Random matrices close to Hermitian or unitary: overview of methods and results

The paper discusses recent progress in understanding statistical properties of eigenvalues of (weakly) non-Hermitian and non-unitary random matrices. The first type of ensembles is of the form (J) over cap = (H) over cap - iGamma, with (H) over cap being a large random N x N Hermitian matrix with independent entries 'deformed' by a certain anti-Hermitian N x N matrix iGamma satisfying in the limit of large dimension N the condition Tr (H) over cap (2) proportional to N Tr Gamma(2). Here Gamma can be either a random or just a fixed given Hermitian matrix. Ensembles of such a type with Gamma greater than or equal to 0 emerge naturally when describing quantum scattering in systems with chaotic dynamics and serve to describe resonance statistics. Related models are used to mimic complex spectra of the Dirac operator with chemical potential in the context of quantum chromodynamics. Ensembles of the second type, arising naturally in scattering theory of discrete-time systems, are formed by N x N matrices (A) over cap with complex entries such that (A) over cap (dagger) (A) over cap = (I) over cap - (T) over cap. For (T) over cap = 0 this coincides with the circular unitary ensemble, and 0 less than or equal to (T) over cap less than or equal to (I) over cap describes deviation from unitarity. Our result amounts to answering statistically the following old question: given the singular values of a matrix (A) over cap describe the locus of its eigenvalues. We systematically show that the obtained expressions for the correlation functions of complex eigenvalues describe a non-trivial crossover from Wigner-Dyson statistics of real/unimodular eigenvalues typical of Hermitian/unitary matrices to Ginibre statistics in the complex plane typical of ensembles with strong non-Hermiticity: <Tr (H) over cap (2)> proportional to <Tr Gamma(2)> when N --> infinity. Finally, we discuss (scarce) results available on eigenvector statistics for weakly non-Hermitian random matrices.