{
  "id": "math/0008191",
  "version": "v2",
  "published": "2000-08-24T22:18:34.000Z",
  "updated": "2001-04-17T20:51:57.000Z",
  "title": "Explicit isoperimetric constants and phase transitions in the random-cluster model",
  "authors": [
    "Olle Haggstrom",
    "Johan Jonasson",
    "Russell Lyons"
  ],
  "journal": "Ann. Probab. 30 (2002), 443--473",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results are obtained for the random-cluster model on nonamenable graphs with cluster parameter $q\\geq 1$. Among these, the main ones are the absence of percolation for the free random-cluster measure at the critical value, and examples of planar regular graphs with regular dual where $\\pc^\\f (q) > \\pu^\\w (q)$ for $q$ large enough. The latter follows from considerations of isoperimetric constants, and we give the first nontrivial explicit calculations of such constants. Such considerations are also used to prove non-robust phase transition for the Potts model on nonamenable regular graphs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2001-04-17T20:51:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20"
    ],
    "keywords": [
      "random-cluster model",
      "explicit isoperimetric constants",
      "first nontrivial explicit calculations",
      "potts model",
      "dependent percolation model"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2000math......8191H"
    }
  }
}