Fluctuations in local quantum unique ergodicity for generalized Wigner matrices

TL;DR

This paper investigates the distribution of eigenvector mass in generalized Wigner matrices, demonstrating convergence to Gaussian behavior across energy levels and establishing quantum ergodicity and mixing bounds.

Contribution

It introduces a four-point decorrelation estimate and a bootstrap method to prove quantum ergodicity and mixing for eigenvectors of generalized Wigner matrices.

Findings

Eigenvector mass distribution converges to Gaussian at all energy levels.

Established high-probability quantum ergodicity bounds.

Proved quantum weak mixing bounds for all eigenvectors.

Abstract

We study the eigenvector mass distribution for generalized Wigner matrices on a set of coordinates $I$, where $N^\varepsilon \le | I | \le N^{1- \varepsilon}$, and prove it converges to a Gaussian at every energy level, including the edge, as $N\rightarrow \infty$. The key technical input is a four-point decorrelation estimate for eigenvectors of matrices with a large Gaussian component. Its proof is an application of the maximum principle to a new set of moment observables satisfying parabolic evolution equations. Additionally, we prove high-probability Quantum Unique Ergodicity and Quantum Weak Mixing bounds for all eigenvectors and all deterministic sets of entries using a novel bootstrap argument.