  We prove a Poisson process approximation result for stabilizing functionals
of a determinantal point process. Our results use concrete couplings of
determinantal processes with different Palm measures and exploit their
association properties. Second, we focus on the Ginibre process and show in the
asymptotic scenario of an increasing window size that the process of points
with a large nearest neighbor distance converges after a suitable scaling to a
Poisson point process. As a corollary, we obtain the scaling of the maximum
nearest neighbor distance in the Ginibre process, which turns out to be
different from its analogue for independent points.
