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We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree d, d . For given interaction parameters J 1 , J 2 > 0 and external field h 1 ∈ , we compute the smallest external fieldh such that the plus measure with parameters J 2 and h dominates the plus measure with parameters J 1 and h 1 for all h ≥h. Moreover, we discuss continuity ofh with respect to the three parameters Jdoi:10.1214/ejp.v15-820 fatcat:tl5tcfvo4fhulmya4utahiipky