Sanov-type large deviations in Schatten classes

TL;DR

This paper establishes a large deviations principle for the spectral measure of random matrices from Schatten classes, showing convergence to the Ullman distribution and extending results to singular values.

Contribution

It introduces a large deviations framework for spectral measures of matrices in Schatten classes, including convergence results and extensions to singular values.

Findings

Spectral measure of scaled random matrices converges to Ullman distribution.

Large deviations principle is established for spectral measures.

Results extend to singular values in Schatten trace classes.

Abstract

Denote by $\lambda_1(A), \ldots, \lambda_n(A)$ the eigenvalues of an $(n\times n)$-matrix $A$. Let $Z_n$ be an $(n\times n)$-matrix chosen uniformly at random from the matrix analogue to the classical $\ell_ p^n$-ball, defined as the set of all self-adjoint $(n\times n)$-matrices satisfying $\sum_{k=1}^n |\lambda_k(A)|^p\leq 1$. We prove a large deviations principle for the (random) spectral measure of the matrix $n^{1/p} Z_n$. As a consequence, we obtain that the spectral measure of $n^{1/p} Z_n$ converges weakly almost surely to a non-random limiting measure given by the Ullman distribution, as $n\to\infty$. The corresponding results for random matrices in Schatten trace classes, where eigenvalues are replaced by the singular values, are also presented.