Random Schrodinger operators on long boxes, noise explosion and the GOE

TL;DR

This paper demonstrates that eigenvalues of certain random Schrödinger operators on long boxes converge to the GOE eigenvalue process, providing a rigorous example supporting Wigner's prediction.

Contribution

It establishes the first rigorous example where eigenvalues of a complex system match the GOE eigenvalue distribution, confirming a long-standing conjecture.

Findings

Eigenvalues in low disorder converge to GOE statistics

Sequences of long boxes exhibit Wigner-like eigenvalue behavior

First rigorous proof of Wigner's prediction in this context

Abstract

It is conjectured that the eigenvalues of random Schrodinger operators at the localization transition in dimensions d>=2 behave like the eigenvalues of the Gaussian Orthogonal Ensemble (GOE). We show that there are sequences of n by m boxes with 1<<m<<n so that the eigenvalues in low disorder converge to Sine1, the limiting eigenvalue process of the GOE. For the GOE case, this is the first example where Wigner's famous prediction is proven rigorously: we exhibit a complex system whose eigenvalues behave like those of random matrices.