{
  "id": "1409.3718",
  "version": "v1",
  "published": "2014-09-12T12:28:25.000Z",
  "updated": "2014-09-12T12:28:25.000Z",
  "title": "Probability distribution of the number of distinct sites visited by a random walk on the finite-size fully-connected lattice",
  "authors": [
    "L. Turban"
  ],
  "comment": "20 pages, 5 figures",
  "journal": "J. Phys. A 47 (2014) 385004",
  "doi": "10.1088/1751-8113/47/38/385004",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "abstract": "The probability distribution of the number $s$ of distinct sites visited up to time $t$ by a random walk on the fully-connected lattice with $N$ sites is first obtained by solving the eigenvalue problem associated with the discrete master equation. Then, using generating function techniques, we compute the joint probability distribution of $s$ and $r$, where $r$ is the number of sites visited only once up to time $t$. Mean values, variances and covariance are deduced from the generating functions and their finite-size-scaling behaviour is studied. Introducing properly centered and scaled variables $u$ and $v$ for $r$ and $s$ and working in the scaling limit ($t\\to\\infty$, $N\\to\\infty$ with $w=t/N$ fixed) the joint probability density of $u$ and $v$ is shown to be a bivariate Gaussian density. It follows that the fluctuations of $r$ and $s$ around their mean values in a finite-size system are Gaussian in the scaling limit. The same type of finite-size scaling is expected to hold on periodic lattices above the critical dimension $d_{\\rm c}=2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-12T12:28:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "distinct sites",
      "probability distribution",
      "random walk",
      "finite-size fully-connected lattice",
      "mean values"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2014,
      "month": "Sep",
      "volume": 47,
      "number": 38,
      "pages": 385004
    },
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014JPhA...47L5004T"
    }
  }
}