Separated-occurrence inequalities for dependent percolation and Ising models

Separated-occurrence inequalities are variants for dependent lattice models of the van den Berg-Kesten inequality for independent models. They take the form $P(A \circ_r B) \leq (1 + ce^{-\epsilon r})P(A)P(B)$, where $A \circ_r B$ is the event that $A$ and $B$ occur at separation $r$ in a configuration $\omega$, that is, there exist two random sets of bonds or sites separated by at least distance $r$, one set responsible for the occurrence of the event $A$ in $\omega$, the other for the occurrence of $B$. We establish such inequalities for certain subcritical FK models, and for certain Ising models which are at supercritical temperature or have an external field, with $A$ and $B$ increasing or decreasing events.