A phase transition for hyperbolic branching processes

F. I. Karpelevich, E. A. Pechersky, Yu M. Suhov

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Abstract

Consider a time-and space-homogeneous random branching Markov process on a d-D Lobachevsky space Hd. Its asymptotic behaviour can be described in terms of the Hausdorff dimension of the (random) set Λ of the accumulation points (on the absolute ∂Hd). 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 ∂Hd, which gives an interesting exmaple of a "geometric" phase transition.

Original languageEnglish (US)
Pages (from-to)627-642
Number of pages16
JournalCommunications In Mathematical Physics
Volume195
Issue number3
DOIs
Publication statusPublished - Jan 1 1998

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All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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