Normal fluctuation in quantum ergodicity for Wigner matrices
Abstract
We consider the quadratic form of a general 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, Yau 2020] and our recent multi-resolvent local laws [Cipolloni, Erdos, Schr\"oder 2020].
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