Level Repulsion for a class of decaying random potentials

TL;DR

This paper investigates the spectral properties of the Anderson model with decaying randomness, showing that near band edges in higher dimensions, the statistics match those of the free operator, especially at small coupling.

Contribution

It demonstrates that in certain regimes, the spectral statistics are universal and independent of randomness, extending understanding of Anderson localization and delocalization phenomena.

Findings

Spectral statistics near band edges are independent of randomness in dimensions d ≥ 3.

At small coupling, the statistics converge to those of the free operator.

Identifies length scales where the transition to free-like statistics occurs.

Abstract

In this paper we consider the Anderson model with decaying randomness and show that statistics near the band edges in the absolutely continuous spectrum in dimensions $d \geq 3$ is independent of the randomness and agrees with that of the free part. We also consider the operators at small coupling and identify the length scales at which the statistics agrees with the free one in the limit when the coupling constant goes to zero.