{
  "id": "math-ph/0307055",
  "version": "v1",
  "published": "2003-07-28T09:28:25.000Z",
  "updated": "2003-07-28T09:28:25.000Z",
  "title": "Random matrices with external source and multiple orthogonal polynomials",
  "authors": [
    "P. M. Bleher",
    "A. B. J. Kuijlaars"
  ],
  "comment": "17 pages",
  "journal": "Int.Math.Res.Not.2004:109-129,2004",
  "categories": [
    "math-ph",
    "hep-th",
    "math.CA",
    "math.MP"
  ],
  "abstract": "We show that the average characteristic polynomial P_n(z) = E [\\det(zI-M)] of the random Hermitian matrix ensemble Z_n^{-1} \\exp(-Tr(V(M)-AM))dM is characterized by multiple orthogonality conditions that depend on the eigenvalues of the external source A. For each eigenvalue a_j of A, there is a weight and P_n has n_j orthogonality conditions with respect to this weight, if n_j is the multiplicity of a_j. The eigenvalue correlation functions have determinantal form, as shown by Zinn-Justin. Here we give a different expression for the kernel. We derive a Christoffel-Darboux formula in case A has two distinct eigenvalues, which leads to a compact formula in terms of a Riemann-Hilbert problem that is satisfied by multiple orthogonal polynomials.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-07-28T09:28:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "multiple orthogonal polynomials",
      "external source",
      "random matrices",
      "average characteristic polynomial",
      "multiple orthogonality conditions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1155/S1073792804132194"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 624472,
      "adsabs": "2003math.ph...7055B"
    }
  }
}