Normal fluctuation in quantum ergodicity for Wigner matrices - Cipolloni, Giorgio; Erdös, László; Schröder, Dominik J (Annals of Probability, 2022)
We consider the quadratic form of a general high-rank deterministic matrix on the eigenvectors of an N×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 and Yau (2021) and our recent multiresolvent local laws (Comm. Math. Phys. 388 (2021) 1005–1048).
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