Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

We establish rapid mixing of the random-cluster Glauber dynamics on random Delta-regular graphs for all q >= 1 and p<pu(q,Delta), where the threshold pu(q,Delta) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) Delta-regular tree. It is expected that this threshold is sharp, and for q>2 the Glauber dynamics on random Delta-regular graphs undergoes an exponential slowdown at pu(q,Delta). More precisely, we show that for every q >= 1, Delta >= 3, and p<pu(q,Delta), with probability 1-o(1) over the choice of a random Delta-regular graph on n vertices, the Glauber dynamics for the random-cluster model has Theta(nlogn) mixing time. As a corollary, we deduce fast mixing of the Swendsen-Wang dynamics for the Potts model on random Delta-regular graphs for every q >= 2, in the tree uniqueness region. Our proof relies on a sharp bound on the "shattering time", i.e., the number of steps required to break up any configuration into O(logn) sized clusters. This is established by analyzing a delicate and novel iterative scheme to simultaneously reveal the underlying random graph with clusters of the Glauber dynamics configuration on it, at a given time.