Spectrum of Large Random Asymmetric Matrices

Abstract

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