BRANCHING RANDOM-WALKS ON TREES

Let p(x, y) be the transition probability of an isotropic random walk on a tree, where each site has d greater-than-or-equal-to 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 lambda-dp(x, y), jump to y at rate nu-dp(x, y), and die at rate delta. Let lambda-2 (respectively, mu-2) be the infimum of lambda 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 lambda-2 and mu-2 exactly, proving that lambda-2 < mu-2: i.e., the process has two phase transitions. We characterize lambda-2 (respectively, mu-2) 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.