{
  "id": "1609.07679",
  "version": "v1",
  "published": "2016-09-24T21:48:14.000Z",
  "updated": "2016-09-24T21:48:14.000Z",
  "title": "Complex Random Matrices have no Real Eigenvalues",
  "authors": [
    "Kyle Luh"
  ],
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "Let $\\zeta = \\xi + i\\xi'$ where $\\xi, \\xi'$ are iid copies of a mean zero, variance one, subgaussian random variable. Let $N_n$ be a $n \\times n$ random matrix with iid entries $\\zeta_{ij} = \\zeta$. We prove that there exists a $c \\in (0,1)$ such that the probability that $N_n$ has any real eigenvalues is less than $c^n$. The bound is optimal up to the value of the constant $c$. The principal component of the proof is an optimal tail bound on the least singular value of matrices of the form $M_n := M + N_n$ where $M$ is a deterministic complex matrix with $\\|M\\| \\leq K n^{1/2}$ for some constant $K$. For this class of random variables, this result improves on the results of Pan and Zhou. In the proof of the tail bound, we develop an optimal small-ball probability bound for complex random variables that generalizes the Littlewood-Offord theory developed by Tao-Vu and Rudelson-Vershynin.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-24T21:48:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex random matrices",
      "real eigenvalues",
      "random matrix",
      "optimal small-ball probability bound",
      "complex random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}