{
  "id": "1206.3200",
  "version": "v1",
  "published": "2012-06-14T17:55:54.000Z",
  "updated": "2012-06-14T17:55:54.000Z",
  "title": "Bounding the partition function of spin-systems",
  "authors": [
    "David Galvin"
  ],
  "comment": "13 pages. Appeared in Electronic Journal of Combinatorics in 2006",
  "categories": [
    "math.CO",
    "math-ph",
    "math.MP"
  ],
  "abstract": "With a graph $G=(V,E)$ we associate a collection of non-negative real weights $\\cup_{v\\in V}{\\lambda_{i,v}:1\\leq i \\leq m} \\cup \\cup_{uv \\in E} {\\lambda_{ij,uv}:1\\leq i \\leq j \\leq m}$. We consider the probability distribution on ${f:V\\rightarrow{1,...,m}}$ in which each $f$ occurs with probability proportional to $\\prod_{v \\in V}\\lambda_{f(v),v}\\prod_{uv \\in E}\\lambda_{f(u)f(v),uv}$. Many well-known statistical physics models, including the Ising model with an external field and the hard-core model with non-uniform activities, can be framed as such a distribution. We obtain an upper bound, independent of $G$, for the partition function (the normalizing constant which turns the assignment of weights on $\\{f:V\\rightarrow{1,...,m\\}}$ into a probability distribution) in the case when $G$ is a regular bipartite graph. This generalizes a bound obtained by Galvin and Tetali who considered the simpler weight collection $\\{\\lambda_i:1 \\leq i \\leq m\\} \\cup \\{\\lambda_{ij}:1 \\leq i \\leq j \\leq m\\}$ with each $\\lambda_{ij}$ either 0 or 1 and with each $f$ chosen with probability proportional to $\\prod_{v \\in V}\\lambda_{f(v)}\\prod_{uv \\in E}\\lambda_{f(u)f(v)}$. Our main tools are a generalization to list homomorphisms of a result of Galvin and Tetali on graph homomorphisms and a straightforward second-moment computation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-14T17:55:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C15",
      "82B20"
    ],
    "keywords": [
      "partition function",
      "probability proportional",
      "spin-systems",
      "probability distribution",
      "well-known statistical physics models"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1206.3200G"
    }
  }
}