The Annals of Applied Probability

The spatial $\Lambda$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination

A. M. Etheridge and A. Véber

Full-text: Open access


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.

Article information

Ann. Appl. Probab., Volume 22, Number 6 (2012), 2165-2209.

First available in Project Euclid: 23 November 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 92D10: Genetics {For genetic algebras, see 17D92} 60J75: Jump processes
Secondary: 60F05: Central limit and other weak theorems

Genealogy recombination coalescent spatial continuum generalized Fleming–Viot process


Etheridge, A. M.; Véber, A. The spatial $\Lambda$-Fleming–Viot process on a large torus: Genealogies in the presence of recombination. Ann. Appl. Probab. 22 (2012), no. 6, 2165--2209. doi:10.1214/12-AAP842.

Export citation


  • [1] Barton, N. H., Etheridge, A. M. and Véber, A. (2010). A new model for evolution in a spatial continuum. Electron. J. Probab. 15 162–216.
  • [2] Barton, N. H., Kelleher, J. and Etheridge, A. M. (2010). A new model for extinction and recolonization in two dimensions: Quantifying phylogeography. Evolution 64 2701–2715.
  • [3] Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • [4] Cox, J. T. (1989). Coalescing random walks and voter model consensus times on the torus in $\mathbb{Z}^{d}$. Ann. Probab. 17 1333–1366.
  • [5] Cox, J. T. and Durrett, R. (2002). The stepping stone model: New formulas expose old myths. Ann. Appl. Probab. 12 1348–1377.
  • [6] Cox, J. T. and Griffeath, D. (1986). Diffusive clustering in the two-dimensional voter model. Ann. Probab. 14 347–370.
  • [7] Cox, J. T. and Griffeath, D. (1990). Mean field asymptotics for the planar stepping stone model. Proc. London Math. Soc. (3) 61 189–208.
  • [8] Etheridge, A. M. (2008). Drift, draft and structure: Some mathematical models of evolution. In Stochastic Models in Biological Sciences. Banach Center Publ. 80 121–144. Polish Acad. Sci. Inst. Math., Warsaw.
  • [9] Felsenstein, J. (1975). A pain in the torus: Some difficulties with the model of isolation by distance. Amer. Nat. 109 359–368.
  • [10] Limic, V. and Sturm, A. (2006). The spatial $\Lambda$-coalescent. Electron. J. Probab. 11 363–393 (electronic).
  • [11] Ridler-Rowe, C. J. (1966). On first hitting times of some recurrent two-dimensional random walks. Z. Wahrsch. Verw. Gebiete 5 187–201.
  • [12] Zähle, I., Cox, J. T. and Durrett, R. (2005). The stepping stone model. II. Genealogies and the infinite sites model. Ann. Appl. Probab. 15 671–699.