Macroscopic asymptotics in discrete beta-ensembles and random tilings

TL;DR

This paper analyzes the macroscopic behavior of discrete beta-ensembles and their applications to random tilings, proving laws of large numbers, large deviations, and Gaussian fluctuations, including extensions of the Kenyon-Okounkov conjecture.

Contribution

It provides a comprehensive asymptotic analysis of discrete beta-ensembles with varying filling fractions, including new results on partition functions, linear statistics, and Gaussian free field fluctuations.

Findings

Law of large numbers and large deviations established for empirical measures.

Asymptotic expansion of partition functions and cumulants, including CLT.

Gaussian free field fluctuations in the liquid region for orientable domains.

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.