We prove an analog of Wegner's estimate for the density of states (DS)
in
finite
volumes for certain families of lattice Schrdinger operators (LSO) with
random potential
generated by a {\it deterministic } dynamical system. We call such
families
"Grand
Ensembles". The main assumption about the underlying dynamics is given in
terms of the
typical rate of returns to initial point, so it is very "mild" and general.
Although our
finite-volume estimates are much weaker than Wegner's estimate for
non-deterministic
potentials and do not imply regularity (or even existence) of the limiting
DS, they allow
to adapt the MSA by von Dreifus -- Klein to {\it generic } deterministic
LSO. While the
localization results are somewhat weaker than those by Bourgain --
Goldstein -- Schlag,
our proof is simpler, modulo existing MSA techniques.

In this talk, we also outline a new adaptation of the von Dreifus --
Klein MSA scheme to
localization in lattice systems of interacting quantum particles in common
external
random potential with independent values (joint project with Yu. Suhov,
Cambridge
University, UK). We believe that our method of Grand Ensembles applies as
well to
multi-particle systems in deterministic external potential.