# Peano modes at the D=2 delocalization transition

V. G. Benza  Dipartimento di Scienza e Alta Tecnologia, Universita’ dell’
Insubria,20064 Como, Italy<EMAIL_ADDRESS>
###### Abstract

We study the elasticity of the plane-filling Peano chain in terms of a spring
network, modeling a flat and thin elastic strip, smoothly curved so as to
follow the chain pattern.
This network sustains normal modes, named Peano modes; in the same way as
acoustic waves in crystals are sustained by the matching of translations with
rotations, these modes arise when discrete dilatations match with $\pi/4$
rotations. The pattern shares this eightfold symmetry with the octagonal
quasicrystal, but is much looser and prone to disruption; its relaxation marks
the saddle point separating the solid from the liquid. In terms of the
equivalent quantum mechanical problem the above corresponds to the D=2 quantum
delocalization transition.
Some experimental observations, regarding a priori uncorrelated systems such
as colloids, bilayer water and chromosomal DNA come together, at least
qualitatively, within the viewpoint proposed here. With reference to the D=2
colloids the model accounts for the bosonic peak found in their phononic
spectrum and for the relaxation of the excitations. The latter typically
evolve towards patterns of increasingly large size, as in the inverse energy
cascade of D=2 turbulence. As for bilayer water, it undergoes a liquid to
quasicrystal transition surprisingly similar to the phase transition found
here. Finally, considering the chromosomal DNA, the model unveils the mixed
nature of its compact conformations, lying at the boundary between a liquid
and a solid, similar to colloids and quasicrystals.

###### pacs:

82.35.Pq,64.60.Al,61.44.Br,63.50.-x,63.20.D-,05.40.Jc

##  1 Introduction

In this work we study the harmonic spectrum and the time relaxation of the
planar Peano-Hilbert curve $PH_{2}$. The curves $PH_{D}$ that fill
D-dimensional spaces are used in ordering multidimensional data [1, 2], as
well as in manufacturing miniature antennas [3]. In the domain of polymer
physics the curve $PH_{3}$ has been proposed as a possible final state of a
scale-invariant collapse [4]. Quite remarkably, given the high degree of
ordering of $PH_{3}$, its pattern has been found to fit with the spatial
organization of the chromosomal DNA in humans over a rather large range of
scales [5]. This resulted from detecting the physical contact frequencies
between pairs of genomic loci, using chromosome conformation capture methods
[6]. Regarding DNA dynamics, significant data emerged from experiments on
bacterial (E.coli) chromosomes [7, 8]. It was found that DNA loci undergo
anomalous diffusion with a rather sharp distribution in the exponents, where
the dominant behavior $(t^{0.4})$ can be connected with a fractal dimension
$D_{f}\leq 3$ [9]. A physical description of the chromosomal conformation
could be of interest for cell biologists and should be based on quantitative
informations regarding the chain topology, the distribution of crosslinks, the
nature of interchain interactions; in addition, this looked-for description
should account for the complex multicomponent medium of the cell. Originally
motivated by this phenomenology, the present paper is limited to analizing the
harmonic properties of the planar curve $PH_{2}$. We consider a quasi one-
dimensional elastic strip, modeled as a mechanical network running along
$PH_{2}$, where deviations from the exact pattern carry a harmonic energy. The
elastic couplings are anisotropic since they vary according with the local
geometry, so that the resulting mechanical network turns out to be rigid:
apart from uniform translations it does not allow for zero-energy modes. In
particular, rigid rotations or uniform dilatations of any portion of the
network have an energetic cost. The model does not account for the interaction
among physically adjacent sites when they are distant along the chain: our aim
is to describe waves propagating along meandering one-dimensional patterns,
not necessarily having the nature of covalent bonds. This description could as
well apply to electromagnetic waves, as it is known that the D=2 elasticity
has strong similarities with the D=2 electromagnetism: the elastic
deformations can be decomposed into irrotational and solenoidal components,
where dilatation $(\mathcal{D})$ and shear $(\mathcal{S})$ have the roles of
electric and magnetic field respectively. The main difference between the two
theories regards the wave propagation properties, because dilatation pays an
higher energy with respect to shear, and as a consequence in the bulk the
longitudinal $(\mathcal{D})$ field propagates with an higher velocity with
respect to the transverse $(\mathcal{S})$ field. The two distinct waves couple
at the medium’s boundary, where they form the Rayleigh waves, which are
characterized by nontrivial boundary-dependent dispersion laws. This
dependence is exploited to extract information during earthquakes, and in the
case of electromagnetism it can be used both to extract and to manipulate
information, as done, e.g., by scattering electromagnetic waves through
photonic crystals [10] or by engineering e.m. wavefronts with metamaterials
[11]. In the elastic model in question here the particular ordering enforced
by the Peano pattern, which is built by inflating quadrupolar partitions of
planar regions, produces non-dispersive Rayleigh waves, named Peano waves.
In order to make clear the physical interpretation of the network, we recall
how contact interactions among rigid disks undergo harmonic modeling, a
procedure going under the name of fluctuation method [12]. In that framework
the notion of contact (harmonic) network [13, 14] can be given the following
precise meaning [15, 16]: ’In a metastable state of a hard sphere system, two
particles are considered to be in contact when they collide during some time
interval $\Delta t$ much smaller than the relaxation time of the structure,
where metastability is lost, and much larger than the (thermal) collision
time’. If the Peano pattern is to be associated with a linear polymer, the
model describes the polymer’s fluctuations within a steep effective potential.
Notice that the potential has its own time evolution, but on the larger scale
of the structural relaxation. At shorter times the line of minima of the
potential follows the topology assigned by the ordered sequence of bendings of
the undeformed pattern. Within this polymeric interpretation, if one were to
add to the picture the contact couplings among adjacent but nonsubsequent
sites, one would end up with an elastic network describing a textured
membrane. The phononic spectrum of such enlarged model would then display in
the density of states, in addition to the contributions described in this
work, the behavior $\rho(\lambda)_{D=2}\approx\lambda$, as expected from
standard D=2 Debye spectra. It is worth mentioning that some experiments on
two-dimensional colloids did in fact display an anomalously high density of
states at low frequencies [17, 18], superimposed to the D=2 Debye behavior. As
a matter of fact we show here that the connection is not limited to the so-
called bosonic peak of colloids, but involves their space-time behavior as
well. In analogy with quasicrystals, whose phononic spectra are marked by
self-similarity [19, 20, 21], we exploit the recursive nature of the curve and
consider the sequence of its periodic approximants having the form of closed
chains on the square lattice. At the lattice sites the harmonic operator acts
on the two-dimensional space of planar deformations; it has a $(2\times 2)$
block tridiagonal structure since it involves, at each site, the two nearest
neigbours along the chain. For the sake of brevity we will call this operator
disordered laplacian, in spite of it being neither scalar nor disordered. In
fact the operator is deterministic, as the sequence of the different vertices
that assigns the chain pattern.
We have found that in the case of $PH_{2}$ the spectral problem for the
disordered laplacian translates into a finite-dimensional trace map, which can
be treated by standard methods. The phononic spectrum of the closed curve
shows a multiplicity of power-law singularities in the density of states, with
exponents covering a whole interval (see Figs.1 and 2). The physics implied by
such an exotic spectrum emerges from the analysis of the relaxation process,
described by the overdamped Langevin equation associated with the disordered
laplacian. The result is that every exponent identifies the scaling behavior
(with respect to length and time) of a specific class of fluctuations.
We then found that the anisotropy of the elastic coupling induces a phase
separation (see Fig.3); the two phases are respectively dominated by
dilatation $(\mathcal{D})$ and shear $(\mathcal{S})$, and can be identified
with solid and liquid. The critical point where the two phases coalesce
corresponds to the isotropic limit; quite remarkably the interference of
$(\mathcal{D})$ and $(\mathcal{S})$ generates not only anomalous fluctuations,
but also phononic modes, the Peano modes. During the relaxation process the
energy flows from localized to extended structures, as in the D=2 turbulence
[22] and as observed in the mentioned experiments on two-dimensional colloids
[17, 18]. From the perspective of the quantum-mechanics determined by the
disordered laplacian, the isotropic limit corresponds to the critical point of
the D=2 quantum delocalization transition. In other words the Peano chain
identifies a two-dimensional extension of the D=1 Aubry-Andre’ class [23]. A
comprehensive exploration of the critical point requires an analysis of the
trace map; here we limit ourselves to some preliminaries regarding its
symmetry properties. As discussed in Section 6 we find an eightfold symmetry,
revealing that $PH_{2}$ is close to the octagonal quasicrystal, but with a
definitely lower connectivity. Interestingly, a liquid to dodecagonal
quasicrystal transition has been reported for bilayer water [24] in a similar
regime. Some preliminaries regarding the properties of the Peano chain are
reported in the Appendix where, more importantly, we derive a substitution
rule for closed chains. In Section 2 we introduce the network model and the
disordered laplacian; we then formulate the eigenvalue problem for this
laplacian in terms of a trace map. In Section 3 we analize the singularities
of the spectral density of states by means of the so-called thermodynamic
formalism of multifractals. Readers uninterested in the spectral properties
can skip Sections 2 and 3 and go over to Section 4, where we examine the
fluctuations. The anomalous diffusion exponents and fractal dimensions are
explicitly obtained for the planar case and estimated by means of a mean-field
argument valid for chains of arbitrary dimension D. The exponents obtained
with $D=3$ appear to be in agreement with measures of anomalous diffusion
performed on bacterial nucleoids [7, 8].
Section 5 is centered on the solid-liquid phase transition and on the
properties of the relaxation process. In Section 6 we discuss the symmetries
of the system at the critical point and illustrate how they relate with the
Peano modes.

##  2 Elastic network and trace map

We consider the vector field of deformations of the plane-filling Peano-
Hilbert chain $PH_{2}$. An example of the patterns described by $PH_{2}$ is
given in Fig.2A; as it is shown in the Appendix, the closed chain can be
constructed by means of an iterative procedure defined over the square
lattice. One starts with the approximant of order $k=1$, taken as the square
path, and inflates the paths with suitable rules. The approximant of order $k$
is a closed chain that covers with no intersection all sites of the
$(2^{k}\times 2^{k}),\,(k=1,2,...)$ lattice. Upon assigning an orientation to
the path the lattice sites can be labeled with the chain coordinate
$n,\,(n=1,2,...4^{k})$, and the vector field of deformations can be written as
$\vec{x}(n)\equiv(x(n),y(n))$. This labeling is appropriate in our context
since we assume that the excitations are transmitted only along the Peano
pattern and that their elastic energy depends on its local geometry. The
dynamics studied here involves the two-dimensional spaces of deformations at
the sites $(n-1,\,n,\,n+1)$. The normal modes of the model solve the following
equation:

$\lambda\,\,\hat{I}\vec{x}(n)=\hat{\Gamma}^{(in)}_{n}\vec{x}(n-1)+\hat{\Gamma}^{(out)}_{n}\vec{x}(n+1)+\hat{V}_{n}\vec{x}(n),$
(1)

where $\hat{I}$ is the identity and
$\hat{\Gamma}^{(in)},\,\hat{\Gamma}^{(out)}$ are the $2\times 2$ matrices
describing the contributions from the (incoming/ outgoing) links attached to
the $n$-th site. The general form of the $\Gamma$’s, when including an
interaction that mixes the $x$ and $y$ components, is given by:

$\Gamma_{x,x}=k_{x},\,\,\,\,\Gamma_{x,y}=\gamma$ (2)
$\Gamma_{y,x}=\gamma,\,\,\,\Gamma_{y,y}=k_{y}$

The operator $\hat{V}$ is the contribution from the vertex that is determined
by requiring the invariance under uniform translations:

$\hat{V}_{n}+\hat{\Gamma}^{(in)}_{n}+\hat{\Gamma}^{(out)}_{n}=\hat{0},$ (3)

this condition thus guarantees that the eigenvalue $\lambda=0$ is in the
spectrum at its upper bound. The network is anisotropic because we assign an
higher energy to the deviations when they are orthogonal to the local
alignement of the pattern. If we label with a $t$ (transverse) and with an $l$
(longitudinal) the corresponding elastic constants we thus have
$k_{t},\,k_{l},\,\,\,k_{t}>k_{l}$. In correspondence with an horizontal link
the diagonal entries of the matrix $\Gamma$ are $k_{x}=k_{l},\,k_{y}=k_{t}$;
the off-diagonal entries $\gamma$ can describe, in the polymeric
interpretation, the elasticity of the chain bendings. One can notice that even
in the absence of this contribution the model has no soft (zero energy) modes
other than the uniform one. On the r.h.s. of the eigenvalue equation one has
an operator with the appearance of a matrix-valued inhomogeneous laplacian
that acts on the two-dimensional vector field of deviations. For the sake of
brevity let us name it disordered laplacian $\Delta_{(PH)}$, with the proviso
that its inhomogeneity is actually deterministic since it is determined by the
Peano pattern. When the off-diagonal contribution is disregarded $(\gamma=0)$
the x and y components are decoupled, and one is left with two scalar
disordered laplacians $\Delta_{x}$ and $\Delta_{y}$:

$\Delta_{(PH)}\equiv Diag(\Delta_{x}(k_{t},k_{l}),\,\Delta_{y}(k_{t},k_{l})).$
(4)

These operators are essentially identical because they turn one into the other
under exchange of the couplings $(k_{t}\to k_{l},\,k_{l}\to k_{t})$:

$\Delta_{y}(k_{t},k_{l})=\Delta_{x}(k_{l},k_{t}),$ (5)

and for this reason we will focus our analysis on $\Delta_{x}.$ Let us thus
write the equation for the component $x(n,t)$:

$(\lambda-V_{n})x(n)=\Gamma^{(in)}_{n}x(n-1)+\Gamma^{(out)}_{n}x(n+1)$ (6)
$(\Gamma^{(out)}_{n}=\Gamma^{(in)}_{n+1})$

Here the coefficients $\Gamma$ are c-numeric and have value $k_{l}(k_{t})$ at
the horizontal (vertical) links; correspondingly the site (vertex) potential
is given by $V_{n}=-(\Gamma^{(in)}_{n}+\Gamma^{(out)}_{n}).$ By following a
standard procedure, this equation can be written in the form of a single step
map acting on a two-dimensional space. Once the transfer matrix $T\equiv T(n)$
has been assigned to the n-th vertex, the vector
$\vec{z}(n)\equiv(x(n),x(n-1))^{\top}$ is mapped by $T$ into $\vec{z}(n+1)$:
$\,\,\vec{z}(n+1)=T(n)\vec{z}(n).$ The transfer matrix $T\equiv T(n)$ has the
form:

$T(n)_{1,1}=\frac{(\lambda-
V_{n})}{\Gamma^{(out)}_{n}},\,\,\,\,T(n)_{1,2}=-\frac{\Gamma^{(in)}_{n}}{\Gamma^{(out)}_{n}}$
(7) $T(n)_{2,1}=1,\,\,\,T(n)_{2,2}=0$

In this way the eigenvalue problem translates into a boundedness condition for
the transfer matrix $T_{tot}$ of the total chain $T_{tot}\equiv T(N_{k})\cdot
T(N_{k}-1)\cdot....\cdot T(2)\cdot T(1)$. It is convenient to make use of
$SL(2,R)$ (i.e. determinant +1) matrices, because the boundedness condition
for $T_{tot}$ can then be written as a single condition for the trace:
$\frac{1}{2}|Trace(T_{tot})|\leq 1.$
In view of that we perform the following transformation:

$T(n)\to\hat{T}(n)\equiv S^{-1}(n+1)\cdot T(n)\cdot
S(n),\,\,\,\,S(n)=\hat{1}/(\Gamma^{(in)}_{n})^{1/2}$

In fact the new matrices $\hat{T}\equiv\hat{T}_{out,in}=\hat{T}(\alpha,\beta)$
belong to $SL(2,R)$ and their entries are:

$\hat{T}_{1,1}=\frac{\lambda}{\beta}+\alpha+\frac{1}{\alpha},\,\,\,\hat{T}_{1,2}=-\alpha$
(8) $\hat{T}_{2,1}=1/\alpha,\,\,\,\hat{T}_{2,2}=0$

The parameters $\alpha,\,\beta$ introduced above are functions of the hopping
coefficients:
$\alpha\equiv\alpha(n)=(\frac{\Gamma^{(in)}_{n}}{\Gamma^{(out)}_{n}})^{1/2}$
and $\beta\equiv\beta(n)=(\Gamma^{(out)}_{n}\cdot\Gamma^{(in)}_{n})^{1/2}.$
The iteration formula obtained in the Appendix (see Eq.A3), that we are going
to represent by means of transfer matrices, involves an operation
$\mathcal{T}$ that describes the inversion of the ordering along the chain.
When applied to a single step transfer matrix this inversion has the effect of
exchanging $\Gamma^{(out)}$ with $\Gamma^{(in)}$ because obviously it must be
$\mathcal{T}(\hat{T}_{out,in})=\hat{T}_{in,out}$; in terms of the parameters
$\alpha,\,\beta$ we have:

$\mathcal{T}(\hat{T}(\alpha,\beta))=\hat{T}(1/\alpha,\beta)=\sigma_{1}\cdot\hat{T}^{-1}(\alpha,\beta)\cdot\sigma_{1},$
(9)

where $\sigma_{1}$ is the Pauli matrix
$[(\sigma_{1})_{12}=(\sigma_{1})_{21}=1,\,(\sigma_{1})_{11}=(\sigma_{1})_{22}=0].$
The action of $\mathcal{T}$ on multiple step transfer matrices $M\in SL(2,R)$
is thus given by $\mathcal{T}(M)=\sigma_{1}\cdot(M)^{-1}\cdot\sigma_{1}.$ We
label the transfer matrices of the vertices with the directions of their links
as follows: $\hat{T}_{xy},\,\hat{T}_{xx},\,\hat{T}_{yx},\,\hat{T}_{yy},$ where
the left (right) label refers to the outgoing (ingoing) link. The inflation
rule for $\hat{T}_{yx}$ and $\hat{T}_{yy}$ has the form:

$\hat{T^{\prime}}_{yx}=\hat{T}_{yy}\cdot\hat{T}_{yx}\cdot\mathcal{T}(\hat{T}_{xy}\cdot\hat{T}_{yx})$
(10)
$\hat{T^{\prime}}_{yy}=\hat{T}_{yx}\cdot\hat{T}_{xy}\cdot\mathcal{T}(\hat{T}_{yx}\cdot\hat{T}_{xy}),$

the ’conjugate’ vertices $\hat{T}_{xy}$ and $\hat{T}_{xx}$ transform according
to the analog of this formula where the labels $x$ and $y$ are exchanged.
By using the representation of $\mathcal{T}$ we have:

$\hat{T^{\prime}}_{xy}=\hat{T}_{yy}\cdot\hat{T}_{yx}\cdot\sigma_{1}\cdot\hat{T}^{-1}_{yx}\cdot\hat{T}^{-1}_{xy}\cdot\sigma_{1}$
(11)
$\hat{T^{\prime}}_{yy}=\hat{T}_{yx}\cdot\hat{T}_{xy}\cdot\sigma_{1}\cdot\hat{T}^{-1}_{xy}\cdot\hat{T}^{-1}_{yx}\cdot\sigma_{1}.$

From this equation we computed the band spectra on closed approximants of
$PH_{2}$. Since the chain pattern is invariant with respect to space
reflexions, it is sufficient to require that the solutions are bounded over a
quarter of the chain. Rather than by diagonalizing products of exponentially
large numbers of matrices, we have studied the spectral problem in the space
of the traces, as this approach allows for a much higher precision. The
boundedness condition quoted above translates into:

$\frac{1}{2}\cdot|Trace(\hat{T}_{yx}\cdot\hat{T}_{xy})|\leq+1,\,\,\,\frac{1}{2}\cdot|Trace(\hat{T}_{yx})|\leq+1.$
(12)

The relevant property to be stressed here is that the space of the traces is
finite-dimensional. This is a direct consequence of the Cayley-Hamilton (CH)
relation

$M^{2}-Trace(M)\cdot M+det(M){\bf 1}=0.$ (13)

In fact it can be verified that (CH) allows to reduce the order of positive
powers of matrices $M^{k},\,(k=2,3,...)$ and to turn negative powers $M^{-1}$
into positive ones. More importantly one can show that the traceless parts
$\mathcal{M}$ of the matrices $M$

$\mathcal{M}\equiv M-\frac{1}{2}\cdot Trace(M)\cdot{\bf 1}$ (14)

satisfy Clifford anticommutation rules. This point can be easily ascertained
by writing the $M$’s in the real representation

$M=m_{0}\cdot\tau_{0}+\vec{m}\cdot\vec{\tau},(\tau_{0}={\bf
1},\tau_{1}=i\cdot\sigma_{2},\,\tau_{2}=\sigma_{3},\,\tau_{3}=\sigma_{1}).$
(15)

This maps the matrices into a four-dimensional space, their coefficients being

$m_{0}\equiv\frac{1}{2}Trace(M),\,\,m_{i},(i=1,2,3).$ (16)

The algebra of the spin operators $\tau_{i},\,(i=1,2,3)$ allows to compute,
for any couple of matrices $A,\,B$, the anticommutator $[A,B]_{+}$. By writing
the result in terms of the traceless parts $\mathcal{A}$ and $\mathcal{B}$ one
finally gets:

$\mathcal{A}\cdot\mathcal{B}+\mathcal{B}\cdot\mathcal{A}=(A;B)\cdot{\bf 1},$
(17)

where $(A;B)\equiv Trace(A\cdot B)-\frac{1}{2}\cdot Trace(A)\cdot Trace(B)$.
These rules allow to collect equal factors within products, so that their
powers can be reduced to order one. In this way we get monomials were the four
types of vertices are raised at the most to power one. The occurrence of their
$\mathcal{T}$-inverted partners introduces in the algebra the matrix
$\sigma_{1}$. We are thus led to operate in the larger space $SL^{\pm}(2,R)$.
In the calculations we renamed the vertex matrices as follows:

$X(1)\equiv\hat{T}_{yx},\,X(\bar{1})\equiv\hat{T}_{xy},\\\
X(2)\equiv\hat{T}_{yy},\,X(\bar{2})\equiv\hat{T}_{xx}.$ (18)

The elements $X(1),\,X(\bar{1})$ correspond to the orthogonal vertices;
products where these factors occur in alternating order are associated with
ladders aligned with the diagonals of the square lattice, as shown in Fig.4.
In general the transfer matrix of a chain with $n$ vertices has the form
$X(\eta_{n},..,\eta_{l},...,\eta_{1})\equiv X(\eta_{n})\cdot..\cdot
X(\eta_{l})\cdot..\cdot X(\eta_{1})$ where the labels $\eta_{l}$ run over the
four values $1,\,\bar{1},\,2,\,\bar{2}.$ We wrote the trace map in terms of
the following variables:

$x(\eta,\eta^{\prime},....)\equiv Trace(X(\eta,\eta^{\prime},..)),$ (19)
$y(\eta,\eta^{\prime},...)\equiv
Trace(X(\eta,\eta^{\prime},....)\cdot\sigma_{1}).$

.

##  3 Multifractal spectrum

We will now illustrate the results obtained with the trace map for the scalar
disordered laplacian $\Delta_{x}$. We considered a very small anisotropy:
$k_{l}=1-\epsilon,\,k_{t}=\frac{1}{k_{l}},\,\,(|\epsilon|<<1).$ When
$\epsilon>0$ the x-variable at the horizontal links undergoes a restoring
force weaker than at the vertical ones. In other words its longitudinal
displacements are slightly softer than the transversal ones.
One finds that after one iteration the initial band is split into four bands.
This branching process repeats itself at the successive steps so that the
number of bands $\mathcal{N}_{k}$ scales as the size of the system
$\mathcal{N}_{k}\approx 4^{k}.$
We have found that upon increasing the anisotropy the bands start to overlap.
The results reported here do not cover this case, as we made sure that the
behavior $\mathcal{N}_{k}\approx 4^{k}$ was preserved up to $k=10$ iterations.
The spectra of ’disordered laplacians’, when characterized by means of the
density of states (DOS) $\rho(\lambda)$, can display a variety of scaling
behaviors $\rho(|\lambda|)\approx(|\lambda|)^{\alpha-1}$. Here instead of
$\alpha$ we use the exponent $\mu$ that gives the scaling of the bandwidths
$\Delta\lambda$ with respect to $\mathcal{N}$:
$\Delta\lambda\approx\mathcal{N}^{-\mu}$. As illustrated in the caption of
Fig.1 $\mu$ and $\alpha$ are simply related: $\mu=\frac{1}{\alpha}$. In the
literature on phononic spectra it is more usual to consider, rather than
$\alpha$ or $\mu$, the spectral dimension $d_{s}$ [25] that gives the scaling
of the density of states with respect to the frequencies
$\rho(\omega)\approx(\omega)^{d_{s}-1}$. Since $\lambda\equiv-\omega^{2}$ the
two exponents are trivially related: $\alpha=\frac{d_{s}}{2}$.

Figure 1: Isotropic limit of the disordered laplacian. We display here the
distribution $S(\mu)$ of the scaling exponents $\mu$ obtained from the
periodic approximant of order k=10. The three plots correspond to increasingly
small values of the anisotropy parameter ($\epsilon=10^{-5}$ red (circles),
$10^{-6}$ green (triangles), $10^{-7}$ blue (squares). It appears that the
isotropic limit is characterized by a spectrum which turns out to be
qualitatively different from what would be expected in the case of periodic
chains (see text). We consider here the distribution for the exponent $\mu$,
which defines the scaling of the bandwidths
$(\Delta\lambda\approx\mathcal{N}^{-\mu})$. We recall that the more frequently
used exponent $\alpha$ refers instead to the scaling of the density of states
(DOS) $\rho(\lambda)\approx\lambda^{\alpha-1}$. The two representations are
equivalent; in fact from the boundedness of the integrated DOS
$H(\lambda):\Delta H(\lambda)\equiv\rho(\lambda)\cdot\Delta\lambda=O(1)$ one
obtains $\mu=\frac{1}{\alpha}$.

Quite surprisingly we have found that close to the isotropic limit the
spectrum has a whole set of power law singularities. In other words the
spectrum is strongly sensitive to disorder even with rather small values of
$\epsilon.$ We did not sistematically explore the isotropic limit $\epsilon\to
0$ but the whole set of singularities [26] found here is qualitatively
different from what would be expected with periodic chains. Here together with
genuine normal modes, herefrom named Peano modes, corresponding to $\mu=1$, we
find singularities with powers below and above $\mu=1$.

Figure 2: Spectrum corresponding to the vector-valued eigenstates: plot of the
distribution $S(\mu)$ obtained with inhomogeneity parameter $\epsilon=10^{-4}$
(green, triangles). For comparison we plot the distribution associated with
the corresponding single variable problem (blue,circles). The interval
$(\mu_{min},\mu_{max})$ appears to be contracted in the two-dimensional
problem. Thus the requirement that the eigenvalues are at the intersection of
the two single-variable spectra $\lambda_{x}(k_{t},k_{l})$ and
$\lambda_{y}(k_{t},k_{l})$ modifies the extremal regions
$\mu\approx\mu_{min},\,\mu\approx\mu_{max}.$

Working on discretized, disordered extensions of laplacians such as the one
examined here, even within a small spectral range $\Delta\lambda$ a variety of
singularities can be found, so that the numerical plots of the DOS versus
$\lambda$ are generally hard to interpret. This difficulty can be surmounted
when the singularities are of power-law type. In that case by collecting
together the portions of the spectrum that share the scaling exponent one can
obtain a smooth distribution. This procedure goes under the name of
thermodynamic formalism [27] for multifractals.
One defines a partition function $Z(\tau)$, where $\tau$ is the analog of an
inverse temperature; the sum is taken over the $\mathcal{N}$ bands of a
periodic approximant, and accounts for their individual widths
$\Delta\lambda_{l},\,(l=1,...,\mathcal{N})$:

$Z(\tau)\equiv\Sigma^{\mathcal{N}}_{l=1}(\Delta\lambda_{l})^{-\tau}\approx(\mathcal{N})^{q(\tau)}.$
(20)

The exponent $q(\tau)$ is the analog of a free energy: in fact it is the sum
of an entropy and an energy contribution. To see this, let us consider the
subset of bands $E(\mu)$ whose widths scale as
$\Delta\lambda\approx\mathcal{N}^{-\mu}.$ If we make the dependence on $\mu$
explicit on the r.h.s. of the equation, we get:

$q(\tau)=S(\mu)+\mu\cdot\tau,\,\,(\mu\equiv\frac{dq}{d\tau}).$ (21)

Fig.1 is the plot of the ’entropy’ function $S(\mu)$ that weights the
different $\mu$-scalings within the spectrum, so that its value corresponds to
the fractal dimension of the subset $E(\mu)$. The extremal values
$\mu_{max,min}\equiv\lim_{\tau\to\pm\infty}\frac{dq}{d\tau}$ of the support of
$S(\mu)$ characterize the ’vacuum’ (temperature T $\to 0_{+}$) and the ’fully
inverted state’ (temperature T $\to 0_{-}$) regimes. Notice that from the
formula relating $q(\tau)$ with $S(\mu)$ one identifies the ’internal energy’
U of the system:

$U\equiv-\frac{dq}{d\tau}$ (22)

So far we have considered the scalar operator $\Delta_{x}$, acting on the
single variable x: this problem describes a ’polarized’ D=1 diffusion, as if a
blind walker experienced a sequence of varying mobilities encoding the path in
its meandering along the plane. The results obtained for the full vector-
valued problem $\Delta_{(PH)}$ are shown in Fig.2, where they are also
compared with the scalar case. The two distributions turn out to be different,
in particular the range of exponents $\mu$ is larger in the scalar case. When
the operator $\Delta_{(PH)}$ is diagonal in x and y, its eigenstates can be
obtained by combining the solutions of the single variable problems in x and
y:

$\vec{\alpha}_{\lambda}(n)\equiv(x_{\lambda_{x}}(n),0)^{\top}\\\
\vec{\beta}_{\lambda}(n)\equiv(0,y_{\lambda_{y}}(n))^{\top},$ (23)

where $\lambda_{x}$ and $\lambda_{y}$ are the eigenvalues of
$\Delta_{x}\equiv\Delta_{x}(k_{t},k_{l})$ and
$\Delta_{y}\equiv\Delta_{y}(k_{t},k_{l})$. In order to have an eigenstate of
$\Delta_{(PH)}$ with eigenvalue $\lambda$ it must be:

$\lambda=\lambda_{y}(k_{t},k_{l})=\lambda_{x}(k_{t},k_{l}).$ (24)

In other words since the spectrum of $\Delta_{(PH)}$ is at the intersection of
the two spectra it is not surprising to find a smaller range of exponents.

##  4 Langevin equation

We proceed now to examine the gaussian relaxation process associated with the
disordered Laplacian $\Delta_{(PH)}$. This allows to understand the nature of
the physical states. It turns out that the behavior of the fluctuations is
determined by the spectral exponents $\mu$. Our discussion relies on the
literature dedicated to dispersion relations of fractals [28] as applied e.g.
to the Generalized Gaussian Structures [29]. We assume that the vector field
of deviations
$\vec{X}(t)\equiv(\vec{x}(1,t),\vec{x}(2,t),....,\vec{x}(n,t),......,\vec{x}(4^{k},t))$
evolves according to the following Langevin equation:

$\frac{d}{dt}\vec{X}(t)=\mathcal{D}\cdot\Delta_{(PH)}\vec{X}(t)+\vec{\xi}(t)$
(25) $<\xi_{i}(n,t)\xi_{j}(n^{\prime},t^{\prime})>=2\cdot\mathcal{D}\cdot
k_{B}T\cdot\delta_{i,j}\cdot\delta(t-t^{\prime})\cdot\delta_{n,n^{\prime}},\,(i,j=x,y).$

This process can be considered as a generalization of the Rouse model,
describing the relaxation of an homogeneous polymer. The critical dynamics of
self-avoiding Rouse chains and membranes has been studied by K.J.Wiese [30].
Here we are studying a purely linear system, but with a nontrivial dependence
on the geometry of the underlying pattern. For the sake of simplicity let us
omit from now on the vector indexes and put $D=1$ and $k_{B}T=1.$ It is well
known that the process defined by a Langevin equation is fully described by
the response function (also called dynamic susceptibility) and by the
correlation function; these two functions are related through the fluctuation-
dissipation theorem [31]. Before examining the case of the disordered
laplacian it is useful to remind some basics on gaussian processes. We
consider the relaxation of a field $\phi(x,t)$ in the presence of white noise
$\xi(x,t)$:

$\frac{d}{dt}\phi(x,t)=-[r-\nabla^{2}]\phi(x,t)+\xi(x,t).$ (26)

In the energy-momentum representation the dynamic susceptibility is
$\chi(\lambda,\mathcal{Q})=(-i\cdot\lambda+r+\mathcal{Q}^{2})^{-1}.$
The correlation function is defined as follows
$C(x,t)\equiv<\phi(x,t)\cdot\phi(0,0)>=\int\frac{d\mathcal{Q}}{2\pi}\int\frac{d\lambda}{2\pi}C(\lambda,\mathcal{Q})e^{i(\mathcal{Q}\cdot
x-\lambda\cdot t)}$ and its energy-momentum form is:

$C(\lambda,\mathcal{Q})=2\cdot(\lambda^{2}+(r+\mathcal{Q}^{2})^{2})^{-1}.$
(27)

We want now to relate the diffusive dispersion in the scale-free regime $r=0$
with the band-edge singularity in the spectrum of the laplacian: more
precisely we show that the $\mathcal{Q}^{2}$ dispersion is the counterpart of
the multiplicity of states at the edges of the spectrum. In order to make this
clear we consider the discretized version of the laplacian, where the counting
of the states is elementary:

$\frac{t}{2}(\psi_{n+1}+\psi_{n-1}-2\cdot\psi_{n})=E\psi_{n},\\\
(\psi(N)=\psi(0))$ (28)

Consider now the map $E\mapsto n$ associated with the set of eigenvalues
$E=E(n)\equiv E(k_{n})=t(cos(k_{n})-1),(k_{n}=\frac{2\pi n}{N})$: this map is
invertible with the exclusion of the band edges $E=0$ and $E=-2t$, where the
DOS displays the D=1 van Hove singularities

$d\rho(E^{\prime})=\frac{1}{2\pi}\frac{dE^{\prime}}{(t^{2}-E^{\prime
2})^{\frac{1}{2}}},\,(E^{\prime}\equiv E+t)].$ (29)

The points $n(E)$, when projected over the E-axis, produce in the interior of
the band the scaling $|dE|\propto|dn|$ and $|dE|\propto|dn|^{2}$ at the edges.
Since one has $|dn|\approx\mathcal{Q}$, the latter scaling is the laplacian’s
dispersion $\mathcal{Q}^{2}$.
We generalize this connection to the case of the disordered laplacian: it is
understood that our argument is merely heuristic because it refers only to the
behavior in the scale-free regime, that is to the region $\lambda\approx 0$
and $\mathcal{Q}\propto\frac{1}{N}$, $N$ being the chain length. Recalling
that within the spectral set $E(\mu)$ the singularity is
$|\Delta\lambda|\approx\mathcal{N}^{-\mu}$, we associate to it a dispersion
relation of the form

$-\lambda(\mathcal{Q})\equiv\omega^{2}(\mathcal{Q})\approx|\mathcal{Q}|^{\mu}$
(30)

This amounts to saying that the contribution arising from $E(\mu)$ is
described by an anomalous gaussian process undergone by a field
$\phi_{\mu}(\lambda,\mathcal{Q})$ with dynamical susceptibility

$\chi_{\mu}(\lambda,\mathcal{Q})=(-i\cdot\lambda+|\mathcal{Q}|^{\mu})^{-1}.$
(31)

The correlation function

$C_{\mu}(\lambda,\mathcal{Q})=\frac{2}{(\lambda^{2}+(|\mathcal{Q}|^{\mu})^{2})}$
(32)

when integrated over $\lambda$ and $\mathcal{Q}$, is represented as follows:

$C_{\mu}(x,t)\equiv<\phi_{\mu}(x,t)\cdot\phi_{\mu}(0,0)>\propto\int\frac{d\mathcal{Q}}{2\pi}\frac{1}{|\mathcal{Q}|^{\mu}}exp^{i(\mathcal{Q}\cdot
x-|\mathcal{Q}|^{\mu}\cdot t)}.$ (33)

The infrared behavior associated with $E(\mu)$ is thus:

$C_{\mu}(n,0)\approx(n)^{\mu-1},\,\,\,n>>1,\,\,(n(L)\approx
L^{2}),\,\,C(0,t)\approx(t)^{1-\frac{1}{\mu}},\,t>>1.$ (34)

Here we replaced the space variable $x$ with the chain’s parameter $n$ and $L$
indicates the linear size of the planar region covered by $n$ subsequent
links.
In conclusion, having shown that the exponent $\mu$ determines both space and
time behaviors in the infrared regime, we can classify the different types of
fluctuations available to the system.
The region $\mu<1$ describes fluctuations that decay in the length $n$ as a
power law. The corresponding eigenfunctions presumably have an oscillating
behavior where the amplitudes are progressively reduced with distance, as in
destructive interference. In addition, when $\mu<1$ the fluctuations are
short-lived since they decay with time as a power law.
In the region $\mu>1$ we find the complementary behavior, where the
fluctuations grow with $n$. They can be characterized in terms of the fractal
dimension $D_{f}=\frac{2}{\mu-1},$ or, in the polymeric interpretation, by
means of the Flory exponent $\nu=\frac{1}{D_{f}}=(\mu-1)/2.$
One can wonder how spatially growing functions could be acceptable
eigenfunctions: the point is that in the presence of oscillations the
boundedness condition can be recurrently fulfilled. In this case there is
presumably some sort of constructive interference. The two regions $\mu<1$ and
$\mu>1$ are complementary also with respect to time dependence: in fact when
$\mu>1$ the chain sites undergo subdiffusion with an exponent
$\gamma_{D}=1-\frac{1}{\mu}$. This behavior can as well be characterized by
the dynamic exponent $z=2/\gamma_{D}$. The scenario found here leads to
associate $\mu>1$ and $\mu<1$ with excitations respectively dominated by shear
and by dilatation.
It is worth mentioning that the fluctuation-dissipation theorem applies to the
anomalous gaussian process described here. As a consequence the well-known
deGennes relation $z=2+D_{f}$ following from that theorem is fulfilled for all
exponents $\mu$. The case $\mu=1$, which marks the separation of the two
regions described above, corresponds to oscillations with stable amplitudes.
The corresponding Peano modes will be discussed in connection with the
symmetries of the system.
We conclude this Section with a comment on the general case of chains
$PH_{D}$. Being always generated by deterministic inflation, it is reasonable
to expect that their self-similarity should be mirrored into nontrivial
spectral properties. The eigenvalue problem in D dimensions, if approached
along the lines presented in this paper, would involve transfer matrices of
dimension $(2D\times 2D).$
This can be difficult to deal with even in the case $D=2$: we remind that it
is possible to improve the numerical resolution, by using traces instead of
matrices, only when the components x and y are decoupled. One can try to
overcome this complexity by relying on a mean-field approach, which in the
present context amounts to disregarding the fluctuations of $\mu$. Taking into
account that the chain is embedded in dimension $D$, self-consistency suggests
that the mean-field exponent $\mu(D)$ should be chosen in such a way as to
make the fractal dimension $D_{f}$ to coincide with the bulk’s dimension:
$D_{f}=D$. In order to determine $\mu(D)$ let us consider a couple of points
(1,2) placed at a distance $r$ in the bulk, and use the formula
$D_{f}=\frac{2}{\mu-1}$. In this way we find:

$<(\vec{x}(1)-\vec{x}(2))^{2}>\approx
r^{2}=n^{\frac{2}{D}}=n^{\mu(D)-1},\,\mu(D)=1+\frac{2}{D}.$ (35)

Notice that in the language of membranes the fluctuating field $\vec{x}(n,t)$
undergoes the roughening transition at $\mu=1,$ hence the mean field sets the
membrane in the rough phase [32].
In the planar case we obtain $\mu(2)=2\equiv D_{f}$. It turns out that
fluctuations of exponent $\mu=2$ would correspond to structures softer than
the elastic network of $PH_{2}$ itself. In fact the maximal growth of the
latter is determined by $\mu_{max}$ and we have found: $\mu_{max}<\mu(2)\equiv
2,\,\,(\gamma=0,\,\epsilon\to 0)$.
In D=3 the mean field exponent is $\mu(3)=\frac{5}{3}$, which implies the
anomalous diffusion $C(0,t)\approx t^{\frac{2}{5}}$. This behavior was
actually found to occur with high probability in experiments on bacterial
chromosomes [7, 33] and is compatible with configurations close to the optimal
packing $D_{f}=3$ [9]. Let us finally notice that fluctuations having a
fractal dimension higher than D cannot be excluded. In fact in some of the
measurements quoted above [7] also rather small anomalous diffusion exponents
$\gamma_{D}$ $(\gamma_{D}<\frac{2}{5})$ were obtained: they would imply
$1<\mu<\mu(3)$ and $D_{f}>3$.

Figure 3: Plot of the function $\frac{dq}{d\tau}(\tau)$ obtained from the
bands of the approximant of order $k=10$. The inhomogeneity parameter is
$\epsilon=10^{-4}.$ The ’internal energy’ U of the system is given by
$U=-\frac{dq}{d\tau}$. The asymptotic values of the function are respectively
$\mu_{min}$, at large negative values of $\tau,\,(\tau\to-\infty)$, and
$\mu_{max},\,(\tau\to+\infty)$ at large positive values of $\tau$. At the
critical (second order) point $\epsilon\to 0$ the function
$\frac{dq}{d\tau}(\tau)$ has a single inflexion at $\tau=0$. The plot
displayed here clearly shows three inflection points, indicating the onset of
the phase separation: the ’internal energy’ $U$ is going to develop two
minima.

##  5 Inverse energy cascade and phase separation

In this Section we describe the global properties of the system’s relaxation
by relying on the results discussed above. We have found that the
thermodynamic formalism of multifractals, which up to now has been used
primarily for technical reasons, involves a physical interpretation of the
’internal energy’ $U=-\frac{dq}{d\tau}(\tau)$ that is consistent with the
scenario of relaxation. The region of negative $\tau$ describes the regime of
negative temperatures (i.e. the inverted system) and in fact, as one can see
from Fig.3, there one finds the higher values of U. The limit $\tau\to+\infty$
corresponds to the ground state of the thermodynamics and there the ’internal
energy’ U has its minimum.
The interesting point, revealing some connection of U with the energies
$\lambda$, is that the infrared region $\lambda\approx 0$ appears to be
dominated by the largest scaling exponent $\mu_{max}>1$: at least this is what
emerged from our numerical checks. As discussed in the previous Section the
values $\mu>1$ are associated with excitations of a large size, presumably
dominated by shear. In the opposite case, i.e. whenever the dilatation
dominates, one expects localized excitations and higher energies, associated
with values $\mu<1$.
If the ’internal energy’ U is monotonic with respect to the energy content of
the fluctuations, then from the plot reported in Fig.3 one gets a clear
picture of the process of relaxation: when prepared on localized states, the
system is led to evolve towards the shear-dominated regime $\mu>1$. This
behavior has been in fact observed in two-dimensional colloids [17, 18] and is
intriguingly similar to the inverse energy cascade of two-dimensional
turbulence [22].
It turns out that the analogy between this model and two-dimensional colloids
is not limited to the space-time behavior of the excitations, but involves
also the spectral properties. In fact if one were to add to the model the
contact couplings among adjacent but nonsubsequent sites, the new network
would describe a sort of textured membrane. The resulting phononic spectrum,
in addition to the contributions arising from the Peano pattern, would display
in the density of states $\rho(\lambda)$ a contribution
$\rho(\lambda)_{D=2}\approx\lambda$, arising from planar phonons, as in the
standard Debye spectra of two-dimensional systems. The quoted experiments on
colloids did in fact display at low frequencies an anomalously high density of
states, superimposed to the D=2 Debye behavior.
So far we have discussed the spectral properties in the closeness of the
isotropic limit, where the spectral bands are predominantly separated. The
question then arises about how the scenario changes when the anisotropy is
large enough to make the bands overlap.
We have found that the anisotropy induces a phase transition [34]: the
spectral measure $S(\mu)$ progressively splits into two separate
distributions. Possibly the simplest example of a complete phase separation in
a spectral measure is provided by the D=1 periodic chain. As we earlier
recalled, in that case there are only two exponents: $\mu=1$ and $\mu=2$.
Since their weights are $S(\mu=1)=1$ and $S(\mu=2)=0$, the periodic chain is
located at the extremal point of a line of phase coexistence, at the
extinction of the phase $\mu=2$, where the system is fully converted into the
$\mu=1$ phase.
A phase coexistence was found [35] in the spectrum of the Quantum Ising
Quasicrystal (QIQX). The QIQX model describes a chain of spins where two Ising
couplings are aligned in quasiperiodic order [36]. When the two couplings are
nearly equal $(J_{0}\approx J_{1})$ the function $S(\mu)$ has support
$\mu_{min}<\mu<\mu_{max}$ with $\mu_{min}=1$ and $\mu_{max}=2$. In other words
the spectrum is at the coalescence of the two mentioned phases of the periodic
chain. As the two couplings are made progressively different $(J_{0}\neq
J_{1})$ the two phases separate.
In general a phase separation can be hardly detected by inspecting $S(\mu)$,
because the numerics naturally converge to the convex envelope of the two
distributions. In fact $S(\mu)$ is the Legendre transform of the object of
relevance in this context, which is the ’internal energy’ U
$(U=-\frac{dq}{d\tau})$. By closely inspecting Fig.3, where we display
$\frac{dq}{d\tau}$ at the value $\epsilon=10^{-4}$ of the anisotropy
parameter, one can notice three inflection points:
$A\approx(-4,.1.),\,B\approx(0.,1.1),\,C\approx(2.,1.3)$. At the (second
order) critical point one expects a single inflexion point, here we are very
close to that point. Upon increasing the parameter $\epsilon$ a local maximum
is expected in the region delimited by $(A,B)$. This maximum identifies the
emergence on the left side of the plot, i.e. in the inverted regime, of a
phase whose dominant scaling evolves towards $\mu_{min}$. On the right side
the scaling of the other phase evolves towards $\mu_{max}$. It is not
difficult to associate the former with the solid, where the high energy
dilatation modes are dominant, and the latter, where the fluctuations grow
both in $n$ and in $t$, with the shear-dominated liquid.

Figure 4: Planar strip associated with products where the two types of
orthogonal vertices $X(1),\,X(\tilde{1})$ are ordered in alternating fashion.
The horizontal (vertical) steps of the ladder correspond to discrete
displacements along the space coordinates $q_{x},\,(q_{y})$. The ladder can be
associated with a trajectory undergoing a sequence of scatterings at the
strip’s boundaries: the corresponding motion, aligned with the direction
depicted in black, would carry a linear momentum $p$ in that direction. This
suggests that the orthogonal vertices $X(1),\,X(\tilde{1})$, which represent
the turning points of the chain, should be associated with displacements in
the plane of momenta $(p_{x},p_{y})$ where the axes form an angle $\pi/4$ with
respect to the orientation of $(q_{x},q_{y})$.

##  6 Critical crystal with eightfold symmetry

In the previous Section we have shown that the model undergoes a phase
transition, and that its isotropic limit is found at the saddle separating two
phases, respectively of solid and liquid type. The Peano network can thus be
considered a critical crystal, marking the point where phase coexistence turns
into coalescence. This situation appears to be the D=2 analog of the well
studied critical regime at the D=1 delocalization transition [23, 37]. It is
thus appropriate to remind some results regarding that case.
The D=1 transition has been described by Wilkinson within a semiclassical
(WKB) approach based on the universal hamiltonian [38]:

$\frac{1}{2}\cdot H\equiv cos(p)+\alpha\cdot cos(q)$ (36)

Exactly at the critical point $\alpha=1$ the hamiltonian has a twofold
symmetry, since it is self-dual in the exchange of the (adimensional) space
and momentum variables. It identifies a square lattice in the $(q,p)$ plane,
the lattice points being at the saddles where the separatrices meet. These
separatrices are soliton trajectories of energy $H=0$; apart from them, every
other orbit is closed, so that when $\alpha=1$ the particles are confined both
in the $q$ and $p$ directions.
When the self-duality is broken, e.g. when $\alpha>1$, aligned solitons merge
into trajectories running parallel to the $p$-axis: in this region the
particles are localized with respect to the $q$-direction. In correspondence
with the complementary coupling $(\frac{1}{\alpha})$ one finds the dual
trajectories, where the particles are instead extended in the $q$-direction.
It resulted from Wilkinson’s analysis that the critical properties obtained
with the universal hamiltonian, which has a twofold symmetry, actually hold
for an entire class of hamiltonians, characterized by fourfold symmetry. The
following is an example of such hamiltonians:

$\frac{1}{2}H\equiv cos(q)+cos(p)+cos(2\cdot q-p)+cos(q+2\cdot p)$ (37)

We suggest that this fourfold symmetry could originate from the operators
representing the discrete translations in the $q$ and $p$ directions, together
with their time-reversed counterparts. Products of these four elements are
identified by the oriented paths on the lattice phase space. In order to see
how the D=1 case is related with our construction it is useful to examine the
quantum hamiltonians based on chains organized according to the two-letter
$(A,\,B)$ Fibonacci substitution rule. The Fibonacci chain is often depicted
as a ladder over a two-dimensional lattice, where the horizontal and vertical
steps correspond to the letters $A$ and $B$. The steps are combined in such a
way as to fit the ladder within a strip having slope $m=\frac{1+\sqrt{5}}{2}$.
The ladder shown in Fig.4 is represented by products where the two operators
$X(1)$ and $X(\bar{1})$, associated with the orthogonal vertices of our
construction, are ordered in alternating fashion. In this elementary
configuration the ladder is contained within a strip of slope $m=1$. One can
notice that since the operators $X(1)$ and $X(\bar{1})$ involve couples of
steps $(AB)$ or $(BA)$, they are more properly associated with translations in
the momenta along the diagonal. The quantum mechanics on the ladder has the
form of a constrained D=2 hamiltonian system where translations in the momenta
are reduced to a single dimension.

Figure 5: Graphical representation of the elementary operations acting on the
four-dimensional lattice. Upper line: the blue (red) square path represents
the loop of discrete translations in the p-plane (q-plane). Lower line, right
side: the eightfold motif obtained by projecting the $p$-loop on the plane of
the $q$-loop.
Lower line, left side: the loop formed by combining the eight operators that
generate the Peano chain. The colour convention is consistent with the rest of
the Figure. The red lines represent paths in the q-plane, hence the red
perimeter describes a loop in the q-plane. The black dots represent the
vertices: the flat vertices $X(2),\,X(\bar{2})$ carry only red lines, the
orthogonal vertices $X(1),\,X(\bar{1})$ are decorated with blue lines. The
blue segments are aligned with the translations on the p-lattice, as
illustrated in Fig.4. If the blue segments are continued up to their
intersections, the eightfold motif shown on the right side is reproduced.

The fact that at the critical point the D=1 quantum system is correctly
described by the WKB approximation translates, in the language of trace maps,
into the condition $Trace(A\cdot B\cdot A^{-1}\cdot B^{-1})=2$, meaning that
the group commutator is the identity. This condition is connected to the
localization properties of the wave functions through the following trace
relation:

$Trace(A\cdot B\cdot A^{-1}\cdot B^{-1})=x^{2}+y^{2}+z^{2}-xyz-2$ (38)
$x\equiv Trace(A),\,y\equiv Trace(B),\,z\equiv Trace(A\cdot B).$

On the r.h.s of the above one recognizes the well-known constant of motion of
the trace map [19] representing the surface $I(x,y,z)\equiv
x^{2}+y^{2}+z^{2}-xyz$. The topology of this surface changes when $I=4$, i.e.
when the group commutator is trivial, and correspondingly two separate types
of map trajectories (localized-delocalized) meet.
We now return to the trace map of our case, leaving to future work [39] a
detailed analysis. The map involves the monomials made out of the four
vertices $(X(1),\,X(\bar{1}),\,X(2),\,X(\bar{2}))$. One has six order-two
monomials, four order-three monomials and one order-four monomial. The degrees
of freedom are doubled since the four basic vertices occur with their
$\mathcal{T}$-reversed copies. The map is thus defined over the traces
$x=Trace(X)$ and $y=Trace(X\cdot\sigma_{1})$ illustrated in Section 2. The
overall space has dimension $15X2$ but this large number can be reduced by
exploiting some symmetries. For instance the invariance under cyclic
permutations generates a triangular symmetry for cubic monomials. Cubic
monomials are further related through some trace identities[39].
On top of that the occurrence of $4\times 2$ basic vertices enforces an
eightfold symmetry. This, we believe, appears as the extension of the D=1 case
that is characterized by a fourfold symmetry. In fact in quantizing an
hamiltonian system over a $(2\times 2)$ phase space one needs $(4\times 2)$
operators representing the discrete $(q_{x},q_{y}),\,(p_{x},p_{y})$
translations and their reversals. These operators act in this case on the
vector of elastic displacements having values in a real two-dimensional vector
space.
In order to understand how this geometry effects the configurations of the
system, we project the lattice of momenta on the plane of configurations. In
Fig.5 we enumerate the discrete translations of the phase space and compare
them with the algebra of the vertices. The motif displayed on the lower left
is obtained by combining the vertices $X(1),\,X(\bar{1}),X(2),\,X(\bar{2})$
with their reversed counterparts. On the lower right we show the effect of
projecting the $(p_{x},p_{y})$ lattice over $(q_{x},q_{y})$. Notice that in
the usual representation of the union of a q-lattice with its dual one assigns
the nodes of the dual to the centers of the q-plaquettes. Under that
convention the tiling of the plane would appear in the form of two equally
oriented square lattices. The eightfold motif obtained here results instead
from the convention of making the centers of the q and p plaquettes to
coincide and to having them rotated of an angle $\pi/4$.
We find that this convention reproduces the geometry associated with the
operators $X(1),\,X(\bar{1}),X(2),\,X(\bar{2})$. In fact the two motifs
displayed on the second line give rise to identical periodic tilings of the
plane.

Figure 6: The octagon and its descendants: this figure illustrates how the
octagonal symmetry is compatible with dilatations. Discrete rotations and
discrete dilatations match, as in regular crystals rotations match with
lattice translations. This ’polar crystal’ property is shared by the Peano
chain with the octagonal quasicrystal. Periodic approximants of the octagonal
quasicrystal are obtained by projecting the 4D hypercube on the plane; the
resulting tiling turns out to have an average connectivity substantially
higher than in the Peano network.

It turns out that in recent experiments on bilayer water, precursors of
crystallization were observed to generate quadratic, hexagonal, pentagonal
patterns, and more importantly, also the dodecagonal quasicrystal [24]. On
very general grounds, since quasicrystals are much softer than solids, it is
somewhat to be expected that in the liquid to solid transition various
structures of intermediate stability should be explored by the system.
Given this eightfold symmetry the question is then: to what extent is the
Peano network related with the octagonal quasicrystal? The effective
coordination number of the network is certainly smaller than the average
coordination number of the octagonal quasicrystal. Periodic approximants of
the latter [40], obtained by projecting the 4D hypercube on the plane, have
square-shaped elementary cells. Each cell has vertices with coordination
numbers $z$ varying from three to eight $(z=3,4,5,6,7,8)$. The free particle
(laplacian) quantum motion defined over such tilings is characterized by
anomalous diffusion, but with exponents much larger than those obtained here
[41]. The reason for that can be easily understood, since in the octagonal
quasicrystal there is a larger number of channels open to hopping.

Figure 7: Peano modes at the matching conditions of the ’polar crystal’. Plot
of the number of solutions $\mathcal{N}$ versus the energy $\lambda$ in the
closeness of $\lambda=0$; the distribution is defined over intervals having
width $\delta\lambda=10^{-8}.$ Apart from the normalization, the plot is a
coarse-graining of the density of states (DOS) $\rho(\lambda)$. The data refer
to the approximant of order k=6 and to the anisotropy parameter
$\epsilon=10^{-6}$: we are thus quite close to the critical point. The points
fall along five curves: $\mathcal{N}(l,\lambda),\,\,(l=1,2,3,4,5)$. The curves
$\sigma\equiv\sigma(\lambda,s)$ orthogonal to the $\mathcal{N}$’s are coloured
in red. The $\sigma$ curve intercepts approximants of increasingly large
orders $l$.

We would like to stress that the octagonal symmetry arising in the Peano
network does not originate in the euclidean metrics, but in the hamiltonian
structure. If the system is integrable the hamiltonian can be made quadratic
and in the isotropic limit the rotational invariance implies the following
form:

$H=\alpha\cdot(p_{x}^{2}+p_{y}^{2})+\beta\cdot(q_{x}^{2}+q_{y}^{2}).$ (39)

At the delocalization transition the system is self-dual, thus at the critical
point it must be $\alpha=\beta$:

$(p_{x}^{2}+p_{y}^{2})=(q_{x}^{2}+q_{y}^{2})=\frac{1}{2\cdot\alpha}\cdot H$
(40)

The above means that the allowed states are points of the $(2\times
2)$-dimensional lattice touched by the $S^{3}$ sphere of the energies and
satisfying the self-duality condition.

Figure 8: Lower order symmetries compatible with the octagon. One can identify
the fourfold loop in the $p$-plane (in blue)
$A^{\prime}B^{\prime}C^{\prime}D^{\prime}$, and, in red, its analog in the
$q$-plane $ABCD$. Triangular loops mix the $q$ and $p$ coordinates. We show
here two such loops: $AB^{\prime}C^{\prime}$ and $CD^{\prime}A^{\prime}$. On
the right side we display two fivefold loops obtained by connecting the
vertices left out of the triangular ones. The loop depicted in green is the
complement of $AB^{\prime}C^{\prime}$. Within a semiclassical description the
loops can be associated with scattering processes adding up to zero momentum
exchange, as required by scale invariance at the critical point.

This allows to understand why the Peano chain can sustain normal modes, in
spite of its intricate structure. As shown in Fig.6 the octagon reproduces
itself at discrete scales where the dilatations match with discrete rotations.
The chain shares with the octagon this ’polar crystal’ property and can thus
support normal (Peano) modes, organized in a hyerarchical fashion. The
numerical results confirm this argument. In fact by inspecting the
multiplicity of solutions as a function of the energies $\lambda$ displayed in
Fig.7 one can find some relevant features enumerated below.
A)The distribution appears as the union of several curves
$\mathcal{N}(l,\lambda),\,(l=1,2,3,4,5)$ having shapes similar to what one
would obtain with periodic chains. The curves can be associated with
approximants of increasing order $l$. In fact they represent solutions of
growing multiplicity, as one can see from their intercepts with the vertical
axis.
B)A second family of curves $\sigma$ (in red) is apparently orthogonal to the
$\mathcal{N}(l,\lambda)$. The $\sigma$’s can be identified by means of their
intercept $\lambda$ with the horizontal axis:
$\sigma\equiv\sigma(\lambda,s),\,|\lambda\to\sigma(\lambda,s=0)$. As the arc
parameter $s$ grows the curve $\sigma(\lambda,s)$ crosses approximants of
increasing order $l$: the multiplicity (degeneracy) of the eigenvalue
$\lambda$ increases accordingly.
The Peano waves can be depicted as Rayleigh waves whose packet is split into
non-dispersive contributions corresponding to the different scales of
approximants. This scale separation actually involves not only the normal
modes $(\mu=1)$, but also the anomalous modes $\mu\neq 1$: one can identify
the $l-$curves down to the edge $\lambda\approx 0$.
In addition to the octagonal symmetry, lower order ones, enumerated in Fig.8,
can as well be of importance. The case of triangular symmetry was shortly
illustrated above, in association with cubic monomials.

##  7 Conclusions

We have studied the elastic deformations of the plane-filling closed Peano
chain, taken as an elastic network. The shear and dilatation modes are
intrinsically coupled, because their waves are constrained to propagate along
a thin planar strip undergoing the sequence of turns dictated by the
deterministic pattern of the chain.
The situation is thus similar to the interferences at the surface of an
elastic material, giving rise to the dispersive Rayleigh waves. The Peano
chain has the property of being self-similar, with periodic approximants. In a
situation of this type the Rayleigh waves can find resonance conditions
necessary for coherent propagation. The corresponding normal modes, named
Peano modes, occur when the rescaling associated with the inflation procedure
matches with the discrete rotational symmetry of the pattern.
We have studied the properties of the spectrum by means of the so-called
thermodynamic formalism for multifractals. In addition to the Peano modes we
have found solutions that can be qualitatively described as oscillating
functions modulated by amplitudes which grow or decay: they are associated
with gaussian relaxation processes characterized by a scaling exponent
$\mu\neq 1$.
These anomalous modes result from the frustrated coupling of shear and
dilatation: when the resonance conditions are not exactly fulfilled, the waves
can be amplified or damped, depending on how interference operates. When
$\mu<1$ the equal time correlation function decays along the chain as
$n^{\mu-1}$, hence these states are of localized type; correspondingly the
chain sites undergo a damping process, with the power law
$t^{1-\frac{1}{\mu}}$. The fluctuations characterized by $\mu>1$ have a
complementary behavior: rather than damping they undergo anomalous diffusion
and their equal time correlation functions grow in amplitude along the chain.
They are predominantly found at low energies $(\lambda\approx 0)$. Taking the
above into account, the relaxation of the system can be described as a flow
from localized high energy structures towards extended low energy ones, where
the latter survive in the long time regime of the process. This picture turns
out to reproduce, at least qualitatively, the behavior observed in two-
dimensional colloids [17, 18]. It may then be not coincidental that also the
presence, observed in the same experiments, of a low-energy bosonic peak can
be readily explained within our model. In our opinion the peak reveals a
contribution to elastic energy arising from closed, large-sized one-
dimensional patterns. Structures of this sort are not easily described within
a membrane-like network model. The density of states should then account for
these excitations, which add up to the standard D=2 phononic part.
Possibly the main result obtained in this work is the proof of the existence
of a critical point in the isotropic limit of the model, where two phases
coalesce. These phases are respectively dominated by the $\mu<1$ and $\mu>1$
regimes: localized versus delocalized, or, in other words, solid versus
liquid. It is thus appropriate to consider the Peano network as a critical
crystal.
Since, as previously mentioned, the dilatation modes have higher energies and
are more localized, the energy flows towards extended configurations and lower
energies, as in the inverse energy cascade of two-dimensional turbulence. We
have argued in addition that this critical point should be characterized by an
eightfold symmetry: from this perspective the Peano chain is closely related
with the octagonal quasicrystal, but has a much lower connectivity. Quite
intriguingly, experiments on bilayer water revealed a transition from the
liquid to the dodecagonal quasicrystal[24]. The eightfold symmmetry that
emerges here is strictly related with us limiting the description to
contributions from large-sized D=1 structures; enlarging this picture could
lead to results of a greater relevance for the phenomenology, but at the
expense of losing the solvability.
From the perspective of the possible connections with the spatial ordering of
chromosomes, the mean-field approach presented in Section 4 gives an estimate
of the anomalous diffusion exponent that fits quite well with the numbers
observed in bacteria.
Let us finally comment on low energy excitations of a different kind that
should as well be expected in Peano-shaped structures. The chain phonons
treated in this work describe the small deviations with respect to a fully
ordered pattern, but in addition one could consider isolated topological
defects, giving rise to deviations from exact space-filling. The motion of
such defects necessarily involves collective reorderings along the chain, i.e.
phasons [42]. We are led to associate with excitations of this kind the
correlated motions observed in chromatin [43]. At even higher energies,
multiple defects should be accounted for, but this goes far beyond the
perspective of our approach; nonetheless, we point out that the chain phonons
should give a relevant contribution in the infrared regime, since they carry
the smaller momenta available to the system, scaling as $L^{-D}.$
Acknowledgements
The Author is indebted to M.Cosentino Lagomarsino for showing him that life
may still be found at the boundaries of maximally connected networks, and
wishes to thank F.Prati and R.Artuso for reading the paper.

## Appendix

In this Appendix we determine the symbolic substitution rule allowing to
generate approximants of the closed Peano-Hilbert chain $PH_{2}$. To the best
of our knowledge the procedure illustrated here is original. It is organized
in such a way as to preserve the correct matching of the four quarters making
up a closed chain, as required in the spectral problem examined in the paper.
The approximant of order k of $PH_{2}$ is a path of length $N_{k}=4^{k}$ that
connects with no intersections all points of the $(2^{k}\times 2^{k})$ square
lattice. We display in Fig.A2 one of the mentioned quarters, corresponding to
the approximant of order k=4. Since the plane has the topology of $S^{2}$, a
closed path cuts the plane in two polar regions. At first sight it is
equivalent to assign a model over the square lattice or over a Peano
approximant, but the two choices are topologically different. In fact the
lattice is a torus while the Peano chain has the topology of a circle (see
also the caption of Fig.A3).
Let us now enumerate all possible vertices, in order to identify chain
patterns with symbolic words. We label the available vertices with the
numerals $n$ and $\tilde{n},\,(n=1,2,..,6),$ where $n$ and $\tilde{n}$ are
oppositely oriented as illustrated in Fig.A1.

Figure 9: The convention used for labeling the different types of vertices is
here specified. As the iteration is based on a quadripartite structure, in
order to assign labels to vertices one can start with the elementary square-
shaped closed path. Since paths can as well go through lattice sites without
bending, in addition to orthogonal vertices also the flat ones (horizontal and
vertical) must be accounted for. With the numerals $(1,2,3,4)$ and $(5,6)$ we
respectively classify the patterns of orthogonal and flat vertices in the
clockwise path; the vertices corresponding to the anticlockwise path are
labeled with $(\tilde{1},\tilde{2},\tilde{3},\tilde{4})$ and
$(\tilde{5},\tilde{6}).$

The approximant of order $k$ is thus written as a symbolic word $W^{k}$ made
out of the $6X2$ letters $l$ of the alphabet defined above:

$W^{(k)}\equiv l(N_{k})l(N_{k}-1).......l(3)l(2)l(1).$ (41)

The alphabet used for the vertices is exceedingly extended, as every vertex is
the image of a flat or orthogonal one under suitable transformations. These
transformations are the discrete rotation $R$, the inversions $P_{x},\,P_{y}$
with respect to the x and y axes and the path inversion $\mathcal{T}$. They
act as listed here:
$\mathcal{T}(m)=\tilde{m},\,\mathcal{T}(m(n)....m(1))\equiv\tilde{m}(1).....\tilde{m}(n)$;
$P_{x}(\tilde{4})=3,\,P_{y}(1)=\tilde{4},\,P_{y}(P_{x}(2))=4$;
$4=R(1),1=R(2)$. We thus single out as basis elements the vertices $m=4$ and
$m=6$.

Figure 10: Closed $PH_{2}$ chain, also known as Moore curve. The lower left
quarter $(L\,\mathcal{L})$ of the approximant of order k=4 is displayed here.
The arrows depict the ingoing and outgoing links connecting this quarter to
its neighbours. The connection to the neighbour lying on the right side is
provided by an horizontal link placed at the uppermost available location. A
complementary situation regards the connection with the upper neighbour: in
this case the link is vertical and placed at the leftmost available location.
The inflation procedure is defined in such a way as to guarantee that this
structure is preserved at all orders.  Figure 11: In the lower line of this
figure are displayed the two choices available when inflating the vertex
$m=4$; the parent (inflated) patterns are depicted in blue (red). With the
left choice, the new vertices formed at the entrance and at the exit of the
$m=4$ motif are respectively flat horizontal $(\tilde{5})$ and orthogonal
$(\tilde{3})$. The convention chosed in the paper is displayed on the right:
an horizontal link enters the orthogonal vertex $(\tilde{1})$, and a vertical
link exits the flat vertical vertex $6$. In the upper part of the figure we
display the two resulting closed patterns after one step of the inflation
procedure; the initial configuration is the squared perimeter depicted in
blue. The two complementary H-shaped portions of the plane enclosed by the red
patterns, when glued together as the two patches of a tennis ball, form a
compact surface $S^{2}$. Hence the approximants separate the compactified
plane in two polar regions. The Peano chain is the maximally extended
’equator’ of this partition.

Our aim is to write the substitution rule under the constraint of closed path.
Quite remarkably the constraint is sufficient to identify the rule. Let us
consider e.g. the choices available for inflating the vertex $m=4$: as shown
in Fig.A3 two choices are available and they would imply twofold branchings at
each inflation step. This ambiguity is eliminated if one requires that the
inflated path must be closed. Every approximant can thus occur in only two
forms; they are displayed in the upper line of Fig.A3 for the first
approximant. In our calculations we have chosen the option shown on the right
column of Fig.A3. With this choice the four quarters making up the chain are
connected through the two external (left and right) vertical links and the two
central horizontal links: the H-shaped pattern displayed on the right side of
Fig.A3 reproduces itself at all orders. Going back to the inflation rule, we
write the convention as follows: $4\to 4^{\prime}\equiv 4\to(643\tilde{1})$
and $6\to 6^{\prime}\equiv(4\tilde{2}\tilde{3}1).$ Considering the symmetries
of the pattern, the following identities must hold at every order:

$1=\mathcal{T}P_{y}(4),\,\,\,\tilde{3}=\mathcal{T}R(4),\,\,\tilde{2}=R^{-1}P_{y}(4),\,\,\tilde{1}=\mathcal{T}R^{-1}(4),\,\,3=\mathcal{T}P_{x}(4).$
(42)

We can thus write the inflation in terms of the two words $Q_{4}$ and $Q_{6}$:

$Q^{(k)}_{6}=Q^{(k-1)}_{4}(R^{-1}P_{y}Q^{(k-1)}_{4})(\mathcal{T}RQ^{(k-1)}_{4})(\mathcal{T}P_{y}Q^{(k-1)}_{4})$
(43)
$Q^{(k+1)}_{4}=Q^{(k)}_{6}Q^{(k)}_{4}(\mathcal{T}P_{x}Q^{(k)}_{4})(\mathcal{T}R^{-1}Q^{(k)}_{4}).$

The word $Q^{(k)}_{4}$ determines the lower left quarter of the closed chain
$W^{(k)}$. The latter is obtained by operating with the symmetries and has the
form:
$W^{(k)}=Q^{(k)}_{4}(\mathcal{T}P_{x}Q^{(k)}_{4})(P_{x}P_{y}Q^{(k)}_{4})(\mathcal{T}P_{y}Q_{4}^{(k)}.)$

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