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 $q\ge 1$ and $p<p_u(q,\Delta)$, where the threshold $p_u(q,\Delta)$ 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 $p_u(q,\Delta)$. More precisely, we show that for every $q\ge 1$, $\Delta\ge 3$, and $p<p_u(q,\Delta)$, 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 $\Theta(n \log n)$ 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 $q\ge 2$, 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.