Reach of Repulsion for Determinantal Point Processes in High Dimensions

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.