A phase transition for hyperbolic branching processes

Consider a time-and space-homogeneous random branching Markov process on a d-D Lobachevsky space H_d. Its asymptotic behaviour can be described in terms of the Hausdorff dimension of the (random) set Λ of the accumulation points (on the absolute ∂H_d). The simplest and most well-known example is the Laplace-Beltrami branching diffusion; in the case d = 2 the Hausdorff dimension of Λ was calculated in [LS]. In this paper we extend the formula for the Hausdorff dimension to d ≥ 3 and a larger class of branching processes. It turns out that the Hausdorff dimension of Λ takes either a value from (O, (d - 1)/2) or equals d - 1, the Euclidean dimension of ∂H_d, which gives an interesting exmaple of a "geometric" phase transition.