Stein's Method, Jack Measure, and the Metropolis Algorithm

The one parameter family of Jack(alpha) measures on partitions is an important discrete analog of Dyson's beta ensembles of random matrix theory. Except for special values of alpha=1/2,1,2 which have group theoretic interpretations, the Jack(alpha) measure has been difficult if not intractable to analyze. This paper proves a central limit theorem (with an error term) for Jack(alpha) measure which works for arbitrary values of alpha. For alpha=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 alpha=2 gives a new central limit theorem for random spherical functions of a Gelfand pair. The proof uses Stein's method and has interesting ingredients: an intruiging construction of an exchangeable pair, properties of Jack polynomials, and work of Hanlon relating Jack polynomials to the Metropolis algorithm.