We consider the problem of sampling from the ferromagnetic Potts and random-cluster models on a general family of random graphs via the Glauber dynamics for the random-cluster model. The random-cluster model is parametrized by an edge probability $p \in (0,1)$ and a cluster weight $q > 0$. We establish that for every $q\ge 1$, the random-cluster Glauber dynamics mixes in optimal $\Theta(n\log n)$ steps on $n$-vertex random graphs having a prescribed degree sequence with bounded average branching $\gamma$ throughout the full high-temperature uniqueness regime $p<p_u(q,\gamma)$. The family of random graph models we consider includes the Erd\H{o}s--R\'enyi random graph $G(n,\gamma/n)$, and so we provide the first polynomial-time sampling algorithm for the ferromagnetic Potts model on Erd\H{o}s--R\'enyi random graphs for the full tree uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the largest degree) for the Potts Glauber dynamics, in the same settings where our $\Theta(n \log n)$ bounds for the random-cluster Glauber dynamics apply. This reveals a novel and significant computational advantage of random-cluster based algorithms for sampling from the Potts model at high temperatures.

翻译：摘要：本文研究通过随机团簇模型的Glauber动力学，从一般随机图族上的铁磁Potts模型和随机团簇模型中采样的问题。随机团簇模型由边概率$p \in (0,1)$和团簇权重$q > 0$参数化。我们证明，对于所有$q\ge 1$，在具有指定度序列且平均分支因子$\gamma$有界的$n$顶点随机图上，随机团簇Glauber动力学在完整高温唯一性区域$p<p_u(q,\gamma)$内以最优的$\Theta(n\log n)$步混合。我们考虑的随机图模型族包括Erdős–Rényi随机图$G(n,\gamma/n)$，从而首次为完整树唯一性区域内的Erdős–Rényi随机图上的铁磁Potts模型提供了多项式时间采样算法。我们将结果与Potts Glauber动力学的混合时间下界（关于最大度指数增长）相结合，该下界适用于我们给出随机团簇Glauber动力学$\Theta(n \log n)$界限的相同设置。这揭示了基于随机团簇的算法在高温下从Potts模型中采样时具有新颖且显著的计算优势。