Asymptotic Behaviour of Poisson-Dirichlet Distribution and Random Energy Model

The family of Poisson-Dirichlet distributions is a collection of two-parameter probability distributions {PD(alpha, theta) : 0 <= alpha < 1, alpha + theta > 0} defined on the infinite-dimensional simplex. The parameters alpha and theta correspond to the stable and gamma component respectively. The distribution PD(alpha, 0) arises in the thermodynamic limit of the Gibbs measure of Derrida's Random Energy Model (REM) in the low temperature regime. In this setting alpha can be written as the ratio between the temperature T and a critical temperature T-c. In this paper, we study the asymptotic behaviour of PD(alpha, theta) as alpha converges to one or equivalently when the temperature approaches the critical value T-c.