Exact asymptotic volume and volume ratio of Schatten unit balls

Abstract

The unit ball Bpn(R) of the finite-dimensional Schatten trace class Spn consists of all real n×n matrices A whose singular values s1(A),…,sn(A) satisfy s1p(A)+…+snp(A)≤1, where p>0. Saint Raymond (1984) showed that the limit limn→∞n1∕2+1∕p(VolBpn(R))1∕n2 exists in (0,∞) and provided both lower and upper bounds. In this manuscript we use the theory of logarithmic potentials in external fields to determine the precise limiting constant and thus the exact asymptotic volume of Bpn(R). The corresponding result for complex Schatten balls is also obtained. 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.