{
  "id": "math-ph/0103025",
  "version": "v1",
  "published": "2001-03-21T03:56:38.000Z",
  "updated": "2001-03-21T03:56:38.000Z",
  "title": "Application of the $τ$-Function Theory of Painlevé Equations to Random Matrices: PIV, PII and the GUE",
  "authors": [
    "P. J. Forrester",
    "N. S. Witte"
  ],
  "comment": "40 pages, Latex2e plus AMS and XY packages. to appear Commun. Math. Phys",
  "doi": "10.1007/s002200100422",
  "categories": [
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of $\\tilde{E}_N(\\lambda;a) := \\Big < \\prod_{l=1}^N \\chi_{(-\\infty, \\lambda]}^{(l)} (\\lambda - \\lambda_l)^a \\Big>$, where $ \\chi_{(-\\infty, \\lambda]}^{(l)} = 1$ for $\\lambda_l \\in (-\\infty, \\lambda]$ and $ \\chi_{(-\\infty, \\lambda]}^{(l)} = 0$ otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of $F_N(\\lambda;a) := \\Big < \\prod_{l=1}^N (\\lambda - \\lambda_l)^a \\Big >$. Of particular interest are $\\tilde{E}_N(\\lambda;2)$ and $F_N(\\lambda;2)$, and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto $\\tau$-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities $\\tilde{E}_N(\\lambda;a)$ and $F_N(\\lambda;a)$ for the other classical matrix ensembles.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-03-21T03:56:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "34A34",
      "34A05",
      "33C45"
    ],
    "keywords": [
      "function theory",
      "random matrices",
      "largest eigenvalue",
      "application",
      "joint eigenvalue distribution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2001,
      "volume": 219,
      "number": 2,
      "pages": 357
    },
    "note": {
      "typesetting": "LaTeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001CMaPh.219..357F"
    }
  }
}