{
  "id": "1604.01299",
  "version": "v1",
  "published": "2016-04-05T15:28:57.000Z",
  "updated": "2016-04-05T15:28:57.000Z",
  "title": "The phase transitions of the random-cluster and Potts models on slabs with $q \\geq 1$ are sharp",
  "authors": [
    "Ioan Manolescu",
    "Aran Raoufi"
  ],
  "comment": "24 pages, 6 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove sharpness of the phase transition for the random-cluster model with $q \\geq 1$ on graphs of the form $\\mathcal{S} := \\mathcal{G} \\times S$, where $\\mathcal{G}$ is a planar lattice with mild symmetry assumptions, and $S$ a finite graph. That is, for any such graph and any $q \\geq 1$, there exists some parameter $p_c = p_c(\\mathcal{S}, q)$, below which the model exhibits exponential decay and above which there exists a.s. an infinite cluster. The result is also valid for the random-cluster model on planar graphs with long range, compactly supported interaction. It extends to the Potts model via the Edwards-Sokal coupling.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-04-05T15:28:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "potts model",
      "phase transition",
      "random-cluster model",
      "mild symmetry assumptions",
      "planar lattice"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160401299M"
    }
  }
}