{
  "id": "1112.1119",
  "version": "v3",
  "published": "2011-12-05T23:23:18.000Z",
  "updated": "2013-06-12T18:27:54.000Z",
  "title": "Asymptotics for products of characteristic polynomials in classical $β$-Ensembles",
  "authors": [
    "Patrick Desrosiers",
    "Dang-Zheng Liu"
  ],
  "comment": "[v3] 35 pages; this is a revised and enlarged version of the article with new references, simplified demonstations, and improved presentation. To be published in Constructive Approximation 37 (2013)",
  "doi": "10.1007/s00365-013-9206-2",
  "categories": [
    "math-ph",
    "math.CA",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi $\\beta$-ensembles of $N\\times N$ random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as $N\\to\\infty$. In the bulk of the spectrum of each $\\beta$-ensemble, the same scaling limit is found to be $e^{p_{1}}{}_1F_{1}$ whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre $\\beta$-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when $\\beta$ is even, scaling limits of the $k$-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson's lemma and the steepest descent method for integrals of Selberg type.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2013-06-12T18:27:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "41A60",
      "05E05",
      "33C70"
    ],
    "keywords": [
      "characteristic polynomials",
      "scaling limit",
      "asymptotic",
      "multivariate airy function",
      "point correlation functions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1112.1119D"
    }
  }
}