Stein’s method and random character ratios

Fulman, Jason

doi:10.1090/S0002-9947-08-04371-7

Stein’s method and random character ratios
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Trans. Amer. Math. Soc. 360 (2008), 3687-3730 Request permission

Abstract:

Stein’s method is used to prove limit theorems for random character ratios. Tools are developed for four types of structures: finite groups, Gelfand pairs, twisted Gelfand pairs, and association schemes. As one example an error term is obtained for a central limit theorem of Kerov on the spectrum of the Cayley graph of the symmetric group generated by $i$-cycles. Other main examples include an error term for a central limit theorem of Ivanov on character ratios of random projective representations of the symmetric group, and a new central limit theorem for the spectrum of certain random walks on perfect matchings. The results are obtained with very little information: a character formula for a single representation close to the trivial representation and estimates on two step transition probabilities of a random walk. The limit theorems stated in this paper are for normal approximation, but many of the tools developed are applicable for arbitrary distributional approximation.

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