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arXiv · 2512.04933

Next-order asymptotics for the volume of Schatten balls

Mathias Sonnleitner
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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 $p\ge 1$.

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Mathias Sonnleitner. 2025-12-04. Next-order asymptotics for the volume of Schatten balls. https://arxiv.org/abs/2512.04933

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