Large deviation eigenvalue density for the soft edge Laguerre and Jacobi $\beta$-ensembles

TL;DR

This paper derives the large deviation eigenvalue density for Laguerre and Jacobi beta-ensembles with extensive exponents, providing asymptotic expansions and tail behaviors of the largest eigenvalue distribution.

Contribution

It offers the first detailed asymptotic expansion for the eigenvalue density in the large deviation regime for these ensembles with extensive exponents.

Findings

Asymptotic expansion of eigenvalue density up to o(1) terms.

Scaling relations to the tail distribution of the largest eigenvalue.

Extension of large deviation results to Laguerre and Jacobi beta-ensembles.

Abstract

We analyze the eigenvalue density for the Laguerre and Jacobi $\beta$-ensembles in the cases that the corresponding exponents are extensive. In particular, we obtain the asymptotic expansion up to terms $o(1)$, in the large deviation regime outside the limiting interval of support. As found in recent studies of the large deviation density for the Gaussian $\beta$-ensemble, and Laguerre $\beta$-ensemble with fixed exponent, there is a scaling from this asymptotic expansion to the right tail asymptotics for the distribution of the largest eigenvalue at the soft edge.