The Segregated Lambda-coalescent

TL;DR

This paper extends the Lambda-coalescent to include spatial structure, revealing a five-phase system with two novel phases influenced by space, including a fractal dust phase and a gradual coming down from infinity.

Contribution

It introduces a spatially extended Lambda-coalescent model and classifies its phase behavior, identifying two new phases caused by spatial effects.

Findings

Identification of five distinct phases in the spatial Lambda-coalescent.

Discovery of a fractal dust phase with a null, non-empty dust set.

Characterization of a critical phase with gradual coming down from infinity.

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 non-empty 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.