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We consider the performance of Glauber dynamics for the random cluster modelwith real parameter $q>1$ and temperature $\beta>0$. Recent work by Helmuth,Jenssen and Perkins detailed the ordered/disordered transition of the model onrandom $\Delta$-regular graphs for all sufficiently large $q$ and obtained anefficient sampling algorithm for all temperatures $\beta$ using clusterexpansion methods. Despite this major progress, the performance of naturalMarkov chains, including Glauber dynamics, is not yet well understood on therandom regular graph, partly because of the non-local nature of the model(especially at low temperatures) and partly because of severe bottleneckphenomena that emerge in a window around the ordered/disordered transition.  Nevertheless, it is widely conjectured that the bottleneck phenomena thatimpede mixing from worst-case starting configurations can be avoided byinitialising the chain more judiciously. Our main result establishes thisconjecture for all sufficiently large $q$ (with respect to $\Delta$).Specifically, we consider the mixing time of Glauber dynamics initialised fromthe two extreme configurations, the all-in and all-out, and obtain a pair offast mixing bounds which cover all temperatures $\beta$, including inparticular the bottleneck window. Our result is inspired by the recent approachof Gheissari and Sinclair for the Ising model who obtained a similar-flavouredmixing-time bound on the random regular graph for sufficiently lowtemperatures. To cover all temperatures in the RC model, we refineappropriately the structural results of Helmuth, Jenssen and Perkins about theordered/disordered transition and show spatial mixing properties "within thephase", which are then related to the evolution of the chain.

Sampling from the random cluster model on random regular graphs at all temperatures via Glauber dynamics