  Let $\Pi$ be a translation invariant point process on the complex plane $\C$
and let $\D \subset \C$ be a bounded open set whose boundary has zero Lebesgue
measure. We study the conditional distribution of the points of $\Pi$ inside
$\D$ given the points outside $\D$. When $\Pi$ is the Ginibre ensemble or the
Gaussian zero process, it been shown in \cite{GP} that this conditional
distribution is mutually absolutely continuous with the Lebesgue measure on its
support. In this paper, we refine the result in \cite{GP} to show that the
conditional density is, roughly speaking, comparable to a squared Vandermonde
density. In particular, this shows that even under spatial conditioning, the
points exhibit repulsion which is quadratic in their mutual separation.
