{
  "id": "1704.05940",
  "version": "v1",
  "published": "2017-04-19T21:46:22.000Z",
  "updated": "2017-04-19T21:46:22.000Z",
  "title": "Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line",
  "authors": [
    "Satya N. Majumdar",
    "Philippe Mounaix",
    "Gregory Schehr"
  ],
  "comment": "20 pages, 3 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We consider a one-dimensional random walk (RW) with a continuous and symmetric jump distribution, $f(\\eta)$, characterized by a L\\'evy index $\\mu \\in (0,2]$, which includes standard random walks ($\\mu=2$) and L\\'evy flights ($0<\\mu<2$). We study the survival probability, $q(x_0,n)$, representing the probability that the RW stays non-negative up to step $n$, starting initially at $x_0 \\geq 0$. Our main focus is on the $x_0$-dependence of $q(x_0,n)$ for large $n$. We show that $q(x_0,n)$ displays two distinct regimes as $x_0$ varies: (i) for $x_0= O(1)$ (\"quantum regime\"), the discreteness of the jump process significantly alters the standard scaling behavior of $q(x_0,n)$ and (ii) for $x_0 = O(n^{1/\\mu})$ (\"classical regime\") the discrete-time nature of the process is irrelevant and one recovers the standard scaling behavior (for $\\mu =2$ this corresponds to the standard Brownian scaling limit). The purpose of this paper is to study how precisely the crossover in $q(x_0,n)$ occurs between the quantum and the classical regime as one increases $x_0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-19T21:46:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "survival probability",
      "lévy flights",
      "semi-infinite line",
      "standard scaling behavior",
      "symmetric jump distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}