A PHASE TRANSITION IN EXCURSIONS FROM INFINITY OF THE "FAST" FRAGMENTATION-COALESCENCE PROCESS
Open Access Publications from the University of California

## 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.

## Published Web Location

https://doi.org/10.1214/16-AOP1150
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

Many UC-authored scholarly publications are freely available on this site because of the UC Academic Senate's Open Access Policy. Let us know how this access is important for you.