Exact asymptotic volume and volume ratioof Schatten unit balls.

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 (1984) showed that the limit lim 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 manuscript we use the theory of logarithmic potentials in external fields to determine the precise limiting constant and thus the exact asymptotic volume of B p n (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. [ABSTRACT FROM AUTHOR]