## Electronic Journal of Probability

### A mixing tree-valued process arising under neutral evolution with recombination

#### Abstract

The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral  recombination graph encodes the genealogies at all loci in one graph.  For a continuous genome $\mathbb G$, we study the tree-valued  process $(T^N_u)_{u\in\mathbb{G}}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure  spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting  process for loci which are far apart.

#### Article information

Source
Electron. J. Probab. Volume 20 (2015), paper no. 94, 22 pp.

Dates
Accepted: 12 September 2015
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465067200

Digital Object Identifier
doi:10.1214/EJP.v20-4286

Mathematical Reviews number (MathSciNet)
MR3399830

Zentralblatt MATH identifier
1371.92096

Rights

#### Citation

Depperschmidt, Andrej; Pardoux, Étienne; Pfaffelhuber, Peter. A mixing tree-valued process arising under neutral evolution with recombination. Electron. J. Probab. 20 (2015), paper no. 94, 22 pp. doi:10.1214/EJP.v20-4286. https://projecteuclid.org/euclid.ejp/1465067200

#### References

• Aldous, David J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5 (1999), no. 1, 3–48.
• Billingsley, Patrick. Convergence of probability measures. Second edition. Wiley Series in Probability and Statistics: Probability and Statistics. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999. x+277 pp. ISBN: 0-471-19745-9.
• Cannings, C. The latent roots of certain Markov chains arising in genetics: a new approach. I. Haploid models. Advances in Appl. Probability 6 (1974), 260–290.
• Chen, G. K., P. Marjoram, and J. D. Wall (2009). Fast and flexible simulation of dna sequence data. Genome research 19/(1), 136–142.
• Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. x+534 pp. ISBN: 0-471-08186-8.
• Ewens, Warren J. Mathematical population genetics. I. Theoretical introduction. Second edition. Interdisciplinary Applied Mathematics, 27. Springer-Verlag, New York, 2004. xx+417 pp. ISBN: 0-387-20191-2.
• Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Convergence in distribution of random metric measure spaces ($\Lambda$-coalescent measure trees). Probab. Theory Related Fields 145 (2009), no. 1-2, 285–322.
• Greven, Andreas; Pfaffelhuber, Peter; Winter, Anita. Tree-valued resampling dynamics martingale problems and applications. Probab. Theory Related Fields 155 (2013), no. 3-4, 789–838.
• Griffiths, R. C. The two-locus ancestral graph. Selected Proceedings of the Sheffield Symposium on Applied Probability (Sheffield, 1989), 100–117, IMS Lecture Notes Monogr. Ser., 18, Inst. Math. Statist., Hayward, CA, 1991.
• Griffiths, Robert C.; Marjoram, Paul. An ancestral recombination graph. Progress in population genetics and human evolution (Minneapolis, MN, 1994), 257–270, IMA Vol. Math. Appl., 87, Springer, New York, 1997.
• Gromov, Misha. Metric structures for Riemannian and non-Riemannian spaces. Based on the 1981 French original. With appendices by M. Katz, P. Pansu and S. Semmes. Translated from the French by Sean Michael Bates. Reprint of the 2001 English edition. Modern BirkhÃ¤user Classics. BirkhÃ¤user Boston, Inc., Boston, MA, 2007. xx+585 pp. ISBN: 978-0-8176-4582-3; 0-8176-4582-9.
• Hudson, R. R. (1983). Properties of a neutral allele model with intragenic recombination. Theoretical Population Biology/~ 23/(2), 183 – 201.
• Kingman, J. F. C. The coalescent. Stochastic Process. Appl. 13 (1982), no. 3, 235–248.
• Leocard, Stephanie; Pardoux, Etienne. Evolution of the ancestral recombination graph along the genome in case of selective sweep. J. Math. Biol. 61 (2010), no. 6, 819–841.
• McVean, G. A. T. and N. J. Cardin (2005). Approximating the coalescent with recombination. Philosophical transactions of the Royal Society of London. Series B, Biological sciences 360/(1459), 1387–1393.
• Pardoux, Etienne; Salamat, Majid. On the height and length of the ancestral recombination graph. J. Appl. Probab. 46 (2009), no. 3, 669–689.
• Rasmussen, M. D., M. J. Hubisz, I. Gronau, and A. Siepel (2014). Genome-wide inference of ancestral recombination graphs. PLoS Genet/~ 10/(5).
• Wiuf, C. and J. Hein (1999). Recombination as a point process along sequences. Theoretical population biology/~ 55/(3), 248–259.