{
  "id": "1205.5355",
  "version": "v1",
  "published": "2012-05-24T07:47:34.000Z",
  "updated": "2012-05-24T07:47:34.000Z",
  "title": "Universality for zeros of random analytic functions",
  "authors": [
    "Zakhar Kabluchko",
    "Dmitry Zaporozhets"
  ],
  "comment": "26 pages, 8 figures, 1 table",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "Let $\\xi_0,\\xi_1,...$ be independent identically distributed (i.i.d.) random variables such that $\\E \\log (1+|\\xi_0|)<\\infty$. We consider random analytic functions of the form $$ G_n(z)=\\sum_{k=0}^{\\infty} \\xi_k f_{k,n} z^k, $$ where $f_{k,n}$ are deterministic complex coefficients. Let $\\nu_n$ be the random measure assigning the same weight $1/n$ to each complex zero of $G_n$. Assuming essentially that $-\\frac 1n \\log f_{[tn], n}\\to u(t)$ as $n\\to\\infty$, where $u(t)$ is some function, we show that the measure $\\nu_n$ converges weakly to some deterministic measure which is characterized in terms of the Legendre--Fenchel transform of $u$. The limiting measure is universal, that is it does not depend on the distribution of the $\\xi_k$'s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-24T07:47:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30B20",
      "26C10",
      "65H04",
      "60G57",
      "60B10",
      "60B20"
    ],
    "keywords": [
      "random analytic functions",
      "universality",
      "deterministic complex coefficients",
      "random polynomial analogue",
      "random measure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.5355K"
    }
  }
}