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Kerov's central limit theorem for Schur-Weyl and Gelfand measures (extended abstract)

Abstract : We show that the shapes of integer partitions chosen randomly according to Schur-Weyl measures of parameter $\alpha =1/2$ and Gelfand measures satisfy Kerov's central limit theorem. Thus, there is a gaussian process $\Delta$ such that under Plancherel, Schur-Weyl or Gelfand measures, the deviations $\Delta_n(s)=\lambda _n(\sqrt{n} s)-\sqrt{n} \lambda _{\infty}^{\ast}(s)$ converge in law towards $\Delta (s)$, up to a translation along the $x$-axis in the case of Schur-Weyl measures, and up to a factor $\sqrt{2}$ and a deterministic remainder in the case of Gelfand measures. The proofs of these results follow the one given by Ivanov and Olshanski for Plancherel measures; hence, one uses a "method of noncommutative moments''.
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Pierre-Loïc Méliot. Kerov's central limit theorem for Schur-Weyl and Gelfand measures (extended abstract). 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.669-680. ⟨hal-01215045⟩

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