Hole Probability for Entire Functions represented by Gaussian Taylor Series

TL;DR

This paper derives precise asymptotic estimates for the probability that a Gaussian entire function, represented by a Taylor series with random Gaussian coefficients, has no zeros within a large disc, revealing how this probability decays with radius.

Contribution

It provides exact asymptotics for the decay rate of hole probabilities in Gaussian entire functions with arbitrary non-random coefficients, without regularity assumptions.

Findings

Exact asymptotics for hole probability decay rate

Dependence of exceptional set on non-random coefficients

No regularity conditions needed on coefficients

Abstract

We study the hole probability of Gaussian entire functions. More specifically, we work with entire functions in Taylor series form with i.i.d complex Gaussian random variables and arbitrary non-random 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. The exceptional set depends only on the non-random coefficients. We assume no regularity conditions on the non-random coefficients.