Reach of Repulsion for Determinantal Point Processes in High Dimensions

TL;DR

This paper investigates how the repulsive behavior of determinantal point processes (DPPs) diminishes in high-dimensional spaces, identifying a critical radius where repulsion is strongest and demonstrating that in many cases, repulsion effects vanish as dimension increases.

Contribution

The paper introduces the concept of the asymptotic reach of repulsion in high-dimensional DPPs and analyzes the decay of repulsive interactions using the first moment measure.

Findings

Number of repulsive points converges to zero in high dimensions.

Existence of a critical radius where repulsion decay is slowest.

Application to high-dimensional Boolean models.

Abstract

Goldman [7] proved that the distribution of a stationary determinantal point process (DPP) $\Phi$ can be coupled with its reduced Palm version $\Phi^{0,!}$ such that there exists a point process $\eta$ where $\Phi = \Phi^{0,!} \cup \eta$ in distribution and $\Phi^{0,!} \cap \eta = \emptyset$. The points of $\eta$ characterize the repulsive nature of a typical point of $\Phi$. In this paper, the first moment measure of $\eta$ is used to study the repulsive behavior of DPPs in high dimensions. It is shown that many families of DPPs have the property that the total number of points in $\eta$ converges in probability to zero as the space dimension $n$ goes to infinity. It is also proved that for some DPPs there exists an $R^*$ such that the decay of the first moment measure of $\eta$ is slowest in a small annulus around the sphere of radius $\sqrt{n}R^*$. This $R^*$ can be interpreted as the asymptotic reach of repulsion of the DPP. Examples of classes of DPP models exhibiting this behavior are presented and an application to high dimensional Boolean models is given.