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The contact process on a tree: Behavior near the first phase transition. (English) Zbl 0821.60093
R. Pemantle [Ann. Probab. 20, No. 4, 2089-2116 (1992; Zbl 0762.60098)] has shown that for the contact process on homogeneous trees \(T\) of degree greater than or equal to 3, there exist critical parameters \(0 < \lambda_ c < \lambda_ c' < \infty\) such that the process dies out when \(\lambda < \lambda_ c\), survives globally but dies out locally when \(\lambda_ c < \lambda < \lambda_ c'\), and survives globally as well as locally when \(\lambda > \lambda_ c'\); he also showed that the survival probability \(\theta (\lambda)\) is a continuous function of \(\lambda\) vanishing at \(\lambda_ c\). Here it is shown (assuming that the degree of \(T\) is greater than or equal to 5) that there exist positive constants \(c_ 1\), \(c_ 2\), \(d_ 1\), \(d_ 2\) such that \[ c_ 1 (\lambda - \lambda_ c) \leq \theta (\lambda) \leq c_ 2 (\lambda - \lambda_ c) \text{ as } \lambda \downarrow \lambda_ c \text{ and } d_ 1(\lambda_ c - \lambda)^{-1} \leq \chi (\lambda) \leq d_ 2 (\lambda_ c - \lambda)^{-1} \text{ as } \lambda \uparrow \lambda_ c. \] Here, \(\chi (\lambda)\) denotes the expected infection time given by \(\chi (\lambda) = E_ \lambda [\int^ \infty_ 0 | \xi_ t | dt]\), where \(| \xi_ t |\) denotes the number of sites in \(\xi_ t\) (the set of points \(x\) in \(T\) such that the origin \((\sigma, 0)\) of \(T \times [0, \infty[\) is connected to \((x,t))\).
Reviewer: K.Schürger (Bonn)

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