外场作用下铁磁Ising模型的Swendsen-Wang动力学
Swendsen-Wang dynamics for the ferromagnetic Ising model with external fields
摘要：
研究了具有一致外场的铁磁伊辛模型的采样问题，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，统一了两个密切相关的模型：子图世界和随机簇模型。通过这种新的观点，我们证明了：（1）Swendsen-Wang动力学的多项式混合时间界，（边缘翻转）随机簇模型的Glauber动力学，推广了Guo和Jerrum（2018）的边界并简化了无场情况的证明;（2）当最大度有界且所有场一致且有界远离$1$时，上述两种动力学的混合时间近似为线性.
英文摘要：
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 $1$.
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