{
  "id": "0811.0979",
  "version": "v1",
  "published": "2008-11-06T16:03:49.000Z",
  "updated": "2008-11-06T16:03:49.000Z",
  "title": "Asymptotic Independence in the Spectrum of the Gaussian Unitary Ensemble",
  "authors": [
    "P. Bianchi",
    "M. Debbah",
    "J. Najim"
  ],
  "comment": "15 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider a $n \\times n$ matrix from the Gaussian Unitary Ensemble (GUE). Given a finite collection of bounded disjoint real Borel sets $(\\Delta_{i,n},\\ 1\\leq i\\leq p)$, properly rescaled, and eventually included in any neighbourhood of the support of Wigner's semi-circle law, we prove that the related counting measures $({\\mathcal N}_n(\\Delta_{i,n}), 1\\leq i\\leq p)$, where ${\\mathcal N}_n(\\Delta)$ represents the number of eigenvalues within $\\Delta$, are asymptotically independent as the size $n$ goes to infinity, $p$ being fixed. As a consequence, we prove that the largest and smallest eigenvalues, properly centered and rescaled, are asymptotically independent; we finally describe the fluctuations of the condition number of a matrix from the GUE.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-06T16:03:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52"
    ],
    "keywords": [
      "gaussian unitary ensemble",
      "asymptotic independence",
      "bounded disjoint real borel sets",
      "asymptotically independent",
      "wigners semi-circle law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.0979B"
    }
  }
}