Distribution of repetitions of ancestors in genealogical trees

Abstract

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 ∞( w), for large g, i.e., for a large number of generations back in the past. For small w, P ∞(w) is a power law ( P ∞( w)∼ w β ), with a non-trivial exponent β 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 ∞( w) are analysed.