Log-gases on a quadratic lattice via discrete loop equations and q-boxed plane partitions
We study a general class of log-gas ensembles on a quadratic lattice. Using a variational principle we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance. We apply our general results to analyze the asymptotic behavior of a -boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field.
Our approach is based on a -analogue of the Schwinger-Dyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to a quadratic lattice.
- 1 Introduction
- 2 General setup
- 3 Law of large numbers
- 4 Nekrasov’s equations
- 5 Central limit theorem: Part I
- 6 Central limit theorem: Part II
- 7 -Racah tiling models and ensembles
- 8 Global asymptotics for the -Racah tiling model
- 9 Connection to log-gases on a quadratic lattice
- 10 Analysis of asymptotic Nekrasov’s equations
Let be a subset of and be a non-negative function. Fix a positive integer and consider the class of probability measures on all -point subsets of of the form
The most studied cases of the above distributions are and , they are usually called discrete and continuous log-gases respectively (if one understands (1.1) as a density on ).
The focus of the present paper is to understand the global asymptotics of for
We refer to the set as a quadratic lattice in the spirit of  and the corresponding distribution as a log-gas on a quadratic lattice.
Our study is motivated by random matrix theory on one side, and by integrable models on the other. We first investigate for a general choice of weights . We prove that these systems obey a law of large large numbers under a certain scaling as goes to infinity and also show that their global fluctuations are asymptotically Gaussian with a universal covariance. The same phenomenon is present in the case of discrete and continuous log-gases. Subsequently, we apply our general results to a class of tiling models that was introduced in  and obtain explicit formulas for their limit shape and global fluctuations.
The probability measures when for some continuous function (typically called potential) have been extensively studied in the random matrix theory world and beyond, see [44, 26, 1, 50] among many others. For example, when the measure describes the joint distribution of the eigenvalues of an matrix from the Gaussian Unitary Ensemble. In the discrete setting, naturally arise in problems of two-dimensional models in statistical mechanics, which include various random tiling models, non-intersecting path ensembles and interacting particle systems, see e.g. [19, 34, 33, 25, 7].
Under weak assumptions on the potential , with or exhibit a law of large numbers as . Specifically, if one forms the (random) empirical measures
then the measures converge weakly in probability to a deterministic measure , called the equilibrium measure. For and this statement goes back to the work of Wigner , and is called Wigner’s semicircle law. The analogous statements for generic were proved much later, see [32, 3, 14]. When similar law of large numbers type results were obtained in [25, 33, 34]. In both the discrete and continuous cases the equilibrium measure is the solution to a suitable variational problem and one establishes the convergence of to by proving large deviation estimates. In essence, maximizes the density (1.1) and the large deviation estimates show that concentrate around that maximum.
The next order asymptotics asks about the fluctuations of as . One natural way to analyze this difference is to form the pairings with smooth test functions and consider the asymptotic behavior of the random variables
There is a very efficient method, which establishes that the limits of (1.2) are Gaussian in a very general setup and its key ingredient is the so-called loop equations (also known as Schwinger-Dyson equations), see [32, 12, 53, 11, 42] and references therein. These are functional equations for certain observables of the log-gases (1.1) that are related to the Stieltjes transforms of the empirical measure and and their cumulants. Since their introduction loop equations have become a powerful tool for studying not only global fluctuations but also local universality for random matrices [13, 2].
In  the authors presented an analogue of the above method in the discrete setting (for ). They introduced discrete loop equations and used them to establish that the limits of (1.2) are Gaussian with a covariance that is the same as in the continuous case for a large class potentials (in fact, they prove such a statement for more general models called discrete -ensembles). These disrete loop equations originate in the work of Nekrasov  and are also called Nekrasov’s equations. The central limit theorem for (1.2) had been previously known for various very specific integrable choices of the potential, see e.g. [7, 52, 15]. The main contribution of  is that it establishes general conditions on the potential that lead to the asymptotic Gaussianity of (1.2). Similarly to the continuous case, discrete loop equations have become a valuable tool to study not only global fluctuations  but also edge universality for discrete -ensembles .
In the present paper we establish the universality type results for the global fluctuations of log-gases on a quadratic lattice. To obtain the law of large numbers we use a similar combination of large deviation estimates and variational problems that were successful in the cases or . In order to study the next order fluctuations we introduce a new version of discrete loop equations for a quadratic lattice, which we also call Nekrasov’s equations, and view the latter as one of the main contributions of this paper. We remark that it is hard to guess that there even exists an analogue of the Nekrasov’s equation in this setting, since it is a very subtle equation which reflects some specific algebraic structure of the system. In fact, we present Nekrasov’s equation for general log-gases on a quadratic lattice but the focus of the present paper is the case Equipped with these new equations, we establish global central limit theorems for log-gases on a quadratic lattice for a multi-cut general potential by adapting the arguments in .
Our main motivation for considering comes from an interesting tiling model introduced in  which we describe next.
1.1.2. The -Racah tiling model
Consider a hexagon drawn on a regular triangular lattice, whose side lengths are given by integers , see Figure 1. We are interested in random tilings of such a hexagon by rhombi, also called lozenges (these are obtained by gluing two neighboring triangles together). There are three types of rhombi that arise in such a way, and they are all colored differently in Figure 1. This model also has a natural interpretation as a boxed plane partition or, equivalently, a random stepped surface formed by a stack of boxes. One can assign to every lattice vertex inside the hexagon an integer , which reflects the height of the stack at that point, see an example in Figure 1. One typically calls the height function and formulates results in terms of it.
The probability measures on the set of tilings that we consider were introduced in  and form a -parameter generalization of the uniform distribution. Denoting the two parameters by and , one defines the weight of a tiling as the product of simple factors over all horizontal rhombi where is the coordinate of the topmost point of the rhombus. The dependence of the factors on the location of the lozenge makes the model inhomogeneous. Note that the uniform measure on tilings is recovered if one sends and . Other interesting cases include , then the weight becomes proportional to (here refers to the number of boxes in the interpretation). In addition, the same way the Hahn orthogonal polynomial ensemble arises in the analysis of uniform lozenge tilings, our measures are related to the -Racah orthogonal polynomials. In this sense, the model goes all the way up to the top of the Askey scheme , and we call it the -Racah tiling model.
We believe that the -Racah tiling model is a source of rich and interesting structures that are worth investigating. The presence of two parameters allows one to consider various limit regimes that lead to quite different behavior of the system as can be seen in Figure 2. One of the central goals of this paper is to understand the asymptotic behavior of the height function of the -Racah tiling model when the sides of the hexagon become large, and simultaneously , where is fixed, see Figure 3 for a sample tiling in this case.
It turns out that one can relate one-dimensional sections of the -Racah tiling model to the log-gas on a quadratic lattice . We will elaborate on this point later in Sections 7 and 9.2, but the identification goes as follows. One places a particle in the center of each horizontal lozenge and takes a vertical section of the model; the resulting “holes” (positions, where there are no particles) form an -point process. Under a suitable change of variables this point process has the same distribution as (1.1) for a set of parameters and weight that depend on the location of the vertical slice. Using this identification, our general results for imply a law of large numbers and central limit theorem for the height function of the tiling model.
Informally, our law of large numbers states that there exists deterministic limit shape and the random height functions concentrate near it with high probability as the parameters of the model scale to their critical values. An important feature of our model is that the limit shape develops frozen facets where the height function is linear. In addition, the frozen facets are interpolated by a connected disordered liquid region. In terms of the tiling a frozen facet corresponds to a region where asymptotically only one type of lozenge is present and in the liquid region one sees lozenges of all three types, see Figure 3.
Similar concentration phenomena for the random height function in the case of the uniform measure and the measure proportional to are well-understood. In particular, in these cases convergence of the random height function to a deterministic function for a large class of domains was established in [31, 18, 20, 21, 38, 51]. The limit shape is given by the unique solution of a suitable variational problem, which in the case of the -Racah tiling model was derived in . Despite setting up the variational problem,  does not prove the concentration phenomenon and we establish it in the main text as Theorem 7.2.2.
The next order asymptotics we obtain show that the one-dimensional fluctuations of the height function around the limit shape are Gaussian with an explicit covariance kernel. An important additional contribution of our work is the introduction of a (rather nontrivial) complex structure on the liquid region. The significance of this map is that the fluctuations of on fixed vertical slices are asymptotically described by the one-dimensional sections of the pullback of the Gaussian free field (GFF for short) on the upper half-plane under the map – see Theorem 7.2.4 for the precise statement. This result admits a natural two-dimensional generalization, which we formulate as Conjecture 8.4.1 in the main text. At this time our methods only provide access to the global fluctuations at fixed vertical sections of the model, and so we cannot establish the full result. Nevertheless, we provide some numerical simulations that give evidence for the validity of the conjecture and hope to address it in the future.
The GFF is believed to be a universal scaling limit for various models of random surfaces in . The appearance of the GFF in tiling models with no frozen zones dates back to [35, 36] and the fluctuations of the liquid region for a random tiling model containing both frozen facets and liquid region were first studied in . In case of the uniform measure on domino and lozenge tilings the convergence to the GFF has been established for a wide class of domains in [52, 16, 17], but there are no results in this direction for more general measures. One possible reason that explains why the GFF was not recognized in the -Racah tiling model is the rather non-trivial change of coordinates that makes the correct covariance structure appear (see Section 8), and already in the (or ) case our result is new. We remark that there is a natural complex coordinate on the liquid region defined by the so-called complex slope, which in the uniform tiling case is known to be intimately related to the complex structure that gives rise to the GFF. For the -Racah tiling model an expression for the complex slope was obtained in  and we believe that it should be connected to our complex structure , but so far we do not know an explicit relationship between them, see Remark 8.2.2.
1.2. Main results
We present here our main results for the log-gas on a quadratic lattice and forgo stating our results on the -Racah tiling model until the main text – Section 7.2 – since it requires the introduction of more notation. Moreover, to simplify our discussion we will only formulate our results for the one-cut case and point to the relevant parts of the paper where their generalizations are given.
Let us first explain our regularity assumptions on the parameters and the weight function. We assume we are given parameters , and . In addition, let , and be sequences of parameters such that
We assume that has the form
for a function that is continuous in the intervals and such that
for some positive constants . We also require that is differentiable and for some there is a bound
We let be as in (1.1) for and with weight function .
Our first result is the law of large numbers for the empirical measures , defined by
There is a deterministic, compactly supported and absolutely continuous probability measure 111Throughout the paper we denote the density of a measure by . such that concentrate (in probability) near . More precisely, for each Lipschitz function defined in a real neighborhood of the interval and each the random variables
converge to in probability and in the sense of moments.
To obtain our central limit theorem we need to impose certain analyticity conditions on the weight that we now detail. We assume that we have an open set , such that for large
In addition, we require that for all sufficiently large there exist analytic functions on such that for and the following hold
where and the constants in the big notation are uniform over in compact subsets of . All the aforementioned functions are holomorphic in .
The assumptions in (1.7) are the analogues of Assumptions 3 and 5 in , and similarly to that paper their importance to the analysis comes from the following observation, which is the starting point for our results. We discuss the general setup and the corresponding Nekrasov’s equation in Section 4.2.
Theorem 1.2.3 (Nekrasov’s equation).
Suppose that (1.7) hold and define
Then is analytic in If are polynomials of degree at most , then so is .
In this sense, the Nekrasov’s equation lead to a functional equation for , and our central limit theorem is a consequence of a careful analysis of the lower order terms of the above limit. We remark that in  the expression that appears in the exponent above is directly the Stieltjes transform and not a modified version of it as in our case, which increases the technical difficulty of our arguments. The appearance of is a novel feature that comes from working on a quadratic lattice and we give some explanation of it in Remark 4.1.3.
We also require that the equilibrium measure has a single band in (see Assumption 5 in Section 2.1). In our context, a band is a maximal interval such that , where and . The parameters that appear in the next Theorem 1.2.5 are then precisely the endpoints of this band.
Then the random variables converge jointly in the sense of moments to an -dimensional centered Gaussian vector with covariance
where are given in Assumption 5 and is a positively oriented contour that encloses the interval .
We emphasize that the covariance in (1.9) depends only on , and is not sensitive to other features of the equilibrium measure . Furthermore, the covariance is the same as for the continuous log-gases, cf. [32, Theorem 2.4] and [50, Chapter 3]. Thus, the discreteness of the model is invisible on the level of the central limit theorem, which is consistent with what was observed for the discrete -ensembles in .
Observe that the covariance has no singularity when , since the RHS of (1.9) has a finite limit when tends to .
In Section 2 we describe the general framework of our study, the scaling regime we consider and the assumptions on the weight . In Section 3 we establish a general law of large numbers as Theorem 3.1.1. Nekrasov’s equation is discussed in Section 4. Sections 5 and 6 contain the proof of Theorem 5.2.7 (our general central limit theorem). A detailed description of the -Racah tiling model is given in Section 7 and we give the proof of our results about its random height function in Section 8. Section 9 provides the verification that the tiling model fits into the general framework of Section 2. Finally, Section 10 collects some technical lemmas we use throughout the paper.
The authors are deeply grateful to Alexei Borodin, Vadim Gorin and Alice Guionnet for very helpful discussions. For the second author the financial support was available through NSF grant DMS:1704186 and the project started when the second author was still a PhD student at Massachusetts Institute of Technology. We also want to thank the hospitality of PCMI during the summer of 2017 supported by NSF grant DMS:1441467.
2. General setup
In this section we describe the general setting of a multi-cut, fixed filling fractions model that we consider and list the specific assumptions we make about it.
2.1. Definition of the system
We begin with some necessary notation. Let , , and be such that . For such parameters we set , which is a finite discrete subset of . We refer to the set as a quadratic lattice in the spirit of . Informally, we want to consider measures of the form
on ordered -tuples , referred to as positions of particles, where for . In the general case, we would also like to have some restrictions on which locations the particles can occupy.
In what follows we formalize the above general description. We fix , which denotes the number of intervals in that contain particles. For each we take integers , such that and disjoint intervals of the real line that are ordered from left to right. We assume that and for and for . Further, the number denotes the number of particles in the th interval. To make the latter statement precise we need to restrict the state space of our point process as follows.
Define the set of indices , , through
The state space consists of -tuples such that for any and we have and if .
To finish the definition of our model we take a weight function , which is assumed to be positive for , where for an element such that , we write to be the largest element in less than . With the above notation we define probability measure on through
where is a normalization constant, typically called the partition function.
2.2. Scaling and regularity assumption
We are interested in obtaining asymptotic statements about as . This requires that we scale our parameters in some way and also impose some regularity conditions for the interval endpoints and the weight functions . We list these assumptions below.
Assumption 1. We assume that we are given parameters , and . For future reference we denote the set of parameters that satisfy the latter conditions by and view it as a subset of with the subspace topology. In addition, we assume that we have a sequence of parameters , and such that
The measures will then be as in (2.1) for and .
Assumption 2. We require that for each as we have for some
In addition, we assume that in the intervals , has the form
for a function that is continuous in the intervals and such that
for some constants .We also require that is differentiable and for some we have the bound
We believe that one can take more general remainders in the above two assumptions, without significantly influencing the arguments in the later parts of the paper. However, we do not pursue this direction due to the lack of natural examples.
Let us denote and observe that the latter is a bijective diffeomorhism from to . Let and note that is positive on the interval .
Assumption 3. Set for . We will often suppress the dependence of on and we assume that for sufficiently large these sequences satisfy
where is some positive constant. In our future results it will be important that the remainders are uniform over , satisfying the above conditions.
The above assumptions will be sufficient to obtain our law of large numbers for . We stated the one-cut case of this law in Theorem 1.2.1. In general, if one assumes that for some positive constants for , then the sequence of empirical measures converges to a deterministic measure , called the equilibrium measure. The precise statement detailing this convergence is given in Theorem 3.1.1, and the equilibrium measure turns out to be the maximizer of a certain variational problem – see Lemma 3.1.2. It depends on , the limiting potential , the endpoints from Assumption 2 and the limiting filling fractions for .
We next isolate the assumptions we require for establishing our central limit theorem, starting with the analytic properties of the weight . Let satisfy
Assumption 4. We assume that we have an open set , such that for large
In addition, we require that for all sufficiently large there exist analytic functions on such that for we have
where and the constants in the big notation are uniform over in compact subsets of . All the aforementioned functions are holomorphic in .
In the case when in Assumption 1 is and uniformly converges to , together with its derivative in a neighborhood of a point , we have
Indeed, if we set and
where we used that by assumption on and .
The next assumption we require is about the equilibrium measure , which was discussed in Remark 2.2.2. A convenient way to encode is through its Stieltjes transform , defined through
The following two functions naturally arise from our discrete loop equations (see Section 4) and play an important role in our further analysis
In Section 3 we show that is analytic, while is a branch of a two-valued analytic function, given by the square root of a holomorphic function on . Below, we will formulate two types of assumptions on the functions and their finite analogues. To obtain our results we will need to assume one or the other. The difference between them is that one relaxes the requirements on the involved functions, but requires stronger assumptions on the parameters of the model.
Assumption 5. For each we let be the equilibrium measure discussed in Remark 2.2.2 for the parameters , , endpoints and filling fractions , .
For each we also let be as in Assumption 4, and define
We assume that is analytic in for all large . In addition, we require that for all large there exist real numbers and functions on such that
, and there are constants such that for .
, where in (recall our square root branch convention from Section 1.2).
The above assumption does not describe a general case for several reasons. Firstly, it implies that has a single interval of support inside each interval . To the authors’ knowledge there are no simple conditions on the potential that ensure that has this property. In addition, as will be discussed in Section 4, one can deduce the analyticity of and in (2.9) under Assumptions 1-4. In Assumption 5, we are making the stronger statement that analyticity is present even for finite . However, it turns out that all of the models we are interested in satisfy such conditions.
Assumption 5’. Assume for all . For each we let be the equilibrium measure discussed in Remark 2.2.2 for the parameters , , endpoints as in Assumptions 1,2 and filling fractions , . Observe that depends on only through the filling fractions, in particular in the one-cut case it does not depend on .
Let be as in (2.9) for the measure . Then for each we require that for all large there exist real numbers and functions on such that
, and there are constants such that for .
, with in .
Assumption 5’ is the analogue of Assumption 4 in [9, Section 3] for our setting. The easiest way to see that the requirements are relaxed compared to Assumption 5 is to consider the one-cut case. In this case, all the dependence on disappears and one is left with a single statement for the equilibrium measure from Remark 2.2.2, as opposed to the countable set of conditions in Assumption 5. The drawback is that to make our arguments go through under these assumptions we require