The complete convergence theorem of the contact process on trees

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 $\lambda_c$