{
  "id": "1901.07614",
  "version": "v1",
  "published": "2019-01-22T21:17:50.000Z",
  "updated": "2019-01-22T21:17:50.000Z",
  "title": "A necessary and sufficient condition for global convergence of the complex zeros of random orthogonal polynomials",
  "authors": [
    "Duncan Dauvergne"
  ],
  "comment": "30 pages",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "Consider random polynomials of the form $G_n = \\sum_{i=0}^n \\xi_i p_i$, where the $\\xi_i$ are i.i.d. non-degenerate complex random variables, and $\\{p_i\\}$ is a sequence of orthonormal polynomials with respect to a regular measure $\\tau$ supported on a compact set $K$. We show that the normalized counting measure of the zeros of $G_n$ converges weakly almost surely to the equilibrium measure of $K$ if and only if $\\mathbb E \\log(1 + |\\xi_0|) < \\infty$. This generalizes the corresponding result of Ibragimov-Zaporozhets in the case when $p_i(z) = z^i$. We also show that the normalized counting measure of the zeros of $G_n$ converges weakly in probability to the equilibrium measure of $K$ if and only if $\\mathbb P (|\\xi_0| > e^n) = o(n^{-1})$. Our proofs rely on results from small ball probability and exploit the structure of general orthogonal polynomials. Our methods also work for sequences of asymptotically minimal polynomials in $L^p(\\tau)$, where $p \\in (0, \\infty]$. In particular, sequences of $L^p$-minimal polynomials and (normalized) Faber and Fekete polynomials fall into this class.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-22T21:17:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "42C05",
      "60B10",
      "60G57"
    ],
    "keywords": [
      "random orthogonal polynomials",
      "sufficient condition",
      "global convergence",
      "complex zeros",
      "non-degenerate complex random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}