Spectrum of large random asymmetric matrices

The average eigenvalue distribution ρ(λ) of N×N real random asymmetric matrices Jij (Jji≠Jij) is calculated in the limit of N→∞. It is found that ρ(λ) is uniform in an ellipse, in the complex plane, whose real and imaginary axes are 1+τ and 1−τ, respectively. The parameter τ is given by τ=N[JijJji]J and N[J2ij]J is normalized to 1. In the τ=1 limit, Wigner’s semicircle law is recovered. The results are extended to complex asymmetric matrices.

Authors: H. J. Sommers, A. Crisanti, H. Sompolinsky, and Y. Stein
Year of publication: 1988
Journal: Phys. Rev. Lett. 60, 1895 – Published 9 May 1988

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