{
  "id": "math/0107140",
  "version": "v3",
  "published": "2001-07-19T21:12:23.000Z",
  "updated": "2003-02-13T17:43:06.000Z",
  "title": "Geometry of the Uniform Spanning Forest: Transitions in Dimensions 4, 8, 12",
  "authors": [
    "Itai Benjamini",
    "Harry Kesten",
    "Yuval Peres",
    "Oded Schramm"
  ],
  "comment": "Current version: added some comments regarding related problems and implications, and made some corrections",
  "journal": "Annals Math.160:465-491,2004",
  "doi": "10.4007/=",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The uniform spanning forest (USF) in Z^d is the weak limit of random, uniformly chosen, spanning trees in [-n,n]^d. Pemantle proved that the USF consists a.s. of a single tree if and only if d <= 4. We prove that any two components of the USF in Z^d are adjacent a.s. if 5 <= d <= 8, but not if d >= 9. More generally, let N(x,y) be the minimum number of edges outside the USF in a path joining x and y in Z^d. Then a.s. max{N(x,y) : x,y in Z^d} is the integer part of (d-1)/4. The notion of stochastic dimension for random relations in the lattice is introduced and used in the proof.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2003-02-13T17:43:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60J15"
    ],
    "keywords": [
      "uniform spanning forest",
      "transitions",
      "weak limit",
      "single tree",
      "stochastic dimension"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 580928,
      "adsabs": "2001math......7140B"
    }
  }
}