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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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