Moderate deviations for spectral measures of random matrix ensembles

TL;DR

This paper establishes a moderate deviation principle for the spectral measures of classical random matrix ensembles, linking eigenvalue distributions to orthogonal polynomial representations and providing a rate function based on density norms.

Contribution

It introduces a moderate deviation principle for spectral measures of Gaussian, Laguerre, and Jacobi ensembles using tridiagonal models and orthogonal polynomial techniques.

Findings

Moderate deviation principle proven for spectral measures.

Rate function expressed via $L^2$-norm of density.

Uses tridiagonal models to connect eigenvalues and orthogonal polynomials.

Abstract

In this paper we consider the (weighted) spectral measure $\mu_n$ of a $n\times n$ random matrix, distributed according to a classical Gaussian, Laguerre or Jacobi ensemble, and show a moderate deviation principle for the standardised signed measure $\sqrt{n/a_n}(\mu_n -\sigma)$. The centering measure $\sigma$ is the weak limit of the empirical eigenvalue distribution and the rate function is given in terms of the $L^2$-norm of the density with respect to $\sigma$. The proof involves the tridiagonal representations of the ensembles which provide us with a sequence of independent random variables and a link to orthogonal polynomials.