{
  "id": "1611.09877",
  "version": "v1",
  "published": "2016-11-29T21:09:21.000Z",
  "updated": "2016-11-29T21:09:21.000Z",
  "title": "Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$",
  "authors": [
    "Hugo Duminil-Copin",
    "Maxime Gagnebin",
    "Matan Harel",
    "Ioan Manolescu",
    "Vincent Tassion"
  ],
  "comment": "43 pages, 10 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show - Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, - Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and - Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\\exp(\\pi^2/\\sqrt{q-4})$ as $q$ tends to 4.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-11-29T21:09:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B20",
      "82B23",
      "82B26"
    ],
    "keywords": [
      "random-cluster model",
      "planar random-cluster",
      "critical potts",
      "discontinuity",
      "correlation length"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}