Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 30 / 3 / a

Order configurations coordinatewise: $\omega \le \eta$ when $\omega (e)\le \eta (e)$ for every coordinate. An increasing event $A$ is an upward-closed set: $\omega \in A$ and $\omega \le \eta$ imply $\eta \in A$.

The finite-lattice FKG inequality states that a probability mass function satisfying the FKG lattice conditionis positively associated. Here meet in a lattice and join in a lattice are coordinatewise minimum and maximum. Thus for increasing observables $f,g$,Taking indicator functions gives $\mu (A\cap B)\ge \mu (A)\mu (B)$ for all increasing events. The strictly positive version suffices for the Ising weights here; the theorem also holds for nonnegative log-supermodular weights of positive total mass.