Random Walk Attracted by Percolation Clusters

Abstract

Starting with a percolation model in $\mathbb{Z}^d$ 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 $f(t)=e^{\beta t}$ we prove that there is a phase transition in $\beta$, i.e., the random walk is subdiffusive for large $\beta$ and is diffusive for small $\beta$.