Abstract
Consider the contact process on a homogeneous tree with degree $d \geq 3$. Denote by
$$\lambda_c = \inf\{\lambda : P(o \in \xi^o_t \text{i.o.}) > 0\}$$
the critical value of local survival probability, where o is the root of the tree. Pemantle and Durrett and Schinazi both conjectured that the complete convergence theorem should hold if $\lambda >\lambda_c$. Here we answer the conjecture affirmatively. Furthermore, we will show that
$$P(o \in \xi^o_t \text{i.o.}) = 0 \quad \text{at $\lambda_c}.$$
Therefore, the conclusion of the complete convergence theorem cannot hold at $\lambda_c$
Citation
Yu Zhang. "The complete convergence theorem of the contact process on trees." Ann. Probab. 24 (3) 1408 - 1443, July 1996. https://doi.org/10.1214/aop/1065725187
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