Random Walk Delayed on Percolation Clusters

Random Walk Delayed on Percolation Clusters

We study a continuous-time random walk on the d-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we ...

Random Walk Attracted by Percolation Clusters

Starting with a percolation model in Z in the subcritical regime, we consider a random walk described as follows: the probability of transition from x to y is proportional to some function f of the size of the cluster of y. This function is supposed to be increasing, so that the random walk is attracted by bigger clusters. For f(t) = e we prove that there is a phase transition in β, i.e., the r...

Quenched Invariance Principle for Simple Random Walk on Percolation Clusters

We consider the simple random walk on the (unique) infinite cluster of super-critical percolation in Z with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate Brownian motion.

Quenched invariance principle for simple random walk on percolation clusters

We consider the simple random walk on the (unique) infinite cluster of supercritical bond percolation in Z with d ≥ 2. We prove that, for almost every percolation configuration, the path distribution of the walk converges weakly to that of non-degenerate, isotropic Brownian motion. Our analysis is based on the consideration of a harmonic deformation of the infinite cluster on which the random w...

Scaling limit of simple random walk on supercritical percolation clusters