We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an 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 and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).
L.E. would like to thank Zhigang Bao for many illuminating discussions in an early stage of this research. The authors are also grateful to Paul Bourgade for his comments on the manuscript and the anonymous referee for several useful suggestions.
"Normal fluctuation in quantum ergodicity for Wigner matrices." Ann. Probab. 50 (3) 984 - 1012, May 2022. https://doi.org/10.1214/21-AOP1552