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November 2017 A phase transition in excursions from infinity of the “fast” fragmentation-coalescence process
Andreas E. Kyprianou, Steven W. Pagett, Tim Rogers, Jason Schweinsberg
Ann. Probab. 45(6A): 3829-3849 (November 2017). DOI: 10.1214/16-AOP1150


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.


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Andreas E. Kyprianou. Steven W. Pagett. Tim Rogers. Jason Schweinsberg. "A phase transition in excursions from infinity of the “fast” fragmentation-coalescence process." Ann. Probab. 45 (6A) 3829 - 3849, November 2017.


Received: 1 February 2016; Revised: 1 September 2016; Published: November 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06838108
MathSciNet: MR3729616
Digital Object Identifier: 10.1214/16-AOP1150

Primary: 60G09 , 60J25

Keywords: Coalescence , Excursion theory , fragmentation

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 6A • November 2017
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