Fluctuations of eigenvector overlaps and the Berry conjecture for Wigner matrices

Lucas Benigni, Giorgio Cipolloni

Published 2022-12-20Version 1

We prove that any finite collection of quadratic forms (overlaps) of general deterministic matrices and eigenvectors of an $N\times N$ Wigner matrix has joint Gaussian fluctuations. This can be viewed as the random matrix analogue of the Berry random wave conjecture.