{
  "id": "math/0605397",
  "version": "v2",
  "published": "2006-05-15T15:40:50.000Z",
  "updated": "2015-06-22T15:05:28.000Z",
  "title": "An explicit bound on the Logarithmic Sobolev constant of weakly dependent random variables",
  "authors": [
    "Katalin Marton"
  ],
  "comment": "This paper has been withdrawn by the author because it was a preliminary version of arXiv:1206.4868",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove logarithmic Sobolev inequality for measures $$ q^n(x^n)=\\text{dist}(X^n)=\\exp\\bigl(-V(x^n)\\bigr), \\quad x^n\\in \\Bbb R^n, $$ under the assumptions that: (i) the conditional distributions $$ Q_i(\\cdot| x_j, j\\neq i)=\\text{dist}(X_i| X_j= x_j, j\\neq i) $$ satisfy a logarithmic Sobolev inequality with a common constant $\\rho$, and (ii) they also satisfy some condition expressing that the mixed partial derivatives of the Hamiltonian $V$ are not too large relative to $\\rho$. \\bigskip Condition (ii) has the form that the norms of some matrices defined in terms of the mixed partial derivatives of $V$ do not exceed $1/2\\cdot\\rho\\cdot(1-\\de)$. The logarithmic Sobolev constant of $q^n$ can then be estimated from below by $1/2\\cdot\\rho\\cdot\\delta$. This improves on earlier results by Th. Bodineau and B. Helffer, by giving an explicit bound, for the logarithmic Sobolev constant for $q^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-05-15T15:40:50.000Z",
      "comment": "31 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-06-22T15:05:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "logarithmic sobolev constant",
      "weakly dependent random variables",
      "explicit bound",
      "logarithmic sobolev inequality",
      "mixed partial derivatives"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......5397M"
    }
  }
}