Eigenvector statistics of the product of Ginibre matrices

TL;DR

This paper introduces a method to compute eigenvector correlations in products of Ginibre matrices, providing explicit formulas, conjectures, and asymptotic results for large matrix sizes, enhancing understanding of non-Hermitian random matrix products.

Contribution

It develops a novel approach to calculate eigenvector correlations in products of Ginibre matrices, including explicit formulas and conjectures supported by analytical and numerical evidence.

Findings

Explicit formulas for small matrix products

Conjecture for the integrated eigenvector overlap

Asymptotic correlation density for large matrices

Abstract

We develop a method to calculate left-right eigenvector correlations of the product of $m$ independent $N\times N$ complex Ginibre matrices. For illustration, we present explicit analytical results for the vector overlap for a couple of examples for small $m$ and $N$. We conjecture that the integrated overlap between left and right eigenvectors is given by the formula $O = 1 + (m/2)(N-1)$ and support this conjecture by analytical and numerical calculations. We derive an analytical expression for the limiting correlation density as $N\rightarrow \infty$ for the product of Ginibre matrices as well as for the product of elliptic matrices. In the latter case, we find that the correlation function is independent of the eccentricities of the elliptic laws.