Spectrum of Large Random Asymmetric Matrices

The average eigenvalue distribution () of N real random asymmetric matrices J₈₉ (J₉₈J₈₉) is calculated in the limit of N. It is found that () is uniform in an ellipse, in the complex plane, whose real and imaginary axes are 1+ and 1-, respectively. The parameter is given by =N{J₈₉J₉₈}₉ and N{J₈₉^2}₉ is normalized to 1. In the =1 limit, Wigner's semicircle law is recovered. The results are extended to complex asymmetric matrices.