# 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.

## Key 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.