Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams
Kerov’s Central Limit Theorem for the Plancherel Measure on Young Diagrams
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年轻图 Plancherel 测度的 Kerov 中心极限定理
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
G. Olshanski
中科院分区：
文献类型：
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作者：
V. Ivanov;G. Olshanski
考虑盒个数为n的随机杨图，按Plcherel测度M_n分布，即图λ的权M_n(λ)等于Dim_2λ/n！，其中Dim_λ表示由λ标度的对称群的不可约表示的维度.作为n→∞，随机形状λ的边界集中在曲线Ω附近(Logan-Shep 1977，Vershik-Kerov 1977)。1993年，科罗夫宣布了一个引人注目的定理，描述了极限形状Ω附近的高斯涨落。在这里，我们建议对他的证明进行重建。它主要是基于科罗夫1999年未发表的工作笔记
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 It is largely based on Kerov’s unpublished work notes, 1999