Vol. 3, No. 2, 2022

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

Evgeni Dimitrov and Alisa Knizel

Vol. 3 (2022), No. 2, 247–342
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 multilevel analogues of loop equations.

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Keywords
loop equations, log-gases, corners process
Mathematical Subject Classification
Primary: 82C41
Secondary: 33D45, 52C20
Milestones
Received: 30 August 2020
Revised: 10 January 2022
Accepted: 15 February 2022
Published: 8 July 2022
Authors
Evgeni Dimitrov
Department of Mathematics
Columbia University
New York, NY
United States
Alisa Knizel
Department of Statistics
University of Chicago
Chicago, IL
United States