Next-order asymptotics for the volume of Schatten balls
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910050107260928 |
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| author | Sonnleitner, Mathias |
| author_facet | Sonnleitner, Mathias |
| contents | 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 $β$-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 $p\ge 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04933 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Next-order asymptotics for the volume of Schatten balls Sonnleitner, Mathias Functional Analysis Metric Geometry Probability 52A23, 47B10 (Primary) 60B20, 82B21 (Secondary) 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 $β$-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 $p\ge 1$. |
| title | Next-order asymptotics for the volume of Schatten balls |
| topic | Functional Analysis Metric Geometry Probability 52A23, 47B10 (Primary) 60B20, 82B21 (Secondary) |
| url | https://arxiv.org/abs/2512.04933 |