{
  "id": "math/0208123",
  "version": "v1",
  "published": "2002-08-15T13:59:24.000Z",
  "updated": "2002-08-15T13:59:24.000Z",
  "title": "Growth and Percolation on the Uniform Infinite Planar Triangulation",
  "authors": [
    "Omer Angel"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "A construction as a growth process for sampling of the uniform infinite planar triangulation (UIPT), defined in a previous paper, is given. The construction is algorithmic in nature, and is an efficient method of sampling a portion of the UIPT. By analyzing the progress rate of the growth process we show that a.s. the UIPT has growth rate r^4 up to polylogarithmic factors, confirming heuristic results from the physics literature. Additionally, the boundary component of the ball of radius r separating it from infinity a.s. has growth rate r^2 up to polylogarithmic factors. It is also shown that the properly scaled size of a variant of the free triangulation of an m-gon converges in distribution to an asymmetric stable random variable of type 1/2. By combining Bernoulli site percolation with the growth process for the UIPT, it is shown that a.s. the critical probability p_c=1/2 and that at p_c percolation does not occur.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-08-15T13:59:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80",
      "05C30",
      "82B43",
      "81T40"
    ],
    "keywords": [
      "uniform infinite planar triangulation",
      "growth process",
      "growth rate",
      "polylogarithmic factors",
      "combining bernoulli site percolation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......8123A"
    }
  }
}