Asymptotics of The Hole Probability for Zeros of Random Entire Functions

TL;DR

This paper investigates the asymptotic decay of the probability that a Gaussian random entire function has no zeros within a large disc, revealing precise exponential decay rates and exploring different coefficient models.

Contribution

It provides the first detailed asymptotic analysis of the hole probability for Gaussian random entire functions, including the flat model and other coefficient distributions.

Findings

Logarithm of hole probability decays like -1/4 * e^2 * r^4 for large r

Asymptotic behavior is established for the flat model

Results extend to other coefficient distributions

Abstract

We study the hole probability of Gaussian random entire functions. More specifically, we work with the flat model (the zero set of this function has a distribution which is invariant with respect to the plane isometries). A hole is the event where the function has no zeros in a disc of radius r. We show that the logarithm of the probability of the hole event decays asymptotically like -1/4 * e^2 * r^4 + o(r^4). We also study the behavior of the hole probability with other types of random coefficients.