On the Wegner orbital model

TL;DR

This paper studies the Wegner orbital model, proving localization at high disorder, and establishing estimates on eigenvalue distributions, with implications for Gaussian band matrices and localization length.

Contribution

It provides new localization results and eigenvalue estimates for the Wegner orbital model and related Gaussian matrices, extending previous theoretical understanding.

Findings

Localization at strong disorder

Wegner-type eigenvalue estimate

Improved localization length bound

Abstract

The Wegner orbital model is a class of random operators introduced by Wegner to model the motion of a quantum particle with many internal degrees of freedom (orbitals) in a disordered medium. We consider the case when the matrix potential is Gaussian, and prove three results: localisation at strong disorder, a Wegner-type estimate on the mean density of eigenvalues, and a Minami-type estimate on the probability of having multiple eigenvalues in a short interval. The last two results are proved in the more general setting of deformed block-Gaussian matrices, which includes a class of Gaussian band matrices as a special case. Emphasis is placed on the dependence of the bounds on the number of orbitals. As an additional application, we improve the upper bound on the localisation length for one-dimensional Gaussian band matrices.