Fluctuations in Local Quantum Unique Ergodicity for Generalized Wigner Matrices

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