UC Berkeley

Let X be a translation invariant point process on the Euclidean space E and let D, a subset of E, be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration O obtained by taking the points of X outside D tell us about the point configuration I obtained from the points of X inside D? We focus mainly on translation invariant point processes on the plane. We show that for the Ginibre ensemble, O determines the number of points in I. We refer to this kind of behaviour as ``rigidity''. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that O determines the number as well as the centre of mass of the points in I. 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. We further show that the conditional density (of the inside points given the outside) 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. We apply these results to the study of continuum percolation on these point processes, and establish the existence of a non-trivial critical radius and the uniqueness of infinite cluster in the supercritical regime. En route, we obtain new estimates on hole probabilities for zeroes of the planar Gaussian Analytic Function. Finally, we apply these ideas to prove completeness properties of random exponential functions originating from "rigid" determinantal point processes. We conclude by establishing miscellaneous other results on determinantal point processes. These include answers to two questions of Lyons and Steif on certain models of stationary determinantal processes on the integers, one involving insertion and deletion tolerance, and the other regarding the recovery of the driving function from the process.