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For a determinantal point process $X$ with a kernel $K$ whose spectrum isstrictly less than one, Andr{\'e} Goldman has established a coupling to itsreduced 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 sharpenthis result, assuming weaker conditions and establishing that $X^u$ can beobtained by removing at most one point from $X$, where we specify thedistribution of the difference $\xi_u:=X\setminus X^u$. This is used fordiscussing the degree of repulsiveness in DPPs in terms of $\xi_u$, includingGinibre point processes and other specific parametric models for DPPs.

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