Rigidity and Tolerance in Gaussian zeroes and Ginibre eigenvalues: quantitative estimates

TL;DR

This paper refines understanding of the conditional distribution of points in Gaussian zeroes and Ginibre eigenvalues, showing they exhibit quadratic repulsion even under spatial conditioning, with the density comparable to a squared Vandermonde.

Contribution

It provides quantitative estimates demonstrating that the conditional density is comparable to a squared Vandermonde, revealing strong quadratic repulsion in these point processes.

Findings

Conditional density is comparable to squared Vandermonde density.

Points exhibit quadratic mutual repulsion under spatial conditioning.

Refines previous results on absolute continuity of conditional distributions.

Abstract

Let $\Pi$ be a translation invariant point process on the complex plane $\C$ and let $\D \subset \C$ be a bounded open set whose boundary has zero Lebesgue measure. We study the conditional distribution of the points of $\Pi$ inside $\D$ given the points outside $\D$. When $\Pi$ is the Ginibre ensemble or the Gaussian zero process, it been shown in \cite{GP} that this conditional distribution is mutually absolutely continuous with the Lebesgue measure on its support. In this paper, we refine the result in \cite{GP} to show that the conditional density 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.