Eigenvalue Distribution of $p$-adic Random Matrices Among Algebraic Extensions, with an Analogue for $p$-adic Random Polynomials
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913147225374720 |
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| author | Shen, Jiahe |
| author_facet | Shen, Jiahe |
| contents | We study the distribution of eigenvalues of Haar-random matrices over $\mathbb{Z}_p$ among algebraic extensions of $\mathbb{Q}_p$. Our results give $p$-adic analogues of the real-eigenvalue counting results of Edelman-Kostlan-Shub for the real Ginibre ensemble, but with a different degree behavior: while real eigenvalues form only a vanishing proportion in the real Ginibre ensemble, $p$-adic eigenvalues are asymptotically evenly distributed among possible extension degrees. We also show that the maximal unramified extension $\mathbb{Q}_p^{\mathrm{un}}$ captures all but a bounded expected number of eigenvalues, and that the expected number of eigenvalues outside $\mathbb{Q}_p^{\mathrm{un}}$ has a finite positive limit with an explicit upper bound.
The proof uses correlation function formulas from the author's previous joint work with Van Peski (arXiv:2601.06283), together with uniform estimates over varying finite extensions. We also prove analogous results for roots of random Haar polynomials over $\mathbb{Z}_p$, using the correlation function formulas of Caruso (arXiv:2110.03942). These polynomial results are $p$-adic analogues of the real-root counting results of Edelman-Kostlan, again with behavior different from the real setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20444 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Eigenvalue Distribution of $p$-adic Random Matrices Among Algebraic Extensions, with an Analogue for $p$-adic Random Polynomials Shen, Jiahe Number Theory Probability We study the distribution of eigenvalues of Haar-random matrices over $\mathbb{Z}_p$ among algebraic extensions of $\mathbb{Q}_p$. Our results give $p$-adic analogues of the real-eigenvalue counting results of Edelman-Kostlan-Shub for the real Ginibre ensemble, but with a different degree behavior: while real eigenvalues form only a vanishing proportion in the real Ginibre ensemble, $p$-adic eigenvalues are asymptotically evenly distributed among possible extension degrees. We also show that the maximal unramified extension $\mathbb{Q}_p^{\mathrm{un}}$ captures all but a bounded expected number of eigenvalues, and that the expected number of eigenvalues outside $\mathbb{Q}_p^{\mathrm{un}}$ has a finite positive limit with an explicit upper bound. The proof uses correlation function formulas from the author's previous joint work with Van Peski (arXiv:2601.06283), together with uniform estimates over varying finite extensions. We also prove analogous results for roots of random Haar polynomials over $\mathbb{Z}_p$, using the correlation function formulas of Caruso (arXiv:2110.03942). These polynomial results are $p$-adic analogues of the real-root counting results of Edelman-Kostlan, again with behavior different from the real setting. |
| title | Eigenvalue Distribution of $p$-adic Random Matrices Among Algebraic Extensions, with an Analogue for $p$-adic Random Polynomials |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2605.20444 |