Almost Hermitian Random Matrices: Crossover from Wigner-Dyson to Ginibre Eigenvalue Statistics

By using the method of orthogonal polynomials, we analyze the statistical properties of complex eigenvalues of random matrices describing a crossover from Hermitian matrices characterized by the Wigner-Dyson statistics of real eigenvalues to strongly non-Hermitian ones whose complex eigenvalues were studied by Ginibre. Two-point statistical measures as, e. g. , spectral form factor, number variance, and small distance behavior of the nearest neighbor distance distribution p (s) are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior p (s) s^5/2 for some parameter values.