Rigidity and Tolerance in point processes: Gaussian zeroes and Ginibre eigenvalues

Let Π be a translation invariant point process on the complex plane C and let D ⊂ C be a bounded open set. We ask what does the point configuration Πout obtained by taking the points of Π outside D tell us about the point configuration Πin of Π inside D? We show that for the Ginibre ensemble, Πout determines the number of points in Πin. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that Πout determines the number as well as the centre of mass of the points in Πin. 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. 2010 Mathematics Subject Classification. Primary 60G55; Secondary 60B20.