Distribution of repetitions of ancestors in genealogical trees

Details Distribution of repetitions of ancestors in genealogical trees Distribution of repetitions of ancestors in genealogical trees

1999-12-03

arxiv.org/abs/cond-mat/9912059v1

Physics

Condensed Matter

20 pages, 5 figures included, to appear in Physica A

Scientific paper

10.1016/S0378-4371(00)00031-5

We calculate the probability distribution of repetitions of ancestors in a genealogical tree for simple neutral models of a closed population with sexual reproduction and non-overlapping generations. Each ancestor at generation g in the past has a weight w which is (up to a normalization) the number of times this ancestor appears in the genealogical tree of an individual at present. The distribution P_g(w) of these weights reaches a stationary shape P_\infty(w) for large g, i.e. for a large number of generations back in the past. For small w, P_\infty(w) is a power law with a non-trivial exponent which can be computed exactly using a standard procedure of the renormalization group approach. Some extensions of the model are discussed and the effect of these variants on the shape of P_\infty(w) are analysed.

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