{
  "id": "0709.0036",
  "version": "v3",
  "published": "2007-09-01T06:57:30.000Z",
  "updated": "2010-05-31T09:08:08.000Z",
  "title": "Circular law for non-central random matrices",
  "authors": [
    "Djalil Chafai"
  ],
  "comment": "accepted in Journal of Theoretical Probability",
  "journal": "Journal of Theoretical Probability 23, 4 (2010) 945-950",
  "doi": "10.1007/s10959-010-0285-8",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(X_{jk})_{j,k\\geq 1}$ be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let $\\la_{n,1},...,\\la_{n,n}$ be the eigenvalues of $(\\frac{1}{\\sqrt{n}}X_{jk})_{1\\leq j,k\\leq n}$. The strong circular law theorem states that with probability one, the empirical spectral distribution $\\frac{1}{n}(\\de_{\\la_{n,1}}+...+\\de_{\\la_{n,n}})$ converges weakly as $n\\to\\infty$ to the uniform law over the unit disc $\\{z\\in\\dC;|z|\\leq1\\}$. In this short note, we provide an elementary argument that allows to add a deterministic matrix $M$ to $(X_{jk})_{1\\leq j,k\\leq n}$ provided that $\\mathrm{Tr}(MM^*)=O(n^2)$ and $\\mathrm{rank}(M)=O(n^\\al)$ with $\\al<1$. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2010-05-31T09:08:08.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "non-central random matrices",
      "strong circular law theorem states",
      "complex random variables",
      "empirical spectral distribution",
      "marchenko-pastur theorems"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0709.0036C"
    }
  }
}