I will consider the random walk on Z^d driven by a field of random i.i.d.
conductances.
The law of the conductances is bounded from above; no restriction is posed on the
lower tail (at zero) except that the bonds with positive conductances percolate.
The presence of very weak bonds allows the random walk in a finite box mix pretty
much arbitrarily slowly. However, when we focus attention on the return probability
to the starting point -- i.e., the heat-kernel -- it turns out that in dimensions
d=2,3 the
decay is as for the simple random walk. On the other hand, in d>4 the heat- kernel
at time 2n may decay as slowly as o(1/n^2) and in d=4 as slowly as O(n^{-2}log n).
These upper bounds can be matched arbitrarily closely by lower bounds in particular
examples. Despite this, the random walk scales to Brownian motion under the usual
diffusive scaling of space and time. Based on joint works with N. Berger, C. Hoffman,
G. Kozma and T. Prescott.