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

Determinantal Point Processes in the Flat Limit: Extended L-ensembles, Partial-Projection DPPs and Universality Classes