{
  "id": "math/0305400",
  "version": "v1",
  "published": "2003-05-28T15:09:04.000Z",
  "updated": "2003-05-28T15:09:04.000Z",
  "title": "Reconstruction thresholds on regular trees",
  "authors": [
    "James B. Martin"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider a branching random walk with binary state space and index set $T^k$, the infinite rooted tree in which each node has k children (also known as the model of \"broadcasting on a tree\"). The root of the tree takes a random value 0 or 1, and then each node passes a value independently to each of its children according to a 2x2 transition matrix P. We say that \"reconstruction is possible\" if the values at the d'th level of the tree contain non-vanishing information about the value at the root as $d\\to\\infty$. Adapting a method of Brightwell and Winkler, we obtain new conditions under which reconstruction is impossible, both in the general case and in the special case $p_{11}=0$. The latter case is closely related to the \"hard-core model\" from statistical physics; a corollary of our results is that, for the hard-core model on the (k+1)-regular tree with activity $\\lambda=1$, the unique simple invariant Gibbs measure is extremal in the set of Gibbs measures, for any k.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-05-28T15:09:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "regular trees",
      "reconstruction thresholds",
      "unique simple invariant gibbs measure",
      "hard-core model",
      "tree contain non-vanishing information"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5400M"
    }
  }
}