Commun. Math. Phys. 195, 627 – 642 (1998) Communications in Mathematical Physics © Springer-Verlag 1998 A Phase Transition for Hyperbolic Branching Processes ⋆ F. I. Karpelevich 1, 2 , E. A. Pechersky 2 , Yu. M. Suhov 2 , 3 1 Moscow Transport University (MIIT), The Russian Ministry of Railways, Moscow 101475, Russia 2 The Dobrushin Mathematical Laboratory, Institute for Problems of Information Transmission, The Russian Academy of Sciences, GSP-4 Moscow 101447, Russia 3 Statistical Laboratory, DPMMS, University of Cambridge, Cambridge CB2 1SB, England, UK Received: 22 September 1997 / Accepted: 19 December 1997 Abstract: 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  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  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  takes either a value from (0, (d − 1)/2) or equals d − 1, the Euclidean dimension of ∂ H d , which gives an interesting exmaple of a “geometric” phase transition. 1. The Main Result Introduction. This paper deals with the Hausdorff dimension of the limiting set  of a homogeneous branching Markov process on the d-dimensional hyperbolic, or Lobachevsky, space H d . The problem was put forward by Lalley and Sellke, and we refer the reader to the introduction to [LS], where it is discussed in the case of a homogeneous branching diffusion on a Lobachevsky plane H 2 . (In this model, the generator of the individual Markov process is one half of the standard Laplace–Beltrami operator.) The main result of Lalley and Sellke is that the Hausdorff dimension of  (which is a subset of the absolute ∂ H 2 ) for the branching diffusion with the offspring number two and fission rate λ equals (1 − √ 1 − 8λ)/2 if λ ≤ 1/8 and 1 if λ> 1/8. A phase transition discovered in [LS] and manifested in the discontinuity of the Hausdorff dimension at λ ⋆ This work was supported in part by the Russian Foundation for Fundamental Research (Grants 96 01 00150 and 97-01-00747); the Russian Ministry of Railways Fund “NIOKR”; The London Mathematical Society; St John’s College, Cambridge; Dublin Institute for Advanced Studies; I.H.E.S., Bures-sur-Yvette; the EC Grant “Training Mobility and Research” (Contracts CHRX-CT 930411 and ERBMRXT-CT 960075A) and the INTAS Grant “Mathematical Methods for Stochastic Discrete Event Systems” (INTAS 93-820).