Gaussian fluctuations of Young diagrams under the Plancherel measure
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Gaussian fluctuations of Young diagrams under the Plancherel measure
We obtain the central limit theorem for fluctuations of Young diagrams around their limit shape in the bulk of the ‘spectrum’ of partitions lwn2N (under the Plancherel measure), thus settling a long-standing problem posed by Logan & Shepp. Namely, under normalization growing like ffiffiffiffiffiffiffiffiffiffi log n p , the corresponding random process in the bulk is shown to converge, in the s...
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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, V...
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The Young-Fibonacci graph YF is an important example (along with the Young lattice) of diierential posets studied by S. Fomin and R. Stanley. For every diierential poset there is a distinguished central measure called the Plancherel measure. We study the Plancherel measure and the associated Markov chain, the Plancherel process, on the Young-Fibonacci graph. We establish a law of large numbers ...
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We study asymptotics of reducible representations of symmetric groups Sq for large q. We decompose such a representation as a sum of irreducible components (or, alternatively, Young diagrams) and we ask what is the character of a randomly chosen component (or, what is the shape of a randomly chosen Young diagram). Our main result is that for a large class of representations the fluctuations of ...
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ژورنال
عنوان ژورنال: Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
سال: 2007
ISSN: 1364-5021,1471-2946
DOI: 10.1098/rspa.2006.1808