 FKG inequality

The FKG inequality turns the FKG lattice condition into positive association of random variables. Let $\mu$ be a strictly positive probability measure on a finite distributive lattice and supposefor every $x,y$. Then for increasing real-valued functions $f,g$,Taking indicator functions gives $\mu (A\cap B)\ge \mu (A)\mu (B)$ for increasing events. The theorem applies to suitable dependent laws as well as product measures.^[ref]

For example, on the Boolean lattice $\{0,1{\}}^2$, give the configurations $00,01,10,11$ unnormalized masses $2,1,1,2$. The only nontrivial lattice check is $2\cdot 2\ge 1\cdot 1$. Each coordinate-one event has probability $1/2$, while their intersection has probability $1/3>1/4$. Thus the coordinates are dependent but positively associated. For a product measure, the FKG lattice condition holds with equality and recovers the Harris-FKG inequality.

The FKG inequality, named after its contributors Fortuin, Kasteleyn, and Ginibre, is a result in probability theory that provides a relationship among joint distributions of certain random variables, particularly in the context of lattice structures, such as spins in statistical mechanics. It is most commonly applied in the study of lattice models in statistical physics, including the Ising model.