{
  "id": "math/0406082",
  "version": "v2",
  "published": "2004-06-04T12:36:23.000Z",
  "updated": "2005-08-30T08:05:42.000Z",
  "title": "Classical and free infinitely divisible distributions and random matrices",
  "authors": [
    "Florent Benaych-Georges"
  ],
  "comment": "Published at http://dx.doi.org/10.1214/009117904000000982 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2005, Vol. 33, No. 3, 1134-1170",
  "doi": "10.1214/009117904000000982",
  "categories": [
    "math.PR",
    "math.OA"
  ],
  "abstract": "We construct a random matrix model for the bijection \\Psi between clas- sical and free infinitely divisible distributions: for every d\\geq1, we associate in a quite natural way to each *-infinitely divisible distribution \\mu a distribution P_d^{\\mu} on the space of d\\times d Hermitian matrices such that P_d^{\\mu}P_d^{\\nu}=P_d^{\\mu*\\nu}. The spectral distribution of a random matrix with distribution P_d^{\\mu} converges in probability to \\Psi (\\mu) when d tends to +\\infty. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko-Pastur distribution. In an analogous way, for every d\\geq1, we associate to each *-infinitely divisible distribution \\mu, a distribution L_d^{\\mu} on the space of complex (non-Hermitian) d\\times d random matrices. If \\mu is symmetric, the symmetrization of the spectral distribution of |M_d|, when M_d is L_d^{\\mu}-distributed, converges in probability to \\Psi(\\mu).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2005-08-30T08:05:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "46L54",
      "60E07",
      "60F05"
    ],
    "keywords": [
      "free infinitely divisible distributions",
      "random matrices",
      "spectral distribution",
      "random matrix model",
      "quite natural way"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6082B"
    }
  }
}