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.

34 pages. This preprint has not undergone peer review, see journal version for the correst list of conditions on $A_n$

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