Central limit theorem for random partitions under the Plancherel measure

A central limit theorem for the Plancherel measure on the ensemble of partitions of asymptotically growing integers is described. It is proved that local fluctuations of the corresponding Young diagrams are asymptotically normal both in the bulk and near the edges of the limiting 'spectrum' of partitions. The domain of asymptotically Gaussian fluctuations, which is described by the theorem extends up to the domain of extreme values where the limit distribution is characterized by the Airy ensemble. The results of this study gives an answer the problem raised by Logan and Shepp and remarkably complements Kerov's theorem on the convergence of integral fluctuations to a generalized Gaussian process.