{
  "id": "math/0508579",
  "version": "v2",
  "published": "2005-08-29T18:44:38.000Z",
  "updated": "2006-03-09T13:28:48.000Z",
  "title": "Valleys and the maximum local time for random walk in random environment",
  "authors": [
    "Amir Dembo",
    "Nina Gantert",
    "Yuval Peres",
    "Zhan Shi"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\xi(n, x)$ be the local time at $x$ for a recurrent one-dimensional random walk in random environment after $n$ steps, and consider the maximum $\\xi^*(n) = \\max_x \\xi(n,x)$. It is known that $\\limsup \\xi^*(n)/n$ is a positive constant a.s. We prove that $\\liminf_n (\\log\\log\\log n)\\xi^*(n)/n$ is a positive constant a.s.; this answers a question of P. R\\'ev\\'esz (1990). The proof is based on an analysis of the {\\em valleys /} in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time $n$ large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2006-03-09T13:28:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60G50",
      "60J55",
      "60F10"
    ],
    "keywords": [
      "maximum local time",
      "random environment",
      "recurrent one-dimensional random walk",
      "uniform exponential tail bound",
      "expected total occupation time"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math......8579D"
    }
  }
}