{
  "id": "1002.3634",
  "version": "v3",
  "published": "2010-02-19T15:57:21.000Z",
  "updated": "2011-02-11T15:48:30.000Z",
  "title": "Asymptotics of the instantons of Painleve I",
  "authors": [
    "Stavros Garoufalidis",
    "Alexander Its",
    "Andrei Kapaev",
    "Marcos Marino"
  ],
  "comment": "29 pages, 8 figures",
  "journal": "International Mathematics Research Notices (2012) Issue 3, pp. 561-606",
  "doi": "10.1093/imrn/rnr029",
  "categories": [
    "math.CA",
    "hep-th"
  ],
  "abstract": "The 0-instanton solution of Painlev\\'e I is a sequence $(u_{n,0})$ of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and 2-dimensional quantum gravity. The asymptotics of the 0-instanton $(u_{n,0})$ for large $n$ were obtained by the third author using the Riemann-Hilbert approach. For $k=0,1,2,...$, the $k$-instanton solution of Painlev\\'e I is a doubly-indexed sequence $(u_{n,k})$ of complex numbers that satisfies an explicit quadratic non-linear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of $(u_{n,k})$ for fixed $k$ and to all orders in $1/n$ using resurgent analysis, and (c) to confirm numerically the predictions of resurgent analysis. We point out that the instanton solutions display a new type of Stokes behavior, induced from the tritronqu\\'ee Painlev\\'e transcendents, and which we call the induced Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new phenomena not seen in 0 and 1-instantons, and their enumerative context is at present unknown.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2011-02-11T15:48:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "asymptotics",
      "explicit quadratic non-linear recursion relation",
      "resurgent analysis",
      "complex numbers",
      "tritronquee painleve transcendents"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 846371,
      "adsabs": "2010arXiv1002.3634G"
    }
  }
}