{
  "id": "1107.4884",
  "version": "v1",
  "published": "2011-07-25T10:22:13.000Z",
  "updated": "2011-07-25T10:22:13.000Z",
  "title": "On $p$-adic Gibbs Measures for Hard Core Model on a Cayley Tree",
  "authors": [
    "D. Gandolfo",
    "U. A. Rozikov",
    "J. Ruiz"
  ],
  "comment": "17 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "In this paper we consider a nearest-neighbor $p$-adic hard core (HC) model, with fugacity $\\lambda$, on a homogeneous Cayley tree of order $k$ (with $k + 1$ neighbors). We focus on $p$-adic Gibbs measures for the HC model, in particular on $p$-adic \"splitting\" Gibbs measures generating a $p$-adic Markov chain along each path on the tree. We show that the $p$-adic HC model is completely different from real HC model: For a fixed $k$ we prove that the $p$-adic HC model may have a splitting Gibbs measure only if $p$ divides $2^k-1$. Moreover if $p$ divides $2^k-1$ but does not divide $k+2$ then there exists unique translational invariant $p$-adic Gibbs measure. We also study $p$-adic periodic splitting Gibbs measures and show that the above model admits only translational invariant and periodic with period two (chess-board) Gibbs measures. For $p\\geq 7$ (resp. $p=2,3,5$) we give necessary and sufficient (resp. necessary) conditions for the existence of a periodic $p$-adic measure. For k=2 a $p$-adic splitting Gibbs measures exists if and only if p=3, in this case we show that if $\\lambda$ belongs to a $p$-adic ball of radius 1/27 then there are precisely two periodic (non translational invariant) $p$-adic Gibbs measures. We prove that a $p$-adic Gibbs measure is bounded if and only if $p\\ne 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-07-25T10:22:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46S10",
      "82B26",
      "12J12"
    ],
    "keywords": [
      "adic gibbs measure",
      "hard core model",
      "cayley tree",
      "translational invariant",
      "adic hc model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.4884G"
    }
  }
}