{
  "id": "1808.09032",
  "version": "v1",
  "published": "2018-08-27T20:43:29.000Z",
  "updated": "2018-08-27T20:43:29.000Z",
  "title": "An upper bound on the number of self-avoiding polygons via joining",
  "authors": [
    "Alan Hammond"
  ],
  "comment": "32 pages and four figures. This article corresponds to Part II of arXiv:1504.05286, a submission giving a unified treatment of three articles",
  "journal": "Ann. Probab. 46 (2018), no. 1, 175-206",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "For $d \\geq 2$ and $n \\in \\mathbb{N}$ even, let $p_n = p_n(d)$ denote the number of length $n$ self-avoiding polygons in $\\mathbb{Z}^d$ up to translation. The polygon cardinality grows exponentially, and the growth rate $\\lim_{n \\in 2\\mathbb{N}} p_n^{1/n} \\in (0,\\infty)$ is called the connective constant and denoted by $\\mu$. Madras [J. Statist. Phys. 78 (1995) no. 3--4, 681--699] has shown that $p_n \\mu^{-n} \\leq C n^{-1/2}$ in dimension $d=2$. Here we establish that $p_n \\mu^{-n} \\leq n^{-3/2 + o(1)}$ for a set of even $n$ of full density when $d=2$. We also consider a certain variant of self-avoiding walk and argue that, when $d \\geq 3$, an upper bound of $n^{-2 + d^{-1} + o(1)}$ holds on a full density set for the counterpart in this variant model of this normalized polygon cardinality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-27T20:43:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper bound",
      "self-avoiding polygons",
      "polygon cardinality grows",
      "full density set",
      "variant model"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}