A Novel Approach to NonHermitian Random Matrix Models
Abstract
In this paper we propose a new method for studying spectral properties of the nonhermitian random matrix ensembles. Alike complex Green's function encodes, via discontinuities, the real spectrum of the hermitian ensembles, the proposed here quaternion extension of the Green's function leads directly to complex spectrum in case of nonhermitian ensembles and encodes additionally some spectral properties of the eigenvectors. The standard twobytwo matrix representation of the quaternions leads to generalization of socalled matrixvalued resolvent, proposed recently in the context of diagrammatic methods [16]. We argue that quaternion Green's function obeys Free Variables Calculus [7,8]. In particular, the quaternion functional inverse of the matrix Green's function, called after [9] Blue's function obeys simple addition law, as observed some time ago [1,3]. Using this law we derive new, general, algorithmic and efficient method to find the nonholomorphic Green's function for all nonhermitian ensembles of the form H+iH', where ensembles H and H' are independent (free in the sense of Voiculescu [7]) hermitian ensembles from arbitrary measure. We demonstrate the power of the method by a straightforward rederivation of spectral properties for several examples of nonhermitian random matrix models.
 Publication:

arXiv eprints
 Pub Date:
 February 2004
 arXiv:
 arXiv:mathph/0402057
 Bibcode:
 2004math.ph...2057J
 Keywords:

 Mathematical Physics;
 Probability;
 Statistical Mechanics
 EPrint:
 33 pages