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Multiple transition points for the contact process on the binary tree. (English) Zbl 0871.60087
Let $$T^d$$ denote the homogeneous (connected) tree in which each vertex has $$d+1$$ neighbours and let $$(A_t)$$ be the finite contact process on $$T_d$$. By definition, we have the transitions $$A\to A\setminus\{x\}$$ for $$x\in A$$ at rate 1, and $$A\to A\cup\{x\}$$ for $$x\notin A$$ at rate $$\lambda\cdot\#\{y\in A: |y-x|=1\}$$ $$(|y-x|$$ denoting the distance between $$x, y\in T^d)$$. $$(A_t)$$ is said to survive strongly if $$P^{\{x\}}(x\in A_t$$ for arbitrarily large $$t) > 0$$. On the other hand, $$(A_t)$$ survives if $$P^{\{x\}}(A_t\neq 0,$$ $$t\geq 0)>0$$. One says that $$(A_t)$$ dies out if it does not survive, and that it survives weakly if it survives, but does not survive strongly. Critical values $$\lambda_1\leq \lambda_2$$ are defined by the requirement that $$(A_t)$$ survives strongly for $$\lambda_1>\lambda_2$$, survives weakly for $$\lambda_1<\lambda< \lambda_2$$ and dies out for $$\lambda < \lambda_1$$. In the case $$d\geq 3$$, R. Pemantle [ibid. 20, No. 4, 2089-2116 (1992; Zbl 0762.60098)] obtained upper bounds on $$\lambda_1$$ and lower bounds on $$\lambda_2$$ implying that $$\lambda_1<\lambda_2$$. In the present paper it is shown that (for homogeneous trees) in the case $$d=2$$, $$\lambda_1\leq 0.605$$, $$\lambda_2\geq 0.609$$ which implies $$\lambda_1< \lambda_2$$.
Reviewer: K.Schürger (Bonn)

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory
##### Keywords:
contact process; survive strongly; survives weakly; lower bounds
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##### References:
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