{
  "id": "1711.02842",
  "version": "v1",
  "published": "2017-11-08T05:53:53.000Z",
  "updated": "2017-11-08T05:53:53.000Z",
  "title": "Random matrices: Probability of Normality",
  "authors": [
    "Andrei Deneanu",
    "Van Vu"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "In this paper, we investigate the following question: How often is a random matrix normal? We consider a random $n\\times n$ matrix, $M_n$, whose entries are i.i.d. Rademacher random variables (taking values $\\{ \\pm1 \\}$ with probability $1/2$) and prove $$2^{-\\left(0.5+o(1)\\right)n^2} \\le P\\left(M_n \\text{ is normal}\\right) \\le 2^{-(0.302+o(1))n^{2}}. $$ We conjecture that the lower bound is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-08T05:53:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "probability",
      "random matrix normal",
      "rademacher random variables",
      "lower bound",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}