Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices

Lucas Benigni

Published 2017-11-19Version 1

We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. Our results apply to a generalized $N\times N$ Rosenzweig-Porter model. We prove that the eigenvectors entries are asymptotically Gaussian with a specific variance, localizing them onto a small, explicit, part of the spectrum. For a well spread initial spectrum, this variance profile universally follows a heavy-tailed Cauchy distribution. The proof relies on a priori local laws for this model as given in [28, 27, 11] and the eigenvector moment flow from [12].