RIGIDITY AND TOLERANCE IN POINT PROCESSES: GAUSSIAN ZEROS AND GINIBRE EIGENVALUES

摘要

Let Pi be a translation-invariant point process on the complex plane C, and let D subset of C be a bounded open set. We ask the following: What does the point configuration Pi(out) obtained by taking the points of Pi outside dJ tell us about the point configuration Pi(in) of Pi inside D? We show that, for the Ginibre ensemble, Pi(out) determines the number of points in Pi(in). For the translation-invariant zero process of a planar Gaussian analytic function, we show that Pi(out) determines the number as well as the center of mass of the points in Pi(in). 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.