{
  "id": "2401.04814",
  "version": "v1",
  "published": "2024-01-09T20:48:08.000Z",
  "updated": "2024-01-09T20:48:08.000Z",
  "title": "Random walks and the \"Euclidean\" association scheme in finite vector spaces",
  "authors": [
    "Charles Brittenham",
    "Jonathan Pakianathan"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "In this paper, we study an association scheme derived from the \"Euclidean\" quadratic form on finite vector spaces over finite fields of odd prime order $q$. We calculate the change of basis matrices $P$, $Q$ between the geometric and spectral basis of the corresponding Bose-Messner algebra as well as the intersection numbers of the scheme. As an application we study the random distance-$t$ walk in finite planes and derive asymptotic formulas (as $q \\to \\infty$) for the probability of return to start point after $\\ell$ steps based on the \"vertical\" equidistribution of Kloosterman sums established by N. Katz.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-09T20:48:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C90",
      "11L05",
      "11T23"
    ],
    "keywords": [
      "finite vector spaces",
      "association scheme",
      "random walks",
      "odd prime order",
      "basis matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}