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Branching Random Walks on Quasi-Transitive Graphs

Published online by Cambridge University Press:  20 May 2003

ALAN STACEY
Affiliation:
Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 OWB, England (e-mail: A.M.Stacey@dpmms.cam.ac.uk)

Abstract

The branching random walk on a regular graph turns out to be particularly easy to analyse using results for the corresponding simple random walk. In this way, one can show that there is an intermediate phase of weak survival if and only if the graph is nonamenable. No such simple analysis holds more generally, and it is known that the nonamenability equivalence does not extend to general connected graphs of bounded degree (although we observe that it does hold for such graphs if the branching random walk is modified in a certain natural way). The most important general class of (bounded degree, connected) graphs for which it is thought that the equivalence may hold is that of quasi-transitive graphs: we show that this is indeed the case.

Type
Research Article
Copyright
© 2003 Cambridge University Press

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