{
  "id": "2205.02025",
  "version": "v1",
  "published": "2022-05-04T12:32:07.000Z",
  "updated": "2022-05-04T12:32:07.000Z",
  "title": "Gibbs measures for HC-model with a countable set of spin values on a Cayley tree",
  "authors": [
    "R. M. Khakimov",
    "M. T. Makhammadaliev",
    "U. A. Rozikov"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper, we study the HC-model with a countable set $\\mathbb Z$ of spin values on a Cayley tree of order $k\\geq 2$. This model is defined by a countable set of parameters (that is, the activity function $\\lambda_i>0$, $i\\in \\mathbb Z$). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained: - Let $\\Lambda=\\sum_i\\lambda_i$. For $\\Lambda=+\\infty$ there are no translation-invariant Gibbs measures (TIGM) and no two-periodic Gibbs measures (TPGM); - For $\\Lambda<+\\infty$, the uniqueness of TIGM is proved; - Let $\\Lambda_{\\rm cr}(k)=\\frac{k^k}{(k-1)^{k+1}}$. If $0<\\Lambda\\leq\\Lambda_{\\rm cr}$, then there is exactly one TPGM that is TIGM; - For $\\Lambda>\\Lambda_{\\rm cr}$, there are exactly three TPGMs, one of which is TIGM.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-04T12:32:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B26",
      "60K35"
    ],
    "keywords": [
      "countable set",
      "cayley tree",
      "spin values",
      "two-periodic gibbs measures",
      "translation-invariant gibbs measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}