Eigenvectors distribution and quantum unique ergodicity for deformed Wigner matrices

TL;DR

This paper studies eigenvector distributions in deformed Wigner matrices, showing they are asymptotically Gaussian with a universal heavy-tailed variance profile, and establishes quantum unique ergodicity under certain conditions.

Contribution

It provides a detailed analysis of eigenvector distributions for deformed Wigner matrices, including universal variance profiles and quantum ergodicity results, extending prior understanding of eigenvector behavior.

Findings

Eigenvector entries are asymptotically Gaussian with a specific variance.

Variance profile universally follows a heavy-tailed Cauchy distribution.

Strong quantum unique ergodicity holds for smooth entries.

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.