{
  "id": "math/0211455",
  "version": "v1",
  "published": "2002-11-29T01:38:44.000Z",
  "updated": "2002-11-29T01:38:44.000Z",
  "title": "Trees and matchings from point processes",
  "authors": [
    "Alexander E. Holroyd",
    "Yuval Peres"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the d-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic isometry-invariant way. For d \\geq 4 our result answers a question posed by Ferrari, Landim and Thorisson. We prove also that any isometry-invariant ergodic point process of finite intensity in Euclidean or hyperbolic space has a perfect matching as a factor graph provided all the inter-point distances are distinct.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-11-29T01:38:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G55",
      "60K35"
    ],
    "keywords": [
      "point processes",
      "factor graph",
      "deterministic isometry-invariant way",
      "isometry-invariant ergodic point process",
      "d-dimensional poisson process"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....11455H"
    }
  }
}