Zeros of Gaussian Analytic Functions and Determinantal Point Processes
The primary objects of study in this book are point processes, which are random variables taking values in the space of discrete subsets of a metric space, where, by a discrete set we mean a countable set with no accumulation points. Precise definitions of relevant notions will be given later.
Many physical phenomena can be modeled by random discrete sets. For example, the arrival times of people in a queue, the arrangement of stars in a galaxy, energy levels of heavy nuclei of atoms etc.
This calls upon probabilists to find point processes that can be mathematically analysed in some detail, as well as capture various qualitative properties of naturally occurring random point sets.
The single most important such process, known as the Poisson process has been widely studied and applied.
The Poisson process is characterized by independence of the process when restricted to disjoint subsets of the underlying space. In this book we focus on two classes of processes where points repel each other: Zeros of random power series with Gaussian coefficients, and determinantal processes. Special attention is given to the zeros of the hyperbolic Gaussian analytic function, which is in the intersection of the two classes. Probabilistic methods are emphasized whenever possible, and an algorithm to generate samples from a determinantal process is included.