{
  "id": "1802.02390",
  "version": "v1",
  "published": "2018-02-07T11:34:49.000Z",
  "updated": "2018-02-07T11:34:49.000Z",
  "title": "Real zeros of random analytic functions associated with geometries of constant curvature",
  "authors": [
    "Hendrik Flasche",
    "Zakhar Kabluchko"
  ],
  "comment": "26 pages, 1 figure",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\xi_0, \\xi_1, \\dots$ be i.i.d. random variables with zero mean and unit variance. We study the following four families of random analytic functions: $$ P_n(z) := \\begin{cases} \\sum_{k=0}^n \\sqrt{\\binom nk} \\xi_k z^k &\\text{ (spherical polynomials)}, \\sum_{k=0}^\\infty \\sqrt{\\frac{n^k}{k!}} \\xi_k z^k &\\text{ (flat random analytic function)}, \\sum_{k=0}^\\infty \\sqrt{\\binom {n+k-1} k} \\xi_k z^k &\\text{ (hyperbolic random analytic functions)}, \\sum_{k=0}^n \\sqrt{\\frac{n^k}{k!}} \\xi_k z^k &\\text{ (Weyl polynomials)}. \\end{cases} $$ We compute explicitly the limiting mean density of real zeroes of these random functions. More precisely, we provide a formula for $\\lim_{n\\to\\infty} n^{-1/2} \\mathbb{E}N_n[a,b]$, where $N_n[a, b]$ is the number of zeroes of $P_n$ in the interval $[a,b]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-07T11:34:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "30C15",
      "26C10",
      "60F99",
      "60F17",
      "60F05",
      "60G15"
    ],
    "keywords": [
      "random analytic functions",
      "real zeros",
      "constant curvature",
      "geometries",
      "random functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}