{
  "id": "1403.5775",
  "version": "v2",
  "published": "2014-03-23T17:13:59.000Z",
  "updated": "2016-04-09T13:42:42.000Z",
  "title": "Fuzzy transformations and extremality of Gibbs measures for the Potts model on a Cayley tree",
  "authors": [
    "C. Kuelske",
    "U. A. Rozikov"
  ],
  "comment": "44 pages. To appear in Random Structures and Algorithms",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on a Cayley tree. In our previous work \\cite{KRK} we gave a full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-23T17:13:59.000Z",
      "abstract": "We continue our study of the full set of translation-invariant splitting Gibbs measures (TISGMs, translation-invariant tree-indexed Markov chains) for the $q$-state Potts model on the Cayley tree. In our previous work we gave the full description of the TISGMs, and showed in particular that at sufficiently low temperatures their number is $2^{q}-1$. In this paper we find some regions for the temperature parameter ensuring that a given TISGM is (non-)extreme in the set of all Gibbs measures. In particular we show the existence of a temperature interval for which there are at least $2^{q-1} + q$ extremal TISGMs. For the Cayley tree of order two we give explicit formulae and some numerical values.",
      "comment": "35 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-04-09T13:42:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B26",
      "60K35"
    ],
    "keywords": [
      "cayley tree",
      "fuzzy transformations",
      "extremality",
      "state potts model",
      "translation-invariant splitting gibbs measures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.5775K"
    }
  }
}