{
  "id": "1803.02867",
  "version": "v1",
  "published": "2018-03-07T20:35:18.000Z",
  "updated": "2018-03-07T20:35:18.000Z",
  "title": "Phase transitions for a model with uncountable spin space on the Cayley tree: the general case",
  "authors": [
    "Golibjon Botirov",
    "Benedikt Jahnel"
  ],
  "comment": "9 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this paper we complete the analysis of a statistical mechanics model on Cayley trees of any degree, started in [EsHaRo12,EsRo10,BoEsRo13,JaKuBo14,Bo17]. The potential is of nearest-neighbor type and the local state space is compact but uncountable. Based on the system parameters we prove existence of a critical value $\\theta_{\\rm c}$ such that for $\\theta\\le \\theta_{\\rm c}$ there is a unique translation-invariant splitting Gibbs measure. For $\\theta_{\\rm c}<\\theta$ there is a phase transition with exactly three translation-invariant splitting Gibbs measures. The proof rests on an analysis of fixed points of an associated non-linear Hammerstein integral operator for the boundary laws.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-07T20:35:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B05",
      "82B20",
      "60K35"
    ],
    "keywords": [
      "uncountable spin space",
      "phase transition",
      "cayley tree",
      "general case",
      "unique translation-invariant splitting gibbs measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}