{
  "id": "1505.04616",
  "version": "v1",
  "published": "2015-05-18T13:03:09.000Z",
  "updated": "2015-05-18T13:03:09.000Z",
  "title": "Records for the number of distinct sites visited by a random walk on the fully-connected lattice",
  "authors": [
    "L. Turban"
  ],
  "comment": "21 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider a random walk on the fully-connected lattice with N sites and study the time evolution of the number of distinct sites s visited by the walker on a subset with n sites. A record value v is obtained for s at a record time t when the walker visits a site of the subset for the first time. The record time t is a partial covering time when v<n and a total covering time when v=n. The probability distributions for the number of records s, the record value v and the record (covering) time t, involving r-Stirling numbers, are obtained using generating function techniques. The mean values, variances and skewnesses are deduced from the generating functions. In the scaling limit the probability distributions for s and v lead to the same Gaussian density. The fluctuations of the record time t are also Gaussian at partial covering. They are distributed according to a Gumbel probability distribution at total covering.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-18T13:03:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distinct sites",
      "random walk",
      "fully-connected lattice",
      "record time",
      "record value"
    ],
    "publication": {
      "doi": "10.1088/1751-8113/48/44/445001",
      "journal": "Journal of Physics A Mathematical General",
      "year": 2015,
      "month": "Nov",
      "volume": 48,
      "number": 44,
      "pages": 445001
    },
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015JPhA...48R5001T"
    }
  }
}