A PHASE TRANSITION IN EXCURSIONS FROM INFINITY OF THE FAST FRAGMENTATION-COALESCENCE PROCESS

• 成果类型: Article
• 署名作者: Kyprianou, Andreas E.; Pagett, Steven W.; Rogers, Tim; Schweinsberg, Jason
• 署名单位: University of Bath; University of California System; University of California San Diego
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1150
• 发表日期: 2017
• 页码: 3829-3849
• 关键词: models

摘要

An important property of Kingman's coalescent is that, starting from a state with an infinite number of blocks, over any positive time horizon, it transitions into an almost surely finite number of blocks. This is known as coming down from infinity. Moreover, of the many different (exchangeable) stochastic coalescent models, Kingman's coalescent is the fastest to come down from infinity. In this article, we study what happens when we counteract this fastest coalescent with the action of an extreme form of fragmentation. We augment Kingman's coalescent, where any two blocks merge at rate c > 0, with a fragmentation mechanism where each block fragments at constant rate, lambda > 0, into its constituent elements. We prove that there exists a phase transition at lambda = c/2, between regimes where the resulting fast fragmentation-coalescence process is able to come down from infinity or not. In the case that lambda < c/2, we develop an excursion theory for the fast fragmentation-coalescence process out of which a number of interesting quantities can be computed explicitly.