Asymptotics of the allele frequency spectrum associated with the Bolthausen-Sznitman coalescent

TL;DR

This paper derives the full asymptotic behavior of the allele frequency spectrum for the Bolthausen-Sznitman coalescent within the infinitely many alleles model, extending previous results known for other coalescents.

Contribution

It provides the first complete asymptotic analysis of the allele frequency spectrum for the Bolthausen-Sznitman coalescent, a key model in population genetics.

Findings

Full asymptotics for the allele frequency spectrum are established.

Results extend understanding of allele distributions in complex coalescent models.

Provides mathematical tools for analyzing genetic diversity in populations.

Abstract

We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it and group together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is the allele frequency spectrum. The celebrated Ewens sampling formula gives precise probabilities for the allele frequency spectrum associated with Kingman's coalescent. This (and the degenerate star-shaped coalescent) are the only Lambda coalescents for which explicit probabilities are known, although they are known to satisfy a recursion due to Moehle. Recently, Berestycki, Berestycki and Schweinsberg have proved asymptotic results for the allele frequency spectra of the Beta(2-alpha,alpha) coalescents with alpha in (1,2). In this paper, we prove full asymptotics for the case of the Bolthausen-Sznitman coalescent.