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 $p\left(s\right)$] are studied in more detail. In particular, we found that the latter function may exhibit unusual behavior $p\left(s\right)\propto s^{5/2}$ for some parameter values.