Couplings for determinantal point processes and their reduced Palm distributions with a view to quantifying repulsiveness

TL;DR

This paper refines the understanding of how reduced Palm processes relate to original determinantal point processes, showing that under weaker conditions, the process can be obtained by removing at most one point, aiding in quantifying repulsiveness.

Contribution

It establishes that the reduced Palm process of a DPP can be obtained by removing at most one point, under weaker conditions, and characterizes the distribution of this point removal.

Findings

Reduced Palm process can be obtained by removing at most one point.

The distribution of the removed point is explicitly characterized.

Application to quantifying the degree of repulsiveness in DPPs.

Abstract

For a determinantal point process $X$ with a kernel $K$ whose spectrum is strictly less than one, Andr{\'e} Goldman has established a coupling to its reduced Palm process $X^u$ at a point $u$ with $K(u,u)>0$ so that almost surely $X^u$ is obtained by removing a finite number of points from $X$. We sharpen this result, assuming weaker conditions and establishing that $X^u$ can be obtained by removing at most one point from $X$, where we specify the distribution of the difference $\xi_u:=X\setminus X^u$. This is used for discussing the degree of repulsiveness in DPPs in terms of $\xi_u$, including Ginibre point processes and other specific parametric models for DPPs.