{
  "id": "1404.4071",
  "version": "v2",
  "published": "2014-04-15T20:36:10.000Z",
  "updated": "2015-01-09T14:46:27.000Z",
  "title": "Phase transition for the dilute clock model",
  "authors": [
    "Inés Armendáriz",
    "Pablo Augusto Ferrari",
    "Nahuel Soprano-Loto"
  ],
  "comment": "14 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We prove that phase transition occurs in the dilute ferromagnetic nearest-neighbour $q$-state clock model in $\\mathbb{Z}^d$, for every $q\\geq 2$ and $d\\geq 2$. This follows from the fact that the Edwards-Sokal random-cluster representation of the clock model stochastically dominates a supercritical Bernoulli bond percolation probability, a technique that has been applied to show phase transition for the low-temperature Potts model. The domination involves a combinatorial lemma which is one of the main points of this article.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-04-15T20:36:10.000Z",
      "title": "Phase transition for the clock model via random-cluster percolation",
      "abstract": "Let $\\phi(\\beta,q,d)$ be the random-cluster probability associated to the $q$-state clock model at inverse temperature $\\beta$ in dimension $d\\ge 2$. We find $\\beta_0(q,d)$ such that for $\\beta>\\beta_0$ the random cluster measure $\\phi(\\beta,q,d)$ stochastically dominates a supercritical Bernoulli bond percolation measure on $\\mathbb{Z}^d$. This provides an upper bound for the critical inverse temperature $\\beta_c$ for the clock model, for every value of $q$ and every dimension.",
      "comment": "13 pages, 1 figure",
      "journal": null,
      "doi": null,
      "authors": [
        "Inés Armendáriz",
        "Pablo Augusto Ferrari",
        "Nahuel Soprano Loto"
      ]
    },
    {
      "version": "v2",
      "updated": "2015-01-09T14:46:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random-cluster percolation",
      "phase transition",
      "inverse temperature",
      "supercritical bernoulli bond percolation measure",
      "random cluster measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1404.4071A"
    }
  }
}