Random normal matrices in the almost-circular regime

TL;DR

This paper investigates the eigenvalue distribution and fluctuations of random normal matrices in a narrow band around the unit circle, revealing new scaling limits and fluctuation laws depending on boundary conditions.

Contribution

It derives the scaling limits of correlation functions for radially symmetric potentials in the almost-circular regime, extending previous results and analyzing boundary condition effects.

Findings

Eigenvalues concentrate in a narrow band around the unit circle

Correlation functions exhibit specific scaling limits in the almost-circular regime

Maximal and minimal modulus fluctuations follow Gumbel or exponential laws

Abstract

We study random normal matrix models whose eigenvalues tend to be distributed within a narrow "band" around the unit circle of width proportional to $\frac1n$, where $n$ is the size of matrices. For general radially symmetric potentials with various boundary conditions, we derive the scaling limits of the correlation functions, some of which appear in the previous literature notably in the context of almost-Hermitian random matrices. We also obtain that fluctuations of the maximal and minimal modulus of the ensembles follow the Gumbel or exponential law depending on the boundary conditions.