  We prove a conjecture raised by the work of Diaconis and Shahshahani (1981)
about the mixing time of random walks on the permutation group induced by a
given conjugacy class. To do this we exploit a connection with coalescence and
fragmentation processes and control the Kantorovitch distance by using a
variant of a coupling due to Oded Schramm. Recasting our proof in the language
of Ricci curvature, our proof establishes the occurrence of a phase transition,
which takes the following form in the case of random transpositions: at time
$cn/2$, the curvature is asymptotically zero for $c\le 1$ and is strictly
positive for $c>1$.
