Limiting eigenvalue distribution of the general deformed Ginibre ensemble

TL;DR

This paper investigates the eigenvalue distribution of a sum of a deterministic or random matrix and a Ginibre matrix, deriving a formula for the limiting distribution's density and estimating convergence rates using supersymmetric integration.

Contribution

It provides a general formula for the density of the limiting eigenvalue distribution of deformed Ginibre matrices under broad conditions, extending previous results.

Findings

Derived explicit density formula for the limiting eigenvalue distribution.

Established convergence rate estimates for the eigenvalue distribution.

Extended known results to more general matrix deformations.

Abstract

Consider the $n\times n$ matrix $X_n=A_n+H_n$, where $A_n$ is a $n\times n$ matrix (either deterministic or random) and $H_n$ is a $n\times n$ matrix independent from $A_n$ drawn from complex Ginibre ensemble. We study the limiting eigenvalue distribution of $X_n$. In arXiv:0807.4898 it was shown that the eigenvalue distribution of $X_n$ converges to some deterministic measure. This measure is known for the case $A_n=0$. Under some general convergence conditions on $A_n$ we prove a formula for the density of the limiting measure. We also obtain an estimation on the rate of convergence of the distribution. The approach used here is based on supersymmetric integration.