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

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.