{
  "id": "1003.3722",
  "version": "v1",
  "published": "2010-03-19T06:32:45.000Z",
  "updated": "2010-03-19T06:32:45.000Z",
  "title": "Stochastic domination for the Ising and fuzzy Potts models",
  "authors": [
    "Marcus Warfheimer"
  ],
  "comment": "22 pages, 5 figures",
  "journal": "Electronic Journal of Probability 2010, 1802-1824",
  "categories": [
    "math.PR"
  ],
  "abstract": "We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\\Td$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\\in\\RR$, we compute the smallest external field $\\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\\geq\\tilde{h}$. Moreover, we discuss continuity of $\\tilde{h}$ with respect to the three parameters $J_1$, $J_2$, $h$ and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on $\\Zd$ the fuzzy Potts measures dominate the same set of product measures while on $\\Td$, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures. For the Ising model, Liggett and Steif proved that on $\\Zd$ the plus measures dominate the same set of product measures while on $\\T^2$ that statement fails completely except when there is a unique phase.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-03-19T06:32:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "fuzzy potts model",
      "stochastic domination",
      "plus measure",
      "external field",
      "product measures"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1003.3722W"
    }
  }
}