Convergence of local eigenvector processes of generalized Wigner matrices

We prove convergence of eigenvector processes of the form [Formula: see text] where uk is a bulk eigenvector of generalized Wigner matrices and ([Formula: see text]) is a family of symmetric matrices with bounded norm and Hölder regularity. We give explicit examples of limiting processes and prove that a large class of Gaussian processes with Hölder-continuous covariance function can be obtained as such a limit using its Karhunen–Loéve expansion. The proof is based on the multi-dimensional convergence proved in [BC24] and a tightness criterion proved using Hölder regularity of the observables.