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We find the total variation mixing time of the interchange process on thedumbbell graph (two complete graphs, $K_n$ and $K_m$, connected by a singleedge), and show that this sequence of chains exhibits the cutoff phenomenonprecisely when the smaller size $m$ goes to infinity. The mixing time undergoesa phase transition at $m\asymp \sqrt{n}$. We also state a conjecture on whenexactly cutoff holds for the interchange process on general graphs.  Our proofs use coupling methods, and they also give the mixing time of thesimple 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). Inparticular, this is a new probabilistic proof for the mixing time of randomtranspositions, first established by Diaconis and Shahshahani (1981).

Mixing time and cutoff phenomenon for the interchange process on dumbbell graphs and the labelled exclusion process on the complete graph