{
  "id": "1207.5570",
  "version": "v2",
  "published": "2012-07-24T01:00:30.000Z",
  "updated": "2014-10-28T11:56:37.000Z",
  "title": "A large deviation principle for Wigner matrices without Gaussian tails",
  "authors": [
    "Charles Bordenave",
    "Pietro Caputo"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/13-AOP866 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)",
  "journal": "Annals of Probability 2014, Vol. 42, No. 6, 2454-2496",
  "doi": "10.1214/13-AOP866",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider $n\\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\\mathbb {P}(|X_{ij}|\\geq t)$ behave like $e^{-at^{\\alpha}}$ for some $a>0$ and $\\alpha \\in(0,2)$. We establish a large deviation principle for the empirical spectral measure of $X/\\sqrt{n}$ with speed $n^{1+\\alpha /2}$ with a good rate function $J(\\mu)$ that is finite only if $\\mu$ is of the form $\\mu=\\mu_{\\mathrm{sc}}\\boxplus\\nu$ for some probability measure $\\nu$ on $\\mathbb {R}$, where $\\boxplus$ denotes the free convolution and $\\mu_{\\mathrm{sc}}$ is Wigner's semicircle law. We obtain explicit expressions for $J(\\mu_{\\mathrm{sc}}\\boxplus\\nu)$ in terms of the $\\alpha$th moment of $\\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-07-24T01:00:30.000Z",
      "title": "A large deviations principle for Wigner matrices without gaussian tails",
      "abstract": "We consider $n\\times n$ hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\\dP(|X_{ij}|\\geq t)$ behave like $e^{-a t^\\alpha}$ for some $a>0$ and $\\alpha\\in(0,2)$. We establish a large deviations principle for the empirical spectral measure of $X/\\sqrt{n}$ with speed $n^{1+\\alpha/2}$ with a good rate function $J(\\mu)$ that is finite only if $\\mu$ is of the form $\\mu=\\mu_{sc} \\boxplus \\nu$ for some probability measure $\\nu$ on $\\dR$, where $\\boxplus $ denotes the free convolution and $\\mu_{sc}$ is Wigner's semicircle law. We obtain explicit expressions for $J(\\mu_{sc} \\boxplus \\nu)$ in terms of the $\\alpha$-th moment of $\\nu$. The proof is based on the analysis of large deviations for the empirical distribution of very sparse random rooted networks.",
      "comment": "31 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-10-28T11:56:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "47A10",
      "15A18",
      "05C80"
    ],
    "keywords": [
      "large deviations principle",
      "gaussian tails",
      "wigner matrices",
      "sparse random rooted networks",
      "wigners semicircle law"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1207.5570B"
    }
  }
}