## The Annals of Probability

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

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

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

#### 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