Two short pieces around the Wigner problem

TL;DR

This paper explores the Wigner semi-circle law through the Stieltjes transform on the real axis, revealing a Cauchy distribution dependence, and extends Dyson's Brownian motion to derive a Langevin equation for the matrix resolvent, linking eigenvector overlaps.

Contribution

It introduces a Coulomb gas perspective on the Stieltjes transform distribution and derives a Langevin equation for the matrix resolvent, extending Dyson's framework.

Findings

Stieltjes transform follows a Cauchy distribution depending on local eigenvalue density.

Derived a Langevin equation for the full matrix resolvent.

Connected eigenvector overlaps with matrix noise dynamics.

Abstract

We revisit the classic Wigner semi-circle from two different angles. One consists in studying the Stieltjes transform directly on the real axis, which does not converge to a fixed value but follows a Cauchy distribution that depends on the local eigenvalue density. This result was recently proven by Aizenman \& Warzel for a wide class of eigenvalue distributions. We shed new light onto their result using a Coulomb gas method. The second angle is to derive a Langevin equation for the full (matrix) resolvent, extending Dyson's Brownian motion framework. The full matrix structure of this equation allows one to recover known results on the overlaps between the eigenvectors of a fixed matrix and its noisy counterpart.