{
  "id": "1705.03104",
  "version": "v1",
  "published": "2017-05-08T22:05:54.000Z",
  "updated": "2017-05-08T22:05:54.000Z",
  "title": "Sharp phase transition for the random-cluster and Potts models via decision trees",
  "authors": [
    "Hugo Duminil-Copin",
    "Aran Raoufi",
    "Vincent Tassion"
  ],
  "comment": "16 pages, 3 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\\beta<\\beta_c$. 2. For the random-cluster model with cluster weight $q\\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 3. For the random-cluster models with cluster weight $q\\geq1$ on planar quasi-transitive graphs $\\mathbb{G}$, $$\\frac{p_c(\\mathbb{G})p_c(\\mathbb{G}^*)}{(1-p_c(\\mathbb{G}))(1-p_c(\\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-08T22:05:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "potts model",
      "sharp phase transition",
      "decision trees",
      "correlations decay exponentially fast",
      "random-cluster model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}