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It is shown that in a large class of disordered systems with non-degeneratedisorder, in presence of non-local interactions, the Integrated Density ofStates (IDS) is at least H\"older continuous in one dimension and universallyinfinitely differentiable in higher dimensions. This result applies also to theIDS in any finite volume subject to the random potential induced by an ambient,infinitely extended disordered media. Dimension one is critical: in theBernoulli case, within the class of exponential interactions, the IDS measureundergoes continuity phase transitions, from absolutely continuous to singularcontinuous behaviour (the singularity in the latter case was known before). Thecontinuity transitions do not occur for sub-exponential or slower decayinginteractions, nor for $d\ge 2$. Technically, the case of polynomial decay isthe simplest one.  The proposed approach provides a complement to the classical Wegner estimatewhich says, essentially, that the IDS in the short-range models is at least asregular as the marginal distribution of the disorder. In the models withnon-local interaction the IDS is actually much more regular than the underlyingdisorder, which can even be discrete, due to the smoothing effect of multipleconvolutions. In turn, smoothness of the IDS is responsable for a mechanismcomplementing the usual Lifshitz tails phenomenon.  It is also shown that the disorder can take various forms (e.g., substitutionor random displacements) and need not be stochastically stationary (as inDelone--Anderson or trim\-med/crooked Hamiltonians, for example); this does notaffect the main phenomena observed already in the simplest setting.  Long-range models have an amazingly large number of connections to severalclassical problems of harmonic analysis, probability theory, dynamical systemsand number theory.

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