Normal fluctuation in quantum ergodicity for Wigner matrices

Summary

We prove a CLT for quadratic forms of eigenvectors of Wigner matrices with arbitrary deterministic matrices, considerably strengthening previous results on quantum unique ergodicity.

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, Erdős, Schröder 2020.

Paper

2103.06730.pdf