  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 $1$.
