An ancestral recombination graph for diploid populations with skewed   offspring distribution
  A large offspring number diploid biparental multilocus population model of Moran type is our object of study. At each timestep, a pair of diploid individuals drawn uniformly at random contribute offspring to the population. The number of offspring can be large relative to the total population size. Similar `heavily skewed' reproduction mechanisms have been considered by various authors recently. Each diploid parental individual contributes exactly one chromosome to each diploid offspring, and hence ancestral lineages can only coalesce when in distinct individuals. A separation of timescales phenomenon is thus observed. A result of M\"{o}hle (1998) is extended to obtain convergence of the ancestral process to an ancestral recombination graph necessarily admitting simultaneous multiple mergers of ancestral lineages. The usual ancestral recombination graph is obtained as a special case of our model when the parents contribute only one offspring to the population each time. Due to diploidy and large offspring numbers, novel effects appear. For example, the marginal genealogy at each locus admits simultaneous multiple mergers in up to four groups, and different loci remain substantially correlated even as the recombination rate grows large. Thus, genealogies for loci far apart on the same chromosome remain correlated. Correlation in coalescence times for two loci is derived and shown to be a function of the coalescence parameters of our model. Extending the observations by Eldon and Wakeley (2008), predictions of linkage disequilibrium are shown to be functions of the reproduction parameters of our model, in addition to the recombination rate. Correlations in ratios of coalescence times between loci can be high, even when the recombination rate is high and sample size is large. 
Matthias Birkner, Jochen Blath, Bjarki Eldon
q-bio.PE math.PR q-bio.QM