{
  "id": "1107.2280",
  "version": "v4",
  "published": "2011-07-12T13:23:34.000Z",
  "updated": "2013-09-18T22:56:25.000Z",
  "title": "Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice",
  "authors": [
    "Daniel Ahlberg"
  ],
  "comment": "23 pages. Together with arXiv:1305.6260, this version replaces the old. The main results have been strengthened and an earlier error in the statement corrected. To appear in J. Theoret. Probab",
  "doi": "10.1007/s10959-013-0521-0",
  "categories": [
    "math.PR"
  ],
  "abstract": "In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\\Z^d$ lattice, where $d\\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2013-09-18T22:56:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "60F15",
      "60F10"
    ],
    "keywords": [
      "asymptotic shape",
      "first-passage percolation",
      "integer lattice",
      "cone-like subgraphs",
      "precise conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1107.2280A"
    }
  }
}