{
  "id": "0810.2994",
  "version": "v3",
  "published": "2008-10-16T19:54:56.000Z",
  "updated": "2009-01-01T17:05:12.000Z",
  "title": "From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices",
  "authors": [
    "Terence Tao",
    "Van Vu"
  ],
  "comment": "25 pages, 8 figures, to appear, Bull. Amer. Math. Soc. Various corrections and referee suggestions incorporated",
  "categories": [
    "math.PR"
  ],
  "abstract": "The famous \\emph{circular law} asserts that if $M_n$ is an $n \\times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the normalized matrix $\\frac{1}{\\sqrt{n}} M_n$ converges almost surely to the uniform distribution on the unit disk $\\{z \\in \\C: |z| \\leq 1 \\}$. After a long sequence of partial results that verified this law under additional assumptions on the distribution of the entries, the full circular law was recently established in \\cite{TVcir2}. In this survey we describe some of the key ingredients used in the establishment of the circular law, in particular recent advances in understanding the Littlewood-Offord problem and its inverse.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-01-01T17:05:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15A52",
      "60G50"
    ],
    "keywords": [
      "spectral distribution",
      "littlewood-offord problem",
      "random matrices",
      "universality",
      "full circular law"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.2994T"
    }
  }
}