Exact asymptotic volume and volume ratio of Schatten unit balls

Abstract

The unit ball $B_p^n(\mathbb{R})$ of the finite-dimensional Schatten trace class $\mathcal S_p^n$ consists of all real $n\times n$ matrices $A$ whose singular values $s_1(A),\ldots,s_n(A)$ satisfy $s_1^p(A)+\ldots+s_n^p(A)\leq 1$, where $p>0$. Saint Raymond [Studia Math.\ 80, 63--75, 1984] showed that the limit $$ \lim_{n\to\infty} n^{1/2 + 1/p} \big(\text{Vol}\, B_p^n(\mathbb{R})\big)^{1/n^2} $$ exists in $(0,\infty)$ 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 $n\to\infty$, thereby extending Saint Raymond's estimate in the case of the nuclear norm ($p=1$) to the full regime $1\leq p \leq \infty$ with exact limiting behavior.