{
  "id": "math/0312043",
  "version": "v1",
  "published": "2003-12-01T23:56:01.000Z",
  "updated": "2003-12-01T23:56:01.000Z",
  "title": "Deviations from the Circular Law",
  "authors": [
    "Brian Rider"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Consider Ginibre's ensemble of $N \\times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\\frac{1}{N}$. As $N \\uparrow \\infty$ the normalized counting measure of the eigenvalues converges to the uniform measure on the unit disk in the complex plane. In this note we describe fluctuations about this {\\em Circular Law}. First we obtain finite $N$ formulas for the covariance of certain linear statistics of the eigenvalues. Asymptotics of these objects coupled with a theorem of Costin and Lebowitz then result in central limit theorems for a variety of these statistics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-12-01T23:56:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "circular law",
      "deviations",
      "non-hermitian random matrices",
      "independent complex gaussians",
      "central limit theorems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math.....12043R"
    }
  }
}