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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
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[13] LOS ANGELES, CALIFORNIA 90024 E-MAIL: tml@math.ucla.edu
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