*Homogenization of Random walk in Asymmetric Random Environment *

**Abstract:**
Consider a nearest neighbor random walk on the integer lattice in d
dimensions. The transition probabilities are themselves random variables,
independent from site to site and with no mean drift. It was shown by Sinai in
1982 that for d=1 the walk is with probability one strongly subdiffusive. In
1983-84 it was conjectured by Derrida and Luck and independently by Fisher, that
for d>2 the walk is diffusive with probability one. In this lecture a series
expansion is presented which may be a candidate for the effective diffusion
constant in d>2. Showing that each term of the series is finite is non-trivial.