Stein's method, Jack measure, and the Metropolis algorithm

Abstract

The one parameter family of Jack α measures on partitions is an important discrete analog of Dyson's β ensembles of random matrix theory. Except for special values of α = 1 2 , 1 , 2 which have group theoretic interpretations, the Jack α measure has been difficult if not intractable to analyze. This paper proves a central limit theorem (with an error term) for Jack α measure which works for arbitrary values of α . For α = 1 we recover a known central limit theorem on the distribution of character ratios of random representations of the symmetric group on transpositions. The case α = 2 gives a new central limit theorem for random spherical functions of a Gelfand pair (or equivalently for the spectrum of a natural random walk on perfect matchings in the complete graph). The proof uses Stein's method and has interesting combinatorial ingredients: an intruiging construction of an exchangeable pair, properties of Jack polynomials, and work of Hanlon relating Jack polynomials to the Metropolis algorithm.