The Eigenvector Bead Process

TL;DR

This paper studies the stability and structure of eigenvector overlaps between a large random matrix and its principal minor, revealing universal behaviors in bulk and edge spectra for Wigner and Wishart matrices.

Contribution

It introduces a detailed analysis of eigenvector overlaps in Wigner and Wishart matrices, uncovering universal laws and the local projection phenomenon at the spectral bulk and edge.

Findings

Eigenvector overlaps follow universal laws expressed via Airy and Sine kernels.

Eigenvectors of large matrices project mainly onto local spectral level eigenvectors.

At spectral edges, overlaps are governed by a random antisymmetric perturbation.

Abstract

We investigate the overlap matrix between the eigenvectors of a Wigner matrix $H_{N+K}$ of size $(N+K)\times(N+K)$ and those of its principal minor $H_N$ of size $N\times N$, for both the real symmetric ($\beta=1$) and complex Hermitian ($\beta=2$) ensembles, in the regime where $N \to \infty$ while $K$ remains fixed. Our analysis yields two main results. (i) In the \emph{bulk} of the spectrum, an eigenvector of $H_{N+K}$ associated with an eigenvalue at energy level $E$ projects primarily onto eigenvectors of $H_N$ located at the same local spectral level. This phenomenon, which we call \emph{local projection}, highlights a robust stability of the eigenbasis under matrix growth. (ii) At the \emph{spectral edge}, the change of basis between the leading eigenspaces of consecutive minors is asymptotically governed by a random antisymmetric perturbation of order $N^{-1/3}$. In both cases, we provide the asymptotic law of the overlaps expressed in terms of the Airy and Sine kernels. We further extend our analysis to the case of Wishart matrices, that is, sample covariance matrices of the form $W = X^{\!\top} X$, where $X \in \mathbb{R}^{T \times N}$ is a matrix with i.i.d.\ random entries. We establish analogous results for the overlaps between eigenvectors of consecutive minors of $W$, both in the bulk and at the spectral edges (soft and hard). The limiting laws share the same universal structure as in the Wigner case, up to explicit constants depending on the aspect ratio $q = N/T$. This demonstrates the universality of the eigenvector overlap process across distinct random matrix ensembles.