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Goldman [7] proved that the distribution of a stationary determinantal pointprocess (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 thispaper, the first moment measure of $\eta$ is used to study the repulsivebehavior of DPPs in high dimensions. It is shown that many families of DPPshave the property that the total number of points in $\eta$ converges inprobability to zero as the space dimension $n$ goes to infinity. It is alsoproved that for some DPPs there exists an $R^*$ such that the decay of thefirst moment measure of $\eta$ is slowest in a small annulus around the sphereof radius $\sqrt{n}R^*$. This $R^*$ can be interpreted as the asymptotic reachof repulsion of the DPP. Examples of classes of DPP models exhibiting thisbehavior are presented and an application to high dimensional Boolean models isgiven.

Reach of Repulsion for Determinantal Point Processes in High Dimensions