{
  "id": "1410.3924",
  "version": "v1",
  "published": "2014-10-15T04:23:06.000Z",
  "updated": "2014-10-15T04:23:06.000Z",
  "title": "Equivalence of decay of correlations, log-Sobolev inequalities, and Poincare inequalities in spin systems with infinite range interactions",
  "authors": [
    "Christopher Henderson",
    "Georg Menz"
  ],
  "comment": "20 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Yoshida proved in the setting of unbounded continuous spins with finite-range interactions the equivalence of a uniform log-Sobolev inequality, a uniform Poincar\\'e inequality, and a suitable decay of spin- spin correlations. We obtain analogous results in the case of infinite-range interactions which decay only algebraically. The results also hold under less restricting conditions on the single-site potentials of the Hamiltonian. The main new technique is to get decay of correlation with the help of a directional Poincar\\'e inequality deduced by an averaging technique. In addition, we show that in the case of ferromagnetic interaction a weaker algebraic decay of correlations can be bootstrapped into a decay of the same order as the spin-spin interactions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-15T04:23:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20",
      "60K35",
      "82C26"
    ],
    "keywords": [
      "infinite range interactions",
      "spin systems",
      "correlation",
      "equivalence",
      "weaker algebraic decay"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.3924H"
    }
  }
}