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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
##### Keywords:
contact process; oriented percolation; critical exponents
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##### References:
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