{
  "id": "0712.3378",
  "version": "v2",
  "published": "2007-12-20T11:23:23.000Z",
  "updated": "2009-05-02T04:50:53.000Z",
  "title": "Scaling Limits for Internal Aggregation Models with Multiple Sources",
  "authors": [
    "Lionel Levine",
    "Yuval Peres"
  ],
  "comment": "74 pages, 4 figures, to appear in J. d'Analyse Math. Main changes in v2: added \"least action principle\" (Lemma 3.2); small corrections in section 4, and corrected the proof of Lemma 5.3 (Lemma 5.4 in the new version); expanded section 6.1",
  "journal": "JOURNAL D'ANALYSE MATH\\'EMATIQUE Volume 111, Number 1, 151--219, 2010",
  "doi": "10.1007/s11854-010-0015-2",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math.AP",
    "math.CO"
  ],
  "abstract": "We study the scaling limits of three different aggregation models on Z^d: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in R^d. In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2009-05-02T04:50:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G50",
      "35R35",
      "31C20"
    ],
    "keywords": [
      "scaling limit",
      "internal aggregation models",
      "multiple sources",
      "pde free boundary problem",
      "internal dla"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.3378L"
    }
  }
}