Random-cluster dynamics in $${{\mathrm{\mathbb {Z}}}}^2$$Z2

The random-cluster model has been widely studied as a unifying framework for random graphs, spin systems and electrical networks, but its dynamics have so far largely resisted analysis. In this paper we analyze the Glauber dynamics of the random-cluster model in the canonical case where the underlying graph is an $$n \times n$$n×n box in the Cartesian lattice $${{\mathrm{\mathbb {Z}}}}^2$$Z2. Our main result is a $$O(n^2\log n)$$O(n2logn) upper bound for the mixing time at all values of the model parameter p except the critical point $$p=p_c(q)$$p=pc(q), and for all values of the second model parameter $$q\ge 1$$q≥1. We also provide a matching lower bound proving that our result is tight. Our analysis takes as its starting point the recent breakthrough by Beffara and Duminil-Copin on the location of the random-cluster phase transition in $${{\mathrm{\mathbb {Z}}}}^2$$Z2. It is reminiscent of similar results for spin systems such as the Ising and Potts models, but requires the reworking of several standard tools in the context of the random-cluster model, which is not a spin system in the usual sense.

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