The Ginibre Ensemble Conditioned on an Overcrowding Event

TL;DR

This paper analyzes the eigenvalue distribution of the complex Ginibre Ensemble conditioned on an overcrowding event, providing asymptotic probability estimates and describing the local eigenvalue behavior in different spatial regions.

Contribution

It introduces a method to estimate the probability of overcrowding events and characterizes the eigenvalue distribution conditioned on such events, including near the boundary of the disk.

Findings

Conditional eigenvalue distribution outside the disk is asymptotically Ginibre.

Near the boundary, the distribution converges to a known determinantal process.

Probability estimates for overcrowding events are obtained asymptotically.

Abstract

We look at the eigenvalues of the complex Ginibre Ensemble of random matrices consisting of $N$ eigenvalues. We study the event that for $ {c \in [0,1]}$, $\lfloor cN \rfloor$ of the eigenvalues are located outside of a disk of radius $ R \in (\sqrt{1-c},1)$. Except for the case $c=1$ the eigenvalue process conditioned on this event is not determinantal. Nevertheless we are able to obtain asymptotic estimates of the probability of the event, and describe the conditional distribution in three spatial regions. For $ \{ \lambda \in \mathbb{C} : \big| \lambda \big| <R\}, \{\lambda \in \mathbb{C} : \big| \lambda \big| > R+\epsilon\} $ the conditional distribution is asymptotically that of a Ginibre ensemble. Meanwhile, near the boundary of the disk, after rescaling by a factor of order $ N$, it tends to the determinantal point process that appears in the limit of the Ginibre ensemble near a hard wall in Seo arXiv:2010.08818 [math-ph] .