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Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Daniela Bertacchi  and
Fabio Zucca
Daniela Bertacchi*
Affiliation:
Università di Milano - Bicocca
Fabio Zucca*
Affiliation:
Politecnico di Milano
*
∗Postal address: Dipartimento di Matematica e Applicazioni, Università di Milano - Bicocca, via Cozzi 53, 20125 Milano, Italy. Email address: daniela.bertacchi@unimib.it
∗∗Postal address: Dipartimento di Matematica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy. Email address: fabio.zucca@polimi.it
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Abstract

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We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.

Keywords

Branching random walkphase transitionmultigraphamenabilitytree

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 2 , June 2008 , pp. 481 - 497
Copyright
Copyright © Applied Probability Trust 2008 

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