Spectrum of Large Random Asymmetric Matrices

TLDR

The authors compute the eigenvalue density of large real asymmetric matrices in the infinite‑size limit, expressing it in terms of a parameter τ defined by the normalized correlation of matrix elements. They find that the eigenvalue density is uniform over an ellipse in the complex plane with axes 1+τ and 1−τ, reduces to Wigner’s semicircle when τ=1, and generalizes to complex 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.