H\"older regularity of the integrated density of states for quasi-periodic long-range operators on $\ell^2(\Z^d)$

TL;DR

This paper establishes the H"older continuity of the integrated density of states for certain quasi-periodic long-range operators on multi-dimensional integer lattices, improving previous results in the perturbative regime.

Contribution

It introduces a new approach combining Aubry duality, Thouless formula, and Lyapunov exponent regularity to prove H"older continuity for these operators.

Findings

Proves H"older continuity of the integrated density of states.

Provides explicit H"older exponent based on potential level sets.

Enhances previous results by Goldstein and Schlag in the perturbative regime.

Abstract

We prove the H\"older continuity of the integrated density of states for a class of quasi-periodic long-range operators on $\ell^2(\Z^d)$ with large trigonometric polynomial potentials and Diophantine frequencies. Moreover, we give the H\"older exponent in terms of the cardinality of the level sets of the potentials, which improves, in the perturbative regime, the result obtained by Goldstein and Schlag \cite{gs2}. Our approach is a combination of Aubry duality, generalized Thouless formula and the regularity of the Lyapunov exponents of analytic quasi-periodic $GL(m,\C)$ cocycles which is proved by quantitative almost reducibility method.