Free Probability and Representations of Large Symmetric Groups

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 Sq. Our analysis considers the case when the Young diagrams rescaled by q 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.