The Annals of Probability

The segregated $\Lambda$-coalescent

Nic Freeman

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 43, Number 2 (2015), 435-467.

Dates
First available in Project Euclid: 2 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.aop/1422885567

Digital Object Identifier
doi:10.1214/13-AOP857

Mathematical Reviews number (MathSciNet)
MR3305997

Zentralblatt MATH identifier
1334.60178

Subjects
Primary: 60G99: None of the above, but in this section
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J85: Applications of branching processes [See also 92Dxx]

Keywords
Lambda coalescent coalescent segregated

Citation

Freeman, Nic. The segregated $\Lambda$-coalescent. Ann. Probab. 43 (2015), no. 2, 435--467. doi:10.1214/13-AOP857. https://projecteuclid.org/euclid.aop/1422885567


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