We study the sampling problem for the ferromagnetic Ising model with consistent external fields, and in particular, Swendsen-Wang dynamics on this model. We introduce a new grand model unifying two closely related models: the subgraph world and the random cluster model. Through this new viewpoint, we show:

• (1)

  polynomial mixing time bounds for Swendsen-Wang dynamics and (edge-flipping) Glauber dynamics of the random cluster model, generalising the bounds and simplifying the proofs for the no-field case by Guo and Jerrum (2018);

• (2)

  near linear mixing time for the two dynamics above if the maximum degree is bounded and all fields are (consistent and) bounded away from 1.

我们研究具有一致外部场的铁磁 Ising 模型的采样问题，特别是该模型上的 Swendsen-Wang 动力学。我们引入了一个新的大模型，它统一了两个密切相关的模型：子图世界和随机聚类模型。通过这个新观点，我们表明：

• (1)

  随机聚类模型的 Swendsen-Wang 动力学和（边缘翻转）Glauber 动力学的多项式混合时间界限，概括了界限并简化了Guo和Jerrum（2018）无场情况的证明；

• (2)

  如果最大程度有界并且所有场（一致且）远离 1，则上述两种动力学接近线性混合时间。