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

Jesper Møller, Eliza O'Reilly

Published 2018-06-19Version 1

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 in distribution $X^u$ is obtained by removing a finite number of points from $X$. The intensity function of the difference $X\setminus X^u$ is known, but apart from special cases the distribution of $X\setminus X^u$ is unknown. Considering the restriction $X_S$ of $X$ to any compact set $S$, we establish a coupling of $X_S$ and its reduced Palm process $X^u_S$ so that the difference is at most one point. Specifically, we assume $K$ restricted to $S\times S$ is either (i) a projection or (ii) has spectrum strictly less than one. In case of (i), we have in distribution that $X^u_S$ is obtained by removing one point from $X_S$, and we can specify the distribution of this point. In case of (ii), in distribution we obtain $X^u_S$ either by moving one point in $X$ or by removing one point from $X_S$, and to a certain extent we can describe the distribution of these points. We discuss how Goldman's and our results can be used for quantifying repulsiveness in $X$.