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The social network model on infinite graphs. (English) Zbl 1459.60145
Summary: Given an infinite connected regular graph $$G=(V,E)$$, place at each vertex $$\operatorname{Poisson}(\lambda)$$ walkers performing independent lazy simple random walks on $$G$$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when $$G$$ is vertex-transitive and amenable, for all $$\lambda>0$$ a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when $$G$$ is nonamenable (not necessarily transitive) there is always a phase transition at some $$\lambda_{\text{c}}(G)>0$$. We give general bounds on $$\lambda_{\text{c}}(G)$$ and study the case that $$G$$ is the $$d$$-regular tree in more detail. Finally, we show that in the nonamenable setup, for every $$\lambda$$ there exists a finite time $$t_{\lambda}(G)$$ such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time $$t_{\lambda}(G)$$.
MSC:
 60J10 Markov chains (discrete-time Markov processes on discrete state spaces) 60G50 Sums of independent random variables; random walks 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics 82B43 Percolation
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