Cutoff phenomenon for the simple exclusion process on the complete graph

We study the time that the simple exclusion process on the complete graph needs to reach equilibrium in terms of total variation distance. For the graph with n vertices and 1<<k<n/2 particles we show that the mixing time is of order (n/2)\log \min(k, \sqrt{n}), and that around this time, for any small positive epsilon the total variation distance drops from 1-epsilon to epsilon in a time window whose width is of order n (i.e. in a much shorter time). Our proof is purely probabilistic and self-contained.