The Poisson Dirichlet Distribution and Related Topics : Models and Asymptotic Behaviors Models The Poisson–Dirichlet Distribution The Two Parameter Poisson–Dirichlet Distribution The Coalescent Stochastic Dynamics Particle Representation Asymptotic Behaviors Fluctuation Theorems Large Deviations for the Poisson–Dirichlet Distribution Large Deviations for the Dirichlet Processes. The Poisson Dirichlet distribution is an infinite dimensional probability distribution. It was introduced by Kingman over thirty years ago and has found applications in a broad range of areas including Bayesian statistics combinatorics differential geometry economics number theory physics and population genetics. This monograph provides a comprehensive study of this distribution and some related topics with particular emphasis on recent progresses in evolutionary dynamics and asymptotic behaviors. One central scheme is the unification of the Poisson Dirichlet distribution the urn structure the coalescent the evolutionary dynamics through the grand particle system of Donnelly and Kurtz. It is largely self contained. The methods and techniques used in it appeal to researchers in a wide variety of subjects.