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.
Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness.Communications in Mathematical Physics, 386(2):1243–1287
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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.