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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.

##### MSC:
 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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