  Moderate deviation principles (MDPs) for random walks on covering graphs with
groups of polynomial volume growth are discussed in a geometric point of view.
They deal with any intermediate spatial scalings between those of laws of large
numbers and those of central limit theorems. The corresponding rate functions
are given by quadratic forms determined by the Albanese metric associated with
the given random walks. We apply MDPs to establish laws of the iterated
logarithm on the covering graphs by characterizing the set of all limit points
of the normalized random walks.
