### Abstract

Original language | English |
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Pages (from-to) | 3829-3849 |

Journal | Annals of Probablitiy |

Volume | 45 |

Issue number | 6A |

DOIs | |

Publication status | Published - 27 Nov 2017 |

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*Annals of Probablitiy*,

*45*(6A), 3829-3849. https://doi.org/10.1214/16-AOP1150

**A phase transition in excursions from infinity of the "fast" fragmentation-coalescence process.** / Kyprianou, Andreas; Rogers, Timothy; Pagett, Steven; Schweinsberg, Jason.

Research output: Contribution to journal › Article

*Annals of Probablitiy*, vol. 45, no. 6A, pp. 3829-3849. https://doi.org/10.1214/16-AOP1150

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TY - JOUR

T1 - A phase transition in excursions from infinity of the "fast" fragmentation-coalescence process

AU - Kyprianou, Andreas

AU - Rogers, Timothy

AU - Pagett, Steven

AU - Schweinsberg, Jason

PY - 2017/11/27

Y1 - 2017/11/27

N2 - 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, λ>0, into its constituent elements. We prove that there exists a phase transition at λ=c/2, between regimes where the resulting “fast” fragmentation-coalescence process is able to come down from infinity or not. In the case that λ<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.

AB - 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, λ>0, into its constituent elements. We prove that there exists a phase transition at λ=c/2, between regimes where the resulting “fast” fragmentation-coalescence process is able to come down from infinity or not. In the case that λ<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.

U2 - 10.1214/16-AOP1150

DO - 10.1214/16-AOP1150

M3 - Article

VL - 45

SP - 3829

EP - 3849

JO - Annals of Probablitiy

JF - Annals of Probablitiy

IS - 6A

ER -