{
  "id": "math/0510237",
  "version": "v1",
  "published": "2005-10-11T22:31:48.000Z",
  "updated": "2005-10-11T22:31:48.000Z",
  "title": "Large deviations for the zero set of an analytic function with diffusing coefficients",
  "authors": [
    "J. Ben Hough"
  ],
  "comment": "24 pages, 5 figures",
  "categories": [
    "math.PR",
    "math.CV"
  ],
  "abstract": "The \"hole probability\" that the zero set of the time dependent planar Gaussian analytic function f(z,t) = sum_(n=0)^infty a_n(t) z^n/sqrt(n!), where a_n(t) are i.i.d. complex valued Ornstein-Uhlenbeck processes, does not intersect a disk of radius R for all 0<t<T decays like exp(-Te^(cR^2)). This result sharply differentiates the zero set of f from a number of canonical evolving planar point processes. For example, the hole probability of the perturbed lattice model {sqrt{\\pi}(m,n) + c zeta_{m,n}: m,n integers} where zeta_(m,n) are i.i.d. Ornstein-Uhlenbeck processes decays like exp(-cTR^4). This stark contrast is also present in the \"overcrowding probability\" that a disk of radius R contains at least N zeros for all 0<t<T.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2005-10-11T22:31:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60G15",
      "60G70"
    ],
    "keywords": [
      "zero set",
      "large deviations",
      "evolving planar point processes",
      "diffusing coefficients",
      "time dependent planar gaussian analytic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2005math.....10237B"
    }
  }
}