{
  "id": "2408.04061",
  "version": "v1",
  "published": "2024-08-07T19:52:43.000Z",
  "updated": "2024-08-07T19:52:43.000Z",
  "title": "Traces of powers of random matrices over local fields",
  "authors": [
    "Noam Pirani"
  ],
  "comment": "72 pages, comments are welocme!",
  "categories": [
    "math.NT",
    "math.PR"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-07T19:52:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local field",
      "independent uniform random variables",
      "random matrices",
      "independent normal random variables",
      "matrix groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 72,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}