  Let X be a translation invariant point process on the complex plane and let D
be a bounded open set whose boundary has zero Lebesgue measure. We ask what
does the point configuration obtained by taking the points of X outside D tell
us about the point configuration inside D? We show that for the Ginibre
ensemble, it determines the number of points in D. For the
translation-invariant zero process of a planar Gaussian Analytic Function, we
show that it determines the number as well as the centre of mass of the points
in D. Further, in both models we prove that the outside says "nothing more"
about the inside, in the sense that the conditional distribution of the inside
points, given the outside, is mutually absolutely continuous with respect to
the Lebesgue measure on its supporting submanifold.

Probability;Statistical Mechanics;Classical Analysis and ODEs