On the large deviations of traces of random matrices

TL;DR

This paper establishes large deviations principles for the spectral measures of Wigner matrices and beta-ensembles across different tail behaviors and potential functions, extending understanding of spectral measure fluctuations.

Contribution

It provides new large deviations results for spectral measures of Wigner matrices with various tail behaviors and for beta-ensembles with polynomial growth potentials.

Findings

Large deviations principles for Wigner matrices with sub-Gaussian tails.

Large deviations results for Gaussian Wigner matrices.

Large deviations for beta-ensembles with polynomial growth potentials.

Abstract

We present large deviations principles for the moments of the empirical spectral measure of Wigner matrices and empirical measure of $\beta$-ensembles in three cases : the case of Wigner matrices without Gaussian tails, that is Wigner matrices whose entries have tail distributions decreasing as $e^{-ct^{\alpha}}$, for some constant $c>0$ and with $\alpha \in (0,2)$, the case of Gaussian Wigner matrices, and the case of $\beta$-ensembles associated with a convex potential with polynomial growth.