A global large deviation principle for discrete $\beta$-ensembles

TL;DR

This paper proves a large deviation principle for discrete beta-ensembles, extending understanding of their spectral measures under broad conditions including variable potentials and infinite support.

Contribution

It establishes a comprehensive large deviation principle for discrete beta-ensembles with general potentials, including those depending on particle number and with slow growth.

Findings

Large deviation principle proven for empirical measures

Applicable to models with potentials depending on particle number

Handles equilibrium measures with infinite support

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 the empirical (or spectral) measures corresponding to these models. Our results apply in the cases when the potential of the model depends on the number of particles, and/or has slow growth near infinity, leading to an equilibrium measure with infinite support.