Exact asymptotic volume and volume ratio of Schatten unit balls release_7ncecl64mrecrlzgeiyqb7vl4e

by Zakhar Kabluchko, Joscha Prochno, Christoph Thaele

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(2018)

Abstract

The unit ball B_p^n(R) of the finite-dimensional Schatten trace class S_p^n consists of all real n× n matrices A whose singular values s_1(A),...,s_n(A) satisfy s_1^p(A)+...+s_n^p(A)≤ 1, where p>0. Saint Raymond [Studia Math. 80, 63--75, 1984] showed that the limit _n→∞ n^1/2 + 1/p(Vol B_p^n(R))^1/n^2 exists in (0,∞) and provided both lower and upper bounds. In this paper we determine the precise limiting constant based on ideas from the theory of logarithmic potentials with external fields. A similar result is obtained for complex Schatten balls. As an application we compute the precise asymptotic volume ratio of the Schatten p-balls, as n→∞, thereby extending Saint Raymond's estimate in the case of the nuclear norm (p=1) to the full regime 1≤ p ≤∞ with exact limiting behavior.
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Date   2018-04-10
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arXiv  1804.03467v1
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