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The critical contact process on a homogeneous tree. (English) Zbl 0798.60091
Consider a homogeneous tree with \(d\) \((\geq 3)\) branches. The contact process on the tree is defined as follows. If there is a particle, it gives birth to a new particle with rate \(\lambda>0\) to each of its \(d\) neighboring sites. If there is a birth in an already occupied site, then the two particles coalesce to one. A particle dies at rate 1. Denote by \(\eta^ x_ t\) the process starting from \(x\) and set \(\lambda_ 1 = \inf \{\lambda : \mathbb{P}_ \lambda [| \eta^ 0_ t | \geq 1\), \(\forall t>0] > 0\}\). The authors prove that the critical value is \(\lambda_ 1\), \(1 \leq \mathbb{E} (| \eta^ 0_ t |) \leq C(d)\), where \(C(d)\) is a constant depending on \(d\) only. Moreover, the critical contact process dies out. It is notable that the proof given here for the last conclusion is much simpler than that for the usual contact process.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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