Limiting eigenvalue distribution of the general deformed Ginibre ensemble

Consider the n n matrix Xₙ=Aₙ+Hₙ, where Aₙ is a n n matrix (either deterministic or random) and Hₙ is a n n matrix independent from Aₙ drawn from complex Ginibre ensemble. We study the limiting eigenvalue distribution of Xₙ. In arXiv: 0807. 4898 it was shown that the eigenvalue distribution of Xₙ converges to some deterministic measure. This measure is known for the case Aₙ=0. Under some general convergence conditions on Aₙ 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.