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Survival of contact processes on the hierarchical group
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  • Published: 16 April 2009

Survival of contact processes on the hierarchical group

  • Siva R. Athreya1 &
  • Jan M. Swart2 

Probability Theory and Related Fields volume 147, pages 529–563 (2010)Cite this article

  • 111 Accesses

  • 13 Citations

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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.

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Authors and Affiliations

  1. Theoretical Statistics and Mathematics Division, Indian Statistical Institute, 8th Mile, Mysore Road, RVCE Post, Bangalore, 560 059, India

    Siva R. Athreya

  2. Institute of Information Theory and Automation of the ASCR (ÚTIA), Pod vodárenskou věží 4, 18208, Prague 8, Czech Republic

    Jan M. Swart

Authors
  1. Siva R. Athreya
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  2. Jan M. Swart
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Correspondence to Jan M. Swart.

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Athreya, S.R., Swart, J.M. Survival of contact processes on the hierarchical group. Probab. Theory Relat. Fields 147, 529–563 (2010). https://doi.org/10.1007/s00440-009-0214-x

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  • Received: 27 August 2008

  • Revised: 02 March 2009

  • Published: 16 April 2009

  • Issue Date: July 2010

  • DOI: https://doi.org/10.1007/s00440-009-0214-x

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Keywords

  • Contact process
  • Survival
  • Hierarchical group
  • Coupling
  • Renormalization group

Mathematics Subject Classification (2000)

  • Primary: 82C22
  • Secondary: 60K35
  • 82C28
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