{
  "id": "math-ph/0306078",
  "version": "v1",
  "published": "2003-06-30T09:39:05.000Z",
  "updated": "2003-06-30T09:39:05.000Z",
  "title": "Universal behavior for averages of characteristic polynomials at the origin of the spectrum",
  "authors": [
    "M. Vanlessen"
  ],
  "comment": "24 pages, 3 figures",
  "categories": [
    "math-ph",
    "math.CA",
    "math.MP"
  ],
  "abstract": "It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble $\\frac{1}{\\hat Z_n}|\\det M|^{2\\alpha}e^{-nV(M)}dM$ of $n\\times n$ Hermitian matrices, these kernels have universal behavior at the origin of the spectrum, as $n\\to\\infty$, in terms of Bessel functions. Our approach is based on the characterization of orthogonal polynomials together with their Cauchy transforms via a matrix Riemann-Hilbert problem, due to Fokas, Its and Kitaev, and on an application of the Deift/Zhou steepest descent method for matrix Riemann-Hilbert problems to obtain the asymptotic behavior of the Riemann-Hilbert problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-06-30T09:39:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "characteristic polynomials",
      "universal behavior",
      "matrix riemann-hilbert problem",
      "hermitian matrices",
      "orthogonal polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.ph...6078V"
    }
  }
}