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

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.

arXiv admin note: text overlap with arXiv:1211.2381

References in corpus (1)

• Zeros of Random Analytic Functions

Cited by in corpus (4)

• Rigidity of Determinantal Point Processes with the Airy, the Bessel and the Gamma Kernel
• Absolute continuity and singularity of Palm measures of the Ginibre point process
• Equivalence of Palm measures for determinantal point processes governed by Bergman kernels
• Rigid stationary determinantal processes in non-Archimedean fields