Next-order asymptotics for the volume of Schatten balls
Mathias Sonnleitner
TL;DR
The paper addresses the problem of determining precise asymptotics for the volumes of unit Schatten-$p$ balls in self-adjoint matrix spaces, known exactly only for $p=2$ and $p=∞$. It develops a framework linking these volumes to the partition function $Z_{n,p,β}$ of a one-dimensional $\beta$-ensemble with potential $V(x)=v_p|x|^p$, and identifies the Ullman distribution $μ_p$ as the equilibrium measure governing the large-$n$ behavior via a large-deviation principle. By verifying regularity conditions and applying LS17, the authors obtain an explicit asymptotic expansion of $\ln \mathrm{vol}(\mathbb{B}_{p,β}^n)$ up to $o(n)$ for $p\ge 3/2$ (and an $O(1)$ refinement for the complex case), with independent concurrent results for certain parameter ranges. The work deepens the connection between high-dimensional convex geometry, random matrix theory, and potential theory, providing refined formulas that unify real, complex, and quaternionic Schatten balls and enabling further applications to related geometric and probabilistic problems.
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 $p\ge\frac{3}{2}$. The proof rests on asymptotics for the partition function of $β$-ensembles due to Leblé 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 $p\ge 1$.