Pseudo-hermitian random matrix theory: a review

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.