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 $n \times n$ box in the Cartesian lattice $\mathbb{Z}^2$. 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 $q\ge 1$. 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 $\mathbb{Z}^2$. 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.
翻译:随机集群模型已被广泛研究,作为随机图形、 旋转系统和电网的统一框架,但其动态迄今基本上抵制了分析。 在本文中,我们分析了在卡通性案例中随机集群模型的Glauber动态, 其基础图形为 $n\times n$n$$ mathbb ⁇ 2$2美元。 我们的主要结果是在模型参数的所有值($p$2\log n) 和第二个模型参数的所有值($p=p_c(q)$1美元除外) 的混合时间上方$O(n%2\log n)$($2美元) 。 在第二个模型参数的所有值($q\ge$1美元) 上方,我们分析了随机集群模型的Grauber动态。 我们还提供了一个匹配的下方约束值, 证明我们的结果很紧。 我们的分析以Beffara和Duminil- Copin最近在随机集群阶段转换位置的突破为 $\\\\\\mathbbb#2美元作为起点。 它令人想起一些类似的结果, 如 Ising 和Potts 模型, 需要重新制作一些标准工具。