  The random-cluster model is a unifying framework for studying random graphs,
spin systems and random networks. The model is closely related to the classical
ferromagnetic Ising and Potts models and is often viewed as a generalization of
these models. In this paper, we study a natural non-local Markov chain known as
the Chayes-Machta dynamics for the mean-field case of the random-cluster model,
where the underlying graph is the complete graph on $n$ vertices. The
random-cluster model is parametrized by an edge probability $p$ and a cluster
weight $q$. Our focus is on the critical regime: $p = p_c(q)$ and $q \in
(1,2)$, where $p_c(q)$ is the threshold corresponding to the order-disorder
phase transition of the model. We show that the mixing time of the
Chayes-Machta dynamics is $O(\log n \cdot \log \log n)$ in this parameter
regime, which reveals that the dynamics does not undergo an exponential
slowdown at criticality, a surprising fact that had been predicted (but not
proved) by statistical physicists. This provides a nearly optimal bound (up to
the $\log\log n$ factor) for the mixing time of the mean-field Chayes-Machta
dynamics in the only regime of parameters where no previous bound was known.
Our proof consists of a multi-phased coupling argument that combines several
key ingredients, including a new local limit theorem, a precise bound on the
maximum of symmetric random walks with varying step sizes, and tailored
estimates for critical random graphs.
