Large-$N$ Eigenvalue Distribution of Randomly Perturbed Asymmetric Matrices

The density of complex eigenvalues of random asymmetric $N\times N$ matrices is found in the large-$N$ limit. The matrices are of the form $H_0+A$ where $A$ is a matrix of $N^2$ independent, identically distributed random variables with zero mean and variance $N^{-1}v^2$. The limiting density $\rho (z,z^*)$ is bounded. The area of the support of $\rho (z,z^*)$ cannot be less than $\pi v^2$. In the case of $H_0$ commuting with its conjugate, $\rho (z,z^*)$ is expressed in terms of the eigenvalue distribution of the non-perturbed part $H_0$.