Open Access
March 2015 The segregated $\Lambda$-coalescent
Nic Freeman
Ann. Probab. 43(2): 435-467 (March 2015). DOI: 10.1214/13-AOP857


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.


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Nic Freeman. "The segregated $\Lambda$-coalescent." Ann. Probab. 43 (2) 435 - 467, March 2015.


Published: March 2015
First available in Project Euclid: 2 February 2015

zbMATH: 1334.60178
MathSciNet: MR3305997
Digital Object Identifier: 10.1214/13-AOP857

Primary: 60G99
Secondary: 60J70 , 60J85

Keywords: Coalescent , Lambda coalescent , segregated

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 2 • March 2015
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