Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams

Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim λ/n!, where dimλ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan– Shepp 1977, Vershik–Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999.