  Determinantal point processes (DPPs) are repulsive point processes where the
interaction between points depends on the determinant of a positive-semi
definite matrix. The contributions of this paper are two-fold. First of all, we
introduce the concept of extended L-ensemble, a novel representation of DPPs.
These extended L-ensembles are interesting objects because they fix some
pathologies in the usual formalism of DPPs, for instance the fact that
projection DPPs are not L-ensembles. Every (fixed-size) DPP is an (fixed-size)
extended L-ensemble, including projection DPPs. This new formalism enables to
introduce and analyze a subclass of DPPs, called partial-projection DPPs.
Secondly, with these new definitions in hand, we first show that
partial-projection DPPs arise as perturbative limits of L-ensembles, that is,
limits in $\varepsilon \rightarrow 0$ of L-ensembles based on matrices of the
form $\varepsilon \mathbf{A} + \mathbf{B}$ where $\mathbf{B}$ is low-rank. We
generalise this result by showing that partial-projection DPPs also arise as
the limiting process of L-ensembles based on kernel matrices, when the kernel
function becomes flat (so that every point interacts with every other point, in
a sense). We show that the limiting point process depends mostly on the
smoothness of the kernel function. In some cases, the limiting process is even
universal, meaning that it does not depend on specifics of the kernel function,
but only on its degree of smoothness.
