{
  "id": "0807.4898",
  "version": "v5",
  "published": "2008-07-30T17:23:48.000Z",
  "updated": "2009-04-23T22:17:37.000Z",
  "title": "Random matrices: Universality of ESDs and the circular law",
  "authors": [
    "Terence Tao",
    "Van Vu",
    "Manjunath Krishnapur"
  ],
  "comment": "45 pages, 8 figures, submitted, Acta Math. The main article is by Tao and Vu, the appendix is by Krishnapur, and the figures are by Phillip Wood. A simplified proof of the replacement principle added; some other corrections",
  "categories": [
    "math.PR"
  ],
  "abstract": "Given an $n \\times n$ complex matrix $A$, let $$\\mu_{A}(x,y):= \\frac{1}{n} |\\{1\\le i \\le n, \\Re \\lambda_i \\le x, \\Im \\lambda_i \\le y\\}|$$ be the empirical spectral distribution (ESD) of its eigenvalues $\\lambda_i \\in \\BBC, i=1, ... n$. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD $\\mu_{\\frac{1}{\\sqrt{n}} A_n}$ of a random matrix $A_n = (a_{ij})_{1 \\leq i,j \\leq n}$ where the random variables $a_{ij} - \\E(a_{ij})$ are iid copies of a fixed random variable $x$ with unit variance. We prove a \\emph{universality principle} for such ensembles, namely that the limit distribution in question is {\\it independent} of the actual choice of $x$. In particular, in order to compute this distribution, one can assume that $x$ is real of complex gaussian. As a related result, we show how laws for this ESD follow from laws for the \\emph{singular} value distribution of $\\frac{1}{\\sqrt{n}} A_n - zI$ for complex $z$. As a corollary we establish the Circular Law conjecture (in both strong and weak forms), that asserts that $\\mu_{\\frac{1}{\\sqrt{n}} A_n}$ converges to the uniform measure on the unit disk when the $a_{ij}$ have zero mean.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2009-04-23T22:17:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52"
    ],
    "keywords": [
      "random matrices",
      "universality",
      "sure convergence sense",
      "circular law conjecture",
      "complex matrix"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 45,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0807.4898T"
    }
  }
}