{
  "id": "1701.09176",
  "version": "v1",
  "published": "2017-01-31T18:41:35.000Z",
  "updated": "2017-01-31T18:41:35.000Z",
  "title": "On the real spectrum of a product of Gaussian random matrices",
  "authors": [
    "Nick Simm"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Let $X_{m} = G_{1}\\ldots G_{m}$ denote the product of $m$ independent random matrices of size $N \\times N$, with each matrix in the product consisting of independent standard Gaussian variables. Denoting by $N_{\\mathbb{R}}(m)$ the total number of real eigenvalues of $X_{m}$, we show that for $m$ fixed \\begin{equation*} \\mathbb{E}(N_{\\mathbb{R}}(m)) = \\sqrt{\\frac{2Nm}{\\pi}}+O(\\log(N)), \\qquad N \\to \\infty. \\end{equation*} This generalizes a well-known result of Edelman et al. \\cite{EKS94} to all $m>1$. Furthermore, we show that the normalized global density of real eigenvalues converges weakly in expectation to the density of the random variable $|U|^{m}B$ where $U$ is uniform on $[-1,1]$ and $B$ is Bernoulli on $\\{-1,1\\}$. This proves a conjecture of Forrester and Ipsen \\cite{FI16}. The results are obtained by the asymptotic analysis of a certain Meijer G-function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-31T18:41:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "15B52",
      "60B20"
    ],
    "keywords": [
      "gaussian random matrices",
      "real spectrum",
      "independent standard gaussian variables",
      "real eigenvalues converges",
      "independent random matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}