{
  "id": "1703.06922",
  "version": "v1",
  "published": "2017-03-20T18:52:48.000Z",
  "updated": "2017-03-20T18:52:48.000Z",
  "title": "Poly-logarithmic localization for random walks among random obstacles",
  "authors": [
    "Jian Ding",
    "Changji Xu"
  ],
  "comment": "22 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Place an obstacle with probability $1-\\mathsf{p}$ independently at each vertex of $\\mathbb Z^d$, and run a simple random walk until hitting one of the obstacles. For $d\\geq 2$ and $\\mathsf{p}$ strictly above the critical threshold for site percolation, we condition on the environment where the origin is contained in an infinite cluster of non-obstacles; we show that for environments with probability tending to 1 as $n\\to \\infty$, conditioning on survival after $n$ steps the simple random walk is localized in a region of size poly-logarithmic in $n$ with probability tending to 1 as $n\\to \\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-03-20T18:52:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60H25",
      "60G70"
    ],
    "keywords": [
      "random obstacles",
      "poly-logarithmic localization",
      "simple random walk",
      "site percolation",
      "environment"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}