Let n≥1 be a natural number, Pn a set of all integer partitions. To each partition λ assign a probability P (λ). Thus we have a finite probability space (Pn, P) and are interested in asymptotic behaviors of a typical partition under P as n→∞. In this talk we particularly focus on random multiplicative measure and Plancherel measure and study the asymptotic fluctuations of the boundary of a Young diagram around its limit shape. It turns out that we need to deal with two cases separately: at the edge and in the bulk. At the edge the fluctuations will asymptotically follow Gumbel and Tracy-Widom extremal laws, while in the bulk asymptotic normality holds.