Poisson–Dirichlet distribution with small mutation rate

A large deviation principle is established for the Poisson–Dirichlet distribution when the mutation rate θ converges to zero. The rate function is identified explicitly, and takes on finite values only on states that have finite number of alleles. This result is then applied to the study of the asymptotic behavior of the homozygosity, and the Poisson–Dirichlet distribution with selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as θ approaches zero.