Large deviations for the zero set of an analytic function with diffusing coefficients

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.