{
  "id": "2004.10574",
  "version": "v1",
  "published": "2020-04-22T13:55:23.000Z",
  "updated": "2020-04-22T13:55:23.000Z",
  "title": "Block factorization of the relative entropy via spatial mixing",
  "authors": [
    "Pietro Caputo",
    "Daniel Parisi"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math.PR",
    "math.FA"
  ],
  "abstract": "We consider spin systems in the $d$-dimensional lattice $Z^d$ satisfying the so-called strong spatial mixing condition. We show that the relative entropy functional of the corresponding Gibbs measure satisfies a family of inequalities which control the entropy on a given region $V\\subset Z^d$ in terms of a weighted sum of the entropies on blocks $A\\subset V$ when each $A$ is given an arbitrary nonnegative weight $\\alpha_A$. These inequalities generalize the well known logarithmic Sobolev inequality for the Glauber dynamics. Moreover, they provide a natural extension of the classical Shearer inequality satisfied by the Shannon entropy. Finally, they imply a family of modified logarithmic Sobolev inequalities which give quantitative control on the convergence to equilibrium of arbitrary weighted block dynamics of heat bath type.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-22T13:55:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B20",
      "82C20",
      "39B62"
    ],
    "keywords": [
      "block factorization",
      "logarithmic sobolev inequality",
      "corresponding gibbs measure satisfies",
      "modified logarithmic sobolev inequalities",
      "strong spatial mixing condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}