Enhanced Wegner and Minami estimates and eigenvalue statistics of random Anderson models at spectral edges

TL;DR

This paper develops enhanced probabilistic estimates for the Anderson model, leading to improved understanding of eigenvalue distributions and spectral statistics at spectral edges, including Lifshitz tail regimes.

Contribution

It introduces refined Wegner and Minami estimates based on the integrated density of states, extending spectral statistics results to spectral edges and Lifshitz tail regimes.

Findings

Enhanced estimates improve eigenvalue and eigenfunction descriptions.

Extended spectral statistics results to spectral edges and Lifshitz tails.

Applicable to higher-dimensional and modified Anderson models.

Abstract

We consider the discrete Anderson model and prove enhanced Wegner and Minami estimates where the interval length is replaced by the IDS computed on the interval. We use these estimates to improve on the description of finite volume eigenvalues and eigenfunctions obtained in a previous paper. As a consequence of the improved description of eigenvalues and eigenfunctions, we revisit a number of results on the spectral statistics in the localized regime and extend their domain of validity, namely : - the local spectral statistics for the unfolded eigenvalues; - the local asymptotic ergodicity of the unfolded eigenvalues; In dimension 1, for the standard Anderson model, the improvement enables us to obtain the local spectral statistics at band edge, that is in the Lifshitz tail regime. In higher dimensions, this works for modified Anderson models.