The Annals of Probability

A phase transition in excursions from infinity of the “fast” fragmentation-coalescence process

Andreas E. Kyprianou, Steven W. Pagett, Tim Rogers, and Jason Schweinsberg

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Abstract

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.

Article information

Source
Ann. Probab., Volume 45, Number 6A (2017), 3829-3849.

Dates
Received: February 2016
Revised: September 2016
First available in Project Euclid: 27 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.aop/1511773665

Digital Object Identifier
doi:10.1214/16-AOP1150

Mathematical Reviews number (MathSciNet)
MR3729616

Zentralblatt MATH identifier
06838108

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60G09: Exchangeability

Keywords
Fragmentation coalescence excursion theory

Citation

Kyprianou, Andreas E.; Pagett, Steven W.; Rogers, Tim; Schweinsberg, Jason. A phase transition in excursions from infinity of the “fast” fragmentation-coalescence process. Ann. Probab. 45 (2017), no. 6A, 3829--3849. doi:10.1214/16-AOP1150. https://projecteuclid.org/euclid.aop/1511773665


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