Sanov-type large deviations in Schatten classes

Abstract

Denote by ${\lambda }_1(A),\dots ,{\lambda }_n(A)$ 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 ${\sum }_{k=1}^n|{\lambda }_k(A){|}^p\le 1$. 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.

31 pages, 4 figures