Limiting eigenvalue distribution of the general deformed Ginibre ensemble

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.