Reach of repulsion for determinantal point processes in high dimensions

Goldman [8] proved that the distribution of a stationary determinantal point process (DPP) $\Phi$ stochastically dominates that of its reduced Palm version $\Phi^{0,!}$. Strassen's theorem then implies the existence of a point process $\eta$ such that $\Phi = \Phi^{0,!} \cup \eta$ in distribution and $\Phi^{0,!} \cap \eta = \emptyset$. The number of points in $\eta$ and their location determine the repulsive nature of a typical point of $\Phi$. In this paper, the repulsive behavior of DPPs in high dimensions is characterized using the measure of repulsiveness defined by the first moment measure of $\eta$. Using this measure, it can be 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. This indicates that these DPPs behave similarly to Poisson point processes in high dimensions. Through a connection with deviation estimates for the norm of high dimensional vectors from their expectation, 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 reach of repulsion of the DPP. The rates for these convergence results can also be computed in several cases. Examples of classes of DPP models exhibiting this behavior are presented and applications to high dimensional Boolean models and nearest neighbor distributions are given.