Random walk in non-elliptic random environments

Much of the literature on random walk in random environment assumes uniformly ellipticity, i.e., that nearest neighbour steps have probabilities bounded away from zero. I’ll describe some work with Mark Holmes (Univ. of Auckland) in which we relax this assumption, and allow some such steps to be forbidden. This leads naturally to percolation models, using which one can in some cases prove ballisticity of the random walks (existence of non-zero asymptotic speeds).