Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

TL;DR

This paper develops a large $N$ theoretical framework to analyze eigenvector correlations in non-Hermitian random matrices, generalizing free probability and revealing universal behaviors in eigenvector statistics.

Contribution

It introduces a diagrammatic approach for two-point eigenvector correlations in non-normal matrices, extending free probability and providing explicit formulas for biunitarily invariant ensembles.

Findings

Derived a simple expression for two-point eigenvector correlation functions.

Conjectured microscopic universality in eigenvector correlations in bulk and at the rim.

Connected results to known universality in the complex Ginibre ensemble.

Abstract

Using large $N$ arguments, we propose a scheme for calculating the two-point eigenvector correlation function for non-normal random matrices in the large $N$ limit. The setting generalizes the quaternionic extension of free probability to two-point functions. In the particular case of biunitarily invariant random matrices, we obtain a simple, general expression for the two-point eigenvector correlation function, which can be viewed as a further generalization of the single ring theorem. This construction has some striking similarities to the freeness of the second kind known for the Hermitian ensembles in large $N$. On the basis of several solved examples, we conjecture two kinds of microscopic universality of the eigenvectors - one in the bulk, and one at the rim. The form of the conjectured bulk universality agrees with the scaling limit found by Chalker and Mehlig [JT Chalker, B Mehlig, PRL, \textbf{81}, 3367 (1998)] in the case of the complex Ginibre ensemble.