The interchange process on dumbbell graphs has a sharp cutoff exactly when the smaller clique size tends to infinity, with the mixing time scaling crossing at m ~ sqrt(n).
The interchange process on high-dimensional products
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abstract
We resolve a long-standing conjecture of Wilson (2004), reiterated by Oliveira (2016), asserting that the mixing-time of the unit-rate Interchange Process on the $n$-dimensional hypercube is of order $n$. This follows from a sharp inequality established at the level of Dirichlet forms, from which we also deduce that macroscopic cycles emerge in constant time, and that the log-Sobolev constant of the exclusion process is of order $1$. Beyond the hypercube, our results apply to cartesian products of arbitrary graphs of fixed size, shedding light on a broad conjecture of Oliveira (2013).
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Mixing time and cutoff phenomenon for the interchange process on dumbbell graphs and the labelled exclusion process on the complete graph
The interchange process on dumbbell graphs has a sharp cutoff exactly when the smaller clique size tends to infinity, with the mixing time scaling crossing at m ~ sqrt(n).