Let \({\mathbb{Y}}_n\) be the set of partitions of \(n\) equipped with the probabilistic Plancherel measure \(M_n\) defined by \(M_n(\lambda)=\dim^2\lambda/n!\), where \(\dim\lambda\) is the dimension of the irreducible \(S_n\)-module associated with \(\lambda\) or, equivalently, the number of standard \(\lambda\)-tableaux. Identifying \(\lambda\) with the corresponding Young diagram, \(\lambda\) is a plane shape of area \(n\) inside the first quadrant \({\mathbb{R}}_+^2\), with row and column coordinates \(r\) and \(s\), respectively. One can change the coordinates to \(x=s-r\), \(y=r+s\), and the boundary \(\partial\lambda\) can be viewed as the graph of a continuous piecewise linear function \(y=\lambda(x)\). Considering the \(\lambda\)’s as points of a probability space, and \(\lambda(x)\) as a random function, the result is that \[ \overline\lambda(x)={1\over \sqrt{n}}\lambda\left(\sqrt{n} x\right)\sim\Omega(x) +{2\over\sqrt{n}}\Delta(x),\quad n\to\infty. \] Here \(\overline\lambda(x)\) is the rescaled version of the function \(\lambda(x)\). As \(n\to\infty\), the graph of \(\overline\lambda(x)\) concentrates near the curve \[ \Omega(x)={2\over\pi}\left(x\arcsin {x\over 2}+\sqrt{4-x^2}\right),\quad |x|\leq 2, \] which corresponds to the law of large numbers [see B. F. Logan and L. A. Shepp, Adv. Math. 26, 206-222 (1977; Zbl 0363.62068) and A. M. Versik and S. V. Kerov, Sov. Math., Dokl. 18, 527-531 (1977); translation from Dokl. Akad. Nauk SSSR 233, 1024-1027 (1977; Zbl 0406.05008)].
The second function \(\Delta(x)\) governs the fluctuations of the random functions \(\overline\lambda\) around the curve \(\Omega\) which corresponds to the central limit theorem. This was announced in S. Kerov [C. R. Acad. Sci., Paris, Sér. I 316, 303-308 (1993; Zbl 0793.43001)] where Kerov outlined the scheme of the proof. This note also contained a number of fruitful ideas. Later A. Hora [Commun. Math. Phys. 195, 405-416 (1998; Zbl 1053.46522)] suggested an elegant proof of the central limit theorem of Kerov.
In 1999 Kerov (1946-2000) found a new approach to his result, which also made apparent the connection with other important combinatorial and probabilistic objects. Unfortunately, he did not publish a detailed paper on the result. The purpose of the paper under review is to give a detailed exposition of the central limit theorem of Kerov and his new approach, reconstructing them from his unpublished work notes.
For the entire collection see [Zbl 0997.00015].