{
  "id": "1208.3366",
  "version": "v1",
  "published": "2012-08-16T13:42:17.000Z",
  "updated": "2012-08-16T13:42:17.000Z",
  "title": "Phase transitions for $P$-adic Potts model on the Cayley tree of order three",
  "authors": [
    "Farrukh Mukhamedov",
    "Hasan Akin"
  ],
  "comment": "27 pages",
  "journal": "J. Stat. Mech. (2013) P07014",
  "doi": "10.1088/1742-5468/2013/07/P07014",
  "categories": [
    "math-ph",
    "math.MP",
    "math.NT",
    "math.PR"
  ],
  "abstract": "In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter $\\rho\\in\\bq_p$. Such a measure is called {\\it generalized $p$-adic quasi Gibbs measure}. When $\\rho$ equals to $p$-adic exponent, then it coincides with the $p$-adic Gibbs measure. When $\\rho=p$, then it coincides with $p$-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of $|\\rho|_p$. Namely, in the first regime, one takes $\\rho=\\exp_p(J)$ for some $J\\in\\bq_p$, in the second one $|\\rho|_p<1$. In each regime, we first find conditions for the existence of generalized $p$-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when $|\\r|_p,|q|_p\\leq p^{-2}$ we prove the existence of a quasi phase transition. It turns out that if $|\\r|_p<|q-1|_p^2<1$ and $\\sqrt{-3}\\in\\bq_p$, then one finds the existence of the strong phase transition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-16T13:42:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46S10",
      "82B26",
      "12J12",
      "39A70",
      "47H10",
      "60K35"
    ],
    "keywords": [
      "phase transition",
      "adic potts model",
      "adic quasi gibbs measure",
      "cayley tree",
      "adic gibbs measure"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Statistical Mechanics: Theory and Experiment",
      "year": 2013,
      "month": "Jul",
      "volume": 2013,
      "number": 7,
      "pages": "07014"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013JSMTE..07..014M"
    }
  }
}