Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

TL;DR

This paper establishes the semicircle law on very short energy scales and demonstrates that eigenvectors of Wigner matrices are typically fully delocalized, with no eigenvector being localized, under certain distribution assumptions.

Contribution

It provides new bounds on the density of states on microscopic scales and proves eigenvector delocalization and non-localization with high probability for Wigner matrices.

Findings

Density of states concentrates around the semicircle law on scales  \, extstyle \\eta \\gg N^{-2/3}

Most eigenvectors are fully delocalized with  \, extstyle \\ell^p-norms comparable to N^{1/p - 1/2}

With high probability, no eigenvector is localized

Abstract

We consider $N\times N$ Hermitian random matrices with i.i.d. entries. The matrix is normalized so that the average spacing between consecutive eigenvalues is of order $1/N$. We study the connection between eigenvalue statistics on microscopic energy scales $\eta\ll1$ and (de)localization properties of the eigenvectors. Under suitable assumptions on the distribution of the single matrix elements, we first give an upper bound on the density of states on short energy scales of order $\eta \sim\log N/N$. We then prove that the density of states concentrates around the Wigner semicircle law on energy scales $\eta\gg N^{-2/3}$. We show that most eigenvectors are fully delocalized in the sense that their $\ell^p$-norms are comparable with $N^{{1}/{p}-{1}/{2}}$ for $p\ge2$, and we obtain the weaker bound $N^{{2}/{3}({1}/{p}-{1}/{2})}$ for all eigenvectors whose eigenvalues are separated away from the spectral edges. We also prove that, with a probability very close to one, no eigenvector can be localized. Finally, we give an optimal bound on the second moment of the Green function.