Poisson-Dirichlet Distribution with Small Mutation Rate

TL;DR

This paper investigates the Poisson-Dirichlet distribution with small mutation rates using large deviations, revealing finite allele numbers at mutation onset and conditions for allele coexistence under strong selection.

Contribution

It provides a large deviation analysis of the distribution's behavior at small mutation rates and explores allele coexistence under intense selection.

Findings

Number of alleles is finite at mutation appearance

Multiple alleles can coexist under high selection as mutation rate approaches zero

Asymptotic behavior of homozygosity analyzed

Abstract

The behavior of the Poisson-Dirichlet distribution with small mutation rate is studied through large deviations. The structure of the rate function indicates that the number of alleles is finite at the instant when mutation appears. The large deviation results are then used to study the asymptotic behavior of the homozygosity, and the Poisson-Dirichlet distribution with symmetric selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as the mutation rate approaches zero.