Paper Title: ISDE with logarithmic interaction and characteristic polynomials.



A Multi-Reference Relaxation Enforced Neighborhood Search Heuristic in SCIP
    
Suresh Bolusani^*10000-0002-5735-3443 Gioni Mexi10000-0003-0964-9802Mathieu Besançon20000-0002-6284-3033 Mark Turner10000-0001-7270-1496

    August 5, 2024
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§ ABSTRACT  We consider certain random matrix eigenvalue dynamics, akin to Dyson Brownian motion, introduced by Rider and Valko <cit.>. This is a multi-dimensional generalisation of a one-dimensional diffusion studied by Matsumoto and Yor <cit.>. We show that from every initial condition, including ones involving coinciding coordinates, the dynamics, enhanced with more information, converge on path-space to an infinite-dimensional Feller-continuous diffusion process. We show that the limiting diffusion solves an infinite-dimensional system of stochastic differential equations (ISDE) with logarithmic interaction. Moreover, we show convergence in the long-time limit of the infinite-dimensional dynamics starting from any initial condition to the equilibrium measure, given by the inverse points of the Bessel determinantal point process. As far as we can tell, this is: (a) the first path-space convergence result of random matrix dynamics starting from every initial condition to an infinite-dimensional Feller diffusion, (b) the first construction of solutions to an ISDE with logarithmic interaction from every initial condition for which the singular drift term can be defined at time 0, (c) the first convergence to equilibrium result from every initial condition for an ISDE of this kind. The argument splits into two parts. The first part builds on the method of intertwiners introduced and developed by Borodin and Olshanski <cit.>. The main new ingredients are a uniform, in a certain sense, approximation theorem of the spectrum of a family of random matrices indexed by an infinite-dimensional space and an extension of the method of intertwiners to deal with convergence to equilibrium. The second part introduces a new approach towards convergence of the singular drift term in the dynamics and for showing non-intersection of the limiting paths via certain “characteristic polynomials" associated to the process. We believe variations of it will be applicable to other infinite-dimensional dynamics coming from random matrices.







§ INTRODUCTION





 


 §.§ Background


The rigorous study of the non-equilibrium statistical mechanics of an infinite system of particles interacting via a potential was initiated in the seminal works <cit.> of Lanford. The study of the stochastic version of this problem, when one adds independent Brownian forces, and the dynamics are now described by an infinite-dimensional system of stochastic differential equations (henceforth abbreviated ISDE) was then first taken up by 
Lang <cit.> who proved well-posedness of the equilibrium dynamics for certain nice potentials. Many authors then contributed to this program, including Doss, Royer, Rost, Lippner and Fritz, and extended such results to non-equilibrium dynamics and more general potentials, see <cit.>. This program essentially culminates with the fundamental paper of Fritz <cit.>, which at least when the dimension of individual particles is one, essentially provides a complete solution theory for potentials which are twice continuously differentiable everywhere and with compact support.

The solution theory of <cit.> however excludes singular potentials and the construction of solutions is stated therein as an important open problem. In particular it excludes, as both assumptions fail, arguably the most famous potential in 1-dimension, namely the logarithm. Formally, such a system of interacting one-dimensional particles is governed by the following singular stochastic differential equations:

    d𝗑_i(t)=√(2𝔞(𝗑_i(t)))d𝗐_i(t)+𝔟(𝗑_i(t))dt+“2𝔞(𝗑_i(t))∑_j≠ i∂_𝗑_i(t)log|𝗑_i(t)-𝗑_j(t)|dt",

for some nice diffusion 𝔞 and drift 𝔟 functions and with the 𝗐_i being independent standard Brownian motions. The singular force of interaction experienced by the i-th particle 𝗑_i which involves all other particles (𝗑_j)_j≠ i is in quotes because it may need to be renormalised in some way in order to make sense.

An important motivation for studying systems of particles interacting via a logarithmic potential comes from the theory of random matrices <cit.> with the most famous example being the following. If (𝐖_t;t≥ 0) is the Brownian motion on N× N Hermitian matrices then its eigenvalues satisfy the closed system of stochastic differential equations (SDE), called Dyson Brownian motion <cit.>: 

    d𝗑_i(t)=d𝗐_i(t)+∑_j=1,j≠ i^N1/𝗑_i(t)-𝗑_j(t)dt,   i=1,…, N.

This SDE has a unique strong solution with almost surely no collisions between particles for all t>0, even if started from initial conditions with coinciding coordinates, see <cit.>. Dyson Brownian motion, and its variants, has been the object of study for decades. Beyond its intrinsic interest from the perspective of stochastic analysis, probability and integrable systems, it has been a key tool in proving universality for random matrices, see <cit.>.

Coming back to our discussion of infinite systems, as far as we are aware, the only existence (and uniqueness) result of solutions to an ISDE with logarithmic interaction from concrete initial conditions is 
 in the tour-de-force work of Tsai <cit.>. This corresponds to the bulk limit (see the seminal work of Spohn <cit.>) of Dyson Brownian motion (<ref>) and the ISDE is given by (it is also possible to include a parameter β≥ 1 in the drift which corresponds to so-called β-ensembles <cit.>): 

    d𝗑_i(t)=d𝗐_i(t)+β/2lim_k →∞∑_j≠ i:|j-i|≤ k1/𝗑_i(t)-𝗑_j(t)dt,   i∈ℤ.

The initial conditions allowed need to satisfy a certain quantitative “balanced condition": particles' positions are approximately uniform in some averaged sense with lower order corrections, see <cit.>. Moreover, the solution constructed in <cit.> satisfies this property almost surely for all times. The same property is also almost surely satisfied by the β-sine point process <cit.> which is the invariant[We note that the labelled ISDE (<ref>) itself does not have an invariant measure. The simplest possible instance of this phenomenon is that of infinitely many independent Brownian motions (𝗐_i)_i∈ℤ. This process does not have an invariant measure on ℝ^ℤ but when viewed as a process on unlabelled configurations the Poisson point process is invariant for it.] measure of the dynamics if we look at (<ref>) as an unlabelled point process. 

It is also important to mention a remarkable, almost thirty year-long program of Osada, later in collaboration with Tanemura on ISDE, see <cit.>. This program develops a general solution theory and can treat a number of examples, also of other singular interactions and in higher dimensions, but it only yields “near-equilibrium" solutions. These are solutions starting from an abstract, and in particular non-explicit, set of allowed configurations in which the solution stays for all times. This set is of full measure, when viewed as a set of unlabelled configurations, with respect to the invariant[This probability measure on unlabelled configurations is given as data for the problem. One should think of it as playing the role the sine point process plays for the Dyson bulk ISDE (<ref>).] probability measure[For a certain parameter range in our model this measure is infinite and this is another novel feature of the present work.] on point processes. The fact that one deals with an abstract set of initial conditions is not a technical restriction but rather an intrinsic feature of the approach which is based on the theory of Dirichlet forms <cit.>. For other applications of Dirichlet form theory on such particle systems, from a geometric viewpoint, see <cit.>.

Although, as far as we can tell there are no other results on ISDE with logarithmic interaction, there is a great number of results on scaling limits of random matrix dynamics. Possibly the most famous being the edge scaling limit of Dyson Brownian motion to the Airy line ensemble  (𝒜ℒℰ_i(·))_i=1^∞, see <cit.>, a central object in the KPZ universality class <cit.>. To do this, a probabilistic technique, that employs a certain Gibbs resampling property of the paths was developed in <cit.>. This led to an enormous amount of activity and has been extremely successful in showing that various models belong to the KPZ universality class <cit.>. Although our goal in this paper is rather orthogonal to these works there are some interesting connections with a new Gibbs resampling property, see Section <ref>. Finally, the way limits of random matrix dynamics, and related discrete models, were first rigorously studied was through the scaling limit of their space-time correlation functions, see <cit.>. This approach provides delicate quantitative information about the models and beautiful explicit formulae for their correlations, but as far as we can tell cannot be used for the type of results we want to prove. We will survey relevant previous works and how they compare with ours in Section <ref>.

The main purpose of this paper is to analyse an ISDE with logarithmic interaction coming from random matrix dynamics for which we can go beyond what was known previously and to introduce some new ideas for studying the singular drift term and non-intersection of the paths via certain “characteristic polynomials" of the process. We believe that these ideas will be useful in proving analogous results for a class of infinite dimensional dynamics coming from random matrices. The main reason we have chosen to start with this specific model is that it enjoys certain integrable properties which allow us to build on a powerful method of Borodin and Olshanski, see <cit.>, originally developed in the discrete setting of dynamics on partitions. It is plausible that this part of the argument may be replaced by more robust techniques in the future. We have not attempted to do this, partly because the intermediate results we need are of independent interest beyond the setting of ISDE.



 §.§ Main results


The model we consider is the eigenvalue evolution of a diffusion on the space of N× N non-negative definite matrices introduced by Rider and Valko in <cit.>, that we recall in Section <ref>. They used this to prove a matrix analogue <cit.> of Dufresne's identity <cit.>, partly motivated by problems in the theory of stochastic operators related to random matrices, see <cit.>. The N=1 case of this process is a celebrated one-dimensional diffusion that comes up in Matsumoto-Yor's study of exponential functionals of Brownian motion <cit.> and is also connected to random polymers <cit.>. It is much-studied in the applied, statistical and financial mathematics literature <cit.> and sometimes goes by the name inhomogeneous geometric Brownian motion.

The dynamics we consider are given by the following system of stochastic equations, with parameter η∈ℝ,

    d𝔵_i(t) = 𝔵_i(t) d𝗐_i(t) -η/2𝔵_i(t)dt+1/2dt+∑_j=1,j≠ i^N𝔵_i(t)𝔵_j(t)/𝔵_i(t)-𝔵_j(t)dt,   i=1,2,…,N,

with the 𝗐_i being independent standard Brownian motions. By general results <cit.> on such systems in finite dimensions we obtain that (<ref>) has a unique strong non-exploding solution starting from any initial condition in the chamber:

    𝕎_N,+ = {𝐱=(x_1,x_2,…,x_N)∈ℝ^N: x_1≥ x_2 ≥…≥ x_N≥ 0 },

and almost surely, for all positive times lives in the interior of 𝕎_N,+, see Lemma <ref>. It is instructive to see that the SDE (<ref>), by rewriting the interaction term, is given by 

    d𝔵_i(t) = 𝔵_i(t) d𝗐_i(t) -η/2𝔵_i(t)dt+1/2dt+[(-N+1) 𝔵_i(t)+𝔵_i(t)^2∑_j=1,j≠ i^N ∂_𝔵_i(t)log|𝔵_i(t)-𝔵_j(t)|]dt,

which elucidates the fact that the stochastic dynamics (<ref>) correspond to independent one-dimensional diffusions interacting via a logarithmic potential.

In order to state our results precisely we need some notation and definitions. Define the following spaces, endowed with the topology of coordinate-wise convergence,

    𝕎_∞,+   ={𝐱=(x_i)_i=1^∞∈ℝ^ℕ:x_1 ≥ x_2 ≥ x_3 ≥⋯≥ 0 },
    𝕎_∞,+^∘   ={𝐱=(x_i)_i=1^∞∈ℝ^ℕ:x_1 > x_2 > x_3 > ⋯ > 0 }.



In fact, our limiting dynamics will live on an enhanced space containing somewhat more information than the above. It is on this enhanced space, and this is essential, that the dynamics will enjoy a Feller-continuity property. The space is defined as follows.


We define the space

    Ω_+def={ω=(𝐱,γ)∈𝕎_∞,+×ℝ_+: ∑_i=1^∞x_i≤γ},

endowed with the topology of coordinate-wise convergence.   

We observe that, Ω_+ is locally compact, metrizable and separable and the topology can be metrised with the metric 𝖽_Ω_+ as follows, with ω=(𝐱,γ),ω̃=(𝐱̃,γ̃) ∈Ω_+,

    𝖽_Ω_+(ω,ω̃)=∑_i=1^∞|x_i-x̃_i|/2^i(1+|x_i-x̃_i|) + | γ-γ̃|.

Moreover, note that Ω_+ is complete and thus it is a Polish space.





Returning to the dynamics, heuristics related to the hard-edge scaling in random matrix theory <cit.> indicate that we should rescale our stochastic processes in space by 1/N to see non-trivial behaviour. In particular, the rescaled SDE (<ref>) becomes:

    d𝗑_i^(N)(t) = 𝗑_i^(N)(t) d𝗐_i(t) -η/2𝗑_i^(N)(t)dt+1/2Ndt+∑_j=1,j≠ i^N𝗑_i^(N)(t)𝗑^(N)_j(t)/𝗑^(N)_i(t)-𝗑^(N)_j(t)dt,    i=1,2,…,N.


We then embed, for different N ∈ℕ, all these processes on the space Ω_+.


Define the process (𝐗^(N)(t);t≥ 0) on Ω_+ by, with (𝗑^(N)(t);t≥ 0) the solution of (<ref>), 

    𝐗^(N)(t)=((𝗑_i^(N)(t))_i=1^∞,∑_i=1^∞𝗑_i^(N)(t)),   ∀ t ≥ 0,

where 𝗑_i^(N)≡ 0, for i>N.


Finally, we recall the definition of a Feller semigroup and Feller process in Definition <ref>, see <cit.>. For a Polish space 𝔛 write C(ℝ_+,𝔛) for the space of continuous functions on ℝ_+ with values in 𝔛, endowed with the topology of locally uniform convergence. Note that, this is again a Polish space, see <cit.>. We can now state our first main result.

 Let η∈ℝ. Then, there exists a unique Feller semigroup (𝔓_∞(t))_t≥ 0 with associated Feller-Markov process (𝐗_t^Ω_+;t≥ 0)=(((𝗑_i(t))_i=1^∞,γ(t));t≥ 0) on Ω_+ satisfying the following. Let 𝐗_0^Ω_+=ω∈Ω_+ be arbitrary and assume 𝐗^(N)(0)→ω in the topology of Ω_+. Then, as N →∞,

    𝐗^(N)d⟶𝐗^Ω_+ in  C(ℝ_+,Ω_+),

with d⟶ denoting convergence in distribution. In particular, (𝐗_t^Ω_+;t ≥ 0) is a diffusion[A strong Markov process with continuous sample paths.]. Moreover, there exists a coupling of the 𝐗^(N) and 𝐗^Ω_+ on a single probability space such that, almost surely, for any T ≥ 0,

    sup_t ∈ [0,T]∑_i=1^∞(𝗑_i^(N)(t)-𝗑_i(t))^2 N →∞⟶  0.




As far as we can tell, this is the first result on path-space convergence of random matrix dynamics from every single initial condition for which the finite-dimensional dynamics are well-defined. Observe that, the limiting diffusion (𝐗_t^Ω_+;t ≥ 0) can start from configurations with coinciding points. In fact, it can even start from the very singular configurations ((0)_i=1^∞,γ) of having all particles at 0!


It is a natural question whether the limiting process solves an ISDE with logarithmic interaction. Our second main result answers this in the affirmative and in fact gives a lot more information. Throughout this paper we use the notation 𝖫𝖺𝗐(𝐘) for the law of a random variable 𝐘 taking values in a Polish space.

 Let η∈ℝ. Let 𝐱∈𝕎_∞,+^∘ and γ≥∑_i=1^∞ x_i. Consider the Feller process on Ω_+, (𝐗_t^Ω_+;t≥ 0)=(((𝗑_i(t))_i=1^∞,γ(t));t≥ 0) from Theorem <ref> with initial condition 𝐗_0^Ω_+=(𝐱,γ). Then, almost surely, for all t ≥ 0, (𝗑_i(t))_i=1^∞∈𝕎_∞,+^∘ and the (𝗑_i(·))_i=1^∞ is a weak[Recall that a weak solution means that on a (filtered) probability space (𝒱,(𝔙_t)_t≥ 0,𝐕) we can define a (𝔙_t)_t≥ 0-adapted sequence of independent standard Brownian motions (𝗐_i)_i=1^∞ and the process (𝗑_i)_i=1^∞  which is adapted with respect to (𝔙_t)_t≥ 0 such 𝐕-a.s. (<ref>) holds for all t≥ 0; in particular the pair ((𝗑_i)_i=1^∞,(𝗐_i)_i=1^∞) is considered a weak solution to (<ref>), see <cit.>. ]
solution to the following ISDE with logarithmic interaction starting from 𝐱, namely it satisfies,


    𝗑_i(t)=x_i+∫_0^t 𝗑_i(s)d𝗐_i(s) -η/2∫_0^t𝗑_i(s)ds+∫_0^t∑_j=1,j≠ i^∞𝗑_i(s)𝗑_j(s)/𝗑_i(s)-𝗑_j(s)ds,   i ∈ℕ,

where the (𝗐_i(t);t≥ 0) are independent standard Brownian motions. Moreover, if we denote by (𝐗(t;γ);t≥ 0) the solution above corresponding to γ, then

    𝖫𝖺𝗐(𝐗(·;γ))≠𝖫𝖺𝗐(𝐗(·;γ̃)), whenever γ≠γ̃.

Finally, out of these solutions there exists a unique one such that almost surely t ↦∑_i=1^∞𝗑_i(t;γ) is continuous for all t ≥ 0 given by the choice γ=∑_i=1^∞ x_i
and this solution is a Markov process.


Observe that, the singular interaction drift in (<ref>) is really a renormalised[Observe that, without subtracting the diverging counterterm the logarithmic interaction is clearly infinite.] logarithmic interaction term, since

    ∑_j=1, j≠ i^∞𝗑_i(t)𝗑_j(t)/𝗑_i(t)-𝗑_j(t)=lim_N→∞[(-N+1)𝗑_i(t)+𝗑_i(t)^2∑_j=1, j≠ i^N 1/𝗑_i(t)-𝗑_j(t)].

In some sense, we could interpret the dynamics (<ref>) as geometric/exponential Brownian motions interacting via a renormalised logarithmic potential. As far as we know, this result is the first construction of solutions to an ISDE with logarithmic interaction from every single initial condition for which the infinitesimal singular drift term makes sense at time t=0. It is optimal since we need the condition ∑_i=1^∞ x_i<∞ for otherwise the drift term is infinite. Also, we note that, the initial configurations, and more generally the distribution of the dynamics at time t≥ 0, do not need to resemble in any qualitative sense the invariant measure (when it exists) to be defined below, like they do, to some extent, in the case of the bulk Dyson ISDE <cit.>. This result is also the first, as far as we can tell[As far as we can tell, even for the class of initial conditions considered in the works <cit.>, a single solution is first constructed which is then shown (under additional assumptions) to be unique.], concrete construction of non-unique (even from a single initial condition) solutions for ISDE with logarithmic interaction. This is an explicit illustration of a phenomenon that is not seen in finite-dimensional random matrix dynamics. 

We now move to our results on the invariant measure and convergence to equilibrium for the dynamics when η>-1. Note that, this restriction is also required for the finite-dimensional dynamics (<ref>) for otherwise there is no invariant probability measure. In fact, the construction of solutions to the ISDE with logarithmic interaction above, when η≤ -1, appears to be the first, of any kind, for which there is no invariant probability measure when viewed as an evolution on unlabelled point processes. We need some more notation and terminology. Let ℐ⊂ℝ be a union of open intervals. Let 𝖢𝗈𝗇𝖿(ℐ) denote the space of configurations over ℐ, namely locally finite collections of points in ℐ, or equivalently the space of ℤ_+-valued measures on ℐ endowed with the vague topology, see <cit.> for details. A determinantal point process on ℐ, with correlation kernel 𝔎, is a probability measure on 𝖢𝗈𝗇𝖿(ℐ) which is determined[Subject to certain mild conditions so that the correlation functions determine the point process, see <cit.>.] by the fact that all its correlation functions (ρ_n)_n=1^∞ with respect to Lebesgue measure are given as ρ_n(x_1,x_2,…,x_n)=(𝔎(x_i,x_j))_i,j=1^n, see <cit.> for rigorous details. 



Let η>-1. The inverse Bessel point process 𝖨𝖡𝖾𝗌^Conf_η with parameter η is the determinantal point process on (0,∞) with correlation kernel 𝖪_η given by 

    𝖪_η(x,y)=
    8/xy𝕁_η(8/x,8/y),   x,y ∈ (0,∞),

where 𝕁_η(x,y) is the so-called the Bessel kernel, with J_η the Bessel function of order η,

    𝕁_η(x,y)=
    x^1/2J_η+1(x^1/2)J_η(y^1/2)-
    y^1/2J_η+1(y^1/2)J_η(x^1/2)
    /2(x-y).



𝖨𝖡𝖾𝗌^Conf_η is called the inverse Bessel point process because it simply consists of the inverse points (up to multiplicative constant) of the Bessel determinantal point process with correlation kernel 𝕁_η <cit.>. The Bessel point process is the universal limit arising at the hard edge of random matrices, see <cit.>.


Since 𝖨𝖡𝖾𝗌^Conf_η consists of distinct points it gives rise to a unique probability measure  𝖨𝖡𝖾𝗌_η on 𝕎_∞,+^∘ by labelling the points of 𝖨𝖡𝖾𝗌^Conf_η in a decreasing order. Define the space

    Ω_+^0def={ω=(𝐱,γ)∈Ω_+:∑_i=1^∞x_i=γ},

and write π:Ω_+ →𝕎_∞,+ for the map given by π((𝐱,γ))=𝐱. 


Let η>-1. Define 𝔐^η to be the unique Borel probability measure on Ω_+ which is supported on Ω_+^0 and satisfies 

    π_*𝔐^η=𝖨𝖡𝖾𝗌_η.



Then, we have the following result on convergence to equilibrium for the process (𝐗_t^Ω_+;t ≥ 0) on Ω_+.

 
Let η>-1 and 𝔎 a Borel probability measure on Ω_+. Consider the Feller process (𝐗_t^Ω_+;t ≥ 0) constructed in Theorem <ref> and suppose 𝖫𝖺𝗐(𝐗_0^Ω_+)=𝔎. Then, as t→∞,

    𝐗_t^Ω_+d⟶𝐙,   where 𝖫𝖺𝗐(𝐙)=𝔐^η.

In particular, if η>-1, 𝔐^η is the unique invariant measure of (𝐗_t^Ω_+;t ≥ 0).


As far as we can tell, the result above is the only convergence to equilibrium result on ISDE with logarithmic interaction. Nevertheless, there is very interesting recent work on ergodicity of unlabelled infinite-dimensional diffusions (namely looking at diffusions on the space 𝖢𝗈𝗇𝖿) with logarithmic interaction. Dyson's model in the bulk is considered in <cit.> and a more general class of models (including the β-ensemble version (<ref>) of Dyson's model) is treated by Suzuki in <cit.>. Both <cit.> and <cit.>  use  a Dirichlet form approach. The paper <cit.> builds on Osada-Tanemura's theory and <cit.> makes use of the DLR equations <cit.> for the β-sine point process <cit.>.



We finally prove the following result which says that although we can construct infinitely many non-equal in law solutions for the ISDE (<ref>) starting from the inverse Bessel points 𝖨𝖡𝖾𝗌_η, there is in some sense only one true equilibrium solution. Another interesting feature is that all of these different solutions start distributed according to 𝖨𝖡𝖾𝗌_η, all but one of them will stop being distributed according to 𝖨𝖡𝖾𝗌_η instantaneously (this does not appear in the statement below but will be clear from the proof), but all of them will converge in distribution back to 𝖨𝖡𝖾𝗌_η as t→∞.


Let η>-1. Let 𝔪 be a Borel probability measure on Ω_+ satisfying π_*𝔪=𝖨𝖡𝖾𝗌_η. Consider the Feller process (𝐗_t^Ω_+;t≥ 0) with random initial condition 𝖫𝖺𝗐(𝐗_0^Ω_+)=𝔪 and write (𝐗_t^Ω_+;t≥ 0)=((𝐗(t;𝔪),γ(t;𝔪));t≥ 0) where 𝐗(t;𝔪)=(𝗑_i(t;𝔪))_i=1^∞. Then, (𝗑_i(·;𝔪))_i=1^∞ solves (<ref>) with initial condition distributed according to 𝖨𝖡𝖾𝗌_η. Out of these solutions (for different 𝔪) there exists a unique solution 𝐗(·;𝔪) satisfying

    𝖫𝖺𝗐(𝐗(t;𝔪))=𝖨𝖡𝖾𝗌_η,   ∀ t ≥ 0,

and it is given by the choice 𝔪=𝔐^η.









 §.§ Strategy of proof

(a) Convergence on path-space and Feller property. Our starting point is a certain non-obvious consistency relation the finite-dimensional dynamics enjoy when N varies. Let (𝔓_N(t))_t≥ 0 be the Markov semigroup associated to the unique strong solution of (<ref>). For 𝐱=(x_1>x_2>⋯>x_N+1), we consider the Markov kernel, 

    Λ_N^N+1(𝐱,d𝐲)=N!∏_1≤ i<j ≤ N(y_i-y_j)/∏_1≤ i<j ≤ N+1(x_i-x_j)1_x_1>y_1>x_2>⋯>x_N>y_N>x_N+1d𝐲.

This Markov kernel has an important interpretation in terms of Hermitian random matrices whose law is invariant under unitary conjugation <cit.>  and can be extended to 𝐱 with coinciding coordinates. We will say more in Section <ref>. The consistency relation we are alluding to, is the following so-called intertwining, between the semigroups:

    𝔓_N+1(t)Λ_N^N+1=Λ_N^N+1𝔓_N(t),   ∀ t≥ 0,  N∈ℕ.

We give a direct proof of this relation but also a different argument using matrix stochastic calculus which gives more intuition regarding where it actually comes from. It is worth mentioning that Dyson Brownian motion also satisfies a relation of this kind, see <cit.>.
 

Now, one can view (𝕎_N,+,Λ_N^N+1)_N=1^∞ as a projective system of measures or equivalently as a Markov chain moving in discrete time, see <cit.> for background. Then, it is a classical theorem of Pickrell <cit.> and Olshanski-Vershik <cit.>, in equivalent form, that the entrance boundary of this Markov chain can be identified with the space Ω_+ and the extremal entrance laws Λ_N^∞(ω,d𝐱), with ω∈Ω_+, have a random matrix interpretation. Moreover, it can be shown that all Markov kernels appearing above are Feller continuous. Then, by the method of intertwiners of Borodin and Olshanski <cit.> the above considerations guarantee the existence of a Feller semigroup (𝔓_∞(t))_t≥ 0 and associated Feller process 𝐗^Ω_+ on Ω_+ which is uniquely determined via the intertwinings:

    𝔓_∞(t)Λ_N^∞=Λ_N^∞𝔓_N(t),  ∀ t ≥ 0,  N ∈ℕ.


To prove convergence to equilibrium in the long-time t limit for 𝐗^Ω_+, Theorem <ref>, we extend the method of intertwiners to deal with such questions by proving the following simple statement. If the finite-dimensional processes converge to equilibrium from any initial condition then the infinite-dimensional process converges to equilibrium from any initial condition. This holds in a general setting and can be applied to other models.


Coming back to describing the Feller process 𝐗^Ω_+, at this stage we know essentially nothing about its trajectories. We do not know if and how it is approximated by the finite dimensional dynamics. We do not know if it has continuous sample paths. It might even be, in some way, degenerate[This may well happen. The most famous example satisfying (<ref>), Dyson Brownian motion itself, also gives rise, by the method of intertwiners, to a Feller process on an extension Ω of the space Ω_+, see Section <ref>, but this “infinite-dimensional stochastic process" is a degenerate deterministic flow on the coordinates of Ω.].

Proving convergence on the path-space C(ℝ_+,Ω_+) would resolve these questions. Since we are dealing with Feller processes one can prove this convergence by proving convergence at the level of the infinitesimal generators for a sufficiently large class of functions. This is in general highly non-trivial though since we are dealing with infinite-dimensional objects. But we have more structure which helps. Let us write 𝖫_N and 𝖫_∞ for the infinitesimal generators of the Feller semigroups (𝔓_N(t))_t≥ 0 and (𝔓_∞(t))_t≥0 respectively. First of all, if 𝒞_N is a core for 𝖫_N for each N ∈ℕ, the space of functions

    span(⋃_K=1^∞Λ_K^∞𝒞_K),

is known to be a core for 𝖫_∞. In our setting, 𝒞_K will be a class of smooth functions with compact support with symmetry, see Section <ref>. Then, for an arbitrary 𝐟 in (<ref>), of the form 𝐟=Λ_K^∞ g, for g ∈𝒞_K, we will show that there exists 𝐟_N in the domain of 𝖫_N such that we have the following convergence in a uniform sense (we are abusing notation here; to be precise we have to embed all of them on Ω_+), as N→∞,

    𝐟_N    ⟶𝐟, 
    𝖫_N 𝐟_N    ⟶𝖫_∞𝐟.

The right choice for 𝐟_N will be nothing but,

    𝐟_N=Λ_N-1^N Λ_N-2^N-1⋯Λ_K^K+1g.

Thus, showing (<ref>) will turn out to be equivalent to proving 

    sup_𝐱^(N)∈𝕎_N,+|Λ_K^∞g(ω(𝐱^(N)))-Λ_N-1^NΛ_N-2^N-1⋯Λ_K^K+1g(𝐱^(N))|N→∞⟶ 0,

where ω(𝐱^(N)) denotes a certain embedding of 𝐱^(N)∈𝕎_N,+ into Ω_+, see Section <ref>. 

This is what we call the uniform approximation theorem and it is arguably our main contribution for this part of the argument. It is inspired by an analogous theorem in the combinatorial setting of the Gelfand-Tsetlin graph <cit.>. There are some important new difficulties which arise in the continuous setting of random matrices however. First of all, the explicit formulae for Λ_K^∞ and Λ_N-1^N⋯Λ_K^K+1 as K× K determinants, due to Olshanski-Vershik <cit.> and Olshanski <cit.>, that we start with, when extended to arbitrary ω∈Ω_+ and 𝐱∈𝕎_N,+ involve as entries derivatives of distributions which is problematic. Fortunately, we can make sense of them by virtue of the fact that we are testing[There is a tension here between the need for the function spaces 𝒞_K to be large enough to deduce convergence from them while at the same time they should only include nice enough functions for the explicit formulae to make sense. We note that, there are alternative, random matrix interpretations of Λ_K^∞ and Λ_N-1^N⋯Λ_K^K+1, see Section <ref>, which can be tested against arbitrary bounded Borel functions but, as far as we can tell, these cannot be used to prove (<ref>), or even guess that a statement like (<ref>) is true.] against compactly supported symmetric (which is important) smooth functions g∈𝒞_K. Moreover, in the combinatorial setting once one gets a good explicit formula as a K× K determinant for the analogues of Λ_K^∞ and Λ_N-1^N⋯Λ_K^K+1 the asymptotic analysis is fairly straightforward: one needs to take the limit of each individual entry in the determinant. In the random matrix setting, despite some post-processing of the formulae of <cit.> to make them look as similar as possible, this does not appear to be the case. For this reason we needed to devise a non-trivial inductive argument in order to establish the uniform asymptotics (<ref>). Finally, to prove (<ref>), by virtue of the intertwining (<ref>), it again boils down to (<ref>).
Hence, given (<ref>) and (<ref>), by general results <cit.>, we obtain the path-space convergence in (<ref>). Some additional arguments then also give convergence in ℓ^2 as in (<ref>). 


It is interesting to note that in the above argument we did not need to use any explicit information about the generator 𝖫_∞. This should not be too surprising. The whole point, and in some sense appeal, of the method of intertwiners is that at the level of dynamics we only need to work with 𝖫_N for fixed N. However, this is also a caveat. If one wants to understand the actual dynamics of 𝐗^Ω_+ new arguments are needed. 

(b) Non-intersection of the paths. The next key step is to show non-intersection of the limiting paths (𝗑_i(·))_i=1^∞. This is essential input to obtain the ISDE. It is also essential input in the approach of Tsai <cit.> and Osada-Tanemura <cit.>; in Tsai's work proving this is one of the major challenges, see discussion around Remark 2.9 therein. As far as we can tell, the method of intertwiners is not useful for proving this type of statement. There is also a new Gibbs resampling property behind the SDE (<ref>), that we attempted to use, but this approach runs into problems that we explain in Section <ref>. So we turned back to make use of the stochastic equations. This cannot be done directly in limit: we have no stochastic equations for the infinite-dimensional process to speak of yet! Instead, we need to control, uniformly in N, the probabilities that the finite-dimensional paths solving (<ref>) come close directly from the SDE. As far as we can tell, this is the first[Both Tsai's <cit.> and Osada-Tanemura's <cit.> approaches for non-intersection do not make direct use of the finite-dimensional SDE. Similarly, the Gibbsian line ensemble approach <cit.> to prove non-intersection is totally different and does not use SDE in any way.] time this has been done. Of course, for fixed N, showing non-intersection is standard and this is the mainstream approach to attack Dyson-like SDE, see <cit.>, but the bounds in the literature blow up with N. There is good reason for this, that we comment on below.


In essence, although the actual proof is presented very differently in terms of multiple stopping times, we establish statements which informally can be interpreted as the following. Let 𝐱^(N) be arbitrary in the interior of 𝕎_N,+ such that ω(𝐱^(N))⟶ω=(𝐱,γ), with 𝐱∈𝕎_∞,+^∘. Then, for any n ∈ℕ, T≥ 0 and ε>0, we can find δ such that,

    sup_N≥ n+1Prob(inf_t∈ [0,T]|𝗑_n^(N)(t)-𝗑_n+1^(N)(t)|<δ|𝗑^(N)(0)=𝐱^(N))< ε.


The reason the bounds, and observables used to obtain them, that are given in the literature for fixed N, are not useful as N →∞, is that they are controlling the collision times of all N paths simultaneously and this cannot work. Instead we perform a double induction argument to prove the statements above one pair of paths at a time. The main ingredients are certain novel observables or Lyapunov functions associated to the dynamics given by variations of the reverse characteristic polynomial of a matrix with eigenvalues (𝗑^(N)_i(t))_i=1^N, see the functions f_n^N and Ψ_n^N and discussion around (<ref>). The crux of the argument, and the reason considering the aforementioned observables is actually useful is the following. When applying Itô's formula to them, the term which comes from the singular drift, which would be hard to control uniformly in N, has negative sign and can simply be dropped to give an N-independent bound. This feature is rather generic. For example the argument works for the β-ensemble version[Include a multiplicative constant β in front of the singular interaction drift.] of the dynamics and variations of it should work for other models. Even if we hadn't known convergence of the finite-dimensional dynamics, the argument gives that any possible subsequential limit would necessarily consist of non-intersecting paths.

(c) The ISDE. Finally, to prove (𝗑_i(·))_i=1^∞ solves the ISDE (<ref>), modulo technicalities, this boils down to convergence of the singular drift term in the finite-dimensional SDE to its infinite-dimensional counterpart. Again, viewing things in terms of certain “characteristic polynomials" of the process will do the trick. In particular, if we write

    Φ_i^N(z;𝐱^(N))=∏_j=1,j≠ i^N(1-x_j^(N)z)^2,  𝐱^(N)∈𝕎_N,+,

then we can observe that,

    ∑_j≠ i^N 𝗑_i^(N)(t)𝗑_j^(N)(t)/𝗑_i^(N)(t)-𝗑_j^(N)(t)=1/2d/d zlogΦ_i^N(z,𝗑^(N)(t))|_z=(𝗑_i^(N)(t))^-1.

Then, making use of the following fact, by virtue of convergence in Ω_+, where convergence holds uniformly on compact sets in z∈ℂ,

    Φ_i^N(z;𝗑^(N)(t)) ⟶  e^-2(γ(t)-𝗑_i(t)) z∏_j=1,j≠ i^∞e^2𝗑_j(t)z(1-𝗑_j(t)z)^2,

combined with some further arguments that make use of the fact that paths are non-intersecting, and no 𝗑_i hits zero, we can show convergence of the drift term.

At this stage the ISDE for the (𝗑_i(·))_i=1^∞ coordinates may still depend on γ(·). Then, the final part of the argument, making use of the structure of the ISDE shows that almost surely, for all t>0, ∑_i=1^∞𝗑_i(t)=γ(t) which completes the proof. Thus, in some sense, the true state space of the diffusion 𝐗^Ω_+ is Ω_+^0 and Ω_+ \Ω_+^0 acts as a kind of entrance boundary for it. At this point all the hard work is done and the proofs of the remaining statements in our main results are fairly straightforward.






 §.§ Previous approaches


The kinds of problems we are considering have been studied intensely for decades. As a result a number of rather sophisticated approaches have been developed to attack them. We survey them below. To different extents, we have attempted to apply each of them to our problem.

(1) Tsai's important paper <cit.> was the only, up until now, work which gave existence of solutions to an ISDE with logarithmic interaction from explicit initial conditions. Tsai uses in a very essential and delicate way the translation-invariance of the dynamics and a certain novel monotonicity property of the corresponding process of gaps between particles. The translation-invariance is definitely not true for our model and it is unclear what the right analogue of the monotonicity property (if one exists) should be. For these fundamental reasons it seems hard to adapt the approach of <cit.> to even give solutions  from a class of initial conditions in our case.

(2) We now move to Osada and Tanemura's program <cit.>. This spans almost three decades and several papers so it is impossible to survey in a few lines. The upshot is the following: one is given as basic data a probability measure 𝖬 on the configuration space 𝖢𝗈𝗇𝖿(𝒵) of some Euclidean space 𝒵 consisting of unlabelled collections of points. From this, under certain assumptions, one first constructs a diffusion on 𝖢𝗈𝗇𝖿(𝒵), starting from any initial condition in an abstract set 𝒮_𝖬⊂𝖢𝗈𝗇𝖿(𝒵),  so that 𝖬(𝒮_𝖬)=1, with 𝖬 as invariant measure. With this as input, and under further assumptions, a weak solution to a labelled ISDE is constructed (always starting and remaining in a labelled version of 𝒮_𝖬), which under additional conditions is shown to be the unique one satisfying certain properties. For some of these conditions[These are for the most part conditions on the correlation functions of 𝖬 and its Campbell measures <cit.>.] black-box theorems already exist to verify them, however some of them do need to be checked by hand. This is in general a highly non-trivial task. For example, the main purpose of the long preprint <cit.> is to verify these conditions in the case of the ISDE associated to the stationary version of the Airy line ensemble in order to apply the theory.

Let us be more precise on how we think the Osada-Tanemura theory relates to our approach. We believe that if one attempts (for η>-1 certain modifications are needed but we will not check the technical conditions here, while for η≤ -1 a fundamental extension of this theory is required since there is no invariant probability measure on 𝖢𝗈𝗇𝖿((0,∞)) to start with) to use this theory to construct “near-equilibrium" solutions for our model, these should match, in a way that needs to be made precise, for initial conditions 𝐱∈𝕎_∞,+^∘ that qualitatively resemble the invariant measure, the solutions that arise from Ω_+^0. It seems plausible that this is a more general phenomenon. The Dirichlet form approach will pick out a distinguished solution from the ones living on an enhanced space. Another related point is that the singular interaction drift of the ISDE in Osada-Tanemura's theory is given by the so-called logarithmic derivative of the point process <cit.>. In some sense this notion associates to the invariant point process a canonical limiting characteristic polynomial (a random analytic function). It would be interesting to understand these potential connections more precisely and rigorously in the future.

(3) The recent work of Landon <cit.> on the edge scaling of β-Dyson Brownian motion at equilibrium is also worth mentioning. Using a different approach to previous works, that takes rigidity estimates for the invariant point process (the β-Airy point process <cit.>) as a key ingredient, Landon can prove convergence under a subsequence for the equilibrium dynamics for all β≥ 1 (the paper <cit.> only deals with β=1,2,4); the fact that the process is in equilibrium is essential for the use of rigidity estimates. However, the limiting process is not shown to solve an ISDE, or even have non-intersecting paths. Nevertheless, it is very natural to expect that these statements should be true.

(4) A different method for proving path-space convergence of random matrix dynamics, having special structure, is the probabilistic approach of Gibbsian line ensembles <cit.>. This type[We will not give the precise definitions here, see for example <cit.>.] of Gibbs property is, in some sense, the right dimension-independent analogue of the fact that the N-dimensional Dyson Brownian motion (<ref>) can equivalently be constructed as N independent Brownian motions conditioned to never intersect <cit.>. This approach is  useful for proving tightness of the paths. To conclude convergence some more information is needed. The current state of the art, in the case of models in the KPZ universality class, is the recent remarkable strong characterisation theorem <cit.> of the Airy line ensemble. This says that, beyond a Brownian resampling property, we only need to know that the top path approximates a parabola to pin down the Airy line ensemble.

(5) Yet another approach to taking limits of random matrix dynamics is to view the N-dimensional interacting SDE as diffusions on 𝖢𝗈𝗇𝖿 and take the limit of their space-time correlation functions. The state of the art, for limits of Dyson Brownian motion in the bulk and non-intersecting squared Bessel processes starting from a class of initial conditions, appears to be the work of Katori-Tanemura <cit.>. As far as we can tell, this approach cannot, on its own at least, be used to obtain results on ISDE.

(6) We now comment on the method of intertwiners[For uses of intertwining relations, with other applications in mind, see for example <cit.>.]. An exposition can be found in the lecture notes and ICM proceedings by Olshanski <cit.>. The method was originally introduced in a discrete setting to construct infinite-dimensional Feller processes coming from continuous-time jump dynamics on partitions which satisfy an intertwining. These Markov dynamics can be thought of as discrete variants of finite-dimensional SDE with logarithmic interaction. The main example the method was developed for in <cit.> is related to the projective system associated to the Gelfand-Tseltin graph <cit.>, whose boundary can be identified with an infinite-dimensional space 𝔊𝔗 with continuous parameters, akin to Ω_+. The invariant measure of the process, the analogue of 𝔐^η, is given by a so-called zw-measure which comes up in representation theory and harmonic analysis <cit.>. One could think of these measures as a more sophisticated version of determinantal point processes that arise in random matrix theory. The current state of the art on this construction is the paper <cit.> of Olshanski, where the generator of the process on 𝔊𝔗 is computed explicitly, not in the natural coordinates of 𝔊𝔗, but rather in a different set of variables and is realised by a second order differential operator. The main idea is that in this other set of variables, which come from abstract algebraic considerations, the generator is not singular. Nevertheless, it is still not known if the process of <cit.> has continuous[Of course, by the Feller property it has a modification with càdlàg paths <cit.>.] sample paths  and this is one of the questions posed in <cit.>. In fact, it is not unreasonable, in analogy with the results of our paper, to expect even more: that in natural coordinates this process should solve some kind of ISDE with singular interaction, at least for some initial conditions. This is probably a hard problem.


In the random matrix setting the method of intertwiners was first applied by one of us in <cit.>. There an abstract construction of an infinite-dimensional Feller process related to the Cauchy ensemble <cit.> is provided. We recall the main result of <cit.> in Theorem <ref> below and we extend it, using the machinery we develop in this paper, to a strong path-space convergence result which also proves that the limiting process has continuous sample paths.


(7) Finally, let us mention that for a wide class of finite-dimensional systems with interactions of the form  (<ref>) a complete solution theory exists, see <cit.>. In finite dimensions one can even construct dynamics when collisions occur between particles (this corresponds to β<1 in the Dyson SDE) using the theory of multivalued SDE, see <cit.>. As far as we can tell, nothing is known for the corresponding question in infinite dimensions.




 §.§ Open problems

We end this introduction with a number of open problems.

    
  * Gibbs property. It would be very interesting if the limiting non-intersecting paths (𝗑_i)_i=1^∞ satisfy a Gibbs resampling property <cit.>. We say more about this problem in Section <ref>.
    
  * Space-time determinantal correlations. A special class of finite-dimensional dynamics with logarithmic interaction, including (<ref>), starting from any deterministic initial condition, is known to have determinantal space-time correlations, see <cit.>. It is an interesting question whether the limiting process we constructed enjoys the space-time determinantal property with an explicit correlation kernel. A somewhat easier question, which nevertheless involves the same fundamental difficulties, is to try to do this for the dynamics starting from the equilibrium measure.
    
  * Rate of convergence to equilibrium. It would be interesting if one could quantify (in any sense) the convergence to equilibrium from Theorem <ref>.
    
  * Existence of solutions for other models. 
    Our approach to construct solutions to ISDE with logarithmic interaction will also work for the dynamical Cauchy model in  Section <ref>. There is an additional and non-trivial complication there in that the pre-limit paths can cross zero. Towards a more general class of models, one needs to replace the first, integrable, part of our argument with more robust techniques coming from stochastic analysis or probability. Then[In fact, most likely an extension of the second part of our argument could be used to replace certain aspects of the first. At least when it comes to tightness of the paths, a uniform bound like (<ref>) is the main ingredient in <cit.>. Note that, if one only cares about the ISDE we do not need the full power of a result like Theorem <ref>.], we believe variations of the second part of our argument can be applied.
    
    
  * Uniqueness of solutions. We proved a certain kind of uniqueness in Theorem <ref>. However, the more natural question is to prove uniqueness, for a general class of ISDE with singular interaction, within a space of paths that take values in certain “rigid configurations", the analogue of “balanced configurations" from the work of Tsai for the Dyson ISDE <cit.>. The precise meaning of “rigid configurations" will need to be model-specific. One would hope that in such spaces of paths the tail-end of the singular drift term can be controlled which is the key requirement for uniqueness. Finally, one needs to ensure solutions taking values in such spaces exist in the first place.




  
Organisation of the paper In Section <ref>, we first recall the method of intertwiners of <cit.> in a general setting. We then extend it to deal with convergence to equilibrium and most importantly we prove the uniform approximation theorem and its various consequences on consistent Feller processes. In Section <ref>, we establish various properties of the solution to the finite-dimensional SDE (<ref>) that allow us to apply the results of Section <ref>. We also briefly discuss a new Gibbs resampling property for (<ref>). In Section <ref>, we first show non-intersection of the limiting paths in Theorem <ref> and then obtain the ISDE in Theorem <ref>. In Section <ref>, we quickly apply our framework to a dynamical model for the Cauchy ensemble, upgrading the abstract results of <cit.>.



  
Acknowledgements TA is grateful to Duncan Dauvergne and Evgeni Dimitrov for a useful discussion regarding the Gibbs resampling property and to Kohei Suzuki for very informative discussions on the Dirichlet form approach to infinite dimensional diffusions with singular interaction. Part of this research was performed while TA was visiting the Institute for Pure and Applied Mathematics (IPAM), which is supported by the National Science Foundation (Grant No. DMS-1925919). ZSM is grateful to EPSRC for funding through a PhD studentship.
   
 

§ CONVERGENCE OF CONSISTENT MARKOV PROCESSES

The main purpose of this section is to establish a general result for path space convergence for a sequence of Markov processes related to random matrices  satisfying  consistency relations. 




 §.§ Method of intertwiners: Construction of the Feller process


We recall the method of intertwiners of <cit.>. Since there is a subtle difference in that we put the weak topology on spaces of probability measures instead of the topology induced by the total variation norm used in <cit.> we give a detailed exposition and some proofs. To set things up, we will need to collect several definitions and facts which can be found, for example, in <cit.>.

The measurable structures on topological spaces below are always understood to be the Borel σ-algebras. A Polish space 𝒳 is a separable completely metrisable topological space. A standard Borel space is a measurable space (𝒳,ℱ) such that there exists a metric on the base set so that the induced Borel σ-algebra is ℱ. A Borel isomorphism is a bi-measurable bijective function between two standard Borel spaces. 

We write ℳ_p(𝒳) for the set of Borel probability measures on a Polish space 𝒳. Unless otherwise stated, we always endow ℳ_p(𝒳) with the weak topology. This makes it a Polish space (and so a standard Borel space). Moreover, the Borel σ-algebra coincides with the σ-algebra generated by the evaluation maps μ↦μ(𝒜), for 𝒜 measurable (but the topology on the base set ℳ_p(𝒳) generated by these maps is different from the weak topology). 

For a Polish space 𝒳 denote by ℬ(𝒳) the space of real-valued bounded Borel functions on 𝒳. A Markov kernel 𝒦: 𝒳_1 →𝒳_2 from 𝒳_1 to 𝒳_2 is a function 𝒦(x,𝒜), where x∈𝒳_1 and 𝒜⊂𝒳_2 is a measurable subset, such that 𝒦(x,·) is a probability measure on 𝒳_2 and x↦𝒦(x,𝒜) is a measurable function on 𝒳_1. Observe that, a Markov kernel 𝒦: ℰ_1 →𝒳_2 can be viewed as a map on probability measures 𝒦:ℳ_p(𝒳_1)→ℳ_p(𝒳_2) or as a map on functions 𝒦:ℬ(𝒳_2) →ℬ(𝒳_1) in the obvious way.

For a locally compact Polish space 𝒳, we write C_0(𝒳) for the space of continuous (real-valued) functions vanishing at infinity, endowed with the supremum norm which makes it a Banach space.
 
 Let 𝒳_1 and 𝒳_2 be locally compact Polish spaces. A Markov kernel 𝒦:𝒳_1 →𝒳_2 is Feller if the induced map from ℬ(𝒳_2) to ℬ(𝒳_1) maps C_0(𝒳_2) to C_0(𝒳_1).


The following is our basic data. Suppose that we are given:

    
  * A sequence of locally compact Polish spaces (ℰ_N)_N=1^∞.
    
  * For each N∈ℕ, a Feller Markov kernel ℒ_N^N+1:ℰ_N+1→ℰ_N.



We would now like to associate a certain limit object to the above data. There are a number of (not always equivalent) ways of doing this <cit.>; see in particular Section 4.3.1 of <cit.> for the connection to entrance boundaries of Markov chains mentioned in the introduction. Below, we discuss the bare minimum we need for our applications.

To begin with, observe that for any N ∈ℕ, the affine maps ℒ_N^N+1:ℳ_p(ℰ_N+1) →ℳ_p(ℰ_N) are continuous. Hence, we can take the projective limit of the convex topological spaces ℳ_p(ℰ_N), which is defined as follows, see Section 3 of <cit.>,

    lim_←Topℳ_p(ℰ_N)def={(μ_N)_N=1^∞∈∏_N=1^∞ℳ_p(ℰ_N): μ_N+1ℒ_N^N+1=μ_N,  ∀ N ∈ℕ}.

This is endowed with the projective limit topology. It is in fact a Polish simplex (and in particular a standard Borel space), see Section 4 of <cit.>. The following terminology will be used throughout the paper.

We call any element of (<ref>) a consistent sequence of (probability) measures.



We then come to the following definition.

Suppose that ℰ_∞ is a locally compact Polish space, equipped with Feller Markov kernels ℒ_N^∞:ℰ_∞→ℰ_N satisfying, for all N ∈ℕ, ℒ_N+1^∞ℒ_N^N+1=ℒ_N^∞, so that the map,

    ℳ_p(ℰ_∞)    →lim_←Topℳ_p(ℰ_N)
    𝔪   ↦(𝔪ℒ_N^∞)_N=1^∞,

is a Borel isomorphism. Then, we say that ℰ_∞ is a Feller boundary of the system (ℰ_N,ℒ_N^N+1)_N=1^∞.


We will not be concerned with existence and uniqueness of such an object for general systems here. In related settings, abstract existence and uniqueness (in an appropriate measurable sense) results exist, see <cit.>. For our ultimate purposes of constructing dynamics on ℰ_∞, the above abstract setting is only useful if we have a nice explicit description for it and the kernels ℒ_N^∞. This is in general a difficult problem but in some cases it admits a good solution. This is the case for the concrete applications to random matrix dynamics we have in mind.

We can now prove the following lemma, which is essentially Lemma 2.3 of <cit.>.

Assume ℰ_∞ is a Feller boundary of (ℰ_N,ℒ_N^N+1)_N=1^∞. Then, the space 

    span(⋃_K=1^∞ℒ_K^∞C_0(ℰ_K))

is dense in C_0(ℰ_∞).



On a locally compact Polish space 𝒳, let ℳ(𝒳) denote the set of signed measures with finite total variation norm ||· ||_tv.
By the Riesz representation theorem the Banach space (ℳ(ℰ_∞),||· ||_tv) is the Banach dual of C_0(ℰ_∞). Hence, in order to prove the result it suffices to show that if μ∈ℳ(ℰ_∞) annihilates all functions in (<ref>) then necessarily μ = 0.
Observe that, this last statement makes no reference to the topology we put on ℳ(ℰ_∞). Assume that μ∈ℳ(ℰ_∞)  annihilates ℒ_N^∞ C_0(ℰ_N), for all N ∈ℕ. This is equivalent, again by the Riesz representation theorem applied to each C_0(ℰ_N), to μℒ_N^∞∈ℳ(ℰ_N) being 0, for all N ∈ℕ. We can write μ(·)=||μ_+||_tvμ_+(·)-||μ_-||_tvμ_-(·) for μ_+,μ_- ∈ℳ_p(ℰ_∞). Since μℒ_N^∞ =0, for all N ∈ℕ we obtain ||μ_+||_tv=||μ_-||_tv and thus μ_+ℒ_N^∞=μ_-ℒ_N^∞, for all N ∈ℕ. Since the map (<ref>) is a bijection we must have μ_+=μ_- and hence μ =0 as desired.


Before constructing Feller Markov dynamics on ℰ_∞, we need to recall the definition of a Feller semigroup, see <cit.>.


 Let 𝒳 be a locally compact Polish space.
A semigroup (𝖯_t)_t≥ 0 of positive contraction operators on C_0(𝒳) is called a Feller (or Feller-Dynkin) semigroup if: (i) 𝖯_t C_0(𝒳)⊂ C_0(𝒳), for all t≥ 0, (ii) 𝖯_tf(x)→ f(x), as t→ 0, for all f∈ C_0(𝒳), x∈𝒳. Associated to (𝖯_t)_t≥ 0 we have a unique Markov process, called a Feller process, on 𝒳.


Now, for each N∈ℕ, let (𝐗^(N)(t);t≥ 0) be a Feller process on ℰ_N with  associated semigroup (𝖯_N(t))_t≥ 0, and suppose these semigroups are consistent 
with (ℒ_N^N+1)_N=1^∞, namely, 

    𝖯_N+1(t)ℒ_N^N+1 = ℒ_N^N+1𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

The following result from <cit.>, essentially Proposition 2.4 therein, gives the existence of a Feller process 𝐗^(∞) on ℰ_∞ consistent with the 𝐗^(N). Because the main idea is beautifully simple and since there are some subtleties with measurability we give some details.


	Let ℰ_∞ be a Feller boundary of (ℰ_N,ℒ_N^N+1)_N=1^∞.  For all N∈ℕ, let (𝖯_N(t))_t≥ 0 be Feller semigroups satisfying (<ref>). Then, there exists a unique Feller Markov semigroup
	(𝖯_∞(t))_t≥ 0 on ℰ_∞, with associated process (𝐗^(∞)(t);t≥ 0), such that 
	
    𝖯_∞(t)ℒ_N^∞
    		= ℒ_N^∞𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

 


To construct the semigroup (𝖯_∞(t))_t≥ 0 we first need to define, for each t ≥ 0, x∈ℰ_∞, a probability measure 𝖯(t)(x,·) on ℰ_∞. Note that, this probability measure needs to satisfy,

    𝖯(t)(x,·)ℒ_N^∞(dy)=ℒ_N^∞𝖯_N(t)(x,dy).

Then, note that, because of (<ref>), and since ℒ_N^∞=ℒ^∞_N+1ℒ_N^N+1, we have

    (ℒ_N^∞𝖯_N(t)(x,·))_N=1^∞∈lim_←Topℳ_p(ℰ_N),

and by virtue of the fact that the map (<ref>) is a Borel isomorphism this defines a unique probability measure 𝖯(t)(x,·) on ℰ_∞ as desired. Observe also that,  since (<ref>) holds for each x∈ℰ_∞, it is equivalent to (<ref>). To see that 𝖯(t):ℰ_∞→ℰ_∞ is a genuine Markov kernel we first observe that the map ℰ_∞→ℳ_p(ℰ_∞) given by x↦𝖯(t)(x,·) is measurable. This is a consequence of the following facts: the map ℰ_∞→ℳ_p(ℰ_∞) which takes x to its delta measure is continuous, the map (<ref>) is a Borel isomorphism and so bi-measurable and the fact that, for each t≥ 0, the map,

    lim_←Topℳ_p(ℰ_N)    →lim_←Topℳ_p(ℰ_N) 
    
    (μ_N)_N=1^∞   ↦(μ_N𝖯_N(t))_N=1^∞,

is continuous, since for each N ∈ℕ, (𝖯_N(t))_t≥ 0 is Feller. Hence, to conclude that, for 𝒜 measurable, x↦𝖯(t)(x,𝒜) is a measurable function, it suffices to observe that the evaluation maps (for measurable 𝒜) ℳ_p(ℰ_∞) → [0,1] given  by μ↦μ(𝒜) are measurable; recall the σ-algebra generated by these maps is the same as the Borel σ-algebra (but the induced topology on the set ℳ_p(ℰ_∞) is different from the weak topology). The semigroup property then follows by virtue of (<ref>) and the semigroup property of the (𝖯_N(t))_t≥ 0. Finally, the Feller continuity property, boils down, by virtue of Lemma <ref>, to the Feller property of the ℒ_N^∞ and (𝖯_N(t))_t≥ 0. See <cit.> for more details.


Note that, the way we obtained (𝖯_∞(t))_t≥ 0 is non-constructive. Thus a direct description of its dynamics becomes a whole separate task. On the other hand, we have obtained a-apriori the highly non-trivial Feller continuity property, which especially when ℰ_∞ turns out to be infinite-dimensional is a major achievement. We also remark that it is easy to see that we can  obtain invariant measures for (𝖯_∞(t))_t≥ 0 from invariant measures for the (𝖯_N(t))_t≥ 0, see <cit.>. Since we will prove a stronger statement in Theorem <ref> below we do not make this explicit. We now move to some important definitions.



 Let 𝒳 be a locally compact Polish space and let (𝖯_t)_t≥ 0 be a Feller semigroup on C_0(𝒳). Its infinitesimal generator 𝒜 is defined to be,
 
    𝒜f=lim_t↓ 0t^-1(𝖯_t f-f),

 whenever the limit exists in C_0(𝒳).
 The domain of 𝒜, that we denote throughout the paper by 𝒟(𝒜), is the set of f∈ C_0(𝒳) for which the limit exists.



In the setting of Theorem <ref>, we denote by 
𝖫_N and 𝖫_∞ the generators, with domains 𝒟(𝖫_N)⊂ C_0(ℰ_N) and 𝒟(𝖫_∞)⊂ C_0(ℰ_∞), associated to the Feller processes (𝐗^(N)(t);t≥ 0) and (𝐗^(∞)(t);t≥ 0) respectively. The following standard definition will be useful.




    
    Let 𝒜 be a Feller generator with domain 𝒟(𝒜). 
A subspace 𝒞⊆𝒟(𝒜) is
    called a core of 𝒜 if the closure of 𝒜|_𝒞 coincides with 𝒜, where, by 𝒜|_𝒞 we mean restriction of 𝒜 to 𝒞.
    
    The generator is clearly uniquely determined by its action on a core. Then, the following very useful lemma, see Proposition 5.2 of <cit.> for a proof, provides a core for 𝖫_∞.

    
    
    In the setting of Theorem <ref>, for each K∈ℕ suppose 𝒞_K⊆𝒟(𝖫_K) is a core for the generator 𝖫_K. The subspace 𝒞_∞⊆ C_0(ℰ_∞) spanned by all subspaces of the form ℒ_K^∞𝒞_K,  K = 1, 2, …, namely

    𝒞_∞def=span(⋃_K=1^∞ℒ_K^∞𝒞_K)

is then a core for the generator 𝖫_∞.




 §.§ Method of intertwiners: Convergence to equilibrium



We now extend the formalism of the method of intertwiners to also deal with convergence to equilibrium for the process on the boundary. The following result, despite its simplicity, appears to be new. Let C_b(𝒳) denote the space of (real-valued) bounded continuous functions on a locally compact Polish space 𝒳.


 
     In the setting of Theorem <ref>, suppose (μ_N)_N=1^∞∈lim_←Topℳ_p(ℰ_N) with  corresponding boundary measure μ_∞∈ℳ_p(ℰ_∞) uniquely determined by,
     
    μ_∞ℒ_N^∞=μ_N,   ∀ N∈ℕ.

     Moreover, suppose that for all (ν_N)_N=1^∞∈lim_←Topℳ_p(ℰ_N), we have
       
    ν_N𝖯_N(t)(f)t→∞⟶μ_N(f),  ∀ f∈ C_b(ℰ_N),  ∀ N ∈ℕ.
  
       Then, for any ν_∞∈ℳ_p(ℰ_∞), we have 
       
    ν_∞𝖯_∞(t)(f)t→∞⟶μ_∞(f),  ∀ f∈ C_b(ℰ_∞).
 
       In particular, μ_∞ is the unique invariant measure of (𝖯_∞(t))_t≥ 0.
    
    
    We show (<ref>) holds for all functions in
𝒞=⋃_N=1^∞ℒ_N^∞C_0(ℰ_N). Observe that, this implies (<ref>) for all f∈ C_0(ℰ_∞) as, by virtue of Lemma <ref>, 𝒞 is a dense subset of C_0(ℰ_∞). This is vague convergence, which together with the fact that the limiting measure μ_∞ is a probability measure, by a standard argument upgrades it to convergence in distribution which proves the theorem.
        
        Fix N∈ℕ and let f_N∈ C_0(ℰ_N). We have using the intertwining relation (<ref>) that
        
    ν_∞𝖯_∞(t)ℒ_N^∞(f_N)=ν_∞ℒ_N^∞𝖯_N(t)(f_N),  ∀ t≥ 0.

        Now, set ν_N=ν_∞ℒ_N^∞. Since (ν_N)_N=1^∞∈lim_←Topℳ_p(ℰ_N), by our assumption ν_N𝖯_N(t)(f_N)→μ_N(f_N) as t→∞. Hence, 
       
    ν_∞ℒ_N^∞𝖯_N(t)(f_N)→μ_N(f_N),   t→∞.

       Recall that μ_N(f_N)=μ_∞(ℒ_N^∞f_N), and thus, display  (<ref>) follows. Then, the fact that μ_∞ is the unique invariant measure follows by a standard argument.
    




 §.§ Projective system from unitarily invariant Hermitian matrices


From now on we focus on two specific projective systems that are related to random matrices. They form the basic integrable structure behind our problem. They come from the study of unitarily invariant measures on infinite Hermitian matrices, see <cit.> for background. We will need several definitions and preliminary results. For the most part, the material in this subsection is standard and has appeared in equivalent forms in <cit.>.

For any N∈ℕ, let 𝕎_N be the Weyl chamber,

    𝕎_N = {𝐱=(x_1,x_2,…,x_N)∈ℝ^N: x_1≥ x_2 ≥…≥ x_N},

The interior of 𝕎_N and 𝕎_N,+ from (<ref>) will be denoted by 𝕎_N^∘ and 𝕎_N,+^∘ respectively.

 For each N∈ℕ, let 
 ℍ(N) denote the space of N× N Hermitian matrices, and ℍ_+(N)⊂ℍ(N) the subspace of non-negative definite ones.
 
 
 We define the eigenvalue maps
 
    𝖾𝗏𝖺𝗅_N : ℍ(N)    →𝕎_N,
    𝐀   ↦𝐱=(x_1≥ x_2 ≥…≥ x_N),

 where 𝐱=(x_1≥ x_2 ≥…≥ x_N) are the ordered eigenvalues of the matrix 𝐀∈ℍ(N). Observe that, 𝖾𝗏𝖺𝗅_N restricts to a map from ℍ_+(N) to 𝕎_N,+.
 
 
 
 We now let, for each N ∈ℕ, 

    Π_N^N+1:ℍ(N+1)   →ℍ(N),
    [𝐀_ij]_i,j=1^N+1   ↦[𝐀_ij]_i,j=1^N,

be the so-called corners map. Also, observe that, Π_N^N+1 restricts to a map from ℍ_+(N+1) to ℍ_+(N). Let 𝕌(N) be the space of N× N unitary matrices. We will denote by 𝐀^† the complex conjugate of a matrix 𝐀. The following definition is one of our basic building blocks.



For any N∈ℕ, we define

    Λ_N^N+1(𝐱,·)def=𝖫𝖺𝗐[𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐔𝖽𝗂𝖺𝗀(𝐱)𝐔^†))],  ∀𝐱∈𝕎_N+1,
   
where 𝐔 is Haar-distributed on 𝕌(N) and 𝖽𝗂𝖺𝗀(𝐱) indicates the diagonal matrix with entries 𝐱.  More generally, we define the composition
 	
    Λ_K^Ndef=Λ_N-1^NΛ_N-2^N-1⋯Λ_K^K+1,  ∀ N>K≥ 1.
 





We write Δ_N for the Vandermonde determinant given by, for 𝐱∈ℝ^N,  
 
    Δ_N(𝐱)=∏_1≤ i<j≤ N (x_i-x_j).

When 𝐱∈𝕎_N+1 and 𝐲∈𝕎_N, 𝐲≺𝐱 denotes interlacing:
	x_1≥ y_1≥…≥ y_N≥ x_N+1. 
It is a classical result by Baryshnikov <cit.> 
 (in fact the computation is implicit in <cit.>, see also <cit.> for an alternative proof) that the following holds. This explicit formula will be useful in the sequel.


 For N∈ℕ, we have
	
    Λ_N^N+1(𝐱,d𝐲)=N!Δ_N(𝐲)/Δ_N+1(𝐱)1_{𝐲≺𝐱}d𝐲,  ∀𝐱∈𝕎_N+1^∘.
 


As expected these Markov kernels are Feller.

 
      Let N>K≥ 1. The Markov kernel Λ_K^N is Feller, that is, for all f∈ C_0(𝕎_K), we have Λ_K^Nf∈ C_0(𝕎_N). Similarly, for all f∈ C_0(𝕎_K,+), we have Λ_K^Nf∈ C_0(𝕎_N,+).
 
 
     The statement follows from Lemma 2.5 of <cit.>, and using the fact that the composition of Feller kernels is Feller. 
 

We now define the infinite dimensional space Ω, an analogue of the space Ω_+ from the introduction, that will allow in the sequel to also consider negative coordinates.


We define the space Ω as follows:

    Ωdef={ω=(𝐱^+,𝐱^-,γ,δ)∈𝕎_∞,+×𝕎_∞,+×ℝ×ℝ_+:
    		∑_i=1^∞(x_i^+)^2+∑_i=1^∞(x_i^-)^2≤δ},
 
endowed with the topology of coordinate-wise convergence. 

 Observe that, Ω like Ω_+ is locally compact, metrizable and separable. Its topology can be metrized with  the following metric 𝖽_Ω, with ω=(𝐱^+,𝐱^-,γ,δ)

    𝖽_Ω(ω,ω̃)=∑_i=1^∞|x_i^+-x̃_i^+|/2^i(1+|x_i^+-x̃_i^+|)+∑_i=1^∞|x_i^–x̃_i^-|/2^i(1+|x_i^–x̃_i^-|)+|γ-γ̃|+|δ-δ̃|.

We note moreover that Ω_+ can be embedded into Ω as follows:

    (𝐱,γ)↦(𝐱,0,γ,∑_j=1^∞ x_j^2),

and observe that that this is a homeomorphism. This embedding is the natural embedding of Ω_+ into Ω and this will be clearer in the sequel.

We now embed 𝕎_N and 𝕎_N,+ into Ω and Ω_+ respectively, in a natural way (again the fact that this is a reasonable thing to do will be clear in what follows).




For each N∈ℕ, we embed the space 𝕎_N into Ω as follows, with 𝐱^(N)∈𝕎_N,
 
    𝐱^(N)↦((N^-1max{x_i^(N),0})_i=1^∞,(N^-1max{-x_N+1-i^(N),0})_i=1^∞,N^-1∑_i=1^Nx_i^(N) ,N^-2∑_i=1^N (x_i^(N))^2),

 where by convention x_i^(N)≡ 0 whenever i∉ 1, N. We denote this embedded point by ω(𝐱^(N)).
Similarly, we can define an embedding of 𝕎_N,+ into Ω_+ as follows, with 𝐱^(N)∈𝕎_N,+,

    𝐱^(N)↦((N^-1x_i^(N))_i=1^∞,N^-1∑_i=1^N x_i^(N)).

Abusing notation, we denote this embedded point in Ω_+ by ω(𝐱^(N)) again.

 
Note that, the embedding of 𝕎_N,+ into Ω_+ matches the embedding of the stochastic dynamics from the introduction.

Now, let 
 ℍ(∞) denotes the space of infinite Hermitian matrices, and write ℍ_+(∞) for the space of non-negative definite infinite Hermitian matrices (defined as the projective limits of the spaces (ℍ(N))_N=1^∞ and (ℍ_+(N))_N=1^∞ respectively under the corners maps Π_N^N+1; we do not make this more precise as we will only need to work with their finite-dimensional projections). Define the map,

    Π_N^∞:ℍ(∞)   →ℍ(N),
    [𝐀_ij]_i,j=1^∞   ↦[𝐀_ij]_i,j=1^N.

Observe that this restricts to a map from ℍ_+(∞) to ℍ_+(N). 

The following definition is important.


 
 
For ω∈Ω, define the infinite Hermitian random matrix  
 
    𝐀_ω^∞def=γ𝐈 + √(δ-∑_i=1^∞[(x_i^+)^2+(x_i^-)^2])𝐆+∑_j=1^∞x_j^+( ξ(j)ξ(j)^†-𝐈) - ∑_j=1^∞x_j^-(ζ(j)ζ^†(j)-𝐈),

where, 𝐆 is an infinite Gaussian Hermitian matrix, namely, 

    𝐆_iid=𝒩(0,1),  ∀ i∈ℕ,  ( 𝐆_ij), ( 𝐆_ij)d=𝒩(0,1/2),  ∀ i<j,

with all independent entries subject to the Hermitian constraint and where 𝒩(0,σ^2) denotes the centered Gaussian random variable of variance σ^2. Moreover, for any j∈ℕ
the ξ(j) and ζ(j) are infinite vectors of i.i.d. (independent, identically distributed) 
 standard complex Gaussian random variables (real and imaginary parts are independent normally distributed random variables with mean zero and variance 1/2).

For ω∈Ω_+, we defined, abusing notation,

    𝐀_ω^∞def=(γ-∑_j=1^∞x_j)𝐈 +∑_j=1^∞x_jξ(j)ξ^†(j),

 where, as before, (ξ(j))_j=1^∞ is a sequence of infinite vectors of i.i.d. complex Gaussian random variables.
 

 We note that the definition of the matrices 𝐀_ω^∞, for ω∈Ω_+, is consistent with viewing ω∈Ω under the embedding (<ref>). We also note that we will never really need to consider the 𝐀_ω^∞ as infinite matrices but rather only consider them through their K× K top-left corner projections Π_K^∞(𝐀_ω^∞). We can now define a family of Markov kernels that play a key role in our argument.


For any K∈ℕ, we define, with (abusing notation) either ω∈Ω or ω∈Ω_+,
  
    Λ_K^∞(ω,·)def=𝖫𝖺𝗐[𝖾𝗏𝖺𝗅_K(Π_K^∞(𝐀_ω^∞))].

    


The following is easy to see.


For any K∈ℕ and ω∈Ω or ω∈Ω_+, Λ_K^∞(ω,·) is a probability measure on 𝕎_K or 𝕎_K,+ respectively. Moreover, we have
    
    Λ_N^∞Λ_K^N = Λ_K^∞,  ∀ N>K.



This first statement is immediate from the definition and the second statement follows by unitary-invariance by virtue of Baryshnikov's result <cit.>.





The kernels Λ_K^∞ are Markov and in fact Feller as expected.

 
    Let K∈ℕ. Then, we have
    
    Λ_K^∞f   ∈ C_0(Ω),    ∀ f∈ C_0(𝕎_K),
    Λ_K^∞f   ∈ C_0(Ω_+),  ∀ f∈ C_0(𝕎_K,+).
 
In particular, the kernels are Feller Markov.
 
 Continuity of ω↦Λ_N^∞(ω,d𝐲), in the sense of weak convergence, was proven in <cit.> (see the proof of Proposition 3.5 therein). From this we also obtain the measurability of ω↦Λ_K^∞(ω,𝒜) for measurable 𝒜. It remains to prove the vanishing at infinity property. In the case of Ω this was shown in <cit.>. We give a more direct argument in the case of Ω_+ (an adaptation will work for Ω as well). Let f∈ C_0(𝕎_K,+) be arbitrary. 
We need to show that for any sequence (ω^(n))_n=1^∞ in Ω_+, with ω^(n)=(𝐱^(n),γ^(n)), such that ω^(n)n→∞⟶∞ (which means γ^(n)→∞), we have (Λ_K^∞f)(ω^(n))n→∞⟶ 0.
To do so, it suffices to show that when γ^(n)n→∞⟶∞, the largest eigenvalue of Π_K^∞(𝐀_ω^(n)^∞) converges to infinity in distribution. Since the eigenvalues of sub-matrices of different orders interlace (Cauchy interlacing theorem), we only need to prove the K=1 case. Note that, from the explicit formula (<ref>), the corresponding random variable 𝗑_n is simply
 
    𝗑_ndef=γ^(n)-∑_j=1^∞x_j^(n) + ∑_j=1^∞Γ_x_j^(n),

 where each Γ_θ denotes a gamma random variable with Laplace transform z↦ (1+θ z)^-1 and they are all independent. Thus, we want to show that 
 𝗑_nn→∞⟶∞
 when γ^(n)n→∞⟶∞. We have two possibilities; either ∑_j=1^∞x_j^(n) remains bounded, in which case the result follows immediately, otherwise, ∑_j=1^∞x_j^(n)n→∞⟶∞  which implies, ∑_j=1^∞Γ_x_j^(n)⟶∞ in distribution. This last claim can easily be seen by virtue of the fact that, for any z,R≥ 0, we have,
 
    ℙ(∑_j=1^∞Γ_x_j^(n)≤ R)   ≤𝔼[exp(-z∑_j=1^∞Γ_x_j^(n))] e^zR≤e^zR/1+z∑_j=1^∞ x_j^(n).

 This completes the proof. 



The following fundamental result is due to Pickrell <cit.> and Olshanski and Vershik <cit.>, in equivalent form (modulo the Feller property of the kernels). In rather less obvious equivalent form, it can be traced back to the work of Schoenberg <cit.> on totally positive functions, see <cit.>. It was translated to the setting of projective systems in <cit.>. A proof, different from <cit.>, directly in this setup and a generalisation is given in <cit.>.


The space Ω is a Feller boundary of (𝕎_N,Λ_N^N+1)_N=1^∞.


The following result is again due to Olshanski and Vershik in equivalent form (modulo the Feller property of the kernels), see Remark 2.11 in <cit.>, .


 The space Ω_+ is a Feller boundary of (𝕎_N,+,Λ_N^N+1)_N=1^∞.


The uniform approximation theorem that we state below, could also be used to prove the results above, by following the argument in Section 3 of <cit.>, modulo an independent proof (we cannot use Lemma <ref> as the argument would be circular) of the density of ∪_N Λ_N^∞ C_0(𝕎_N) and ∪_N Λ_N^∞ C_0(𝕎_N,+) in C_0(Ω) and C_0(Ω_+) respectively. In the combinatorial setting of the Gelfand-Tsetlin graph this is not too difficult, see Section 2.7 in <cit.>, but in the continuous setting there are additional subtleties. We will not pursue this here.




 §.§ Uniform Approximation theorem
 



 
Let (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞ or (𝕎_N,+)_N=1^∞. Let C_c^∞
(ℝ^K) denote the space of (real-valued) smooth and compactly supported functions on ℝ^K. It will be very convenient in what follows to consider the symmetric extension of the kernels given in (<ref>) and (<ref>) to ℝ^K, which we denote by the same notation. Thus, both Λ_K^N(𝐱,d𝐲) and Λ_K^∞(ω,d𝐲) can be viewed as positive measures on ℝ^K with total mass K! (we choose not to normalise them to be probability measures as it will be more convenient) and can act on functions defined on ℝ^K (this will be clear from context). The following is the main technical result of this part of the paper.
    
 Let K∈ℕ.
For all g∈ C_c^∞(ℝ^K),
we have 

    sup_𝐱^(N)∈ℰ_N|Λ_K^∞g(ω(𝐱^(N)))-Λ_K^Ng(𝐱^(N))|→ 0,
      as  N→∞.


The proof of this theorem will be given in Subsection <ref>. The following function spaces will appear frequently below.

Define C_c,sym^∞(ℝ^K) to be the space of smooth compactly supported functions on ℝ^K that are moreover symmetric in their variables. Then, we define 
C_c,sym^∞(ℰ_K) to be the space of functions on ℰ_K which are restrictions of functions C_c,sym^∞(ℝ^K) to ℰ_K.  


Observe that, by definition
 
    ∫_ℰ_Kg(𝐲)Λ_K^N(𝐱,d𝐲)   =
         1/K!∫_ℝ^Kg(𝐲)Λ_K^N(𝐱,d𝐲),
    ∫_ℰ_Kg(𝐲)Λ_K^∞(ω,d𝐲)   =
         1/K!∫_ℝ^Kg(𝐲)Λ_K^∞(ω,d𝐲),   ∀ g∈ C_c,sym^∞(ℝ^K).

 We have the following corollary of Theorem <ref>.
 
Let K ∈ℕ. For all g ∈ C_c,sym^∞(ℰ_K), we have

    sup_𝐱^(N)∈ℰ_N|∫_ℰ_Kg(𝐲)Λ_K^∞(ω(𝐱^(N)),d𝐲)-∫_ℰ_Kg(𝐲)Λ_K^N(𝐱^(N),d𝐲)|→ 0,
      as  N→∞.

 




 §.§ Approximation of Markov processes


In this subsection we show how using the uniform approximation theorem above and an underlying intertwining we can prove a path-space convergence result to the abstract Feller process associated to (𝖯_∞(t))_t≥ 0. 

To be precise, the running assumption is the following. In the setting of Subsection <ref>, let (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞ or (𝕎_N,+)_N=1^∞ and 
(ℒ_N^N+1)_N=1^∞=(Λ_N^N+1)_N=1^∞. Namely, we consider the systems (𝕎_N,Λ_N^N+1)_N=1^∞ and (𝕎_N,+,Λ_N^N+1)_N=1^∞, 
with Λ_N^N+1 given by (<ref>).  As we have seen in Theorems <ref> and <ref>, the Feller boundaries of these systems are given by Ω and Ω_+ respectively. 



We moreover consider, for each N∈ℕ, a Feller Markov process (𝔵^(N)(t);t≥ 0), with continuous sample paths,
on ℰ_N with transition semigroup (𝖯_N(t))_N=1^∞ and we assume that these semigroups are intertwined:

    𝖯_N+1(t)Λ_N^N+1 = Λ_N^N+1𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

Let (𝐗_t^Ω;t≥ 0) and
(𝐗_t^Ω_+;t≥ 0)
be the corresponding Feller processes on Ω and Ω_+ constructed as in Theorem <ref>,  respectively. We denote the Feller semigroup of the process on the boundary, abusing notation, for either Ω or Ω_+, by (𝖯_∞(t))_t≥ 0.
Moreover, recall that we have 

    𝖯_∞(t)Λ_N^∞
    		= Λ_N^∞𝖯_N(t),  ∀ t≥ 0, ∀ N∈ℕ.

 
We need two abstract definitions that we will specialise to our setting shortly. 


    Suppose (ℒ_n)_n=1^∞ is a sequence of Banach spaces, with the norm of ℒ_n denoted by ·_n. Let also ℒ be another Banach space with norm denoted by ·. 
    We say that a sequence of vectors (𝐟_n)_n=1^∞ with 𝐟_n∈ℒ_n approximates a vector 𝐟∈ℒ, and we write 𝐟_n→𝐟, if and only if there exist bounded linear operators π_n: ℒ→ℒ_n, satisfying sup_n∈ℕ|π_n|_n<∞, where |·|_n denotes the operator norm of linear operators from ℒ to ℒ_n, such that,
    
    lim_n→∞𝐟_n-π_n 𝐟_n=0.





Let 𝒜_N and 𝒜_∞ be the generators (defined in an analogous way to Definition <ref>) corresponding to strongly continuous semigroups of contractions on the Banach spaces ℒ_N and ℒ_∞, respectively. We say that 𝒜_∞ is approximated by (𝒜_N)_N=1^∞
  if for any 𝐟 in a core (defined in an analogous way to Definition <ref>) of 𝒜_∞, there exist a sequence  𝐟_N∈𝒟(𝒜_N) such that 𝐟_N→𝐟 and 𝒜_N𝐟_N →𝒜_∞𝐟.





The following proposition is key for our purposes. Here, we take the Banach spaces to be ℒ_N=C_0(ℰ_N) and ℒ=C_0(ℰ_∞) and the implicit maps π_N: C_0(ℰ_∞) → C_0(ℰ_N) to be the ones induced by embeddings of ℰ_N into ℰ_∞, namely (<ref>) or (<ref>).



In the setting described above, we let 𝖫_N and 𝖫_∞ denote the generators of the Feller semigroups  (𝖯_N(t))_t≥ 0 and (𝖯_∞(t))_t≥ 0, respectively. Let 𝒞_N=C_c,sym^∞(ℰ_N) and 𝒞_∞ be as in (<ref>). Moreover, assume that, for all N ∈ℕ, 𝒞_N⊆𝒟(𝖫_N) and 𝖫_N𝒞_N ⊂𝒞_N. Then, for any 𝐟∈𝒞_∞, there exists a sequence (𝐟_N)_N=1^∞ with 𝐟_N∈𝒞_N 
such that 𝐟_N→𝐟 and 𝖫_N𝐟_N →𝖫_∞𝐟.


An immediate consequence of Proposition <ref> and Lemma <ref> is the following result.

 
    In the setting of Proposition <ref>, suppose moreover, that for each N∈ℕ,
    𝒞_N
    is a core for 𝖫_N. Then, 𝖫_∞ is approximated by  𝖫_N in the sense of Definition <ref>.


The two main ingredients to prove Proposition <ref> is
Theorem <ref>, in the form of Corollary <ref>, and the intertwining relation (<ref>). Before giving the proof we obtain as an immediate consequence the convergence of the corresponding Markov dynamics on path space.



In the setting described above, let ℰ_N=𝕎_N and 
(𝐗^(N)_t; t≥ 0) be the process on Ω corresponding to the Feller process (𝔵^(N)(t); t≥ 0), having continuous trajectories, with transition semigroup (𝖯_N(t))_t≥ 0,
under the embedding (<ref>) of 𝕎_N into Ω. Suppose moreover that, for each N ∈ℕ, 𝒞_N=C_c,sym^∞(𝕎_N) 
is a core, and invariant under 𝖫_N, the generator of (𝖯_N(t))_t≥ 0. Then, if 𝐗^(N)_0dN→∞⟶𝐗^Ω_0, we have that,

    𝐗^(N)dN→∞⟶𝐗^Ω,   in  C(ℝ_+,Ω).

   
        Using Corollary <ref> (recall that the implicit maps π_N therein are induced by the embeddings (<ref>) of 𝕎_N into Ω), it follows from <cit.>[Chapter 4, Theorem 2.11] that 𝐗^(N)⟶𝐗^Ω, in distribution, in D(ℝ_+,Ω)
        the space of càdlàg functions on ℝ_+ with values in Ω endowed with the Skorokhod topology, see <cit.>. 
       The desired result is then concluded from the well-known fact that the space C(ℝ_+,Ω) is closed in D(ℝ_+,Ω) in the Skorokhod topology and recalling that, for each N∈ℕ, 𝐗^(N)∈ C(ℝ_+,Ω) .
   
   

An analogous statement holds when the processes live in (𝕎_N,+)_N=1^∞. The proof is the same.


 
In the setting described above, let ℰ_N=𝕎_N,+ and 
(𝐗^(N)_t; t≥ 0) be the process on Ω_+ corresponding to the Feller process (𝔵^(N)(t); t≥ 0), having continuous trajectories, with semigroup (𝖯_N(t))_t≥ 0
under the embedding (<ref>) of 𝕎_N,+ into Ω_+. Suppose moreover that, for each N ∈ℕ, 𝒞_N=C_c,sym^∞(𝕎_N,+) 
is a core, and invariant under 𝖫_N, the generator of (𝖯_N(t))_t≥ 0. Then, if 𝐗^(N)_0dN→∞⟶𝐗^Ω_+_0, we have that,

    𝐗^(N)dN→∞⟶𝐗^Ω_+,   in  C(ℝ_+,Ω_+).

  

  Let us write 𝐗_·^Ω_+=((𝗑_i(·))_i=1^∞,γ(·)) below for the process on Ω_+ in coordinates and similarly 𝐗_·^(N)=((𝗑_i^(N)(·))_i=1^∞,γ^(N)(·)) for the embedded process (recall γ^(N)=∑𝗑_i^(N)).
  
  
Observe that, Theorem <ref>, in particular implies that

    𝗑_i^(N)dN→∞⟶𝗑_i,  in  C(ℝ_+,ℝ_+),   ∀ i ∈ℕ,   and ∑_i=1^N𝗑_i^(N)dN→∞⟶γ,  in  C(ℝ_+,ℝ_+).

  

  
  
    In the setting of Theorem <ref>, there exists a coupling of the 𝐗^(N) and 𝐗^Ω_+ on a single probability space such that, almost surely, for any T ≥ 0,
    
    sup_t∈[0,T]∑_i=1^∞(𝗑_i^(N)(t)-𝗑_i(t))^2 N→∞⟶ 0.





     
    We use the Skorokhod representation theorem to couple 𝐗^(N) and 𝐗^Ω_+ on a single probability space, and, abusing notation, denote the new processes by the same symbol. One can then write for all M∈ℕ,
    
    sup_t∈[0,T]∑_i=1^∞(𝗑_i^(N)(t)-𝗑_i(t))^2≤     
    2sup_t∈[0,T]∑_i=M+1^∞(𝗑_i^(N)(t))^2+2sup_t∈[0,T]∑_i=M+1^∞(𝗑_i(t))^2
       +sup_t∈[0,T]∑_i=1^M(𝗑_i^(N)(t)-𝗑_i(t))^2.

    Let ε>0 be given. Observe first that as sup_t∈[0,T]∑_i=1^∞(𝗑_i(t))^2<∞, there exists M_0∈ℕ, such that for all M≥ M_0, 
    
    2sup_t∈[0,T]∑_i=M+1^∞(𝗑_i(t))^2≤ε/4.

    Then, using Theorem <ref> we have 
    
    sup_t∈[0,T]∑_i=1^M_0(𝗑_i^(N)(t)-𝗑_i(t))^2≤ε/4,

    for large enough N. 
    Note moreover that,
    
    sup_t∈[0,T]∑_i=M_0+1^∞(𝗑_i^(N)(t))^2≤sup_t∈[0,T]𝗑_M_0^(N)(t)sup_t∈[0,T]∑_i=M_0^∞𝗑_i^(N)(t).

Now, from (<ref>), we have 
 
    lim_N→∞sup_t∈[0,T]𝗑_M_0^(N)(t)=sup_t∈[0,T]𝗑_M_0(t),  sup_N∈ℕsup_t∈[0,T]∑_i=M_0^∞𝗑_i^(N)(t)def=c<∞.

    Hence, one can pick a possibly larger M_0 such that sup_t∈[0,T]𝗑_M_0(t)<ε/4c also holds, and thus,
    
    lim sup_N ≥ 1sup_t∈[0,T]∑_i=M_0+1^∞(𝗑_i^(N)(t))^2≤ε/4.

    Putting everything together, we conclude (<ref>).
    



    
   When considering processes on 𝕎_N and Ω instead, a very similar argument can be used to show the following convergence in ℓ^3 as a consequence of Theorem <ref>. Below we write 𝐗_·^Ω=((𝗑^+_i(·))_i=1^∞,(𝗑^-_i(·))_i=1^∞,γ(·),δ(·))  for the process on Ω in coordinates and similarly 𝐗_·^(N)=((𝗑^(N),+_i(·))_i=1^∞,(𝗑^(N),-_i(·))_i=1^∞,γ^(N)(·),δ^(N)(·)) for the embedded process.



   
   
    In the setting of Theorem <ref>,  there exists a coupling of the 𝐗^(N) and 𝐗^Ω on a single probability space such that, almost surely, for any T ≥ 0,
    
    sup_t∈[0,T](∑_i=1^∞|𝗑_i^(N),+(t)-𝗑_i^+(t)|^3+∑_i=1^∞|𝗑_i^(N),-(t)-𝗑_i^-(t)|^3)N→∞⟶ 0.



We now prove Proposition <ref>.


   
   We prove the proposition in the case (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞. The exact same argument works for (ℰ_N)_N=1^∞=(𝕎_N,+)_N=1^∞.
   

We then take the Banach spaces (ℒ_N)_N=1^∞=(C_0(𝕎_N))_N=1^∞ and ℒ=C_0(Ω) in the Definition <ref>.
    Considering the definition of 𝒞_∞ from Lemma <ref>,  it only suffices to prove the statement for functions of the form 𝐟=Λ_K^∞g  with some K∈ℕ and g∈𝒞_K.
       We thus fix a positive integer K, and take an arbitrary function g∈𝒞_K. For any N>K, let 
       𝐟_Ndef=Λ_K^Ng and 𝐟def=Λ_K^∞g. Observe that, by the Feller property of the kernels, 
       𝐟_N∈ C_0(𝕎_N) and 𝐟∈ C_0(Ω), respectively. We want to show that 
       
    𝐟_N→𝐟,   as  N→∞,

        in the sense of Definition <ref>. For each N∈ℕ, we then define the operator π_N:C_0(Ω)→ C_0(𝕎_N) as follows
       
    (π_Nf)(𝐱^(N))=f(ω(𝐱^(N))),  𝐱^(N)∈𝕎_N,

       where by ω(𝐱^(N)) we mean the embedding given in (<ref>). It is easy to see  that the operator norm |π_N|_N≤ 1, uniformly in N, and so the conditions given in Definition <ref> are satisfied. We thus need to check that 
       
    π_N𝐟-𝐟_N_N=sup_𝐱^(N)∈𝕎_N|(π_N𝐟)(𝐱^(N))-𝐟_N(𝐱^(N))|N→∞⟶ 0.

       Note that this is exactly the expression in (<ref>) from Corollary <ref>, and so (<ref>) is verified. We next want to show that
              
    𝖫_N𝐟_N→𝖫_∞𝐟,   as  N→∞,

       also holds, that is, 
            
    π_N(𝖫_∞𝐟)-𝖫_N𝐟_N_N→ 0,   as  N→∞.

       
    First, it follows from the intertwining relations (<ref>) and (<ref>) between the semigroups that for all f∈𝒟(𝖫_K), 
    
    𝖫_NΛ_K^Nf   =Λ_K^N𝖫_Kf,  ∀ N>K,
    𝖫_∞Λ_K^∞f   =Λ_K^∞𝖫_Kf.

    In particular, one can check that

    𝐟_N∈𝒟(𝖫_N). We can then write
    
    π_N(𝖫_∞𝐟)-𝖫_N𝐟_N_N=
             π_N(𝖫_∞Λ_K^∞g)-𝖫_NΛ_K^Ng_N
             =π_N(Λ_K^∞𝖫_Kg)-Λ_K^N𝖫_Kg_N.

       Note moreover that, by assumption, the space 𝒞_K is invariant under the action of 𝖫_K  and thus 𝖫_Kg∈𝒞_K. Hence, (<ref>) follows exactly in the same way as (<ref>) from Corollary <ref>. 
   


 §.§ Proof of the uniform approximation theorem


We only consider the case (ℰ_N)_N=1^∞=(𝕎_N)_N=1^∞ throughout this section. It is not hard to see, by inspecting (<ref>) and the relevant definitions carefully, that in the case (ℰ_N)_N=1^∞=(𝕎_N,+)_N=1^∞ we are simply taking the supremum over 𝕎_N,+ instead of 𝕎_N while the quantity we are taking the supremum of is exactly the same as for 𝕎_N. Thus, the result for 𝕎_N,+ is a special case of the one over 𝕎_N. We need some preliminaries. 




  §.§.§ Explicit formulae of Markov kernels

We first present an explicit formula for Λ_K^N(𝐱^(N),d𝐲) which is the starting point in our analysis. For any N≥ 2 and 𝐱^(N)∈𝕎_N^∘, we consider the fundamental spline function ℳ(y;𝐱^(N)):ℝ→ℝ, with knots 𝐱^(N), given by 

    ℳ(y;𝐱^(N))= (N-1)∑_i=1^N
            (x_i^(N)-y)_+^N-2/∏_j=1,j≠ i^N(x_i^(N)-x_j^(N)),
 
   where, for z∈ℝ, (z)_+=max{z,0}. One can check that, for all 𝐱^(N)∈𝕎_N^∘, y ↦ℳ(y;𝐱^(N)) is a piecewise polynomial function 
   with N-2 continuous derivatives. Moreover, it satisfies
   
    ∫_ℝℳ(y;𝐱^(N))dy=1.

 
The next theorem, proven by Olshanski in <cit.>, gives an expression for the Markov kernel Λ_K^N defined in (<ref>) in terms of the spline function introduced above.



    Let N≥ 2 and 𝐱^(N)∈𝕎_N^∘. Then, for any 1≤ K≤ N-1, we have

    Λ_K^N(𝐱^(N),d𝐲)   =
    ∏_l=1^K-1N-K+ll[ ℳ(y_K-j+1;x_N-i+1^(N),…,x_K-i+1^(N)) ]_i,j=1^K/∏_j-i≥ N-K+1(x_i^(N)-x_j^(N))Δ_K(𝐲)d𝐲,  ∀𝐲∈𝕎_K.







We note that, the case K=1, boils down to the following identity

    Λ_1^N(𝐱^(N),dy)=ℳ(y;𝐱^(N))dy,  ∀𝐱^(N)∈𝕎_N^∘,

which is an observation first due to Okounkov <cit.>.


The formula from Theorem <ref> is not yet amenable to asymptotic analysis, partly due to the denominator, which may vanish and lead to singularities. Thankfully, we can use a trick based on a formula for derivatives of spline functions to get rid of the denominator entirely. The new formula for Λ_K^N will however involve derivatives of the spline function ℳ which when we extend in the sequel the formula to 𝐱^(N) with coinciding coordinates will need to be interpreted in a weak sense.



In the setting of Theorem <ref>, we have
    
    Λ_K^N(𝐱^(N),d𝐲)=
    (∏_l=1^K-11/l!)
    (ℳ^(K-i)(dy_K-j+1;x_1^(N),…,x_N-i+1^(N)))_i,j=1^KΔ_K(𝐲).

Here, for any j=1,…,K-1,  ℳ^(j) denotes the j-th derivative of the function ℳ with the convention ℳ^(0)=ℳ, 
and, ℳ^(j)(dy,·)=ℳ^(j)(y,·)dy


    We have following relation for the derivative of the spline function in (<ref>), obtained by direct computation,

    ℳ^(1)(y;𝐱^(N))=N-1/x_1^(N)-x_N^(N)(ℳ(y;x_2^(N),…,x_N^(N))-ℳ(y;x_1^(N),…,x_N-1^(N))).

We now consider the formula given in Theorem (<ref>) and repeatedly apply elementary row operations to the determinant in (<ref>) and use (<ref>) to obtain (<ref>).



Let 
g∈  C_c^∞(ℝ^K)
and consider
Λ_K^Ng(𝐱^(N)) for 𝐱^(N)∈𝕎_N^∘. We have
from (<ref>) that

    Λ_K^Ng(𝐱^(N))
       =∫_ℝ^KΛ_K^N(𝐱^(N),d𝐲)g(𝐲)
    
       =
    ∫_ℝ^K(∏_l=1^K-11/l!)
    (ℳ^(K-i)(dy_K-j+1;x_1^(N),…,x_N-i+1^(N)))_i,j=1^KΔ_K(𝐲)g(𝐲).

 Next, one can expand the determinant and use integration by parts for each term in the expansion to rewrite the above expression as 

    Λ_K^Ng(𝐱^(N))   =(-1)^K(K-1)/2∫_ℝ^K∑_σ∈𝕊_Ksgn(σ)∏_j=1^KΛ_1^N-K+j(dy_σ(j);x_1^(N),…,x_N-K+j^(N))
       ×∂_𝐲^σ(Δ_K(𝐲)g(𝐲)),

where 
we used (<ref>) and for σ in 𝕊_K, the symmetric group of degree K, the differential operator ∂^σ acts on functions h∈ C_c^∞(ℝ^K) as follows:

    ∂_𝐲^σh(𝐲)=∂^K(K-1)/2/∂ y_1^σ(1)-1∂ y_2^σ(2)-1⋯∂ y_K^σ(K)-1h(𝐲).

 Now suppose 𝐱∈𝕎_N is arbitrary. Take a sequence (𝐱_n)_n=1^∞ in 𝕎_N^∘ such that 𝐱_nn→∞⟶𝐱. It then follows from Lemma <ref> that both the left and right hand side of equality (<ref>) above, when evaluating at 𝐱_n,  converge to the corresponding terms for 𝐱, as n→∞. This gives an explicit expression of Λ_K^Ng(𝐱) for all g∈ C_c^∞(ℝ^K) and general 𝐱∈𝕎_N.
Note that, this expression matches the one we obtain by formally expanding the following determinant
 
    ∫_ℝ^K(∏_l=1^K-11/l!)
    (ℳ^(K-i)(dy_K-j+1;x_1,…,x_N-i+1))_i,j=1^KΔ_K(𝐲)g(𝐲),

 where, the derivatives correspond to the distributional derivative
 of the kernel ℳ(dy,·). In particular, it is consistent with the formal expansion of the determinant along rows and columns.  From now on, all appearances of these distributional determinants will be understood to be interpreted in the weak sense explained above when applied to C_c^∞(ℝ^K) functions (and the functions we need to apply them to will always be in this space).




We next present an analogous explicit formula for the kernel Λ_K^∞. We do this in the case of Ω. The case of Ω_+ is in fact a special case under the embedding (<ref>). So, suppose ω∈Ω is arbitrary and written in coordinates as ω=(𝐱^+,𝐱^-,γ,δ). We will make very frequent use of the following notation in everything that follows.



 For ω∈Ω, or, abusing notation, for ω∈Ω_+, we define the probability measure ϕ_ω(dy) on ℝ by,

    ϕ_ω(dy)def=Λ_1^∞(ω,dy).



Given arbitrary ω∈Ω as above, we define, for ϵ>0, ω_ϵ∈Ω by:

    ω_ϵ=(𝐱^+,𝐱^-,γ,δ+ϵ).

We observe that, since ϕ_ω_ϵ involves a Gaussian component, it has a smooth density with respect to Lebesgue measure in ℝ, see <cit.>, which we denote by, abusing notation, ϕ_ω_ϵ(y). We write ϕ_ω_ϵ^(ℓ)(dy)=ϕ_ω_ϵ^(ℓ)(y)dy for its ℓ-th derivative. We have the following result.






Let ϵ>0. Then, the Markov kernel Λ_K^∞(ω_ϵ,d𝐲) has an explicit density with respect to Lebesgue measure in 𝕎_K and is given by,

    Λ_K^∞(ω_ϵ,d𝐲)=
              (∏_l=1^K-11/l!)(ϕ_ω_ϵ^(j-1)(dy_i))_i,j=1^KΔ_K(𝐲)d𝐲.
 




This is implicit in the proof of Theorem 7.7 in <cit.>.


Recall (or simply observe) that, as ϵ→ 0, we have the weak convergence

    Λ^∞_K(ω_ϵ,d𝐲)⟶Λ^∞_K(ω,d𝐲).

Now, given g∈ C_c^∞(ℝ^K), we expand the determinant in (<ref>) and use integration  by parts as before and then take ϵ→ 0 by virtue of (<ref>) to make sense of the potentially distributional determinant, for general ω∈Ω,

    (∏_l=1^K-11/l!)(ϕ_ω^(j-1)(dy_i))_i,j=1^KΔ_K(𝐲),

as follows, when acting on g∈ C_c^∞(ℝ^K),
 

    ∫_ℝ^KΛ_K^∞(ω,d𝐲)g(𝐲)
       =(-1)^K(K-1)/2∫_ℝ^K∏_j=1^Kϕ_ω(dy_j)∑_σ∈𝕊_Ksgn(σ)∂_𝐲^σ(Δ_K(𝐲)g(𝐲)),

where the differential operator is the same as in (<ref>).
 
As before, this weak-sense definition is consistent with the formal expansion of the determinant along rows and columns.  From now on, all appearances of these distributional determinants will be understood to be interpreted in the weak sense explained above when applied to C_c^∞(ℝ^K) functions (all functions we need to apply them to will always be in this space).







  §.§.§ Proof of Theorem <ref>

 For any fixed K∈ℕ, N>K, and 𝐱^(N)∈𝕎_N, we consider the following variant of Λ_K^∞,
 
    Λ̃_K^∞(ω(𝐱^(N)), d𝐲)   =
    (∏_l=1^K-11/l!)
    (ϕ_ω(x_1^(N),…,x_N-K+j^(N))^(j-1)(dy_i))_i,j=1^KΔ_K(𝐲).

We note that we only need (and in fact only define) Λ̃_K^∞ for ω∈Ω of the form ω(𝐱^(N)) for 𝐱^(N)∈𝕎_N. Again the determinant in equation (<ref>) above is interpreted in the weak sense when tested against smooth functions of compact support as explained previously. We first prove the desired result with
Λ_K^∞ replaced by
Λ̃_K^∞ in (<ref>).






In the setting of Theorem <ref>, we have
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^Ng(𝐱^(N))|→ 0,  as  N→∞.
 

After establishing Theorem <ref>, we prove that Λ̃_K^∞g(ω(𝐱^(N)))  and Λ_K^∞g(ω(𝐱^(N))) are asymptotically uniformly close as N→∞, see Proposition <ref> for the exact statement, which concludes the desired result.
The proof of Theorem <ref> is by an inductive argument in K. The variant Λ̃_K^∞ of Λ_K^∞ is very well-adapted to proving the rather complicated inductive step and this is its raison d'être.  Before giving the proof, we need  some more preliminaries and notations. 



Denote the Fourier and inverse Fourier transforms of an integrable function f on ℝ^n by f̂ and f̌ respectively given by:

    f̂(ξ)=∫_ℝ^ne^i∑_j=1^ny_jξ_jf(𝐲)d𝐲, f̌(ξ)= (2π)^-n∫_ℝ^ne^-i∑_j=1^ny_jξ_jf(𝐲)d𝐲,  ξ∈ℝ^n.


 We define the following functions
 
    F_+(z;ω)   def=e^iγ z∏_j=1^∞e^-ix_jz/1-ix_jz,  ∀ z∈ℝ,  ω∈Ω_+,
            
    
            F(z;ω)   def=e^iγ z-δ-∑_i=1^∞[(x_i^+)^2+(x_i^-)^2]/2z^2∏_j=1^∞e^-ix_j^+z/1-ix_j^+ze^ix_j^-z/1+ix_j^-z,  ∀ z∈ℝ,  ω∈Ω.

Note that, the infinite products are well-defined by virtue of ω∈Ω_+ and ω∈Ω respectively.



It is easy to check that F_+(·;ω) and F(·;ω) are 
the characteristic functions (i.e. Fourier transforms) of the probability law ϕ_ω when ω∈Ω_+ and Ω, respectively. In particular, we have

    ∫_ℝ F(z;ω)g(z)dz=
         ∫_ℝϕ_ω(dz)ĝ(z),  ∀ g∈𝒮(ℝ),

where 𝒮(ℝ) denotes the space of Schwartz functions on ℝ.  We will also need the following formula in the argument.

    Let 𝐱^(N)∈𝕎_N. Then, we have
    
    ∏_j=1^N(1-izx_j^(N)/N)^-1
        = ∫_-∞^∞(1-izy/N)^-Nℳ(dy;𝐱^(N)).


    
    See Section 8 of <cit.>, stated there for 𝐱^(N)∈𝕎_N^∘ which can be extended to coinciding coordinates by continuity.
    




    Let K=1 and fix g∈ C_c^∞(ℝ). Observe that Λ̃_1^∞=Λ_1^∞. 
    We have by definition that
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_1^∞g(ω(𝐱^(N)))-Λ_1^Ng(𝐱^(N))|   = sup_𝐱^(N)∈𝕎_N|∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)-∫_-∞^∞ℳ(dy;𝐱^(N))g(y)|.

 We have, again by definition,
 
    ∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)
    = 1/2π∫_-∞^∞F(z;ω(𝐱^(N)))(z)ǧ(z)dz.

Using (<ref>) and then (<ref>) we obtain
 
    ∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)   =1/2π∫_-∞^∞∏_j=1^N(1-izx_j^(N)/N)^-1ǧ(z)dz
       = 1/2π∫_-∞^∞∫_-∞^∞(1-izw/N)^-Nℳ(dw;𝐱^(N))
    ǧ(z)dz.

Note now that for any fix w∈ℝ, (1-izw/N)^-N converges pointwise to e^iwz. Also, since
g∈ C_c^∞(ℝ),

the function ǧ is integrable on ℝ. Thus, by dominated convergence theorem,

    𝖦_N(w)   def=1/2π∫_-∞^∞(1-izw/N)^-Nǧ(z)dz-g(w)
    
       =1/2π∫_-∞^∞[(1-izw/N)^-N-e^izw]ǧ(z)dz → 0,   as  N→∞.

 We show in the Lemma <ref> below that the above convergence is uniform over ℝ which proves the result for the case K=1. This is because by the uniform convergence, we have for N large enough that sup_w∈ℝ|𝖦_N(w)|<ε, for a given ε>0,   
 and accordingly,
    
    sup_𝐱^(N)∈𝕎_N|∫_-∞^∞ϕ_ω(𝐱^(N))(dy)g(y)-∫_-∞^∞ℳ(dy;𝐱^(N))g(y)|=sup_𝐱^(N)∈𝕎_N|∫_-∞^∞𝖦_N(w)ℳ(dw;𝐱^(N))|
         =sup_w∈ℝ|𝖦_N(w)|sup_𝐱^(N)∈𝕎_N|∫_-∞^∞ℳ(dw;𝐱^(N))|<ε.

 Here, we used the fact that ∫_-∞^∞ℳ(dw;𝐱^(N))=1 for every 𝐱^(N)∈𝕎_N.





  Let 𝖦_N be the function defined in (<ref>). Then, we have,
  
    sup_w∈ℝ|𝖦_N(w)|N →∞⟶ 0.

  

    Suppose by contradiction that the assertion is not true. Thus, there exist δ>0 and a sequence (w_N_n)_n=1^∞ such that 
    
    |𝖦_N_n(w_N_n)|>δ,  ∀ n∈ℕ.

    There are two possibilities:
    
        
  (i)
        There exists some positive constant L
        such that |w_N_n|≤ L for all n∈ℕ.
        
  (ii)
        The sequence (w_N_n)_n=1^∞ is unbounded.
    
    We show both of these cases lead to a contradiction.
    First we assume (i) occurs. Consider the sequence of functions 
    (𝖦_N_n)_n=1^∞
    on the interval [-L,L]. One can see that this is a sequence of uniformly bounded and 
    
    equicontinuous functions
    on [-L,L], and hence by the Arzelà–Ascoli theorem, there exists a subsequence (𝖦_N_n_k)_k∈ℕ converging uniformly to 0 on [-L,L]. This implies that for large enough k we have
    
    |𝖦_N_n_k(w)|<δ,  ∀ w∈ [-L,L],

    which contradicts (<ref>).
    
    Next, we consider (ii), namely |w_N_n|→∞ as n→∞. 
    Then, there exists a non-decreasing, in absolute value, subsequence of (w_N_n)_n=1^∞  diverging to infinity. We denote this subsequence using the same index, and thus, we have 
    |w_N_1|≤|w_N_2|≤…⟶∞.
    Suppose first that
    lim_N_n→∞|w_N_n|/√(N_n)=0,
    
    and let
    
    f_n(z)def=(
         1-izw_N_n/N_n)^-N_n-e^izw_N_n,    z∈ℝ.

    One can easily check that 
   f_n converges to 0 pointwise,
    and so does 𝖦_N_n(w_N_n).  
    This again violates (<ref>).
    Finally, we assume that 
    
    lim_N_n→∞|w_N_n|/√(N_n)> 0, namely it is equal to a positive constant ℓ, or ∞. 
    Note that in this case, g(w_N_n)=0 for large enough n.
Let 

    ℐ_ndef=∫_-∞^∞(1-izw_N_n/N_n)^-N_nǧ(z)dz,  n∈ℕ,

    We show in the following that ℐ_n→ 0 as n→∞ in both cases.  
    As a result, we conclude in view of (<ref>), that |𝖦_N_n(w_N_n)|< δ for sufficiently large n, which contradicts (<ref>) as well, and completes the proof of the lemma. 
    
    Assume first that lim_n→∞|w_N_n|/√(N_n)= ∞.
     Observe that in this case, 
     (1-izw_N_n/N_n)^-N_nn→∞⟶ 0, 
   and hence, ℐ_nn→∞⟶ 0 (with the same reasoning as before) as desired. Next,  suppose lim_n→∞|w_N_n|/√(N_n)=ℓ>0. Since ǧ is integrable, one can choose M_δ>0 such that

    ∫_ℝ∖ (-M_δ,M_δ)ǧ(z)dz ≤δ/8.

Note also that |(1-izw_N_n/N_n)^-N_n|≤ 1 for all n∈ℕ. So 
we have
 
    |ℐ_n|≤|∫_-M_δ^M_δ(1-izw_N_n/N_n)^-N_nǧ(z)dz|+δ/8.

 Now, since lim_n→∞|w_N_n|/√(N_n)=ℓ, there exists n_0∈ℕ such that |w_N_n|/√(N_n)≤ℓ+1 for all n≥ n_0. We pick such an n_0 so that N_n_0> M_δ^2(ℓ+1)^2 also holds, on account of which we can write a Taylor expansion for the function (1-izw_N_n/N_n)^-N_n on the interval (-M_δ,M_δ). 
 Accordingly, we have for all z∈(-M_δ,M_δ) and n≥ n_0,
 
    (1-izw_N_n/N_n)^-N_n= e^izw_N_n-z^2w_N_n^2/N_n+
        R_n(z),

 where R_n(z)= z^3w_N_n^3/3N_n^2+
    z^4w_N_n^4/4N_n^3+ …=
    O(1/√(N_n)).
 Then, 

    |ℐ_n|   ≤|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_n+R_n(z)ǧ(z)dz|+δ/8
       ≤|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|+|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_n(e^R_n(z)-1)ǧ(z)dz|+δ/8.

 The second integral in the above expression converges to 0 as n→∞, since (e^R_n(z)-1)n→∞⟶ 0 and the integrand is bounded by ǧ up to a multiplicative constant on [-M_δ,M_δ]. Thus, for large enough n∈ℕ, we get  
 
    |ℐ_n|   ≤|∫_-M_δ^M_δe^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|+2δ/8
    ≤|∫_-∞^∞e^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|+3δ/8.

Note now that for each n∈ℕ, the function 
 e^-z^2w_N_n^2/N_nǧ(z)
 is a Schwartz function and the Fourier transform maps the Schwartz space to itself.
It thus follows that for n large enough, we have

    |∫_-∞^∞e^izw_N_n-z^2w_N_n^2/N_nǧ(z)dz|<δ/8,

 and hence,
|ℐ_n|<δ/2. 
This finally concludes the proof of the lemma.
 
  

Suppose (<ref>) holds for K∈ℕ, namely, for all g∈ C_c^∞(ℝ^K), 
we have 
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^Ng(𝐱^(N))|⟶ 0,   as  N→∞.
 
 We need to show that for all g∈ C_c^∞(ℝ^K+1),
 
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|⟶ 0,   as  N→∞.
 
 Let g∈ C_c^∞(ℝ^K+1). We have by definition that 
 
 
    (Λ̃_K+1^∞g)(ω(𝐱^(N)))   =
    (∏_l=1^K1/l!)
    ∫_ℝ^K+1(ϕ_ω(x_1^(N),…,x_N-K-j+1^(N))^(j-1)(dy_i))_i,j=1^K+1Δ_K+1(𝐲)g(𝐲)
    
       = (∏_l=1^K1/l!)
    ∫_ℝ^K+1(ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i,j=1^K+1Δ_K+1(𝐲)g(𝐲).

Also, from (<ref>) we have

    (Λ_K+1^Ng)(𝐱^(N))
    =(∏_l=1^K1/l!)∫_ℝ^K+1(ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i,j=1^K+1Δ_K+1(𝐲)g(𝐲).

Thus, we obtain

    |   Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|
    =
    |∫_ℝ^K+1(∏_l=1^K1/l!)[(ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i,j=1^K+1
       -
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i,j=1^K+1]
    Δ_K+1(𝐲)g(𝐲)|

Now, by expanding both determinants along the first row (recall the definition of the determinant in the weak sense is consistent with expanding along a row or column), we get 

    |   Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|
       =|∫_ℝ^K+1∑_k=1^K+1(∏_l=1^K1/l!)(-1)^1+k[ϕ_ω(𝐱^(N))^(K)(dy_K+2-k)(ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i=2,j=1,j≠ k^K+1
       -ℳ^(K)(dy_K+2-k;𝐱^(N))
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1]Δ_K+1(𝐲)g(𝐲)|
    
       =1/K!|
    ∑_k=1^K+1(-1)^1+k∫_ℝ^K(∏_l=1^K-11/l!)
    [
    (ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i=2,j=1,j≠ k^K+1
       -
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1]
    ∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)
       ×(-1)^K+1-k∫_ℝϕ_ω(𝐱^(N))^(K)(dy_K+2-k)∏_j≠ K+2-k1≤ j≤ K+1(y_K+2-k-y_j)g(𝐲)
    _(-1)^K∫_ℝϕ_ω(𝐱^(N))(dy_K+2-k)h_k(𝐲)
          +∑_k=1^K+1(-1)^1+k∫_ℝ^K(∏_l=1^K1/l!)
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_j≠ ki=2,j=1^K+1∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)
    
       ×(-1)^K+1-k∫_ℝ[ℳ^(K)(dy_K+2-k;𝐱^(N))-ϕ_ω(𝐱^(N))^(K)(dy_K+2-k)]
    ∏_j≠ K+2-k1≤ j≤ K+1(y_K+2-k-y_j)g(𝐲)
    _(-1)^K∫_ℝ[ℳ(dy_K+2-k;𝐱^(N))-ϕ_ω(𝐱^(N))(dy_K+2-k)]h_k(𝐲)|,
 
 where h_k∈ C_c^∞(ℝ^K+1) is defined by
 
    h_k(𝐲)=d^K/dy_K+2-k^K[∏_j≠ K+2-k1≤ j≤ K+1(y_K+2-k-y_j)g(𝐲)].

Thus, we have

    |   Λ̃_K+1^∞g(ω(𝐱^(N)))-Λ_K+1^Ng(𝐱^(N))|≤1/K!∑_k=1^K+1∫_ℝϕ_ω(𝐱^(N))(dy_K+2-k)
    
         |∫_ℝ^K(∏_l=1^K-11/l!)[
    (ϕ_ω(x_1^(N),…,x_N+1-i^(N))^(K+1-i)(dy_K+2-j))_i=2,j=1,j≠ k^K+1
       -
    (ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1]
    ∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)h_k(𝐲)|
    
         +1/K!∑_k=1^K+1(∏_l=1^K1/l!)
    ∫_ℝ^K(ℳ^(K+1-i)(dy_K+2-j;x_1^(N),…,x_N+1-i^(N)))_i=2,j=1,j≠ k^K+1∏_i,j≠ K+2-k1≤ i<j≤ K+1(y_i-y_j)
    
       ×|∫_ℝ[ℳ(dy_K+2-k;𝐱^(N))-ϕ_ω(𝐱^(N))(dy_K+2-k)]h_k(𝐲)|.
 
To complete the proof, we need to show that  
 the above expression becomes arbitrarily small, uniformly in 𝐱^(N), for large enough N.
Clearly, it suffices to prove this for each term of the above two series individually. We prove that this is indeed the case for the first series. A completely analogous argument works for the second too.


Let ε>0 be given.
Fix 1≤ k≤ K+1 and consider the k-th term of the first series. For each a∈ℝ, let 𝗁_a∈ C_c^∞(ℝ^K)  be the function defined by

    𝗁_a(𝐲)def=h_k(y_1,…,y_k-1,a,y_k,…,y_K),   𝐲∈ℝ^K.



Since h_k∈ C_c^∞(ℝ^K+1), there exists a compact interval ℑ_k such that 𝗁_a≡ 0 
for all a∉ℑ_k.
Moreover, by virtue of the induction hypothesis, for every a∈ℑ_k, there exist a smallest 𝖭(a)∈ℕ (depending on a), such that for all N≥𝖭(a), we have 

    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞𝗁_a(ω(𝐱^(N)))-Λ_K^N𝗁_a(𝐱^(N))|
    <ε.

Note now that if 𝖬_kdef=sup_a∈ℑ_k𝖭(a) is finite,
then the above inequality holds for all a∈ℑ_k and N≥𝖬_k, and so, the result follows immediately owing to the fact that ϕ_ω(𝐱^(N)) is a probability measure on ℝ for every 𝐱^(N)∈𝕎_N.

We thus wish to show that 𝖬_k<∞. Suppose by contradiction that it is not true. Accordingly, one can find a sequence (𝔞_n)_n=1^∞∈ℑ_k with 𝔞_nn→∞⟶𝔞_*∈ℑ_k such that 
𝖭(n)↗∞.
Let 𝗁_*def=𝗁_𝔞_* and
𝖭_*∈ℕ 
be such that 

    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞𝗁_*(ω(𝐱^(N)))-Λ_K^N𝗁_*(𝐱^(N))|< ε/2
 
for all N≥𝖭_* (note that 𝖭_* is also well-defined by the induction hypothesis).

Now, since h_k∈ C_c^∞(ℝ^K+1) and 𝔞_nn→∞⟶𝔞_*, we have 𝗁_ndef=𝗁_𝔞_nn→∞⟶𝗁_* pointwise. Moreover, 
it follows from the Arzelà–Ascoli theorem that there exists a uniformly convergent sub-sequence of this sequence which we denote by the same notation. 
Hence, we can pick 𝗆∈ℕ large enough so that for all n≥𝗆, we have 𝖭(n)> 𝖭_* and 

    |𝗁_n(𝐲)-𝗁_*(𝐲)|<ε/4,  ∀𝐲∈ℝ^K.
 
Therefore, we obtain 
 
    |Λ̃_K^∞𝗁_𝗆(ω(𝐱^(N)))-Λ_K^N𝗁_𝗆(𝐱^(N))|   ≤|Λ̃_K^∞(𝗁_𝗆-𝗁_*)(ω(𝐱^(N)))|+|Λ_K^N(𝗁_𝗆-𝗁_*)(𝐱^(N))|
         +|Λ̃_K^∞𝗁_*(ω(𝐱^(N)))-Λ_K^N𝗁_*(𝐱^(N))|
       <
         ε/4+ε/4+ε/2=ε,

 for all N≥𝖭_*, and 𝐱^(N)∈𝕎_N. This implies that
 
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞𝗁_𝗆(ω(𝐱^(N)))-Λ_K^N𝗁_𝗆(𝐱^(N))|<ε,  ∀ N≥𝖭_*.

 Thus, we must have 𝖭(𝗆)≤𝖭_* which is a contradiction in view of the particular choice of 𝖭(𝗆) and 𝗆.

We finally conclude Theorem <ref>  by proving the following proposition.



Let K∈ℕ. Then, for all g∈ C_c^∞(ℝ^K) we have:

    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^∞g(ω(𝐱^(N)))|→ 0,
      as  N→∞.





    We have by definition
    
    sup_𝐱^(N)∈𝕎_N|Λ̃_K^∞g(ω(𝐱^(N)))-Λ_K^∞g(ω(𝐱^(N)))|
         =sup_𝐱^(N)∈𝕎_N|
    ∫_ℝ^K(∏_l=1^K-11/l!)
    ((ϕ_ω(x_K-j+1^(N),…,x_N^(N))^(j-1)(dy_K+1-i))_i,j=1^K -
    (ϕ_ω(𝐱^(N))^(j-1)(dy_K+1-i))_i,j=1^K)h(𝐲)|,

with 
h(𝐲)=Δ_K(𝐲)g(𝐲).
We show that the above expression is arbitrarily small for large enough N. Define the following K × K matrices,

    𝐅̃(𝐳;𝐱^(N))=
        [   (-iz_K)^K-1∏_j=1^N(1-iz_Kx_j^(N)/N)^-1                                        …   (-iz_1)^K-1∏_j=1^N(1-iz_1x_j^(N)/N)^-1; (-iz_K)^K-2∏_j=2^N(1-iz_Kx_j^(N)/N-1)^-1                                        … (-iz_1)^K-2∏_j=2^N(1-iz_1x_j^(N)/N-1)^-1;                                        ⋮                                        ⋱                                        ⋮;          ∏_j=K^N(1-iz_Kx_j^(N)/N-K+1)^-1                                        …          ∏_j=K^N(1-iz_1x_j^(N)/N-K+1)^-1 ],
    𝐅(𝐳;𝐱^(N))=
        [ (-iz_K)^K-1∏_j=1^N(1-iz_Kx_j^(N)/N)^-1                                      … (-iz_1)^K-1∏_j=1^N(1-iz_1x_j^(N)/N)^-1; (-iz_K)^K-2∏_j=2^N(1-iz_Kx_j^(N)/N)^-1                                      … (-iz_1)^K-2∏_j=2^N(1-iz_1x_j^(N)/N)^-1;                                      ⋮                                      ⋱                                      ⋮;            ∏_j=K^N(1-iz_Kx_j^(N)/N)^-1                                      …            ∏_j=K^N(1-iz_1x_j^(N)/N)^-1 ].

We now use (<ref>) repeatedly to rewrite the determinants in (<ref>) in terms of the determinants of the corresponding Fourier transforms.  
Then, we do elementary row operations repeatedly for the second determinant.
We finally use (<ref>) and expand both determinants in order to obtain

    ∫_ℝ^K   [(ϕ_ω(x_K-j+1^(N),…,x_N^(N))^(j-1)(dy_K+1-i))_i,j=1^K -
    (ϕ_ω(𝐱^(N))^(j-1)(dy_K+1-i))_i,j=1^K]Δ_K(𝐲)g(𝐲)
    
       =(2π)^-K∫_ℝ^K[(𝐅̃(𝐳;𝐱^(N)))
    -(𝐅(𝐳;𝐱^(N)))]
    ȟ(𝐳)d𝐳
       =(2π)^-K∑_σ∈𝕊_Ksgn(σ)∫_ℝ^K∫_ℝ^K[∏_l=1^K(1-iz_σ(l)w_σ(l)/N-l+1)^-(N-l+1)-∏_l=1^K(1-iz_σ(l)w_σ(l)/N)^-(N-l+1)] 
    
       ×(∏_l=1^K(-iz_σ(l))^K-lȟ(𝐳)d𝐳)∏_l=1^Kℳ(dw_σ(l);x_l^(N),…,x_N^(N)).

Note now that for any fixed w_σ(l), the expression in the brackets on the RHS goes to 0 pointwise in z_σ(l), and moreover, it is bounded. Therefore, since the function in the parentheses is integrable, the inner integral converges to 0 by the dominated convergence theorem. One can then use a similar argument as in the proof of Lemma <ref> to verify that the convergence is indeed uniform 
and hence, the expression can be made arbitrarily small by picking large enough N. Finally, the conclusion follows by fact that ℳ(dw;𝐱) is a probability measure on ℝ.  








§ PROPERTIES OF THE FINITE-DIMENSIONAL DIFFUSIONS
  


 §.§ Well-posedness and Feller property


In this subsection we present several, for the most part standard, results on the dynamics (<ref>) that we will need.
 

 
The SDE (<ref>) has a unique strong non-exploding solution (𝔵^(N)(t);t≥ 0) for any initial condition 𝔵^(N)(0)=𝐱∈𝕎_N,+. Moreover, almost surely, for all t>0, 𝔵^(N)(t)∈𝕎_N,+^>, where

    𝕎_N,+^>={𝐱∈ℝ^N:x_1>x_2>⋯ >x_N≥ 0}.

Finally, if 𝔵^(N)(0)=𝐱∈𝕎_N,+^∘, then almost surely for all t≥ 0, 𝔵^(N)(t)∈𝕎_N,+^∘.



All statements, except the last one, follow from the main theorem of <cit.> by verifying the elementary conditions therein. To establish the last statement, observe that since 𝐱∈𝕎_N,+^∘, using Itô's formula we have, for all t≥ 0,

    log𝔵_N^(N)(t)    = log x_N+ 𝗐_N(t) -1+η/2t+∫_0^t1/2𝔵_N^(N)(s)ds+∫_0^t∑_j=1^N-1𝔵_j^(N)(s)/𝔵_N^(N)(s)-𝔵_j^(N)(s)ds
       ≥log x_N+ 𝗐_N(t) -1+η/2t-(N-1)t(inf_s∈[0,t]|1-𝔵_N^(N)(s)/𝔵_N-1^(N)(s)|
    )^-1.

The conclusion then follows since the RHS is finite.



 The Markov semigroup associated with (𝔵^(N)(t);t≥ 0), the unique strong solution of (<ref>), is denoted by (𝔓_N(t))_t≥ 0.




We next introduce certain matrix-valued dynamics (𝐇_t^(N);t≥ 0) from <cit.> whose eigenvalue evolution is described by the SDE (<ref>).
Recall that ℍ_+(N) denotes the space of N× N non-negative definite Hermitian matrices.
We denote by Tr(𝐀) the trace of  a matrix 𝐀. Consider the following ℍ_+(N)-valued SDE introduced by Rider-Valko in <cit.>,

    d𝐇_t^(N)=1/2(dΓ_t^(N)𝐇_t^(N)+𝐇_t^(N)d(
     Γ_t^(N))^†)+(-η+N/2𝐇_t^(N)+1/2(1+Tr(𝐇_t^(N)))𝐈)dt,

where 
   (Γ_t^(N);t ≥ 0) is a matrix-valued complex Brownian motion; namely, an N× N matrix with the entries independent complex Brownian motions (the real and imaginary parts are independent standard real Brownian motions), and
𝐈 is the N× N identity matrix.

For any 𝐇_0^(N)∈ℍ_+(N), the SDE (<ref>) has a unique strong solution.


    It is easy to see that the coefficients of the equation are Lipschitz continuous in  matrix norm, and thus existence and uniqueness of the strong solution follows from well-known results, see for example <cit.>. 


 
Let 𝐱∈𝕎_N,+ and 𝐇∈ℍ_+(N) be such that 𝖾𝗏𝖺𝗅_N(𝐇)=𝐱. 
Let (𝐇_t^(N);t≥ 0) be the solution to (<ref>) starting from 𝐇 and (𝔵^(N)(t);t≥ 0) be the solution to (<ref>) starting from 𝐱. Then,
    
    (𝖾𝗏𝖺𝗅_N(𝐇^(N)_t);t≥ 0)d=(𝔵^(N)(t);t≥ 0).




The statement is a consequence of the general result of <cit.>, which relates matrix diffusions solving SDE generalising (<ref>) to their eigenvalues evolutions given in terms of SDE generalising (<ref>). The computation is also implicit in <cit.>.





By looking at the stochastic equation satisfied by the process t↦(𝐇_t^(N)), see proof of Proposition 12 in 
<cit.>, we can improve the statement of Lemma <ref> regarding never reaching the origin, to initial conditions 𝐱 with coinciding coordinates.




 Let η∈ℝ and N∈ℕ. The semigroup (𝔓_N(t))_t≥ 0 is a Feller semigroup.



    The semigroup associated to (𝐇_t^(N);t≥ 0), as a unique strong solution to a SDE with Lipschitz coefficients, is Feller by classical results, see <cit.>. The Feller property of the eigenvalue process follows from that of the matrix process. More precisely,
    let (𝖰_N(t))_t≥ 0 denote the semigroup associated to (𝐇_t^(N);t ≥ 0). Note that, if 𝐱=𝖾𝗏𝖺𝗅_N(𝐇), one has from Lemma <ref>,
    
    (𝔓_N(t)f)(𝐱)=(𝖰_N(t)f∘𝖾𝗏𝖺𝗅_N)(𝖽𝗂𝖺𝗀(𝐱)),

   and moreover, note that if f ∈ C_0(𝕎_N,+) then f ∘𝖾𝗏𝖺𝗅_N ∈ C_0(ℍ_+(N)). From this, and the Feller property of (𝖰_N(t))_t≥ 0, one immediately checks the Feller property for (𝔓_N(t))_t≥ 0. 



By hypoellipticity <cit.> of the generator of (𝖰_N(t))_t≥ 0, shown in Proposition 12 of <cit.>, we have that 𝖰_N(t)(𝐀,d𝐁)=𝖰_N(t)(𝐀,𝐁)d𝐁, with d𝐁 Lebesgue measure on ℍ_+(N), where the density 𝖰_N(t)(𝐀,𝐁) is smooth, viewed as a function in the matrix entries, see <cit.>.  



Our next goal is to give a description of the N-dimensional dynamics (𝔓_N(t))_t≥ 0, in some sense, in terms of one-dimensional diffusions. For this, we define for fixed N∈ℕ and η∈ℝ, consider unique strong solution to the following SDE in [0,∞),

    d𝗓^(N)(t)    = 𝗓^(N)(t) d𝗐(t) + [(1-η/2-N)𝗓^(N)(t)+1/2]dt,

with 𝗐 as a usual a standard Brownian motion, with infinitesimal generator,

    𝖫_x^(N) = x^2/2∂_x^2+((1-η/2-N)x+1/2)∂_x
 
and transition density with respect to Lebesgue measure in [0,∞) that we denote by 𝔭^(N)_t(x,y). It is known that ∞ is a natural boundary and 0 an entrance boundary in the terminology of Feller <cit.>. In particular, almost surely, for all strictly positive times the diffusion 𝗓^(N) is in (0,∞). 

We then have the following proposition. Analogous results are well-known for certain generalisations of Dyson Brownian motion: the radial Dunkl <cit.> and Heckman-Opdam <cit.> processes.


  Let η∈ℝ and N∈ℕ. Then, C^∞_c,sym(𝕎_N,+) forms a core for the generator of the semigroup (𝔓_N(t))_t≥ 0  and it is moreover invariant under the action of the generator. Finally, the kernel 𝔓_N(t)(𝐱,d𝐲) of 𝔓_N(t) has an explicit density with respect to the Lebesgue measure in 𝕎_N,+ given by (abusing notation), for 𝐱∈𝕎_N,+^∘, t>0,

    𝔓_N(t)(x,y)= e^-λ_NtΔ_N(𝐲)/Δ_N(𝐱)(𝔭_t^(N)(x_i,y_j))_i,j=1^N,

with λ_N=N(N-1)(1-3η/2-2N)/6.
   


 Let us denote by (𝖫_N,𝒟(𝖫_N)) the generator of (𝔓_N(t))_t≥ 0. Define the formal differential operator 𝐋_N by,

    𝐋_N=∑_i=1^N x_i^2/2∂_x_i^2+∑_i=1^N[-η/2x_i+1/2+∑_j=1,j≠ i^Nx_ix_j/x_i-x_j]∂_x_i.

We will identify shortly 𝖫_N with 𝐋_N on C^∞_c,sym(𝕎_N,+).

We first check that C_c,sym^∞(𝕎_N,+) is invariant under 𝐋_N. Let g∈ C_c,sym^∞(𝕎_N,+) be arbitrary and let f be a function in C_c,sym^∞(ℝ^N) that restricts to g∈ C_c,sym^∞(𝕎_N,+). Consider 𝐋_Nf, where we view 𝐋_N as acting on functions on ℝ^N in the obvious way, and observe that on 𝕎_N,+ this restricts to 𝐋_N g. It then suffices to show 𝐋_Nf ∈ C_c,sym^∞(ℝ^N). 𝐋_Nf is clearly of compact support and moreover it is symmetric in its arguments. Showing smoothness boils down to showing that the only part which is possibly singular,

    (x_1,…, x_N)   ↦∑_1≤ i< j≤ N x_i x_j ∂_x_if(x_1,…,x_N)-∂_x_jf(x_1,…,x_N)/x_i-x_j,

is actually smooth on the whole of ℝ^N. This moreover boils down to proving the following claim. For a smooth symmetric function H on ℝ^2, the set of which we denote by C_sym^∞(ℝ^2), the function 

    (x,y) ↦∂_x H(x,y)-∂_yH(x,y)/x-y

is smooth on the whole of ℝ^2, which is a direct consequence of known results. Namely, for H∈ C_sym^∞(ℝ^2) let ℌ be the unique function such that 

    H(x,y)=ℌ(S_1(x,y),S_2(x,y)),

where S_1(x,y)=-(x+y) and S_2(x,y)=xy, see <cit.>.
Then, we have, see <cit.>,

    ∂_x H(x,y)-∂_yH(x,y)/x-y=-∂ℌ/∂S_2.

Finally, from Corollary 3.3 of <cit.>, ℌ is a smooth function and this establishes the claim. 

We now show that 𝖫_N extends 𝐋_N on C_0(𝕎_N,+) and C^∞_c,sym(𝕎_N,+) is a core for it. First, for f so that 𝐋_Nf makes sense, define the process t↦𝖬_f(t) by, 

    𝖬_f(t)def=f(𝗓(t))-f(𝗓(0))-∫_0^t 𝐋_N f(𝗓(s))ds,

where 𝖫𝖺𝗐(𝗓)=𝖫𝖺𝗐(𝔵), with 𝔵 the unique strong solution to (<ref>), namely the process with generator 𝖫_N. Observe that, for f ∈ C^∞_c,sym(𝕎_N,+), 𝖬_f is bounded since 𝐋_N C^∞_c,sym(𝕎_N,+) ⊂ C^∞_c,sym(𝕎_N,+). Hence, an application of  Itô's formula <cit.> to the solution of the SDE (<ref>) gives that t↦𝖬_f(t) is a martingale (with respect to the natural filtration of the coordinate process). In particular, from Theorem 17.23 of <cit.> we obtain that 𝖫_N extends (we do not know yet if this extension is unique) 𝐋_N|_C^∞_c,sym(𝕎_N,+) to C_0(𝕎_N,+).  

It remains to show that C^∞_c,sym(𝕎_N,+) is actually a core for the generator 𝖫_N. This is done by connecting the SDE (<ref>) to a well-posed martingale problem. Define C_𝐱(ℝ_+,𝕎_N,+)={f∈ C(ℝ_+,𝕎_N,+):f(0)=𝐱}.
We say that a family of probability measures {𝐏_𝐱}_𝐱∈𝕎_N,+ on C(ℝ_+,𝕎_N,+), endowed with the filtration generated by the coordinate process, is a solution to the martingale problem for (𝐋_N,C^∞_c,sym(𝕎_N,+)) if 𝐏_𝐱(C_𝐱(ℝ_+,𝕎_N,+))=1, and under 𝐏_𝐱, for every f∈ C^∞_c,sym(𝕎_N,+), the process t↦𝖬_f(t) defined in (<ref>), where 𝖫𝖺𝗐(𝗓)=𝐏_𝐱,
 is a martingale. We now show that this martingale problem is well-posed, namely has a unique solution. First, the law of any weak solution, to the SDE (<ref>) starting from 𝐱, gives rises to a solution to the martingale problem by an application of Itô's formula <cit.>. We then show that this solution to the martingale problem must be unique. Consider an arbitrary solution to the martingale problem. For R>0, define the stopping times,
 
    ζ_R=inf{t≥ 0;||𝗓(t)||≥ R   or ∑_1=i≠ j=N|∫_0^t𝗓_i(s)𝗓_j(s)/𝗓_i(s)-𝗓_j(s)ds |≥ R },

 where, for 𝐱∈ℝ^N, ||𝐱|| is its Euclidean norm. For i,j=1,…,N, we choose f_i,f_ij∈ C_c^∞(𝕎_N,+) satisfying f_i(𝐱)=x_i, f_ij(𝐱)=x_ix_j, for ||𝐱||≤ R. We now claim that, for all i,j=1,…,N, the processes defined by,

    t ↦𝖭_i(t∧ζ_R)def=𝖬_f_i^R(t∧ζ_R)=𝗓_i(t∧ζ_R)-𝗓_i(0)-∫_0^t∧η_R(-η/2𝗓_i(s)+1/2+∑_j=1,j≠ i^N𝗓_i(s) 𝗓_j(s)/𝗓_i(s)-𝗓_j(s))ds

and t ↦𝖭_ij(t∧ζ_R)def=𝖬_f_ij^R(t∧ζ_R) are martingales. We approximate f_i^R and f_ij^R by functions in C_c,sym^∞(𝕎_N,+), for which we know the martingale property of 𝖬_f. By the Stone-Weirstrass theorem the space C_c,sym^∞(𝕎_N,+) is dense in C_0(𝕎_N,+) but we need slightly more in convergence of the derivatives.
More generally, let us take a function g∈ C_c^∞(𝕎_N) (note that we can view the functions f_i^R,f_ij^R as in C_c^∞(𝕎_N)). Then, extend g by symmetry to the whole of ℝ^N to obtain a function 𝖦. Observe that, by construction 𝖦 is symmetric, has compact support, is smooth except on the hyperplanes {x_i_1=x_i_2=⋯=x_i_ℓ}, is Lipschitz on the whole of ℝ^N and 𝖦|_𝕎_N=g. Let 𝔲 be a symmetric approximation of the identity and define, for each m ∈ℕ,

    𝔤_m(𝐱)def=∫_ℝ^N𝖦(𝐲) m^N𝔲(m(𝐱-𝐲))d𝐲.

Then, observe that, 𝔤_m ∈ C^∞_c,sym(ℝ^N) and that 𝔤_m approximates, as m →∞, g in C_0(𝕎_N) and all its derivatives approximate uniformly on compact sets in 𝕎_N^∘ the derivatives of g. From the above we get that there exist sequences (𝔣_i^m)_m=1^∞, (𝔣_ij^m)_m=1^∞ in C_c,sym^∞(𝕎_N,+) so that, as m →∞, for all i,j=1,…,N,

    𝖬_𝔣_i^m(·∧ζ_R) ⟶𝖬_f_i^R(·∧ζ_R),  𝖬_𝔣_ij^m(·∧ζ_R) ⟶𝖬_f_ij^R(·∧ζ_R),

uniformly on compact sets, in distribution under 𝐏_𝐱. Hence, we easily obtain that for all i,j=1,…,N, the processes t ↦𝖬_f_i^R(t ∧ζ_R), t ↦𝖬_f_ij^R(t ∧ζ_R) are martingales as desired. Then, we can follow the computation in Theorem 32.7 of <cit.> or Proposition 5.4.6 of <cit.> to obtain independent standard Brownian motions (𝗐_i(·))_i=1^N (with respect to possibly an extension of the original filtration), such that, for all i=1,…,N, 
 
    𝖭_i(t∧ζ_R)=∫_0^t∧ζ_R𝗓_i(s)d𝗐_i(s),   ∀ t≥ 0.

 In particular, we obtain that (𝗓_i(·∧ζ_R),𝗐_i(·∧ζ_R))_i=1^N is a weak solution to the SDE (<ref>) up until time ζ_R. But by pathwise uniqueness shown in <cit.>, this solution must be equal to the unique strong solution of (<ref>) up until ζ_R. Observe that, under the law of the solution to (<ref>), ζ_R→∞, as R →∞. In particular, ζ_R →∞, as R→∞, under 𝐏_𝐱 and from a solution to the martingale problem we constructed a solution to (<ref>) for all times. Hence, the solution to the martingale problem for (𝐋_N,C^∞_c,sym(𝕎_N,+)) must be unique (for otherwise we would have constructed two different in law solutions to (<ref>)). The associated diffusion process to it, see <cit.>, is clearly nothing but the solution to the SDE (<ref>), namely the Feller process with semigroup (𝔓_N(t))_t≥ 0. By virtue of Theorem 2.5 of <cit.>, since C_c,sym^∞(𝕎_N,+) is dense C_0(𝕎_N,+) and the martingale problem for (𝐋_N,C^∞_c,sym(𝕎_N,+)) is well-posed, we obtain that operator 𝐋_N is closable in C_0(𝕎_N,+) and the closure is nothing but the generator 𝖫_N of the semigroup (𝔓_N(t))_t ≥ 0. Namely 𝖫_N|_C_c,sym^∞(𝕎_N,+)=𝐋_N|_C_c,sym^∞(𝕎_N,+)=𝖫_N or in other words, C_c,sym^∞(𝕎_N,+) is a core for (𝖫_N,𝒟(𝖫_N)).



The final statement in the proposition is a direct consequence of the following observation. The generator 𝐋_N acting on g∈ C_c,sym^∞(𝕎_N,+) can be written, after an elementary computation, as

    𝐋_Ng(x_1,…,x_N)= [Δ_N(𝐱)^-1∘(∑_i=1^N 𝖫_x_i^(N))∘Δ_N(𝐱)-λ_N]g(x_1,…,x_N),

which is exactly the Doob h-transform, see <cit.>, of N independent one-dimensional diffusions with generator 𝖫^(N) killed when they intersect, by the Vandermonde determinant Δ_N. This is the Markov process which has transition kernel having density (<ref>), by virtue of the Karlin-McGregor formula <cit.>, and the conclusion follows. 






 §.§ Consistency


The following consistency or intertwining relation between (𝔓_N(t))_t≥ 0 and (𝔓_N+1(t))_t≥0 is  the main algebraic ingredient for the method of intertwiners.

         We have the intertwining relations:
         
    𝔓_N+1(t)Λ_N^N+1=Λ_N^N+1𝔓_N(t),  ∀ t≥0, ∀ N∈ℕ.

     

We present two arguments for a proof. The first argument is a little more sophisticated and uses matrix stochastic calculus. It is conceptually more natural as it somehow explains the origin of the intertwining: the intertwining (<ref>) simply follows from consistency with respect to corners projection at the level of the matrix diffusions. On the other hand, we only give a full proof for the special initial condition 𝐇^(N)_0=0 which allows for important simplifications (in the case of Dyson Brownian motion which is also consistent with Λ_N^N+1 the adaptation of the argument works for any initial condition).  The second argument is a direct verification, which works for all initial conditions, and thus is a complete proof, using the explicit formula from (<ref>).

We begin with a little formalism giving consistency at the level of the eigenvalues from consistency at the level of the matrix process.

    
        Let (𝐂_t^(N);t≥ 0) and (𝐂_t^(N+1);t≥ 0) be stochastic processes on ℍ(N) and ℍ(N+1) respectively. Suppose that (𝖾𝗏𝖺𝗅_N(𝐂_t^(N));t≥ 0) and (𝖾𝗏𝖺𝗅_N+1(𝐂_t^(N+1));t≥ 0) are Markovian with semigroups (𝖯_N(t))_t≥ 0 and (𝖯_N+1(t))_t≥ 0 respectively. Assume that, for all t≥ 0,
       
    (𝐔^(K))^†𝐂_t^(K)𝐔^(K)d=𝐂_t^(K),  ∀𝐔^(K)∈𝕌(K),   K=N,N+1.

        Suppose moreover that there exists a set 𝒜⊆ℍ(N+1) such that for all 𝐂_0^(N+1)∈𝒜,
        
    Π_N^N+1(𝐂_t^(N+1))d=𝐂_t^(N),   ∀ t ≥ 0.

        Then, we have
        
    𝖯_N+1(t)Λ_N^N+1(𝐱,·)=Λ_N^N+1𝖯_N(t)(𝐱,·),  ∀𝐱∈𝖾𝗏𝖺𝗅_N+1(𝒜),  ∀ t≥ 0.

     
    
    Fix 𝐱 so that 𝐱=𝖾𝗏𝖺𝗅_(N+1)(𝐂_0^(N+1)), with 𝐂_0^(N+1)∈𝒜. We claim that, the LHS of (<ref>) is  given by,
    
    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐂_t^(N+1)))),

     while the RHS by,
     
    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐂_t^(N))).

     These are equal by virtue of (<ref>) and the conclusion follows. To see the claim, first observe that, 
     
    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐂_t^(N)))(d𝐲)=μ𝖯_N(t)(d𝐲),

     where, by unitary invariance (<ref>), μ is given by,
     
    μ(d𝐲)def=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐂_0^(N)))(d𝐲)=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐂_0^(N+1))))(d𝐲)=Λ_N^N+1(𝐱,d𝐲).

     On the other hand, again by unitary invariance (<ref>), we also have, 

    𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(Π_N^N+1(𝐂_t^(N+1))))(d𝐲)=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N+1(𝐂_t^(N+1)))Λ_N^N+1(d𝐲)=ν𝖯_N+1(t)Λ_N^N+1(d𝐲),

where ν is given by,

    ν(d𝐲)def=𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N+1(𝐂_0^(N+1)))=1_𝐱(d𝐲).

 This completes the proof.
    
    
     We will apply this proposition (with the obvious notational identifications) to prove Theorem <ref> in the special case 𝐇_0^(N)=0, namely for 𝐱=0.
     
     
     Let (𝐇_t^(N); t≥ 0) be the unique strong solution of the equation (<ref>) starting from 𝐇_0^(N)=0, where 0 denotes N× N zero matrix. We have
    
    𝐇_t^(N)d=Π_N^N+1(𝐇_t^(N+1)),  ∀ t≥ 0.



We first  show that the solution to the SDE (<ref>) can be solved explicitly in terms of the matrix exponential Brownian motion (𝐌_t^(θ);t≥ 0), which we introduce next. This is given by the unique strong solution of the matrix SDE: 
    
    d𝐌_t^(θ) =1/2𝐌_t^(θ)d𝐖_t^(N) -θ𝐌_t^(θ)dt,

where θ∈ℝ and (𝐖^(N)_t;t≥ 0) is an N× N complex matrix-valued Brownian motion.  Below we use the notation 𝐀^-† for (𝐀^†)^-1.





The unique strong solution (𝐇_t^(N);t≥ 0) of (<ref>), starting from 𝐇^(N)_0∈ℍ_+(N) is given by

    ((𝐌_t^(-ν_N))^-1(𝐇_0 + 1/2∫_0^t𝐌_s^(-ν_N)(𝐌_s^(-ν_N))^†ds)(𝐌_t^(-ν_N))^-†;t≥ 0).

where ν_N=1/4(η+N), with η being the same parameter as throughout this paper.








First, note that from (<ref>) we have
    
    d(𝐌_t^(θ))^† =1/2d(𝐖_t^(N))^†(𝐌_t^(θ))^† -θ(𝐌_t^(θ))^†dt.

Moreover, using Itô's formula we obtain 
    
    d(𝐌_t^(θ))^-1   = - 1/2d𝐖_t^(N)(𝐌_t^(θ))^-1 +θ(𝐌_t^(θ))^-1dt
    d(𝐌_t^(θ))^-†   = -1/2(𝐌_t^(θ))^-†d(𝐖_t^(N))^† +θ(𝐌_t^(θ))^-†dt.

One can then check directly that (<ref>) solves the SDE (<ref>) by applying Itô's formula to the expression in (<ref>) and using the equations above.





In the setting of Theorem <ref>, we have, for fixed t≥ 0,

    𝐇_t^(N)d=1/2∫_0^t𝐌_s^(ν_N)(𝐌_s^(ν_N))^†ds,

where ν_N=1/4(η+N), with η being the same parameter as throughout this paper.








For any fixed t≥ 0, define 

    (𝐍_s^(ν_N);0≤ s≤ t)def=((𝐌_t^(-ν_N))^-1𝐌_t-s^(-ν_N);0≤ s≤ t).

Then, making the change of variable u→ t-u in (<ref>), one can write

    𝐇_t^(N)d=(𝐌_t^(-ν_N))^-1𝐇_0 (𝐌_t^(-ν_N))^-†+ 1/2∫_0^t𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du.

From (<ref>), we have when 𝐇_0^(N)=0,

    𝐇_t^(N)d=1/2∫_0^t𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du.

Moreover, it is shown in Lemma 2.1 of <cit.> (essentially the matrix analogue of the time-reversal property of Brownian motion) that,

    (𝐍_u^(ν_N);0≤ u≤ t)d=(𝐌_u^(ν_N);0≤ u≤ t),

    which implies the result.




By virtue of (<ref>), it suffices to show that 

    ∫_0^t𝐌_s^(ν_N)(𝐌_s^(ν_N))^†dsd=Π_N^N+1(∫_0^t𝐌_s^(ν_N+1)(𝐌_s^(ν_N+1))^†ds),  ∀ t≥ 0,   N∈ℕ.

By applying Itô's formula we have from (<ref>) with θ=ν_N,

    d(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)   = d𝐌_t^(ν_N)(𝐌_t^(ν_N))^†+𝐌_t^(ν_N)d(𝐌_t^(ν_N))^†+
       d𝐌_t^(ν_N)d(𝐌_t^(ν_N))^†
       =1/2𝐌_t^(ν_N)(d𝐖_t^(N)+d(𝐖_t^(N))^†)(𝐌_t^(ν_N))^†+(N/2-2ν_N)𝐌_t^(ν_N)(𝐌_t^(ν_N))^†dt.

Now, using Lévy's characterization theorem, see <cit.>, one 
can easily check that

    d𝐖̂_t^(N)def=(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)^-1/2𝐌_t^(ν_N)d𝐖_t^(N)(𝐌_t^(ν_N))^†(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)^-1/2,

 is an N× N matrix complex Brownian motion. Thus, the equation (<ref>) can be written in the following closed form:

    d(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)   =1/2√((𝐌_t^(ν_N)(𝐌_t^(ν_N))^†))(d𝐖̂_t^(N)+d(𝐖̂_t^(N))^†)√(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†)
         +(N/2-2ν_N)𝐌_t^(ν_N)(𝐌_t^(ν_N))^†dt.

We next consider the SDE for the (N+1) × (N+1) matrix process, that we can write as follows
 
    d𝐌_t^(ν_N+1)   = 1/2𝐌_t^(ν_N+1)d𝐖_t^(N+1) -ν_N+1𝐌_t^(ν_N+1)dt 
       = 1/2[ 𝐌̃_t^(N)      𝐚_t;    𝐛_t^T      c_t ]d[ 𝐖_t^(N)     𝐮_t;   𝐯_t^T     w_t ] 
        - ν_N+1[ 𝐌̃_t^(N)      𝐚_t;    𝐛_t^T      c_t ]dt,

where 𝐌̃_t^(N) is the N× N top-left corner of 𝐌_t^(ν_N+1), 𝐚_t and 𝐛_t are 
ℂ^N-valued, and c_t∈ℂ. Here,
𝐖̃_t^(N), 𝐮_t,𝐯_t, and w_t are complex matrix, complex vector, and ℂ-valued Brownian motions and 𝐀^T denotes the transpose of 𝐀.  One can see from the above equation that

    d[ 𝐌̃_t^(N)      𝐚_t ]=1/2[ 𝐌̃_t^(N)      𝐚_t ]d𝐖_t^(N+1)
        - ν_N+1[ 𝐌̃_t^(N)      𝐚_t ]dt.

We also have that for the N× N top-left corner of 𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†: 

    Π_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†)= [ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†.


Therefore,
by
Itô's formula we get 

    dΠ_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†)   = 
        d[ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†+
      [ 𝐌̃_t^(N)      𝐚_t ]d[ 𝐌̃_t^(N)      𝐚_t ]^†
          + d[ 𝐌̃_t^(N)      𝐚_t ]d[ 𝐌̃_t^(N)      𝐚_t ]
        ^†.

Using (<ref>), we then obtain

    dΠ_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†)     = 1/2[ 𝐌̃_t^(N)      𝐚_t ]( d𝐖_t^(N+1)+ d(𝐖_t^(N+1))
    ^†)
        [ 𝐌̃_t^(N)      𝐚_t ]^†
         + (N+1/2-2ν_N+1)[ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†dt.

Similarly to (<ref>), by virtue of Levy's characterization theorem, we can define the N× N matrix-valued complex Brownian motion (𝐖̂_t^(N))_t≥ 0 (different from (<ref>)) by the formula

    d𝐖̂_t^(N)def=([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)^-1/2[ 𝐌̃_t^(N)      𝐚_t ]d𝐖_t^(N+1)×
    [ 𝐌̃_t^(N)      𝐚_t ]^†([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)^-1/2.

Using these we represent the SDE (<ref>) in the following closed form, recalling (<ref>),

    dΠ_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†) = 1/2√([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)(d𝐖̂_t^(N)+ d(𝐖̂_t^(N))
    ^†)×
       √([ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†)
        + (N+1/2-2ν_N+1)[ 𝐌̃_t^(N)      𝐚_t ][ 𝐌̃_t^(N)      𝐚_t ]^†dt.

Now, comparing the SDEs (<ref>) and 
(<ref>)
and considering the definition of the parameter ν_N, it follows that, both 
(𝐌_t^(ν_N)(𝐌_t^(ν_N))^†;t ≥ 0)
and
(Π_N^N+1(𝐌_t^(ν_N+1)(𝐌_t^(ν_N+1))^†);t ≥ 0)
satisfy the same equation, having a unique weak solution, and hence, these processes have the same distribution. This in particular implies 
that 

    ∫_0^t𝐌_s^(ν_N)(𝐌_s^(ν_N))^†dsd=Π_N^N+1(∫_0^t𝐌_s^(ν_N+1)(𝐌_s^(ν_N+1))^†ds),   N∈ℕ,

 for any fixed t>0, as desired.

We now give a direct proof for Theorem <ref> for any initial condition. 



      We will show  
        
    𝔓_N+1(t)Λ_N^N+1(𝐱,·)=Λ_N^N+1𝔓_N(t)(𝐱,·),  ∀𝐱∈𝕎_N+1,+^∘,  ∀ t≥ 0.

        The case of general 𝐱∈𝕎_N+1,+ then follows by the Feller property, see Proposition <ref>. We have using the explicit expressions in (<ref>) and (<ref>), that (<ref>) boils down to showing:

    ∫_𝕎_N+1,+e^-λ_N+1t(𝔭^N+1_t(x_i,y_j))_i,j=1^N+11_𝐳≺𝐲d𝐲=
            ∫_𝕎_N,+e^-λ_Nt1_𝐲≺𝐱(𝔭^N_t(y_i,z_j))_i,j=1^Nd𝐲.

We write 𝐱≺̃𝐲 to denote y_1≥ x_1> y_2≥ x_2>…≥ x_N> y_N+1, when 𝐱∈𝕎_N and 𝐲∈𝕎_N+1.
Then, we recall the well-known identity
1_𝐱≺̃𝐲= (φ(x_i,y_j))_i,j=1^N+1 with 


    φ(x,y)=
    -1_y>x,       x,y∈ℝ_+,
    
     1,       x=virt,

where we use the convention x_N+1=virt for a “virtual variable". Note that, we can replace 1_𝐲≺𝐱 with 1_𝐲≺̃𝐱 in (<ref>).
Accordingly, with this replacement, the LHS of (<ref>), by the Andreief identity, is equal to 

    e^-λ_N+1t[      [𝔭_t^(N+1)φ^*](x_1,z_1)      [𝔭_t^(N+1)φ^*](x_1,z_2)                            ⋯                            1;      [𝔭_t^(N+1)φ^*](x_2,z_1)      [𝔭_t^(N+1)φ^*](x_2,z_2)                            ⋯                            1;                            ⋮                            ⋮                                                         ⋮; [𝔭_t^(N+1)(t)φ^*](x_N+1,z_1)    [𝔭_t^(N+1)φ^*](x_N+1,z_2)                            ⋯                            1 ],

where φ^*(x,y)=φ(y,x), and by definition [𝔭_t^(N+1)φ^*](x,y) denotes compositition of kernels so that the entries of the matrix above are as follows:

    [𝔭_t^(N+1)φ^*](x_i,z_j)=∫_0^∞𝔭_t^(N+1)(x_i,y)φ^*(y,z_j)dy=-∫_z_j^∞𝔭_t^(N+1)(x_i,y)dy,  ∀ i,j=1,…,N+1.

For the RHS, we first expand the (N+1)×(N+1) determinant corresponding to 1_𝐱≺̃𝐲. We next use the Andreief identity, and then again recombine terms to obtain 

    ∑_l=0^N+1(-1)^N+1-l e^-λ_Nt   ([φ^*𝔭_t^(N)](x_i,z_j))_i,j=1^N+1=
       
         e^-λ_Nt[   [φ^*𝔭_t^(N)](x_1,z_1)   [φ^*𝔭_t^(N)](x_1,z_2)                       ⋯                       1;   [φ^*𝔭_t^(N)](x_2,z_1)   [φ^*𝔭_t^(N)](x_2,z_2)                       ⋯                       1;                       ⋮                       ⋮                                               ⋮; [φ^*𝔭_t^(N)](x_N+1,z_1) [φ^*𝔭_t^(N)](x_N+1,z_2)                       ⋯                       1 ],

where 

    [φ^*𝔭_t^(N)](x_i,z_j)=∫_0^∞φ^*(x_i,y)𝔭_t^(N)(y,z_j)dy=-∫_0^x_i𝔭_t^(N)(y,z_j)dy,  ∀ i,j=1,…,N+1.

We now claim that

    𝔭_t^(N)(x,z)=e^(η/2+N)t∂_x∫_z^∞𝔭_t^(N+1)(x,y)dydef=q_t(x,y).

 We first note, the key algebraic consistency relation
∂_x𝖫_x^(N+1)=𝖫_x^(N)∂_x+(-η/2-N)∂_x.
Making use of this, it can be checked that q_t(x,y) is the fundamental solution to the Kolmogorov backward equation associated to the one-dimensional diffusion
𝖫_x^(N), which is unique, and hence, (<ref>) holds. 
Thus, we have

    -∫_0^x_i𝔭_t^(N)(y,z_j)dy=e^(η/2+N)t[∫_z_j^∞𝔭_t^(N+1)(0,y)dy-∫_z_j^∞𝔭_t^(N+1)(x_i,y)dy], ∀ i,j=1,…,N+1.

One can now substitute this in the determinant in (<ref>) and use column operations to show the desired identity, by noting that λ_N+1-λ_N=N(-η/2-N).






 §.§ Invariant measure and convergence to equilibrium

The inverse Laguerre ensemble 𝔐_N^η
of parameter η>-1 is the probability measure on (𝕎_N,+)_N=1^∞ given by 
	
    𝔐_N^η(d𝐱)=1/𝒵_N^(η)Δ_N^2(x)∏_i=1^Nx_i^-η-2Ne^-1/x_id𝐱,

for some normalisation constant 𝒵_N^(η). We now show that the solution (𝔵^(N)(t);t≥ 0) of (<ref>) started from arbitrary initial condition, converges to 𝔐_N^η.


    Let η>-1 and consider the unique strong solution (𝔵^(N)(t);t≥ 0) of (<ref>). Suppose 𝔎∈ℳ_p(𝕎_N,+)  and that 𝖫𝖺𝗐(𝔵^(N)(0))=𝔎. Then,  as t→∞,

    𝔵^(N)(t) d⟶𝐙,   where 𝖫𝖺𝗐(𝐙)=𝔐_N^η.
 
In particular, 𝔐_N^η is the unique invariant measure of (𝔓_N(t))_t≥ 0.


We use the corresponding matrix process to prove the result.
Recall that from (<ref>), the unique strong solution (𝐇_t^(N);t≥ 0) 
starting from 𝐇_0^(N)∈ℍ_+(N) can be written as 

    (𝐌_t^(ν_N))^-1𝐇^(N)_0 (𝐌_t^(-ν_N))^-†+ 1/2∫_0^t𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du.

Let us take 𝐇_0^(N) such that 𝖫𝖺𝗐(𝖾𝗏𝖺𝗅_N(𝐇_0^(N)))=𝔎 (and independent of the driving Brownian motions). Then, from Lemma 2.2 in <cit.> the first term converges to 0 almost surely as t→∞. Hence, we obtain

    𝐇_t^(N)d⟶1/2∫_0^∞𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du,  as  t→∞.

Now, by the matrix Dufresne identity proven in <cit.>, see also <cit.>, we know that

    𝖫𝖺𝗐(1/2∫_0^∞𝐍_u^(ν_N)(𝐍_u^(ν_N))^†du) =ℑ𝔏_N^η,

where ℑ𝔏_N^η is the inverse Laguerre measure on ℍ_+(N)  given by the formula,

    ℑ𝔏_N^η(d𝐀)=(𝐀)^-η-2Nexp(-Tr(𝐀^-1))  d𝐀,

with d𝐀 denoting the Lebesgue measure on ℍ_+(N), for some normalisation constant 𝒵̃_N^(η).
By Weyl's integration formula we know that 

    (𝖾𝗏𝖺𝗅_N)_*ℑ𝔏_N^η=𝔐_N^η,

and this concludes the proof of (<ref>). The final statement follows by a standard argument.


As a consequence of Theorem <ref> and uniqueness of the invariant measures from Proposition <ref> we obtain the consistency relation between the inverse Laguerre ensembles; see also <cit.> for a direct proof, using orthogonal polynomials.

     
        Let η>-1. Then, we have
         
    𝔐^η_N+1Λ_N^N+1=𝔐^η_N,  ∀ N∈ℕ.



We also have the following equivalent abstract description of the measure 𝔐^η from the introduction.


Let η>-1. The unique 𝔪∈ℳ_p(Ω_+) satisfying:

    𝔪Λ_N^∞=𝔐_N^η,   ∀ N∈ℕ,

is given by 𝔐^η.



This is the main result of <cit.>.





 §.§ Proof of Theorem <ref> and Theorem <ref>



This follows by taking the semigroups (𝖯_N(t))_t≥ 0 to be (𝔓_N(t))_t≥ 0 in Theorem <ref>, Theorem <ref> and Proposition <ref> all of whose conditions are satisfied by virtue of Proposition <ref> and Proposition <ref> and Theorem <ref> and Theorem <ref>.



By taking (𝖯_N(t))_t≥ 0 to be (𝔓_N(t))_t≥ 0 this is a consequence of Theorem <ref> by virtue of Proposition <ref> and Proposition <ref>. 





 §.§ On the Gibbs property


We briefly discuss a new Gibbs resampling property, see <cit.>, underlying our model. We assume the reader is familiar with the basic terminology of Gibbsian line ensembles, see <cit.>.  This subsection is not part of our argument and can be skipped. 

For η>-1, there exists an explicit spectral expansion of 𝔭_t^(N)(x,y) in terms of special functions, see <cit.>. This expansion is well-adapted to looking at the t→∞ asymptotics of 𝔭_t^(N)(x,y) and using it one can prove that the semigroup (𝔓_N(t))_t≥ 0 is exactly the transition semigroup of N independent one-dimensional diffusions with generator 𝖫^(N) from (<ref>) conditioned to never intersect[We note that this statement does not follow directly from the Doob h-transform structure of (𝔓_N(t))_t≥ 0 in (<ref>). One needs to compute the asymptotics of the first collision time of the N independent one-dimensional diffusions 𝖫^(N) in order show that these two structures coincide and this is in general a delicate problem.]. In particular, (𝔵^(N)(t);t≥ 0), viewed as a line ensemble with N lines, see <cit.>, enjoys a certain Gibbs resampling property with respect to the law of conditioned 𝖫^(N)-bridges, see <cit.> for the rigorous definition of diffusion bridges; in the framework of <cit.> we replace Brownian bridges with 𝖫^(N)-bridges. Similarly, the rescaled process (𝗑^(N)(t);t≥ 0) from (<ref>) enjoys a Gibbs resampling property with respect to 𝖫_rsc^(N)-bridges, where 𝖫_rsc^(N) is the generator of the rescaled by N^-1 diffusion 𝖫^(N) given by:

    𝖫_rsc^(N)=x^2/2∂_x^2+((1-η/2-N)x+1/2N)∂_x,

with transition density denoted by 𝔮_t^(N)(x,y). It is then a natural question whether, the limiting paths ((𝗑_i(t))_i=1^∞;t≥ 0), which we do not yet know if they are non-intersecting, enjoy a Gibbs resampling property as well. If they did this would give an alternative, to the one in Section <ref> below, albeit more complicated, way to prove non-intersection.

One can show, that given the convergence of the paths, that we already know from Theorem <ref>, this question really boils down, modulo assumptions of lesser importance to avoid pathological situations, to convergence of the free 𝖫_rsc^(N)-bridge, whose time-inhomogeneous generator, in case of the bridge ending at point y at time T>0, is given by, for 0≤ t<T, see <cit.>,

    x^2/2∂_x^2+((1-η/2-N)x+1/2N+x^2∂_x log𝔮_T-t^(N)(x,y))∂_x,

to a limiting diffusion bridge as N →∞. This question turns out to be complicated. The spectral expansion in <cit.> is not well-adapted for looking at the N →∞ limit of the above expression.  Nevertheless, using some judicious transformations, it is possible to connect this problem to a problem of asymptotics of the logarithmic derivative of the transition kernel of a Schrödinger semigroup with an explicit N-dependent Morse potential 𝖵_N(x), see <cit.>. What we gain by doing this is that all N-dependence of the problem is now in the potential 𝖵_N(x) and it is  easy to see that it converges to an explicit potential 𝖵_∞(x). Alas, the limiting potential 𝖵_∞(x) has the “wrong" sign and the associated Schrödinger semigroup blows up! In particular, the transition kernel of the Schrödinger semigroup with potential 𝖵_N(x) diverges as N→∞. However, its logarithmic derivative may still well converge, due a subtle cancellation, and so this does not contradict the  existence of a limiting Gibbs property. 









§ THE ISDE


In this section we first prove that (𝗑_i(·))_i=1^∞ almost surely consists of non-intersecting paths in Theorem <ref> and then in Theorem <ref> that it solves the ISDE (<ref>). The main idea will be to use various “characteristic polynomial"-like objects associated to the finite-dimensional dynamics (𝗑^(N)_i(·))_i=1^N, first as Lyapunov functions, to control uniformly in N, the times two paths come close and then to show convergence of the singular drift term. We need some preliminaries.

We recall a result from <cit.> on convergence of polynomials with real zeros to analytic functions that we use as one of the ingredients in the proofs of 
Theorem <ref> and Theorem <ref> below. Given, 𝐱^(N)∈𝕎_N define the polynomial

    𝔘_N(z;𝐱^(N))def=∏_j=1^N(1-x_j^(N)z).

This is the reverse characteristic polynomial of a matrix with eigenvalues 𝐱^(N). We also have the following definition.






Define the entire functions 𝖤(z;ω) and 𝖤_+(z;ω), indexed by ω∈Ω and ω∈Ω_+ respectively,
given by their Hadamard products:

    𝖤(z;ω)   def=e^-γ z-δ-∑_i=1^∞[(x_i^+)^2+(x_i^-)^2]/2z^2∏_j=1^∞e^x_j^+ z(1-x_j^+z)∏_j=1^∞e^-x_j^- z(1+x_j^- z),  ω∈Ω,
    𝖤_+(z;ω)   def=e^-γ z∏_j=1^∞e^x_j z(1-x_jz),  ω∈Ω_+.

  


Then, we have the following result, see
Proposition 2.5 and Remark 2.7 of <cit.>.


Suppose (𝐱^(N))_N=1^∞∈(𝕎_N)_N=1^∞ such that ω(𝐱^(N))N→∞⟶ω∈Ω. Then, the following convergence holds uniformly on compact sets in ℂ:

    𝔘_N(z;𝐱^(N))→𝖤(z;ω),    as  N→∞.

If instead (𝐱^(N))_N=1^∞∈(𝕎_N,+)_N=1^∞ such that ω(𝐱^(N))N→∞⟶ω∈Ω_+, then the following convergence holds uniformly on compact sets in ℂ:

    𝔘_N(z;𝐱^(N))→𝖤_+(z;ω),    as  N→∞.


Note that this convergence statement is consistent with the embedding (<ref>) of Ω_+ into Ω. Below we will only use the result for 𝕎_N,+ and Ω_+.






 §.§ Non-intersection via characteristic polynomials




We use the following notation convention throughout. For (𝐱,γ)∈Ω_+ we write ℙ_(𝐱,γ) for the law of (𝐗_t^Ω_+;t≥ 0) from Theorem <ref> starting from (𝐱,γ) and abusing notation (as we do not assume that all these processes are defined on the same probability space unless otherwise stated), for 𝐱^(N)∈𝕎_N,+, we write ℙ_𝐱^(N) for the law of the process solving (<ref>) starting from 𝐱^(N). We write 𝔼_(𝐱,γ) and 𝔼_𝐱^(N) for the corresponding expectations.


   Let η∈ℝ, 𝐱∈𝕎_∞,+^∘, γ∈ℝ_+, with ∑_i=1^∞ x_i≤γ. Consider the Feller process on Ω_+, (𝐗_t^Ω_+;t≥ 0)=(((𝗑_i(t))_i=1^∞,γ(t));t≥ 0) from Theorem <ref> with initial condition 𝐗_0^Ω_+=(𝐱,γ). Then, for all T≥ 0 and n∈ℕ we have
    
    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_n(t)>0
    )   =1,
    ℙ_(𝐱,γ)(inf_t∈[0,T]|
    𝗑_n(t)-𝗑_n+1(t)
    |>0)   =1.




  We prove the result using an inductive argument. For each N∈ℕ, let 𝐱^(N)∈𝕎_N,+^∘ converge to (𝐱,γ)∈Ω_+ under the embedding (<ref>). 
Let 𝗑^(N) be the solution to the equation (<ref>) started at 𝐱^(N).
We will make use of the log transform of (<ref>), which we consider in its integral form:

    log𝗑_i^(N)(t)    = log x_i^(N)+ 𝗐_i(t) -1+η/2t+∫_0^t1/2N𝗑_i^(N)(s)ds
         +
    ∫_0^t∑_j=1, j≠ i^N𝗑_j^(N)(s)/𝗑_i^(N)(s)-𝗑_j^(N)(s)ds, i=1,…,N.

Let us continue with some definitions. We always use the convention inf∅ =∞ in what follows. 
Define, 
for any N∈ℕ, the following stopping times (with respect to the natural filtration) associated to (𝗑_i^(N))_i=1^N,

    σ_n^N(ϵ)   def=inf{t≥ 0;𝗑_n^(N)(t)≤ϵ},  ϵ>0,  n=1,…,N
    τ_n^N(δ)   def=inf{t≥ 0;|1-𝗑_n+1^(N) (t)/𝗑_n^(N) (t)|≤δ},  δ>0,  n=1,…,N-1,
    θ_n^N(r)   def=inf{t≥ 0;𝗑_n^(N)(t)≥ r},  r>0,  n=1,…,N,

and 
analogously, the stopping times (with respect to the natural filtration) associated to (𝗑_i)_i=1^∞,

    σ_n(ϵ)   def=inf{t≥ 0;𝗑_n(t)≤ϵ},  ϵ>0,  n∈ℕ,
    τ_n(δ)   def=inf{t≥ 0;|1-𝗑_n+1 (t)/𝗑_n (t)|≤δ},  δ>0,  n∈ℕ,
    θ_n(r)   def=inf{t≥ 0;𝗑_n(t)≥ r},  r>0,  n∈ℕ.

Note that, by Lemma <ref>, τ_n^N(δ) is well-defined for any N∈ℕ and n≤ N-1. 
It will be justified later on that τ_n(δ) is also well-defined for all n∈ℕ.


Now, let N∈ℕ. We define for any 𝐱^(N)∈𝕎_N,+^∘, the following variant of the characteristic polynomial associated to 𝐱^(N),

    Ψ_n^N(𝐱^(N))   def=∏_i=1^n∏_j=n+1^N(1-x_j^(N)/x_i^(N)),   n=1,…,N-1,
 
and consider the Lyapunov function
given as:

    f_n^N(𝐱^(N))def=-log(Ψ_n^N(𝐱^(N))),   n=1,…,N-1.

It is easy to see that f_n^N(𝐱^(N))>0 for all 𝐱^(N)∈𝕎_N,+^∘ and n=1,…,N-1.
    We next define
 
    κ_n^N(R)def=inf{t≥ 0;f_n^N(𝗑^(N)(t))≥ R},    R>0,   n=1,…,N-1.

    Observe that,

    {τ_n^N(δ)≤ a}⊂{κ_n^N(log(δ^-1))≤ a},  for δ<1.

We are going to prove by induction on n that 
for all T≥ 0 and n∈ℕ, 

    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_n(t)>0
    )   =1,
    ℙ_(𝐱,γ)(inf_t∈[0,T]|1-𝗑_n+1(t)/𝗑_n(t)|>0)   =1.

Note that, this implies non-collision, since given (<ref>) and (<ref>), we have almost surely,

    inf_t∈[0,T]|
    𝗑_n(t)-𝗑_n+1(t)
    |≥inf_t∈[0,T]𝗑_n(t)
    inf_t∈[0,T]|1-𝗑_n+1(t)/𝗑_n(t)|,

and hence,

    ℙ_(𝐱,γ)(inf_t∈[0,T]|
    𝗑_n(t)-𝗑_n+1(t)
    |>0)=1.


The base case.
Using (<ref>), we have for any t≥ 0 that
	
    log𝗑_1^(N)(t∧σ_1^N(ϵ)∧θ_1^N(r))   ≥log x_1^(N)
    		-|1+η/2|t+𝗐_1(t∧σ_1^N(ϵ)∧θ_1^N(r)).

	Therefore, since t ↦𝗐_1(t∧σ_1^N(ϵ)∧θ_1^N(r)) is a mean-zero martingale, taking expectation of both sides we obtain
	
    log r+logϵℙ_𝗑^(N)(σ_1^N(ϵ)< t∧θ_1^N(r))   ≥log x_1^(N)-|1+η/2|t,

and thus, for any ϵ<1,

    ℙ_𝐱^(N)(σ_1^N(ϵ)< t∧θ_1^N(r))   ≤log r-log x_1^(N)+|1+η/2|t/|logϵ|.

Observe that, the bound (<ref>) is uniform in N (recall that x_1^(N)→ x_1>0). We would like to conclude that the same bound (with x_1 in the formula instead) holds for the stopping times σ_1(ϵ) and θ_1(r) corresponding to the limit paths. A little argument is needed though. These stopping times, viewed as functionals on paths are not continuous, but fortunately they are not far from continuous. For more details see Chapter VI, Sections 2,3 of <cit.> which we follow closely. For a ∈ℝ_+, f∈ C(ℝ_+,ℝ_+), we define

    𝖲_≥^a(f)=inf{t≥ 0; f(t)≥ a},  𝖲_≤^a(f)=inf{t≥ 0; f(t)≤ a}.

 
 Write ℝ_+,ext=ℝ_+∪{∞}. For fixed f∈ C(ℝ_+,ℝ_+) define the map 𝖡_f by,
 
    𝖡_f:ℝ_+^2    ⟶ℝ_+,ext^2
    
        (a,b)   ↦(𝖲_≤^a(f),𝖲_≥^b(f)),

 which from Lemma 2.10 of Chapter VI of <cit.> only has jump discontinuities, while for fixed (a,b)∈ℝ_+^2 define the map 𝖠_(a,b) by,
 
    𝖠_(a,b):C(ℝ_+,ℝ_+)    ⟶ℝ_+,ext^2
    
     f   ↦(𝖲_≤^a(f),𝖲_≥^b(f)).

 Now, if we let 𝔇_(a,b) denote the set of discontinuities of the map 𝖠_(a,b) and 𝖣(f) denote the set of discontinuities of the map 𝖡_f, then from Proposition 2.11 of Chapter VI of <cit.>, since 𝖠_(a,b) is continuous at every f so that (a,b)∉𝖣(f), we obtain that,
 
    𝔇_(a,b)⊂{f∈ C(ℝ_+,ℝ_+):(a,b)∈𝖣(f)}.

 Let us write ℙ^(1)_(𝐱,γ) for 𝖫𝖺𝗐(𝗑_1) on C(ℝ_+,ℝ_+) under ℙ_(𝐱,γ). Then, define

    𝒱={(a,b)∈ℝ_+^2:ℙ_(𝐱,γ)^(1)(𝔇_(a,b))>0}⊂{(a,b)∈ℝ_+^2:ℙ_(𝐱,γ)^(1)((a,b)∈𝖣(𝗑_1))>0}.
 
Then, by virtue of the fact that the map 𝖡_f only has jump discontinuities, we obtain, see Lemma 3.12 of Chapter VI of <cit.>, that the sets 𝒱_1={a∈ℝ_+:(a,b)∈𝒱} and 𝒱_2={b∈ℝ_+:(a,b)∈𝒱} are at most countable. In particular, we can find sequences (ϵ_m)_m=1^∞ and (r_ℓ)_ℓ=1^∞, with ϵ_m→ 0 and r_ℓ→∞, as m,ℓ→∞, so that for any m,ℓ∈ℕ, ℙ_(𝐱,γ)^(1)(𝔇_(ϵ_m,r_ℓ))=0. Now, observe that,

    (σ_1^N(ϵ),θ_1^N(r))=𝖠_(ϵ,r)(𝗑_1^(N))
    ,   (σ_1(ϵ),θ_1(r))=𝖠_(ϵ,r)(𝗑_1).

Hence, by the continuous mapping theorem, we obtain that for all m,l∈ℕ,  as N →∞,

    (σ_1^N(ϵ_m),θ_1^N(r_ℓ))d⟶(σ_1(ϵ_m),θ_1(r_ℓ)).

By the Portmanteau theorem for convergence in distribution, we get that, for any open set 𝒪⊆ℝ^2,  
 
    ℙ_(𝐱,γ)((σ_1(ϵ_m),θ_1(r_ℓ))∈𝒪)
         ≤lim inf_N→∞ℙ_𝐱^(N)((σ_1^N(ϵ_m),θ_1^N(r_ℓ))∈𝒪).

 From this and  (<ref>) we conclude that (for ϵ_m<1),

    ℙ_(𝐱,γ)(σ_1(ϵ_m)< t∧θ_1(r_ℓ))≤lim inf_N→∞ℙ_𝗑^(N)(σ_1^N(ϵ_m)< t∧θ_1^N(r_ℓ))≤log r_ℓ-log x_1+|1+η/2|t/|logϵ_m|,

and hence, by the monotone convergence theorem,

    ℙ_(𝐱,γ)(lim_m→∞σ_1(ϵ_m)< t∧θ_1(r_ℓ))=0.

Therefore, by taking ℓ,t→∞, it follows, by virtue of the fact that (recall that r_ℓ→∞), θ_1(r_ℓ)a.s.⟶∞, that ℙ_(𝐱,γ)(lim_m →∞σ_1(ϵ_m)=∞)=1, namely, since ϵ_m → 0,

    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_1(t)>0
    )=1.



We now proceed to prove the second assertion. 
Using Itô's formula, we have 

    df_1^N(𝗑^(N)(t))   =
        d𝖬_1^(N)(t)+
        N-1/2N𝗑_1^(N)(t)dt-∑_j=2^N(𝗑_j^(N)(t))^2/(𝗑_1^(N)(t)-𝗑_j^(N)(t))^2dt 
         
    +∑_j=2^N∑_ℓ=2,ℓ≠ j^N𝗑_j^(N)(t)𝗑_ℓ^(N)(t)/(𝗑_1^(N)(t)-𝗑_ℓ^(N)(t))(𝗑_j^(N)(t)-𝗑_ℓ^(N)(t))dt.

where

    𝖬_1^(N)(t)def=(N-1) 𝗐_1(t)+∑_ℓ=2^N∫_0^t𝗑_1^(N)d𝗐_1(s)-𝗑_ℓ^(N)d𝗐_ℓ(s)/𝗑_1^(N)-𝗑_ℓ^(N),   ∀ t ≥ 0,

is a local martingale. Now, we observe, and this is key, that for any t≥ 0, the last term on the RHS of (<ref>) is non-positive, by virtue of the elementary inequality, for 𝐲∈𝕎_N,+^∘,

    ∑_j=2^N∑_ℓ=2,ℓ≠ j^Ny_j y_ℓ/(y_1-y_ℓ)(y_j-y_ℓ)
    =∑_j<ℓj,ℓ=2^Ny_j y_ℓ/y_j-y_ℓ(1/y_1-y_ℓ-1/y_1-y_j)≤ 0.

Thus, we can drop this term and write 

    f_1^N(𝗑^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)))   ≤ f_1^N(𝐱^(N))+𝖬_1^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)) 
         +∫_0^t∧κ_1^N(R)∧σ_1^N(ϵ)1/2𝗑_1^(N)(s)ds
       ≤ f_1^N(𝐱^(N))+𝖬_1^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ))+ 
        t/ϵ.

Now, since the stopped process t ↦𝖬_1^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)) is a mean-zero martingale, we get

    𝔼_𝐱^(N)[f_1^N(𝗑^(N)(t∧κ_1^N(R)∧σ_1^N(ϵ)))]   ≤ f_1^N(𝐱^(N))+t/ϵ.

Therefore, we have

    Rℙ_𝐱^(N)(κ_1^N(R)< t∧σ_1^N(ϵ))≤sup_N∈ℕf_1^N(𝐱^(N))+
        t/ϵ.

Note that, since
ω(𝐱^(N))N→∞⟶(𝐱,γ), it follows from Proposition <ref> that 
Ψ_1^N(𝐱^(N)) converges to a strictly positive limit, 
and hence, 
Cdef=sup_N∈ℕf_1^N(𝐱^(N))<∞. We thus have,

    ℙ_𝐱^(N)(κ_1^N(R)< t∧σ_1^N(ϵ))≤C+t/ϵ/R.

Let δ<1 be arbitrary. We have in view of (<ref>) that 

    ℙ_𝐱^(N)(τ_1^N(δ)< t∧σ_1^N(ϵ))≤C+t/ϵ/|logδ|,

and note that this bound is independent of N. Observe that,

    τ_1^N(δ)=𝖲_≤^δ(|1-𝗑_2^(N)/𝗑_1^(N)|),   τ_1(δ)=𝖲_≤^δ(|1-𝗑_2/𝗑_1|),

and that by virtue of (<ref>), t↦ |𝗑_1(t)|^-1|𝗑_1(t)-𝗑_2(t)| is almost surely, a well-defined continuous function. We now argue similarly to before to obtain the existence of sequences, abusing notation (the sequence (ϵ_ℓ)_ℓ=1^∞ here may have nothing to do with the sequence (ϵ_m)_m=1^∞ from earlier), (δ_m)_m=1^∞, (ϵ_ℓ)_ℓ=1^∞, with δ_m, ϵ_ℓ→ 0, as m,ℓ→∞, so that, for all m,ℓ∈ℕ, as N →∞,

    (τ_1^N(δ_m),σ_1^N(ϵ_ℓ))d⟶(τ_1(δ_m),σ_1(ϵ_ℓ)).

Hence, by taking N→∞ in (<ref>), one obtains, for δ_m<1,

    ℙ_(𝐱,γ)(τ_1(δ_m)< t∧σ_1(ϵ_ℓ))≤lim inf_N →∞ℙ_𝐱^(N)(τ_1^N(δ_m)< t∧σ_1^N(ϵ_ℓ))≤C+t/ϵ_ℓ/|logδ_m|.

It thus follows that,

    ℙ_(𝐱,γ)(lim_m→∞τ_1(δ_m)< t∧σ_1(ϵ_ℓ))=0.

Now, we take t→∞,
 and then ℓ→∞, to finally get

    ℙ_(𝐱,γ)(lim_m→∞τ_1(δ_m)≥lim_ℓ→∞σ_1(ϵ_ℓ))=1.

But ℙ_(𝐱,γ)(lim_ℓ→∞σ_1(ϵ_ℓ )=∞)=1 by virtue of (<ref>).
This implies

    ℙ_(𝐱,γ)(lim_m →∞τ_1(δ_m)=∞)=1.

That is, for all T≥ 0, we have

    ℙ_(𝐱,γ)(inf_t∈[0,T]|1-𝗑_2(t)/𝗑_1(t)|>0)=1,

which completes the proof of the case n=1.

The inductive step. We next assume that (<ref>) and  (<ref>) hold for n=k, and prove them for n=k+1.
From (<ref>), we can write

    log𝗑_k+1^(N)(t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r))    ≥log x_k+1^(N)
        +𝗐_k+1(t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r))
     
     -|1+η/2|   t+∫_0^t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r)∑_j=1, j≠ k+1^k𝗑_j^(N)(t)/𝗑_k+1^(N)(t)-𝗑_j^(N)(t)dt
    ≥log x_k+1^(N)   
        -|1+η/2|t+𝗐_k+1(t∧σ_k+1^N(ϵ)∧τ_k^N(δ)∧θ_k+1^N(r))-kt/δ,

 and thus, we have,

    log r+logϵℙ_𝗑^(N)(σ_k+1^N(ϵ)< t∧τ_k^N(δ)∧θ_k+1^N(r))   ≥log x_k+1^(N)-|1+η/2|t-kt/δ.

 One can check, by making use of the induction hypothesis, that as previously, there exist sequences (ϵ_m)_m=1^∞, (δ_ℓ)_ℓ=1^∞, (r_q)_q=1^∞, with ϵ_m,δ_ℓ→ 0, r_q →∞, as m,ℓ,q →∞, so that the corresponding stopping times converge in distribution, as N →∞, to their limiting counterparts, and in particular we obtain,
 
    ℙ_(𝐱,γ)(lim_m→∞σ_k+1(ϵ_m)< t∧τ_k(δ_ℓ)∧θ_k+1(r_q))=0.

 Hence, it follows that ℙ_(𝐱,γ)(lim_m →∞σ_k+1(ϵ_m)=∞)=1, if we take m →∞ first, and then, t,ℓ,q→∞ whereby τ_k(δ_ℓ),θ_k+1(r_q)a.s.⟶∞. Equivalently,

    ℙ_(𝐱,γ)(inf_t∈[0,T]𝗑_k+1(t)>0
    )=1.

 
We next apply Itô's formula to f_k+1^N to obtain

    df_k+1^N(𝗑^(N)(t))   =
        d𝖬_k+1^(N)(t)+
        ∑_i=1^k+1N-k-1/2N𝗑_i^(N)(t)dt-∑_i=1^k+1∑_j=k+2^N(𝗑_j^(N)(t))^2/(𝗑_i^(N)(t)-𝗑_j^(N)(t))^2dt  
         +
       ∑_i=1^k+1∑_j=k+2^N∑_ℓ≠ i,jℓ=1^N
    𝗑_j^(N)(t)𝗑_ℓ^(N)(t)/(𝗑_i^(N)(t)-𝗑_ℓ^(N)(t))(𝗑_j^(N)(t)-𝗑_ℓ^(N)(t))dt,

where t↦𝖬_k+1^(N)(t) stands for the corresponding local martingale term.
We note, and this is again important, that the last term in the above expression is non-positive for any t≥ 0, by virtue of the inequality, for 𝐲∈𝕎_N,+^∘,

    ∑_i=1^k+1∑_j=k+2^N∑_ℓ≠ i,jℓ=1^N y_j y_ℓ/(y_i-y_ℓ)(y_j-y_ℓ)≤ 0,

which is easily seen to be true since the LHS can be written as,

    ∑_i=1^k+1∑_j=k+2^N∑_ℓ=1,ℓ≠ i^k+1y_j y_ℓ/(y_i-y_ℓ)(y_j-y_ℓ) +∑_i=1^k+1∑_j=k+2^N∑_ℓ≠ jℓ=k+2^N y_j y_ℓ/(y_i-y_ℓ)(y_j-y_ℓ)
       =
    ∑_j=k+2^N∑_i<ℓi,ℓ=1^k+1y_j/y_i-y_ℓ(y_ℓ/y_j-y_ℓ-y_i/y_j-y_i) +∑_i=1^k+1∑_j<ℓj,ℓ=k+2^Ny_j y_ℓ/y_j-y_ℓ(1/y_i-y_ℓ-1/y_i-y_j) ≤ 0.

Thus, by dropping it, we have

    f_k+1^N(𝗑^(N)(t∧κ_k+1^N(R)∧σ_k+1^N(ϵ)))   ≤ f_k+1^N(𝐱^(N))+𝖬_k+1^(N)(t∧κ_k+1^N(R)∧σ_k+1^N(ϵ))
         +∑_i=1^k+1∫_0^t∧κ_k+1^N(R)∧σ_k+1^N(ϵ)1/2𝗑_n^(N)(t)dt
       ≤  f_k+1^N(𝐱^(N))+𝖬_k+1^(N)(t∧κ_k+1^N(R)∧σ_k+1^N(ϵ))+ 
        k/ϵt.

One can now follow the argument for the case n=1 to establish the desired result in the exact same way.



We note that in the argument above we only needed to take as input convergence of the paths and not necessarily convergence in C(ℝ_+,Ω_+). Had we not even known convergence of the paths, the argument would in any case give that any possible subsequential limits would need to consist of non-intersecting paths.





 §.§ The ISDE via characteristic polynomials


We now prove that (𝗑_i(·))_i=1^∞ solves the ISDE (<ref>). Beyond the use of characteristic polynomials to show convergence of the singular drift, it is interesting to note the following. In the construction of weak solutions to finite-dimensional SDE, see <cit.>, the driving Brownian motions can usually[The upshot is that one wants to construct the driving Brownian motions from the solution, see <cit.>. If the noise degenerates, one needs an auxiliary, independent from the solution, Brownian motion, see <cit.>. However, in our setting the noise does not degenerate as 𝗑_i>0 and this is not the source of the additional randomness. ] be taken to be measurable with respect to the natural filtration generated by the solution. Here, the driving Brownian motions (𝗐_i(·))_i=1^∞ will in fact be measurable not with the filtration generated by (𝗑_i(·))_i=1^∞ on its own but rather with the filtration generated by the enhanced process 𝐗_·^Ω_+=((𝗑_i(·))_i=1^∞,γ(·)). This is another illustration of the phenomenon that in the infinite-dimensional limit additional information is created.


    Let η∈ℝ, 𝐱∈𝕎_∞,+^∘, γ∈ℝ_+ with ∑_i=1^∞x_i≤γ.
    Let 𝐗_·^Ω_+ and 𝗑(·)=(𝗑_i(·))_i=1^∞ be as in Theorem <ref>. Then, 𝗑(·) is a weak solution, starting from 𝐱, to the ISDE (<ref>). Moreover, we have that,
 
    ℙ_(𝐱,γ)(𝐗^Ω_+_t∈Ω_+^0, ∀ t>0)=1,   ∀ (𝐱,γ)∈Ω_+.






We first prove that there exists a filtered probability space, on which independent standard Brownian motions (𝗐̃_i)_i=1^∞ adapted to the filtration and an equal in law copy (𝗑̃,γ̃) of (𝗑,γ), also adapted to the filtration, are defined, such that almost surely, for all t≥ 0:
   	
    𝗑̃_i(t) = x_i+ ∫_0^t𝗑̃_i(s) d𝗐̃_i(s) +∫_0^t (-η/2𝗑̃_i(s)+γ̃(s)-∑_j=1^∞𝗑̃_j(s)+∑_j=1,j≠ i^∞𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s))ds,    i∈ℕ.

The filtration will be the natural filtration of (𝗑̃,γ̃). Recall that the natural filtration (ℱ_t)_t≥ 0 of a process (ℜ_t;t≥ 0) is given by, for all t≥ 0,

    ℱ_t=σ(ℜ_t; 0 ≤ s ≤ t),

 where for a collection of random variables 𝒞, we write σ (𝒞) for the sigma algebra generated by them. 
 
We now prove this claim. For each N∈ℕ, let 𝐱^(N)∈𝕎_N,+^∘ be such that it converges, under the embedding (<ref>), to (𝐱,γ).  Let 𝗑^(N) be the solution to the equation (<ref>) starting at 𝐱^(N). 
Writing the equation in the integral form, one has for i=1,2,…,N,

    𝗑_i^(N)(t)  =	x_i^(N) +∫_0^t𝗑_i^(N)(s) d𝗐_i(s) - η/2∫_0^t𝗑_i^(N)(s)ds
    	+t/2N
    	+∫_0^t∑_j=1,j≠ i^N𝗑_i^(N)(s)𝗑_j^(N)(s)/𝗑_i^(N)(s)-𝗑_j^(N)(s)ds.
   
  By Theorem <ref>, 𝐗^(N)d⟶(𝗑,γ) on C(ℝ_+, Ω_+) where as usual, 𝐗^(N) corresponds to the embedded process on Ω_+. Now, by the Skorokhod
 representation theorem, there exists a probability space (Ω̃,𝐏̃) on which 
 𝗑̃^(N)d=𝗑^(N), i.e.  𝐗̃^(N)d=𝐗^(N), and 𝐗̃^Ω_+d=𝐗^Ω_+, i.e. (𝗑̃,γ̃)d=(𝗑,γ), 
  are defined such that 𝐏̃-a.s.,
  
    𝐗̃^(N)N→∞⟶ (𝗑̃,γ̃),  on   C(ℝ_+, Ω_+).

Let us fix N ∈ℕ. Let us write (ℱ_t^(N))_t≥ 0 for the natural filtration of 
𝗑̃^(N). We first show that there exist  (ℱ_t^(N))_t≥ 0-adapted independent standard Brownian motions (𝗐̃_i^(N))_i=1^N so that 𝐏̃-a.s., for all t≥ 0 and  i=1,2,…,N, we have
 
    𝗑̃_i^(N)(t)  =	x_i^(N)+∫_0^t𝗑̃_i^(N)(s) d𝗐̃_i^(N)(s) - η/2∫_0^t𝗑̃_i^(N)(s)ds
      +t/2N
       +∫_0^t∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)ds.
 
Let us define the process ((𝖸_i^(N)(t))_i=1^N;t ≥ 0) coordinate-wise, as follows,

    (𝖸_i^(N)(t);t≥ 0)   def=(𝗑̃_i^(N)(t) -	x_i^(N) + η/2∫_0^t𝗑̃_i^(N)(s)ds
      -t/2N
       -∫_0^t∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)ds;t≥ 0)
       d=(∫_0^t 𝗑_i^(N) (s)d𝗐_i(s) t≥ 0).

The equality in distribution is by virtue of the fact that 𝗑̃^(N)d=𝗑^(N) and that 𝗑^(N) is a solution to the SDE (<ref>). Observe that, 𝖸^(N) is adapted with respect to (ℱ_t^(N))_t≥ 0. It is also easy to check that 𝖸^(N) is an N-dimensional continuous local martingale with respect to 
(ℱ_t^(N))_t≥ 0, with quadratic variation[For a continuous N-dimensional local martingale (𝗓_i(·))_i=1^N we write ⟨𝗓_i, 𝗓_j ⟩_t, i,j=1,…,N for its quadratic (co-)variation.]  <cit.> given by,

    ⟨𝖸_i^(N),𝖸_j^(N)⟩_t=1_i=j∫_0^t 𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)ds,   ∀ t ≥ 0,  i,j=1,2,…,N.

Now, we can define, for i=1,…,N, by virtue of the fact that 𝐏̃-a.s. 𝗑̃_i^(N)>0, the stochastic integral,

    𝗐̃_i^(N)(t)def=∫_0^t 1/𝗑̃_i^(N)(s)d𝖸_i^(N)(s),   ∀ t ≥ 0.

Observe that, (𝗐̃_i^(N))_i=1^N is adapted with respect to (ℱ_t^(N))_t≥ 0 and moreover it is in fact an N-dimensional continuous local martingale, with respect to this filtration, satisfying,

    ⟨𝗐̃_i^(N),𝗐̃_j^(N)⟩_t=1_i=jt,   ∀ t ≥ 0,  i,j=1,2,…,N.

Hence, by virtue of Levy's characterisation theorem <cit.>, (𝗐̃_i^(N))_i=1^N is a sequence of independent standard Brownian motions. Moreover, from (<ref>), we have

    𝖸_i^(N)(t)=∫_0^t 𝗑̃_i^(N)(s)d𝗐̃_i^(N)(s),   ∀ t ≥ 0,  i=1,2,…,N,

from which (<ref>) follows.








We next prove (<ref>) holds. 
Clearly, proving (<ref>) amounts to showing the convergence of the integrals to the corresponding terms in the limit. Let i∈ℕ be fixed. We proceed to show the convergence of the drift terms first. Observe that, 𝐏̃-a.s., for all t≥ 0,

    ∫_0^t𝗑̃_i^(N)(s)ds→∫_0^t𝗑̃_i(s)ds,    as  N→∞.

We thus need to prove convergence of the interaction terms, namely, 𝐏̃-a.s., for all t ≥ 0,

    ∫_0^t∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)dsN→∞⟶1/2∫_0^t(γ̃(s)-∑_j=1^∞𝗑̃_j(s))ds
       +∫_0^t∑_j=1,j≠ i^∞𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

To this end, we will use Proposition <ref> on convergence of the corresponding characteristic polynomials.
Let N∈ℕ and 1≤ i≤ N. We consider the following entire functions: 
    Φ_i^N(z;𝐱^(N))   def=∏_j=1,j≠ i^N(1-x_j^(N)z)^2,  𝐱^(N)∈𝕎_N,+,
    𝖤_i(z;ω)   def=e^-2(γ-x_i) z∏_j=1,j≠ i^∞e^2x_jz(1-x_jz)^2,  ω=(𝐱,γ)∈Ω_+.

Observe that, the integrands in (<ref>) are determined in terms of the functions Φ_i^N(z;𝐱^(N)) and 𝖤_i(z;ω) respectively. Indeed, 
the corresponding terms, at time t≥ 0, are equal to 

    1/2d/d zlogΦ_i^N(z,𝗑̃^(N)(t))|_z=(𝗑̃_i^(N)(t))^-1,
       1/2d/d zlog𝖤_i(z;𝐗̃_t^Ω_+)|_z=(𝗑̃_i(t))^-1.

Note that, the expressions above are well-defined by virtue of the fact that the paths are 𝐏̃-a.s. non-intersecting and never hit zero. We now prove that, 𝐏̃-a.s., for all t≥ 0,

    d/d zlogΦ_i^N(z,𝗑̃^(N)(t))|_z=(𝗑̃_i^(N)(t))^-1N→∞⟶d/dzlog𝖤_i(z;𝐗̃_t^Ω_+)|_z=(𝗑̃_i(t))^-1.

 Since we have convergence on Ω_+, as in (<ref>), it follows from
 Proposition <ref>, in particular equation (<ref>), that 𝐏̃-a.s., for all t≥ 0,

    Φ_i^N(z,𝗑̃^(N)(t))⟶𝖤_i(z;𝐗̃_t^Ω_+),   as  N→∞,

uniformly on compact sets in ℂ, and the same is true for the corresponding derivatives by analyticity.
Note that, by virtue of Theorem <ref> we have 𝐏̃-a.s., for all t≥ 0

    (𝗑̃_i^(N)(t))^-1⟶(𝗑̃_i(t))^-1,   as N→∞.

From the above, and the fact that all processes we are considering are 𝐏̃-a.s. non-intersecting, see Lemma <ref> and Theorem <ref>, the convergence in (<ref>) is concluded. This shows convergence of the integrands for all times t≥ 0. We now prove that the corresponding integrals also converge. First, note that as the paths are ordered and 𝗑̃_i^(N)> 0, we can write, 𝐏̃-a.s., for all t≥ 0,
     
    𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)≤𝗑̃_i+1^(N)(s)/𝗑̃_i^(N)(s),  ∀ j,N≥ i+1,  s∈[0,t].

    Also, using convergence of the coordinates and since almost surely 𝗑̃_i> 0, we have that 𝐏̃-a.s., for all t≥ 0,
    
    inf_s∈[0,t]|1-𝗑̃_i+1^(N)(s)/𝗑̃_i^(N)(s)|→inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)| ,   as  N→∞.

    On the other hand, by the non-intersection property of the limiting paths, 𝐏̃-a.s., for all t≥ 0,
    
    inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|>0.

    Hence, there exists a random N_0∈ℕ such that, 𝐏̃-a.s., for all t≥ 0,
        
    inf_s∈[0,t]|1-𝗑̃_i+1^(N)(s)/𝗑̃_i^(N)(s)|≥1/2inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|,   ∀ N≥ N_0.

    Observe now that for all N≥ N_0, we have 𝐏̃-a.s., for all t≥ 0,
    
    𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)≤ 2(inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|)^-1𝗑̃_j^(N)(s),  ∀ j≥ i+1,  s∈[0,t],
 
    and therefore, 𝐏̃-a.s., for all t≥ 0
       
    ∑_j=1,j≠ i^N𝗑̃_i^(N)(s)𝗑̃_j^(N)(s)/𝗑̃_i^(N)(s)-𝗑̃_j^(N)(s)≤ 2(inf_s∈[0,t]|1-𝗑̃_i+1(s)/𝗑̃_i(s)|)^-1∑_j=1^N𝗑̃_j^(N)(s),   ∀ N≥ N_0.

This, together with the fact that 𝐏̃-a.s., ∑_j=1^N𝗑̃_j^(N)(·)N→∞⟶γ̃(·) in C(ℝ_+,ℝ_+) implies (<ref>) by applying the generalized Lebesgue dominated convergence theorem. 

We now deal with the stochastic integral term. From now on, let us denote by (ℱ_t)_t≥ 0 the natural filtration of the process (𝐗̃^Ω_+_t; t≥ 0). We deduce from the convergence of both the LHS and the drift term in (<ref>) that 𝐏̃-a.s., for all i∈ℕ,
𝖸_i^(N)(·) converges, as N →∞, uniformly on compact sets,
to the continuous process 𝖸_i(·) given by,

    𝖸_i(t)= 𝗑̃_i(t) -x_i-∫_0^t (-η/2𝗑̃_i(s)+γ̃(s)-∑_j=1^∞𝗑̃_j(s)+∑_j=1,j≠ i^∞𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s))ds,   ∀ t ≥ 0.

Observe that, 𝖸(·)=(𝖸_i(·))_i=1^∞ is adapted with respect to (ℱ_t)_t≥ 0. We now claim that for any K ∈ℕ, (𝖸_i(·))_i=1^K is a K-dimensional continuous local martingale with quadratic variation given by,

    ⟨𝖸_i,𝖸_j⟩_t=1_i=j∫_0^t 𝗑̃_i(s)𝗑̃_j(s)ds,   ∀ t ≥ 0,  i,j ∈ℕ.

Let us assume this claim momentarily. Then, we can define, by virtue of the fact that 𝐏̃-a.s. 𝗑̃_i>0, for all i ∈ℕ, the stochastic integral,

    𝗐̃_i(t)def=∫_0^t 1/𝗑̃_i(s)d𝖸_i(s),   ∀ t ≥ 0.

Observe that (𝗐̃_i(·))_i=1^∞ is adapted with respect to (ℱ_t)_t≥ 0 and moreover, for each K ∈ℕ, (𝗐̃_i(·))_i=1^K is a K-dimensional continuous local martingale with quadratic variation,

    ⟨𝗐̃_i,𝗐̃_j⟩_t=1_i=jt,   ∀ t ≥ 0,  i,j ∈ℕ.

Thus, making use of Levy's characterisation theorem <cit.> we get that (𝗐̃_i)_i=1^∞ is a sequence of independent standard Brownian motions. Finally, from (<ref>) we get, for all i∈ℕ,

    𝖸_i(t)=∫_0^t 𝗑̃_i(s)d𝗐̃_i(s),   ∀ t ≥ 0.

This concludes the proof of (<ref>) modulo the claim. We now prove it. Fix K∈ℕ. Define the stopping times ζ_K^N(R) and ζ_K(R), with respect to the corresponding filtrations (ℱ_t^(N))_t≥ 0 and (ℱ_t)_t≥ 0, for R>0, N ≥ K,

    ζ_K^N(R)def=inf{t≥ 0;∑_i=1^K |𝖸_i^(N)(t)|≥ R},   ζ_K(R)def=inf{t≥ 0;∑_i=1^K |𝖸_i(t)|≥ R}.

Observe that, 𝐏̃-a.s., ζ_K(R) R →∞⟶∞. Moreover, arguing similarly to the proof of Theorem <ref> we get that there exists a sequence (R_m)_m=1^∞, with R_m→∞ so that for any fixed m ∈ℕ, ζ_K^N(R_m) ⟶ζ_K(R_m), in distribution, as N →∞. Let t≥ s and R_m be fixed. Let 𝔊 be an arbitrary bounded continuous functional on C([0,s],Ω_+), the space of continuous functions on [0,s] with values in Ω_+. Note that, we have, as N →∞,

    (𝖸^(N)_i(t∧ζ^N_K(R_m))-𝖸^(N)_i(s∧ζ^N_K(R_m)))_i=1^K𝔊[(𝐗̃^(N)(u);0≤ u ≤ s)] 
       d⟶(𝖸_i(t∧ζ_K(R_m))-𝖸_i(s∧ζ_K(R_m)))_i=1^K𝔊[(𝐗̃^Ω_+(u);0≤ u ≤ s)].

Moreover, observe that, if for fixed N ∈ℕ, we define the functional 𝖦 on C([0,s],𝕎_N,+) by,

    𝖦[((f_i(u))_i=1^N;0≤ u ≤ s )] =𝔊[(((N^-1f_i(u))_i=1^∞,N^-1∑_i=1^Nf_i(u)); 0 ≤ u ≤ s)],

with the convention f_i≡ 0, for i>N, then 𝖦 is bounded and continuous. Hence, putting everything together, by virtue of the local martingale property of 𝖸^(N), we obtain,

    𝔼_(𝐱,γ)[(𝖸_i(t∧ζ_K(R_m))-𝖸_i(s∧ζ_K(R_m)))_i=1^K𝔊[(𝐗̃^Ω_+_u;0≤ u ≤ s)]]
       =lim_N →∞𝔼_𝐱^(N)[(𝖸^(N)_i(t∧ζ^N_K(R_m))-𝖸^(N)_i(s∧ζ^N_K(R_m)))_i=1^K𝔊[(𝐗̃^(N)(u);0≤ u ≤ s)]]
       =lim_N →∞𝔼_𝐱^(N)[(𝖸^(N)_i(t∧ζ^N_K(R_m))-𝖸^(N)_i(s∧ζ^N_K(R_m)))_i=1^K𝖦[(𝗑̃^(N)(u);0≤ u ≤ s)]]=(0)_i=1^K.

This proves that (𝖸_i(·))_i=1^K is a continuous local martingale, with (ζ_K(R_m))_m=1^∞ a localising sequence, with respect to (ℱ_t)_t≥ 0. Finally note that, 𝐏̃-a.s. the quadratic variations of the 𝖸^(N)'s also converge, as N →∞, to the desired expression and this concludes the proof of the claim and thus the proof of (<ref>).




We now proceed to show that 𝗑̃ actually solves (<ref>). It will suffice to show that 𝐏̃-a.s., for all t≥ 0, we have  

    ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds= 0.

Observe that, for any M∈ℕ and t≥ 0, we have from (<ref>),
     
    ∑_i=1^M𝗑̃_i(t)=   ∑_i=1^Mx_i+∫_0^t∑_i=1^M𝗑̃_i(s) d𝗐̃_i(s)-η/2∫_0^t∑_i=1^M𝗑̃_i(s)ds+M∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds
       +∫_0^t∑_i=1^M∑_j>M𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

 Therefore, since the first and last terms in the above display are non-negative, one has
     
    M∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds≤∑_i=1^M𝗑̃_i(t)-∫_0^t∑_i=1^M𝗑̃_i(s) d𝗐̃_i(s)+η/2∫_0^t∑_i=1^M𝗑̃_i(s)ds.

  Note that, all the terms on the RHS are finite uniformly as M →∞. Thus, dividing both sides by M, and then, taking M to infinity, one obtains that 𝐏̃-a.s., for all t≥ 0,
 
    ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds≤ 0.

 But clearly, ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds≥ 0 and thus we get 𝐏̃-a.s., ∫_0^t(γ̃(s)-∑_i=1^∞𝗑̃_i(s))ds= 0, as desired.


 We now want to prove the second statement in the theorem. First observe that, the above implies that 𝐏̃-a.s., for all t≥ 0, 
    γ̃(s)=∑_i=1^∞𝗑̃_i(s),   for almost all  s∈[0,t].

Furthermore, we have, 𝐏̃-a.s., for all M∈ℕ and t≥ 0,
  
    ∑_i=1^M𝗑̃_i(t)=   ∑_i=1^Mx_i+∫_0^t∑_i=1^M𝗑̃_i(s) d𝗐̃_i(s)-η/2∫_0^t∑_i=1^M𝗑̃_i(s)ds+∫_0^t∑_i=1^M∑_j>M𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

Now, by making use of the Burkholder-Davis-Gundy inequality for local martingales, see <cit.>, or directly using Problem 1.5.25 of <cit.>, by virtue of the fact that ∑_i=K^∞𝗑̃_i(·)^2 → 0, uniformly on compact sets, as K →∞, we obtain the existence of a sequence (M_ℓ)_ℓ=1^∞ with M_ℓ→∞, as ℓ→∞, such that 𝐏̃-a.s. the following limit exists uniformly on compact sets,

    𝖴(t)def=lim_ℓ→∞∫_0^t ∑_i=1^M_ℓ𝗑̃_i(s)d𝗐̃_i(s),   ∀ t ≥ 0.

In particular, 𝐏̃-a.s. the function t ↦𝖴(t) is continuous. Then, putting M=M_ℓ and taking the limit ℓ→∞ in both sides of (<ref>), we obtain that 𝐏̃-a.s., for all t≥ 0,
 
    ∑_i=1^∞𝗑̃_i(t)=   ∑_i=1^∞x_i+𝖴(t)-η/2∫_0^t∑_i=1^∞𝗑̃_i(s)ds+lim_ℓ→∞∫_0^t∑_i=1^M_ℓ∑_j>M_ℓ𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds.

Observe now that the second and third terms on the RHS of (<ref>) are continuous functions of t, and thus, any possible discontinuity of t ↦∑_i=1^∞𝗑̃_i(t) arises from the last term. Note also that since the integrand is strictly positive, one has 
 
    lim_ℓ→∞∫_0^𝗎∑_i=1^M_ℓ∑_j>M_ℓ𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds>lim_t↗𝗎lim_ℓ→∞∫_0^t∑_i=1^M_ℓ∑_j>M_ℓ𝗑̃_i(s)𝗑̃_j(s)/𝗑̃_i(s)-𝗑̃_j(s)ds,

 at any point of discontinuity 𝗎>0 of t ↦∑_i=1^∞𝗑̃_i(t), and hence, from (<ref>) we get 
 
    ∑_i=1^∞𝗑̃_i(𝗎)> lim_t_k↗𝗎∑_i=1^∞𝗑̃_i(t_k)=lim_t_k↗𝗎γ̃(t)=γ̃(𝗎),

where, by virtue of (<ref>), the sequence (t_k)_k=1^∞ is picked so that γ̃(t_k)=∑_i=1^∞𝗑̃_i(t_k) holds for all k∈ℕ and recall that t↦γ̃(t) is continuous.
But, this is a contradiction as we know that
∑_i=1^∞𝗑̃_i(t)≤γ̃(t), for all t≥ 0. Therefore, it follows that
 t ↦∑_i=1^∞𝗑̃_i(t) is, in fact, a continuous function on (0,∞) and in particular, we have that 𝐏̃-a.s.,
 
    ∑_i=1^∞𝗑̃_i(t)=γ̃(t),   ∀ t>0,
 which is equivalent to (<ref>). This completes the proof.
 


 
 Observe that, in the argument above we did not need the full power of convergence in C(ℝ_+,Ω_+) to show convergence of the singular drift term. Nevertheless, one will always need somewhat more information than simply convergence of the paths for this.    
 

 
 It is interesting to note that Theorem <ref> combined with Theorem <ref> give an immediate, alternative, and non-computational, proof of the fact that if, instead of Definition <ref>, we define 𝔐^η as the unique probability measure on Ω_+ satisfying 𝔐^ηΛ^∞_N=𝔐^η_N for all N∈ℕ, then 𝔐^η must be supported on Ω_+^0. The original proof in <cit.> uses the underlying determinantal point process structure and some hard estimates on the correlation kernel of the Laguerre unitary ensemble. The dynamical proof we just mentioned avoids these considerations entirely.
 

We now prove that solutions corresponding to different parameters γ are different in law.
 
  
 In the setting of Theorem <ref>, we denote by (𝐗(t;γ);t≥ 0) the solution to (<ref>) corresponding to γ, then,

    𝖫𝖺𝗐(𝐗(·;γ))≠𝖫𝖺𝗐(𝐗(·;γ̃)), whenever γ≠γ̃.


  
  Based on (<ref>) we have that a.s.

    ∑_i=1^∞𝗑_i(t;γ)
        =γ(t),  ∑_i=1^∞𝗑_i(t;γ̃)
    =γ̃(t),  ∀ t>0.

Note now that γ and γ̃ do not have same distribution, as they are continuous functions with γ(0)≠γ̃(0), and thus, by virtue of (<ref>),

    𝖫𝖺𝗐(∑_i=1^∞𝗑_i(·;γ))≠𝖫𝖺𝗐(∑_i=1^∞𝗑_i(·;γ̃)),

 from which the conclusion follows immediately. 
  
 
 We have the following result which completes the statement of Theorem <ref>.

 
Out of all solutions to the ISDE (<ref>) constructed as in Theorem <ref>, there exists a unique one such that almost surely t ↦∑_i=1^∞𝗑_i(t;γ) is continuous for all t ≥ 0 given by the choice γ=∑_i=1^∞ x_i.
Moreover, this solution is a Markov process. 
 

 We know from Theorem <ref> that 𝐗^Ω_+_·=(𝐗(·;γ),γ(·)) has continuous sample paths. 
 It then follows from continuity of t↦γ(t) and (<ref>) that 
almost surely t ↦∑_i=1^∞𝗑_i(t;γ) is continuous for all t ≥ 0 if and only if γ=∑_i=1^∞ x_i. 


Towards the second assertion, note that by the Markov property of 𝐗^Ω_+, for all 
t≥ s ≥ 0 and all bounded, measurable functions 𝖥:Ω_+→ℝ, we have
 
    𝔼[𝖥(𝐗(t;γ),γ(t))| ℱ_s]=𝔼[𝖥(𝐗(t;γ),γ(t))|σ(𝐗(s;γ),γ(s))],

 where, as before,
(ℱ_t)_t≥ 0
 is the natural filtration of the process (𝐗^Ω_+_t; t≥ 0) and 𝔼[·|σ(𝒞)] denotes conditional expectation with respect to σ(𝒞).

Now, consider the distinguished solution 𝐗(·;γ_*)  of the ISDE with γ_*=∑_i=1^∞ x_i.
Note that, since ∑_i=1^∞𝗑_i(s;γ_*)=γ(s), for all s≥ 0, we have
 
    σ(𝐗(s;γ_*),γ(s))=σ(𝐗(s;γ_*)),  ∀ s≥ 0,

 and thus, in this case the natural filtration (ℱ_t^𝐗(·;γ_*))_t≥ 0 generated by (𝐗(t;γ_*);t≥ 0) is equal to (ℱ_t)_t≥ 0. Denote by 𝒲^∞ the state space of the solution:
 
    𝒲^∞={𝐱∈𝕎_∞,+^∘:∑_i=1^∞ x_i<∞}.

 Now, let 𝖿:𝒲^∞→ℝ be an arbitrary bounded measurable function and take 𝖥 to be its obvious lift on Ω_+. Thus summarising, by virtue (<ref>) and (<ref>), we have
  
    𝔼[𝖿(𝐗(t;γ_*))|ℱ_s^𝐗(·;γ_*)]=𝔼[𝖿(𝐗(t;γ_*))|σ(𝐗(s;γ_*))],  ∀ s≥ 0,

 which implies that  (𝐗(t;γ_*))_t≥ 0 is a Markov process and concludes the proof. 
 


    This follows from Theorem <ref>, Theorem <ref>, Theorem <ref>, and Proposition <ref>.





    By Theorem <ref>, the probability measure 𝔪 corresponds to the unique sequence of probability measures (𝔪_N)_N=1^∞ on (𝕎_N,+)_N=1^∞ given by,
    
    𝔪_N=𝔪Λ_N^∞,   ∀ N∈ℕ.

    Let 𝗑^(N) be the solution of (<ref>) started at 𝗑^(N)(0)d=𝔪_N. By combining Theorem <ref> and Lemma <ref>, a standard argument gives that, under the embedding (<ref>) of 𝕎_N,+ into Ω_+, 𝔪_N converges weakly to 𝔪. Namely, 𝐗^(N)_0d⟶𝐗^Ω_+_0, and hence,
 𝐗^(N)d⟶𝐗^Ω_+ by Theorem <ref>. One can then follow the argument of Theorem <ref> to prove that (𝐗(t;𝔪);t≥ 0) is a weak solution of the ISDE (<ref>) with initial condition 𝐗(0;𝔪)d=𝖨𝖡𝖾𝗌_η. Note that, as 𝔐^η is the invariant measure of 𝐗^Ω_+, we have 
 
    𝖫𝖺𝗐(𝐗(t;𝔐^η))=𝖨𝖡𝖾𝗌_η,   ∀ t ≥ 0,

 On the other hand, if 𝔪≠𝔐^η, we must have
 
    ℙ_𝔪(γ(0;𝔪)>∑_i=1^∞𝗑_i(0;𝔪))>0,

 where ℙ_𝔪 denotes 𝖫𝖺𝗐(𝐗^Ω_+) if 𝖫𝖺𝗐(𝐗_0^Ω_+)=𝔪. Thus, based on the arguments in Theorem <ref>, since γ(t;𝔪)=∑_i=1^∞𝗑_i(t;𝔪) for all t>0, and recall that t↦γ(t;𝔪) is continuous, we obtain that the stochastic processes t ↦∑_i=1^∞𝗑_i(t;𝔪) and t↦∑_i=1^∞𝗑_i(t;𝔐^η) are not equal in law, for any 𝔪≠𝔐^η, and so, 

    𝖫𝖺𝗐(𝐗(·;𝔐^η))≠𝖫𝖺𝗐(𝐗(·;𝔪)).
 
That is, the following holds if and only if 𝔪=𝔐^η, 
 
    𝖫𝖺𝗐(𝐗(t;𝔪))=𝖨𝖡𝖾𝗌_η,   ∀ t ≥ 0,

and this completes the proof.

 












§ DYNAMICAL CAUCHY MODEL



In this section, we consider stochastic dynamics related to the Cauchy <cit.> or so-called Hua-Pickrell ensemble <cit.>. This is the following probability measure on 𝕎_N,
depending on a parameter s∈ℂ, with (s)>-1/2,

    𝔑_N^(s)(d𝐱)=1/𝒵_N^(s)Δ_N^2(𝐱)
        ∏_j=1^N(1 + x_j^2)^-(s)-Ne^2(s)tan^-1(x_j)d𝐱,
 
where,
𝒵_N^(s) is a normalizing constant.

The associated N-particle dynamics, that we call the dynamical Cauchy model, introduced in <cit.> (they were Hua-Pickrell diffusions therein),  is given by the unique solution to the following system of stochastic equations
 
    d𝔵_i(t) = √(2[𝔵_i(t)^2+1])d𝗐_i(t) + 2[(1-N-(s))𝔵_i(t)+(s)]dt+2∑_j=1:j≠ i^N𝔵_i(t)^2+1/𝔵_i(t)-𝔵_j(t)dt,
 
for i=1,…, N, where as usual the 𝗐_i are independent standard Brownian motions.
It is proven in <cit.>, making use of the results of <cit.>,  that
(<ref>) has a unique strong solution for any initial condition 𝐱∈𝕎_N and almost surely for all t>0, 𝗑(t)∈𝕎_N^∘ and moreover the associated semigroup (𝔔^(s)_N(t))_t≥ 0 is Feller. The following result is the main theorem of <cit.>.


Let s∈ℂ. Then, the semigroups (𝔔_N^(s)(t))_t≥ 0 are intertwined:

    𝔔_N+1^(s)(t)Λ_N^N+1=
    Λ_N^N+1𝔔_N^(s)(t),  ∀ t≥ 0,    N∈ℕ.

In particular, there exists a unique Feller semigroup (𝔔_∞^(s)(t))_t≥ 0, with associated stochastic process (𝐗_t^Ω;t≥ 0) on Ω, defined via the intertwining

    𝔔_∞^(s)(t)Λ_N^∞=
    Λ_N^∞𝔔_N^(s)(t),  ∀ t≥ 0,    N∈ℕ.
   
Moreover, if (s)>-1/2 then (𝔔_∞^(s)(t))_t≥ 0 has a unique invariant measure 𝔑_∞^(s) uniquely determined via the relation

    𝔑_∞^(s)Λ_N^∞ = 𝔑_N^(s),   ∀ N ∈ℕ.




Before giving our main result on the dynamical Cauchy model it is worth pointing out, that for real s>-1/2, the invariant measure associated to the dynamics (𝔔^s_∞(t))_t≥ 0 is known to admit an explicit description. We briefly spell this out.


Let s>-1/2. The Hua-Pickrell point process 𝖧𝖯^Conf_s with parameter s is the determinantal point process on (-∞,0)×(0,∞) with correlation kernel 𝔎_s given by 
    
    𝔎_s(z,w) = 1/2πΓ(s+1)^2/Γ(2s+1)Γ(2s+2)𝒴_1^(s)(z)𝒴_2^(s)(w)-𝒴_1^(s)(w)𝒴_2^(s)(z)/z-w,

where the functions 𝒴_1^(s)(z), 𝒴_2^(s)(z) are given by 

    𝒴_1^(s)(z) = 2^2s-1/2Γ(s+1/2) ·1/|z|^1/2J_s-1/2(1/|z|),   𝒴_2^(s)(z) = 2^2s+1/2Γ(s+3/2) ·1/|z|^1/2 J_s+1/2(1/|z|),

where as before J_r denotes the Bessel function with parameter r.


For s=0 and under the transformation x↦ x^-1, 𝖧𝖯^Conf_0 becomes the sine determinantal point process, the universal limit of random matrices in the bulk of the spectrum <cit.>. Since 𝖧𝖯^Conf_s consists of distinct points it gives rise to a unique probability measure  𝖧𝖯_s on 𝕎_∞,+^∘×𝕎_∞,+ by labelling its positive and negative points respectively according to their absolute values. Define the space

    Ω^0def={ω=(𝐱^+,𝐱^-,γ,δ)∈Ω:γ=lim_k→∞∑_i=1^∞[x_i^+1_x_i^+≥ k^-2- x_i^-1_x_i^-≥ k^-2] , δ=∑_i=1^∞[(x^+_i)^2+(x^-_i)^2]},

and write π:Ω→𝕎_∞,+×𝕎_∞,+ for the map π((𝐱^+,𝐱^-,γ,δ))=(𝐱^+,𝐱^-). Then, it turns out that 𝔑_∞^(s) is the unique probability measure on Ω which is supported on Ω^0 and moreover satisfies (see the discussion in <cit.> and the references therein),

    π_*𝔑^(s)_∞=𝖧𝖯_s.





We now upgrade the abstract construction of (𝐗^Ω_t;t≥ 0) from Theorem <ref> to a strong approximation theorem from the finite dimensional dynamics which moreover implies that (𝐗^Ω_t;t≥ 0) has continuous sample paths. Finally, we prove convergence to equilibrium from any initial condition. We leave the very interesting problem of constructing an ISDE, the drift of which will most likely require renormalisation to even make sense, to future work. We define 𝐗^(N)=((𝗑^(N),+_i)_i=1^∞,(𝗑_i^(N),-)_i=1^∞,γ^(N),δ^(N)) to be the process on Ω obtained from (<ref>) under the embedding (<ref>) of 𝕎_N into Ω.





Let s∈ℂ. Consider the Feller process (𝐗_t^Ω;t≥ 0) with semigroup (𝔔_∞^(s)(t))_t≥ 0 constructed in Theorem <ref> and the embedded processes 𝐗^(N) on Ω defined above. Let 𝐗_0^Ω=ω∈Ω be arbitrary and assume 𝐗^(N)(0)→ω in the topology of Ω. Then, as N →∞,

    𝐗^(N)d⟶𝐗^Ω in  C(ℝ_+,Ω).

Moreover, there exists a coupling of the 𝐗^(N) and 𝐗^Ω such that, almost surely, for all T≥0, we have:
    
    sup_t∈[0,T](∑_i=1^∞|𝗑_i^(N),+(t)-𝗑_i^+(t)|^3+∑_i=1^∞|𝗑_i^(N),-(t)-𝗑_i^-(t)|^3)N→∞⟶ 0.

   Finally, for (s)>-1/2 and 𝔎 an arbitrary probability measure on Ω, if 𝖫𝖺𝗐(𝐗_0^Ω)=𝔎, then as t→∞,

    𝐗_t^Ωd⟶𝐙,   where 𝖫𝖺𝗐(𝐙)=𝔑_∞^s.




All the statements in the theorem, except for convergence to equilibrium, follow by combining Theorem <ref> and the fact that C^∞_c,sym(𝕎_N) is a core, and invariant under the generator of  (𝔔_N^(s)(t))_t≥ 0 (this can be proven in exactly the same way as Proposition <ref>), by virtue of Theorem <ref> and Proposition <ref>. The only thing that remains to note is convergence to equilibrium of the finite-dimensional processes which itself follows from that of the corresponding matrix process shown in <cit.>. Then, the remaining statement in the theorem follows from Theorem <ref>.



acm




School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.mailto:theo.assiotis@ed.ac.uktheo.assiotis@ed.ac.uk



School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.mailto:theo.assiotis@ed.ac.ukZ.S.Mirsajjadi@sms.ed.ac.uk


 Paper Title: Integrability of the multi-species ASEP with long-range jumps on $\mathbb{Z}$.







Integrability of the multi-species ASEP with long-range jumps on ℤ
    Eunghyun Lee[eunghyun.lee@nu.edu.kz]
 Department of Mathematics,
                                            
 School of Sciences and Humanities, 
 Nazarbayev University, 
 Kazakhstan    
    
===============================================================================================================================================================================================

§ ABSTRACT 
Let us consider a two-sided multi-species stochastic particle model with finitely many particles on ℤ defined as follows. Suppose that each particle is labelled by a positive integer l and waits a random time exponentially distributed with rate 1. It then chooses the right direction to jump with probability p or the left direction with probability q=1-p. If the particle chooses the right direction, it jumps to the nearest site occupied by a particle l'<l (with the convention that an empty site is considered as a particle with labelled 0). If the particle chooses the left direction, it follows the rule of the multi-species totally asymmetric simple exclusion process (mTASEP). We show that this model is integrable and provide the exact formula of the transition probability using the Bethe ansatz. 
 

Key words:
Exactly solvable models, Bethe ansatz, ASEP, TASEP, PushTASEP, Integrability, Symmetry




§ INTRODUCTION

In this paper, we treat a multi-species stochastic particle model with finitely many particles on the infinite lattice ℤ, which has the exclusion rule that each site cannot be occupied by more than one particle. Each particle is labelled by a positive integer l representing its species. A state of the system  can be described by specifying each particle's species and its location.  To be more specific, if  X=(x_1,…, x_N)∈ℤ^N with x_1<…<x_N represents the positions of N particles and the species of the particle at x_i is denoted by π_i, then we denote the state of the system by (X,π)=(x_1,…, x_N, π_1⋯π_N) with π = π_1⋯π_N. In a single-species model, of course, a state is simply denoted by (x_1,…, x_N).

The asymmetric simple exclusion process (ASEP) is the most well-known single-species stochastic particle model. In the ASEP on ℤ, each particle waits an exponential time with rate 1 to choose a direction to jump. It chooses the right direction with probability p or the left direction with probability q=1-p. After choosing a direction, a particle at x immediately jumps to x± 1 if the  site is empty. However, if the target site is already occupied by another particle, the particle gives up jumping and waits a random time again to jump. The technique to find the transition probability of the ASEP on ℤ using the Bethe ansatz is well known <cit.>.

In the multi-species ASEP, a higher-numbered particle has priority over a lower-numbered particle in jumping. To be more specific, if a particle l at x chooses to jump to x ± 1  already occupied by l'<l, then the particle l can indeed jump to the site by switching their positions.  However, if l'≥ l, then the particle  l gives up jumping to the site occupied by l'. This model has been extensively studied in many directions <cit.>.

Another stochastic particle model with rules as simple as those of the ASEP is the drop-push model <cit.>. In the drop-push model, each particle jumps to the next site on the right after an exponential random time with rate 1. Unlike the ASEP, a particle can jump even if the target site is already occupied by pushing the particle occupying the site. For example, if  a particle at x jumps to x+1 occupied by another particle, the particle at x jumps to x+1, pushing the particle at x+1 to x+2. In this case, if x+2 is also occupied, then the particle pushed to x+2 also pushes the particle at x+2 to x+3 (Figure <ref> illustrates the definition of the drop-push model).



 A multi-species version of the (one-sided) drop-push model was introduced as the frog model in <cit.>. Additionally, the stationary distribution of the frog model  was recently studied <cit.>. In particular, <cit.> treated various multi-species particle models in a unified way to study the stationary distribution. Another direction of extending to a multi-species version of the (one-sided) drop-push model is found in <cit.>. A straightforward extension of the  one-sided drop-push model to the one where a particle jumps to the right with rate p or to the left with rate q=1-p is not integrable in the sense that the Bethe ansatz is not applicable. However, it is known that if the jumping rates satisfy certain conditions, the two-sided model is integrable <cit.>.  A natural question arises as to whether there exists an integrable multi-species version of the model in <cit.>. Unfortunately, the authors' best efforts to find such a version have not been fruitful. Nevertheless, in this paper, we introduce a two-sided integrable multi-species model incorporating  the rule of the long-range jumps partially.

 

Definition of the model
 

In this paper, we show that if particles follow the rule of the frog model when they jump to the right and follow the rule of the multi-species TASEP when they jump to the left, then the corresponding two-sided model is integrable. Hence, we can use the Bethe ansatz to find the transition probability. For our task, overall, we will take a similar approach to the method in <cit.>. The precise definition of the model is as follows.

A particle  l at x waits an exponential time with rate 1 to jump to the right with probability p or to the left with probability q=1-p. If l chooses the right direction, it jumps to the nearest right site occupied by  l'<l (including  l'=0 which corresponds to an empty site), and  l' immediately jumps to the nearest right site occupied by l”<l'. If  l at x chooses the left direction to jump and x-1 is occupied by l'<l, then l jumps to x-1 and l'  moves to x. However, if  l'≥ l, then l gives up jumping to x-1.

We will call this model the multi-species ASEP  with long-range jumps. The definition of our model with p=1 can be motivated by the basic coupling  of two drop-push models with different configurations. The idea of viewing a low-numbered particle in the two-species ASEP as a discrepancy between the configurations of two ASEPs is well known <cit.>. Using the same idea, let η_t be a drop-push model and η_t(x) represent the state of site x at time t; that is, η_t(x) = 0 if x is unoccupied and η_t(x) = 1 if it is occupied. Let (η_t, ζ_t) be the coupled process of the drop-push models η_t and ζ_t with η_0(x) ≤ζ_0(x) for all x. If we think of a site x with η_t(x) = ζ_t(x) = 1 as being occupied by a particle of species 2, and  a site x with η_t(x) =0, ζ_t(x) = 1 as being occupied by a particle of species 1, then the evolution of the coupled process is  the same as that of our model p=1 in Definition <ref>. Similarly, the two-sided model with general p in Definition <ref> can be interpreted via the basic coupling of  two-sided corresponding single species models. The two-sided single-species model where a particle follows the rule of the drop-push model when it moves to the right and follows the rule of the TASEP when it moves to the left was studied in <cit.>, so our model can be seen as a multi-species version of the model in <cit.>.  Figure <ref> illustrates the interpretation by the basic coupling.



Notations

 

Suppose that there are N particles labelled by positive integers a_1,…, a_N, and let ℳ=[a_1,…,a_N] be the multi-set of those N integers. Let P_(Y,ν)(X,π;t) be the probability that the system will be in state (X,π) at time t given that the system is in state (Y,ν) at time 0. We view π and ν  as permutations of  ℳ of N elements. Of course, if π and ν are permutations of two different multi-sets, then  P_(Y,ν)(X,π;t)=0.

We will consider an arbitrary multi-set [a_1,…,a_N] with each a_i ∈{1,…, N} to cover  all possible compositions of species. For this purpose, let 𝐏_Y(X;t) be  the N^N × N^N matrix such that  rows and columns are labelled with 1⋯ 1, …, N⋯ N and the (π,ν)-element is P_(Y,ν)(X,π;t).  According to the definition of the model, 𝐏_Y(X;t) is a lower triangular matrix when rows and columns are labelled lexicographically from 1 ⋯ 1 to N ⋯ N.  For example, for N=2,

    𝐏_Y(X;t) =
        11    12    21    22 
    
        11     P_(Y,11)(X,11;t)      0      0     0 
    
        12     0       P_(Y,12)(X,12;t)     0      0 
    
        21     0       P_(Y,12)(X,21;t)      P_(Y,21)(X,21;t)      0 
    
        22    0      0    0      P_(Y,22)(X,22;t)
        .

Throughout this paper, 𝐈 denotes the identity matrix and 0 denotes the zero matrix. If needed, we write   𝐈_n and 0_n to specify that their dimensions are n by n. We write  𝐌_(π,ν) for  the (π,ν)-element of a matrix 𝐌.

 

Organization of the paper
 

In Section <ref>, we investigate the case of N=2 as a preliminary building block for the case of general N.  In Section <ref>, we investigate the case of N=3. The arguments for N=3  build upon the results for N=2. However, in the case of N=3, two-body reducibility and the Yang-Baxter equation are discussed for integrability. Moving from N=2 to N=3 is not a trivial extension, but if the integrability of N=3 is proved, then the integrability of general N is trivially confirmed. Hence, we provide detailed  arguments for the case of N=3 in Section  <ref> and only the results for general N  in Section <ref>. This approach also helps us understand the integrability of the model better, reducing the heavy notations for general N. In Section <ref>, we provide the exact formula for the transition probability for an arbitrary N-particle system.


§ N=2



 §.§ Master equations

Each element P_(Y,ν)(X,π;t) in (<ref>) satisfies the master equation  and the initial condition:

    P_(Y,ν)(X,π;0) =
    
    1      if (Y,ν)=(X,π)
    
    0     otherwise.

For a fixed Y, the form of the master equation for P_(Y,ν)(X,π;t) depends on X, π,ν. The master equation for P_(Y,ν)(X,π;t) when X=(x_1,x_2) with  x_1< x_2-1 is the (π,ν)-element of the matrix equation:

    d/dt𝐏_Y(x_1,x_2;t) =      p 𝐏_Y(x_1-1,x_2;t) + p 𝐏_Y(x_1,x_2-1;t) 
    
                                          +      q 𝐏_Y(x_1+1,x_2;t) + q 𝐏_Y(x_1,x_2+1;t) - 2𝐏_Y(x_1,x_2;t)

where the differentiation on the left side in  (<ref>) denotes the differentiation of each matrix element.

The master equation for P_(Y,ν)(X,π;t)  when  X=(x,x+1) is
the (π,ν)-element of the matrix equation:

    d/dt𝐏_Y(x,x+1;t) = p𝐏_Y(x-1,x+1;t) +   p𝐁 𝐏_Y(x-1,x;t) - 2p𝐏_Y(x,x+1;t) 
       + q𝐏_Y(x,x+2;t)  + q𝐁_1 𝐏_Y(x,x+1;t)- q𝐁_2𝐏_Y(x,x+1;t)

where

    𝐁 =     11     12     21     22  
    
        11     1     0     0     0  
    
        12     0     0     0     0  
    
        21     0     1     1     0  
    
        22     0     0     0     1
      ,


    𝐁_1 =     11     12     21     22  
    
        11     0     0     0     0  
    
        12     0     0     0     0  
    
        21     0     1     0     0  
    
        22     0     0     0     0
      ,

and

    𝐁_2 =     11     12     21     22  
    
        11     1     0     0     0  
    
        12     0     2     0     0  
    
        21     0     0     1     0  
    
        22     0     0     0     1
      .

Note that the (ij,lm)-element of p𝐁 represents the instantaneous rate of transition from state (x-1,x,lm) to state (x,x+1, ij), while the (ij,lm)-element of q𝐁_1 represents the instantaneous rate of transition from state (x,x+1,lm) to state (x,x+1, ij). Moreover,  (<ref>) can be expressed as:

    d/dt𝐏_Y(x,x+1;t) =     p𝐏_Y(x-1,x+1;t) +   p𝐁 𝐏_Y(x-1,x;t)  
         + q𝐏_Y(x,x+2;t)  + q𝐁 𝐏_Y(x,x+1;t)- 2𝐏_Y(x,x+1;t)

due to the relation 𝐁 - 𝐁_1 +  𝐁_2  = 2𝐈.

The matrix form of the initial condition (<ref>)  is given by:

    𝐏_Y(X;0) =
    𝐈    if X = Y, 
    0    if X≠Y.



 §.§ Bethe ansatz

In this section, we extend the  Bethe ansatz method <cit.> to find solutions of  the matrix equations  (<ref>) and (<ref>).

Recall  that 𝐏_Y(X;t) in (<ref>) was defined for the physical region, meaning for Y=(y_1,y_2) and X=(x_1,x_2) with y_1<y_2 and x_1<x_2, respectively. Let 𝐔(X;t) be the 4× 4 matrix obtained by replacing P_(Y,ν)(X,π;t) in (<ref>) with some functions U_(Y,ν)(X,π;t) defined for all  X,Y ∈ℤ^2. Suppose that 𝐔(X;t)= 𝐔(x_1,x_2;t) satisfies

    d/dt𝐔(x_1,x_2;t) =      p𝐔(x_1-1,x_2;t) + p𝐔(x_1,x_2-1;t) 
    
    +     q𝐔(x_1+1,x_2;t) + q𝐔(x_1,x_2+1;t) - 2𝐔(x_1,x_2;t)

for all x_1,x_2 ∈ℤ and
 
    𝐔(x,x;t)= 𝐁 𝐔(x-1,x;t)

for all x ∈ℤ. Then, for x_1=x and x_2 = x+1, (<ref>) equals

    d/dt𝐔(x,x+1;t) =     p𝐔(x-1,x+1;t) +   p𝐁 𝐔(x-1,x;t)  
         + q𝐔(x,x+2;t)  + q𝐁 𝐔(x,x+1;t)- 2𝐔(x,x+1;t)

which is in form of  (<ref>). Thus, if 𝐔(x_1,x_2;t)  satisfies  (<ref>) and  the so called, the boundary condition (<ref>), then it obviously satisfies (<ref>) when x_1< x_2-1, and satisfies (<ref>) when x_1=x and x_2 = x+1.

By  the ansatz of the separation of variables U_(Y,ν)(X,π;t)= U_(Y,ν)(X,π)T(t) and letting 𝐔(X) = 𝐔(x_1,x_2) be the matrix such that 𝐔(X;t) =𝐔(X) T(t) and then substituting it into (<ref>), we obtain T(t) = e^ε t  for some constant  ε (a constant with respect to t,x_1,x_2). The equations for spatial variables of (<ref>) and (<ref>) are

    ε𝐔(x_1,x_2) =      p𝐔(x_1-1,x_2) + p𝐔(x_1,x_2-1)  
          + q𝐔(x_1+1,x_2) + q𝐔(x_1,x_2+1) - 2𝐔(x_1,x_2)

 and
 
    𝐔(x,x)= 𝐁 𝐔(x-1,x).


The Bethe ansatz solution of the matrix equation (<ref>) is given by

    𝐔(x_1,x_2)= 𝐀_12ξ_1^x_1ξ_2^x_2 +𝐀_21ξ_2^x_1ξ_1^x_2

for any  nonzero complex numbers ξ_1 and ξ_2. Here, 𝐀_12 and 𝐀_21 are 4× 4 matrices  similar in structure to 𝐔(x_1,x_2), meaning that the positions of the zero elements in  𝐀_12, 𝐀_21 and 𝐔(x_1,x_2) correspond. The nonzero elements of 𝐀_12 and 𝐀_21 are constants (with respect to t, x_1,x_2). By substituting (<ref>) in (<ref>), we  find

    ε =p/ξ_1 + p/ξ_2 +  qξ_1 + qξ_2- 2.

Additionally, since (<ref>) must satisfy (<ref>), substituting (<ref>) into (<ref>) yields

    𝐀_21 =  -(𝐈 - 𝐁1ξ_2)^-1(𝐈 - 𝐁1ξ_1)𝐀_12.

It is important to note that  𝐈 - 𝐁1ξ is  block-diagonal, allowing us to  compute

    -(𝐈 - 𝐁1ξ_2)^-1(𝐈 - 𝐁1ξ_1)

block-wise.  Specifically, for the 2× 2 blocks in the center of (<ref>),
we have

    -[𝐈_2 -     12     21
    
    12    0    0 
    
    21   1ξ_2   1ξ_2]^-1[𝐈_2 -     12     21
    
    12    0    0 
    
    21   1ξ_1   1ξ_1] =     12     21
    
    12    -1    0 
    
    21    T_21     S_21

where

    S_21 =  -1-1/ξ_1/1-1/ξ_2  and  T_21 = 1/ξ_1-1/ξ_2/1-1/ξ_2.

Hence, we obtain

    𝐀_21 =
       11    12    21    22 
    
                                      11         S_21    0     0     0 
    
                                      12         0     -1     0      0
    
                                      21         0    T_21    S_21    0 
    
                                      22         0     0     0      S_21
    𝐀_12


as a condition for (<ref>) to be satisfied. Thus, 𝐔(x_1,x_2)e^ε t with  (<ref>) and  (<ref>) satisfies  (<ref>) when x_1<x_2-1, and  (<ref>) when x_1=x and x_2 = x+1. (For now, 𝐔(x_1,x_2)e^ε t is not the transition probability yet because we have not shown that it satisfies the initial condition.)

A key idea of defining our two-sided model  as in Definition <ref>  comes from the observation that the boundary condition for the one-sided multi-species model with long-range jumps (to the right direction) and the boundary condition for the multi-species TASEP (to the left direction) are the same, as  given by (<ref>).



§ N=3

For N=3, 𝐏_Y(X;t)= 𝐏_Y(x_1,x_2,x_3;t) is a 27 × 27 matrix whose rows and columns are labelled by 111,112,…, 333.  For X=(x_1,x_2,x_3) with x_1< x_2-1<x_3-2, we immediately obtain the master equation in matrix form:

    d/dt𝐏_Y(x_1,x_2,x_3;t)=     p𝐏_Y(x_1-1,x_2,x_3;t) + p𝐏_Y(x_1,x_2-1,x_3;t) + p𝐏(x_1,x_2,x_3-1;t)
    
    +     q𝐏_Y(x_1+1,x_2,x_3;t) + q𝐏_Y(x_1,x_2+1,x_3;t) + q𝐏(x_1,x_2,x_3+1;t)
        - 3𝐏_Y(x_1,x_2,x_3;t).



 §.§ Master equation for X=(x_1,x_2,x_3) with x_1=x, x_2=x+1 and x_2<x_3-1

In this case, we could find  the master equation in matrix form by finding all master equations for the elements of  𝐏_Y(X;t), but we will use another approach. Considering the positions of particles and the definition of the model, we see that the master equation for (x,x+1, x_3)  should be in the form of

    d/dt𝐏_Y(x,x+1,x_3;t) =       p𝐏_Y(x-1,x+1,x_3;t) + 𝐂 𝐏_Y(x-1,x,x_3;t)
    
    +     p𝐏_Y(x,x+1,x_3-1;t)-3p𝐏_Y(x,x+1,x_3;t)
    
    +    𝐂_1𝐏_Y(x,x+1,x_3;t) + q𝐏_Y(x,x+2,x_3;t)
    
    +     q𝐏_Y(x,x+1,x_3+1;t)-𝐂_2𝐏_Y(x,x+1,x_3;t)

for some 27 × 27 matrices 𝐂, 𝐂_1, 𝐂_2.


  §.§.§ Matrix 𝐂

The  (ijk,lmn)-element of 𝐂 is the instantaneous rate of transition from state (x-1,x,x_3,lmn) to state (x,x+1,x_3,ijk). In fact, if the transition

    (x-1,x,x_3,lmn)  ⟶  (x,x+1,x_3,ijk)

is permissible, then it occurs with rate p by the definition of the model, and if (<ref>) is not permissible, its rate is obviously 0. The permissibility of (<ref>) depends on ijk and lmn. Since x_3 is kept the same during the transition, it means that k=n. If k≠ n,  (<ref>) is not possible so its rate is 0. In the case that k=n, we may say that the permissibility of  (<ref>) depends on the permissibility of the transition from (x-1,x,lm) to (x,x+1,ij), which is actually  encoded in  (<ref>). Now, let us view 𝐂 as a 9 × 9 partitioned matrix whose rows and columns are labelled by 11·, 12·, 13·, 21·, 22·, 23·, 31·, 32·, 33·, and the (ij·, lm·)-element is a 3× 3 matrix whose rows are labelled by ij1,ij2,ij3 and columns are labelled by lm1,lm2,lm3. According to the permissibility of the transition from (x-1,x,lm) to (x,x+1,ij)  available in (<ref>) and the requirement k=n for  (<ref>), we obtain

    𝐂=     11·    12·    13·    21·    22·    23·    31·    32·    33·
    
         11·    p𝐈   0   0   0   0   0   0   0   0
    
         12·   0   0   0   0   0   0   0   0   0
    
         13·   0   0   0   0   0   0   0   0   0
    
          21·   0     p𝐈   0     p𝐈   0   0   0   0   0
    
         22·   0   0   0   0     p𝐈   0   0   0   0
    
         23·   0   0   0   0   0   0   0   0   0
    
         31·   0   0     p𝐈   0   0   0     p𝐈   0   0
    
        32·   0   0   0   0   0     p𝐈   0     p𝐈   0
    
         33·   0   0   0   0   0   0   0   0    p𝐈
    .

If we let

    𝐁=     11     12     13     21     22    23     31     32     33
    
         11     1    0     0     0     0     0     0     0    0
    
         12      0     0      0      0      0      0    0       0    0
    
         13      0     0      0     0     0      0     0     0     0
    
          21   0      1      0      1     0      0      0     0    0
    
         22      0      0      0      0      1      0      0     0    0
    
         23     0      0      0      0    0      0      0      0    0
    
         31     0     0     1     0      0     0     1     0     0
    
        32       0      0     0      0     0      1      0     1   0
    
         33     0      0     0     0      0      0     0      0    1
    ,

then we have a succinct expression:

    𝐂= p𝐁⊗𝐈.


We have already used the notation 𝐁 in (<ref>) for N=2, but here for N=3 we use it again because later the notation 𝐁 will be defined for general N consistently with (<ref>) and  (<ref>).



  §.§.§ Matrices 𝐂_1 and 𝐂_2

The  (ijk,lmn)-element of 𝐂_1 is the instantaneous rate of transition

    (x,x+1,x_3,lmn)  ⟶  (x,x+1,x_3,ijk)

which is due to the interchange of positions of particles at x and x+1. Hence, the instantaneous rate of (<ref>) is q if the transition is permissible, and otherwise, the rate is 0. Since x+1< x_3-1 in (<ref>), it means that k=n, and we see that the permissibility of (<ref>) depends on the permissibility of

    (x,x+1,lm)  ⟶  (x,x+1,ij)

which is actually encoded in  (<ref>). Using a similar argument as in Section <ref>, if we let

    𝐁_1=     11     12     13     21     22    23     31     32     33
    
         11     0    0     0     0     0     0     0     0    0
    
         12      0     0      0      0      0      0    0       0    0
    
         13      0     0      0     0     0      0     0     0     0
    
          21   0      1      0      0     0      0      0     0    0
    
         22      0      0      0      0      0      0      0     0    0
    
         23     0      0      0      0    0      0      0      0    0
    
         31     0     0     1     0      0     0     0     0     0
    
        32       0      0     0      0     0      1      0     0   0
    
         33     0      0     0     0      0      0     0      0    0
    ,

 we have

    𝐂_1 = q𝐁_1 ⊗𝐈.


We have already used the notation 𝐁_1 in (<ref>) for N=2, but here for N=3 we use it again because later the notation 𝐁_1 will be defined for general N consistently with (<ref>) and  (<ref>).

The matrix 𝐂_2 is a diagonal matrix such that the (ijk,ijk)-element is the instantaneous rate of jumping out of the state (x,x+1,x_3,ijk) due to particles' jumps to the left. The rate of the jump of the particle k at x_3 to the left direction is q and the rate of the jump of i at x and j at x+1 is encoded in (<ref>). Hence, if we let

    𝐁_2=     11     12     13     21     22    23     31     32     33
    
         11     1    0     0     0     0     0     0     0    0
    
         12      0     2      0      0      0      0    0       0    0
    
         13      0     0      2     0     0      0     0     0     0
    
          21   0      0      0      1     0      0      0     0    0
    
         22      0      0      0      0      1      0      0     0    0
    
         23     0      0      0      0    0      2      0      0    0
    
         31     0     0     0     0      0     0     1     0     0
    
        32       0      0     0      0     0      0      0     1   0
    
         33     0      0     0     0      0      0     0      0    1
    ,

then we obtain

    𝐂_2 = q𝐁_2 ⊗𝐈 + q𝐈_27.


We have already used the notation 𝐁_2 in (<ref>) for N=2, but here for N=3 we use it again because later the notation 𝐁_2 will be defined for general N consistently with (<ref>) and  (<ref>).

Moreover, observing that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_9,  we see that (<ref>) is equivalent to

    d/dt𝐏_Y(x,x+1,x_3;t)  =  p𝐏_Y(x-1,x+1,x_3;t) + p(𝐁⊗𝐈) 𝐏_Y(x-1,x,x_3;t) 
        + p𝐏_Y(x,x+1,x_3-1;t) +q(𝐁⊗𝐈)𝐏_Y(x,x+1,x_3;t) + q𝐏_Y(x,x+2,x_3;t)
        +q𝐏_Y(x,x+1,x_3+1;t)-3𝐏_Y(x,x+1,x_3;t).



 §.§ Master equation for X=(x_1,x_2,x_3) with x_1+1<x_2, x_2=x and x_3=x+1

Considering the positions of particles and the definition of the model, we see that the master equation  should be in the form:

    d/dt𝐏_Y(x_1,x,x+1;t) =       p𝐏_Y(x_1-1,x,x+1;t) + p𝐏_Y(x_1,x-1,x+1;t)
    
    +    𝐄 𝐏_Y(x_1,x-1,x;t)-3p𝐏_Y(x_1,x,x+1;t)
    
    +      q𝐏_Y(x_1+1,x,x+1;t)+ 𝐄_1𝐏_Y(x_1,x,x+1;t)
    
    +     q𝐏_Y(x_1,x,x+2;t)-𝐄_2𝐏_Y(x_1,x,x+1;t)

for some 27 × 27 matrices 𝐄, 𝐄_1, 𝐄_2.

First, we find the matrix 𝐄. The (ijk,lmn)-element of 𝐄 is the instantaneous rate of  the transition

    (x_1,x-1,x,lmn)  ⟶  (x_1,x,x+1,ijk).

If the transition (<ref>) is permissible, then it occurs with rate p by the definition of the model. If (<ref>) is not permissible, its rate is obviously 0.     Since x_1 is kept the same during the transition (<ref>), it means that i=l. If  i≠ l, then  (<ref>) is not possible, so its rate is  0. In the case that i=l,  we may say that the permissibility of   (<ref>) depends on the permissibility of the transition from (x-1,x,mn) to (x,x+1,jk), which is actually encoded in the matrix 𝐁 in (<ref>).

Now, let us view 𝐄 as a 3× 3 partitioned matrix whose rows and columns are labelled by 1**, 2**, 3*. The (i**, l**)-element is a 9 × 9 matrix whose rows and columns are labelled by i11,i12,…, i33 and l11,l12,…, l33, respectively. Then, using 𝐁 in (<ref>) and the requirement i =l for  the transition (<ref>), we obtain

    𝐄=    1**     2**     3** 
    
        1**     p𝐁   0   0
    
        2**     0    p𝐁   0
    
        3**     0   0     p𝐁
     = 𝐈⊗ p𝐁.

Using similar arguments as in Section <ref>, we obtain:
 
    𝐄_1 =  𝐈⊗ q𝐁_1

 with  𝐁_1 in (<ref>), and
 
    𝐄_2 = 𝐈⊗ q𝐁_2  + q𝐈_27

with 𝐁_2 in (<ref>). Moreover, observing that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_9, we see that (<ref>) is equivalent to

    d/dt𝐏_Y(x_1,x,x+1;t) =   p𝐏_Y(x_1-1,x,x+1;t) + p𝐏_Y(x_1,x-1,x+1;t)
       + p(𝐈⊗𝐁)  𝐏_Y(x_1,x-1,x;t)+  q𝐏_Y(x_1+1,x,x+1;t)
        +q(𝐈⊗𝐁)𝐏_Y(x_1,x,x+1;t)+ q𝐏_Y(x_1,x,x+2;t)-3𝐏_Y(x_1,x,x+1;t).



 §.§ Master equation for X=(x,x+1,x+2)

Considering all possible states that can transition to states with (x,x+1,x+2), we see that  the master equation should be in the form:

    d/dt𝐏_Y(x,x+1,x+2;t) =     p𝐏_Y(x-1,x+1,x+2;t) + 𝐀𝐏_Y(x-1,x,x+2;t) 
    
    +    𝐀_1 𝐏_Y(x-1,x,x+1;t) -3p 𝐏_Y(x,x+1,x+2;t) 
    
    +     q𝐏_Y(x,x+1,x+3;t) + 𝐀_2 𝐏_Y(x,x+1,x+2;t)
    
    -    𝐀_3 𝐏_Y(x,x+1,x+2;t)

for some  27 × 27 matrices  𝐀,𝐀_1, 𝐀_2, 𝐀_3.

The (ijk, lmn)-element of 𝐀 is the instantaneous rate of transition from  (x-1,x,x+2,lmn) to  (x,x+1,x+2,ijk).  Using the same argument as in Section <ref>, we find that 𝐀 = 𝐂 =  p𝐁⊗𝐈.

The (ijk, lmn)-element of 𝐀_1 is the instantaneous rate of transition

    (x-1,x,x+1,lmn)   ⟶  (x,x+1,x+2,ijk)

We could find all matrix elements of  𝐀_1 by checking the permissibility of (<ref>), but we will take a different approach by considering an alternative perspective on the definition of our model. Specifically, we will interpret the rule of the long-range jump as consecutive two-body interactions.

 (equivalent to Definition <ref>)   A particle  l at position x waits an exponential time with rate 1 and then jumps to x+1 with probability p or to x-1 with probability q=1-p. When the particle jumps to x ± 1, if x ± 1 is already occupied by a particle l', then x ± 1 instantaneously accommodates both l and l'  such that max(l,l')  is positioned to the left of min(l,l') at x ± 1, and min(l,l') immediately jumps to (x ± 1) +1.

Figure <ref> and <ref> illustrate some transitions as described in Definition <ref>.




According to Definition <ref>, the transition (<ref>) can be interpreted as

    (x-1,x,x+1,lmn)   (x,x,x+1,l'm'n')
         (x,x+1,x+1,l”m”n”)  (x,x+1,x+2,ijk)

where  and  occur instantaneously. We see that  for (<ref>) to be possible it must be that n = n', l' = l” and l”m”n”=ijk. Using the arguments in Section <ref> and <ref>, the permissibility of   is represented by 𝐁⊗𝐈 and  the permissibility of   is represented by 𝐈⊗𝐁. It is obvious that  is represented by the identity matrix.  Hence, we conclude that L

    𝐀_1 =  p(𝐈⊗𝐁)(𝐁⊗𝐈).

The (ijk,lmn)-element of 𝐀_2 is the instantaneous rate of transition from  (x,x+1,x+2,lmn) to  (x,x+1,x+2,ijk). This transition is possible when either m>l or n>m according to the rule of the mTASEP to the left direction. Hence, using the arguments in Section <ref>, we have

    𝐀_2 = q(𝐁_1 ⊗𝐈) +  q(𝐈⊗𝐁_1).

𝐀_3 is a diagonal matrix such that the (ijk,ijk)-element represents the rate of jumping out of the state (x,x+1,x+2,ijk) due to particles' jumps to the left direction. Recall that q𝐁_2 ⊗𝐈 describes the rate of jumping out of the state (x,x+1,x_3,ijk) with x+1 < x_3-1 due to the first particle's jump  to the left  and the second particle's jump  to the left (which is possible only when j>i) and 𝐈⊗ q𝐁_2  describes the rate of jumping out of the state (x_1,x,x+1,ijk) with x_1<x-1 due to the second particle's jump to the left  and third particle's jump to the left (which is possible only when k>j). Hence, it must be that

    𝐀_3 = q𝐁_2 ⊗𝐈  + 𝐈⊗ q𝐁_2 - q𝐈_27.

Again, using that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_9, we see that (<ref>) is equivalent to

    d/dt𝐏_Y(x,x+1,x+2;t) = p𝐏_Y(x-1,x+1,x+2;t) + p(𝐁⊗𝐈) 𝐏_Y(x-1,x,x+2;t) 
       +p(𝐈⊗𝐁)(𝐁⊗𝐈) 𝐏_Y(x-1,x,x+1;t) +q𝐏_Y(x,x+1,x+3;t) 
       +  q(𝐁⊗𝐈 +  𝐈⊗𝐁) 𝐏_Y(x,x+1,x+2;t)- 3 𝐏_Y(x,x+1,x+2;t).



 §.§ Two-body reducibility

We obtained four different forms  of the master equation  (<ref>), (<ref>), (<ref>), and (<ref>) for 𝐏_Y(X;t), depending on X=(x_1,x_2,x_3). Now, we will show that (<ref>), (<ref>), and (<ref>) can be derived from the equation in the form of  (<ref>) and the boundary conditions that describe two-particle interactions.

We naturally extend the notations 𝐔(X;t) and  U_(Y,ν)(X,π;t) for N=2 in Section <ref>  to N=3. Suppose that 𝐔(X;t) = 𝐔(x_1,x_2,x_3;t) satisfies
 
    d/dt𝐔(x_1,x_2,x_3;t)=     p𝐔(x_1-1,x_2,x_3;t) + p𝐔(x_1,x_2-1,x_3;t) + p𝐔(x_1,x_2,x_3-1;t)
    
    +    q𝐔(x_1+1,x_2,x_3;t) + q𝐔(x_1,x_2+1,x_3;t) + q𝐔(x_1,x_2,x_3+1;t)
        - 3𝐔(x_1,x_2,x_3;t)

for all x_1,x_2,x_3 ∈ℤ. If 𝐔(x_1,x_2,x_3;t) also satisfies

    (𝐁⊗𝐈) 𝐔(x-1,x,x_3;t) =  𝐔(x,x,x_3;t)   for all x,x_3 ∈ℤ,

then for x_1=x, x_2 = x+1, x_2<x_3-1, (<ref>) becomes
  
    d/dt𝐔(x,x+1,x_3;t)  =  p𝐔(x-1,x+1,x_3;t) + p(𝐁⊗𝐈) 𝐔(x-1,x,x_3;t) 
        + p𝐔(x,x+1,x_3-1;t) +q(𝐁⊗𝐈)𝐔(x,x+1,x_3;t) + q𝐔(x,x+2,x_3;t)
        +q𝐔(x,x+1,x_3+1;t)-3𝐔(x,x+1,x_3;t)

which is in the form of (<ref>). Similarly, if  𝐔(x_1,x_2,x_3;t) satisfies

    ( 𝐈⊗𝐁)𝐔(x_1,x-1,x;t) =𝐔(x_1,x,x;t)   for all x_1,x ∈ℤ,

then (<ref>) becomes
 
    d/dt𝐔(x_1,x,x+1;t) =   p𝐔(x_1-1,x,x+1;t) + p𝐔(x_1,x-1,x+1;t)
       + p(𝐈⊗𝐁)  𝐔(x_1,x-1,x;t)+  q𝐔(x_1+1,x,x+1;t)
        +q(𝐈⊗𝐁)𝐔(x_1,x,x+1;t)+ q𝐔(x_1,x,x+2;t)-3𝐔(x_1,x,x+1;t).

which is in the form of (<ref>).

Now, we find a condition for (<ref>) to be in the form of (<ref>). For this purpose, we substitute x_1=x,x_2 = x+1, x_3 = x+2 in (<ref>) and compare it with (<ref>). Then, we can see that

    p𝐔(x,x,x+2;t) + p𝐔(x,x+1,x+1;t) +  q𝐔(x+1,x+1,x+2;t)  + q𝐔(x,x+2,x+2;t) 
       = p(𝐁⊗𝐈) 𝐔(x-1,x,x+2;t) +p(𝐈⊗𝐁)(𝐁⊗𝐈) 𝐔(x-1,x,x+1;t) 
       + q(𝐁⊗𝐈 +  𝐈⊗𝐁)𝐔(x,x+1,x+2;t)

for all x ∈ℤ if and only if  (<ref>) is in the form of (<ref>). To satisfy (<ref>), we set:

    p𝐔(x,x,x+2;t) + p𝐔(x,x+1,x+1;t) = 
       p(𝐁⊗𝐈) 𝐔(x-1,x,x+2;t) +p(𝐈⊗𝐁)(𝐁⊗𝐈) 𝐔(x-1,x,x+1;t)

and

    q𝐔(x+1,x+1,x+2;t)  + q𝐔(x,x+2,x+2;t) = 
        q(𝐁⊗𝐈 +  𝐈⊗𝐁)𝐔(x,x+1,x+2;t).

Now, we will show that  (<ref>) and (<ref>)  are satisfied if the boundary conditions (<ref>) and (<ref>)  are satisfied. It is easy to see that (<ref>) is satisfied by  (<ref>) and (<ref>). Let us show that  (<ref>) is satisfied by (<ref>) and (<ref>). The first term on the left side of (<ref>) is the same as the first term on the right side of (<ref>) by (<ref>). Also, the second term on the left side of (<ref>) is the same as the second term on the right side of (<ref>) because

    ( 𝐈⊗𝐁)(𝐁⊗𝐈)𝐔(x-1,x,x+1;t) = (𝐈⊗𝐁)𝐔(x,x,x+1;t)

by (<ref>), and

    (𝐈⊗𝐁)𝐔(x,x,x+1;t) = 𝐔(x,x+1,x+1;t)

by (<ref>).


 §.§ Bethe ansatz solution

In the previous section, we showed that if 𝐔(X;t) satisfies (<ref>), (<ref>) and (<ref>), then it satisfies the master equation  (<ref>), (<ref>), (<ref>), or (<ref>) for each X in the corresponding physical region. In this section, we find such a solution 𝐔(X;t) that satisfies (<ref>), (<ref>) and (<ref>). Extending the idea and the notations in Section <ref>,  we separate variables in the form U(X;t) = U(X)T(t). This leads to the following spatial equations derived from (<ref>), (<ref>) and (<ref>):

    ε𝐔(x_1,x_2,x_3) =     p𝐔(x_1-1,x_2,x_3) + p𝐔(x_1,x_2-1,x_3) + p𝐔(x_1,x_2,x_3-1) 
    
    +     q𝐔(x_1+1,x_2,x_3) + q𝐔(x_1,x_2+1,x_3) + q𝐔(x_1,x_2,x_3+1)
    
    -     3𝐔(x_1,x_2,x_3)

and

    (𝐁⊗𝐈) 𝐔(x-1,x,x_3)    =    𝐔(x,x,x_3),
    
    ( 𝐈⊗𝐁)𝐔(x_1,x-1,x)    =    𝐔(x_1,x,x).

The Bethe ansatz solution for  (<ref>) is given by

    𝐔(x_1,x_2,x_3) = ∑_σ∈ S_3𝐀_σξ_σ(1)^x_1ξ_σ(2)^x_2ξ_σ(3)^x_3

where ξ_1,ξ_2,ξ_3 are nonzero complex numbers. The sum is taken over all permutations σ=σ(1)σ(2)σ(3) in the symmetric group S_3, and 𝐀_σ is a 27 × 27 matrix with the same structure as  𝐔(x_1,x_2,x_3) (that is, zero elements in  𝐀_σ and  𝐔(x_1,x_2,x_3) occupy the same positions), whose nonzero elements are constants with respect to t,x_1,x_2,x_3. Substituting (<ref>) into (<ref>), we find that

    ε = p/ξ_1 + p/ξ_2 + p/ξ_3 + qξ_1 +qξ_2 + qξ_3 - 3.


We now substitute (<ref>) into (<ref>) and (<ref>) to derive conditions on 𝐀_σ that ensure (<ref>) and (<ref>) are satisfied. First, substituting (<ref>) into (<ref>) and simplifying,  we obtain

    (𝐀_123+𝐀_213)+ (𝐀_132+𝐀_312)+ (𝐀_231+𝐀_321)  
       =(𝐁⊗𝐈)((𝐀_1231ξ_1+𝐀_2131ξ_2)+ (𝐀_1321ξ_1+𝐀_3121ξ_3)+ (𝐀_2311ξ_2+𝐀_3211ξ_3)).

We set

    𝐀_123+𝐀_213 =     (𝐁⊗𝐈)(𝐀_1231ξ_1+𝐀_2131ξ_2), 
    𝐀_132+𝐀_312 =     (𝐁⊗𝐈)(𝐀_1321ξ_1+𝐀_3121ξ_3),
    𝐀_231+𝐀_321 =     (𝐁⊗𝐈)(𝐀_2311ξ_2+𝐀_3211ξ_3),

to satisfy (<ref>). Equivalently, we have

    𝐀_213 =     -(𝐈_27 - 1ξ_2𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_1𝐁⊗𝐈)𝐀_123,  
    𝐀_132 =     -(𝐈_27 - 1ξ_1𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_3𝐁⊗𝐈)𝐀_312,  
    𝐀_321 =     -(𝐈_27 -1ξ_3𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_2𝐁⊗𝐈)𝐀_231.

Recall that  in (<ref>) 𝐈 = 𝐈_3.  Performing the matrix multiplications in (<ref>), we obtain

    -(𝐈_27 - 1ξ_β𝐁⊗𝐈)^-1(𝐈_27 - 1ξ_α𝐁⊗𝐈)
      =    -(𝐈_9⊗𝐈 - 1ξ_β𝐁⊗𝐈)^-1(𝐈_9⊗𝐈 - 1ξ_α𝐁⊗𝐈) 
     =    -((𝐈_9 - 1ξ_β𝐁)⊗𝐈)^-1((𝐈_9 - 1ξ_α𝐁)⊗𝐈)
     =     -((𝐈_9 - 1ξ_β𝐁)^-1⊗𝐈)((𝐈_9 - 1ξ_α𝐁)⊗𝐈) 
     =    -((𝐈_9 - 1ξ_β𝐁)^-1(𝐈_9 - 1ξ_α𝐁))⊗𝐈.

Note that 𝐁 can be made block-diagonal by reordering labels of rows and columns in the order of 11,22,33,12,21,13,31,23,32. Using the results from (<ref>), we then obtain

    -(𝐈_9 - 1ξ_β𝐁)^-1(𝐈_9 - 1ξ_α𝐁)  = 𝐑_βα

where 𝐑_βα is a 9 × 9 matrix with the (ij,lm)-element given by

    (𝐑_βα)_(ij,lm) =
    
    S_βα      if ij = lm and i≥ j
    
    -1      if ij = lm and i<j
    
    T_βα      if ij = ml and i>j
    
    0      otherwise.

Here, S_βα and T_βα  are defined in (<ref>).
Thus, (<ref>) can be  more succinctly written as

    𝐀_βαγ =  (𝐑_βα⊗𝐈)𝐀_αβγ.

Therfore, if 𝐀_σ satisfies (<ref>), then the Bethe ansatz solution (<ref>) will also satisfy (<ref>). Similarly, we can demonstrate that if

    𝐀_αγβ = (𝐈⊗𝐑_γβ)𝐀_αβγ,

then the Bethe ansatz solution (<ref>) will satisfy (<ref>). However, to ensure consistency between the expressions in (<ref>) and (<ref>), it must be verified that
 
    (𝐑_γβ⊗𝐈)(𝐈⊗𝐑_γα)(𝐑_βα⊗𝐈) = (𝐈⊗𝐑_βα)(𝐑_γα⊗𝐈)(𝐈⊗𝐑_γβ)

 and
 
    (𝐑_βα⊗𝐈)(𝐑_αβ⊗𝐈) = 𝐈_27=( 𝐈⊗𝐑_βα)( 𝐈⊗𝐑_αβ).

The proofs of (<ref>) and (<ref>) are provided in Appendix <ref>, using the approach outlined in Section 2.4 of <cit.>.


§ GENERAL N≥ 2

In this section, we extend the results for N=2,3 to general N≥ 2.


 §.§ Master equations

First, we generalize the matrices 𝐁 in (<ref>) and   (<ref>) for N=2,3   to general N.

We define 𝐁 as the N^2 × N^2 matrix with rows and columns labelled lexicographically by 11,12,…, NN,  such that the (ij,lm)-element is given by

    𝐁_(ij,lm)
    = 
    1   if ij = lm and i≥ j,
    
    1   if ij = ml and i>j,
    
    0   otherwise,


Now, we define a matrix that describes the permissibility of transition for a two-particle sector consisting of the l-th and (l+1)-th particles.

For l=1,…, N-1, we define N^N × N^N matrix

    𝐁^(l)   :=   𝐈  ⊗  ⋯  ⊗  𝐈_(l-1) times  ⊗  𝐁  ⊗  𝐈  ⊗  ⋯  ⊗  𝐈_(N-l-1) times;

where 𝐈  is the N × N identity matrix.

Then, 𝐁^(l) is an N^N × N^N matrix with rows and column labelled lexicographically  by π and ν, respectively, with π, ν = 1⋯1, …, N⋯ N. The (π,ν)-element of 𝐁^(l) describes the permissibility of  transition

    (x_1,…, x_l-1,x-1, x,x_l+2,…, x_N, ν)   ⟶    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, π)

where π_i = ν_i for i ≠  l,l+1.

We generalize 𝐁_1 in (<ref>) and (<ref>), and 𝐁_2 in (<ref>) and (<ref>) to general N.

We define 𝐁_1 and 𝐁_2 as  N^2 × N^2 matrices with rows and columns labelled lexicographically by 11,12,…, NN,  such that their (ij,lm)-elements are given by

    (𝐁_1)_(ij,lm)
    = 
    1   if ij = ml and i> j,
    
    0   otherwise.

and

    (𝐁_2)_(ij,lm)
    = 
    1   if ij = lm and i≥ j,
    
    2   if ij = lm and i< j,
    
    0   otherwise,

respectively. In a same manner to Definition <ref>, we define 𝐁_1^(l) and 𝐁_2^(l).

The (π,ν)-element of 𝐁_1^(l) describes the permissibility of  transition

    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, ν)   ⟶    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, π)

where π_i = ν_i for i ≠  l,l+1.  The (π,π)-element of the N^N × N^N diagonal matrix  q𝐁_2^(l)   represents  the rate of jumping out of the state

    (x_1,…, x_l-1,x, x+1,x_l+2,…, x_N, π)

due to the jumps of π_l and π_l+1 to the left when x-1 is empty.

We extend the notations 𝐔(X;t) and U_(Y,ν)(X,π;t)  used for N=2,3 to general N. If 𝐔(X;t) satisfies
 
    d/dt𝐔(X;t) =    ∑_i=1^N p𝐔(x_1,⋯, x_i-1,x_i-1,x_i+1,⋯, x_N;t)  
    
                                 +    ∑_i=1^N q𝐔(x_1,⋯, x_i-1,x_i+1,x_i+1,⋯, x_N;t)
    
                                 -    N 𝐔(X;t)

and the boundary conditions
 
    𝐔(x_1,⋯,x_l-1,x,x,x_l+2,⋯, x_N;t) =𝐁^(l) 𝐔(x_1,⋯,x_l-1,x-1,x,x_l+2,⋯, x_N;t)

 for all l=1,…, N-1, then 𝐔(X;t) satisfies the master equation corresponding to any given X in the physical region. For example,  for X=(x,x+1,…,x+N-1), (<ref>) becomes

    d/dt𝐔(X;t) =     p𝐔(x-1,x+1,…, x+N-1;t) 
       +  p𝐁^(1)𝐔(x-1,x,x+2,…,x+N-1 ;t)  
       +  p𝐁^(2)𝐁^(1)𝐔(x-1,x,x+1,x+3,…,x+N-1 ;t)  
       + ⋯ +  p(𝐁^(N-1)⋯𝐁^(1))𝐔(x-1,x,…,x+N-2 ;t) 
       + q𝐔(x,x+1,…, x+N-2, x+N;t) 
       +  q(𝐁^(1) + ⋯+ 𝐁^(N-1))𝐔(X;t)- N𝐔(X;t)

by the boundary conditions (<ref>). If we use the fact that 𝐁_2 + 𝐁 - 𝐁_1 = 2𝐈_N^2, then (<ref>) can be rewritten as   the master equation involving  𝐁^(l), 𝐁_1^(l), 𝐁_2^(l).


 §.§ Bethe ansatz solution

   The Bethe ansatz solution to (<ref>) is given by
 
    𝐔(X;t) = ∑_σ∈ S_N𝐀_σ∏_i=1^N(ξ_σ(i)^x_i e^ε(ξ_i)t)

 where
 
    ε(ξ_i) = p/ξ_i +qξ_i - 1.

 To ensure that  (<ref>) satisfies (<ref>) for each l=1,⋯, N-1,   we substitute (<ref>) into (<ref>) and simplify to obtain
 
    ∑_σ∈ S_N(𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ = 0

 for each l.

 Let A_N be the subgroup of all even permutations of S_N, and  let T_i ∈ S_N be the simple transposition that exchanges the i-th and the (i+1)-th elements. We can then express  (<ref>) as
 
    0 =  ∑_σ∈ A_N(𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ
     + ∑_σ∈ A_N(𝐈_N^N - 1ξ_σ(l+1)𝐁^(l)) 𝐀_T_lσ.

 To satisfy (<ref>), we set
 
    0 = (𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ
     + (𝐈_N^N - 1ξ_σ(l+1)𝐁^(l)) 𝐀_T_lσ.

for each σ∈ A_N. Consequently, we obtain
 
    𝐀_T_lσ =     -(𝐈_N^N - 1ξ_σ(l+1)𝐁^(1))^-1 (𝐈_N^N - 1ξ_σ(l)𝐁^(l)) 𝐀_σ
    
     =     ( 𝐈 ⊗ ⋯ ⊗ 𝐈_(l-1) times ⊗ 𝐑_σ(l+1)σ(l) ⊗ 𝐈 ⊗ ⋯ ⊗ 𝐈_(N-l-1) times  )𝐀_σ

 where the second equality in (<ref>) is derived similarly to  (<ref>), and  𝐑_σ(l+1)σ(l) is the N^2 × N^2 matrix whose rows and columns are labelled by 11,…, NN, with its (ij,lm)-elements given as  in (<ref>).
  
Let 𝐑_βα be N^2 × N^2 matrix with rows and columns  labelled lexicographically by 11,…, NN, where  the (ij,lm)-element is given by (<ref>). We define
 
    𝐓_l,βα:= 𝐈  ⊗  ⋯  ⊗  𝐈_(l-1) times  ⊗  𝐑_βα  ⊗  𝐈  ⊗  ⋯  ⊗  𝐈_(N-l-1) times

where 𝐈 is the N× N matrix.
 
The following  proposition follows from the arguments presented above.

If

    𝐀_T_lσ =  𝐓_l,σ(l+1)σ(l)𝐀_σ

for each σ∈ S_N, then the boundary condition (<ref>) is satisfied.

Since {T_1,⋯, T_N-1} generates S_N, for any σ∈ S_N, there exists an expression

    σ = T_l_k⋯ T_l_2T_l_1

for some l_1,…, l_k ∈{1,…, N-1}.  The following lemma provided a condition for (<ref>) to be satisfied, which is evident  from the construction.

Let σ be given by (<ref>) and let σ^(i) =  T_l_i⋯ T_l_1, i=1,…, k with σ^(0) denoting the identity permutation. If

    𝐀_σ = 𝐀_ T_l_k⋯ T_l_2T_l_1 := 𝐓_l_k,σ^(k-1)(l_k+1)σ^(k-1)(l_k) × ⋯ × 𝐓_l_1,σ^(0)(l_1+1)σ^(0)(l_1),

then (<ref>) is satisfied.

For 𝐀_σ in (<ref>) to be well-defined,  the following relations  must also be satisfied:

    𝐓_i,αβ𝐓_i,βα = 𝐈;
       𝐓_i,βγ𝐓_i+1,αγ𝐓_i,αβ = 𝐓_i+1,αβ𝐓_i,αγ𝐓_i+1,βγ.

In fact, Lemma <ref> and Lemma <ref> demonstrate that these relations are satisfied.


 §.§ Transition probabilities

In the previous section, we claimed that 𝐔(X;t) in (<ref>) with 𝐀_σ defined by (<ref>) satisfies the master equation for each X in the corresponding physical region. This holds true when performing contour integrals over all ξ_i variables with appropriate contours. This approach is well known for various integrable particle models <cit.>. The  transition probability P_(Y,ν)(X,π;t) of our model can also be expressed as a contour integral, analogous to the formula for the multi-species ASEP in <cit.>.

Let C_R_i be a counterclockwise circle centered at the origin with radius R_i and suppose that 1<R_1< ⋯ <R_N. Let 𝐀_σ be given by (<ref>) and ε(ξ_i) by (<ref>). For any arbitrary initial state (Y,ν), the transition probability P_(Y,ν)(X,π;t) is given by:

    P_(Y,ν)(X,π;t) = (1/2π i)^N∫_C_R_N⋯∫_C_R_1∑_σ∈ S_N(𝐀_σ)_(π,ν)(∏_i=1^Nξ_σ(i)^x_i - y_σ(i) - 1e^ε(ξ_i)t)dξ_1 … dξ_N.



It is sufficient to verify the initial condition. Since the denominators of the nontrivial terms in the matrix 𝐑_βα that is contained in the expression of 𝐀_σ match those of S_βα^† with μ=1 and λ = 0 in Section 4 of <cit.>, the proofs of Corollary 2 and Theorem 1 in <cit.> essentially imply that (<ref>) satisfies the initial condition.



§ DISCUSSION

We have defined a multi-species stochastic particle model (distinct from the multi-species ASEP) on ℤ, where particles move either  to the right or the left: a particle jumping to the right moves to the nearest site occupied by a lower-numbered particle (or an empty site) and pushes the lower-numbered particle  forward (if any);  a particle jumping  to the left follows the rules of the conventional multi-species TASEP. We have shown that  this model is integrable and  derived the formula for the transition probability for an arbitrary initial state. A key motivation of defining this model such a way that the model can be integrable is the symmetry between the one-sided model with long-range jumps and the multi-species TASEP in the opposite direction, in the sense that their boundary conditions are compatible when using the Bethe ansatz.

Given the form of the matrices in (<ref>), it is anticipated that P_(Y,ν)(X,π;t) with π = ν can be expressed as a determinant (See <cit.> for related results in the multi-species TASEP context). Generally, the integrand in (<ref>) can be a sum of numerous products due to  𝐀_σ being a product of matrices as described in (<ref>).  However, for certain special initial permutations ν of species, such as 12⋯ N, 12⋯2, 1⋯ 12 and so on, it seems that (𝐀_σ)_(π,ν) can be expressed as a single product  (See <cit.> for related results for the multi-species ASEP). Also, it would be interesting to investigate whether the methods used in this paper could serve as a machinery to develop other new integrable multi-species particle models as well as multi-species versions of other known single-species models such as <cit.>. Furthermore, these methods might help find the transition probabilities for some known multi-species particle models mentioned in <cit.>.

Finally, we have not yet identified an integrable multi-species model where particles make long-range jumps in both directions, that is, a multi-species version of the model in <cit.>. Developing such a multi-species version and finding its transition probability remain an area for future research.







§ PROOF OF THE CONSISTENCY CONDITIONS

We use the same approach as in Section 2.4 of <cit.>.

If 𝐑_βα is the N^2 × N^2 matrix defined by (<ref>), then  𝐑_βα𝐑_αβ is the identity matrix.


Let ij and kl be labels of rows and columns with ij, kl = 11,…, NN. For i =j,
 
    (𝐑_βα𝐑_αβ)_(ii,kl) =    ∑_(mn)(𝐑_βα)_(ii,mn)(𝐑_αβ)_(mn,kl)
    
     =    (𝐑_βα)_(ii,ii)(𝐑_αβ)_(ii,kl) = 
     S_βαS_αβ=1    if kl = ii,
    
     S_βα· 0=0    if kl ≠ ii.

For i>j,

    (𝐑_βα𝐑_αβ)_(ij,kl) =    ∑_(mn)(𝐑_βα)_(ij,mn)(𝐑_αβ)_(mn,kl)
    
     =    (𝐑_βα)_(ij,ij)(𝐑_αβ)_(ij,kl) + (𝐑_βα)_(ij,ji)(𝐑_αβ)_(ji,kl)
    
     =    
      S_βα· S_αβ + T_βα· 0 =1     if kl = ij,
    
     S_βα· T_αβ+T_βα·(-1) = 0    if kl = ji
    
     S_βα·0 + T_βα· 0 = 0    if kl ≠ ij,ji.

 For i<j,
 
    (𝐑_βα𝐑_αβ)_(ij,kl) =    ∑_(mn)(𝐑_βα)_(ij,mn)(𝐑_αβ)_(mn,kl)
    
     =    (𝐑_βα)_(ij,ij)(𝐑_αβ)_(ij,kl)+(𝐑_βα)_(ij,ji)(𝐑_αβ)_(ji,kl)
    
     =    
      (-1)·(-1)+  0· T_βα =1     if kl = ij,
    
      (-1)· 0+0· S_αβ = 0    if kl = ji
    
     (-1)· 0 +0·  0= 0    if kl ≠ ij,ji.


 (Yang-Baxter equation)
Let N ≥ 3. If 𝐑_βα is the N^2 × N^2 matrix defined by (<ref>) and 𝐈 be the N × N matrix,

    (𝐑_γβ⊗𝐈)(𝐈⊗𝐑_γα)(𝐑_βα⊗𝐈) = (𝐈⊗𝐑_βα)(𝐑_γα⊗𝐈)(𝐈⊗𝐑_γβ).



Both (𝐈⊗𝐑_βα) and (𝐑_βα⊗𝐈) are N^3 × N^3 matrices with rows and columns labelled with 111,⋯, NNN, and each label is a permutation of a certain multi-set [i,j,k]. If ijk and lmn are permutations from two different multi-sets, then

    (𝐈⊗𝐑_βα)_(ijk,lmn) = 0 = (𝐑_βα⊗𝐈)_(ijk,lmn).

 Hence, if we reorder the columns and rows so that all permutations  of a given multi-set are grouped, then both (𝐈⊗𝐑_βα) and (𝐑_βα⊗𝐈) become block-diagonal. Let

    (𝐑_βα⊗𝐈)_[i,j,k]

be the block on the diagonal of the matrix 𝐑_βα⊗𝐈  whose  rows are columns are labelled with permutations of [i,j,k]. We similarly define  (𝐈⊗𝐑_βα)_[i,j,k]. It suffices to show  (<ref>) blockwise, that is,

    (𝐑_γβ⊗𝐈)_[i,j,k](𝐈⊗𝐑_γα)_[i,j,k](𝐑_βα⊗𝐈)_[i,j,k] = (𝐈⊗𝐑_βα)_[i,j,k](𝐑_γα⊗𝐈)_[i,j,k](𝐈⊗𝐑_γβ)_[i,j,k]

for each multi-set [i,j,k].  Each multi-set [i,j,k] of integers 1,…, N is one of the forms [i,i,i], [i,i,j] with i< j, [i,i,j] with i> j and [i,j,k] with i<j<k.  First, for the multi-set [i,i,i], (<ref>) is simply

    S_γβS_γαS_βα  = S_βαS_γαS_γβ

which is obviously true because

    (𝐑_βα⊗𝐈)_[i,i,i] = S_βα = (𝐈⊗𝐑_βα)_[i,i,i].

For [i,i,j] with i<j, we can obtain

    (𝐑_βα⊗𝐈)_[i,i,j]  =
        iij     iji     jii 
    
    iij     S_βα    0     0  
    
    iji    0    -1     0
    
    jii     0    T_βα    S_βα

and

    (𝐈⊗𝐑_βα)_[i,i,j]=    iij     iji     jii 
    
    iij     -1    0     0  
    
    iji    T_βα   S_βα    0
    
    jii     0    0     S_βα.

Using these matrices, we can verify (<ref>) by direct matrix computation. Similarly, (<ref>) is verified for [i,i,j] with i>j. Finally, for [i,j,k] with i<j<k, we obtain

    (𝐑_βα⊗𝐈)_[i,j,k]  =     ijk     ikj     jik     jki     kij     kji 
    
        ijk     -1     0     0     0     0     0 
    
        ikj     0     -1     0     0     0     0 
    
       jik      T_βα    0     S_βα    0    0      0 
    
       jki     0     0     0     -1     0      0
    
       kij     0      T_βα    0     0    S_βα    0 
    
       kji     0     0     0     T_βα    0     S_βα

and

    (𝐈⊗𝐑_βα)_[i,j,k]  =     ijk     ikj     jik     jki     kij     kji 
    
        ijk     -1     0     0     0     0     0 
    
        ikj     T_βα    S_βα    0     0     0     0 
    
       jik     0      0     -1     0    0      0 
    
       jki     0     0     T_βα    S_βα    0      0
    
       kij     0      0     0     0    -1      0 
    
       kji     0     0     0     0      T_βα    S_βα
    .

Again, we can verify (<ref>) by  directly performing matrix multiplication.






Acknowledgement This research is funded by Nazarbayev University under the faculty-development competitive research grants program
for 2024 – 2026 (grant number 021220FD4251). The author is grateful to Axel R. Saenz for valuable discussions, and to Temirlan Raimbekov and Nazgul Tileukabyl for their assistance in writing the manuscript.

 

Conflict of Interest The author declares that there is no conflict of interest.
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 Paper Title: A sharp lower bound on the small eigenvalues of surfaces.


GOProteinGNN: Leveraging Protein Knowledge Graphs for Protein Representation Learning
    Kira Radinsky
    August 5, 2024
=====================================================================================



§ ABSTRACT 
    Let  be a compact hyperbolic surface of genus g≥ 2 and let () = 1/()∫_1/x^2 ∧ 1 dx, where x is the injectivity radius at x.
    We prove that for any k∈{1,…, 2g-3}, the k-th eigenvalue λ_k of the Laplacian satisfies
    
    λ_k ≥c k^2/() g^2  ,
    
    where c>0 is some universal constant. We use this bound to prove the heat kernel estimate
    
    1/()∫_| txx -1/() |  dx ≤  C √(()/t)    ∀ t ≥ 1   ,
    
    where C<∞ is some universal constant. These bounds are optimal in the sense that for every g≥ 2 there exists a compact hyperbolic surface of genus g satisfying the reverse inequalities with different  constants.




§ INTRODUCTION
 
Let  be a compact hyperbolic surface of genus g≥ 2. Due to compactness, the Laplacian has a discrete spectrum 0=λ_0 < λ_1 ≤λ_2 ≤… with λ_j →∞ as j →∞. 
The eigenvalues below 1/4 are usually called small, see <cit.> and <cit.>.
A classical construction of Buser <cit.> shows that for any >0 and any g≥ 2 there exists such a compact hyperbolic surface  of genus g with λ_k ≤ for all k=1,…, 2g-3. Otal and Rosas <cit.> showed that λ_2g-2 > 1/4 for any , hence only the first 2g-3 eigenvalues can be small. Here we provide a sharp estimate as to how small the small eigenvalues can be in terms of k and g and the quantity () =1/()∫_1/x^2∧ 1 dx, where x is the injectivity radius at x.

 There exists a universal constant c>0 such that for any compact hyperbolic surface  of genus g≥ 2 and every k∈{1, …, 2g-3},   

    λ_k ≥c k^2/() g^2  .



We use this estimate to provide a uniform bound on the heat kernel trace on .

For any compact hyperbolic surface  
     
    1/()∫_| txx -1/() |  dx ≤ C √(()/t)    ∀ t ≥ 1   .

where txy is the heat kernel of the Laplacian on  and C<∞ is a universal constant.


We also show that <ref> and <ref> are optimal in the sense that for any g≥ 2 and I≥ 1, there exists a compact hyperbolic surface  of genus g and I(S)
≥ I such that the reversed inequalities of (<ref>) and (<ref>)
 holds with a different constant.

We remark that () has an additional simple geometric interpretation. Let γ_1,…, γ_s be the set of all simple closed geodesics in  of length at most 2sinh^-1(1) and denote by {ℓ(γ_i)}_i=1^s their lengths. Then it is not hard to prove (see <ref>) that 
        
    () ≍ 1 + 1/()∑_i=1^s 1/ℓ(γ_i)  .





 §.§ Related work
 A classical paper of Schoen, Wolpert and Yau <cit.> provides the bound λ_k ≥α(g) L_k(), where α(g) is an unknown function of g, and L_k() is the minimal possible sum of the lengths of simple closed geodesics in  that cut  into k + 1 components. In order to improve the dependence on g of this bound, it is shown in <cit.> that λ_1 ≳ L_1()/g^2 and in <cit.> that λ_2 ≳ L_2()/g^2. 

We remark that while both quantities L_k() and 1/() measure the amount of bottlenecks in a surface (see <ref>), they are in general incomparable. Whenever the injectivity radius is uniformly bounded below, <ref> implies that λ_k ≳ k^2/g^2 for k∈{1,…,2g-3} improving a result obtained in <cit.> regarding such lower bounds in any thick part of the moduli space. 

On-diagonal heat kernel bounds such as (<ref>) are a well studied topic going back to Nash, Moser and Varopoulos, see  <cit.>. They are classically proved using a suitable isoperimetric inequality and typically yield a uniform estimate in x. In this paper, however, one can see that isoperimetric inequalities are irrelevant; for example, one can take O(1) primitive closed geodesic of length Θ(1), pinch them so that their length is Θ(1/g) and () will remain of the same order, leaving the bounds of Theorems <ref> and <ref> unaltered. Additionally, one cannot hope for a bound on the on-diagonal heat kernel that is uniform in x without some global isoperimetric inequality. 



  §.§.§ Discrete analogue
 
Laplacian eigenvalues and heat kernel bounds are a well studied topic in the discrete setting as well. Let G be a simple, connected, regular graph with n vertices and consider the lazy random walk on it, that is, the walker moves to a uniformly drawn neighbor with probability 1/2 or stays put otherwise. Let P denote its transition probability matrix. The graph Laplacian ℒ=I-P has eigenvalues 0=λ_0 < λ_1 ≤…λ_n-1≤ 1. The bound 

    λ_k ≳k^2/n^2  ,

for all k∈{1,…,n} is well known, and follows directly from the estimate (see <cit.>): 

    1/n |∑_x P^t(x,x) - 1  | ≲1/√(t)  ,

for any integer t≥ 1 (this is the discrete analogue of (<ref>)). Indeed, recalling that ∑_x P^t(x,x) = ∑_i=0^n-1 (1-λ_i)^t and 
plugging in t=⌈λ_k^-1⌉ to the above immediately yields (<ref>). In this setup it is also known that the t^-1/2 upper bound on |P^t(x,x)-n^-1| holds uniformly in x. Of course we cannot hope for the bound (<ref>) to be valid for all surfaces, nor can we expect a uniform bound in x for same reason, namely, the possibility of arbitrarily small bottlenecks.

Thus, the more appropriate discrete analogue are finite weighted graphs. In this setting, bounds on the spectral measure at a vertex were given by Lyons and Oveis Gharan <cit.>. Similar bounds were obtained by Lyons and Judge <cit.> in the setup of homogenous Riemannian manifolds. We were greatly inspired by the use of the spectral kernel in <cit.>; as done there, we too bound the eigenvalues by the Dirichlet energy of a function associated with the spectral kernel, but the bulk of this paper is dedicated to bounding below this energy using a novel geometric argument based on extremal length. 



 §.§ Extremal length
 

Extremal length is a geometric method in complex analysis that has had a profound influence on the theory of conformal and quasi-conformal mappings. In this paper we obtain an explicit geometric bound on the extremal length between two  small geodesic discs placed at arbitrary locations in . Given a Borel measurable function ρ : → [0,∞) and a rectifiable curve γ⊆, the ρ-length of γ is
    
    L(γ, ρ) = ∫_γρ |dz|   .


 Given a collection Γ of rectifiable curves in  we define the extremal length of Γ as 
    
    Γ = sup_ρinf_γ∈Γ L(γ,ρ)^2/∫_ρ^2  dμ  ,

    where the supremum is taken over all Borel measurable functions ρ : → [0,∞) and μ is the Riemannian area measure of .


For a general compact Riemannian manifold M, the injectivity radius of a point x ∈ M, denoted x, M, is the supremum radius such that the exponential map exp_x is a diffeomorphism. We often write x when it is clear what M is from the context. The injectivity radius of the manifold M itself is M = inf_x ∈ Mx,M. When  is a Riemannian surface of constant curvature -1, i.e., a hyperbolic surface, x, is the supremum  r so that the ball of radius r around x is isometric to the ball of radius r in the hyperbolic plane .

 
    For x ∈, let 
    
    x= x,  1   .

    For a rectifiable curve γ⊆, let
    
    γ = ∫_γz^-1dz  .
    
    This lets us define the weighted distance between x and y as 
    
    xy = inf_γγ  ,

    where the infimum ranges over all rectifiable curves γ which connect x to y.




Our main geometric result is the following.

 
    Let x,y ∈ and r_x, r_y be positive numbers satisfying 
    
    d_(x,y)≥ r_x+r_y   ,   r_x ≤x/2   ,    r_y ≤y/2   .
        
    Let Γ be the set of all curves in  \( B(x, r_x)  B(y,r_y) ) from ∂ B(x,r_x) to ∂ B(y,r_y). Then 
    
    Γ≲xy + log(x/r_x) + log(y/r_y)   .
            


 Let us provide a rough, yet guiding, intuition. It is well known that any compact hyperbolic surface  has  a pair of pants decomposition in which all simple closed geodesics of length at most 2sinh^-1(1) are boundary geodesics where two pairs of pants were glued. This is the content of the so-called collar lemma (<ref>). If a path γ traverses from one side of a narrow collar of width ℓ to the other, then the contribution of this part of the curve to its weight γ is proportional to 1/ℓ. Thus, γ is the sum of the reciprocal lengths of the narrow collars through which it passes, up to a multiplicative constant.

We may think of this pair of pants decomposition of  as defining a 3-regular graph in which the pants are vertices and edges are formed between two pairs of pants if they are glued at one of their three boundary geodesics. Assign to each edge a weight c_e equal to the length of the corresponding boundary geodesic. The weighted distance between two vertices x,y in this analogous discrete setting is the minimal sum of reciprocals edge weights 1/c(e) over all paths from x to y.

If the weights {c_e}  are thought of as  electric conductances assigned to the edges, then by the parallel law together with Rayleigh's monotonicity law, the effective electric resistance between x and y is at most the sum of edge resistances, i.e. 1/c(e), over the edges of any path from x to y. So the weighted distance between two vertices x and y gives an upper bound on the effective electric resistance (this bound is far from sharp in many cases). Extremal length is the continuous analogue of effective electric resistance and so <ref> is a very rough analogue of the aforementioned bound. 




 §.§ Notation and organization

For two non-negative functions f(x) and g(x), we write f ≲ g to mean that there exists a universal constant C > 0, such that f(x) ≤ C g(x) for all x. We write f(x) ≍ g(x) if both f ≲ g and g ≲ f. We denote the distance between two points in  by xy. If γ is a rectifiable curve in S, we denote its length by ℓ(γ).

The rest of this paper is organized as follows. In the next section, we provide the necessary geometric background. In <ref> we prove <ref> and in 
<ref> we use it to prove <ref>. 
<ref> is proved in <ref> and optimality of <ref> and <ref> is shown in <ref>.



§ GEOMETRIC PRELIMINARIES

An important geometric characterization of hyperbolic surfaces is the so-called collar lemma, which describes the parts of the surface with small injectivity radius. The following is a combination of Theorem 4.1.1 and Theorem 4.1.6 from <cit.>.
 
    Let γ_1, …, γ_s be the set of all simple closed geodesics of length ≤ 2sinh^-1(1) on a hyperbolic surface . Let W(γ_i) =sinh^-1(1/sinh(1/2γ_i)) and C(γ_i) = {x∈|dist(x,γ_i) ≤ W(γ_i)}. Then 
            
        
  * The sets C(γ_i)_i=1^s are pairwise disjoint.
        
  * x≥sinh^-1(1) for all x ∉_i C(γ_i).
        
  * Each C(γ_i) is isometric to the cylinder [-W(γ_i), W(γ_i)] × ( / 2π)
        with the Riemannian metric ds^2 = dρ^2 + γ_i^2cosh(ρ)^2 dθ^2.
        
  * If x ∈ C(γ_i) and d = dist(x,∂ C(γ_i)), then
        
    sinh(x) = cosh(1/2γ_i)cosh(d) - sinh(d).

    


The following proposition provides a formula and an estimate of the injectivity radius of points in the collars in terms of their cylindrical coordinates.

Let T be an infinite hyperbolic collar whose shortest closed geodesic γ has length ℓ. Denote by r(ρ,ℓ) the injectivity radius at a point of distance ρ from γ. Then

    r(ρ, ℓ) = 1/2cosh^-1(1 + (cosh(ℓ) -1 )cosh(ρ)^2 )   .

In particular, 

    r(ρ, ℓ) ≍ℓcosh(ρ)

for all ℓ≤ 2sinh^-1(1) and all ρ≤ W(γ) where W(γ) is defined in <ref>.


In the upper-half plane model, an infinite collar can be realized as the quotient of  by the relation x ∼λ x, where λ∈ is chosen such that the hyperbolic distance satisfies d_(i,λ i) = ℓ. Since i and λ i are on the imaginary axis, a simple integration gives log(λ) = ℓ. The injectivity radius is then given by 1/2d_(x,e^ℓ x). To calculate this quantity, let x be a point in the upper-half plane whose hyperbolic distance from the imaginary axis is ρ. Let r=|x|, so that d_(x,ir)=ρ. The distance between two points in the upper-half plane is given by

    cosh(d_(z,w)) = 1 + |z-w|^2/2 zw  .
 
This lets us find the imaginary part of x. Using the distance formula, we have

    cosh(ρ) -1    =cosh(d_(x,ir))-1
       =|x-ir|^2/2xr=x^2+(x-r)^2/2xr=2r^2-2xr/2xr=r/x-1   ,

and so x=r/cosh(ρ). Hence

    cosh(d_(x,e^ℓ x))=1 + |x|^2(e^ℓ-1)^2/2e^ℓx^2 = 1 + (e^ℓ-1)^2 cosh(ρ)^2/2e^ℓ,

yielding

    d_(x,e^ℓ x)= cosh^-1(1 + (cosh(ℓ) -1 )cosh(ρ)^2 )   .

The asymptotic result then follows from the asymptotic expansions cosh^-1(1+x) = √(x) + O(x^3/2) and cosh(ℓ)-1 = ℓ^2/2 + O(ℓ^4), and observing that W(γ)≤log(1/ℓ) + C for some absolute constant C as long as ℓ≤ 2. 




        Let  be a compact hyperbolic surface, and let γ_1, …, γ_s be the set of all simple closed geodesics of length ≤ 2sinh^-1(1) in . Then 
        
    () ≍ 1 + 1/()∑_i=1^s 1/ℓ(γ_i)  .

    
 We apply <ref> and obtain that 
    
    ∫_∖_i C(γ_i)1/x^2 1 dx ≍()   .

    For points x ∈_i C(γ_i), the integrand is proportional to 1/x^2. For every collar C(γ_i), by <ref> we therefore have
    
    ∫_C(γ_i)1/x^2 1 dx    ≍∫_C(γ_i)1/x^2 dx 
       ≍∫_0^2π∫_0^∞1/ℓ(γ_i)^2 cosh(ρ)^2ℓ(γ_i)cosh(ρ)  dρ dθ≍1/ℓ(γ_i)  ,

    and the result follows.




    Let  be a compact hyperbolic surface, and let x,y∈ with xy≤ 1. Then
    
    x≍y  .



We appeal to <ref>. We use the term thin part for the set _i=1^s C(γ_i) and thick part for its complement. There are four cases for the relative location of x and y. Firstly, if x and y are both in the thick part, then both x and y are bounded from below by sinh^-1(1), so x≍y. Secondly, if x is in the thick part and y is in the thin part, then y is at distance at most 1 from ∂ C(γ_i) for some i, and so by the fourth item in <ref>, with d≤ 1,
    
    y,   = sinh^-1(cosh(1/2γ_i)cosh(d) - sinh(d) ) 
       ≥sinh^-1(cosh(d) - sinh(d) ) ≥sinh^-1(cosh(1) - sinh(1) )   .

We learn that in this case the injectivity radius at y is bounded from below by a universal constant, and so y≍x follows. Thirdly, if x and y are both in the thin part but belong to different collars, then both x and y are at distance at most 1 from the boundary of some collar, and the same calculation as above apply again to show that both injectivity radii are bounded below. Lastly, suppose x and y are both in the thin part and belong to the same collar C(ℓ(γ)). Writing x = (ρ_1,θ_1) and y = (ρ_2, θ_2) in the cylindrical coordinates of <ref>, we have by <ref> that 
    
    x,/y,≍ℓ(γ) cosh(ρ_1)/ℓ(γ) cosh(ρ_2)  .

    Since cosh(ρ) ≍ e^ρ and ρ_1 - ρ_2≤ 1, the result follows. 


Let M be a Riemannian surface and γ :[a,b] → M a unit speed geodesic in M. For every r ∈ and t ∈ [a,b], let φ_γ(r,t) be the point obtained by going a signed distance r along the geodesic which is perpendicular to γ at the point γ(t). If a point z ∈ M has exactly one pair (r,t) such that z = φ_γ(r,t), then the pair (r,t) are called the Fermi coordinates of z with respect to γ.

When γ is a geodesic in the hyperbolic plane , the map φ_γ is a bijection from ^2 →, and hence it is just a coordinate change; the metric tensor of  in Fermi coordinates is given by 

    ds^2 = dr^2 + cosh(r)^2 dt^2   ,

see <cit.>. For a general surface, the map φ_γ is usually not one-to-one; however, we can still use Fermi coordinates as long as φ_γ is restricted to domains where it is one-to-one. 

  Let M be a Riemannian surface whose curvature is bounded from above by a nonnegative constant K and let γ : [0, ℓ] → M be a unit speed minimal geodesic. Consider the set  

    D_γ = { (r,t) : t∈ [0,ℓ]   , |r| ≤min{γ(t)/4,  π/2 √(K)}}  .

Then the map φ_γ:D_γ→ M is one-to-one.

    
Assume by contradiction that z = φ_γ(t,r) = φ_γ(s,ρ) for t ≠ s. Since zγ(t)≤γ(t)/4 and · is 1-Lipschitz, we deduce that z≥ 3 γ(t)/4; similarly, z≥ 3 γ(s)/4. Thus, the geodesic triangle whose points are z, γ(t) and γ(s) is contained in the ball B(z, z). By the triangle inequality, all sides of this triangle have lengths ≤π / √(K). By a standard triangle comparison theorem <cit.>, the angles of this triangle are smaller than the corresponding angles of a model triangle with the same side-lengths in the sphere of curvature K. This is a contradiction, since the lengths of the segments connecting z to γ(t) and z to γ(s) are smaller than π/2√(K), so the comparison triangle must have an acute angle on the segment connecting γ(t) to γ(s).





§ EXTREMAL LENGTH AND THE INVERSE INJECTIVITY RADIUS


The goal of this section is to prove <ref>. Let us briefly explain the intuition. For any  ρ:S → [0,∞) we need to find a curve from ∂ B(x,r_x) to ∂ B(y,r_y) which has small ρ-length. An adversarial choice of ρ would put significant weight on the narrow regions of the surface (that is, inside thin collars) since there it can capture all the curves passing through the collar for a relatively small price to pay to the contribution to  ∫_ρ^2. Hence we wish to find curves that try to avoid thin collars. To that aim, the reciprocal injectivity weight (<ref>) is useful, since it “punishes” curves that go through thin collars. Our main effort is to prove that if a curve γ minimizes this weight among curves connecting x to y, then the expected ρ-length of a curve randomly chosen from a carefully prescribed tubular neighborhood of γ is bounded above by the right hand side of (<ref>).

In order to study curves in a tubular neighborhood of γ, it is convenient to use Fermi coordinates. Since the reciprocal-injectivity weight-minimizing curve γ is in general not a geodesic and not even smooth (unless S≥ 1), we will first conformally stretch S so that a collar of width ℓ turns to a cylinder of constant width and length 1/ℓ while incurring only a constant additive error to the curvature at any point. In this new surface  the calculation of extremal length becomes natural. Since extremal length is a conformal invariant, the bound we obtain holds also for the original manifold . Let us now give details.

Any conformal coordinate change can be expressed by multiplying the metric by a smooth function f:→ (0,∞). We would have liked to define f simply as 1/z for all z ∈. However, this function is not differentiable, and moreover, it is hard to control the curvature change that it induces. 

Instead, let γ_1, …, γ_s be the set of all simple closed geodesics of length ≤ 2sinh^-1(1) in . Recall from the collar lemma (<ref>) that  can be partitioned into parts, where one part is “thick” and has injectivity radius bounded below by sinh^-1(1), while the others (the “collars”) are isometric to cylinders of length 2W(γ_i) defined in <ref>. For every i = 1 …, s, define the function f_i :→ by    
    
    f_i(x) = 1/ℓ(γ_i) cosh(x)   x < W(γ_i) - 2 
    
                1    x≥ W(γ_i) - 2  .

It is a straightforward calculation that W(γ_i) ∈(log(1/ℓ(γ_i)) + 1, log(1/ℓ(γ_i))+2), and hence the jump discontinuity in f_i is bounded above by a constant. <ref> then implies that if z is in a collar C(γ_i) and has Fermi coordinates (ρ, θ) relative to the geodesic γ_i, then f_i(ρ) ≍ 1/z.
    
Let ψ(x):→ be the standard smooth mollifier
    
    ψ(x) = 
                ce^-1/1-x^2   x≤ 1 
    
                0    otherwise ,

where c is chosen so that ∫_-1^1ψ(x)dx = 1. We now define f: → (0,∞) as follows: 
    
    f(z) = 
    			1    if z ∉_iC(γ_i)
    ∫_-W(γ_i)^W(γ_i) f_i(x)ψ(ρ - x)dx    if z∈ C(γ_i) and has Fermi coordinate (ρ, θ)  .

It is immediate to check that f is smooth. Hence, if g is the hyperbolic metric on , the surface  defined on the same point-set as  equipped with the metric  g̃ = f(z)g is a Riemannian surface conformal to . We denote the length of curves on  by ℓ̃, the integral over curves on  by ∫dz̃, the area of sets by , and the Fermi coordinate map by φ̃. 

  The function f and the resulting surface  satisfy the following properties.     
    
        
  *  f(z) ≍ 1/z for all z ∈.
        
        
  *  There exist universal constants K_1, K_2 ≥ 0 so that  
        K(z) ∈ [-K_1, K_2] 
        for all z ∈, where K : → is the Gaussian curvature in .
        
        
  *  ≳ 1.
        
        
  *  There exists a universal constant r_0>0 such that the following holds. Let (r,t) be the Fermi coordinates relative to a minimal geodesic γ in , and assume r≤ r_0. Then the metric tensor under these coordinates is given by
        
    ds^2 = dr^2 + G(r,t)^2 dt^2   ,

        where
        
    cos(r √(K_2)) ≤ G(r,t) ≤cosh(r √(K_1))   .

    


    
  
 
Firstly, since ψ has finite support and f_i(x) ≍ f_i(y) for all x-y≤ 2 and all i, the smoothing cannot change the value of of f_i by more than a constant factor. Hence f(z) ≍ 1/z for all z ∈ by <ref>.
        
Secondly, the function f is constant and equal to 1 in the thick part of , as well as in all collars C(γ_i) with W(γ_i) ≤ 2, and so the curvature there remains -1. It therefore suffices to calculate the new curvature in the collars C(γ_i) with W(γ_i)>2. Recall that the curvature of  is given by 
        
    K = 1/f(-1 - Δlog(f) )  .

(see e.g. <cit.>). Let (ρ, θ) be the cylindrical coordinates of a point z ∈ C(γ_i) as in <ref>. Observe that f(z) does not depend on θ; so by slight abuse of notation, we may therefore write f(z) = f(ρ) inside the collar. The Laplacian is then given by 
        
    Δlog f = 1/cosh(ρ)∂/∂ρ( cosh(ρ) ∂/∂ρlog(f) )  ,

        which yields, after a straightforward calculation,
        
    K = -1/f(f^2 + f(tanh(ρ)f' + f”) - (f')^2 /f^2)   .

Since ψ(x) has bounded first and second derivatives, by our choice of f we have that f(ρ) ≍ f'(ρ) ≍ f”(ρ), and so the absolute value of the expression in the parenthesis is bounded above by a constant. Since f is bounded from below by a universal constant, this gives the desired curvature bounds.

Thirdly, by a result of Klingenberg (see e.g. <cit.>), if a Riemannian surface M has curvature bounded above by a positive constant K_2, then  
        
    M≥min{π/√(K_2), 1/2· (length of shortest closed geodesic) } .

We have just proved that  has curvature bounded above by a universal constant, so the first term in the minimum above is bounded away from 0. Let γ⊆ be a shortest closed geodesic in . Assume first that γ is contractible (note that we cannot immediately rule out this case because  is no longer of non-positive curvature) and denote its interior by Ω. If ℓ̃(γ) ≥ 1 there is nothing to prove, so we also assume that ℓ̃(γ) ≤ 1. Since f(z) ≳ 1, it follows that ℓ(γ) ≲ 1 as well. By a standard isoperimetric inequality <cit.>, we have ℓ(γ) ≳(Ω)^1/2 and since ℓ(γ) ≲ 1, <ref> implies that the values of f(z) for all z ∈γΩ differ by a universal multiplicative constant. Hence ℓ̃(γ) ≳(Ω)^1/2. On the other hand, since γ is a closed geodesic in  and Ω is a topological disc, by the Gauss-Bonnet theorem, ∫_Ω K(z) dz̃ = 2π which implies that (Ω) ≥ 2π/K_2. This shows that ℓ̃(γ) ≳ 1 as wanted.

The case that γ is non-contractible is easier. Then γ's length can be calculated by integrating over points in :
        
    ℓ̃(γ) = ∫_γ f(z) dz≍∫_γ1/zdz  .

Since the length of γ in S is larger than z for all z∈γ, we obtain ℓ̃(γ) ≳ 1.

Lastly, for the fourth assertion, for every t let η_t be the geodesic perpendicular to γ at γ(t). For a fixed t_0, the vector field ∂/∂ t is a Jacobi field along η_t_0 and is perpendicular to it. Hence, the function G(r) = ∂/∂ t_(r,t_0) satisfies the Jacobi equation ∂_r^2 G + KG = 0, with initial conditions G(0) = 1 and ∂_r G(0) = 0 and so the function h = ∂_r G/G satisfies the Riccati equation ∂_r h + h^2 +K = 0. The result then follows from standard comparison estimates (see e.g. <cit.>); for instance, for the upper bound, the real function f(r) satisfying f'=-f^2 +K_1 with initial condition f(0)=0 is easily seen to be (log(cosh(√(K_1)r)))'. Since we proved that K ≥ -K_1, standard comparison arguments yield that h(r) ≤ f(r) which, by integration, gives the upper bound in (<ref>). 




 §.§ Proof of 

Let f:S→ (0,∞) be the function defined  in (<ref>) and let  be the resulting surface defined below it. Since extremal length is conformally invariant, we can compute Γ on the new manifold  instead of on .

We write γ for a fixed minimal geodesic in  connecting x to y, let β = 1/4(1 ℓ̃(γ)), and let α > 0 be a universal small constant that will be chosen later. 
Our goal is to find, for every ρ, an explicit curve in Γ with small ρ-length, by restricting ourselves to curves contained in a smaller domain D ⊆. This domain will be a tubular neighborhood of γ; since it is best to use as wide a domain as possible, the radius of this tubular neighborhood, while forced to start small, will quickly expand to constant width. Formally, for every q ∈ [-1,1], let γ_q : [0, ℓ̃(γ)] → be given by γ_q(t) = φ̃_γ(q · r(t),t), where φ̃_γ is the Fermi coordinate map in , and 

    r(t) = α(t/β + (1-t/β) r_x/x)     t ∈ [0, β] 
    α    t ∈ [β, ℓ̃(γ) - β] 
    α( (ℓ̃(γ)-t)/β + (1-(ℓ̃(γ)-t)/β)r_y/y)     t ∈ [ℓ̃(γ) - β, ℓ̃(γ)]   ,

see <ref>. Since we assume that r_x ≤x and r_y ≤y  we have that r(t) ≤α for all t, so using the second and third assertions of <ref> together with <ref> allows us to choose α>0 to be a small enough universal constant so that φ̃_γ is one-to-one in the region [-α, α]× [0,ℓ̃(γ)]. We further impose that α≤ r_0 (where r_0 is from the fourth assertion of <ref>) is small enough so that for all t ∈ [0, ℓ̃(γ)] we have

    cos(r(t) √(K_2)) ≳ 1   .
 



Let D = {γ_q(t) | q∈ [-1,1], t ∈ [0, ℓ̃(γ)] } be the colored region depicted in <ref>. Denote by μ the Riemannian area measure on D according to the metric g̃, and by ν the measure on D of γ_q(t), obtained by taking q to be a uniform random variable in [-1,1] and t an independent uniform random variable in [0,ℓ̃(γ)]. Let ρ : → [0,∞) be a Borel measurable function. Integrating ρ according to ν, we have

    ∫_Dρ dν = 1/2 ℓ̃(γ)∫_-1^1 ∫_0^ℓ̃(γ)ρ(γ_q(t)) dt  dq   .

Thus, there exists a q^*∈ [-1,1] such that

    ∫_0^ℓ̃(γ)ρ(γ_q^*(t)) dt ≤ℓ̃(γ) ∫_D ρ  dν  .

By <ref> of <ref>, we have that for every q ∈ [-1,1],

    ∫_γ_qρ dz̃ = ∫_0^ℓ̃(γ)ρ(γ_q(t)) γ_q'(t) dt ≲∫_0^ℓ̃(γ)ρ(γ_q(t))  dt

and so 

    ∫_γ_q^*ρ dz̃≲ℓ̃(γ) ∫_D ρ  dν  .

We bound ∫_Dρ dν by comparing it to μ using the Cauchy-Schwarz inequality,

    ∫_Dρ dν = ∫_Dρdν/dμ dμ≤√(∫_Dρ^2  dμ) √(∫_D( dν/dμ)^2  dμ)  .

Since every curve γ_q contains a curve in Γ, we obtain 

    Γ = sup_ρinf_η∈ΓL(η, ρ)^2/∫_Dρ^2  dμ≤sup_ρinf_q ∈ [-1,1](∫_γ_qρ dz̃)^2/∫_Dρ^2  dμ≲ℓ̃(γ)^2 ∫_D(dν/dμ)^2 dμ  .

We now bound the Radon-Nikodym derivative in the right-hand side. For a point z∈ D, the Radon-Nikodym derivative is given by 

    dν/dμ(z) = lim_→ 0ν( B(z,))/μ( B(z,) )≍lim_→ 0Leb({(q,t) ∈ [-1,1] × [0,ℓ̃(γ)] |γ_q(t) ∈ B(z, ) })/ 2 ℓ̃(γ) ^2  ,

where we use the fact that μ(B(z,)) ≍^2 since the curvature of  is in [-K_1,K_2] (so for small enough , μ(B(z,)) is larger than the area of a spherical cap of radius  on a sphere of radius 1/√(K_2), but smaller than the volume of a hyperbolic disc of curvature -K_1 of radius  (see e.g. <cit.>).

 
    Let z_1, z_2 ∈ D have Fermi coordinates (r_1, t_1) and (r_2, t_2) with respect to γ under the metric g̃.
    For any >0 small enough, if the distance between z_1 and z_2 is at most , then t_1 - t_2≲. 



    Let η:[0, z_1z_2] → be a unit speed minimal geodesic from z_1 to z_2. As long as >0 is small,  B(z,) is convex so η⊆ B(z,). Denoting the Fermi coordinates of η(s) by (r(s), t(s)), we have
    
    ≥z_1z_2 = ∫_0^z_1z_2√((dr/ds)^2 + G(r(s), t(s))^2(dt/ds)^2) ds 
       ≥∫_0^z_1z_2cos(r(s)√(K_2)) dt/dsds  ≳t_2 - t_1  ,

where the second inequality is by <ref> of <ref> and the last inequality is by (<ref>). 


Let t be such that there exists a q ∈ [-1,1] with γ_q(t) ∈ B(z,). For every such t, the curve η_t:[-1,1] → given by η_t(q) = φ̃_γ(q· r(t), t) is a minimal geodesic with speed r(t). Since it is minimal, the length of η_t  B(z, ) is bounded by 2, and so the Lebesgue measure of all q such that η_t(q) ∈ B(z,) is bounded by 2 / r(t). Denoting the Fermi coordinates of z by (r_z, t_z) in  relative to γ, by <ref> and the fact that r(t) is continuous we have

    Leb({(q,t) ∈ [-1,1] × [0,ℓ̃(γ)] |γ_q(t) ∈ B(z, ) }) ≲∫_t_z - C^t_z + C/r(t) dt ≍^2/r(t_z)  .

Combining this with (<ref>) and (<ref>) and <ref> of <ref> we get

    Γ   ≲∫_D 1/r(t_z)^2 dμ(z) ≤∫_0^ℓ̃(γ)∫_-r(t)^r(t)1/r(t)^2cosh(r(t)√(K_1))  ds  dt 
           ≲∫_0^ℓ̃(γ)1/r(t) dt  .

We partition the integral over t into three parts as in the definition of r(t), see <ref>. Firstly, when t∈ [β, ℓ̃(γ)-β] we have that r(t)=α, and so
    
    ∫_β^ℓ̃(γ)-β1/r(t) dt ≲ℓ̃(γ) = ∫_γf(z)dz≍xy  ,
    
where we used the fact that γ minimizes the distance in  from x to y and that f(z) ≍z^-1. Secondly, when t ∈ [0,β], we get
    
    ∫_0^β1/r(t) dt = ∫_0^βα/t/β + (1-t/β) r_x/x dt =  αβlog(x/r_x)/1-r_x/x≲log(x/r_x)  ,

since r_x ≤x/2. A similar calculation shows that when t ∈ [ℓ̃(γ)-β, ℓ̃(γ)], we have  
    
    ∫_ℓ̃(γ)-β^ℓ̃(γ)1/r(t) dt ≍log(y/r_y)  .

Combining (<ref>), (<ref>) and (<ref>) into (<ref>) proves the desired inequality. 



§ PROOF OF THE EIGENVALUES LOWER BOUND


The goal of this section is to prove <ref>. For λ > 0, define N(λ) to be the number of eigenvalues in (0,λ] of the Laplacian on  . Our main goal is to show that there exists a universal constant > 0 such that N(λ) ≲ g √(λ(S)) for all λ≤. Indeed, this would show that 

    λ_k ≥c k^2/(S)g^2

for all k such that c k^2/(S)g^2≤. Since (S) ≥ 1, this condition holds when k =1, …, ⌊ g √(/c)⌋ so <ref> then follows by decreasing the value c.

Let  be a constant whose value is to be determined, fix λ <, and let {ϕ_0, ϕ_1, …} be a smooth orthonormal basis of L^2() where ϕ_k is a unit eigenvector of the Laplacian corresponding to λ_k (see e.g. <cit.> for the existence of such a basis). Define the function e_λ : ^2 → by

    e_λ(x,z) = ∑_k ≥ 1 : λ_k ≤λϕ_k(x)  ϕ_k(z)   ;

the function e_λ is called the spectral kernel. For x∈, we define 

    μ_x(λ) = e_λ(x,x) = ∑_k ≥ 1 : λ_k ≤λϕ_k(x)^2   ,
 
and observe that

    ∫_μ_x(λ)  dx = N(λ)   .

Hence, (<ref>) would follow if we could show that μ_x(λ) ≲√(λ I(S)). However, unlike the case of a finite regular connected graph described in <ref> (in which μ_x(λ) ≲√(λ)) we cannot hope to bound μ_x(λ) uniformly in x. Thus, our goal is to bound μ_x(λ) for each x∈ separately. 

For x with μ_x > 0, define the function f_x :→ by

    f_x(z) = e_λ(x,z)/√(μ_x(λ))  ,

and observe that f_x(x) = √(μ_x(λ)) and that

    f_x_2^2 = ∫_ e_λ(x,z)^2 dz/μ_x(λ) =  ∫_(∑_k ≥ 1 : λ_k ≤λϕ_k(x) ·ϕ_k(z) )^2  dz/μ_x(λ) = μ_x(λ)/μ_x(λ) = 1   .



We will require an upper bound on the gradient of f similar to Bernstein inequalities for Laplacian eigenfunctions. We will use the following bound on the gradient of f_x; it follows from known methods (such as <cit.>) but we were unable to find a reference that implies it, therefore we provide its short proof in <ref>.


 For all z ∈ we have
    
    f_x(z)≲  e^2√(λ)z^-1  .
        


Recall <ref> for the weighted distance ··, and for any x∈ and R>0 let 

    xR = { y ∈|xy≤ R}  .



Let R ≥ 1 and x ∈. Suppose that either (xR) ≥16/μ_x(λ) or xR =. Then  

    μ_x(λ) ≲λ (R + log (1/μ_x(λ)∧ 1 ) )    .


 We first find a point y∈xR such that f_x(y) ≤1/4f_x(x). Indeed, if xR =, then it is immediate that there exists y∈xR with f_x(y)=0 since ∫_y f_x(y)d=0. Otherwise, we have that

    ({z ∈| f_x(z) > 1/4f_x(x) }) < ||f_x||^2/f_x(x)^2/16 = 16/μ_x(λ)  ,
 
hence in the case (xR) ≥16/μ_x(λ) we again obtain such a point y. 

Next, using <ref> we can choose a universal constant c>0 small enough so that setting 

    r_x = c (f_x(x) ∧ 1) x  ,    r_y = c (f_x(x)∧ 1) y  ,

and since λ < is bounded from above, this yields that

    inf{ f_x(z) : z∈ B(x,r_x) }≥3 f_x(x) / 4  ,

and 

    sup{ f_x(z) : z ∈ B(y,r_y) }≤ f_x(x) /2   .

In particular, the balls B(x,r_x) and B(y,r_y) are disjoint. Let Γ denote the set of all rectifiable curves between the boundaries of the balls which do not enter the balls. <ref> states that

    Γ≲ R + log( 1/c(f_x(x)∧ 1))   .

It is well known that given two disjoint open sets A and B of , the extremal length of the family of curves connecting ∂ A and ∂ B in ∖ (A∪ B) is the inverse of the Dirichlet energy ∫_S  f^2 of the unique harmonic function  f in ∖ (A∪ B) taking 1 on ∂ A and 0 on ∂ B, see <cit.>. Since harmonic functions minimize the Dirichlet energy, after rescaling the boundary values we obtain that

    ∫_ f_x(z)^2  dz ≥f_x(x)^2/16/Γ≳f_x(x)^2/R + log( 1/c(f_x(x)∧ 1))  .

Note that we have used <cit.> with the level sets { z : f_x(z) ≤ f_x(x)/2 } and {z : f_x(z) ≥ 3f_x(x)/4} taking the role of A and B above and used a trivial monotonicity of the Dirichlet energy. Since f_x is a linear combination of eigenfunctions with corresponding eigenvalues at most λ we may bound

    λ≥<Δ f_x, f_x >  = < f_x,  f_x > = ∫_ f_x(z)^2  dz   .

We put this and f_x(x)^2=μ_x(λ) in (<ref>) and obtain that 

    μ_x(λ) ≲λ (R + log (1/μ_x(λ)∧ 1 ) )   .




    Let x ∈ and R ≥ 1 be such that xR≠. Then
    
    ∫_xR1/z^2 dz ≳ R   .



Since xR≠ the boundary ∂xR is non-empty. Let y be a point in ∂xR which minimizes the original distance xy. We denote by γ a minimal geodesic in  between x and y, which, by our choice of y, is contained in xR.

We proceed by analyzing two cases. First, if ℓ=xy≤ 1, then all points z∈ B_S(x,ℓ) have z≍x uniformly by <ref>. Since y ∈∂xR we must have that γ≥ R, hence x^-1≳ R/ℓ. We get

    ∫_xR1/z^2 dz ≥∫_B_S(x,ℓ)1/z^2 dz ≳ℓ^2/x^2≳ R^2 ≥ R   ,

where the first and last inequalities are since R ≥ 1.

Next we handle the case in which ℓ>1. Let c∈(0,1/4) be a small universal constant that will be chosen later. Parameterizing γ by length, we first note that for any t ∈ [0,ℓ-c] we must have  

    B_S(γ(t), cγ(t)) ⊆xR  ,

since otherwise the ball B_S(γ(t), cγ(t)) contains some other point of ∂ B_w(x,R) which we can connect to γ(t) in a path of length at most c, contradicting the minimality of ℓ.

Consider the set D = { (t,r) : t ∈ [0,ℓ-c], |r|≤ cγ(t)} which is a subset of D_γ defined above <ref>. Since γ is a unit speed minimal geodesic connecting two points, <ref> implies that the Fermi coordinate map φ_γ is one-to-one. By (<ref>) we have φ_γ(D) ⊆ B_w(x,R) so

    ∫_xR1/z^2 dz    ≥   ∫_φ_γ(D)1/z^2 dz  = ∫_0^ℓ-c∫_-cγ(t)^cγ(t)1/φ_γ(t,r)^2cosh(r) dr dt 
       ≳   ∫_0^ℓ-cc/γ(t)dt = c ·(γ - γ[0,ℓ-c])  .

where we used (<ref>) and the fact that φ_γ(t,r)≍γ(t).

We argue that if z∈xR and d_S(x,z) ≥ 1/2, then z≥/R for some constant >0. Otherwise, by <ref>,  any curve connecting x to z, just by considering its last tip of length 1/2, must have weight ≳z^-1≥ R/, which is a contradiction for > 0 small enough.

We apply this with z=γ(ℓ-c) and obtain that γ(ℓ-c)≥√(c)/R as long as c>0 is small enough. By <ref> we have γ(t)≳√(c)/R for all t∈ [ℓ-c,ℓ]. We deduce that γ[ℓ-c,ℓ]≲√(c) R, which together with (<ref>) and that γ≥ R gives

    ∫_xR1/z^2 dz ≳ c(γ - γ[ℓ-c,ℓ]) ≳ c(R - √(c)R) ≳ R   ,

concluding the proof. 


    For every ∈ (0,1) and R ≥ 1, let 
    
    R = {x∈|( xR) <  R xR≠}  .

    Then
    
    (R) ≲() ()   .



Let x_1, …, x_k be a maximal set of points in R such that the interiors of x_iR/2 are pairwise disjoint. By maximality, for every x ∈R the set xR/2 intersects some x_iR/2, so the triangle inequality implies that xx_i≤ R and we deduce that the balls {x_iR}_i=1^k cover R. As the volume of each ball is at most R, we have that 

    R k ≥(R)   .

On the other hand, 

    () ·() = ∫_1/z^2 dz ≥∑_i=1^k ∫_x_iR/21/z^2 dz   ≳   k R,
    
by <ref>. This gives the desired result.




 §.§ Proof of 


For each j ≥ 0 we set the numbers

    μ_j = 2^j √(λ I(S))  ,      R_j = c 2^j 
    √(I(S)/λ) = c μ_j / λ  ,

where c>0 is a small constant which will be chosen later. We take  to be small enough so that R_0 ≥ 1 for all λ <. We further set 

    0 = { x ∈| 0 < μ_x(λ) ≤μ_0 }  ,

and for every integer j ≥ 1,

    j = { x ∈|μ_j-1 < μ_x(λ) ≤μ_j }  .

We proceed by bounding (G_j) from above. If x∈ G_j we obtain by <ref> that either 
(xR_j) ≤ 32/μ_j and xR_j≠ or that

    μ_j ≲λ R_j + λlog (1/(μ_j/2∧ 1)) ≲λ R_j + λ (μ_j ∧ 1)^-1/2  .

The first term on the right-hand side is equal to cμ_j, and cannot be ≳μ_j for c small enough. The second term on the right-hand side is bounded above by λ^3/4 since μ_j ≥μ_0 ≥√(λ), hence it also cannot be greater than μ_j as long as  is small enough. Thus (xR_j) ≤ 32/μ_j and xR_j≠. By <ref> with =32/μ_j R_j = 32/c4^j (S), we get

    (G_j) ≲ 4^-j()   .

We put this in (<ref>), giving

    N(λ) = ∫_μ_x(λ)  dx ≤∑_j ≥ 0μ_j (G_j) ≲∑_j ≥ 0 2^-j√(λ(S))  ,

which implies (<ref>) and concludes the proof. 



 §.§ Proof of the gradient upper bound
 

We follow the harmonic extension method of <cit.>.
Let ϕ_0, ϕ_1, … ϕ_k be the orthonormal eigenvectors of the Laplacian with eigenvalue smaller than or equal to λ (as described in the beginning of this section). Then f_x can be written as the linear combination  f_x(z) = ∑_i=1^k β_i ϕ_i(z), with β_i=ϕ_i(x)/√(μ_x(λ)). Let h(z,t) be the harmonic extension of f_x to the manifold M = ×, i.e. 

    h(z,t) = ∑_i=1^k β_i ϕ_i(z)e^√(λ_i) t  .

Observe that h(z,0) agrees with f_x on the coordinates corresponding to , and has an additional coordinate equal to ∑_i=1^k β_i ϕ_i(z) √(λ_i).  Thus, | f_x(z)|^2 ≤ | h(z,0)|^2. Since h  is harmonic, by a classical theorem of Schoen and Yau (<cit.> with K=1 and a=1, see also  (3.3) in <cit.>) we have

    | h(z,0)|^2 ≤ C sup{h(y)^2 | y ∈B_M((z,0), 1 )}  .

Suppose that the supremum is attained at the point (z^*,t^*) ∈ B_M((z,0),1 ). Since h is harmonic, h^2 is subharmonic, and for all r < (z^*,t^*), M we have by <cit.> that

    h(z^*,t^*)^2 ≤1/(B_M((z^*, t^*),r ) )∫_B_M((z^*, t^*),r )  h(y)^2  dM   .

 By the triangle inequality, the ball B_M((z^*,t^*),r ) contains the set B_(z^*, r/2) × [t^*-r/2, t^*+r/2], thus, its volume is at least 4πsinh(r/2)^2 · r/2 ≳ r^3. Together with (<ref>), this gives

    | f_x(z)|^2 ≲1/r^3∫_B_M((z^*, t^*),r )  h(y)^2 dy   .

Now take r = min{1, (z^*,t^*), M} = z^*. By <ref>, since z^*z≤ 1, we have z^-1≍z^*^-1, and so 1/r ≍z^-1. To bound the integral, observe that B_M((z^*, t^*),r ) ⊆× [t^*-r,t^*+r].  We then have

    ∫_B_M((z^*, t^*),r )  h(y)^2  dM    ≤∫_t^*-r^t^*+r∫_ h(z,t)^2  dz dt 
       = ∫_t^*-r^t^*+r∫_(∑_i=1^k β_i ϕ_i(z) e^√(λ_i)t)^2  dz dt 
       = ∫_t^*-r^t^*+r∑_i=1^k β_i^2 e^2√(λ_i)t dt 
       ≤ 2r e^4√(λ)∑_i=1^k β_i^2 = 2r e^4√(λ)f_2^2 = 2re^4√(λ)  ,

where the last equality is due to (<ref>). Using (<ref>) we deduce that

    f_x(z)≲ e^2√(λ) r^-1≲ e^2√(λ)z^-1

as needed.




§ HEAT KERNEL BOUND
 
The goal of this section is to prove <ref>. 
Let γ_1, …, γ_s be the set of all simple closed geodesics of length ≤ 2sinh^-1(1) and let C(γ_i) be their collars, as described in <ref>. It will be convenient to chop off the parts of the collar at distance at most 1 to its boundary, that is, we define C(γ_i) := {x ∈ C(γ_i) |dist(x, ∂ C(γ_i)) ≥ 1}. 

Let := _i{C(γ_i)} and := \. By <ref> we have that x≳ 1 for every x∈ and x≲ 1 for every x∈.

 
For any x∈ and any t ∈ [1/2,1] we have 

    txx≲t00 + 1   .




For a hyperbolic surface , recall that the heat kernel txy is given by txy = ∑_T ∈ΓtxTy, where Γ is a group red of isometries acting on  so that = Γ\, and x and y are arbitrary inverse images of x and y of the projection to . To estimate txx, we partition the group translations by distance of T x̃ from x̃.  Denoting

    Γ(m) := {T ∈Γ| m < d_(x̃, T x̃) ≤ m+1 }  ,
 
we have

    txx   = t00 + ∑_m=0^∞∑_T ∈Γ(m)tx̃Tx̃  .

We bound |Γ(m)| by an explicit version of <cit.>. Let D be the disc around x̃ of radius x. Every disc T(D) with T∈Γ(m) is contained in the disc of radius m+1+x around x̃ in  which has volume 2π( cosh(m+1+x)-1 ) ≤ 2π e^m+1+x. On the other hand, the volume of T(D) is 2π( cosh(x)-1 ) and {T(D)}_T∈Γ(m) are non-intersecting. Thus,

    #Γ(m) ≤e^x+1/cosh(x)-1 e^m ≲(1+1/x^2) e^m ≲ e^m   .

where in the second inequality we relied on the asymptotics of the function cosh(r) for r → 0 and r →∞, and in the third inequality we relied on our assumption that x∈. Next, a well-known bound on p_t^ (see <cit.>) states that  

    txy≲1/texp(-d_(x,y)^2/8t)   .

We use this to bound the sum on the right hand side of (<ref>) by

    ∑_m=1^∞∑_T ∈Γ(m)tx̃T x̃≲1/t∑_m=1^∞exp( m - m^2 / 8t ) ≲ 1   ,

since the term over m is convergent. For the sum over T ∈Γ(0) we note that d_(x̃,Tx̃) ≥x≳ 1 for every T ∈Γ(0) so by (<ref>) we obtain a contribution of another constant. 


We next bound the behavior of txx in the thin part.

 
    Let γ be a simple closed geodesic of length ℓ≤ 2sinh^-1(1). Then for any t ∈ [1/2,1] we have     
    ∫_C(γ)txx dx ≲log(1/ℓ)   .



 We apply the classical Li and Yau theorem (see Corollary 3.1 in <cit.> with α=3/2 and ε=1/2) which asserts that if  is a complete Riemannian manifold without boundary and with Ricci curvature bounded from below by -K, then 

    txy≲(B(x,√(t)))^-1/2(B(y,√(t)))^-1/2exp(C Kt-d_(x,y)^2/4.5t)   ,

where C<∞ is a universal constant. Using the above bound yields 

    txx≲1/(B(x,√(t)))  ,

as long as t≤ 1. Let x∈C(γ) have Fermi coordinates (ρ, θ) relative to γ, where ρ≥ 0 without loss of generality. The ball B(x,√(t)) is contained in the collar C(γ). Hence, if we let η be the unit-speed geodesic given by η(s) = φ_γ(ρ +s, θ) with s∈[0,√(t)/2] (that is, η  extends outwards from x, is perpendicular to γ has length √(t)/2), then B(x, √(t)) contains the tubular neighborhood of η given by φ_η([0, √(t)/2] ×[0, r]) where r≍x (which is possible since x≲ 1). We thus obtain that (B(x,√(t))) ≳x. Plugging this into  above volume estimate into the Li-Yau bound (<ref>), by <ref> we thus obtain 
 
    ∫_C(γ)txx dx     ≲∫_0^2π∫_0^W(ℓ)1/xℓcosh(ρ)  dρ dθ≲ W(ℓ) ≲log(1/ℓ)   .





 §.§ Proof of 

It is well known that p_t(x,x)≥ 1/(S) so the expression in the absolute value is always non-negative. By the spectral theorem <cit.>, 

    ∫_[ txx - 1/()]  dx = ∑_k=1^∞e^-t λ_k =  ∑_k=1^2g-3e^-t λ_k + ∑_k=2g-2^∞e^-t λ_k  .

Applying <ref> and recalling that g≍() we bound the first sum on the right-hand side by

    ∑_k=1^2g-3e^-t λ_k≤∑_k=1^∞e^-ct k^2/()()^2≲∫_0^∞exp(-ct x^2/()()^2)dx ≍() √(()/t)  .
    
For the second sum on the right-hand side, a result by Otal and Rosas <cit.> states that λ_2g-2 > 1/4. Denoting t = τ + 1/2 for τ≥ 1/2, we get

    ∑_k=2g-2^∞ e^-t λ_k = ∑_k=2g-2^∞ e^ -(τ+1/2) λ_k≤ e^-τ/4∑_k=0^∞e^-1/2λ_k≲ e^-t/4∫_1/2xx dx   .


Let γ_1, …, γ_s be the set of all simple closed geodesics of length ≤ 2sinh^-1(1) in . We appeal to  <ref> and <ref> with t = 1/2; since 1/200≍ 1, we get

    ∫_1/2xx dx    ≲() + ∑_i=1^slog(1/ℓ(γ_i)) ≲() √(())  ,

since ∑_i=1^slog(1/ℓ(γ_i))≲∑_i=1^sℓ(γ_i)^-1/2 which by Cauchy-Schwartz is at most √(s∑_i=1^sℓ(γ_i)^-1) and we get the above inequality using <ref>. All this gives that

    ∑_k=2g-2^∞e^-t λ_k≲ e^-t/4() √(())  ,

which together with (<ref>) and (<ref>) concludes the proof.    


The choice of t=1 in <ref> is arbitrary since it is the large values of t that are interesting. The same proof shows that for any t_0>0 there exists C(t_0)<∞ such that (<ref>) holds for all t≥ t_0 and C=C(t_0).




§ SHARPNESS
 
Let I≥ 1 be real and g ≥ 2 be an integer.  Set = min{1/I, sinh^-1(1)} and n=g-1. Let Y be a pair of pants with boundary geodesics of lengths (1,1,), and let X be the surface obtained by gluing 
two identical copies of Y along their two boundary geodesics of length 1, without twists. Note that X has two boundary components, each of length . Finally, let  be the surface of genus g obtained by gluing together without twists n copies of X in a cycle. See <ref>.



It is straightforward to check that () ≍ 1/, and so () ≳ I. We claim that the eigenvalues of the Laplacian on  satisfy 

    λ_k≤C  k^2/g^2  ,

where k∈{1,…, n } and C<∞ is a universal constant. We will do this by constructing appropriate test functions for the minimax principle.

Let k ≤ n be fixed. Consider  k disjoint connected subsets _1, …, _k of , where each _i is a concatenation of n/k consecutive copies of X. We assume for simplicity that n/k is odd and write n/k = 2m+1; the even case is handled similarly. For each i we define a function f_i(x) whose support is _i, so that the functions {f_i}_i=1^k have disjoint supports. Let us describe f_1; the other functions are constructed similarly. Assume that S_1 is the concatenation of X_1,…, X_2m+1. If x ∈ X_j for some j∈{1,…, 2m+1} and x∂ X_j≥ 1, then we set f_1(x) = j/m+1 if j ≤ m+1 and f_1(x) = 1 - j-(m+1)/m+1 if j > m+1. When x∂ X_j < 1, we set the values of f_i so that they interpolate linearly between the two values in the boundary {x : x∂ X_j = 1}. 

It is straightforward to verify that f_i_2 ≍√(m) and ∫_ f_i^2  d≲/m. The functions {f_i} are not smooth, but one can easily make them smooth (say, by mollifying, as we did in (<ref>)) and still have the same bounds on the norm and the Dirichlet energy as above. Thus, the minimax principle <cit.> and the fact that m ≍ n/k gives that λ_k≲ k^2/n^2. This is the required bound for k∈{1,… g-1} and it implies the bound for g≤ k≤ 2g-3 by properly adjusting the constant C.

Lastly, this upper bound on λ_k for k∈{1,…,n} immediately shows that the reverse inequality in <ref> holds. Indeed, we plug in this upper bound for the first n eigenvalues in (<ref>) and all other terms we bound below by 0.



§ ACKNOWLEDGEMENTS
 The authors are supported by ERC consolidator grant 101001124 (UniversalMap) as well as ISF grants 1294/19 and 898/23. We thank Rotem Assouline for his help with the proof of <ref> of <ref> and also Bo'az Klartag, Ze'ev Rudnick and Perla Sousi for useful discussions.





