Optimal transport, determinantal point processes and the Bergman kernel

TL;DR

This paper analyzes the Bergman determinantal point process, exploring its properties under restrictions, optimal transport inequalities, and deviations in point counts, providing new bounds and insights into its behavior.

Contribution

It introduces truncated variants of the Bergman kernel, establishes optimal transport inequalities, and offers bounds on point count deviations, addressing open questions in the field.

Findings

Derived bounds on the deviation of point counts in restricted Bergman processes

Established optimal transport inequalities involving the Wasserstein distance

Provided general deviation results for determinantal point processes

Abstract

We study the Bergman determinantal point process from a theoretical point of view motivated by its simulation. We construct restricted and restricted-truncated variants of the Bergman kernel and show optimal transport inequalities involving the Kantorovitch-Rubinstein Wasserstein distance to show to what extent it is fair to truncate the restriction of this point process to a compact ball of radius $1 - \varepsilon $. We also investigate the deviation of the number of points of the restricted Bergman determinantal point process, indicate which number of points looks like an optimal choice, and provide upper bounds on its deviation, providing an answer to an open question asked in [5]. We also consider restrictions to other regions and investigate the choice of such regions for restriction. Finally, we provide general results as to the deviation of the number of points of any determinantal point process.