Exact asymptotic volume and volume ratio of Schatten unit balls

The unit ball $B_p^n(mathbb{R})$ of the finite-dimensional Schatten trace class $mathcal S_p^n$ consists of all real $ntimes n$ 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 $$ lim_{ntoinfty} n^{1/2 + 1/p} big(text{Vol}, B_p^n(mathbb{R})big)^{1/n^2} $$ 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 $ntoinfty$, thereby extending Saint Raymonds estimate in the case of the nuclear norm ($p=1$) to the full regime $1leq p leq infty$ with exact limiting behavior.