Exploiting the coherent medium approximation, I investigate a random walk on objects distributed randomly in continuous space when jump rate depends distance between two adjacent objects. In one dimension, it is shown that decays exponentially long limit, non-diffusive to diffusive transition occurs as density of sites increased. three dimensions, exists has super Gaussian decay.

In this note we give three short results concerning the elliptic Harnack inequality (EHI), in the context of random walks on graphs. The first is that the EHI implies polynomial growth of the number of points in balls, and the second that the EHI is equivalent to an annulus type Harnack inequality for Green’s functions. The third result uses the lamplighter group to give a counterexample concer...