Normal Convergence for Random Partitions with Multiplicative Measures

Abstract

Let ${\cal P}_n$ be the space of partitions of integer $n\ge 0$, let ${\cal P}$ be the space of all partitions, and define a class of multiplicative measures induced by ${\cal F}_\beta (z)=\prod_{k}(1-z^k)^{k^\beta}$ with $\beta>-1$. Based on limit shapes and other asymptotic properties studied by Vershik, we establish normal convergence for the size and parts of random partitions.