Random characteristics for Wigner matrices

Per Von Soosten,Simone Warzel

doi:10.1214/19-ecp278

Abstract

We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.

Highlights

Starting with the seminal works of Erdos, Schlein, and Yau [15, 16], a large portion of recent progress in random matrix theory rests on strong concentration of measure phenomena for the resolvent on almost microscopic scales
We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution
The purpose of this paper is to show that this approach to local resolvent estimates is not limited to Wigner matrices with Gaussian entries

Summary

Introduction

Starting with the seminal works of Erdos, Schlein, and Yau [15, 16], a large portion of recent progress in random matrix theory rests on strong concentration of measure phenomena for the resolvent on almost microscopic scales. The purpose of this paper is to show that this approach to local resolvent estimates is not limited to Wigner matrices with Gaussian entries The basis of this is the construction of a matrix martingale H(t) whose rescaled entries follow a given density at time t = 1. We will illustrate our method by giving a new proof of the weak bulk local semicircle law for the normalized resolvent trace of Wigner matrices whose entries are drawn from densities satisfying Assumption 1.1. To state this result, we will make use of the stochastic domination language of [12].

The self-energy correction

Fluctuations along characteristic curves

Proof of the local semicircle law