Nonstandard limit theorems and large deviation for beta -Jacobi ensembles with a different scaling

TL;DR

This paper studies the spectral behavior of beta-Jacobi ensembles under new scaling conditions, establishing weak convergence, a central limit theorem, and large deviation principles for the spectral measure.

Contribution

It extends previous results by relaxing the scaling condition and provides new limit theorems for the spectral measure of beta-Jacobi ensembles.

Findings

Weak convergence to a modified Watcher law

Central limit theorem for spectral measure

Large deviation principles established

Abstract

We consider $\beta$-Jacobi ensembles with parameters $p_1, p_2\geq n.$ We prove that the empirical measure of the rescaled Jacobi ensembles converges weakly to a modified Watcher law via the spectral measure method, which revisits the weak limits obtained in \cite{MaLDPJ} while replacing the condition $\beta n\!>\!> \log n$ by $\beta n\!>\!>1.$ We also provide the central limit theorem and the large deviation for the corresponding rescaled spectral measure.