Traces of powers of random matrices over local fields

TL;DR

This paper extends known results about the distribution of traces of powers of random matrices from classical compact groups to matrices over local fields, showing convergence to uniform distributions and independence in the limit.

Contribution

It proves that traces of powers of random matrices over local fields converge to independent uniform variables, generalizing previous finite and compact group results.

Findings

Traces of powers converge to independent uniform variables on the ring of integers.

Results hold for fixed and growing number of powers with respect to matrix size.

Similar distributional results are established for other matrix groups over local fields.

Abstract

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 $\pi$, 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 $\pi$ 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.