# Complete diagrammatics of the single ring theorem

## TL;DR

This paper develops diagrammatic methods to connect different free probability concepts, rederives key theorems, and extends understanding of eigenvector correlations in large random matrices.

## Contribution

It introduces explicit diagrammatic relations for cumulant generating functions and unifies Hermitian and non-normal operator frameworks in free probability.

## Key findings

- Rederived the Haagerup-Larsen theorem
- Extended the theorem to eigenvector correlation functions
- Mapped Hermitian and non-normal operator areas

## Abstract

Using diagrammatic techniques, we provide explicit functional relations between the cumulant generating functions for the biunitarily invariant ensembles in the limit of large size of matrices. The formalism allows to map two distinct areas of free random variables: Hermitian positive definite operators and non-normal R-diagonal operators. We also rederive the Haagerup-Larsen theorem and show how its recent extension to the eigenvector correlation function appears naturally within this approach.