Random-Cluster Dynamics in ℤ 2

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 × n box in the Cartesian lattice ℤ 2. Our main result is a O ( n 2 log n ) upper bound for the mixing time at all values of the model parameter p except the critical point p = p c ( q ), and for all values of the second model parameter 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 ℤ 2. 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.

Authors

• Antonio Blanca
• Alistair Sinclair

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