Lectures on the local semicircle law for Wigner matrices

TL;DR

This paper introduces the local semicircle law for Wigner matrices, demonstrating how eigenvalue distributions closely follow the semicircle law at small scales and exploring its applications in eigenvector delocalization, eigenvalue rigidity, and spectral statistics.

Contribution

It provides a rigorous proof of the local semicircle law for Wigner matrices and applies it to eigenvector delocalization, eigenvalue rigidity, and comparison of local eigenvalue statistics.

Findings

Eigenvalue distribution closely follows the semicircle law at small spectral scales.

Eigenvectors are approximately flat with high probability.

Eigenvalue locations exhibit large deviation bounds (rigidity).

Abstract

These notes provide an introduction to the local semicircle law from random matrix theory, as well as some of its applications. We focus on Wigner matrices, Hermitian random matrices with independent upper-triangular entries with zero expectation and constant variance. We state and prove the local semicircle law, which says that the eigenvalue distribution of a Wigner matrix is close to Wigner's semicircle distribution, down to spectral scales containing slightly more than one eigenvalue. This local semicircle law is formulated using the Green function, whose individual entries are controlled by large deviation bounds. We then discuss three applications of the local semicircle law: first, complete delocalization of the eigenvectors, stating that with high probability the eigenvectors are approximately flat; second, rigidity of the eigenvalues, giving large deviation bounds on the locations of the individual eigenvalues; third, a comparison argument for the local eigenvalue statistics in the bulk spectrum, showing that the local eigenvalue statistics of two Wigner matrices coincide provided the first four moments of their entries coincide. We also sketch further applications to eigenvalues near the spectral edge, and to the distribution of eigenvectors.