  Following [5], we analyze regularity properties of single-site probability
distributions of the random potential and of the Integrated Density of States
(IDS) in the Anderson models with infinite-range interactions. In the present
work, we study in detail a class of polynomially decaying interaction
potentials of rather artificial (piecewise-constant) form, and give a complete
proof of infinite smoothness of the IDS in an arbitrarily large finite domain
subject to the fluctuations of the entire, infinite random environment. A
variant of this result, based as in [5] on the harmonic analysis of probability
measures, results in a proof of spectral and dynamical Anderson localization in
the considered models.
