Reach of repulsion for determinantal point processes in high dimensions

Abstract

Abstract Goldman (2010) proved that the distribution of a stationary determinantal point process (DPP) Φ can be coupled with its reduced Palm version Φ 0,! such that there exists a point process η where Φ=Φ 0,! ∪η in distribution and Φ 0,! ∩η=∅. The points of η characterize the repulsive nature of a typical point of Φ. In this paper we use the first-moment measure of η to study the repulsive behavior of DPPs in high dimensions. We show that many families of DPPs have the property that the total number of points in η converges in probability to 0 as the space dimension n →∞. We also prove that for some DPPs, there exists an R ∗ such that the decay of the first-moment measure of η is slowest in a small annulus around the sphere of radius √ 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.

Locations

• Journal of Applied Probability - View - PDF
• arXiv (Cornell University) - View - PDF