{
  "id": "math/0410311",
  "version": "v1",
  "published": "2004-10-13T07:53:40.000Z",
  "updated": "2004-10-13T07:53:40.000Z",
  "title": "Branching Processes, and Random-Cluster Measures on Trees",
  "authors": [
    "Geoffrey Grimmett",
    "Svante Janson"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Random-cluster measures on infinite regular trees are studied in conjunction with a general type of `boundary condition', namely an equivalence relation on the set of infinite paths of the tree. The uniqueness and non-uniqueness of random-cluster measures are explored for certain classes of equivalence relations. In proving uniqueness, the following problem concerning branching processes is encountered and answered. Consider bond percolation on the family-tree $T$ of a branching process. What is the probability that every infinite path of $T$, beginning at its root, contains some vertex which is itself the root of an infinite open sub-tree?",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-10-13T07:53:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J80",
      "82B20"
    ],
    "keywords": [
      "random-cluster measures",
      "infinite path",
      "equivalence relation",
      "infinite open sub-tree",
      "infinite regular trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10311G"
    }
  }
}