Fluctuations for Zeros of Gaussian Taylor Series

TL;DR

This paper investigates the fluctuations in the number of zeros of Gaussian Taylor series, providing sharp bounds on variance growth and employing Wiman-Valiron theory to handle general coefficient variances.

Contribution

It introduces sharp bounds for zero count fluctuations in Gaussian Taylor series and extends results without assumptions on coefficient variances using Wiman-Valiron theory.

Findings

Sharp bounds for variance growth in entire Gaussian Taylor series

Applicability of results under no assumptions on coefficient variance

Validation of bounds through analysis of admissible covariance kernels

Abstract

We study fluctuations in the number of zeros of random analytic functions given by a Taylor series whose coefficients are independent complex Gaussians. When the functions are entire, we find sharp bounds for the asymptotic growth rate of the variance of the number of zeros in large disks centered at the origin. To obtain a result that holds under no assumptions on the variance of the Taylor coefficients we employ the Wiman-Valiron theory. We demonstrate the sharpness of our bounds by studying well-behaved covariance kernels, which we call admissible (after Hayman).