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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