A phase transition for hyperbolic branching processes
type: Item
instance of: scholarly article
title: A phase transition for hyperbolic branching processes
zbMATH Open document ID: 0933.60084
author: F. I. Karpelevich
author: Yuri Suhov
author: Eugene Pechersky
published in: Communications in Mathematical Physics
publication date: 2000-03-27T00:00:00Z
review text: 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 \(d-1\).
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 82B26
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 28A78
zbMATH DE Number: 1295346
zbMATH Keywords: branching
zbMATH Keywords: diffusion
zbMATH Keywords: branching Markov process
zbMATH Keywords: hyperbolic space
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: phase transition
MaRDI profile type: Publication
full work available at URL: s002200050405
OpenAlex ID: W2011152726
DOI: 10.1007/S002200050405
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description: scientific article; zbMATH DE number 1295346