  Determinantal point processes (DPPs) have recently proved to be a useful
class of models in several areas of statistics, including spatial statistics,
statistical learning and telecommunications networks. They are models for
repulsive (or regular, or inhibitive) point processes, in the sense that nearby
points of the process tend to repel each other. We consider two ways to
quantify the repulsiveness of a point process, both based on its second-order
properties, and we address the question of how repulsive a stationary DPP can
be. We determine the most repulsive stationary DPP, when the intensity is
fixed, and for a given $R>0$ we investigate repulsiveness in the subclass of
$R$-dependent stationary DPPs, that is, stationary DPPs with $R$-compactly
supported kernels. Finally, in both the general case and the $R$-dependent
case, we present some new parametric families of stationary DPPs that can cover
a large range of DPPs, from the stationary Poisson process (the case of no
interaction) to the most repulsive DPP.
