{
  "id": "2212.01702",
  "version": "v1",
  "published": "2022-12-03T22:31:16.000Z",
  "updated": "2022-12-03T22:31:16.000Z",
  "title": "Asymptotics for the Number of Random Walks in the Euclidean Lattice",
  "authors": [
    "Dorin Dumitraşcu",
    "Liviu Suciu"
  ],
  "comment": "26 pages, 6 figures",
  "categories": [
    "math.CO",
    "math.PR"
  ],
  "abstract": "We give precise asymptotics to the number of random walks in the standard orthogonal lattice in $\\mathbb{R}^d$ that return to the starting point at step $2n$, both for all such walks and for the ones that return for the first time. The first set of asymptotics is obtained in an elementary way, by using a combinatorial and geometric multiplication principle together with the classical theory of Legendre polynomials. As an easy consequence we obtain a unified proof of P\\'{o}lya's theorem. By showing that the relevant generating functions are $\\Delta$-analytic, we use the deeper Tauberian theory of singularity analysis to obtain the asymptotics for the first return paths.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-03T22:31:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A16",
      "05A10",
      "40E99",
      "33C45"
    ],
    "keywords": [
      "random walks",
      "euclidean lattice",
      "first return paths",
      "deeper tauberian theory",
      "geometric multiplication principle"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}