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Survival of contact processes on the hierarchical group. (English) Zbl 1191.82028

Summary: We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
92D30 Epidemiology
60J25 Continuous-time Markov processes on general state spaces
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