{
  "id": "1505.04159",
  "version": "v1",
  "published": "2015-05-15T18:57:16.000Z",
  "updated": "2015-05-15T18:57:16.000Z",
  "title": "Continuity of the phase transition for planar random-cluster and Potts models with $1\\le q\\le4$",
  "authors": [
    "Hugo Duminil-Copin",
    "Vladas Sidoravicius",
    "Vincent Tassion"
  ],
  "comment": "66 pages, 15 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "This article studies the planar Potts model and its random-cluster representation. We show that the phase transition of the nearest-neighbor ferromagnetic $q$-state Potts model on $\\mathbb Z^2$ is continuous for $q\\in\\{2,3,4\\}$, in the sense that there exists a unique Gibbs state, or equivalently that there is no ordering for the critical Gibbs states with monochromatic boundary conditions. The proof uses the random-cluster model with cluster-weight $q\\ge1$ (note that $q$ is not necessarily an integer) and is based on two ingredients: 1. The fact that the two-point function for the free state decays sub-exponentially fast for cluster-weights $1\\le q\\le 4$, which is derived studying parafermionic observables on a discrete Riemann surface. 2. A new result proving the equivalence of several properties of critical random-cluster models: - the absence of infinite-cluster for wired boundary conditions, - the uniqueness of infinite-volume measures, - the sub-exponential decay of the two-point function for free boundary conditions, - a Russo-Seymour-Welsh type result on crossing probabilities in rectangles with arbitrary boundary conditions. The result leads to a number of consequences concerning the scaling limit of the random-cluster model with $1\\le q \\le 4$. It shows that the family of interfaces (for instance for Dobrushin boundary conditions) are tight when taking the scaling limit and that any sub-sequential limit can be parametrized by a Loewner chain. We also study the effect of boundary conditions on these sub-sequential limits. Let us mention that the result should be instrumental in the study of critical exponents as well.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-15T18:57:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B27"
    ],
    "keywords": [
      "boundary conditions",
      "potts model",
      "phase transition",
      "planar random-cluster",
      "random-cluster model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 66,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150504159D"
    }
  }
}