Asymptotic theory of Schatten classes

The study of Schatten classes has a long tradition in geometric functional analysis and related fields. In this paper we study a variety of geometric and probabilistic aspects of finite-dimensional Schatten classes of not necessarily square matrices. Among the main results are the exact and asymptotic volume of the Schatten-$\mathrm{\infty }$ unit ball, the boundedness of its isotropy constant, a Poincaré-Maxwell-Borel lemma for the uniform distribution on the Schatten-$\mathrm{\infty }$ ball, and Sanov-type large deviations principles for the singular values of matrices sampled uniformly from the Schatten-$p$ unit ball for any $0<p\le \mathrm{\infty }$.