{
  "id": "1806.09717",
  "version": "v1",
  "published": "2018-06-25T22:27:40.000Z",
  "updated": "2018-06-25T22:27:40.000Z",
  "title": "Bounds on multiple self-avoiding polygons",
  "authors": [
    "Kyungpyo Hong",
    "Seungsang Oh"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A self-avoiding polygon is a lattice polygon consisting of a closed self-avoiding walk on a square lattice. Surprisingly little is known rigorously about the enumeration of self-avoiding polygons, although there are numerous conjectures that are believed to be true and strongly supported by numerical simulations. As an analogous problem of this study, we consider multiple self-avoiding polygons in a confined region, as a model for multiple ring polymers in physics. We find rigorous lower and upper bounds of the number $p_{m \\times n}$ of distinct multiple self-avoiding polygons in the $m \\times n$ rectangular grid on the square lattice. For $m=2$, $p_{2 \\times n} = 2^{n-1}-1$. And, for integers $m,n \\geq 3$, $$2^{m+n-3} \\left(\\frac{17}{10}\\right)^{(m-2)(n-2)} \\ \\leq \\ p_{m \\times n} \\ \\leq \\ 2^{m+n-3} \\left(\\frac{31}{16}\\right)^{(m-2)(n-2)}.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-06-25T22:27:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57M25",
      "82B20",
      "82B41",
      "82D60"
    ],
    "keywords": [
      "square lattice",
      "distinct multiple self-avoiding polygons",
      "rectangular grid",
      "multiple ring polymers",
      "upper bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}