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We study the sampling problem for the ferromagnetic Ising model withconsistent external fields, and in particular, Swendsen-Wang dynamics on thismodel. We introduce a new grand model unifying two closely related models: thesubgraph world and the random cluster model. Through this new viewpoint, weshow: (1) polynomial mixing time bounds for Swendsen-Wang dynamics and(edge-flipping) Glauber dynamics of the random cluster model, generalising thebounds 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 maximumdegree is bounded and all fields are (consistent and) bounded away from $1$.

Swendsen-Wang dynamics for the ferromagnetic Ising model with external fields