Eigenvector Distribution of Wigner Matrices

TL;DR

This paper proves that the eigenvector distributions of generalized Wigner matrices match those of Gaussian ensembles near the spectral edge and in the bulk under certain moment-matching conditions.

Contribution

It establishes universality of eigenvector distributions for generalized Wigner matrices based on moment matching and level repulsion assumptions.

Findings

Eigenvector distributions near spectral edge match Gaussian ensembles.

Bulk eigenvector distributions agree under four-moment matching.

Results hold for generalized Wigner matrices with certain spectral properties.

Abstract

We consider $N\times N$ Hermitian or symmetric random matrices with independent entries. The distribution of the $(i,j)$-th matrix element is given by a probability measure $\nu_{ij}$ whose first two moments coincide with those of the corresponding Gaussian ensemble. We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution $\nu_{ij}$ coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector-eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.