{
  "id": "2412.19762",
  "version": "v1",
  "published": "2024-12-27T17:42:34.000Z",
  "updated": "2024-12-27T17:42:34.000Z",
  "title": "Can one hear the shape of a random walk?",
  "authors": [
    "Michael J. Larsen"
  ],
  "comment": "16 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "To what extent is the underlying distribution of a finitely supported unbiased random walk on $\\mathbb{Z}$ determined by the sequence of times at which the walk returns to the origin? The main result of this paper is that, in various senses, most unbiased random walks on $\\mathbb{Z}$ are determined up to equivalence by the sequence $I_1,I_2,I_3,\\ldots$, where $I_n$ denotes the probability of being at the origin after $n$ steps. The proof depends on the classification of finite simple groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-12-27T17:42:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "58J53"
    ],
    "keywords": [
      "finite simple groups",
      "finitely supported unbiased random walk",
      "main result",
      "walk returns",
      "equivalence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}