Random matrices close to Hermitian or unitary: overview of methods and results
Abstract
The paper discusses recent progress in understanding statistical properties of eigenvalues of (weakly) nonHermitian and nonunitary 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 antiHermitian N × N matrix ihat Gamma satisfying in the limit of large dimension N the condition Tr hat H^{2} propto N Tr hat Gamma^{2}. 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 discretetime 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 nontrivial crossover from WignerDyson statistics of real/unimodular eigenvalues typical of Hermitian/unitary matrices to Ginibre statistics in the complex plane typical of ensembles with strong nonHermiticity: langleTr hat H^{2}rangle propto langleTr hat Gamma^{2}rangle when N rightarrow infty. Finally, we discuss (scarce) results available on eigenvector statistics for weakly nonHermitian random matrices.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 March 2003
 DOI:
 10.1088/03054470/36/12/326
 arXiv:
 arXiv:nlin/0207051
 Bibcode:
 2003JPhA...36.3303F
 Keywords:

 Nonlinear Sciences  Chaotic Dynamics;
 Condensed Matter  Mesoscale and Nanoscale Physics
 EPrint:
 Published version, with a few more misprints corrected