Universality of the second correlation function of the deformed Ginibre ensemble

TL;DR

This paper proves that the local eigenvalue correlations in the bulk of the deformed Ginibre ensemble are universal and match those of the pure Ginibre ensemble under broad conditions.

Contribution

It establishes the universality of the second eigenvalue correlation function for a wide class of deformed Ginibre matrices.

Findings

Asymptotic local behavior matches the pure Ginibre ensemble

Universality holds under general assumptions on the deformation matrix

Results extend understanding of eigenvalue statistics in non-Hermitian random matrices

Abstract

We study the deformed complex Ginibre ensemble $H=A_0+H_0$, where $H_0$ is the complex matrix with iid Gaussian entries, and $A_0$ is some general $n\times n$ matrix (it can be random and in this case it is independent of $H_0$). Assuming rather general assumptions on $A_0$, we prove that the asymptotic local behavior of the second correlation function of the eigenvalues of such matrices in the bulk coincides with that for the pure complex Ginibre ensemble.