  We study the asymptotic behavior of the long cycles of a random permutation
of $n$ objects with respect to multiplicative measures with polynomial growing
cycle weights. We show that the longest cycle and the length differences
between the longest cycles converge, after suitable normalisation, in
distribution to iid random variables $(Z_j)_{j\in\mathbb{N}}$ such that
$\exp(-Z_j)$ is exponentially distributed. Our method is based on generating
functions and the saddle point method.
