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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:
  M. Aizenman, (1993). In preparation.  Aizenman, M.; Newman, C.M., Tree graph inequalities and critical behavior in percolation models, J. statist. phys., 44, 393-454, (1984) · Zbl 0586.60096  Barsky, D.J.; Aizenman, M., Percolation critical exponents under the triangle condition, Ann. probab., 19, 1520-1536, (1991) · Zbl 0747.60093  D.J. Barsky and C.C. Wu (1993). In preparation.  Bezuidenhout, C.E.; Grimmett, G.R., The critical contact process dies out, Ann. probab., 18, 1462-1482, (1990) · Zbl 0718.60109  Bezuidenhout, C.E.; Grimmett, G.R., Exponential decay for subcritical contact and percolation processes, Ann. probab., 19, 984-1009, (1991) · Zbl 0743.60107  Durrett, R., Lecture notes on particle systems and percolation, (1988), Wadsworth Belmont CA · Zbl 0659.60129  Griffeath, D., Additive and cancellative interacting particle systems, () · Zbl 0412.60095  Griffeath, D., The basic contact process, Stochastic process. appl., 11, 151-168, (1981)  Grimmett, G.R.; Newman, C.M., Percolation in ∞ + 1 dimensions, (), 167-190 · Zbl 0721.60121  Harris, T.E., Contact interactions on a lattice, Ann. probab., 2, 969-988, (1974) · Zbl 0334.60052  Harris, T.E., Additive set-valued Markov processes and graphical methods, Ann. probab., 6, 355-378, (1978) · Zbl 0378.60106  Liggett, T., Interacting particle systems, (1985), Springer New York · Zbl 0559.60078  Morrow, G.; Schinazi, R.; Zhang, Y., The critical contact process on a homogeneous tree, (1993), preprint  Nguyen, B.G., Gap exponents for percolation processes, J. statist. phys., 49, 235-243, (1987) · Zbl 0962.82521  Nguyen, B.G.; Yang, W.S., Triangle condition for oriented percolation in high dimensions, Ann. probab., 21, 1809-1844, (1993) · Zbl 0806.60097  Pemantle, R., The contact process on trees: a beginning, Ann. probab., 20, 2089-2116, (1992) · Zbl 0762.60098  Wu, C.C., Critical behavior of percolation and Markov fields on branching planes, J. appl. probab., 30, 538-547, (1993) · Zbl 0787.60124
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