Free probability and representations of large symmetric groups

Abstract

We study the asymptotic behavior of the free cumulants (in the sense of free probability theory of Voiculescu) of Jucys--Murphy elements--or equivalently--of the transition measure associated with a Young diagram. We express these cumulants in terms of normalized characters of the appropriate representation of the symmetric group S_q. Our analysis considers the case when the Young diagrams rescaled by q^{-1/2} converge towards some prescribed shape. We find explicitly the second order asymptotic expansion and outline the algorithm which allows to find the asymptotic expansion of any order. As a corollary we obtain the second order asymptotic expansion of characters evaluated on cycles in terms of free cumulants, i.e. we find explicitly terms in Kerov polynomials with the appropriate degree.