The density of zeros of random power series with stationary complex Gaussian coefficients

TL;DR

This paper investigates the distribution of zeros of random power series with stationary Gaussian coefficients, revealing how spectral properties influence zero density near the convergence boundary.

Contribution

It provides a detailed asymptotic analysis of zero density for stationary Gaussian power series and links spectral measure support to boundary analytic continuation.

Findings

Zero density decreases with coefficient dependence compared to i.i.d. case

Spectral measure's density affects zero distribution near boundary

Support of spectral measure relates to analytic continuation at boundary

Abstract

We study the zeros of random power series with stationary complex Gaussian coefficients, whose spectral measure is absolutely continuous. We analyze the precise asymptotic behavior of the radial density of zeros near the boundary of the circle of convergence. The dependence of the coefficients generally reduces the density of zeros compared with that of the hyperbolic Gaussian analytic function (the i.i.d. coefficients case), where the spectral density and its zeros plays a crucial role in this reduction. We also show the relationship between the support of the spectral measure and the analytic continuation at the boundary of the circle of convergence.