{
  "id": "1511.05387",
  "version": "v1",
  "published": "2015-11-17T13:08:48.000Z",
  "updated": "2015-11-17T13:08:48.000Z",
  "title": "Crossing probability for directed polymers in random media: exact tail of the distribution",
  "authors": [
    "Andrea De Luca",
    "Pierre Le Doussal"
  ],
  "comment": "16 pages + 6 pages of appendices, 2 figures",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn"
  ],
  "abstract": "We study the probability $p \\equiv p_\\eta(t)$ that two directed polymers in a given random potential $\\eta$ and with fixed and nearby endpoints, do not cross until time $t$. This probability is itself a random variable (over samples $\\eta$) which, as we show, acquires a very broad probability distribution at large time. In particular the moments of $p$ are found to be dominated by atypical samples where $p$ is of order unity. Building on a formula established by us in a previous work using nested Bethe Ansatz and Macdonald process methods, we obtain analytically the leading large time behavior of {\\it all moments} $\\overline{p^m}\\simeq \\gamma_m/t$. From this, we extract the exact tail $\\sim \\rho(p)/t$ of the probability distribution of the non-crossing probability at large time. The exact formula is compared to numerical simulations, with excellent agreement.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-17T13:08:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact tail",
      "directed polymers",
      "random media",
      "crossing probability",
      "broad probability distribution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}