Sharp estimates for the integrated density of states in Anderson tight-binding models

TL;DR

This paper provides sharp estimates for the integrated density of states in 1D and 2D Anderson models, showing that simple formulas with energy shifts can accurately approximate the IDOS across the spectrum.

Contribution

It introduces simple, accurate formulas for approximating the IDOS in Anderson models, extending the landscape law with practical energy shift adjustments.

Findings

High-accuracy IDOS approximation in 1D using a simple multiplicative energy shift.

In 2D, the same approximation applies with different prefactors at spectrum edges.

Bounds from the landscape law are validated and refined for Anderson models.

Abstract

Recent work [G. David, M. Filoche, and S. Mayboroda, arXiv:1909.10558[Adv. Math. (to be published)]] has proved the existence of bounds from above and below for the Integrated Density of States (IDOS) of the Schr\"odinger operator throughout the spectrum, called the landscape law. These bounds involve dimensional constants whose optimal values are yet to be determined. Here, we investigate the accuracy of the landscape law in 1D and 2D tight-binding Anderson models, with binary or uniform random distributions. We show, in particular, that in 1D, the IDOS can be approximated with high accuracy through a single formula involving a remarkably simple multiplicative energy shift. In 2D, the same idea applies but the prefactor has to be changed between the bottom and top parts of the spectrum.