Dynamics for the Mean-field Random-cluster Model

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and random spanning trees, but its dynamics have so far largely resisted analysis. In this work we study a natural non-local Markov chain known as the Chayes-Machta dynamics for the mean-field case of the random-cluster model, and identify a critical regime (λs, λS) of the model parameter λ in which the dynamics undergoes an exponential slowdown. Namely, we prove that the mixing time is Θ(logn) if λ 6∈ [λs, λS ], and exp(Ω( √ n)) when λ ∈ (λs, λS). These results hold for all values of the second model parameter q > 1. In addition, we prove that the local heat-bath dynamics undergoes a similar exponential slowdown in (λs, λS). Joint work with Alistair Sinclair.