R-transforms for non-Hermitian matrices: a spherical integral approach

TL;DR

This paper links the formalism of R-transforms for non-Hermitian matrices to spherical integrals via the replica method, offering a new way to compute these transforms beyond special cases.

Contribution

It reveals that non-Hermitian R-transforms derive from a single scalar function, generalizing previous restricted cases and simplifying their computation.

Findings

Established a connection between R-transforms and spherical integrals for non-Hermitian matrices.

Demonstrated that R-transforms originate from a single scalar function of two variables.

Provided a new method to compute R-transforms beyond bi-invariant and elliptic ensembles.

Abstract

In this paper, we establish a connection between the formalism of $\mathcal{R}$-transforms for non-Hermitian random matrices and the framework of spherical integrals, using the replica method. This connection was previously proved in the Hermitian setting and in the case of bi-invariant random matrices. We show that the $\mathcal{R}$-transforms used in the non-Hermitian context in fact originate from a single scalar function of two variables. This provides a new and transparent way to compute $\mathcal{R}$-transforms, which until now had been known only in restricted cases such as bi-invariant, Hermitian, or elliptic ensembles.