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Dominik Schröder

Normal fluctuation in quantum ergodicity for Wigner matrices

Giorgio Cipolloni, László Erdős, Dominik Schröder

Ann. Probab.Vol. 50 (2022)


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


We consider the quadratic form of a general deterministic matrix on the eigenvectors of an N×NN\times N Wigner matrix and prove that it has Gaussian fluctuation for each bulk eigenvector in the large NN 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.