Local eigenvalue statistics for higher-rank Anderson models after Dietlein-Elgart

TL;DR

This paper proves that the local eigenvalue statistics for higher-rank Anderson models on d are Poisson distributed near spectral band edges, improving previous results by showing a simpler Poisson process rather than a compound Poisson process.

Contribution

It extends the eigenvalue level spacing method to higher-rank Anderson models, demonstrating Poisson eigenvalue statistics near band edges, simplifying prior complex results.

Findings

Local eigenvalue statistics are Poisson distributed near spectral edges.

The method applies to higher-rank models with uniform perturbations.

Simplifies the proof of eigenvalue distribution in Anderson models.

Abstract

We use the method of eigenvalue level spacing developed by Dietlein and Elgart (arXiv:1712.03925) to prove that the local eigenvalue statistics (LES) for the Anderson model on $Z^d$, with uniform higher-rank $m \geq 2$, single-site perturbations, is given by a Poisson point process with intensity measure $n(E_0)~ds$, where $n(E_0)$ is the density of states at energy $E_0$ in the region of localization near the spectral band edges. This improves the result of Hislop and Krishna (arXiv:1809.01236), who proved that the LES is a compound Poisson process with L\'evy measure supported on the set $\{1, 2, \ldots, m \}$. Our proofs are an application of the ideas of Dieltein and Elgart to these higher-rank lattice models with two spectral band edges, and illustrate, in a simpler setting, the key steps of the proof of Dieltein and Elgart.