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

TL;DR

This paper studies the probabilities of large deviations in the zero set of a time-dependent Gaussian analytic function with diffusing coefficients, revealing sharp contrasts with other planar point processes.

Contribution

It provides precise asymptotics for hole and overcrowding probabilities of zeros in a Gaussian analytic function with Ornstein-Uhlenbeck coefficients, highlighting unique large deviation behaviors.

Findings

Hole probability decays like exp(-T e^{c R^2})

Overcrowding probability decay differs significantly from lattice models

Zero set behavior sharply contrasts with canonical point processes

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.