{
  "id": "math/0404043",
  "version": "v1",
  "published": "2004-04-02T22:12:37.000Z",
  "updated": "2004-04-02T22:12:37.000Z",
  "title": "Choosing a Spanning Tree for the Integer Lattice Uniformly",
  "authors": [
    "Robin Pemantle"
  ],
  "comment": "24 pages",
  "journal": "Ann. Probab., 19, 1559 - 1574 (1991)",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the nearest neighbor graph for the integer lattice Z^d in d dimensions. For a large finite piece of it, consider choosing a spanning tree for that piece uniformly among all possible subgraphs that are spanning trees. As the piece gets larger, this approaches a limiting measure on the set of spanning graphs for Z^d. This is shown to be a tree if and only if d=<4. In this case, the tree has only one topological end, i.e. there are no doubly infinite paths. When d>=5 the spanning forest has infinitely many components almost surely, with each component having one or two topological ends.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-04-02T22:12:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60K35"
    ],
    "keywords": [
      "spanning tree",
      "integer lattice",
      "nearest neighbor graph",
      "large finite piece",
      "dimensions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......4043P"
    }
  }
}