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

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.