{
  "id": "math/0401072",
  "version": "v1",
  "published": "2004-01-08T10:56:23.000Z",
  "updated": "2004-01-08T10:56:23.000Z",
  "title": "Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms",
  "authors": [
    "Remco van der Hofstad",
    "Gordon Slade"
  ],
  "comment": "18 pages, 3 figures",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $p_c(\\mathbb{Q}_n)$ and $p_c(\\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\\mathbb{Q}_n = \\{0,1\\}^n$ and on $\\Z^n$, respectively. Let $\\Omega = n$ for $\\mathbb{G} = \\mathbb{Q}_n$ and $\\Omega = 2n$ for $\\mathbb{G} = \\mathbb{Z}^n$ denote the degree of $\\mathbb{G}$. We use the lace expansion to prove that for both $\\mathbb{G} = \\mathbb{Q}_n$ and $\\mathbb{G} = \\mathbb{Z}^n$, $p_c(\\mathbb{G}) & = \\cn^{-1} + \\cn^{-2} + {7/2} \\cn^{-3} + O(\\cn^{-4}).$ This extends by two terms the result $p_c(\\mathbb{Q}_n) = \\cn^{-1} + O(\\cn^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for $\\mathbb{Z}^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-01-08T10:56:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "60K35",
      "82B43"
    ],
    "keywords": [
      "percolation critical values",
      "nearest-neighbour bond percolation",
      "van der hofstad",
      "lace expansion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......1072V"
    }
  }
}