  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.
