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, \Td. For given interaction parameters J1,
J2>0 and external field h_1\in\RR, we compute the smallest external field
h~ such that the plus measure with parameters J2 and h dominates
the plus measure with parameters J1 and h1 for all h≥h~.
Moreover, we discuss continuity of h~ with respect to the three
parameters J1, J2, h and also how the plus measures are stochastically
ordered in the interaction parameter for a fixed external field. Next, we
consider the fuzzy Potts model and prove that on \Zd the fuzzy Potts measures
dominate the same set of product measures while on \Td, for certain parameter
values, the free and minus fuzzy Potts measures dominate different product
measures. For the Ising model, Liggett and Steif proved that on \Zd the plus
measures dominate the same set of product measures while on \T^2 that
statement fails completely except when there is a unique phase.Comment: 22 pages, 5 figure