Universality of smoothness of Density of States in arbitrary higher-dimensional disorder under non-local interactions I. From Viéte--Euler identity to Anderson localization

Abstract

It is shown that in a large class of disordered systems with non-degenerate disorder, in presence of non-local interactions, the Integrated Density of States (IDS) is at least Hölder continuous in one dimension and universally infinitely differentiable in higher dimensions. This result applies also to the IDS in any finite volume subject to the random potential induced by an ambient, infinitely extended disordered media. Dimension one is critical: in the Bernoulli case, within the class of exponential interactions, the IDS measure undergoes continuity phase transitions, from absolutely continuous to singular continuous behaviour (the singularity in the latter case was known before). The continuity transitions do not occur for sub-exponential or slower decaying interactions, nor for $d\ge 2$. Technically, the case of polynomial decay is the simplest one. The proposed approach provides a complement to the classical Wegner estimate which says, essentially, that the IDS in the short-range models is at least as regular as the marginal distribution of the disorder. In the models with non-local interaction the IDS is actually much more regular than the underlying disorder, which can even be discrete, due to the smoothing effect of multiple convolutions. In turn, smoothness of the IDS is responsable for a mechanism complementing the usual Lifshitz tails phenomenon. It is also shown that the disorder can take various forms (e.g., substitution or random displacements) and need not be stochastically stationary (as in Delone--Anderson or trim\-med/crooked Hamiltonians, for example); this does not affect the main phenomena observed already in the simplest setting. Long-range models have an amazingly large number of connections to several classical problems of harmonic analysis, probability theory, dynamical systems and number theory.