{
  "id": "1904.11149",
  "version": "v1",
  "published": "2019-04-25T04:01:19.000Z",
  "updated": "2019-04-25T04:01:19.000Z",
  "title": "Self-avoiding walk on the complete graph",
  "authors": [
    "Gordon Slade"
  ],
  "comment": "10 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "There is an extensive literature concerning self-avoiding walk on infinite graphs, but the subject is relatively undeveloped on finite graphs. The purpose of this paper is to elucidate the phase transition for self-avoiding walk on the simplest finite graph: the complete graph. We make the elementary observation that the susceptibility of the self-avoiding walk on the complete graph is given exactly in terms of the incomplete gamma function. The known asymptotic behaviour of the incomplete gamma function then yields a complete description of the finite-size scaling of the self-avoiding walk on the complete graph. As a basic example, we compute the limiting distribution of the length of a self-avoiding walk on the complete graph, in subcritical, critical, and supercritical regimes. This provides a prototype for more complex unsolved problems such as the self-avoiding walk on the hypercube or on a high-dimensional torus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-25T04:01:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B41",
      "33B20",
      "82B27",
      "60K35"
    ],
    "keywords": [
      "complete graph",
      "incomplete gamma function",
      "simplest finite graph",
      "elementary observation",
      "infinite graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}