  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.
