  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.
