Optimal decay of eigenvector overlap for non-Hermitian random matrices

TL;DR

This paper proves that the overlap between eigenvectors of large non-Hermitian random matrices decays as the inverse square of the eigenvalue distance, extending previous results to more general ensembles.

Contribution

It establishes a universal decay rate for eigenvector overlaps in i.i.d. non-Hermitian matrices, generalizing prior Gaussian-specific findings.

Findings

Eigenvector overlaps decay as the inverse square of eigenvalue distance.

Established a two-resolvent local law with optimal decay and spectral edge dependence.

Extended decay results from Gaussian to general i.i.d. matrix ensembles.

Abstract

We consider the standard overlap $\mathcal{O}_{ij}: =\langle \mathbf{r}_j, \mathbf{r}_i\rangle\langle \mathbf{l}_j, \mathbf{l}_i\rangle$ of any bi-orthogonal family of left and right eigenvectors of a large random matrix $X$ with centred i.i.d. entries and we prove that it decays as an inverse second power of the distance between the corresponding eigenvalues. This extends similar results for the complex Gaussian ensemble from Bourgade and Dubach [arXiv:1801.01219], as well as Benaych-Georges and Zeitouni [arXiv:1806.06806], to any i.i.d. matrix ensemble in both symmetry classes. As a main tool, we prove a two-resolvent local law for the Hermitisation of $X$ uniformly in the spectrum with optimal decay rate and optimal dependence on the density near the spectral edge.