{
  "id": "0911.5668",
  "version": "v2",
  "published": "2009-11-30T15:30:39.000Z",
  "updated": "2010-01-28T16:17:34.000Z",
  "title": "Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits",
  "authors": [
    "Nicholas Crawford",
    "Allan Sly"
  ],
  "comment": "47 pages. Minor Revision",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study limit laws for simple random walks on supercritical long range percolation clusters on $\\Z^d, d \\geq 1$. For the long range percolation model, the probability that two vertices $x, y$ are connected behaves asymptotically as $\\|x-y\\|_2^{-s}$. When $s\\in(d, d+1)$, we prove that the scaling limit of simple random walk on the infinite component converges to an $\\alpha$-stable L\\'evy process with $\\alpha = s-d$ establishing a conjecture of Berger and Biskup. The convergence holds in both the quenched and annealed senses. In the case where $d=1$ and $s>2$ we show that the simple random walk converges to a Brownian motion. The proof combines heat kernel bounds from our companion paper, ergodic theory estimates and an involved coupling constructed through the exploration of a large number of walks on the cluster.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-01-28T16:17:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F17",
      "82B41",
      "82B43"
    ],
    "keywords": [
      "scaling limit",
      "simple random walk converges",
      "supercritical long range percolation clusters",
      "long range percolation model",
      "study limit laws"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0911.5668C"
    }
  }
}