A Novel Approach to Non-Hermitian Random Matrix Models

In this paper we propose a new method for studying spectral properties of the non-hermitian 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 non-hermitian ensembles and encodes additionally some spectral properties of the eigenvectors. The standard two-by-two matrix representation of the quaternions leads to generalization of so-called matrix-valued resolvent, proposed recently in the context of diagrammatic methods [1-6]. 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 non-holomorphic Green's function for all non-hermitian 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 non-hermitian random matrix models.