Large deviations for near-extreme eigenvalues in the beta-ensembles

TL;DR

This paper establishes a large deviation principle and fluctuation convergence for the empirical spectral distribution near the maximum eigenvalue in beta ensembles with convex polynomial potentials, extending understanding of eigenvalue crowding.

Contribution

It introduces a large deviation principle for the spectral distribution from the rightmost particle in beta ensembles, generalizing previous GUE results.

Findings

Proved large deviation principle for the spectral distribution near the maximum eigenvalue.

Established convergence of fluctuations in the spectral distribution.

Extended results to beta ensembles with convex polynomial potentials.

Abstract

For beta ensembles with convex poynomial potentials, we prove a large deviation principle for the empirical spectral distribution seen from the rightmost particle. This modified spectral distribution was introduced by Perret and Schehr (J. Stat. Phys. 2014) to study the crowding near the maximal eigenvalue, in the case of the GUE. We prove also convergence of fluctuations.