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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 strongerspatial 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 some interesting consequences of our results, with applications to continuum percolation, reconstruction of Gaussian entire
functions, and others. This is based on joint work with Fedor Nazarov, Yuval Peres and Mikhail Sodin.
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