{
  "id": "1602.06475",
  "version": "v1",
  "published": "2016-02-20T23:44:14.000Z",
  "updated": "2016-02-20T23:44:14.000Z",
  "title": "Inequalities for critical exponents in $d$-dimensional sandpiles",
  "authors": [
    "Sandeep Bhupatiraju",
    "Jack Hanson",
    "Antal A. Járai"
  ],
  "comment": "58 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider the Abelian sandpile measure on $\\mathbb{Z}^d$, $d \\ge 2$, obtained as the $L \\to \\infty$ limit of the stationary distribution of the sandpile on $[-L,L]^d \\cap \\mathbb{Z}^d$. When adding a grain of sand at the origin, some region topples during stabilization. We prove bounds on the behaviour of various avalanche characteristics: the probability that a given vertex topples, the radius of the toppled region, and the number of vertices toppled. Our results yield rigorous inequalities for the relevant critical exponents. In $d = 2$ we show that for any $1 \\le k < \\infty$, the last $k$ waves of the avalanche have an infinite volume limit, satisfying a power law upper bound on the radius.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-20T23:44:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82C20"
    ],
    "keywords": [
      "dimensional sandpiles",
      "power law upper bound",
      "infinite volume limit",
      "results yield rigorous inequalities",
      "abelian sandpile measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 58,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160206475B"
    }
  }
}