Traces of powers of random matrices over local fields
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909281457012736 |
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| author | Pirani, Noam |
| author_facet | Pirani, Noam |
| contents | Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group $U_n$, the unitary symplectic group $USp_{2n}$ or the orthogonal group $O_n$. Diaconis and Shashahani proved that the traces $\mathrm{tr}(M),\mathrm{tr}(M^2),\ldots,\mathrm{tr}(M^k)$ converge in distribution to independent normal random variables as $k$ is fixed and $n\to\infty$. Recently, Gorodetsky and Rodgers proved analogs for these results for matrices chosen from certain finite matrix groups. For example, let $M$ be chosen uniformly at random from $U_n(\mathbb{F}_q)$. They show that $\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^{k}$ converge in distribution to independent uniform random variables in $\mathbb{F}_{q^2}$ as $k$ is fixed and $n\to\infty$.
We prove analogs for these results over local fields. Let $\mathcal{F}$ be a local field with a ring of integers $\mathcal{O}$, a uniformizer $π$, and a residue field of odd characteristic. Let $\mathcal{K}/\mathcal{F}$ be an unramified extension of degree $2$ with a ring of integers $\mathcal{R}$. Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group $U_n(\mathcal{O})$, and fix $k$. We prove that the traces of powers $\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^k$ converge to independent uniform random variables on $\mathcal{R}$, as $n\to\infty$. We also consider the case where $k$ may tend to infinity with $n$. We show that for some constant $c$ (coming from the mod $π$ distribution), the total variation distance from independent uniform random variables on $\mathcal{R}$ is $o(1)$ as $n\to\infty$, as long as $k<c\cdot n$. We also consider other matrix groups over local fields and prove similar results for them. Moreover, we consider traces of powers $M^{pi}$ and traces of negative powers, and show that apart from certain necessary modular restrictions, they also equidistribute in the limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_04061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Traces of powers of random matrices over local fields Pirani, Noam Number Theory Probability Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group $U_n$, the unitary symplectic group $USp_{2n}$ or the orthogonal group $O_n$. Diaconis and Shashahani proved that the traces $\mathrm{tr}(M),\mathrm{tr}(M^2),\ldots,\mathrm{tr}(M^k)$ converge in distribution to independent normal random variables as $k$ is fixed and $n\to\infty$. Recently, Gorodetsky and Rodgers proved analogs for these results for matrices chosen from certain finite matrix groups. For example, let $M$ be chosen uniformly at random from $U_n(\mathbb{F}_q)$. They show that $\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^{k}$ converge in distribution to independent uniform random variables in $\mathbb{F}_{q^2}$ as $k$ is fixed and $n\to\infty$. We prove analogs for these results over local fields. Let $\mathcal{F}$ be a local field with a ring of integers $\mathcal{O}$, a uniformizer $π$, and a residue field of odd characteristic. Let $\mathcal{K}/\mathcal{F}$ be an unramified extension of degree $2$ with a ring of integers $\mathcal{R}$. Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group $U_n(\mathcal{O})$, and fix $k$. We prove that the traces of powers $\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^k$ converge to independent uniform random variables on $\mathcal{R}$, as $n\to\infty$. We also consider the case where $k$ may tend to infinity with $n$. We show that for some constant $c$ (coming from the mod $π$ distribution), the total variation distance from independent uniform random variables on $\mathcal{R}$ is $o(1)$ as $n\to\infty$, as long as $k<c\cdot n$. We also consider other matrix groups over local fields and prove similar results for them. Moreover, we consider traces of powers $M^{pi}$ and traces of negative powers, and show that apart from certain necessary modular restrictions, they also equidistribute in the limit. |
| title | Traces of powers of random matrices over local fields |
| topic | Number Theory Probability |
| url | https://arxiv.org/abs/2408.04061 |