Normal fluctuation in quantum ergodicity for Wigner matrices

TL;DR

This paper proves that quadratic forms of deterministic matrices on eigenvectors of Wigner matrices exhibit Gaussian fluctuations in the large size limit, combining energy methods and local laws.

Contribution

It introduces a novel proof combining Dyson Brownian motion energy methods with multi-resolvent local laws to analyze fluctuations in Wigner matrices.

Findings

Quadratic forms have Gaussian fluctuations in the bulk eigenvectors.

The proof combines energy methods with local laws.

Results hold in the large N limit.

Abstract

We consider the quadratic form of a general deterministic matrix on the eigenvectors of an $N\times N$ Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large $N$ limit. The proof is a combination of the energy method for the Dyson Brownian motion inspired by [Marcinek, Yau 2020] and our recent multi-resolvent local laws [Cipolloni, Erd\H{o}s, Schr\"oder 2020].