{
  "id": "1803.05249",
  "version": "v1",
  "published": "2018-03-14T13:06:14.000Z",
  "updated": "2018-03-14T13:06:14.000Z",
  "title": "The skeleton of the UIPT, seen from infinity",
  "authors": [
    "Nicolas Curien",
    "Laurent Ménard"
  ],
  "comment": "34 pages, 14 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "We prove that geodesic rays in the Uniform Infinite Planar Triangulation (UIPT) coalesce in a strong sense using the skeleton decomposition of random triangulations discovered by Krikun. This implies the existence of a unique horofunction measuring distances from infinity in the UIPT. We then use this horofunction to define the skeleton \"seen from infinity\" of the UIPT and relate it to a simple Galton--Watson tree conditioned to survive, giving a new and particularly simple construction of the UIPT. Scaling limits of perimeters and volumes of horohulls within this new decomposition are also derived, as well as a new proof of the $2$-point function formula for random triangulations in the scaling limit due to Ambj{\\o}rn and Watabiki.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-14T13:06:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random triangulations",
      "uniform infinite planar triangulation",
      "simple galton-watson tree",
      "unique horofunction measuring distances",
      "scaling limit"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}