Conditional Association and Spin Systems

A 1977 theorem of T. Harris states that an attractive spin system preserves the class of associated probability measures. We study analogues of this result for measures that satisfy various conditional positive correlations properties. In particular, we show that a spin system preserves measures satisfying the FKG lattice condition (essentially) if and only if distinct spins flip independently. The downward FKG property, which has been useful recently in the study of the contact process, lies between the properties of lattice FKG and association. We prove that this property is preserved by a spin system if the death rates are constant and the birth rates are additive (e.g., the contact process), and prove a partial converse to this statement. Finally, we introduce a new property, which we call downward conditional association, which lies between the FKG lattice condition and downward FKG, and find essentially necessary and sufficient conditions for this property to be preserved by a spin system. This suggests that the latter property may be more natural than the downward FKG property.