{
  "id": "1609.06325",
  "version": "v1",
  "published": "2016-09-20T20:00:19.000Z",
  "updated": "2016-09-20T20:00:19.000Z",
  "title": "Exact distributions of cover times for $N$ independent random walkers in one dimension",
  "authors": [
    "Satya N. Majumdar",
    "Sanjib Sabhapandit",
    "Gregory Schehr"
  ],
  "comment": "5+2 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "We study the probability density function (PDF) of the cover time $t_c$ of a finite interval of size $L$, by $N$ independent one-dimensional Brownian motions, each with diffusion constant $D$. The cover time $t_c$ is the minimum time needed such that each point of the entire interval is visited by at least one of the $N$ walkers. We derive exact results for the full PDF of $t_c$ for arbitrary $N \\geq 1$, for both reflecting and periodic boundary conditions. The PDFs depend explicitly on $N$ and on the boundary conditions. In the limit of large $N$, we show that $t_c$ approaches its average value $\\langle t_c \\rangle \\approx L^2/(16\\, D \\, \\ln N)$, with fluctuations vanishing as $1/(\\ln N)^2$. We also compute the centered and scaled limiting distributions for large $N$ for both boundary conditions and show that they are given by nontrivial $N$-independent scaling functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-20T20:00:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independent random walkers",
      "cover time",
      "exact distributions",
      "independent one-dimensional brownian motions",
      "probability density function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 2,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}