Non-hermitian random matrix theory: Method of hermitian reduction

Abstract

We consider random non-hermitian matrices in the large- N limit. The power of analytic function theory cannot be brought to bear directly to analyze non-hermitian random matrices, in contrast to hermitian random matrices. To overcome this difficulty, we show that associated to each ensemble of non-hermitian matrices there is an auxiliary ensemble of random hermitian matrices which can be analyzed by the usual methods. We then extract the Green function and the density of eigenvalues of the non-hermitian ensemble from those of the auxiliary ensemble. We apply this “method of hermitization” to several examples, and discuss a number of related issues.