  In this work, we obtain the central limit theorem for fluctuations of Young
diagrams around their limit shape in the bulk of the "spectrum" of partitions
of a large integer n (under the Plancherel measure). More specifically, we show
that, under the suitable normalization (growing as the square root of log n),
the corresponding random process converges, in the sense of finite dimensional
distributions, to a Gaussian process with independent values. The proof uses
heavily the determinantal structure of the correlation functions and is based
on the application of the Costin-Lebowitz-Soshnikov central limit theorem. At
the spectrum edges, the fluctuation asymptotics is expressed in terms of the
largest members of the Airy ensemble; in particular, at the upper edge the
limit distribution appears to be discrete (without any normalization). These
results admit an elegant symmetric reformulation under the rotation of Young
diagrams by 45 degrees, where the normalization no longer depends on the
location of the spectrum point. We also discuss the link of our central limit
theorem with an earlier result by S.V. Kerov on the convergence to a
generalized Gaussian process.
