Asymptotic sampling formulae for Lambda-coalescents

TL;DR

This paper develops a method to derive sampling formulae for populations with genealogies modeled by Lambda-coalescents, providing asymptotic insights into genetic diversity measures as sample size grows.

Contribution

It introduces a robust approach linking the speed of coming down from infinity in Lambda-coalescents to sampling formulae, with detailed results under regular variation assumptions.

Findings

Exact asymptotic formulas for site and allele frequency spectra

Results on the number of segregating sites in large samples

Identification of a phase transition at alpha=3/2

Abstract

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Lambda-coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to infinity. Some of our results hold in the case of a general Lambda-coalescent that comes down from infinity, but we obtain more precise information under a regular variation assumption. In this case, we obtain results of independent interest for the time at which a mutation uniformly chosen at random was generated. This exhibits a phase transition at \alpha=3/2, where \alpha \in(1,2) is the exponent of regular variation.