{
  "id": "1302.2452",
  "version": "v1",
  "published": "2013-02-11T11:26:41.000Z",
  "updated": "2013-02-11T11:26:41.000Z",
  "title": "Exact distributions of the number of distinct and common sites visited by N independent random walkers",
  "authors": [
    "Anupam Kundu",
    "Satya N. Majumdar",
    "Gregory Schehr"
  ],
  "comment": "5 pages, 3 figures",
  "journal": "Phys. Rev. Lett. 110, 220602 (2013)",
  "doi": "10.1103/PhysRevLett.110.220602",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math.PR"
  ],
  "abstract": "We study the number of distinct sites S_N(t) and common sites W_N(t) visited by N independent one dimensional random walkers, all starting at the origin, after t time steps. We show that these two random variables can be mapped onto extreme value quantities associated to N independent random walkers. Using this mapping, we compute exactly their probability distributions P_N^d(S,t) and P_N^d(W,t) for any value of N in the limit of large time t, where the random walkers can be described by Brownian motions. In the large N limit one finds that S_N(t)/\\sqrt{t} \\propto 2 \\sqrt{\\log N} + \\widetilde{s}/(2 \\sqrt{\\log N}) and W_N(t)/\\sqrt{t} \\propto \\widetilde{w}/N where \\widetilde{s} and \\widetilde{w} are random variables whose probability density functions (pdfs) are computed exactly and are found to be non trivial. We verify our results through direct numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-02-11T11:26:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.Fb",
      "02.50.-r",
      "05.40.Jc"
    ],
    "keywords": [
      "independent random walkers",
      "common sites",
      "exact distributions",
      "random variables",
      "probability density functions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review Letters",
      "year": 2013,
      "month": "May",
      "volume": 110,
      "number": 22,
      "pages": 220602
    },
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013PhRvL.110v0602K"
    }
  }
}