On the Fluctuations for Multiplicative Ensembles of Random Integer Partitions with Equiweighted Parts

Let Λ n be the space of partitions of integer n ≥ 1. We discuss the multiplicative ensembles on Λ n with equiweighted parts. Based on the limit shape and other asymptotic properties studied by Leonid V. Bogachev (Random Structures and Algorithms, 2015, 47(2): 227–266), we establish limit theorems for the fluctuations of parts around the limit shape. Our proof consists of two parts: attack the corresponding problems with independence in a proper grand ensemble and transfer the results to a small ensemble by conditional probability.