Asymptotics of discrete $\beta$-corners processes via two-level discrete loop equations

TL;DR

This paper investigates the asymptotic behavior of a class of discrete particle ensembles related to random matrix theory, revealing universal Gaussian fluctuations with novel covariance structures through new algebraic identities.

Contribution

It introduces discrete multi-level loop equations and demonstrates their use in analyzing fluctuations of discrete beta-corners processes, highlighting differences from classical random matrix results.

Findings

Global fluctuations are asymptotically Gaussian.

Covariance structures differ from classical random matrix theory.

Novel algebraic identities serve as discrete loop equations.

Abstract

We introduce and study a class of discrete particle ensembles that naturally arise in connection with classical random matrix ensembles, log-gases and Jack polynomials. Under technical assumptions on a general analytic potential we prove that the global fluctuations of these ensembles are asymptotically Gaussian with a universal covariance that remarkably differs from its counterpart in random matrix theory. Our main tools are certain novel algebraic identities that we have discovered. They play a role of discrete multi-level analogues of loop equations.

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