Local semicircle law and complete delocalization for Wigner random matrices

TL;DR

This paper proves that Wigner random matrices exhibit the semicircle law at very small energy scales and that their eigenvectors are completely delocalized, with components uniformly small.

Contribution

It establishes the local semicircle law at near-optimal scales and confirms complete delocalization of eigenvectors for Wigner matrices under broad conditions.

Findings

Eigenvalue density concentrates around the semicircle law at scales  \\gg N^{-1} (\,log N)^8

Eigenvectors are completely delocalized with components of order O(N^{-1/2})

Results hold away from spectral edges with broad distribution assumptions

Abstract

We consider $N\times N$ Hermitian random matrices with independent identical distributed entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order 1/N. Under suitable assumptions on the distribution of the single matrix element, we prove that, away from the spectral edges, the density of eigenvalues concentrates around the Wigner semicircle law on energy scales $\eta \gg N^{-1} (\log N)^8$. Up to the logarithmic factor, this is the smallest energy scale for which the semicircle law may be valid. We also prove that for all eigenvalues away from the spectral edges, the $\ell^\infty$-norm of the corresponding eigenvectors is of order $O(N^{-1/2})$, modulo logarithmic corrections. The upper bound $O(N^{-1/2})$ implies that every eigenvector is completely delocalized, i.e., the maximum size of the components of the eigenvector is of the same order as their average size. In the Appendix, we include a lemma by J. Bourgain which removes one of our assumptions on the distribution of the matrix elements.