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

Fluctuations in local quantum unique ergodicity for generalized Wigner matrices