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 hat J = hat H - ihat Gamma, with hat H being a large random N × N Hermitian matrix with independent entries 'deformed' by a certain anti-Hermitian N × N matrix ihat Gamma satisfying in the limit of large dimension N the condition Tr hat H2 propto N Tr hat Gamma2. Here hat Gamma can be either a random or just a fixed given Hermitian matrix. Ensembles of such a type with hat Gamma geq 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 × N matrices hat A with complex entries such that hat Adaggerhat A = hat I - hat T. For hat T = 0 this coincides with the circular unitary ensemble, and 0 leq hat T leq hat I describes deviation from unitarity. Our result amounts to answering statistically the following old question: given the singular values of a matrix hat A 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: langleTr hat H2rangle propto langleTr hat Gamma2rangle when N rightarrow infty. Finally, we discuss (scarce) results available on eigenvector statistics for weakly non-Hermitian random matrices.
Journal of Physics A Mathematical General
- Pub Date:
- March 2003
- Nonlinear Sciences - Chaotic Dynamics;
- Condensed Matter - Mesoscale and Nanoscale Physics
- Published version, with a few more misprints corrected