{
  "id": "2208.00067",
  "version": "v1",
  "published": "2022-07-29T20:41:11.000Z",
  "updated": "2022-07-29T20:41:11.000Z",
  "title": "Statistical properties of sites visited by independent random walks",
  "authors": [
    "E. Ben-Naim",
    "P. L. Krapivsky"
  ],
  "comment": "11 pages, 12 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math.PR"
  ],
  "abstract": "The set of visited sites and the number of visited sites are two basic properties of the random walk trajectory. We consider two independent random walks on a hyper-cubic lattice and study ordering probabilities associated with these characteristics. The first is the probability that during the time interval (0,t), the number of sites visited by a walker never exceeds that of another walker. The second is the probability that the sites visited by a walker remain a subset of the sites visited by another walker. Using numerical simulations, we investigate the leading asymptotic behaviors of the ordering probabilities in spatial dimensions d=1,2,3,4. We also study the evolution of the number of ties between the number of visited sites. We show analytically that the average number of ties increases as $a_1\\ln t$ with $a_1=0.970508$ in one dimension and as $(\\ln t)^2$ in two dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-29T20:41:11.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "independent random walks",
      "statistical properties",
      "visited sites",
      "probability",
      "random walk trajectory"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}