{
  "id": "2106.04019",
  "version": "v1",
  "published": "2021-06-08T00:10:21.000Z",
  "updated": "2021-06-08T00:10:21.000Z",
  "title": "Random walks on SL_2(C): spectral gap and local limit theorems",
  "authors": [
    "Tien-Cuong Dinh",
    "Lucas Kaufmann",
    "Hao Wu"
  ],
  "comment": "49 pages",
  "categories": [
    "math.PR",
    "math.CV",
    "math.DS"
  ],
  "abstract": "We obtain new limit theorems for random walks on SL_2(C) under low moment conditions. For non-elementary measures with a finite second moment we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of \\'E. Le Page. We also obtain a Local Limit Theorem for the matrix coefficients under a third moment condition, improving a recent result of Grama-Quint-Xiao. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space W^{1,2} that provides uniform estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-08T00:10:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B15",
      "60B20",
      "37C30"
    ],
    "keywords": [
      "local limit theorem",
      "random walks",
      "spectral gap",
      "function space",
      "third moment condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}