Gaussian fluctuations of Young diagrams under the Plancherel measure

We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the `spectrum' of partitions lambda proves n is an element of N (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like root log n, the corresponding random process in the bulk is shown to converge, in the sense of finite-dimensional distributions, to a Gaussian process with independent values, while local correlations in the vicinity of each point, measured on various power scales, possess certain self-similarity. The proofs are based on the Poissonization techniques and use Costin Lebowitz Soshnikov's central limit theorem for determinantal random point processes. Our results admit a striking reformulation after the rotation of Young diagrams by 45 degrees, whereby the normalization no longer depends on the location in the spectrum. In addition, we explain heuristically the link with an earlier result by Kerov on the convergence to a generalized Gaussian process.