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December 2012 The spatial $\Lambda$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination
A. M. Etheridge, A. Véber
Ann. Appl. Probab. 22(6): 2165-2209 (December 2012). DOI: 10.1214/12-AAP842


We extend the spatial $\Lambda$-Fleming–Viot process introduced in [Electron. J. Probab. 15 (2010) 162–216] to incorporate recombination. The process models allele frequencies in a population which is distributed over the two-dimensional torus $\mathbb{T} (L)$ of sidelength $L$ and is subject to two kinds of reproduction events: small events of radius $\mathcal{O} (1)$ and much rarer large events of radius $\mathcal{O} (L^{\alpha})$ for some $\alpha\in(0,1]$. We investigate the correlation between the times to the most recent common ancestor of alleles at two linked loci for a sample of size two from the population. These individuals are initially sampled from “far apart” on the torus. As $L$ tends to infinity, depending on the frequency of the large events, the recombination rate and the initial distance between the two individuals sampled, we obtain either a complete decorrelation of the coalescence times at the two loci, or a sharp transition between a first period of complete correlation and a subsequent period during which the remaining times needed to reach the most recent common ancestor at each locus are independent. We use our computations to derive approximate probabilities of identity by descent as a function of the separation at which the two individuals are sampled.


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A. M. Etheridge. A. Véber. "The spatial $\Lambda$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination." Ann. Appl. Probab. 22 (6) 2165 - 2209, December 2012.


Published: December 2012
First available in Project Euclid: 23 November 2012

zbMATH: 1273.60092
MathSciNet: MR3024966
Digital Object Identifier: 10.1214/12-AAP842

Primary: 60J25 , 60J75 , 92D10
Secondary: 60F05

Keywords: Coalescent , genealogy , generalized Fleming–Viot process , recombination , spatial continuum

Rights: Copyright © 2012 Institute of Mathematical Statistics


Vol.22 • No. 6 • December 2012
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