Eigenvector decorrelation for random matrices

TL;DR

This paper investigates how small perturbations in random matrices cause their eigenvectors to become nearly orthogonal, revealing insights into eigenvector sensitivity and extending the Eigenstate Thermalization Hypothesis.

Contribution

It demonstrates that eigenvectors of deformed Wigner matrices become orthogonal under small perturbations and generalizes the Eigenstate Thermalization Hypothesis to different spectral families.

Findings

Eigenvectors become asymptotically orthogonal with small perturbations.

Quadratic forms of eigenvectors are of size N^{-1/2}.

Generalization of Eigenstate Thermalization Hypothesis.

Abstract

We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their bulk eigenvectors become asymptotically orthogonal as soon as $\mathrm{Tr}(D_1-D_2)^2\gg 1$, or their respective energies are separated on a scale much bigger than the local eigenvalue spacing. Furthermore, we show that quadratic forms of eigenvectors of $W+D_1$, $W+D_2$ with any deterministic matrix $A\in\mathbf{C}^{N\times N}$ in a specific subspace of codimension one are of size $N^{-1/2}$. This proves a generalization of the Eigenstate Thermalization Hypothesis to eigenvectors belonging to two different spectral families.