{
  "id": "math/0512378",
  "version": "v2",
  "published": "2005-12-15T20:13:02.000Z",
  "updated": "2006-11-14T07:23:49.000Z",
  "title": "Gibbs distributions for random partitions generated by a fragmentation process",
  "authors": [
    "Nathanael Berestycki",
    "Jim Pitman"
  ],
  "comment": "38 pages, 2 figures, version considerably modified. To appear in the Journal of Statistical Physics",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math.CO"
  ],
  "abstract": "In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w\\_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w\\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent process with affine collision rate K\\_{i,j}=a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton-Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-11-14T07:23:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J10",
      "60K35",
      "05A15",
      "05A19"
    ],
    "keywords": [
      "random partitions",
      "fragmentation process",
      "gibbs distributions",
      "suitable offspring distribution",
      "weight sequence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....12378B"
    }
  }
}