Zeros of Random Analytic Functions

TL;DR

This thesis explores zeros of random analytic functions, including Gaussian and non-Gaussian types, analyzing their distribution, stationarity, asymptotic behavior, and large deviations across different geometric domains.

Contribution

It introduces methods to generate non-Gaussian stationary zero sets and studies their distribution, asymptotic properties, and large deviations, extending the understanding of random analytic functions.

Findings

Zero sets can be stationary in various symmetric spaces.

Distribution of zeros can belong to determinantal point processes.

Asymptotic normality holds for smooth statistics of zeros.

Abstract

The dominant theme of this thesis is that random matrix valued analytic functions, generalizing both random matrices and random analytic functions, for many purposes can (and perhaps should) be effectively studied in that level of generality. We study zeros of random analytic functions in one complex variable. It is known that there is a one parameter family of Gaussian analytic functions with zero sets that are stationary in each of the three symmetric spaces, namely the plane, the sphere and the unit disk, under the corresponding group of isometries. We show a way to generate non Gaussian random analytic functions whose zero sets are also stationary in the same domains. There are particular cases where the exact distribution of the zero set turns out to belong to an important class of point processes known as determinantal point processes. Apart from questions regarding the exact distribution of zero sets, we also study certain asymptotic properties. We show asymptotic normality for smooth statistics applied to zeros of these random analytic functions. Lastly, we present some results on certain large deviation problems for the zeros of the planar and hyperbolic Gaussian analytic functions.