Point processes, hole events, and large deviations: random complex zeros and Coulomb gases

TL;DR

This paper surveys the study of hole events in particle systems, focusing on large deviation principles and recent techniques to understand the likelihood and configurations of large empty regions in complex zeros and Coulomb gases.

Contribution

It reviews recent advances in applying large deviation principles to analyze hole probabilities and configurations in point processes, especially in complex zeros and Coulomb gases.

Findings

Confirmed predictions for large fluctuations in the number of points in Ginibre ensembles.

Highlighted the use of LDP techniques to determine most likely configurations for large holes.

Discussed potential future research directions in the field.

Abstract

We consider particle systems (also known as point processes) on the line and in the plane, and are particularly interested in "hole" events, when there are no particles in a large disk (or some other domain). We survey the extensive work on hole probabilities and the related large deviation principles (LDP), which has been undertaken mostly in the last two decades. We mainly focus on the recent applications of LDP-inspired techniques to the study of hole probabilities, and the determination of the most likely configurations of particles that have large holes. As an application of this approach, we illustrate how one can confirm some of the predictions of Jancovici, Lebowitz, and Manificat for large fluctuation in the number of points for the (two-dimensional) $\beta$-Ginibre ensembles. We also discuss some possible directions for future investigations.