Mathematics – Probability
Scientific paper
2011-09-20
Mathematics
Probability
Updated to submitted article. 44 pages, 2 figures. Results which appear in v1 but not in v2 will form part of a future article
Scientific paper
We construct a natural 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, and obtain 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 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 total number of blocks becomes a.s. finite at a deterministic, positive time.
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