The segregated $\Lambda$-coalescent

Abstract

We construct an extension of the $\Lambda$-coalescent to a spatial continuum and analyse its behaviour. Like the $\Lambda$-coalescent, the individuals in our model can be separated into (i) a dust component and (ii) large blocks of coalesced individuals. We identify a five phase system, where our phases are defined according to changes in the qualitative behaviour of the dust and large blocks. We completely classify the phase behaviour, including necessary and sufficient conditions for the model to come down from infinity.

We believe that two of our phases are new to $\Lambda$-coalescent theory and directly reflect the incorporation of space into our model. Firstly, our semicritical phase sees a null but nonempty set of dust. In this phase the dust becomes a random fractal, of a type which is closely related to iterated function systems. Secondly, our model has a critical phase in which the coalescent comes down from infinity gradually during a bounded, deterministic time interval.