# Ancestries of a recombining diploid population

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## Abstract

We derive the exact one-step transition probabilities of the number of lineages that are ancestral to a random sample from the current generation of a bi-parental population that is evolving under the discrete Wright–Fisher model with \(n\) diploid individuals. Our model allows for a per-generation recombination probability of \(r\). When \(r=1\), our model is equivalent to Chang’s (Adv Appl Probab 31:1002–1038, 1999) model for the karyotic pedigree. When \(r=0\), our model is equivalent to Kingman’s (Stoch Process Appl 13:235–248, 1982) discrete coalescent model for the cytoplasmic tree or sub-karyotic tree containing a DNA locus that is free of intra-locus recombination. When \(0 < r <1\) our model can be thought to track a sub-karyotic ancestral graph containing a DNA sequence from an autosomal chromosome that has an intra-locus recombination probability \(r\). Thus, our family of models indexed by \(r\in [0,1]\) connects Kingman’s discrete coalescent to Chang’s pedigree in a continuous way as \(r\) goes from \(0\) to \(1\). For large populations, we also study three properties of the ancestral process corresponding to a given \(r\in (0,1)\): the time \(\fancyscript{T}_n\) to a most recent common ancestor (MRCA) of the population, the time \(\fancyscript{U}_n\) at which all individuals are either common ancestors of all present day individuals or ancestral to none of them, and the fraction of individuals that are common ancestors at time \(\fancyscript{U}_n\). These results generalize the three main results of Chang’s (Adv Appl Probab 31:1002–1038, 1999). When we appropriately rescale time and recombination probability by the population size, our model leads to the continuous time Markov chain called the ancestral recombination graph of Hudson (Theor Popul Biol 23:183–201, 1983) and Griffiths (The two-locus ancestral graph, Institute of Mathematical Statistics 100–117, 1991).

## Keywords

Kingman’s coalescent Chang’s pedigree Griffiths’ and Hudson’s ancestral recombination graphs Zygotic/karyotic/cytoplasmic/sub-karyotic ancestral graphs## Mathematics Subject Classification

60C05 60J85 92D15 60J05## Notes

### Acknowledgments

B.T. was partly supported by post-doctoral fellowships at Department of Statistics, University of Oxford, UK (EPSRC grant EP/E05885X/1) and at Instituto de Matemática e Estatística, Universidade de São Paulo, Brasil (CNPq Processo 151782/2010-5 & MaCLinC). B.T. would like to thank Professor Yoshiharu Kohayakawa for many useful discussions related to Theorems 2, 3 and Corollary 1. R.S. was partly supported by a visiting scholarship at Department of Mathematics, Cornell University, Ithaca, NY, USA, a sabbatical grant from College of Engineering, University of Canterbury, and consulting revenues from Wynyard Group, Christchurch, NZ. R.S. thanks Robert C. Griffiths for an introduction to his ARGs, Alison Etheridge for discussions on stationary behaviour of the ancestral size chain, Jae Young Choi and Neil Gemmell for discussions on cytoplasmic inheritance, and Krithika Yogeeswaran for discussions on the nested embeddings in Fig. 3. A.V. was supported by the ANR project MANEGE (ANR-09-BLAN-0215) and R.S. and A.V. were supported in part by the chaire Modélisation Mathématique et Biodiversité of Veolia Environnement–École Polytechnique–Museum National d’Histoire Naturelle–Fondation X. The authors would like to thank the Referees and Associate Editor for their helpful suggestions to improve the presentation of the paper.

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