Eigenvalue distributions on a single ring.

Abstract

In 1965 J. Ginibre introduced an ensemble of random matrices with no symmetry conditions imposed as the mathematical counterpart to hermitian random matrix theory. In his original paper he treats the case of matrices with i.i.d. normally distributed real, complex or quaternion entries. Since then, mainly due to interest from applications, the development of non-hermitian random matrix theory has further evolved, though the eigenvalue statistics of non-hermitian random matrices are far from being as thoroughly understood as their hermitian counterpart. A characteristic of non-hermitian random matrices are eigenvalue distributions in the complex plane. Real asymmetric random matrices have the additional caveat of having real and complex eigenvalues and thus are technically more challenging. In the following work a new three-fold family of non-hermitian random matrices is introduced via a quadratization procedure. As a consequence the entries of these matrices are highly dependent. For all three ensembles the joint eigenvalue probability density functions and eigenvalue correlations are derived for β = 1, 2. In the limit of large matrix dimensions a classification of eigenvalue correlation functions for different asymptotic regimes is undertaken. In tune with the title of this work for all three ensembles there exists an asymptotic regime, in which the eigenvalues are supported on an annulus around the origin. Thus the induced family of non-hermitian random matrix ensembles serves as an example, for ensembles of the Feinberg-Zee type with logarithmic potential.