The limit shape of Young diagrams under the Plancherel
measure was found by Vershik \& Kerov (1977) and Logan \& Shepp
(1977). We obtain a central limit theorem for fluctuations of Young
diagrams in the bulk of the partition '`spectrum''. More
specifically, under a suitable (logarithmic) normalization, the
corresponding random process converges (in the FDD sense) to a
Gaussian process with independent values. We also discuss a link
with an earlier result by Kerov (1993) on the convergence to a
generalized Gaussian process. The proof is based on poissonization
of the Plancherel measure and an application of a general central
limit theorem for determinantal point processes. (Joint work with
Zhonggen Su.)