Pseudo-hermitian random matrix theory: a review

TL;DR

This review discusses pseudo-hermitian random matrix theory, focusing on models with indefinite metrics, eigenvalue distributions, phase transitions, and analytical methods, supported by numerical simulations.

Contribution

It introduces pseudo-hermitian random matrices, derives explicit eigenvalue density expressions, and analyzes phase transitions related to eigenvalue support merging.

Findings

Eigenvalues are real or form complex-conjugate pairs.

Explicit eigenvalue density formulas are derived.

Eigenvalue support undergoes phase transitions as parameters vary.

Abstract

We review our recent results on pseudo-hermitian random matrix theory which were hitherto presented in various conferences and talks. (Detailed accounts of our work will appear soon in separate publications.) Following an introduction of this new type of random matrices, we focus on two specific models of matrices which are pseudo-hermitian with respect to a given indefinite metric B. Eigenvalues of pseudo-hermitian matrices are either real, or come in complex-conjugate pairs. The diagrammatic method is applied to deriving explicit analytical expressions for the density of eigenvalues in the complex plane and on the real axis, in the large-N, planar limit. In one of the models we discuss, the metric B depends on a certain real parameter t. As t varies, the model exhibits various "phase transitions" associated with eigenvalues flowing from the complex plane onto the real axis, causing disjoint eigenvalue support intervals to merge. Our analytical results agree well with presented numerical simulations.