  We prove that the distribution of eigenvectors of generalized Wigner matrices
is universal both in the bulk and at the edge. This includes a probabilistic
version of local quantum unique ergodicity and asymptotic normality of the
eigenvector entries. The proof relies on analyzing the eigenvector flow under
the Dyson Brownian motion. The key new ideas are: (1) the introduction of the
eigenvector moment flow, a multi-particle random walk in a random environment,
(2) an effective estimate on the regularity of this flow based on maximum
principle and (3) optimal finite speed of propagation holds for the eigenvector
moment flow with very high probability.
