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Branching random walks on trees. (English) Zbl 0763.60042
Consider contact processes, branching random walks, and biased voter models, when the underlying graph is an infinite regular tree of degree \(d\) \((\geq 3)\). Continuing work of the reviewer and C. Newman [Disorder in physical systems, Vol. in Honour of J. M. Hammersley 70th Birthday, 167-190 (1990; Zbl 0721.60121)] and R. Pemantle [Ann. Probab. 20, 2089-2116 (1992)], the authors of the current paper prove the existence of three phases for the problems of branching random walks (BRW) and biased voter models. The BRW result may be summarized as follows. Let \(p(x,y)\) be the transition matrix of an isotropic random walk on the tree. A particle at \(x\) gives birth to a new particle at \(y\) at rate \(\lambda dp(x,y)\), or alternatively it jumps to \(y\) at rate \(\nu dp(x,y)\); a third possibility is that the particle at \(x\) dies 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). Then \(\lambda_ 2\) and \(\mu_ 2\) are computed exactly, and it is found that \(\lambda_ 2<\mu_ 2\). A similar result is obtained for a biased voter model. For all three processes it is proved that the probability a given vertex is occupied for arbitrarily large times is a discontinuous function of \(\lambda\) at the point \(\lambda=\mu_ 2\). This conclusion is false for hypercubic lattices.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G50 Sums of independent random variables; random walks
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