  We study the random walk on the symmetric group $S_n$ generated by the
conjugacy class of cycles of length $k$. We show that the convergence to
uniform measure of this walk has a cut-off in total variation distance after
$\frac{n}{k} log n$ steps, uniformly in $k = o(n)$ as $n \to \infty$. The
analysis follows from a new asymptotic estimation of the characters of the
symmetric group evaluated at cycles.
