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).
Time to reach stationarit y in the Bernoulli–Laplace urn model, SIAM Journal on Mathematical Analysis 18 (1987), No
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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).