The social network model on infinite graphs
  Given an infinite connected regular graph $G=(V,E)$, place at each vertex Pois($\lambda$) 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 non-amenable (not necessarily transitive) there is always a phase transition at some $\lambda_{c}(G)>0$. We give general bounds on $\lambda_{c}(G)$ and study the case that $G$ is the $d$-regular tree in more details. Finally, we show that in the non-amenable 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 $t_{\lambda}(G)$. 
Jonathan Hermon, Ben Morris, Chuan Qin and Allan Sly
math.PR