The branching random walk and contact process on Galton-Watson and nonhomogeneous trees

Abstract

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 Madras and Schinazi. 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 Pemantle. Key words and phrases. Tree, branching random walk, contact process, phase transition, spectral radius.