{
  "id": "0707.2159",
  "version": "v1",
  "published": "2007-07-14T16:21:31.000Z",
  "updated": "2007-07-14T16:21:31.000Z",
  "title": "The spectrum of heavy-tailed random matrices",
  "authors": [
    "Gerard Ben Arous",
    "Alice Guionnet"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $X_N$ be an $N\\ts N$ random symmetric matrix with independent equidistributed entries. If the law $P$ of the entries has a finite second moment, it was shown by Wigner \\cite{wigner} that the empirical distribution of the eigenvalues of $X_N$, once renormalized by $\\sqrt{N}$, converges almost surely and in expectation to the so-called semicircular distribution as $N$ goes to infinity. In this paper we study the same question when $P$ is in the domain of attraction of an $\\alpha$-stable law. We prove that if we renormalize the eigenvalues by a constant $a_N$ of order $N^{\\frac{1}{\\alpha}}$, the corresponding spectral distribution converges in expectation towards a law $\\mu_\\alpha$ which only depends on $\\alpha$. We characterize $\\mu_\\alpha$ and study some of its properties; it is a heavy-tailed probability measure which is absolutely continuous with respect to Lebesgue measure except possibly on a compact set of capacity zero.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-07-14T16:21:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "60E07"
    ],
    "keywords": [
      "heavy-tailed random matrices",
      "finite second moment",
      "random symmetric matrix",
      "corresponding spectral distribution converges",
      "capacity zero"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0707.2159B"
    }
  }
}