{
  "id": "1212.3816",
  "version": "v2",
  "published": "2012-12-16T18:48:15.000Z",
  "updated": "2013-01-04T22:22:58.000Z",
  "title": "Tail decay for the distribution of the endpoint of a directed polymer",
  "authors": [
    "Thomas Bothner",
    "Karl Liechty"
  ],
  "comment": "24 pages, 2 figures",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR",
    "nlin.SI"
  ],
  "abstract": "We obtain an asymptotic expansion for the tails of the random variable $\\tcal=\\arg\\max_{u\\in\\mathbb{R}}(\\mathcal{A}_2(u)-u^2)$ where $\\mathcal{A}_2$ is the Airy$_2$ process. Using the formula of Schehr \\cite{Sch} that connects the density function of $\\tcal$ to the Hastings-McLeod solution of the second Painlev\\'e equation, we prove that as $t\\rightarrow\\infty$, $\\mathbb{P}(|\\tcal|>t)=Ce^{-4/3\\varphi(t)}t^{-145/32}(1+O(t^{-3/4}))$, where $\\varphi(t)=t^3-2t^{3/2}+3t^{3/4}$, and the constant $C$ is given explicitly.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-04T22:22:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "37A50",
      "35Q15"
    ],
    "keywords": [
      "tail decay",
      "directed polymer",
      "distribution",
      "second painleve equation",
      "density function"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1088/0951-7715/26/5/1449",
      "journal": "Nonlinearity",
      "year": 2013,
      "month": "May",
      "volume": 26,
      "number": 5,
      "pages": 1449
    },
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1207679,
      "adsabs": "2013Nonli..26.1449B"
    }
  }
}