  Let a simple random walk run inside a torus of dimension three or higher for
a number of steps which is a constant proportion of the volume. We examine
geometric properties of the range, the random subgraph induced by the set of
vertices visited by the walk. Distance and mixing bounds for the typical range
are proven that are a $k$-iterated log factor from those on the full torus for
arbitrary $k$. The proof uses hierarchical renormalization and techniques that
can possibly be applied to other random processes in the Euclidean lattice. We
use the same technique to bound the heat kernel of a random walk on random
interlacements.
