{
  "id": "1206.6184",
  "version": "v1",
  "published": "2012-06-27T06:43:26.000Z",
  "updated": "2012-06-27T06:43:26.000Z",
  "title": "Number of Common Sites Visited by N Random Walkers",
  "authors": [
    "Satya N. Majumdar",
    "Mikhail V. Tamm"
  ],
  "comment": "5 pages, 3 .eps figures included",
  "journal": "Phys. Rev. E 86, 021135 (2012)",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We compute analytically the mean number of common sites, W_N(t), visited by N independent random walkers each of length t and all starting at the origin at t=0 in d dimensions. We show that in the (N-d) plane, there are three distinct regimes for the asymptotic large t growth of W_N(t). These three regimes are separated by two critical lines d=2 and d=d_c(N)=2N/(N-1) in the (N-d) plane. For d<2, W_N(t)\\sim t^{d/2} for large t (the N dependence is only in the prefactor). For 2<d<d_c(N), W_N(t)\\sim t^{\\nu} where the exponent \\nu= N-d(N-1)/2 varies with N and d. For d>d_c(N), W_N(t) approaches a constant as t\\to \\infty. Exactly at the critical dimensions there are logaritmic corrections: for d=2, we get W_N(t)\\sim t/[\\ln t]^N, while for d=d_c(N), W_N(t)\\sim \\ln t for large t. Our analytical predictions are verified in numerical simulations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-06-27T06:43:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.40.Fb",
      "02.50.Cw",
      "05.40.Jc",
      "24.60.-k"
    ],
    "keywords": [
      "common sites",
      "independent random walkers",
      "mean number",
      "asymptotic large",
      "dimensions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review E",
      "doi": "10.1103/PhysRevE.86.021135",
      "year": 2012,
      "month": "Aug",
      "volume": 86,
      "number": 2,
      "pages": "021135"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012PhRvE..86b1135M"
    }
  }
}