Universality for Diagonal Eigenvector Overlaps of non-Hermitian Random Matrices

TL;DR

This paper proves the universal behavior of eigenvalue and eigenvector overlap distributions in non-Hermitian random matrices, covering bulk and edge cases, with bounds on singular values and singular vectors.

Contribution

It establishes the universality of joint eigenvalue and eigenvector overlap distributions for complex and real non-Hermitian matrices, including new bounds on singular values and vectors.

Findings

Universal distribution of eigenvalue and eigenvector overlaps proven

Bounds obtained for least non-zero singular value at the edge

Bounds established for inner products of singular vectors near the spectral edge

Abstract

We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the proof we obtain a bound for the least non-zero singular value of $X-z$ when $z$ is an edge eigenvalue and a bound for the inner product between left and right singular vectors of $X-z$ when $|z|=1+O(N^{-1/2})$.