A Universality Property of Gaussian Analytic Functions

TL;DR

This paper demonstrates a universality property of zero sets of random analytic functions on the unit disk, showing that their distribution near the boundary converges to a universal form regardless of coefficient distribution.

Contribution

It proves that the zero set distribution of general random analytic functions near the boundary matches the Gaussian case, revealing a universal behavior.

Findings

Zero sets form a determinantal point process near the boundary

Universality holds for a broad class of coefficient distributions

Elementary and general proof technique

Abstract

We consider random analytic functions defined on the unit disk of the complex plane as power series such that the coefficients are i.i.d., complex valued random variables, with mean zero and unit variance. For the case of complex Gaussian coefficients, Peres and Vir\'ag showed that the zero set forms a determinantal point process with the Bergman kernel. We show that for general choices of random coefficients, the zero set is asymptotically given by the same distribution near the boundary of the disk, which expresses a universality property. The proof is elementary and general.