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In a Poisson point process we have independence between disjoint spatial domains, so the points outside a disk give us no information on the points inside. The story gets a lot more interesting for processes with stronger spatial correlation. In the case of Ginibre ensemble, a process arising from eigenvalues of random matrices, we prove that the outside points determine exactly the number of points inside, and further, we demonstrate that they determine nothing more. In the case of zero ensembles of Gaussian power series, we prove that the outside points determine exactly the number and the centre of mass of the inside points, and nothing further. These phenomena suggest a certain hierarchy of point processes according to their rigidity; Poisson, Ginibre and the Gaussian power series fit in at levels 0, 1 and 2 in this ladder. Time permitting, we will also look at several interesting consequences of our results, with applications to continuum percolation, reconstruction of Gaussian entire functions, completeness of random exponentials, and others. Based on joint works with Manju Krishnapur, Fedor Nazarov, Yuval Peres and Misha Sodin.