Branching random walks on trees

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Abstract

Let 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 λ22: 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 similar results for the biased voter model. Finally, for the contact process, branching random walk and biased voter model on the tree, we prove that the second phase transition has a discontinuity which is absent in Euclidian space

Keywords

branching random walk
tree
biased voter model
contact process
phase transition

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Partially supported by the Natural Sciences and Engineering Research Council of Canada

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Partially supported by FAPESP Brazil. Current address: Department of Mathematics, University of Colorado at Colorado Springs, Colorado Springs, CO 80933, USA