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

  • Author(s): Kyprianou, AE
  • Pagett, SW
  • Rogers, T
  • Schweinsberg, J
  • et al.
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 it's 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

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