{
  "id": "0708.2895",
  "version": "v5",
  "published": "2007-08-21T17:49:18.000Z",
  "updated": "2008-02-29T00:52:07.000Z",
  "title": "Random Matrices: The circular Law",
  "authors": [
    "Terence Tao",
    "Van Vu"
  ],
  "comment": "46 pages, no figures, submitted. More minor corrections",
  "categories": [
    "math.PR",
    "math.SP"
  ],
  "abstract": "Let $\\a$ be a complex random variable with mean zero and bounded variance $\\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\\a$. Let $\\lambda_{1}, ..., \\lambda_{n}$ be the eigenvalues of $\\frac{1}{\\sigma \\sqrt n}N_{n}$. Define the empirical spectral distribution $\\mu_{n}$ of $N_{n}$ by the formula $$ \\mu_n(s,t) := \\frac{1}{n} # \\{k \\leq n| \\Re(\\lambda_k) \\leq s; \\Im(\\lambda_k) \\leq t \\}.$$ The Circular law conjecture asserts that $\\mu_{n}$ converges to the uniform distribution $\\mu_\\infty$ over the unit disk as $n$ tends to infinity. We prove this conjecture under the slightly stronger assumption that the $(2+\\eta)\\th$-moment of $\\a$ is bounded, for any $\\eta >0$. Our method builds and improves upon earlier work of Girko, Bai, G\\\"otze-Tikhomirov, and Pan-Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2008-02-29T00:52:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B25"
    ],
    "keywords": [
      "circular law conjecture asserts",
      "sparse random matrices",
      "mean zero",
      "earlier work",
      "method builds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0708.2895T"
    }
  }
}