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Collisions of random walks in reversible random graphs. (English) Zbl 1329.60357
Summary: We prove that in any recurrent reversible random rooted graph, two independent simple random walks started at the same vertex collide infinitely often almost surely. This applies to the uniform infinite planar triangulation and quadrangulation and to the incipient infinite cluster in \(\mathbb{Z}^2\).

MSC:
60K37 Processes in random environments
60G50 Sums of independent random variables; random walks
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
05C81 Random walks on graphs
05C80 Random graphs (graph-theoretic aspects)
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