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.
Categories: math.NT math.PR