1 Introduction

# Finite-particle approximations for interacting Brownian particles with logarithmic potentials

## Abstract.

We prove the convergence of -particle systems of Brownian particles with logarithmic interaction potentials onto a system described by the infinite-dimensional stochastic differential equation (ISDE). For this proof we present two general theorems on the finite-particle approximations of interacting Brownian motions. In the first general theorem, we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy a uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem. In the second general theorem, we restrict to the case of reversible particle systems and simplify the sufficient condition. We deduce the second theorem from the first. We apply the second general theorem to Airy interacting Brownian motion with , and the Ginibre interacting Brownian motion. The former appears in the soft-edge limit of Gaussian (orthogonal/unitary/symplectic) ensembles in one spatial dimension, and the latter in the bulk limit of Ginibre ensemble in two spatial dimensions, corresponding to a quantum statistical system for which the eigen-value spectra belong to non-Hermitian Gaussian random matrices. The passage from the finite-particle stochastic differential equation (SDE) to the limit ISDE is a sensitive problem because the logarithmic potentials are long range and unbounded at infinity. Indeed, the limit ISDEs are not easily detectable from those of finite dimensions. Our general theorems can be applied straightforwardly to the grand canonical Gibbs measures with Ruelle-class potentials such as Lennard-Jones 6-12 potentials and and Riesz potentials.

## 1. Introduction

Interacting Brownian motion in infinite dimensions is prototypical of diffusion processes of infinitely many particle systems, initiated by Lang [12, 13], followed by Fritz [3], Tanemura [30], and others. Typically, interacting Brownian motion with Ruelle-class (translation invariant) interaction and inverse temperature is given by

 (1.1) dXit=dBit−β2∞∑j;j≠i∇Ψ(Xit−Xjt)dt(i∈N).

Here an interaction is called Ruelle-class if is super stable in the sense of Ruelle, and integrable at infinity [28].

The system is a diffusion process with state space , and has no natural invariant measures. Indeed, such a measure , if exists, is informally given by

 (1.2) ˇμ=1Ze−β∑∞(i,j);i

which cannot be justified as it is because of the presence of an infinite product of Lebesgue measures. To rigorize the expression (1.2), the Dobrushin–Lanford–Ruelle (DLR) framework introduces the notion of a Gibbs measure. A point process is called a -canonical Gibbs measure if it satisfies the DLR equation: for each and -a.s.

 (1.3) μmr,ξ(ds)=1Zmr,ξe−β{∑mi

where , , , and is the outer condition. Furthermore, denotes the regular conditional probability:

 μmr,ξ(ds)=μ(πr(s)∈ds|s(Sr)=m,πcr(s)=πcr(ξ)).

Then is a reversible measure of the delabeled dynamics such that .

If the number of particles is finite, say, then SDE (1.1) becomes

 (1.4) dXN,it=dBit−β2{∇ΦN(XN,it)+N∑j;j≠i∇Ψ(XN,it−XN,jt)}dt(1≤i≤N),

where is a confining free potential vanishing zero as goes to infinity. The associated labeled measure is then given by

 (1.5) ˇμN=1Ze−β{∑Ni=1ΦN(xi)+∑N(i,j);i

The relation between (1.4) and (1.5) is as follows. We first consider the diffusion process associated with the Dirichlet form with domain on , called the distorted Brownian motion, such that

 EˇμN(f,g)=∫(Rd)N12N∑i=1∇if⋅∇igˇμN(dxN),

where , , and denotes the inner product in . The generator of is then given by

 EˇμN(f,g)=(−LˇμNf,g)L2((Rd)N,ˇμN).

Integration by parts yields the representation of the generator of the diffusion process such that

 LˇμN=12Δ−β2N∑i=1{∇ΦN(xi)+N∑j;j≠i∇Ψ(xi−xj)}⋅∇i,

which together with It formula yields SDE (1.4).

For a finite or infinite sequence , we set and call a delabeling map. For a point process , we say a measurable map defined for -a.s. with value is called a label with respect to if . Let be a label with respect to . We denote by and the first -components of these labels, respectively. We take such that the associated point process converges weakly to :

 (1.6) limN→∞μN=μ weakly.

The associated delabeling is reversible with respect to . The labeled process and can be recovered from and by taking suitable initial labels and , respectively. Choosing the labels in such a way that for each

 (1.7) limN→∞μN∘ℓ−1N,m=μ∘ℓ−1m weakly,

we have the convergence of labeled dynamics to such that for each

 (1.8) limN→∞(XN,1,…,XN,m)=(X1,…,Xm) in law in C([0,∞);(Rd)m).

We expect this convergence because of the absolute convergence of the drift terms in (1.1) and energy in the DLR equation (1.3) for well-behaved initial distributions although it still requires some work to justify this rigorously even if [12].

If we take logarithmic functions as interaction potentials, then the situation changes drastically. Consider the soft-edge scaling limit of Gaussian (orthogonal/unitary/symplectic) ensembles. Then the -labeled density is given by

 (1.9) ˇμNAiry,β(dxN)=1Z{N∏i

and the associated -particle dynamics described by SDE

 (1.10) dXN,it=dBit+β2N∑j=1,j≠i1XN,it−XN,jtdt−β2{N1/3+12N1/3XN,it}dt.

The correspondence between (1.9) and (1.10) is transparent and same as above. Indeed, we first consider distorted Brownian motion (Dirichlet spaces with as a common time change and energy measure), then we obtain the generator of the associated diffusion process by integration by parts. SDE (1.10) thus follows from the generator immediately.

It is known that the thermodynamic limit of the associated point process exists for each [27]. Its -point correlation function is explicitly given as a determinant of certain kernels if [1, 15]. Indeed, if , then the -point correlation function of the limit point process is

 ρmAi,2(xm)=det[KAi,2(xi,xj)]mi,j=1,

where is the continuous kernel such that, for ,

 KAi,2(x,y)=Ai(x)Ai′(y)−Ai′(x)Ai(y)x−y.

We set here and denote by the Airy function given by

 Ai(z)=12π∫Rdkei(zk+k3/3),z∈R.

For similar expressions in terms of the quaternion determinant are known [1, 15].

From the convergence of equilibrium states, we may expect the convergence of solutions of SDEs (1.10). The divergence of the coefficients in (1.10) and the very long-range nature of the logarithmic interaction however prove to be problematic. Even an informal representation of the limit coefficients is nontrivial but has been obtained in [26]. Indeed, the limit ISDEs are given by

 (1.11) dXit=dBit+β2limr→∞{∑|Xjt|

Here , which is the shifted and rescaled semicircle function at the right edge.

As an application of our main theorem (Theorem 2.2), we prove the convergence (1.8) of solutions from (1.10) to (1.11) for with . We also prove that the limit points of solutions of (1.10) satisfy ISDE (1.11) with .

For general , the existence and uniqueness of solutions of (1.11) is still an open problem. Indeed, the proof in [26] relies on a general theory developed in [18, 19, 20, 21, 25], which reduces the problem to the quasi-Gibbs property and the existence of the logarithmic derivative of the equilibrium state. These key properties are proved only for at present. We refer to [20, 21] for the definition of the quasi-Gibbs property and Definition 2.1 for the logarithmic derivative.

Another typical example is the Ginibre interacting Brownian motion, which is an infinite-particle system in (naturally regarded as ), whose equilibrium state is the Ginibre point process . The -point correlation function with respect to Gaussian measure on is then given by

 ρmgin(xm)=det[exi¯xj]mi,j=1.

The Ginibre point process is the thermodynamic limit of -particle point process whose labeled measure is given by

 Extra open brace or missing close brace

The associated -particle SDE is then given by

 (1.12) dXN,it= dBit−XN,itdt+N∑j=1,j≠iXN,it−XN,jt|XN,it−XN,jt|2dt(1≤i≤N).

We shall prove that the limit ISDEs are

 (1.13) dXit= dBit+limr→∞∑|Xit−Xjt|

In [19, 25], it is proved that these ISDEs have the same pathwise unique strong solution for -a.s. , where is a label and is an initial point. As an example of applications of our second main theorem (Theorem 2.2), we prove the convergence of solutions of (1.12) to those of (1.13) and (1.14). This example indicates again the sensitivity of the representation of the limit ISDE. Such varieties of the limit ISDEs are a result of the long-range nature of the logarithmic potential.

The main purpose of the present paper is to develop a general theory for finite-particle convergence applicable to logarithmic potentials, and in particular, the Airy and Ginibre point processes. Our theory is also applicable to essentially all Gibbs measures with Ruelle-class potentials such as the Lennard-Jones 6-12 potential and Riesz potentials.

In the first main theorem (Theorem 2.1), we present a sufficient condition for a kind of tightness of solutions of stochastic differential equations (SDE) describing finite-particle systems, and prove that the limit points solve the corresponding ISDE. This implies, if in addition the limit ISDE enjoy uniqueness of solutions, then the full sequence converges. We treat non-reversible case in the first main theorem.

In the second main theorem (Theorem 2.2), we restrict to the case of reversible particle systems and simplify the sufficient condition. Because of reversibility, the sufficient condition is reduced to the convergence of logarithmic derivative of with marginal assumptions. We shall deduce Theorem 2.2 from Theorem 2.1 and apply Theorem 2.2 to all examples in the present paper.

If , and , there exists an algebraic method to construct the associated stochastic processes [7, 8, 9, 10], and to prove the convergence of finite-particle systems [24, 23]. This method requires that interaction is the logarithmic function with and depends crucially on an explicit calculation of space-time determinantal kernels. It is thus not applicable to even if .

As for Sine point processes, Tsai proved the convergence of finite-particle systems for all [31]. His method relies on a coupling method based on monotonicity of SDEs, which is very specific to this model.

The organization of the paper is as follows: In Section 2, we state the main theorems (Theorem 2.1 and Theorem 2.2). In Section 3, we prove Theorem 2.1. In Section 4, we prove Theorem 2.2 using Theorem 2.1. In Section 5, we present examples.

## 2. Set up and the main theorems

### 2.1. Configuration spaces and Campbell measures

Let be a closed set in whose interior is a connected open set satisfying and the boundary having Lebesgue measure zero. A configuration on is a Radon measure on consisting of delta masses. We set . Let be the set consisting of all configurations of . By definition, is given by

 S={s=∑iδsi; s(Sr)<∞ for each r∈N}.

By convention, we regard the zero measure as an element of . We endow with the vague topology, which makes a Polish space. is called the configuration space over and a probability measure on is called a point process on .

A symmetric and locally integrable function is called the -point correlation function of a point process on with respect to the Lebesgue measure if satisfies

 ∫Ak11×⋯×Akmmρn(x1,…,xn)dx1⋯dxn=∫Sm∏i=1s(Ai)!(s(Ai)−ki)!dμ

for any sequence of disjoint bounded measurable sets and a sequence of natural numbers satisfying . When , according to our interpretation, by convention. Hereafter, we always consider correlation functions with respect to Lebesgue measures.

A point process is called the reduced Palm measure of conditioned at if is the regular conditional probability defined as

 μx=μ(⋅−δx|s({x})≥1).

A Radon measure on is called the 1-Campbell measure of if is given by

 (2.1) μ[1](dxds)=ρ1(x)μx(ds)dx.

### 2.2. Finite-particle approximations (general case)

Let be a sequence of point processes on such that . We assume:
(H1) Each has a correlation function satisfying for each

 (2.2) limN→∞ρN,n(x)=ρn(x) uniformly on Snr for all n∈N, (2.3) supN∈Nsupx∈SnrρN,n(x)≤cn???nc???n,

where and are constants independent of .

It is known that (2.2) and (2.3) imply weak convergence (1.6) [20, Lemma A.1]. As in Section 1, let and be labels of and , respectively. We assume:

(H2) For each , (1.7) holds. That is,

 (\ref{:10h}) limN→∞μN∘ℓ−1N,m=μ∘ℓ−1m weakly in Sm.

We shall later take as an initial distribution of a labeled finite-particle system. Hence (H2) means convergence of the initial distribution of the labeled dynamics. There exist infinitely many different labels , and we choose a label such that the initial distribution of the labeled dynamics converges. (H2) will be used in Theorem 2.2 and Theorem 2.1.

For and , we set

 X⋄it=∞∑j≠iδXjt, and XN,⋄it=N∑j≠iδXN,jt,

where denotes the zero measure for . Let and be measurable functions. We introduce the finite-dimensional SDE of with these coefficients such that for

 (2.4) dXN,it =σN(XN,it,XN,⋄it)dBit+bN(XN,it,XN,⋄it)dt (2.5) XN0 =s.

We assume:

(H3) SDE (2.4) and (2.5) has a unique solution for -a.s. for each : this solution does not explode. Furthermore, when is non-void, particles never hit the boundary.

We set and assume:

(H4) are bounded and continuous on , and converge uniformly to on for each . Furthermore, are uniformly elliptic on for each and are uniformly bounded on .

From (H4) we see that converge uniformly to on each compact set , and that and are bounded and continuous on . There thus exists a positive constant such that

 (2.6) ||a||S×S, ||∇xa||S×S, supN∈N||aN||S×S, supN∈N||∇xaN||S×S≤c???.

Here denotes the uniform norm on . Furthermore, we see that is uniformly elliptic on each . From these, we expect that SDEs (2.4) have a sub-sequential limit.

 limN→∞{XN,it−XN,i0} =limN→∞∫t0σN(XN,it,XN,⋄it)dBiu+limN→∞∫t0bN(XN,it,XN,⋄it)du =∫t0σ(XN,it,XN,⋄it)dBiu+limN→∞∫t0bN(XN,it,XN,⋄it)du.

To identify the second term on the right-hand side and to justify the convergence, we make further assumptions. As the examples in Section 1 suggest, the identification of the limit is a sensitive problem, which is at the heart of the present paper.

We set the maximal module variable of the first -particles by

 ¯¯¯¯¯XN,m=mmaxi=1supt∈[0,T]|XN,it|.

and by the maximal label with which the particle intersects ; that is,

 LNr=max{i∈N∪{∞};|XN,it|≤r for some 0≤t≤T}.

We assume the following.
(I1) For each

 (2.7) lima→∞liminfN→∞PμN∘ℓ−1N(¯¯¯¯¯XN,m≤a)=1

and there exists a constant such that for

 (2.8) supN∈Nm∑i=1EμN∘ℓ−1N[|XN,it−XN,iu|4;¯¯¯¯¯XN,m≤a]≤c???|t−u|2.

Furthermore, for each

 (2.9) limL→∞liminfN→∞PμN∘ℓ−1N(LNr≤L)=1.

Let be the one-Campbell measure of defined as (2.1). Set . Then by (2.3) for each . Without loss of generality, we can assume that for all . Let . Let be the probability measure defined as . Let be a map from to itself such that , where . Let be the sub--field of generated by . Because is a subset of , we can and do regard as a -field on , which is trivial outside .

We set a tail-truncated coefficient of and their tail parts by

 (2.10) bNr,s=E¯μN,[1]r[bN|Fr,s],bN=bNr,s+bN,tailr,s.

We can and do take a version of such that

 (2.11) bNr,s(x,y)=0 for x∉Sr, (2.12) bNr,s(x,y)=bNr+1,s(x,y) for x∈Sr.

We next introduce a cut-off coefficient of . Let be a continuous and -measurable function on such that

 (2.13) bNr,s,p(x,y) =0 for x∉Sr (2.14) bNr,s,p(x,y) =bNr+1,s,p(x,y),% for x∈Sr−1

and that, for , where ,

 (2.15) bNr,s,p(x,y)=0 for (x,y)∈(S×S)r,p+1, (2.16) bNr,s,p(x,y)=bNr,s(x,y) for (x,y)∉(S×S)r,p.

The main requirements for and are the following:

(I2) There exists a such that and that for each

 (2.17) limsupN→∞∫Sr×S|bN|^pdμN,[1]<∞.

Furthermore, for each , there exists a constant such that

 (2.18) supp∈NsupN∈NEμN∘ℓ−1N[∫T0|bNr,s,p(XN,it,XN,⋄it)|^pdt]≤c???.

We set . Let denote the uniform norm on and set . For a function on we denote by , where is a function on such that is symmetric in for each and . We decompose as

 (2.19) bNr,s=bNr,s,p+bNr,s−bNr,s,p

and we assume:

(I3) For each such that , there exists such that

 (2.20) limN→∞∥bNr,s,p−br,s,p∥S×Smr=0. Moreover, bNr,s,p are differentiable in x and satisfying the bounds: (2.21) supN∈N∥∇bNr,s,p∥S×Smr<∞, (2.22) limp→∞supN∈N∥bNr,s,p−bNr,s∥L^p(μN,[1]r)=0.

Furthermore, we assume for each

 (2.23) limp→∞limsupN→∞EμN∘ℓ−1N[∫T0|{bNr,s,p−bNr,s}(XN,it,XN,⋄it)|^pdt]=0, (2.24) limp→∞Eμ∘ℓ−1[∫T0|{br,s,p−br,s}(Xit,X⋄it)|^pdt]=0,

where is such that

 (2.25) br,s(x,y)=limN→∞bNr,s(x,y) for each (x,y)∈⋃p∈N(S×S)cr,p.
###### Remark 2.1.

We see that by definition and for by (2.11). The limit in (2.25) exists because of (2.15), (2.16), and (2.20).

(I4) There exists a independent of and such that

 (2.26) lims→∞limsupN→∞∥bN,tailr,s−btail∥L^p(μN,[1]r)=0.

Furthermore, for each :

 (2.27) lims→∞limsupN→∞EμN∘ℓ−1N[∫T0|(bN,tailr,s−btail)(XN,it,XN,⋄it)|^pdt]=0.

We remark that is automatically independent of for consistency (2.16). By assumption, is a function of . From (2.10) and (2.19) we have

 (2.28) bN=bNr,s,p+btail+{bNr,s−bNr,s,p}+{bN,tailr,s−btail}.

In (I3) and (I4), we have assumed that the last two terms and in (2.28) are asymptotically negligible.

Under these assumptions, we prove in Lemma 3.1 that there exists such that for each

 (2.29) lims→∞∥br,s−b∥L^p(μN,[1]r)=0.

We assume:
(I5) For each

 (2.30)