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 uniqueness regime $p<p_u(q,\gamma)$. Notably, this uniqueness threshold does not decay with the maximum degree and only depends on the average degree. In particular, $p_u(q,\gamma)$ is expected to be sharp, in that for general $q$ and $\gamma$, the random-cluster Glauber dynamics should slow down dramatically at $p_u(q,\gamma)$. The family of random graph models we consider include 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 the Erd\H{o}s--R\'enyi random graphs that works for all $q$ in the full uniqueness regime. We accompany our results with mixing time lower bounds (exponential in the maximum 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 significant computational advantage of random-cluster based algorithms for sampling from the Potts Gibbs distribution at high temperatures in the presence of high-degree vertices.

翻译：我们考虑从铁磁波和随机集群模型中取样的问题。 在随机集成模型中, 随机集成模型通过 Glauber 动态, 以普通的随机图表组别来取样。 随机集成模型以边缘概率 $ p = in (0, 1美元) 和组重量 q > 0 美元。 我们确定, 每1美元中, 随机集聚色动态混合为最佳 $\ Theta (n\log n) 美元 和 $\ gamma 的随机集成图案。 在整个独特系统 $p\ p_ u( q,\ gamma) 中, 以限定的平均分数为单位 。 随机集的基数基数( 美元和 美元) 随机集的基数( 美元) 基数- 基数( 美元) 基数( 美元) 随机集色变色( 美元) 的基数- 基数- 基数( 美元) 基数- 数( 数- 数组数数数- 数字模型中, 我们的数数数数数数数组 的基数数- 数- 数数- 基数- 数- 基数- 基数- 基数- 基数- 基数- 基数- 数- 基数- 基数- 基数- 数- 基数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数- 数-