{
  "id": "1407.6523",
  "version": "v1",
  "published": "2014-07-24T10:53:24.000Z",
  "updated": "2014-07-24T10:53:24.000Z",
  "title": "Asymptotic distribution of complex zeros of random analytic functions",
  "authors": [
    "Zakhar Kabluchko",
    "Dmitry Zaporozhets"
  ],
  "comment": "Published in at http://dx.doi.org/10.1214/13-AOP847 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: substantial text overlap with arXiv:1205.5355",
  "journal": "Annals of Probability 2014, Vol. 42, No. 4, 1374-1395",
  "doi": "10.1214/13-AOP847",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\xi_0,\\xi_1,\\ldots$ be independent identically distributed complex- valued random variables such that $\\mathbb{E}\\log(1+|\\xi _0|)<\\infty$. We consider random analytic functions of the form \\[\\mathbf{G}_n(z)=\\sum_{k=0}^{\\infty}\\xi_kf_{k,n}z^k,\\] where $f_{k,n}$ are deterministic complex coefficients. Let $\\mu_n$ be the random measure counting the complex zeros of $\\mathbf{G}_n$ according to their multiplicities. Assuming essentially that $-\\frac{1}{n}\\log f_{[tn],n}\\to u(t)$ as $n\\to\\infty$, where $u(t)$ is some function, we show that the measure $\\frac{1}{n}\\mu_n$ converges in probability to some deterministic measure $\\mu$ which is characterized in terms of the Legendre-Fenchel transform of $u$. The limiting measure $\\mu$ 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": "2014-07-24T10:53:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random analytic functions",
      "complex zeros",
      "asymptotic distribution",
      "deterministic complex coefficients",
      "random polynomial analogue"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.6523K"
    }
  }
}