{
  "id": "1602.03107",
  "version": "v1",
  "published": "2016-02-09T18:10:36.000Z",
  "updated": "2016-02-09T18:10:36.000Z",
  "title": "Range of (1,2) random walk in random environment",
  "authors": [
    "Hua-Ming Wang"
  ],
  "comment": "14 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider $(1,2)$ random walk in random environment $\\{X_n\\}_{n\\ge0}.$ In each step, the walk jumps at most a distance $2$ to the right or a distance $1$ to the left. For the walk transient to the right, it is proved that almost surely $\\lim_{x\\rightarrow\\infty}\\frac{\\#\\{X_n:\\ 0\\le X_n\\le x,\\ n\\ge0\\}}{x}=\\theta$ for some $0<\\theta<1.$ The result shows that the range of the walk covers only a linear proportion of the lattice of the positive half line. For the nearest neighbor random walk in random or non-random environment, this phenomenon could not appear in any circumstance.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-09T18:10:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "60K05"
    ],
    "keywords": [
      "nearest neighbor random walk",
      "walk covers",
      "non-random environment",
      "walk jumps",
      "positive half line"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203107W"
    }
  }
}