Hole probability for zeroes of Gaussian Taylor series with finite radii of convergence

TL;DR

This paper investigates the probability that certain Gaussian Taylor series with specific coefficient behaviors do not vanish within the unit disk, providing bounds that depend on the growth or decay of their coefficients.

Contribution

It offers new bounds on hole probabilities for Gaussian Taylor series with coefficients exhibiting power-law behavior, extending understanding of zero distributions.

Findings

Bounds depend on coefficient growth or decay

Results apply to power-law coefficient behavior

Zero set invariance under disk isometries

Abstract

We study a family of random Taylor series $$F(z) = \sum_{n\ge 0} \zeta_n a_n z^n$$ with radius of convergence almost surely $1$ and independent identically distributed complex Gaussian coefficients $(\zeta_n)$; these Taylor series are distinguished by the invariance of their zero sets with respect to isometries of the unit disk. We find reasonably tight upper and lower bounds on the probability that $F$ does not vanish in the disk $\{|z|\le r\}$ as $r\uparrow 1$. Our bounds take different forms according to whether the non-random coefficients $(a_n)$ grow, decay or remain of the same order. The results apply more generally to a class of Gaussian Taylor series whose coefficients $(a_n)$ display power-law behavior.