Anisotropic branching random walks on homogeneous trees

Abstract

Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundary Q of the tree. The random subset ∧ of the boundary consisting of all ends of the tree in which the population survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure μ supported by Ω such that the Hausdorff dimension of ∧ ∩ Ωμ, where Ωμ, is the set of μ-generic points of Ω, converges to one half the Hausdorff dimension of Ωμ at the phase separation point. Exact formulas are obtained for the Hausdorff dimensions of ∧ and ∧ ∩ Ωμ, and it is shown that the log Hausdorff dimension of ∧ has critical exponent 1/2 at the phase separation point.