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◦rB) ≤ (1 + ce−ǫr)P (A)P (B), where A ◦r B is the event that A and B occur at separation r in a configuration ω, 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 ω, the other for the occurrence of B. We establish such inequalities for subcritical FK models, and for Ising models which are at supercritical temperature or have an external field, with A and B increasing or decreasing events.