Spectral density asymptotics for Gaussian and Laguerre $\beta$-ensembles in the exponentially small region

TL;DR

This paper derives asymptotic expansions for spectral densities and the largest eigenvalue distribution tails in Gaussian and Laguerre beta-ensembles, especially in the exponentially small region outside the main spectral support.

Contribution

It provides the first two terms of the large N asymptotic expansion of the beta moment of the characteristic polynomial for these ensembles and connects this to the tail behavior of the largest eigenvalue.

Findings

Asymptotic expansion of spectral density outside the main support.

Leading form of the right tail of the largest eigenvalue distribution.

Scaling relation between density asymptotics and tail distribution.

Abstract

The first two terms in the large $N$ asymptotic expansion of the $\beta$ moment of the characteristic polynomial for the Gaussian and Laguerre $\beta$-ensembles are calculated. This is used to compute the asymptotic expansion of the spectral density in these ensembles, in the exponentially small region outside the leading support, up to terms $o(1)$ . The leading form of the right tail of the distribution of the largest eigenvalue is given by the density in this regime. It is demonstrated that there is a scaling from this, to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.