Universality of Correlations for Random Analytic Functions

TL;DR

This paper demonstrates that the zero sets of certain random analytic functions exhibit universal behavior near the domain boundary, converging to those of Gaussian analytic functions regardless of the distribution of coefficients.

Contribution

It proves the universality of zero set distributions for a broad class of random analytic functions with non-Gaussian coefficients.

Findings

Zero sets converge in distribution to GAF zero sets near the boundary.

Universality holds for non-Gaussian coefficients under certain conditions.

Results extend understanding of random analytic functions beyond Gaussian cases.

Abstract

We review a result obtained with Andrew Ledoan and Marco Merkli. Consider a random analytic function $f(z) = \sum_{n=0}^{\infty} a_n X_n z^n$, where the $X_n$'s are i.i.d., complex valued random variables with mean zero and unit variance, and the coefficients $a_n$ are non-random and chosen so that the variance transforms covariantly under conformal transformations of the domain. If the $X_n$'s are Gaussian, this is called a Gaussian analytic function (GAF). We prove that, even if the coefficients are not Gaussian, the zero set converges in distribution to that of a GAF near the boundary of the domain.