{
  "id": "2009.13661",
  "version": "v1",
  "published": "2020-09-28T22:24:54.000Z",
  "updated": "2020-09-28T22:24:54.000Z",
  "title": "Infinite System of Random Walkers: Winners and Losers",
  "authors": [
    "P. L. Krapivsky"
  ],
  "comment": "5 pages",
  "categories": [
    "cond-mat.stat-mech",
    "math.PR",
    "physics.soc-ph"
  ],
  "abstract": "We study an infinite system of particles initially occupying a half-line $y\\leq 0$ and undergoing random walks on the entire line. The right-most particle is called a leader. Surprisingly, every particle except the original leader may never achieve the leadership throughout the evolution. For the equidistant initial configuration, the $k^{\\text{th}}$ particle attains the leadership with probability $e^{-2} k^{-1} (\\ln k)^{-1/2}$ when $k\\gg 1$. This provides a quantitative measure of the correlation between earlier misfortune (represented by $k$) and eternal failure. We also show that the winner defined as the first walker overtaking the initial leader has label $k\\gg 1$ with probability decaying as $\\exp\\!\\left[-\\tfrac{1}{2}(\\ln k)^2\\right]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-28T22:24:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite system",
      "random walkers",
      "equidistant initial configuration",
      "original leader",
      "leadership"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}