{
  "id": "cond-mat/0210501",
  "version": "v1",
  "published": "2002-10-22T17:05:34.000Z",
  "updated": "2002-10-22T17:05:34.000Z",
  "title": "Ordering of Random Walks: The Leader and the Laggard",
  "authors": [
    "D. ben-Avraham",
    "B. M. Johnson",
    "C. A. Monaco",
    "P. L. Krapivsky",
    "S. Redner"
  ],
  "comment": "7 pages, 4 figures, 2-column revtex 4 format",
  "journal": "J. Phys. A 36, 1789-1799 (2003)",
  "doi": "10.1088/0305-4470/36/7/301",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We investigate two complementary problems related to maintaining the relative positions of N random walks on the line: (i) the leader problem, that is, the probability {\\cal L}_N(t) that the leftmost particle remains the leftmost as a function of time and (ii) the laggard problem, the probability {\\cal R}_N(t) that the rightmost particle never becomes the leftmost. We map these ordering problems onto an equivalent (N-1)-dimensional electrostatic problem. From this construction we obtain a very accurate estimate for {\\cal L}_N(t) for N=4, the first case that is not exactly soluble: {\\cal L}_4(t) ~ t^{-\\beta_4}, with \\beta_4=0.91342(8). The probability of being the laggard also decays algebraically, {\\cal R}_N(t) ~ t^{-\\gamma_N}; we derive \\gamma_2=1/2, \\gamma_3=3/8, and argue that \\gamma_N--> ln N/N$ as N-->oo.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-10-22T17:05:34.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random walks",
      "probability",
      "leftmost particle remains",
      "first case",
      "leader problem"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "journal": "Journal of Physics A Mathematical General",
      "year": 2003,
      "month": "Feb",
      "volume": 36,
      "number": 7,
      "pages": 1789
    },
    "note": {
      "typesetting": "RevTeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003JPhA...36.1789B"
    }
  }
}