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) proportional to s(5/2) for some parameter values.