Here we have made use of the fact that the covariant derivative simplifies when acting on a scalar field.",
"Four-scalar interactions are then generated by the following loop diagrams: Figure: The scalar-gravity-interaction vertices that are generated by the kinetic term for the scalar field allow to construct these two one-loop diagrams that generate scalar self-interactions.Similar diagrams induce higher $\\phi ^{2n}$ interactions.",
"It follows from dimensional analysis that such diagrams will generate momentum-dependent scalar self-interactions: Depending on the convention one chooses where to have the scale-dependent Newton coupling $G_N(k)$ appear, either the graviton propagator will be $\\sim G_N$ , or each vertex will be $\\sim G_N$ for the two-vertex diagram, and $\\sim \\sqrt{G_N}$ for the four-vertex diagram.",
"Thus the contribution will always generate a term $\\sim G^2_N$ , so four powers of momenta are necessary to compensate the dimensionality of the Newton coupling.",
"Thereby the first term that is induced in the effective action will be quartic in the scalar field and in the momentum.",
"The associated coupling will later be called $\\bar{\\rho }$ .",
"Thus gravity fluctuations do not directly induce a running in the scalar potential, as has been observed in [53], [54].",
"However, as we will point out, there is an indirect effect: The metric-induced momentum-dependent scalar self interactions contribute to the running of the scalar potential, and induce a non-vanishing $\\lambda _{\\phi } \\phi ^4$ term.",
"The crucial point about the diagrams in fig.",
"REF is, that they yield contributions $\\sim G_N^{m}$ where m is the number of metric propagators, and are independent of the coupling that they generate.",
"In the example above, we thus get the following contribution to the $\\beta $ function of the dimensionful scalar coupling $\\bar{\\rho }$ (see eq.",
"(REF )): $\\beta _{\\bar{\\rho }} = G_N^2 f(\\lambda )+...,$ where $f(\\lambda )$ is a function fo the cosmological constant and further terms are $\\sim \\bar{\\rho }$ , $\\sim \\bar{\\rho }^2$ and proportional to further matter couplings.",
"The main point is, that, even if we set all matter couplings to zero, i.e.",
"we start with a free theory, then metric fluctuations generate interaction terms for the matter, see also fig.",
"REF .",
"Figure: Here we exemplify a matter β\\beta function for the coupling ρ\\rho : Without coupling to gravity (blue line), this β\\beta function will admit a Gaußian fixed point, as well as possibly a non-Gaußian one.",
"Including gravity fluctuations, a term ∼G\\sim G will be added to the β\\beta function, thus shifting the Gaußian to an interacting, non-Gaußian fixed point (red dashed line).",
"A second non-Gaußian fixed point may or may not exist.Most importantly, metric fluctuations can add terms to the matter $\\beta $ functions which lead to a vanishing of all fixed points.",
"If such a finding where to be confirmed in untruncated theory space, the asymptotic-safety scenario would not be compatible with the existence of this particular type of matter.",
"Within a truncation, such a definite conclusion can obviously not be drawn."
],
[
"Set-up of the calculation",
"To evaluate the $\\beta $ functions of the running couplings we require a tool that is applicable in the perturbative as well as the non-perturbative regime.",
"We also aim for an analytic continuum calculation.",
"We employ the functional Renormalisation Group, where the Wetterich equation [4] allows to evaluate $\\beta $ functions even in the non-perturbative regime.",
"Introducing an infrared (IR) mass-like regulator function $R_k(p)$ suppresses IR modes (with $p^2 <k^2$ ) in the generating functional.",
"The scale-dependent effective action $\\Gamma _k$ then contains the effect of quantum fluctuations above the scale $k$ only.",
"Its scale-dependence is given by the following functional differential equation: $\\partial _t \\Gamma _k= \\frac{1}{2} {\\rm STr} \\left(\\Gamma _k^{(2)}+R_k \\right)^{-1}\\partial _t R_k.$ Herein $\\partial _t = k\\, \\partial _k$ and $\\Gamma _k^{(2)}$ is matrix-valued in field space and denotes the second functional derivative of the effective action with respect to the fields.",
"Adding the mass-like regulator and taking the inverse yields the full propagator.",
"The supertrace contains a trace over all indices; in the case of a continuous momentum variable it thus implies an integration over the momentum.",
"On the technical side, the main advantage of this equation is its one-loop form, since it can be written as the supertrace over the full propagator, with the regulator insertion $\\partial _t R_k$ in the loop.",
"Nevertheless it is crucial to stress that it also yields higher terms in a perturbative expansion, since it depends on the full, field- and momentum-dependent propagator, and not just on the perturbative propagator, see, e.g.",
"[55].",
"For the gravitational part we work in the background field formalism [56], where the full metric is split according to $g_{\\mu \\nu }= \\bar{g}_{\\mu \\nu }+ h_{\\mu \\nu },$ where this split does not imply that we consider only small fluctuations around, e.g.",
"a flat background.",
"Within the FRG approach we have access to physics also in the fully non-perturbative regime.",
"The background-field formalism, being highly useful in non-abelian gauge theories (see, e.g.",
"[7], [8]), is mandatory in gravity, since the background metric allows for a meaningful notion of \"high-momentum\" and \"low-momentum\" modes as implied by the spectrum of the background covariant Laplacian.",
"Within Yang-Mills theories the $\\beta $ functions are independent of the choice of background, as long as it serves to distinguish the different operators in the truncation.",
"Within gravity, an exception exists, as the topology of the background can alter the spectrum of fluctuations in the infrared.",
"Thus only the UV behaviour of the running couplings is independent of the choice of background, whereas topologically distinct backgrounds can lead to a different IR behaviour [57].",
"As we are only interested in the $\\beta $ functions as they pertain to the UV, we can choose different backgrounds for the gravitational and the matter $\\beta $ functions.",
"In the context of an interacting theory, the Wetterich equation is usually applied to a truncated theory space.",
"We thus do not keep all infinitely many terms that are part of the full effective action, but concentrate on a (typically) finite subset.",
"In our case, where we are interested in demonstrating that from a free matter theory coupled to gravity matter self-interactions will be generated, we choose a truncation of the following form: $\\Gamma _k &=& \\Gamma _{k\\,\\mathrm {EH}} + \\Gamma _{k\\,\\mathrm {gf}}+\\frac{1}{2}Z_{\\phi }(k) \\int d^4x \\sqrt{g} \\,g^{\\mu \\nu }\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi \\nonumber \\\\&+&\\bar{\\rho }_a(k)\\int d^4x \\sqrt{g} \\,g^{\\mu \\nu }g^{\\kappa \\lambda }\\,\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi \\, \\partial _{\\kappa }\\phi \\partial _{\\lambda }\\phi \\nonumber \\\\&+& \\bar{\\rho }_b (k) \\int d^4x \\sqrt{g}\\, \\phi ^2 (g^{\\mu \\nu }\\nabla _{\\mu }\\partial _{\\nu } \\phi )\\, (g^{\\kappa \\lambda } \\nabla _{\\kappa } \\partial _{\\lambda } \\phi ) \\nonumber \\\\&+& \\bar{\\rho }_c(k) \\int d^4x \\sqrt{g} \\,(g^{\\mu \\nu }\\partial _{\\mu } \\phi \\partial _{\\nu }\\phi )\\, (\\phi g^{\\kappa \\lambda } \\nabla _{\\kappa } \\partial _{\\lambda } \\phi )\\nonumber \\\\&+& \\bar{\\lambda }_{\\phi } (k)\\int d^4x \\sqrt{g}\\, \\phi ^4,$ where $\\nabla _{\\mu }$ denotes the usual covariant derivative.",
"It is important to note that we have included all independent operators of fourth order in derivatives and fields.",
"All other operators of the same order can be rewritten as a linear combination of the above ones, and possibly additional terms that depend on the curvature.",
"In order to demonstrate that metric fluctuations also remove the Gaußian fixed point in the scalar potential, we include the $\\bar{\\lambda }_{\\phi }$ coupling, which is to be distinguished from the cosmological constant $\\bar{\\lambda }$ by its subscript.",
"The Einstein-Hilbert and the gauge-fixing term read: $\\Gamma _{k\\,\\mathrm {EH}}&=& 2 \\bar{\\kappa }^2 Z_{\\text{N}} (k)\\int d^4 x \\sqrt{g}(-R+ 2 \\bar{\\lambda }(k)),\\\\\\Gamma _{k\\,\\mathrm {gf}}&=& \\frac{Z_{\\text{N}}(k)}{2\\alpha }\\int d^4 x\\sqrt{\\bar{g}}\\, \\bar{g}^{\\mu \\nu }F_{\\mu }[\\bar{g}, h]F_{\\nu }[\\bar{g},h],$ with $F_{\\mu }[\\bar{g}, h]= \\sqrt{2} \\bar{\\kappa } \\left(\\bar{D}^{\\nu }h_{\\mu \\nu }-\\frac{1+\\rho _{\\rm gh}}{4}\\bar{D}_{\\mu }h^{\\nu }{}_{\\nu } \\right).$ Herein, $\\bar{\\kappa }= (32 \\pi G_{\\text{N}})^{-\\frac{1}{2}}$ is related to the bare Newton constant $G_{\\text{N}}$ and $\\bar{\\lambda }$ is the cosmological constant.",
"The gauge-fixing term is supplemented by an appropriate ghost term.",
"As within this truncation ghosts do not couple to the scalar fieldNote however that in general the existence of such coupling should be expected, as it is induced by diagrams similar to those in fig.",
"REF ., we can neglect the ghost term in the calculation of the scalar $\\beta $ functions.",
"We work in Landau deWitt gauge $\\rho _{\\rm gh} \\rightarrow \\alpha $ , $\\alpha \\rightarrow 0$ here.",
"In the following, we employ a decomposition of the metric fluctuations into irreducible components under the Poincare group.",
"In our choice of gauge, only the transverse traceless mode $h_{\\mu \\nu }^{TT}$ (with $\\bar{D}^{\\mu }h_{\\mu \\nu }^{TT}=0$ and $\\bar{g}^{\\mu \\nu } h_{\\mu \\nu }^{TT}=0$ ) and the trace mode $h = \\bar{g}^{\\mu \\nu }h_{\\mu \\nu }$ can contribute to the running in the matter sector.",
"To evaluate matter $\\beta $ functions, it is useful to proceed as follows: Splitting $\\Gamma _k^{(2)}+R_k =\\mathcal {P}_k+\\mathcal {F}_k$ , where all scalar-field dependent terms enter the fluctuation matrix $\\mathcal {F}_k$ , we may now expand the right-hand side of the flow equation as follows: $\\partial _t \\Gamma _k&=& \\frac{1}{2}{\\rm STr} \\lbrace [\\Gamma _k^{(2)}+R_k]^{-1}(\\partial _t R_k)\\rbrace \\\\&=& \\frac{1}{2} {\\rm STr}\\, \\tilde{\\partial }_t\\ln \\mathcal {P}_k+\\frac{1}{2}\\sum _{n=1}^{\\infty }\\frac{(-1)^{n-1}}{n} {\\rm STr}\\,\\tilde{\\partial }_t(\\mathcal {P}_k^{-1}\\mathcal {F}_k)^n,\\nonumber $ where the derivative $\\tilde{\\partial }_t$ in the second line by definition acts only on the $k$ dependence of the regulator, $\\tilde{\\partial }_t=\\int \\partial _t R_k\\frac{\\delta }{\\delta R_k}$ , see appendix .",
"Since each factor of $\\mathcal {F}_k$ contains a coupling to external fields, this expansion simply corresponds to an expansion in the number of vertices.",
"Thus we can straightforwardly write down the diagrammatic expansion of a $\\beta $ function, see fig.",
"REF .",
"Herein, the internal propagator lines contain the mass-like regulator-function, and in the case of the graviton, also depend on the cosmological constant.",
"In order to project onto the anomalous dimension $\\eta _{\\phi } = - \\partial _t \\ln Z_{\\phi }(k)$ , we can terminate the expansion at second order in the vertices, since all further terms in the expansion must have more than two external scalar fields.",
"Similarly, the evaluation of the $\\partial _t \\bar{\\rho }_i(k)$ for $i= a,b,c$ and $\\partial _t \\bar{\\lambda }_{\\phi }$ requires terms up to $\\left(\\mathcal {P}^{-1}\\mathcal {F}\\right)^4$ , see fig.",
"REF .",
"Figure: Here we show the diagrammatric expansion of the Wetterich equation, projected onto the φ 2 \\phi ^2 and φ 4 \\phi ^4 subspace of theory space, respectively.",
"Diagrams 1a, 1b and 1c contribute to η φ =-∂ t lnZ φ \\eta _{\\phi }= - \\partial _t \\ln Z_{\\phi }, whereas all diagrams 2 in principle contribute to β ρ i \\beta _{\\rho _i} and β λ \\beta _{\\lambda }.Each loop contains a regulator insertion ∂ t R k \\partial _t R_k that can be found on any of the internal lines.Clearly a flat background, $\\bar{g}_{\\mu \\nu }= \\delta _{\\mu \\nu }$ is fully sufficient to project onto the matter $\\beta $ functions, and thus preferable for reasons of technical simplicity.",
"By using a regulator of the form $R_k = \\Gamma _k^{(2)}\\Big |_{\\phi =0}r\\left(\\frac{ \\Gamma _k^{(2)}\\Big |_{\\phi =0}(p^2)}{k^2}\\right),$ it is then straightforward to find which type of contribution to the $\\beta $ function a particular diagram corresponds to: Each graviton propagator comes with a factor of $G_N(k)$ .",
"Thus, e.g.",
"the two-vertex contribution 2B, see fig.",
"REF yields a contribution $\\sim Z_{\\phi }(k)^2\\, G_N^2$ , since the vertex is $\\sim Z_{\\phi }(k)$ , and each propagator $\\sim G_N(k)$ .",
"Next we introduce the anomalous dimension $\\eta _{\\phi }= - \\partial _t \\ln Z_{\\phi },$ as well as dimensionless and renormalised couplings $\\rho _i (k)&=& \\bar{\\rho }_i(k) \\frac{k^4}{Z_{\\phi }^2(k)}\\, \\quad \\quad i = a,b,c,\\nonumber \\\\\\lambda _{\\phi }(k)&=& \\bar{\\lambda }_{\\phi }\\frac{1}{Z_{\\phi }(k)^2},\\nonumber \\\\g(k)&=&\\frac{G_{\\text{N}} k^2}{Z_{\\text{N}}(k)}= \\frac{k^2}{32\\pi \\bar{\\kappa }^2\\, Z_{\\text{N}}},\\nonumber \\\\\\lambda (k)&=& \\frac{\\bar{\\lambda }}{k^2}.$ It is then straightforward to see that the $\\beta _{\\rho _i}$ have the following form $&{}&\\partial _t \\rho _i =\\beta _{\\rho _i}\\nonumber \\\\&=& 4 \\rho _i + 2 \\eta _{\\phi } \\rho _i + c_1\\, g^2 \\, f_1(\\lambda )+ \\sum _{j= a,b,c} c_2\\, g \\, \\rho _j \\, f_2 (\\lambda ) \\nonumber \\\\&+& \\!\\!\\!\\!\\sum _{j,k= a,b,c} c_3\\, \\rho _j \\rho _k, + c_4 \\, g\\, \\lambda _{\\phi } \\, f_3(\\lambda ) + \\sum _{j=a,b,c} c_5\\, \\rho _j \\lambda _{\\phi },$ where the first term arises from the canonical dimensionality.",
"The constants $c_i$ are regulator and gauge dependent, as are the functions $f_i(\\lambda )$ .",
"These quantities remain to be determined.",
"Crucially, we see that depending on the $c_i$ and the behaviour of $f_i(\\lambda )$ , the point $\\rho _i =0$ may not be a fixed point of the RG flow, whereas it clearly is in the case $g=0$ .",
"In contrast, the $\\beta $ function for $\\lambda _{\\phi }$ will not have a contribution $\\sim g^2$ , since the diagrams 2B and 2G in fig.",
"REF vanish if projected onto vanishing external momenta.",
"We thus have that $\\beta _{\\lambda _{\\phi }}&=& 2 \\eta _{\\phi } \\lambda _{\\phi } + \\sum _{j= a,b,c} d_1\\, g \\, \\rho _j \\, h_1 (\\lambda ) + \\sum _{j,k= a,b,c} d_2\\, \\rho _j \\rho _k, \\nonumber \\\\&+& d_3 \\, g\\, \\lambda _{\\phi } \\, h_2(\\lambda ) + \\sum _{j=a,b,c} d_4\\, \\rho _j \\lambda _{\\phi },$ where $h_i(\\lambda )$ are scheme-dependent functions of $\\lambda $ and $d_i$ are scheme-dependent constants.",
"Here, the contributions $\\sim \\rho _i^2$ play an important role, since for $d_2 \\ne 0$ , these imply that $\\beta _{\\lambda _{\\phi }} \\ne 0$ at $\\lambda _{\\phi }=0.$ Having said that the $c_i, d_i$ and $f_i(\\lambda ), h_i(\\lambda )$ have a gauge- as well as regulator-dependence, let us note that this will induce quantitative variations in fixed-point values and critical exponents, however the existence of fixed points and the relevance of these couplings are universal properties.",
"For the gravitational $\\beta $ functions, it is crucial to note that there cannot be a direct contribution $\\sim \\rho _i, \\lambda _{\\phi }$ , since $\\Gamma _k^{(2)}$ evaluated at vanishing $\\phi _i$ (in order to project onto the terms $\\int d^4x \\, \\sqrt{g}$ and $\\int d^4x\\, \\sqrt{g}\\, R$ ) does not contain a term $\\sim \\rho _i$ or $\\sim \\lambda _{\\phi }$ .",
"Thus the matter contribution to $\\partial _t g$ and $\\partial _t \\lambda $ arises from the kinetic matter term only.",
"Since $\\eta _{\\phi }= \\eta _{\\phi }(\\rho _a, \\rho _b,\\rho _c)$ , we do however get a backcoupling of the $\\rho _i$ into the gravitational $\\beta $ functions in this indirect way.",
"Here we can use that a maximally symmetric background with positive curvature suffices to evaluate these $\\beta $ functions.",
"For the metric and ghost contributions, we proceed as in [21], [22].",
"For the matter contribution, we use that for an exponential shape function it is straightforward to evaluate the functional traces via an inverse Laplace transform; for details, see appendix .",
"In the following, we will use a spectrally and RG adjusted regulator [58], [59] of the form (REF ) with exponential shape function $r(y)= \\left( e^{y}-1\\right)^{-1}$ .",
"Therein $p^2$ is understood to be a placeholder for the eigenvalues of the differential operator constituting the kinetic term, evaluated on the appropriate background.",
"As an example, for the evaluation of $\\beta _{\\rho _i}$ , $p^2$ is to be understood as the momentum, whereas $p^2 = - \\bar{D}^2$ with $\\bar{D}^2$ being the Laplacian on a maximally symmetric space of positive curvature for the evaluation of $\\beta _{g}$ and $\\beta _{\\lambda }$ .",
"Note that our choice of regulator implies that the right-hand side of the flow equation will depend on $\\eta _{\\phi }$ , $\\eta _N$ and $\\partial _t \\lambda $ , but not on $\\partial _t \\rho _i$ and $\\partial _t \\lambda _{\\phi }$ .",
"To obtain the anomalous dimension, we apply the following projection rule: $\\eta _{\\phi }= -\\frac{1}{8}\\left(Z_{\\phi } \\right)^{-1}\\Bigl ( \\frac{\\partial }{\\partial p_{1\\mu }}\\frac{\\partial }{\\partial p_1^{\\mu }} \\int \\frac{d^4 p_2}{(2 \\pi )^4} \\frac{\\delta }{\\delta \\phi (p_1)} \\frac{\\delta }{\\delta \\phi (p_2)} \\partial _t \\Gamma _k \\Bigr )\\Big |_{\\phi = 0, p_1=0}.$ To evaluate the $\\beta $ functions for the matter couplings we employ the following projection rules, where the numerical coefficients arise due to the differing tensor structure of the four matter couplings: $\\partial _t \\bar{\\rho }_a &=& \\left(\\left(\\frac{1}{384} \\partial _{l_{1\\mu }} \\partial _{l_{1\\mu }} \\partial _{l_{2\\nu }} \\partial _{l_{2\\nu }} -\\frac{11}{1152}\\partial _{l_{1 \\mu }} \\partial _{l_{1\\mu }}\\partial _{l_{2\\nu }}\\partial _{l_{3\\nu }}+\\frac{1}{144}\\partial _{l_{1\\mu }}\\partial _{l_{2\\mu }}\\partial _{l_{1\\nu }}\\partial _{l_{3\\nu }} \\right)\\int \\frac{d^4 l_4}{(2 \\pi )^4}\\frac{\\delta ^4}{\\delta \\phi (l_1) \\delta \\phi (l_2)\\delta \\phi (l_3)\\delta \\phi (l_4)} \\Gamma _k\\right)\\Big |_{\\phi =0, l_i =0},\\nonumber \\\\\\partial _t \\bar{\\rho }_b &=& \\left(\\left(\\frac{1}{384} \\partial _{l_{1\\mu }} \\partial _{l_{1\\mu }} \\partial _{l_{2\\nu }} \\partial _{l_{2\\nu }} -\\frac{1}{192}\\partial _{l_{1 \\mu }} \\partial _{l_{1\\mu }}\\partial _{l_{2\\nu }}\\partial _{l_{3\\nu }}+\\frac{1}{192}\\partial _{l_{1\\mu }}\\partial _{l_{2\\mu }}\\partial _{l_{1\\nu }}\\partial _{l_{3\\nu }} \\right)\\int \\frac{d^4 l_4}{(2 \\pi )^4}\\frac{\\delta ^4}{\\delta \\phi (l_1) \\delta \\phi (l_2)\\delta \\phi (l_3)\\delta \\phi (l_4)} \\Gamma _k\\right)\\Big |_{\\phi =0, l_i =0},\\nonumber \\\\\\partial _t \\bar{\\rho }_c &=& \\left(\\left(\\frac{1}{192} \\partial _{l_{1\\mu }} \\partial _{l_{1\\mu }} \\partial _{l_{2\\nu }} \\partial _{l_{2\\nu }} -\\frac{1}{48}\\partial _{l_{1 \\mu }} \\partial _{l_{1\\mu }}\\partial _{l_{2\\nu }}\\partial _{l_{3\\nu }}+\\frac{1}{48}\\partial _{l_{1\\mu }}\\partial _{l_{2\\mu }}\\partial _{l_{1\\nu }}\\partial _{l_{3\\nu }} \\right)\\int \\frac{d^4 l_4}{(2 \\pi )^4}\\frac{\\delta ^4}{\\delta \\phi (l_1) \\delta \\phi (l_2)\\delta \\phi (l_3)\\delta \\phi (l_4)} \\Gamma _k\\right)\\Big |_{\\phi =0, l_i =0},\\nonumber \\\\\\partial _t \\bar{\\lambda }_{\\phi }&=& \\frac{1}{\\Omega }\\frac{1}{24}\\int \\frac{d^4 l_4}{(2 \\pi )^4}\\left(\\frac{\\delta ^4}{\\delta \\phi (l_1) \\delta \\phi (l_2)\\delta \\phi (l_3)\\delta \\phi (l_4)} \\Gamma _k\\right)\\Big |_{\\phi =0, l_i =0}.$ For further details, and the appropriate vertices for the evaluation of the $\\beta $ functions, see appendix ."
],
[
"Matter sector without gravity",
"Let us first set $g=0$ and thereby switch off metric fluctuations to briefly examine the pure matter truncation.",
"As expected, it admits a Gaußian fixed point with critical exponents $0,-4,-4,-4$ as corresponding to the canonical dimensionality.",
"Further, we also find several non-Gaußian fixed points in the system.",
"These have in common, that at all of these, at least one of the couplings $\\rho _i, \\lambda _{\\phi }$ has a negative value, thus corresponding to an unstable direction of the Euclidean action.",
"This does however not imply that these fixed points should be discarded, since clearly higher-order operators beyond our truncation, such as $\\phi ^6$ operators will be induced.",
"Thus, the Euclidean action in an extended truncation could be bounded from below even at negative values of the $\\rho _i$ and $\\lambda _{\\phi }$ .",
"In tab.",
"REF , we list the fixed points in this truncation; discarding further spurious fixed points since their rather large anomalous dimension suggests that terms beyond our truncation will crucially alter these.",
"Table: Here we list the fixed-point values, as well as the value of the anomalous dimension η φ \\eta _{\\phi } and the critical exponents at the various fixed points at g=0g=0.",
"We find the Gaußian fixed point, as well as two non-Gaußian ones, where the first has two relevant directions.Thus, the pure matter system admits several fixed points to construct a possible UV completion.",
"Note that the critical exponents at the non-Gaußian fixed points deviate significantly from the canonical dimensionality of the couplings.",
"To determine the relevance of a coupling, only the real part of the critical exponent matters.",
"The imaginary parts indicate that the flow approaches the fixed point in a spiral-type shape.",
"In particular, the second of the three fixed points even shows two relevant directions, see tab.",
"REF .",
"Hence, using this fixed point as a UV completion for the scalar theory in this truncation implies a lower level of predictivity than expected from a simple dimensional analysis.",
"In the following, we will discuss the fate of these fixed points under the coupling to gravity.",
"We first focus on the Gaußian fixed point, which becomes interacting for $g\\ne 0$ ."
],
[
"Fixed point analysis in the matter sector: shifted Gaußian fixed point",
"Our first result is that the contribution from metric fluctuations that is independent of the matter couplings $\\bar{\\rho }_{a,b,c}$ and $\\bar{\\lambda }_{\\phi }$ , is indeed non-zero.",
"For unspecified regulator shape function $r(p^2/k^2)$ , the contribution takes the following form: $\\beta _{\\bar{\\rho }_a}\\Big |_{\\rho _{a,b,c},\\lambda _{\\phi }=0}&=&\\!\\frac{575}{1728} Z_{\\phi }^2\\int \\frac{d^4p}{(2 \\pi )^4}\\tilde{\\partial }_t \\frac{1}{\\left(\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}\\left(1+r \\left( \\frac{\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}(p^2)}{k^2} \\right) \\right)\\Big |_{\\phi =0}\\right)^2 }\\nonumber \\\\&{}&+ \\frac{1}{1024} Z_{\\phi }^4\\int \\frac{d^4p}{(2 \\pi )^4}\\tilde{\\partial }_t \\frac{1}{\\left(\\Gamma _{k\\, hh}^{(2)}\\left(1+r \\left( \\frac{\\Gamma _{k\\, hh}^{(2)}(p^2)}{k^2} \\right) \\right)\\Big |_{\\phi =0}\\right)^2 }\\frac{1}{\\left(\\Gamma _{k\\, \\phi \\phi }^{(2)}\\left(1+r \\left( \\frac{\\Gamma _{k\\, \\phi \\phi }^{(2)}(p^2)}{k^2} \\right) \\right)\\Big |_{\\phi =0}\\right)^2 }\\nonumber \\\\\\beta _{\\bar{\\rho }_b}\\Big |_{\\rho _{a,b,c},\\lambda _{\\phi }=0}&=&\\!\\frac{85}{576} Z_{\\phi }^2\\int \\frac{d^4p}{(2 \\pi )^4}\\tilde{\\partial }_t \\frac{1}{\\left(\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}\\left(1+r \\left( \\frac{\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}(p^2)}{k^2} \\right) \\right)\\Big |_{\\phi =0}\\right)^2 }\\nonumber \\\\\\beta _{\\bar{\\rho }_c}\\Big |_{\\rho _{a,b,c},\\lambda _{\\phi }=0}&=&\\!\\frac{55}{96} Z_{\\phi }^2\\int \\frac{d^4p}{(2 \\pi )^4}\\tilde{\\partial }_t \\frac{1}{\\left(\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}\\left(1+r \\left( \\frac{\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}(p^2)}{k^2} \\right) \\right)\\Big |_{\\phi =0}\\right)^2}$ Here, $\\Gamma _{k\\, h^{TT}h^{TT}}^{(2)}$ is the scalar part of the inverse transverse traceless propagator and similarly, $\\Gamma _{k\\, hh}^{(2)}$ is the inverse propagator of the trace mode $h$ .",
"Herein, all $\\beta _{\\bar{\\rho }_i}$ receive a non-vanishing contribution from the two-vertex diagram 2B, see fig.",
"REF , whereas only $\\beta _{\\bar{\\rho }_a}$ receives a contribution from the four-vertex diagram 2G.",
"Due to the different tensor structures of the operators assciated to the couplings $\\bar{\\rho }_i$ , the projection of this diagram onto $\\bar{\\rho }_b$ and $\\bar{\\rho }_c$ vanishes.",
"The two-vertex diagram only has a contribution from the transverse traceless graviton mode, due to the form of the vertices, see app. .",
"Furthermore, the four-vertex diagram with internal transverse traceless gravitons has a vanishing projection onto the operators under consideration here, with four external momenta.",
"In fact, this diagram contributes to the running of couplings of higher-order operators, i.e.",
"operators with four scalar fields and more than four external momenta.",
"The effect of these contributions is simple: At vanishing matter couplings, the contributions eq.",
"(REF ) remain non-zero.",
"Therefore, the right-hand side of the $\\beta $ functions of the couplings $\\rho _{i}$ is not zero at vanishing matter coupling.",
"Accordingly there cannot be a Gaußian fixed point in the three momentum-dependent couplings $\\rho _{a,b,c}$ .",
"We conclude that as soon as a free scalar theory is coupled to gravity, non-vanishing matter self-interactions are induced.",
"This has a crucial effect on the $\\beta $ function for the momentum-independent coupling $\\bar{\\lambda }_{\\phi }$ : Whereas metric fluctuations do not directly induce this term, there is a contribution arising from the purely scalar diagram 2c in fig.",
"REF , which is $\\sim \\rho _b$ : $&{}&\\beta _{\\bar{\\lambda }_{\\phi }}\\Big |_{\\lambda _{\\phi }=0}\\\\&=&4 \\bar{\\rho }_b^2\\int \\frac{d^4p}{(2 \\pi )^4} \\tilde{\\partial }_t \\frac{1}{\\left(\\Gamma _{k\\, \\phi \\phi }^{(2)}\\left(1+r \\left( \\frac{\\Gamma _{k\\,\\phi \\phi }^{(2)}(p^2)}{k^2} \\right) \\right)\\Big |_{\\phi =0}\\right)^2 }\\nonumber $ Herein, $\\Gamma _{k\\, \\phi \\phi }^{(2)}$ is the inverse scalar propagator.",
"This contribution arises, as the contribution $\\sim \\rho _b$ to the four-scalar vertex clearly contains a part which is proportional to the momenta of two of the four scalar fields only, see app. .",
"Constructing the diagram 2c in fig.",
"REF , there will accordingly be a non-vanishing contribution if the momenta of the external fields are taken to zero.",
"This is precisely the non-vanishing contribution to $\\beta _{\\lambda _{\\phi }}$ .",
"We thus observe that at $\\lambda _{\\phi }=0$ , the right-hand side of $\\beta _{\\lambda _{\\phi }}$ is non-zero, and thus this $\\beta $ function does not admit a Gaußian fixed point, see fig.",
"REF .",
"To summarise, metric fluctuations yield a non-vanishing contribution to momentum-dependent self-interactions parameterised by the couplings $\\rho _{a,b,c}$ .",
"Depending on the tensor structure, these can actually yield contributions to $\\beta _{\\lambda _{\\phi }}$ .",
"This implies that $\\lambda _{\\phi }=0$ will not be a fixed point of the matter $\\beta $ functions in the presence of metric fluctuations, since then $\\rho _b=0$ is not a fixed point and accordingly $\\beta _{\\lambda } \\ne 0 $ at $\\lambda _{\\phi } =0$ .",
"Thus our results imply that metric fluctuations induce a non-vanishing potential for scalar matter, since momentum-dependent as well as momentum-independent couplings will be induced.",
"Most importantly, this means that canonical power-counting arguments in the matter sector will not hold: If the system of $\\beta $ functions admits a fixed point, it can only be a non-Gaußian one.",
"Therefore the critical exponents will not be given by the canonical dimensionality of the couplings, but will receive additional contributions from quantum fluctuations and have non-vanishing anomalous dimensions.",
"This is particularly interesting in the case of power-counting marginal couplings such as $\\lambda _{\\phi }$ , which will be turned into a relevant or irrelevant coupling.",
"In the following, we will illustrate this, by looking at the full system of $\\beta $ functions for $\\rho _{a,b,c}$ and $\\lambda _{\\phi }$ .",
"Here, we treat $g$ and $\\lambda $ as parameters, and set $\\eta _{N}=-2$ and $\\partial _t \\lambda =0$ .",
"We investigate the behaviour of the Gaußian fixed point as a function of $g$ .",
"For increasing $g$ and $\\lambda $ it will clearly be shifted.",
"Indeed we find two fixed points which are connected to the Gaußian one continuously as a function of $g$ ; for the examination of further non-Gaußian fixed points, see section REF .",
"As expected, at $g=0$ , where metric fluctuations are switched off, the system $\\beta _{\\rho _{a,b,c}}, \\beta _{\\lambda _{\\phi }}$ admits a Gaußian fixed point with critical exponents $\\theta _{a,b,c}=-4$ and $\\theta _{\\lambda _{\\phi }}=0$ , corresponding precisely to the power-counting dimensionality of these couplings.",
"The situation changes at $g>0$ : As is clearly visible in fig.",
"REF , the fixed-point values increase in their absolute value as a function of $g$ .",
"Similarly, the anomalous contributions to the critical exponents increases with $g$ , see fig.",
"REF .",
"At $g\\ne 0$ there is no fixed point with vanishing matter couplings.",
"We conclude that the so-called Gaußian matter fixed point is shifted and becomes a fully non-Gaußian fixed point, where both gravitational and matter couplings are non-vanishing.",
"This confirms our claim that metric fluctuations generically induce matter self-interactions even if a free matter theory is coupled to gravity.",
"Figure: Here we plot the fixed-point values of the matter self-couplings as a function of gg for a value of the cosmological constant λ=1/10\\lambda = 1/10.",
"The blue circles show the first fixed point, the orange squares the second fixed point.",
"The fixed point values approach each other for increasing values of gg.",
"Note that in fact, the values of λ φ \\lambda _{\\phi } at the second fixed point are small, but non-zero for g>0g>0.Beyond g=0.35g =0.35, we do not find any fixed points, i.e.",
"the two fixed points annihilate.The mechanism behind this is simple: The most important contribution to the $\\beta $ functions of the $\\rho _i$ comes from diagrams, where all vertices are $\\sim Z_{\\phi }$ .",
"These generate a contribution to the $\\beta $ functions for $\\rho _i$ , which is non-vanishing if all matter self-interactions are set to zero.",
"Thereby the $\\beta $ functions in eq.",
"(REF ) will have a non-vanishing contribution with $c_1 \\ne 0$ .",
"When taking $\\rho _i\\rightarrow 0$ , this term remains.",
"Thus $\\rho _i=0$ and $\\lambda _{\\phi }=0$ is not a zero of the $\\beta $ functions and thus does not correspond to a fixed point.",
"Most importantly, we note that the critical exponents in the matter sector deviate very significantly from their canonical values, which they would have at a Gaußian fixed point, see fig.",
"REF .",
"Thus metric fluctuations not only induce a shift in the $\\beta $ functions of matter couplings, such that these are non-vanishing at the fixed point, most importantly they considerably alter the scaling behaviour.",
"This change in the scaling is due to diagrams such as a tadpole diagram and further mixed metric-matter diagrams which yield contributions $\\sim g \\, \\rho _i$ and $\\sim g\\, \\lambda _{\\phi }$ and thus change the critical exponents in comparison to the free theory.",
"A further contribution arises from the purely scalar two-vertex diagram, which yields contribution $\\sim \\rho _i \\rho _j$ and $\\sim \\rho _i \\lambda _{\\phi }$ to $\\beta _{\\rho _i}$ and $\\beta _{\\lambda _{\\phi }}$ .",
"Once metric fluctuations induce a non-zero value for the matter couplings, these in turn contribute to change the critical exponents from the power-counting values.",
"We point out that such diagrams are expected to contribute to the $\\beta $ functions of all matter couplings, even those which are not induced by metric fluctuations, i.e.",
"where a contribution of the type $\\sim c_1$ (see eq.",
"(REF )) is absent.",
"Furthermore the $\\rho _i$ also couple into the flow of other matter couplings.",
"We might thus speculate, that a similar shift in the critical exponents as observed here, could also occur for other operators.",
"It is thus possible that power-counting marginal operators are shifted into relevance at the shifted GFP.",
"An example of this is clearly given by the first fixed point, where one critical exponent is actually positive, see fig.",
"REF .",
"Furthermore we make the important observation, that at both fixed points the critical exponents are actually shifted towards larger values.",
"We might thus conjecture that operators of canonical dimensionality -2, which we have not included in our study, such as two-momentum four-scalar operators, will also be shifted into relevance at these fixed points.",
"Figure: Here we plot the four critical exponents at the two fixed points, for λ=1/10\\lambda =1/10 and as a function of gg.",
"Clearly, at g=0g=0, the critical exponents are given by the canonical dimensionality of the couplings ρ ¯\\bar{\\rho } and λ\\lambda .",
"At g>0g>0, they deviate significantly.",
"Most importantly, there is one relevant critical exponent at the first fixed point (upper panel).Further, our results suggest, that quantum effects will also be non-negligible for couplings beyond our truncation.",
"Diagrams as shown in fig.",
"REF will generate further interaction terms.",
"In fact our calculation is a first step in a more general direction.",
"It is directly clear from the expression for the vertices, that metric fluctuations will generate all terms of the form $\\left(g^{\\mu \\nu }\\partial _{\\mu }\\phi \\partial _{\\nu }\\phi \\right)^n =: \\mathbf {\\Phi },$ with $n$ integer, and similar powers of the other tensor structures.",
"Thus the effective action will be of the form $\\Gamma _{k\\, \\phi }= V_k(\\mathbf {\\Phi })+...$ with some function $V_k$ , which we have assumed to be expandable in a Taylor series and have studied the first two coefficients in the Taylor expansion here.",
"Further terms with higher powers of momenta can also exist.",
"Figure: Metric loops will generically induce non-vanishing scalar self-interactions with an arbitrary even number of scalar fields.A full study of scalar fields coupled to gravity should include the full momentum-dependent potential $V(\\mathbf {\\Phi })$ .",
"Let us comment on two related issues, namely the question of unitarity, i.e.",
"absence of unphysical degrees of freedom, and the question if the effective equation of motion for $\\phi $ , which can be derived from eq.",
"(REF ), will allow to specify initial data with only a finite number of free parameters.",
"As we have explained, one should generically expect that infinitely many terms of the form $\\mathbf {\\Phi }^n$ will acquire a non-zero coupling through metric fluctuations.",
"Accordingly the effective equations of motion will generically contain infinitely many derivatives, thus naively requiring an infinite number of initial conditions.",
"This naively corresponds to a loss of predictivity at the level of the effective equations of motionThere are two potential sources of loss of predictivity in a scenario with a non-Gaußian fixed point: The first is, that infinitely many non-zero couplings only lead to a predictive theory if only a finite subset of them is relevant.",
"This seems to be the case in asymptotically safe quantum gravity, and implies, that it is possible to write down effective equations of motion which contain an infinite number of terms, but just a finite number of free parameters.",
"Having arrived at this step, the next question is if these effective equations of motion permit a solution with only a finite number of initial conditions, else predictivity is lost at this step.. As explained in detail in [60], differential equations with an infinite number of derivatives do not generically allow to freely specify an infinite number of initial conditions.",
"In fact, each pole in the propagator comes with 2 initial conditions, at least in the case of a free field theory.",
"Thus the question of predictivity is closely linked to the existence of unphysical degrees of freedom: Whenever the propagator acquires a non-physical pole, further initial conditions are necessary to specify the solution to the equations of motion in the free field theory case.",
"If asymptotically safe quantum gravity is a physically permissable theory in the sense of being ghost free, then $\\Gamma _{k\\rightarrow p}(p^2)$ must not have a ghost pole, see [20] for indications that RG trajectories of this type exist.",
"The precise relation between ghost poles and initial conditions must be examined in the fully interacting theory, but assuming that the analysis of the free field case carries over, the avoidance of further propagator poles at the same time implies that the solution to the equations of motion will not require infinitely many initial conditions [60], thus the theory is predictive.",
"Let us emphasise that this question of predictivity cannot be addressed from our calculation, as we truncate the effective action at order $\\phi ^4$ .",
"It is interesting to observe that the fact that infinitely many couplings are non-zero at the fixed point, which at a first glance seems to constitute a problem, really might be the mechanism to precisely avoid this problem and render the theory ghost-free.",
"Let us now consider the anomalous dimension $\\eta _{\\phi }$ .",
"It is of interest for several reasons, the first being, that it enters the scaling relation of any scalar coupling in the vicinity of the fixed point.",
"To see this, consider any coupling $g_i \\int d^4 x \\sqrt{g} \\mathcal {O}_n$ , where $\\mathcal {O}_n$ is an operator containing $n$ powers of the scalar field, an arbitrary power of derivatives and any further operators that depend on the metric, such as, e.g.",
"the curvature scalar $R$ .",
"Redefining the scalar field such that the kinetic term has standard normalisation, $\\phi \\rightarrow \\frac{\\phi }{Z_{\\phi }^{\\frac{1}{2}}}$ , yields a renormalised coupling $\\hat{g}_n= \\frac{g_n}{Z_{\\phi }^{\\frac{n}{2}}}$ .",
"Accordingly the $\\beta $ function for $\\hat{g}_n$ contains a term of the form $\\beta _{\\hat{g}} = n \\eta _{\\phi } \\hat{g}_n+...$ .",
"To calculate the critical exponent associated with this coupling (for simplicity we assume that the stability matrix is already diagonal, but our argument carries through to the case where different operators mix) we take the derivative $-\\frac{\\partial \\beta _{\\hat{g}_n}}{\\partial \\hat{g}_n}=- n \\eta _{\\phi }+...$ .",
"Thus we conclude, that the question, if a coupling is relevant, depends on the anomalous dimension of the corresponding field, see also [22] for the application of this argument to the Faddeev-Popov ghost sector.",
"To obtain the complete set of relevant directions we need to consider the anomalous dimension.",
"In the case of the scalar field coupled to gravity, we observe that increasing values of $g$ induce a change of sign in $\\eta _{\\phi }$ at both fixed points, such that $\\eta _{\\phi }$ becomes negative, see fig.",
"REF .",
"Thus further matter couplings besides those included in our truncation, may become relevant.",
"Figure: Here we plot the value of η φ \\eta _{\\phi } as a function of gg for λ=1/10\\lambda = 1/10 at the fixed points as in fig.",
".",
"Towards larger values of gg, η φ \\eta _{\\phi } becomes increasingly negative at both fixed points, implying that further matter couplings could be shifted into relevance.The anomalous dimension is also of interest for a second reason: Asymptotic safety is assumed to yield a fractal spacetime in the sense that one manifold is endowed with a family of metrics, labelled by the RG scale $k$ [16], [61], [42].",
"In particular, indications are found that in the fixed-point regime, an effective dimensional reduction to $d=2$ occurs.",
"There are several possible interpretations for this result: The first implies that the effective dimensional reduction is visible to all fields, i.e.",
"metric as well as matter fluctuations behave as in 2 dimensions.",
"The second interpretation is, that this effective dimensional reduction is felt by the gravitational degrees of freedom only, but matter fields cannot be described as propagating in 2 dimensions at high momentum scales.",
"The first interpretation would require $\\eta \\rightarrow -2$ for all fields, since then the propagator scales logarithmically with distance, i.e.",
"as in 2 dimensions.",
"Naturally, we do not expect to obtain the exact value $-2$ in any truncation.",
"On the other hand we do expect, that the sign of $\\eta $ is not dependent on the truncation, since the sign crucially decides on the number of relevant operators, which is generally expected to be a truncation-independent quantity in the sense that the relevance of any operator should not depend on the truncation in which its $\\beta $ function is evaluated.",
"Thus a positive anomalous dimension would imply, that the dimensional reduction is a mechanism which applies to the propagation of gravitational fluctuations, but not necessarily to matter fluctuations.",
"Here, we observe that for increasing $g$ , the values of $\\eta _{\\phi }$ become negative.",
"Extensions of the truncation can induce changes $\\mathcal {O}(1)$ in these quantities, so whether a values of $\\eta _{\\phi } \\sim -2$ will be approached in a more complete truncation remains to be investigated.",
"We conclude that the observed value of $\\eta _{\\phi }$ at the shifted Gaußian fixed point in this truncation is not incompatible with some form of dimensional reduction.",
"A crucial consequence of the anomalous dimension is the momentum-dependence of the propagator, $\\left(\\Gamma _k^{(2)}\\right)^{-1}= \\left( Z_k p^2\\right)^{-1}$ .",
"In a single-scale setting, where the momentum of the scalar field sets the physical scale, the identification $k^2=p^2$ is reasonable, its justification being that in such a setting a tree level evaluation of the effective action suffices to evaluate the leading order contribution to physical quantities such as scattering cross sections.",
"Thus we have that $\\left(\\Gamma _k^{(2)}\\right)^{-1}=\\left( p^2\\right)^{1-\\eta /2}$ .",
"This type of RG improvement can be used to deduce experimental consequences of asymptotically safe quantum gravity (or, within the effective field-theory framework, of other UV completions for gravity).",
"Here we conclude that the observed values of $\\eta _{\\phi }<0$ for larger $g$ imply a suppression of scalar fluctuations in comparison to the perturbative expectation.",
"Thus, the contribution of high-momentum scalar fluctuations in loop diagrams is suppressed, which can have observable consequences.",
"Note that $\\eta _{\\phi }>0$ makes the scalar propagator even more divergent than in perturbation theory.",
"However this does not imply that physical quantities will be divergent at this fixed point.",
"It is to be expected that anomalous scaling of vertices yields finite values for scattering amplitudes etc."
],
[
"Fixed-point analysis for the full truncation",
"As discussed, we observe two interating fixed points in the matter sector at small values of $g$ and $\\lambda $ , which are connected continuously to the Gaußian fixed point.",
"These vanish if we go beyond certain critical values of $g$ and $\\lambda $ , e.g.",
"along the line $\\lambda = 1/10$ , they vanish for $g>0.35$ .",
"Let us note that within an extended truncation, this can very well change: On the one hand, further contributions to the matter beta functions exist, e.g.",
"from further matter couplings, such as $\\phi ^6$ couplings.",
"These can in principle balance the effect of metric fluctuations and allow the shifted Gaußian fixed points to exist beyond the values found in this truncation.",
"Furthermore, the fixed-point values in the gravitational sector depend on the inclusion of further operators, and further matter fields.",
"Finally, we work within a single-metric approximation, and do not take the bimetric structure of the RG flow in the background-field formalism into account in our calculation.",
"Thus our results do not imply that within extended truncations, a shifted Gaußian fixed point cannot exist in the full truncation.",
"Here we make the following observation: The full system of $\\beta $ functions for the matter and gravity sector (see app.",
"for the matter $\\beta $ functions for general shape function) does not admit a Gaußian matter fixed point (where gravitational couplings are non-zero, but matter couplings are zero), and it also does not admit the existence of an interacting fixed point which is connected continuously to the Gaußian one for $g \\rightarrow 0$ .",
"We find a non-Gaußian fixed points at considerably larger values of the matter couplings, see tab.",
"REF .",
"This fixed point seems to be the continuation of the third fixed point in the pure matter system, see tab.",
"REF .",
"Note that the effect of metric fluctuations is to considerably change the values of the critical exponents, albeit keeping them negative thus corresponding to irrelevant couplings.",
"Table: Here we give the fixed-point values of all, gravitational and matter couplings, at the non-Gaußian fixed point.",
"We also list the value of the anomalous dimension, and the critical exponents.",
"We find two critical exponents with positive real part, as well as for critical exponents with negative real part, corresponding to to irrelevant directions.Since the momentum-independent coupling $\\lambda _{\\phi }$ is negative at the fixed point, the Euclidean effective potential seems unbounded from below.",
"Here it is important to realise that higher-order terms in the scalar field will be induced by metric and scalar fluctuations.",
"These can result in a bounded effective potential, even for $\\lambda _{\\phi }<0$ .",
"Thus we cannot discard this fixed point on the grounds of stability.",
"The mechanism, why the shifted Gaußian fixed point in the matter couplings vanishes completely for larger values of $g$ and $\\lambda $ , but not the non-Gaußian one, is simple to understand: Whereas the effect of metric fluctuations is to shift the matter $\\beta $ function in such a direction as to induce the vanishing of fixed points, matter fluctuations have the opposite effect: The pure matter system admits the existence of non-trivial fixed points.",
"Thus the only scenario in which a full fixed point can survive the onset of metric fluctuation is one where the matter sector itself is strongly interacting, since then clearly the effect of metric fluctuations is less dominant than at the Gaußian matter fixed point.",
"We thus arrive at the following conclusion: In a scalar theory coupled to asymptotically safe quantum gravity, there is no Gaußian matter fixed point.",
"For very small values of $g$ it becomes shifted into an interacting fixed point.",
"Finally, at larger values of $g$ , where the gravitational $\\beta $ functions have fixed points, the shifted Gaußian fixed point in the matter sector does not exist any more (i.e.",
"the zeros of the $\\beta $ function lie in the complex plane away from the real axis).",
"There is a fully non-Gaußian fixed point at larger values of the matter couplings.",
"Canonical power-counting does not hold at this fixed point, due to the large fixed-point values.",
"We conclude that in constructing a quantum theory of gravitational and matter degrees of freedom based on asymptotic safety the fact that the gravitational sector remains interacting in the UV implies that there will be a strongly-coupled matter sector.",
"Within our truncation we find no evidence for a picture where the far UV is dominated by metric fluctuations only, and the matter sector becomes trivial.",
"In contrast, the UV behaviour of the theory is determined by an interacting matter and gravity theory, where metric fluctuations induce non-vanishing matter couplings, and accordingly matter fluctuations become important and drive the running couplings in both sectors.",
"Interestingly, none of the matter couplings is shifted into relevance, although quantum fluctuations induce large departures from canonical scaling.",
"This is also reflected in the large anomalous dimension $\\eta _{\\phi }$ .",
"Note however, that two of the critical exponents are rather close to zero.",
"It is thus conceivable, that similar shifts in couplings of canonical dimensionality $-2$ result in these couplings becoming relevant.",
"Therefore, extended truncations of scalar matter coupled to gravity might exhibit more relevant couplings than the sum of relevant couplings in the gravitational and the matter sector considered separately.",
"We observe that since this fixed point is dominated by matter fluctuations, the anomalous dimension for the scalar is positive.",
"Let us discuss the dependence of our results on the choice of regulator: Here, we employ a spectrally adjusted cutoff, that implies the existence of terms $\\sim \\eta _{\\phi }, \\eta _N, \\partial _t \\lambda $ on the right-hand side of the flow-equation.",
"These terms result in a clearly non-perturbative structure of the flow equation, since $\\eta _{\\phi }$ will also depend on the inverse of the matter couplings, see app. .",
"In Yang-Mills theory, this structure is crucial to uncover the existence of an infrared attractive non-Gaußian fixed point, see [58].",
"A regulator of this type has also been employed in [22] to evaluate the ghost anomalous dimension in asymptotic safety, as compared to a non-spectrally adjusted one for the evaluation of the same quantity in [23].",
"In the same choice of gauge, the difference in regularisation scheme yielded $\\eta _c = -1.3$ vs. $\\eta _c = -1.8$ .",
"This exemplifies that numerical differences follow from the use of one versus the other regularisation scheme, but important conclusions, in particular about the relevance of couplings, are not affected.",
"Most importantly, as has been pointed out in [63], there exists a choice of non-spectrally adjusted cutoff which reproduces the fixed-point values found with a spectrally adjusted cutoff.",
"We thus conclude that our main results on the existence of fixed points and most importantly the critical exponents should not be affected by a change in the cutoff procedure.",
"Since new matter-couplings are induced by metric fluctuations, it is interesting to investigate their backreaction onto the gravitational sector.",
"This is another test of the consistency of the asymptotic-safety scenario: To decide whether a fixed point is an artifact of a truncation, or exists in full theory space, it is useful to investigate its stability under extensions of the truncation.",
"Furthermore, the numerical values of the critical exponents are a good measure of the stability of the fixed point under such extensions.",
"Numerous results exist showing the stability of the fixed point under various extensions of the truncation [12], [13], [14], [29], [30], [15], [18], [19], [20], [35], [36], [23], [22], [24], [25], [26].",
"Here, we add further evidence, by including the couplings $Z_{\\phi }$ and $\\rho _i$ into the flow of $g$ and $\\lambda $ .",
"At a first glance, one would not expect a backcoupling of $\\rho _i$ into the gravitational $\\beta $ functions $\\beta _g$ and $\\beta _{\\lambda }$ , since the terms $\\sim \\rho _i$ in the effective action do not contribute to the scalar two-point function at vanishing scalar field, see also sect. .",
"However the $\\rho _i$ contribute to $\\eta _{\\phi }$ , which does of course enter the gravitational $\\beta $ functions.",
"Since $Z_{\\phi }(k)$ corresponds to an inessential coupling, it can be eliminated from the other $\\beta $ functions, by simply inserting $\\eta _{\\phi }(g, \\lambda , \\rho _a,\\rho _b,\\rho _c)$ .",
"In this fashion the $\\rho _i$ enter $\\beta _{g}$ and $\\beta _{\\lambda }$ , and can thus alter the RG flow of $g$ and $\\lambda $ .",
"As a result, we point out that the RG flow projected onto the Einstein-Hilbert subsector $\\lbrace \\beta _g, \\beta _{\\lambda }\\rbrace $ is very stable under this extension of the truncation.",
"As can be seen in fig.",
"REF , changing the value of the matter couplings considerably only leads to a very slight change in the fixed-point values and the structure of the RG flow in the Einstein-Hilbert sector.",
"In particular, our extended truncation again permits to choose initial conditions for the RG flow within the UV-critical surface of the NGFP which give a trajectory with a long classical regime, see fig.",
"REF , and, e.g.",
"[62].",
"Figure: Here we plot the RG flow towards the infrared, projected onto the Einstein-Hilbert plane for ρ a =47.094\\rho _a=47.094, ρ b =24.771\\rho _b=24.771, ρ c =66.120\\rho _c=66.120 and λ φ =-127.502\\lambda _{\\phi }= -127.502, cf.",
"tab. .",
"Clearly the flow resembles the flow in the pure Einstein-Hilbert truncation at vanishing matter couplings to a very high degree and admits trajectories passing very close to the Gaußian fixed point and yielding a constant value of the dimensionful Newton coupling and cosmological constant, in accordance with observations.Taken together, we take this as further evidence for the NGFP in the gravity theory not being an artifact of the truncation, but existing in full theory space."
],
[
"The possibility of observable effects in the CMB",
"Inflation with a scalar field within asymptotically safe quantum gravity has been studied in [64].",
"The main conclusion is that inflation and a scale-invariant spectrum of scalar perturbations are possible in such a setting.",
"As is known, derivative interactions do not alter the slow-roll conditions for inflation, since within the standard scenario, the classical background value of the scalar field is constant over space, and $\\frac{1}{2}\\dot{\\phi }^2<< V(\\phi )$ .",
"Clearly adding any type of derivative-interaction to the potential respects these conditions.",
"Momentum-dependent interaction terms have been studied in the context of inflation, see, e.g.",
"[65].",
"Thus our result, that metric fluctuations generate momentum-dependent scalar interactions, will not affect the conclusion that asymptotically safe quantum gravity can admit inflation with a scalar field.",
"Interestingly, one of the operators that we investigate here, namely $\\Phi ^2$ has been shown [66] to potentially give larger non-Gaußianities in the spectrum of the scalar field fluctuations than in the standard slow-roll inflation without this additional term [67].",
"Following [66], the result for the non-Gaußianity parameter $f_{NL}$This parameter is defined as follows: The power spectrum $\\mathcal {P}(p_1, p_2)$ and the bispectrum $\\mathcal {B}(p_1, p_2,p_3)$ of the scalar perturbations are defined via the Fouriertransform of the two- and the three-point function, modulo a $\\delta $ function for momentum-conservation.",
"Then the simplest measure for non-Gaußianity is $f_{NL}= \\frac{5}{18}\\frac{\\mathcal {B}(p,p,p)}{\\mathcal {P}(p,p)^2}$ .",
"is $f_{NL} \\sim \\bar{\\rho }_a$ .",
"The value for $f_{NL}$ thus depends on the identification of $k$ : Evaluating the three-point function of the scalar field fluctuations during inflation requires the identification of the RG scale $k$ with a physical scale of the system.",
"Here, a possible choice is $k \\sim H$ where $H$ is the Hubble scale during inflation, see, e.g.",
"[68] and references therein.",
"The precise scale-identification is crucial for a quantitatively meaningful prediction of the non-Gaussianity.",
"Since asymptotically safe quantum gravity necessitates the existence of such a term, this effect could in principle be an observable consequence of asymptotic safety with experiments.",
"For a prediction of the numerical value of $f_{NL}$ , the flow of $\\bar{\\rho }_a$ on a trajectory potentially describing our universe is necessary.",
"In particular this will crucially depend on terms beyond our truncation, such as $\\mathcal {O}(\\phi ^6)$ terms, which give a direct contribution to $\\beta _{\\rho _a}$ .",
"In a more general setting, fluctuations of the metric generate this term within the effective-field theory setting.",
"Since it leaves an observable imprint in the CMB, measurements of the non-Gaußianity allow us in principle to access the RG flow of Newton's coupling and the cosmological constant in other quantum gravity proposals.",
"In more detail, the idea is the following one: Different quantum gravity proposals differ in the choice of their fundamental variables, the realisation of fundamental symmetries etc.",
"Thus the \"pure quantum gravity regime\", in which even the notion of a continuous spacetime is often postulated to break down, looks widely different in distinct quantum gravity proposals.",
"They agree on the necessity to recover classical Einstein gravity at small momentum (large length) scales.",
"Presumably there is a regime between a \"pure quantum gravity regime\" and the classical regime, in which quantum gravity fluctuations start to be important, but they can be calculated within a framework, in which the symmetries of Einstein gravity hold, and the metric is used as an effective degree of freedom.",
"This regime is a semiclassical regime, where our calculation holds, and it presumably dominates energy scales at which inflation takes place.",
"Within this regime, the RG framework can be applied.",
"The crucial point to realise is that different proposals for quantum gravity will translate into different initial (i.e.",
"high-energy) values for the couplings, at which the RG flow will start.",
"In order for the quantum gravity proposal to be realised in our universe, the RG flow must reach a region where $G_N = \\rm const$ on small momentum scales.",
"On scales slightly below the initial UV scale, the values of different couplings can be very different for different quantum gravity proposals.",
"What they all have in common is the generation of matter-interaction terms such as the one that we have investigated here.",
"Thus many proposals for quantum gravity, different as they may be in the very far UV, will all generate a non-zero value for $\\rho _a$ at energy scales presumably below the Planck scales.",
"Bridging the gap between the fundamental description and the effective-field theory description then allows to predict as specific RG trajectory $\\rho _a(k)$ for a specific UV completion.",
"As $\\rho _a$ is in principle accessible to observations in the non-Gaußianity of the CMB, we can in principle use this observation to infer the RG trajectory of the Newton coupling and the cosmological constant.",
"Thus different UV completions for gravity can be distinguished at the level of observation."
],
[
"Conclusions and outlook",
"We have studied quantum gravity coupled to matter, and shown that in the case of a scalar field, quantum gravity fluctuations, which we parameterise as metric fluctuations, generate scalar self-interactions when coupled to gravity.",
"We point out that this is a generic feature of quantum gravity fluctuations: Whenever a free matter field theory is coupled to gravity, the $\\sqrt{g}$ generates matter-graviton vertices, and graviton loops generate matter self-interactions, in particular in the non-perturbative regime.",
"Here we have shown that in the case of a scalar field, the induced interactions are momentum dependent, and thus have the form $\\left(\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi \\right)^{2n}$ , $\\phi ^n (\\nabla ^2 \\phi )^n$ and $\\left(\\partial _{\\mu }\\phi \\partial ^{\\mu }\\phi \\right)^n (\\phi \\nabla ^2 \\phi )^n$ with $n \\ge 1$ .",
"Most importantly, these matter self-interactions also couple into the flow of the scalar effective potential, and remove the Gaußian fixed point in the scalar momentum-independent coupling $\\lambda _{\\phi }$ .",
"Thus we conclude that quantum gravity fluctuations alter the properties and the dynamics of matter systems.",
"Let us stress that this result holds within asymptotically safe quantum gravity, but also in an effective-field-theory setting, where the UV completion for gravity is not specified.",
"We thus conclude, that quantum gravity effects generically alter the properties and dynamics of matter theories in comparison to the gravity-free case.",
"As shown in a previous work [27], [28], a similar mechanism is at work when fermions are coupled minimally to gravity: Starting with only a minimally coupled kinetic term, four-fermion interactions are induced, even if these four-fermion couplings are set to zero initially.",
"Due to the dimensionality of $\\frac{3}{2}$ of a fermion field, the induced fermionic self-interactions are not momentum-dependent in this case.",
"Let us comment on the interesting case of gauge fields coupled to gravity, studied in [69], [70], [71], [72], [73], [32], [31], [34], which is the subject of future work: In a similar way to what we observed here, higher-order gauge-field self-interactions will presumably be generated.",
"Since there is no symmetry to forbid the generation of these terms, only a highly non-trivial cancellation mechanism between diagrams could avoid the generation of some of the infinitely many $\\left(F_{\\mu \\nu }F^{\\mu \\nu }\\right)^n$ terms.",
"In principle, quantum gravity effects could even lead to an unstable matter action, thus excluding this type of UV completion for gravity.",
"Let us also note that a similar mechanism is expected to be at work in a first-order formulation of gravity, as examined in [74], [75] in the context of asymptotic safety.",
"Noting that the metric-induced matter interactions can alter the dynamics of matter fields at high momenta, and also potentially increase the number of relevant couplings, and thus the free parameters of the theory, one might wonder if there is a way to avoid this mechanism.",
"At a first glance, unimodular quantum gravity [76], [77], where $\\sqrt{g}$ is held fixed, and which, at the classical level is equivalent to General Relativity, might provide a solution: One might think that holding $\\sqrt{g}$ fixed avoids the generation of matter-graviton-interaction vertices from a free theory.",
"However this is only partially right, since kinetic terms usually contain further factors of the metric.",
"Even if the covariant derivative reduces to a partial derivative, as in the case of a scalar field, further factors of the inverse metric exist (as in $g^{\\mu \\nu } \\partial _{\\mu }\\phi \\partial _{\\nu }\\phi $ ).",
"These can be expanded in an infinite series in the fluctuation metric, thus again generating metric-matter interactions, which induce matter self-interactions through loop diagrams.",
"We conclude that the mechanism which generates matter self-interactions when a free matter theory is coupled to gravity, seems to be rather generic, and can presumably not be avoided for any type of matter, be it scalar, fermionic or spin-1.",
"Within our truncation, we find the following interesting results: Metric fluctuations shift the so-called Gaußian matter fixed point to make it an interacting one, where momentum-dependent interaction terms have non-zero couplings and also induce a non-vanishing value for the momentum-independent coupling $\\lambda _{\\phi }$ .",
"First treating the (dimensionless) Newton coupling $g$ and the cosmological constant $\\lambda $ as external parameters, we find two fixed points in the matter $\\beta $ functions which are connected continuously to the Gaußian fixed point for $g \\rightarrow 0$ and which exist for small values of $\\lambda $ and $g$ .",
"Most importantly, the critical exponent of the matter couplings at the fixed points are changed considerably in comparison to the Gaußian case, and in fact $\\lambda _{\\phi }$ can turn into a relevant coupling.",
"Studying the full system of $\\beta $ functions for $g, \\lambda , \\rho _{a}, \\rho _{b}, \\rho _c, \\lambda _{\\phi }, Z_{\\phi }$ we note that the shfited Gaußian fixed points cease to exist: They exist for small values of $g$ , but since metric fluctuations have the effect shifting the matter $\\beta $ functions such that fixed points tend to be removed, values around $\\lbrace g \\sim 0.25, \\lambda \\sim 0.35\\rbrace $ are too big in order for the shifted Gaußian fixed point to exist.",
"In the full truncation, only a non-Gaußian fixed point with large values of the matter couplings exist.",
"In the limit $g \\rightarrow 0$ , where metric fluctuations are turned off, it is not continuously connected to the Gaußian fixed point.",
"At this non-Gaußian fixed points, the matter couplings remain irrelevant, but the critical exponents deviate from the power-counting dimensionality of the couplings.",
"The gravitational couplings correspond to relevant directions.",
"Thus our 6-coupling theory space admits a UV completion which is fully interacting and has two free parameters.",
"We therefore conclude that within asymptotically safe quantum gravity, it is crucial to take into account the metric-induced matter self-interactions, since their existence implies that in the far UV, both gravitational as well as matter couplings are non-zero and therefore both sector contribute to the RG flow.",
"Most importantly, the critical exponents deviate significantly from the canonical scaling dimensions.",
"We thus conclude, that within a more extended truncation, matter couplings with negative canonical dimensionality could even be shifted into relevance.",
"Note furthermore that matter self-interactions generically couple back into the flow of the gravitational couplings.",
"In our case, this happens since the anomalous dimension $\\eta _{\\phi }$ , which couples into $\\beta _g$ and $\\beta _{\\lambda }$ , depends on the $\\rho _i$ .",
"Potentially, gravity-induced matter self-interactions can then crucially alter the RG flow in the gravitational sector.",
"In the truncation investigated here, the change of the gravitational RG flow in comparison to the matter-free case is only mild, but a complete study of the gravitational RG flow must take into account the metric-induced matter self-interactions.",
"Future directions of this work should obviously include extension of the truncation in the matter sector, but also resolve the single-metric approximation and account for the bimetric nature of the RG flow in the backgrounf-field formalism in gravity.",
"Finally, further matter fields, that in more realistic toy models of the standard model obviously couple to both gravity as well as the scalar sector, can also have a crucial effect on the fixed-point structure in the UV.",
"Note also that the mechanism that we explained here, also applies to Faddeev-Popov ghosts fields, which arise in the context of gauge-fixing.",
"Metric fluctuations can be expected to generate further terms involving ghost fields beyond the simple exponential of the Faddeev-Popov determinant.",
"It should be expected that the ghost sector of asymptotically safe quantum gravity differs significantly from a perturbative ghost sector.",
"Whether this allows for a solution of the Gribov problem in quantum gravity remains to be investigated.",
"We point out that momentum-dependent scalar interaction terms as the one investigated here have been examined in the context of inflation: Assuming that the slow-roll conditions for a scalar field are satisfied, this type of term is also allowed.",
"However it does have interesting consequences for possible observations, since it can induce larger non-Gaußianities in the CMB than within a standard slow-roll scenario.",
"We conclude that, although quantitative precision cannot be expected from our truncation, asymptotically safe quantum gravity might be expected to leave a potentially observable imprint in large non-Gaußianities in the CMB, which might be observable.",
"Acknowledgements I would like to thank H. Gies for helpful comments on the manuscript and N. Afshordi for helpful discussions.",
"Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation."
],
[
"Vertices",
"In the following we list all vertices which can be derived from our truncation.",
"Here, $h = \\bar{g}^{\\mu \\nu }h_{\\mu \\nu }$ and $\\bar{g}^{\\mu \\nu }h_{\\mu \\nu }^{TT}=0$ and $\\bar{D}^{\\mu }h_{\\mu \\nu }^{TT}=0$ .",
"Further components of a decomposition of the fluctuation metric do not couple into the matter $\\beta $ functions.",
"We follow the conventions $\\Phi _a(x)= \\int \\frac{d^4p}{(2 \\pi )^4}\\Phi _a(p)e^{- i \\, p \\cdot x}$ and $\\Gamma ^{(2)}_{\\Phi _a \\Phi _b }(p,q)= \\frac{\\delta ^2 \\Gamma _{k}}{\\delta \\Phi _a(p)\\delta \\Phi _b(-q)}.$ Herein $\\Phi _a$ denotes the components of a superfield, thus $\\Phi _a= \\left(\\phi , h_{\\mu \\nu }^{TT}, h \\right)$ .",
"With this conventions in mind our truncation yields the following vertices: $\\Gamma _{k\\, h \\phi }^{(2)}(p,q)&=& \\frac{1}{4}Z_{\\phi }(k) \\left(-p \\cdot q +q^2\\right) \\phi (q-p)\\nonumber \\\\&{}&+ \\int _{l_1,l_2} \\phi (l_1)\\phi (l_2)\\phi (q-p-l_1-l_2)\\Bigl (\\frac{\\bar{\\rho }_b}{2}\\left( - p \\cdot l_1 l_1^2- p^2 l_1^2+p \\cdot l_1 q^2\\right) + 2 \\bar{\\lambda }_{\\phi } \\nonumber \\\\&{}&+\\frac{\\bar{\\rho }_c}{4}\\left(-l_1 \\cdot q\\, p \\cdot l_2 - l_1 \\cdot l_2 p \\cdot q - p \\cdot l_1 p \\cdot l_2 - l_1 \\cdot l_1 p \\cdot l_2 - l_1 \\cdot l_2 p \\cdot l_2\\right)\\Bigr ).$ Here we use the following notation: $\\int _q = \\int \\frac{d^4 q}{(2 \\pi )^4}$ .",
"$\\Gamma _{k\\, \\phi \\, h}^{(2)}(p,q)&=&\\frac{1}{4}Z_{\\phi }(k) \\left( -p\\cdot q+p^2\\right) \\phi (q-p)\\nonumber \\\\&{}&+ \\int _{l_1,l_2} \\phi (l_1)\\phi (l_2)\\phi (q-p-l_1-l_2)\\Bigl (\\frac{\\bar{\\rho }_b}{2}\\left( q \\cdot l_1 l_1^2- q^2 l_1^2-q \\cdot l_2 p^2\\right) +2 \\bar{\\lambda }_{\\phi }\\nonumber \\\\&{}&+\\frac{\\bar{\\rho }_c}{4}\\left(-l_1 \\cdot p\\, q \\cdot l_2 - l_1 \\cdot l_2 p \\cdot q - q \\cdot l_1 q \\cdot l_2 + l_1 \\cdot l_1 q \\cdot l_2 + l_1 \\cdot l_2 q \\cdot l_2 \\right)\\Bigr ).$ We make the following useful observation: From the kinetic term, there is no vertex with two internal scalar gravitons, so the vertex is $\\sim \\rho _i$ : $\\Gamma _{k\\, h\\, h}^{(2)}(p,q)&=&\\int _{l_1,l_2,l_3} \\phi (l_1)\\phi (l_2)\\phi (l_3)\\phi (q-p-l_1-l_2-l_3)\\Bigl (\\frac{\\bar{\\rho }_b}{8}\\left( q \\cdot l_1 l_2^2-p \\cdot l_1 l_2^2-\\frac{1}{2} p \\cdot l_1 q \\cdot l_2 -\\frac{1}{2} q \\cdot l_1 p \\cdot l_2\\right) \\nonumber \\\\&{}&- \\frac{\\bar{\\rho }_c}{16} \\left(-l_1 \\cdot l_2 q \\cdot l_3 + l_1 \\cdot l_2 p \\cdot l_3\\right) + \\frac{1}{8}\\bar{\\lambda }_{\\phi }\\Bigr ).$ The vertices coupling the scalar to the transverse traceless graviton component read as follows: $\\Gamma _{k\\, h_{\\mu \\nu }^{TT}\\, \\phi }(p,q)&=& -\\frac{Z_{\\phi }(k)}{2}\\phi (q-p)\\left(2 q_{\\mu }q_{\\nu }-q_{\\mu }p_{\\nu }-q_{\\nu }p_{\\mu }+p_{\\mu }p_{\\nu } \\right)\\nonumber \\\\&+&\\int _{l_1,l_2} \\phi (l_1)\\phi (l_2)\\phi (q-p-l_1-l_2)\\Bigl (-2 \\bar{\\rho }_b(k) \\left(2l_{2\\mu }l_{2\\nu } l_1^2 +q_{\\mu }q_{\\nu }l_2^2+ l_{2\\mu }l_{2\\nu }q^2 \\right)\\nonumber \\\\&{}&-2 \\bar{\\rho }_a(k) \\left(\\left(-q_{\\mu }l_{1\\, \\nu }-q_{\\nu }l_{1\\, \\mu }\\right) (-l_2 \\cdot p +l_2 \\cdot q-l_1\\cdot l_2 -l_2^2)+\\left(l_{1\\, \\mu }l_{2\\, \\nu }+l_{1\\, \\nu }l_{2\\, \\mu } \\right) (q\\cdot p+q\\cdot l_1+q \\cdot l_2-q^2)\\right) \\nonumber \\\\&{}& - \\frac{\\bar{\\rho }_c}{2} \\Bigl (l_{2\\mu }l_{2\\nu }\\left(-2 l_1 \\cdot q-2l_1 \\cdot p-2l_1^2-2 l_1 \\cdot l_2 \\right)+q^2 \\left(l_{1\\mu }l_{2\\nu }+l_{1\\nu }l_{2\\mu } \\right)+2 q_{\\mu }q_{\\nu }l_1 \\cdot l_2 \\nonumber \\\\&{}&\\quad \\quad + l_2^2 \\left(-l_{1\\mu }q_{\\nu }-l_{1\\nu }q_{\\mu }-p_{\\mu }l_{1\\nu }-p_{\\nu }l_{1\\mu } -2 l_{1\\mu }l_{1\\nu }-l_{1\\mu }l_{2\\nu }-l_{1\\nu }l_{2\\mu }\\right)\\Bigr )\\Bigr ),$ $\\Gamma _{k\\, \\phi \\, h_{\\mu \\nu }^{TT}}&=& -\\frac{Z_{\\phi }(k)}{2}\\phi (q-p) \\left(2 p_{\\mu }p_{\\nu }-q_{\\mu }p_{\\nu } -q_{\\nu }p_{\\mu }+q_{\\mu }q_{\\nu }\\right)\\nonumber \\\\&+& \\int _{l_1, l_2}\\phi (l_1)\\phi (l_2)\\phi (q-p-l_1-l_2)\\Bigl (-2 \\bar{\\rho }_b(k) \\left(2 l_{1\\mu }l_{1\\nu }l_{2}^2+p_{\\mu }p_{\\nu }l_2^2+l_{2\\mu }l_{2\\nu }p^2 \\right)\\nonumber \\\\&{}&-2 \\bar{\\rho }_a(k)\\left(\\left(p_{\\mu }l_{1\\, \\nu }+p_{\\nu }l_{1\\, \\mu }\\right)(l_2 \\cdot q - l_2\\cdot p -l_1 \\cdot l_2 -l_2^2)+\\left(l_{1\\, \\mu }l_{2\\, \\nu }+l_{1\\, \\nu }l_{2\\, \\mu } \\right)\\left( p \\cdot q - p^2-p \\cdot l_1 -p \\cdot l_2\\right) \\right)\\nonumber \\\\&{}&-\\frac{\\bar{\\rho }_c}{2} \\Bigl (l_{2\\mu }l_{2\\nu }\\left(2 l_1 \\cdot p+2l_1 \\cdot q-2l_1^2-2 l_1 \\cdot l_2 \\right)+p^2 \\left(l_{1\\mu }l_{2\\nu }+l_{1\\nu }l_{2\\mu } \\right)+2 p_{\\mu }p_{\\nu }l_1 \\cdot l_2 \\nonumber \\\\&{}&\\quad \\quad + l_2^2 \\left(l_{1\\mu }p_{\\nu }+l_{1\\nu }p_{\\mu }+q_{\\mu }l_{1\\nu }+q_{\\nu }l_{1\\mu } -2 l_{1\\mu }l_{1\\nu }-l_{1\\mu }l_{2\\nu }-l_{1\\nu }l_{2\\mu }\\right)\\Bigr )\\Bigr ),$ $&{}&\\Gamma _{k\\, h_{\\mu \\nu }^{TT}\\, h_{\\kappa \\lambda }^{TT}}^{(2)}(p,q)\\nonumber \\\\&=&\\frac{Z_{\\phi }(k)}{8} \\int _{l_1}\\phi (l_1)\\,\\phi (q-p-l_1)\\Bigl (l_1^{\\gamma }(-p_{\\gamma }+q_{\\gamma }-l_{1\\, \\gamma })\\left(\\delta _{\\mu \\kappa }\\delta _{\\nu \\lambda }+\\delta _{\\mu \\lambda }\\delta _{\\nu \\kappa } \\right) \\nonumber \\\\&{}& \\phantom{xxxx}+ \\Bigl ( \\left(l_{1\\, \\mu }\\delta _{\\nu \\lambda }+l_{1\\, \\nu }\\delta _{\\mu \\lambda } \\right)(p_{\\kappa }+l_{1\\, \\kappa })+\\left(l_{1\\, \\mu }\\delta _{\\nu \\kappa }+l_{1\\, \\nu }\\delta _{\\mu \\kappa } \\right)(p_{\\lambda }+l_{1\\, \\lambda })\\nonumber \\\\&{}&\\phantom{xxxxxx}+\\left(l_{1\\, \\kappa }\\delta _{\\lambda \\mu }+ l_{1\\, \\lambda }\\delta _{\\kappa \\mu } \\right)\\left(-q_{\\nu }+l_{1\\,\\nu } \\right)+\\left(l_{1\\, \\kappa }\\delta _{\\lambda \\nu }+ l_{1\\, \\lambda }\\delta _{\\kappa \\nu } \\right)\\left(-q_{\\mu }+l_{1\\,\\mu } \\right)\\Bigr )\\Bigr )\\nonumber \\\\&+& \\int _{l_1,l_2,l_3} \\phi (l_1)\\,\\phi (l_2)\\,\\phi (l_3)\\,\\phi (q-p-l_1-l_2-l_3)\\Bigl (\\frac{\\bar{\\rho }_a(k)}{2}\\Bigl [ \\frac{-1}{2}\\left(\\delta _{\\mu \\kappa }\\delta _{\\nu \\lambda }+ \\delta _{\\mu \\lambda }\\delta _{\\nu \\kappa } \\right)l_1 \\cdot l_2 (l_3 \\cdot q - l_3 \\cdot p -l_1\\cdot l_3 -l_2 \\cdot l_3 -l_3^2)\\nonumber \\\\&{}& \\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3} }+ \\Bigl ( l_{1\\, \\mu }l_{2\\, \\kappa }\\delta _{\\nu \\lambda }+l_{1\\, \\mu }l_{2\\, \\lambda }\\delta _{\\nu \\kappa }+l_{1\\, \\nu }l_{2\\, \\kappa }\\delta _{\\mu \\lambda }+l_{1\\, \\nu }l_{2\\, \\lambda }\\delta _{\\mu \\kappa }+l_{1\\, \\kappa }l_{2\\, \\nu }\\delta _{\\mu \\lambda }+l_{1\\, \\kappa }l_{2\\, \\mu }\\delta _{\\nu \\lambda }+ l_{1\\,\\lambda }l_{2\\, \\mu }\\delta _{\\nu \\kappa }+ l_{1\\, \\lambda }l_{2\\, \\nu }\\delta _{\\mu \\kappa } \\Bigr )\\cdot \\nonumber \\\\&{}& \\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}} \\cdot (l_3 \\cdot q - l_3 \\cdot p -l_1 \\cdot l_3 -l_2 \\cdot l_3 -l_3^2)\\nonumber \\\\&{}& \\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3} }+ \\frac{1}{2}\\Bigl (\\left(l_{1\\, \\mu }l_{2\\, \\nu }+l_{1\\, \\nu }l_{2\\, \\mu } \\right)\\left(l_{3\\, \\kappa }\\left(-p_{\\lambda }-l_{1\\, \\lambda }-l_{2\\, \\lambda }-l_{3\\, \\lambda } \\right)+l_{3\\, \\lambda }\\left( -p_{\\kappa }-l_{1\\, \\kappa }-l_{2\\, \\kappa }-l_{3\\, \\kappa }\\right) \\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}}+\\left(l_{1\\, \\kappa }l_{2\\, \\lambda }+ l_{1\\, \\lambda }l_{2\\, \\kappa } \\right)\\left(l_{3\\, \\mu }\\left( q_{\\nu }-l_{1\\, \\nu }-l_{2\\, \\nu }-l_{3\\, \\nu }\\right) +l_{3\\, \\nu }\\left( q_{\\mu }-l_{1\\, \\mu }-l_{2\\, \\mu }-l_{3\\, \\mu }\\right)\\right) \\Bigr )\\Bigr ]\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}}+ \\frac{\\bar{\\rho }_b}{2} \\Bigl [ \\left(-l_1^2 l_2^2 +2 q \\cdot l_1 l_2^2-2 p \\cdot l_1 l_2^2\\right)\\frac{1}{2}\\left(\\delta _{\\mu \\kappa }\\delta _{\\nu \\lambda }+ \\delta _{\\mu \\lambda }\\delta _{\\nu \\kappa } \\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}}+l_2^2 \\left(\\left(p_{\\lambda }l_{1\\mu }-q_{\\mu }l_{1\\lambda }\\right)\\delta _{\\nu \\kappa }+\\left(p_{\\kappa }l_{1\\mu }-q_{\\mu }l_{1\\kappa }\\right)\\delta _{\\nu \\lambda }+\\left(p_{\\lambda }l_{1\\nu }-q_{\\nu }l_{1\\lambda }\\right)\\delta _{\\mu \\kappa }+\\left(p_{\\kappa }l_{1\\nu }-q_{\\nu }l_{1\\kappa }\\right)\\delta _{\\mu \\lambda } \\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}} +2l_2^2 \\left(l_{1\\mu }l_{1\\lambda }\\delta _{\\nu \\kappa }+ l_{1\\mu }l_{1\\kappa }\\delta _{\\nu \\lambda }+ l_{1\\nu }l_{1\\lambda }\\delta _{\\kappa \\mu }+ l_{1\\nu }l_{1\\kappa }\\delta _{\\lambda \\mu } \\right) + 2 \\left(l_{1\\mu }l_{1\\nu }l_{2\\kappa }l_{2\\lambda }+l_{1\\kappa }l_{1\\lambda }l_{2\\mu }l_{2\\nu } \\right) \\Bigr ]\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}}+ \\frac{\\bar{\\rho }_c}{2} \\Bigl [ \\left( - l_1 \\cdot l_2 l_3^2 + l_1 \\cdot l_2 q \\cdot l_3 - l_1 \\cdot l_2 p \\cdot l_3\\right))\\frac{1}{2}\\left(\\delta _{\\mu \\kappa }\\delta _{\\nu \\lambda }+ \\delta _{\\mu \\lambda }\\delta _{\\nu \\kappa } \\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}}+\\frac{1}{2} l_3^2 \\left( \\left(l_{1\\mu }l_{2\\lambda }+l_{1\\lambda }l_{2\\mu }\\right)\\delta _{\\nu \\kappa }+ \\left(l_{1\\mu }l_{2\\kappa }+l_{1\\kappa }l_{2\\mu } \\right)\\delta _{\\lambda \\nu }+ \\left( l_{1\\nu }l_{2 \\lambda }+ l_{1\\lambda }l_{2\\nu }\\right)\\delta _{\\mu \\kappa }+ \\left(l_{1}\\nu l_{2\\kappa }+l_{1\\kappa }l_{2\\nu }\\right)\\delta _{\\lambda \\mu }\\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}}+ l_1 \\cdot l_2 \\left( l_{3\\mu }l_{3\\lambda }\\delta _{\\nu \\kappa }+ l_{3\\mu }l_{3\\kappa }\\delta _{\\nu \\lambda }+ l_{3\\nu }l_{3\\lambda }\\delta _{\\mu \\kappa }+ l_{3\\nu }l_{3\\kappa }\\delta _{\\mu \\lambda }\\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}}+\\frac{1}{2} l_1 \\cdot l_2 \\left(\\left(p_{\\lambda }l_{3\\mu }-q_{\\mu }l_{3\\lambda } \\right)\\delta _{\\nu \\kappa }+ \\left(p_{\\lambda }l_{3\\nu }-q_{\\nu }l_{3\\lambda } \\right)\\delta _{\\mu \\kappa } +\\left(p_{\\kappa }l_{3 \\mu }-q_{\\mu }l_{3\\kappa } \\right)\\delta _{\\lambda \\nu }+ \\left(p_{\\kappa }l_{3\\nu }-q_{\\nu }l_{3\\kappa } \\right)\\delta _{\\mu \\lambda }\\right)\\nonumber \\\\&{}&\\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}\\int _{l_1,l_2,l_3}}+ \\left(l_{1\\mu }l_{2\\nu } +l_{1\\nu }l_{2\\mu }\\right)l_{3\\kappa }l_{3\\lambda }+ \\left( l_{1\\kappa }l_{2\\lambda }+l_{1\\lambda }l_{2\\kappa }\\right)l_{3\\mu }l_{3\\nu }\\Bigr ]\\nonumber \\\\&{}& \\phantom{\\frac{\\bar{\\rho }_{\\phi }(k)}{2}} - \\frac{\\bar{\\lambda }_{\\phi }}{4} \\left(\\delta _{\\mu \\kappa }\\delta _{\\nu \\lambda }+ \\delta _{\\mu \\lambda }\\delta _{\\nu \\kappa } \\right)\\Bigr ).$ Finally $\\Gamma _{k\\, \\phi \\phi }^{(2)}(p,q)&=& \\int _{l_1} \\phi (l_1)\\,\\phi (q-p-l_1) \\Bigl ( \\bar{\\rho }_a(k)\\left(-8 p \\cdot q\\, l_1 \\cdot q +8 p\\cdot q\\, l_1 \\cdot p + 4 p \\cdot q\\, l_1^2+ 8 p \\cdot l_1 \\,q \\cdot l_1 -4 p \\cdot l_1\\, q^2 +4 l_1 \\cdot q \\,p^2 \\right)\\nonumber \\\\&{}&\\bar{\\rho }_b(k)\\left(2 l_1^2 \\left(q^2+p^2+l_1^2-2 q \\cdot p -2 q \\cdot l_1+ 2 p \\cdot l_1\\right) +4 q^2 l_1^2+4 p^2 l_1^2+2 p^2 q^2\\right)\\nonumber \\\\&{}&\\bar{\\rho }_c(k) \\left(-3 p^2 l_1^2-2 p\\cdot l_1 l_1^2+p \\cdot l_1 q^2-3 q^2 l_1^2+2 q \\cdot l_1 l_1^2 -q \\cdot l_1 p^2+2p\\cdot q l_1^2+q \\cdot l_1 q^2 -p \\cdot l_1 p^2 \\right)\\nonumber \\\\&{}& +12 \\bar{\\lambda }_{\\phi }\\Bigr ).$ For the graviton propagators, see [22].",
"The scalar propagator is given by $\\mathcal {P}_{\\phi \\phi }(p,q)= Z_{\\phi }(k) p^2 \\delta ^{4}(p-q)$ ."
],
[
"Matter contribution to gravitational $\\beta $ functions",
"To obtain the matter contribution to the gravitational $\\beta $ functions, we evaluate the flow equation on a spherical background, where $\\bar{D}^2$ is the Laplacian acting on scalar fields, and the expansion of the heat-kernel trace is well-known: $e^{-s \\left( -\\bar{D}^2\\right)}=\\left( \\frac{1}{4 \\pi s}\\right)^2 \\int d^4x \\sqrt{g}\\, \\left(1+ \\frac{1}{6}s\\, R+ ... \\right).$ The first two terms in the expansion suffice to project $\\partial _t \\Gamma _k$ onto $\\beta _{g}$ and $\\beta _{\\lambda }$ .",
"To use this, we rewrite $\\partial _t \\Gamma _{k\\, \\phi }$ , using inverse Laplace transformations for shape functions as obtained in [58], where the subscript $\\phi $ implies that we only consider scalar fluctuation fields, as follows $\\phantom{x}$ $\\phantom{x}$ $\\partial _t \\Gamma _{k\\, \\phi } &=& \\frac{1}{2} {\\rm Tr} \\frac{\\partial _t R_k}{\\Gamma _k^{(2)}+ R_k} \\nonumber \\\\&=&\\frac{1}{2} {\\rm Tr} \\left(\\frac{\\partial _t \\Gamma _k^{(2)}}{\\Gamma _k} \\frac{r(y)}{1+r(y)} \\right)+ \\frac{1}{2}{\\rm Tr} \\frac{r^{\\prime }(y) (-2 y)}{1+r(y)}\\nonumber \\\\&=& - \\frac{\\eta _{\\phi }}{2} \\int _0^{\\infty } ds\\, \\delta (s-1) \\,{\\rm Tr}\\, e^{-s\\frac{-\\bar{D}^2}{k^2}} \\nonumber \\\\&{}&-\\int _0^{\\infty }ds\\, \\sum _{m=1}^{\\infty } \\delta (s-m)\\frac{\\rm d}{{\\rm d}s}\\, {\\rm Tr} \\,e^{-s\\frac{-\\bar{D}^2}{k^2}} ,$ where we explicitly used $r(y)= \\left( e^{y}-1\\right)^{-1}$ .",
"In the next step, we use the heat-kernel expansion eq.",
"(REF ) to obtain the matter contributions to the $\\beta $ functions $\\partial _t g\\vert _{\\rm matter}&=& 16 \\pi \\, g^2 \\left(-\\frac{1}{12 \\,(4 \\pi )^2}\\eta _{\\phi } +\\frac{1}{16 \\cdot 36}\\right)\\nonumber \\\\\\partial _t \\lambda \\vert _{\\rm matter}&=& g \\frac{\\zeta (3)}{ \\pi }+ 16 \\pi \\, g \\, \\lambda \\,\\left( -\\frac{1}{12\\cdot 16 \\pi ^2}\\eta _{\\phi }+ \\frac{1}{16 \\cdot 36}\\right)\\nonumber \\\\&{}&- \\frac{\\eta _{\\phi }\\, g}{4 \\pi }.$"
],
[
"Scale-derivative of the regulator",
"After projecting the right-hand side of the Wetterich equation on the appropriate power of external momenta, it contains up to the fourth derivative of the regulator shape function $r(y)$ .",
"Extending [22], we observe that for a function $f$ depending on the regulator shape function $r_i$ (here $i$ is an index that labels the different fields) and its derivatives the following holds: $&{}&\\tilde{\\partial }_t f(r_i,r_i^{(1)},...,r_i^{(4)})(p)\\nonumber \\\\&=& \\int _{p^{\\prime }} \\partial _t R_i(p^{\\prime }) \\frac{\\delta }{\\delta R_i(p^{\\prime })} f(r_i,r_i^{(1)},...,r_i^{(4)})(p)\\nonumber \\\\&=&\\int _{p^{\\prime }} \\frac{\\partial _t R_i(p^{\\prime })}{Z_i\\left(p^{\\prime 2}-\\lambda _i \\right)}\\left( f^{(1,0,0,0,0)}(r_i,...,r_i^{(4)})(p) \\delta _{ij}\\delta _{p,p^{\\prime }}+...+f^{(0,0,0,0,1)}(r_i,...,r_i^{(4)})(p) \\delta _{ij}\\partial _{\\tilde{y}}^4\\delta _{p,p^{\\prime }}\\right)\\nonumber \\\\&=&f^{(1,0,0,0,0)}(r_i,r_i^{(1)},...,r_i^{(4)})(p) \\frac{\\partial _t R_i(p)}{Z_i\\left(p^2-\\lambda _i \\right)}+...+f^{(0,0,0,0,1)}(r_i,r_i^{(1)},...,r_i^{(4)})(p) \\partial _{\\tilde{y}}^4 \\frac{\\partial _t R_i(p)}{Z_i\\left(p^2-\\lambda _i \\right)},$ wherein $\\tilde{y}= \\frac{p^2- \\lambda _i}{k^2}$ is the argument of the regulator shape function.",
"Herein $\\lambda _i=0$ for the scalar matter shape function.",
"Rewriting $\\partial _{\\tilde{y}}= \\frac{k^2}{2p^2}p_{\\mu }\\frac{\\partial }{\\partial p_{\\mu }}$ allows to deduce the form of $\\partial _{\\tilde{y}}^n \\frac{\\partial _t R)i(p)}{Z_i\\left(p^2-\\lambda _i \\right)}$ straightforwardly."
],
[
"$\\beta $ functions for unspecified shape function",
"Here we present the $\\beta $ functions for the dimensionfull (unrenormalised) quantities for a regulator of the form $\\Gamma _k^{(2)}\\Big |_{\\phi =0}r(\\frac{p^2}{k^2})$ .",
"All numerical results in the main text are obtained using an exponential shape function.",
"$\\beta _{\\bar{\\rho }_a}&=&\\frac{1}{2}\\int \\frac{d^4p}{(2 \\pi )^4} \\Bigl [\\left( \\frac{10}{3} \\bar{\\rho }_a+ \\frac{10}{9} \\bar{\\rho }_b-\\frac{5}{9}\\bar{\\rho }_c\\right)\\tilde{\\partial }_t\\mathcal {G}_{TT} -\\frac{1}{2} \\frac{575}{1728} Z_{\\phi }^2\\tilde{\\partial }_t\\mathcal {G}_{TT}^2\\nonumber \\\\&{}& -\\frac{1}{2}\\Bigl [ \\left(40 (p^2)^2 \\bar{\\rho }_a^2- 80 (p^2)^2 \\bar{\\rho }_a\\bar{\\rho }_b+ 16 (p^2)^2 \\bar{\\rho }_b^2+ 48 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_c- \\frac{40}{3}(p^2)^2 \\bar{\\rho }_a\\bar{\\rho }_c+ 32 (p^2)^2 \\bar{\\rho }_b\\bar{\\rho }_c+ 2 (p^2)^2 \\bar{\\rho }_c^2 \\right)\\tilde{\\partial }_t\\mathcal {G}_{\\phi }^2\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( 480 p^2\\bar{\\lambda }_{\\phi }\\bar{\\rho }_a-192 p^2 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b+ \\frac{400}{3} (p^2)^3 \\bar{\\rho }_a\\bar{\\rho }_b- 80 (p^2)^3 \\bar{\\rho }_b^2- 144 \\bar{\\lambda }_{\\phi }p^2 \\bar{\\rho }_c- 40 (p^2)^3 \\bar{\\rho }_b\\bar{\\rho }_c\\right) \\tilde{\\partial }_t\\mathcal {R}^{(1)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+\\left( -288 \\bar{\\lambda }_{\\phi }^2 + 160 (p^2)^2 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_a- 384 \\bar{\\lambda }_{\\phi }(p^2)^2 \\bar{\\rho }_b-48 (p^2)^2 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_c+\\frac{80}{3}(p^2)^4\\bar{\\rho }_a\\bar{\\rho }_b-72 (p^2)^4 \\bar{\\rho }_b^2- 8 (p^2)^4 \\bar{\\rho }_b\\bar{\\rho }_c\\right) \\tilde{\\partial }_t\\mathcal {R}^{(2)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( -288 p^2 \\bar{\\lambda }_{\\phi }^2-160 (p^2)^3 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b-\\frac{56}{3}(p^2)^5 \\bar{\\rho }_b^2\\right)\\tilde{\\partial }_t\\mathcal {R}^{(3)} \\mathcal {G}_{\\phi }^3 +\\left(-48 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 - 16 \\bar{\\lambda }_{\\phi }(p^2)^4 \\bar{\\rho }_b- \\frac{4}{3} (p^2)^6 \\bar{\\rho }_b^2\\right) \\tilde{\\partial }_t\\mathcal {R}^{(4)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+ \\left(\\!576 \\bar{\\lambda }_{\\phi }^2-\\!320 (p^2)^2 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_a+\\!768 \\bar{\\lambda }_{\\phi }(p^2)^2 \\bar{\\rho }_b-\\!",
"\\frac{160}{3} (p^2)^4 \\bar{\\rho }_a\\bar{\\rho }_b+\\!",
"144 (p^2)^4 \\bar{\\rho }_b^2 +\\!96 \\bar{\\lambda }_{\\phi }(p^2)^2 \\bar{\\rho }_c+ \\!16 (p^2)^4 \\bar{\\rho }_b\\bar{\\rho }_c\\!\\right)\\tilde{\\partial }_t(\\mathcal {R}^{(1)} )^2\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( 288 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 +96 \\bar{\\lambda }_{\\phi }(p^2)^4 \\bar{\\rho }_b+ 8 (p^2)^6 \\bar{\\rho }_b^2\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(2)} \\right)^2\\mathcal {G}_{\\phi }^4+\\left(1728 \\bar{\\lambda }_{\\phi }^2 p^2 +960 (p^2)^3 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b+ 112 (p^2)^5 \\bar{\\rho }_b^2\\right)\\tilde{\\partial }_t\\mathcal {R}^{(1)} \\mathcal {R}^{(2)}\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\, +\\left( 384 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 + 128 \\bar{\\lambda }_{\\phi }(p^2)^4 \\bar{\\rho }_b+ \\frac{32}{3} (p^2)^6 \\bar{\\rho }_b^2\\right)\\tilde{\\partial }_t\\mathcal {R}^{(1)} \\mathcal {R}^{(3)}\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( -1728 \\bar{\\lambda }_{\\phi }^2 p^2 -960 \\bar{\\lambda }_{\\phi }(p^2)^3 \\bar{\\rho }_b- 112 (p^2)^5 \\bar{\\rho }_b^2\\right) \\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^3 \\mathcal {G}_{\\phi }^5\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+ \\left(-1728 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 -576 \\bar{\\lambda }_{\\phi }(p^2)^4 \\bar{\\rho }_b- 48 (p^2)^6 \\bar{\\rho }_b^2 \\right) \\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^2 \\mathcal {R}^{(2)} \\mathcal {G}_{\\phi }^5\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(1152 \\bar{\\lambda }_{\\phi }^2 (p^2)^2+384 \\bar{\\lambda }_{\\phi }(p^2)^4 \\bar{\\rho }_b+32 (p^2)^6 \\bar{\\rho }_b^2 \\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^4\\mathcal {G}_{\\phi }^6\\Bigr ]\\nonumber \\\\&{}&- \\frac{1}{2} Z{\\phi }\\left( \\frac{20}{9} \\bar{\\rho }_bp^2 - \\frac{5}{9} \\bar{\\rho }_cp^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT} \\mathcal {G}_{\\phi \\phi }- \\frac{1}{2} Z{\\phi }\\left( -\\frac{1}{96} \\bar{\\rho }_cp^4 \\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(1)}\\mathcal {G}_{\\phi \\phi }^2\\nonumber \\\\&{}& +\\frac{1}{3}Z_{\\phi }^2\\Bigl [ \\left( \\frac{3}{8} \\bar{\\rho }_a(p^2)^2 + \\frac{1}{8}\\bar{\\rho }_b(p^2)^2- \\frac{1}{16} \\bar{\\rho }_c(p^2)^2\\right)\\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^2+ \\left( -\\frac{1}{4} \\bar{\\rho }_b(p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^3 \\mathcal {R}^{(1)}\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+ \\left( \\frac{3}{4} \\bar{\\lambda }_{\\phi }(p^2)^2 + \\frac{1}{8} \\bar{\\rho }_b(p^2)^4\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^4 \\left(\\mathcal {R}^{(1)} \\right)^2\\Bigr ]+\\frac{1}{3} Z_{\\phi }^2 \\left( 20 \\bar{\\lambda }_{\\phi }+ \\frac{10}{3} \\bar{\\rho }_b(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT} \\mathcal {G}_{\\phi \\phi }^2\\nonumber \\\\&{}& -\\frac{1}{4} \\frac{1}{1024}Z_{\\phi }^4 \\tilde{\\partial }_t\\mathcal {G}_{\\phi \\phi }^2 \\mathcal {G}_{hh}^2\\Bigr ].$ $\\beta _{\\bar{\\rho }_b}&=&\\frac{1}{2}\\int \\frac{d^4p}{(2 \\pi )^4} \\Bigl [ \\frac{10}{3} \\bar{\\rho }_b \\tilde{\\partial }_t\\mathcal {G}_{TT}-\\frac{1}{2} \\frac{85}{576} Z_{\\phi }^2\\tilde{\\partial }_t\\mathcal {G}_{TT}^2\\nonumber \\\\&{}& -\\frac{1}{2}\\Bigl [ \\left(240 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b+216 (p^2)^2 \\bar{\\rho }_b^2 - 72 (p^2)^2 \\bar{\\rho }_b\\bar{\\rho }_c+10 \\bar{\\rho }_c^2 (p^2)^2 \\right)\\tilde{\\partial }_t\\mathcal {G}_{\\phi }^2\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+\\left( -768 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_bp^2-240 \\bar{\\rho }_b^2 (p^2)^3+144 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_cp^2+40 \\bar{\\rho }_b\\bar{\\rho }_c(p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {R}^{(1)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+\\left( -288 \\bar{\\lambda }_{\\phi }^2-576 (p^2)^2 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b-104 \\bar{\\rho }_b^2 (p^2)^4 +48 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_c(p^2)^2+8 \\bar{\\rho }_b\\bar{\\rho }_c(p^2)^4\\right) \\tilde{\\partial }_t\\mathcal {R}^{(2)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+\\left( -288 \\bar{\\lambda }_{\\phi }^2 p^2-160 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^3 - \\frac{56}{3} \\bar{\\rho }_b^2 (p^2)^5\\right) \\tilde{\\partial }_t\\mathcal {R}^{(3)}\\mathcal {G}_{\\phi }^3+\\left( -48 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 -16 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4 - \\frac{4}{3} \\bar{\\rho }_b^2 (p^2)^6\\right) \\tilde{\\partial }_t\\mathcal {R}^{(4)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+ \\left( 576 \\bar{\\lambda }_{\\phi }^2+1152 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^2+208 \\bar{\\rho }_b^2 (p^2)^4-96 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_c(p^2)^2-16 \\bar{\\rho }_b\\bar{\\rho }_c(p^2)^4\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)} \\right)^2\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(288 \\bar{\\lambda }_{\\phi }^2 (p^2)^2+96 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4+8 \\bar{\\rho }_b^2 (p^2)^6\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(2)} \\right)^2\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( 1728 \\bar{\\lambda }_{\\phi }^2 p^2+960 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^3+112 \\bar{\\rho }_b^2 (p^2)^5\\right)\\tilde{\\partial }_t\\mathcal {R}^{(1)} \\mathcal {R}^{(2)}\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\, +\\left(384 \\bar{\\lambda }_{\\phi }^2 (p^2)^2+128 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4+\\frac{32}{3} \\bar{\\rho }_b^2 (p^2)^6\\right)\\tilde{\\partial }_t\\mathcal {R}^{(1)} \\mathcal {R}^{(3)}\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( -1728 \\bar{\\lambda }_{\\phi }^2 p^2-960 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^3-112 \\bar{\\rho }_b^2 (p^2)^5\\right) \\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^3 \\mathcal {G}_{\\phi }^5\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+ \\left( -1728 \\bar{\\lambda }_{\\phi }^2 (p^2)^2-576 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4-48 \\bar{\\rho }_b^2 (p^2)^6\\right) \\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^2 \\mathcal {R}^{(2)} \\mathcal {G}_{\\phi }^5\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(1152 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 + 384 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4 + 32 \\bar{\\rho }_b^2 (p^2)^6\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^4\\mathcal {G}_{\\phi }^6\\Bigr ]\\nonumber \\\\&{}&- \\frac{1}{2} Z{\\phi }\\left( \\frac{5}{3} \\bar{\\rho }_bp^2 \\right) \\tilde{\\partial }_t\\mathcal {G}_{TT} \\mathcal {G}_{\\phi \\phi }\\nonumber \\\\&{}&- \\frac{1}{2}Z{\\phi } \\Bigl [\\left( -\\frac{1}{16} \\bar{\\rho }_bp^2\\right)\\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }+\\left( -3 \\bar{\\lambda }_{\\phi }- \\frac{3}{32} \\bar{\\rho }_b(p^2)^2 \\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(1)}\\mathcal {G}_{\\phi \\phi }^2+\\left( -3 \\bar{\\lambda }_{\\phi }p^2- \\frac{1}{32} \\bar{\\rho }_b(p^2)^3 \\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(2)}\\mathcal {G}_{\\phi \\phi }^2\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( -\\frac{1}{2}\\bar{\\lambda }_{\\phi }(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(3)}\\mathcal {G}_{\\phi \\phi }^2+\\left(+6 \\bar{\\lambda }_{\\phi }p^2 +\\frac{1}{16}\\bar{\\rho }_b\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\left(\\mathcal {R}^{(1)}\\right)^2\\mathcal {G}_{\\phi \\phi }^3+\\left( 3 \\bar{\\lambda }_{\\phi }(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(1)}\\mathcal {R}^{(2)}\\mathcal {G}_{\\phi \\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(-3 \\bar{\\lambda }_{\\phi }(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\left(\\mathcal {R}^{(1)}\\right)^3\\mathcal {G}_{\\phi \\phi }^4\\Bigr ]\\nonumber \\\\&{}& +\\frac{1}{3}Z_{\\phi }^2\\Bigl [ \\left( \\frac{9}{4} \\bar{\\lambda }_{\\phi }+ \\frac{3}{2} \\bar{\\rho }_b(p^2)^2\\right)\\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^2+ \\left( - \\frac{45}{8}\\bar{\\lambda }_{\\phi }p^2 - \\frac{27}{16} \\bar{\\rho }_b(p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^3 \\mathcal {R}^{(1)}\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+ \\left( - \\frac{9}{8} \\bar{\\lambda }_{\\phi }(p^2)^2- \\frac{3}{16} \\bar{\\rho }_b(p^2)^4\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^3 \\mathcal {R}^{(2)}+ \\left(\\frac{27}{8} \\bar{\\lambda }_{\\phi }(p^2)^2 + \\frac{9}{16} \\bar{\\rho }_b(p^2)^4 \\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^4 \\left(\\mathcal {R}^{(1)} \\right)^2\\Bigr ]\\nonumber \\\\&{}&\\frac{1}{3} Z_{\\phi }^2 \\left( 15 \\bar{\\lambda }_{\\phi }+ \\frac{5}{2} \\bar{\\rho }_b(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT} \\mathcal {G}_{\\phi \\phi }^2\\Bigr ].$ $\\beta _{\\bar{\\rho }_c}&=&\\frac{1}{2}\\int \\frac{d^4p}{(2 \\pi )^4} \\Bigl [ \\left( \\frac{10}{3} \\bar{\\rho }_b+\\frac{5}{3} \\bar{\\rho }_c\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT}-\\frac{1}{2} \\frac{55}{96} Z_{\\phi }^2\\tilde{\\partial }_t\\mathcal {G}_{TT}^2\\nonumber \\\\&{}& -\\frac{1}{2}\\Bigl [ \\left(192 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b- 144 \\bar{\\rho }_a\\bar{\\rho }_b(p^2)^2 +160 \\bar{\\rho }_b^2 (p^2)^2+144 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_c+40 \\bar{\\rho }_a\\bar{\\rho }_c(p^2)^2+40 \\bar{\\rho }_b\\bar{\\rho }_c(p^2)^2-4 \\bar{\\rho }_c^2 (p^2)^2\\right)\\tilde{\\partial }_t\\mathcal {G}_{\\phi }^2\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( 288 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_ap^2 -960 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_bp^2 +80 \\bar{\\rho }_a\\bar{\\rho }_b(p^2)^3 -320 \\bar{\\rho }_b^2 (p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {R}^{(1)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+\\left(-176 \\bar{\\rho }_b^2 (p^2)^4 +16 \\bar{\\rho }_a\\bar{\\rho }_b(p^2)^4 -960 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^2 -576 \\bar{\\lambda }_{\\phi }^2 +96 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_a(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {R}^{(2)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+\\left(-576 \\bar{\\lambda }_{\\phi }^2 p^2 -320 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^3 -\\frac{112}{3} \\bar{\\rho }_b^2 (p^2)^5\\right) \\tilde{\\partial }_t\\mathcal {R}^{(3)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( - \\frac{8}{3} \\bar{\\rho }_b^2 (p^2)^6-96 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 -32 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4\\right) \\tilde{\\partial }_t\\mathcal {R}^{(4)}\\mathcal {G}_{\\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+ \\left( 1152 \\bar{\\lambda }_{\\phi }^2 -192 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_a(p^2)^2+1920 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^2 -32 \\bar{\\rho }_a\\bar{\\rho }_b(p^2)^4 +352 \\bar{\\rho }_b^2 (p^2)^4\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)} \\right)^2\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(576 \\bar{\\lambda }_{\\phi }^2 (p^2)^2+192 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4 +16 \\bar{\\rho }_b^2 (p^2)^6\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(2)} \\right)^2\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(224 \\bar{\\rho }_b^2 (p^2)^5+1920 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^3 +3456 \\bar{\\lambda }_{\\phi }^2 p^2\\right)\\tilde{\\partial }_t\\mathcal {R}^{(1)} \\mathcal {R}^{(2)}\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\, +\\left(+\\frac{64}{3} \\bar{\\rho }_b^2 (p^2)^6+256 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4+768 \\bar{\\lambda }_{\\phi }^2 (p^2)^2\\right)\\tilde{\\partial }_t\\mathcal {R}^{(1)} \\mathcal {R}^{(3)}\\mathcal {G}_{\\phi }^4\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( -3456 \\bar{\\lambda }_{\\phi }^2 p^2-224 \\bar{\\rho }_b^2 (p^2)^5-1920 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^3\\right) \\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^3 \\mathcal {G}_{\\phi }^5\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+ \\left(-96 \\bar{\\rho }_b^2 (p^2)^6-3456 \\bar{\\lambda }_{\\phi }^2 (p^2)^2 -1152 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4\\right) \\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^2 \\mathcal {R}^{(2)} \\mathcal {G}_{\\phi }^5\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left(2304 \\bar{\\lambda }_{\\phi }^2 (p^2)^2+768 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^4+64 \\bar{\\rho }_b^2 (p^2)^6\\right)\\tilde{\\partial }_t\\left(\\mathcal {R}^{(1)}\\right)^4\\mathcal {G}_{\\phi }^6\\Bigr ]\\nonumber \\\\&{}&- \\frac{1}{2}Z{\\phi } \\left( \\frac{20}{3} \\bar{\\rho }_bp^2 - \\frac{5}{6} \\bar{\\rho }_cp^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT} \\mathcal {G}_{\\phi \\phi }\\nonumber \\\\&{}&- \\frac{1}{2} Z{\\phi }\\Bigl [\\left( \\frac{1}{8} \\bar{\\rho }_bp^2- \\frac{3}{32} \\bar{\\rho }_cp^2\\right)\\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }+\\left( -6 \\bar{\\lambda }_{\\phi }- \\frac{5}{16}\\bar{\\rho }_b(p^2)^2 +\\frac{1}{32} \\bar{\\rho }_c(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(1)}\\mathcal {G}_{\\phi \\phi }^2\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( -6 \\bar{\\lambda }_{\\phi }p^2- \\frac{1}{16} \\bar{\\rho }_b(p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(2)}\\mathcal {G}_{\\phi \\phi }^2- \\bar{\\lambda }_{\\phi }(p^2)^2 \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(3)}\\mathcal {G}_{\\phi \\phi }^2+\\left(12 \\bar{\\lambda }_{\\phi }p^2+ \\frac{1}{8} \\bar{\\rho }_b(p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\left(\\mathcal {R}^{(1)}\\right)^2\\mathcal {G}_{\\phi \\phi }^3\\nonumber \\\\&{}&\\, \\, \\, \\,\\,\\,+\\left( 6 \\bar{\\lambda }_{\\phi }(p^2)^2 \\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {R}^{(1)}\\mathcal {R}^{(2)}\\mathcal {G}_{\\phi \\phi }^3+\\left(-6 \\bar{\\lambda }_{\\phi }(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\left(\\mathcal {R}^{(1)}\\right)^3\\mathcal {G}_{\\phi \\phi }^4\\Bigr ]\\nonumber \\\\&{}& +\\frac{1}{3}Z_{\\phi }^2\\Bigl [ \\left( \\frac{9}{8}\\bar{\\rho }_b(p^2)^2+\\frac{3}{16}\\bar{\\rho }_c(p^2)^2\\right)\\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^2+ \\left( - \\frac{27}{4} \\bar{\\lambda }_{\\phi }p^2 - \\frac{21}{8} \\bar{\\rho }_b(p^2)^3\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^3 \\mathcal {R}^{(1)}\\nonumber \\\\&{}& \\, \\, \\, \\,\\,\\,+ \\left(- \\frac{9}{4} \\bar{\\lambda }_{\\phi }(p^2)^2- \\frac{3}{8} \\bar{\\rho }_b(p^2)^4 \\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^3 \\mathcal {R}^{(2)}+ \\left( \\frac{27}{4} \\bar{\\lambda }_{\\phi }(p^2)^2 + \\frac{9}{8} \\bar{\\rho }_b(p^2)^4\\right) \\tilde{\\partial }_t\\mathcal {G}_{hh} \\mathcal {G}_{\\phi \\phi }^4 \\left(\\mathcal {R}^{(1)} \\right)^2\\Bigr ]\\nonumber \\\\&{}&\\frac{1}{3} Z_{\\phi }^2 \\left( 60 \\bar{\\lambda }_{\\phi }+ 10 \\bar{\\rho }_b(p^2)^2\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT} \\mathcal {G}_{\\phi \\phi }^2\\Bigr ].$ $\\beta _{\\bar{\\lambda }_{\\phi }}&=&\\frac{1}{2}\\int \\frac{d^4p}{(2 \\pi )^4} \\Bigl [ \\left( -\\frac{5}{2} \\bar{\\lambda }_{\\phi }\\right) \\tilde{\\partial }_t\\mathcal {G}_{TT}+ \\frac{1}{8} \\bar{\\lambda }_{\\phi } \\tilde{\\partial }_t\\mathcal {G}_{hh}-\\frac{1}{2} \\left(144 \\bar{\\lambda }_{\\phi }^2 + 48 \\bar{\\lambda }_{\\phi }\\bar{\\rho }_b(p^2)^2 + 4 \\bar{\\rho }_b^2 (p^2)^4\\right)\\tilde{\\partial }_t\\mathcal {G}_{\\phi }^2\\Bigr ].$ Herein, $\\mathcal {G}_{xx} = \\frac{1}{\\Gamma _{k\\, xx}^{(2)}+R_k}$ and $\\mathcal {R}^{(n)}= \\frac{1}{(k^2)^{n-1}} \\left(n r^{\\prime }(y)+ r^{\\prime \\prime }(y) y \\right)$ for $n>1$ and $\\mathcal {R}^{(1)}=1+r(y)+ r^{\\prime }(y)y$ .",
"Herein $r(y)$ is the regulator shape function with $y= \\frac{p^2}{ k^2}$ .",
"This structure arises, since all diagrams where the external derivative with respect to the momentum hits the propagator, can be organised in a way that only derivatives of the scalar propagator, and not the graviton propagator, need to be taken."
],
[
"Scalar anomalous dimension",
"Here we give our result for the anomalous dimension $\\eta _{\\phi }$ for a spectrally adjusted cutoff with exponential shape function: $\\eta _{\\phi }&=&\\frac{1}{96 \\pi g \\lambda \\left(e^{4 \\lambda /3}\\text{Ei}\\left(\\frac{4 \\lambda }{3}\\right)-\\left(e^{4 \\lambda /3}+1\\right)\\text{Ei}\\left(\\frac{8 \\lambda }{3}\\right)+\\text{Ei}(4 \\lambda )\\right)+12 \\pi ge^{4 \\lambda }-9 e^{8 \\lambda /3} (4 \\pi (g+24 \\pi )+9 (4 \\rho _b-2\\rho _c+3 \\rho _a))} 4 e^{4 \\lambda /3} \\cdot \\nonumber \\\\&{}&\\Bigl (8 \\pi g \\left(\\text{Ei}\\left(\\frac{8 \\lambda }{3}\\right) (\\partial _t \\lambda (3-12 \\lambda )+\\lambda (3 \\eta _N-32\\lambda +6))+3 \\text{Ei}(4 \\lambda ) (\\partial _t \\lambda (4 \\lambda -1)+\\lambda (-\\eta _N+8 \\lambda -2))+8 \\lambda ^2 \\text{Ei}\\left(\\frac{4 \\lambda }{3}\\right)\\right)\\nonumber \\\\&{}&+4 e^{4 \\lambda /3} \\left(2 \\pi g \\left(\\text{Ei}\\left(\\frac{4\\lambda }{3}\\right)-\\text{Ei}\\left(\\frac{8 \\lambda }{3}\\right)\\right)(\\partial _t \\lambda (8 \\lambda -3)+\\lambda (-3 \\eta _N+16 \\lambda -6))-3(\\pi g (4 \\lambda +3)+9 (4 \\rho _b-2 \\rho _c+3 \\rho _a))\\right)\\nonumber \\\\&{}&-3 \\pi g e^{8 \\lambda /3} (4 \\partial _t \\lambda -3 \\eta _N+6)-3 \\pi g \\eta _N e^{4 \\lambda }\\Bigr )$"
]
] | 1204.0965 |
[
[
"Andreev current induced by ferromagnetic resonance"
],
[
"Abstract We study charge transport through a metallic dot coupled to a superconducting and a ferromagnetic lead with a precessing magnetization due to ferromagnetic resonance.",
"Using the quasiclassical theory, we find that the magnetization precession induces a dc current in the subgap regime even in the absence of a bias voltage.",
"This effect is due to the rectification of the ac spin currents at the interface with the ferromagnet; it exists in the absence of spin current in the superconductor.",
"When the dot is strongly coupled to the superconductor, we find a strong enhancement in a wide range of parameters as compared to the induced current in the normal state."
],
[
"Andreev current induced by ferromagnetic resonance Caroline Richard Manuel Houzet Julia S. Meyer SPSMS, UMR-E 9001 CEA/UJF-Grenoble 1, INAC, Grenoble, F-38054, France We study charge transport through a metallic dot coupled to a superconducting and a ferromagnetic lead with a precessing magnetization due to ferromagnetic resonance.",
"Using the quasiclassical theory, we find that the magnetization precession induces a dc current in the subgap regime even in the absence of a bias voltage.",
"This effect is due to the rectification of the ac spin currents at the interface with the ferromagnet; it exists in the absence of spin current in the superconductor.",
"When the dot is strongly coupled to the superconductor, we find a strong enhancement in a wide range of parameters as compared to the induced current in the normal state.",
"74.45.+c, 75.76.+j, 72.25.-b Spin-transfer torque allows one to manipulate the magnetization of a ferromagnetic (F) layer by means of a spin-polarized current [1], [2].",
"Random-access memories using this effect in order to induce magnetization reversal of the active elements are on their way to commercialization.",
"The reverse effect, namely the generation of a spin current in a normal metal (N) by means of a dynamically precessing ferromagnetic metal, has also been predicted [3].",
"In the absence of direct spin probes, this effect may be measured by using a second ferromagnet as an analyzer that converts the spin current into a charge current.",
"However, it was pointed out theoretically [4] and measured experimentally [5], [6] that a single F/N junction is enough to both generate and detect the spin current through the generation of a dc voltage at ferromagnetic resonance (FMR) in an open-circuit geometry.",
"At the origin of this phenomenon is the spin accumulation on the normal side of the junction – due to the precession-induced spin current – which is typically different from that on the ferromagnetic side.",
"If transmissions for the majority and minority electron species through the junction are different, the difference in spin accumulation generates a net charge current which must be compensated by a difference in electrochemical potentials such that no charge accumulation occurs.",
"Spin relaxation inhibits spin accumulation and, thus, suppresses the effect.",
"The aim of our work is to explore how this effect is modified in a ferromagnet/superconductor junction.",
"The combination of ferromagnetic and superconducting (S) materials has been shown to lead to a variety of interesting spin phenomena [7], [8].",
"However, the study of the interplay between magnetization dynamics and superconductivity is a relatively new topic.",
"Experimentally, a narrowing of the FMR width at the superconducting transition was observed in an F/S bilayer [9].",
"Theoretically, it was proposed that a dynamically precessing ferromagnet may generate a long-range proximity effect [10].",
"This effect would manifest itself in the enhancement of the critical current in a phase-biased ferromagnetic Josephson junction under FMR conditions.",
"Related signatures in the tunneling density of states of the F layer have also been investigated [11].",
"However, these works disregard interface effects and, therefore, do not take into account the possible generation of an FMR-induced dc voltage.",
"Finally, let us note that the first experiments on voltage generation by FMR of Refs.",
"[5], [6] were performed with Al as the normal metal, which becomes superconducting at low temperatures.",
"As the FMR-generated charge current in an F/N junction is typically associated with a spin current, one may wonder what happens in an F/S junction in the subgap regime, where transport is mediated by Andreev processes [12].",
"We show that the generation of charge current in the absence of a spin current in a conventional singlet superconductor is possible.",
"In fact, the absence of spin currents in the superconductor may even lead to a strong enhancement of the induced charge current as compared to the normal state.",
"Figure: Setup of the junction.",
"A metallic dot is coupled to a ferromagnetic lead with precessing magnetization m(t){m}(t) on the left and to a normal or superconducting lead on the right.",
"The left barrier is characterized by the conductances G l G_l and G m G_m, defined above Eq.",
"(), whereas the right barrier is characterized by the conductance G r G_r.As in the normal state, the two main ingredients necessary to generate the effect are spin-dependent transmissions through the junction and a spin accumulation region The effect is absent in the theory of Ref.",
"[18], which did not include such a region..",
"The simplest setup meeting these requirements is a metallic dot coupled through tunnel barriers to a ferromagnet and to a superconductor (see Fig.",
"REF ).",
"Coulomb blockade effects are neglected, assuming that the conductances of the barriers largely exceed the conductance quantum.",
"The magnetization precession in the ferromagnetic lead is described by a time-dependent exchange field, ${J}(t)=J{m}(t)$ with ${m}(t)=(\\sin \\theta \\cos \\Omega t,\\sin \\theta \\sin \\Omega t,\\cos \\theta ),$ acting on the spin of the conduction electrons.",
"Here the precession frequency, $\\Omega $ , and the tilt angle, $\\theta $ , are both tunable with external dc and rf fields under standard FMR conditions [13].",
"We will consider them as externally fixed parameters.",
"The precession of the magnetization drives the system out of equilibrium and, thus, may generate a current.",
"To describe the system, we use the quasiclassical Keldysh theory [14].",
"In particular, the current through the junction can be expressed in terms of the quasiclassical Green function $\\check{g}$ of the dot.",
"Here $\\check{g}$ is a matrix in Keldysh, Nambu, and spin space.",
"In Keldysh space, it has a triangular structure with retarded ($\\hat{g}^R$ ), advanced ($\\hat{g}^A$ ), and Keldysh ($\\hat{g}^K$ ) components.",
"Furthermore, it satisfies the normalization condition $\\check{g}^2=1$ .",
"The equations determining the Green functions are most conveniently written in the rotational frame for the magnetization precession, where the problem is stationary [10].",
"Assuming that the conductance of the dot largely exceeds the conductances of the junctions with the leads, the equation determining $\\check{g}$ may be cast in the form $-i\\frac{2\\pi G_Q}{\\delta }\\left[\\left(E+\\frac{\\Omega }{2}\\sigma _z \\right)\\tau _z, \\check{g}\\right]+\\check{I}_l+ \\check{I}_r=0.$ Here $\\sigma _i$ and $\\tau _i$ are Pauli matrices in Nambu and spin space, respectively ($i=x,y,z$ ).",
"Furthermore, $G_Q=e^2/\\pi $ is the conductance quantum (in units where $\\hbar =1$ ), and $\\delta $ is the mean level spacing in the dot.",
"The spin-dependent energy shift $\\pm \\Omega /2$ is a spin-resolved chemical potential induced by the transformation from the laboratory to the rotational frame.",
"The boundary conditions with the ferromagnetic ($l$ = left) and superconducting ($r$ = right) leads are represented by the matrix currents $\\check{I}_{l/r}$ and depend on the Green functions $\\check{g}_{l/r}$ describing the non-equilibrium state in the leads due to the magnetization precession.",
"Tunneling through an F/N interface is generally spin-dependent.",
"The relevant processes can be characterized by the total conductance of the junction, $G_l$ , and the difference between the conductances for the majority and minority electrons, $G_m$ In addition one may take into account the imaginary part of the spin-mixing conductance, $G_\\phi $ , which accounts for a spin-dependent phase shift upon reflection.",
"In good F/N contacts with large Fermi velocity mismatch, $G_\\phi $ is typically small and therefore neglected.",
"It may, however, be large in tunnel junctions.",
"As $G_\\phi $ leads to spin relaxation, it would suppress the effect studied in this work..",
"The matrix current at the tunnel interface between the dot and the ferromagnet then takes the form [15] $\\check{I}_l=\\frac{G_l}{2}[ \\check{g}_l,\\check{g}]+\\frac{G_m}{4}[ \\lbrace {m}\\cdot {{\\sigma }}\\tau _z,\\check{g}_l\\rbrace ,\\check{g}],$ where ${m}\\equiv {m}(0)$ .",
"Within the quasiclassical approximation, we assume $|G_m|\\ll G_l$ .",
"Thus, $G_m$ can be treated perturbatively.",
"The Green function in the F lead, $\\check{g}_l$ , is determined by $\\left[\\left(E+\\frac{\\Omega }{2}\\sigma _z+J{{m}}\\cdot {\\sigma }\\right)\\tau _z-\\check{\\Sigma }, \\check{g}_l \\right]=0,$ where the self-energy $\\check{\\Sigma }=-i\\Gamma \\check{g}_N(E+(\\Omega /2)\\sigma _z)$ accounts for inelastic scattering in the relaxation time approximation.",
"Here, $1/\\Gamma $ is the inelastic scattering time and $\\check{g}_N$ is the equilibrium Green function in a normal metal.",
"Namely, $\\hat{g}_N^{R(A)}(E)=\\pm \\tau _z$ and $\\hat{g}_N^{K}(E)=2\\tau _z f(E)$ , where $f(E)=\\tanh (E/2T)$ is related to the Fermi distribution at temperature $T$ .",
"For a large exchange field, $J\\gg \\Omega ,\\Gamma $ , the solution of Eq.",
"(REF ) takes the form $\\hat{g}_l^{R(A)}=\\pm \\tau _z$ and $\\hat{g}_l^{K}=2\\tau _z(f_++f_-\\cos \\theta {m}\\!\\cdot \\!",
"{{\\sigma }})$ , where $f_\\pm (E)=[f(E+\\Omega /2)\\pm f(E-\\Omega /2)]/2$ .",
"The matrix current at the dot-superconductor tunnel interface is given as $\\check{I}_r=\\frac{G_r}{2}[ \\check{g}_r,\\check{g}],$ where $G_r$ is the conductance of the junction.",
"The Green function in the S lead reads $\\check{g}_r=\\check{g}_S(E+(\\Omega /2)\\sigma _z)$ , where $\\check{g}_S$ is the equilibrium Green function in a superconductor.",
"Namely, $\\hat{g}^{R(A)}_S(E)=(-iE \\tau _z+\\Delta \\tau _x)/\\sqrt{\\Delta ^2-(E\\pm i0^+)^2}$ and $\\hat{g}^{K}_S(E)=[\\hat{g}^{R}_S(E)-\\hat{g}^{A}_S(E)]f(E)$ , where $\\Delta $ is the superconducting order parameter (taken to be real).",
"Now we have all the ingredients necessary to determine the Green function in the dot and subsequently the spin and charge currents at both interfaces.",
"The charge currents are given by $I_{l/r}=\\frac{1}{16e}\\int dE\\; \\mathrm {Tr}[\\tau _z \\hat{I}_{l/r}^K].$ Current conservation ensures that $I\\equiv I_l=-I_r$ .",
"The spin currents in the rotational frame are given by ${I}_{l/r}=-\\frac{1}{32e^2}\\int dE \\;\\mathrm {Tr}[ {\\sigma } \\hat{I}_{l/r}^K].$ In the laboratory frame, they decompose into a dc contribution along the precession axis, $I_{\\alpha ,z}$ , and ac components in the perpendicular plane, $I_{\\alpha ,x/y}(t)=I_{\\alpha ,x/y}\\cos \\Omega t\\mp I_{\\alpha ,y/x}\\sin \\Omega t$ .",
"Contrarily to the charge current, the spin currents do not need to be conserved.",
"Eq.",
"(REF ) yields ${I}_{l}+{I}_{r}-\\frac{\\Omega }{16\\delta }\\int dE\\;\\mathrm {Tr}[(\\hat{\\bf z}\\times {\\sigma })\\tau _z\\hat{g}^K]=0.$ Thus, only the dc spin current along $\\hat{\\bf z}$ is conserved.",
"While our main interest are the FMR-induced currents in the subgap regime of an F-dot-S junction, we first study the simpler case of an F-dot-N junction for comparison.",
"For better readibility, in the following, we will normalize conductances by $G_\\Sigma =G_l+G_r$ and energies by the Thouless energy $E_g=G_\\Sigma \\delta /(4\\pi G_Q)$ .",
"In particular, we introduce the dimensionless conductances $\\gamma _\\alpha =G_\\alpha /G_\\Sigma $ ($\\alpha =l,r,m$ ) as well as the dimensionless energies $\\epsilon =E/E_g$ and $\\omega =\\Omega /(2E_g)$ .",
"A normal lead is described by setting $\\Delta =0$ in the above equations for $\\check{g}_r$ .",
"In the absence of superconductivity, the retarded and advanced Green functions in the dot are trivial, $\\hat{g}^{R(A)}=\\pm \\tau _z$ .",
"The Keldysh component is obtained with the help of Eqs.",
"(REF ), (REF ), and (REF ).",
"There is a remarkable relation between the spin current at the left contact with the F lead and the charge current to lowest order in $\\gamma _m$ , $I=\\frac{2e\\gamma _m\\gamma _r}{\\gamma _l}{m}(t)\\cdot {I}_l(t)=\\frac{2e\\gamma _m\\gamma _r}{\\gamma _l}(I_{l,x}\\sin \\theta +I_{l,z}\\cos \\theta ),$ namely the charge current is proportional to the projection of the spin current onto the instantaneous magnetization axis of the barrier due to the spin-dependent conductance $G_m$ .",
"That is, the charge current originates from two effects: (i) the rectification of the ac in-plane spin current pumped from the ferromagnet, and (ii) the conversion of the dc spin current along the $\\hat{\\bf z}$ -axis into a charge current.",
"It turns out that the two effects have opposite sign, and that the former dominates over the latter.",
"Namely, we find $I_{l,x}&=&\\frac{G_lE_g}{2e^2}\\frac{\\omega (\\gamma _r+\\omega ^2)}{1+\\omega ^2}\\sin \\theta \\cos \\theta ,\\\\I_{l,z}&=&-\\frac{G_lE_g}{2e^2}\\omega \\gamma _r\\sin ^2\\theta .$ Note that, in the limit $\\gamma _r\\ll \\gamma _l$ , the spin current along the $\\hat{\\bf z}$ -axis is negligible.",
"By contrast, in the limit $\\gamma _l\\ll \\gamma _r$ , the two components are of comparable magnitude and almost complete cancellation between the competing effects takes place.",
"The charge current reads $I=\\frac{G_m E_g}{e}\\frac{\\gamma _r\\gamma _l\\omega ^3\\sin ^2\\theta \\cos \\theta }{1+\\omega ^2}.$ At large precession frequency, $\\omega \\gg 1$ , the current scales linearly with frequency, $I\\simeq (G_lG_rG_m/2eG_\\Sigma ^2)\\Omega \\sin ^2\\theta \\cos \\theta $ .",
"In particular, in an open-circuit geometry, this would correspond to an FMR-induced dc voltage $eV=(G_m/2G_\\Sigma )\\Omega \\sin ^2\\theta \\cos \\theta $ in accordance with Refs.",
"[4], [5], [6].",
"At $\\omega \\ll 1$ , spin-relaxation mechanisms induced by the tunnel coupling of the dot to the leads tend to suppress the effect.",
"We now turn to the F-dot-S junction.",
"In the subgap regime, the spin current at the interface with the superconductor vanishes, ${I}_r=0$ .",
"Thus, $I_{z,l}=0$ .",
"However, an ac spin current is present at the interface with the ferromagnet.",
"Then, the Andreev charge current originates entirely from the rectification of this ac spin current.",
"Restricting ourselves to energy scales much smaller than $\\Delta $ , the Green function in the superconducting lead takes the simple form $\\hat{g}_r^{R(A)}=\\tau _x$ and $\\hat{g}_r^K=0$ .",
"Taking $\\gamma _m$ as a small parameter, we search for a perturbative solution of equation (REF ) in the form $\\check{g} =\\check{g}_0+\\gamma _m\\check{g}_1+\\dots $ Due to the proximity effect, now the retarded and advanced Green functions of the dot are modified as well.",
"To zeroth order in $\\gamma _m$ , an explicit solution is given by $\\hat{g}_0^{R(A)}=\\frac{\\gamma _r\\tau _x+[-i(\\epsilon +\\omega \\sigma _z)\\pm \\gamma _l]\\tau _z}{\\sqrt{\\gamma _r^2-(\\epsilon +\\omega \\sigma _z\\pm i\\gamma _l)^2}}.$ Here, $\\gamma _r$ is the effective minigap due to the coupling with the S lead [16], $\\omega $ an effective exchange field, and $\\gamma _l$ yields a broadening of the energy levels due to the coupling with the F lead.",
"The Keldysh Green function can be cast in the form $\\hat{g}_0^K=\\hat{g}^R\\hat{\\varphi }-\\hat{\\varphi }\\hat{g}^A$ with $\\hat{\\varphi }=f_++f_-\\cos \\theta \\left[\\frac{\\gamma _l\\sin \\theta }{\\omega ^2+\\gamma _l^2}(\\gamma _l\\sigma _x\\!-\\!\\omega \\sigma _y)+\\cos \\theta \\sigma _z\\right]\\!\\!.$ The function $\\hat{\\varphi }$ can be interpreted as a matrix-distribution function.",
"Note that it does not depend on $\\gamma _r$ as subgap electrons only thermalize with the F lead.",
"To first order in $\\gamma _m$ , a solution which satisfies the normalization condition, $\\check{g}^2=1$ , is obtained in the form $\\check{g}_1=\\check{g}_0 \\check{X} -\\check{X} \\check{g}_0$ .",
"For the advanced and retarded components, one finds $\\hat{X}^{R(A)}=\\mp (\\sin \\theta /2 \\omega )[i\\gamma _r/(\\epsilon \\pm i\\gamma _l) \\tau _x+\\tau _z]\\sigma _y$ .",
"The Keldysh component can be decomposed as $\\hat{X}^K=X^K_x\\tau _x+X^K_z\\tau _z$ , where $X^K_x$ and $X^K_z$ solve the coupled equations $&&2 \\gamma _l X^K_z -i\\omega [\\sigma _z,X^K_z]\\\\&=&2 \\sin \\theta [ \\cos \\theta \\sin \\theta f_-+ f_+(\\sigma _x-\\frac{ \\gamma _l}{ \\omega }\\sigma _y)],\\nonumber \\\\&&2\\epsilon X^K_x-2i \\gamma _r X^K_z+\\omega \\lbrace \\sigma _z,X^K_x\\rbrace \\\\&=&2i\\frac{\\gamma _l\\gamma _r\\sin \\theta }{ \\omega (\\gamma _l^2+ \\epsilon ^2)}[\\gamma _l f_+\\sigma _y-\\epsilon f_- \\cos \\theta (\\cos \\theta \\sigma _x-\\sin \\theta \\sigma _z) ].\\nonumber $ Evaluating the current at the right interface, Eq.",
"(REF ) yields $I=-i\\gamma _m G_r/(16e)\\int dE\\; \\mathrm {Tr}[\\tau _y \\hat{g}_1^K]$ .",
"Inserting the solution for $\\hat{g}_1^K$ and using the property $\\hat{g}_0^R(-\\epsilon )=-\\sigma _x\\tau _z\\hat{g}_0^A(\\epsilon )\\sigma _x\\tau _z$ , we obtain the current $I&=& \\frac{1}{2}I_0\\frac{\\gamma _r^2\\omega }{\\gamma _l^2+\\omega ^2}\\int d\\epsilon \\;\\frac{\\epsilon f_-}{(\\gamma _l^2+\\epsilon ^2)(\\epsilon +\\omega )}\\\\&&\\qquad \\qquad \\qquad \\times \\sum _\\pm \\frac{-\\gamma _l(\\epsilon +\\omega )\\pm i(\\gamma _l^2-\\epsilon \\omega )}{\\sqrt{\\gamma _r^2-(\\epsilon +\\omega \\pm i \\gamma _l)^2}},\\nonumber $ where $I_0= (G_mE_g/e)\\sin ^2\\theta \\cos \\theta $ .",
"The current as a function of frequency for different values of $\\gamma _l=1-\\gamma _r$ is shown in Fig.",
"REF .",
"Simple analytic expressions can be found in different asymptotic regimes.",
"Figure: Andreev current induced by ferromagnetic resonance as a function of precession frequency for different values of γ l =G l /(G r +G l )\\gamma _l=G_l/(G_r+G_l).Here I 0 =(G m E g /e)sin 2 θcosθI_0=(G_mE_g/e)\\sin ^2\\theta \\cos \\theta .In particular, at temperature $T=0$ and low frequency $|\\omega |\\ll \\gamma $ , where $\\gamma =(\\gamma _l^2+\\gamma _r^2)^{1/2}$ , the FMR-induced current is given as $I\\simeq \\frac{10}{3}I_0\\frac{\\gamma _r^2\\gamma _l}{\\gamma ^7}\\omega ^5.$ The large power $\\omega ^5$ indicates the strong suppression of the effect.",
"At large frequencies $|\\omega |\\gg \\gamma $ , the current saturates.",
"The frequency-independent value is given by $I\\simeq \\frac{\\pi }{2}I_0\\,\\mathrm {sign}(\\omega )\\times \\left\\lbrace \\begin{array}{lll}\\gamma _r^2, & \\,& \\gamma _r\\ll \\gamma _l, \\\\1, & \\,& \\gamma _l\\ll \\gamma _r.\\end{array}\\right.$ The saturation can be understood as Andreev processes become inefficient at energies larger than the minigap [17].",
"Depending whether the dot is more strongly coupled to the ferromagnet or to the superconductor, the crossover between these asymptotic regimes is different.",
"If the dot is weakly coupled to the superconductor, $\\gamma _r\\ll \\gamma _l$ , a smooth crossover happens at $\\omega \\sim 1\\gg \\gamma _r$ with a typical current $I/I_0\\sim \\gamma _r^2$ .",
"By contrast, if the dot is weakly coupled to the ferromagnet, $\\gamma _l\\ll \\gamma _r$ , the crossover in the region $\\omega \\sim 1/2$ is described by $I\\simeq I_0\\times \\!\\left\\lbrace \\begin{array}{lll}\\gamma _l/(2\\sqrt{-\\delta \\omega }),&\\,&\\!-1\\ll \\delta \\omega \\ll -\\gamma _l,\\\\2\\sqrt{\\delta \\omega },&\\,&\\!\\gamma _l\\ll \\delta \\omega \\ll 1,\\end{array}\\right.$ where $\\delta \\omega \\equiv \\omega -1/2$ , with a typical current $I/I_0\\sim \\sqrt{\\gamma _l}$ at $\\omega =1/2$ .",
"While in the asymptotic regimes of $\\omega $ very small or very large, the current is suppressed as compared to the normal state, there is in fact a wide intermediate regime where it may be strongly enhanced.",
"Comparing Eqs.",
"(REF ) and (REF ), one notices that, if the dot is strongly coupled to the superconductor, in the regime $1/2<\\omega <\\gamma _l^{-1}$ the induced current in the superconducting state exceeds the induced current in the normal state.",
"This effect may be understood due to the absence of a dc spin current along along the $\\hat{\\bf z}$ -axis which leads to a strong suppression of the effect in the normal state.",
"Fig.",
"REF shows the current in the superconducting and normal state as well the contribution due to rectification only in the normal state.",
"The ratio between the current in the superconducting state and the latter contribution in the normal state reflects the ratio between Andreev and normal state conductances in an N-dot-S junction.",
"Figure: Induced current in the superconducting (dotted line) and normal state for γ l =0.4\\gamma _l=0.4.The thin line shows the contribution to the normal state current due to rectification only.So far we assumed that the magnetization in the ferromagnet is uniform.",
"However, boundary effects may lead to a suppression of the magnetization in the vicinity of the F/N interface.",
"This would result in a different resonance frequency at the barrier than in the ferromagnetic reservoir and, consequently, in a tilt angle $\\theta _B\\ne \\theta $ at FMR.",
"The effect can be accounted for by replacing ${m}$ with ${m}_B=(\\sin \\theta _B,0,\\cos \\theta _B)$ in Eq.",
"(REF ).",
"In particular, at $\\theta _B=0$ , the spin dependent conductance $G_m$ refers to the constant axis $\\hat{\\bf z}$ .",
"In the normal case, the relation between spin and charge currents, Eq.",
"(REF ), now reads $I=(2e\\gamma _m\\gamma _r/\\gamma _l){m}_B(t)\\cdot {I}_l(t)$ .",
"While at $\\theta _B=\\theta $ the rectification of the in-plane ac spin currents always dominates over the conversion of the dc spin current along $\\hat{\\bf z}$ into a charge current, this effect is completely suppressed at $\\theta _B=0$ .",
"As a consequence, at $\\theta _B=0$ , the charge current, $I=-({G_m E_g}/{e})\\omega \\gamma _r^2\\sin ^2\\theta $ , has the opposite sign compared to Eq.",
"(REF ).",
"In general, both effects are important.",
"The sign reversal occurs at $\\tan \\theta _B=[1-\\gamma _l\\omega ^2/(\\gamma _r+\\omega ^2)]\\tan \\theta $ .",
"In the superconducting case, the dc spin current along $\\hat{\\bf z}$ is always zero, and the charge current is due entirely to the rectification of the in-plane ac spin currents.",
"As a consequence, we find that the charge current vanishes at $\\theta _B=0$ .",
"The general result is obtained from Eq.",
"(REF ) by replacing $I_0$ with $I_0^B=(G_mE_g/e)\\sin \\theta _B\\sin \\theta \\cos \\theta $ .",
"In summary, we demonstrate that a subgap charge current in an F/S junction may be induced by ferromagnetic resonance.",
"The effect is due to the rectification of ac spin currents generated by the precessing magnetization in the ferromagnet.",
"In the normal case, a competing effect of conversion of a dc spin current into a charge current exists.",
"This effect is absent in an F/S junction as the superconductor cannot carry a subgap spin current.",
"As a consequence, the induced current in the superconducting state may be strongly enhanced as compared to the normal state.",
"Interesting non-equilibrium phenomena should be expected in ferromagnetic Josephson junctions under ferromagnetic resonance conditions.",
"We acknowledge funding through an ANR grant (ANR-11-JS04-003-01) and an EU-FP7 Marie Curie IRG."
]
] | 1204.1491 |
[
[
"The Galactic Center Weather Forecast"
],
[
"Abstract In accretion-based models for Sgr A* the X-ray, infrared, and millimeter emission arise in a hot, geometrically thick accretion flow close to the black hole.",
"The spectrum and size of the source depend on the black hole mass accretion rate $\\dot{M}$.",
"Since Gillessen et al.",
"have recently discovered a cloud moving toward Sgr A* that will arrive in summer 2013, $\\dot{M}$ may increase from its present value $\\dot{M}_0$.",
"We therefore reconsider the \"best-bet\" accretion model of Moscibrodzka et al., which is based on a general relativistic MHD flow model and fully relativistic radiative transfer, for a range of $\\dot{M}$.",
"We find that for modest increases in $\\dot{M}$ the characteristic ring of emission due to the photon orbit becomes brighter, more extended, and easier to detect by the planned Event Horizon Telescope submm VLBI experiment.",
"If $\\dot{M} \\gtrsim 8 \\dot{M}_0$ this \"silhouette of the black hole will be hidden beneath the synchrotron photosphere at 230 GHz, and for $\\dot{M} \\gtrsim 16 \\dot{M}_0$ the silhouette is hidden at 345 GHz.",
"We also find that for $\\dot{M} > 2 \\dot{M}_0$ the near-horizon accretion flow becomes a persistent X-ray and mid-infrared source, and in the near-infrared Sgr A* will acquire a persistent component that is brighter than currently observed flares."
],
[
"Introduction",
"The recent discovery of a cloud moving towards the Galactic Center [18] creates a potential opportunity for testing models of Sgr A*.",
"The $> 3 M_{\\oplus }$ cloud will interact strongly with gas near nominal pericenter at $r_p \\simeq 300 {\\rm AU} \\simeq 8000GM/c^2$ ($M \\equiv $ black hole mass), and may change the black hole accretion rate $\\dot{M}$ .",
"Since the structure of the cloud and the surrounding medium are uncertain, possible outcomes range from very small changes in the accretion rate over timescales of decades to rapid, large changes in the accretion rate.",
"The dynamical timescale at $r_p$ is $t_d = (r_p^3/(G M))^{1/2} = 0.5$ yr, and the viscous timescale $t_{vis} = (\\alpha \\Omega )^{-1} (R/H)^2\\approx 100$ yr $\\gg t_{PhD}$ , assuming $\\alpha = 0.05$ and $H/R = 0.3$ , i.e.",
"a hot, radiatively inefficient accretion flow.",
"After an initial transient phase while the flow circularizes—accompanied by transient emission—it is natural to think the flow will settle into a steady state.",
"The settling timescale could be as little as a few $t_d$ , and so the steady state may arrive as soon as mid-2014.",
"If the resulting flow can be modeled as a steady disk, the excess mass will drain away on the viscous timescale, i.e.",
"the source will remain bright well into the 22nd century.",
"It is therefore interesting to ask how changes in $\\dot{M}$ will manifest themselves observationally.",
"Current observations of Sgr A* show $F_\\nu = 0.5-1$ Jy at 1-50 GHz with a nearly flat spectral slope [10]; $F_\\nu \\sim \\nu ^{p=0.17-0.3}$ .",
"The spectral slope becomes flatter and variable at 230-690 GHz, with $p= -0.46 - 0.08$ , [22]), which is commonly interpreted as signaling a transition from optically thick to optically thin synchrotron emission.",
"The discovery of polarized emission (polarization fraction at level of a few per cent) at $\\lambda =1.3$ mm and subsequent measurement of Faraday rotation imply a model dependent limit $2 \\times 10^{-7} > \\dot{M} > 2 \\times 10^{-9}{\\rm M_\\odot yr^{-1}}$ in the inner accretion flow ([4], [23]).",
"Sgr A* is resolved by VLBI at $1.3$ mm where interstellar scattering is comparable to the intrinsic source FWHM $37^{+16}_{-10} \\mu {\\rm as}$ [7].",
"This is smaller than the apparent diameter of the event horizon $\\sim 55 \\mu as$ .",
"Sgr A* fluctuates rapidly in the near-infrared (NIR), with average $F_\\nu \\sim 1 {\\rm mJy}$ ([6]).",
"It is not yet detected at mid-infrared (MIR) wavelengths ($\\lambda \\le 8.6 \\mu {\\rm m}$ , e.g.",
"[33]).",
"In the X-ray Sgr A* exhibits flares with characteristic duration of order an hour and a duty cycle of order $5\\%$ .",
"The upper limit for quiescent state X-ray emission is $\\nu L_\\nu < 2.4 \\times 10^{33} {\\rm erg s^{-1}}$ [3].",
"We are not aware of any secular trends in these observed properties of Sgr A*.",
"One model of Sgr A* that fits most observational constraints is our relativistic accretion model [27], where submillimeter, IR, and X-ray emission arise in an optically thin, geometrically thick disk close to the event horizon; radio emission is assumed to arise nonthermally in a synchrotron photosphere at larger radius [30], but is not predicted by our model, which focuses on the inner parts of the accretion flow.",
"The underlying flow model is a general relativistic magnetohydrodynamic (GRMHD) simulation ([14], [28], [36]).",
"The emerging radiation is calculated using Monte Carlo [8] and ray-tracing schemes.",
"The $1.3$ mm flux originates as Doppler boosted synchrotron emission from the approaching side of the disk between $\\sim 10 GM/c^2$ and the innermost stable circular orbit (ISCO).",
"Our relativistic disk model assumes a thermal electron distribution function, neglects thermal conduction, and assumes a constant ratio of ion to electron temperature $T_i/T_e$ .",
"The model does not produce the observed IR flaring at the correct amplitude, but can with the addition of a small nonthermal component in the distribution function [19].",
"If we fix the black hole mass ($M_{BH}=4.5 \\times 10^6 M_{\\odot }$ , [16]) and distance ($D=8.4\\, {\\rm kpc}$ , [17]), the remaining model parameters are the source inclination $i$ , black hole spin $a_*$ ($0 \\le a_* \\le 1$ ), $T_i/T_e$ , and $\\dot{M}$ .",
"We fix $\\dot{M}$ so that the $1.3$ mm flux matches the observed $\\simeq 3$ Jy.",
"The relativistic accretion models are not tightly constrained by the data, but they reveal the following: (1) face-on models that reproduce the millimeter flux would look like rings and therefore, in VLBI data, have dips in visibilities on fixed intermediate baselines, while existing observations suggest that the ring radius would need to vary to fit the data ([13]; see also the discussion of [5]).",
"More nearly edge-on models are therefore favored; (2) models with $a_* \\gtrsim 0.98$ that reproduce the millimeter flux have a hot, dense inner disk that would overproduce X-rays via inverse Compton scattering.",
"Lower spin, $a_* \\sim 0.9$ , models are therefore favored; (3) the observed source size and flux fix the temperature of the emitting electrons $T_e = F_\\nu c^2/(4\\pi k \\nu ^2\\sigma ^2)$ ($\\sigma \\equiv $ the RMS size of the source on the sky) and this turns out to favor $T_i/T_e \\simeq 3$ models.",
"The “best-bet” model from [27] has $a_* \\simeq 0.94$ , $i = 85$ deg, $\\dot{M} \\equiv \\dot{M}_0 \\simeq 2 \\times 10^{-9} {\\rm M_{\\odot }yr^{-1}}$ , and $T_i/T_e = 3$ .",
"The relativistic disk model uses self-consistent dynamics and radiative transfer but is not unique.",
"The electron distribution function is particularly poorly constrained.",
"It is likely anisotropic, may contain multiple temperature components ([32]) and power-law components, and vary in basic functional form with time and position.",
"Alternative accretion models (e.g.",
"[5], [35]) make different assumptions about the flow and/or distribution function and favor slightly different $\\dot{M}, a_*,$ and $i$ .",
"These models may respond differently to an increase in $\\dot{M}$ .",
"Indeed, radically different models may also fit the data.",
"The model dynamics depends on the initial magnetic field distribution, particularly the distribution of vertical flux through the accretion disk; models with large vertical magnetic flux (e.g.",
"[24]) are likely to respond quite differently to variations in the mass flux.",
"Also, jet models for Sgr A* ([11], [21], [12]) posit a luminous jet and a comparatively dim accretion disk.",
"Again, these may respond differently to an influx of mass.",
"How, then, does our relativistic disk model respond to an increase in $\\dot{M}$ ?",
"In this Letter we calculate the $1.3$ and $0.87$ mm flux and source size as well as the spectrum that would result for the best-bet model over a range of $\\dot{M}$ .",
"One key question we seek to answer is whether a small increase in $\\dot{M}$ would hide the event horizon (and the signature ring-like appearance of the photon orbit, also known as the shadow or silhouette of the event horizon) underneath a synchrotron photosphere.",
"This might prevent detection of the photon orbit by the planned Event Horizon Telescope [7].",
"Another key question is whether the increased $\\dot{M}$ would make Sgr A* detectable in its quiescent state in the IR and X-ray.",
"Below we describe variation of the flux and source morphology at 230GHz ($1.3$ mm) and 345GHz ($0.87$ mm) (§ ), describe variation of the spectrum (§ ), and finally discuss which features of the results are likely to be most robust (§ )."
],
[
"Change of Sgr A* sub-mm luminosity and size for enhanced $\\dot{M}$",
"How do we naively expect the mm disk image size and flux to respond to changes in $\\dot{M}$ ?",
"In our model, $1.3$ mm emission in Sgr A* is thermal synchrotron emission from plasma with optical depth $\\tau _\\nu \\sim 1$ , near the ISCO.",
"The true electron distribution undoubtedly contains nonthermal components (see, Riquelme et al.",
"2012).",
"Models with thermal + power-law distribution functions (e.g.",
"[5]) contain an O(1/3) nonthermal contribution to the flux at $1.3$ mm, which hints at how uncertainty in the distribution function translates into uncertainties in the spectrum.",
"The thermal synchrotron absorptivity is $\\alpha _{\\nu ,a}=j_\\nu /B_\\nu $ , where $j_{\\nu } = \\frac{\\sqrt{2}\\pi e^2 n_e \\nu _s}{3cK_2(\\Theta _e^{-1})}(X^{1/2} + 2^{11/12} X^{1/6} )^2 \\exp (-X^{1/3})$ , $X = \\nu /\\nu _s$ , $\\nu _s=2/9 (eB/2\\pi m_ec) \\Theta _e^2 \\sin \\theta $ , $\\theta $ is an angle between the magnetic field vector and emitted photon, $K_2$ is a modified Bessel function of the second kind [20] and $B_\\nu \\simeq 2 \\nu ^2 \\Theta _e m_e$ .",
"Near $1.3$ mm, $X \\sim 1$ , and the emissivity is nearly independent of frequency, so $j_\\nu \\propto \\nu ^0n_e B$ and $\\alpha _{\\nu ,a} \\propto \\nu ^{-2} n_e B \\Theta _e^{-1}$ .",
"We will assume that $n_e \\propto \\dot{M} r^{-3/2}$ , $\\Theta _e \\propto 1/r$ , and $\\beta \\sim $ const., so that $B \\propto \\dot{M}^{1/2}r^{-5/4}$ for $r > G M/c^2$ , and ignore relativistic corrections.",
"Then for $\\tau _\\nu \\ll 1$ (or $\\dot{M} \\ll \\dot{M}_0$ ) the source has size $\\sim G M/c^2$ and the flux $F_\\nu \\sim (4/3)\\pi (G M/c^2)^3 j_\\nu \\propto \\dot{M}^{3/2}$ .",
"For $\\tau _\\nu \\gg 1$ the source size is set by the photosphere radius $r_{ph}$ where $\\int _{r_{ph}}^\\infty \\alpha _a(r)dr = 1$ (i.e.",
"for $\\dot{M} \\gg \\dot{M}_0$ , but not so large that $\\Theta _e(r_{ph}) < 0.5$ so that our emissivity approximation fails).",
"Then $r_{ph} \\propto \\dot{M}^{3/2}/\\nu ^2$ , and the flux $F_\\nu \\propto r_{ph}^2 B_\\nu (r_{ph}) \\sim \\dot{M}^{9/4}/\\nu $ .",
"These simple scaling laws, unfortunately, are not a good description of the variation of source size and flux with $\\dot{M}$ .",
"There are at least three reasons for this.",
"First, relativistic effects are very important; for $\\dot{M} \\sim \\dot{M}_0$ emission comes from close to the photon orbit and the source size is determined by Doppler beaming and gravitational lensing.",
"Second, for the best-bet model at $1.3$ mm $\\tau _\\nu \\sim 1$ , so in a turbulent flow there is a complicated variation of the size of the effective photosphere with $\\dot{M}$ .",
"Third, the emissivity is not precisely frequency independent near peak.",
"We therefore need to turn to numerical models.",
"The best-best model is taken from a survey of 2D models.",
"Here we adopt the best-bet model parameters ($a_*=0.94$ , $i=85 \\deg $ and $T_i/T_e=3$ ) and use them to set parameters for a fully 3D model [36] A 3D GRMHD model parameter survey is still too computationally expensive.",
"It also has a poor return on investment given the electron distribution function uncertainties.. We use the same data set as [9], and choose three representative snapshots taken at times when the flow is quiescent ($t=5000, 9000$ and $13000 GM/c^3$ , where $GM/c^3 = 20$ s).",
"We then recalculate disk images and spectra for $\\dot{M} = (0.5, 1, 2, 4, 8, 16,32,64) \\dot{M}_0$ .",
"The 3D model with $\\dot{M} = \\dot{M}_0$ is broadly consistent with observational data but is slightly more luminous at higher energies than 2D models ($\\beta $ is lower in the 3D models, and this changes the X-ray to millimeter color).",
"The model is self-consistent only for $\\dot{M} <64 \\dot{M}_0$ .",
"At higher $\\dot{M}$ the efficiency of the flow is $>0.1$ and therefore our neglect of cooling in the underlying GRMHD model is not justified.",
"At higher $\\dot{M}$ the $1.3$ mm photosphere also lies outside the limited range in radius where $\\langle d\\dot{M}/dr \\rangle =0$ , so the flow is not in a steady state.",
"The images and total fluxes emitted by the disk at 230 and 345GHz are calculated using a ray tracing scheme [29].",
"To estimate the size of the emitting region we calculate the eigenvalues of the matrix formed by taking the second angular moments of the image on the sky (the principal axis lengths).",
"The major and minor axis eigenvalues, $\\sigma _1$ and $\\sigma _2$ respectively, are related by $\\sigma ={\\rm FWHM}/2.3$ to the FWHM of the axisymmetric Gaussian model used to interpret the VLBI observations.",
"We use $\\langle \\sigma \\rangle = (\\sigma _1+\\sigma _2)/2$ to measure the average radius of the emitting spot.",
"Figures REF and REF show the variation of 230 and 345 GHz Sgr A* model images with $\\dot{M}$ (based on a single snapshot from the 3D GRMHD simulation).",
"Evidently modest increases $\\dot{M}$ will make the ring-like feature that is the observational signature of the photon orbit easier to detect.",
"For $\\dot{M} \\gtrsim 8 \\dot{M}_0$ , however, the ring (or black hole silhouette) is hidden beneath the synchrotron photosphere at 230 GHz.",
"The silhouette survives for higher $\\dot{M}$ at 345 GHz, disappearing only at $\\dot{M} \\gtrsim 16\\dot{M}_0$ .",
"For low $\\dot{M}$ the silhouette is also difficult to detect because the emitting region is too small.",
"Figure: Horizon silhouette detectability at 230 GHz for variousM ˙\\dot{M}.",
"Panels from a) to h) show the imagesof Sgr A* calculated for M ˙=(0.5,1,2,4,8,16,32,64)M ˙ 0 \\dot{M} =(0.5, 1, 2, 4, 8, 16, 32, 64) \\dot{M}_0 ,respectively.",
"The center of the circle is positioned at the image centroidand its radius, r=(σ 1 +σ 2 )/2r=(\\sigma _1+\\sigma _2)/2 is the RMS radius ofthe emitting region.Figure: Same as in Figure but for ν=345\\nu =345 GHz.Figure: Sizes (in terms of σ\\sigma ) of the emitting region and the total flux as afunction of M ˙\\dot{M} at 230 (upper panels) and 345GHz (lowerpanels).",
"Different point types correspond to three different snapshots fromthe 3D GRMHD simulation.",
"The dashed/dotted lines show the best fit to the databy Equations , , , and.Figure REF , shows the accretion flow image size ($\\langle \\sigma \\rangle $ , circle radii in Figure REF and REF ) and flux at 230 and 345GHz as a function of $\\dot{M}$ .",
"Different point types correspond to three different snapshots from the 3D GRMHD simulation.",
"The size of the emission region and flux increase with $\\dot{M}$ .",
"The 345/230 GHz flux ratio increases with the increasing $\\dot{M}$ ; this is caused by the shift of the synchrotron peak towards higher energies at higher $\\dot{M}$ .",
"The following simple fitting formulas describe how $\\sigma $ and $F_\\nu $ depend on $\\dot{M}$ : $\\langle \\sigma \\rangle _{230{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}15.2 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{0.38},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\21.1 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 13,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.",
"}\\, \\, {\\rm [\\mu as]}$ $\\langle \\sigma \\rangle _{345{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}12 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{0.31},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\19.7 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 8.3,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.",
"}\\, \\, {\\rm [\\mu as]}$ $F_{230{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}3.17 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{1.03},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\10.62 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 3.8,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.",
"}\\, \\, {\\rm [Jy]} $ $F_{345{\\rm GHz}}={\\left\\lbrace \\begin{array}{ll}3.2 \\times (\\frac{\\dot{M}}{\\dot{M}_0})^{1.3},& \\mbox{for}\\frac{\\dot{M}}{\\dot{M}_0} < 2\\\\25.13 \\times \\log _{10} (\\frac{\\dot{M}}{\\dot{M}_0})+ 0.54,& \\mbox{for} \\frac{\\dot{M}}{\\dot{M}_0} \\ge 2\\end{array}\\right.",
"}\\, \\, {\\rm [Jy]} $ The above constants are nontrivial to interpret because they encapsulate the complexities of the accretion flow structure and relativistic effects in the radiation transport.",
"The advantage of the above formulas is their simplicity.",
"The fitting functions are shown in Figure REF as dashed and dotted lines.",
"Figure: Relation between two observables: the flux and the size of theimage at ν=\\nu =230 and 345 GHz.",
"The dashed and dotted lines are thebest fits to the data points.Figure REF shows the relation between two observables, $\\sigma $ and $F_\\nu $ .",
"The size is a linear function of the flux and increases more steeply at 230 GHz than at 345 GHz.",
"We also provide two phenomenological scaling laws fitted to the data: $\\langle \\sigma \\rangle _{230GHz} = 1.72 (F_{230{\\rm GHz}}/{\\rm Jy}) + 9.3 \\,\\,\\,{\\rm [\\mu as]} $ $\\langle \\sigma \\rangle _{345GHz} = 0.73 (F_{345{\\rm GHz}}/{\\rm Jy}) + 9.2 \\,\\,\\,{\\rm [\\mu as]} $ Notice that these fits apply to the best-bet model only.",
"For other $a_*$ , $T_i/T_e$ , or $i$ the scalings will be slightly different, although if the source is optically thick the linear scaling follows directly from $T_e \\propto 1/r$ .",
"Notice that Equation REF predicts the change of Sgr A* size by 9 per cent when 230 GHz flux changes from 2 to 2.7 Jy.",
"Taking into account observational and theoretical uncertainties this is consistent with the observed variations of the size of the source which increases by a few per cent as the flux increases from 2 to 2.7 Jy [13]."
],
[
"Spectra",
"Our model spectra are generated by thermal synchrotron emission in the submillimeter/far-IR bump, and by Compton scattering in the X-rays.",
"The spectral slope of flaring NIR emission, and its high degree of linear polarization [6], imply that it is synchrotron from a small, nonthermal tail of high energy electrons that is not (but can be; see [19]) included in our best-bet model.",
"What are the expected scaling laws?",
"Again, $j_\\nu \\sim n_e B \\sim \\dot{M}^{3/2}$ .",
"The luminosity around the synchrotron peak is $L_{peak}\\sim 4 \\pi \\nu _{peak} j_{\\nu _{peak}} (GM/c^2)^3 \\sim \\dot{M}^{9/4}$ , where $\\nu _{peak} \\sim \\dot{M}^{3/4}$ is such that $\\alpha _\\nu GM/c^2 =1$ .",
"The emission rightward of the MIR/NIR is produced by Compton up-scattered synchrotron radiation.",
"The Thomson depth $\\tau _{sc} =\\sigma _{TH} n_e GM/c^2 \\sim \\dot{M}$ , the X-ray luminosity is expected to scale as: $\\nu L_\\nu (\\nu \\approx 10^{18} {\\rm Hz} ) \\sim L_{peak}\\tau _{sc} \\sim \\dot{M}^{13/4}$ , assuming that X-rays are produced primarily by singly scattered synchrotron photons.",
"What do the numerical models show?",
"Figure REF shows spectra emitted from the 3D disk model as observed at $i\\approx 85 \\deg $ .",
"The SEDs are calculated using a general relativistic Monte Carlo scheme [8].",
"The NIR luminosity ${\\nu }L_{\\nu }(\\nu =10^{14} {\\rm Hz}) \\sim \\dot{M}^{2.5}$ , which is only slightly steeper than the expected dependence for the synchrotron peak.",
"${\\nu }L_{\\nu }(\\nu \\approx 10^{18} {\\rm Hz}) \\sim \\dot{M}^{3.25}$ agrees well with the expected scaling.",
"We conclude that Sgr A* would become a persistent MIR and X-ray source (above the present upper limits of $84 {\\rm mJy}$ in MIR and $2.4 \\times 10^{33} {\\rm erg s^{-1}}$ in X-rays) if $\\dot{M} > 2 \\dot{M}_0$ .",
"This is conservative in the sense that our models are strictly thermal.",
"The addition of a high energy nonthermal tail would only increase the MIR/NIR and X-ray flux.",
"We do not consider higher accretion rate models because for $\\dot{M} =64 \\dot{M}_0$ (${\\nu }L_{\\nu }(\\nu =10^{18} {\\rm Hz})=10^{39} {\\rm ergs^{-1}}$ ) the model becomes radiatively efficient, $\\epsilon =L_{Bol}/\\dot{M} c^2 > 0.1$ , and our neglect of radiative cooling in the underlying 3D GRMHD model is unjustified.",
"Finally, notice that the MeV flux increases sharply with $\\dot{M}$ .",
"This suggests that electron-positron pair production by photon-photon collisions in the funnel over the poles of the hole would increase sharply (the pair production $\\dot{n}_{e^\\pm } \\sim L_{\\gamma }^2$ , [26]).",
"If this pair production is connected to jet production, it is reasonable to think that a high $\\dot{M}$ Sgr A* might also produce a jet.",
"Figure: Spectrum emitted by 3D disk model for various M ˙\\dot{M}.",
"TheM ˙/M ˙ 0 \\dot{M}/\\dot{M}_0 is shown on the righthand side.",
"Observational pointsand upper limits are taken from: , ,, , ,.",
"The black symbols in the NIR showing the flaringstate are from .",
"An example of X-ray flare is taken from ."
],
[
"Discussion",
"In summary, we have used general relativistic disk models and relativistic radiative transfer to recalculate millimeter images and spectra for Sgr A* at a range of $\\dot{M} > \\dot{M}_0$ .",
"Our models predict the following: (1) if the 230 GHz flux increases by more than a factor of 2, corresponding to an increase of $\\dot{M}$ by more than a factor of 2, the central accretion flow will become a persistent, detectable, MIR and X-ray source; (2) the photon orbit, which produces the narrow ring of emission visible in Figures 1 and 2, becomes easier to detect for modest increases in the 230 GHz flux; (3) the photon orbit is cloaked beneath the synchrotron photosphere at 230 GHz for $\\dot{M}\\gtrsim 8 \\dot{M}_0$ , or 230 GHz flux $\\gtrsim 13$ Jy; (4) the photon orbit is cloaked at 345 GHz only at higher $\\dot{M} \\gtrsim 16\\dot{M}_0$ , or 230 GHz flux $\\gtrsim 17$ Jy; (5) the size of the source increases in proportion to the flux at both 230 and 345 GHz.",
"We suspect that almost any accretion model for Sgr A* with a spatially uniform model for the plasma distribution function will reach qualitatively similar conclusions, but that jet models may differ significantly.",
"There are order-unity uncertainties in our model predictions due to uncertainties in the plasma model.",
"What range of changes in $\\dot{M}$ are reasonable?",
"In our best-bet model the mass at radii within a factor of two of the pericenter radius is $\\approx 10^{-2.5}$ $M_{\\oplus }$ , assuming steady mass inflow from $r_p$ to the event horizon.",
"The addition of even a fraction of the inferred cloud mass to the accretion flow in a ring near $r_p$ would (eventually) increase $\\dot{M}$ by a factor of $\\sim 300$ .",
"On the other hand, stellar winds supply mass in the neighborhood of the central black hole at $\\sim 10^{-3} M_{\\odot } yr^{-1}$ .",
"Models by [31] and [34] suggest that most of this mass is ejected in the form of a wind, and that $\\sim 10^{-4.5}M_{\\odot } yr^{-1}$ to $10^{-7.3} M_{\\odot } yr^{-1}$ flows inward.",
"A reasonable extrapolation of these models suggest the accretion flow at $r < r_p$ has a mass of $\\sim 2 M_{\\oplus }$ ; this is comparable to estimates of the mass of the inflowing cloud, so in this case we might expect a factor of 2 increase in $\\dot{M}$ .",
"This work was supported by the National Science Foundation under grant AST 07-09246 and by NASA under grant NNX10AD03G, through TeraGrid resources provided by NCSA and TACC."
]
] | 1204.1371 |
[
[
"On the longest length of arithmetic progressions"
],
[
"Abstract Suppose that $\\xi^{(n)}_1,\\xi^{(n)}_2,...,\\xi^{(n)}_n$ are i.i.d with $P(\\xi^{(n)}_i=1)=p_n=1-P(\\xi^{(n)}_i=0)$.",
"Let $U^{(n)}$ and $W^{(n)}$ be the longest length of arithmetic progressions and of arithmetic progressions mod $n$ relative to $\\xi^{(n)}_1,\\xi^{(n)}_2,..., \\xi^{(n)}_n$ respectively.",
"Firstly, the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ are given.",
"Simultaneously, the errors are estimated by using Chen-Stein method.",
"Next, the almost surely limits are discussed when all $p_n$ are equal and when considered on a common probability space.",
"Finally, we consider the case that $\\lim_{n\\to\\infty}p_n=0$ and $\\lim_{n\\to\\infty}{np_n}=\\infty$.",
"We prove that as $n$ tends to $\\infty$, the probability that $U^{(n)}$ takes two numbers and $W^{(n)}$ takes three numbers tends to 1."
],
[
"Introduction and main results",
"Limit distributions for the longest length of runs with respect to Bernoulli sequence have been investigated for a long time, see, e.g., [5], [6],[7], [8] and [9].",
"But what about the longest length of arithmetic progressions?",
"Problems connected to arithmetic progressions are very important in number theory, see [10].",
"For example, Roth's theory says that every set of integers of positive density contained infinitely many progressions of length three.",
"Suppose that $\\xi _1,\\xi _2,\\cdots $ is a Bernoulli sequence with $P(\\xi _i=1)=p=1-q$ , where $0<p<1$ .",
"Let $\\Sigma _n=\\lbrace 1\\le i\\le n:\\xi _i=1\\rbrace $ be the random subset of $\\lbrace 1,2,\\cdots n\\rbrace $ decided by $\\xi _1,\\xi _2,\\cdots ,\\xi _n$ .",
"For any $1\\le a,s\\le n$ , define $U^{(n)}_{a,s}=\\max \\lbrace 1\\le m\\le 1+[\\frac{n-a}{s}]:\\xi _a=1,\\xi _{a+s}=1,\\cdots ,\\xi _{a+(m-1)s}=1 \\rbrace ,$ which is the maximum length of arithmetic progressions in $\\Sigma _n$ starting at $a$ , with difference $s$ .",
"Let $U^{(n)}=\\max _{1\\le a,s\\le n} U^{(n)}_{a,s},$ which is the length of the longest arithmetic progression in $\\Sigma _n$ .",
"We call $U^{(n)}$ the longest length of arithmetic progressions relative to $\\xi _1,\\xi _2,\\cdots ,\\xi _n$ .",
"For any $1\\le a,s\\le n$ , the numbers $a, a+s \\mod {n}, a+2s \\mod {n}, \\cdots , a+({n}/{\\gcd (s,n)}-1) s\\mod {n}$ are different while $a+\\big ( {n}/{\\gcd (s,n)}\\big ) s \\mod {n}=a,$ where $\\gcd (s,n)$ denotes the greatest common divisor of $s$ and $n$ .",
"For convenience, let $kn \\mod {n}=n$ for any integer $k$ .",
"Define $W^{(n)}_{a,s}=\\max \\lbrace 1\\le m\\le \\frac{n}{\\gcd (s,n)} :\\xi _a=1,\\xi _{a+s \\mod {n}}=1,\\cdots ,\\xi _{a+(m-1)s\\mod {n}}=1 \\rbrace ,$ and $W^{(n)}=\\max _{1\\le a,s\\le n} W^{(n)}_{a,s}.$ We call $W^{(n)}$ the the longest length of arithmetic progressions mod $n$ relative to $\\xi _1,\\xi _2,\\cdots ,\\xi _n$ .",
"Note that $U^{(n)}$ is an increasing of $n$ while $W^{(n)}$ is not.",
"In [4], the authors discussed the limit distribution of $U^{(n)}$ and $W^{(n)}$ in the case that $p=1/2$ .",
"The results can be easily extended to the case that $p\\ne 1/2$ .",
"Set $C= {-2}/{\\ln p}$ and let $\\ln $ denote the logarithm of base $e$ .",
"In [4], they proved that as $n$ tends to $\\infty $ , $\\frac{U^{(n)}}{C\\ln n}\\rightarrow 1 $ in probability and $\\frac{W^{(n)}}{C\\ln n}\\rightarrow 1 $ in probability.",
"Furthermore, $\\lim _{n\\rightarrow \\infty } \\frac{U^{(n)}}{C\\ln n}=1 \\,\\, a.s.$ and $1=\\liminf _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}<\\frac{3}{2}\\le \\limsup _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}\\,\\, a.s.$ The authors also conjectured that $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}=\\frac{3}{2} \\,\\,a.s.$ In this paper, we will use Chen-Stein method to study the asymptotic distributions of $U^{(n)}$ and $W^{(n)}$ more carefully.",
"In addition, the errors are also given.",
"The limit distributions we get are a bit different from that in [4].",
"Next, we prove the conjecture and give more description about almost surely limits.",
"Set $D=1/{\\ln p}$ .",
"For any integer $1<r\\le n$ , let $\\lambda _{n,r}=\\lambda _{n,r,p}=\\frac{n^2p^r(p+qr)}{2r(r-1)}.$ and $\\mu _{n,r}=\\mu _{n,r,p}=\\frac{qn^2p^r}{2}.",
"$ Theorem 1.1 (1) It holds that $\\max _{1< r\\le n} |P(U^{(n)}<r)-e^{-\\lambda _{n,r}}|\\le O(\\frac{\\ln ^{4} n\\ln \\ln n}{n}).$ (2) Let $h_n=C\\ln n+ D\\ln \\ln n$ .",
"For any $a<1$ , $\\max _{x\\ge aD\\ln \\ln n,h_n+x\\in \\mathbb {Z}}|\\exp (\\frac{-q\\ln p}{4} p^{x}) P(U^{(n)}<h_n +x)-1|\\le O(\\frac{\\ln \\ln n}{\\ln ^{1-a} n}).$ (3) As $n$ tends to $\\infty $ , $\\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}\\rightarrow D \\text{ in probability }.$ (4)Almost surely, $&&D=\\liminf _{n\\rightarrow \\infty }\\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}<\\limsup _{n\\rightarrow \\infty }\\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}=0,\\\\&&\\lim _{n\\rightarrow \\infty }\\frac{1}{n\\ln n}(\\sum _{k=1}^n U^{(2^k)}+ \\frac{\\ln 2}{\\ln p} n^2)=D$ and $\\lim _{n\\rightarrow \\infty }\\frac{1}{\\ln n\\ln \\ln n}(\\sum _{k=1}^n \\frac{U^{(k)}}{k}+ D\\ln ^2 n)=D.$ Theorem 1.2 (1)It holds that $\\max _{1< r\\le n} |P(W^{(n)}<r)-e^{-\\mu _{n,r}}|\\le O(\\frac{\\ln ^7 n}{n}).",
"$ (2) For any $a$ , $\\max _{x\\ge D\\ln \\ln n+a,C\\ln n+x\\in \\mathbb {Z}}|\\exp (\\frac{q}{2} p^{x}) P(W^{(n)}<C\\ln n +x)-1|\\le O(\\frac{\\ln ^7 n}{n^{1-\\frac{qp^a}{2}}}).$ (3)As $n$ tends to $\\infty $ , $\\frac{W^{(n)}-C\\ln n}{\\ln \\ln n}\\rightarrow 0 \\text{ in probability }.$ (4) That (REF ) holds and $\\liminf _{n\\rightarrow \\infty }\\frac{W^{(n)}-C\\ln n}{\\ln \\ln n}=D \\,\\,a.s.$ (5) Almost surely, $0=\\liminf _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}<\\limsup _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}=-D $ and $\\lim _{n\\rightarrow \\infty }\\frac{1}{n\\ln n}(\\sum _{k=1}^n W^{(2^k)}+ \\frac{\\ln 2}{\\ln p} n^2)=0.",
"$ Next, we shall consider the case that the success probability is not fixed.",
"Suppose that $\\xi ^{(n)}_1,\\xi ^{(n)}_2,\\cdots , \\xi ^{(n)}_n$ are i.i.d with $P(\\xi ^{(n)}_i=1)=p_n=1-P(\\xi ^{(n)}_i=0)$ .",
"Use $U^{(n,p_n)}$ and $W^{(n,p_n)}$ to denote the the longest length of arithmetic progressions or of arithmetic progressions mod $n$ relative to $\\xi ^{(n)}_1,\\xi ^{(n)}_2,\\cdots ,\\xi ^{(n)}_n$ respectively.",
"We have the following results.",
"Theorem 1.3 Assume that $\\lim _{n\\rightarrow \\infty } p_n=0, \\lim _{n\\rightarrow \\infty }np_n=\\infty {\\text{\\,}\\, and \\,\\,}\\lim _{n\\rightarrow \\infty } \\frac{2\\ln n}{-\\ln p_n}=b.",
"$ (i) If $b=\\infty $ , then $\\lim _{n\\rightarrow \\infty } P(U^{(n,p_n)}\\in \\lbrace [\\frac{2\\ln n}{-\\ln p_n}+\\frac{\\ln \\ln n}{\\ln p_n}],[\\frac{2\\ln n}{-\\ln p_n}+\\frac{\\ln \\ln n}{\\ln p_n}]+1\\rbrace )=1$ and $\\lim _{n\\rightarrow \\infty } P(W^{(n,p_n)}\\in \\lbrace [\\frac{2\\ln n}{-\\ln p_n}]-1,[\\frac{2\\ln n}{-\\ln p_n}],[\\frac{2\\ln n}{-\\ln p_n}]+1\\rbrace =1.$ (ii)If $b=2$ , or if $2<b<\\infty $ and $b$ is not an integer, then $\\lim _{n\\rightarrow \\infty } P(W^{(n,p_n)}=[b])=\\lim _{n\\rightarrow \\infty } P(U^{(n,p_n)}=[b])=1.$ (iii) If $b\\ge 3$ and $b$ is an integer, then $\\lim _{n\\rightarrow \\infty } P(W^{(n,p_n)}\\in \\lbrace b, b-1\\rbrace )=\\lim _{n\\rightarrow \\infty } P(U^{(n,p_n)}\\in \\lbrace b,b-1\\rbrace )=1.$ If in addition $u=\\lim _{n\\rightarrow \\infty } n^2p^{b}_n\\le \\infty $ exists, then $\\lim _{n\\rightarrow \\infty } P(U^{(n,p_n)}= b-1)=e^{-\\frac{u}{2(b-1)}}=1-\\lim _{n\\rightarrow \\infty } P(U^{(n,p_n)}=b)$ and $\\lim _{n\\rightarrow \\infty } P(W^{(n,p_n)}=b-1)=e^{-\\frac{u}{2}}=1-\\lim _{n\\rightarrow \\infty } P(W^{(n,p_n)}=b),$ where $[x]$ denotes the integer part of $x$ .",
"The paper is organized as follows.",
"In §2, the equivalent statements of () and (REF ) are given.",
"The proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3 are given in §2, §3 and §4 respectively."
],
[
"Auxiliary Results",
"For clarify, we give a simple lemma that will be used.",
"Lemma 2.1 Suppose that $b_n>0,\\sum _n b_n=\\infty , \\lim _{n\\rightarrow \\infty }\\frac{\\sum _{k=1}^n b_k}{n\\max _{1\\le k\\le n} b_k}=1 $ and $a\\le \\liminf _{n\\rightarrow \\infty } \\frac{a_n}{b_n}\\le \\limsup _{n\\rightarrow \\infty } \\frac{a_n}{b_n}<\\infty .$ Then $\\lim _{n\\rightarrow \\infty } \\frac{\\sum _{k=1}^n a_k}{\\sum _{k=1}^n b_k}=a$ if and only if for all $c>a$ , $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n I_{\\lbrace {a_k}/{b_k}<c\\rbrace }=1.$ At first, we shall prove the sufficiency.",
"By (REF ), there is $d$ such that ${a_k}/{b_k}<d$ for all $k$ .",
"For any $c>a$ , let $A_{n}=\\lbrace 1\\le k\\le n: {a_k}/{b_k}\\ge c\\rbrace $ .",
"Then (REF ) implies that $\\lim _{n\\rightarrow \\infty } {|A_{n}|}/{n}=0$ .",
"Hence $\\frac{\\sum _{k=1}^n a_k}{\\sum _{k=1}^n b_k}\\le c+ \\frac{(d-c)\\sum _{k\\in A_{n}}b_k}{\\sum _{k=1}^n b_k}\\le c +\\frac{(d-c)|A_{n}|\\max _{1\\le k\\le n} b_k}{\\sum _{k=1}^n b_k}\\rightarrow c.$ The arbitrary of $c>a$ yields that $\\limsup _{n\\rightarrow \\infty } {\\sum _{k=1}^n a_k}/{\\sum _{k=1}^n b_k}\\le a$ .",
"This, together with (REF ) and (REF ), gives (REF ).",
"Next, we shall show the necessity.",
"By (REF ), for any $\\varepsilon >0$ , there is $K$ such that $ {a_k}/{b_k}>a-\\varepsilon $ for all $k\\ge K$ .",
"For any $c>a$ , let $B_n=\\lbrace K<k\\le n: {a_k}/{b_k}<c\\rbrace $ .",
"Then for $n>K$ , $\\frac{\\sum _{k=1}^n a_k}{\\sum _{k=1}^n b_k}&\\ge & c+\\frac{\\sum _{k=1}^{K} (a_k-cb_k)+(a-\\varepsilon -c)\\sum _{k\\in B_n} b_k}{\\sum _{k=1}^n b_k}\\\\&\\ge & c+\\frac{\\sum _{k=1}^{K} (a_k-cb_k)+(a-\\varepsilon -c)|B_{n}|\\max _{1\\le k\\le n} b_k}{\\sum _{k=1}^n b_k}.$ Letting $n$ tends to $\\infty $ , we get that $\\liminf _{n\\rightarrow \\infty }\\frac{|B_{n}|\\max _{1\\le k\\le n} b_k}{\\sum _{k=1}^n b_k}\\ge \\frac{c-a}{c+\\varepsilon -a}.$ The arbitrary of $\\varepsilon >0$ , together with (REF ), implies (REF ) and completes our proof.",
"$\\Box $ As an application of Lemma REF .",
"Suppose that (REF ) holds.",
"In addition, suppose that $b_nc_n>0$ , $\\sum _{n} b_nc_n=\\infty $ and $ \\lim _{n\\rightarrow \\infty }\\frac{\\sum _{k=1}^n b_kc_k}{n\\max _{1\\le k\\le n} b_kc_k}=1$ .",
"Then $\\lim _{n\\rightarrow \\infty } {\\sum _{k=1}^n a_kc_k}/{\\sum _{k=1}^n b_kc_k}=a$ if and only if (REF ) holds for all $c>a$ .",
"Particularly, by letting $c_n=1/b_n$ , we see that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n \\frac{a_k}{b_k}=a$ if and only if (REF ) holds for all $c>a$ .",
"Therefore if (REF ) and (REF ) hold, then $\\lim _{n\\rightarrow \\infty } {\\sum _{k=1}^n a_k}/{\\sum _{k=1}^n b_k}=a$ if and only if $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n \\frac{a_k}{b_k}=a$ , if and only if (REF ) holds for all $c>a$ .",
"Note that $n\\ln n-n\\le \\sum _{k=1}^n \\ln k \\le n\\ln n$ .",
"By Lemma REF , we have the following propositions.",
"Proposition 2.1 If (REF ) holds, then () holds if and only if $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} {\\sum _{k=2}^n \\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}}=D \\,\\,a.s, $ and also if and only if for any $1>\\varepsilon >0$ , $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\sum _{k=1}^n I_{\\lbrace U^{(2^k)}<C\\ln 2^k+D(1-\\varepsilon )\\ln \\ln 2^k\\rbrace }=1 \\,\\, a.s. $ Proposition 2.2 If (REF ) holds, then (REF ) holds if and only if $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} {\\sum _{k=2}^n \\frac{W^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}}=0 \\,\\,a.s, $ and also if and only if for any $\\varepsilon >0$ , $\\lim _{n\\rightarrow \\infty }\\frac{1}{n} \\sum _{k=1}^n I_{\\lbrace W^{(2^k)}<C\\ln 2^k-\\varepsilon D\\ln \\ln 2^k\\rbrace }=1 \\,\\, a.s. $"
],
[
"The asymptotic distribution of $U^{(n)}$",
"Suppose that $2\\le r\\le n$ .",
"Let $B_{n}&=&B^{(r)}_{n}=\\lbrace (a,s):1\\le a,s\\le n, a+(r-1)s\\le n\\rbrace \\\\&=&\\lbrace (a,s):1\\le s\\le [\\frac{n-1}{r-1}],1\\le a\\le n-(r-1)s\\rbrace .$ For any $(a,s)\\in B_n$ , let $A_{a,s}=A^{(r)}_{a,s}=\\lbrace \\xi _a=1,\\xi _{a+s}=1,\\cdots , \\xi _{a+(r-1)s}=1\\rbrace \\cap \\lbrace a-s\\le 0 \\text{ or }\\xi _{a-s}=0\\rbrace .$ Then $P(U^{(n)}\\ge r)=P(\\bigcup \\limits _{(a,s)\\in B_n} A_{a,s}).$ Let $I=I_{n,r}=\\sum \\limits _{(a,s)\\in B_n} P(A_{a,s})$ .",
"Then we have $P(U^{(n)}\\ge r)\\le I.",
"$ Set $B_{a,s}=B^{(r)}_{a,s}=\\left\\lbrace \\begin{array}{ll}\\lbrace a,a+s,\\cdots , a+(r-1)s\\rbrace , & \\hbox{if $a\\le s$;} \\\\\\lbrace a-s,a,a+s,\\cdots , a+(r-1)s\\rbrace , & \\hbox{otherwise.",
"}\\end{array}\\right.$ Let $G$ be the graph with vertex set $B_n$ and edges defined by $(a,s) \\sim (b,t)$ if and only if $B_{a,s}\\cap B_{b,t}\\ne \\emptyset $ .",
"Then $G$ is a dependency graph of $\\lbrace I_{A_{a,s}}: (a,s)\\in B_n\\rbrace $ , where $I_{A_{a,s}}$ is the indicator function of $A_{a,s}$ .",
"The notion of dependency graphs can be found in [1],[3] or in §2.1 of [4].",
"Set $e^{(n,r)}=\\sum _{(a,s)\\in B_n}\\sum _{(b,t)\\sim (a,s)} P(A_{a,s})P(A_{b,t})+ \\sum _{(a,s)\\in B_n}\\sum _{(b,t)\\sim (a,s),(b,t)\\ne (a,s)} P(A_{a,s}A_{b,t}).$ Note that $P(U^{(n)}<r)=P(\\sum _{(a,s)\\in B_n} I_{A_{a,s}}=0)$ .",
"Applying the Chen-Stein method, (see [2], [3] or Theorem 3 of [4]), we get that $|P(U^{(n)}<r)-e^{-I}|\\le e^{(n,r)}.",
"$ Lemma 3.1 It holds that $p^r\\frac{(n-r)^2}{2(r-1)}-p^{r+1}\\frac{n^2}{2r}\\le I\\le p^r\\frac{n^2}{2(r-1)}-p^{r+1}\\frac{(n-r)^2}{2r}.$ Clearly, $|B_n|=\\frac{[\\frac{n-1}{r-1}]}{2} (2n-r+1-(r-1)[\\frac{n-1}{r-1}])$ .",
"It implies that $\\frac{(n-r)^2}{2(r-1)}\\le |B_n|\\le \\frac{n^2}{2(r-1)}.",
"$ Note that $B_n\\cap \\lbrace (a,s):a>s\\rbrace =\\lbrace (a,s):1\\le s\\le [ {n-1\\over r}], s<a\\le n-(r-1)s\\rbrace $ .",
"We have $\\frac{(n-r)^2}{2r}\\le |B_n\\cap \\lbrace (a,s):a>s\\rbrace |\\le \\frac{n^2}{2r}.",
"$ Clearly, $I=p^r|B_n|-p^{r+1}|B_n\\cap \\lbrace (a,s):a>s\\rbrace |$ .",
"This, together with (REF ) and (REF ), gives (REF ).",
"$\\Box $ Lemma 3.2 It holds that $e^{(n,r)}\\le 9(n^3 p^{2r-1}+n^2 r^3 p^{\\frac{5}{3} r-1}+n^2 p^{\\frac{3}{2} r-1})$ .",
"Let $c_1&=&|\\lbrace (a,s,b,t)\\in B_n\\times B_n: (a,s)\\sim (b,t)\\rbrace |,\\\\c_2&=&|\\lbrace (a,s,b,t)\\in B_n\\times B_n: |B_{a,s}\\cap B_{b,t}|\\ge 2\\rbrace |$ and $c_3=|\\lbrace (a,s,b,t)\\in B_n\\times B_n: (a,s)\\sim (b,t),t=2s\\text{ or } s=2t\\rbrace |.$ Then $c_1\\le &&|\\lbrace (a,s,b,t): (a,s)\\in B_n, 1\\le t\\le [\\frac{n-1}{r-1}], b=a+is-jt,\\\\&&-1\\le i,j\\le r-1\\rbrace |\\le 9n^3/2$ and $c_3&\\le & 2|\\lbrace (a,s,b,t): (a,s)\\in B_n, t=2s , b=a+is-jt,-1\\le i,j\\le r-1\\rbrace |\\\\&=&2|\\lbrace (a,s,b):(a,s)\\in B_n, b=a+ks,-2r+1\\le k\\le r+1\\rbrace |\\le 7n^2.$ Suppose that $|B_{a,s}\\cap B_{b,t}|\\ge 2$ and $x_0$ is the minimal number of the set $B_{a,s}\\cap B_{b,t}$ .",
"Then $x_0=a+is=b+jt$ for some $-1\\le i,j\\le r-1$ .",
"If $x\\in B_{a,s}\\cap B_{b,t}$ and $x>x_0$ , then $x=a+i^{\\prime }s=b+j^{\\prime }t$ for some $-1\\le i^{\\prime },j^{\\prime }\\le r-1$ .",
"It follows that $x-x_0=(i^{\\prime }-i)s=(j^{\\prime }-j)t$ .",
"Thus $t=\\frac{k_1}{k_2}s$ for some $1\\le k_1,k_2 \\le r$ .",
"In addition, there is positive integer $k$ such that $i^{\\prime }-i=kt_0, j^{\\prime }-j=ks_0$ and $x-x_0=kst_0$ , where $s_0={s}/{\\gcd (s,t)}$ and $t_0={t}/{\\gcd (s,t)}$ .",
"Since $i^{\\prime }-i\\le r$ , $k\\le {r}/{t_0}$ .",
"Similarly, $k\\le {r}/{s_0}$ .",
"Therefore $|B_{a,s}\\cap B_{b,t}|\\le {r}/\\max (s_0, t_0)+1.$ Consequently, $|B_{a,s}\\cap B_{b,t}|\\le {r}/{3}+1$ whenever $\\max (s_0, t_0)\\ge 3$ .",
"When $\\max (s_0, t_0)=2$ , $|B_{a,s}\\cap B_{b,t}|\\le {r}/{2}+1$ .",
"Actually, in this case, $s=2t$ or $t=2s$ .",
"When $\\max (s_0, t_0)=1$ , $s=t$ .",
"We shall show that $A_{a,s}\\cap A_{b,s}=\\emptyset $ whenever $a\\ne b$ and $B_{a,s}\\cap B_{b,s}\\ne \\emptyset $ .",
"Assume that $b>a$ without loss of generality.",
"Since $B_{a,s}\\cap B_{b,s}\\ne \\emptyset $ , $a+is=b+js$ for some $-1\\le i,j\\le r-1$ and hence $b=a+ks$ for some $1\\le k \\le r$ .",
"Thus $A_{a,s}\\subseteq \\lbrace \\xi _{a+(k-1)s}=1\\rbrace $ and $A_{b,s}\\subseteq \\lbrace \\xi _{b-s}=0\\rbrace =\\lbrace \\xi _{a+(k-1)s}=0\\rbrace $ .",
"It implies that $A_{a,s}\\cap A_{b,s}=\\emptyset $ as desired.",
"In view of the discussion above, we have $c_2\\le && |\\lbrace (a,s,b,t): (a,s)\\in B_n, t= {sk_1}/{k_2}, b=a+is-jt,\\\\&&-1\\le i,j\\le r-1,1\\le k_1,k_2\\le r\\rbrace |\\le 9n^2r^3/4$ and $e^{(n,r)}\\le 2c_1p^{2r-1}+c_2p^{\\frac{5r}{3}-1}+c_3p^{\\frac{3r}{2}-1}\\le &9(n^3 p^{2r-1}+n^2 r^3 p^{\\frac{5}{3} r-1}+n^2 p^{\\frac{3}{2} r-1})$ as desired.",
"$\\Box $ Similar as the proof of Lemma REF , we may show that for any $m$ , $n$ and $2\\le r_m\\le r_n$ , with $H=\\lbrace (a,s,b,t):(a,s)\\in B^{(r_m)}_m, (b,t)\\in B^{(r_n)}_n\\rbrace $ , $|\\lbrace (a,s,b,t)\\in H:|B^{(r_m)}_{a,s}\\cap B^{(r_n)}_{b,t}|\\ge 1\\rbrace |\\le 9m^2n/2 $ and $|\\lbrace (a,s,b,t)\\in H:|B^{(r_m)}_{a,s}\\cap B^{(r_n)}_{b,t}|\\ge 2\\rbrace |\\le 9m^2r_mr^2_n/4.$ Also we can show that if $|B^{(r_m)}_{a,s}\\cap B^{(r_n)}_{b,t}|>\\frac{r_n}{2}+1$ and $A^{(r_m)}_{a,s}\\cap A^{(r_n)}_{b,t}\\ne \\emptyset $ , then $(b,t)=(a,s)$ .",
"Therefore $|\\lbrace (a,s,b,t)\\in H:|B^{(r_k)}_{a,s}\\cap B^{(r_m)}_{b,t}|>\\frac{r_n}{2}+1,A^{(r_m)}_{a,s}\\cap A^{(r_n)}_{b,t}\\ne \\emptyset \\rbrace |\\le k^2/r_k.$ Proof of Theorem 1.1 (1) Lemma REF and (REF ) imply that $|e^{-I_{n,r}}-e^{-\\lambda _{n,r}}|\\le |I_{n,r}- \\lambda _{n,r}|\\le 2np^r.$ Let $r_n=[\\frac{-2\\ln n}{\\ln p}+2 \\frac{\\ln \\ln n}{\\ln p}+\\frac{\\ln \\ln \\ln n}{2\\ln p}]$ and $R_n=[\\frac{-3\\ln n}{\\ln p}]$ .",
"By (REF ), (REF ) and Lemma REF , $&&\\max _{r_n\\le r\\le R_n}|P(U^{(n)}<r)-e^{-\\lambda _{n,r}}|\\nonumber \\\\&&\\le \\max _{r_n\\le r\\le R_n}\\big (|P(U^{(n)}<r)-e^{-I_{n,r}}|+|e^{-I_{n,r}}-e^{-\\lambda _{n,r}}|\\big )\\nonumber \\\\&&\\le \\max _{r_n\\le r\\le R_n} (e^{(n,r)}+ 2np^{r})\\nonumber \\\\&&\\le 9(n^3 p^{2r_n-1}+n^2 R^3_n p^{\\frac{5}{3} r_n-1}+n^2 p^{\\frac{3}{2} r_n-1})+2np^{r_n}\\nonumber \\\\&&=O(\\frac{\\ln ^{4} n\\ln \\ln n}{n}).$ On the other hand, it's easy to check that $e^{-\\lambda _{n,r_n}}=e^{-O(\\ln n\\sqrt{\\ln \\ln n})}=o(n^{-1})$ and $1-e^{-\\lambda _{n,R_n}}=1-e^{-O(\\frac{1}{n\\ln n})}=o(n^{-1})$ .",
"Note that $P(U^{(n)}<r)$ and $e^{-\\lambda _{n,r}}$ are both increasing functions of $r$ .",
"Hence when $r<r_n$ , $&&|P(U^{(n)}<r)-e^{-\\lambda _{n,r}}|\\le P(U^{(n)}<r)+e^{-\\lambda _{n,r}}\\le P(U^{(n)}<r_n)+e^{-\\lambda _{n,r_n}}\\\\&\\le & |P(U^{(n)}<r_n)-e^{-\\lambda _{n,r_n}}|+2e^{-\\lambda _{n,r_n}}\\le O(\\frac{\\ln ^{4} n\\ln \\ln n}{n}).$ Similarly, when $r>R_n$ , $&&|P(U^{(n)}<r)-e^{-\\lambda _{n,r}}|\\le 1-P(U^{(n)}<R_n)+1-e^{-\\lambda _{n,R_n}}\\\\&\\le & |P(U^{(n)}<R_n)-e^{-\\lambda _{n,R_n}}|+2(1-e^{-\\lambda _{n,R_n}})\\le O(\\frac{\\ln ^{4} n\\ln \\ln n}{n}).$ This completes the proof of (REF ).$\\Box $ (2) Let $r=h_n+x$ .",
"For convenience, set $\\epsilon _{n,x}=|\\frac{-q\\ln p}{4} p^{x}-\\lambda _{n,r}|= p^{x}\\big |\\frac{-q\\ln p}{4}-\\frac{q\\ln n}{2(r-1)}-\\frac{p \\ln n}{2r(r-1)}\\big |.$ Then $&&|\\exp (\\frac{-q\\ln p}{4} p^{x}) P(U^{(n)}<h_n +x)-1|\\nonumber \\\\&\\le & \\exp (\\frac{-q\\ln p}{4} p^{x})|P(U^{(n)}<r)-e^{-\\lambda _{n,r}}|+|\\exp (\\epsilon _{n,x})-1|$ Since $a<1$ , $\\exp (\\frac{-q\\ln p}{4} p^{aD\\ln \\ln n})=\\exp (\\frac{-q\\ln p}{4}\\ln ^a n)=o(n^{\\frac{1}{3}}).$ Thus $&&\\max _{x\\ge aD\\ln \\ln n,h_n+x\\in \\mathbb {Z}}\\exp (\\frac{-q\\ln p}{4} p^{x})|P(U^{(n)}<r)-e^{-\\lambda _{n,r}}|\\nonumber \\\\&\\le & o(n^{\\frac{1}{3}})O(n^{-1}\\ln ^{4} n\\ln \\ln n))=o(n^{-\\frac{1}{2}}).$ It's easy to verify that $\\max _{ x > -D\\ln \\ln n+1}\\epsilon _{n,x}\\le p^{-D\\ln \\ln n}O(1)=O(\\frac{1}{\\ln n})$ and $\\max _{ aD\\ln \\ln n\\le x \\le -D\\ln \\ln n+1}\\epsilon _{n,x}\\le p^{ aD\\ln \\ln n}O(\\frac{\\ln \\ln n}{\\ln n})=O(\\frac{ \\ln \\ln n}{\\ln ^{1-a} n}).$ Now (REF ) follows by (REF )–(REF ).",
"(3) For any $\\varepsilon >0$ , (REF ) and (REF ) imply that $P(U^{(n)}\\ge C\\ln n +(1-\\varepsilon )D\\ln \\ln n)\\le O(\\ln ^{-\\varepsilon } n).$ On the other hand, by (REF ) and (REF ), $P(U^{(n)}< C\\ln n +(1+\\varepsilon )D\\ln \\ln n)\\le e^{-O(\\ln ^{\\varepsilon } n)}+O(\\frac{\\ln ^{4} n\\ln \\ln n}{n}).$ Hence (REF ) holds.",
"(4) By (REF ), $\\sum _{k=1}^\\infty P(U^{(2^k)}< C\\ln 2^k +(1+\\varepsilon )D\\ln \\ln 2^k)<\\infty $ for any $\\varepsilon >0$ .",
"One then deduces from the Borel-Cantelli Lemma that $P(U^{(2^k)}<C\\ln 2^k +(1+\\varepsilon )D \\ln \\ln 2^k \\,\\,\\,\\,i.o.",
")=0.$ It follows that for almost surely $\\omega $ , there is $K(\\omega )$ such that for $k\\ge K(\\omega )$ , $U^{(2^k)}(\\omega )\\ge C\\ln 2^k +(1+\\varepsilon )D\\ln \\ln 2^k.$ If $n>2^{K(\\omega )}$ , then $2^k\\le n<2^{k+1}$ for some $k\\ge K(\\omega )$ .",
"Hence $U^{(2^k)}(\\omega )\\le U^{(n)}(\\omega )\\le U^{(2^{k+1})}(\\omega )$ .",
"This, together with (REF ), gives that $U^{(n)}(\\omega )\\ge C\\ln n-C\\ln 2+(1+\\varepsilon )D\\ln \\ln n.$ Now the arbitrary of $\\varepsilon >0$ yields that $\\liminf _{n\\rightarrow \\infty } \\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}\\ge D$ .",
"Therefore $\\liminf _{n\\rightarrow \\infty } \\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}= D$ by considering (REF ).",
"Let $T_k$ be the longest length of arithmetic progressions relative to $\\xi _{2^{k-1}+1},\\\\ \\xi _{2^{k-1}+2},\\cdots ,\\xi _{2^{k}}$ .",
"Then $T_1,T_2,\\cdots $ are independent.",
"In addition, $T_k$ has the same distribution with $U^{(2^{k-1})}$ .",
"By (REF ), $P(T_k\\ge C\\ln 2^k)=P(U^{(2^{k-1})}\\ge C\\ln 2^k)\\le O(1/k).$ Hence $\\sum _{k=1}^\\infty P(T_k\\ge C\\ln 2^k)=\\infty $ .",
"By Borel-Cantelli Lemma, $P(T_k\\ge C\\ln 2^k\\,\\,\\, i.o.",
")=1$ .",
"Consequently, $\\limsup _{k\\rightarrow \\infty } \\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}\\ge 0$ by noting that $U^{(2^k)}\\ge T_k$ .",
"On the other hand, (REF ) yields that $\\sum _{k=1}^\\infty P(U^{(2^k)}\\ge C\\ln 2^k +(1-\\varepsilon )D \\ln \\ln 2^k)<\\infty $ whenever $\\varepsilon >1$ .",
"Hence $\\limsup _{k\\rightarrow \\infty } \\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}\\le 0$ .",
"Therefore $\\limsup _{k\\rightarrow \\infty } \\\\\\frac{U^{(2^k)}-C\\ln 2^k}{\\ln \\ln 2^k}= 0$ .",
"Furthermore, we can deduce that $\\limsup _{n\\rightarrow \\infty } \\frac{U^{(n)}-C\\ln n}{\\ln \\ln n}= 0$ by the fact that $U^{(n)}$ is increasing.",
"This completes the proof of (REF ).",
"Now we come to prove ().",
"By Proposition REF , we need only to show (REF ).",
"Let $r_k= [C\\ln 2^k+D(1-\\varepsilon )\\ln \\ln 2^k]$ , $V_n=\\lbrace (a,s,k):1\\le k\\le n, (a,s)\\in B_{2^k}^{(r_k)}\\rbrace $ and $\\Lambda (n)=\\sum _{(a,s,k)\\in V_n}I_{ A_{a,s}^{(r_k)}}$ .",
"Then it suffices to show that $\\lim _{n\\rightarrow \\infty }\\Lambda (n)/n=0$ a.s. Clearly, $E\\Lambda (n)=\\sum _{k=1}^n I_{2^k,r_k}=O(n^{1-\\varepsilon })$ and $D\\Lambda (n)$ is less than the sum of $p^{r_k+r_m-|B^{(r_k)}_{a,s}\\cap B^{(r_m)}_{b,t}|}$ with $(a,s,k),(b,t,m)\\in V_n$ , $B^{(r_k)}_{a,s}\\cap B^{(r_m)}_{b,t}\\ne \\emptyset $ and $A^{(r_k)}_{a,s}\\cap A^{(r_m)}_{b,t}\\ne \\emptyset $ .",
"By (REF )–(REF ), $D\\Lambda (n)\\le &&\\sum _{1\\le i\\le j\\le n} O(2^{-j}ij+2^{-j}i^{2}j^3+2^{2i-2j}j^{1-\\varepsilon }i^{-1})\\\\\\le && O(\\sum _{j=1}^\\infty 2^{-j} j^{6})+O(\\sum _{j=1}^n j^{-\\varepsilon }\\sum _{k=0}^{j-1} 2^{-2k}j(j-k)^{-1})\\\\\\le &&O(1)+\\sum _{k=0}^{\\infty } 2^{-2k}(k+1)O(\\sum _{j=1}^n j^{-\\varepsilon })=O(n^{1-\\varepsilon }).$ Then by Tchebychev's inequality, for any $\\delta >0$ , $\\sum _{n=1}^\\infty P(|\\Lambda _n/n-E(\\Lambda _n/n)|>\\delta )\\le \\sum _{n=1}^\\infty O(\\frac{1}{n^{1+\\varepsilon }\\delta ^2})<\\infty .$ The Borel-Cantelli Lemma yields that $\\Lambda _n/n\\rightarrow 0\\,\\,a.s.$ .",
"Hence (REF ) holds as desired.",
"Finally, we shall prove (REF ).",
"Let $c_n=[\\ln n/\\ln 2]$ .",
"Then $2^{c_n}\\le n<2^{c_n+1}$ .",
"For any integers $1\\le a\\le b$ , $\\ln \\frac{b+1}{a}=\\int _{a}^{b+1} \\frac{1}{x}\\,dx\\le \\sum _{i=a}^b\\frac{1}{i}\\le \\int _{a-1}^b \\frac{1}{x}\\,dx=\\ln \\frac{b}{a-1}.$ Thus $\\sum _{k=1}^n \\frac{U^{(k)}}{k}\\ge \\sum _{i=0}^{c_n-1} U^{(2^i)}\\sum _{j=2^i}^{2^{i+1}-1}\\frac{1}{j} \\ge \\ln 2\\sum _{i=0}^{c_n-1} U^{(2^i)}$ and $\\sum _{k=2}^n \\frac{U^{(k)}}{k}\\le \\sum _{i=0}^{c_n} U^{(2^{i+1})}\\sum _{j=2^i+1}^{2^{i+1}}\\frac{1}{j} \\le \\ln 2\\sum _{i=1}^{c_n+1} U^{(2^i)}.$ Hence by (), $\\lim _{n\\rightarrow \\infty }\\frac{1}{c_n\\ln c_n}(\\frac{1}{\\ln 2}\\sum _{k=1}^n \\frac{U^{(k)}}{k}+\\frac{\\ln 2}{\\ln p} c^2_n)=D\\,\\,a.s.$ It follows (REF ) immediately and completes the proof of Theorem 1.1.",
"$\\Box $"
],
[
"The asymptotic distribution of $W^{(n)}$",
"Suppose that $2\\le r\\le n$ .",
"For any $1\\le a,s \\le n$ , let $\\tilde{A}_{a,s}=\\tilde{A}^{(n,r)}_{a,s}=\\lbrace \\xi _a=0,\\xi _{a+s \\mod {n}}=1,\\cdots , \\xi _{a+rs \\mod {n}}=1\\rbrace .$ Let $\\tilde{B}_n=\\tilde{B}^{(r)}_n=\\lbrace (a,s): 1\\le a\\le n,1\\le s\\le [ n/2],\\gcd (n,s)<n/r\\rbrace $ and $A_1=\\bigcup \\limits _{(a,s)\\in \\tilde{B}_n} \\tilde{A}_{a,s}.",
"$ Set $C_{a,s}=\\lbrace a,a+s \\mod {n},a+2s \\mod {n},a+3s \\mod {n},\\cdots \\rbrace $ .",
"Then $C_{a,s}=\\lbrace a,a+s \\mod {n},\\cdots , a+ (\\frac{n}{\\gcd (s,n)}-1)s\\mod {n} \\rbrace $ and $|C_{a,s}|=n/\\gcd (s,n)$ .",
"Set $A_2=\\bigcup \\limits _{s|n,s\\le n/r,1\\le a\\le s} \\lbrace \\xi _i=1,\\forall i\\in C_{a,s}\\rbrace ,$ where $s|n$ means that $s$ is a divisor of $n$ .",
"Lemma 4.1 It holds that $\\lbrace W^{(n)}\\ge r\\rbrace =A_1\\cup A_2.$ Put $W^{(n)}_s=\\max \\limits _{1\\le i \\le n} W^{(n)}_{i,s},$ which is the maximum length of arithmetic progressions mod $n$ in $\\Sigma _n$ with difference $s$ .",
"For any $m \\ge 0$ , $\\lbrace \\xi _a=1,\\xi _{a+s \\mod {n}}=1,\\cdots ,\\xi _{a+ms \\mod {n}}=1\\rbrace $ if and only if $\\lbrace \\xi _b=1,\\xi _{b+(n-s) \\mod {n}}=1,\\cdots ,\\xi _{b+m(n-s) \\mod {s}}=1\\rbrace $ , where $b=a+ms \\mod {n}$ .",
"In addition, $\\gcd (s,n)=\\gcd (n-s,n)$ .",
"Hence $W_{s}^{(n)}=W_{n-s}^{(n)}$ for all $1\\le s \\le n$ .",
"Consequently, $W^{(n)}=\\max _{1\\le s \\le n} W^{(n)}_s=\\max _{1\\le s \\le [ n/2]} W^{(n)}_s.$ For any $1\\le a,b\\le n$ , $C_{a,s}\\cap C_{b,s}=\\emptyset $ or $C_{a,s}=C_{b,s}$ .",
"In addition $C_{a,s}=C_{b,s}$ if and only if $b=a+\\gcd (s,n)\\cdot k$ for some integer $k$ .",
"Thus $\\lbrace 1,2,\\cdots ,n\\rbrace $ is the disjoint union of $C_{a,s}$ with $1\\le a\\le \\gcd (s,n)$ .",
"It follows that $W^{(n)}_s=\\max _{1\\le a \\le \\gcd (s,n)}\\tilde{W}^{(n)}_{a,s}.$ where $\\tilde{W}^{(n)}_{a,s}=\\max \\limits _{i\\in C_{a,s}} W^{(n)}_{i,s}$ .",
"Note that $\\lbrace \\tilde{W}^{(n)}_{a,s}\\ge r\\rbrace = \\lbrace \\xi _i=1,\\forall i\\in C_{a,s}\\rbrace $ when $n/\\gcd (s,n)= r$ , and $\\lbrace \\tilde{W}^{(n)}_{a,s}\\ge r\\rbrace =(\\bigcup \\limits _{i\\in C_{a,s}} \\tilde{A}_{i,s})\\bigcup \\lbrace \\xi _i=1,\\forall i\\in C_{a,s}\\rbrace .$ provided $n/\\gcd (s,n)> r$ .",
"Therefore $\\lbrace W^{(n)}\\ge r\\rbrace = \\big (\\bigcup _{(i,s)\\in \\tilde{B}_n}\\tilde{A}_{i,s}\\big )\\bigcup \\big (\\bigcup _{1\\le a\\le \\gcd (s,n),1\\le s \\le [ \\frac{n}{2}], \\frac{n}{\\gcd (s,n)}\\ge r}\\lbrace \\xi _i=1,\\forall i\\in C_{a,s}\\rbrace \\big ).$ This, together with the fact that $C_{a,s}=C_{a,\\gcd (s,n)}$ , yields (REF ).",
"$\\Box $ Let $\\tilde{I}=\\tilde{I}_{n,r}=\\sum _{(a,s)\\in \\tilde{B}_n} P(\\tilde{A}_{a,s})$ .",
"By Lemma (REF ), we have $P(A_1)\\le P(W^{(n)}\\ge r)\\le P(A_1)+P(A_2)\\le \\tilde{I}+P(A_2).$ Set $\\tilde{B}_{a,s}=\\tilde{B}^{(n,r)}_{a,s}=\\lbrace a,a+s \\mod {n},\\cdots , a+rs \\mod {n}\\rbrace .$ Define $\\tilde{G}$ to be the graph with vertex set $\\tilde{B}_n$ and edges defined by $(a,s) \\sim (b,t)$ if and only if $\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}\\ne \\emptyset $ .",
"Then $\\tilde{G}$ is a dependency graph of $\\lbrace I_{\\tilde{A}_{a,s}}: (a,s)\\in \\tilde{B}_n\\rbrace $ .",
"Put $\\tilde{e}^{(n,r)}=\\sum _{(a,s)\\in \\tilde{B}_n}\\sum _{(b,t)\\sim (a,s)} P(\\tilde{A}_{a,s})P(\\tilde{A}_{b,t})+\\sum _{(a,s)\\in \\tilde{B}_n}\\sum _{(b,t)\\sim (a,s),(b,t)\\ne (a,s)} P(\\tilde{A}_{a,s} \\tilde{A}_{b,t}).$ Then $|P(A^c_1)-e^{-\\tilde{I}}|\\le \\tilde{e}^{(n,r)}.$ The estimations of $P(A_2)$ , $\\tilde{I}$ and $\\tilde{e}^{(n,r)}$ are given in the following two lemmas.",
"Lemma 4.2 We have $P(A_2)\\le \\frac{np^r}{qr}$ and $(1 -\\frac{(r+1)^2}{2n})\\frac{qn^2p^r}{2}\\le \\tilde{I} \\le \\frac{qn^2 p^r}{2}.$ Obviously, $P(A_2)&\\le & \\sum _{s|n,s\\le n/r}sp^{\\frac{n}{s}}\\le \\sum _{i=r}^n \\frac{n}{i}p^i\\le \\frac{n}{r} \\sum _{i=r}^n p^i\\le \\frac{np^r}{qr}.$ Clearly, $\\lbrace 1\\le s\\le [\\frac{n}{2}]: \\gcd (n,s)\\ge \\frac{n}{r}\\rbrace \\subseteq \\lbrace s=\\frac{n}{i} j:2\\le i\\le r,1\\le j\\le [\\frac{i}{2}]\\rbrace $ .",
"It implies that $\\frac{n^2}{2}\\ge |B_n|\\ge n([\\frac{n}{2}]-\\sum _{i=2}^r [\\frac{i}{2}])\\ge \\frac{n^2}{2}(1 -\\frac{(r+1)^2}{2n}).$ Now the fact that $\\tilde{I}=|\\tilde{B}_n|qp^r$ yields (REF ) immediately.",
"$\\Box $ Lemma 4.3 It holds that $\\tilde{e}^{(n,r)}\\le 4(n^3r^2p^{2r-1}+n^2r^5p^{\\frac{3}{2} r-1}+nr^6 p^r)$ .",
"Let $H=\\lbrace (a,s,b,t)\\in \\tilde{B}_n\\times \\tilde{B}_n: (a,s)\\sim (b,t)\\rbrace $ , $\\tilde{c}_1=|H|$ , $\\tilde{c}_2&=&|\\lbrace (a,s,b,t)\\in H: |\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|\\ge 2\\rbrace |$ and $\\tilde{c}_3=|\\lbrace (a,s,b,t)\\in H: (a,s)\\ne (b,t),|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|>\\frac{r}{2}+1,\\tilde{A}_{a,s}\\cap \\tilde{A}_{b,t}\\ne \\emptyset \\rbrace |.$ Then $\\tilde{c}_1\\le && |\\lbrace (a,s,b,t): (a,s)\\in \\tilde{B}_n, 1\\le t\\le [\\frac{n}{2}], b=a+is-jt \\mod {n},\\nonumber \\\\&&0\\le i,j\\le r\\rbrace |\\le n^3r^2.$ Suppose that $|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|\\ge 2$ .",
"Then there is $0\\le j_1<j_2\\le r$ and $x,y\\in \\tilde{B}_{a,s}$ such that $b+j_1t\\mod {n}=x$ and $b+j_2t\\mod {n}=y$ .",
"Hence $(j_2-j_1)t-kn=y-x$ for some $0\\le k \\le j_2-j_1$ and $b=x-j_1t\\mod {n}$ .",
"Therefore $\\tilde{c}_2\\le && |\\lbrace (a,s,b,t): (a,s)\\in \\tilde{B}_n, t=(kn+y-x)/i, b=x-jt\\mod {n},\\nonumber \\\\&&1\\le i\\le r, 0\\le j,k\\le r,x,y\\in \\tilde{B}_{a,s}\\rbrace |\\le 3n^2r^5.$ We shall show that $\\tilde{A}_{a,s}\\cap \\tilde{A}_{b,s}=\\emptyset $ when $b\\ne a$ and $(a,s)\\sim (b,s)$ .",
"Let $j^*=\\min \\lbrace 0 \\le j \\le r: b+j s \\mod {n} \\in \\tilde{B}_{a,s}\\rbrace .$ Then there is $0\\le i \\le r$ such that $b+j^* s \\mod {n} =a+ is \\mod {n}$ .",
"It follows that $b+(j^*-1) s \\mod {n} =a+ (i-1)s \\mod {n}$ .",
"Hence $j^*-1<0$ or $i-1<0$ .",
"If $j^*-1<0$ , then $j^*=0$ and hence $b=a+is \\mod {n}$ .",
"Since $b\\ne a$ , $i\\ne 0$ .",
"We have $\\tilde{A}_{a,s}\\cap \\tilde{A}_{b,s}\\subseteq \\lbrace \\xi _{a+is \\mod {n}}=1,\\xi _{b}=0\\rbrace =\\lbrace \\xi _{b}=1,\\xi _{b}=0\\rbrace =\\emptyset $ .",
"Similarly $\\tilde{A}_{a,s}\\cap \\tilde{A}_{b,s}=\\emptyset $ when $i-1<0$ .",
"Suppose that $s\\ne t$ , $|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|> r/2+1$ and $|\\tilde{B}_{a,s}\\cap \\tilde{B}_{b,t}|=\\lbrace a+i_0s,a+i_1s,\\cdots ,a+i_ks\\rbrace $ with $0\\le i_0<i_1<\\cdots <i_k\\le r$ .",
"Then there is $ l$ such that $i_{l+1}-i_l=1$ .",
"Since $a+i_ls \\mod {n}=b+j_1 \\mod {n}$ and $a+i_{l+1}s\\mod {n} =b+j_2 \\mod {n}$ for some $0\\le j_1,j_2\\le r$ and $j_1\\ne j_2$ , $s=it \\mod {n}$ with $i=j_2-j_1$ .",
"If $i=1$ , then $s=t$ .",
"If $i=-1$ , then $s=n-t$ and hence $s=t=n/2$ by the fact that $1\\le s,t\\le n/2$ .",
"The contradiction shows that $1<|i|\\le r$ .",
"Similarly, $t=js \\mod {n}$ for some $1< |j|\\le r$ .",
"It follows that $s=ijs \\mod {n}$ , that is $(ij-1)s=\\jmath n$ for some $|\\jmath |\\le r^2/2$ .",
"Consequently, $\\tilde{c}_3 &\\le & |\\lbrace (a,s,b,t): s=\\jmath n/(ij-1), t=js \\mod {n}, b=a+ls-mt\\mod {n},\\nonumber \\\\&&1\\le a\\le n, 1<|i|,|j|\\le r, |\\jmath |\\le r^2/2, 0\\le l,m\\le r\\rbrace |\\le 4nr^6.",
"$ Thus our result holds by noting that $\\tilde{e}^{(n,r)}\\le 2\\tilde{c}_1 p^{2r-1}+\\tilde{c}_2 p^{\\frac{3r}{2}-1}+\\tilde{c}_3 p^{r}$ .",
"$\\Box $ Similarly, we may show that for any fixed $m, n$ , $|\\lbrace (a,s,b,t)\\in \\tilde{B}_m^{(r_m)}\\times \\tilde{B}_n^{(r_n)}:|\\tilde{B}^{(m,r_m)}_{a,s}\\cap \\tilde{B}^{(n,r_n)}_{b,t}|\\ge 1\\rbrace |\\le m^2nr_mr_n $ and $|\\lbrace (a,s,b,t)\\in \\tilde{B}_m^{(r_m)}\\times \\tilde{B}_n^{(r_n)}:|\\tilde{B}^{(m,r_m)}_{a,s}\\cap \\tilde{B}^{(n,r_n)}_{b,t}|\\ge 2\\rbrace |\\le 3m^2r^2_mr^3_n.",
"$ Lemma 4.4 Suppose that $(a,s)\\in \\tilde{B}_{m}^{(r_m)}$ and $(b,t)\\in \\tilde{B}_{n}^{(r_n)}$ .",
"If $n\\ge 2m$ , $r_n\\ge 36$ and $t>3n/r_n$ , then $|\\tilde{B}^{(m,r_m)}_{a,s}\\cap \\tilde{B}^{(n,r_n)}_{b,t}|<\\frac{3}{4} r_n$ .",
"If $1\\le x\\le m$ , $A=\\lbrace x,x+t,\\cdots ,x+kt\\rbrace \\subseteq \\lbrace 1,2,\\cdots ,n\\rbrace $ and $x+(k+1)t>n$ , then $k\\ge 1$ and $x+\\frac{k+1}{2}t>n/2\\ge m$ .",
"Thus $|A\\cap \\lbrace 1,2,\\cdots ,m\\rbrace |\\le (k+1)/2$ when $k$ is odd, or $|A\\cap \\lbrace 1,2,\\cdots ,m\\rbrace |\\le k/2+1$ when $k$ is even.",
"Hence $|A\\cap \\lbrace 1,2,\\cdots ,m\\rbrace |/|A|\\le 2/3$ .",
"Since $t>3n/r_n$ , there is $h\\ge 3$ , $0\\le i_1<i_2<\\cdots i_h<r_n$ such that $b+(i_1+1)t>n\\ge b+i_1 t, b+(i_2+1)t>2n\\ge b+i_2 t, \\cdots , b+(i_h+1)t>hn\\ge b+i_h t$ and $b+r_nt\\le (h+1)n$ .",
"Let $i_0=-1$ , $i_{h+1}=r_n$ and $A_j=\\lbrace b+(i_{j}+1)t \\mod {n}, b+(i_{j}+2)t\\mod {n},\\cdots , b+i_{j+1} t\\mod {n}\\rbrace .$ Then $|A_j\\cap \\lbrace 1,2,\\cdots m\\rbrace |\\le \\frac{2}{3}|A_j|$ when $0\\le j\\le h-1$ , and $\\jmath =|A_{h+1}\\cap \\lbrace 1,2\\cdots ,m\\rbrace |\\le m/t+1$ .",
"Thus $&&|\\lbrace b,b+t\\mod {n},\\cdots , b+r_n t\\mod {n}\\rbrace \\cap \\lbrace 1,2,\\cdots , m\\rbrace |/r_n\\\\&&\\le \\frac{\\frac{2}{3} \\sum _{j=0}^{h-1}|A_j|+\\jmath }{r_n}\\le \\frac{2(i_h+1+\\jmath )}{3r_n}+\\frac{\\jmath }{3r_n}\\le \\frac{2}{3}+\\frac{m}{3r_nt}+\\frac{1}{r_n}<\\frac{3}{4}.\\Box $ Proof of Theorem 1.2 (1)Firstly, by (REF ) and (REF ), $|e^{-\\tilde{I}_{n,r}}-e^{-\\mu _{n,r}}|\\le |\\tilde{I}_{n,r}- \\mu _{n,r}|\\le qnp^r r^2.$ Next, let $r_n=[\\frac{-2\\ln n}{\\ln p}+ \\frac{\\ln \\ln n}{\\ln p}+\\frac{\\ln 2-\\ln q}{\\ln p}-1]$ and $R_n=[\\frac{-3\\ln n}{\\ln p}]$ .",
"Lemma REF , together with (REF ), (REF ), (REF ) and (REF ), yields that $&&\\max _{r_n\\le r\\le R_n}|P(W^{(n)}<r)-e^{-\\mu _{n,r}}|\\nonumber \\\\&&\\le \\max _{r_n\\le r\\le R_n}|P(W^{(n)}<r)-e^{-\\tilde{I}_{n,r}}|+\\max _{r_n\\le r\\le R_n}|e^{-\\tilde{I}_{n,r}}-e^{-\\mu _{n,r}}|\\nonumber \\\\&&\\le \\max _{r_n\\le r\\le R_n} (\\tilde{e}^{(n,r)}+\\frac{np^r}{qr}+ qnp^{r}r^2)\\nonumber \\\\&&= O(n^{-1}\\ln ^{7} n).$ Furthermore, it's easy to check that $e^{-\\mu _{n,r_n}}=o(n^{-1})$ and $1-e^{-\\mu _{n,R_n}}=O(n^{-1})$ .",
"Therefore(REF ) holds by noting that $P(W^{(n)}<r)$ and $e^{-\\mu _{n,r}}$ are both increasing functions of $r$ .",
"(2) Let $r=C\\ln n+x$ .",
"Then $\\mu _{n,r}= {qp^x}/{2} $ .",
"Hence (REF ) implies that $&&\\max _{x\\ge D{\\ln \\ln n}+a,C\\ln n+x\\in \\mathbb {Z}}|\\exp ( {qp^x}/{2} ) P(W^{(n)}<C\\ln n +x)-1|\\\\&\\le & \\exp (qp^{D \\ln \\ln n+a}/2) O(\\frac{\\ln ^7 n}{n})=O(\\frac{\\ln ^7 n}{n^{1-\\frac{qp^a}{2}}}).$ (3) In view of (REF ), it holds that $P(W^{(n)}<C\\ln n+\\varepsilon D\\ln \\ln n)=e^{-O(\\ln ^\\varepsilon n)}+O(\\frac{\\ln ^7 n}{n})\\rightarrow \\left\\lbrace \\begin{array}{ll}1, & \\hbox{$\\varepsilon <0$;} \\\\0, & \\hbox{$\\varepsilon >0$.",
"}\\end{array}\\right.$ It follows (REF ) immediately.",
"(4) In view of (REF ), to prove (REF ), it remains only to show that $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}\\le \\frac{3}{2} \\,\\, a.s. $ For any $\\varepsilon >0$ , by (REF ),(REF ) and (REF ), $P(W^{(n)}>(1+\\varepsilon )C\\ln n) \\le O(n^{-2\\varepsilon })+O(\\frac{n^{-1-2\\varepsilon }}{\\ln n}).$ Hence $\\sum _{n=1}^\\infty P(W^{(n)}>(1+\\varepsilon )C\\ln n) <\\infty $ whenever $\\varepsilon >\\frac{1}{2}$ .",
"Therefore, $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(n)}}{C\\ln n}\\le \\frac{3}{2} \\,\\,\\,a.s$ .",
"as desired.",
"Since $W^{(n)}\\ge U^{(n)}$ , by (REF ), to prove (REF ), we need only to show that $\\liminf _{n\\rightarrow \\infty } \\frac{W^{(n)}-C\\ln n}{\\ln \\ln n}\\le D\\,\\,\\, a.s. $ Fix any $0<\\varepsilon <1$ .",
"Let $r_n=[C\\ln n+\\varepsilon D\\ln \\ln n]$ , $H_n=\\lbrace W^{(n)}<r_n\\rbrace $ and $X_k=\\sum _{n=k+1}^{2k} I_{H_n}$ .",
"Then $P(\\bigcup _{n=k+1}^{2k}H_n)=P(X_k>0)\\ge \\frac{(EX_k)^2}{EX^2_k}=\\frac{(\\sum _{n=k+1}^{2k} P(H_n))^2}{\\sum _{m,n=k+1}^{2n} P(H_mH_n)}.",
"$ Clearly, $\\mu _{n,r_n}=O(\\ln ^\\varepsilon n)=o(\\ln n/4)$ .",
"Together with (REF ), it implies that $\\sum _{n=k+1}^{2k} P(H_n)\\ge \\sum _{n=k+1}^{2k}e^{-\\mu _{n,r_n}}-O(\\ln ^7k)\\ge O(k^{3/4}).",
"$ Set $E_n=\\lbrace (a,s):1\\le a\\le n, [\\frac{3n}{C\\ln n}]+1\\le s\\le [\\frac{n}{2}],\\gcd (n,s)<\\frac{n}{r_n}\\rbrace $ .",
"By (REF ) and (REF ), $P((H_mH_n)^c)\\ge P(\\cup _{(a,s)\\in E_m}\\tilde{A}^{(m,r_m)}_{a,s}\\cup _{(b,t)\\in E_n}\\tilde{A}^{(n,r_n)}_{b,t}).$ Let $V$ be the graph with vertex set $V_{k}=\\lbrace (a,s,n):k+1\\le n\\le 2k, (a,s)\\in E_n\\rbrace $ and edges defined by $(a,s,m)\\sim (b,t,n)$ if and only if $\\tilde{B}^{(m,r_m)}_{a,s}\\cap \\tilde{B}^{(n,r_n)}_{b,t}\\ne \\emptyset $ .",
"Then $V$ is a dependency graph of $\\lbrace I_{\\tilde{A}^{(n,r_n)}_{a,s}}:(a,s,n)\\in V_k\\rbrace $ .",
"Let $J_{m}=\\sum _{(a,s)\\in E_m} P(\\tilde{A}^{(m,r_m)}_{a,s})$ .",
"Then $P(H_mH_n)\\le e^{-J_m-J_n}+\\tilde{e}^{(m,r_m)}+\\tilde{e}^{(n,r_n)}+2 \\tilde{e}^{(m,r_m,n,r_n)}, $ where $\\tilde{e}^{(m,r_m,n,r_n)}=\\sum _{(a,s,m)\\sim (b,t,n)}\\big (P(\\tilde{A}^{(m,r_m)}_{a,s})P(\\tilde{A}^{(n,r_n)}_{b,t})+P(\\tilde{A}^{(m,r_m)}_{a,s}\\tilde{A}^{(n,r_n)}_{b,t})\\big )$ .",
"It follows that $\\sum _{m,n=k+1}^{2k} P(H_mH_n)\\le (\\sum _{n=k+1}^{2k} e^{-J_n})^2+{\\cal L}_k, $ where ${\\cal L}_k=2k\\sum _{n=k+1}^{2k}\\tilde{e}^{(n,r_n)}+2\\sum _{m,n=k+1}^{2k}\\tilde{e}^{(m,r_m,n,r_n)} $ .",
"One deduces from (REF ) that $J_n\\ge \\mu _{n,r_n}(1-\\frac{(r_n+1)^2}{2n}-\\frac{6}{C\\ln n})$ .",
"Hence $\\sum _{n=k+1}^{2k} e^{-J_n}\\le e^{O(\\ln ^{\\varepsilon -1}k)} \\sum _{n=k+1}^{2k} e^{- \\mu _{n,r_n}} $ by noting that $\\mu _{n,r_n}=O(\\ln ^\\varepsilon n)$ .",
"If we have showed that ${\\cal L}_k\\le O(k\\ln ^8 k)$ , then $\\lim _{k\\rightarrow \\infty }P(\\bigcup _{n=k+1}^{2k}H_n)=1$ and hence (REF ) holds, in view of (REF )–(REF ).",
"Now we shall show that ${\\cal L}_k\\le O(k \\ln ^8 k)$ .",
"By Lemma REF , $ k\\sum _{n=k+1}^{2k}\\tilde{e}^{(n,r_n)}\\le O(k\\ln ^7k).$ Fix any $(a,s,m)\\in V_k$ , define $\\gamma (a,s,m)$ to be the set of all triples $(b,t,n)\\in V_k$ such that $|\\tilde{B}^{(n,r_n)}_{b,t}\\cap \\tilde{B}^{(m,r_m)}_{a,s}|> {C\\ln {(2k)}}/2$ .",
"Suppose that $k$ is sufficiently large and $(b,t,n)\\in \\gamma (a,s,m)$ .",
"Let $i_l=\\min \\lbrace j\\ge 0:b+jt>ln\\rbrace $ , $\\jmath =\\max \\lbrace l:i_l\\le r_m\\rbrace $ and $Z_l=\\lbrace b+it-ln:i_l\\le i\\le min(i_{l+1}-1,r_n)\\rbrace $ .",
"Then $\\tilde{B}^{(n,r_n)}_{b,t}$ is the disjoint union of $Z_i$ with $0\\le i\\le \\jmath $ .",
"Since $[\\frac{3n}{C\\ln n}]+1\\le t\\le [\\frac{n}{2}]$ , $|Z_i|\\le C\\ln n/3+1$ for all $i\\le \\jmath $ , and $|Z_i|\\ge 2$ for $0<i<\\jmath $ .",
"Thus $|\\tilde{B}^{(n,r_n)}_{b,t}|=r_n+1\\ge 2(\\jmath -1)$ .",
"It follows that $ \\jmath < C\\ln { (2k)}/2-3$ .",
"Since $|\\tilde{B}^{(n,r_n)}_{b,t}\\cap \\tilde{B}^{(m,r_m)}_{a,s}|> {C\\ln {(2k)}}/2$ and $|Z_i|\\le C\\ln n/3+1$ , there are $l$ and $\\imath $ such that $\\imath \\ne l$ , $|Z_l\\cap \\tilde{B}^{(m,r_m)}_{a,s}|\\ge 2$ and $|Z_\\imath \\cap \\tilde{B}^{(m,r_m)}_{a,s}|\\ge 1$ .",
"That is to say, there is $0\\le i,j,\\ell \\le r_n$ and $x,y,z\\in \\tilde{B}_{a,s,m}$ such that $b+it-ln=x$ , $b+jt-ln=y$ and $b+\\ell t-\\imath n=z$ .",
"It leads to $(j-i)t=y-x$ and $(\\imath -l)n=(\\ell -i)t+x-z$ .",
"Therefore $|\\gamma (a,s,m)|\\le &&|\\lbrace (b,t,n): t=(y-x)/i,n=(\\ell t+x-z)/j, b+\\imath t \\mod {n}=x,\\\\&&|i|,|j|,|\\ell |,|\\imath |\\le C\\ln 2k, i,j\\ne 0,x,y,z\\in \\tilde{B}_{a,s,m}\\rbrace |=O(\\ln ^7 k).$ Combining with (REF ) and (REF ), one then deduces that $\\sum _{m,n=k+1}^{2k}\\tilde{e}^{(m,r_m,n,r_n)} &\\le & \\sum _{m,n=k+1}^{2k}\\big (2p^{r_m+r_n-1}m^2nr_mr_n+3p^{r_m+r_n-C\\ln (2k)/2}m^2r^2_m r^3_n\\big )\\\\&&+\\sum _{(a,s,m)\\in V_k}p^{r_m}|\\gamma (a,s,m)|\\le O(k\\ln ^8 k)$ as desired.",
"It completes the proof of (REF ).",
"(5) Now we come to the proof of (REF ).",
"By (REF ), for any $0<\\varepsilon <1$ , $P(W^{(2^n)}<C\\ln 2^n+\\varepsilon D\\ln \\ln 2^n)=e^{-O(n^\\varepsilon )}$ and $P(W^{(2^n)}>C\\ln 2^n-(1+\\varepsilon ) D\\ln \\ln 2^n)=O(n^{-(1+\\varepsilon )}).$ Hence $\\liminf _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}\\ge 0$ and $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}\\le -D$ by the Borel-Cantelli Lemma.",
"In view of (REF ), it remains only to show that $\\limsup _{n\\rightarrow \\infty }\\frac{W^{(2^n)}-C\\ln 2^n}{\\ln \\ln 2^n}\\ge -D. $ For any $0<\\varepsilon <1$ , let $r_n=[C\\ln n-\\varepsilon D\\ln \\ln n]$ and $F_{n}=\\lbrace (a,s):1\\le a\\le n, 3n/r_n<s\\le [n/2], \\gcd (n,s)<n/r_n\\rbrace $ .",
"To show (REF ), it suffices to show that $\\lim _{n\\rightarrow \\infty }P(\\bigcup _{n=k+1}^{2k} \\lbrace W^{(2^n)}\\ge r_{2^n}\\rbrace )=1$ .",
"It need only to show that $\\lim _{n\\rightarrow \\infty }P(\\bigcup _{(a,s,m)\\in G_k}\\tilde{A}^{(m,r_m)}_{a,s})=1$ , where $G_k=\\lbrace (a,s,2^n):k+1\\le n\\le 2k, (a,s)\\in F_{2^n}\\rbrace $ .",
"Let $X_k=\\sum _{(a,s,m)\\in G_k} I_{\\tilde{A}^{(m,r_m)}_{a,s}}$ .",
"Clearly $EX_k=O(k^{1-\\varepsilon })$ .",
"By Lemma REF , Lemma REF , (REF ) and (REF ), $DX_k &&\\le \\sum _{(a,s,m)\\in G_k }P(\\tilde{A}^{(m,r_m)}_{a,s})+2\\sum _{n=k+1}^{2k} \\tilde{e}^{(2^n,r_{2^n})}\\\\&&+2\\sum _{k+1\\le m<n\\le 2k} (p^{r_{2^m}+r_{2^n}-1}2^{2m}2^nr_{2^m}r_{2^n}+3p^{r_{2^m}+\\frac{1}{4} r_{2^n}}2^{2m}r^2_{2^m}r^3_{2^n})\\\\&&= O(k^{1-\\varepsilon }).$ Hence $P(\\bigcup _{(a,s,m)\\in G_k}\\tilde{A}^{(m,r_m)}_{a,s})=P(X_k>0)\\ge \\frac{(EX_k)^2}{DX_k+(EX_k)^2}\\rightarrow 1.$ As to (REF ), by Proposition REF , we need only to show (REF ).",
"It suffices to show that $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n \\sum _{(a,s)\\in F_{2^k}}I_{\\tilde{A}^{(2^k,r_{2^k})}_{a,s}}=0 \\,\\, a.s. $ and $\\lim _{n\\rightarrow \\infty } \\frac{1}{n} \\sum _{k=1}^n I_{H_k}=0 \\,\\, a.s.,$ where $H_k=\\lbrace W^{(2^k)}\\ge r_{2^k}\\rbrace \\setminus (\\bigcup _{(a,s)\\in F_{2^k}}\\tilde{A}^{(2^k,r_{2^k})}_{a,s})$ .",
"Similar as the proof of (REF ), we may show (REF ).",
"Note that $P(H_k)=O(k^{-1-\\varepsilon })$ .",
"By Borel-Cantelli Lemma, $\\lim _{k\\rightarrow \\infty }I_{H_k}=0 \\,\\,a.s$ .",
"It follows (REF ) and completes our proof.",
"$\\Box $"
],
[
"The asymptotic distributions when $p_n=o(1)$",
"Proof of Theorem 1.3 (i) Set $q_n=1-p_n$ and $r_n=[\\frac{2\\ln n}{-\\ln p_n}+\\frac{\\ln \\ln n}{\\ln p_n}]$ .",
"Clearly, $p^{r_n+1}_n\\le n^{-2}\\ln n \\le p^{r_n}_n$ .",
"Similar as the proof of (REF ), we may show that there is a constant $c>0$ such that $\\max _{0\\le k\\le 2}|P(U^{(n,p_n)}<r_n+k)-e^{-\\lambda _{n,r_n+k,p_n}}|\\le c p^{-3}_nn^{-1}\\ln ^2 n. $ Since $\\lim _{n\\rightarrow \\infty }\\frac{2\\ln n}{-\\ln p_n}=\\infty $ , $\\frac{\\ln n}{-\\ln p_n}\\ge 10$ and hence $p^{-1}_n\\le n^{0.1}$ for sufficiently large $n$ .",
"This, together with (REF ) implies that $\\lim _{n\\rightarrow \\infty }\\max _{0\\le k\\le 2}|P(U^{(n,p_n)}<r_n+k)-e^{-\\lambda _{n,r_n+k,p_n}}|=0.",
"$ In addition, $\\lim _{n\\rightarrow \\infty } \\lambda _{n,r_n,p_n}\\ge \\lim _{n\\rightarrow \\infty }\\frac{q_n\\ln n}{2(r_n-1)}=\\lim _{n\\rightarrow \\infty }\\frac{-\\ln p_n}{4}=\\infty $ by noting that $p_n\\rightarrow 0$ .",
"Similarly, $\\lim _{n\\rightarrow \\infty } \\lambda _{n,r_n+2,p_n}\\le \\lim _{n\\rightarrow \\infty } (\\frac{p^2_n\\ln ^2 p_n}{8\\ln n}-\\frac{p_n\\ln p_n}{4})=0.",
"$ Therefore (REF ) holds by (REF )–(REF ).",
"Use the same method, we may prove (REF ) and complete the proof of (i).",
"(ii)(iii): Since $\\lim _{n\\rightarrow \\infty } np_n=\\infty $ , $\\lim _{n\\rightarrow \\infty }P(\\sum _{i=1}^n \\xi ^{(n)}_i\\ge 2)=1$ and hence $\\lim _{n\\rightarrow \\infty }P(U^{(n,p_n)}\\ge 2)=1.$ Choose an $0<\\varepsilon <0.1$ such that $[b-\\varepsilon , b)\\cup (b,b+\\varepsilon )$ contains no integers.",
"There is $N$ such that $\\frac{b-\\varepsilon }{2}< \\ln n/(-\\ln p_n)<\\frac{b+\\varepsilon }{2}$ for all $n>N$ .",
"It follows that for all $n>N$ , $p^{\\frac{-b+\\varepsilon }{2}}_n<n<p^{\\frac{-b-\\varepsilon }{2}}_n.",
"$ By (REF ), (REF ) and Lemma REF , for $n>N$ , $P(W^{(n,p_n)}\\ge [b]+1)\\le \\frac{p_n^{[b]+1-b/2-\\varepsilon /2}}{q_n([b]+1])}+\\frac{q_n p_n^{[b]+1-b-\\varepsilon }}{2}\\rightarrow 0.$ Suppose that $r<b\\le r+1$ where $r$ is a positive integer.",
"By (REF ), $n^2p^r_n\\ge p^{r-(b-\\varepsilon )}_n\\rightarrow \\infty $ .",
"Let $X_n=\\sum _{(a,s)\\in B^{(r)}_n}I_{A^{(r)}_{a,s}}$ .",
"By Lemma REF and Lemma REF , $EX_n=O(n^2p^r_n)$ and $DX_n\\le EX_n+O(n^3p^{2r-1}_n)=o\\big ((EX_n)^2\\big )$ .",
"Consequently, $P(U^{(n,p_n)}\\ge r)=P(X_n\\ge 0)\\ge \\frac{(EX_n)^2}{DX_n+(EX_n)^2}\\rightarrow 1.$ Similar as the proof of (REF ) and (REF ), by using (REF ),we may show that $\\lim _{n\\rightarrow \\infty }|P(U^{(n,p_n)}<b)-e^{-\\lambda _{n,b,p_n}}|=\\lim _{n\\rightarrow \\infty }|P(W^{(n,p_n)}<b)-e^{-\\mu _{n,b,p_n}}|=0 $ when $b\\ge 3$ and $b$ is an integer.",
"Furthermore, if $u=\\lim _{n\\rightarrow \\infty } n^2p^b_n\\le \\infty $ exists, then $\\lim _{n\\rightarrow \\infty }P(U^{((n,p_n)}<b)=\\lim _{n\\rightarrow \\infty }e^{-\\lambda _{n,b,p_n}}=e^{-\\frac{u}{2(b-1)}}$ and $\\lim _{n\\rightarrow \\infty }P(W^{(n,p_n)}<b)=\\lim _{n\\rightarrow \\infty }e^{-\\mu _{n,b,p_n}}=e^{-\\frac{u}{2}}.$ Thus our result holds by (REF ),(REF ),(REF ),(REF ), (REF ) and by noting that $W^{(n,p_n)}\\ge U^{(n,p_n)}$ .",
"$\\Box $"
]
] | 1204.1149 |
[
[
"Observations of Low Frequency Solar Radio Bursts from the Rosse\n Solar-Terrestrial Observatory"
],
[
"Abstract The Rosse Solar-Terrestrial Observatory (RSTO; www.rosseobservatory.ie) was established at Birr Castle, Co. Offaly, Ireland (53 05'38.9\", 7 55'12.7\") in 2010 to study solar radio bursts and the response of the Earth's ionosphere and geomagnetic field.",
"To date, three Compound Astronomical Low-cost Low-frequency Instrument for Spectroscopy and Transportable Observatory (CALLISTO) spectrometers have been installed, with the capability of observing in the frequency range 10-870 MHz.",
"The receivers are fed simultaneously by biconical and log-periodic antennas.",
"Nominally, frequency spectra in the range 10-400 MHz are obtained with 4 sweeps per second over 600 channels.",
"Here, we describe the RSTO solar radio spectrometer set-up, and present dynamic spectra of a sample of Type II, III and IV radio bursts.",
"In particular, we describe fine-scale structure observed in Type II bursts, including band splitting and rapidly varying herringbone features."
],
[
"Introduction",
"The Sun is an active star that produces large-scale energetic events, such as solar flares and coronal mass ejections (CMEs).",
"These phenomena are observable across the electromagnetic spectrum, from gamma rays at hundreds of MeV to radio waves with wavelengths of tens of metres.",
"Solar flares and CMEs can excite plasma oscillations which can emit radiation at metric and decametric wavelengths.",
"These bursts are classified in five main types.",
"Type I bursts are short-duration narrowband features that are associated with active regions [16].",
"Type II bursts are thought to be excited by magnetohydrodynamic (MHD) shockwaves associated with CMEs [18], while Type IIIs result from energetic particles escaping along open magnetic field lines.",
"Type IV bursts show broad continuum emission with rapidly-varying fine structures [15].",
"The smooth short-lived continuum following a Type III burst is called a Type V. Several radio telescope designs have been developed to observe solar radio activity, including interferometers, spectrometers and imaging-spectrometers.",
"The Culgoora Radioheliograph was a 96-element, 3 km diameter radio synthesis telescope and operated at 80 MHz [20].",
"Operations began in 1968, but were discontinued in 1986.",
"However, Culgoora still operates a radio spectrograph at 18–1800 MHz.",
"Developed in the early 1980s, the Nançay Decameter Array consists of two phased arrays producing dynamic spectra at 10–80 MHz [9].",
"This operates alongside a radioheliograph, which produces solar images at a number of frequencies between 150–420 MHz.",
"During the 1980s and 1990s, ETH Zurich developed a number of solar radio spectrometers [2], [17].",
"Their latest instrument, Phoenix II, is a Fourier-based spectrograph operating at 0.1–4 GHz, with 2000 channels and a temporal resolution of better than 1 s. In recent years, the Green Bank Solar Radio Burst Spectrometer (GBRBS) has been developed in the US, composed of three swept-frequency systems that support observations at 18–1100 MHz, with a temporal resolution of approximately 1 ms [23].",
"Similarly, the Artemis spectrograph in Greece, observes at 20 to 650 MHz, and operates with a 7 m moving parabolic antenna at 110–650 MHz and a stationary antenna for the 20–110 MHz range [13].",
"A new milestone in low frequency radio instrumentation was reached with the establishment of the Dutch-lead LOw Frequency ARray [7].",
"This is the latest development in software-based radio interferometry, and provides the capability of simultaneously recording dynamic spectra and images of solar phenomena [10].",
"The CALLISTO (Compound Astronomical Low-cost Low-frequency Instrument for Spectroscopy and Transportable Observatory) spectrograph is a new concept for solar radio spectrographs, designed by ETH Zurich [1].",
"This is a low-cost radio spectrometer used to monitor metric and decametric radio bursts, and which has been deployed to a number of sites around the world to allow for 24 hour monitoring of solar radio activity.",
"In order to monitor solar activity and its effects on the Earth, we set up an autonomous solar radio observing station, the Rosse Solar-Terrestrial Observatory (RSTO), which has been operating since September 2010.",
"RSTO is located in the grounds of Birr Castle, Co. Offaly, Ireland, and was named for the 3rd Earl of Rosse, Sir William Parsons, who constructed the 6-feet diameter “Leviathan Telescope\" in the 1840s [12].",
"RSTO is part of the e-CALLISTO networkwww.e-CALLISTO.org.",
"The network consists of a number of spectrometers located around the globe, and designed to monitor solar radio emission in the metre and decametre bands ([3]; Figure REF ).",
"Each of the instruments observes automatically, and data is collected each day via the Internet and stored in a central database at Fachhochschule Nordwestschweiz (FHNW), and operated by ETH Zurichsoleil.i4ds.ch/solarradio/CALLISTOQuicklooks/.",
"One of the important features of RSTO is the particularly low radio frequency interference (RFI) of the site, which is further described in Section REF .",
"In this paper, we describe the suite of CALLISTO spectrographs at RSTO (Section 2).",
"Examples of observations are then reported in Section 3, while plans for future instruments at RSTO and conclusions are given in Section 4.",
"Figure: Worldwide distribution of a portion of the radio spectrographs in the e-CALLISTO network.",
"The network also includes spectrographs in Australia (Perth and Melbourne), Hawaii, Germany, Kazakhstan, Sri Lanka, Italy, Slovakia and Malaysia.CALLISTO spectrometers are designed to monitor solar radio bursts in the frequency range 10–870 MHz.",
"CALLISTO is composed of standard electronic components, employing a Digital Video Broadcasting-Terrestrial (DVB-T) tuner assembled on a single printed circuit board.",
"The number of channels per frequency sweep can vary between 1 and 400, with a maximum of 800 measurements per second.",
"An individual channel has 300 kHz bandwidth during a typical frequency sweep of 250 ms, and can be tuned by the control software in steps of 62.5 kHz to obtain a more detailed spectrum of the radio environment.",
"The narrow channel width allows for the measurement of selected channels that avoid known bands of radio interference from terrestrial sources."
],
[
"CALLISTO Spectrometer Set-up at RSTO",
"RSTO operates three CALLISTO receivers fed by a broadband log-periodic antenna and a biconical antenna (Figure REF ).",
"Nominally, the RSTO set-up operates at 600 channels with a sampling time of 250 ms seconds per sweep.",
"CALLISTO 1 observes at 10–100 MHz, CALLISTO 2 at 100–200 MHz, and CALLISTO 3 at 200–400 MHz.",
"The system has been optimised to measure the dynamic spectra of Type II radio bursts produced by coronal shock waves, and Type III radio bursts produced by near-relativistic electrons streaming along open magnetic field lines.",
"It can also record other radio bursts, such as Type IV bursts and Type I noise storms.",
"The log-periodic antenna has a frequency band of 50 to 1300 MHz with a $\\sim $ 50 degree half-power beam-width (HPBW).",
"The antenna is fixed to an alt-azimuth drive which tracks the Sun to optimize its response.",
"The biconical antenna is 4 m long and has a nominal frequency sensitivity from 10 to 300 MHz.",
"It is also mounted on a motorized rotator to track the Sun.",
"CALLISTO 2 and 3 operates with a pre-amplifier that has a frequency range of 5–1500 MHz, and a typical noise figure of 1.2 dB, while a similar pre-amplifier is separately connected to CALLISTO 1.",
"The system set-up is optimized to reach the ionospheric cutoff frequency at $\\sim $ 10 MHz.",
"In order to do this, the receiver with a nominal operational band between 45 to 870 MHz has to operate with a frequency up-converter, shifting the range between 10–100 MHz to 220–310 MHz.",
"The observed frequencies are then down-converted in software.",
"Figure: Radio frequency survey of the RSTO in Birr Castle Demesne (blue), Bleien Radio Observatory in Switzerland (red; offset by 10 dB) and from Potsdam Bornim (green; offset by 20 dB).",
"The RSTO spectrum is quiet at all frequencies that were tested, except for the FM band covering 88–108 MHz.",
"The surveys were conducted using the same equipment.",
"Note, LOFAR operates at ∼\\sim 20–240 MHz."
],
[
"RSTO Radio Frequency Interference Survey",
"A survey of RFI at RSTO was performed in June 2009www.rosseobservatory.ie/presentations/birr_radio_survey.pdf.",
"The detected spectrum is shown in Figure REF .",
"A commercial DVB-T antenna covering the range from 20 MHz up to 900 MHz was used for the survey, which was directly connected via a low-loss coaxial cable to a CALLISTO receiver with a sensitivity of 25 mV/dB.",
"The channel resolution was 62.5 kHz, while the radiometric bandwidth was about 300 kHz.",
"The sampling time was 1.25 ms per frequency interval, while the integration time was about 1 ms.",
"Figure REF shows the RFI radio surveys of RSTO, Bleien Observatory in Switzerland, and the Potsdam LOFAR station in Germany.",
"There is an high level of interference at 20–200 MHz for the Bleien and Potsdam sites, while the RSTO site has a low level of RFI.",
"The radio spectrum at RSTO is extremely quiet compared to the majority of e-CALLISTO sites around the world.",
"FM-radio and DVB-T are less intense than other sites, making the RSTO site an ideal location for low–frequency solar radio observations.",
"Indeed, Birr Castle Demesne is a near–ideal site for frequency-agile or Fourier-based spectrometers.",
"All protected frequencies for radio astronomy are free from interference, and could be used for single frequency observations to determine solar radio flux using broadband antennas.",
"As a result of this survey, the Irish astronomy community are considering Birr Castle Demesne as a site for a LOFAR stationwww.lofar.ie.",
"Figure: Dynamic spectrum of the 22 September 2011 Type II radio burst and related GOES–15 light curve showing an X1.4 flare.",
"This burst shows fundamental (F) and harmonic (H) emission.",
"Band splitting of the order of 10 MHz can also be seen in the harmonic backbone at times around 10:42 UT."
],
[
"Sample RSTO Dynamic Spectra",
"Observations started in September 2010, and first light was achieved on 17 November 2010.",
"Since then, a large number of radio bursts have been recorded.",
"In this section, we present a number of observations and give a brief description of each.",
"All RSTO data is provided to the community at www.rosseobservatory.ie.",
"Figure: Dynamic spectrum of the 07June 2011 Type II radio burst.",
"In the inset of panel (a) a short-lived fundamental and harmonic backbones are visible.",
"Multiple herringbone structure are visible at ∼\\sim 40–80 MHz in panel (b)."
],
[
"Type II bursts",
"The appearance of Type II radio bursts can vary significantly in dynamic spectra.",
"The 22 September 2011 Type II radio burst shown in Figure REF was associated with an X1.4 class flare which started at 10:29:00 UT.",
"The flare was identified to have originated in NOAA active region 11302 and was associated with a CME.",
"The burst started at 10:39:06 UT, and shows both fundamental (F) and harmonic (H) bands of emission.",
"The fundamental emission is visible between 20 and 60 MHz, while the harmonic backbone lies between 60 and 90 MHz.",
"Emission higher then 88 MHz is attenuated by the presence of the FM band.",
"This structure is typical of the majority of Type II burst, i.e., when the harmonic backbone is present, it is almost always stronger than the fundamental.",
"The two backbones show a drift rate of $\\sim $ 0.22 MHz s$^{-1}$ , drifting towards lower frequencies as the plasma becomes less dense at larger distances from the Sun.",
"A shock velocity of 1240 km s$^{-1}$ was estimated using the 1–fold Newkirk model [19].",
"Type II bursts typically last 5–10 minutes, but bursts exceeding 30 minutes have been known.",
"Furthermore, short–duration bursts under one minute have been identified.",
"A short–lived Type II observed on 2011 June 7 is shown in the inset on the top right of Figure REF (a).",
"It is much more difficult to interpret short–duration Type II bursts, particularly if they occur at similar times and frequencies as other radio activity.",
"In addition, the fundamental/harmonic structure is split into two roughly parallel bands as evident in Figure REF (a).",
"The band-splitting phenomenon has two interpretations.",
"It could be due to either the shape of the electron density distribution in the corona [14] or the emission ahead and behind the shock front [21].",
"Another peculiarity of Type II burst is the sporadic presence of herringbones, small features similar to Type III bursts that straddle the backbones emission.",
"In Figure REF (b) and Figure REF , herringbone features are evident.",
"On 22 September 2011, about 50 herringbones drifting upwards in frequency between 40 and 80 MHz and downwards between 40 and 15 MHz are clearly visible between 10:51:00 UT and 10:53:00 UT.",
"These are believed to be due to electron beams ejected out from the shock front moving through the upper and lower corona [11].",
"The drift in frequency is very fast ($\\sim $ 22 MHz s$^{-1}$ ), with a corresponding velocity of $\\sim $ 0.1c.",
"Figure: Dynamic spectrum of herringbones following a Type II radio burst on 22 September 2011.",
"These are thought to be due to electron beams shooting out from the shock front moving through the upper and lower corona.Figure: Several Type III radio bursts observed on 21 October 2011.",
"Broad band emission is superimposed on the bursts.",
"Also shown a Type II burst between 140 and 330 MHz."
],
[
"Type III bursts",
"Figure REF shows a series of Type III bursts starting at 12:56:05 UT on 21 October 2011.",
"The emission drifts from 400 to 20 MHz in frequency.",
"A broad-band emission following the Type IIIs, called a Type V radio burst, is also evident from 20 to 150 MHz.",
"The event was associated with an M1 flare which occurred in NOAA 11319.",
"Furthermore, an associated Type II burst starts at 12:58:12 UT, drifting from frequencies of 350 MHz to 170 MHz over 45 s. We determined a velocity drift using the 1–fold Newkirk Model of 790 km s$^{-1}$ .",
"This Type II burst was possibly associated with a CME that appeared in the SOHO/LASCO C2 field–of–view at 13:36:00 UT.",
"Type III bursts can occur in groups, recurrently over an extended period, and continuously in the form of storms.",
"Type III bursts are very common features in the metric range [4].",
"Since the first-light of the CALLISTO-based spectrometer at RSTO, a large number of Type III bursts were detected.",
"Type III bursts are produced by relativistic electrons traveling along open magnetic fields and therefore the drift in frequency detected in the radio spectra is very steep as the electrons travel fast in the corona and density becomes more and more tenuous.",
"Similarly to Type II bursts, Type IIIs can often show a harmonic component.",
"Since the drift rate is very fast, the two components usually merge, making their detection difficult [15].",
"Figure: Type II and Type IV radio bursts observed on 07June 2011.",
"The darker feature starting at 06:25:30 between 150 and 45 MHz is a Type II radio burst.",
"The spectrum shows a continuum emission starting at 06:31:10 UT between 400-130 MHz and a moving Type IV starting at 290 MHz drifting downwards in frequency."
],
[
"Type IV bursts",
"In Figure REF , a Type IV burst observed on 07June 2011 is shown.",
"The emission is related to an M2 flare which occurred in NOAA 11226.",
"A halo CME with an estimated plane-of-sky speed of 1155 km s$^{-1}$ was also detected by LASCO.",
"The broad emission started at 06:32:10 UT from 400 to 200 MHz and then spread from 400 to 100 MHz as it gained intensity.",
"The burst can be seen to extend to 400 MHz for its duration, which is the upper frequency limit of the spectrograph.",
"A drifting feature is also evident.",
"This starts at about the same time as the Type IV, but which moves to lower frequencies over time.",
"This is called a moving Type IV and has a bandwidth of approximately 100 MHz.",
"Type IV continua can exhibit a wide range of forms [22].",
"They are broadband, usually 500 MHz wide, but some can exceed two or three times this.",
"Type IV bursts can be very uniform in intensity, or they can fluctuate with complex underlying internal structures.",
"Small fine structure are superimposed in the drifting continua.",
"They are believed to be generated by emission of electrons trapped in post-flare loops and the drift is linked with the formation of the loops at successively higher altitudes [8]."
],
[
"Discussion and Future Directions",
"Three CALLISTO spectrographs are currently installed at RSTO, sampling the radio spectrum between 10 and 400 MHz with 600 frequency channels and a temporal resolution of 250 ms per sweep.",
"The system can potentially measure between 10 and 870 MHz.",
"This unique configuration of CALLISTO receivers and the use of a biconical antenna and a frequency up-converter allows a good sampling of the spectra with a low-cost system.",
"Since September 2010, a number of radio bursts have been detected, enabling us to identify fine-scale features in Type II and Type III radio bursts, including Type II band splitting and herringbones.",
"One important characteristic of the site is its extremely low RFI.",
"In 2012, a fourth CALLISTO receiver will be added to RSTO set-up in order to extend the current observational mode to 10–870 MHz, and increasing the number of channels to 800 per sweep.",
"We have also installed a ionispheric monitoring instrument as a complimentary instrument to CALLISTO at RSTO.",
"The Atmospheric Weather Electromagnetic System of Observation, Modeling, and Education [6].",
"AWESOME is a ionospheric monitoring sensor designed by Stanford University, which is used to detect ionospheric disturbances.",
"The Very Low Frequency (VLF) and Extremely Low Frequency (ELF) sensors of AWESOME monitor radio frequencies in the region of 3-30 kHz and 0.3-3 kHz, respectively.",
"The AWESOME receiver uses wire-loop antennas, each sensitive to the component of the magnetic field in the direction orthogonal to the plane of the loop.",
"ELF/VLF remote sensing enables study of a broad set of phenomena, each of which impacts the ionosphere in a unique way, including solar flares, cosmic gamma ray bursts, lightning strikes and lightning-related effects, earthquakes, electron precipitation, and the aurora.",
"Sudden ionospheric disturbances (SIDs) occur in association with solar flares and have a very strong and relatively long-lasting effect on the ionosphere [5].",
"We are now in the process of testing a flux-gate magnetometer to measure fluctuations in the Earth's magnetic field in Ireland.",
"Using e-CALLISTO, we will measure the on-set time of solar radio bursts near the Sun, measure their effects on the Earth's ionosphere with AWESOME, and ultimately determine their effects to the geomagnetic field in Ireland.",
"As part of a worldwide network of observatories, RSTO will provide an extensive capability to monitor solar activity and its effects on Earth in a real-time, continuous manner.",
"This is made possible by the low RFI of the Birr Castle Demesne site.",
"RSTO has been earmarked as an ideal location for a LOFAR station (www.lofar.ie).",
"When combined with data from space-based observatories, such as NASA's STEREO and Solar Dynamic Observatory (SDO) spacecraft, this will contribute towards a capability to track and understand the propagation of storms from the surface of the Sun to their local effects on Earth.",
"The authors are indebted to the 7th Earl of Rosse and the Birr Castle Demesne staff, particularly George Vaugh, for their support during the development of the RSTO.",
"We would also like to thank the TCD Centre for Telecommunications Value-chain Research (CTVR) and the School of Physics Mechanical Workshop.",
"PZ is supported by a TCD Innovation Bursary.",
"EC is a Government of Ireland Scholar supported by the Irish Research Council for Science, Engineering and Technology.",
"We would also like to thank the Alice Barklie Bequest to the TCD School of Physics.",
"RSTO was established under the auspices of International Heliophysical Year 2007 and the International Space Weather Initiative, supported by the United Nations Basic Space Science Initiative."
]
] | 1204.0943 |
[
[
"The Transitivity of Trust Problem in the Interaction of Android\n Applications"
],
[
"Abstract Mobile phones have developed into complex platforms with large numbers of installed applications and a wide range of sensitive data.",
"Application security policies limit the permissions of each installed application.",
"As applications may interact, restricting single applications may create a false sense of security for the end users while data may still leave the mobile phone through other applications.",
"Instead, the information flow needs to be policed for the composite system of applications in a transparent and usable manner.",
"In this paper, we propose to employ static analysis based on the software architecture and focused data flow analysis to scalably detect information flows between components.",
"Specifically, we aim to reveal transitivity of trust problems in multi-component mobile platforms.",
"We demonstrate the feasibility of our approach with Android applications, although the generalization of the analysis to similar composition-based architectures, such as Service-oriented Architecture, can also be explored in the future."
],
[
"Introduction",
"Powerful and well-connected smartphones are becoming increasingly common with the availability of affordable devices and data plans.",
"Increasingly, the smartphones' features are provided by focused applications that users can easily install from application market places.",
"On the other hand, with tens of thousands of applications available, there is only limited control over the quality and intent of those applications.",
"Mobile code and extensibility is one of the key issues that increase the complexity of information security [1].",
"To counter this threat, mobile operating systems impose security restrictions for each application.",
"The Android mobile operating system is one of the major systems on mobile phones.",
"In case of the Android security model, the least-privilege principle is enforced through application-level permissions that can be requested by the applications.",
"End users need to grant the permissions at install time and thereby decide on the adequacy of the required permissions and the trustworthiness of the individual application.",
"The permission granting procedure places a burden on the end users, who need to reason about how the application might employ the permissions.",
"In particular, the end user has little knowledge about the consequences regarding the transitivity of permission granting.",
"As depicted in Figure REF , a malicious application (1) with only local permissions (2) may proxy sensitive data (3) through third party applications and services (4) to external destinations (5).",
"We describe further attack scenarios in Section REF .",
"The inter-component cooperation is an important concept on the Android platform, but the user needs to be able to differentiate between legitimate and malicious information flows.",
"Figure: Example of malicious information flowThe above-described issue of missing transparency poses a risk with the spreading of smartphones, the large numbers of available applications and the prevalent custom of installing applications from untrustworthy sources.",
"The high likelihood of the threat can be deduced from the wealth of sensitive data that is stored on mobile phones, ranging from online banking and business application credentials to communication data and location information.",
"On the Android platform, the first attacks have already been conducted through malicious online banking applications [2].",
"The threat is further increased by the number of data channels, such as the short message service, E-mail or Web access that allow the flow of information out of the device context.",
"Moreover, attack scenarios are not limited to confidentiality breaches.",
"The integrity of the device may also be endangered through similar attack vectors.",
"For example, permissions to use expensive services may be abused.",
"In this paper, we describe an approach to detecting illegitimate information flows between different applications and out of the platform.",
"This way, problems can be revealed that are induced by interacting applications and permission transitivity.",
"We demonstrate the feasibility of our approach with applications running on the Android platform, although other mobile platforms and application markets (iPhone Store, Blackberry World, Windows Mobile Market) are similarly threatened.",
"The transitivity-of-trust problem is not only restricted to mobile platforms.",
"A similar threat can be seen in other multi-component environments, such as Social Networking applications and Service-oriented Architecture (SOA).",
"In this sense, we see our work as a starting point for research, namely, analyzing the consequences of the extensibility of systems with respect to security and mitigating possible risks induced by this paradigm.",
"Our approach to the information flow analysis spanning multiple applications is as follows.",
"The information on data sources such as location services, databases, or contact lists are combined with information on the data sinks (outgoing channels).",
"These input data are used in a two-layer information flow analysis: First, we identify Android components and the respective inter-process communication (IPC) points with the help of the reverse engineering tool-suite Bauhaus [3].",
"This part of the analysis is carried out at the level of the software architecture, reducing the analysis effort.",
"In the second step, we use these architectural information to slice the code and conduct focused data flow analyses on the software slices, resulting in the actual information flows that are used to construct an information flow graph at the architectural layer.",
"The information flow graph can then be used by developers and security experts to identify malicious flows and the graph can be checked against a policy of legitimate information flows.",
"Moreover, an abstract representation can help end users in assessing the risk from a new application.",
"The advantage of the proposed two-layer approach is its scalability.",
"In addition, the approach is practically relevant and has real-world environments as the benchmark.",
"In summary, our analysis method can be considered complementary to other static code analysis approaches that can dectect implementation bugs such as SQL injection and Cross-site scripting vulnerablities [4].",
"Our approach, however, is focused more on the aspect of program comprehension for security and makes transparent interactions between different applications.",
"The remainder of this paper is structured as follows.",
"In Section , we briefly describe the background on Android, Android's security concepts and the software analysis tools used for our analyses.",
"We, then, list possible attack scenarios on the Android platform and show the relevance of the transitivity of trust problem before discussing the data sources and sinks of Android applications.",
"In Section , we present our approach to the security analysis of Android applications in detail, followed by a case study in Section .",
"We discuss the advantages but also limitations of our approach in Section .",
"After listing the related work, we conclude in Section .",
"Applications on the Android platform are developed using the Java programming language.",
"Android applications are not executed on traditional Java Virtual Machines, but are converted into the custom DEX bytecode and interpreted with the Dalvik virtual machine.",
"The Android SDK supports most of the Java Platform, Standard Edition and contains, in addition, Android-specific extensions, including telephony functions and a UI framework.",
"Android applications consist of four basic component types: activities, services, content providers, and broadcast receivers.",
"Activities constitute the presentation layer of an application and allow users to directly interact with the application.",
"Services represent background processes without a user interface.",
"Content providers are data stores that allow developers to share databases across application boundaries.",
"Broadcast receivers receive and react to broadcast messages, for example the “battery low” message from the Android OS or messages from other applications.",
"For communication between the individual components of applications, inter-process communication (IPC) provides a means to pass messages between different applications, activities, and services [5].",
"Android uses messages that contain meta information and arbitrary data, called intents, for IPC.",
"Android components follow a lifecycle that is managed by the OS.",
"As a consequence, there is no main() method from which the applications are started.",
"Instead, the Android OS calls the lifecycle methods, such as onCreate(), whenever e.g.",
"an activity is started for the first time or a new message is received by a service.",
"Further information about component lifecycles is available from the Android Developer's Guide [6]."
],
[
"Android Security Concepts",
"Android has two basic methods of security enforcement [5].",
"Firstly, applications run as Linux processes with individual Unix users and thus are separated from each other.",
"This way, a security hole in one application does not affect other applications.",
"However, as mentioned above, Android provides IPC mechanisms that need to be secured.",
"The Android middleware implements a reference monitor to mediate the access to application components based upon permission labels defined for the component being accessed.",
"If an application intends to access a component of another application, the end user has to grant an appropriate permission.",
"The requested permissions are specified in the application's policy file.",
"All permissions requested by an application are granted by the end user at installation time, i.e., the permission assignment cannot be changed at runtime.",
"During the installation process, a list of dangerous permissions is presented to the end user in a dialog window and needs to be confirmed.",
"Furthermore, the security model has several refinements that increase the model's complexity.",
"One example is the concept of shared user IDs that allow different applications to share the same user ID if the applications are issued by the same developer.",
"Another refinement are protected APIs: Several security-critical system resources can be accessed directly rather than using components.",
"Examples of such resources are Internet services that allow an application to open arbitrary network sockets and have full access to the Internet and outgoing call APIs that allow an application to monitor, modify, or abort outgoing calls.",
"In order to mediate access to such resources, additional in-code security checks have been implemented.",
"Moreover, permissions are assigned protection levels such as “normal”, “dangerous”, and “signature”.",
"The Android security model also supports delegation concepts such as pending intents and URI permissions that can only be checked at the code level rather than at the policy level [5]."
],
[
"Architecture-Based Analysis with the Bauhaus Tool",
"The Bauhaus tool-suite is a reverse engineering tool-suite that has been developed for more than ten years and has been used in several industry projects [3].",
"Bauhaus allows one to deduce two abstractions from the source code, namely the Intermediate Language (IML) and the resource flow graph (RFG).",
"The IML representation is, in essence, an attributed syntax tree (an enhanced AST) that contains the detailed program structure information such as loop statements, variable definitions and name bindings.",
"The RFG representation works at a higher abstraction level and represents architecturally relevant information of the software.",
"The RFG is a hierarchical graph that consists of typed nodes and edges representing elements like types, components and routines and their relations.",
"The information that is stored in the RFG is structured in views, where each view represents a different aspect of the architecture, e.g.",
"a call graph.",
"Technically, views are subgraphs of the RFG.",
"Bauhaus supports a meta-model and thus allows one to define new node and edge types.",
"Currently, RFG profiles exist for C/C++, C#, and Java, representing syntactical elements of the respective language and their relations.",
"For example, typical node types for Java are Class, Method, and Member; edge types are Member Set, Member Use, and Dispatching_Call among others.",
"In particular, an extension of the Java-based RFG model with Android-specific node and edge types is also possible.",
"For visualizing the different views of RFGs, the Graphical Visualiser (Gravis) has been implemented [7].",
"Gravis facilitates high-level analysis of the system and provides a rich functionality to produce new views by RFG analyses or to manipulate generated views."
],
[
"Low-Level Analysis with the Soot Tool",
"The Bauhaus RFG represents the software architecture, but this abstract representation lacks detailed program information that is needed for data flow analysis.",
"Thus, for our goal of tracing the data flow through the application, we need an enhanced AST.",
"The Bauhaus IML-generator supports Java program code below version 1.5, but for developing and analyzing Android applications, we need to analyze Java 1.5 code and above [8].",
"To deal with this issue, we chose Soot, a well-established Java analysis tool [9], as a basis for performing the data flow analysis.",
"Soot was designed as a Java bytecode optimization framework in 1999.",
"In the following years, this framework has been enhanced with several other analysis methods, like points-to analysis [10] or dynamic inter-procedural analysis [11].",
"Soot provides the ability to inject self-written analyses into the existing analysis chain on every intermediate representation [9].",
"Our analysis takes place on the Jimple representation, a 3-address code representation.",
"The built-in call graph generation and flow analysis framework does not facilitate our analysis, since, to take advantage of the Android framework semantics as described in Section REF , a custom static data flow analysis is required."
],
[
"The Transitivity of Trust Problem",
"We first describe different attack scenarios that may lead to undesirable information flows on the Android platform.",
"In particular, these scenarios show that real attacks are possible that exploit transitive trust issues.",
"Thereafter, we argue that the transitivity problem is more general, not restricted to Android or other smartphone platforms."
],
[
"Threats from Android Applications",
"We identified three classes of attack scenarios through Android applications against the confidentiality of user data, depicted in Figure REF .",
"In the simplest case, scenario (a), a maliciously crafted application is published through the Android market.",
"While the application may provide a useful function on the surface, behind the back of the user, it transfers sensitive data (1) to a Web service on the Internet (2).",
"The Android security concept requires the end user to notice the combination of permissions to read sensitive data and access the Internet and cancel the installation.",
"There are several reasons why this assumption may fail: End users are used to accepting permission requests with every installation of applications, thus tempted to just acknowledge the shown list; Many applications require rather broad permissions, for example, Internet access for update checks; The dangerous permissions may be “hidden” between less critical or irrelevant permissions, such as controlling the display backlight; In a subconscious risk assessment, the end user may deem the usefulness of the application so high that the risk may be acceptable despite the unusual combination of permissions.",
"Moreover, as seen in a recent attack, applications can access sensitive data without explicitly requiring a permission, for example the phone serial number [12].",
"One way to counter these kinds of attacks is to make the information flows from critical sources to sinks transparent.",
"For end users, the permissions listed in the affirmation dialog could be enriched with indications how these relate to information flows.",
"Scenario (b) is considerably more complex since the attacker takes advantage of application interaction mechanisms in Android.",
"Similar to scenario (a), the goal is to compromise the end users' confidentiality by disclosing sensitive data.",
"To hide the critical combination of permissions to read sensible data and to send it to remote services, reading and sending are split into separate applications.",
"The first application appears to be “local-only” and has read access to sensitive data (1).",
"This application interacts using the Android IPC mechanisms, for example, a service binding, with a second application without the end users' knowledge (2).",
"The second application requires Internet access permissions and can thus forward the data to a remote service (3).",
"The interaction between the two applications can be completely hidden from the end user.",
"There are two approaches to prevent this attack: either to make the information flows within each application explicit, in this case from a data source to the Android IPC and from Android IPC to a data sink.",
"Alternatively, to analyze all installed applications to identify combinations of applications that can interact to create an information flow from a critical data source to a remote data sink.",
"Scenario (c) is a variant of (b), but with malicious intentions by either the first or the second application.",
"If the first one is malicious as shown in Figure REF , it will read sensitive data as in scenario (b).",
"It will then abuse an erroneous application into tricking it to transfer the data to a remote target.",
"In the second case, not shown, an erroneous application that may read sensitive data offers this data through Android IPC and a malicious application retrieves the data to forward it to a remote service.",
"Repositories, such as OpenIntentshttp://www.openintents.org, that offer interfaces for several inter-component interactions may facilitate this kind of attack.",
"Apart from making the critical information flows transparent, it is helpful to validate whether adequate permission enforcement is enacted on critical information flow paths within applications, either manifest-based or programmatic to counter this threat.",
"If permissions are enforced, end users have a chance of noticing unusual combinations of permission requirements that do not match with the claimed application purpose."
],
[
"Transitivity Issues on Different Platforms",
"The transitivity of trust problem as discussed before can be regarded as an instance of a more general security problem in software, namely, undesirable interaction of different applications and components, respectively.",
"This problem has been described in the literature, e.g., by Piessens [13] and Anderson [14].",
"Transitivity issues between applications are not limited to the Android platform or in general to mobile platforms.",
"Android, however, is a classical example of such systems.",
"First, it supports the mobile code paradigm which supports dynamically loading new applications.",
"Furthermore, although there has been implemented a separation mechanism for applications (or Android components), access between the separated applications is still possible via IPC in order to allow the development of practically useful applications.",
"Similar remarks hold for other systems such as multifunction smartcards as stated, for example, by Anderson [14].",
"These cards allow one to install different applications, e.g., one application for electronic passport functionality and another one for digital signature to allow for legally binding business.",
"In particular, the Javacard technology provides the possibility to dynamically load new Java applets [15].",
"In order to protect the applications from each other, the concept of “application firewalls” has been introduced.",
"Similarly to Android, however, there exists a mechanism to share Java objects between applications.",
"As a consequence, transitivity issues stemming from interacting applications are again conceivable as first discussed by McGraw and Felten [16].",
"In this scenario, an application A allows a trusted application B to access a sensitive object x via a virtual method x.foo().",
"Application B then gives a third application (not necessarily trusted by A) access to a method y.bar(), which calls x.foo().",
"This way, C indirectly has access to object x, although A has not explicitly given that permission.",
"The interaction problem and specifically transitivity issues also exist in SOA, which is based on extensible systems such as JavaEE and .NET.",
"In particular, Web services, which often implement SOA, aim at coupling and composing services depending on the needs of an organization.",
"As Karp et al.",
"point out, service chaining leads to transitivity problems [17].",
"We now briefly discuss this point in the context of Enterprise Resource Planning (ERP) systems, which often extensively support SOA.",
"SAP, for example, makes the NetWeaver platform available to integrate different applications such as the Human Resources module or Material Management.",
"In particular, SAP NetWeaver can be used to allow different SAP modules or even external applications access to centrally administered data such as master data (e.g., about customers or vendors), which are sensitive for an organization.",
"A business process such as a loan origination workflow, for example, may then access or manipulate master data of a customer via a Web service.",
"This loan origination workflow itself might be exposed as a service within the organization.",
"If this service is not secured appropriately (e.g., the role-based access control policy is erroneous), then it is conceivable that these sensitive data might be accessed by unauthorized actors through the service.",
"In summary, application interaction w.r.t.",
"transitivity is a prevalent problem on many platforms.",
"In this paper, we focus on the Android platform, although the techniques we use can be applied to other systems as well.",
"Specifically, we then need to map the system-specific programming concepts to our analysis infrastructure and, e.g., introduce specific modeling elements at the RFG level."
],
[
"Critical Information Sources and Sinks in Android",
"Our approach to information flow analysis is to analyze inter-component flows from information sources, such as contact lists, to channels through which information leaves the device context.",
"Thus, we must identify communication mechanisms between components as well as critical incoming and outgoing channels on the Android platform.",
"The incoming channels are referred to as “data sources”, outgoing channels as “data sinks”.",
"We identified a list of inter-component communication mechanisms, sources and sinks by exploring of the Android application framework and the provided samples.",
"Table: Inter-component communication mechanismsInter-component communication takes place between the Android component types as described in Section REF .",
"Table REF lists the primary communication types on the Android platform.",
"For the sake of brevity, only individual examples of the API calls are given.",
"Table: Data sources (incoming channels)Table: Data sinks (outgoing channels)The origin of the data in a information flow needs to be known to effectively analyze the flows' criticality.",
"Table REF provides a list of data sources that allow the flow of information into the device and application context.",
"Enck et al.",
"similarly identified data sources for the placement of security hooks in their dynamic analysis, categorizing sources into sensors, such as location sensors and camera, information databases and device identifiers [18].",
"The criticality of a data source is determined by the data that the source makes potentially accessible.",
"As the criticality depends on various factors, we only evaluate the criticality for average users at this point to give a risk estimation.",
"We will conduct an in-depth analysis of source criticality as part of our future work.",
"For the criticality values in Table REF , high criticality indicates that the impact is potentially significant.",
"Medium criticality is assigned for observation scenarios, where consequences resulting from attacks are limited for average users, depending on the monitored person in a given case.",
"Low criticality indicates that there is only little impact that most users might accept the annoyance.",
"An example of a data source with medium criticality is Android's Location Manager, which provides access to the device's geographical location and is used in this paper's case study.",
"Bluetooth is a data source with high criticality because of the possibility to access critical data like contacts, files and images on paired devices through this channel.",
"In Table REF , we list data sinks of Android applications with possible attack scenarios and the requirements for the realization of this scenario as well as a valuation of the attack complexity.",
"While there are severity metrics for software vulnerabilities, the existing models do not match the requirements of the evaluation of information flow data sinks.",
"We assess the attack complexity through the complexity of possible attack scenarios.",
"For medium attack complexity, it is sufficient for a malicious application to trick a single application into proxying sensitive data to external destinations.",
"In cases that require several applications to be coordinated for an attack, we rate the attack complexity as high.",
"Very high attack complexity indicates that it is, in addition, necessary to modify the operation system and/or external services, such as the Google Maps Web service.",
"An example of a data sink with medium attack complexity is a WebView which displays web pages as part of the UI.",
"To channel data out of the device context through this sink, a malicious application has to manipulate the target URI.",
"In contrast, content providers are data sinks with high attack complexity because a malicious application must mislead one application into writing into a content provider and another component afterwards into using this content provider to channel data out.",
"Following the standard risk assessment approach of $\\mathit {Risk} =\\mathit {Probability} \\times \\mathit {Impact}$ , the risk of an information flow can be approximated from the source criticality (impact) and the sinks attack complexity (probability).",
"Thus, a low attack complexity of data sinks combined with a high criticality of data sources results in a maximum risk."
],
[
"Information Flow Analysis of Android Applications",
"To improve the transparency with respect to the transitivity of trust problem on the Android platform, we propose to analyze the information flows between the applications.",
"We first introduce the analysis method on a high level before we describe our prototype implementation of the analysis."
],
[
"Analysis Method",
"Our analysis approach aims to identify undesirable information flows between different Android applications and components, respectively.",
"In order to analyze a larger set of applications (as it usually exists on an end user's phone), we did not directly employ the AST for the entire analysis.",
"We rather employ two layers of abstraction in the analysis, beginning at the level of the software architecture to identify the Android components, before diving into the AST details to enrich the architecture and, lastly, deriving the final results from the architecture.",
"In this last step, we employ the architecture to compose information flows through single Android components into information flows spanning an entire application, and thereafter compose these intra-application information flows to inter-application information flows.",
"All architectural-level analyses are conducted on a hierarchical architecture graph that represents architectural elements, such as methods and classes, as nodes and relations between the elements, such as calls, as edges.",
"In the following, we explain our analysis algorithm in more detail.",
"truefalsefalse17pt5pt4ptfalseb language=ruby,label=pseudocode,caption=High-level analysis algorithm,float]codeexamples/pseudocode.rb A listing of the high-level algorithm is shown in Ruby-style pseudo-code in Listing .",
"The algorithm starts from a set of Android applications that should be considered for inter-application information flows.",
"In the first architectural analysis phase, Android components are identified for each application.",
"Components are the basic entities in the Android programming model that communicate through IPC, including services, activities and broadcast receivers (see Section REF ).",
"In further architecture-level analyses, we search for the IPC entry and exit points for each component using architectural patterns.",
"These points are the basis for the detailed analysis of intra-component information flows at the AST level.",
"As shown in Listing , backward slicing is conducted on the AST, starting from each of the exit points.",
"The goal is to identify information flows that reach one of the entry points of the component, representing an intra-component information flow.",
"truefalsefalse17pt5pt4ptfalseb language=ruby,label=pseudocode-component,caption=Information flows inside each individual component,float]codeexamples/pseudocode-component-flows.rb At the architectural level, the intra-component flows are used to enrich the information flow graph with communication links within each application, resulting in a component-level flow graph.",
"Next, we identify information flows on the level of individual applications.",
"We focus on flows that originate outside the application context, pass through it and leave the application again.",
"As depicted in Listing , we conduct a reachability search on the flow graph to find the information flows within each application.",
"Searching the flow graph significantly reduces the search space since we only consider the identified flows and not the entire AST.",
"We start out from selected entry and exit points: sources and sinks.",
"Sources are entry points of components that connect to the outside of the application, for example, receiving broadcasts.",
"Sinks are the opposite, those exit points that leave from the application, for example, starting application-external activities or accessing Web pages.",
"From the reachability analysis on the component-level flow graph, we identify all flows between sources and sinks within the application.",
"At this point, we found information flows that pass through applications, but only for individual applications.",
"For inter-application flows, in the last phase, we identify the information flows that involve critical sources and sinks as described in Section .",
"An application-level flow graph is constructed from the intra-application flows as the basis for the identification of critical flows.",
"Again, a reachability search is conducted, starting from critical sinks and searching for flows to one of the critical sources.",
"As a result, all information flows are known that originate at critical sources and terminate at critical sinks.",
"truefalsefalse17pt5pt4ptfalseb language=ruby,label=pseudocode-application,caption=Information flows inside each individual application,float]codeexamples/pseudocode-application-flows.rb"
],
[
"Prototype Implementation",
"We implemented the analysis method described above as a prototype that identifies information flows between Android applications and the Android platform.",
"The prototype uses two distinct tools to implement the analysis.",
"As depicted in Figure REF , we employ the Bauhaus tool suite at the architectural level and the Soot tool for AST-based analyses.",
"Figure: Analysis workflow"
],
[
"Identify components and IPC points",
"In the first phase of the analysis, we identify the components as well as the associated entry and exit points on the architectural level.",
"The architecture-level analyses are based upon the RFG that the Bauhaus tool generates from the Java bytecode of the analyzed applications.",
"From the global RFG, we create a reduced information flow subgraph (view), containing only the relevant parts of the studied components.",
"The relevant parts are identified through the search for relevant Android framework patterns that are described in Section .",
"The Bauhaus tool-suite provides Python language bindings to read and modify RFGs.",
"We developed a Python script that prototypically identifies relevant parts through pattern matching and marks the related methods, classes and calls by adding the nodes and edges to an information flow view.",
"To interface with the further, AST-based data flow analysis, the identified methods and classes are listed together with the critical calls in an XML-based exchange format that is passed to the next analysis stage."
],
[
"Identify component-level flows",
"In order to find information flows between entry and exit points in components, an intra-component data flow analysis is carried out at the AST level.",
"We use the previously identified entry and exit points to focus the data-flow analysis and significantly reduce the analysis effort at this level.",
"We developed analysis algorithms for the Soot tool that utilize the known Android framework semantics.",
"For each class of entry and exit points that is supported by the prototype, a corresponding analysis building block has been implemented.",
"The behavior of the Android platform prevents the Soot framework from generating a sufficient call graph for our analyses.",
"One reason is that there is no single entry point to the Android applications, such as the traditional main() method, but several, depending on the IPC mechanism.",
"More importantly, there are several, partly dynamic framework semantics that need to be part of the call graph, such as UI event handlers, but are difficult to be statically analyzed.",
"To identify the intra-component information flows, we search for all program points that affect a given exit point in a component.",
"Therefore, we chose a static backward slicing technique as described by Weiser [19].",
"If the backward slicing reaches an entry point of the component under investigation, we consider this an information flow for the specific entry and exit points."
],
[
"Identify application-level flows",
"The component-level flow data from the AST-based analysis is now employed to enrich the information flow graph.",
"The primary purpose of the information flow graph is to allow developers and security experts to quickly identify flows and determine the risks related to the flows.",
"The information flow graph is represented as an RFG view in the Bauhaus tool suite and is extended as follows.",
"For each information flow that has been identified in the previous analysis step, the intra-component flow edge is drawn between the entry and exit point and the corresponding nodes are added to the view.",
"If the origin of the current flow is of type “source”, an information flow edge is inserted from the origin to the point of entry inside the current information flow.",
"Additionally, for all types of destinations, an edge is inserted between the point of exit and the destination's point of entry.",
"We derive the application-level information flows by conducting a reachability analysis based on the information flow graph, starting from exit points that leave the application, backwards to entry points that enter the application.",
"In a last step, we combine the RFGs of multiple applications to identify the critical flows between sources and sinks in the application ecosystem again through a reachability analysis.",
"The resulting view for the case study below is shown in Figure REF as it is displayed in Bauhaus' Gravis tool."
],
[
"Evaluation",
"We evaluate our approach by means of a case study of a public transport-related application and thereafter show how the analysis results can be displayed in a usable manner.",
"Figure: Case study setup for public transport application"
],
[
"Public Transport Application",
"As a case study, we chose to analyze two real-world Android applications that are available on the Android Market with installations on more than 2000 devices.",
"There are several reasons for selecting these two applications.",
"One pragmatic aspect is that one of the authors developed the applications so that we had access to the applications' bytecode as the basis for our analysis.",
"The case study did not affect the design or implementation of the applications since the development of the applications were finished when our research work on the analysis started.",
"A more important criterion for choosing this application was that it encompassed different frequently-used Android concepts such as starting activities or binding to services and a multi-component design (see Section REF ).",
"With this case study, we also demonstrate that our analysis approach supports more complex semantics of the Android framework such as registering and executing remote service callbacks.",
"The two applications also demonstrate that the transitivity of trust is a necessary concept, although the missing transparency may cause the concept to be misused.",
"The first application is called “PubTrans”To preserve the anonymity of the submission, we employ pseudonyms for the applications and removed the original package names., an interface to a public transport routing Web application, primarily improving the input form to take advantage of the context (current location and time as well as previous searches).",
"PubTrans takes parameters such as origin, destination and desired arrival time and sends a query to the routing Web application.",
"Thus, the PubTrans application requires unrestricted Web access privileges.",
"When entering the public transport routing parameters, the user may choose to take the current address as the origin.",
"Since using detailed location data is not strictly necessary for the application's main goal, location queries have been factored out into a separate Android component that is installed as a separate application.",
"As shown in Figure REF , the PubTransLocation application thus requires location data permissions.",
"With two separate applications, a user may choose whether she would like to grant location access.",
"Still, as shown in the figure, there is an information flow between both applications.",
"This information flow is required to fulfill the intended goals, but it was not explicitly granted at installation time.",
"Although not harmful, this information flow is an example of the missing transparency with regard to the transitivity of trust on the Android platform.",
"truefalsefalse17pt5pt4ptfalseb language=xml,label=lst:exchange-format-example,caption=Excerpt of analysis input,float]codeexamples/analysis-input-components-poe.xml We now describe our analysis approach in more detail with the help of this case study.",
"In Figure REF , we can see that our system consists of two applications with three Android components.",
"We can identify the entry and exit points of the components by means of the architecture-level analysis (“Identify components and IPC points” step in Section REF ).",
"Taking a closer look at the ResultWebView activity, we obtain as an entry point the onCreate() method and as an exit point the call to the Android WebView UI element, which is at the same time a possible data sink as described in Section .",
"Thereafter, we interface with the Soot tool to perform the detailed analysis on the AST.",
"We use an XML-based exchange format to pass on the architectural information to Soot.",
"The component description excerpt in Listing displays two IPC points in the ResultWebView component.",
"Figure: Data flow analysis of the ResultWebView componentFigure: Generated information flow view in the Gravis toolOn the AST level, we carry out the Soot analysis with the backward slicing algorithm (“Identify component-level flows” step), based on the component and IPC point information.",
"For the ResultWebView component, we need to verify, for example, whether an intra-component information flow exists between the IPC points onCreate() and WebView.",
"Figure REF depicts a didactically simplified excerpt of the backward slice production corresponding to the information flow from the ResultWebView entry point onCreate to WebView.",
"In the figure, the backward slice starting point is the exit point of the component, calling loadUrl() on a WebView component.",
"Beginning here, we look up all variables passed with the method call and move backward along the statements inside the method loadResults() to identify each point that affects the exit point.",
"When the beginning of this method is reached, we have a set of variables that affect the exit point and we evaluate whether any of these are method parameters to do further analysis on affected points, maybe, in other methods of the ResultWebView class.",
"In the shown case, the variable extras, marked green in Figure REF , is a parameter, so we trace where the current method was called.",
"The method was called by onCreate() that was described as a starting point for activities in Section REF .",
"Inside onCreate(), loadResults() is called with the returned value of the inherited method getIntent().",
"This inherited method is another artifact of Android activities and returns the intent with a set of parameters that started the activity.",
"Thus, we identified an intra-component information flow between the entry point onCreate and the exit point WebView.",
"truefalsefalse17pt5pt4ptfalseb language=xml,label=lst:exchange-output-example,caption=Component-level flows as output from analysis,float]codeexamples/exchange-output-example.xml The results from the AST-level analysis are passed back to the architectural analysis through an exchange file, shown in Listing .",
"In the last step, the information flow data are used to draw appropriate edges in the information flow view on the architectural level (“Identify application-level flows” step).",
"The information flow view of the resulting RFG is shown in Figure REF as displayed in the Bauhaus Gravis visualization.",
"When comparing the information flow graph to the case study set up in Figure REF , one can follow the information flow from the location provider source through the three components to the Internet.",
"Thus, the developed method successfully identified the potentially harmful, at least intransparent information flow.",
"While this detailed visualization is helpful for developers who need to find out which architectural elements are related to potentially unexpected information flows, security engineers and end users need significantly more abstract visualizations.",
"For this reason, further graph searches are conducted to identify the critical flows that are to be displayed at higher levels of abstraction."
],
[
"Visualization for Information Flow Transparency",
"As indicated in the previous section, the developer-oriented visualization in Figure REF is too detailed to be of use for end users.",
"To provide an adequate level of abstraction, we generate a more abstract visualization from the analysis results, depicted in Figure REF .",
"The goal is to provide insights into the potentially malicious information flows between the applications and critical sources and sinks.",
"We display those information flows that take advantage of the transitivity of trust.",
"First, we show all information flows that start out at a critical source and lead to a critical sink.",
"Additionally, to prevent false negatives, we also display flows to the sink from applications on the path.",
"One option is for end users to employ this visualization on-demand to gain an overview of hidden information flows on their devices.",
"Arguably even more effectively, the visualization might serve as an addition to the existing installation process.",
"In this case, additional information flows that are facilitated by the new application are shown after the user has accepted the permissions that the application requires but before the actual installation.",
"The latter case is what Figure REF depicts.",
"Figure: End-user information flow visualizationWhile currently implemented as a separate application, the information flow transparency view could be integrated into the installer at a later time.",
"Also, the flow transparency application currently reads the information flow data from a file that has been previously generated on a PC.",
"We may port the analysis to the Android platform as part of our future work.",
"Alternatively, the analysis may be conducted by the Android Market owner when the application is uploaded to the market and provided at installation time in addition to the application package.",
"The generated information flow data may additionally be used by security engineers to assess the security of a set of applications.",
"For example, the market supervision could use an appropriate visualization to identify potentially harmful applications.",
"Information security staff at companies might also be interested in analyzing the security of applications on their employees' smartphones."
],
[
"Discussion",
"We discuss the advantages as well as the limitations of our approach.",
"In particular, this discussion covers aspects such as false positives, scalability and the relationship to dynamic analysis."
],
[
"Case Study Results",
"The positive result from the case study is promising, in particular, when considering that the studied applications are real-world applications that have not been modified for the developed method.",
"However, the proposed method depends on framework semantics and thus is limited to those sources, sinks and IPC mechanisms that are implemented.",
"As of now, we have discovered the critical mechanisms that are listed in Section .",
"Currently, the algorithm is implemented for the location provider source, the WebView sink as well as startActivity and bindService IPC.",
"We are planning to add the other mechanisms as part of our future work.",
"In general, unsupported source or sink types will result in false negatives.",
"Moreover, while evaluating the proposed method, we discovered two aspects that are difficult to automatically analyze and may lead to additional false negatives.",
"Firstly, user inputs can contain credentials and need to be taken into account as a data source.",
"Also, information flows involving content providers as intermediate step are difficult to follow without deeper knowledge of content provider URI semantics."
],
[
"False Positives",
"Similar to other static analyses, our approach may lead to false positives that can be produced on different levels of our analyses.",
"The first source of false positives is that we find all information flows from critical data sources to data sinks irrespective of whether they are intended or unintended.",
"In our case study, it is intended by the programmer that the location information is passed to the Web page because it is the only way to retrieve the requested information.",
"Therefore, this information flow is a false positive from the programmer's viewpoint.",
"And it may also be a false positive for an end user if she is aware that the feature necessarily needs the location data.",
"The second source of false positives is the underlying data flow analysis.",
"This analysis may find data flows that are not existent due to overestimation.",
"In such a case, we may identify a connection between a data source and a data sink within an application that may not occur in practice.",
"This false positive on the lower level of our analysis may lead to false positives in the set of possible inter-component information flows."
],
[
"Scalability",
"The highest complexity of the proposed algorithm lies in the backward slicing algorithm.",
"Specifically, the construction of the internal data structure for slicing, i.e., the AST or to be more precise, the program dependence graph (PDG) is the time-consuming step [20].",
"By isolating the backslicing runs for each component, we are confident that the whole algorithm will scale well in relation to the number of examined components.",
"By selecting only potential paths between the components' entry and exit points with the help of the RFG (see the step “Identify components and IPC points”), the number of the nodes of the PDG will be reduced.",
"We do not have to build a complete PDG for all Android components.",
"The effort for the backward slicing algorithm then is linear in the size of the PDG [20]."
],
[
"Dynamic versus Static Analysis",
"The focus of our work lies on static code analysis, which in principle can be carried out offline, e.g., on other hardware as done in our current prototype implementation.",
"Dynamic analysis is another approach to address the problem of undesirable information flows on Android.",
"Specifically, the TaintDroid tool implements dynamic monitoring of privacy-relevant flows by modifying the Dalvik VM and the Android kernel [18].",
"Instead of static analysis before installation, TaintDroid complements our approach by offering analyses at runtime.",
"While TaintDroid aims to minimize the performance overhead, static analyses can also afford to carry out more detailed analyses.",
"For example, the tools employed by our approach allow us to even detect indirect information flows induced by control flows [21] although this is not the topic of this paper.",
"One argument in favor of the dynamic analysis of Android applications is that no source code is needed [18].",
"Android uses a different distribution format called DEX, which is a customized bytecode format being register-based rather than employing an operand stack.",
"However, we have obtained promising early results with the dex2jar tool, which translates DEX to Java bytecode [22].",
"Since our analyses work on Java bytecode (see Section REF ), we also applied our tools to DEX code for the case study.",
"However, applying the dex2jar tool to larger application sets remains future work.",
"Not all properties can be inferred from the code statically.",
"One example is the implicit intent decomposition mechanism in the Android framework.",
"It decides at runtime among all registered components which component offers the requested intent features and accordingly suits to the implicit intent call.",
"At this point, static analysis cannot determine which components will be connected at runtime and which component the platform will choose, if there exists more than one suitable component.",
"In the end, a hybrid approach consisting of both static and dynamic analyses will be reasonable.",
"This way, static analyses can be improved by information gained from runtime analyses.",
"Furthermore, users who cannot afford to use static analysis tools can rely on the TaintDroid approach, whereas in business or governmental scenarios as well as at market entry, the static approach can be employed additionally."
],
[
"Related Work",
"There exist a plethora of works for the static security analysis of software, e.g., discussed by Chess and West [23].",
"The works on static information flow analysis for security often resides at the source code level.",
"Some approaches deal with programmer-written annotations for information flows that permit static code checks [24], [25], [26].",
"Moreover, the language-based security extensions in JFlow [24] support the modeling of access control.",
"This allows one to statically check code privileges, but all modeled access controls will be removed after the JFlow compiler processing.",
"This kind of language-based security analysis is limited to the use of annotations by programmers at the source-code level.",
"Our analysis methods, however, works without code annotations aiming to detect undesirable information flow between different applications and components, respectively.",
"In another approach, type-based security combines annotations with dependence graph-based information flow control [27].",
"Hammer's proposed analysis uses the Java bytecode and a succinct security policy specification that is inserted as annotations in code comments.",
"Although both approaches aim to detect information flow violations of Java-based applications, they differ in the analysis techniques they use.",
"We employ an analysis approach using the RFG to restrict the search space and thereafter carrying out a more focused analysis at the detail level.",
"Hammer uses the complete dependence graph to directly conduct the information flow analysis.",
"In addition, Hammer's method requires code annotations for the security labeling, similar to JFlow.",
"This way, this approach can only be applied by the developer, but neither by the Android Market owner nor the end user.",
"Chandra and Franz implemented an information flow framework for Java applications using static as well as dynamic checks [21].",
"Static checks are needed to improve the dynamic analysis such that information about alternative control paths is also available.",
"The approach works at the bytecode level and is fully compatible with the class file format.",
"In general, all aforementioned approaches tackle the problem of indirect information flow induced by the control flow of applications, whereas we currently only analyze direct flows.",
"The focus of our work lies on an inter-application analysis.",
"Furthermore, our techniques are tailored towards the Android platform by considering Android-specific programming concepts as well as data sources and sinks.",
"Certainly, it is worthwhile to address indirect information flows in future work.",
"Important research prototypes from static security analysis are e.g.",
"MOPS [28], Eau Claire [29], BLAST [30], and LAPSE [4].",
"MOPS uses temporal logics as formalism and model checking to discover issues such as race conditions in C programs.",
"Eau Claire allows the formulation of pre- and postconditions as code annotations and is based on a theorem prover.",
"Eau Claire detects general security problems such as buffer overflows.",
"BLAST uses (lazy) abstraction to find safety properties in C/C++ code.",
"The tool xg++ by Ashcraft and Engler was used to detect vulnerabilities in the Linux Kernel [31].",
"The LAPSE approach by Livshits and Lam resembles our approach in also targeting Java applications.",
"In contrast, we focus on interactions between applications and specifically consider Android's framework semantics for our analyses, whereas LAPSE aims to detect implementation bugs such as SQL injection and Cross-Site scripting vulnerabilities.",
"However, the techniques employed by the LAPSE approach can also be used for our analyses at the source code level instead of the backward slicing algorithm.",
"Some of the research prototypes have developed into commercial tools such as Fortify Source Code Analyzer [32] and Coverity Prevent [33].",
"Our approach is complementary to all those works because we utilize architectural information to focus the analysis at the code level.",
"In addition, those tools are designed to detect common low-level security bugs such as buffer overflows, SQL injection and Cross-site scripting vulnerabilities.",
"We, however, focus on information flow analysis, and more generally aim to detect security problems induced by software extensibility.",
"After the release of the Android platform, some works have addressed the built-in application security mechanisms of this platform [5], [34].",
"Moreover, Chaudhuri et al.",
"define a formal language to describe Android application behavior and the application's permission usage [35].",
"As discussed above the TaintDroid approach is close to our work.",
"Beyond TaintDroid, there are also other approaches with similar goals.",
"KIRIN is an alternative application installer for Android with a built-in security framework that enforces policy invariants on the phone [36].",
"The tool checks at application install time for issues such as unchecked interfaces.",
"When problems are found, the application installation is canceled.",
"For the analysis, KIRIN only uses the Android Manifest file (containing the permissions) and does not look at the program code.",
"In addition, the interplay between different applications was not considered.",
"Nauman et al.",
"present an Android permission framework and a policy configuration user interface that allows the user to dynamically limit application permissions at install time [37].",
"Similarly to KIRIN, no inter-component relations are taken into consideration for the policy enforcement and only conditions according to time and usage count are described.",
"On a lower level, Shabtai et al.",
"[38] enable the SELinux feature in the Android kernel to explore additional protection opportunities and benchmark the system performance with activated SELinux on a HTC G1 smartphone running Android.",
"Another research approach is the SAINT architecture [39].",
"It inserts enforcement hooks into Android's middleware layer to improve the currently limited Android security architecture.",
"This work takes semantics such as location and time into account, but strictly focuses on the developer view of permissions and does not account for transitive data flows."
],
[
"Conclusion and Outlook",
"In this paper, we discussed how the transitivity of trust problem affects dynamic multi-component systems.",
"Focusing on the Android platform, we presented a two-layer approach to the static security analysis of information flows for composite Android applications and thus approached the transitivity of trust problem in this context.",
"On the upper layer, we use the software architecture to slice the applications.",
"Thereafter, the actual data flow analysis is carried out at the AST level.",
"The results are integrated into the architecture to derive information flows at the architectural layer.",
"We demonstrated the effectiveness of our analysis method with the help of two real-world applications, which use advanced Java and Android programming concepts such as inner classes, GUI handling, and Android service binding.",
"There are several directions for further research.",
"First, we aim to support a more complete set of data sources and sinks as well as other concepts of the Android framework such as pending intents, URI permissions, and service hooks.",
"Furthermore, our static analysis can be combined with dynamic analyses into a hybrid approach in order to improve on the precision of the analyses.",
"Lastly, we will analyze larger sets of applications.",
"For example, it would be interesting to investigate (at least) parts of the Android market and develop information flow policies that the applications should adhere to."
]
] | 1204.1458 |
[
[
"Design of an optical reference cavity with low thermal noise and\n flexible thermal expansion properties"
],
[
"Abstract An ultrastable optical reference cavity with re-entrant fused silica mirrors and a ULE spacer structure is designed through finite element analysis.",
"The designed cavity has a low thermal noise limit of $1\\times10^{-16}$ and a flexible zero crossing temperature of the effective coefficient of thermal expansion (CTE).",
"The CTE zero crossing temperature difference between a composite cavity and a pure ULE cavity can be tuned from $-10\\ ^{\\circ}$C to $23\\ ^{\\circ}$C, which enables operation of the designed reference cavity near room temperature without worrying about the CTE zero crossing temperature of the ULE spacer.",
"The design can be applied to cavities with different lengths.",
"Vibration immunity of the cavity is also achieved through structure optimization."
],
[
"Introduction",
"Ultra-stable lasers [1] are essential in high-resolution laser spectroscopy [2], optical frequency standards [3], [4], [5], [6], [7], [8], [9], [10], [11], gravitational wave detection [12], and fundamental tests of physics [13], [14].",
"To achieve such a frequency stabilized local oscillator, the laser is typically servo locked to a high-finesse reference cavity by using Pound-Drever-Hall (PDH) technique [15].",
"Within the servo locking bandwidth, the stability of the locked laser is determined by the stability of the optical path-length of the reference cavity.",
"At low frequencies (below 100 Hz) the most severe fluctuations of the cavity length are caused by environmental disturbances such as seismic and acoustic noise.",
"At higher frequencies these disturbances are usually damped by active vibration isolation.",
"It can be shown that the sensitivity of the reference cavity to the low frequency environmental disturbances can be minimized by optimizing cavity geometry and mounting method [16], [17], [18], [19], [20], [21], [22], [23].",
"While the sensitivity of a reference cavity to mechanical vibrations can be suppressed such that the influence of these perturbations can be ignored, there still exists a fundamental thermal noise limit to the level of stability achievable with the reference cavity [24].",
"This limit is attributed to Brownian motion of the reference cavity spacer, mirror substrates, and optical coatings [24].",
"As have been widely reported before [25], [26], [27], [28], [29], using short length cavities ($\\sim 10$ cm) constructed entirely from ULE glass, the relative frequency stability characterized by Allan deviation was limited by thermal noise to the level of $10^{-15}$ .",
"Several methods can be used to lower the thermal noise limit, such as increasing the reference cavity spacer length, increasing the laser spot size, and using high mechanical $Q$ materials.",
"Of these methods, using high $Q$ materials offer the biggest improvements.",
"To lower the thermal noise limit of reference cavities, a commonly used high $Q$ material is fused silica (FS).",
"Because FS has a much larger coefficient of thermal expansion (CTE) (500 ppb/K) than that of ULE ($\\pm 50$ ppb/K) [30], it is not appropriate to use FS as cavity spacer material.",
"Since the dominant thermal noise contribution comes from the mirror substrates, it would be more suitable to just use FS as mirror substrates, and keep ULE as cavity spacer material.",
"Replacing the ULE mirrors of a short ULE cavity by FS mirrors can lower the thermal noise limit approximately by a factor of two [28].",
"Combining with a longer length cavity spacer, it is then possible to reach a thermal noise limit of $10^{-16}$ .",
"However, the CTE mismatch between FS and ULE can cause a problem.",
"Temperature changes inside the reference cavity can create stress which bends the mirror substrates, and introduces a larger effective CTE for the composite cavity.",
"This problem can be alleviated by setting the operating temperature of the reference cavity at the zero crossing temperature of the effective CTE.",
"If a ULE spacer has its zero crossing temperature of CTE at room temperature, then the zero crossing temperature of the effective CTE will be much lower.",
"However, this is not very convenient, because cooling a reference cavity is technically more challenging than simply heating the reference cavity.",
"If the temperature is set too low, it can introduce extra problems, like water condensation on cavity housing, etc.",
"For practical purpose, it is preferable to stabilize the reference cavity temperature a little bit above room temperature.",
"Presently there are several approaches to achieve this objective.",
"One is to find a ULE piece that has negative CTE at room temperature, and in combination with positive CTE of FS, it is possible to make the zero crossing temperature of the effective CTE to be above room temperature [30].",
"Another method is to contact ULE rings to the back side of FS mirrors to prevent the mirror bending due to CTE mismatch [31].",
"As a consequence, the zero crossing temperature of the effective CTE can be above room temperature.",
"Both methods are used by Jiang et al., and a fractional frequency instability of $2\\times 10^{-16}$ was achieved [32].",
"The above two approaches have their limitations.",
"The first approach relies on the availability of ULE material with negative CTE at room temperature.",
"This requires the good will of the ULE manufacturer, and might not be always available.",
"The second approach in principle can be used to tune the zero crossing temperature over a range of about 30 $^{\\circ }$ C, but it only works for a smaller diameter ULE spacer [31].",
"To lower the thermal noise of the reference cavity, it is common to choose a longer cavity length with bigger diameter.",
"The tuning range then is limited to only about $5\\sim 6$ $^{\\circ }$ C for this geometry according to our calculations.",
"Normal ULE has the zero crossing temperature of CTE varying between 5 $^\\circ $ C and 35 $^\\circ $ C [30], [33], so with this limited tuning range it is not always possible to raise the zero crossing temperature of the effective CTE to above room temperature.",
"Another approach is to use a re-entrant mirror design for the reference cavity [34].",
"By using a re-entrant mirror design, the direction of FS mirror thermal expansion is opposite to that of ULE spacer, so in principle they can be compensated better, and introduce less stress to the mirrors.",
"In reality the tuning range of the zero crossing temperature difference is about 20 $^\\circ $ C $\\sim $ 30 $^\\circ $ C. Consequently, this cavity design is only suitable for cases when ULE spacers have a zero crossing temperature lower than 5 $^\\circ $ C. In this paper, we analyze a modified re-entrant mirror design for the reference cavity, where the ULE rings for the re-entrant FS mirrors are replaced with FS rings.",
"With this design modification, the zero crossing temperature of the effective CTE can be tuned to much greater range, and it is very easy to design a composite cavity so that the zero crossing temperature of the effective CTE is above room temperature.",
"Since a low thermal noise reference cavity design is central to this paper, a detailed discussion of the thermal noise limit of a ULE cavity is presented in Sec.",
"2.",
"Based on our calculation results, a particular geometry of the reference cavity is chosen, and re-entrant FS mirrors are used to replace the traditional ULE mirrors.",
"With this geometry, in Sec.",
"3 we analyze the zero crossing temperature of the composite cavity with finite elements analysis (FEA), and compare the results with that of the other approaches.",
"For completeness, in Sec.",
"4 we also design the cavity to be vibration-insensitive.",
"Finally we concluded in Sec.",
"5."
],
[
"Calculation of Thermal Noise Limits",
"The stability of a reference cavity is limited by an inevitable mechanical thermal fluctuation due to the Brownian motions of materials.",
"From the fluctuation-dissipation theorem, the most significant parameters in the calculation of the thermal noise spectrum for a given structure are the mechanical quality $Q$ factors of materials.",
"Materials with higher $Q$ factors generate less mechanical dissipation, thus contribute smaller thermal noise.",
"To get an order of magnitude estimation of their contributions, using the formulas given in [24], the power spectral density of mirror displacement due to thermal noise, $G_m(f)$ , which typically dominates the spacer contribution, is given by $G_m(f)=\\frac{4k_BT}{2\\pi f}\\frac{1-\\sigma _{sub}^2}{\\sqrt{\\pi }E_{sub} w_0}\\frac{1}{Q_{sub}}(1+\\frac{2}{\\sqrt{\\pi }}\\frac{1-2\\sigma _{sub}}{1-\\sigma _{sub}}\\frac{Q_{sub}}{Q_{coating}}\\frac{d}{w_0}),$ where $k_B$ is the Boltzmann's constant, $T$ is the temperature, $\\sigma _{sub}$ is the Poisson's ratio of mirror substrate, $E_{sub}$ is the Young's modulus of mirror substrate, $w_0$ is the beam radius, $Q_{sub}(Q_{coating})$ is the mechanical quality factor of the mirror substrate (coating), and $d$ is the coating thickness ($\\sim 5$ $\\mu $ m).",
"In addition, the spacer contribution can be written as $G_{sp}(f)=\\frac{4k_BT}{2\\pi f}\\frac{L}{3\\pi R^2E_{sp}}\\frac{1}{Q_{sp}},$ where $L$ and $R$ are the spacer length and radius, respectively, $E_{sp}$ and $Q_{sp}$ is the Young's modulus and mechanical quality factor of the spacer, respectively.",
"The total reference cavity displacement power spectral density due to thermal noise is $G_t(f)=2G_m(f)+G_{sp}(f).$ From which we can calculate the Allan deviation of the relative stability of the cavity as $\\sigma _y=\\frac{\\sqrt{2\\ln 2f}\\cdot \\sqrt{G_t(f)}}{L}.$ From Eq.",
"(REF ) and Eq.",
"(REF ), we can see that by choosing materials of higher $Q$ factors, the thermal noise limit can be efficiently lowered.",
"Among the low CTE materials that are commonly used in ultra-stable cavities, ULE has a $Q$ factor of $6.1\\times 10^4$ that is more than one order of magnitude larger than that of Zerodur ($3.1\\times 10^3$ ).",
"Furthermore, FS has an even larger $Q$ factor of about $10^6$ [35].",
"In cases where thermal noise limits are the ultimate limiting factor of achievable cavity stability, it is very effective to use a higher $Q$ factor material (like FS) as mirror substrate while keeping ULE as spacer material.",
"Figure: Variation of vibration sensitivity of a 30 cm long cavity with respect to spacer diameter.",
"The vibration sensitivity becomes much smoother at diameter larger than 15 cm.",
"Consequently, a spacer diameter of 15 cm is chosen in all following simulationsAccording to Eq.",
"(REF ), using longer cavity can also have the benefit of suppressing the thermal noise.",
"In addition, the total thermal noise limit also depends on the laser beam sizes on the reflecting mirror surfaces.",
"Larger beam radius contributes a lower thermal noise.",
"Taking into account of all these effects, we calculated the thermal noise limits of cavities with different dimensions and configurations, expressed in both Allan deviation and frequency spectral density, and the data are listed in Table REF .",
"By replacing ULE mirrors with FS mirrors and increasing the cavity length from 10 cm to 30 cm, the thermal noise limit can be suppressed by nearly one order of magnitude.",
"Consequently in the following we will consider the design of a long, FS mirror substrate cavity.",
"Notice here that for longer cavity, the spacer radius is also increased.",
"This is due to the fact that longer cavity is more susceptible to vibration influence, and larger cavity can help to suppress this effect.",
"The result is shown in Fig.",
"REF .",
"The detail is explained in Sec.",
"4.",
"Table: Thermal noise limits of optical reference cavities with different configurations and dimensions.",
"LL, RR, and w 01 /w 02 w_{01}/w_{02} are the spacer length, spacer radius and approximate mode spot sizes (intensity 1/e 2 1/e^2 radius) on the plano-confocal cavity mirrors, respectively.",
"σ\\sigma is the Allan deviation of cavity stability and S ν (1Hz)\\sqrt{S_{\\nu }(1Hz)} is the root cavity frequency noise spectral density at 1 Hz for the 1064 nm light."
],
[
"Compensation of Thermal Distortion",
"By replacing the ULE mirrors with FS mirrors, CTE mismatch between the two materials will cause structure deformation when environmental temperature changes, and thus change the effective CTE of the composite cavity, $\\alpha _{eff}$ , and zero crossing temperature of the effective CTE, $T_0(eff)$ .",
"To tune this zero crossing temperature to near room temperature, strategies include attaching ULE rings to FS mirrors, and using re-entrant FS mirrors inside the spacer [31], [34].",
"In this section, both strategies are analyzed with FEA for a 30 cm long reference cavity with a diameter of 15 cm.",
"Based on the analysis, we make modifications to the design and also give the simulation results.",
"In our calculations, we assume the ULE spacer has length of $L$ , and optical contacted FS mirrors have radius of $R$ .",
"Due to the CTE difference, a temperature variation $dT$ results in a difference of the radial expansion between $dR=(\\alpha _{FS}-\\alpha _{ULE})RdT$ .",
"This results in an axial expansion $dA$ of the mirrors if we assume the optical contact to be perfectly rigid.",
"Assuming a linear stress-strain relation, the radial expansion $dR$ and the axial expansion $dA$ are related by a constant $\\delta $ to be $dA=\\delta dR$ .",
"The effective cavity CTE is then $\\alpha _{eff}&=&\\frac{\\alpha _{ULE}LdT+2dA}{LdT}\\nonumber \\\\&=&\\alpha _{ULE}+2\\delta \\frac{R}{L}(\\alpha _{FS}-\\alpha _{ULE}).$ We are primarily interested in the difference of zero crossing temperatures, $\\Delta T=T_0(eff)-T_0(ULE),$ between the effective CTE of cavity and the original CTE of ULE material.",
"We aim to find a structure design which can tune the effective zero crossing temperature of the cavity CTE, $T_0(eff)$ , to be slightly higher than room temperature (25 $^\\circ $ C $\\sim $ 28 $^\\circ $ C) for better thermal control of the system.",
"Since normal ULE material has zero crossing temperature of CTE, $T_0(ULE)$ , in the range between 5 $^\\circ $ C and 35 $^\\circ $ C [30], [33], this means that the tuning ability of an optimal design should at least achieve $-10\\ ^\\circ \\mathrm {C}\\le \\Delta T\\le 23\\ ^\\circ \\mathrm {C}$ .",
"Following Ref.",
"[31], we add ULE rings to the back side of FS mirrors to form a “sandwich” structure and analyze the thermal properties of the combined cavity using both Comsol multiphysics and ANSYS FEA packages.",
"Both analyses give consistent results.",
"Numerical analysis indicates that the temperature difference $\\Delta T$ to be only slightly higher than zero for a 30 cm long cavity, even if many efforts have been paid to optimize the cavity structures, including the introduction of a tapered structure to reduce the areas of the ends of spacer and the adoption of larger mirrors (1.5 inch diameter).",
"Cavity geometries and FEA analysis results are shown in Fig.",
"REF and Fig.",
"REF , respectively.",
"As shown in Fig.",
"REF , for cavities with simple cylindrical structures, it is impossible to tune the temperature difference $\\Delta T$ to be greater than zero.",
"For tapered cavities, an upper limit of 3 $^\\circ $ C of temperature difference in the case of 1.5 inch mirrors structure can be achieved.",
"Although this might be good enough for some cases when the ULE spacer has a $T_0(ULE)$ near or higher than room temperature, it is not flexible enough because you are not normally able to choose the very batch of material with the characteristics you want.",
"It should be noticed that although smaller inner diameter of the ULE ring $d$ can further increase the range of $\\Delta T$ , it has a practical limit and should not be too small, otherwise the entrance of the laser beam will be affected, and the scattered light caused by laser diffraction at small hole will introduce unwanted noise.",
"Figure: Geometries of sandwich cavities and FEA simulations of the deformations caused by thermal expansion mismatch when temperature is increased by 1 ∘ ^\\circ C. (a) cavity with a cylindrical spacer.",
"(b) cavity with a tapered spacer.",
"Color scale shows the axial displacement.",
"dd and hh are the inner diameter and thickness of the ULE rings, respectively.",
"The notch structure is designed for vibration suppression purpose, and will be explained in Sec.",
"4.Figure: FEA simulations of the CTE zero crossing temperature tuning capabilities of sandwich cavities with different dimensions and configurations, including cylindrical and tapered structures (upper panel) as well as designs of 1.5 inch mirror structures (lower panel).",
"dd and hh are the inner diameter and thickness of the ULE rings, respectively, other parameters of D ' D^{\\prime } and L ' L^{\\prime } are defined in Fig.1.",
"Right vertical axis shows the coupling coefficient δ\\delta as defined in Ref.",
"that directly connects the axial mirror displacement with the radial expansion between mirror and spacer.Cavities with FS mirrors re-entrant within the bores of ULE spacers have the advantage of compensation of opposite thermal expansion between spacers and mirrors inside.",
"Geometry of the cavity is illustrated in Fig.",
"REF .",
"In this structure, FS mirrors are optically contacted to ULE rings which are in turn optically contacted to the ends of ULE spacer.",
"Simulations show that temperature difference between $T_0(eff)$ and $T_0(ULE)$ can be as large as 30 $^\\circ $ C, and the results are shown in Fig.",
"REF .",
"Again, the inner diameter of the ring $d_1$ cannot be too small to affect the entrance of laser beam.",
"In contrast with that of the “sandwich” structure, it is difficult to tune the $T_0(eff)$ lower.",
"The tuning range of the zero crossing temperature difference, $\\Delta T$ , is about 20 $^\\circ $ C $\\sim $ 30 $^\\circ $ C. Consequently, this cavity design is only suitable for cases when ULE materials have a $T_0(ULE)$ lower than 5 $^\\circ $ C. Figure: Geometry and dimensions of cavity with re-entrant FS mirrors optically contacted to ULE rings.",
"The ULE ring has an inner diameter of d 1 d_1, an outer diameter of d 2 d_2, and hh is the thickness of rings.Figure: FEA simulations of the CTE zero crossing temperature tuning range of re-entrant cavities with FS mirrors optically contacted to outer ULE rings with different structure parameters.",
"d 1 d_1 and d 2 d_2 are the inner and outer diameters of ULE rings, respectively, and hh is the thickness of rings.",
"Definitions of these dimensional parameters are shown in Fig.",
".One of the reasons for such a high temperature difference is due to huge difference of CTE between FS and ULE.",
"FS has a CTE of about $5.2\\times 10^{-7}$ K$^{-1}$ at room temperature while ULE has CTE of less than $10^{-8}$ K$^{-1}$ (depending on temperature $T$ ) [30], [33].",
"However, the more important reason is the deformation of mirrors caused by CTE mismatch.",
"This deformation makes the FS mirrors bulge in the same direction of its thermal expansion direction, as shown in Fig.",
"REF (a), leading to a result that only long cavities are suitable for this design.",
"This process can be understood theoretically using Eq.",
"REF .",
"For the re-entrant cavity structure, the effective CTE of the cavity can be expressed as: $\\alpha _{eff}&=&\\frac{\\alpha _{ULE}L-2\\alpha _{FS}h_m}{L-2h_m}-2\\delta \\frac{R}{L}(\\alpha _{FS}-\\alpha _{ULE})\\nonumber \\\\&\\approx &\\alpha _{ULE}-2\\alpha _{FS}\\frac{h_m}{L}-2\\delta \\frac{R}{L}(\\alpha _{FS}-\\alpha _{ULE}),$ where $h_m$ is the thickness of the FS mirror substrates, and is assumed to be $6.2$ mm in our calculations.",
"Here we have used the assumption that $L >> h_m$ .",
"In first order approximation we can write the CTE of ULE around its zero crossing temperature $T_0(ULE)$ to be $\\alpha _{ULE}(T)=\\beta (T-T_0(ULE)),$ where $\\beta $ is a linear temperature coefficient and is taken to be $2\\times 10^{-9}$ K$^{-2}$ [31].",
"Thus the effective CTE can be expressed as: $\\alpha _{eff}&\\approx &\\beta [T-T_0(ULE)-\\frac{2\\alpha _{FS}h_m+2\\delta R(\\alpha _{FS}-\\overline{\\alpha _{ULE}})}{\\beta L}]\\nonumber \\\\&\\equiv &\\beta [T-T_0(eff)].$ Here we have used the fact that $2\\delta R << L$ to replace the $\\alpha _{ULE}$ in the numerator with temperature independent value of $\\overline{\\alpha _{ULE}}$ .",
"The effective zero crossing temperature of CTE of the cavity can then be expressed as $T_0(eff)=T_0(ULE)+\\frac{2\\alpha _{FS}h_m+2\\delta R(\\alpha _{FS}-\\overline{\\alpha _{ULE}})}{\\beta L}.$ According to the above equations, to make sure the temperature difference of $\\Delta T=T_0(eff)-T_0(ULE)$ to be smaller than 23 $^\\circ $ C, the spacer length $L$ must be at least 14 cm even without considering the effect of deformation coefficient $\\delta $ .",
"If $\\delta $ is positive, then the mirrors bulge in the same direction of its thermal expansion direction, as shown in Fig.",
"REF (a), the required cavity length will be even longer, typically longer than 30 cm.",
"Currently spacer length longer than 30 cm is very difficult to obtain, and has higher vibration sensitivity.",
"To obtain a more flexible tuning range of $\\Delta T$ with a moderate cavity length, the coefficient $\\delta $ in Eq.",
"(REF ) needs to be negative to compensate the huge difference between the CTE of FS and ULE.",
"Therefore, a modified design of re-entrant cavity is applied, in which the FS mirrors are optically contacted to FS rings instead of ULE rings.",
"In this configuration, the deformation of CTE mismatch is between the FS rings and ULE spacer, causing the FS rings bulge in the opposite direction of the mirror thermal expansion direction, as shown in Fig.",
"REF (b).",
"In another word, this cavity structure has a negative $\\delta $ .",
"Simulations of cavities with this new structure are shown in Fig.",
"REF .",
"Figure: FEA simulations of the deformations caused by CTE mismatch of re-entrant cavities; (a) deformation between FS mirror and ULE ring causing the mirror bulges in the same direction as its thermal expansion direction; (b) deformation between FS ring and ULE spacer causing the ring bulges in the opposite direction of the mirror thermal expansion direction.",
"Color scale shows the axial displacement.Figure: Tuning range of CTE zero crossing temperature of re-entrant cavities with FS mirror optically contacted to outer FS rings instead of ULE rings.",
"The definitions of dimensional parameters are the same as those in Fig.",
".According to the simulations, the modified re-entrant cavity yields a larger tuning range of $\\Delta T$ from -7 $^\\circ $ C to 25 $^\\circ $ C simply by varying the thickness $h$ of FS rings, and this range can be adjusted further by altering the outer diameter $d_2$ according to the characteristics of ULE batch used and the set-point temperature of the thermal control system.",
"With this design, the flexibility requirement of $-10\\ ^\\circ \\mathrm {C}\\le \\Delta T\\le 23\\ ^\\circ \\mathrm {C}$ is easily achieved.",
"Through comparison, cavities with FS rings of relatively smaller outer diameter $d_2$ are more suitable because of the less area of optical contact as well as the compatibility with vibration immunity design which will be discussed in Sec.",
"4.",
"As a result, in the following simulations and analysis, an outer diameter of 80 mm of FS ring is adopted.",
"Figure REF shows the zero crossing temperature tuning capabilities with different cavity lengths between the original and the modified re-entrant cavity.",
"The zero crossing temperature tuning ranges are much narrower for the original design (only about 5 $^\\circ $ C) than those of the new design (more than 30 $^\\circ $ C).",
"Furthermore, to obtain a moderate temperature tuning (less than 25 $^\\circ $ C), the cavity length must be longer than 30 cm with the original design.",
"In comparison, for the new design, variation of ULE spacer lengths from 10 cm to any longer length can achieve a moderate temperature tuning range.",
"It is interesting to note that all curves except $L=10$ cm in Fig.",
"REF (b) intersect and cross zero at the same position where the ring thickness $h$ is 11.2 mm.",
"Referring to Eq.",
"(REF ), this intersection with zero reveals the fact that when the ring thickness is at this particular value ($h$ =11.2 mm) the thermal expansions of FS mirrors are exactly compensated by the deformation strain of FS rings no matter how long the cavity is.",
"In this case, the effective CTE of the cavity is the same as that of the ULE spacer material.",
"For $L=10$ cm case, the requirement $2\\delta R << L$ is not valid, which causes the zero crossing point to be different with that of longer cavities.",
"Figure: Comparison of the zero crossing temperature tuning capabilities for different cavity lengths between the original re-entrant cavity (a) and the modified cavity with FS rings (b).",
"The inner diameters of rings d 1 d_1 are 15 mm for both cases.In this work, the linear temperature coefficient of the effective CTE and mirror deformations of the modified cavity are also studied.",
"The results are shown in Fig.",
"REF and listed in Table REF .",
"According to Eq.",
"(REF ), the linear temperature coefficient $\\beta $ of the effective CTE also depends on cavity structure.",
"Figure REF (a) shows the variation of $\\beta $ with respect to ring thickness $h$ of different cavity designs, including the sandwich cavity and the two re-entrant cavities.",
"The coefficients are all near $2\\times 10^{-9}$ K$^{-2}$ , which is the original linear temperature coefficient of ULE material, and the exact values depend on the detailed structure parameters.",
"In addition, the deformations of mirror caused by CTE mismatch are also analyzed and are compared between different cavity designs.",
"The results are shown in Fig.",
"REF (b).",
"Mirrors of the “sandwich” design and re-entrant designs all deform in a convex shape at the mirror centers, and the added radius of curvature (negative in these cases) has a magnitude of approximately $10^4$ m to $10^5$ m, which are 4 to 5 orders of magnitude larger than those of the cavity mirrors (0.5 m or 1 m), and thus has negligible contribution.",
"As can be seen from the figure, “sandwich” structure causes more abrupt distortions within the mirror.",
"Details of the properties of cavities with different structure designs are listed in Table REF , including their thermal noise limits.",
"It should be noted here that in the “sandwich” design and re-entrant designs, the introduction of ULE or FS rings increases the thermal noise limit by a very small amount.",
"This effect is taken into account in the thermal noise limits calculations in Table REF .",
"Figure: Comparison of properties between different designs of composite cavities: (a) linear temperature coefficient β\\beta of the effective CTE; (b) mirror deformations due to CTE mismatch of sandwich design (upper) and re-entrant designs (below), respectively.Table: Properties of reference cavities with different structure designs.",
"Relative deformation is defined as the ratio between the original mirror radius of curvature and the extra radius of curvature caused by mirror deformations (both horizontal and vertical directions have similar results)."
],
[
"Design of Vibration Insensitive Cavity",
"The cavities designed with re-entrant FS mirrors and FS rings have an excellent CTE zero crossing temperature tuning property with a very low thermal noise limit.",
"In order to reach this thermal noise limit, the cavity should be further designed to be vibration insensitive.",
"A quantitative analysis of the elastic deformation of reference cavities provides valuable guidance for cavity design.",
"Since the time-dependent vibration perturbation consists of a broad spectrum, in principle a full dynamic analysis of the cavity deformation is required to correctly predict the cavity vibration sensitivity.",
"This dynamic analysis to our knowledge is not available in the literature.",
"However, we can greatly simplify the process by performing static analysis.",
"A static analysis can be used to approximate dynamic analysis because the following two conditions are satisfied.",
"First, only the vibration modes with frequency lower than 100 Hz are of interest, because the modes with higher frequencies will be significantly damped by either passive or active vibration isolations used in all reference cavities systems.",
"Second, the considered cavity dimension is much smaller than the cavity vibration characteristic wavelength, meaning that all particles move in-phase inside each eigenmode.",
"Thus the dynamic response of the cavity can be mimicked by applying a static force with the random acceleration frozen at that moment.",
"For example, consider ULE spacer material (with $\\rho $ of 2.21 g/cm$^3$ and $E$ of 67.6 GPa), its dispersion relation is [17] $f=\\sqrt{\\frac{E}{\\rho }}\\frac{1}{\\lambda }.$ For oscillation frequency of 100 Hz, the characteristic wavelength $\\lambda $ of 55 m is much greater than the simulated 0.3 m cavity length.",
"Consequently, at these low frequencies considered here the static analysis is a reliable substitute for the dynamic analysis.",
"In static analysis the elastic deformation of cavities caused by a gravity-like acceleration are studied.",
"To understand the principle, for simplicity we assume the cavity is held in a plane instead of several mounting points.",
"If the cavity is horizontally held on a plane from bottom, compression caused by acceleration normal to the supporting plane will cause expansion in length because of the non-zero Poisson's ratio; on the other hand, if the cavity is hung on the plane from top, the deformation is reversed, the cavity elongates in height and shortens in length.",
"When the cavity is held on its symmetry plane, the lengthening of the upper part and the shortening of the bottom part compensate each other and the axial length of the cavity remains unchanged under vertical accelerations.",
"In practice, cavities are held on discrete points instead of the complete plane, which will cause additional bending deformation as well.",
"The bending effect along with the simple compression or expansion, contribute to the total cavity length variation.",
"In this case, the optimal supporting plane is no-longer the symmetry plane and the cavity length variation under acceleration is dependent on both the discrete supporting positions and the mounting plane of those discrete points.",
"The bending of cavity will cause mirror tilting as well.",
"If the cavity is supported on points locating at the two opposite sides along the horizontal axial length, it will bend downwards under vertical acceleration and the mirrors at both sides tilt correspondingly; if the cavity is supported on the middle of its length, as an extreme, it will bend upwards, and the mirrors tilt in an opposite way.",
"There must be a critical position along the length of the cavity (Airy points), on which the cavity is supported, so that the mirror tilting is eliminated by the compensation of both bending processes.",
"To eliminate both cavity length variation and mirrors tilting simultaneously in the case of our cavity design, we apply a notched structure [17], [18], [19], [22] to the cavities as shown in Fig.",
"REF .",
"In simulations, we implement FEA analysis by applying a gravity-like force on the cavity vertically with $g=9.8$ m$/\\mathrm {s}^2$ , and the support with $1\\times 1$ mm$^2$ square on the notch surface is constrained to simulate the mounting condition.",
"Optimal dimensions of the notched structure are searched so that the two kinds of critical mounting positions with zero length variation and zero mirrors tilting of the cavity overlap, while maintaining a less rigorous requirement of mounting accuracy and machining tolerance.",
"The definitions of parameters about the dimensional and mounting configurations are also illustrated in Fig.",
"REF .",
"Figure: (a) Dimensions (C x C_x and C y C_y) of the notched structure, and positions of mounting supports (X p X_p and Z p Z_p) on the notch surface.",
"(b) FEA simulations of the deformation of a quarter section of the cavity with optimal notch sizes and supporting positions under vertical vibration.During simulations, cavities with different notch sizes ($C_x$ and $C_y$ ) are examined to search for an optimal dimension with which an overlap of the above two critical mounting positions is available.",
"In most cases, the mounting positions satisfying those two conditions do not overlap, which means only one of the effects can be eliminated.",
"Analysis results are shown in Fig.",
"REF in which the mounting positions with zero length variations and zero mirrors tilting of different notch sizes are compared.",
"Calculations reveal that the mounting positions with zero mirror tilting angles are not sensitive to the changes of the notch sizes, while the zero crossing mounting positions (where the length variations of cavities are immune to vertical vibrations) are much more sensitive to the notch sizes, as shown in Fig.",
"REF (a) and its inset.",
"It is also found that at a particular value of $C_y$ (or $C_x$ ) there is a corresponding $C_x$ (or $C_y$ ) that can result in the overlap of these two critical positions.",
"Another important parameter is the slope of the cavity length variation curve, which reflects the sensitivity of vibration immunity to mounting positions accuracy and machining tolerance.",
"Different slopes of cavities with different notch sizes are shown in Fig.",
"REF (b).",
"It indicates that the sensitivity of the frequency variation as mounting positions changes in the transverse ($X$ ) direction is not sensitive to the notch sizes, neither $C_x$ nor $C_y$ , and has a value of approximately 42 kHz/g/mm.",
"In contrast, the sensitivity with mounting positions changes in the axial ($Z$ ) direction steadily decreases with decreasing notch sizes of both $C_x$ and $C_y$ .",
"As a result, smaller notches are much more preferred when optimizing the cavity dimensions.",
"We choose dimensions of $C_x$ =28 mm and $C_y$ =49 mm and the corresponding optimized vibration sensitivity of the cavity are shown in Fig.",
"REF .",
"Figure: (a) Comparison of critical positions, Z p Z_p, for cavity mounting with different notch sizes.",
"(b) the slopes near the zero crossing positions of curves related to the dependence of resonance frequency deviation on mounting positions under vertical vibrations with different notch sizes.Figure: (a) FEA simulations of the axial length variations at the mirror center and mirror tilting angles at different mounting positions of Z p Z_p.",
"(b) FEA simulations of the axial length variations at the mirror center and mirror tilting angles at different mounting positions of X p X_p.",
"The cavity has notch sizes of C x C_x=28 mm and C y C_y=49 mm, under a gravitational acceleration of g=9.8g=9.8 m/s 2 /\\mathrm {s}^2.With the optimized notch dimensions, both the length variation and the mirror tilting have a value of zero at the same mounting position.",
"As can be seen in Fig.",
"REF , at the zero crossing point, the frequency variation has a very small slope against mounting position deviation in $Z$ direction, which is 13.1 kHz/g/mm, corresponding to a relative frequency instability of $4.3\\times 10^{-11}$ /g/mm at a laser wavelength of 1064 nm.",
"The slope of mirror tilting in the axial direction is about $2.3$ nrad/g/mm, corresponding to about $4.5$ kHz/g/mm, thus the relative frequency instability of $1.5\\times 10^{-11}$ /g/mm if the optical axis is displaced from the mechanical axis by $\\pm 1$ mm.",
"Comparably, the slope of frequency variation in transverse ($X$ ) direction is four times larger than that in $Z$ direction, as shown in Fig.",
"REF (b), indicating that the cavity length variation is more sensitive and critical to the accuracy of mounting positions in $X$ direction.",
"The slope of $42.7$ kHz/g/mm corresponds to a relative frequency instability of $1.4\\times 10^{-10}$ /g/mm at 1064 nm.",
"The sensitivity of mirror tilting to mounting accuracy in $X$ direction is very small ($6.1\\times 10^{-12}$ /g/mm at $\\pm 1$ mm optical axis displacement) and can be neglected."
],
[
"Conclusion",
"In conclusion, we have proposed a new ultra-stable optical reference cavity design with ULE spacer, re-entrant FS mirrors and FS rings based on extensive FEA simulations for different cavity structures.",
"The designed cavity has an ultra-low thermal noise limit at the levels of $1\\times 10^{-16}$ .",
"The zero crossing temperature of the effective CTE of the designed cavity can be easily tuned to above room temperature with any ULE spacer materials, as long as the CTE zero crossing temperature of the selected ULE spacer is measured beforehand.",
"Currently the CTE zero crossing temperature of the ULE spacer can be measured with an accuracy of $\\pm 1$ $^\\circ $ C at reasonable cost (Stable Laser Systems).",
"This accuracy level is more than enough for our design purpose.",
"No matter what the zero crossing temperature of the selected ULE spacer is measured to be, one can always design a cavity with compatible FS rings thickness such that the zero crossing temperature of the effective CTE is above room temperature.",
"The design is applicable for cavities with different lengths.",
"In addition, we have designed the reference cavity to be vibration insensitive through FEA structure optimization.",
"Ultra-low thermal noise limit reference cavities play a key role in realizing ultra-narrow linewidth lasers.",
"Reference cavities based on ULE and FS materials are still very popular choices for lab-based or portable ultra-narrow linewidth lasers.",
"To further reduce the thermal noise limit to below $1\\times 10^{-16}$ , ULE based reference cavities are no longer adequate.",
"To this end, there are currently strong interests in the developments of cryogenic cavities based on single crystal silicon or sapphire [36], [37].",
"We hope the technique discussed in this paper can also stimulate ideas in the design of these cryogenic cavities."
],
[
"Acknowledgments",
"We thank Yan-Yi Jiang and Long-sheng Ma for helpful discussions, and M. Notcutt for technical suggestions.",
"The project is partially supported by the National Basic Research Program of China (Grant No.2012CB821300), the National Natural Science Foundation of China (Grant Number 61108025 and 11174095), and Program for New Century Excellent Talents by the Ministry of Education."
]
] | 1204.1209 |
[
[
"Transverse-Momentum Dependence of the J/psi Nuclear Modification in d+Au\n Collisions at sqrt(s_NN)=200 GeV"
],
[
"Abstract We present measured J/psi production rates in d+Au collisions at sqrt(s_NN) = 200 GeV over a broad range of transverse momentum (p_T=0-14 GeV/c) and rapidity (-2.2<y<2.2).",
"We construct the nuclear-modification factor R_dAu for these kinematics and as a function of collision centrality (related to impact parameter for the R_dAu collision).",
"We find that the modification is largest for collisions with small impact parameters, and observe a suppression (R_dAu<1) for p_T<4 GeV/c at positive rapidities.",
"At negative rapidity we observe a suppression for p_T<2 GeV/c then an enhancement (R_dAu>1) for p_T>2 GeV/c.",
"The observed enhancement at negative rapidity has implications for the observed modification in heavy-ion collisions at high p_T."
],
[
"Introduction ",
"Modifications of quarkonia yields when production takes place in a nuclear target, often termed cold-nuclear-matter (CNM) effects, give insight into the production and evolution of $q\\bar{q}$ pairs.",
"A number of effects are predicted to occur in the presence of nuclear matter (for a recent review, see [1]).",
"These include nuclear breakup, modification of the parton-distribution functions, initial-state parton-energy loss and, more recently, coherent gluon saturation.",
"Measuring the production rate of quarkonia in a nuclear environment over a broad range of collision energies, and as a function of all kinematic variables, is the best way to disentangle these different mechanisms.",
"The measurement of $J/\\psi $ production rates over a broad range of rapidity ($y$ ) and transverse momentum ($p_T$) samples a wide range of parton momentum fraction ($x$ ) and energy transfer ($Q^{2}$ ), providing a simultaneous constraint on the modification of parton-distribution functions inside nuclei (nPDF's).",
"The production of $J/\\psi $ mesons, which at RHIC occurs mainly through gluon fusion, can provide critical input on the modification of the gluon distribution, which is probed only indirectly by the deep-inelastic scattering (DIS) data that forms the bulk of the current constraints on the nPDF parametrizations.",
"Measuring the $p_T$ distribution of $J/\\psi $ production allows access to $p_T$-broadening effects, which are not constrained by measurements of the rapidity dependence alone.",
"The $p_T$-broadening effects on quarkonia production at high energies are not well constrained by current data.",
"New data for $J/\\psi $ production over a broad range in $p_T$ is necessary to provide guidance for theoretical calculations.",
"The CNM effects on $J/\\psi $ production have been studied in fixed-target $p+A$ experiments at SPS, FNAL, and HERA [2], [3], [4], [5], [6], [7], [8] spanning the center of mass energy range $\\mbox{$\\sqrt{s}$} \\approx 17-42$ GeV.",
"The fixed-target results at midrapidity show greater suppression of $J/\\psi $ production at lower collision energy [6].",
"This has been interpreted [9] as an increase of the nuclear breakup of the $J/\\psi $ through collisions with nuclei.",
"At lower collision energy the crossing time of the nuclei is long enough for the $J/\\psi $ to fully form.",
"The fully formed $J/\\psi $ has an increased probability of interacting with other nucleons in the collision, which can cause the breakup of the $J/\\psi $ into heavy-meson pairs.",
"At higher collision energies it is likely that the time required for the $J/\\psi $ to fully evolve is as long, or longer than the crossing time of the collision.",
"This may result in a decrease in the probability of collisions with other nucleons, leading to less suppression of the $J/\\psi $ production.",
"The E866 [7] and HERA-B [8] experiments have measured $J/\\psi $ production as a function of $p_T$ in fixed target $p+A$ experiments.",
"Results are presented in terms of the nuclear-suppression factor, $\\alpha $ , which is obtained assuming that the cross section for $p+A$ collisions scales as $\\sigma _{pA}=\\sigma _{pN}\\times A^{\\alpha }$ , where $\\sigma _{pN}$ is the proton-nucleon cross section and $A$ is the mass number.",
"They find a $p_T$ dependence of $\\alpha $ , which is similar across a range of Feynman-$x$ ($x_F$ ) and $p_T$.",
"At $\\mbox{$p_T$} <2$ GeV/$c~$they find a suppression in the $J/\\psi $ production that transitions to an excess in the $J/\\psi $ production at higher $p_T$, which is characteristic of multiple scattering of the incident parton [7].",
"It is crucial to test these conclusions at the higher energies provided by $d+$ Au collisions at RHIC in order to better understand the $J/\\psi $ production mechanisms.",
"Measuring, and understanding, the CNM effects on quarkonia production is critical to interpreting the results for $J/\\psi $ production in nucleus-nucleus ($A+A$ ) collisions.",
"In 1986 Matsui and Satz predicted that the suppression of $J/\\psi $ production in heavy-ion collisions would be a clear signature of the formation of a quark-gluon plasma [10].",
"The Debye color screening of the dense medium produced is expected to cause the dissociation of bound states, thereby causing a decrease in the observed production.",
"Since then suppression of quarkonia production has been observed for a number of states, including the $J/\\psi $ and $\\Upsilon $ , over a wide range in collision energy [11], [12], [13], [14].",
"However, the interpretation of these results is still unclear.",
"Before the modification due to the produced medium can be determined, the CNM effects must first be corrected.",
"This has been done at lower energies [11], but accurate data on CNM effects are still absent at the higher energies of RHIC and the LHC.",
"Here we report new high-precision measurements of the $J/\\psi $ production as a function of $p_T$ and collision centrality in $d+$ Au collisions at $\\sqrt{s_{_{NN}}}$ = 200 GeV.",
"We also present measurements of the $J/\\psi $ $R_{d{\\rm Au}}$ as a function of $p_T$, rapidity, and collision centrality using data for $J/\\psi $ production in $p$ +$p$ collisions published in [15].",
"PHENIX has previously measured the $J/\\psi $ yield in $d+$ Au collisions [16], [17] with data recorded in 2003.",
"The data presented here, recorded in 2008, feature an increase in statistics of 30–50 times over those used in the previously published results, as well as a significant reduction of the systematic uncertainties.",
"The rapidity dependence of $J/\\psi $ production in $d+$ Au collisions from this data set has been previously published in [18].",
"This paper presents results for the $p_T$ dependence of the $J/\\psi $ yield from the same data set."
],
[
"Experimental Apparatus & Data Sets ",
"The PHENIX detector [19] comprises three separate spectrometers in three pseudorapidity ($\\eta $ ) ranges.",
"Two central arms at midrapidity cover $\\left|\\eta \\right|<0.35$ and have an azimuthal coverage ($\\phi $ ) of $\\pi /2$ rad each, while muon arms at backward/forward rapidity cover $-2.2<\\eta <-1.2$ (Au going direction) and $1.2<\\eta <2.4$ ($d$ going direction), with full azimuthal coverage.",
"In the central arms the $J/\\psi $ yield is measured via dielectron decays.",
"Charged particle tracks are reconstructed using the drift chamber and pad chambers.",
"Electron candidates are selected by matching charged tracks to hits in the ring imaging Čerenkov (RICH) counters and clusters in the Electromagnetic Calorimeters (EMCal).",
"In $d+$ Au collisions, a charged track is identified as an electron by requiring at least two matching RICH phototube hits within a radius of $3.4<R[cm]<8.4$ with respect to the center defined by the track projection at the RICH.",
"It is also required that the position of the EMCal cluster associated to the track projection match within $\\pm 4\\sigma $ , and that the ratio of the energy deposited in the EMCal cluster to the momentum of the tracks matches unity within $\\pm 2.5\\sigma $ , where $\\sigma $ characterizes the momentum dependent width of the matching distributions.",
"A further cut of 200 MeV/$c~$on the momentum of the electron is added to reduce the combinatorial background, since the yield of electrons from $J/\\psi $ decays observed in data and simulations is negligible below 200 MeV/$c~$.",
"At forward and backward rapidity, the $J/\\psi $ yield is measured via dimuon decays.",
"Muons are identified by matching tracks measured in cathode-strip chambers, referred to as the muon tracker (MuTr), to hits in alternating planes of Iarocci tubes and steel absorbers, referred to as the muon identifier (MuID).",
"Each muon arm is located behind a thick copper and iron absorber that is meant to stop most hadrons produced during the collisions, so that the detected muons must penetrate 8 to 11 interaction lengths of material in total.",
"Beam interactions are selected with a minimum-bias (MB) trigger requiring at least one hit in each of two beam-beam counters (BBCs) located at positive and negative pseudorapidity $3<|\\eta |<3.9$ .",
"The MB selection covers $88 \\pm 4\\%$ of the total $d+$ Au inelastic cross section of 2260 mb [20].",
"The $d+$ Au data sample used in this analysis requires the MB trigger to be in coincidence with an additional Level-1 trigger.",
"For electrons, this is a single electron EMCal RICH trigger (ERT), which requires a minimum energy deposited in any 2$\\times $ 2 group of EMCal towers, plus an associated hit in the RICH.",
"Two thresholds on the minimum EMCal energy, 600 MeV and 800 MeV, were used, each for roughly half of the data sample.",
"For muons, the level 1 trigger requires two tracks identified as muon candidates.",
"The trigger logic for a muon candidate requires a “road” of fired Iarocci tubes in at least 4 planes, including the most downstream plane relative to the collision point.",
"Additionally, collisions are required to be within $\\pm 30$ cm of the center of the interaction region.",
"Collisions in that range see the full geometric acceptance of the central arms, and this cut also provides a reduction of the systematic uncertainties on the centrality selection needed for the data from the muon arms.",
"The data sets sampled via the Level-1 triggers represent analyzed integrated luminosities of 62.7 nb$^{-1}$ (electrons) and 54.0 nb$^{-1}$ (muons) and nucleon-nucleon integrated luminosities of 24.7 pb$^{-1}$ and 21.3 pb$^{-1}$ respectively.",
"Table: Characterization of the collision centrality for d+d+Au collisions alongwith the correction factor cc (see text for details).The centrality, which is related to the impact parameter, $b$ , of the $d+$ Au collision is determined using the total charge deposited in the BBC located at negative rapidity (Au-going direction).",
"The centrality is defined as a percentage of the total charge distribution referenced to the greatest charge, $i.e.$ 0–20% refers to the 20% of the total charge distribution with the greatest charge.",
"On average the 0–20% centrality corresponds to collisions with the smallest $b$ .",
"Figure: (Color online)Nucleon-nucleon collision (N coll N_{\\rm coll}) distributions for each centrality binobtained using a Glauber model for d+d+Au collisions described in thetext.For each centrality bin the mean number of nucleon-nucleon collisions ($\\left< N_{\\rm coll}\\right>$) is determined using a Glauber calculation [21] combined with a simulation of the BBC response (as described in [18]).",
"The resulting $\\left< N_{\\rm coll}\\right>$ values for the centrality categories used in this analysis are shown in Table REF .",
"The $N_{\\rm coll}$ distributions within each centrality bin are shown in Fig.",
"REF .",
"There is a significant overlap between the $N_{\\rm coll}$ distributions for different centralities.",
"Also shown in Table REF is the correction factor $c$ , which accounts for the correlation between the detection of a $J/\\psi $ in the final state and an increase in the total charge collected in the BBC [16].",
"This correlation affects both the MB-trigger efficiency and the determination of the centrality of a given collision.",
"The correction factors for each centrality bin are obtained within the same Glauber framework as the $\\left< N_{\\rm coll}\\right>$ values by assuming that one of the N binary collisions produces a charge in the BBC that is characteristic of a hard-scattering process (the remaining N-1 binary collisions maintain a BBC charge distribution characteristic of soft scattering processes).",
"The increase in the BBC charge from a hard process is tuned using real data.",
"Since both $c$ and $\\left< N_{\\rm coll}\\right>$ are calculated in the same Glauber framework there are correlations between their uncertainties.",
"These correlations are removed in the ratio of $c/\\mbox{$\\left< N_{\\rm coll}\\right>$} $ , which occurs in the calculation of $R_{d{\\rm Au}}$.",
"The resulting values and uncertainties are given in the third column of Table REF .",
"The correction factor for 0–100% centrality contains an additional factor to extrapolate the measured yield, which covers only 88% of all $d+$ Au collisions, to 100% of the $d+$ Au inelastic cross section, essentially correcting for the efficiency of the BBC trigger.",
"This correction is again determined within the Glauber framework using the parametrization of the BBC trigger efficiency."
],
[
"The procedure for analyzing the $\\mbox{$J/\\psi $} \\rightarrow e^+e^-$ signal and the results in the central arms are discussed in this section.",
"The extraction of the correlated $e^+e^-$ yield is discussed in Sec.",
"REF .",
"The estimation of the correlated background and losses due to the radiative tail in the $J/\\psi $ mass distribution is discussed in Sec.",
"REF .",
"The estimation of the detector efficiencies is described in Sec.",
"REF .",
"The calculation of the $J/\\psi $ invariant yield is detailed in Sec.",
"REF .",
"The $p$ +$p$ baseline used in calculating $R_{d{\\rm Au}}$ is described in Sec.",
"REF ."
],
[
"Correlated $e^+e^-$ Signal extraction ",
"The $\\mbox{$J/\\psi $} \\rightarrow e^+e^-$ yield is measured using the invariant mass spectrum for all dielectron pairs where at least one of the electrons fired the ERT trigger.",
"This selection is necessary to match the conditions under which the $J/\\psi $ trigger efficiency is calculated (see Sec.",
"REF ).",
"An example of the dielectron mass spectrum is shown in Fig.",
"REF for 0–20% central collisions.",
"In a given bin of $p_T$, rapidity, and collision centrality, the correlated $e^+e^-$ yield ($N_{e^+e^-}$ ) is determined by counting over a fixed mass window of $2.8 < M_{ee}\\,[\\mbox{GeV/$c^{2}~$}] < 3.3$ the number of unlike-sign dielectrons, after the subtraction of the like-sign dielectrons, which arise by random association and so are representative of the combinatorial background within the unlike sign dielectron distribution.",
"This method assumes that the acceptance is the same for $e^-$ and $e^+$ , which, while untrue at lower masses, is a good assumption in the $J/\\psi $ mass range.",
"At higher $p_T$, where statistical precision is limited, the yield, along with the statistical uncertainties, are derived from Poisson statistics.",
"Assuming both the unlike-sign (foreground) and like-sign (background) are independent, and assuming no negative signal, the combined distribution $ P(s) =\\sum _{k=0}^{fg}\\frac{(bg+fg-k)!}{bg!(fg-k)!}\\frac{1}{2}\\left(\\frac{1}{2}\\right)^{bg+fg-k}\\frac{s^ke^{-s}}{k!",
"}, $ represents the probability of a signal ($s$ ) given a number of unlike-sign dielectrons ($fg$ ) and a number of like-sign dielectrons ($bg$ ) (see [15] for derivation).",
"The mean and standard deviation of Eq.",
"REF are then used as the yield and uncertainties.",
"A correlated $e^{+}e^{-}$ yield in the mass window $2.8 < M_{ee}\\,[\\mbox{GeV/$c^{2}~$}] < 3.3$ of approximately 8600 is obtained across all $p_T$ and collision centralities.",
"Figure: (Color online)(top) Invariant mass distribution of unlike-sign (filled circles) andlike-sign (filled boxes) dielectron pairs in central d+d+Au collisions,integrated over p T p_T and rapidity.",
"Dashed vertical lines represent themass range used to determine the correlated e + e - e^+e^- yield.",
"(bottom)Correlated dielectron invariant mass distribution for MB d+d+Au collisions.",
"The line shapes are those used to extract the continuum andradiative tail contributions to the correlated e + e - e^+e^- yield in the massrange 2.8<M ee [GeV/c 2 ]<3.32.8 < M_{ee}\\,[\\mbox{GeV/$c^{2}~$}] < 3.3 ."
],
[
"Estimation of the Correlated Background and Losses Due to the\nRadiative Tail in the ",
"When using the like-sign subtraction method there remains a correlated background under the observed $J/\\psi $ peak.",
"This background comes mainly from open-heavy-flavor decays and Drell-Yan pairs, and must be separated from the $J/\\psi $ signal of interest.",
"Counting the dielectron signal only over a fixed mass window also causes an underestimate of the $J/\\psi $ yield due to the fraction of the $J/\\psi $ line shape that falls outside the mass window of choice.",
"These two effects are quantified by using simulated particle line shapes fitted to the real data distribution.",
"The $J/\\psi $ and $\\psi ^{\\prime }$ mesons with uniform distributions in $p_T$ ($0<\\mbox{$p_T$} \\,[\\mbox{GeV/$c~$}]<12$ ) and rapidity ($\\left|y\\right|<0.35$) are decayed to $e^+e^-$ and the external radiation effects are evaluated using a geant-3 based model of the PHENIX detector (described in Sec.",
"REF ).",
"While a uniform distribution in $p_T$ is unrealistic, the $J/\\psi $ rapidity distribution is roughly constant within $\\left|y\\right|<0.35$.",
"When used here, the $J/\\psi $ and $\\psi ^{\\prime }$ line shapes will be compared to $p_T$ integrated data as a function of invariant mass only, with a mass resolution fitted to the data, and therefore the effect of using a uniform $p_T$ distribution is negligible.",
"The line shape for $J/\\psi $ radiative decays ($\\mbox{$J/\\psi $} \\rightarrow e^+e^-\\gamma $ ), also called internal radiation, is based on calculations of the mass distribution from QED [22] convoluted with the detector resolution.",
"Line shapes for the correlated background from heavy-flavor decays along with Drell-Yan pairs are simulated using pythia [23].",
"The correlated background from heavy-flavor decays comes from semi-leptonic decays of correlated $D\\bar{D}$ and $B\\bar{B}$ (i.e.",
"$D\\rightarrow e^{+}+X$ and $\\bar{D}\\rightarrow e^-+X$ ).",
"The decay electrons from pythia are then run through the same geant simulation of the PHENIX detector to evaluate the external radiation effects.",
"These line shapes are generated assuming $p+p$ collisions, and no corrections for CNM effects (i.e.",
"application of nPDF modifications) are applied to the distributions.",
"We assume that the CNM effects on these distributions are likely small and roughly constant over the narrow mass window used due to the $x$ values probed.",
"No suppression of heavy-flavor production has been observed in $d+$ Au collisions, and we assume that any suppression, if it exists, does not significantly effect the overall line shapes.",
"The line shapes are then fitted to the $p_T$ and collision centrality integrated invariant mass spectrum over the mass range $2<M_{ee}\\,[\\mbox{GeV/$c^{2}~$}]<8$ where the normalizations on the $J/\\psi $, $\\psi ^{\\prime }$, correlated heavy flavor, and DY are free to vary independently.",
"The best fit is shown in the quarkonium mass region in Fig.",
"REF , where the continuum line shape is the combination of correlated $e^+e^-$ pairs from $D\\bar{D}$ , $B\\bar{B}$ , and DY decays, and the $J/\\psi $ and $\\psi ^{\\prime }$ line shapes are the combinations of the line shapes from both internal and external radiation effects.",
"Within the mass window $2.8<M_{ee}\\,[\\mbox{GeV/$c^{2}~$}]<3.3$ the correlated continuum contribution ($\\epsilon _{\\rm cont}$ ) is found to be 6.6 $\\pm $ 0.2% and the fraction of the $J/\\psi $ line shape contained within the mass window ($\\epsilon _{\\rm rad}$ ) to be 94.3 $\\pm $ 0.2%, where the uncertainties are derived from the uncertainty in the fit.",
"The disagreement between the fit and the data in the $3.7<M_{ee}\\,[\\mbox{GeV/$c^{2}~$}]<4.5$ mass range is likely due to the inability of the $D\\bar{D}$ and $B\\bar{B}$ line shapes to match the shape of the data at higher mass.",
"However, large changes in the ratio of their contributions have only a small effect on the extracted values of $\\epsilon _{\\rm cont}$ and $\\epsilon _{\\rm rad}$ , and this is accounted for in the quoted uncertainties."
],
[
"Acceptance and Efficiency Studies ",
"The $J/\\psi $ acceptance is investigated using a geant-3 [24] based Monte Carlo model of the PHENIX detector.",
"Dead and malfunctioning channels in the detector are removed from both the detector simulation and real data.",
"The accuracy of the simulations is tested by comparing simulated single electron distributions with those from real data.",
"The agreement across the detector and data taking period is determined to be within 3.2%.",
"A conservative estimate, which assumes that the uncertainty is correlated for both electrons in a pair, of $2\\times 3.2\\%=6.4\\%$ is assigned as a systematic uncertainty on the $J/\\psi $ acceptance based on the quality of the matching between simulations and data.",
"To determine the $J/\\psi $ acceptance, $\\mbox{$J/\\psi $} \\rightarrow e^+e^-$ decays are simulated with uniform distributions in $p_T$, rapidity ($|y|<0.5$ ) and collision vertex.",
"While distributions uniform in $p_T$ are not realistic, the corrections are made over a small $p_T$ bin where the real distribution can be approximated as linear.",
"This assumption, and the effect of bin sharing, is tested later and taken into account when assigning systematic uncertainties.",
"The fraction of $J/\\psi $ decays that are reconstructed corresponds to the combination of the geometric acceptance and the electron ID efficiency ($A\\times \\epsilon _{\\rm eID}$ ).",
"The resulting $A\\times \\epsilon _{\\rm eID}$ is shown as a function of $p_T$ in Fig.",
"REF .",
"It has an average value of 1.5% in 1 unit of rapidity.",
"The dip in $A\\times \\epsilon _{eID}$ followed by a continual increase with $p_T$ marks the transition from the $e^+e^-$ pair at low $p_T$ being produced back to back and being detected one in each of the PHENIX central arms, to the pair at high $p_T$ being produced in a collinear manner and being detected both in the same PHENIX central arm.",
"The low point at $\\mbox{$p_T$} \\approx 3$ GeV/$c~$corresponds to the $e^+e^-$ being produced at roughly 90$^{\\rm o}$ relative to each other in the lab frame, which due to the PHENIX geometry has the lowest probability for detection.",
"The electron ID efficiency, which is mainly due to track reconstruction cuts used to avoid the misidentification of hadrons as electrons, was cross checked using electrons from $\\pi ^0$ Dalitz decays and $\\gamma $ conversions as described in [15], and a systematic uncertainty of 1.1% is assigned based on that comparison.",
"The effect of momentum smearing on the electrons in simulations, which can cause a $J/\\psi $ to be reconstructed into a different $p_T$ bin than the one it was generated in, was also investigated.",
"The effect was found to be minimal for all but the highest $p_T$ bins and an uncertainty of 0.2% was assigned based on a Monte-Carlo study effect and a parametrization of the measured momentum resolution for electrons.",
"A combined uncertainty of 6.5% is assigned to the $J/\\psi $ $A\\times \\epsilon _{eID}$ by adding the simulation/data matching, eID, and momentum smearing uncertainties in quadrature.",
"The ERT trigger efficiency is evaluated using simulations of $J/\\psi $ decays and parametrizations of the single electron trigger efficiencies in each trigger tile.",
"A MB data sample of single electrons is used to measure the $p_T$ dependent efficiency of each 2x2 EMCal trigger tile and each RICH trigger tile independently by calculating the fraction of electrons that fired the trigger tile compared to all those passing through it.",
"The resulting distributions are then fitted with an error(uniform) function for each trigger tile in the EMCal(RICH).",
"These functions are then used with simulated $J/\\psi $ decays to estimate the efficiency of the ERT trigger for triggering on $e^+e^-$ pairs from $J/\\psi $ decays ($\\epsilon _{\\rm ERT}^{J/\\psi }$ ).",
"The trigger efficiency is evaluated only for simulated $J/\\psi $ decays for which both electrons passed an acceptance and trigger check in order to avoid double counting the acceptance efficiency.",
"This procedure is repeated independently for each of the two EMCal trigger thresholds used during the run.",
"The $p_T$ dependence of $\\epsilon _{\\rm ERT}^{J/\\psi }$ is shown in Fig.",
"REF , where both ERT trigger thresholds have been combined using the relative luminosities of each data sample.",
"It has an average value of 77%.",
"The dip seen at $\\mbox{$p_T$} \\approx 3$ GeV/$c~$is due to the kinematics of the $J/\\psi $ decays.",
"In that $p_T$ range there is a high probability for the decay electrons to have unbalanced momenta, where one of the electrons will have a momentum below or near the trigger threshold, resulting in a lower probability for triggering on the $J/\\psi $.",
"The effect of the fit function used in the EMCal trigger tile efficiencies is investigated by replacing the error function with a double-Fermi function.",
"This gives an average change in the $J/\\psi $ ERT efficiency of 0.31%.",
"The statistical uncertainty in the trigger tile efficiency leads to an uncertainty in the $J/\\psi $ ERT efficiency of 1.6%.",
"Summing these uncertainties in quadrature gives a total uncertainty on $\\epsilon _{\\rm ERT}^{J/\\psi }$ of 1.6%, which is heavily dominated by the uncertainty in the efficiency of each ERT trigger tile.",
"The detector occupancy effect is negligible, even in 0–20% central $d+$ Au collisions (a finding consistent with previous embedding studies in peripheral Cu+Cu [25] with similar multiplicities).",
"A 1% systematic uncertainty was assigned based on studies where simulated $J/\\psi $ decays were embedded into real events.",
"This result agrees well with the studies done in [16], where a slightly larger systematic uncertainty was assigned because of the lower statistical precision of the simulations used.",
"Figure: The J/ψJ/\\psi acceptance ×\\times electron ID efficiency (a) and J/ψJ/\\psi ERTtrigger efficiency (b) as a function of p T p_T for y<0.35\\left|y\\right|<0.35 where the shaded boxes represent the systematic uncertainties."
],
[
"Invariant Yield Results ",
"The $J/\\psi $ invariant yield in a given rapidity, transverse-momentum, and centrality bin is $\\frac{B_{ll}}{2\\pi \\mbox{$p_T$}} \\frac{d^{2}N}{dyd\\mbox{$p_T$}}= \\frac{1}{2\\pi \\mbox{$p_T$} \\Delta \\mbox{$p_T$} \\Delta {y}}\\frac{c N_{J/\\psi }}{N_{\\rm EVT}\\epsilon _{\\rm tot}},$ where $B_{ll}$ is the $J/\\psi $ $\\rightarrow l^+l^-$ branching ratio, $N_{J/\\psi }$ is the measured $J/\\psi $ yield, $N_{\\rm EVT}$ is the number of sampled MB events in the given centrality bin, $\\Delta {y}$ is the width of the rapidity bin, $\\Delta {\\mbox{$p_T$}}$ is the width of the $p_T$ bin, $\\epsilon _{\\rm tot}=A\\times \\epsilon _{\\rm eID}\\ \\epsilon _{\\rm ERT}^{J/\\psi }\\ \\epsilon _{\\rm rad}$ and $c$ is the BBC bias correction factor described in Sec. .",
"At midrapidity $N_{J/\\psi }=N_{e^+e^-}(1-\\epsilon _{\\rm cont})$ , where $\\epsilon _{\\rm cont}$ is the correlated dielectron continuum contribution in the $J/\\psi $ mass range.",
"The 0–100% centrality integrated $J/\\psi $ invariant yield is shown as a function of $p_T$ in Fig.",
"REF , and for four centrality bins in Fig.",
"REF .",
"Here the values shown represent the average over the $p_T$ bin and are plotted at the center of the bin, as this provides the measured information without introducing further systematic uncertainties.",
"Table: A summary of the systematic uncertainties at y<0.35\\left|y\\right|<0.35.A summary of all the relevant systematic uncertainties at midrapidity is shown in Table REF , along with their classification into Type A, B, or C uncertainties.",
"Type A represents uncertainties that are uncorrelated from point to point, Type B represents uncertainties that are correlated from point to point, and Type C represents uncertainties in the overall normalization.",
"Figure: (Color Online)J/ψJ/\\psi invariant yield as a function of p T p_T for pp+pp and0–100% centrality integrated d+d+Au collisions.",
"The type C systematicuncertainty for each distribution is given as a percentage in the legend.The midrapidity d+d+Au and pp+pp results are discussed inSecs.",
"& while the forward/backwardrapidity results are discussed inSecs.",
"& .Figure: (Color Online)J/ψJ/\\psi invariant yield as a function of p T p_T for eachcentrality at y<0.35\\left|y\\right|<0.35.",
"The type C systematic uncertainty for eachdistribution is given as a percentage in the legend."
],
[
"The $p$ +$p$ baseline used to calculate $R_{d{\\rm Au}}$ is extracted from 2006 data published in [15].",
"The integrated luminosity was 6.2$\\pm $ 0.6 pb$^{-1}$ .",
"In the analysis, described in detail in [15], the effect of the $J/\\psi $ polarization on the $J/\\psi $ acceptance is included.",
"This effect is not included in the $d+$ Au result presented here due to a lack of knowledge of the effects of a nuclear target on the $J/\\psi $ polarization.",
"The $J/\\psi $ polarization is therefore assumed to be zero.",
"To remain consistent, this effect is removed from the $p$ +$p$ baseline as well, so that, assuming the polarization does not change drastically between $p$ +$p$ and $d+$ Au, the effects will cancel in the nuclear ratio, $R_{d{\\rm Au}}$.",
"The $p$ +$p$ invariant yields as a function of $p_T$ used in this work, shown in Fig.",
"REF , have been converted from the invariant cross sections published in [15] using an inelastic cross section of 42 mb."
],
[
"The procedure for analyzing the $\\mbox{$J/\\psi $} \\rightarrow \\mu ^+\\mu ^-$ signal at backward and forward rapidity in the muon arms is discussed in this section.",
"The procedures are similar to those detailed in [15], with only a brief summary presented here, except where there are differences.",
"As in [15], the rapidity region of the forward muon arm used in the analysis was truncated to $1.2<y<2.2$ to match the rapidity coverage of the backward muon arm.",
"The extraction of the raw $J/\\psi $ yield is discussed in Sec.",
"REF .",
"The estimation of the detector efficiencies is described in Sec.",
"REF .",
"The calculation of the $J/\\psi $ invariant yield is detailed in Sec.",
"REF .",
"The $p$ +$p$ baseline used in calculating $R_{d{\\rm Au}}$ is described in Sec.REF ."
],
[
"$\\mbox{$J/\\psi $} \\rightarrow \\mu ^+\\mu ^-$ Signal extraction",
"At forward and backward rapidity, the invariant mass distribution is calculated for all unlike-sign dimuons in events that pass the trigger requirements described in Sec. .",
"The combinatorial background is estimated from the invariant mass distribution formed by pairing unlike-sign muon candidates from different events.",
"This is done to reduce the background statistical uncertainty below what is possible by subtracting like sign pairs from the same event, and is needed because the signal to background present at forward/backward rapidity is smaller than at midrapidity.",
"The mixed event muon pairs are required to have vertices that differ by no more than 3 cm in the beam direction.",
"The mixed event spectrum is normalized by the factor $\\alpha =\\frac{\\sqrt{(N_{\\mu ^+\\mu ^+}^{\\rm same})(N_{\\mu ^-\\mu ^-}^{\\rm same})}}{\\sqrt{(N_{\\mu ^+\\mu ^+}^{mixed})(N_{\\mu ^-\\mu ^-}^{mixed})}},$ where $N_{\\mu \\mu }^{\\rm same}$ and $N_{\\mu \\mu }^{mixed}$ are the number of pairs formed from two muons in the same or in mixed events, respectively.",
"The remaining correlated dimuon mass distribution after the subtraction of the mixed event combinatorial background contains dimuons from $J/\\psi $ and $\\psi ^{\\prime }$ decays, as well as correlated dimuons from heavy-flavor decays and Drell-Yan pairs.",
"Due to the momentum resolution of the detector, there is no clean discrimination between the $J/\\psi $ and $\\psi ^{\\prime }$ in the mass distribution.",
"However the $\\psi ^{\\prime }$ contribution is expected to be negligible in the mass window of interest.",
"A function consisting of an exponential component combined with two Gaussian distributions, which are used to better reproduce the mass resolution present in the muon arms, was used to fit the dimuon mass distribution, convolved with a function to account for the variation in acceptance over the invariant mass range.",
"An example of the fitted mass distribution is shown in Fig.",
"REF .",
"Both the $J/\\psi $ component of the fit, and direct counting after the subtraction of the fitted exponential background, are used to evaluate the $J/\\psi $ yield.",
"The difference between the two methods is taken as a Type A systematic uncertainty.",
"This uncertainty is typically small ($\\approx 2$ %) but can be significantly larger at high $p_T$ where there are fewer counts.",
"Measured $J/\\psi $ yields of approximately 38000 and 42000 are obtained at backward and forward rapidity, respectively.",
"Figure: (Color Online)(top) Invariant mass distribution of unlike-sign (filled circles) andlike-sign (filled boxes) dimuon pairs for 2<p T <2.252<\\mbox{$p_T$} <2.25 GeV/cc~at forwardrapidity and 60–88% central events.",
"(bottom) Invariant mass distributionof correlated dimuon pairs after the subtraction of the combinatorialbackground.",
"The solid line represents the fit to the invariant massdistribution, which includes the double Gaussian signal component(dot-dashed line) and exponential background (dotted line)."
],
[
"Acceptance and Efficiency Studies ",
"Studies of the response of the muon arm spectrometers to dimuons from $J/\\psi $ decays are performed using a tuned geant3-based simulation of the muon arms, coupled with a MuID trigger emulator.",
"The MuID panel-by-panel efficiencies are estimated using the fraction of reconstructed roads in real data.",
"Where statistics are limited, the operational history of each channel recorded during the run was used to estimate the efficiency.",
"A systematic uncertainty of 4% is assigned to the MUID efficiency based on this comparison.",
"Charge distributions in each part of the MuTr observed in real data, along with dead channels and their variation with time over the run, are used to give an accurate description of the MuTr efficiency within the detector simulation.",
"The $J/\\psi $ acceptance $\\times $ efficiency ($A\\times \\epsilon $ ) evaluation uses a pythia simulation with several parton distributions as input to account for the unknown underlying rapidity dependence of the $J/\\psi $ yield.",
"A 4% systematic uncertainty is assigned based on changes in the input parton distributions.",
"A systematic uncertainty of 6.4(7)% on the $J/\\psi $ yield is assigned to the backward(forward) rapidity due to the uncertainties in the acceptance x efficiency determination method itself."
],
[
"Invariant Yield Results ",
"The $J/\\psi $ invariant yield at backward/forward rapidity is calculated using Eq.",
"REF , where $\\epsilon _{\\rm tot}=A\\times \\epsilon $ .",
"A summary of the systematic uncertainties is given in Table REF .",
"The backward and forward 0–100% centrality-integrated $J/\\psi $ invariant yields are shown as a function of $p_T$ in Fig.",
"REF , while the $J/\\psi $ invariant yields are shown as a function of $p_T$ in each centrality bin in Fig.",
"REF .",
"Figure: (Color Online)J/ψJ/\\psi invariant yield as a function of p T p_T for eachcentrality for a) -2.2<y<-1.2-2.2<y<-1.2 and b) 1.2<y<2.21.2<y<2.2.",
"The type C systematic uncertaintyfor each distribution is given as a percentage in the legend."
],
[
"The $p$ +$p$ baseline used to calculate $R_{d{\\rm Au}}$ is extracted from a combined analysis of data taken in 2006 and 2008, published in [15].",
"The combined integrated luminosity was 9.3$\\pm $ 0.9 pb$^{-1}$ .",
"As discussed in Sec.",
"REF , the effect of the $J/\\psi $ polarization on the $J/\\psi $ acceptance is removed from the results used here.",
"The $J/\\psi $ invariant yield in $p$ +$p$ collisions at forward/backward rapidity used here is shown as a function of $p_T$ in Fig.",
"REF for convenience, where we have converted from the invariant cross sections published in [15] using an inelastic cross section of 42 mb."
],
[
"Calculation of ",
"The $\\langle p_T^2 \\rangle $ is calculated for each of the $J/\\psi $ invariant yields presented in Secs.",
"REF and REF , and the resulting values are shown in Table REF .",
"Table: 〈p T 2 〉\\langle p_T^2 \\rangle results for pp+pp and d+d+Au collisions where the first quoteduncertainty corresponds to the type A uncertainties and the secondcorresponds to the type B uncertainties.Unlike in previous analyses [16], where the $\\langle p_T^2 \\rangle $ was calculated for $\\mbox{$p_T$} \\le 5$ GeV/$c~$due to statistical limitations at high $p_T$, here we have calculated the $\\langle p_T^2 \\rangle $ over the full $p_T$ range.",
"First the $\\langle p_T^2 \\rangle $ was calculated numerically up to the $p_T$ limits of the measured distribution $\\left(\\mbox{$\\langle p_T^2 \\rangle |_{p_T\\le p_T^{\\rm max}}$} \\right)$ .",
"The correlated uncertainty was propagated to $\\langle p_T^2 \\rangle |_{p_T\\le p_T^{\\rm max}}$ by sampling the type B uncertainty distributions of the first and last $p_T$ point of the invariant yield, and assuming a linear correlation in between.",
"For a more detailed description of this procedure see Appendix REF .",
"To account for the differences in the $p_T$ limits of the various distributions, the $\\langle p_T^2 \\rangle |_{p_T\\le p_T^{\\rm max}}$ value was corrected to the $p_T$ range from zero to infinity.",
"This was done by fitting the distribution with a modified Kaplan function of the form $f(\\mbox{$p_T$})=p_0\\left(1-\\left(\\frac{\\mbox{$p_T$}}{p_1}\\right)^2\\right)^{p_2}$ where each parameter was free to vary.",
"The ratio $k=\\frac{\\mbox{$\\langle p_T^2 \\rangle $} [0,\\infty ]}{\\mbox{$\\langle p_T^2 \\rangle $} [0,\\mbox{$p_T$} ^{\\rm max}]}$ was then calculated from the fit and applied to the numerically calculated $\\langle p_T^2 \\rangle |_{p_T\\le p_T^{\\rm max}}$.",
"In all cases the correction factor was small ($k<1.03$ ), and an uncertainty in the correction factor based on the fit uncertainty is included in the Type B uncertainties shown in Table REF .",
"For a more detailed description of this procedure, including the fit results and the calculated values of $k$ see Appendix .",
"The $\\langle p_T^2 \\rangle $ for $p$ +$p$ collisions was previously published in [15].",
"But we report the result here with the effect of the $J/\\psi $ polarization on the acceptance removed.",
"The results are in good agreement with those presented in [15], and are shown in Table REF .",
"Figure REF shows $\\Delta \\mbox{$\\langle p_T^2 \\rangle $} =\\mbox{$\\langle p_T^2 \\rangle $} _{dAu}-\\mbox{$\\langle p_T^2 \\rangle $} _{pp}$ as a function of $N_{\\rm coll}$.",
"There is a broadening in the $p_T$ distribution with respect to $p$ +$p$, which increases with $N_{\\rm coll}$, and is similar at forward and backward rapidities.",
"We observe a larger increase in the $p_T$ broadening at midrapidity.",
"However, this observation is tempered by the relatively large uncertainties present in the data.",
"Figure: (Color Online)The difference between the J/ψJ/\\psi 〈p T 2 〉\\langle p_T^2 \\rangle in d+d+Au and pp+pp collisions as afunction of N coll N_{\\rm coll} in d+d+Au collisions.",
"The boxes drawn atΔ〈p T 2 〉=0\\Delta \\mbox{$\\langle p_T^2 \\rangle $} =0 represent the combined statistical and systematicuncertainties from the pp+pp calculation."
],
[
"The ",
"To quantify the $d+$ Au cold nuclear matter effects, the $J/\\psi $ $R_{d{\\rm Au}}$ is calculated for a given $p_T$, $y$ , and centrality bin as: $\\mbox{$R_{d{\\rm Au}}$} (i) = \\frac{c}{\\mbox{$\\left< N_{\\rm coll}(i)\\right>$}}\\frac{d^2N_{J/\\psi }^{d+\\rm Au}(i)/dyd\\mbox{$p_T$}}{d^2N_{J/\\psi }^{p+p}/dyd\\mbox{$p_T$}},$ where $d^2N_{J/\\psi }^{d+\\rm Au}(i)/dyd\\mbox{$p_T$} $ is the $d+$ Au invariant yield for the $i^{\\rm th}$ centrality bin, $d^2N_{J/\\psi }^{p+p}/dyd\\mbox{$p_T$} $ is the $p$ +$p$ invariant yield for the same $p_T$ and $y$ bin, and $\\left< N_{\\rm coll}(i)\\right>$ is the average number of binary collisions for the given centrality bin, as listed in Table REF .",
"The 0–100% centrality integrated $J/\\psi $ $R_{d{\\rm Au}}$ as a function of $p_T$ is shown in Fig.",
"REF for each of the three rapidity regions.",
"The numerical values can be found in Table REF , REF , and REF for backward, mid and forward rapidity, respectively.",
"Figure REF shows a different behavior for $R_{d{\\rm Au}}$ at backward ($-2.2<y<-1.2$) as opposed to mid ($\\left|y\\right|<0.35$) and forward ($1.2<y<2.2$) rapidities.",
"At backward rapidity, the $R_{d{\\rm Au}}$ is suppressed only at the lowest $p_T$, with a rapid increase to $\\mbox{$R_{d{\\rm Au}}$} =1.0$ at $\\mbox{$p_T$} \\approx 1.5$ GeV/$c~$.",
"The mid and forward rapidity data, on the other hand, exhibit a similar level of suppression at the lowest $p_T$, but a much more gradual increase in $R_{d{\\rm Au}}$ with $p_T$, increasing to $\\mbox{$R_{d{\\rm Au}}$} =1.0$ only at $\\mbox{$p_T$} \\approx 4.0$ GeV/$c~$.",
"Figure REF shows the same 0–100% $R_{d{\\rm Au}}$ vs $p_T$ at all rapidities overlaid.",
"It is striking that the shape and absolute scale for the mid and forward rapidity data is nearly consistent across the entire $p_T$ range of the data.",
"Figure: (Color Online)J/ψJ/\\psi R d Au R_{d{\\rm Au}}, as a function of p T p_T for 0–100% centrality integrated d+d+Au collisions at eachrapidity.",
"The Type C systematic uncertainty for each distribution isgiven as a percentage in the legend.Due to the statistical limitations of the data at high $p_T$, it is unclear from Fig.",
"REF whether the $R_{d{\\rm Au}}$ increases significantly above one.",
"To investigate the high-$p_T$ behavior of the $R_{d{\\rm Au}}$ at each rapidity, the average $R_{d{\\rm Au}}$ was calculated for $\\mbox{$p_T$} >4$ GeV/$c~$by fitting each distribution with a constant.",
"The results are shown in Table REF along with the fit uncertainties, which take into account only the type A uncertainties on the data.",
"Since the type B uncertainties are roughly consistent in the fit range, we have chosen here to add the average type B uncertainty for $\\mbox{$p_T$} >4$ GeV/$c~$in quadrature with the type C uncertainty.",
"We find that at mid and forward rapidity the average $R_{d{\\rm Au}}$ for $\\mbox{$p_T$} >4$ GeV/$c~$is consistent with 1.0, while at backward rapidity the average $R_{d{\\rm Au}}$ is greater than 1.0.",
"The production of a $J/\\psi $ at forward rapidity in $A+A$ collisions involves a low-$x$ gluon colliding with a high-$x$ gluon.",
"The symmetry due to identical colliding nuclei results, essentially, in the folding of the forward and backward rapidity $R_{d{\\rm Au}}$.",
"The production of a $J/\\psi $ at midrapidity results, essentially, in the folding of the midrapidity $R_{d{\\rm Au}}$ with itself.",
"This picture is simplistic and leaves out many details, but it gives some expectation for the result of the modification of $J/\\psi $ production in $A+A$ collisions due to CNM effects.",
"If we extrapolate the observed behavior of $R_{d{\\rm Au}}$ to the modification of $J/\\psi $'s produced at forward rapidity in $A+A$ collisions, we would expect a $R_{\\rm AA}$ contribution from CNM effects to be similar to, or greater than, 1.0 at high $p_T$ and a modification similar to 1.0 at midrapidity.",
"The observation at midrapidity of a $J/\\psi $ $R_{\\rm AA}$ in Cu+Cu collisions that exceeds, but is consistent with, 1.0 at high $p_T$ [26] may therefore be largely accounted for by the contribution from CNM effects.",
"Further work is needed to understand the detailed propagation of measured results in $d+$ Au collisions to an expected CNM contribution in $A+A$ collisions before this can be fully understood.",
"Table: The average 0–100% R d Au R_{d{\\rm Au}} for p T >4\\mbox{$p_T$} >4 GeV/cc~where the first quoteduncertainty corresponds to the fit uncertainty and the second correspondsto the combined type B and C systematic uncertainties.Figures REF , REF , and REF show $R_{d{\\rm Au}}$ vs $p_T$ in four centrality bins for backward rapidity, midrapidity, and forward rapidity, respectively.",
"Numerical values can be found in Tables REF , REF , REF and REF for 0–20%, 20–40%, 40–60% and 60–88% central collisions respectively.",
"For peripheral collisions the $R_{d{\\rm Au}}$ remains consistent with 1.0 within statistical and systematic uncertainties across all $p_T$ in all rapidity regions.",
"Figure: (Color Online)J/ψ→μ + μ - J/\\psi \\rightarrow \\mu ^+\\mu ^- R d Au R_{d{\\rm Au}}, as afunction of p T p_T for a) central, b) midcentral, c)midperipheral, and d) peripheral events at -2.2<y<-1.2-2.2<y<-1.2.",
"The 60–88% R d Au R_{d{\\rm Au}} point at p T =5.75\\mbox{$p_T$} =5.75 GeV/cc~has been left off the plot, as it is above theplotted range and has very large uncertainties, however it is included inTable .",
"Curves are calculations by Lansberg etal.",
"discussed in the text.Figure: (Color Online)J/ψ→e + e - J/\\psi \\rightarrow e^+e^- R d Au R_{d{\\rm Au}}, as afunction of p T p_T for a) central, b) midcentral, c)midperipheral, and d) peripheral events at y<0.35\\left|y\\right|<0.35.",
"Curves are calculationsby Lansberg et al.",
"discussed in the text.Figure: (Color Online)J/ψ→μ + μ - J/\\psi \\rightarrow \\mu ^+\\mu ^- R d Au R_{d{\\rm Au}}, as afunction of p T p_T for a) central, b) midcentral, c)midperipheral, and d) peripheral events at 1.2<y<2.21.2<y<2.2.",
"Curves are calculationsby Lansberg et al.",
"discussed in the text."
],
[
"Comparison with Model Predictions ",
"As mentioned previously, various models have been suggested to describe the cold nuclear matter effects on $J/\\psi $ production.",
"The models that will be discussed here include a combination of physical effects such as shadowing, nuclear breakup, and the Cronin effect.",
"Shadowing, the modification of the parton distributions within a nucleus, is calculated using parametrizations of deep inelastic scattering data in the form of nuclear modified parton distribution functions (nPDF's).",
"There are a number of nPDF sets available, including nDSg [28], EKS98 [29] and EPS09 [30], which provide distributions of this modification based on different parametrizations of the available data.",
"For $J/\\psi $ production in $d+$ Au collisions the relevant distributions are those providing the modification of the gluon distribution within a Au nucleus, as $J/\\psi $'s are produced primarily through gluon fusion at $\\sqrt{s_{_{NN}}}$ =200 GeV.",
"The nPDF's provide modifications as a function of parton momentum fraction ($x$ ) and energy transfer ($Q^2$ ).",
"Knowledge of the $J/\\psi $ production kinematics is then needed to produce a modification to $J/\\psi $ production in $d+$ Au collisions.",
"For $J/\\psi $ production at backward rapidity and $0<\\mbox{$p_T$} <8$ GeV/$c~$, a range of roughly $0.051<x<0.39$ in the Au nucleus is probed, assuming simple 2$\\rightarrow $ 1 kinematics.",
"While 2$\\rightarrow $ 1 kinematics are inadequate to describe the production of a $J/\\psi $ with nonzero $p_T$, they are used here to provide a simple estimation of the $x$ and $Q^{2}$ ranges covered.",
"Likewise, midrapidity covers roughly $0.0094<x<0.071$ and forward rapidity covers roughly $0.0017<x<0.013$ .",
"A range of roughly $10<Q^2[{\\rm GeV}^2/c^2]<74$ is probed at each rapidity under the same assumptions.",
"The data thus provide a strong constraint to shadowing models over a wide range of $x$ and $Q^2$ .",
"Nuclear breakup, the dissociation of $c\\bar{c}$ pairs that would have formed $J/\\psi $'s through collisions with nucleons, is often parametrized through a breakup cross section.",
"Little theoretical or experimental guidance currently exists on the exact nature of this effect due to the many complications and competing effects involved in $J/\\psi $ production in $p(d)+A$ collisions.",
"Often this effect is modeled by a simple “effective” cross section, which remains constant with $p_T$, however there are a number of models, including a dynamic breakup cross section that changes based on the kinematics of the produced $J/\\psi $.",
"The broadening of the $p_T$ distribution, termed the Cronin effect [31], is typically attributed to multiple elastic scattering of the incoming parton before the hard collision that produces the $J/\\psi $.",
"This modifies the $p_T$ dependence of the $J/\\psi $ production by adding $p_T$ vectorially to the incoming parton.",
"This generally causes a decrease in $J/\\psi $ production at low $p_T$ and a compensating increase at higher $p_T$ ($\\mbox{$p_T$} \\approx 5-10$ GeV/$c~$), which eventually falls off at yet higher $p_T$ ($\\mbox{$p_T$} \\approx 10$ GeV/$c~$).",
"The first set of model calculations that we discuss is by Kopeliovich et al.",
"[32], [33] calculates the effects on a $c\\bar{c}$ dipole propagating through a nucleus.",
"The $J/\\psi $ production is calculated based on 2$\\rightarrow $ 1 kinematics, $x=\\frac{\\sqrt{\\langle M_{c\\bar{c}}^2\\rangle +\\langle \\mbox{$p_T$} ^2\\rangle }}{\\sqrt{s}}e^{-y},$ where $\\langle M_{c\\bar{c}}^2\\rangle =2M_{J/\\psi }^2$ is fixed based on the $c\\bar{c}$ invariant mass distribution predicted by the color singlet model.",
"The calculation includes shadowing, taken from the nDSg nPDF set, as well as nuclear breakup and the Cronin effect.",
"The nuclear breakup is calculated using a parametrization of the dipole cross section fitted to measurements of the proton structure function at HERA [34], yielding a breakup cross section that is dependent on kinematics of the $J/\\psi $.",
"The results from this calculation are shown for the 0–100% $R_{d{\\rm Au}}$ at all rapidities in Fig.",
"REF as the dot-dashed curves.",
"This is a parameter free calculation, with no overall normalization or fits to the data presented here.",
"The $p_T$ shape is in good agreement with the data at mid and forward rapidity, but the theory shows a greater overall level of suppression than is seen in the data.",
"At backward rapidity there is a disagreement with the shape of the $p_T$ distribution.",
"While the theory predicts a similar $p_T$ shape at all rapidities, the data show a much faster rise in $R_{d{\\rm Au}}$ with increasing $p_T$ at backward rapidity.",
"It is also worth noting, as shown in [35], that this model cannot simultaneously describe the rapidity dependence of the PHENIX $R_{d{\\rm Au}}$ and $R_{\\rm CP}$, which is the ratio of $R_{d{\\rm Au}}$ in central collisions to the $R_{d{\\rm Au}}$ in peripheral collisions, for $J/\\psi $ production and therefore may not have an accurate description of the geometric dependence of the modification.",
"A second set of model calculations, performed by Lansberg et al.",
"[36], [27], are shown in Figure REF .",
"This model uses a Monte-Carlo approach within a Glauber model of $d+$ Au collisions.",
"The $J/\\psi $ production is calculated using the color singlet model that utilizes $2\\rightarrow 2$ kinematics, namely $g+g\\rightarrow \\mbox{$J/\\psi $} +g$ , where the majority of the $J/\\psi $ $p_T$ is balanced by the emission of a hard gluon in the final state, rather than $2\\rightarrow 1$ processes, where the $J/\\psi $ $p_T$ comes entirely from the transverse momentum carried by the colliding gluons.",
"The $J/\\psi $ production is modified in $d+$ Au collisions by shadowing effects parametrized using various nPDF sets.",
"The calculations shown in Fig.",
"REF utilize the nDSg nPDF set.",
"Similar calculations using the EKS98 and EPS08 [37] nPDF sets can be found in [27].",
"Nuclear breakup of the $J/\\psi $ is taken into account through the use of an effective, $p_T$-independent, absorption cross section of 4.2 mb.",
"Results using $\\sigma _{abs}=$ 0, 2.6, and 6 mb can also be found in [27].",
"We have chosen to highlight only $\\sigma _{abs}=4.2$ mb here as it reproduces the rapidity dependence of the 60–88% $R_{d{\\rm Au}}$ reasonably well [27] where shadowing corrections are expected to be small.",
"The results of this calculation, shown in Fig.",
"REF for 0–100% $R_{d{\\rm Au}}$ at all rapidities, shows reasonable agreement with the overall level of modification seen at low $p_T$ in the data at mid and forward rapidities while the calculation predicts a flatter distribution with increasing $p_T$ than is seen in the data.",
"The shape of the distribution at backward rapidity is markedly different than the data.",
"While the data rapidly increase to $\\mbox{$R_{d{\\rm Au}}$} \\approx 1$ at low $p_T$, the calculation shows a $R_{d{\\rm Au}}$ that is essentially constant with increasing $p_T$.",
"When comparing the two sets of model calculations in Fig.",
"REF the calculations from Kopeliovich et al.",
"[32], [33] have a different and more pronounced shape when compared to the calculations from Lansberg, et al [36], [27].",
"Both sets of calculations utilize the nDSg nPDF set, suggesting a common contribution from shadowing.",
"However, the $J/\\psi $ production kinematics are calculated differently, which will lead to some difference in the shadowing contribution.",
"The calculations from Kopeliovich et al.",
"include the Cronin effect, which provides a decrease in $J/\\psi $ production at low $p_T$ and an increase at higher $p_T$, creating an $R_{d{\\rm Au}}$ that exhibits less suppression at high $p_T$ than at low $p_T$.",
"The calculations from Lansberg et al.",
"do not include the Cronin effect, and therefore the $p_T$ shape of $R_{d{\\rm Au}}$ should be dominated by the effect of shadowing, and therefore the choice of nPDF set.",
"The spatial dependence of the shadowing has been taken into account in [27], where it is assumed that the shadowing is proportional to the local density.",
"This assumption allows for calculation of the $R_{d{\\rm Au}}$ vs $p_T$ in different centrality bins.",
"The results of the calculation in the four PHENIX centrality bins are shown in Figures REF , REF and REF for backward, mid, and forward rapidity, respectively.",
"Here we have chosen to include calculations using the EKS98 nPDF set along with those using the nDSg nPDF set as this will provide a direct comparison between the effects due to different nPDF sets, since the $J/\\psi $ production kinematics and $\\sigma _{abs}$ values are identical between the two calculations.",
"At mid and forward rapidity the calculations are similar to each other and show reasonable agreement with the $R_{d{\\rm Au}}$ distributions within the current statistical and systematic uncertainties, although the calculation appears to predict a slightly larger average suppression for peripheral collisions at forward rapidity than is seen in the data.",
"This could be due to the value of $\\sigma _{abs}$ used at forward rapidity, as the value of 4.2 mb was chosen by eye rather than fitted to the data, and it may not be independent of $y$ .",
"At backward rapidity the calculations are in disagreement with the data for all but the most peripheral collisions, where both the calculations and the data show an $R_{d{\\rm Au}}$ consistent with 1.0 at all $p_T$.",
"While the calculations at backward rapidity using the nDSg nPDF set are roughly constant with $p_T$ for each centrality, the calculations using the EKS98 nPDF set show an enhancement in the suppression of $R_{d{\\rm Au}}$ with increasing $p_T$ for central and midcentral collisions, whereas the data shows the opposite trend.",
"At backward rapidity and low $p_T$ (Bjorken $x\\approx 0.1$ for the parton in the Au nucleus) production occurs in the anti-shadowing region, while at high $p_T$ ($x\\approx 0.3$ ) production begins to move towards what is termed the EMC [38] region.",
"In 1986, a suppression of the quark distributions within nuclei was discovered in the range $0.35<x<0.7$ by the European Muon Collaboration (EMC) [38] in deep inelastic scattering.",
"While there is still debate about the source of this suppression in the quark distributions, no direct evidence of an EMC effect has yet been reported in the gluon distributions.",
"Few constraints exist in this region, and there is large disagreement in the modification of the gluon density between nPDF's.",
"The nDSg nPDF set includes no suppression in the EMC region, and only a small anti-shadowing effect, while the EKS98 nPDF exhibits a suppression in the EMC region similar to that observed in the quark distributions, and a larger anti-shadowing effect (see [30] for a comparison of nPDF sets).",
"The larger anti-shadowing combined with the inclusion of an EMC effect in the EKS98 nPDF set cause a decrease in the calculated $R_{d{\\rm Au}}$ as $p_T$ (and correspondingly, $x$ ) increases.",
"The lack of a strong anti-shadowing effect combined with the absence of an EMC effect in the nDSg nPDF causes the calculation of $R_{d{\\rm Au}}$ to remain roughly constant with increasing $p_T$.",
"In [39] the authors infer from measurements of $\\Upsilon $ production at RHIC that a strong EMC effect must be present to explain the observed modification.",
"Depending on the mapping of the $J/\\psi $ $y$ and $p_T$ to $x$ , which is model dependent, the high $p_T$ data at backward rapidity may allow us to probe this region.",
"The large uncertainties present in the high $p_T$ $R_{d{\\rm Au}}$, along with complications from competing physics effects in this region, however, prevent any strong conclusions from being drawn at this time.",
"Figure: (Color Online)J/ψJ/\\psi R d Au R_{d{\\rm Au}}, as a function of p T p_T momentum for midrapidity 0–100%centrality integrated d+d+Au collisions.",
"The curves are theoreticalcalculations from described in the text.A third set of model calculations by Sharma and Vitev [40] is compared with the midrapidity 0–100% centrality integrated $R_{d{\\rm Au}}$ in Fig.",
"REF .",
"This model describes $J/\\psi $ production using nonrelativistic quantum chromodynamics (NRQCD).",
"The effect of nuclear shadowing is calculated using EKS98 in the EMC region ($x>0.25$ ), while for lower values, power suppressed coherent final-state scattering leads to a modification of parton $x$ [41].",
"Initial state energy loss, which accounts for the radiative energy loss of the incoming particles through multiple interactions with the target nucleus is included.",
"This effect reduces the energy of the incoming parton, so, to achieve the same final-state kinematics the parton must have a greater momentum, and therefore a larger value of $x$ .",
"This effectively shifts the portion of the gluon distribution sampled to higher $x$ .",
"Also included is a calculation of the Cronin effect.",
"The solid curve in Fig.",
"REF shows the full calculation including the Cronin effect.",
"The dashed curve in Figure REF is the same calculation without the Cronin effect.",
"This comparison gives a direct indication of the contribution from the Cronin effect, which is evidently over predicted when compared to the data.",
"The results presented here will hopefully provide a much needed constraint on the Cronin effect at RHIC energies.",
"The calculation including the Cronin effect indicate an $R_{d{\\rm Au}}$ that decreases at higher $p_T$.",
"This is consistent with data, however the current statistical and systematic uncertainties make determining the precise trend of $R_{d{\\rm Au}}$ difficult at high $p_T$.",
"Better data with a larger $p_T$ coverage is needed to determine the $J/\\psi $ modification at higher $p_T$."
],
[
"Summary & Conclusions ",
"We have measured the $J/\\psi $ invariant yield and $R_{d{\\rm Au}}$ , as a function of $p_T$ over three rapidity ranges in $d+$ Au collisions at $\\mbox{$\\sqrt{s_{_{NN}}}$} =200$ GeV using the PHENIX detector.",
"These measurements provide a large improvement in statistical precision and $p_T$ reach over the previously published PHENIX $d+$ Au results [16], [17], and are the first measurements of the centrality dependence of the $J/\\psi $ $p_T$ distribution in $d+$ Au collisions by PHENIX.",
"The $\\Delta \\mbox{$\\langle p_T^2 \\rangle $} $ values determined from the data show a marked increase with $N_{\\rm coll}$ that is similar at all rapidities.",
"The $R_{d{\\rm Au}}$ vs $p_T$ displays similar behavior at mid and forward rapidity, showing suppression at low $p_T$ with a gradual increase to a value consistent with 1.0.",
"The $R_{d{\\rm Au}}$ at backward rapidity has a different distribution with $p_T$, showing a more rapid increase from suppression to a value of 1.0, and transitioning to $\\mbox{$R_{d{\\rm Au}}$} >1.0$ above 2 GeV/$c~$.",
"These trends are greater for central collisions, while the peripheral collisions show $R_{d{\\rm Au}}$ consistent with 1.0 across all rapidities.",
"We find an average $R_{d{\\rm Au}}$ for $\\mbox{$p_T$} >4$ GeV/$c~$of $1.27\\pm 0.06\\pm 0.11$ at backward rapidity, and an $R_{d{\\rm Au}}$ consistent with 1.0 at mid and forward rapidity.",
"This implies a CNM contribution in $A+A$ collisions that is likely consistent with 1.0 at high $p_T$ across all rapidity.",
"This could potentially explain the reported increase in $R_{\\rm AA}$ with increasing $p_T$ [26].",
"However more data and further work to understand the propagation of $R_{d{\\rm Au}}$ to $R_{\\rm AA}$ is needed to confirm this.",
"A comparison of the measured $R_{d{\\rm Au}}$ with three types of theoretical calculations was shown.",
"The parameter independent dipole model of $J/\\psi $ production in $p+A$ collisions agrees well with the shape of the data at mid and forward rapidities, while the shape of the predicted $p_T$ dependence is different from the data at backward rapidity.",
"However the suppression is over-predicted at all rapidities.",
"The second model uses $2\\rightarrow 2$ $J/\\psi $ production kinematics coupled with shadowing taken from both EKS98 and nDSg nPDF sets as well as an effective absorption cross section of 4.2 mb.",
"The calculations with both EKS98 and nDSg show good agreement with the data at midrapidity in each centrality bin, as well as the centrality integrated case.",
"At forward rapidity the shape of the distribution is in reasonable agreement with the data, while the overall level of suppression seems to be greater in the model calculations than the data.",
"At backward rapidity, the model calculations using both EKS98 and nDSG nPDF sets are in strong disagreement with the data.",
"At backward rapidity calculations using the nDSg nPDF set show a suppression that is constant with $p_T$, while those using the EKS98 nPDF set predict an increase of suppression with increasing $p_T$.",
"The data show the opposite trend.",
"The third model, an NRQCD calculation of high $p_T$ $J/\\psi $ production show a Cronin effect, which although generally consistent with the data, is significantly larger than observed in the data, and a suppression at high $p_T$ that cannot be confirmed due to the large uncertainties at high $p_T$ and the limited $p_T$ reach of the current data.",
"In summary, the data presented here cover a large range in $x$ and $Q^2$ , providing a further constraint on the modification of the gluon distribution in nuclei, as well as providing constraints on the size of the Cronin effect on $J/\\psi $ production at RHIC."
],
[
"ACKNOWLEDGMENTS",
"We thank the staff of the Collider-Accelerator and Physics Departments at Brookhaven National Laboratory and the staff of the other PHENIX participating institutions for their vital contributions.",
"We thank Jean-Philippe Lansberg, Nicolas Matagne, Boris Kopeliovich, Ivan Vitev, and Rishi Sharma for useful discussions and theoretical calculations.",
"We acknowledge support from the Office of Nuclear Physics in the Office of Science of the Department of Energy, the National Science Foundation, Abilene Christian University Research Council, Research Foundation of SUNY, and Dean of the College of Arts and Sciences, Vanderbilt University (U.S.A), Ministry of Education, Culture, Sports, Science, and Technology and the Japan Society for the Promotion of Science (Japan), Conselho Nacional de Desenvolvimento Científico e Tecnológico and Fundação de Amparo à Pesquisa do Estado de São Paulo (Brazil), Natural Science Foundation of China (P. R. China), Ministry of Education, Youth and Sports (Czech Republic), Centre National de la Recherche Scientifique, Commissariat à l'Énergie Atomique, and Institut National de Physique Nucléaire et de Physique des Particules (France), Ministry of Industry, Science and Tekhnologies, Bundesministerium für Bildung und Forschung, Deutscher Akademischer Austausch Dienst, and Alexander von Humboldt Stiftung (Germany), Hungarian National Science Fund, OTKA (Hungary), Department of Atomic Energy and Department of Science and Technology (India), Israel Science Foundation (Israel), National Research Foundation and WCU program of the Ministry Education Science and Technology (Korea), Ministry of Education and Science, Russian Academy of Sciences, Federal Agency of Atomic Energy (Russia), VR and the Wallenberg Foundation (Sweden), the U.S.",
"Civilian Research and Development Foundation for the Independent States of the Former Soviet Union, the US-Hungarian Fulbright Foundation for Educational Exchange, and the US-Israel Binational Science Foundation."
],
[
"Fitting The ",
"The $J/\\psi $ invariant yields as a function of $p_T$ were fitted with a modified Kaplan function of the form $f(\\mbox{$p_T$})=p_0\\left(1-\\left(\\frac{\\mbox{$p_T$}}{p_1}\\right)^2\\right)^{p_2}.$ The data points were compared to the integral of the function over the $p_T$ bin when calculating the $\\chi ^2$ .",
"The fit results, along with the ratio of the data to the fit are shown in Fig.",
"REF for $p$ +$p$ collisions and Figure REF for 0–100% centrality integrated $d+$ Au collisions.",
"The fit results for each centrality bin are shown in Figs.",
"REF , REF , and REF for backward, mid, and forward rapidities, respectively."
],
[
"Calculating the Correction Factor $k$",
"To account for the fact that the experimental upper $p_T$ limits on the $J/\\psi $ invariant yield distributions vary with rapidity and centrality, a correction factor was calculated using the fits described in Sec.",
"REF .",
"The ratio $k=\\frac{\\mbox{$\\langle p_T^2 \\rangle $} [0,\\infty ]}{\\mbox{$\\langle p_T^2 \\rangle $} [0,\\mbox{$p_T$} ^{\\rm max}]}$ was calculated from the fit and applied to the numerically calculated $\\langle p_T^2 \\rangle |_{p_T\\le p_T^{\\rm max}}$.",
"The correction factors are shown in Table REF , and are in all cases small ($k<1.03$ ).",
"The uncertainty on $k$ is derived from the fit uncertainty by varying the data points within their statistical uncertainties, refitting, and thereby finding the variation in $k$ .",
"Figure: (Color Online)Results of modified Kaplan fits to backward rapidity 0–100% d+d+Au (Left), midrapidity 0-100% d+d+Au (Center), and forward rapidity 0-100%d+d+Au (Right).Figure: (Color Online)Results of modified Kaplan fits to midrapidity d+d+Au collisions for eachcentrality.Figure: (Color Online)Results of modified Kaplan fits to forward rapidity d+d+Au collisions foreach centrality.Table: 〈p T 2 〉\\langle p_T^2 \\rangle correction factors, kk, for pp+pp andd+d+Au collisions."
],
[
"Propagating the Type B uncertainties to ",
"When propagating the Type B systematic uncertainties on the $J/\\psi $ invariant yields to the calculated $\\langle p_T^2 \\rangle $ values, the type B uncertainties are assumed to be normally distributed.",
"With this assumption we independently sample the uncertainty distribution of the first and the last data point of the $p_T$ distribution.",
"We then assume the Type B uncertainties are linearly correlated between these two values.",
"The resulting distribution of the $\\langle p_T^2 \\rangle $ values that arises from this procedure gives an estimate of the effect of the Type B uncertainties on the value of $\\langle p_T^2 \\rangle $.",
"The Type C systematic uncertainties on the $J/\\psi $ invariant yields do not affect the calculation of $\\langle p_T^2 \\rangle $.",
"The Type C uncertainties are a global uncertainty, which cancels in the calculation."
]
] | 1204.0777 |
[
[
"The roles of charge exchange and dissociation in spreading Saturn's\n neutral clouds"
],
[
"Abstract Neutrals sourced directly from Enceladus's plumes are initially confined to a dense neutral torus in Enceladus's orbit around Saturn.",
"This neutral torus is redistributed by charge exchange, impact/photodissociation, and neutral-neutral collisions to produce Saturn's neutral clouds.",
"Here we consider the former processes in greater detail than in previous studies.",
"In the case of dissociation, models have assumed that OH is produced with a single speed of 1 km/s, whereas laboratory measurements suggest a range of speeds between 1 and 1.6 km/s.",
"We show that the high-speed case increases dissociation's range of influence from 9 to 15 Rs.",
"For charge exchange, we present a new modeling approach, where the ions are followed within a neutral background, whereas neutral cloud models are conventionally constructed from the neutrals' point of view.",
"This approach allows us to comment on the significance of the ions' gyrophase at the moment charge exchange occurs.",
"Accounting for gyrophase: (1) has no consequence on the H2O cloud; (2) doubles the local density of OH at the orbit of Enceladus; and (3) decreases the oxygen densities at Enceladus's orbit by less than 10%.",
"Finally, we consider velocity-dependent, as well as species-dependent cross sections and find that the oxygen cloud produced from charge exchange is spread out more than H2O, whereas the OH cloud is the most confined."
],
[
"Introduction",
"The Enceladus plumes directly produce a dense H$_2$ O torus centered on Enceladus's orbit, within which charge exchange and dissociation subsequently produce neutrals that either feed Saturn's extended neutral clouds, collide (absorb) with Saturn and its rings, or leave the system altogether on escape orbits.",
"This paper is a report on the results of a sensitivity study of low-velocity charge exchange and dissociation within the neutral torus.",
"Several decades before Cassini arrived at Saturn and the Enceladus water plumes were discovered [12], neutral hydrogen was observed in Saturn's magnetosphere, both from Earth [43] and from Voyagers 1 and 2 [33].",
"Hydroxyl was later discovered by [34] with HST, and more recently, [8] detected atomic oxygen.",
"These observations collectively hinted at the presence of a source of water, and models predicted its location to be near the orbit of Enceladus (c.f., [21]).",
"After identifying the Enceladus plumes as the dominant source of the water-group neutrals (O, OH, H$_2$ O)—and indeed the co-rotating plasma itself via electron impact and photoionization [46], [35], [36]—researchers have been attempting to understand how neutrals are transported from Enceladus to 20 Saturn radii ($\\textrm {R}_\\textrm {S}$ = 6 $\\times 10^9$ cm) and beyond, as observed by [34], [8], and most recently by [25].",
"Early on, [21] mentioned the role of charge exchange in this inflation process.",
"[18] later showed that if magnetospheric plasma is slowed sufficiently with respect to neutrals in the Enceladus torus, then charge exchange produces a sufficient number of particles with velocities capable of spreading the dense H$_2$ O Enceladus torus into the cloud observed by [34].",
"[9] pointed out the importance of dipole–dipole interactions in collisions involving H$_2$ O molecules.",
"She showed that collisions inside the dense Enceladus torus (parameterized by macroscopic viscosity) are alone sufficient for the creation of the extended component of Saturn's neutral cloud.",
"[5] later argued that Farmer's fluid treatment is inappropriate for neutral–neutral collisions in the Enceladus torus, where the mean free path is on the order of the torus size itself.",
"Instead, [5] approached the problem with a direct simulation Monte Carlo (DSMC) model.",
"Their model self-consistently included losses due to charge exchange, dissociation, and ionization, whereas [9] accounted for losses to charge exchange and ionization by evolving the neutral cloud for the time scales (months to a few years) found in [36].",
"Both studies agree that neutral–neutral collisions are necessary for the inflation of Saturn's neutral cloud.",
"Collisions between neutrals occur at a rate proportional to the square of the neutral density.",
"Thus, where neutral densities peak near the orbit of Enceladus, neutral–neutral collisions occur more often than either charge exchange or dissociation, whereas near 6 $\\textrm {R}_\\textrm {S}$ , neutral densities drop and all three processes become comparable (see Fig.",
"3, [5]).",
"Models involving neutral collisions have recently been validated with Herschel observations by [14], who attribute a warm and broadened Enceladus torus to heating via neutral–neutral collisions; the effect of these interactions should therefore be included in any attempt to fully model Saturn's neutral clouds.",
"Nevertheless, several first-order conclusions can be drawn by revisiting charge exchange and dissociation.",
"Previous neutral cloud models approach charge exchange from the neutrals' point of view, whereas we follow the ion along its trajectory, thus allowing us to identify the gyrophase at which an ion undergoes charge exchange.",
"We find that including the phase dependence doubles OH densities at the orbit of Enceladus, decreases oxygen density by $\\lesssim $ 10%, and has no effect on H$_2$ O (section REF ).",
"Also, the velocity-dependence of charge exchange varies by species.",
"Previous studies ($e.g.$ , [18], [5]) have considered velocity-dependence, but have used a single cross section to represent all charge exchanges.",
"We show in section REF that symmetric reactions such as $\\mathrm {H_2O+H_2O^+\\rightarrow H_2O^++H_2O^*}$ (the asterisk identifies a neutral released with the speed of the reacting ion) tend to distribute neutrals closer to Saturn, while asymmetric exchanges such as $\\mathrm {H_2O+O^+\\rightarrow H_2O^++O^*}$ populate a more extended cloud, with less absorption on Saturn.",
"With regard to dissociation, OH produced by impact/photodissociation of H$_2$ O has previously been modeled with an initial speed of 1 $\\textrm {km\\ s}^{-1}$ [19], [5], whereas recent laboratory measurements span 1 to 1.6 $\\textrm {km\\ s}^{-1}$ , depending on the molecule's internal energy [45], [24].",
"We model this parameter space and find that, relative to charge exchange, most OH found inside 9 $\\textrm {R}_\\textrm {S}$ is produced by dissociation when OH is dissociated from H$_2$ O at 1 $\\textrm {km\\ s}^{-1}$ , with that location extended to 15 $\\textrm {R}_\\textrm {S}$ when OH is dissociated from H$_2$ O at 1.6 $\\textrm {km\\ s}^{-1}$ instead.",
"This paper is organized as follows.",
"The model for the production of neutrals via dissociation and three illustrative charge exchanges is explained in section .",
"Our results are found in section , followed by a discussion in section .",
"The important points are summarized in section ."
],
[
"Model",
"We begin with a few words on nomenclature.",
"The neutral torus in this paper pertains to the primary neutral torus (not plasma torus) supplied directly by Enceladus's plumes.",
"The neutral clouds refer to the secondary neutrals produced from charge exchange and dissociation in the neutral torus.",
"The production of Saturn's neutral cloud is modeled in two steps.",
"We first construct a dense H$_2$ O torus from a plume positioned at Enceladus's south pole with specifications based on several Cassini Enceladus flybys ([39]; see also [38], [37], and [40]).",
"Secondary neutrals are then produced from the primary neutral torus by charge exchange and dissociation, some of which remain gravitationally bound to Saturn and form the neutral clouds.",
"On the basis that they spend most of the time outside the neutral torus and plasma sheet, we assume their lifetimes to be determined solely by photo-processes, though in section REF , we consider the effects of charge exchange and electron impact."
],
[
"Neutral torus model",
"The neutral torus and Enceladus plume models are described in the following subsections.",
"Our aim is to study the effects of several important reactions occurring in Enceladus's orbit.",
"The primary neutral torus, fed directly by Enceladus, is produced in the model by evolving water molecules released from Enceladus into a dense neutral torus centered on Enceladus's orbit (3.95 $\\textrm {R}_\\textrm {S}$ ).",
"The assumption is that all H$_2$ O is initially produced by a single plume at Enceladus's south pole.",
"In reality, more than one plume has been observed [26], and researchers such as [31] and [39] have studied their signature on flyby observations.",
"For our purposes, the detailed influence of multiple plumes can be neglected.",
"The plume particles' radial speed distribution is prescribed as a one-dimensional Maxwellian with temperature $T=180$ K [41], [12]: $f(v)=\\left(\\frac{m_\\mathrm {H_2O}}{2\\pi kT}\\right)^{1/2}\\exp \\left[-\\frac{m_\\mathrm {H_2O}}{2kT}(v-v_\\mathrm {bulk})^2\\right],$ where $v_\\mathrm {bulk}$ is the bulk speed, equal to 720 m s$^{-1}$ , 1.8$\\times $ the thermal speed estimated by [39] ($v_\\mathrm {therm}=\\sqrt{2kT/m_\\mathrm {H_2O}}=400$ $\\textrm {m\\ s}^{-1}$ ).",
"Additionally, a raised cosine distribution is used to determine where the molecules are released: $g(\\theta )=\\left\\lbrace \\begin{array}{ll}\\frac{1}{\\theta _0}\\left[1+\\cos \\left(\\frac{\\theta }{\\theta _0}\\pi \\right)\\right]&\\mbox{ if $\\theta < \\theta _0=30^\\circ $} \\\\0 &\\mbox{ otherwise.",
"}\\end{array} \\right.$ Co-latitude theta is measured from Enceladus's south pole, and $\\theta _0=30^\\circ $ is the plume half-width, based on INMS in situ observations [39].",
"We assume no azimuthal dependence.",
"Enceladus's gravity is ignored since the escape velocity, $v_\\mathrm {esc}\\,=\\,240 \\mathrm {\\ m\\ s^{-1}}$ , is greatly exceeded for most molecules (99%), where $v_\\mathrm {bulk}>v_\\mathrm {therm}>v_\\mathrm {esc}$ ; our results differ by less than one percent whether Enceladus's gravity is considered or not.",
"Particles released from Enceladus are thus assumed to move on Keplerian orbits with respect to Saturn.",
"Each water molecule is allowed to orbit inside the torus for a period determined by the collective lifetimes against photodissociation, electron-impact dissociation, and charge exchange.",
"To be clear, the molecules forming the neutral torus are subject to all of the losses stated, while the neutral cloud is subject to photo-processes only (section REF ).",
"Further reaction details are important for modeling plasma characteristics but only the timescales given below are required to model the neutral torus.",
"The photodissociation lifetime for H$_2$ O, $\\tau _\\mathrm {phot}=9.1\\times 10^6$ s, comes directly from [15], scaled to Saturn's distance from the Sun.",
"At peak solar activity, neutral abundances attributed to dissociation double [16]."
],
[
"Impact dissociation",
"In the case of impact dissociation, suprathermal (hot) electrons dominate [11].",
"Assuming the conditions near Enceladus's orbit apply throughout the neutral torus, we estimate the hot electron density and temperature to be 160 eV and 0.3 cm$^{-3}$ from [11], which fit within a range of recent observations (cf., [46], [36]).",
"The rate coefficient for impact dissocation of water is found by convolving $\\sigma (v)v$ with a 160 eV Maxwellian distribution to find $\\kappa _\\mathrm {imp}=1.5\\times 10^{-6}\\mbox{ cm$^3$ s$^{-1}$}$ ([11], table S9).",
"Above 100 eV, $\\kappa _\\mathrm {imp} $ is insensitive to temperature, making this estimate valid for a range of observations.",
"Assuming the hot electron density, $n_\\mathrm {eh}$ , is constant over the neutral torus, the lifetime against impact dissociation is $\\tau _\\mathrm {imp}\\approx [\\kappa _\\mathrm {imp}n_\\mathrm {eh}]^{-1}=2.2 \\times 10^6$ s. Dissociation $via$ thermal electrons—whose temperature and density are 2 eV and 60 cm$^{-3}$ —is also expected, but such collisions occur 5$\\times $ less often ([11], table S9) and are thus ignored here."
],
[
"Charge exchange",
"The following three reactions are included in this study: $\\mathrm {H_2O+H_2O^+} &\\rightarrow \\mathrm {H_2O^+ +H_2O^*} \\\\\\mathrm {H_2O+H_2O^+} &\\rightarrow \\mathrm {H_3O^+ +OH^*} \\\\\\mathrm {H_2O+O^+} &\\rightarrow \\mathrm {H_2O^++O^*} .$ Other charge exchanges are important in the neutral torus, some of which involve ions reacting with secondary neutrals such as H, O, and OH.",
"To model their effect properly, one would calculate these neutral densities as in a conventional time-dependent neutral cloud model.",
"We estimate that including all such reactions would increase our estimates on neutral cloud densities by approximately a factor of two.",
"The primary purpose of choosing this combination of reactions is to study three classes of charge exchanges, for which the collision frequency decreases with, increases with, or is independent of the relative speed of the reacting pair (reactions , , and REF , respectively).",
"We return to this point in section REF .",
"An estimate of the charge exchange lifetime can be made by adding the rate coefficients for reactions REF –.",
"Multiplying by the observed H$_2$ O$^+$ and O$^+$ densities near the orbit of Enceladus (6 and 12 cm$^{-3}$ , [36]), we find $\\tau _\\mathrm {chex}=[\\kappa _\\mathrm {exch}^\\mathrm {H_2O^+}n_\\mathrm {H_2O^+}+\\kappa _\\mathrm {exch}^\\mathrm {O^+}n_\\mathrm {O^+}]^{-1}=[6.0+2.8]^{-1}\\times 10^8 \\mathrm {\\ s\\ }=1.1\\times 10^7$ s. The reaction rates are from [23] and [1] for $\\kappa ^\\mathrm {H_2O^+}_\\mathrm {exch}$ and $\\kappa ^\\mathrm {O^+}_\\mathrm {exch}$ , respectively.",
"The lifetime of H$_2$ O against the sum of these processes is then $\\tau _\\mathrm {torus}=\\left[\\frac{1}{\\tau _\\mathrm {phot}} +\\frac{1}{\\tau _\\mathrm {imp}} +\\frac{1}{\\tau _\\mathrm {chex}}\\right]^{-1} &=&(1.1+4.5+0.91)^{-1}\\times 10^7 \\mathrm {\\ s} \\nonumber \\\\&=& 1.6\\times 10^6 \\mathrm {\\ s}\\approx 20 \\mbox{ days.}",
"$ These estimates can be compared to Fig.",
"3 of [5]: for example, our lifetime against dissociation is 1.7 $\\times 10^6$ s, while they use $7\\times 10^6$ s near 4 $\\textrm {R}_\\textrm {S}$ .",
"The discrepancy comes mostly from the impact dissociation timescale.",
"Cassidy's dissociation rate was calculated using [32], wherein CAPS ELS data were fitted and extapolated down from 5.5 $\\textrm {R}_\\textrm {S}$ , while our own estimate hinges on a hot electron density derived from our chemistry model [11].",
"For charge exchange, we have a lifetime of 1.1 $\\times 10^7$ s, and [5] have a comparable $8\\times 10^6$ s. Particles are created and tracked in each of our model runs, and the results are scaled to the number of water molecules in the real neutral torus.",
"The total number is estimated from an assumed neutral source rate from Enceladus of $\\dot{M}=200\\ \\textrm {kg\\ s}^{-1}$ [19], [12], [13] and lifetime, $\\tau _\\mathrm {torus}$ (Eq.",
"REF ): $N_\\mathrm {torus}=\\dot{M}\\tau _\\mathrm {torus}/m_\\mathrm {H_{2}O}\\approx 1.1\\times 10^{34} \\mathrm {\\ H_2O\\ molecules}.$ We bin and azimuthally average the results to find a 2-D density function, $n_\\mathrm {torus}(r,\\theta )$ (radius and latitude), through which to introduce ions for charge exchange.",
"This function also determines from where dissociated neutrals are produced."
],
[
"Plume model",
"Before describing dissociation and charge exchange within the torus, we address a calculation with the purpose of comparing the neutral production near Enceladus with that from the entire torus.",
"In doing so, we prescribe a plume whose density is consistent with Eqs.",
"REF and REF .",
"In this case, the ambient neutral density can be ignored compared to the neutrals leaving the surface of Enceladus directly.",
"The densities are determined everywhere by imposing integrated flux ($\\int n(r,\\theta )v(r,\\theta ) dA$ ) and energy ($mv^2/2-mM_\\mathrm {E}G/r$ ) conservation at a given distance $r$ .",
"The picture can be simplified, however, since most neutrals have at least twice Enceladus's escape velocity.",
"The speeds are thus independent of $r$ in the immediate vicinity of Enceladus.",
"By equating the integrated flux at the surface of Enceladus to the same integral at another distance $r>\\mathrm {R_E}$ , we find a familiar $1/r^2$ dependence: $n_\\mathrm {plume}(r,\\theta )=n(\\theta )\\left(\\frac{\\mathrm {R_E}}{r}\\right)^2\\exp \\left[-\\left(\\frac{r-\\mathrm {R_E}}{H_r}\\right)\\right].$ The trailing exponential factor is imposed $ad$ $hoc$ to keep the total plume content finite, and reflects Saturn's influence as the molecules leave Enceladus.",
"Consistent with [31], $H_r$ is set at 4 times the Hill radius of 948 km.",
"The angular dependence is consistent with our velocity distribution in the previous section (Eq.",
"REF ), $n(\\theta )=\\left\\lbrace \\begin{array}{ll}\\frac{n_0}{2}\\left[1+\\cos \\left(\\frac{\\theta }{\\theta _0}\\pi \\right)\\right]&\\mbox{ if $\\theta < \\theta _0=30^\\circ $} \\\\0 &\\mbox{ otherwise,}\\end{array} \\right.$ which is normalized such that $n(0)=n_0$ , where the plume strength $n_0$ is found to be $5.9\\times 10^8\\ \\mathrm {cm}^{-3}$ by integrating $n(\\theta )v_\\mathrm {bulk}$ over the area spanning the south pole of Enceladus from $\\theta =0^\\circ $ to $\\theta =\\theta _0=30^\\circ $ , and setting that result equal to the plume production rate of $\\dot{M}/m_\\mathrm {H_2O}=(200\\ \\textrm {kg\\ s}^{-1})/m_\\mathrm {H_2O}=6.7\\times 10^{27}$ molecules per second.",
"In section , we compare the results of charge exchange with the plume ($n_\\mathrm {plume}$ ) to that with the entire torus ($n_\\mathrm {torus}$ ).",
"Production of Saturn's neutral clouds entails following the neutrals produced by dissociation and charge exchange occurring within the neutral torus.",
"The treatment of each of these processes are described below."
],
[
"Dissociation",
"The hydroxyl radical, OH, produced largely by dissociated H$_2$ O, has previously been modeled with a single speed of 1 $\\textrm {km\\ s}^{-1}$ (i.e., [19], [5]).",
"Dissociated OH has been measured however with speeds between 1 and 1.6 $\\textrm {km\\ s}^{-1}$ [45], [24].",
"Here we bound this range by modeling the OH neutral clouds produced from an azimuthally-symmetric source (with respect to Saturn) with velocities drawn from Maxwellian distributions with temperatures $T=\\frac{1}{2}m_\\mathrm {OH}v_\\mathrm {mp}^2$ , where the most probable speed, $v_\\mathrm {mp}$ , is set to 1 and 1.6 $\\textrm {km\\ s}^{-1}$ , representing the low- and high-speed limits.",
"The initial locations of the ejected OH are determined by the spatial distribution of neutrals in the Enceladus torus ($n_\\mathrm {torus}(r,\\theta )$ , section REF ), and the directions of their release are chosen randomly and isotropically.",
"The molecules orbit Saturn until they are photodissociated and removed from the system.",
"By assuming a volume over which dissociations occur, the number of modeled OH molecules can be scaled to a realistic value.",
"We take the volume to be a torus centered on Enceladus (3.95 $\\textrm {R}_\\textrm {S}$ ), with a minor radius of 1 $\\textrm {R}_\\textrm {S}$ : $ V\\approx 2\\pi (4\\,\\textrm {R}_\\textrm {S})(2\\,\\textrm {R}_\\textrm {S})^2=2\\times 10^{31}\\,\\mathrm {cm}^3.$ For impact dissociation, we then expect a contribution of $ N^\\mathrm {imp}_\\mathrm {cloud}=k_\\mathrm {imp}\\tau _\\mathrm {phot}^\\mathrm {OH}V=2.8\\times 10^{34} \\mbox{\\ OH molecules},$ where $k_\\mathrm {imp}=7.9\\times 10^{-5}$ cm$^{-3}$ s$^{-1}$ is the rate (per volume) of impact dissociations occurring between suprathermal electrons and H$_2$ O molecules in the torus [11] and $\\tau _\\mathrm {phot}^\\mathrm {OH}=1.8\\times 10^7$ s is the photodissociation lifetime of OH at Saturn [15].",
"The number of OH molecules produced by photodissociation in the torus is similarly given by $ N^\\mathrm {phot}_\\mathrm {cloud}=k_\\mathrm {phot}\\tau _\\mathrm {phot}^\\mathrm {OH}V=7.6\\times 10^{33} \\mbox{\\ OH molecules},$ where $k_\\mathrm {phot}=2.1\\times 10^{-5}$ cm$^{-3}$ s$^{-1}$ is the rate (per volume) of H$_2$ O photodissociations occurring in the Enceladus torus ([11], Table S9).",
"The total abundance attributed to dissociation is then given by the sum of Eqs.",
"REF and REF .",
"[5] constrained their study with HST observations [25] and found a similar OH content (see comparison in Fig.",
"REF c, this paper).",
"That neutral production by photo- and impact dissociation are comparable in magnitude is itself noteworthy.",
"This condition is not shared by systems with hotter and denser plasma.",
"For example, electron impact dissociation and ionization dominate over photon-driven processes in Jupiter's Io torus, where the plasma is warmer where the pick-up energies are four times higher than at Enceladus [6], [11].",
"We also note that unlike with Io, long neutral lifetimes in the Enceladus neutral torus inhibit the response of Saturn's neutral clouds to short-term plume variability, though variability on the order of months has been studied by [39]."
],
[
"Charge exchange",
"We now describe the model for producing and following neutrals from charge exchange.",
"[5] and [19] also considered velocity-dependent charge exchange, but unlike these previous studies, we capture the gyrophase at which the reactions occur by following ions along their trajectories (section REF ).",
"We also prescribe cross sections specific to each reaction, being particularly interested in the effects of low-velocity charge exchange.",
"At very high speeds, the cross sections go to zero for all charge exchanges [17].",
"At low relative velocities, however (few $\\textrm {km\\ s}^{-1}$ ), the details of the collision are determined by the nature of the reacting species.",
"If the reactants and products are identical, apart from an electron ($i.e.$ , $\\mathrm {H_2O + H_2O^+}\\rightarrow \\mathrm {H_2O^+ + H_2O^*}$ ), the reaction is termed resonant, or symmetric, and the cross sections grow as the inverse of the relative speed.",
"If the reactants differ, as with $\\mathrm {H_2O + O^+}\\rightarrow \\mathrm {H_2O^+ + O^*}$ , the cross sections are likely to vanish at low speeds [27]—the difference being that the energy of the electron configurations is unchanged for symmetric-type charge exchanges [17].",
"Neutrals produced from resonant charge exchange therefore tend to have lower velocities than do neutrals produced from non-resonant (asymmetric) charge exchange.",
"This is a key point central to much of our discussion in section .",
"Individual ions are followed as they traverse the neutral torus (section REF ).",
"This approach allows their gyrophase to be determined the instant that charge exchanges occur (see Fig.",
"REF ).",
"The implicit assumption is that the collision is elastic, and that the neutral product has an initial velocity given by the ion velocity just before the exchange takes place.",
"The ions are introduced into the model from two Maxwellian speed distributions, $f_\\perp (v_\\perp )&=\\frac{m_\\mathrm {ion}}{kT_\\perp }v_\\perp \\exp \\left[-\\frac{m_\\mathrm {ion}v_\\perp ^2}{2kT_\\perp } \\right]\\mbox{ (speeds perpendicular to $B$)}\\\\f_\\parallel (v_\\parallel )&=\\sqrt{\\frac{m_\\mathrm {ion}}{2\\pi kT_\\parallel }}\\exp \\left[-\\frac{m_\\mathrm {ion}v_\\parallel ^2}{2kT_\\parallel } \\right]\\mbox{ (speeds parallel to $B$)}$ with a temperature anisotropy of $\\frac{kT_\\perp }{kT_\\parallel }=\\frac{27\\,\\mathrm {eV}}{5.4\\,\\mathrm {eV}}=5$ for both O$^+$ and H$_2$ O$^+$ [36].",
"The perpendicular temperature is derived from the pick-up ion velocity at the orbit of Enceladus, determined from CAPS data by [44] [$kT_\\perp =\\frac{1}{2}m_\\mathrm {W^+}(v_\\phi -v_\\mathrm {Kep})^2$ ].",
"The ions also rotate around a guiding center (field line) moving at $v_\\phi =18$ $\\textrm {km\\ s}^{-1}$ in a frame rotating with the neutrals.",
"For the component of our study aimed at estimating local neutral production (section REF ), ions passing near Enceladus are diverted (treating Enceladus as a rigid cylinder) and are slowed to 10% of the ambient flow speed to account for the effects of mass-loading (see [10]).",
"Time steps are taken at less than 1% of an ion's gyroperiod: $\\Delta t=R\\,\\times \\,T_\\mathrm {gyro}=R\\,\\times \\,\\frac{2\\pi m}{qB},$ where $R$ is a random number between 0 and 0.01, $T_\\mathrm {gyro}$ is the ion's gyroperiod (3.6 s for H$_2$ O$^+$ ), $B$ = 325 nT, and $q$ and $m$ are the charge and mass, respectively, of the reacting ion.",
"Such resolution is necessary in order to capture the significance of the energy dependence at low relative speeds.",
"After each time step, the collision frequency $\\nu $ is calculated from $\\nu (r,\\theta ,v_\\mathrm {rel})=n(r,\\theta ) \\sigma (v_\\mathrm {rel})v_\\mathrm {rel},$ where $n(r,\\theta )$ is the local H$_2$ O density (section REF ), $v_\\mathrm {rel}$ is the relative velocity between the reacting ion and neutral, and $\\sigma (v_\\mathrm {rel})$ is the velocity-dependent cross section.",
"Poisson statistics are used to test the likelihood of one or more reactions having occurred within $\\Delta t$ .",
"If $\\exp (-\\nu \\Delta t)$ is less than a second random number between 0 and 1, then a reaction occurs.",
"The possibility of multiple reactions occurring over $\\Delta t$ is taken into account, but it is neglectable (appendix A).",
"As with OH produced by dissociation (section REF ), neutrals produced by charge exchange are followed under the influence of Saturn's gravity until they are photodissociated or photoionized.",
"Their initial location and velocity are taken to be that of the reacting ion, pre-transfer.",
"The model runs are centered on the orbit of Enceladus spanning 10 R$_\\mathrm {E}$ in the direction of corotation (R$_\\mathrm {E}$ = 250 km = radius of Enceladus) and $\\pm $ 120 R$_\\mathrm {E}$ (0.5 $\\textrm {R}_\\textrm {S}$ ) in both the radial and $z$ directions to adequately sample the H$_2$ O torus (section REF ).",
"Ions are introduced into the model on the upstream boundary, and their guiding centers flow downstream at a speed $v_\\mathrm {plasma}=18$ $\\textrm {km\\ s}^{-1}$ relative to the neutrals.",
"Their starting location in ($r,z$ ) is chosen randomly."
],
[
"Scaling",
"The neutral clouds formed $via$ charge exchange are done so in our model by following a relatively small number of ions, and must thus be scaled to facilitate comparison with observations and other models.",
"The number of neutrals in our modeled clouds have been scaled by accounting for the following.",
"First, the number of representative ions used to produce the neutral clouds $via$ charge exchange falls short of, and must be scaled to, the number of ions present in the actual plasma torus, $n_\\mathrm {ion}V$ .",
"The volume of the plasma torus, $V$ , is given in Eq.",
"REF , and $n=12$ and 6 cm$^{-3}$ for O$^+$ and H$_2$ O$^+$ , respectively [36].",
"Second, we have argued that photo-processes are more likely to occur than either charge exchange or electron-impact processes throughout the neutral clouds with the exception of very near the neutral torus.",
"In keeping with this assumption, the plasma torus thus feeds the extended neutral clouds $via$ charge exchange for a photodissociation (photoionization in the case of oxygen) time scale before equilibrium of the neutral cloud is achieved: $\\tau _\\mathrm {phot}= $ 14, 0.6, 0.3 years for O, OH, and H$_2$ O, respectively [11].",
"Our model runs followed 10$^5$ ions for 100 seconds, and the resulting neutral clouds were scaled as described."
],
[
"Results",
"In the following sections, we present and discuss the neutral clouds resulting from dissociation and charge exchange in our model."
],
[
"Charge exchange",
"In the neutrals' reference frame, ions oscillate between $\\approx $ 0 $\\textrm {km\\ s}^{-1}$ and twice the local pick-up speed ($v_\\phi \\approx 18\\ \\textrm {km\\ s}^{-1}$ ) due to gyro-motion.",
"A cartoon of this can be seen in Fig.",
"REF , where $v_\\mathrm {rel}\\approx 0$ at the cusp of the ion trajectory and reaches a maximum of $v_\\mathrm {rel}\\approx 2 v_\\phi $ along the flow direction.",
"Shown are several trajectories for which $v_\\perp $ is either less than, greater than, or approximately equal to the bulk flow velocity.",
"The neutrals formed $via$ charge exchange follow the trajectories indicated in red.",
"The velocity dependence of reactions REF – are determined by the details of the reacting species [17].",
"Essentially, symmetric charge exchanges have cross sections that increase monotonically with decreasing velocity, whereas cross sections for asymmetric exchange peak and then vanish at low relative speeds.",
"The implication is that symmetric exchanges produce lower velocity neutrals and a more compact neutral cloud than do asymmetric reactions.",
"With symmetric charge exchange, the cross sections go as ${v_\\mathrm {rel}}^{-1}$ , so that the collision frequency ($n\\sigma v$ ) is independent of $v$ , as with reaction REF , whereas asymmetric exchanges are defined by cross sections (and collision frequencies) which tend rapidly toward zero at low relative velocities ($\\sim {v_\\mathrm {rel}}^4$ , [27]).",
"The cross sections (10$^{-16}$ cm$^2$ ) used in this paper to study reactions REF – plotted in Fig.",
"REF a are given by $\\mbox{Reaction \\ref {h2o_chex}} :&\\ \\mathrm {H_2O + H_2O^+} \\rightarrow \\mathrm {H_2O^+ + H_2O^*} \\nonumber \\\\\\sigma _\\mathrm {H_2O} =&\\ 38E_\\mathrm {rel}^{-0.5} \\\\\\mbox{Reaction \\ref {oh_chex}} :&\\ \\mathrm {H_2O + H_2O^+} \\rightarrow \\mathrm {H_3O^+ + OH^*} \\nonumber \\\\\\sigma _\\mathrm {OH} =&\\ 38E_\\mathrm {rel}^{-0.88}-0.39\\exp \\left[{-\\frac{1}{2}\\left(\\frac{E_\\mathrm {rel}-57}{12}\\right)^2}\\right] \\\\\\mbox{Reaction \\ref {o_chex}} :&\\ \\mathrm {H_2O + O^+} \\rightarrow \\mathrm {H_2O^+ + O^*} \\nonumber \\\\\\sigma _\\mathrm {O} =&\\ 69E_\\mathrm {rel}^{-0.29}+30\\exp \\left[{-\\frac{1}{2}\\left(\\frac{E_\\mathrm {rel}-65}{18}\\right)^2}\\right].$ The Gaussian terms in Eqs.",
"and account for downward and upward trends in the associated data sets near 30 $\\textrm {km\\ s}^{-1}$ , but have little consequence on the neutral cloud, given that most bound particles are produced at lower velocities.",
"Symmetric exchanges occur between like species by definition, although unlike species also exhibit symmetric behavior on occasion.",
"Therefore, we explore several hypothetical behaviors for the OH$^*$ -producing reaction at low energies.",
"This test is separate from, but related to, the comparison between reactions REF – themselves, and it motivates the point that both high and low energy behaviors have an important effect on the neutral cloud.",
"With $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ , we have extropolated the best-fit curve (Eq. )",
"to the lowest energies.",
"Symmetric and asymmetric behaviors are explored with $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ and $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ [27], [17].",
"$\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ is the same as $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ except that below 1.5 eV, $\\sigma ^\\mathrm {symmetric}_\\mathrm {OH}=30E_\\mathrm {rel}^{-0.5}\\times 10^{-16}$ cm$^2$ .",
"Notice that a similar energy dependence also applies to Eq.",
"REF , consistent with symmetric charge exchange.",
"$\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ is the same as $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ except that below 1.5 eV, $\\sigma ^\\mathrm {asymmetric}_\\mathrm {OH}=11E_\\mathrm {rel}^2\\times 10^{-16}$ cm$^2$ .",
"Although it could be argued that $\\sigma _\\mathrm {OH}^\\mathrm {symmetric}$ better fits the data if the two measurements at 2 eV are ignored, our results for reaction were obtained with $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ unless noted otherwise.",
"We will discuss the implications of choosing $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ over $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ and $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ shortly.",
"The collision frequencies ($n\\sigma v$ ) are plotted in Fig.",
"REF b for a given neutral density—in this case for $n_\\mathrm {H_2O}=10^3$ cm$^{-3}$ .",
"The collision frequency for oxygen increases with relative speed, while it is constant for water, and peaks at low velocities for OH.",
"The significance is that the oxygen cloud tends to be more extended than either the OH or H$_2$ O clouds.",
"The average collision frequency is also much higher for oxygen ($\\times $ 10) than for either OH or H$_2$ O, resulting in greater oxygen abundance.",
"The equatorial neutral cloud densities resulting from reactions REF – are plotted in Fig.",
"REF .",
"Only neutrals produced from charge exchange are shown; neither the Enceladus neutral torus, nor the neutrals produced via dissociation have been included.",
"Oxygen is two orders of magnitude more abundant than either OH or H$_2$ O because of the higher rate of production, but also because oxygen has a longer lifetime against photoionization than either OH or H$_2$ O have against photodissociation.",
"Unlike [5], dissociated neutrals from the latter processes are not tracked in our model.",
"Beyond the scope of the present study, this additional heating source would serve to further inflate the oxygen and OH clouds.",
"Fig.",
"REF b is the same as REF a, except that the profiles are normalized to the peak density at the orbit of Enceladus.",
"The oxygen cloud is seen to be the most extended, followed by water, and finally by OH, with an order of magnitude separating the three species at 20 $\\mathrm {R_S}$ .",
"The effects of low-velocity charge exchange are shown in Fig.",
"REF .",
"In Fig.",
"REF a, we see that the peak density (as well as the total neutral cloud content) is the highest with $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ because more low-velocity neutrals are produced than with either $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ or $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ .",
"Conversely, fewer low-velocity neutrals are available to populate the region near Enceladus's orbit with $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ when compared to either $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ or $\\sigma ^\\textrm {symmetric}_\\textrm {OH}$ .",
"Stated another way, $\\sigma ^\\textrm {extrapolated}_\\textrm {OH}$ yields a neutral cloud with the steepest slope, and $\\sigma ^\\textrm {asymmetric}_\\textrm {OH}$ , the shallowest.",
"Fig.",
"REF b is identical to Fig.",
"REF a, apart from normalization.",
"In this case, the slope of the density profile should not be confused with the effect of inflating (spreading) the OH cloud.",
"It should be viewed, rather, as the enhancement or depletion of low velocity neutrals to fill the region inside of $\\approx 10$ $\\textrm {R}_\\textrm {S}$ .",
"In other words, neutrals beyond 10 $\\textrm {R}_\\textrm {S}$ are mostly formed in charge exchanges at high velocities, for which all $\\sigma _\\mathrm {OH}$ converge to the same curve (Fig.",
"REF ).",
"We have assumed to this point that the plasma is sub-corotating in Enceladus's orbit (18 $\\textrm {km\\ s}^{-1}$ , [44]).",
"One might expect, however, that the neutral clould would be affected in a measurable way if instead, the plasma corotates at 26 $\\textrm {km\\ s}^{-1}$ .",
"The H$_2$ O cloud would be least affected, given that the collision frequency of reaction REF is independent of speed (Fig.",
"REF b), but what about reactions such as and , whose collision frequencies are velocity-dependent?",
"Increasing the plasma speed amounts to shifting the spread of ion velocities in Fig.",
"REF to the right, which would on average increase the speed of the neutral products.",
"This is indeed the case, and in such a test where we increased the plasma speed from 18 to 26 $\\textrm {km\\ s}^{-1}$ , the oxygen cloud increased in abundance and became even more extended.",
"The OH cloud also expanded somewhat, but decreased in total abundance.",
"Unfortunately, the differences were less than 10% in both the slope of the distribution and in total oxygen abundance, suggesting that neutral cloud observations are in this way unlikely to predict plasma speeds in the torus.",
"We described in section REF the production in our model of the neutral H$_2$ O torus from the Enceladus plumes.",
"The plumes themselves have also been prescribed as a separate background density ($n_\\mathrm {plume}$ , section REF ) so that we can compare charge exchange occurring throughout the neutral torus to that occurring only within the Enceladus plumes.",
"The results are shown in Fig.",
"REF , where we have plotted the oxygen clouds produced from charge exchange within both the Enceladus plumes (local) and the entire neutral torus (global).",
"The results are for reaction , but the same test with reactions REF and produces similar results.",
"Immediately noticable is that the local production is $\\approx 0.1$ % of the overall neutral production.",
"The torus's dominance of neutral production can be explained as follows.",
"First, the volume of the torus where reactions are occurring can be estimated as $2\\pi (4\\mathrm {R_S})(0.2\\mathrm {R_S})^2$ , where $0.1\\mathrm {R_S}$ is roughly the torus's scale height.",
"The volume of the plume can be estimated from Eq.",
"REF , where the dimensions are on the order of a cylinder with width 2$\\mathrm {R_E}$ and height $H_r\\approx 16\\mathrm {R_E}$ .",
"Dividing these volumes gives roughly $250(\\mathrm {R_E}/\\mathrm {R_S})^3\\approx 10^{-5}$ .",
"Further, the collision frequencies are proportional to the neutral density, which in the plume are on the order of 10$^7$ cm$^{-3}$ , whereas typical torus densities are 10$^5$ cm$^{-3}$ , making collisions in the plume 100$\\times $ more frequent per volume than in the torus.",
"All told, the ratio of the volumes ($10^{-5}$ ) combined with the ratio of densities ($10^2$ ) explain the local-to-global neutral production ratio of $10^{-3}$ shown in Fig.",
"REF a.",
"A similar pattern has been shown to exist at Jupiter by [2] and [7], where the majority of plasma is produced throughout the neutral torus, rather than near the interaction at Io itself.",
"The slopes of the neutral clouds from the plume and torus are most easily compared in Fig.",
"REF b, in which the density profiles have been normalized.",
"The local source produces a more confined neutral cloud because the ions from which they originate have been slowed near the plume to account for the effect of mass-loading ([10]).",
"Nevertheless, such a signature would be difficult to untangle in the data since global exceeds local production so overwhelmingly."
],
[
"Dissociation",
"A major component of the OH cloud is produced by dissociation within the neutral torus, whereby the initial velocities of the OH products range from 1 to 1.6 $\\textrm {km\\ s}^{-1}$ [45], [24].",
"In Fig.",
"REF , the clouds resulting from the high- and low-speed cases are plotted along with the result from velocity-dependent charge exchange in section REF .",
"First note that dissociation contributes 100$\\times $ more OH than does charge exchange at the orbit of Enceladus (4 $\\textrm {R}_\\textrm {S}$ ); the total cloud mass is almost 100$\\times $ greater as well.",
"Second, dissociation dominates over charge exchange from the Enceladus torus out to 9 and 15 $\\textrm {R}_\\textrm {S}$ in the low- and high-speed cases, respectively.",
"The OH cloud content will only be marginally affected by variable solar activity [16], given that impact dissociation contributes 4$\\times $ more neutrals than does photodissociation, by virtue of the respective reaction rates (section REF ).",
"In both cases, few neutrals are absorbed by the rings, and even less by Saturn itself.",
"The same is not true of charge exchange, where $\\approx 50$ % of the neutrals are absorbed by Saturn (section REF ).",
"Fig.",
"REF c is a two-dimensional version of Fig.",
"REF , where the dissociation results have been averaged and added to the results from charge exchange.",
"Saturn is at the left, and the Enceladus's orbit is located on the equator at 4 $\\textrm {R}_\\textrm {S}$ .",
"In addition to being confined radially, the dissociated neutrals are also bound tightly to the equator, while neutrals from charge exchange tenuously fill the magnetosphere.",
"Fig.",
"REF a shows the hydrogen cloud that accompanies the dissociated OH clouds ($\\mathrm {H_2O + e,\\gamma }\\rightarrow \\mathrm {OH^*+H^*}$ ).",
"To conserve momentum, the hydrogen atoms have 17$\\times $ the speed of the dissociated OH molecules, and thus range between 17 and 27 $\\textrm {km\\ s}^{-1}$ , with a relatively large, diffuse neutral cloud.",
"Shown is the result for the low-speed case, which produces more bound particles and thus a more substantial neutral cloud.",
"Charge exchange from reactions such as $\\mathrm {H_2O+H^+}\\rightarrow \\mathrm {H_2O^++H^*}$ are also responsible for H-cloud production, and deserve attention in future studies."
],
[
"Fates of neutral atoms and molecules",
"In our model, neutrals created by dissociation and charge exchange are eventually either absorbed by Saturn, escape the system, or orbit until they are destroyed (ionized) by photons.",
"In Fig.",
"REF a the fates for each species are given by percentage.",
"In the case of hydrogen, the results are from the dissociation model, described in section REF .",
"The enormous amount of escape (84%) is due to the high velocities ($\\approx 17$ $\\textrm {km\\ s}^{-1}$ ) with which hydrogen is created following H$_2$ O dissociation, and the 8% absorption is largely comprised of hydrogen which would escape the system otherwise.",
"Oxygen is produced purely from charge exchange in our model (reaction ).",
"About one-half escapes, one-third is absorbed, and the remaining 13% contributes to the neutral cloud before being photoionized.",
"Water is also produced purely by charge exchange (reaction REF ) with 18% contributing to the neutral cloud.",
"Percentage-wise, more water is absorbed than oxygen because oxygen is produced with higher speeds and generally larger orbits (section REF ).",
"The fate of OH is dominated by dissociation: 96% feed the neutral cloud (ultimately ionized), 4% are absorbed, and virtually none escape.",
"The reason for the large percentage of bound and unabsorbed neutrals is that dissociated OH has a velocity spread of 1 to 1.6 $\\textrm {km\\ s}^{-1}$ in the neutral frame, compared to the escape speed of $\\approx 5$ $\\textrm {km\\ s}^{-1}$ in the same frame.",
"Looking only at OH produced by charge exchange (minor compared to dissociation), 58% are absorbed, 23% supply the neutral cloud, and 20% escape.",
"Compared to H$_2$ O, an even greater percentage of charge-exchanged OH is absorbed because the cross sections favor production of low-velocity OH molecules (Fig.",
"REF b).",
"The production of oxygen via dissociation of H$_2$ O has been ignored in this paper on the grounds that, unlike OH, oxygen is largely produced by charge exchange.",
"The cross section for oxygen-producing charge exchange is an order of magnitude higher than that for the OH-producing reaction near the plasma flow speed of $v_\\mathrm {plasma}=18$ $\\textrm {km\\ s}^{-1}$ (Fig.",
"REF a), while the photodissociation rates are an order of magnitude smaller [15].",
"We estimate that including oxygen produced from dissociation would increase the total oxygen cloud content by less than 20%.",
"Charge exchange and dissociation play a large role in creating Saturn's neutral clouds from the plume-fed neutral torus.",
"The reactions we have included have been chosen to demonstrate the effects of low velocity charge exchange and dissociation, but they are also among the most important.",
"The neutral cloud densities presented in this paper are expected to undershoot the results from models which include the additional reactions found in Fig.",
"3, of [11] by no more than a factor of two.",
"With this caveat in mind, we now compare the present results with several other recent models."
],
[
"Comparison with other models",
"Fig.",
"REF b: J06 is the work of [18], where they also investigated the neutral clouds created from low-velocity charge exchange in the stagnated flows in Enceladus's orbit.",
"Fig.",
"REF b: J07 is from [20], where the authors were primarily interested in the interaction between the neutral cloud and Saturn's rings.",
"The most recent model comes from [5] (C10), where they investigated the spreading of the neutral cloud from neutral–neutral collisions.",
"To compare with these studies, we first had to weight our H, O, OH, and H$_2$ O clouds.",
"We did this for two limiting cases.",
"In the first case ($\\tau _\\mathrm {phot}$ , Fig.",
"REF b), we assume, as we have thus far, that the neutral clouds evolve until destroyed by either photoionization or photodissociation: H, O, OH, H$_2$ O = 40, 14, 0.6, 0.3 years, respectively.",
"These lifetimes yield an upper limit since charge exchange and electron impact are not included as losses.",
"In the second case ($\\tau _\\mathrm {all}$ ) , we derived a lower limit to the lifetimes from Table 2 of [11] by summing the additional losses due to charge exchange and electron impact, finding: H, O, OH, H$_2$ O = 0.4, 0.4, 0.2, 0.03 years, respectively.",
"Notice in particular the drastically different times scales for H and O, where including the additional sinks reduce the size of the H cloud by a factor of 40/0.4 = 100, and the oxygen cloud by 14/0.4 = 35.",
"This case represents an extreme limit, given that the neutrals spend almost all of their time orbiting outside of the Enceladus torus, where compared to photo-processes, the chances of charge exchange and electron impact are relatively unlikely.",
"We mention, however, that [30], [29] has shown that circulation patterns inside of 12 $\\textrm {R}_\\textrm {S}$ at Saturn gives rise to `butterfly' hot electron pitch angle distributions, related to low temperature anisotropy ($T_\\perp /T_\\parallel $ ), on which proton field-aligned distributions depend [36].",
"The individual clouds (excluding hydrogen) were weighted by the stated time scales and totaled in Fig.",
"REF b.",
"When only losses to photodissociation/ionization are considered ($\\tau _\\mathrm {phot}$ ), the neutral cloud is dominated by oxygen, whose fate thus determines that of the neutral cloud.",
"When charge exchange and electron impact are also included ($\\tau _\\mathrm {all}$ ), dissociated OH contributes significantly, driving the neutral cloud (ionized) percentage up, and the escape percentage down.",
"We note that the neutral fates presented in [3] (escape = 44%, ionized = 17%, absorbed = 39%) were based on an earlier version of our model which only included H$_2$ O.",
"The particles that are neither absorbed nor lost by escape make up the neutral clouds.",
"In the case where the cloud evolves for $\\tau _\\mathrm {phot}$ , oxygen and hydrogen dominate since they are far less likely to be photoionized than are OH and H$_2$ O to be photodissociated.",
"With charge exchange and electron impact included ($\\tau _\\mathrm {all}$ ), however, more oxygen and hydrogen are removed from the system, which then tends to favor a molecular OH–H$_2$ O cloud.",
"In terms of total mass the same applies, although hydrogen accounts for only a few percent at most.",
"We find that the total cloud mass is bounded between $\\approx 1$ and 10 Mtons, for $\\tau _\\mathrm {all}$ and $\\tau _\\mathrm {phot}$ , respectively.",
"It is worth pausing to re-emphasize that the system is in reality better represented by the $\\tau _\\mathrm {phot}$ case, from which all neutral clouds in this paper have been derived.",
"The $\\tau _\\mathrm {all}$ case is strictly valid only for neutrals within the Enceladus torus, though reactions with electrons and protons may also prove important, as discussed above.",
"What is illustrated, however, is that Saturn's magnetosphere is less oxygen-dominated than suggested by looking at losses from photo-processes alone.",
"These results suggest that our oxygen abundances are somewhat overestimated, likely by less than a factor of two."
],
[
"Neutral absorption",
"The particles absorbed by Saturn and its rings are plotted by species and latitude in Fig.",
"REF .",
"In Fig.",
"REF b, we see that most absorption comes from oxygen (74%), followed by H$_2$ O (11%), OH (9%), and finally by hydrogen (6%).",
"Absorption is equally divided between Saturn and its rings except in the case of OH, where twice as much falls on Saturn's rings.",
"This is because OH is largely produced by impact dissociation, which creates slower neutrals than does charge exchange, whereby in our model, H$_2$ O and oxygen arise exclusively.",
"In Fig.",
"REF b, absorption is plotted against Saturn's latitude.",
"Because the model is symmetric about the equator, the results apply to either hemisphere.",
"Oxygen, water, and OH follow the same trends because they all originate from charge exchange (dissociated OH is slow and does not reach Saturn), and have been created from ions with similar velocity distributions.",
"Any second-order differences due to the velocity-dependence of the respective cross sections are not immediately apparent.",
"Hydrogen, on the other hand, is produced entirely by dissociation in the model and exhibits a more uniform flux across Saturn.",
"The explanation is that the velocity distribution from which hydrogen is produced is isotropic, whereas that which produces charge-exchanged neutrals is bi-Maxwellian (section REF ).",
"The fluxes shown in Fig.",
"REF b are consistent with [14], who modeled recent Herschel observations of Saturn's water torus and found an average flux of $6\\times 10^5$ cm$^{-2}$ s$^{-1}$ for H$_2$ O + OH impinging on Saturn."
],
[
"Discussion",
"Some useful conclusions can be drawn by further contrasting our results with [5] (C10).",
"It is important that we first mention a profound difference between our models.",
"The model of C10 effectively carries out resonant charge exchange only, which does not chemically alter the neutral population; neutrals in their model are produced either directly from Enceladus or from subsequent dissociations.",
"Neutrals in our model, on the other hand, originate from Enceladus (H$_2$ O).",
"OH is then created $via$ dissociation (as with C10), but secondary O, OH, and H$_2$ O populations are $created$ from H$_2$ O $via$ charge exchange with the dense plume-fed Enceladus torus.",
"The C10 model redistributes neutrals around Saturn, while we redistribute and chemically re-assign neutral abundances by allowing for asymmetric charge exchanges.",
"Thus, it may well be a coincidence that our models are similar in total abundance.",
"While it may be difficult to compare our total abundances, the slope of our radial density profiles can be contrasted directly because our redistribution mechanisms (charge exchange and dissociation) are similar.",
"Differences are due largely to C10's inclusion of neutral collisions and our prescribing unique velocity-dependent charge exchange for each of the O-, OH-, and H$_2$ O-producing reactions (reactions REF –).",
"Our neutral clouds are compared with C10 in Fig.",
"REF .",
"All of our clouds include contributions from charge exchange, but the H$_2$ O cloud is mostly comprised of water sourced directly from Enceladus (3.95 $\\textrm {R}_\\textrm {S}$ ), and OH includes the additional source from dissociation.",
"In the C10 model, the water molecules were spread due to neutral–neutral collisions, which explains our higher H$_2$ O densities near Enceladus's orbit (Fig.",
"REF a).",
"The slope of the oxygen profile agrees best with C10 because their charge exchange cross section most resembles our own (Eq.",
").",
"Our H$_2$ O profiles agree less, and our OH slopes, the least, due mainly to the strong effect that neutral collisions have on those more polar molecules.",
"In particular, C10 used a much larger cross section for neutral collisions involving H$_2$ O and OH [42] than for atomic oxygen [4].",
"This helps to further explain our agreement with their oxygen profile since we exclude neutral–neutral collisions from our model altogether.",
"We conclude that neutral–neutral collisions appear to play a less significant role with atomic species, such as oxygen and hydrogen.",
"The column densities (Fig.",
"REF b) are similar to C10, who constrained their O and OH clouds with the most recent Cassini UVIS results of [25].",
"Our oxygen density—as well as our total oxygen content (Fig.",
"REF c)—is higher for two reasons.",
"First, we use a larger cross section than does C10 for reaction , and second, the clouds presented here have been limited only by photoionization.",
"Charge exchange and electron impact are second order losses beyond 6 $\\mathrm {R_S}$ , but including them would favorably reduce the oxygen content more than OH and H$_2$ O (section REF ), bringing our models into better agreement.",
"Our total H$_2$ O content is 4$\\times $ less than C10 found (Fig.",
"REF c).",
"This is partly because we have subjected H$_2$ O molecules in the primary (plume-fed) neutral torus to the shortest lifetimes possible (section REF ), whereas C10 tracks molecules that get kicked out of the densest plasma $via$ neutral collisions, and thus survive longer, being less susceptible to both charge exchange and electron impact.",
"That their total H$_2$ O content is higher than ours (Fig.",
"REF c), does not contradict the fact that their H$_2$ O column density is lower; neutral–neutral collisions would spread out the torus, lowering the column density, while allowing neutrals to survive longer, increasing the total abundance.",
"Our model would benefit by including the redistribution attributed to neutral collisions by allowing particles to interact in a direct simulation Monte Carlo (DSMC) model such as in C10.",
"Likewise, DSMC models would benefit by including charge exchange cross sections specific to each reaction.",
"Such models should also take into account asymmetric charge exchanges, which affects neutral cloud composition.",
"The reactions modeled in this study were chosen in order to measure the effect of symmetric and asymmetric charge exchange at low velocities.",
"Building upon our findings, future studies should include additional neutral-producing charge exchanges, such as $\\mathrm {OH^+ + H_2O} \\rightarrow \\mathrm {OH^* + H_2O^+}$ , $\\mathrm {H^+ + H_2O} \\rightarrow \\mathrm {H^* + H_2O^+}$ , and $\\mathrm {OH^+ + H_2O} \\rightarrow \\mathrm {O^* + H_3O^+}$ , as well as dissociative recombination of H$_2$ O$^+$ ."
],
[
"Conclusions",
"We have modeled low-velocity charge exchange from the point of view of the ions, allowing us to study the effects of velocity as well as gyrophase.",
"With reactions REF –, we have been able to offer an estimate on the size and shape of the neutral clouds at Saturn, while simultaneously exploring the sensitivity of the neutral clouds to a variety of velocity-dependent reactions.",
"We have also re-visited the production of OH following H$_2$ O dissociation in the primary neutral torus.",
"Previous models have used 1 $\\textrm {km\\ s}^{-1}$ as the initial velocity for OH, while measurements suggest a range of speeds from 1 to 1.6 $\\textrm {km\\ s}^{-1}$ .",
"In our model, the higher speed increases the range within which dissociation dominates neutral production from 9 to 15 $\\textrm {R}_\\textrm {S}$ .",
"Additional findings are: (1.)",
"Charge exchange cross sections that increase steeply at low speeds tend to produce neutral clouds more confined to the orbit of Enceladus, implying the most spreading for oxygen, moderate spreading for H$_2$ O, and the least for OH (Fig.",
"REF ).",
"Accounting for gyrophase doubles the local OH density within Encelacus's orbit, has $\\approx $ no effect on H$_2$ O, and decreases oxygen density by less than 10%.",
"(2.)",
"Enceladus is solely responsible for the creation of the neutral H$_2$ O torus $via$ thermal ejection from its plumes.",
"However, Saturn's neutral clouds are overwhelmingly produced by charge exchange and dissociation occurring throughout the torus (99%), and not near Enceladus itself (Fig.",
"REF ).",
"(3.)",
"We estimate that roughly half of all neutrals escape the system, with the remaining equally divided between absorption by the rings/planet and the neutral clouds (Fig.",
"REF ).",
"Less than 50 $\\textrm {kg\\ s}^{-1}$ is thus ionized and transported out of the system as plasma.",
"This number is expected to represent an upper limit, given we have assumed that all particles forming the neutral clouds are ultimately ionized; a more accurate result would require modeling the detailed effects of charge exchange and neutral–neutral collisions within the neutral clouds.",
"This estimate can be compared to [36], whose Figs.",
"14 and 17 give roughly $([NL^2]_\\mathrm {W^+}/L^2)\\times m_\\mathrm {W^+}/\\tau _\\mathrm {transport}\\approx 3\\times 10^{31}\\times m_\\mathrm {W^+}/10^5$ s $\\approx 10$ $\\textrm {kg\\ s}^{-1}$ at $L=10$ .",
"(4.)",
"Saturn's neutral cloud has a total mass of at least 1 Mton, but likely much closer to 10 Mtons.",
"The primary plume-fed neutral torus (0.3 Mtons) is comprised entirely of water in our model, while the secondary neutral clouds are broken down into H ($\\lesssim 5$ %), O ($\\lesssim 82$ %), OH ($\\gtrsim 13$ %), and H$_2$ O ($\\approx 1$ %).",
"Atomic oxygen dominates the composition both because of a high production rate from charge exchange as well as a long lifetime against photoionization.",
"Charge exchange and reactions with electrons favorably remove hydrogen and oxygen, but are secondary loss mechanisms throughout the majority of the magnetosphere.",
"(5.)",
"Our model predicts fluxes on Saturn from charge exchange of $\\approx 6\\times 10^5$ cm$^{-2}$ s$^{-1}$ for both OH and H$_2$ O (consistent with Herschel observations by [14]), and oxygen is about 5$\\times $ higher.",
"Absorption is divided equally between Saturn and its rings (Fig.",
"REF a).",
"(6.)",
"Our total neutral abundances are similar to [5] (C10) for both OH and H$_2$ O, and 4$\\times $ higher for oxygen (Fig.",
"REF ).",
"Differences in the slopes of our equatorial density profiles are in part due to our not including neutral–neutral collisions, while this fact appears to have no effect on the oxygen profile.",
"On the other hand, C10 did not include the effects on neutral chemistry following asymmetric charge exchanges, nor did they use velocity-dependent cross sections particular to each reaction.",
"Herschel observations by [14] confirm the importance of neutral–neutral collisions for H$_2$ O, but if oxygen is the dominant neutral species in Saturn's magnetosphere, as our model predicts, neutral–neutral collisions may play a smaller role in Saturn's neutral cloud than previously expected.",
"Given the effect on both the size and shape of the neutral clouds, we suggest that future neutral cloud models include charge exchange cross sections unique to each reaction.",
"Asymmetric charge exchange also has an important effect on neutral chemistry that should be implemented.",
"Regarding the ions' gyrophase, Monte Carlo models can account for its effect by using phase-dependent probability distributions.",
"Finally, the range of OH velocities studied here should be considered when modeling dissociation.",
"Moving forward, we plan to implement these suggestions into the neutral cloud model of C10 and to couple that model with the chemistry model of [11].",
"Constrained by Cassini plasma observations, the chemistry model uses C10's neutrals as input, and provides ion temperatures and densities throughout the magnetosphere ($<$ 20 $\\textrm {R}_\\textrm {S}$ ), which C10 in turn uses to update neutral densities.",
"An improved understanding of two issues is planned: (1) Where does plasma transport become important?",
"(2) What is the role of hot electrons with regard to ion–neutral chemistry inside 20 $\\textrm {R}_\\textrm {S}$ ?"
],
[
"Collision probability",
"The average number of collisions occurring during a time interval $\\Delta t$ is given by $\\lambda \\equiv \\nu \\Delta t$ , where $\\nu =n_\\mathrm {neutrals}\\sigma (v_\\mathrm {rel})v_\\mathrm {rel}$ is the local collision frequency, assumed to be constant during $\\Delta t$ .",
"Statistics are applied to determine if and how many reactions occur during $\\Delta t$ .",
"The Poisson distribution [47], [28] gives the probability of suffering exactly $n$ collisions for a given $\\lambda $ : $ f(n;\\lambda )=\\frac{e^{-\\lambda }\\lambda ^n}{n!",
"}.$ Notice that Eq.",
"REF peaks at $n=\\lambda $ if one treats $n$ as a continuous variable.",
"Summing Eq.",
"REF discretely from $n=k$ to $n=\\infty $ gives the probability of suffering at least $k$ collisions during $\\Delta t$ , $ P_k(\\lambda )=e^{-\\lambda }\\sum _{n=k}^\\infty \\frac{\\lambda ^n}{n!",
"}.$ Because Eq.",
"REF is normalized ($e^{-\\lambda }\\sum _{n=0}^\\infty \\lambda ^n/n!=e^{-\\lambda }e^\\lambda =1$ ), Eq.",
"REF can be conveniently rewritten as $ P_k(\\lambda )=1-e^{-\\lambda }\\sum _{n=0}^{k-1} \\frac{\\lambda ^n}{n!",
"}.$ A random number ($0<N<1$ ) is compared to each $P_k$ at each timestep.",
"The largest $k$ for which $P_k > N$ determines how many fast neutrals (collisions), $k$ , are produced during $\\Delta t$ .",
"In practice, it is only necessary to compare to the first few $P_k$ when $\\lambda \\ll 1$ , made evident by the leading terms in Eq.",
"REF for $k+1$ and $k$ : $ \\frac{P_{k+1}}{P_k} \\approx \\frac{f(k+1;\\lambda )}{f(k;\\lambda )}= \\frac{\\lambda ^{k+1}/(k+1)!",
"}{\\lambda ^k/k!",
"}=\\frac{\\lambda }{k+1}\\xrightarrow[\\lambda \\rightarrow 0]{}0.$ Multiple collisions are thus increasingly unlikely when $\\lambda \\ll 1$ .",
"In such cases, comparison with $P_1=1-e^{-\\lambda }\\approx \\lambda = \\nu \\Delta t$ is sufficient.",
"This work was supported under the NESSF program, fellowship number 11-Planet11R-0005.",
"BF thanks two reviewers for their feedback and many useful suggestions.",
"Figure: Sketch of gyrating ions in the neutral framewith guiding centers moving along a prescribedflow field, shown here near Enceladus for scale.Warm ions (v ⊥ >v flow v_\\perp > v_\\mathrm {flow}) move on trajectories that coilaround themselves and do not reach zero relative velocity with respect to theneutrals at any point.Cool ions (v ⊥ <v flow v_\\perp < v_\\mathrm {flow}) essentially tracetheir guiding centers with `snake-like' trajectories,and also do not obtain zero relative velocity.",
"Fresh pick-up ions(v ⊥ ≈v flow v_\\perp \\approx v_\\mathrm {flow}) do, however, obtain zero relative velocity at thecusps of their cycloidal trajectories.",
"Neutrals produced by charge exchange(whose trajectories are indicated by the red lines) tend to be createdwith velocities at which the respective reaction rates peak (Fig.",
").Figure: (a.)",
"Cross sections for the reactions listed in the legend.Data for reactions and are from , anddata for reaction is from .σ OH extrapolated \\sigma ^\\textrm {extrapolated}_\\textrm {OH}, σ OH symmetric \\sigma ^\\textrm {symmetric}_\\textrm {OH}, and σ OH asymmetric \\sigma ^\\textrm {asymmetric}_\\textrm {OH} are hypothetical fits applying to the OH * ^*-producing reaction, and are explored inFig.",
".Ions oscillate between ≈0\\approx 0 and 36 km s -1 \\textrm {km\\ s}^{-1}in theEnceladus torus.(b.)",
"Collision frequency, nσ(v)vn\\sigma (v)v, for a given density ofn H 2O=10 3 cm -3 n_\\mathrm {H_2O}=10^3\\ \\mathrm {cm}^{-3}plotted over the same energy range.The collision frequency increases withenergy in the oxygen-forming reaction, while the water-formingreaction is independent of energy and the OH-formingreaction declines with energy.Figure: Neutral clouds produced by the reactions shown in Fig.",
".",
"(a.)",
"Oxygen is the most abundant because the cross section section is10×\\times higher than than with O and OH.",
"The lifetime of oxygenagainst photoionizationis also much longer than the lifetime for either OH or H 2 _2O againstphotodissociation.",
"(b.)",
"Same as above, but normalized to peak.",
"Oxygenshows the most spreading because reactants are produced with higher velocities(Fig.",
"b), which expands the cloud.",
"The same trendholds with H 2 _2O and OH, where OH tendsto be created with the lowest velocities (Fig.",
", σ OH extrapolated \\sigma ^\\textrm {extrapolated}_\\textrm {OH}).Figure: Neutral OH clouds produced from three hypothetical charge exchange crosssections: σ OH extrapolated \\sigma ^\\textrm {extrapolated}_\\textrm {OH}, σ OH symmetric \\sigma ^\\textrm {symmetric}_\\textrm {OH}, andσ OH asymmetric \\sigma ^\\textrm {asymmetric}_\\textrm {OH} (Fig.",
").",
"(a.)",
"σ OH extrapolated \\sigma ^\\textrm {extrapolated}_\\textrm {OH} produces the highest density(σ OH asymmetric \\sigma ^\\textrm {asymmetric}_\\textrm {OH}, the lowest) at Enceladus because of the creation of additionallow-velocity particles.(b.)",
"Same as above, but normalized to peak.",
"Thedifferences in density in the tail is not an indication of spreading, but ratherfurther illustrates the deficiency in the peak density, going from σ OH extrapolated \\sigma ^\\textrm {extrapolated}_\\textrm {OH} toσ OH asymmetric \\sigma ^\\textrm {asymmetric}_\\textrm {OH}.Figure: (a) Comparison between charge exchanged neutrals produced near the Enceladus plumeand those produced from the neutral torus as a whole—in this case for oxygen.(b.)",
"Though shown here for oxygen, all chargeexchange reactions near the plume result in a cloud with less spreading thantheir global counterpart due to the imposed slowing of the plasma (and hence, therelease of slower neutral products) near the plume inresponse to mass-loading .Figure: OH clouds produced from charge exchangeand high- and low-speeddissociation.",
"Dissociationdominates neutral cloud production inside 9–15 R S \\textrm {R}_\\textrm {S}, at which point chargeexchange becomes the dominant contributor.Figure: Neutral cloud densities in the rr–zz plane.(a.)",
"Hydrogen produced purely from H 2 _2O dissociation.(b.)",
"Oxygen produced purely from charge exchange (reaction).(c.)",
"Hydroxyl produced from the combination of charge exchange anddissociation.",
"Dissociation dominates inward of 9–15 R S \\textrm {R}_\\textrm {S} along theequator, while charge exchange (reaction )tenuously fills the magnetosphere elsewhere.(d.)",
"Water produced entirely by charge exchange (reaction ).(e.)",
"Dense torus fed directly by the Enceladus plumes (section).Figure: The fates of neutrals in our model along with the results from othermodels.",
"(a.)",
"Dissociation produces low-velocity neutrals andOH is thus not likely to escape or to be absorbed.Conversely, dissociation also produces hydrogen which largely leaves thesystem.",
"(b.)",
"The results ofJ06 , J07 , and C10, along with our own weighted totals (excluding hydrogen;see section).In the case of τ phot \\tau _\\mathrm {phot}, the lifetimeof the cloud is determined by photoionization/dissociation only, whereaswith τ all \\tau _\\mathrm {all}, welimit the lifetimes by also including electron impact and charge exchange.These limiting cases bound the previous studies, exceptthat C10 has more absorption attributed to neutral–neutral collisions.Figure: (a.)",
"Neutrals absorbed by Saturn, plotted by species.Partitions with horizontal linesindicate percentages absorbed by Saturn's rings.",
"(b.)",
"Neutral fluxon Saturn as a function of latitude.",
"Neutrals produced by charge exchange(H 2 _2O, OH, and O) peak in flux at low latitudes due to the nature of the iondistributions from which they orginate, which have initial velocity vectorspredominantly in the ring plane.",
"Conversely, hydrogen flux is constantacross Saturn because it originates from dissociation, whose velocitydistribution is prescribed as isotropic.Note that OH produced by dissociation is not energeticenough to reach Saturn.Figure: (a.)",
"Total neutral clouds from our model, compared with (C10).All clouds include contributions from charge exchange (reactions–), while H 2 _2O is largelycomprised of water sourced directly from Enceladus, and OH includescontributions from dissociation.The cloud densities are limited by photodissociation for OH andH 2 _2O and by photoionization for O.Including chargeexchange as a loss for cloud neutrals would reduce the lifetime for Omore than for either the OH or H 2 _2O, and would lower the relative oxygenabundance accordingly.(b.)",
"Equatorial column densities found by integrating theplotted equatorial densities.The H 2 _2O column density is similar to C10,despite their having a very differentradial distributions.(c.)",
"Total neutral cloud content.",
"Our total H 2 _2O contentis less than C10 found, while our H 2 _2O column density is higherbecause our H 2 _2O cloud is not subjected to neutral collisions and is thus moreconfined."
]
] | 1204.0979 |
[
[
"Experimenting with Component-Based Middleware for Adaptive Fault\n Tolerant Computing"
],
[
"Abstract This short paper describes early experiments to validate the capabilities of a component-based platform to observe and control a software architecture in the small.",
"This is part of a whole process for resilient computing, i.e.",
"targeting the adaptation of fault-tolerance mechanisms at runtime."
],
[
"Problem statement",
"Dependable systems designers must ensure that dependability properties are not violated by the target system evolution.",
"This leads to the notion of resilient computing [1].",
"Dependable systems consist of a functional layer to which are associated one or several fault tolerance mechanisms (FTMs).",
"The choice of an appropriate FTM is based on several criteria: fault model, application assumptions such as determinism and state accessibility, available resources.",
"Should the evolution of the system during its operational lifetime entail one or several changes in the values of these parameters, the initial FTM could become useless and, more importantly, inappropriate wrt the actual fault model that may evolve during system's lifetime.",
"A complete restart with new FTMs is not a solution for systems which must not stop for a long period of time.",
"Our work aims at enabling the adaptation of FTMs through a differential approach minimizing the modifications to perform a transition from one FTM to another one.",
"To reach our goal, we need software engineering tools enabling modular design and exploring and manipulating the software architecture at runtime, namely component-based middleware.This paper reports on the design and the experimental component-based implementation of a FTM, a Primary-Backup Replication (PBR) and its manipulation at runtime."
],
[
"Component-based Design",
"Service Component Architecture (SCA) provides a set of specifications for building and composing loosely-coupled, tailorable applications encompassing a wide range of technologies.",
"The main idea of this paradigm is that applications are built from bricks (i.e., components) exposing their functionalities in the form of services.",
"Such an approach facilitates reuse and evolutivity as components consume services provided by other components without being aware of how they are implemented.",
"The component-based middleware on which we develop our mechanisms is OW2 FraSCAti [2], a platform providing runtime support for SCA.",
"FraSCAti offers support for runtime reconfigurations of the application's architecture in several ways, one of which being FScript [3], a script language for exploring and modifying component-based systems.",
"This support is necessary to fulfill the requirements we identified for building a resilient computing framework: access to components' state and properties; control over components' lifecycle (start, stop); control over interactions between components, i.e.",
"for destroying or creating bindings."
],
[
"Current Implementation",
"Figure REF shows the component-based design of PBR, with a detailed view of the primary entity.",
"All components are implemented in Java.",
"Our application consists of three components and their interactions: the client, the primary processing the requests, the backup processing the checkpoints sent by the primary.",
"In parallel, a failure detector periodically checks liveness of the primary using a heartbeat mechanism.",
"Should the primary crash, the backup substitutes for it.",
"The component-based design emphasizes the separation of concerns between the functional layer of the application (the actual server) and the non-functional one (the FTM protocol, primary-side and backup-side).",
"This kind of separation, together with a careful design of the FTM supported by the freedom to build components as fine-grained as we want, give us a strong degree of control over the architecture of our application.",
"We exercise this control at runtime for manipulating the components by modifying the values of their properties, their connections, stopping, restarting, removing, replacing them, etc."
],
[
"Experimenting with dynamics at runtime",
"In this first experiment, the objective is not the to fully adapt a given fault tolerance mechanism, but to experiment the capabilities of the component-based middleware to manipulate a component-based software architecture at runtime.",
"To this aim, we target the above-described component-based implementation of the PBR strategy.",
"In our scenario, the implementation of the recovery, i.e., when backup becomes the primary due to the primary failing, is not built-in; we show that this recovery procedure can be done by manipulating components on-the-fly.",
"This can be regarded as a dynamic reconfiguration in the small, involving a single mechanism.",
"The performed actions are the following: the client is stopped; a reference to the requested operation is obtained by introspecting the client component; the binding between the client and the former primary is disconnected; a new connection between the client and the backup is established Figure: The script for reconfiguring our component-based PBRThe script in Figure REF implements the previously defined actions.",
"This script can be executed step-by-step using the the OW2 FraSCAti interactive Explorer (a sort of testing phase), but can also be directly executed."
],
[
"Overall Development Process and Future Works",
"Through this experimental implementation, we explored the functionalities of OW2 FraSCAti and assessed its suitability for our resilient computing framework.",
"The work described in this paper is part of a development process aiming to provide both a methodology and a tool-box for building resilient systems.",
"The overall process is described in [4].",
"The main steps identified in this process are: building a classification of FTMs based on parameters such as the ones enumerated in section ; designing FTMs for adaptation, i.e.",
"foreseeing future transitions between them and possible combinations; mapping these designs on components and bindings; writing and executing scripts performing the desired transitions at runtime, in response to different changes in the environment and in the application.",
"A typical scenario to validate our approach can be the following: at initial time, a duplex strategy was selected for a given function to comply with the crash fault model, as requested in the specification; at a later time, the monitoring of the system reveals a high number of transient physical faults impacting the results of the function.",
"It is decided to combine a time redundancy strategy with the duplex strategy to comply with this new situation.",
"This involves manipulating the component architecture of a duplex strategy to insert components performing time redundancy.",
"Our current experiments show that the facilities provided by the component-based middleware enable this reconfiguration to be done at runtime."
],
[
"Conclusion",
"Our approach to resilient computing relies on several steps, from a design for adaptation down to the verification of the consistency of distributed updates of a real system.",
"A cornerstone in this process is the manipulation of software architecture at runtime.",
"A component-based middleware is an essential piece of this process.",
"The simple experiments briefly described in this paper show that the provided observation and control capabilities enable resilient computing to be addressed."
]
] | 1204.1232 |
[
[
"Direct Observation of the Hyperfine Transition of the Ground State\n Positronium"
],
[
"Abstract We report the first direct measurement of the hyperfine transition of the ground state positronium.",
"The hyperfine structure between ortho-positronium and para-positronium is about 203 GHz.",
"We develop a new optical system to accumulate about 10 kW power using a gyrotron, a mode converter, and a Fabry-P\\'{e}rot cavity.",
"The hyperfine transition has been observed with a significance of 5.4 standard deviations.",
"The transition probability is measured to be $A = 3.1^{+1.6}_{-1.2} \\times 10^{-8}$ s$^{-1}$ for the first time, which is in good agreement with the theoretical value of $3.37 \\times 10^{-8}$ s$^{-1}$."
],
[
"Direct Observation of the Hyperfine Transition of the Ground State Positronium T. Yamazaki,$^1$ A. Miyazaki,$^1$ T. Suehara,$^1$ T. Namba,$^1$ S. Asai,$^1$ T. Kobayashi,$^1$ H. Saito,$^2$ I. Ogawa,$^3$ T. Idehara,$^3$ and S. Sabchevski$^4$ $^1$ Department of Physics, Graduate School of Science, and International Center for Elementary Particle Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan $^2$ Institute of Physics, Graduate School of Arts and Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8902, Japan $^3$ Research Center for Development of Far-Infrared Region, University of Fukui, 3-9-1 Bunkyo, Fukui-shi, Fukui 910-8507, Japan $^4$ Bulgarian Academy of Sciences, 72 Tzarigradsko Shose Blvd., 1784 Sofia, Bulgaria We report the first direct measurement of the hyperfine transition of the ground state positronium.",
"The hyperfine structure between ortho-positronium and para-positronium is about 203 GHz.",
"We develop a new optical system to accumulate about 10 kW power using a gyrotron, a mode converter, and a Fabry-Pérot cavity.",
"The hyperfine transition has been observed with a significance of 5.4 standard deviations.",
"The transition probability is measured to be $A = 3.1^{+1.6}_{-1.2} \\times 10^{-8}$ s$^{-1}$ for the first time, which is in good agreement with the theoretical value of $3.37 \\times 10^{-8}$ s$^{-1}$ .",
"Positronium (Ps) [1], a bound state of an electron and a positron, is a purely leptonic system and is a good target to study quantum electrodynamics (QED) in bound state.",
"The triplet ($1^{3}S_{1}$ ) state of Ps, ortho-positronium (o-Ps), decays into three gamma rays with a lifetime of $\\tau _{\\text{o}} = 142$ ns [2], [3].",
"On the other hand, the singlet ($1^{1}S_{0}$ ) state of Ps, para-positronium (p-Ps), decays into two gamma rays in $\\tau _{\\text{p}} = 125$ ps [4].",
"The energy level of the ground state o-Ps is higher than that of the ground state p-Ps due to the spin-spin interaction between the electron and the positron.",
"This difference is called the hyperfine structure of the ground state positronium (Ps-HFS), which is about 203 GHz.",
"Although precise measurements of Ps-HFS have been performed in 1970s and 1980s [5], [6], all of them are indirect measurements using Zeeman splitting of about 3 GHz caused by a static magnetic field of about 1 T. There is a discrepancy of 3.9 standard deviations (15 ppm) between the measured and the theoretical value [7].",
"The largest systematic uncertainty common to all previous measurements is the non-uniformity of the static magnetic field.",
"It is important to directly measure Ps-HFS, in order to avoid the systematic uncertainty of the static magnetic field.",
"Here we present a direct observation of the hyperfine transition between Ps-HFS, which is the first great step toward a direct measurement of Ps-HFS.",
"The hyperfine transition of the ground state Ps, which is $M1$ transition, has not yet been observed directly, since the transition probability (Einstein's $A$ coefficient is $A = 3.37 \\times 10^{-8}$ s$^{-1}$ [8]) is $10^{14}$ times smaller than the decay rate of o-Ps ($7.040 1(6) \\times 10^{6}$ s$^{-1}$ [2], [3]).",
"In order to cause sufficient amount of stimulated emission from o-Ps to p-Ps, we develop a new optical system which consists of a gyrotron as a sub-THz radiation source, a mode converter to convert the gyrotron output to a Gaussian beam, and a Fabry-Pérot cavity to accumulate high power sub-THz radiation.",
"The gyrotron is a novel high power radiation source for sub-THz to THz region, which enables us to perform a direct measurement of the hyperfine transition.",
"High power 203 GHz radiation in the Fabry-Pérot cavity causes the hyperfine transition from the ground state o-Ps to p-Ps, and p-Ps promptly decays into two back-to-back 511 keV gamma rays.",
"Consequently, the transition signal ($\\text{o-Ps}\\rightarrow \\text{p-Ps}\\rightarrow 2\\gamma $ ) has distinctive features that it has a lifetime of o-Ps and decays into two back-to-back 511 keV gamma rays as p-Ps.",
"Figure: Schematic diagrams of our experimental setup.Top view of the gas chamber is shown in the box.M1 and M2 are parabolic mirrors made of aluminum.We use a gold mesh plane mirror with a transmittance of about 3 % as a beam splitter (BS).Three pyroelectric detectors (PY) are used to monitor the incident, the reflected and the transmitted power.Figure REF shows a schematic view of our experimental setup.",
"We use Gyrotron FU CW V [9], which produces 202.89 GHz (140.06 GHz) radiation in TE$_{03}$ (TE$_{02}$ ) mode in 15 ms pulses at 20 Hz.",
"The power is monitored with a pyroelectric detector, which is fed back to voltage of the heater of the electron gun.",
"As a result, it can operate stably with about 300 W power within 10 % fluctuation.",
"In order to enhance the output power of the gyrotron, the radiation is accumulated in a Fabry-Pérot cavity.",
"The gyrotron output (TE$_{0n}$ mode) is converted to a Gaussian beam so as to obtain good coupling with the Fabry-Pérot cavity.",
"Main components of a mode converter are a step-cut waveguide and a large parabolic mirror made of aluminum (Vlasov antenna).",
"They convert TE$_{0n}$ mode to a bi-Gaussian beam geometrically if the axis of the step-cut waveguide and the focal point of the parabola are matched [10].",
"Two mirrors (M1 and M2) are used to convert the bi-Gaussian beam into a Gaussian beam.",
"In order to improve the beam quality, we insert an aperture (diam.",
"$= 50$ mm) as a spatial filter to block out side lobes of the beam.",
"Spatial distribution of the beam is measured by exposing a PVC sheet to the beam and taking its picture by an infrared camera.",
"Power conversion efficiency, which is estimated from the spatial distribution, is $28\\pm 2$ % due to a limitation of a purity of wave mode in the gyrotron output.",
"The Fabry-Pérot cavity is made with a gold mesh plane mirror (diam.",
"$= 50$ mm) and a copper concave mirror (diam.",
"$= 50$ mm, curvature $= 300$ mm).",
"The incident Gaussian beam resonates within the Fabry-Pérot cavity when the cavity length (136 mm) is equal to a half-integer multiple of the wavelength of the radiation (about 1.5 mm).",
"The cavity length is controlled by moving the copper concave mirror mounted on an X-axis stage (NANO CONTROL TS102-G).",
"The gold mesh plane mirror is a key component of the Fabry-Pérot cavity, and is made on a SiO$_{2}$ plate using photolithography and liftoff technique.",
"The line width and separation are 200 $\\mu $ m and 160 $\\mu $ m, respectively.",
"The mesh parameters are designed to obtain high reflectivity (99.38 %) and reasonable transmittance (0.39 %), which are simulated with CST Microwave Studio [11].",
"As a result, the finesse of the Fabry-Pérot cavity attains ${\\mathcal {F}} = 623 \\pm 29$ , which is estimated from the width of the resonance peak while changing cavity length.",
"The power accumulated in the Fabry-Pérot cavity reaches about 10 kW.",
"The power accumulated in the Fabry-Pérot cavity is estimated with the power transmitted through a hole (diam.",
"$= 0.6$ mm) at the center of the copper concave mirror.",
"The transmitted power is monitored with a pyroelectric detector.",
"A ratio between accumulated power and transmitted power is obtained from independent measurements using the Gaussian beam as follows.",
"First, the beam undergoes total absorption in water.",
"Its total power is estimated from a temperature increase of the water.",
"Next, the copper concave mirror is exposed to the beam, and power transmitted through the hole is measured with the pyroelectric detector.",
"From these measurements, the ratio of the Gaussian beam power to transmitted power is obtained.",
"However, the spatial distribution of the Gaussian beam is different from that of the beam inside the Fabry-Pérot cavity.",
"Correcting the difference of the beam shapes and considering that only the beam going to the copper mirror direction can be transmitted through the hole, the ratio of accumulated power to transmitted power is obtained.",
"The Gaussian beam shape is measured with a PVC sheet, and the spatial distribution in the Fabry-Pérot cavity is calculated from the cavity length, the curvature of the copper mirror, and the radiation wavelength.",
"The uncertainty of the ratio of accumulated power to transmitted power is $^{+33}_{-30}$ % because of the fluctuation of the beam shape between the exposure on the copper mirror and that on the PVC sheet.",
"The uncertainty of power does not affect the direct measurement of the hyperfine transition, but contributes to the accuracy of the transition probability, which is also measured in our experiment.",
"Positronium formation assembly shown in Fig.",
"REF is as follows: A 780 kBq $^{22}$ Na positron source is placed above a thin plastic scintillator (NE-102, thickness $= 0.1$ mm).",
"Emitted positrons pass through the scintillator and produce light pulses that are directed to two 1.5-inch fine-mesh photomultipliers (Hamamatsu R5924-70) by the light guide.",
"Positrons form Ps when stopped in the mixed gas (1.9 atm N$_{2}$ and 0.1 atm i-C$_{4}$ H$_{10}$ ) [12].",
"About 5 % of positrons are tagged by the plastic scintillator and stop in the gas, and then, about 1/4 of them form Ps.",
"Therefore, the Ps formation rate is about 10$^{4}$ s$^{-1}$ .",
"Ps has kinematic energy of about 1 eV just after its formation.",
"It becomes thermalized after $O(10\\text{~ns})$ with elastic collisions with gas molecules and the kinetic energy becomes about 1/30 eV.",
"Since we use delayed coincidence as shown in Fig.",
"REF , the width of the Doppler broadening due to motion of thermalized Ps is only about $\\Delta f_{\\text{D}} = 0.08$ GHz, which is much smaller than the natural linewidth $\\Delta f_{\\text{n}} \\sim 1/2\\pi \\tau _{\\text{p}} = 1.27$ GHz.",
"Gamma rays emitted from Ps decay are observed in four LaBr$_{3}$ (Ce) crystals (Saint-Gobain Crystals, diam.",
"$= 1.5$ inch and length = 2.0 inch).",
"The four detectors are placed as shown in Fig.",
"REF to make four back-to-back pairs.",
"The scintillation pulses of the LaBr$_3$ (Ce) crystals are detected with 1.5-inch fine-mesh photomultipliers (Hamamatsu R5924-70).",
"The energy resolution of the LaBr$_{3}$ (Ce) detectors is 4 % (FWHM) at 511 keV.",
"The primary decay time is 16 ns.",
"These are advantages for tagging monochromatic 511 keV gamma rays and avoiding pileup of gamma ray signals.",
"In order to select Ps decay events, data acquisition logic is set up as follows: When at least one back-to-back signal from the LaBr$_3$ (Ce) scintillator pairs is coincident within 40 ns, and then when this coincidence is within $-$ 100 ns to 1100 ns of the timing of the plastic scintillator, data acquisition is triggered.",
"A charge ADC (PHILLIPS 7167) and another charge ADC (REPIC RPC-022) are used to measure the energy information of the plastic scintillator with short and long gate, respectively.",
"The energy difference between short and long gates is used to suppress accidental background as mentioned later.",
"The outputs of the LaBr$_3$ (Ce) detectors are recorded with a charge ADC (CAEN C1205).",
"The time information between the plastic and LaBr$_3$ (Ce) scintillators is recorded using a direct clock (2 GHz) count type TDC (KEK GNC-060) [2].",
"Four runs have been performed.",
"In three runs (Run I, III, and IV), 202.89 GHz radiation (TE$_{03}$ mode) is used, and different powers are accumulated in the Fabry-Pérot cavity (11.0 kW, 0.0 kW, and 5.6 kW).",
"In another run (Run II), off-resonance frequency of 140.06 GHz in TE$_{02}$ mode is used to check systematic effects due to the absorption of the radiation in the mixed gas.",
"Total period of data acquisition is about two weeks.",
"During the data acquisition, energy and time calibrations are performed every 30 minutes.",
"Trigger rates are about 1 kHz.",
"The $\\gamma $ -ray peak at 511 keV and the zero energy peak are used to calibrate the LaBr$_3$ (Ce) detectors.",
"The room temperature is maintained within $26 \\pm 1\\ ^{\\circ }$ C in order to maintain good stability during the data acquisition.",
"Figure REF shows the time difference between the plastic scintillator signal and the coincidence signal of the LaBr$_3$ (Ce) detectors.",
"A sharp peak from prompt annihilation is followed by the exponential curve of transition signals and o-Ps decay signals, and then the constant spectrum due to accidental overlaps of a triggered positron and uncorrelated gamma rays.",
"A good timing resolution ($\\sigma = 0.8$ ns) is obtained.",
"After selecting a time window from 50 ns to 350 ns to enhance the transition signals and o-Ps decay events, accidental events remain as the dominant source of back-to-back 511 keV gamma rays.",
"In the case of accidental events, there is another plastic scintillator hit at the timing of $\\gamma $ -ray hit.",
"The energy deposit on the plastic scintillator measured with long gate becomes larger than that measured with short gate.",
"To reject accidental events, the energy difference between long gate and short gate is limited from $-2.5$ pe (photoelectron) to 1.7 pe.",
"This cut is applied on both PMTs of the plastic scintillators.",
"Figure: Time difference between the plastic scintillator and the coincidence signal of the LaBr 3 _3(Ce) scintillators.Solid line and hatched histogram show the time spectrum before and after accidental rejection cut, respectively.The time window for delayed coincidence is shown as a dashed line.Finally, we count the number of events in which back-to-back 511 keV gamma rays are observed.",
"Figure REF shows the energy spectra measured with the LaBr$_3$ (Ce) scintillator in the highest power on-resonance run (Run I, $11.0^{+3.6}_{-3.3}$ kW).",
"The delayed coincidence and the accidental rejection are applied.",
"In addition, a 511 keV $\\gamma $ -ray hit on the LaBr$_3$ (Ce) scintillator at the opposite side of the back-to-back pair is required, where the energy window is set from 494 keV to 536 keV.",
"Remaining accidental background is estimated from the events in another time window set from 850 ns to 900 ns, and is subtracted.",
"Circles and triangles show “beam ON” and “beam OFF” spectra, respectively.",
"The data taken during “beam OFF” period in the pulse beam are used to estimate background.",
"The “beam OFF” spectrum consists of pick-off annihilation ($\\text{o-Ps}+e^{-}\\rightarrow 2\\gamma +e^{-}$ ) and $3\\gamma $ decay ($\\text{o-Ps}\\rightarrow 3\\gamma $ ) of o-Ps.",
"Transition signals ($\\text{o-Ps}\\rightarrow \\text{p-Ps}\\rightarrow 2\\gamma $ ) increase when o-Ps are exposed to high power sub-THz radiation during “beam ON” period.",
"The signal rate in the energy window from 494 keV to 536 keV is $R_{\\text{ON}}-R_{\\text{OFF}} = 15.1 \\pm 2.7 ({\\text{stat.",
"}})$ mHz, where $R_{\\text{ON}}$ ($R_{\\text{OFF}}$ ) is the “beam ON” (“beam OFF”) event rate after all event selections are applied.",
"Figure: Energy spectra of the LaBr 3 _3(Ce) scintillator in the highest power on-resonance run (Run I, 11.0 -3.3 +3.6 11.0^{+3.6}_{-3.3} kW) after all event selections are applied.Circles and triangles show “beam ON” and “beam OFF” spectra, respectively.Systematic errors are summarized in Table REF .",
"The largest contribution is the uncertainty in Ps formation probability.",
"Ps formation probabilities of the “beam ON” and the “beam OFF” data are different because of absorption of the sub-THz radiation in the mixed gas, which is enhanced when the beam resonates with the Fabry-Pérot cavity.",
"The difference is estimated by counting the number of events in the time window before the energy cut is applied, since Ps formation probability is independent of the $\\gamma $ -ray energy cut.",
"The difference in Ps formation probability is the largest in off-resonance run (Run II).",
"Another dominant systematic error is uncertainty in the efficiency of the accidental rejection cut.",
"Inefficiency of the accidental rejection depends on the rates of the plastic scintillator signals which go over the discriminator threshold ($\\sim 1$ pe).",
"This systematic effect is estimated from the difference of the efficiency of the accidental rejection between “beam ON” and “beam OFF”, which is independent of the $\\gamma $ -ray energy cut.",
"In addition, if the energy resolution and energy scale of the LaBr$_3$ (Ce) scintillator are different between “beam ON” and “beam OFF”, fake signals appear because of the back-to-back 511 keV energy selection.",
"This effect is estimated using Monte Carlo simulation with Geant4 [13] where the energy resolution and energy scale taken from data are used as input.",
"The last dominant source is the uncertainty of background normalization.",
"The background is estimated from “beam OFF” events.",
"Its normalization is performed using the number of events in the prompt time window set from $-3$ ns to 1.5 ns, where the usual $e^{+}$ annihilation is dominant (77 %).",
"Statistical accuracy determines the normalization uncertainty.",
"Table: Summary of the systematic errors.",
"The values are ratios to the background.The systematic errors discussed above are independent, and the total systematic error can be calculated as their quadrature sum.",
"Final result with the systematic errors is $R_{\\text{ON}} - R_{\\text{OFF}} = 15.1 \\pm 2.7 ({\\text{stat.}})",
"^{+0.5}_{-0.8} ({\\text{sys.",
"}})\\ \\text{mHz}.$ This is the first direct observation of the hyperfine transition of the ground state positronium with a significance of 5.4 standard deviations.",
"In addition, the fraction of the transition signals is proportional to the power accumulated in the Fabry-Pérot cavity (Fig.",
"REF ), and the off-resonance data (Run II) gives a null result as expected, despite the relatively large difference in Ps formation probability as seen in Table REF .",
"Figure: Power dependence of the amount of the transition signals.",
"The vertical axis shows signal to background ratio.",
"The horizontal axis shows the power accumulated in the Fabry-Pérot cavity.",
"The error bars contain statistical uncertainty as well as systematic uncertainties summarized in Table .",
"The dashed line shows the result of a linear fit.The transition probability (or Einstein's $A$ coefficient) between the ground state Ps-HFS is also measured for the first time.",
"It can be estimated from the observed transition rate, the power accumulated in the Fabry-Pérot cavity, and $2\\gamma / 3\\gamma $ detection efficiency estimated from Monte Carlo Simulation with Geant4.",
"The estimated result is $A = 3.1^{+1.6}_{-1.2} \\times 10^{-8}\\ \\text{s}^{-1},$ which is consistent with the theoretical value of $3.37 \\times 10^{-8}$ s$^{-1}$ [8].",
"The largest uncertainty is the estimation of the absolute power accumulated in the Fabry-Pérot cavity.",
"Our next target is to directly measure Ps-HFS for the first time.",
"Output frequency of gyrotron can be changed with cavities of different sizes.",
"In Ps-HFS measurement, relative accuracy of the power estimation at different frequency points is necessary.",
"In addition, in order to perform precise measurement of Ps-HFS, we need more statistics.",
"A possible way to increase statistics is to use a slow positron beam and make positroniums in vacuum using a thin metal foil [14].",
"It also eliminates systematic uncertainty and beam power loss due to absorption of the sub-THz radiation.",
"In summary, the hyperfine transition of the ground state positronium has been observed directly for the first time with a significance of 5.4 standard deviations.",
"We develop a new optical system to accumulate about 10 kW power using a gyrotron, a mode converter, and a Fabry-Pérot cavity, in order to cause observable amount of stimulated emission from o-Ps to p-Ps.",
"The transition probability (or Einstein's $A$ coefficient) is also measured to be $A = 3.1^{+1.6}_{-1.2} \\times 10^{-8}$ s$^{-1}$ for the first time, which is in good agreement with the theoretical value.",
"This research was funded in part by the Japan Society for the Promotion of Science."
]
] | 1204.1129 |
[
[
"Associated production of $J/\\psi$-mesons and open charm and double open\n charm production at the LHC"
],
[
"Abstract Theoretical predictions of cross sections and properties of the $J/\\psi$-meson production in association with an open charm hadron and formation of two open charm hadrons from two $c\\bar{c}$ pairs in the LHC conditions are presented.",
"Processes in both single and in double parton scattering mechanisms are included into consideration.",
"Special attention is paid to the kinematic limits of the LHCb detector for which comparison with the newest experimental data is carried out."
],
[
"Introduction",
"In recent work of the LHCb collaboration [1] data on the double $J/\\psi $ -meson production at $7~\\mathrm {TeV}$ energy is presented.",
"At first blush value of the double $J/\\psi $ production cross section reported ($5.1\\pm 1.1~\\mathrm {nb}$ ) is accordant within uncertainty limits with the predictions obtained in the leading order (LO) QCD calculations [2], [3].",
"These calculations lead to the total cross section value of $10 \\div 27~\\mathrm {nb}$ and to $3 \\div 5~\\mathrm {nb}$ in the kinematic limits of the LHCb detector.",
"It is well known that such calculations include big uncertainties connected with the hard scale selection, next to the LO (NLO) contributions and allowing for the relative motion of $c$ -quarks in the $J/\\psi $ -meson.",
"It is known that the last of this factors increases cross section of double quarkonia production in $e^+e^-$ -annihilation in several times [4], [5], [6], [7].",
"Apart from uncertainties in the partonic cross section of double $J/\\psi $ production a new problem arises in the LHC conditions.",
"The hadronic cross section appears to be three orders of magnitude higher than the cross section of the partonic subprocess.",
"This phenomenon dues to the high luminosity of low-$x$ gluons with fraction of proton momenta of about $10^{-4}\\div 10^{-3}$ , which contribute most to the processes in question.",
"Such an enhancement gave rise to discussion of double parton interactions in a single $pp$ -collision (DPS) [8], [9], [10], [11] with independent production of particles considered in each interaction.",
"In works [12], [13], [14] mechanism of double $J/\\psi $ production in DPS approach was considered and it was shown that DPS can give significant contribution to the channel in question in the LHCb detector conditions.",
"Although SPSWe will address formation of considered final states in a single parton interaction as SPS.",
"and DPS models predict somewhat different kinematic distributions for the $J/\\psi $ pairs produced, the question if enhancement of statistics gained allows to distinguish this mechanisms remains open.",
"On the other hand at least in the LO there is a qualitative difference between predictions for the $J/\\psi +\\chi _c$ production obtained in the SPS and DPS models.",
"Moreover, additional DPS contribution should obviously express itself in other channels of the four $c$ -quark domain: in the associated production of $J/\\psi +D$In the following we will refer to the $J/\\psi +D$ production for the production of $J/\\psi $ and a $c\\bar{c}$ pair, from which at least one $c$ -quark hadronize into an observed open charm hadron.",
"and in the four $D$ -meson productionIn the following we will refer to the four $D$ -meson production for the production of a $c\\bar{c}c\\bar{c}$ configuration, from which at least two $c$ -quarks hadronize into observed open charm hadrons..",
"In the beginning of 2012 first LHCb results for the channels listed were presented [15].",
"It is interesting to understand the interplay of the SPS and DPS mechanisms in these channels.",
"Currently there are estimations of cross sections of the SPS processes contributing these final states in the LO perturbative QCD formalism [16], [17], [18], [19], [20], [21].",
"In the current work we review results obtained in the LO perturbative QCD for the SPS contribution and estimate DPS contributions for the channels mentioned."
],
[
"Four heavy quark production in the single gluon-gluon interaction",
"One of the first research, in which the possibility to observe four heavy quark production at colliders was discussed is [16].",
"In this work cross sections of subprocesses $gg \\rightarrow Q_1\\bar{Q}_1 Q_2 \\bar{Q}_2$ and $q \\bar{q}\\rightarrow Q_1\\bar{Q}_1 Q_2 \\bar{Q}_2$ was estimated within LO perturbative QCD approach for the kinematical conditions of the LHC and SSC.",
"Slightly later the analogous processes in which quark and antiquark are bind in a doubly-heavy meson were investigated.",
"Production of the $S$ -wave $B_c$ -meson in the $gg \\rightarrow B_c + b + \\bar{c}$ and $q \\bar{q}\\rightarrow B_c + b + \\bar{c}$ processes was estimated in the works [22], [23], [24], [25], [26], [27], [28], [29].",
"Calculation of the $P$ -wave $B_c$ -meson production cross section was done in the studies [30], [31], [32].",
"These researches continued in the works [33], [34], [35].",
"Associated $J/\\psi $ and $D$ -meson production, as well as $\\Upsilon $ and $B$ -meson production, was also estimated within the same technique in [17], [18], [19], [20], [21].",
"It is worth to note that doubly heavy baryon production implies production of two heavy quarks.",
"Therefore, assuming that the doubly heavy baryon is created in the heavy diquark hadronization, one can study the doubly heavy baryon production by analogy with the $B_c$ -meson production [36], [37], [17], [38], [39].",
"Calculations show that gluon-gluon interactions provide the main contribution into the four heavy quark production in the LHC experiments.",
"Quark-antiquark annihilation amounts to about 10%.",
"That is why production in the gluon-gluon interactions is mainly discussed in this paper.",
"Usually calculations are made under an assumption that initial gluons are real and their transverse momenta are negligible (the collinear approach).",
"To simulate real distribution over the transverse momenta of initial gluons in our studies we use the Pythia 6.4 MC generator [40].",
"In this connection it is worth noting researches [18], [19] where transverse momenta and virtualities of gluons are taken into account in the framework of the $k_T$ -factorization approach."
],
[
"Pair production of $J/\\psi $ -mesons in the LHCb detector",
"Production of two charmonia in SPS can be described within perturbative QCD by the fourth order in $\\alpha _S$ Feynman diagrams.",
"For the $J/\\psi $ -mesons pair formation invariant masses and quantum numbers ($1^{--}$ ) of two $c\\bar{c}$ pairs are fixed.",
"Cross section of the hard subprocess of two $c\\bar{c}$ pair formation in the color-singlet ($1_\\mathrm {C}$ ) state with $m_{c\\bar{c}}\\approx m_{J/\\psi }$ is proportional toFor the precise expression see [41], [42], [2].",
"$\\hat{\\sigma }(gg \\rightarrow J/\\psi J/\\psi )\\sim \\frac{\\alpha _S^4 |\\psi (0)|^4}{m_{J/\\psi }^8},$ where $|\\psi (0)|$ is the value of the $c\\bar{c}$ wave function in the $J/\\psi $ -meson at the origin.",
"Emergence of this factor dues to the approximation in which momenta of $c$ and $\\bar{c}$ quarks are parallel and their relative momentum is neglected in the matrix element of the subprocess in question ($\\delta $ -approximation).",
"At large invariant masses of the $J/\\psi $ -meson pair cross section (REF ) decreases with the rise of the full energy squared $\\hat{s}$ as $\\hat{\\sigma }(gg \\rightarrow J/\\psi J/\\psi )\\sim \\frac{\\alpha _S^4 |\\psi (0)|^4}{\\hat{s}^4}.$ Numerical result of $4.1\\pm 1.2~\\mathrm {nb}$ [2] derived in the assumptions listed was obtained using the hard subprocess scale equal to the transverse mass of one of the $J/\\psi $ -mesons produced and using the CTEQ5L proton pdfs [43].",
"As mentioned above this value agrees within uncertainty limit with the experimental value of $5.1\\pm 1.1~\\mathrm {nb}$ measured in [1].",
"Variation of the hard scale from the one half to two transverse masses of the $J/\\psi $ -meson produced changes the cross section value from $5.1~\\mathrm {nb}$ to $3~\\mathrm {nb}$ .",
"If CTEQ6LL pdfs [44] are used, cross section has maximum at the scale of about one transverse mass of $J/\\psi $ and amounts to $3.2~\\mathrm {nb}$ .",
"Cross sections are less at both half and double scales and are $2.8~\\mathrm {nb}$ and $2.6~\\mathrm {nb}$ respectively.",
"Presence of extremum dues to the fact that with rise of the scale strong coupling constant decreases while gluon density grows.",
"As well as in the manuscript [2] we include contribution from the production and decay of the $\\psi (2S)$ state into the $J/\\psi $ -mesons yield.",
"Fig.",
"REF Figure: Distribution over the invariant mass of the J/ψJ/\\psi -meson pairs compared with the LHCb measurement.Solid curves were obtained with m T J/ψ m_T^{J/\\psi } as the hard scale, dashed — 2·m T J/ψ 2 \\cdot m_T^{J/\\psi } and dotted —0.5·m T J/ψ 0.5 \\cdot m_T^{J/\\psi }.",
"For every scale choice upper curve corresponds to the CTEQ5L, lower — to the CTEQ6LL pdfused.shows distributions over the invariant mass of the $J/\\psi $ pairs calculated within the assumptions mentioned for the different hard scale choices and pdf sets in comparison with the experimental data reported by LHCb in [1].",
"One can see that the shape of the distributions predicted is nearly the same.",
"What concerns experimental distribution, it looks tilted to the bigger invariant mass values.",
"We would like to notice that this fact can be accounted for by the relative $c$ -quarks motion in the $J/\\psi $ -meson.",
"With this aim we calculated cross section of the process in question averaged by some “duality” region of the $c\\bar{c}$ invariant mass: $\\hat{\\sigma }^{\\rm {dual}} (gg \\rightarrow J/\\psi (\\psi ^{\\prime }) J/\\psi (\\psi ^{\\prime })) \\approx \\\\ \\approx \\int \\!\\!\\!",
"\\int _{2m_c}^{2m_{D}+\\Delta }\\frac{d^2\\sigma \\left( gg \\rightarrow (c\\bar{c})^{S=1}_{\\rm {1_C}}+(c\\bar{c})^{S=1}_{\\rm {1_C}}\\right)}{dm_{{c\\bar{c}}_1}dm_{{c\\bar{c}}_2}}dm_{{c\\bar{c}}_1}dm_{{c\\bar{c}}_2},$ where $m_D$ is the $D$ -meson mass.",
"The $c$ -quark mass was taken equal to $m_c=1.25 \\mbox{ GeV}.$ The $\\Delta $ parameter can be selected in such a way that the value of the pair production cross section obtained coincides with the total production cross section of the $J/\\psi J/\\psi $ , $J/\\psi \\psi ^{\\prime }$ and $\\psi ^{\\prime } \\psi ^{\\prime }$ final states calculated in the $\\delta $ -approximation.",
"If one takes $\\sqrt{\\hat{s}}/2$ for the hard scale of the subprocesses considered the correspondence is reached at $\\Delta =0.3 \\mbox{ GeV}$ : $\\hat{\\sigma }^{\\rm {dual}} (gg \\rightarrow J/\\psi (\\psi ^{\\prime }) J/\\psi (\\psi ^{\\prime }), \\Delta =0.3 {\\mbox{ GeV}} ) \\approx 4.4 \\mbox{ nb.",
"}$ At $\\Delta =0.5 \\mbox{ GeV}$ the cross section estimated in the duality approach is close to the value reported by the LHCb experiment: $\\sigma ^{\\rm {dual}} (gg \\rightarrow J/\\psi (\\psi ^{\\prime }) J/\\psi (\\psi ^{\\prime }), \\Delta =0.5 {\\mbox{ GeV}}) \\approx 5.8 \\mbox{ nb.",
"}$ Increase of $\\Delta $ leads to the growth of the total cross section on the one hand, and improves agreement in the $m_{J/\\psi J/\\psi }$ distribution on the other (Fig.",
"REF ).",
"Figure: Distribution over the invariant mass of the J/ψJ/\\psi -meson pairs in the “duality” approachcompared with the LHCb measurement.",
"Solid curve was obtained with Δ=0.5\\Delta =0.5 GeV, dashed —with Δ=0.3\\Delta =0.3 GeV and dotted — in the δ\\delta -approximation.In the LHC environment huge density of low-$x$ gluons leads to the increase of the multiple gluon-gluon interactions probability within one proton-proton collision.",
"In the DPS approach, which implies production of particles concerned in two independent subprocesses, the cross section is written down as following: $\\sigma ^{ A B }_{\\rm DPS} = \\frac{m}{2} \\frac{\\sigma ^{ A}_{\\rm SPS}\\sigma ^{ B}_{\\rm SPS}}{\\sigma _{\\rm eff}}.$ where the $\\sigma _{\\rm eff}=14.5~\\mathrm {mb}$ parameter was measured in the four jets and three jets plus photon modes by the CDF and D0 detectors [45], [46].",
"The $m$ parameter equals 1 for identical subprocesses and 2 for different.",
"For the $J/\\psi $ pairs production in the LHCb conditions expression (REF ) leads to the following cross section value: $\\sigma _{\\rm DPS}^{pp\\rightarrow J/\\psi J/\\psi +X} &=& 4~\\mathrm {nb}.$ Known inclusive production cross section of the $J/\\psi $ meson in the LHCb kinematic limits, $\\sigma _{J/\\psi } = 10.5~\\mathrm {\\mu b}$ , was used.",
"In the work [14] authors note that the DPS contribution can be located at bigger $J/\\psi $ pair invariant masses than the SPS one.",
"One of the proposed methods to distinguish the DPS signal from the SPS one is to study correlations between azimuthal angles of two mesons produced or between their rapidities [14], [13].",
"However analysis involving modelling in the Pythia generator shows that correlations presenting in collinear approach completely go out when including transverse momenta of the initial gluons into consideration (Fig.",
"REF ).",
"Figure: Distribution over the difference of the azimuthal angles of the J/ψJ/\\psi -mesons produced.",
"Curve designationscoincide with Fig.",
".To be more precise, depending on the scale choice collinear or anticollinear directions of the $J/\\psi $ momenta can dominate.",
"Moreover, at the standard scale of one $J/\\psi $ transverse mass the relative angle correlation is absent at all.",
"It is the model implemented in the Pythia generator which is completely responsible for the distribution over the transverse momenta of the initial gluons, so model-independent prediction on the $\\Delta \\phi $ distribution can not be derived.",
"Investigation of the rapidity correlation appears to be more fruitful.",
"In spite of narrowness of LHCb rapidity window ($2.0<y<4.5$ ) it appears to be sufficient to test QCD predictions which state that the difference in rapidity between the $J/\\psi $ -mesons produced does not exceed 2 units of rapidity (Fig.",
"REF ).",
"Figure: Distribution over the difference of rapidities of the J/ψJ/\\psi -mesons produced.",
"Curve designationscoincide with Fig.",
".At the current stage DPS predicts no correlations between products of two partonic interactions at all.",
"Apart from the correlation studies investigation of the $P$ -wave state contributions to the total $J/\\psi $ production can be fruitful.",
"Indeed, section rules emerging in the CS LO pQCD consideration [47] imply significant limitation on the final states which can appear in the gluon fusion process.",
"According to the $C$ -parity conservation occurrence of the $C$ -odd $J/\\psi \\chi _C$ final state should be suppressed in the SPS.",
"As in DPS formation of charmonia occurs independently, it does not have any suppression in this channel.",
"That is why DPS should dominate in the $J/\\psi \\chi _C$ state production (possibly followed by the $\\chi _C \\rightarrow J/\\psi \\gamma $ decay).",
"It was pointed out in the work [13] that similar situation takes place in the $J/\\psi \\Upsilon $ mode in which SPS and DPS lead to different hierarchy of the $pp \\rightarrow J/\\psi J/\\psi $ , $pp \\rightarrow J/\\psi \\Upsilon $ and $pp \\rightarrow \\Upsilon \\Upsilon $ cross sections.",
"As it was noted, we have taken into account contribution of the $pp \\rightarrow J/\\psi \\psi (2S)$ and $pp \\rightarrow \\psi (2S) \\psi (2S)$ processes followed by the $\\psi (2S) \\rightarrow J/\\psi X$ decays to the total $J/\\psi $ pairs yield.",
"$\\psi (2S)$ -mesons originating from these processes can also be detected by a leptonic decay, just like $J/\\psi $ .",
"It is interesting to compare ratios of $J/\\psi J/\\psi $ and $J/\\psi \\psi (2S)$ yields predicted by SPS and DPS models.",
"Both in SPS and in DPS approaches ratio of different meson pair yields can be estimated using values of the $c\\bar{c}$ wave functions in the charmonia at the origin: $&\\sigma (pp \\rightarrow J/\\psi J/\\psi ) : \\sigma (pp \\rightarrow J/\\psi \\psi (2S)) : \\sigma (pp \\rightarrow \\psi (2S) \\psi (2S))& \\approx \\nonumber \\\\\\approx &\\psi _{J/\\psi }(0)^4 : 2 \\cdot \\psi _{J/\\psi }(0)^2 \\psi _{\\psi (2S)}(0)^2 : \\psi _{\\psi (2S)}(0)^4& \\approx \\nonumber \\\\\\approx &1 : 1 : 0.3.&$ where $\\psi _{J/\\psi }(0) = 0.21~\\mathrm {GeV}^{3/2}$ , $\\psi _{\\psi (2S)}(0) = 0.16~\\mathrm {GeV}^{3/2}$ .",
"A more accurate estimate which allows for different meson masses leads to the relation $&\\sigma (pp \\rightarrow J/\\psi J/\\psi ) : \\sigma (pp \\rightarrow J/\\psi \\psi (2S)) : \\sigma (pp \\rightarrow \\psi (2S) \\psi (2S))& \\approx \\nonumber \\\\\\approx &1.7 : 1 : 0.15.&$ Accounting contributions from the $\\psi (2S)$ decays in the channels discussed one gets finally $&\\sigma (pp \\rightarrow J/\\psi J/\\psi ) : \\sigma (pp \\rightarrow J/\\psi \\psi (2S)) : \\sigma (pp \\rightarrow \\psi (2S) \\psi (2S))& \\approx \\nonumber \\\\\\approx &2.2 : 1 : 0.13.&$ What concerns DPS, using inclusive $J/\\psi $ and $\\psi (2S)$ production cross sections equal to $10.5~\\mathrm {\\mu b}$ [48] and $1.88~\\mathrm {\\mu b}$ [49] respectively, one gets $&\\sigma (pp \\rightarrow J/\\psi J/\\psi ) : \\sigma (pp \\rightarrow J/\\psi \\psi (2S)) : \\sigma (pp \\rightarrow \\psi (2S) \\psi (2S))& =\\nonumber \\\\= &\\sigma _{J/\\psi }^2 : 2 \\cdot \\sigma _{J/\\psi } \\sigma _{\\psi (2S)} : \\sigma _{\\psi (2S)}^2& =\\nonumber \\\\= &2.8 : 1 : 0.9.&$ It can be seen that DPS predicts slightly larger suppression of the $J/\\psi \\psi (2S)$ production compared to SPS.",
"The main reason of it is that inclusive $J/\\psi $ production cross section already includes contribution from the $\\chi _C$ decays, which can amount up to $20 \\div 30\\%$ [50], [51].",
"If one excludes expected contribution of $\\chi _C$ decays by taking $J/\\psi $ production cross section equal $0.8 \\times 10.5~\\mathrm {\\mu b}= 8.4~\\mathrm {\\mu b}$ , then DPS prediction amounts to $&\\sigma (pp \\rightarrow J/\\psi J/\\psi ) : \\sigma (pp \\rightarrow J/\\psi \\psi (2S)) : \\sigma (pp \\rightarrow \\psi (2S) \\psi (2S))& =\\nonumber \\\\= &\\sigma _{J/\\psi }^2 : 2 \\cdot \\sigma _{J/\\psi } \\sigma _{\\psi (2S)}^2 : \\sigma _{\\psi (2S)}^2& =\\nonumber \\\\= &2.2 : 1 : 0.11.& $ Up to the uncertainties this relation coincides with the SPS prediction (REF ).",
"Uncertainties in the cross section ratios predicted by DPS can be estimated by the largest relative uncertainty in the measurement of the cross sections involved.",
"This uncertainty is maximum for the $\\psi (2S)$ measurement and reaches $20\\%$ [49].",
"Unfortunately difference between relation (REF ), which suspects feed-down from the $\\chi _C$ production, and relations (REF ), (REF ), which do not, is of the same order.",
"Nonetheless it would be interesting to measure ratio of the $J/\\psi J/\\psi $ and $J/\\psi \\psi (2S)$ yields experimentally."
],
[
"Associated production of $J/\\psi $ and {{formula:cbc72be0-c83b-4033-84a3-24794eaae22b}} meson in the LHCb detector",
"To compare predictions for the $gg \\rightarrow J/\\psi c \\bar{c}$ and $gg \\rightarrow c \\bar{c} c \\bar{c}$ processes with experiment some model of the $c$ -quark transition into a specific hadron should be used.",
"The most common hardonization model is based on the assumption that charm hadron moves approximately in the same direction that the initial $c$ quark does and obtains some fraction $z$ of the quark momentum with the probability $D_{c \\rightarrow H}(z)$ (so called fragmentation function).",
"At the scale of about $c$ -quark mass the mean $z$ value is about 0.7.",
"Two following parametrizations are used in our calculations: the standard parametrization of Pythia 6.4 and a pQCD motivated parametrization of BCFY [52] with the parameter values obtained in [53].",
"It is worth mentioning here that as it was shown in [54], [55], [56], [57], [58], there are models in which hadronization is not described by simple fragmentation.",
"For example, it is reasonable to suppose that $c$ -quark can pull a light quark from the sea without loosing any momentum.",
"In this case it can be formally assumed that $D_{c \\rightarrow H}(z)=\\delta (z)$ .",
"Moreover it can be supposed that in some cases the final hadron momentum is even larger (by a quantity of about $\\frac{m_q}{m_c}p_c$ ) than the initial $c$ -quark momentum.",
"All mentioned possibilities have been considered in the present estimations of the cross section values.",
"Nevertheless it should be stressed that these estimations are too rough to give preference to some particular hadronization model.",
"Recently cross section value of the associated production of $J/\\psi $ together with a $D$ -meson has been measured by the LHCb collaboration for the following kinematical region [15]: $J/\\psi $ meson is produced in the rapidity region $2.0<y_{J/\\psi }<4.0$ ; one charmed hadron is produced in rapidity region $2.0<y_{D}<4.0$ and has transverse momenta $3 \\mbox{ GeV} < p_T^D < 12 \\mbox{ GeV}$ .",
"Calculations within the LO of pQCD lead to the cross section value $20\\div 60 \\mbox{ nb}$ depending on the scale choice and the $c$ -quark hadronization model [17], [18], [20], [21] (the scale value was varied from $\\sqrt{\\hat{s}}/4$ to $\\sqrt{\\hat{s}}$ ).",
"Nevertheless, as it was shown is paper [27], interaction of the sea $c$ -quark from one proton with gluon from the other can essentially contribute to the $J/\\psi $ -meson and $c$ -quark associated production, i.e.",
"the subprocess $cg \\rightarrow J/\\psi c$Form now on summation with charge conjugate mode is implied.",
"should also be taken into account, as well as the main subprocess $gg \\rightarrow J/\\psi c \\bar{c}$ .",
"It is natural for such an approach, that problems connected with double counting and non-zero $c$ -quark mass essentially impede an accurate estimation of the calculation uncertainties.",
"It can be assumed that this method is already valid at the transverse momenta of the charmed hadron $p_T^D>3\\mbox{ GeV}\\approx 2m_c$ and that interference contributions are small.",
"Also one can try to avoid double counting by subtracting the part due to the direct gluon splitting from the total $c$ -quark structure function: $\\tilde{f}_c(x,Q^2)=f_c(x,Q^2)-\\frac{\\alpha _s(Q^2)}{4\\pi }\\int _x^1 \\frac{dz}{z}\\left[\\Bigl (\\frac{x}{z}\\Bigr )^2+\\Bigl (1-\\frac{x}{z}\\Bigr )^2 + \\frac{2m_c^2 x (z-x)}{z^2Q^2}\\right] f_g(z,Q^2),$ where splitting function is taken from [59].",
"The cross section value of the subprocesses $cg \\rightarrow J/\\psi c$ was found to be about $10\\div 40~\\mathrm {nb}$ .",
"Therefore the contribution of such corrections to the $J/\\psi +D$ associated production is of the same order as the contribution of the main subprocess $gg\\rightarrow J/\\psi c \\bar{c}$ .",
"Thus the calculations within pQCD lead to the cross section value of about $30\\div 100$ nb for the $J/\\psi +D$ production in the LHCb fiducial region.",
"It should be noticed that in contrast to the charmonia pairs production in the associated charm production there are no $C$ -parity selection rules.",
"So one should expect not only feed-down from the $J/\\psi +\\psi (2S)$ production but also from the $J/\\psi +\\chi _C$ one.",
"This contributions can increase observed $J/\\psi + D$ cross section by up to 50%.",
"In the Fig.",
"REF Figure: Distribution over the transverse momentum of the J/ψJ/\\psi -meson in the J/ψ+DJ/\\psi + D productioncompared with the LHCb measurement (points for J/ψJ/\\psi produced together with D 0 D^0 or D + D^+-meson are shown).Solid curves were obtained with the hard scale value of 1·m T J/ψ 1\\cdot m_T^{J/\\psi }, dashed — 2·m T J/ψ 2 \\cdot m_T^{J/\\psi } and dotted —0.5·m T J/ψ 0.5 \\cdot m_T^{J/\\psi }.",
"Dot-dashed curve corresponds to the collinear gluon approximation.cross section distribution over the $J/\\psi $ meson transverse momentum in the $gg \\rightarrow J/\\psi +c\\bar{c}$ subprocess is shown in comparison with the LHCb experimental data.",
"The $d \\ln \\sigma /d p_T$ distributions are plotted, i.e.",
"spectra are normalized to unity.",
"Both $J/\\psi $ and associated charmed hadron produced in the events plotted are limited to the LHCb fiducial region.",
"It can be seen that at least in the high $p_T^{J/\\psi }$ region the predicted slope is in a good agreement with the experimental data.",
"It should be noticed that $p_T^{J/\\psi }$ distribution in the inclusive $J/\\psi $ production measured by LHCb exhibits significantly more rapid decrease with the $p_T^{J/\\psi }$ growth.",
"Cross section distribution over the $D$ -meson transverse momentum for the same $gg \\rightarrow J/\\psi +c\\bar{c}$ subprocess is given in the Fig.",
"REF Figure: Distribution over the transverse momentum of the DD-meson in the J/ψ+DJ/\\psi + D productioncompared with the LHCb measurement (points for D 0 D^0 and D + D^+-mesons are shown).Curve designations coincide with Fig.",
".and demonstrates good agreement with the LHCb measurement.",
"As in the previous figure, both spectra are normalized to unity.",
"In contrast to the $J/\\psi $ signal, both predicted and measured spectra are similar to those in the inclusive $D$ -meson production at LHCb [48].",
"As in double $J/\\psi $ production, essential angle and rapidity correlations in the $gg \\rightarrow J/\\psi c \\bar{c}$ process are predicted by pQCD.",
"Within collinear approach $J/\\psi $ and $D$ mesons move in the opposite directions in most cases.",
"However no concrete prediction can be made when taking into account transverse gluon motion in the framework of the Pythia generator as the distribution is highly sensitive on the scale selection (see Fig.",
"REF ).",
"Figure: Distribution over the difference of the azimuthal angles of the J/ψJ/\\psi and DD-meson in the J/ψ+DJ/\\psi + D production.Curve designations coincide with Fig.",
".What concerns distribution over the rapidity difference between $J/\\psi $ and $D$ -meson produced, from Fig.",
"REF Figure: Distribution over the difference of rapidities of the J/ψJ/\\psi and DD-meson in the J/ψ+DJ/\\psi + D production.Solid curve corresponds to the LHCb kinematic limits imposed, dashed — to the absence of kinematic limits.one can see that contrary to the two $J/\\psi $ -meson production, LHCb rapidity window appears to be too narrow to observe rapidity correlations predicted in the $gg \\rightarrow J/\\psi c \\bar{c}$ subprocess.",
"We omit discussion of correlations between $D$ and $\\bar{D}$ mesons in the $J/\\psi + D$ associated production as LHCb analysis focuses on events in which one co-produced $D$ meson is observed.",
"The cross section value obtained by LHCb collaboration, $\\sigma ^\\mathrm {exp}(pp \\rightarrow J/\\psi + D^0(D^+,D^+_s,\\Lambda ^+_c)+X)\\approx 300 \\mbox{ nb},$ is several times larger than the SPS prediction of $30\\div 100~\\mathrm {nb}$ .",
"Let us now address to the simultaneous production of $J/\\psi $ and open charm in two gluon-gluon interactions.",
"Within the DPS approach cross section of the associated $J/\\psi $ and $D$ -meson production can be expressed as follows: $\\sigma ^{\\rm DPS}_{J/\\psi D} = \\frac{\\sigma _{J/\\psi } \\sigma _D}{\\sigma _{\\rm eff.",
"}},$ where $\\sigma _{J/\\psi }$ and $\\sigma _D$ are cross sections of the inclusive $J/\\psi $ and $D$ -meson production in the LHCb acceptance correspondingly.",
"Recalculated for the fiducial region discussed ($2<y<4$ , $p_T^D>3~\\mathrm {GeV}$ ), these values are $9~\\mathrm {\\mu b}$ and $380~\\mathrm {\\mu b}$ respectively [60], [48].",
"As always summation with the charge conjugate state is assumed.",
"Unpublished cross section of the $\\Lambda _C$ inclusive production is not included in consideration.",
"Thus the associated $J/\\psi $ and $D$ meson production cross section for the LHCb kinematical region within the DPS model can be estimated as $\\sigma ^{\\rm DPS}_{J/\\psi D} = 240~\\mathrm {nb}.$ As earlier, numerical value of $\\sigma _{\\rm eff.",
"}=14.5~\\mathrm {mb}$ [45], [46] was used.",
"One can see that DPS prediction is several times larger than the SPS one and within uncertainty limits agrees with the experimental value (REF )."
],
[
"Four $D$ -meson production in the LHCb detector",
"In the same LHCb studies [15] production of four $c$ -quarks is investigated.",
"Events in which two open charm hadrons both containing $c$ -quark (or both containing $\\bar{c}$ -quark) are produced in the fiducial region $2.0<y<4.0$ , $3 \\mbox{ GeV} < p_T < 12 \\mbox{ GeV}$ were selected.",
"The calculation within LO of QCD in SPS approach gives for this kinematical region cross section value of $\\sigma ^\\mathrm {pQCD} (gg\\rightarrow c \\bar{c} c \\bar{c})\\sim 50 \\div 500~\\mathrm {nb}$ depending on the scale selection and the $c$ -quark hadronization model used.",
"There is an indication that interaction with sea $c$ -quarks contribute essentially into this process, as well as into the associated production of $J/\\psi $ and $c$ .",
"According to our preliminary estimation, cross section value for the process $c g \\rightarrow c c\\bar{c}$ (plus charge conjugate) is about $\\sigma ^\\mathrm {pQCD} (c g\\rightarrow c c \\bar{c}) \\sim 200 \\div 500~\\mathrm {nb}$ depending on the scale selection and the $c$ -quark hadronization model used.",
"The $c$ -quark structure function has been taken in the form (REF ).",
"The interactions between two sea $c$ -quarks can also be considered.",
"Our estimations show that this process can give a contribution comparable to the two processes mentioned above: $\\sigma ^\\mathrm {pQCD} (c c\\rightarrow c c) \\sim 40 \\div 200~\\mathrm {nb}.$ Thus one can conclude that predictions obtained in the LO pQCD within SPS approach underestimate the experimental value of about $3\\mathrm {\\mu b}$ [15].",
"Also it is worth mentioning that the experimental spectra shapes also can not be exactly reproduced.",
"Nevertheless some futures of the experimental spectra can be understood from such calculations using different kinematical cuts.",
"For example the local minimum near 6 GeV in the experimental cross section distribution over the invariant mass $m_{cc}$ of two charmed particles is probably connected with the cut on the minimum transverse momenta at the LHCb data analysis [15]: $m_{cc}^\\mathrm {loc.min}\\approx 2p_T^\\mathrm {min}.$ Also the rapid decrease of the cross section at $m_{cc}>20$ GeV can be explained by cut on the maximum transverse momenta: $m_{cc}^\\mathrm {cut}\\approx 2p_T^\\mathrm {max}.$ Let us now turn to the DPS contribution to the different $D$ -meson pairs production.",
"Expression (REF ) has to be modified as experimentally observed quantities are inclusive production cross sections of particular types of $D$ -mesons summed together with anti-mesons of the same type.",
"These cross sections can be written down as follows: $\\sigma ^{incl.",
"}_i = \\sigma _1 p^{c\\vee \\bar{c}}_i + \\sigma _2 (2 p^{c\\vee \\bar{c}}_i - (p^{c\\vee \\bar{c}}_i)^2),$ where $\\sigma _1$ and $\\sigma _2$ are cross sections of one and two $c\\bar{c}$ pair production in a single proton-proton collision respectively and $p^{c\\vee \\bar{c}}_i$ is probability that $c$ or $\\bar{c}$ quark transits into detected hadron of type $i$ .",
"In the following we will be interested in events in which both $c$ and $\\bar{c}$ quarks form two $D$ -mesons of particular type in the detector acceptance, or it is done by pairs of identical quarks — $cc$ or $\\bar{c}\\bar{c}$ .",
"In the first case cross section of the $i$ type meson pair production can be written down as $\\sigma ^{diff.",
"}_{i,i} = \\sigma _1 p^{c\\wedge \\bar{c}}_{i,i} + \\sigma _2 (2 p^{c\\wedge \\bar{c}}_{i,i} - (p^{c\\wedge \\bar{c}}_{i,i})^2+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,i})^2/2 ),$ and in the second — as $\\sigma ^{same}_{i,i} = \\sigma _2 ((p^{c\\wedge \\bar{c}}_{i,i})^2+ 2 (p^{c\\wedge \\bar{c}}_{i,i})(p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,i})+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,i})^2/2 ).$ Here $p^{c\\wedge \\bar{c}}_{i,j}$ stands for the probability for $c$ and $\\bar{c}$ quarks from one pair to transit into mesons of type $i$ and $j$ observed in the detector and probabilities for quarks from the different pairs are assumed independent.",
"For the different $i$ and $j$ types of mesons analogous quantities are written down as $\\sigma ^{diff.",
"}_{i,j} &=& \\sigma _1 p^{c\\wedge \\bar{c}}_{i,j} + \\sigma _2 (2 p^{c\\wedge \\bar{c}}_{i,j}- (p^{c\\wedge \\bar{c}}_{i,j})^2+ 2 p^{c\\wedge \\bar{c}}_{i,i}p^{c\\wedge \\bar{c}}_{j,j}+ 2 p^{c\\wedge \\bar{c}}_{i,i} (p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j}) +\\nonumber \\\\&+& 2 p^{c\\wedge \\bar{c}}_{j,j} (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})(p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j})),\\\\\\sigma ^{same}_{i,j} &=& \\sigma _2 (0.5 (p^{c\\wedge \\bar{c}}_{i,j})^2+ 2 p^{c\\wedge \\bar{c}}_{i,i}p^{c\\wedge \\bar{c}}_{j,j}+ 2 p^{c\\wedge \\bar{c}}_{i,i} (p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j}) +\\nonumber \\\\&+& 2 p^{c\\wedge \\bar{c}}_{j,j} (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})+ (p^{c\\vee \\bar{c}}_i - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{i,i})(p^{c\\vee \\bar{c}}_j - p^{c\\wedge \\bar{c}}_{i,j} - p^{c\\wedge \\bar{c}}_{j,j})).$ To solve the equations adduced we will use known inclusive production cross sections of particular $D$ -meson types [48].",
"As LHCb collaboration presents these cross sections in bins of rapidity and transverse momenta, they can be recalculated into the fiducial region discussed ($2<y<4$ , $3\\mathrm {GeV}<p_T^D<12\\mathrm {GeV}$ ).",
"We will also assume that the total $c\\bar{c}$ production cross section in the $7~\\mathrm {GeV}$ proton-proton collisions is known.",
"It was obtained using the Pythia generator calibrated by known inclusive open charm production cross sections in the LHCb acceptance and is equal to $6.1\\pm 0.9\\mathrm { mb}$ [48].",
"According to the DPS approach, cross section of two $c\\bar{c}$ pairs production in a single proton-proton scattering is given by expression (REF ): $\\sigma _2 = \\frac{\\sigma _1^2}{2 \\sigma _{\\rm eff.}}",
"= 1.3\\pm 0.4\\mbox{ mb}.$ However until $\\sigma ^{diff.",
"}_{i,j}$ or $\\sigma ^{same}_{i,j}$ cross sections are measured there is no sufficient information to derive the $p^{c\\wedge \\bar{c}}_{i,j}$ probabilities.",
"So we will assume that rather rigid kinematic cuts imposed result in the smallness of probability to observe both particles produced from a $c\\bar{c}$ pair in the detector.",
"Then neglecting double counting one can write down $p^{c\\wedge \\bar{c}}_{i,i} \\approx (p^{c\\vee \\bar{c}}_i)^2, \\qquad p^{c\\wedge \\bar{c}}_{i,j} \\approx 2 p^{c\\vee \\bar{c}}_i p^{c\\vee \\bar{c}}_j.$ Under the assumptions listed equations (REF ) — (REF ) can be solved.",
"Obtained cross sections of pair production of $D^0$ , $D^+$ and $D_s^+$ mesons are given in Table REF Table: Cross sections of different DD-meson pairs production compared with the LHCb results.together with the values measured by the LHCb.",
"We would like to stress one more time here that summation with the charge conjugate states is everywhere assumed.",
"Generally speaking, good agreement between the DPS predictions and the experimental results is observed.",
"Nonetheless, it is mentioned in [15] that $p_T$ -spectra of $D$ -mesons in pair production significantly differ from those in inclusive open charm production, while similar $p_T$ -behaviour could be expected in the DPS model."
],
[
"Conclusion",
"It is well known that the particle production multiplicity increases with the energy of hadronic interactions.",
"Therefore phenomenon of multiple production should be observed for charmed and beauty particles as well, but at the higher energies due to the larger masses.",
"At the LHC energy yield of charm particles (6.1 mb [48]) is comparable to the common light particle yields, so production of two, three and so on pairs should be expected as well as single $c\\bar{c}$ pair production.",
"Recently the first data on the four $c$ -quark production in the proton-proton interactions have been obtained by the LHCb Collaboration [15].",
"From the theoretical point of view processes in single gluon-gluon interactions (such as $gg \\rightarrow c\\bar{c}c\\bar{c}$ ) are the natural source of multiple charm production.",
"The calculations within LO of pQCD in SPS approach had been done earlier for the process of $J/\\psi $ pair production [41], [42], [2], $J/\\psi +c\\bar{c}$ associated production [16], [17], [18], [19], [20], [21], and for the four $c$ -quarks production.",
"The main conclusion to be drawn from these theoretical studies and from the recent LHCb results is that SPS model used together with the LO pQCD can not describe all the data on multiple charm production.",
"The presented analysis shows that only data on $J/\\psi $ pair production is in satisfactory agreement with SPS LO pQCD predictions.",
"The predictions obtained for the $J/\\psi + D$ associated production, as well as for the four $D$ -meson production underestimate the experimental data in several times.",
"As alternative model we consider the simplest model of double parton scattering (DPS).",
"In the frame work of this approach it is assumed that two $c\\bar{c}$ pairs are produced independently in two different partonic collisions.",
"DPS predictions on the cross section values fairly agree with the experimental data.",
"As it was shown in [61], cross section of pair charm production becomes equal to the ordinary $c\\bar{c}$ cross section at the energy of about 20 TeV.",
"It is interesting to note, that for the double $J/\\psi $ production predictions of SPS and DPS models are fairly close, for the $J/\\psi + D$ associated production the DPS prediction exceeds the SPS one in several times and for the four $D$ -meson production excess is even higher.",
"At first glance it seems amazing as an attempt to explain advantage of the DPS model by combinatorial factor only does not lead to distinction in the channels discussed.",
"Infinitesimality connected with the $\\alpha _S$ constant is same for both SPS and DPS: in SPS the factor is $\\alpha _S^4$ and in DPS — $\\alpha _S^2 \\times \\alpha _S^2 = \\alpha _S^4$ .",
"From our point of view the reasonable explanation lies in the different phase volumes for the SPS and DPS production: in SPS final state contains three particles for the $J/\\psi + c \\bar{c}$ production and four for the $c \\bar{c} c \\bar{c}$ production, so differential cross sections of these processes peak at the larger $\\sqrt{\\hat{s}}$ values at the expense of phase volume factors.",
"By-turn this leads to the smaller gluon luminosity as compared to the $2 \\rightarrow 2$ processes which take place in the DPS model.",
"Authors would like to thank Vanya Belyaev for fruitful discussions.",
"The work was financially supported by Russian Foundation for Basic Research (grant #10-02-00061a), grant of the president of Russian Federation (#MK-3513.2012.2) and by the non-commercial foundation “Dynasty”."
]
] | 1204.1058 |
[
[
"Bell nonlocality in quantum-gravity induced minimal-length quantum\n mechanics"
],
[
"Abstract Different approaches to quantum gravity converge in predicting the existence of a minimal scale of length.",
"This raises the fundamental question as to whether and how an intrinsic limit to spatial resolution can affect quantum mechanical observables associated to internal degrees of freedom.",
"We answer this question in general terms by showing that the spin operator acquires a momentum-dependent contribution in quantum mechanics equipped with a minimal length.",
"Among other consequences, this modification induces a form of quantum nonlocality stronger than the one arising in ordinary quantum mechanics.",
"In particular, we show that violations of the Bell inequality can exceed the maximum value allowed in ordinary quantum mechanics by a positive multiplicative function of the momentum operator."
],
[
"[2]1 2 Bell nonlocality in quantum-gravity induced minimal-length quantum mechanics Pasquale Bosso[][email protected] University of Lethbridge, Department of Physics and Astronomy, 4401 University Drive, Lethbridge, Alberta, Canada, T1K 3M4 Luciano Petruzziello[][email protected] Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy INFN, Sezione di Napoli, Gruppo collegato di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy Fabian Wagner [][email protected] Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland CP3-Origins, University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark Fabrizio Illuminati[][email protected] Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy INFN, Sezione di Napoli, Gruppo collegato di Salerno, Via Giovanni Paolo II, 132 I-84084 Fisciano (SA), Italy Different approaches to quantum gravity converge in predicting the existence of a minimal scale of length.",
"This raises the fundamental question as to whether and how an intrinsic limit to spatial resolution can affect quantum mechanical observables associated to internal degrees of freedom.",
"We answer this question in general terms by showing that the spin operator acquires a momentum-dependent contribution in quantum mechanics equipped with a minimal length.",
"Among other consequences, this modification induces a form of quantum nonlocality stronger than the one arising in ordinary quantum mechanics.",
"In particular, we show that violations of the Bell inequality can exceed the maximum value allowed in ordinary quantum mechanics by a positive multiplicative function of the momentum operator.",
"Introduction – The development of a coherent and self–consistent theory of quantum gravity appears as one of the most ambitious and controversial research goals in contemporary physics.",
"Despite the existence of promising candidate theories, progress in their development and assessment is hampered by the difficulty of probing genuine signatures of quantum gravitational effects at the scales of energy, length and time required in laboratory tests or in astrophysical observations.",
"In this context, much effort has been devoted to the construction of physically motivated effective models with the aim of identifying, quantifying and ideally testing specific features expected to emerge in the regime where quantum and gravitational effects can be simultaneously relevant [1].",
"A significant number of such models imply the existence of a minimal measurable length, thus forbidding the possibility of localizing quantum systems with arbitrary precision [2], [3], [4], [5], [6], [7], [8], [9], [10], [11].",
"On the other hand, a theme of central interest that has stimulated an intense and ongoing debate concerns the relations between quantum gravity and the defining properties of quantum entanglement and quantum nonlocality, since specific aspects of the latter might provide indirect evidence of the former [12], [13], [14], [15], [16], [17].",
"Given this framework, it is important to analyze how the existence of a minimal length can affect the structure of quantum mechanical observables and nonlocal quantum correlations when moving from the high-energy regimes to non-relativistic quantum mechanics.",
"Thus motivated, in the present work we investigate the above issue, showing that in minimal-length quantum mechanics (MLQM) the internal degrees of freedom are modified by acquiring a functional dependence on the motional ones.",
"In turn, this generalization significantly impacts quantum nonlocality and Bell's theorem of ordinary quantum mechanics (QM) [18], [19], [20].",
"In particular, we find that the Tsirel'son upper bound to the violation of Bell inequalities holding in ordinary QM [21], [22], [23], [24], [25] is enhanced in MLQM by a positive multiplicative function of the motional degrees of freedom, thereby yielding a form of quantum nonlocality stronger than in ordinary QM.",
"In principle, such a quantitative prediction is experimentally testable and could be exploited to prove or disprove the existence of a minimal scale of length.",
"Moreover, since the generalized quantum mechanical observables and quantum nonlocal correlations depend on the model of choice, they give rise to different modifications of the Tsirel'son bound, thus potentially allowing for the comparison and discrimination of distinct theoretical approaches to quantum gravity.",
"Deformed commutation relations and generalized uncertainty principle – Assuming the existence of a minimal length, in the non-relativistic limit several phenomenological models of quantum gravity reduce to ordinary quantum mechanics with deformed canonical commutation relations (DCCRs) and, correspondingly, a generalized uncertainty principle (GUP).",
"A physically and mathematically consistent starting point is the Robertson–Schrödinger prescription for non-commuting observables [26], [27], which allows to infer modifications of the Heisenberg algebra from an uncertainty relation featuring a minimal scale of length [28], [29], [30], [31], [32], [33], [34], [35], [36], [37].",
"In particular, concerning the most general scheme that includes coordinate commutativity, isotropy and rotational invariance, it can be shown that the physical phase-space variables satisfy the algebra [38], [29] [xi,xj]=[i,j]=0 , [xi,j]= i[f(2)ij+f(2)ij2] , with $f$ and $\\bar{f}$ being two arbitrary, analytic, dimensionless functions and where $\\hat{\\pi }^2\\equiv \\delta ^{ij}\\hat{\\pi }_i\\hat{\\pi }_j$ (here and in the following, unless otherwise stated, the use of natural units is understood throughout).",
"Verification of the only nontrivial Jacobi identity, $\\left[\\hat{x}^i,\\left[\\hat{x}^j,\\hat{\\pi }_k\\right]\\right]+\\mathrm {cycl.\\,perm.",
"}=0$ , constrains the above functions to satisfy the relation ${f}=\\frac{2\\left(\\log f\\right)^{\\prime }\\hat{\\pi }^2}{1-2\\left(\\log f\\right)^{\\prime }\\hat{\\pi }^2}f \\, ,$ where the prime denotes derivation with respect to $\\hat{\\pi }^2$ .",
"This condition is consistent with the requirement of spatial commutativity only if $2(\\log f)^{\\prime }\\hat{\\pi }^2<1, \\qquad \\forall \\, \\, \\, \\hat{\\pi }^2 \\, .",
"$ In order to define proper observables associated to the internal degrees of freedom (such as the spin) and the corresponding notion of nonlocality in a MLQM equipped with DCCRs and a GUP, we need to verify that the classical limit of such extended quantum mechanics coincides with the one of ordinary quantum mechanics, so as to guarantee the possibility of local hidden variable theories and the validity of Bell inequalities in both settings.",
"The classical limit of effective schemes encompassing DCCRs and a GUP leads to standard, unmodified Poisson brackets at the perturbative level of small corrections to the canonical commutation relations [39].",
"Here, the same result applies to the general case of corrections behaving as a function $f(\\hat{\\pi }^2)$ of the momentum operator squared.",
"Since the functions $f$ and $\\bar{f}$ must be dimensionless, their argument must be made dimensionless by introducing a characteristic energy scale.",
"Such quantum gravitational threshold is commonly identified by the Planck mass $m_\\text{p}=\\sqrt{\\hbar \\,c/G_N}$ , with $\\hbar $ being the reduced Planck constant, $G_N$ the gravitational constant and $c$ the speed of light in vacuum, so that the argument of the function $f$ is rescaled accordingly $f = f\\left(\\frac{\\hat{\\pi }^2}{m_\\text{p}^2c^2}\\right) \\, .$ Ordinary QM is recovered in the low-energy limit, yielding $\\lim _{\\hat{\\pi }^2\\rightarrow 0}f=1$ and $\\lim _{\\hat{\\pi }^2\\rightarrow 0}\\bar{f}=0$ .",
"Due to the functional dependence of Eq.",
"(REF ), the above limit is equivalent to the formal one $G_N\\rightarrow 0$ , which amounts to neglect the presence of gravity.",
"Similarly, a classical dynamical theory (CT) is retrieved in the formal limit $\\hbar \\rightarrow 0$ .",
"Since in Eq.",
"(REF ) $G_N$ comes in pair with the inverse of $\\hbar $ , the resulting CT is independent of the gravitational constant.",
"Indeed, for $G_N\\rightarrow 0$ and $\\hbar \\rightarrow 0$ simultaneously, the Poisson brackets stemming from the DCCRs can only either be divergent/vanishing (two ill-defined scenarios) or exhibit a constant correction.",
"In the latter case, they are related to the Poisson brackets of classical mechanics by a canonical transformation; correspondingly, a well-defined classical limit is necessarily trivial.",
"The above discussion may be summarized by the following illustrative diagram: $\\begin{tikzcd}\\text{MLQM} [rd,\"\\hbar \\rightarrow 0\"^{\\prime }] [r, \"G_N\\rightarrow 0\"] & \\text{QM} [d,\"\\hbar \\rightarrow 0\"] \\\\& \\text{CT}\\end{tikzcd}$ which underpins the non-commutativity of the limits $G_N\\rightarrow 0$ and $\\hbar \\rightarrow 0$ .",
"After having established the well-defined behavior of the classical limit, we move to discuss how Eqs.",
"(REF ) entail the presence of a minimal scale of length.",
"Recalling the Robertson–Schrödinger derivation of the uncertainty relation for two non-commuting observables $\\hat{x}^i$ and $\\hat{\\pi }_i$ , namely $\\Delta x^i\\ge \\frac{|\\langle [\\hat{x}^i,\\hat{\\pi }_i]\\rangle |}{2(\\Delta \\pi _i)}= \\frac{1}{2\\Delta \\pi _i} \\left|\\langle f(\\hat{\\pi }^2) \\rangle + \\left\\langle {f}(\\hat{\\pi }^2) \\frac{(\\hat{\\pi }^i)^2}{\\hat{\\pi }^2}\\right\\rangle \\right| \\, ,$ we note that, depending on the choice of $f$ , the r.h.s.",
"of Eq.",
"(REF ) may feature a global minimum, thus implying a nonvanishing minimal length.",
"We obtain an immediate visualization of this occurrence by expanding $f$ and $\\bar{f}$ to leading order in $\\hat{\\pi }^2/m_\\text{p}^2$ in a generic scheme of MLQM f(2) 1 + 2mp2 , f(2) 2 2mp2 , where the dimensionless deformation parameter $\\beta $ characterizes the specific model considered.",
"Then, to leading order: $\\left|\\langle f(\\hat{\\pi }^2) \\rangle + \\left\\langle {f}(\\hat{\\pi }^2) \\frac{(\\hat{\\pi }^i)^2}{\\hat{\\pi }^2}\\right\\rangle \\right|\\\\\\gtrsim \\left|1 + \\beta \\frac{\\sum _j\\Delta \\pi ^j\\Delta \\pi _j}{m_\\text{p}^2} + 2 \\beta \\frac{(\\Delta \\pi ^i)^2}{m_\\text{p}^2}\\right| \\, ,$ but since $\\sum _j\\Delta \\pi _j\\Delta \\pi ^j\\ge (\\Delta \\pi _i)^2$ , it follows that $\\Delta x^i\\ge \\frac{1}{\\Delta \\pi _i}\\left[1+3\\beta \\left(\\frac{\\Delta \\pi _i}{m_p}\\right)^2\\right] \\, .$ This expression features a global minimum for $\\Delta x^i = \\sqrt{3 \\beta }/m_p$, which encompasses the existence of a limited spatial resolution.",
"Returning to Eqs.",
"(REF ) and resorting to the Einstein convention on repeated indices, we may introduce suitable auxiliary operators in order to recover the canonical symplectic structure.",
"We thus define the canonical variables $\\hat{X}^i$ and $\\hat{\\Pi }^i$ as $\\hat{X}^i=\\hat{x}^i,\\hspace{28.45274pt}\\hat{\\Pi }_i=\\frac{\\hat{\\pi }_i}{f} \\, ,$ which satisfy the Heisenberg algebra $\\left[\\hat{X}^i,\\hat{X}^j\\right]=\\left[\\hat{\\Pi }_i,\\hat{\\Pi }_j\\right]=0 \\, ,\\hspace{28.45274pt}\\left[\\hat{X}^i,\\hat{\\Pi }_j\\right]=i\\delta ^i_j \\, .$ The construction of canonical and physical operators extends to angular momentum [40].",
"In the context of a modified quantum mechanics with DCCRs, the physical orbital angular momentum operator is defined as $\\hat{l}_i\\equiv \\epsilon _{ijk}\\hat{x}^j\\hat{\\pi }^k$ .",
"This operator satisfies the deformed algebra $\\left[\\hat{l}_i,\\hat{l}_j\\right]=i\\varepsilon _{ijk}\\hat{l}^kf\\left(\\hat{\\pi }^2\\right) \\, .$ The ordinary SO(3) algebra is recovered by introducing the canonical angular momentum operator $\\hat{L}_i=\\frac{\\hat{l}_i}{f} \\, .$ It is important to observe that, since $\\hat{\\pi }^2$ is the Casimir invariant of the algebra, there is no operator ordering ambiguity.",
"These considerations suggest an analogous reformulation of the intrinsic angular momentum in MLQM.",
"Deformed spin operator – Let us consider a spinor field interacting with a classical magnetic field.",
"In momentum space, the Dirac equation describing a relativistic spin-$1/2$ particle reads $i\\partial _t\\psi =\\left(\\gamma ^i\\hat{\\pi }_i+\\gamma ^0m\\right)\\psi \\, ,$ with the gamma matrices in the Dirac representation $\\gamma ^i=\\begin{pmatrix}0&\\sigma ^i\\\\\\sigma ^i&0\\end{pmatrix},\\hspace{28.45274pt}\\gamma ^0=\\begin{pmatrix}{1}&0\\\\0&-{1}\\end{pmatrix}.$ In order to account for an external magnetic field such that $A_\\mu =\\left(0,A_i\\right)$ [41] and by means of the minimal coupling prescription, the physical momenta in Eq.",
"(REF ) are replaced by the operators $\\hat{p}_i=\\hat{\\pi }_i-eA_i$ that preserve the $U(1)$ gauge invariance.",
"This transformation modifies the underlying gauge symmetry in such a way that the theory remains invariant under the modified gauge transformations.",
"Splitting the spinor field into its particle/antiparticle components $\\psi =(\\varphi ,\\chi )$ , we can consider the non-relativistic limit by singling out a mass-dependent phase $\\psi \\rightarrow e^{-imt}\\psi $ and employing the non-relativistic limit $|m\\psi |\\gg |\\partial _t\\psi |.$ Within this framework, the generalized Schrödinger equation governing the dynamics of the particle wave function $\\varphi $ in MLQM reads $i\\partial _t\\varphi =\\frac{\\sigma ^i\\hat{p}_i\\sigma ^j\\hat{p}_j}{2m}\\varphi \\, .$ Upon introducing the magnetic field $B^i=\\epsilon ^{ijk}\\partial _jA_k$ along with the spin operator $\\hat{S}_i=\\sigma _i/2,$ we obtain that, up to $\\mathcal {O}(e)$ , the effective state vector dynamics in MLQM reads $i\\partial _t\\varphi =\\frac{f^2}{2m}\\hat{\\Pi }^2-ef\\left(\\hat{L}_i+2\\hat{S}_i\\right)B^i=\\frac{\\hat{\\pi }^2}{2m}-e\\left(\\hat{l}_i+2\\hat{s}_i\\right)B^i \\, ,$ where we have defined the deformed spin operator as $\\hat{s}_i\\equiv f \\hat{S}_i.$ It is worth observing that, when written in terms of the physical operators, no correction proportional to $f$ appears in the dynamical equation.",
"In light of the above, the physical intrinsic and orbital angular momentum operators that the external magnetic field couples to are, respectively, $\\hat{s}_i$ and $\\hat{l}_i$ .",
"Consequently, the deformation of the spin algebra coincides with the deformation of the orbital angular momentum algebra (REF ), namely $\\left[\\hat{s}_i,\\hat{s}_j\\right]=i\\hbar \\varepsilon _{ijk}\\hat{s}^kf\\left(\\hat{\\pi }^2\\right) \\, .$ This observation implies that expectation values of spin-dependent observables on any quantum state must account for the presence of the momentum operator.",
"Therefore, given a quantum state that is a tensor product of spin- and momentum-dependent components, i. e.: $|\\Psi \\rangle =|\\psi _s\\rangle \\otimes |\\psi _\\pi \\rangle $ , we have that in MLQM $\\hat{s}_i|\\Psi \\rangle =\\hat{S}_i|\\psi _s\\rangle \\otimes f(\\hat{\\pi }^2)|\\psi _\\pi \\rangle \\, ,$ which yields $\\langle \\hat{s_i}\\rangle = {f(\\hat{\\pi }^2)}\\langle \\hat{S}_i\\rangle \\, .$ Quantum nonlocality – A most fundamental question arising in MLQM concerns how momentum-dependent state expectations of the form of Eq.",
"(REF ) modify quantum nonlocality as ruled by Bell's theorem [18], [19], [20].",
"In the standard framework of Bell nonlocality, two parties, $A$ and $B$ , perform causally separated experiments on a bipartite system.",
"Let $\\hat{A}$ and $\\hat{B}$ denote two dichotomous observables associated with party $A$ and party $B$ , $(a,a^{\\prime })$ and $(b,b^{\\prime })$ two sets of different experimental settings associated with party $A$ and party $B$ , respectively, and $C( \\cdot , \\cdot )$ the correlation functions between outcomes corresponding to the four available experimental configurations.",
"Within a local hidden variable theory, a suitable combination $\\mathcal {S}$ of such correlations must be bounded according to the following Clauser-Horne-Shimony-Holt (CHSH) inequality [42]: $\\mathcal {S}=\\left|C\\left(a,b\\right)-C\\left(a,b^{\\prime }\\right)+C\\left(a^{\\prime },b\\right)+C\\left(a^{\\prime },b^{\\prime }\\right)\\right|\\le 2 \\, .$ In the context of ordinary QM one has $C(a,b)=\\langle \\hat{A}(a)\\hat{B}(b)\\rangle $ (and analogously for the remaining correlations) and the inequality Eq.",
"(REF ) is violated.",
"Remarkably, although the highest possible, absolute algebraic value for $\\mathcal {S}$ is $\\mathcal {S}^{max} = 4$ , within QM the maximum achievable value is $\\mathcal {S}^{max}_{QM} = 2\\sqrt{2}$ , as first proved by Tsirel'son [21], [22].",
"This bound places a strong limit to the degree of nonlocality featured by standard QM.",
"Let us now consider an experimental realization involving two spin-$1/2$ observables.",
"We can choose any one of the four maximally entangled, two-qubit Bell states as the initial state, for instance $|\\Psi \\rangle =\\left(|0,1\\rangle -|1,0\\rangle \\right)/\\sqrt{2}$ .",
"Next, consider the four dichotomous observables A(a) = Sz1 , A(a') = Sx1 , B(b) = -1(Sz + Sx)2 , B(b') = 1(Sz - Sx)2 .",
"With this choice, one finds that $\\mathcal {S}$ saturates exactly the Tsirel'son bound $2\\sqrt{2}$ , thereby realizing the largest degree of nonlocal correlation achievable within standard QM and showing that maximal entanglement is equivalent to maximal Bell nonlocality on pure states.",
"Various physical principles have been invoked to justify the existence of such a strict upper limit to nonlocal quantum mechanical correlations between spatially-separated events in ordinary QM [23], [24], [25].",
"In the framework of MLQM, the basic structure of quantum nonlocality stands firm; in particular, Bell's theorem and the CHSH inequality of standard QM continue to hold unmodified by virtue of the non-commutativity of the classical and non-gravitational limits, as summarized in (REF ).",
"On the other hand, Tsirel'son bound must be generalized, as the spin correlation functions in CHSH-type experiments become momentum-dependent.",
"Indeed, by virtue of Eq.",
"(REF ) and upon replacing the canonical spin operators with the physical ones $(\\hat{S}_x,\\hat{S}_z)\\rightarrow (\\hat{s}_x,\\hat{s}_z)$ in Eqs.",
"(REF ) and (REF ), one has $C(a,b)={f^2(\\hat{\\pi }^2)}\\langle \\hat{A}(a)\\hat{B}(b)\\rangle $, and similarly for the other correlations.",
"Therefore, after some algebra, one finds: $\\mathcal {S}_{MLQM}^{max} = {f^2\\left(\\hat{\\pi }^2\\right)} \\, \\mathcal {S}^{max}_{QM} = 2\\sqrt{2} \\,{f^2\\left(\\hat{\\pi }^2\\right)} \\, .$ Equation (REF ) shows that non-relativistic MLQM derived from effective, low-energy models of quantum gravity features nonlocal quantum correlations that become scale-dependent via functions of the square of the momentum operator and that are generically stronger than the ones holding in standard, non-relativistic QM without minimal-length induced deformations.",
"We might speculate that the modified Tsirel'son bound $\\mathcal {S}_{MLQM}^{max}$ might saturate the absolute algebraic limit of $\\mathcal {S}^{max} = 4$ as one approaches the Planck regime; however, the question must remain open, since the validity of the existing low-energy effective models of quantum gravity breaks down at the Planck scale.",
"Furthermore, it is likely that in the context of MLQM the very definition of the absolute algebraic limit might need to be generalized accordingly for each different space-time scale.",
"Given the above words of caution, we can provide some explicit (albeit very preliminary and extremely conservative) estimates of the correcting factor $f(\\hat{\\pi }^2)$ in Eq.",
"(REF ).",
"Let us first consider the linear correction deduced from models of the Doubly Special Relativity type [43], [44]: ${f^2(\\hat{\\pi }^2)}\\simeq 1+2\\alpha \\frac{{\\sqrt{\\hat{\\pi }^2}}}{m_{\\text{p}}}=1+\\alpha \\frac{\\sqrt{2ME_{\\text{kin}}}}{m_{\\text{p}}} \\, ,$ where $M$ is the particle mass, $E_\\text{kin}$ the kinetic energy, and $\\alpha $ the linear deformation parameter which is typically considered to be of order unity, i. e., $\\alpha \\sim \\mathcal {O}(1)$ .",
"Assuming for instance that a CHSH-type experiment is performed at the scales of systems of muons with mass $M \\sim \\mathcal {O}(10^{-1})$ GeV and kinetic energy $E_{\\text{kin}}\\sim \\mathcal {O}(10^{-2})$ GeV so that we are well within the non-relativistic regime, we find for the relative correction ratio in orders of magnitude $\\Delta \\mathcal {S} = \\frac{\\mathcal {S}_{MLQM}^{max} - \\mathcal {S}^{max}_{QM}}{\\mathcal {S}^{max}_{QM}} \\sim \\mathcal {O}(10^{-20}) \\, .$ On the other hand, if we consider a string-induced MLQM, we have ${f^2(\\hat{\\pi }^2)}\\simeq 1+2\\beta \\frac{{\\hat{\\pi }^2}}{m_{\\text{p}}^2} \\, .$ Assuming again a deformation parameter $\\beta \\sim \\mathcal {O}(1)$ and the same experiment at the same scale as in the previous example, the relative correction is $\\Delta \\mathcal {S}\\sim \\mathcal {O}(10^{-40})$ .",
"In all cases, considering larger values of the deformation parameter and/or higher energy scales leads obviously to significantly larger corrections.",
"Finally, it is worth emphasizing that the DCCRs between position and momentum operators do not contradict any fundamental axiom of quantum theory.",
"Therefore, Eq.",
"(REF ) represents a generalization rather than a violation of the Tsirel'son bound because it provides the correct prescription to account for the physical observables in the presence of quantum gravitational effects.",
"Discussion and outlook – In this work, we have discussed how the existence of a minimal scale of length predicted by models of quantum gravity implies a momentum-dependent modification of the internal degrees of freedom, and thus a minimal-length induced modification of non-relativistic quantum mechanics.",
"We have explored the consequences of this quantum-gravity induced deformation of ordinary quantum mechanics on Bell's theorem and quantum nonlocality.",
"We have found that the Tsirel'son upper bound on the violation of the CHSH inequalities that holds in ordinary quantum mechanics is exceeded in minimum-length quantum mechanics, thus leading to stronger forms of nonlocal quantum correlations.",
"Besides its relevance for fundamental quantum theory and quantum information science, this enhanced, scale-dependent quantum nonlocality could be exploited to put to the test distinct effective models of quantum gravity.",
"Our results seem to suggest that the existence of a minimal length might be at odds with one or more of the principles that have been recently formulated to motivate the very existence of the original value of the Tsirel'son bound in ordinary quantum mechanics [23], [24], [25].",
"Either the axioms of no-advantage for nonlocal computation postulate [23], information causality [24] and macroscopic locality [25] are amenable to further generalizations, or they might be intimately tied to the structure of ordinary quantum mechanics with no minimal length.",
"In this context, further in-depth investigations will be needed in order to clarify the relation between quantum nonlocality and quantum gravity phenomenology encompassing the existence of a minimal scale of length.",
"Acknowledgements – F.I.",
"and L.P. acknowledge support by MUR (Ministero dell'Università e della Ricerca) under the project PRIN 2017 “Taming complexity via QUantum Strategies: a Hybrid Integrated Photonic approach” (QUSHIP) Id.",
"2017SRNBRK.",
"F.W.",
"thanks the quantum gravity group at the University of Southern Denmark for its kind hospitality and the Polish National Research and Development Center (NCBR) for support under the project ”UNIWERSYTET 2.0.",
"– STREFA KARIERY”, POWR.03.05.00-00-Z064/17-00 (2018-2022).",
"P.B., L.P. and F.W.",
"acknowledge networking support by the COST Action CA18108."
]
] | 2207.10418 |
[
[
"A Wavelet Transform and self-supervised learning-based framework for\n bearing fault diagnosis with limited labeled data"
],
[
"Abstract Traditional supervised bearing fault diagnosis methods rely on massive labelled data, yet annotations may be very time-consuming or infeasible.",
"The fault diagnosis approach that utilizes limited labelled data is becoming increasingly popular.",
"In this paper, a Wavelet Transform (WT) and self-supervised learning-based bearing fault diagnosis framework is proposed to address the lack of supervised samples issue.",
"Adopting the WT and cubic spline interpolation technique, original measured vibration signals are converted to the time-frequency maps (TFMs) with a fixed scale as inputs.",
"The Vision Transformer (ViT) is employed as the encoder for feature extraction, and the self-distillation with no labels (DINO) algorithm is introduced in the proposed framework for self-supervised learning with limited labelled data and sufficient unlabeled data.",
"Two rolling bearing fault datasets are used for validations.",
"In the case of both datasets only containing 1% labelled samples, utilizing the feature vectors extracted by the trained encoder without fine-tuning, over 90\\% average diagnosis accuracy can be obtained based on the simple K-Nearest Neighbor (KNN) classifier.",
"Furthermore, the superiority of the proposed method is demonstrated in comparison with other self-supervised fault diagnosis methods."
],
[
"Introduction",
"Rotating machinery plays a vital role in modern industries, so its condition monitoring and health management are of great importance.",
"According to statistics, about 45%-55% of rotating machinery and equipment failure is caused by damage to the bearing part [1].",
"Therefore, timely and accurate bearing fault diagnosis has always been highly demanded to enhance machine reliability [2], [3].",
"Traditional bearing fault diagnosis methods are based on the physical model, and the specific fault components are analyzed by various signal processing techniques [4], [5], [6].",
"However, these physical models are not universal in complex, high-dimensional systems.",
"With the rapid development of industrial technology, the structure of rotating machinery becomes more and more complicated, and the limitations of conventional fault diagnosis methods are gradually highlighted.",
"In recent years, benefited from the advances in industrial computer and sensor technology, data-driven intelligent fault diagnosis method has attracted more and more attention from researchers due to the great merits of high accuracy and low requirement for prior knowledge [7].",
"Data-driven fault diagnosis methods can be divided in two categories: Machine Learning (ML) based approach and Deep Learning (DL) based approach.",
"Currently, many ML models such as Support Vector Machine (SVM) [8], [9], [10], Self-Organized Map (SOM) [11], [12], Auto-Encoder (AE) [13], [14], and Radial Basis Function (RBF) neural network [15], [16] have been extensively employed in the field of fault diagnosis.",
"Nevertheless, most of these approaches still require manual design features, which means limitations in adaptive feature extraction.",
"DL based techniques can automatically learn the feature representation from mass data according to the given task and have a strong feature extraction ability.",
"These approaches have been widely developed, and numerous promising results have been acquired [17].",
"Assorted neural network architectures and enhanced techniques, such as Deep Belief Network (DBN) [18], Convolutional Neural Network (CNN) [19], [20], [21], Recurrent Neural Network (RNN) [22], [23], Attention mechanism [24], Transformer [25], [26] and their variants [27], [28] have also been generally exploited for the fault diagnosis.",
"Jie et al.",
"[29] established a novel Gaussian-Bernoulli deep belief network (GDBN) model for intelligent fault diagnosis, where the graph regularization and sparse features learning are embedded.",
"Combining the advantages of attention mechanism, Squeeze-and-Excitation Network (SENet) [30], and soft threshold, Zhao et al.",
"[31] developed the Deep Residual Shrinkage Networks (DRSN).",
"This new deep learning model can achieve a high fault diagnosis accuracy under strong noise interference.",
"Aiming at the problem that the connection of the local fragments (namely quasi-periodicity) in the measured vibration signals is easy to be neglected, Gao et al.",
"[32] performed a novel weak fauld diagnosis method for the rolling bearings based on the Long Short Term Memory (LSTM) network and multichannel Continuous Wavelet Transform (MCCWT).",
"Compared to the traditional CWT, MCCWT converts the original sample space into a multichannel representation, which improves the feature extraction capability of the LSTM.",
"In recent work, Ding et al.",
"[33] applied the Transformer architecture to fault diagnosis of rolling bearings.",
"Based on the view that the time-frequency map obtained by signal processing is formed by splicing the instantaneous spectrum of the signal over a period of time, the time-frequency Transformer (TFT) model is presented.",
"Through experiments, the effectiveness and superiority of the proposed TFT are validated.",
"In brief, the advances in DL based fault diagnosis methods fully prove the potency of the emerging data-driven algorithm for processing complex mechanical systems [34].",
"Satisfactory fault diagnosis results reported in the above literature are premised on the sufficient labeled data.",
"These DL based approaches are based on the paradigm of supervised learning for end-to-end training, thus acquiring the features with generalization ability [35].",
"However, in real engineering environment, labeling large amounts of measured data may be time-consuming and costly.",
"Besides, considering the complexity and uncertainty of the machinery system, in many cases, the fault mode corresponding to the acquired vibration signal is virtually unidentified, which leads that supervised learning is not effective enough.",
"Therefore, the problem of fault diagnosis with limited labeled data has aroused extensive attention from researchers.",
"Currently, two main approaches to this problem are dataset extension and exploring unlabeled data.",
"Dataset extension aims to generate the additional \"labeled\" data based on the limited labeled samples through a technique similar to interpolation, including various data augmentation methods [36], [37], Generative Adversarial Network (GAN) [38], [39], [40], etc.",
"Li et al.",
"[41] designed a data augmentation method that combined different signal processing techniques such as masking noise, signal translation, stretching, etc.",
"The experimental results show that the diagnosis performance of the DL model can get a promotion from more generated samples.",
"By employing the Sparse Auto-Encoder (SAE) to reduce the dimension of the original data, Ma et al.",
"[42] improved the traditional GAN and proposed the Sparsity-Constrained Generative Adversarial Network (SCGAN), which better converges to Nash equilibrium, and higher diagnosis accuracy can be achieved with limited labeled data.",
"Different from the above research, Wang et al.",
"came up with a novel dataset extension method based on the Sub-Pixel Convolutional Neural Network (ESPCN) [43].",
"Through this method, generated data with high-resolution can be acquired.",
"Experimental results of gearbox and bearing datasets show that the proposed method has strong feasibility to carry out data augmentation for fault diagnosis of rotating machines under the speed fluctuation condition.",
"Overall, sample size can be enlarged using dataset extension, and the lack of labeled data is alleviated.",
"However, fake samples generated based on the dataset extension technology will inevitably be similar to the real samples, resulting in lacking data diversity, which will increase the risk of overfitting.",
"In addition, mass unlabeled data without precise machine health conditions, which can be easily collected in general, does not get effective utilization in these methods.",
"To solve these problems, fault diagnosis methods under the limited labeled samples case by exploring unlabeled data are presented, known as unsupervised learning [2], [44], self-supervised learning [45], [46], [47], and semi-supervised learning [48], [49], [50].",
"Zhang et al.",
"[51] offered an unsupervised framework with Reconstruction Sparse Filtering (RSF) for rolling bearing diagnosis.",
"The basis vectors are constrained explicitly by a Soft-Reconstruction Penalty (SRP), enabling RSF to learn a group of independent basis vectors to extract dissimilar features without applying any labeled sample.",
"Li et al.",
"[52] leveraged deep InfoMax (DIM) to improve the generalization ability of the features extracted by a CNN encoder, which can alleviate the overfitting problem when the labeled fault samples are limited.",
"Li et al.",
"[53] designed a three-stage semi-supervised fault diagnosis conjoin unsupervised clustering and supervised fine-tuning.",
"Even only one labeled sample for each class, a high testing accuracy is obtained by this method.",
"The above research results demonstrate the feasibility of exploring unlabeled data to address the challenging diagnostic tasks with limited labeled samples.",
"This paper focuses on the problem of bearing fault diagnosis with availabilities of limited labeled data and sufficient unlabeled data.",
"Under this proposition, a Wavelet Transform and self-supervised learning-based framework is proposed.",
"Herein, the Vision Transformer is adopted as an encoder to extract the feature vectors, which possesses the preponderance of great parallelism and strong scalability.",
"Accordingly, the projector head is introduced on the top of the encoder to establish the model network for self-supervised learning, whose output is pseudo labels.",
"Based on this, a pretext task called \"local to global correspondence\" is constructed by initializing the teacher and student networks with the same architecture.",
"Moreover, the centering and sharpening operations included in the self-distillation with no labels algorithm are integrated into the proposed framework to train the model networks, which enable us effectively avoid the mode collapse during the training procedure.",
"Furthermore, the Wavelet Transform technique is used to convert the original time-domain vibration signals into time-frequency maps.",
"This preprocessing method helps the encoder learn a better feature representation.",
"Experiments on two bearing fault datasets where only 1% labeled data is contained are implemented to validate the proposed method, and over 90% average testing accuracy is achieved."
],
[
"Proposed method",
"The proposed method mainly comprises three major parts: (1) preprocessing pipeline of the raw vibration signals; (2) model network for feature representation; (3) training algorithm for the model network.",
"In this section, we will illustrate our proposed methodology in detail."
],
[
"Wavelet Transform and preprocessing pipeline",
"In the fault diagnosis of rotating machinery, background noise is pervasive due to sensors' noise input and environmental factors, which will interfere with the time-domain signals.",
"Additionally, the acquired vibration signals are usually non-stationary due to speed fluctuation and fault.",
"In this case, neither time-domain analysis nor frequency-domain analysis is suitable.",
"Therefore, it is necessary to convert the raw vibration signals into time-frequency representation (TFR).",
"In this paper, we utilize the Wavelet Transform (WT) as the time-frequency domain analysis method.",
"The WT of a given vibration signal $x(t)$ can be defined as $WT(a, \\tau ) = \\frac{1}{\\sqrt{a}}\\int _{-\\infty }^{+\\infty }x(t)\\psi (\\frac{t-\\tau }{a})dt = \\int _{-\\infty }^{+\\infty }x(t)\\psi _{a, \\tau }(t)dt$ where $a$ is scaling parameter, $\\tau $ is time translation parameter.",
"$\\psi _{a, \\tau }(t) = \\frac{1}{\\sqrt{a}}\\psi (\\frac{t-\\tau }{a})$ is wavelet basis function of the WT.",
"Fig.REF shows the TFR of the raw vibration signals based on the WT.",
"We can find that the frequency-domain signals of each time step are well extracted.",
"Nevertheless, it should be noted that the size of TFR is related to the number of sampling points, which means that the shape of TFR may be uncertain.",
"Besides, numerous sampling points can result in a huge TFR, making subsequent calculations difficult.",
"To fix the shape of TFR, as shown in Fig.REF , we introduce a resize method.",
"The specific approach is as follows: first, the amplitude range of TFR is scaled to [0, 1] by normalization.",
"Then, the amplitude values of TFR are mapped to color values.",
"Finally, the size of TFR is fixed at $224 \\times 224$ by cubic spline interpolation.",
"The TFR after the above treatments is denoted as time-frequency map (TFM), whose shape is $224 \\times 224 \\times 3$ .",
"TFM can be directly used as input to the model network.",
"Figure: Preprocessing pipeline of the original vibration signalWe need two networks in the following training process: teacher network and student network.",
"In this paper, both of them are collectively called the model network.",
"As shown in Fig.REF , the model network is consists of two stages: an encoder and a projector head.",
"The encoder provides a feature extraction and representation from the input TFM.",
"A variety of DL models, such as Residual Neural Network (ResNet) and Vision Transformer (ViT) [54], can be used as encoder's backbone, and in this paper, we employ the ViT as our backbone.",
"The impact of different backbones on the fault diagnosis performance will be discussed later.",
"For the sake of presentation, the input TFM is denoted as $x \\in \\mathbb {R}^{H \\times W \\times C}$ ignoring the batch size, where $(H, W)$ is the shape of the TFM, $C$ is the number of channels.",
"As described previously, $H = W = 224$ , and $C$ is 3.",
"To conform the input form of the standard Transformer, we reshape the TFM into a sequence of flattened 2D patches, denoted as $x_{p} \\in \\mathbb {R}^{{N}\\times ({P^2 \\times C})}$ ， where $(P, P)$ is the resolution of each TFM patch, $N$ is the number of patches, and $N = HW/P^2$ .",
"This process is visually illustrated in Fig.REF , which divides the input TFM evenly into patches of a specified size.",
"Then, we reorder these patches into a sequence, as shown in Eq.",
"(REF ) $x \\in \\mathbb {R}^{H \\times W \\times C} \\stackrel{\\text{reshape}}{\\longrightarrow } x_{p} = [x_{p}^{1}, x_{p}^{2}, ..., x_{p}^{N}] \\in \\mathbb {R}^{{N}\\times ({P^2 \\times C})}$ where $x_{p}^{i}$ is the $i$ th patch, and $i = 1, 2, 3, ..., N$ .",
"Then, like the procedure of word embedding, we flatten each patch and map them to the high-dimensional embedding space through a learnable linear projection, as given in Eq.",
"(REF ) $z_{0} = [x_{p}^{1}, x_{p}^{2}, ..., x_{p}^{N}] \\cdot W_{emd} \\in \\mathbb {R}^{N \\times d}$ where $W_{emd} \\in \\mathbb {R}^{(P^{2} \\times C ) \\times d}$ is a trainable embedding matrix, $d$ is the embedding dimension.",
"It should be noted that all patches share the same embedding matrix, so Eq.",
"(REF ) is mathematically equivalent to a 2D convolution operation.",
"Then, since the Transformer does not contain the position information, position encoding is added to retain the absolute and relative position information of the patches.",
"And beyond that, similar to BERT's class token, we present a learnable vector (denoted as $x_{class}$ ) to serve as the feature representation, which can be obtained by Eq.",
"(REF ) $z_{0} \\leftarrow [x_{class}; z_{0}] + E_{pos}$ where $x_{class} \\in \\mathbb {R}^{d}$ , $E_{pos} \\in \\mathbb {R}^{(N+1) \\times d}$ .",
"Both of them are learnable parameters.",
"Eq.",
"(REF ) to Eq.",
"(REF ) are collectively called patch embedding, as shown in Fig.REF .",
"$z_{0}$ is referred to as \"embedding sequence.\"",
"Figure: The architecture of the model network.The following part of the encoder can be regarded as a feature extraction structure, which takes the embedding sequence $z_{0}$ as input.",
"In the ViT architecture, the core part of feature extraction is the Transformer layer, which comprises multiple Transformer basic blocks stacked on top of each other.",
"Fig.REF introduces the structure of the Transformer base blocks, including Multi-head Self-attention mechanism (MSA) and Multilayer perceptron blocks (MLP).",
"MSA is the most critical definition in Transformer basic blocks, which is based on the attention mechanism.",
"The attention mechanism can be explained in terms of soft addressing.",
"Consider that an element in the memory is composed of a key ($K$ ) and value ($V$ ).",
"Currently, there is a query ($Q$ ), and we need to pick out $V$ based on the similarity between $Q$ and $K$ .",
"Note that for the same $Q$ value, instead of hard addressing, we may take all the $V$ s in the memory and sum them weighted based on their importance.",
"The importance of $V$ is measured by the comparability between $Q$ and its corresponding $K$ , which is denoted as the Attention Distribution (AD).",
"The calculation method of AD is called the score function, such as Scaled Dot-Product Attention, Bahdanau Attention, and Content-based Attention.",
"In this paper, we adopt the Scaled Dot-Product Attention in our backbone, which is described as $attn(Q, K, V) = softmax(\\frac{QK^{T}}{\\sqrt{d_{K}}})V$ where $d_{K}$ is the embedding dimension of $K$ .",
"$\\frac{1}{d_{K}}$ is a scaling factor for stabilizing the gradient.",
"However, the attention mechanism in Eq.",
"(REF ) cannot extract the information in the embedding sequence under the different subspace.",
"Then, to avoid this deficiency, similar to the group convolution operation, researchers have introduced the Multi-head attention (MHA)mechanism, which can be given as $\\begin{aligned}MHA(Q, K, V) & = [head_{1}, head_{2},..., head_{h}] \\cdot W_{O} \\\\\\text{where} \\; head_{i} & = attn(QW_{Q}^{i}, KW_{K}^{i}, VW_{V}^{i})\\end{aligned}$ where $W_{O} \\in \\mathbb {R}^{hd_{V} \\times d}$ , $W_{Q}^{i} \\in \\mathbb {R}^{d \\times d_{K}}$ , $W_{K}^{i} \\in \\mathbb {R}^{d \\times d_{K}}$ , $W_{V}^{i} \\in \\mathbb {R}^{d \\times d_{V}}$ , $h$ is the number of $head$ and $d_{V}$ is the dimension of values.",
"Furthermore, when $Q = K = V$ , Eq.",
"(REF ) is also called Multi-head Self-attention mechanism, namely MSA.",
"The output of MSA in $l$ th Transformer basic block (denoted as $z_{l}^{MSA}$ ) with residual connection and Layer normalization (LN) can be described as $\\begin{aligned}z_{l}^{MSA} & = MSA(LN(z_{l-1})) + z_{l-1} \\\\& = MHA(LN(z_{l-1}), LN(z_{l-1}), LN(z_{l-1})) + z_{l-1}\\end{aligned}$ where $z_{l}$ , $l = 1, 2, 3, ..., depth$ is the output of the $l$ th Transformer basic block in Transformer layer, and $depth$ is the number of Transformer blocks.",
"$z_{0}$ is the embedding sequence.",
"Moreover, MLP blocks are applied after MSA in every Transformer basic block to achieve more complex nonlinear mapping.",
"The MLP block in the $l$ th Transformer basic block includes a nonlinear projection layer with an activation function and a linear projection layer, which is given in Eq.",
"(REF ) $MLP(z_{l}^{MSA}) = GeLU(z_{l}^{MSA}W_{1}^{l}+b_{1}^{l})W_{2}^{l}+b_{2}^{l}$ where $W_{1}^{l} \\in \\mathbb {R}^{d \\times d_{MLP}}$ , $b_{1}^{l} \\in \\mathbb {R}^{d_{MLP}}$ , $W_{2}^{l} \\in \\mathbb {R}^{d_{MLP} \\times d}$ , $b_{2}^{l} \\in \\mathbb {R}^{d}$ .",
"$d_{MLP}$ represents the embedding dimension of the nonlinear projection layer, and $GeLU(x)$ indicates the Gaussian error Linear Unit (GeLU), which can be shown as Eq.",
"(REF ) $GeLU(x) = x \\phi (x) = x[1 + erf(x/{\\sqrt{2}})]/2 \\approx 0.5x(1 + tanh[\\sqrt{2/\\pi }(x + 0.045x^3)])$ where $\\phi (x)$ is the standard Gaussian distribution function.",
"GeLU is smoother over the entire input range than Rectified Linear Unit (ReLU) and Leaky ReLU, with no discontinuous gradient at 0.",
"Combining the residual connection and LN, the output of MLP in the $l$ th Transformer basic block (namely $z_l^{MLP}$ ) can be obtained as $z_{l} = MLP(LN(z_{l}^{MSA})) + z_{l}^{MSA}$ Based on Eq.",
"(REF ) and Eq.",
"(REF ), the final output of the Transformer layer and the feature representation (denoted as $y$ ) in the ViT are shown in $\\begin{aligned}z_{l}^{MSA} & = MSA(LN(z_{l-1})) + z_{l-1}, & l = 1, 2, 3,..., depth \\\\z_{l} & = MLP(LN(z_{l}^{MSA})) + z_{l}^{MSA}, & l = 1, 2, 3,..., depth \\\\y & = LN(z_{depth}[0])\\end{aligned}$ where $z_{depth}$ is the final output of the Transformer layer, and $z_{depth}^{0}$ means the class token in $z_{depth}$ .",
"To facilitate the expression in the following, we denote the set of learnable parameters in the model network as $\\theta $ and denote the calculation process of the encoder part as $f_{\\theta }$ .",
"Then the feature representation can be expressed as $y \\triangleq f_{\\theta }(x)$ , where $x$ is the input TFM."
],
[
"Projector head",
"The architecture of the projection head is straightforward, consisting of an $n$ -layer MLP, an L2-normalization layer, and a weight normalized linear layer.",
"The calculation of the $n$ -layer MLP and linear layer is almost precisely the same as Eq.",
"(REF ), not tired in words here.",
"The L2-Normalization layer stabilizes the training process.",
"Overall, the projector head plays almost the same role as the last layers of the various supervised learning backbones, which transforms the feature representation of the encoder into the probability distribution over $K$ dimension.",
"The difference is that this probability distribution is meaningless, which means that the projector head outputs the pseudo labels.",
"The hidden and bottleneck dimension of the MLP in the projector head is expressed as $d_{MLP}^{head}$ .",
"Finally, similar to the expressive method in the encoder part, the calculation process of the pseudo labels is denoted as $g_{\\theta }(y) = g_{\\theta }(f_{\\theta }(x)) = q_{\\theta }(x)$ , where $q = g \\circ f$ , and $y$ is the feature representation."
],
[
"Self-distillation with no labels",
"When the sample data has no labels or only a few labels, traditional supervised learning cannot effectively train the network.",
"We adopt the self-distillation with no labels (DINO) algorithm [55] to train the model network for the fault diagnosis in the case of few labels.",
"DINO is a self-supervised learning framework that can be interpreted as a form of knowledge distillation.",
"DINO's goal is to make the network learn a feature representation that can be used for downstream tasks by exploring the information of the unlabeled data.",
"As described previously, DINO uses two model networks to learn: the teacher and student networks, as shown in Fig.REF .",
"The teacher and student networks share the same architecture (shown in Fig.REF ), but parameterized by $\\theta _{s}$ and $\\theta _{t}$ respectively.",
"Then, we design a self-supervised learning task called \"local to global correspondence\" to train the teacher and student networks.",
"As shown in Fig.REF , given a TFM, we can construct different views, or called crops, of the original image through the data augmentation (including random crop, Gaussian blur, color jittering, and solarization).",
"Then, it should be noted that two different random crop scale parameters are utilized to obtain the global and local views separately.",
"Each local crop contains only a small area (scale range [0.05, $s$ ]), while the global crops cover a large area (scale range [$s$ , 1]) of the original TFM.",
"$s$ is the scale parameter in the random crop process.",
"In a random crop process, we can obtain a set of different crops, namely $X^{\\prime }$ , which contains 2 global views (denoted as $X^{g} = \\left\\lbrace x_{1}^{g}, x_{2}^{g}\\right\\rbrace $ ) and $N$ local views (denoted as $X^{l} = \\left\\lbrace x_{1}^{l}, x_{2}^{l}, ..., x_{N}^{l} \\right\\rbrace $ ), $X^{\\prime } = X^{l}\\cup X^{g}$ .",
"The teacher network only accepts global crops ($X^{g}$ ) as input, while the student network passes all crops ($X^{\\prime }$ ).",
"Figure: DINO's overall process.Next, we will discuss the other computational details in DINO's algorithm.",
"We want the student network's output to match the given teacher network.",
"From this perspective, DINO is very similar to knowledge distillation [56].",
"The difference lies in the teacher network in knowledge distillation is an extensive pre-trained network.",
"In DINO, the teacher and student networks share the same structure and have not undergone any pre-training.",
"Then, the probability of pseudo labels $P$ can be obtained by normalizing the output of the model network through a softmax function with temperature $P(x)^{(i)} = \\frac{\\text{exp}(q_{\\theta }(x)^{(i)} / \\tau )}{\\sum _{k = 1}^{K} \\text{exp}(q_{\\theta }(x)^{(k)} / \\tau )}$ where $P$ is the probability distribution with a given input $x$ , $\\tau $ is a temperature parameter which can control the sharpness of $P$ .",
"A larger $\\tau $ smoothes $P$ , while a smaller $\\tau $ encourages the sharper output distribution.",
"For example, if we set $\\tau = 0$ , then Eq.",
"(REF ) is equivalent to the One-hot encoding.",
"Here, we adopt different temperature parameters in the teacher and student networks, denoted as $\\tau _{t}$ and $\\tau _{s}$ , respectively.",
"In addition, we require that $\\tau _{t}$ must be lower than $\\tau _{s}$ , and this design is called \"sharpening.\"",
"Sharpening is to avoid the mode collapse in the training process, which will be discussed in detail later.",
"Moreover, as shown in Fig.REF , compared with the student network, another design, called \"centering\", has been added to the teacher network.",
"The specific approach is to add a bias item $c \\in \\mathbb {R}^{K}$ to the output of the teacher network and update it among different batches based on the Exponential Moving Average (EMA) approach in the training process, which can be described as Eq.",
"(REF ) $\\begin{aligned}q_{\\theta _{t}} (x) & \\leftarrow q_{\\theta _{t}} (x) + c \\\\c & \\leftarrow m_{c}c + (1 - m_{c}) \\frac{1}{B} \\sum _{i = 1}^{B} q_{\\theta _{t}} (x[i]) \\\\\\end{aligned}$ where $m_{c} \\in [0, 1]$ is the momentum parameter in the centering, $B$ is batch size.",
"In the same way as sharpening, centering can also avoid the mode collapse, whose impact will be presented later.",
"Based on the above statement, the goal of DINO is to minimize the loss function shown in Eq.",
"(REF ) $\\mathcal {L}_{\\theta _{s}, \\theta _{t}} = \\sum _{x_{t} \\in X^{g}} \\sum _{{x_{s} \\in X^{\\prime } \\\\ x_{s} \\ne x_{t}}} \\mathcal {L}_{ce} (P_{s} (x_{s}), P_{t} (x_{t}))$ where $\\mathcal {L}_{ce}$ is the cross-entropy between the $P_{s}$ and $P_{t}$ , and $\\mathcal {L}_{ce} (a, b) = -a \\text{ln} b$ .",
"We call the $\\mathcal {L}_{\\theta _{s}, \\theta _{t}}$ as \"target entropy\".",
"Finally, we explain the parameter updating process of the teacher and student networks.",
"As shown in Fig.REF , $\\theta _{s}$ is iteratively updated by the backpropagation.",
"Furthermore, when constructing the computation graph, we apply the stop-gradient in the teacher network, that is, $\\theta _{t}$ is not updated based on the loss function's gradient.",
"In each training epoch, the iteration form of $\\theta _{t}$ is similar to Eq.",
"(REF ), EMA approach can be described as $\\theta _{t} \\leftarrow m \\theta _{t} + (1 - m) \\theta _{s}$ where $m \\in [0, 1]$ is the momentum parameter.",
"The process of DINO is shown in Algorithm.REF .",
"As the training process goes on, $\\theta _{t}$ and $\\theta _{s}$ keep iterating and updating.",
"When the training is over, all the encoder parameters in the teacher network and student networks are frozen, and the feature representation $q_{\\theta } (x)$ of the given TFM can be obtained.",
"[h] The process of DINO algorithm Input: training TMF dataset (without labels), Dataloader with batch size $B$ Initialization: teacher network $q_{\\theta _{t}}$ , student network $q_{\\theta _{s}}$ , bias term in centering $c$ Define: data augmentation strategy for global views $Aug_{g}(\\cdot )$ , data augmentation strategy for local views $Aug_{l}(\\cdot )$ [1] $epoch$ in range(max_epoch) $x$ in Dataloader Obtain the global crops: $X^{g} = Aug_{g}(x)$ Obtain the local crops: $X^{l} = Aug_{l}(x)$ , and let $X^{\\prime } = X^{g} \\cup X^{l}$ Centering operation: $q_{\\theta _{t}} (x) \\leftarrow q_{\\theta _{t}} (x) + c$ Calculate the target entropy: $\\mathcal {L}_{\\theta _{s}, \\theta _{t}} = \\sum _{x_{t} \\in X^{g}} \\sum _{{x_{s} \\in X^{\\prime } \\\\ x_{s} \\ne x_{t}}} \\mathcal {L}_{ce} (P_{s} (x_{s}), P_{t} (x_{t}))$ Apply the stop-gradient in the teacher network: $\\text{stop-gradient} (q_{\\theta _{t}})$ Backpropagation for the student network: $\\theta _{s} \\leftarrow \\text{optimizer}(\\theta _{s}, \\nabla \\mathcal {L}_{\\theta _{s}, \\theta _{t}})$ Update the teacher network by EMA: $\\theta _{t} \\leftarrow m \\theta _{t} + (1 - m) \\theta _{s}$ Update the bias term by EMA: $c \\leftarrow m_{c}c + (1 - m_{c}) \\frac{1}{B} \\sum _{i = 1}^{B} q_{\\theta _{t}} (x[i])$"
],
[
"Overview",
"We propose a new intelligent fault diagnosis method for the rolling bearings based on the DINO algorithm and ViT model to solve the fault diagnosis problem under the limited labeled data condition.",
"The process framework is illustrated in Fig.REF , and its specific stages can be described as follows 1) Converting the original vibration signals into TFMs through the WT and resize operation.",
"2) Constructing the model network based on the ViT.",
"3) Combining the limited labeled and unlabeled data and conducting the self-supervised learning via the ViT model and DINO algorithm.",
"4) Freezing and saving the model networks' parameters.",
"Adopting the encoder part of the teacher network to extract the feature representation from the input TFMs.",
"5) Outputting the fault diagnosis results based on the limited labeled data and K-nearest neighbor (KNN) classifier.",
"Figure: The overall framework of the proposed method."
],
[
"Datasets description",
"In this paper, we adopt the CWRU dataset [57], which is collected by the Bearing Data Center of Case Western Reserve University (CWRU), and the XJTU dataset [58], which is provided by the Xi'an Jiaotong University (XJTU), to train the model network and evaluate the effectiveness of the proposed method.",
"Both are publicly available rolling bearing fault datasets widely used by many researchers.",
"The following is a brief Introduction to the datasets.",
"1) The CWRU dataset is a rolling bearing prefabricated fault dataset.",
"The dataset is composed of multivariate vibration signals generated by a bearing test-rig, as presented in Fig.REF (a).",
"The vibration signals are measured by the acceleration sensor with 12kHz sampling rates.",
"We adopt the bearing data from the drive end of the motor (bearing type 6205-2RS JEM SKF) in this study, and three fault types are processed with the Electro-Discharge Machining (EDM), including inner race fault (IR), outer race fault (OR), and rolling ball fault (RB).",
"Besides, various fault diameters ranging from 7 inches to 21 mils are introduced in each fault type, separately.",
"In summary, the CWRU dataset contains ten failure modes (including the normal condition, denoted as NC).",
"Then, based on the resampling method, the specific composition of the CWRU dataset in this paper is shown in Tab.REF , where only 1% limited labeled data is involved.",
"2) The XJTU dataset comprises complete run-to-failure data of 15 rolling element bearings (same type, LDK UER204) acquired by conducting many accelerated degradation experiments, whose experimental device is shown in Fig.REF (b).",
"Vibration signals of the tested bearings are obtained by two acceleration sensors in horizontal and vertical directions with 25.6kHz sampling rates.",
"In this paper, the recorded data from Bearing 3_1, Bearing 3_2, and Bearing 2_3 are chosen as the analysis data, which contains four fault modes: Outer race (OR), Inner race, ball, cage and outer race (IBCO), Inner race (IR), and Cage.",
"Then, similar to the CWRU dataset, the XJTU dataset after resampling is shown in Tab.REF , which also contains only 1% labeled data.",
"Figure: Test rig of the datasets.",
"(a) CWRU dataset.",
"(b) XJTU dataset.Table: Composition of the datasets"
],
[
"Parameters setup and other details",
"During the training process, the structural parameters of the model network adopted in this study are shown in Tab.REF .",
"The backbone of the encoder follows a lightweight ViT architecture, where the patch size of the input TFM $P$ is 16, the embedding dimension $d$ is 192, the number of $head$ in Eq.",
"(REF ) is 3, the dimension of the keys and values is 64, the embedding dimension of the nonlinear projection layer in the MLP block $d_{MLP}$ satisfies $d_{MLP} = 4 d$ , and the number of stacked Transformer basic blocks $depth$ is 12.",
"Besides, for the projector head, the dimension of the pseudo labels $K$ is 1024, and there are 3-layer MLP, where the dimension of the hidden and bottleneck layer $d_{MLP}^{head}$ is (2048, 2048, 256), respectively.",
"Then, the parameters of the DINO algorithm are shown in Tab.REF .",
"The momentum parameter of the teacher network $m$ in Eq.",
"(REF ) follows the cosine schedule from 0.996 to 1 during the training process.",
"The momentum parameter in the centering is 0.9.",
"The temperature of the teacher and student networks ($\\tau _{t}$ and $\\tau _{s}$ ) are 0.04 and 0.1, separately.",
"As described previously, $\\tau _{t}$ should be lower than $\\tau _{s}$ for sharpening.",
"Moreover, the scale parameter $s$ is 0.4, which means that the scale range of the local views is $[0.05, s]$ , and the scale range of the global views is $[s, 1]$ .",
"The number of local crops $N$ is 8.",
"The above parameters are adjustable hyperparameters, and we will discuss their impact on fault diagnosis performance in detail later.",
"Table: Structural parameters of the model network.Table: Parameters of the DINO algorithm.Finally, we adopt the Adam optimizer to minimize the target entropy in Eq.",
"(REF ), and set the weight decay equal to 0.04 to introduce the regularization.",
"The batch size $B$ is 64, and the max training epoch is set to 100.",
"Then, we use the warm-up strategy combined with cosine schedule to adjust the learning rate.",
"At the first ten training epochs, the learning rate increases linearly from $1 \\times 10^{-6}$ to $1.25 \\times 10^{-4}$ , and the subsequent learning rate follows the cosine schedule.",
"The hardware environment is Ryzen 3995WX, NVIDIA RTX 3090, and we adopt Python 3.8, Pytorch 1.8.1, and CUDA 10.2 for the deep learning framework."
],
[
"Mode collapse problem",
"In self-supervised learning, mode collapse is a crucial problem.",
"Mode collapse refers to the situation in which the network gradually converges to trivial solutions due to the abnormal training process.",
"For example, the trained adversarial generation network's generator can only generate one kind of image.",
"As shown in Fig.REF and Fig.REF , DINO's goal is to make the $K$ -dimensional pseudo labels of the student network match the teacher network.",
"Based on this, there are two forms of the collapse in the proposed method: 1) the outputs of the teacher and student networks are evenly distributed in each dimension (namely over-alignment), that is, the probability value of each pseudo label is $\\frac{1}{K}$ .",
"2) the output of the teacher and student networks is 1 in one dimension and 0 in all others (namely over-uniformity), such as $[0, 1, 0, 0, ..., 0]$ .",
"Then, as described previously, we introduce centering and sharpening to avoid collapse, and we will study their respective roles in this section.",
"Firstly, the cross entropy in Eq.",
"(REF ) can be decomposed into Kullback-Leibler (KL) divergence and entropy $\\sum _{x_{t} \\in X^{g}} \\sum _{{x_{s} \\in X^{\\prime } \\\\ x_{s} \\ne x_{t}}} \\mathcal {L}_{ce} (P_{s} (x_{s}), P_{t} (x_{t})) = \\sum _{x_{t} \\in X^{g}} \\sum _{{x_{s} \\in X^{\\prime } \\\\ x_{s} \\ne x_{t}}} D_{KL} (P_{s} (x_{s}) | P_{t} (x_{t})) + \\sum _{x_{t} \\in X^{g}} h(P_{t} (x_{t}))$ where $D_{KL}$ is the KL divergence between $P_{t}$ and $P_{s}$ , and $h$ is the entropy of the teacher network's output $P_{t}$ .",
"$D_{KL}$ can calculate the match between $P_{t}$ and $P_{s}$ .",
"If $P_{t}$ and $P_{s}$ are identical, then $D_{KL}$ equals zero, which means the mode collapse.",
"Besides, $h$ can measure the uncertainty of $P_{t}$ , which can be given as $h(P_{t}) = \\sum _{i = 1}^{K} -P_{t}^{(i)} ln P_{t}^{(i)}$ where $P_{t}^{(i)}$ is $i$ th element in $P_{t}$ .",
"As mentioned above, the value of $D_{KL}$ can be used to judge whether the mode collapse occurs.",
"Furthermore, we can determine the form of the mode collapse based on the value of $h$ .",
"It can be seen from Eq.",
"(REF ) that $h$ will converge to $\\text{ln} K$ for over-uniformity case, and will converge to zero for over-alignment case.",
"Subsequently, we present the evolution of $\\mathcal {L}_{\\theta _{s}, \\theta _{t}}$ , $D_{KL}$ and $h$ under different datasets and various designs in Fig.REF .",
"To analyze the respective roles of the centering and sharpening, we set up four different designs: 1) The output of the teacher network goes through both centering and sharpening operations (namely Both), as shown in Fig.REF .",
"2) There is only centering in the DINO framework.",
"3) There is only sharpening in the DINO framework.",
"4) The output of the teacher network is identical to the student network without any additional modifications (namely Neither).",
"As shown in Fig.REF (a) and (e), in the case of only centering, the target entropy will stay at an enormous value during the entire training process.",
"Conversely, in the case of only sharpening, the target entropy converges rapidly to zero.",
"Both cases indicate that the model networks have not been trained normally.",
"Further, it can be seen from Fig.REF (b) and (f) that the KL divergence between the teacher and student outputs degrades to zero promptly in all cases except Both, which means $P_{t} = P_{s}$ , indicating the mode collapse.",
"However, the entropy converges to different values under the diverse designs.",
"It can be seen from the data in Fig.REF (c) and (g) that if there is only centering operation, the entropy of the teacher network output $h$ will converge to $\\text{ln} K$ , indicating the over-uniformity.",
"And $h$ will converge to 0 if there is only sharpening in the DINO framework, which means the over-alignment case.",
"In the absence of centering and sharpening, the result is the same as that of only centering, and over-uniformity occurs.",
"Finally, from the blue lines in Fig.REF we can see that $\\mathcal {L}_{\\theta _{s}, \\theta _{t}}$ and $h$ show a tendency to decline gradually during training, and $D_{KL}$ do not converge to 0, indicating that there is no mode collapse.",
"These results suggest that the teacher network and the student network can carry out normal training only in the case of Both.",
"Figure: Mode collapse study in four designs under the different datasets.",
"(a)-(c) CWRU dataset.",
"(e)-(g) XJTU dataset.",
"(a) and (e) target entropy.",
"(b) and (f) KL divergence.",
"(c) and (g) entropy.In summary, for the informants in this section, it can be observed that the centering and sharpening operations are complementary in avoiding the mode collapse.",
"Missing either operation will cause the mode collapse in the proposed method.",
"Centering can inhibit over-alignment, but encourage over-uniformity.",
"Sharpening has the opposite effect, which can avoid over-uniformity but promote over-alignment.",
"In addition, mode collapse will also occur if there is no centering or sharpening.",
"Only by applying both operations simultaneously can the model network be appropriately trained."
],
[
"Fault diagnosis results based on the proposed method",
"Based on the DINO algorithm, the model networks are trained on the CWRU and XJTU datasets without any labels.",
"The parameters in the encoder part of the teacher network are frozen, and we can use it to extract the feature representation of the input TFM.",
"The extracted feature representation is called feature vectors.",
"Then, to diagnosis the input TFM, we calculate its feature vectors and compare it against the labeled data.",
"Adopting the above process, we can make fault diagnose for many unlabeled data with the KNN classifier, even if the labeled data is minimal.",
"Tab.REF provides the fault diagnosis results obtained by our proposed method.",
"We select multiple the number of nearest neighbors (denoted as $N_{k}$ ), ranging from 10 to 100, to optimize the classifier parameters.",
"In addition, to enhance the robustness of the classifier to $N_{k}$ , we introduce a temperature parameter $\\tau _{k}$ as a divisor when calculating the similarity, similar to Eq.",
"(REF ).",
"The effect of $\\tau _{k}$ is also set out in Tab.REF .",
"We find that $N_{k}$ has a significant impact on the fault diagnosis accuracy of the KNN classifier if without $\\tau _{k}$ .",
"On the CWRU dataset, the accuracy of the classifier is 92.65% with $N_{k} = 10$ , but when $N_{k}$ equals 40, the accuracy drops to 79.19%.",
"Analogously, the fault diagnosis accuracy of KNN is 83.08% with $N_{k} = 10$ and 74.07% with $N_{k} = 40$ on the XJTU dataset.",
"These results show that the fault diagnosis performance of the classifier will be sensitive to $N_{k}$ if there is no temperature parameter.",
"Furthermore, the first and third rows in Tab.REF illustrate that $\\tau _{k}$ can obviously improve the robustness of the classifier to $N_{k}$ , and increase the fault diagnosis accuracy.",
"Taking the CWRU dataset as an example, for the different values of $N_{k}$ , the top and minimum accuracy of the classifier are 95.27% and 93.44%, and their fluctuation range is significantly reduced compared with the data without $\\tau _{s}$ .",
"Then, by applying $\\tau _{s}$ , the accuracy of KNN on the CWRU and XJTU datasets achieve 95.27% and 89.33% with $N_{k} = 10$ , respectively.",
"This is a brilliant fault diagnosis result under the limited labeled data condition.",
"Moreover, it should be pointed out that the feature vectors extracted by the trained encoder do not undergo any additional fine-tuning, and great fault diagnosis accuracy shows that feature vectors possess good generalization ability.",
"Table: Fault diagnosis results with only 1% labeled data (unit: %).To analyze the detailed fault diagnosis results, Fig.REF provides the confusion matrices of the proposed method on the CWRU and XJTU datasets, where rows represent the actual fault mode and columns is the predicted fault mode.",
"Specifically, on the CWRU dataset, promising fault diagnosis accuracy can be obtained in most fault modes, and the misidentification results mainly focus on the class RB014.",
"This result may be explained by the fact that the features of rolling ball fault condition is general weak.",
"However, we can still obtain higher than 77% identification accuracy with only 1% labeled data.",
"In contrast, due to the earlier fault stage of the rolling bearing during the run-to-failure process, the proposed method achieves relatively lower diagnosis accuracy on the XJTU dataset.",
"About 89.33% testing accuracy with the unlabeled data can be acquired, and 28.74% OR samples are misclassified as IR.",
"This result indicates that the outer race fault in early stage may be a challenging fault mode.",
"But, over 70% accuracy for OR can be still achieved.",
"That shows that the proposed method has satisfactory fault diagnosis performance in the case of limited labeled data.",
"Figure: Confusion matrix results.",
"(a) CWRU dataset.",
"(b) XJTU dataset.Overall, through the results of fault diagnosis accuracy and confusion matrices under the different datasets, the effectiveness of the proposed method is validated in this section.",
"Feature vectors with strong generalization ability can be acquired utilizing the model networks and DINO algorithm, and high diagnosis accuracy is obtained with only 1% labeled data."
],
[
"Influence of the hyperparameters",
"In this section, we will investigate the influence of the hyperparameters in the proposed method, including the encoder form, pseudo labels dimension $K$ , temperature parameters in the teacher and student networks ($\\tau _{t}$ and $\\tau _{s}$ ), scale parameter $s$ , momentum parameter in centering $m_{c}$ , and the number of local crops $N$ .",
"The impact of the hyperparameters is evaluated from two aspects: accuracy and peak memory when running.",
"The accuracy directly reflects the fault diagnosis performance, while the peak memory can represent the model scale and computational complexity.",
"These results are summarized in Tab.REF .",
"Firstly, the result in group A shows that we can obtain a higher accuracy based on the feature vectors extracted by the teacher network.",
"A possible explanation for this might be that the outputs of the teacher network are sharper ($\\tau _{t} < \\tau _{s}$ ), which helps the model learn features with more generalization ability.",
"Next, as one of the critical hyperparameters in the proposed method, the effects of the pseudo labels dimension $K$ are discussed in group B in Tab.REF .",
"One unanticipated finding is that a larger $K$ may not imply better performance.",
"For example, the fault diagnosis accuracy of $K = 64$ is higher than that of $K = 512$ .",
"This result suggests that the performance of the proposed method may be further improved by fine-tuning $K$ .",
"Within the range that we set in this study, the accuracy is highest when $K$ equals 1024.",
"Meanwhile, the data in group B also show that the size of $K$ has no significant impact on the peak memory.",
"This is because $K$ is associated only with the last layer of the projector head, which is a negligible value compared to the entire model network.",
"Besides, the relationship between the temperature parameter of teacher network and the testing accuracy is presented in group C in Tab.REF .",
"We observe that the model is susceptible to $\\tau _{t}$ , and inappropriate $\\tau _{t}$ will significantly reduce the model performance, even leading to a failed training process.",
"Too small $\\tau _{t}$ will make the teacher output too sharp, which causes the network easily fall into the local minimum, resulting in the decline of diagnosis accuracy.",
"Nevertheless, as mentioned in Section 4.1, we need sharpening to avoid the mode collapse, so a too large $\\tau _{t}$ is also unbefitting.",
"Actually, through the experiment, we find that when the teacher temperature is higher than 0.06 (such as 0.8), the entropy of teacher output consistently converges to $\\text{ln} K$ , indicating over-uniformity.",
"Therefore, the fault diagnosis performance is inferior with an overlarge $\\tau _{t}$ .",
"Next, group D in Tab.REF compares the testing accuracy and peak memory of the proposed method under diverse student temperature $\\tau _{s}$ .",
"As described previously, the design of sharpening demands that $\\tau _{t}$ must be lower than $\\tau _{s}$ to avoid the mode collapse, further confirmed by the result in group D. If $\\tau _{t}$ is higher than or close to $\\tau _{s}$ , such as 0.03 or 0.05, sharpening will fail, and the poor diagnosis performance will be obtained.",
"In our experiment, the student temperature below 0.05 is not recommended.",
"In addition, similar to $\\tau _{t}$ , oversize $\\tau _{s}$ is able to reduce the fault diagnosis accuracy.",
"Hence, a moderate $\\tau _{s}$ is suggested in practical implementations.",
"Scale range $s$ is also an essential parameter that affects the fault diagnosis performance of the proposed method, and the testing results with different $s$ are presented in group E. A general pattern summarized from group E is that higher diagnosis accuracy can be acquired with a medium size of $s$ .",
"A lower $s$ means smaller areas of the original TFM adopted to generate the global and local crops, which encourages the model networks to concentrate on the local features of the input.",
"On the contrary, a larger $s$ helps our proposed method to learn the global features of the TFM.",
"Both local and global features are necessary for the fault diagnosis problem, so a moderate $s$ should be selected to balance their effects.",
"In our study, the highest accuracy can be achieved with $s = 0.4$ .",
"Moreover, since the local and global crops are resized to a fixed size, $s$ does not influence the computational complexity.",
"Per our earlier discussion, there are two additional designs in the teacher output to prevent the mode collapse: sharpening and centering.",
"$\\tau _{t}$ and $\\tau _{s}$ are related to sharpening, and their effects have been investigated in group C and D. $m_{c}$ is the momentum parameter in centering, and its impact is provided in group F. It is somewhat surprising that, compared with the temperature parameters, the convergence of the proposed method is robust to a wide range of $m_{c}$ .",
"We do not notice the over-uniformity or over-alignment with $m_{c} \\in [0, 0.99]$ in our trials.",
"The model network only occurs the model collapse when the update of bias term in centering is too slow, such as $m_{c} = 0.999$ .",
"Simultaneously, the maximum testing accuracy can be procured with $m_{c}$ equals 0.9 in our datasets.",
"Finally, the influence of the number of local crops $N$ is shown in group G in Tab.REF .",
"A universal display pattern is observed that a larger $N$ can improve the diagnosis performance.",
"Nonetheless, the computational complexity of the proposed method also increases with the expansion of $N$ , so we should weigh it according to the specific hardware environment in the actual application.",
"Table: Influence of the hyperparameters"
],
[
"Attention maps visualization",
"Great interpretability is a significant advantage of the attention mechanism.",
"This section employs attention maps (denoted as AM) visualization to explore the feature representation process qualitatively.",
"Here, three dissimilar forms of AM are introduced: 1) Class token AM (CAM), which is calculated by the class token and embedding sequence on the heads of the last Transformer basic block in the model network.",
"CAM can represent the importance of each patch in the input TFM to the feature representation.",
"2) Threshold class token AM (TAM).",
"We visualize the marked patches obtained by thresholding the CAM to keep 90% of the attention, illustrating the areas that primarily affect the feature vectors.",
"3) Embedding sequence AM (EAM), computing from the embedding sequence in the last Transformer basic block.",
"Note that there are three $head$ in the MSA, so the EAM also can be described in three different subspaces, denoted as $\\text{EAM}_1$ , $\\text{EAM}_2$ , and $\\text{EAM}_3$ respectively.",
"The results of the attention maps visualization obtained from the untrained and trained teacher network are compared in Fig.REF .",
"As shown in Fig.REF (a), for the network adopting the random weight without any training, no salient attention value is found in CAM, and marked patches in TAM almost cover the whole input region.",
"Additionally, the attention values in EAM also present a consistent distribution form, indicating a poor feature extraction ability.",
"In contrast, the teacher network trained by self-supervised learning demonstrates an entirely different pattern.",
"As can be seen from Fig.REF (b), the prominent parts in TFM have more significant attention values in CAM, and the marked patches in TAM are also concentrated on the smaller areas.",
"That is, the trained teacher network pays more attention to the patches where the amplitude in the TFM is more pronounced, which adheres to our intuition.",
"Then, most of the attention values in EMA are centralized in a few patches, suggesting that the teacher network grasps the crucial information for fault diagnosis from the input TFM.",
"Figure: Attention maps visualization results.",
"(a) Without training.",
"(b) Self-supervised learning.Taken together, through the attention maps visualization, the feature representation process in our proposed method is analyzed.",
"Based on self-supervised learning, the teacher network spontaneously learns the fault-specific features without any labels, and it pays more attention to the patches where the amplitude in the TFM is more obvious."
],
[
"Feature vectors visualization",
"In addition to the testing accuracy, the distribution form of feature vectors is also a meaningful indicator to evaluate the proposed fault diagnosis method.",
"Therefore, we will focus on discussing the visualization results of feature vectors in this section.",
"In this study, the dimension of the feature vectors extracted by the model network is 192, which is a higher-dimensional space that cannot be visualized directly.",
"Here, we adopt the t-distributed Stochastic Neighbor Embedding (t-SNE) technique as a dimension reduction method.",
"As a comparison, visualization results of the input TFMs are also necessary.",
"However, it should be noted that the scale of the TFM is $224 \\times 224 \\times 3$ , so processing it immediately with t-SNE will be very time-consuming and not effective.",
"To address this issue, an approach combined the Principal Component Analysis (PCA) and t-SNE method is utilized.",
"Firstly, PCA is used to decompose the first 192 principal components of the TFM and integrate them into a feature vector.",
"Then, t-SNE is applied to reduce its dimension further and realize visualization.",
"Based on the above analysis, Fig.REF shows the feature vectors visualization results on the CWRU and XJTU datasets.",
"A large amount of overlapping parts are observed for different fault modes in Fig.REF (a) and (c), demonstrating that it is not feasible to use the input TFM for fault diagnosis directly.",
"Then, turning now to the visualization results of the feature vectors obtained by our proposed method.",
"As shown in Fig.REF (b) and (d), most instances of the same fault mode are projected into the same region, and different fault modes are separated.",
"That shows that the proposed method can extract the feature vectors with inter-class separability by exploring information in unlabeled data.",
"However, we should point out that confusion still occurs between a few different samples due to the lack of supervised learning with labeled data, such as RB007 and RB021 in Fig.REF (b), and OR and IR in Fig.REF (d).",
"These results are compatible with the above confusion matrix analysis results in Fig.REF , revealing some more challenging fault diagnosis tasks.",
"Figure: Feature vectors visualization results.",
"(a) and (b) CWRU dataset.",
"(c) and (d) XJTU dataset.",
"(a) and (c) The input TFMs.",
"(b) and (d) Feature vectors extracted by our proposed method"
],
[
"Comparison with other methods",
"To further verify the superiority of the proposed method, comparisons with other fault diagnosis methods in practical application are necessary.",
"Our method consists of two main parts: the training paradigm (DINO) and the backbone of the encoder (ViT).",
"Therefore, the contrast experiments also include different training paradigms and backbones.",
"In terms of the training paradigm, conventional supervised learning and other self-supervised learning methods are involved.",
"The supervised learning technique only considers the cross-entropy loss of the limited labeled data, and the self-supervised learning approaches adopted for comparisons incorporate the SimCLR [59] and BYOL [60].",
"Then, in the aspect of the encoder backbone, ResNet18 [61] and DenseNet121 [62] are considered.",
"These training paradigms and encoder backbones are used for the CWRU and XJTU datasets, respectively, and their results are summarized in Tab.REF .",
"Note that these fault diagnosis methods share the same data pre-processing pipeline shown in Fig.REF .",
"Among them, due to the lack of labeled data, the self-supervised learning paradigm can not effectively train the networks.",
"It is difficult to obtain the feature vectors with preferable generalization ability through self-supervised learning, so its testing accuracy is relatively low.",
"By contrast, better fault diagnosis performance can be acquired by training the model network with the self-supervised learning method.",
"Regardless of the encoder backbone, the testing accuracy obtained by DINO are generally higher than those of other self-supervised learning method, suggesting that DINO is a more powerful training paradigm.",
"Besides, surprisingly, adopting the ResNet18 rather than ViT as the encoder backbone can get a higher testing accuracy on the XJTU dataset.",
"But overall, among all these solutions, ViT trained by DINO achieves the highest average diagnosis accuracy on the given datasets, which further proves the effectiveness and superiority of the proposed method for the rolling bearing fault diagnosis with limited labeled data.",
"Table: Testing accuracy of the comparison methods on the CWRU and XJTU datasets"
],
[
"Conclusions",
"Faced with the contradiction between the conventional supervised fault diagnosis methods' dependency on massive labeled data and engineering practice, a Wavelet Transform (WT) and self-supervised learning-based fault diagnosis framework for bearing fault diagnosis with limited labeled data has been presented.",
"In this method, the original vibration signals are pre-processed by WT and cubic spline interpolation to obtain the time-frequency maps (TFMs) with a specific scale.",
"The teacher and student networks are established based on the Vision Transformer (ViT) encoder and projector head, and a pretext task called \"local to global correspondence\" is introduced for self-supervised learning.",
"Adopting the Self-distillation with no labels (DINO) algorithm to train the teacher and student networks, fault-specific feature representation can be obtained.",
"The main conclusions are summarized as follows.",
"1) Through the DINO algorithm, effective feature vectors from the complex TFMs can be extracted by the trained ViT encoder, and involving the centering and sharpening operations in the teacher network can avoid the problem of mode collapse efficaciously during the self-supervised learning procedure.",
"The complementary effect of the centering and sharpening is observed, where centering encourages over-uniformity but inhibits over-alignment, while sharpening has the opposite function.",
"Only by applying both operations simultaneously can the mode collapse be avoided.",
"2) In the situation that only 1% labeled samples are included in the CWRU and XJTU datasets, adopting the feature vectors extracted by the trained encoder without any fine-tuning, 95.27% and 89.33% testing accuracy can be obtained based on the simple K-Nearest Neighbor (KNN) classifier.",
"3) The influence of hyperparameters is discussed in detail.",
"Teacher temperature and student temperature can significantly affect the fault diagnosis performance.",
"Inappropriate values of them may directly cause the mode collapse, so a careful adjustment is recommended.",
"Then, the scale range and momentum parameter in centering are also crucial, and there is an optimal value for both hyperparameters.",
"The computational complexity and diagnosis accuracy of the proposed framework increase with the expansion of the number of local crops, so it should be weighed according to the specific hardware environment in the actual application.",
"Finally, the explicit law of the influence of the pseudo labels dimension is not observed, but on the whole, it has a relatively unapparent impact on the fault diagnosis accuracy of the proposed framework.",
"4) Different training paradigms, such as supervised learning, SimCLR, BYOL, DINO, and various encoder backbones, including ResNet18, DenseNet121, and ViT, are considered in twelve comparative fault diagnosis approaches.",
"Among them, the proposed method has the highest average accuracy on the CWRU and XJTU datasets, demonstrating its effectiveness and superiority.",
"Since sufficient unlabeled data is needed in this study, the main limitation lies in that the proposed method is not suitable for the small sample.",
"In future work, further research should focus on the improved diagnosis approach for small-sample learning."
],
[
"Acknowledgements",
"It is very grateful for the financial supports from the National Major Science and Technology Projects of China (No.",
"2017-IV-0008-0045), the National Natural Science Foundation of China (Nos.",
"11972129, 11732005) and the Fundamental Research Funds for the Central Universities."
]
] | 2207.10432 |
[
[
"The Unscented Transform Controller: a new model predictive control law\n for highly nonlinear systems"
],
[
"Abstract The Unscented Transform which is the basis of the Unscented Kalman Filter, UKF, is used here to develop a novel predictive controller for non-linear plants, called the Unscented Transform Controller, UTC.",
"The UTC can be seen as the dual of the UKF, the same way as the LQG regulator and the Kalman Filter are related.",
"The UTC is demonstrated on the control of complex maneuvers in free fall of a virtual skydiver the model of which was verified in wind tunnel and free fall experiments."
],
[
"INTRODUCTION",
"Even after more than one century of research, non-linear control is still more of an art than an orderly engineering discipline.",
"A plethora of methods and approaches exist, [5], from various linearization approaches to more recent neural network controllers, and non-linear model predictive control, [3].",
"As a general rule, the control designer must profoundly understand the controlled plant and its particular non-linear structure in order to succeed in choosing an appropriate control method.",
"In a similar way, non-linear filtering and estimation, [7], is an art in contrast to the systematic way linear estimation is done with the Kalman filter, [9].",
"For non-linear processes, developments of the Kalman filter include the Extended Kalman Filter (EKF), particle filters, and in particular the numerically efficient Unscented Kalman Filter, UKF, [6], [8].",
"A well known feature of linear control and estimation is the duality of the LQG controller and the Kalman filter, [10].",
"In this paper, a novel controller dual of the UKF, called the Unscented Transform Controller, UTC, is proposed for the control of nonlinear plants.",
"The UTC is model based, and can be seen as an original model predictive controller.",
"The inspiration behind the UTC is the successful use of the UKF for the estimation of inputs to a highly non-linear skydiver model in complex free-fall maneuvers, and the attempt to control that system.",
"While simple maneuvers for such a model can be efficiently followed using robust linear control, [11], such an approach fails for complex maneuvers.",
"Another less than successful attempt was the use of a Recurrent Neural Network Controller, as reported in [4].",
"The UTC, however, accurately tracks the desired maneuvers, as demonstrated in this paper, and in [4] for additional maneuvers such as front and back layouts: transitions between belly-to-earth and back-to-earth equilibria through the head-to-earth orientation.",
"The paper is organized as follows: Section describes the skydiver model, the control input, and the control task.",
"In Section the Unscented Transform Controller is presented in detail together with some illuminating simulations.",
"Section contains the conclusion with suggestions for future work, and the remark that [4] contains a rudimentary stability proof for the case the UTC is used to control a linear plant."
],
[
"SYSTEM UNDER INVESTIGATION",
"The system under investigation is a skydiver performing aerial maneuvers in free-fall.",
"A model of skydiver's body, aerodynamic forces and moments acting on it at terminal velocity (around 60 (m/s)), and equations of motion were developed in [1].",
"The model was validated in free-fall and wind tunnel experiments with different skydivers performing a variety of maneuvers.",
"The model is driven by the following inputs: 1. body posture; 2. roll, pitch, and yaw damping moment coefficients; and 3. input moment coefficients, associated with body segments.",
"The damping moment coefficients reflect an approximate body resistance (e.g.",
"muscle stiffness) to the developing rotation rates.",
"The input moment coefficients reflect the consciously applied physical resistance of each limb to the aerodynamic force acting on it.",
"The body posture is defined by the relative orientation of body limbs, while each joint has 3 rotational degrees-of-freedom (DOFs).",
"During skilled actions the human central nervous system organizes these DOFs into movement patterns: combinations of DOFs that are activated synchronously and proportionally, as a single unit.",
"These movement patterns can be extracted from free-fall experiments by recording a sequence of measured postures during the maneuver time and conducting Principal Component Analysis (PCA).",
"It was discovered that experienced skydivers can perform most maneuvers by utilizing just one movement pattern [2].",
"These maneuvers can be reconstructed in simulation by actuating the skydiver model with the chosen movement pattern, the amplitude of which is the control variable.",
"For example, in [4] the aerial rotations maneuver (360 degrees right and left turns in a belly-to-earth position) is reconstructed in simulation by applying a proportional controller with a feed-forward part, as shown in (REF ).",
"${\\begin{array}{c}u(t) =1.5 \\cdot (yr_{ref}(t)-yr_{real}(t))+0.1\\cdot yr_{ref}(t) \\\\-1.5 \\le u(t) \\le 1.5 (rad), \\quad -3.5 \\le \\frac{du(t)}{dt} \\le 3.5 (rad/s) \\\\pose(t) = NeutralPose + u(t) \\cdot TurningPattern\\end{array}}$ where $yr_{ref}(t)$ $(rad/s)$ is the reference yaw rate profile; $yr_{real}(t)$ $(rad/s)$ is the yaw rate of the virtual skydiver in simulation; $u(t)$ $(rad)$ is the turning pattern angle command; $NeutralPose$ $(rad)$ is the trimmed pose of a skydiver falling straight down, defined by a vector containing values of body DOFs for all modeled joints; and $TurningPattern$ is an eigenvector defining the movement pattern utilized for turning.",
"If the reference signal has a slow dynamics, the damping moment coefficients may remain constant during the maneuver, and zero input moments can be assumed.",
"However, as the amplitude and/or the frequency of the desired yaw rate increases it becomes impossible to track the reference signal just by means of posture adjustments.",
"The reason is that the magnitude of the movement pattern is limited due to the natural constraints of the human body.",
"Additionally, the range of orientations of each limb relative to the airflow, within which the posture preserves aerodynamic efficiency, is also limited.",
"Therefore, the adjustment of input and damping moments becomes necessary.",
"Consider, for example, the yaw rate reference signal in (REF ).",
"$yr_{ref}(t)=400\\cdot \\frac{\\pi }{180}sin(2\\cdot \\pi \\cdot 0.2 \\cdot t)$ Figure: Tracking the yaw rate reference profile with constant yaw damping moment coefficient (0.5) and pose controller in ().This signal can not be accurately tracked by utilizing only a turning movement pattern, while the yaw damping moment coefficient remains constant, see Fig.",
"REF .",
"Thus, for the purpose of an illustrating example of the control method described in the following section, three additional control variables are utilized: the yaw damping moment coefficient, and two input moment coefficients.",
"For the purpose of simulating aerial rotations, the following simplified model for the input moments is used: ${\\begin{array}{c}Mz_{hands} = K_{N}\\cdot lx_{hand} \\cdot (k_{right}-k_{left}) \\\\Fy_{hands} = -K_{N}\\cdot (k_{right}+k_{left}) \\end{array}}$ where $Mz$ (Nm) is the roll moment acting about the longitudinal body axis, $Fy$ (N) is the force acting along the sagittal axis, $lx_{hand}$ =0.35 (m) is the characteristic (for neutral pose) distance of hands from the longitudinal axis projected to the frontal plane, $K_{N}$ =100 (N) is a scaling coefficient, and $k_{right}$ , $k_{left}$ are the user input dimensionless coefficients related to the right and left hands, accordingly.",
"They represent pressure applied on the airflow with the hands.",
"This creates a roll moment, which will cause the skydiver to develop a roll angle and therefore partially expose his torso to the airflow during the turn.",
"This will create a force along a frontal axis, which will create a yaw moment, desired for the turning maneuver."
],
[
"UNSCENTED TRANSFORM CONTROLLER",
"This control law was inspired by a modification to the Unscented Kalman Filter developed in [1] for estimation of user inputs from experiments of advanced skydiving maneuvers in free-fall.",
"At the prediction step the sigma points and the skydiver state were propagated during the prediction horizon $t_{pred}=0.25$ (s).",
"Allowing each sigma point to drive the skydiver dynamics during the prediction window was essential in order to determine what influence a certain combination of user inputs has on the skydiver state.",
"The prediction time reflected the skydiver model time constant.",
"Notice, that estimating the user inputs that explain the measured plant dynamics is a very similar problem to predicting the user inputs that will provide the desired maneuvers.",
"The latter is basically the definition of the control problem under investigation: designing a control algorithm, which will compute commands for body posture, input moment coefficients, and damping moment coefficients in order to track the reference angular/linear velocity signals.",
"Therefore, we suggest the novel control scheme outlined below, which was termed the Unscented Transform Controller (UTC).",
"It has much in common with the concept of standard Model Predictive Control, MPC, [12].",
"However it has some important differences: The control output is not a result of optimization of a cost function, as in MPC, but a weighted average of sigma points.",
"This average also represents an optimal solution in the sense of Minimum Mean Square Error (MMSE), given the sigma points.",
"Propagation of the plant for 4 control variables (i.e.",
"9 sigma points) is less computationally intensive than solving a nonlinear optimization problem at each step for the same prediction horizon.",
"Additionally, there are no convergence problems that optimization engines have to deal with.",
"The control outputs do not need to be parameterized as in non-linear MPC [3], what also contributes to the simplicity of the problem formulation, tuning, and computations.",
"It is possible to incorporate the desired/expected dynamics of the control signals into the prediction step, when the sigma points are propagated.",
"For example, it can be enforced that the turning pattern angle command has the dynamics of Proportional-Integral (PI) control.",
"These features are demonstrated subsequently."
],
[
"Formulation",
"The structure of this controller is shown in block diagram in Fig.",
"REF : it includes the definition of initial conditions, prediction step, propagation of the plant model up to the prediction horizon, and update step.",
"Figure: Block diagram of the Unscented Transform Controller, comprising the computation of sigma points, prediction step, observation model, and update step.The control variables include the input moment coefficients, damping moment coefficients, and angles of the involved movement patterns.",
"Specifically, in the case of rotation maneuvers: $\\vec{U}_k=[Cm_{damp}^{yaw}, k_{right}, k_{left}, \\alpha ]^T \\in \\textbf {R}^m,\\; m=4$ where $k$ is the simulation step such that $t=k \\cdot dt$ ; $k_{right}, k_{left}$ are the input moment coefficients, see (REF ); $Cm_{damp}^{yaw}$ is the yaw damping moment coefficient; and $\\alpha $ (rad) is the angle of the turning movement pattern, so that the body posture is then computed as: $pose_k=NeutralPose+\\alpha _k \\cdot TurningPattern$ If no a-priori knowledge is assumed about the dynamics of these control variables, the controller dynamics becomes: $\\vec{U}_{k+1}=\\vec{U}_k+\\vec{w_u}_k, \\qquad \\vec{w_u}_k \\sim N(0,Qu)$ where $\\vec{w_u}_k$ is the process noise, which will be discussed later, as it is the primary tuning parameter used for obtaining different solutions to the tracking problem."
],
[
"Reference Signal",
"The desired angular/linear velocities play in UTC the same role as the measurement vector in a conventional UKF.",
"In this section it is the yaw rate reference signal (dimension $l=1$ ), which has to be tracked, defined in (REF ).",
"Thus, $ Y_{ref}(k \\cdot dt + t_{pred})=yr_{ref}(k \\cdot dt + t_{pred})$ where $dt$ is the time between step $k$ and $k+1$ , and $t_{pred}$ is the prediction horizon.",
"The measurement noise covariance matrix present in UKF, is denoted in UTC by $P_{err}$ and defines how accurately the reference signal will be tracked, i.e.",
"the trade-off between accuracy and control effort."
],
[
"Initial Conditions",
"The initial conditions (step $k=0$ ) are chosen as follows: ${\\begin{array}{c}\\vec{U}_{k/k}=[0.5, 0, 0, 0]^T \\\\Pu_{k/k}=\\begin{bmatrix}6.25 & s_{12}\\cdot 0.74 & s_{13}\\cdot 0.74 & s_{14}\\cdot 0.56 \\\\ s_{12}\\cdot 0.74 & 0.01 & 0 & 0 \\\\ s_{13}\\cdot 0.74 & 0 & 0.01 & 0 \\\\ s_{14}\\cdot 0.56 & 0 & 0 & 0.01\\end{bmatrix}\\end{array}}$ where $s_{12}, s_{13}, s_{14}$ are the signs of the relevant covariance elements.",
"There are several options for defining $s_{12}, s_{13}$ as a function of $k$ , which will be discussed later.",
"The initial $s_{14}=-1$ , meaning that in order to start turning the yaw damping moment must be reduced.",
"Equation (REF ) below defines $s_{14}$ as a function of $k$ : ${\\begin{array}{c}\\Delta (k)=|yr_{ref}(k+1)|-|yr_{ref}(k)| \\\\\\Delta _{hyst}(k)=|yr_{ref}(k+k_{hyst}+1)|-|yr_{ref}(k+k_{hyst})|\\\\s_{14}(k) = {\\left\\lbrace \\begin{array}{ll}-sign(\\Delta (k)), & \\quad s_{14}(k-1)>0 \\\\-sign(\\Delta _{hyst}(k)), & \\quad s_{14}(k-1) \\le 0\\end{array}\\right.",
"}\\end{array}}$ where $k_{hyst}$ is set around $\\frac{1 (s)}{dt}$ and it is used to provide some hysteresis: the yaw damping moment is increased about 1 second before the turning rate command starts slowing down.",
"Additionally, the following parameters are chosen: ${\\begin{array}{c}P_{err}=0.01^2 \\\\Qu=Pu_{k=0} \\cdot 0.1\\end{array}}$"
],
[
"Sigma Points",
"At each simulation step we choose $2m+1$ sigma points $\\vec{U^i}$ and their associated weights $W^i$ in the following way, where the value of $W^0$ was chosen by tuning: ${\\begin{array}{c}\\vec{U^0}=\\vec{U}_{k/k} \\\\\\vec{U^i}=\\vec{U}_{k/k} \\pm \\sqrt{\\frac{n}{1-W^0}}\\vec{S}_j, \\qquad i=1,..2m, \\quad j=1,..m \\\\W^i=\\frac{1-W^0}{2m}, \\qquad W^0=0.25\\end{array}}$ where $\\vec{S}_j$ is column $j$ of matrix $S$ that satisfies $S \\cdot S = Pu_{k/k}$ ."
],
[
"Actuation Constraints",
"After the sigma points are selected they are bounded by the minimum and maximum values of the actuation constraints: $ {\\begin{array}{c}\\vec{U^i}_{min}=[0.01, 0, 0, -1.5]^T \\\\\\vec{U^i}_{max}=[6, 5, 5, 1.5]^T\\end{array}}$ Additionally, the turning pattern angle command $\\alpha $ has a slew rate limit, according to (REF ).",
"This is taken into account when the plant model is propagated: the turning pattern angle at each step is computed from $\\vec{U^i}$ and the slew rate constraint.",
"The sigma points are prevented from being identical, what can happen after imposing the constraints, by modifying the relevant elements of sigma points.",
"If an element $j$ of $ \\vec{U}_{k/k}$ equals to its min or max boundary and $\\vec{U^i}(j)<\\vec{U^i}_{min}(j)$ or $\\vec{U^i}(j)>\\vec{U^i}_{max}(j)$ , respectively, then this element is moved to the other side of the boundary: $ \\vec{U^i}(j)=1.5 \\cdot \\vec{U}_{k/k}(j)-0.5 \\cdot \\vec{U^i}(j)$"
],
[
"Observation Model: Propagation of the plant dynamics",
"Each sigma point drives a skydiver model, since the control variables contained in $\\vec{U^i}$ are the plant actuators.",
"The plant equations are propagated from the plant's state at step $k$ ($\\vec{X_k}$ ) until the prediction horizon ($t_{pred}$ ).",
"The plant's state contains the linear and angular velocities of the skydiver.",
"The final value of yaw rate $Y^i_{k+\\frac{t_{pred}}{dt}}$ , computed for each sigma point $i$ and saved as $Y^i$ , is then weighted, (REF ), and used for the update step.",
"${\\begin{array}{c}Y^i \\text{ is } Y^i_{k+\\frac{t_{pred}}{dt}}\\\\Ypred_{k}= \\displaystyle \\sum _{i=0}^{2m}W^iY^i \\\\ W^i=\\frac{1-W^0}{2m}\\end{array}}$ The prediction horizon $t_{pred}$ is a tuning parameter and will be discussed later in this section."
],
[
"Prediction Step",
"The computation flow is schematically shown in Figure REF .",
"Sigma points are selected according to (REF ), constrained according to (REF ), (REF ), propagated until the prediction horizon $\\vec{U^i}_{k+\\frac{t_{pred}}{dt}}$ , saved as $\\vec{U^i}$ , and summarized as: ${\\begin{array}{c}\\vec{U^i} \\text{ is } \\vec{U^i}_{k+\\frac{t_{pred}}{dt}} \\\\\\vec{U}_{k+1/k}=\\displaystyle \\sum _{i=0}^{2m}W^i\\vec{U^i}\\end{array}}$ $Pu_{k+1/k}=Qu+\\displaystyle \\sum _{i=0}^{2m}W^i(\\vec{U^i}-\\vec{U}_{k+1/k})\\cdot (\\vec{U^i}-\\vec{U}_{k+1/k})^T$"
],
[
"Update Step",
"The updated controller $\\vec{U}_{k+1/k+1}$ is bounded according to (REF ), and the slew rate constraint (REF )."
],
[
"Simulation Results",
"The fast turning maneuver defined in Sect.",
"is now tracked utilizing the UTC, while different solutions are obtained by the means of configuring the controller parameters.",
"First of all, the input moment coefficients $k_{right}$ , $k_{left}$ are considered.",
"Their desired dynamics is not known and, therefore, the behavior is driven by the process noise $Qu$ .",
"Thus, in order to get qualitatively different solutions the off-diagonal values of $Qu$ are modeled in different ways, e.g.",
"(REF ), (REF ).",
"${\\begin{array}{c}Qu_{1,2} = Qu_{2,1} = q_{1,2} \\cdot sign(yr_{ref}) \\\\Qu_{1,3} = Qu_{3,1} = -q_{1,3} \\cdot sign(yr_{ref})\\end{array}}$ where $1,2,3$ are the indices of $Cm_{damp}^{yaw}$ , $k_{right}$ , $k_{left}$ , respectively, and $q_{1,2}, q_{1,3}$ are tuning parameters that can be assumed equal.",
"${\\begin{array}{c}Qu_{1,2} = Qu_{2,1} = q_{1,2} \\cdot sign(\\frac{d}{dt} yr_{ref}) \\\\Qu_{1,3} = Qu_{3,1} = q_{1,3} \\cdot sign(\\frac{d}{dt}yr_{ref})\\end{array}}$ The coupling in (REF ) means that when $Cm_{damp}^{yaw}$ is reducing, making the turn faster, the appropriate input moment is increased to further accelerate the turn.",
"Thus, the coupling expressed by $Qu$ should be negative for $ k_{left}$ during left turns ($sign(yr_{ref})>0$ ) and for $ k_{right}$ during right turns, and positive otherwise.",
"The desired dynamics of the turning pattern angle command ($\\alpha $ ) can be assumed unknown, and left to be resolved by the UTC.",
"Optionally, the dynamics given in (REF ) can be utilized.",
"The gains of the proportional and the feed-forward parts can be tuning parameters.",
"The computation of the controller command during the prediction window can be implemented as shown in (REF ).",
"This enforces the slew rate constraint on $\\alpha (t)$ , see (REF ).",
"$\\begin{split}\\alpha (t)= \\alpha (t-dt)+ & K_p \\cdot (yr_{ref}(t)-yr_{ref}(t-dt))- \\\\ -&K_p \\cdot (yr_{real}(t)-yr_{real}(t-dt))+ \\\\ +&K_{FF} \\cdot (yr_{ref}(t+\\tau )-yr_{ref}(t+\\tau -dt))\\end{split}$ Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming unknown controller dynamics and correlation between input moment and yaw damping moment coefficients according to ().Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming controller dynamics according to () and correlation between input moment and yaw damping moment coefficients according to ()Figure: Tracking the yaw rate reference profile with three user input coefficients and body posture controlled by the Unscented Transform Controller, assuming controller dynamics according to () and correlation between input moment and yaw damping moment coefficients according to ()When a desired dynamics for the turning pattern angle is incorporated in the prediction step, a better accuracy is achieved: compare Figures REF , REF .",
"Also a smaller prediction horizon can be used.",
"For $dt=0.0042 \\, (s)$ 20 prediction steps were used (i.e.",
"$t_{pred}\\approx 0.08 \\, (s)$ ) when assuming unknown dynamics for $\\alpha (t)$ , versus 5 steps ($t_{pred}\\approx 0.02 \\, (s)$ ) when incorporating (REF ).",
"Notice the qualitatively different behavior of the input moment coefficients, shown in Figs.",
"REF , REF .",
"If $\\alpha (t)$ dynamics is unknown, the best accuracy is achieved when the correlation between the yaw damping moment and the input moment coefficients is defined according to (REF ), see Fig.",
"REF .",
"Notice, that (REF ) is a more intuitive model than (REF ).",
"If $\\alpha (t)$ dynamics is reasonably modeled, tuning the correlation between these variables is less significant.",
"Accurate tracking is achieved even if no correlation is assumed.",
"This, and other interesting solutions, are shown in [4].",
"The minimal control effort is achieved when combining (REF ) and (REF ), see Fig.",
"REF ."
],
[
"CONCLUSIONS AND FUTURE WORK",
"From simulations we observe the following advantages of the UTC: 1.",
"Fast convergence (around 2 (s)); 2.",
"Not much sensitivity to the tuning parameters: if the covariances are set to reasonable values, i.e.",
"the same order of magnitude as the allowed ranges of the actuators involved, UTC always converges.",
"3.",
"Convenient tuning: achieved by the means of setting the elements of $Pu_0$ and $Qu$ , which have a clear physical meaning.",
"Therefore, it is possible to obtain different solutions that implement the desired/ expected behavior of the control variables.",
"Further investigating the UTC properties remains an issue for future work: sensitivity to model uncertainty and measurement noise, and operation with only partial state feedback.",
"Additionally, it is important to formulate and prove conditions for the global/ local stability of systems controlled by the UTC.",
"Initial efforts in this regard are presented in [4]: for linear systems a stability condition takes the form of an Algebraic Riccati Equation, while for the number of prediction steps $n>1$ it has one additional term."
]
] | 2207.10496 |
[
[
"Towards Efficient Adversarial Training on Vision Transformers"
],
[
"Abstract Vision Transformer (ViT), as a powerful alternative to Convolutional Neural Network (CNN), has received much attention.",
"Recent work showed that ViTs are also vulnerable to adversarial examples like CNNs.",
"To build robust ViTs, an intuitive way is to apply adversarial training since it has been shown as one of the most effective ways to accomplish robust CNNs.",
"However, one major limitation of adversarial training is its heavy computational cost.",
"The self-attention mechanism adopted by ViTs is a computationally intense operation whose expense increases quadratically with the number of input patches, making adversarial training on ViTs even more time-consuming.",
"In this work, we first comprehensively study fast adversarial training on a variety of vision transformers and illustrate the relationship between the efficiency and robustness.",
"Then, to expediate adversarial training on ViTs, we propose an efficient Attention Guided Adversarial Training mechanism.",
"Specifically, relying on the specialty of self-attention, we actively remove certain patch embeddings of each layer with an attention-guided dropping strategy during adversarial training.",
"The slimmed self-attention modules accelerate the adversarial training on ViTs significantly.",
"With only 65\\% of the fast adversarial training time, we match the state-of-the-art results on the challenging ImageNet benchmark."
],
[
"Introduction",
"Vision Transformers with the self-attention mechanism have been broadly studied and become de facto state-of-the-art models for many benchmarks.",
"Recent works broadly investigated the traits of this new genre of architectures on computer vision tasks.",
"Meanwhile, the adversarial robustness of Vision Transformers has also been intensively studied [11], [51], [9], [43], [7], [1], [59], [83], [29], [45], [44], [47], [63], [32], [22].",
"To build robust Vision Transformers, an intuitive way is to apply adversarial training [62], [25] since it has been shown to be one of the most effective ways to achieve robust CNNs [23], [42], [5].",
"However, one major limitation of adversarial training is its expensive computational cost.",
"Adversarial training is known for requiring no extra cost during testing but greatly increasing the training cost.",
"Huge efforts have been devoted to overcome this deficit [61], [80], [75], [3], [65], [50], [30], [31].",
"However, the philosophy of designing stronger ViTs has largely lifted up the computation intensity and weakened the performance of previously-proposed techniques.",
"The specialty of the newly-proposed self-attention design [19], [13] in ViTs also introduces new challenges for accelerating adversarial training.",
"In this paper, we study the problem of how to efficiently carry out adversarial training on Vision Transformers.",
"We first apply the state-of-the-art Fast Adversarial Training (Fast AT) algorithm [61], [80] on a variety of vision transformers and analyze how factors like attention mechanism, computational complexity, and parameter size influence the training quality.",
"To the best of our knowledge, we are the first to accomplish a broad investigation on this topic.",
"Our survey shows that, although ViTs outperform CNNs by a great margin on robustness, they have hugely increased the computational complexity.",
"The self-attention mechanism adopted by ViTs is a computationally intense operation whose cost increases quadratically with the number of input patches.",
"This newly-emerged module hampers the utilization of several techniques for accelerating adversarial training.",
"Meanwhile, we find that large ViTs models also suffer from obvious catastrophic over-fitting problems [55].",
"This eventually leads to the degradation of robustness on ViTs with increasingly large capacity.",
"Figure: (a) AGAT chooses to drop a certain proportion of image embeddings based on the attention information at each self-attention layer.",
"(b) We plot the robust accuracy against the training time (hours) for various ViTs.",
"AGAT substantially accelerates adversarial training, while maintaining or improving the robustness at the same time.To make adversarial training efficient on the heavy-weight vision transformers, we investigate in accelerating adversarial training for vision transformers.",
"Particularly, we leverage the specialty that the self-attention mechanism of transformers is capable of processing variable-length inputs.",
"This specialty of ViTs has been utilized in a wide range of applications, including processing variable-length word sequences for translations [19], mining graphs with unlimited edges [74], etc.",
"Recently, on vision tasks, several works [69], [54], [16] explored the possibility of dropping input image patches during training or testing for acceleration purposes.",
"However, randomly dropping a certain number of input patches will inevitably hurt the training quality.",
"Thus, many works have proposed adaptive designs [49], [82] in the scenarios of a variety of targeted tasks.",
"Enlightened by the above works, we propose an Attention-Guided Adversarial Training (AGAT) mechanism, where we drop the patches based on the attention information.",
"As illustrated by Fig.",
"REF , our method intends to drop image embeddings after each layer of self-attention.",
"Note that the self-attention layer is non-parametric and thus is not limited to a static number of inputs.",
"We drop the embeddings with lower attention and keep the higher ones.",
"Such a design will better preserve the feed-forward process and therefore guard the backward gradient computation of generating the adversarial examples.",
"As shown in Fig.",
"REF , AGAT gets to keep the training quality mostly unchanged or be improved by taking only $65\\%$ training time.",
"Our work matches the state-of-the-art results of adversarial robustness on the challenging benchmark of ImageNet."
],
[
"Relate Works",
"Adversarial Training.",
"Adversarial attacks [66], [23], [42], [79], [37], [38] intend to endanger the performance of deep networks via repeatedly optimizing the input images with repect to the output of the model.",
"To counter this unwanted deficit, various defensive approaches were proposed [46], [20], [39], [81], [27], [64], [60], [40].",
"Among these defenses, the methodology of adversarial training withstands most kinds of examinations and has become one of a few defenses that can consistently improve the robustness of deep networks when facing most attacks [5].",
"However, adversarial training is known to suffer from complexity issues [61], [80], [53].",
"Particularly, Fast AT [80] enhances the single-step adversarial training with random initialization.",
"Fast AT shows promising results on benchmark datasets.",
"Later works [4], [33], [75], [3], [65], [50] also proposed improved variants of Fast AT.",
"Vision Transformer.",
"The Transformer architecture and its self-attention mechanism were first proposed in the field of natural language processing (NLP) [73], [19], [13] and then adopted in the scenario of computer vision [84], [17], [78].",
"After the huge efforts of a surge of explorations [21], [70], [76], the Vision Transformer (ViT) has shown the potential to surpass the traditional convolutional neural networks.",
"Then, researchers keep pushing this new philosophy of model design into a wide range of fields like high-resolution vision tasks [41], [77].",
"Meanwhile, to reduce the huge computational expense that is brought by the densely modeled self-attention mechanism, various techniques have been proposed [76], [17].",
"Adversarial Robustness of ViT.",
"The adversarial robustness of ViT has also achieved great attention due to its impressive performance [12], [62], [52], [48], [26], [8], [10], [63].",
"Some works [12], [62], [10] first reported positive results where they showed that standard ViTs perform more robust than standard CNNs under adversarial attacks.",
"The later works [8], [26] revealed that ViTs are not more robust than CNNs if both are trained in the same training framework.",
"By adopting Transformers’ training recipes, CNNs can become as robust as Transformers on defending against adversarial attacks.",
"In both sides, we can observe that the clean accuracy of standard models can be easily reduced to near zero under standard attack protocols.",
"In addtion, Fu et al.",
"[22] studied attacking ViTs in a patch-wise approach, which reveals the unique vulnerability of ViTs.",
"To boost the adversarial robustness of ViTs, recent works [67], [6] explored multiple-step adversarial training to ViTs.",
"Shao et al.",
"[62], tested the vanilla adversarial training on CIFAR10.",
"However, multi-step adversarial training is computationally expensive.",
"And in this work, we take the step of exploring fast single-step adversarial training on ViT models."
],
[
"Fast Adversarial Training on Vision Transformers",
"We first comprehensively study the Fast AT [80] algorithm on vision transformers.",
"Fast AT is designed to be efficient so that it can be applied to large models (e.g., ResNet-101 [28]) on large-scale datasets (e.g., ImageNet [57]).",
"Specifically, Fast AT refines the standard single-step FGSM [23] algorithm by adopting a large random perturbation as the starting point for searching adversarial examples.",
"By doing so, Fast AT is supposed to effectively resist the catastrophic overfitting [80] of training with the plain FGSM.",
"Thus, Fast AT preserves the effectiveness while significantly improves the efficiency over the multi-step PGD [42] algorithm.",
"To better understand the robustness of the newly-developed ViT models, in this section, we apply Fast AT to a wide range of ViTs.",
"We select nineteen models of different sizes from five vision transformer families, including ViT [21], CaiT [72], LeViT [24], SwinTransformer [41], and CrossFormer [77].",
"The selected models cover a wide range of model designs, including hybrid models [24], slim models [24], [77], constrained attention [41], [77], and multi-scale attention [24], [77].",
"Following the settings of Fast AT [80], we set the perturbation radius to $2/255$ for ImageNet and test the adversarially trained models with the 100-step PGD attack.",
"Our training schedule aligns with SwinTransformer [41] and DeiT [71].",
"We keep all hyper-parameters, e.g., image size, training epoch, and data augmentation, identical for all models.",
"We also present the results of Fast AT on the CNN models of ResNet-50 and ResNet-101 for comparison.",
"As shown by Fig.",
"REF , ViTs are consistently more robust than CNNs.",
"This aligns with concurrent researches on model robustness [68].",
"We conclude draw novel observations as follows: Figure: (a) Robust accuracy for various ViTs.",
"(b) Large ViTs like CaiT-S36 may suffer from obvious unstable training and over-fitting, while designs like window attention can alleviate the issue.",
"(c) The perfromance of ViTs on Fast AT may not align with their natural accuracy of natural training.",
"(d) In contrast, the natural and robust accuracies of ViTs align with each other when both are under Fast AT.",
"(e) Large ViTs greatly increase the computational complexity.",
"Within the same transformer family, larger transformers do not always result in better robustness.",
"In Fig.",
"REF , the transformer families of LeViT, CaiT, and ViT exhibit a pattern of over-fitting.",
"Namely, as the models get larger, the network robustness learned by Fast AT degrades conversely.",
"For instance, Cait-S36 performs worse than CaiT-XXS24.",
"Fig.",
"REF provides more details of the above over-fitting issue.",
"The optimization of the CaiT-S36 model gradually degrades after a certain point.",
"In contrast, the CaiT-XXS24 model possesses a monotonically increasing training curve.",
"This aligns with previous findings that ViTs may suffer from more severe over-fitting on natural tasks and need the assistance of aggressive data augmentation schedules [71].",
"Moreover, transformers with a constrained attention mechanism can alleviate the over-fitting problem.",
"In Fig.",
"REF , unlike CaiT-S36, the large model of Swin-Base shows a steady training curve.",
"Among different transformer families, the attention mechanism designed for better natural performance not necessarily results in better robustness.",
"In Fig.",
"REF , for each transformer architecture, we plot its natural performance under natural training against its robust performance under Fast AT.",
"Among different transformer architectures, the two metrics of natural accuracy and robust accuracy approximately form a line, indicating the close relation between natural and robust performances.",
"Compared with the line formed by CNNs, vision transformers consistently achieve better robustness on models with similar natural performance.",
"However, we can observe a few outliers of models from the LeViT and CaiT families.",
"Specifically, the hybrid design of LeViT can achieve high robustness with very small models.",
"Large CaiTs, despite being effective on the natural task, result in obvious inferiority on robustness.",
"Notice that, for each point in Fig.",
"REF , the two metrics of the vertical axis and the horizontal axis are evaluated on two models, either adversarially-trained or naturally-trained, of the same architecture.",
"When we plot the natural accuracy and robust accuracy, both of which are under Fast AT, as shown by Fig.",
"REF , the two metrics consistently align with each other without any outlier, revealing the difference between Fast AT and standard training.",
"SOTA ViTs suffer from a severe efficiency issue and require much more training time than SOTA CNNs.",
"The above over-fitting problem mainly shows on large ViTs like Cait-S36 or LeViT-384, but not on the small ones.",
"This is reasonable since models like Cait-S36 are consistently larger than commonly-used CNNs.",
"However, the over-fitting is not the only problem of adopting larger and larger ViTs.",
"These increasingly large models have hugely lifted up the computation intensity.",
"In Fig.",
"REF , we plot robustness (Robust Accuracy) against computation intensity (GFLOPs).",
"State-of-the-art ViTs can be a few times larger than CNNs.",
"This makes utilizing Fast AT even harder since the self-attention module is more intricate to accelerate."
],
[
"Efficient Adversarial Training on Vision Transformers",
"As discussed in the last section, one major problem that hinders the deployment of adversarial training on vision transformers is the efficiency issue.",
"Adversarial training is known to be computationally intensive.",
"This greatly hampers its usage on large-scale models or datasets.",
"Various techniques have been proposed to mitigate this problem.",
"However, with the revolution brought by vision transformers, many existing techniques such as variable-resolution training [80] have been unusable.",
"More importantly, there is a rising trend of adopting increasingly large ViTs for better performance.",
"The complexity of these enormous ViTs is too large to afford, even for efficient algorithms like Fast AT.",
"In this section, we first analyze the computational complexity of popular ViTs.",
"Our analysis shows that ViT requires a much longer time to finish adversarial training, which is caused by the large computational cost of ViT brought by a large number of input patches.",
"Then, we explore the input patches to reduce the brought computational cost.",
"Given the fact that the flexibility of self-attention allows ViTs to process an arbitrary length of image patches, we explore a random patch dropping strategy to reduce the computation.",
"The dropping operation with the reduced number of patches can accelerate adversarial training, as expected.",
"However, the naive dropping strategy will also hurt robustness.",
"To address the above issues, we propose our Attention-Guided Adversarial Training algorithm, which selectively drops patches based on attention magnitude."
],
[
"Computation Intensity of ViTs",
"We first formally formulate our task.",
"For each matrix, we present its shape in the lower right corner and its index in the upper right corner.",
"Denote the input feature as $_{p \\times d}$ , which consists of a sequence of $p$ embeddings with the dimension being $d$ : $=[X^1_{d}, X^2_{d}, ..., X^p_d]$ .",
"Each embedding relates to a specific non-overlapped patch of the input image.",
"Vision transformer consists of a list of blocks, each of which consists of two kinds of computation, i.e., the Multi-head Self-Attention layer (MSA) and the Multi-Layer Perceptron layer (MLP).",
"In the MSA module, $$ is first normalized via Layer Normalization and then transformed to the query, key, and value matrices ($,\\ ,\\ $ ).",
"$[_{p\\times d},\\ _{p\\times d},\\ _{p\\times d} ] = \\text{LayerNorm}(_{p\\times d})^{1}_{d\\times 3d} .$ For the multi-head design, we partition the $,\\ ,\\ $ matrices of shape $p\\times d$ into $h$ heads, with each part having a shape of $p\\times \\frac{d}{h}$ .",
"Then, taking the first head as an example, $^1$ will be re-weighted by $^1$ with the following form: $\\text{Attn}(^1_{p\\times \\frac{d}{h}}, ^1_{p\\times \\frac{d}{h}}, ^1_{p\\times \\frac{d}{h}})=\\text{SoftMax}(^1{^1}^{\\top } / \\sqrt{d} + )^1=^1_{p\\times p}^1 .$ $$ is a learnable bias.",
"Note that all the column vectors of $^1$ are normalized by Softmax and thus have a summation of 1 in each.",
"Then, $$ value of each head will be concatenated and transformed to the output of MSA.",
"$^{\\prime }_{p\\times d} = \\text{Concat}(^1^1, ^2^2,..., ^h^h)_{p\\times d} ^2_{d\\times d} .$ A following MLP module will take in the output of MSA, $^{\\prime }$ , and transform each embedding with Layer Normalization and GELU activation.",
"$^{\\prime \\prime }_{p \\times c} =\\text{GELU}\\Big [\\text{LayerNorm}(^{\\prime }_{p \\times d})^{3}_{d\\times 4d}\\Big ]^{4}_{4d\\times d} .$ The computational complexity of the above process is: $\\mathrm {\\Omega }(\\text{MSA}) = 4pd^2+2p^2d+pd ;\\quad \\mathrm {\\Omega }(\\text{MLP}) = 8pd^2+pd .$ A typical vision transformer will consecutively conduct the above process to generate the final image representation for prediction.",
"Take the ViT-Base model as an example.",
"Each layer has $d=768$ .",
"Therefore, we have $\\mathrm {\\Omega }(\\text{MSA})+\\mathrm {\\Omega }(\\text{MLP})=7\\times 10^6p+1.5\\times 10^3p^2$ .",
"Since $7\\times 10^6\\gg 1.5\\times 10^3$ and $p$ is mostly around $2\\times 10^2$ , the computational complexity of the entire ViT is approximately linear to $p$ ."
],
[
"Dropping Patch: The Flexibility of Self-Attention",
"Different from the convolutional operation where the hyperparameters (e.g., kernel size, padding size) are supposed to be fixed, the self-attention operation does not require the inputs with fixed length.",
"For instance, this flexibility of self-attention is leveraged to process an arbitrary length of words in NLP tasks.",
"Similarly, the flexibility makes its adaption to graph data feasible, in which different nodes can have a different number of connected edges [74].",
"When transformers with self-attention mechanisms have been introduced into computer vision tasks, researchers also investigate dynamically dropping the patches or the embeddings in the forward pass of a ViT model [69], [54], [16].",
"It is found that, when a constrained quantity of patches are dropped, the forward inference can be significantly accelerated.",
"Meanwhile, the performance of the model will be only slightly degraded [49], [82].",
"Several works utilize this feature to design new mechanisms for their own unique purposes.",
"In this work, we also explore a patch dropping strategy to accelerate adversarial training given the excellent trade-off it achieves.",
"We first test the scheme of randomly dropping a certain number of input patches to see how it influences the training quality of Fast AT.",
"We report the results in Fig REF , where we plot the robustness against the training epoch for different ratios of dropping.",
"Note that no patches will be dropped during inference in the testing stage.",
"When the number of input patches is reduced, the forward inference of ViT can be accelerated.",
"Surprisingly, from the figure, we also observe that the dropping operation also stabilizes the adversarial training and alleviates the phenomenon of catastrophic over-fitting [80], [58], [33].",
"As shown by Fig REF , we repeat Fast AT without any dropping for three times.",
"The training procedure can be very unstable and occasionally drop to zero accuracy.",
"We conjecture that it is the regularization effect brought by the patch dropping operation that stabilizes Fast AT.",
"To further verify this conjecture, we test ViTs equipped with the dropout operation as in DeiT [71].",
"The dropout module is applied right after the self-attention module.",
"As shown in Fig REF , like dropping patches, the attention dropout module also stabilizes Fast AT.",
"Unlike dropping patches, the dropout module cannot save computation.",
"However, the final robust accuracy can be reduced in both cases when dropping is applied.",
"The random patch dropping strategy poses a dilemma.",
"Namely, it brings both acceleration and performance degradation.",
"In the following section, we will present our attention-guided patch dropping strategy, where we achieve a better trade-off between efficiency and effectiveness.",
"Figure: (a) We repeat the baseline without any dropping for three times.",
"We can observe the training is very unstable.",
"(b) Training curve when we drop various rates of input patches.",
"(c)Training curve when we adopt various rates of attention dropout.Figure: The illustration of the plain self-attention layer and the attention layer of AGAT.",
"The embedding mask is directly generated from and is non-parametric.",
"Thus, for the same model, one can adopt our AGAT during training and the plain self-attention during testing, respectively."
],
[
"Attention-Guided Adversarial Training",
"It is known that adversarial training utilizes adversarial attacks to generate examples so that the network can learn to fit the generated adversarial examples.",
"This is a typical hard example mining framework.",
"The more powerful the adversarial examples are, the more robust the learned network will be.",
"The quality of the adversarial examples relies on the adversarial attacking algorithm.",
"And the attack algorithm depends on accurately estimating the gradient of the input pixels with respect to the loss function.",
"Thus, a major insight of achieving our goal is to drop the patches that will barely hamper the gradient estimation.",
"A similar philosophy has been utilized for sparse attacks or black-box attacks.",
"Recent works studied the learned attention and found that the magnitude of the attention can reveal how salient an embedding is [15].",
"This enlightens us that we can utilize this ready-made information to filter salient embeddings.",
"Thus, we sum $\\frac{1}{h}\\sum _{i=1}^{h} ^i$ by row and generate an index vector $$ .",
"Note that the column of $\\frac{1}{h}\\sum _{i=1}^{h} ^i$ is the weighted-average parameter and thus always equal to 1.",
"It indicates how much a generated embedding receives the information of each input embedding.",
"In contrast, the row of $\\frac{1}{h}\\sum _{i=1}^{h} ^i$ reveals how much each input embedding influences the output embeddings.",
"This value differs from embedding to embedding.",
"Thus, we choose to select the top-k embeddings based on their magnitude in $$ .",
"Then, the formulation of (REF ) becomes: $^{\\prime }_{k\\times d} =\\text{MaskBy}\\Big [\\text{Concat}(^1^1, ..., ^h^h),\\text{Topk}( )\\Big ]_{k\\times d}^2_{d\\times d} .$ We drop the embeddings after the weighted average calculation of $$ so that the magnitude of embeddings will be kept stable.",
"The number of embeddings will reduce from $p$ to $k$ .",
"To fully utilize the attention information in each layer, we propose a layer-wise exponential dropping scheme.",
"Namely, in each layer, we drop a constant proportion of patches.",
"Thus, this scheme will drop more embeddings on deeper layers, where the embeddings are consistently more redundant [54].",
"We set the dropping rate to $0.9$ .",
"On a 12-layer ViT-Base model, the final layer will process only $31\\%$ number of embeddings and save more than $40\\%$ FLOPs of the entire model.",
"A detailed implementation of our Attention-Guided Adversarial Training is shown in Algorithm REF .",
"Our AGAT only modifies the training process.",
"During testing, we use the original model for prediction.",
"The class token will not be dropped when it involves the feed-forward procedure.",
"[t] AGAT attention code (PyTorch-like) whitefullflexiblebcodeblue def init(num_heads, dim, drop_rate): head_dim = dim // num_heads # num_heads: the number of heads, dim: embedding dimension scale = head_dim ** -0.5 qkv = nn.Linear(dim, dim * 3) proj = nn.Linear(dim, dim) def forward(self, x): # x: input tensor with the shape of (b, p, d); b, p, d = x.shape # b: batch size; p: patch number q, k, v = qkv(x).reshape(b, p, 3, num_heads, head_dim).permute(2, 0, 3, 1, 4).unbind(0) attn = (q @ k.transpose(-2, -1)) * scale attn = attn.softmax(dim=-1) # b num_heads p p own_attn = torch.sum(torch.sum(attn, dim=1), dim=-2) # b p kept_num = int(p * drop_rate) - 1 # compute the number of kept embeddings _, rank_indices = torch.topk(own_attn[:,1:], k=kept_num, dim=-1) # b k rank_indices = rank_indices.unsqueeze(-1).repeat(1,1,dim) # for the API of torch.gather x = (attn @ v).transpose(1, 2).reshape(B, N, C) x = torch.gather(x, dim=1, index=rank_indices) # drop embeddings return proj(x)"
],
[
"Experimental Setup",
"Dataset We evaluate our method on the challenging ImageNet [56] dataset.",
"Due to the huge computational expense of adversarial training, most adversarial training approaches are only verified on relatively small datasets such as CIFAR10 [35] or MNIST [36].",
"Our efforts on improving the efficiency of adversarial training allow us to apply adversarial training to large-scale datasets.",
"The input image size is 224 on all models for a fair comparison.",
"Training Schedule Following previous works [71], [77], we use the AdamW [34] optimizer for training for 300 epochs with a cosine decay learning rate schedule.",
"The initial learning rate is set to $0.001$ .",
"The first 20 epochs adopt the linear warm-up strategy.",
"The batch size is 1024 split on 8 NVIDIA A100 GPUs.",
"Evaluation Metrics We report our major results on two metrics, robust accuracy and GFLOPs.",
"We mainly focus on improving the speed of training and keeping the learned robustness unchanged at the mean time.",
"For a direct impression of training speed, we also record the training time for each method.",
"Note that the training time is not only determined by the efficiency of the training algorithm, but also the IO speed and many other nonnegligible factors.",
"Adversarial Attack We choose the powerful multi-step PGD attack [42] with the perturbation radius being $2/255$ or $4/255$ and the optimization step being 20 or 100 [81].",
"We also test different kinds of attacks, including black-box attacks, to rule out the possibility of obfuscated gradient [5].",
"Table: Adversarial Training on The ImageNet Dataset.",
"In most cases, our Attention-Guided Adversarial Training on ViTs achieves comparable clean performance and robust accuracy to Fast-AT with much less time.",
"The conclusion still holds when different perturbation ranges are applied.Vision Transformers Our AGAT can be directly used on the self-attention model and most of its variants.",
"We apply our AGAT to three commonly-used models ViT [21], CaiT [72], and LeViT [24].",
"All the three models are built on the original self-attention module and can fully reveal the effectiveness of our AGAT.",
"In future work, we will explore combining our AGAT with more sophisticated attention mechanism like window attention [41] or multi-scale attention [77].",
"Training Algorithm For vision transformers, we compare our AGAT with FastAT and the random dropping strategy (Random).",
"we also provide results of Free AT [61] and Grad Align [4] algorithms on the ResNet [28] models."
],
[
"Improved Efficiency of Adversarial Training on ImageNet",
"We present the performance of AGAT in Table REF .",
"Because AGAT drops a static rate of embeddings for each self-attention layer, the depth of the ViTs will decide the total number of dropped features.",
"Thus, we adjust the dropping rate for ViTs with different numbers of blocks so that the complexity of the feed-forward process will be approximately reduced by $40\\%$ of the baseline.",
"For instance, we set the dropping rate to $0.1$ for models with 12 blocks but $0.03$ for models with 36 blocks.",
"For the baseline of randomly dropping input patches, we can always set the dropping rate to $0.4$ since the total amount of saved computation will not be affected by the number of blocks.",
"As shown by Table REF , for each vision transformer, the Fast AT baseline achieves high robustness but is time-consuming, while the Random dropping strategy saves training time but achieves inferior robustness.",
"In contrast, AGAT achieves comparable robustness with Fast AT using much less training time.",
"Particularly, on the ViT-Base model, AGAT achieves similar robustness to Fast AT but only takes $65\\%$ of training time.",
"Meanwhile, since slim models such as ViT-Tiny and LeViT-128 do not possess the same level of model redundancy as their large-sized counterparts, the robustness of these slim models degrades more dramatically than larger ones when we randomly drop patches.",
"When trained with AGAT, the robustness of slim models matches the plain Fast AT .",
"Table: Evaluation of Adversarially Trained Models under Various Attacks.",
"Robust accuracy of adversarially trained models is reported in this table.",
"The robust accuracy achieved by our AGAT is comparable to that by Fast-AT under various attack evaluations and different perturbation ranges."
],
[
"Results under Various Attacks.",
"Adversarial robustness is known to be hard to examine.",
"Several defensive algorithms were found to be vulnerable to tailored attacks.",
"One of the most important and typical representatives of such phenomena is the obfuscated gradient problem.",
"Our AGAT does not fall into this category, considering the algorithm neither utilizes any stochastic process nor hampers the gradient computation.",
"In fact, our AGAT only takes effect on the training stage and does not modify any procedure during evaluation.",
"To further show the robustness of the learned ViTs, we present the robust accuracy of our learned models under various different attacks in Table REF .",
"We select the attacking criteria of C&W [14], Square [2], APGD-CE [18], and APGD-DLR [18].",
"The AGAT models achieve the same level of robustness as Fast AT."
],
[
"Robustness and Dropping Rate.",
"We cross-validate the rate of dropping of our AGAT in Table REF .",
"To provide a clear comparison with the random dropping strategy, we compare their learned robust accuracy when both dropping strategies reduce approximately the same amount of computation.",
"Due to the difference between the two algorithms (layer-wise vs input-wise), the actual learning rates are different across the two methods.",
"It can be told that AGAT can maintain the learned robustness in a wide range of dropping rates.",
"In contrast, the random dropping strategy significantly degrades the performance, especially for the slim model of ViT-Tiny.",
"Dropping more than $40\\%$ computation will bring obvious degradation on robustness, even for AGAT.",
"Thus, we consider this dropping rate as a good trade-off between effectiveness and efficiency.",
"Table: Ablation Study on Dropping Strategy.",
"We compare our attention-guided dropping strategy with random dropping.",
"Ours outperforms random dropping constantly in different dropping rates."
],
[
"Visualization.",
"To better get an insight of how AGAT takes effect, we visualize the internal results of the ViT-Base model.",
"For each of the 12 blocks in ViT-Base, we show the position of the dropped embeddings by masking out the corresponding image patches.",
"In Fig.",
"REF , the position of the dropped embedding mainly concentrates on the relatively unimportant positions like background, while the patches of the main object are mostly kept.",
"This indicates that our AGAT successfully guards the feed-forward procedure and thus secures the generation of adversarial examples.",
"We also visualize the corresponding value of attention for each patch.",
"The darkness of each patch position indicates how much the corresponding embedding of this patch influences the other embeddings.",
"For each block, we normalize all the values of attention by dividing the maximum value of attention.",
"This visualization also demonstrates that the dropping rate of AGAT gets to cover the embeddings on the position of the main object.",
"Therefore, dropping rates larger than the chosen value may lose crucial information for inference and thus hamper the generation of adversarial examples for training.",
"Figure: Visualizations of dropped image patches and the distribution of attention.",
"AGAT mainly cuts off the computation of non-object embeddings and thus maintains the performance and gradient computation."
],
[
"Conclusions",
"Adversarial training is one of the most effective defense methods to boost the adversarial robustness of models.",
"However, it is computationally expensive, even after many efforts have been made to address it.",
"The emergence of ViTs, whose computational cost increases quadratically with the number of input patches, makes adversarial training more challenging.",
"In this work, we first thoroughly examined the most popular fast adversarial training on various ViTs.",
"Our investigation shows that ViT achieves higher robust accuracy than ResNet, while it does suffer from a large computation burden, as expected.",
"Our further exploration showed that random input patch dropping can accelerate and stabilize the adversarial training, which, however, sacrifices the final robust accuracy.",
"To overcome the dilemma, we proposed an Attention-Guided Adversarial Training (AGAT) mechanism based on the specialty of the self-attention mechanism.",
"Our AGAT leverages the attention to guide the patch dropping process, which accelerates the adversarial training significantly and maintains the high robust accuracy of ViTs.",
"We hope that this work can serve the community as a baseline for research on efficient adversarial training on vision transformers.",
"Acknowledgements.",
"This work was supported in part by The National Key Research and Development Program of China (Grant Nos: 2018AAA0101400), in part by The National Nature Science Foundation of China (Grant Nos: 62036009, U1909203, 61936006, 62133013), in part by Innovation Capability Support Program of Shaanxi (Program No.",
"2021TD-05)."
]
] | 2207.10498 |
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