{
  "id": "1710.00937",
  "version": "v1",
  "published": "2017-10-02T23:10:15.000Z",
  "updated": "2017-10-02T23:10:15.000Z",
  "title": "Natural boundary and zero distribution of random polynomials in smooth domains",
  "authors": [
    "Igor Pritsker",
    "Koushik Ramachandran"
  ],
  "comment": "9 pages",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "We consider the zero distribution of random polynomials of the form $P_n(z) = \\sum_{k=0}^n a_k B_k(z)$, where $\\{a_k\\}_{k=0}^{\\infty}$ are non-trivial i.i.d. complex random variables with mean $0$ and finite variance. Polynomials $\\{B_k\\}_{k=0}^{\\infty}$ are selected from a standard basis such as Szeg\\H{o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ whose boundary is $C^{2, \\alpha}$ smooth. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$. We also show that if $\\{a_k\\}_{k=0}^{\\infty}$ are i.i.d. random variables, and the domain $G$ has analytic boundary, then for a random series of the form $f(z) =\\sum_{k=0}^{\\infty}a_k B_k(z),$ $\\partial{G}$ is almost surely a natural boundary for $f(z).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-02T23:10:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60F15",
      "30B40"
    ],
    "keywords": [
      "natural boundary",
      "zero distribution",
      "random polynomials",
      "smooth domains",
      "complex random variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}