Asymptotic Properties of Random Restricted Partitions

TL;DR

This paper investigates the asymptotic behavior of integer partitions under two probability measures, deriving limiting distributions for parts and the largest part as the number of parts and total sum grow large.

Contribution

It introduces new asymptotic results for restricted partitions under two probability measures, including cases with the Dirichlet and uniform distributions.

Findings

Largest part satisfies the central limit theorem under certain growth conditions.

Derived joint distributions of parts for large n and m.

Unified approach for different probability measures on partitions.

Abstract

We study two types of probability measures on the set of integer partitions of $n$ with at most $m$ parts. The first one chooses the random partition with a chance related to its largest part only. We then obtain the limiting distributions of all of the parts together and that of the largest part as $n$ tends to infinity while $m$ is fixed or tends to infinity. In particular, if $m$ goes to infinity not too fast, the largest part satisfies the central limit theorem. The second measure is very general. It includes the Dirichlet distribution and the uniform distribution as special cases. We derive the asymptotic distributions of the parts jointly by taking limits of $n$ and $m$ in the same manner as that in the first probability measure.