Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices

Abstract: 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. In the case of smooth entries, we also obtain a strong form of quantum unique ergodicity as an overwhelming probability bound on the eigenvectors probability mass. The proof relies on a priori local laws for this model and the eigenvector moment flow.