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

By V. Ivanov, Grigori Olshanski. 2003

182 citations recorded by the source corpus.

Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure M n. That is, the weight M n(λ) of a diagram λ equals dim2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group 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 It is largely based on Kerov’s unpublished work notes, 1999

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/fa5060c9712b67346c4b5936f91e3d0a27a6a695

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Citations in the corpus

Characters of symmetric groups and free cumulants (2003, methodology): “I would like to thank A. Okounkov and R. Stanley for useful communication, as well as G. Olshanski for providing me a copy of [ IO ].”
Explicit combinatorial interpretation of Kerov character polynomials as numbers of permutation factorizations (2008, result): “These basic functionals S2, S3, . . . of shape are not new; they already appeared (possibly with a slightly modified normalization) in the work of Ivanov and Olshanski [ IO02 ] and implicitly in the work of Kerov [Ker98, Ker99, Ker03].”
Exponential Approximation by Stein's Method and Spectral Graph Theory (2006, background): “But together with the large body of work on Kerov’s central limit theorem for the spectrum of the random transposition walk (Kerov, 1993; Fulman, 2006b,a, 2008; Śniady, 2006; Ivanov and Olshanski, 2002; Hora, 1998a), it does make the problems studied in the current paper very natural.”
Mod-ϕ Convergence (2016, background): “It is shown in [IO02] that for any partition λ of size n and for any k, pk(λ) = Σk(λ) + remainder,”
Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, background): “From there one can then derive (67) by expanding the exponential function (just as in [IO02] Proposition 3.”