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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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