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The branching random walk and contact process on Galton-Watson and nonhomogeneous trees. (English) Zbl 1013.60078

Summary: We show that the branching random walk on a Galton-Watson tree may have one or two phase transitions, depending on the relative sizes of the mean degree and the maximum degree. We show that there are some Galton-Watson trees on which the branching random walk has one phase transition while the contact process has two; this contradicts a conjecture of N. Madras and R. Schinazi [Stochastic Processes Appl. 42, No. 2, 255-267 (1992; Zbl 0763.60042)]. We show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree, contradicting a conjecture of R. Pemantle [Ann. Probab. 20, No. 4, 2089-2116 (1992; Zbl 0762.60098)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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