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C. M. Fortuin coauthored the FKG inequality with P. W. Kasteleyn and J. Ginibre. The theorem makes a precise sufficient condition for increasing observables to be positively correlated. Its applications include Ising models and the random-cluster model, connecting ordered probability spaces with statistical mechanics.^[ref]

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.

J. Ginibre coauthored the FKG inequality with C. M. Fortuin and P. W. Kasteleyn. They proved positive correlation of increasing functions on a finite distributive lattice under a condition on its probability weights, with applications to Ising models and the random-cluster model.^[ref]

P. W. Kasteleyn coauthored the FKG inequality with C. M. Fortuin and J. Ginibre. Their theorem gives positive association for increasing functions on a finite distributive lattice when the probability weights satisfy the required lattice condition. It provides a rigorous correlation tool for Ising models and the random-cluster model.^[ref]