A phase transition for hyperbolic branching processes
The asymptotic behaviour of a branching Markov process on hyperbolic space may be described by the random set of accumulation points on the boundary. In the case of the Laplace-Beltrami branching diffusion in two-dimensional hyperbolic space, \textit{S. P. Lalley} and \textit{T. Sellke} [Probab. Theory Relat. Fields 108, No. 2, 171-192 (1997; Zbl 0883.60092)] found a surprising discontinuity in the Hausdorff dimension of this set in dependence on the fission rate of the process. In the present paper this result is generalized in two directions: a more general class of branching processes (in discrete and continuous time) and also hyperbolic spaces of higher dimensions are considered. The following geometric phase transition is shown: either the Hausdorff dimension of the limiting set is in the interval \((0,(d-1)/2)\), namely in the subcritical case, or it is full, i.e. equal to \(d-1\), namely in the supercritical case. Pictorially, if one fixes a process and increases the fission rate from \(0\) to infinity, the Hausdorff dimension increases continuously from \(0\) to \((d-1)/2\) and at the critical value jumps to \(d-1\).
scientific article; zbMATH DE number 1295346
scientific article