{
  "id": "1508.03923",
  "version": "v1",
  "published": "2015-08-17T05:00:19.000Z",
  "updated": "2015-08-17T05:00:19.000Z",
  "title": "Boundaries of Planar Graphs: A Unified Approach",
  "authors": [
    "Tom Hutchcroft",
    "Yuval Peres"
  ],
  "comment": "21 pages, 2 figures",
  "categories": [
    "math.PR"
  ],
  "abstract": "We give a new proof that the Poisson boundary of a planar graph coincides with the boundary of its square tiling and with the boundary of its circle packing, originally proven by Georgakopoulos and Angel, Barlow, Gurel-Gurevich and Nachmias respectively. Our proof is robust, and also allows us to identify the Poisson boundaries of graphs that are rough-isometric to planar graphs. We also prove that the boundary of the square tiling of a bounded degree plane triangulation coincides with its Martin boundary. This is done by comparing the square tiling of the triangulation with its circle packing.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-08-17T05:00:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unified approach",
      "square tiling",
      "bounded degree plane triangulation coincides",
      "poisson boundary",
      "planar graph coincides"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150803923H"
    }
  }
}