Exact asymptotic volume and volume ratio of Schatten unit balls release_7ncecl64mrecrlzgeiyqb7vl4e

by Zakhar Kabluchko, Joscha Prochno, Christoph Thaele

Released as a article .



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.
In text/plain format

Archived Files and Locations

application/pdf  206.0 kB
web.archive.org (webarchive)
arxiv.org (repository)
Read Archived PDF
Type  article
Stage   submitted
Date   2018-04-10
Version   v1
Language   en ?
arXiv  1804.03467v1
Work Entity
access all versions, variants, and formats of this works (eg, pre-prints)
Catalog Record
Revision: 925546ab-d09e-4b26-97e0-297efe28badc