Gaussian asymptotics of discrete $\beta$-ensembles

TL;DR

This paper establishes that a broad class of discrete particle ensembles, including those modeling random tilings and non-intersecting paths, exhibit Gaussian fluctuations with universal covariance in the large particle limit.

Contribution

It introduces a unified framework for analyzing discrete $eta$-ensembles and proves their global fluctuations are asymptotically Gaussian with universal covariance.

Findings

Global fluctuations are asymptotically Gaussian as N→∞.

The covariance of fluctuations is universal, matching that of continuous $eta$-ensembles.

The method employs a discrete Schwinger-Dyson equation approach.

Abstract

We introduce and study stochastic $N$-particle ensembles which are discretizations for general-$\beta$ log-gases of random matrix theory. The examples include random tilings, families of non-intersecting paths, $(z,w)$-measures, etc. We prove that under technical assumptions on general analytic potential, the global fluctuations for such ensembles are asymptotically Gaussian as $N\to\infty$. 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.