The Hole Probability for Gaussian Entire Functions

TL;DR

This paper investigates the probability that a Gaussian entire function has no zeros within a large disc, providing precise asymptotic decay rates for this probability as the radius increases.

Contribution

It derives exact asymptotics for the decay rate of the hole probability for Gaussian entire functions, extending understanding of zero distributions in random entire functions.

Findings

Exact asymptotics for hole probability decay rate

Asymptotic results hold outside a small deterministic exceptional set

Provides insights into zero-free regions of Gaussian entire functions

Abstract

We study the hole probability of Gaussian random entire functions. More specifically, we work with entire functions in Taylor series form with i.i.d complex Gaussian coefficients. A hole is the event where the function has no zeros in a disc of radius r. We find exact asymptotics for the rate of decay of the hole probability for large values of r, outside a small exceptional set (which is deterministic).