FKG Inequality - The Inequality

The Inequality

Let be a finite distributive lattice, and μ a nonnegative function on it, that is assumed to satisfy the (FKG) lattice condition (sometimes a function satisfying this condition is called log supermodular) i.e.,

for all x, y in the lattice .

The FKG inequality then says that for any two monotonically increasing functions ƒ and g on, the following positive correlation inequality holds:

The same inequality (positive correlation) is true when both ƒ and g are decreasing. If one is increasing and the other is decreasing, then they are negatively correlated and the above inequality is reversed.

Similar statements hold more generally, when is not necessarily finite, not even countable. In that case, μ has to be a finite measure, and the lattice condition has to be defined using cylinder events; see, e.g., Section 2.2 of Grimmett (1999).

For proofs, see the original Fortuin, Kasteleyn & Ginibre (1971) or the Ahlswede–Daykin inequality (1978). Also, a rough sketch is given below, due to Holley (1974), using a Markov chain coupling argument.