Non-Hermitean Random Matrix Theory: method of hermitization

TL;DR

This paper introduces the 'method of hermitization' to analyze non-Hermitian random matrices by associating them with Hermitian ensembles, enabling the use of traditional analysis techniques for eigenvalue distributions.

Contribution

The paper proposes a novel 'method of hermitization' that transforms non-Hermitian matrices into Hermitian ones for easier spectral analysis, extending existing methods.

Findings

Successfully applied the method to multiple examples

Derived eigenvalue densities from auxiliary Hermitian ensembles

Provided insights into non-Hermitian spectral properties

Abstract

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