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:我们考虑在随机集聚模型的Glauber动态中,从铁磁波和随机集成模型对随机图的普通组别进行取样的问题。随机集集模型以边缘概率$p = in (0,1美元) 和组重量 q = 0美元来对准。 我们确定,对于每1美元,随机集聚色动态组合以美元为最佳 $Theta(n\log n), 美元- 顶端随机图中带有一定度序列的随机图解图。 在整个高温集模型中,随机组群模型以美元=u( q,\ gamma) 美元来对随机图进行匹配。 我们认为随机图模型的组合包括Erd\H{o}- R\ enyy 随机图 $G(n,\ gammama/n), 因此我们为Erd\ H{rg- gammamamamal 模型的定序模型- 值模型- Remmamamamex 和我们最高级的底层的直流流数据, 与我们最高级的直径的直径的直径的直径的直色的直径的直径的直径的直径的直径的直图结果。</s>