Random-Cluster Dynamics on Random Regular Graphs in Tree Uniqueness

We establish rapid mixing of the random-cluster Glauber dynamics on random Δ-regular graphs for all q≥ 1 and p< p_u(q, Δ) , where the threshold p_u(q, Δ) corresponds to a uniqueness/non-uniqueness phase transition for the random-cluster model on the (infinite) Δ-regular tree. It is expected that this threshold is sharp, and for q> 2 the Glauber dynamics on random Δ-regular graphs undergoes an exponential slowdown at p_u(q, Δ). More precisely, we show that for every q≥ 1 , Δ≥ 3 , and p< p_u(q, Δ) , with probability 1 - o(1) over the choice of a random Δ-regular graph on n vertices, the Glauber dynamics for the random-cluster model has Θ(nlog n) mixing time. As a corollary, we deduce fast mixing of the Swendsen–Wang dynamics for the Potts model on random Δ-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(log n) 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.