The joint mixing time of the random walk and dynamical random-cluster process is Θ(log n) when edge updates are fast enough in the subcritical regime on random regular graphs.
Cutoff phenomena for random walks on random regular graphs.Duke Mathematical Journal, 153(3), June 2010
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Logarithmic Mixing of Random Walks on Dynamical Random Cluster Models
The joint mixing time of the random walk and dynamical random-cluster process is Θ(log n) when edge updates are fast enough in the subcritical regime on random regular graphs.