{
  "id": "1610.05082",
  "version": "v1",
  "published": "2016-10-17T12:40:43.000Z",
  "updated": "2016-10-17T12:40:43.000Z",
  "title": "Weighted dependency graphs and the Ising model",
  "authors": [
    "Jehanne Dousse",
    "Valentin Féray"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We consider the Ising model on the $d$-dimensional square lattice at either very high temperature, very low temperature or in a strong magnetic field. In each of these three regimes, we give a simple proof that the joint cumulants of spins decay exponentially fast with respect to the tree-length of the set of spins. With the recent terminology of weighted dependency graphs introduced by the second author, this means that we have a weighted dependency graph structure on the spins. We use this structure to reprove Newman's central limit theorem for the magnetization in a growing box, and to extend it to the number of occurrences of a given spin pattern.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-17T12:40:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20",
      "60F05"
    ],
    "keywords": [
      "ising model",
      "reprove newmans central limit theorem",
      "weighted dependency graph structure",
      "spins decay exponentially fast",
      "strong magnetic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}