Survival of contact processes on the hierarchical group

• 成果类型: Article
• 署名作者: Athreya, Siva R.; Swart, Jan M.
• 署名单位: Czech Academy of Sciences; Institute of Information Theory & Automation of the Czech Academy of Sciences; Indian Statistical Institute; Indian Statistical Institute Bangalore
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-009-0214-x
• 发表日期: 2010
• 页码: 529-563
• 关键词: phase-transition; models

摘要

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.