The second lowest extremal invariant measure of the contact process

• 成果类型: Article
• 署名作者: Salzano, M; Schonmann, RH
• 署名单位: University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1023481114
• 发表日期: 1997
• 页码: 1846-1871
• 关键词: complete convergence theorem; trees

摘要

We study the ergodic behavior of the contact process on infinite connected graphs of bounded degree. We show that the fundamental notion of complete convergence is not as well behaved as it was thought to be. In particular there are graphs for which complete convergence holds in any number of separated intervals of values of the infection parameter and fails for the other values of this parameter. We then introduce a basic invariant probability measure related to the recurrence properties of the process, and an associated notion of convergence that we call partial convergence. This notion is shown to be better behaved than complete convergence, and to hold;in certain cases in which complete convergence fails. Relations between partial and complete convergence are presented, as well as tools to verify when these properties hold. For homogeneous graphs we show that whenever recurrence takes place (i.e., whenever local survival occurs) there are exactly two extremal invariant measures.