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

TL;DR

This paper investigates how the configuration of points outside a bounded region in certain translation-invariant point processes determines specific internal properties, revealing rigidity and tolerance phenomena in Gaussian zeros and Ginibre eigenvalues.

Contribution

It establishes that the outside configuration determines the number and center of mass of points inside a region for specific point processes, and shows the conditional distribution is mutually absolutely continuous.

Findings

Ginibre ensemble outside configuration determines the number of points inside.

Gaussian Analytic Function zeros outside determine both number and center of mass.

Conditional distribution of inside points given outside is mutually absolutely continuous.

Abstract

Let X be a translation invariant point process on the complex plane and let D be a bounded open set whose boundary has zero Lebesgue measure. We ask what does the point configuration obtained by taking the points of X outside D tell us about the point configuration inside D? We show that for the Ginibre ensemble, it determines the number of points in D. For the translation-invariant zero process of a planar Gaussian Analytic Function, we show that it determines the number as well as the centre of mass of the points in D. 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.