Continuity of higher-order derivatives for integrated density of states of the discrete Anderson model with respect to the disorder parameter

TL;DR

This paper establishes quantitative bounds on the higher-order derivatives of the integrated density of states for the Anderson model, demonstrating their continuity with respect to the disorder parameter in the localized regime.

Contribution

It provides the first rigorous estimates on the continuity of higher-order derivatives of the IDS for the Anderson model under strong disorder conditions.

Findings

Bounds on differences of higher-order derivatives of the IDS

Results valid on $ abla^d$ lattice and Bethe lattice

Continuity established in the localized regime

Abstract

We derive quantitative continuity estimates for the higher-order derivatives of the integrated density of states (IDS) with respect to the disorder parameter for the Anderson model on $\ell^2(\mathbb{G})$. Here $\mathbb{G}=\mathbb{Z}^d$ or $\mathbb{B}$, where $\mathbb{B}$ denotes the Bethe lattice. Our results hold in the regime of strong disorder, where entire spectrum is localized. We assume sufficient smoothness of the density of the single site distribution so that the IDS admits higher-order derivatives. More precisely, we establish bounds on the difference between higher-order derivatives of the IDS in terms of the differences in the disorder parameters.