Stein’s method and Plancherel measure of the symmetric group

By Jason E. Fulman. 2003

53 citations recorded by the source corpus.

We initiate a Stein's method approach to the study of the Plancherel measure of the symmetric group. A new proof of Kerov's central limit theorem for character ratios of random representations of the symmetric group on transpositions is obtained; the proof gives an error term. The construction of an exchangeable pair needed for applying Stein's method arises from the theory of harmonic functions on Bratelli diagrams. We also find the spectrum of the Markov chain on partitions underlying the construction of the exchangeable pair. This yields an intriguing method for studying the asymptotic decomposition of tensor powers of some representations of the symmetric group.

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

View Paper: https://www.semanticscholar.org/paper/449f0347ef4bcf50c1bdfb24ec3aef0669cf1889

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Exponential Approximation by Stein's Method and Spectral Graph Theory (2006, methodology): “Section 4 of the paper Fulman (2008) used Stein’s method to study random spherical functions of Gelfand pairs.”
Two-parameter family of infinite-dimensional diffusions on the Kingman simplex (2007, background): “They were studied in [2], [8], and [9].”
A short survey of Stein's method (2014, methodology): “In 2005, Fulman [46] extended the method of exchangeable pairs to study Plancherel measures on symmetric groups.”
Infinite-dimensional diffusions as limits of random walks on partitions (2007, background): “hains does not pretend to originality. The second named author learnt it long ago from Sergei Kerov (but Kerov never published it). In a diﬀerent context, such Markov chains appeared in Fulman’s work [F]: he was interested in the Plancherel distributions Mn which correspond to the case when P is the”
MULTIVARIATE NORMAL APPROXIMATION USING EXCHANGEABLE PAIRS (2007, unknown): “The method of exchangeable pairs is a general technique whose applicability is not restricted to sums of random variables; for a nontrivial modern example, see the work of Jason Fulman [18] on central limit theorems for complicated objects arising from the representation theory of permutation groups”