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Asymptotic theory of Schatten classes

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arxiv 2404.07145 v1 pith:MXXNJZUL submitted 2024-04-10 math.FA math.OAmath.PR

Michael Juhos , Zakhar Kabluchko , Joscha Prochno This is my paper · ORCID
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keywords ballclassesinftyschattenschatten-asymptoticgeometricmatrices
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abstract

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-$\infty$ unit ball, the boundedness of its isotropy constant, a Poincar\'e-Maxwell-Borel lemma for the uniform distribution on the Schatten-$\infty$ ball, and Sanov-type large deviations principles for the singular values of matrices sampled uniformly from the Schatten-$p$ unit ball for any $0 < p \leq \infty$.

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  1. Limit Theorems for the Volume of Random Projections and Sections of $\ell_p^N$-balls

    math.PR 2024-12 conditional novelty 7.0 of 10

    For fixed m, the rescaled volume of a random m-dimensional projection or section of an ell_p ball satisfies a CLT, an MDP, and an LDP with explicit limits as N grows.

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