# Examples of measures that satisfy FKG, but not the FKG lattice condition

Let a percolation measure be a measure on $$\{0,1\}^n$$. We have a natural partial order on $$\{0,1\}^n$$ given by comparing all coordinates. An event $$A$$ is called increasing if for all $$\omega \in A$$ we have for every $$\eta \geq \omega$$ that $$\eta \in A$$. Now a percolation measure $$\mu$$ satisfies the FKG inequality if for all increasing events $$A, B$$ we have that $$\mu(A \cap B) \geq \mu(A) \mu(B).$$

A criterion for FKG is the FKG-lattice condition. To state it for $$v \in \{0,1\}^n$$ and $$e, f \in \{1, \dots , n \}$$ with $$e \neq f$$ let the configurations $$v_{ef}, v^{ef}, v_{e}^f,v^{e}_f$$ be the configurations that coincide with $$v$$ for all but the two entries $$e,f$$ and there $$v_{ef}(e) = 0 = v_{ef}(f)$$ , $$v_{e}^f(e) = 0, v_{e}^f(f) = 1$$ etc.

If we always have that $$\mu(v_{ef}) \mu(v^{ef}) \geq \mu(v_{e}^f) \mu(v^{e}_f)$$ and $$\mu(v) > 0$$ for all $$v$$ (i.e. strictly positive). Then this implies FKG (see Ex. 11.2 here https://www.ihes.fr/~duminil/publi/2017PIMS.pdf).

Now, what are (interesting) examples of percolation measures that do not satisfy the FKG lattice condition, but that do satisfy FKG?

• One example is the Wolff measure, as described in this paper; see the remark following the statement of Theorem 3.3. Jul 12, 2020 at 18:03