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

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 thefastest' 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 resultingfast' 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.