{
  "id": "math/0404046",
  "version": "v1",
  "published": "2004-04-02T22:47:03.000Z",
  "updated": "2004-04-02T22:47:03.000Z",
  "title": "The Contact Process on Trees",
  "authors": [
    "Robin Pemantle"
  ],
  "comment": "41 pages",
  "journal": "Ann. Probab., 20, 2089 - 2116 (1992)",
  "categories": [
    "math.PR"
  ],
  "abstract": "The contact process on an infinite homogeneous tree is shown to exhibit at least two phase transitions as the infection parameter lambda is varied. For small values of lambda a single infection eventually dies out. For larger lambda the infection lives forever with positive probability but eventually leaves any finite set. (The survival probability is a continuous function of lambda, and the proof of this is much easier than it is for the contact process on d-dimensional integer lattices.) For still larger lambda the infection converges in distribution to a nontrivial invariant measure. For an n-ary tree, with n large, the first of these transitions occurs when lambda~1/n and the second occurs when 1/2 sqrt{n}<lambda<e/sqrt{n}. Nonhomogeneous trees whose vertices have degrees varying between 1 and n behave essentially as homogeneous n-ary trees, provided that vertices of degree n are not too rare. In particular, letting n go to infty, Galton-Watson trees whose vertices have degree n with probability that does not decrease exponentially with n may have both phase transitions occur together at lambda=0. The nature of the second phase transition is not yet clear and several problems are mentioned in this regard.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-04-02T22:47:03.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35"
    ],
    "keywords": [
      "contact process",
      "n-ary tree",
      "larger lambda",
      "nontrivial invariant measure",
      "probability"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4046P"
    }
  }
}