Random-Cluster Dynamics in $\mathbb{Z}^2$
نویسندگان
چکیده
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. Our main result is a O(n log n) upper bound for the mixing time at all values of the model parameter p except the critical point p = pc(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. 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. ∗Computer Science Division, U.C. Berkeley, Berkeley, CA 94720. Email: [email protected]. Research supported in part by an NSF Graduate Research Fellowship and NSF grant CCF-1420934. †Computer Science Division, U.C. Berkeley, Berkeley, CA 94720. Email: [email protected]. Research supported in part by NSF grant CCF-1420934. ar X iv :1 51 0. 06 76 2v 1 [ cs .D M ] 2 2 O ct 2 01 5
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