ALMOST HERMITIAN RANDOM MATRICES - CROSSOVER FROM WIGNER-DYSON TO GINIBRE EIGENVALUE STATISTICS

By using the method of orthogonal polynomials, we analyze the statisti cal properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner-Dyson st atistics of real eigenvalues to strongly non-Hermitian ones whose comp lex eigenvalues were studied by Ginibre. Two-point statistical measure s [as, e.g., spectral form factor, number variance, and small distance behavior of the nearest neighbor distance distribution p(s)] are stud ied in more detail. In particular, we found that the latter function m ay exhibit unusual behavior p(s) proportional to s(5/2) for some param eter values.