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Macroscopic asymptotics in discrete beta-ensembles and random tilings

Gaëtan Borot, Vadim Gorin, Alice Guionnet

TL;DR

The paper develops a comprehensive macroscopic asymptotic theory for discrete multi-group beta-ensembles, establishing the law of large numbers, large deviations, and finite-size corrections in the large-$N$ regime. It introduces Nekrasov equations as a robust discrete analogue of Dyson–Schwinger equations, analyzes their solvability via inversion of a master operator tied to spectral-curve geometry, and derives all-order correlator expansions along with partition-function asymptotics, including discrete Gaussian components for fluctuating fillings. The framework applies to a broad class of models, notably uniformly random lozenge tilings on general domains (planar or non-orientable), and yields Gaussian free field fluctuations in orientable liquid regions while providing a modified Kenyon–Okounkov picture in non-orientable settings. The results unify and extend prior work (BGG, BG_multicut) and offer a systematic method to connect one-dimensional discrete ensembles to two-dimensional fluctuation fields through Riemann–Hilbert problems and spectral curves, with concrete tiling applications and explicit asymptotics for partition functions in several solvable cases (e.g., zw-measures, Gaussian weights).

Abstract

We carry out the asymptotic analysis of repulsive ensembles of N particles which are discrete analogues of continuous 1d log-gases or beta-ensembles of random matrix theory. The ensembles that we study have several groups of particles which can have different intensities of repulsion. They appear naturally in models of random domino and lozenge tilings, random partitions, supersymmetric gauge theory, asymptotic representation theory, discrete orthogonal polynomial ensembles, etc. We allow filling fractions to be either fixed, or free, or to vary while respecting affine constraints. We are interested in the macroscopic behavior of the distribution of particles, captured by linear statistics, partition functions, and their finite-size corrections as N is large. We prove the law of large numbers and large deviations for the empirical measure around the equilibrium measure. To reach finite-size correction we assume off-criticality. For fixed filling fractions, we prove an asymptotic expansion for the partition function and for the cumulants of linear statistics, in particular establishing a central limit theorem. For varying filling fractions, we prove that the central limit theorem is perturbed by an additional discrete Gaussian component oscillating with N. We apply our general results to the study of uniformly random lozenge tilings on a large class of domains -- not necessarily planar, simply-connected, nor orientable. When the analogues of filling fractions are fixed and this domain is orientable, we show that the Gaussian fluctuations on the vertical extend to the whole liquid region and are governed there by the Gaussian free field, as predicted by the Kenyon-Okounkov conjecture. We also establish a modification of the Kenyon-Okounkov conjecture in the non-orientable case. Complementarily, we prove discrete Gaussian fluctuations for filling fractions, when they are not fixed.

Macroscopic asymptotics in discrete beta-ensembles and random tilings

TL;DR

The paper develops a comprehensive macroscopic asymptotic theory for discrete multi-group beta-ensembles, establishing the law of large numbers, large deviations, and finite-size corrections in the large- regime. It introduces Nekrasov equations as a robust discrete analogue of Dyson–Schwinger equations, analyzes their solvability via inversion of a master operator tied to spectral-curve geometry, and derives all-order correlator expansions along with partition-function asymptotics, including discrete Gaussian components for fluctuating fillings. The framework applies to a broad class of models, notably uniformly random lozenge tilings on general domains (planar or non-orientable), and yields Gaussian free field fluctuations in orientable liquid regions while providing a modified Kenyon–Okounkov picture in non-orientable settings. The results unify and extend prior work (BGG, BG_multicut) and offer a systematic method to connect one-dimensional discrete ensembles to two-dimensional fluctuation fields through Riemann–Hilbert problems and spectral curves, with concrete tiling applications and explicit asymptotics for partition functions in several solvable cases (e.g., zw-measures, Gaussian weights).

Abstract

We carry out the asymptotic analysis of repulsive ensembles of N particles which are discrete analogues of continuous 1d log-gases or beta-ensembles of random matrix theory. The ensembles that we study have several groups of particles which can have different intensities of repulsion. They appear naturally in models of random domino and lozenge tilings, random partitions, supersymmetric gauge theory, asymptotic representation theory, discrete orthogonal polynomial ensembles, etc. We allow filling fractions to be either fixed, or free, or to vary while respecting affine constraints. We are interested in the macroscopic behavior of the distribution of particles, captured by linear statistics, partition functions, and their finite-size corrections as N is large. We prove the law of large numbers and large deviations for the empirical measure around the equilibrium measure. To reach finite-size correction we assume off-criticality. For fixed filling fractions, we prove an asymptotic expansion for the partition function and for the cumulants of linear statistics, in particular establishing a central limit theorem. For varying filling fractions, we prove that the central limit theorem is perturbed by an additional discrete Gaussian component oscillating with N. We apply our general results to the study of uniformly random lozenge tilings on a large class of domains -- not necessarily planar, simply-connected, nor orientable. When the analogues of filling fractions are fixed and this domain is orientable, we show that the Gaussian fluctuations on the vertical extend to the whole liquid region and are governed there by the Gaussian free field, as predicted by the Kenyon-Okounkov conjecture. We also establish a modification of the Kenyon-Okounkov conjecture in the non-orientable case. Complementarily, we prove discrete Gaussian fluctuations for filling fractions, when they are not fixed.
Paper Structure (191 sections, 190 theorems, 928 equations, 50 figures, 1 table)

This paper contains 191 sections, 190 theorems, 928 equations, 50 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background, motivations, and objectives
  3. Paradigm and continuous ensembles
  4. Random tilings and discrete ensembles
  5. Discrete analogues of beta-ensembles: our objects and goals
  6. Illustration of results
  7. Lozenge tilings of the hexagon with a hole
  8. Lozenge tilings of C-shaped domains
  9. The zw-measures
  10. The discrete ensemble with Gaussian weight
  11. Methodology
  12. Law of large numbers and concentration of measures
  13. Dyson--Schwinger equations and Nekrasov equations
  14. Expansion of the correlators and central limit theorem for fixed filling fractions
  15. Expansion of the partition function for fixed filling fractions
  16. ...and 176 more sections

Key Result

Theorem 1

Assume that $A$, $B$, $C$, $D$, $\mathfrak t$ grow linearly with a large parameter $\mathcal{N}$. Then, under a non-degeneracy condition (cf. Definition Definition_hex_hole_non_degenerate), the random variable $N_1$ is asymptotically discrete Gaussian, in the sense that there exist three constants $

Figures (50)

  • Figure 1: Left: A lozenge tiling of $3\times 4 \times 5$ hexagon and its vertical section. Right: Simulation of a uniformly random lozenge tiling of $100\times 100 \times 100$ hexagon (three types of lozenges shown in colors)
  • Figure 2: Left panel: Lozenge tiling of the $4\times 7\times 5$ hexagon with a rhombic $2\times 2$ hole (shown in blue). The horizontal lozenges outside the hole and on the $\mathfrak{t}$-th vertical line are shown in gray. Right panel: $100\times 100\times 100$ hexagon with a hole. We thank Leonid Petrov for his help with this and other simulations.
  • Figure 3: Tilings of two different $C$-shaped regions. Gray horizontal lozenges are distributed as \ref{['eq_measure_general_intro']}.
  • Figure 4: Uniform lozenge tilings of domains in Figure \ref{['Fig:Cshapeint']} resized $8x$. Simulation using Leo_ultimate.
  • Figure 5: Difference between heights of two independent samples as in Figure \ref{['Fig:Cshape_simulation']}, divided by $\sqrt{2}$: for Gaussian fields, this has the same distribution as a single sample recentered around its mean.
  • ...and 45 more figures

Theorems & Definitions (285)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 1.1: Law of large numbers
  • Theorem 1.2: Concentration of empirical measures
  • Lemma 1.3: Concentration of linear statistics
  • Theorem 1.4: Nekrasov equation
  • Corollary 1.5: Holomorphicity of $R(z)$
  • Theorem 1.6: Central limit theorem for fixed filling fractions
  • ...and 275 more