Single-crossover recombination and ancestral recombination trees. (English) Zbl 1284.92063

Summary: We consider the Wright-Fisher model for a population of \(N\) individuals, each identified with a sequence of a finite number of sites, and single-crossover recombinations between them. We trace back the ancestry of single individuals from the present population. In the \(N\to\infty\) limit without rescaling of parameters or time, this ancestral process is described by a random tree, whose branching events correspond to the splitting of the sequence due to recombination. With the help of a decomposition of the trees into subtrees, we calculate the probabilities of the topologies of the ancestral trees. At the same time, these probabilities lead to a semi-explicit solution of the deterministic single-crossover equation. The latter is a discrete-time dynamical system that emerges from the Wright-Fisher model via a law of large numbers and has been waiting for a solution for many decades.


92D15 Problems related to evolution
92D10 Genetics and epigenetics
60J28 Applications of continuous-time Markov processes on discrete state spaces
60F15 Strong limit theorems
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