On The Eigenvalue Rigidity of the Jacobi Unitary Ensemble

TL;DR

This paper establishes the optimal global eigenvalue rigidity for the Jacobi unitary ensemble by linking eigenvalue counting measures to Gaussian multiplicative chaos, advancing understanding of spectral behavior.

Contribution

It introduces a novel approach connecting eigenvalue measures to Gaussian chaos and proves an optimal rigidity estimate for the Jacobi ensemble.

Findings

Eigenvalue counting measures converge to Gaussian multiplicative chaos.

Established optimal global eigenvalue rigidity.

Applied asymptotic analysis of exponential moments.

Abstract

In this paper, we prove an optimal global rigidity estimate for the eigenvalues of the Jacobi unitary ensemble. Our approach begins by constructing a random measure defined through the eigenvalue counting function. We then prove its convergence to a Gaussian multiplicative chaos measure, which leads to the desired rigidity result. To establish this convergence, we apply a sufficient condition from Claeys et al. \cite{CFL2021} and conduct an asymptotic analysis of the related exponential moments.