{
  "id": "hep-th/9211141",
  "version": "v1",
  "published": "1992-12-01T01:21:00.000Z",
  "updated": "1992-12-01T01:21:00.000Z",
  "title": "Level-Spacing Distributions and the Airy Kernel",
  "authors": [
    "Craig A. Tracy",
    "Harold Widom"
  ],
  "comment": "35 pages, LaTeX document using REVTEX macros",
  "journal": "Commun.Math.Phys. 159 (1994) 151-174",
  "doi": "10.1007/BF02100489",
  "categories": [
    "hep-th",
    "cond-mat",
    "math-ph",
    "math.MP",
    "nlin.SI",
    "solv-int"
  ],
  "abstract": "Scaling level-spacing distribution functions in the ``bulk of the spectrum'' in random matrix models of $N\\times N$ hermitian matrices and then going to the limit $N\\to\\infty$, leads to the Fredholm determinant of the sine kernel $\\sin\\pi(x-y)/\\pi (x-y)$. Similarly a scaling limit at the ``edge of the spectrum'' leads to the Airy kernel $[{\\rm Ai}(x) {\\rm Ai}'(y) -{\\rm Ai}'(x) {\\rm Ai}(y)]/(x-y)$. In this paper we derive analogues for this Airy kernel of the following properties of the sine kernel: the completely integrable system of P.D.E.'s found by Jimbo, Miwa, M{\\^o}ri and Sato; the expression, in the case of a single interval, of the Fredholm determinant in terms of a Painlev{\\'e} transcendent; the existence of a commuting differential operator; and the fact that this operator can be used in the derivation of asymptotics, for general $n$, of the probability that an interval contains precisely $n$ eigenvalues.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1992-12-01T01:21:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "airy kernel",
      "fredholm determinant",
      "sine kernel",
      "random matrix models",
      "interval contains"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Commun. Math. Phys."
    },
    "note": {
      "typesetting": "RevTeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 33739
    }
  }
}