Gaussian asymptotics of discrete –ensembles
We introduce and study stochastic –particle ensembles which are discretizations for general– log-gases of random matrix theory. The examples include random tilings, families of non–intersecting paths, –measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as . The covariance is universal and coincides with its counterpart in random matrix theory.
Our main tool is an appropriate discrete version of the Schwinger–Dyson (or loop) equations, which originates in the work of Nekrasov and his collaborators.
- 1 Introduction
- 2 Toy example: binomial weight
- 3 General setup
- 4 Nekrasov’s equation
- 5 Law of Large Numbers
- 6 Second order expansion.
- 7 Central Limit Theorem
- 8 Non–vanishing weights
- 9 Examples
- 10 Exponential bound on the support.
1.1. Continuous log–gases
A general– log-gas is a probability distribution on –tuples of reals with density proportional to
where is a continuous function called potential. For and , the density (1) describes the joint distribution of the eigenvalues of random matrices from Gaussian Orthogonal/Unitary/Symplectic Ensemble; much more general potentials are widespread and extensively studied in random matrix theory and beyond, see the books [Me], [Fo], [AGZ], [Ox], [PS].
Under weak assumptions on the potential, the ensembles (1) exhibit a Law of Large Numbers as , which means that the (random) empirical measure defined via
converges (weakly, in probability) to a non-random equilibrium measure . For and , this statement dates back to the work of Wigner [Wi], and is known in this case as the Wigner semicircle law. The results for generic were established much later, see [BPS], [BeGu], [J1].
The next order asymptotics asks about global fluctuations, i.e. how the functional behaves as . A natural approach here is to take (sufficiently smooth) functions and consider the asymptotic behavior of random variables
For quite general potentials the limits of (2) are Gaussian with universal covariance depending only on the support of the equilibrium measure . In the breakthrough paper [J1] Johansson proved such a statement for general and wide class of potentials under the assumption that has a single interval of support. Further developments have led to establishing such results for generic analytic potentials, see [KS], [BoGu1], [Shch], [BoGu2]. Note that when the support of has several intervals one needs to be careful as an additional discrete component might appear. This does not happen if one deterministically fixes the filling fractions, which are the numbers of particles in each interval of the support. In the one-interval case the limiting covariance can be identified with that of a section of the two–dimensional Gaussian Free Field, see [B2], [BoGo2] for the details.
While for certain specific choices of potentials as well as for special values there are several different methods for establishing central limit theorems for global fluctuations, all the developments for generic and rely on the analysis of loop equations (also known as Schwinger–Dyson equations). These are equations for certain observables of the log–gases (1), which originated and have been widely used in physics literature, cf. [Mi], [AM], [Ey1], [CE] and references therein. More precisely, the observables of interest are the Stieltjes transforms of the empirical measure and the Schwinger–Dyson equations involve their cumulants. Formally assuming that these cumulants have a converging expansion as , one can use the equations to derive recursively the asymptotics of the cumulants starting from the equilibrium measure. These recursive relations are sometimes called the topological recursion because, at least when , they mimic the relations between maps of different genus, see e.g. [EO] and allow to recover the result from ’t Hooft and Brézin-Itzykson-Parisi-Zuber [BIPZ] showing that matrix integrals can be seen as generating functions of maps; for the general recursion see [CE].
Loop equations were introduced to the mathematical community by Johansson in [J1] to derive the Gaussian behavior; his results were significantly extended in the later work [KS], [BoGu1], [Shch], [BoGu2]. The loop equations, or rather “their spirit”, also have further applications far beyond the central limit theorems for global fluctuations, e.g. they were used in the recent work on local universality for random matrices, see [BEY], [BFG]. We would like to emphasize an important distinction between the approaches of physics and mathematics literature: the latter operates not with formal expansions, but with converging asymptotic expansions. This requires a priori estimates, whose derivations rely on a set of tools different from the loop equations themselves.
The Schwinger-Dyson equations for general– log–gases are obtained by integration by parts and derivation with respect to the potential. In fact, they can be derived in a much more general continuous setting, for instance when the underlying measure is the Haar measure on a compact Lie group, see e.g. [AGZ, (5.4.29)], and further used to derive the topological asymptotic expansions of related matrix models, see [CGM], [GN]. For a recent application of the Schwinger-Dyson equations to the lattice gauge theory see [Cha].
1.2. Integrable discretization of log–gases
The discrete versions of the distribution (1) (with ’s living on a lattice) at arise in numerous problems of statistical mechanics. Examples include random tilings (cf. Figure 1 and [CLP], [J4], [G], [BKMM]), stochastic systems of non–intersecting paths (cf. [KOR], [BBDT]), last passage percolation (cf. [J2]), interacting particle systems (cf. [J3], [BF]).
The law of large numbers for such systems (even for general values of ) can be established by essentially the same methods as for the continuous ones, cf. [J3], [J4], [Fe]. However, the situation is drastically different for the study of the asymptotics for global fluctuations. While for some very specific integrable choices of the potential at the central limit theorem was proven (cf. [BF], [P2], [BD]), until now no general approach that would work for generic and existed. We note however the works [HO, Chapter 12], [DF], [Mo] where the global Gaussian asymptotics was proven for certain (very concrete) discrete general probabilistic models originating in the representation theory. Moreover, it was not clear whether the CLT in the case of general potential should be the same as in the continuous case, or there should be some significant differences (indeed, for instance the local limits in the bulk must be different). The main technical difficulty lied in the absence of a nice generalization of the loop equations to the discrete setting.
In [Ey2], [Ey3], Eynard proposed to interpret laws of random partitions, which include discrete analogues of (3), as matrix models. One of the key points in his interpretation is to view discrete sums as highly oscillatory continuous integrals. Based on this identification, he conjectured that the asymptotic expansions in this case are described by the same Schwinger-Dyson equations with initial step given by the equilibrium measure of the model, see e.g [Ey2, Section 2.4, Section 2.7.1]. In particular, the fluctuations in the context of the central limit theorem should be universal. However, so far this approach has not yet progressed much beyond the predictions.
The central goal of this article is to study global fluctuations of the empirical measure for discrete analogues of general- log–gases. One outcome is that indeed these fluctuations are universal and described by the same covariance as for their continuous counterparts. Our analysis is based on appropriate discrete versions of the Schwinger-Dyson equations, which (unlike in the continuous setting) do not appear as a direct consequence of integration by parts or perturbative arguments.
The search for the discrete loop equations starts with identifying a good discrete analogue of the general distribution (1). For that we fix a parameter and a positive real–valued function 111 should decay at least as with as .. Consider the probability distribution
on ordered –tuples , such that and are integers. We refer to ’s as the positions of particles. Let us note that if , then do not sit on the fixed lattice.
Note that if , then (3) has the same form as (1) with . Similarly, leads to (1) with 222However, observe that the lattice where particles sit at is not .. More generally, if we set , then as , the ratio of Gamma-functions in (3) behaves as and mimics (1) with .
The most important reason to view (3) as a correct integrable discretization of the continuous log–gas (1) is the following observation which is the starting point of the results in the present paper.
Theorem 1.1 (Nekrasov’s equation).
Consider the probability distribution (3), and assume that
If are holomorphic in a domain , then so is . Moreover, if are polynomials of degree at most , then so is .
It is reasonable to ask how one could guess the form of the measure (3), which would lead to Theorem 1.1. The integrability properties of such measures, including the product of the ratios of Gamma functions as in (3), can be traced to their connections to representation theory and symmetric functions. The same products of Gamma functions appear in the evaluation formulas for Jack symmetric polynomials (cf. [Ma, Chapter VI, Section 10]) and in problems of asymptotic representation theory (cf. [O2]). Another trace of integrability is the existence of discrete Selberg integrals, which are evaluation formulas for the partition function in (3) for special choices of the weight . Examples of such evaluations can be found in [O2, Section 2], [GS, Section 2.2]. The latter reference also explains the degeneration to the conventional Selberg integral, which is the computation of the normalization constants for continuous log–gases (1) with specific choices of the potential , see [Me, Chapter 17], [Fo, Chapter 4]. On the other hand, for a “naive” discrete version, when one takes the same formula (1) as a definition of a discrete log–gas, we are not aware of the existence of similar evaluation formulas (outside , when (1) and (3) coincide).
1.3. Global fluctuations of discrete log–gases: one cut case
We proceed to our results on global fluctuations for the distributions (3) as . For that we need to postulate how the weight changes as . Our methods work for a variety of possibilities here, but to simplify the exposition we stick to the following assumption in this section:
where is an analytic function of real argument such that for large enough, is monotone and satisfies
Our first result is the law of large numbers for the empirical measures defined via
There exists a deterministic absolutely continuous compactly supported probability measure with , such that converges to as , in the sense that for any compactly supported Lipshitz function the following convergence in probability holds:
In fact, we prove a more general statement where, in particular, does not have to be analytic, see Theorems 5.3 and 10.1 below. The measure is the equilibrium measure, and it can be found as a solution to a variational problem. At , Theorem 5.3 reduces to results of [J3], [J4], [Fe]. For general values of , additional arguments are required, and we present them.
Note the condition , which is not present in the continuous log–gases, and arises from the fact that the minimal distance between adjacent particles in (3) is at least . This is a specific feature of the discrete models.
At this moment we need to make certain assumptions on the equilibrium measure. A band333We follow the terminology from [BKMM]. of is a maximal interval such that on . From the random matrix literature (cf. [BDE], [Shch], [BoGu2]) one expects that the global fluctuations are qualitatively different depending on whether has one or more bands. Here we stick to the one band case.
converge (in distribution and in the sense of moments) to centered Gaussian random variables with explicit covariance depending on and , and given in (127) below. In particular, if , , , then
The second order asymptotic expansion of the mean can be also analyzed, cf. Theorem 6.1 with , .
The technical Assumption 4 from Section 3.2 is a statement that a certain function produced from the equilibrium measure has no zeros. In Section 9.3 we show that this assumption always holds when is a convex function. A somewhat similar assumption appears in the work of Johansson [J1] on the central limit theorem for global fluctuations of continuous log–gases.
Let us emphasize that the limiting covariance in Theorem 1.3 depends only on the support of the equilibrium measure, but it is not sensitive to other features. The same phenomenon is known in the random matrix setting. Moreover, comparing (7) with expressions in [J1, Theorem 4.2], [PS, Chapter 3], we conclude that the covariance is precisely the same as for the continuous log–gases. Thus, the discreteness of the model is invisible on the level of the central limit theorem.
The covariance in Theorem 1.3 can be related to that of a section of the 2d Gaussian Free Field in the upper half–plane with Dirichlet boundary conditions. One way to predict that is by noticing that sections in lozenge tilings models with several specific boundary conditions yield distributions of the form (3) with , see Section 9.2 for one example. On the other hand, there exist several results on the appearance of the Gaussian Free Field in the asymptotics for global fluctuations in lozenge tilings, cf. [K], [BF], [P2], [BuGo2].
Subsequent work [KS], [BoGu1] for continuous log–gases showed that Johansson’s technical assumption on the equilibrium measure can be relaxed and replaced by certain weaker assumptions, which hold for generic analytic potentials. We hope that a similar thing can also be done in the present discrete setting, but this would require further investigations.
1.4. Weight supported on several finite intervals
In several applications the weight is not defined on the whole real line, but instead it is supported by a union of several disjoint intervals , . In other words, the particles are now confined to the union of these intervals . For instance, this happens in tilings models, see Section 9.2.
At this point we have an additional choice: One could either fix the filling fractions , which are the numbers of particles in each of the intervals , or not. In the present paper we stick to the former and fix .
We refer to Sections 3 and 5 below for details of the exact assumptions that we impose on the weight , intervals , and filling fractions . Here we will only briefly summarize the results obtained in such a framework.
The first result is the law of large numbers, which is the exact analogue of Theorem 1.2, see Theorem 5.3 for the details. As before, for the asymptotics of the global fluctuations we need to have some information about the bands of the equilibrium measure . We assume that there is one band per interval.
Theorem 1.6 (Theorem 7.1).
Assume that all the data specifying the model satisfies Assumptions 1–5 in Section 3.2 below. In particular, the equilibrium measure has bands , , one per interval of the support of the model. Take functions , which are analytic in a neighborhood of for large . Then as the joint moments of the random variables
approximate those of centered Gaussian random variables with asymptotic covariance depending only on and given by (116).
For , the limiting covariance coincides with that of Theorem 1.3. For it is no longer expressible through elementary functions, e.g. for elliptic functions appear in the formulas. We again emphasize that the covariance depend only on the end–points of the bands and is not sensitive to other features of the equilibrium measure. Furthermore, the covariance is the same as for the continuous log–gases, cf. [Shch], [BoGu2].
Similarly to the case, one could try to identify the covariance with that of a canonically defined random field. We believe that there should be a link to sections of the Gaussian Free Field in a domain with holes, but we postpone this discussion till a future publication.
1.5. Organization of the paper
In Section 2 we present our approach and results: Nekrasov’s equation, Law of Large Numbers and Central Limit Theorem — in the simplest yet non-trivial case when and in Theorem 1.1 are linear functions. The measure in this case is known as the Krawtchouk orthogonal polynomial ensemble; it appeared and was studied in numerous previous articles. In particular, the Central Limit Theorem for global fluctuations in this specific case can be also obtained by other methods: see [CJY], [BD], and [BuGo2]444We remark that while the approaches of these articles admit various generalizations, to the best of our knowledge, they do not extend further in the direction of the present paper..
In Sections 3–8 we explain a much more general framework, in which the same ideas work and lead to the Law of Large Numbers and Central Limit Theorem. In these sections is an arbitrary positive number, and the weight and functions are general (subject to certain technical assumptions).
In Section 9 we specialize the general framework to certain specific examples, which include lozenge tilings and –measures from asymptotic representation theory. We explain how all the technical assumptions are checked in all these examples.
We are very grateful to Nikita Nekrasov for numerous fruitful discussions which led us to Theorem 1.1 in its present form. We also would like to thank Andrei Okounkov for drawing our attention to the existence of discrete loop equations. We are grateful to two anonymous referees for their useful comments.
A. B. was partially supported by the NSF grant DMS-1056390. V. G. was partially supported by the NSF grant DMS-1407562. A. G. was partially supported by the Simons Foundation and by the NSF Grant DMS-1307704.
2. Toy example: binomial weight
The aim of this section is to describe our method in the simplest, yet non-trivial case of the binomial weight and . The resulting –particle ensemble is known as the Krawtchouk orthogonal polynomial ensemble. It appeared in the literature before in the connection with uniformly random domino tilings of the Aztec diamond (cf. [J4]), last passage percolation (cf. [J2]), stochastic systems of non–intersecting paths (cf. [KOR], [BBDT]), and with representation theory of the infinite–dimensional unitary group (cf. [BO2, Section 5], [B1, Section 4]).
In the subsequent sections we will extend our approach to more general cases, but the methodology and many key ideas remain the same.
Fix two integers and consider the space of –tuples of integers
We define a probability distribution on through
We remark that the partition function is explicitly known in this case:
However, in a generic situation there are typically no simple formulas for the partition function and we are not going to use its explicit form.
Our analysis of the distribution is based on the following version of Theorem 4.1. This is essentially due to [N], [NP], [NS] and we will call the main statement the Nekrasov’s equation. It should be viewed as a discrete space analogue of the Schwinger–Dyson (also known as “loop”) equations.
Let be a probability distribution on –tuples such that
and are analytic functions in a complex neighborhood of which are positive on and satisfy . Define
Then is analytic in the same complex neighborhood of . If are polynomials of degree at most , then is also a polynomial of degree at most .
The possible singularities of are simple poles at points . Let us compute a residue at such a point.
The expectation in (9) is a sum over all possible configurations . Such a configuration contributes to the residue at if either or for some .
We separately analyze the contributions appearing from each , which we now fix. Given a particle configuration , let denote the configuration with th coordinate increased by and let denote the configuration with th coordinate decreased by . Note that, in principle (similarly ) might fail to be in , as the coordinates might coincide. However, in this case the formula for still applies and gives zero555This is where we need the condition of , which translates into ..
The contribution to the residue of at , arising from the th coordinate of is
But using the definition of we see that
We conclude that for each , the terms with and (or and ) in the first and second sum in (10) cancel out and the total residue is zero.
For the polynomiality statement it suffices to notice that if are polynomials of degree at most , then is an entire function which grows as as . Hence, by Liouville’s theorem is a polynomial. ∎
Note that for the distribution (8) the functions can be chosen to be linear and we set
In this section we aim to study the asymptotics of the distributions as . The parameter will also depend on . We fix and set .
2.1. Law of Large Numbers
As , a certain law of large numbers holds for the measures . Let us introduce a random probability measure on via
The measure is often referred to as the empirical measure of the point configuration . Note that our definitions imply the condition , which shows that for any interval , its –measure is bounded from above by .
The idea for the proof of the law of large numbers for as is to establish the large deviations principle for the measures , which would show that the measure is concentrated on which maximize the probability density (8). This was done rigorously in [J3], [J4], [Fe], see also Section 2.4 below for some details. The explicit formula for is then obtained as a solution of a variational problem, this solution for our weight was found earlier in [DS, Example 4.2]. The developments of these articles are summarized in the following proposition.
The measures converge (weakly, in probability) to a deterministic absolutely continuous measure , which is called equilibrium measure. For , the density of the measure is
and for ,
where is the inverse cotangent function.
A convenient way of working with the equilibrium measure is through its Stieltjes transform defined through
We introduce two functions
for and consider their analytic continuations.
For any ,
with the branch of the square root chosen so that as .
We first observe that is, in fact, a linear polynomial. Indeed, this is the limit of in Proposition 2.1 for the measure . The fact that as , implies , and thus . Further, observe that the definition of and implies
2.2. Second order expansion
Define the Stieltjes transform of the prelimit empirical measure (12) through
We aim to study how approximate as . For that we introduce a deformed version of the same function. Take parameters , such that for all and all and let the deformed distribution be defined through
If , then is the undeformed measure. In general, may be a complex–valued measure. The normalizing constant in (15) is chosen so that the total mass of is , i.e. . Let us note that the numbers are always chosen small enough, which guarantees that .
Observe that satisfies the assumptions of Proposition 2.1 with
As above, we set , and omit it from the notations. We also define
We further define as the empirical distribution of and set
Note that we often omit the dependence on from the notations, to keep them concise.
For any , the th mixed derivative
is the joint cumulant of random variables , , …, with respect to the measure .
We first note that the joint cumulants are invariant under addition of constants. Therefore, we can replace by for the purpose of computing the cumulants.
Further recall that one way to define the joint cumulant of bounded random variables is through
Taking the derivative with respect to explicitly, we can rewrite this also as
Setting , , and observing that
we get the desired statement. ∎
Fix and choose complex numbers . Then, as ,
where is a simple positively–oriented contour enclosing the segment (the points and are outside the contour). The remainder is uniform over in compact subsets of the unbounded component of . The case is that we take no derivatives in (21).
The only cases, where the right–hand side of (21) is meaningful are , since its limit is zero for . Indeed, only depends on , and moreover it is a sum of functions of single variables , ; therefore, all mixed partial derivatives vanish. However, we present Theorem 2.5 in this way, since that’s the form which appears in our proofs. For we will compute the limit of (21) below in Theorem 2.8. For and generic the right–hand side of (21) has no limit, as in the definition of oscillates.
It might seem a bit unexpected that for the right–hand side of (21) does not vanish when . Indeed, in the context of random matrices this corresponds to case, where the mean is known to vanish (cf. [J1]). For our model the non-zero mean can be traced back to two features. First, due to discreteness we can not adjust so that becomes zero for all . Second, the logarithm of the weight, , has a non-trivial asymptotic expansion, which affects the result.
Before providing a proof of Theorem 2.5, let us give its important corollaries. Note that if we set in Theorem 2.5, then it gives the limit behavior for . In fact, it also gives the Central Limit Theorem for .
The random field , , converges as (in the sense of joint moments, uniformly in in compact subsets of ) to a centered complex Gaussian random field with second moment
Note that the covariance has no singularity at , since the right–hand side of (22) has a finite limit.
Since , the formula (22) is sufficient for determining the asymptotic covariance of the random field .
Proof of Theorem 2.8.
We start from the result of Lemma 2.4 for the joint cumulant of random variables , , …, . In particular, if , then we get the covariance of and .
The term with integrates to , as it has no singularities inside . The term with is computed as the sum of the residues at and at , which gives the desired covariance formula. ∎
Theorem 2.8 implies the central limit theorem for general analytic linear statistics.
Let the distribution be given by (8) with . Take real valued functions on , which can be extended to holomorphic functions in a complex neighborhood of . Then as the random variables
converge in the sense of moments to centered Gaussian random variables with covariance
where is a positively oriented contour in which encloses , and is given by (22).
Proof of Corollary 2.11.
Therefore, all the moments of are obtained from the centered moments of by integration. Since the latter converge uniformly in on the integration contour, so do the former. It remains to use the fact that the integrals of jointly Gaussian random variables are also Gaussian. ∎
The rest of this section is devoted to the proof of Theorem 2.5.
2.3. Heuristic argument for Theorem 2.5
In this section we present a sketch of the proof for Theorem 2.5 in which we omit crucial bounds on remainders in the asymptotic formulas. These bounds will be established further on.
We start from the statement of Proposition 2.1 for the measures