Proof of the Weak Local Law for Wigner Matrices using Resolvent Expansions

TL;DR

This paper introduces a new, shorter proof of the Weak Local Semicircle Law for Wigner matrices by leveraging resolvent expansions, identities, and concentration inequalities, improving understanding of spectral properties.

Contribution

It provides a novel proof method for the Local Semicircle Law using algebraic resolvent techniques and a bootstrapping approach, distinct from previous methods.

Findings

Shorter proof of the Weak Local Law for Wigner matrices

Effective use of resolvent expansions and identities

Enhanced understanding of spectral bounds at small scales

Abstract

The aim of this paper is to provide a novel proof for the Local Semicircle Law for the Wigner ensemble. The core of the proof is the intensive use of the algebraic structure that arises, i.e. resolvent expansions and resolvent identities. On the analytic side, concentration of measure results and high probability bounds are used. The conclusion is obtained using a bootstrapping argument that provides information about the change of the bounds from large to small scales. This approach leads to a new and shorter proof of the Weak Local Law for Wigner Matrices, that exploits heavily the algebraic structure appearing in the setting.