Fluctuations in local quantum unique ergodicity for generalized Wigner matrices

Abstract

We study the eigenvector mass distribution for generalized Wigner matrices on a set of coordinates $I$, where $N^{\epsilon }\le |I|\le N^{1-\epsilon }$, and prove it converges to a Gaussian at every energy level, including the edge, as $N\to \infty$. The key technical input is a four-point decorrelation estimate for eigenvectors of matrices with a large Gaussian component. Its proof is an application of the maximum principle to a new set of moment observables satisfying parabolic evolution equations. Additionally, we prove high-probability Quantum Unique Ergodicity and Quantum Weak Mixing bounds for all eigenvectors and all deterministic sets of entries using a novel bootstrap argument.

arXiv comment44 pages. Minor revisions

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