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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?

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

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