{
  "id": "2001.02084",
  "version": "v1",
  "published": "2020-01-07T14:56:35.000Z",
  "updated": "2020-01-07T14:56:35.000Z",
  "title": "Asymptotic counts for the walk multiples of self-avoiding polygons on lattices using sieves",
  "authors": [
    "P. -L. Giscard"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We use combinatorial sieves to prove exact, explicit and compact formulas for the fraction of all closed walks on any finite or infinite vertex-transitive graph whose last erased loop is any chosen self-avoiding polygon (SAP). In stark contrast with approaches based on probability theory, we proceed via purely deterministic arguments relying on Viennot's theory of heaps of pieces seen as a semi-commutative extension of number theory. This approach sheds light on the origin of the discrepancies between exponents stemming from loop-erased walk and self-avoiding polygon models, and suggests a natural route to bridge the gap between both. Our results are illustrated by calculations on the infinite square lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-07T14:56:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C38",
      "05A15",
      "11N36"
    ],
    "keywords": [
      "walk multiples",
      "asymptotic counts",
      "approach sheds light",
      "infinite square lattice",
      "number theory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}