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Random walk attracted by percolation clusters. (English) Zbl 1112.60089
Summary: 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 $$\beta$$.

##### MSC:
 60K37 Processes in random environments 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60K35 Interacting random processes; statistical mechanics type models; percolation theory
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