{
  "id": "1901.08294",
  "version": "v1",
  "published": "2019-01-24T09:09:09.000Z",
  "updated": "2019-01-24T09:09:09.000Z",
  "title": "Renormalization of crossing probabilities in the planar random-cluster model",
  "authors": [
    "Hugo Duminil-Copin",
    "Vincent Tassion"
  ],
  "comment": "28 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The study of crossing probabilities - i.e. probabilities of existence of paths crossing rectangles - has been at the heart of the theory of two-dimensional percolation since its beginning. They may be used to prove a number of results on the model, including speed of mixing, tails of decay of the connectivity probabilities, scaling relations, etc. In this article, we develop a renormalization scheme for crossing probabilities in the two-dimensional random-cluster model. The outcome of the process is a precise description of an alternative between four behaviors: - Subcritical: Crossing probabilities, even with favorable boundary conditions, converge exponentially fast to 0. - Supercritical: Crossing probabilities, even with unfavorable boundary conditions, converge exponentially fast to 1. - Critical discontinuous: Crossing probabilities converge to 0 exponentially fast with unfavorable boundary conditions and to 1 with favorable boundary conditions. - Critical continuous: Crossing probabilities remain bounded away from 0 and 1 uniformly in the boundary conditions. The approach does not rely on self-duality, enabling it to apply in a much larger generality, including the random-cluster model on arbitrary graphs with sufficient symmetry, but also other models like certain random height models.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-24T09:09:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "crossing probabilities",
      "planar random-cluster model",
      "probabilities remain bounded away",
      "renormalization",
      "unfavorable boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}