{
  "id": "1503.07926",
  "version": "v1",
  "published": "2015-03-26T23:13:39.000Z",
  "updated": "2015-03-26T23:13:39.000Z",
  "title": "What is the probability that a large random matrix has no real eigenvalues?",
  "authors": [
    "Eugene Kanzieper",
    "Mihail Poplavskyi",
    "Carsten Timm",
    "Roger Tribe",
    "Oleg Zaboronski"
  ],
  "comment": "23 pages, 1 figure",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study the large-$n$ limit of the probability $p_{2n,2k}$ that a random $2n\\times 2n$ matrix sampled from the real Ginibre ensemble has $2k$ real eigenvalues. We prove that, $$\\lim_{n\\rightarrow \\infty}\\frac {1}{\\sqrt{2n}} \\log p_{2n,2k}=\\lim_{n\\rightarrow \\infty}\\frac {1}{\\sqrt{2n}} \\log p_{2n,0}= -\\frac{1}{\\sqrt{2\\pi}}\\zeta\\left(\\frac{3}{2}\\right),$$ where $\\zeta$ is the Riemann zeta-function. Moreover, for any sequence of non-negative integers $(k_n)_{n\\geq 1}$, $$\\lim_{n\\rightarrow \\infty}\\frac {1}{\\sqrt{2n}} \\log p_{2n,2k_n}=-\\frac{1}{\\sqrt{2\\pi}}\\zeta\\left(\\frac{3}{2}\\right),$$ provided $\\lim_{n\\rightarrow \\infty} \\left(n^{-1/2}\\log(n)\\right) k_{n}=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-26T23:13:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20"
    ],
    "keywords": [
      "large random matrix",
      "real eigenvalues",
      "probability",
      "riemann zeta-function",
      "real ginibre ensemble"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1357230
    }
  }
}