Large deviations for discrete $\beta$-ensembles

TL;DR

This paper establishes a large deviation principle for the rightmost particle in discrete beta-ensembles, extending understanding of their probabilistic behavior and applying results to measures linked to Jack symmetric functions.

Contribution

It introduces a general large deviation principle for discrete beta-ensembles and applies it to specific measures related to Jack symmetric functions.

Findings

Large deviation principle proven for the rightmost particle

Results applicable to measures connected to Jack symmetric functions

Provides a theoretical framework for analyzing extreme particles in discrete beta-ensembles

Abstract

We consider discrete $\beta$-ensembles as introduced by Borodin, Gorin and Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017). Under general assumptions, we establish a large deviation principle for their rightmost particle. We apply our general results to two classes of measures that are related to Jack symmetric functions.