ALMOST-HERMITIAN RANDOM MATRICES - EIGENVALUE DENSITY IN THE COMPLEX-PLANE

We consider an ensemble of large non-Hermitian random matrices of the form (H) over cap + i (A) over cap(s), where (H) over cap and (A) over cap(s) are Hermitian statistically independent random N x N matrices. We demonstrate the existence of a new nontrivial regime of weak non-H ermiticity characterized by the condition that the average of NTr (A) over cap(s)(2) is of the same order as that of Tr (H) over cap(2) when N --> infinity. We find explicitly the density of complex eigenvalues for this regime in the limit of infinite matrix dimension. The densit y determines the eigenvalue distribution in the crossover regime betwe en random Hermitian matrices whose real eigenvalues are distributed ac cording to the Wigner semi-circle law and random complex matrices whos e eigenvalues are distributed in the complex plane according to the so -called ''elliptic law''.