{
  "id": "1811.02226",
  "version": "v1",
  "published": "2018-11-06T08:50:24.000Z",
  "updated": "2018-11-06T08:50:24.000Z",
  "title": "The heavy range of randomly biased walks on trees",
  "authors": [
    "Pierre Andreoletti",
    "Roland Diel"
  ],
  "comment": "34 pages (version 2)",
  "categories": [
    "math.PR"
  ],
  "abstract": "We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16]. Here we study the heavy range: the number of edges visited at least $\\alpha$ times for some real $\\alpha$. The asymptotic behavior of this process when $\\alpha$ is a power of the number of steps of the walk is given for all the recurrent cases. It turns out that this heavy range plays a crucial role in the rate of convergence of an estimator of the environment from a single trajectory of the RWRE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-06T08:50:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "randomly biased walks",
      "recurrent random walks",
      "heavy range plays",
      "random environment",
      "galton-watson trees"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}