{
  "id": "1501.07258",
  "version": "v1",
  "published": "2015-01-28T20:15:39.000Z",
  "updated": "2015-01-28T20:15:39.000Z",
  "title": "The divisible sandpile at critical density",
  "authors": [
    "Lionel Levine",
    "Mathav Murugan",
    "Yuval Peres",
    "Baris Evren Ugurcan"
  ],
  "comment": "33 pages",
  "categories": [
    "math.PR",
    "cond-mat.stat-mech",
    "math.AP"
  ],
  "abstract": "The divisible sandpile starts with i.i.d. random variables (\"masses\") at the vertices of an infinite, vertex-transitive graph, and redistributes mass by a local toppling rule in an attempt to make all masses at most 1. The process stabilizes almost surely if m<1 and it almost surely does not stabilize if m>1, where $m$ is the mean mass per vertex. The main result of this paper is that in the critical case m=1, if the initial masses have finite variance, then the process almost surely does not stabilize. To give quantitative estimates on a finite graph, we relate the number of topplings to a discrete biLaplacian Gaussian field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-28T20:15:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J45",
      "60G15",
      "82C20",
      "82C26"
    ],
    "keywords": [
      "critical density",
      "divisible sandpile",
      "discrete bilaplacian gaussian field",
      "random variables",
      "mean mass"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150107258L"
    }
  }
}