Spectrum of Large Random Asymmetric Matrices

Abstract

The average eigenvalue distribution $\rho \left(\lambda \right)$ of $N\times N$ real random asymmetric matrices $J_{\mathrm{ij}} \left(J_{\mathrm{ji}}\ne J_{\mathrm{ij}}\right)$ is calculated in the limit of $N\to \infty$. It is found that $\rho \left(\lambda \right)$ is uniform in an ellipse, in the complex plane, whose real and imaginary axes are $1+\tau$ and $1-\tau$, respectively. The parameter $\tau$ is given by $\tau =N{\left[J_{\mathrm{ij}}{J}_{\mathrm{ji}}\right]}_J$ and $N{\left[J_{\mathrm{ij}}^2\right]}_J$ is normalized to 1. In the $\tau =1\mathrm{}$ limit, Wigner's semicircle law is recovered. The results are extended to complex asymmetric matrices.