{
  "id": "1602.05598",
  "version": "v1",
  "published": "2016-02-17T21:16:32.000Z",
  "updated": "2016-02-17T21:16:32.000Z",
  "title": "Isoperimetry in supercritical bond percolation in dimensions three and higher",
  "authors": [
    "Julian Gold"
  ],
  "comment": "65 pages, comments welcome",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the isoperimetric subgraphs of the infinite cluster $\\textbf{C}_\\infty$ for supercritical bond percolation on $\\mathbb{Z}^d$ with $d\\geq 3$. Specifically, we consider the subgraphs of $\\textbf{C}_\\infty \\cap [-n,n]^d$ which have minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs, obtaining that when suitably rescaled, these subgraphs converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for $\\textbf{C}_\\infty \\cap [-n,n]^d$. This settles a conjecture of Benjamini for the version of the Cheeger constant defined here.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-17T21:16:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "52B60"
    ],
    "keywords": [
      "supercritical bond percolation",
      "dimensions",
      "isoperimetry",
      "cheeger constant",
      "deterministic shape"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 65,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}