  We find the total variation mixing time of the interchange process on the
dumbbell graph (two complete graphs, $K_n$ and $K_m$, connected by a single
edge), and show that this sequence of chains exhibits the cutoff phenomenon
precisely when the smaller size $m$ goes to infinity. The mixing time undergoes
a phase transition at $m\asymp \sqrt{n}$. We also state a conjecture on when
exactly cutoff holds for the interchange process on general graphs.
  Our proofs use coupling methods, and they also give the mixing time of the
simple exclusion process of $k$ labelled particles in the complete graph $K_n$,
for any $k\leq n$, with cutoff, as conjectured by Lacoin and Leblond (2011). In
particular, this is a new probabilistic proof for the mixing time of random
transpositions, first established by Diaconis and Shahshahani (1981).

Probability;Combinatorics;Group Theory