this direction (in the multi-dimensional context) were obtained by von Dreifus and Klein [@DK91].

The proofs of all statements not given in the main text can be found in Appendix.

Basic definitions and assumptions.  {#sec:basic.notions}
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Assumptions
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We consider the lattice Anderson Hamiltonians of the form \[eq:H\] H() = -+ gV(x;), where $V:\DZ^d\times\Omega\to\DR$ is a random field relative to some probability space $(\Omega, \CF, \DP)$, and $\Delta$ is the nearest-neighbor lattice Laplacian.

For the sake of clarity, our method is presented first in a simpler situation where the random field $V$ is IID, with common marginal probability distribution function (PDF) $F_{V}(t) = \pr{ V(0;\omega) \leq t}$. In this case, we assume:

1.  The marginal PDF $F_V$ is uniformly Hölder-continuous: for some $b>0$, \[eq:V\] t, F\_V(t+) - F\_V(t) \^b.

Hölder continuity can be relaxed to log-Hölder continuity: $F_V(t+\epsilon) - F_V(t) \leq \Const \ln^{-A} \epsilon^{-1}$ with sufficiently large $A>0$. This is sufficient for the proof of a polynomial decay of eigenfunction (EF) correlators and for dynamical localization at a polynomial rate. In order to prove decay of EF correlators faster than polynomial, one needs a regularity condition for $F_V$ slightly stronger that log-Hölder continuity; this is why we make the assumption 