Electronic Journal of Probability

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

Andrej Depperschmidt, Étienne Pardoux, and Peter Pfaffelhuber

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 92D15: Problems related to evolution 60G10: Stationary processes
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Ancestral recombination graph Kingman coalescent tree-valued process Gromov-Hausdorff metric

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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

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