Next-order asymptotics for the volume of Schatten balls

TL;DR

This paper derives an asymptotic expansion for the volume of Schatten balls in high dimensions for all p>1, extending known results and using advanced asymptotic analysis of partition functions.

Contribution

It provides the first asymptotic expansion of the logarithmic volume of Schatten balls for general p>1, including complex cases, based on beta-ensemble asymptotics.

Findings

Asymptotic expansion of volume for Schatten balls for all p>1

Extension of known results to complex Schatten classes

Connection to beta-ensemble asymptotics

Abstract

The volume of the unit balls of self-adjoint finite-dimensional Schatten $p$-classes of $n\times n$-matrices, $1\le p\le \infty$, is only known exactly for $p=2$ and $p=\infty$. We give an asymptotic expansion of the logarithmic volume to order $o(n)$ for general $p>1$. The proof rests on asymptotics for the partition function of $\beta$-ensembles due to Lebl\'e and Serfaty [Invent. Math. 210(3):645--757, 2017]. Independently, the case $p\ge 2$ was obtained by Dworaczek Guera, Memin and Pain [arXiv:2511.05386]. In the complex case the asymptotic expansion is continued to order $O(1)$ for all $p\ge 1$.