Nonstandard limit theorems and large deviations for the Jacobi beta ensemble

TL;DR

This paper investigates the asymptotic behavior of eigenvalue distributions in the Jacobi beta ensemble under nonstandard scaling, revealing convergence to classical laws and establishing large deviation principles.

Contribution

It demonstrates weak convergence of eigenvalue distributions under new scaling regimes and derives large deviation principles with classical ensemble rate functions.

Findings

Eigenvalue distribution converges to Marchenko-Pastur law when parameters grow faster than dimension.

Eigenvalue distribution converges to semicircle law under alternative scaling.

Large deviation principles are established with rate functions matching classical ensembles.

Abstract

In this paper we show weak convergence of the empirical eigenvalue distribution and of the weighted spectral measure of the Jacobi ensemble, when one or both parameters grow faster than the dimension $n$. In these cases the limit measure is given by the Marchenko-Pastur law and the semicircle law, respectively. For the weighted spectral measure we also prove large deviation principles under this scaling, where the rate functions are those of the other classical ensembles.