Probability Theory and Related Fields

, Volume 147, Issue 3, pp 529–563

Survival of contact processes on the hierarchical group

Article

DOI: 10.1007/s00440-009-0214-x

Cite this article as:
Athreya, S.R. & Swart, J.M. Probab. Theory Relat. Fields (2010) 147: 529. doi:10.1007/s00440-009-0214-x

Abstract

We consider contact processes on the hierarchical group, where sites infect other sites at a rate depending on their hierarchical distance, and sites become healthy with a constant recovery rate. If the infection rates decay too fast as a function of the hierarchical distance, then we show that the critical recovery rate is zero. On the other hand, we derive sufficient conditions on the speed of decay of the infection rates for the process to exhibit a nontrivial phase transition between extinction and survival. For our sufficient conditions, we use a coupling argument that compares contact processes on the hierarchical group with freedom two with contact processes on a renormalized lattice. An interesting novelty in this renormalization argument is the use of a result due to Rogers and Pitman on Markov functionals.

Keywords

Contact processSurvivalHierarchical groupCouplingRenormalization group

Mathematics Subject Classification (2000)

Primary: 82C22Secondary: 60K3582C28

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics DivisionIndian Statistical InstituteBangaloreIndia
  2. 2.Institute of Information Theory and Automation of the ASCR (ÚTIA)Prague 8Czech Republic