Sanov-type large deviations in Schatten classes

Denote by lambda(1) (A),...,lambda(n)(A) the eigenvalues of an (n x n)-matrix A. Let Z(n) be an (n x n)-matrix chosen uniformly at random from the matrix analogue to the classical l(p)(n)-ball, defined as the set of all self-adjoint (n x n)-matrices satisfying Sigma(n)(k=1) vertical bar lambda(k)(A)vertical bar(p) <= 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 -> infinity. The corresponding results for random matrices in Schatten trace classes, where eigenvalues are replaced by the singular values, are also presented.