{
  "id": "1908.03872",
  "version": "v1",
  "published": "2019-08-11T08:42:31.000Z",
  "updated": "2019-08-11T08:42:31.000Z",
  "title": "New scaling laws for self-avoiding walks: bridges and worms",
  "authors": [
    "Bertrand Duplantier",
    "Anthony J Guttmann"
  ],
  "comment": "10 pages, 3 figures Dedicated to the memory of Vladimir Rittenberg",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We show how the theory of the critical behaviour of $d$-dimensional polymer networks gives a scaling relation for self-avoiding {\\em bridges} that relates the critical exponent for bridges $\\gamma_b$ to that of terminally-attached self-avoiding arches, $\\gamma_{1,1},$ and the {correlation} length exponent $\\nu.$ We find $\\gamma_b = \\gamma_{1,1}+\\nu.$ We provide compelling numerical evidence for this result in both two- and three-dimensions. Another subset of SAWs, called {\\em worms}, are defined as the subset of SAWs whose origin and end-point have the same $x$-coordinate. We give a scaling relation for the corresponding critical exponent $\\gamma_w,$ which is $\\gamma_w=\\gamma-\\nu.$ This too is supported by enumerative results in the two-dimensional case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-11T08:42:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "self-avoiding walks",
      "scaling laws",
      "critical exponent",
      "scaling relation",
      "dimensional polymer networks"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}