DOIAbstract
AbstractLet p(x, y) be the transition probability of an isotropic random walk on a tree, where each site has d ⩾3 neighbors. We define a branching random walk by letting a particle at site x give birth to a new particle at site y at rate λdp(x, y), jump to y at rate vdp(x, y), and die at rate δ. Let λ2 (respectively, μ2) be the infimum of λ such that the process starting with one particle has positive probability of surviving forever (respectively, of having a fixed site occupied at arbitrarily large times). We compute λ2 and μ2 exactly, proving that λ2<μ2: i.e., the process has two phase transitions. We characterize λ2 (respectively, μ2) in terms of the expected number of particles on the tree (respectively, at a fixed site). We also prove...
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