Nonhermitean Random Matrix Models

TL;DR

This paper extends diagrammatic methods in random matrix theory to nonhermitean models, analyzing eigenvalue distributions and correlations with 1/N approximation, and providing explicit conditions for singularities.

Contribution

It introduces a generalized diagrammatic approach for nonhermitean random matrices and derives universal forms for two-point functions in this context.

Findings

Derived explicit conditions for eigenvalue singularities.

Generalized macroscopic universality to nonhermitean matrices.

Analytical results agree with numerical simulations.

Abstract

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to nonhermitean random matrix models using the 1/N approximation. A number of one- and two-point functions are evaluated on their holomorphic and nonholomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic correlations. Generic form for the two-point functions are obtained, generalizing the concept of macroscopic universality to nonhermitean random matrices. We show that the holomorphic and nonholomorphic one- and two-point functions condition the behavior of pertinent partition functions to order $O(1/N)$. We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.