# FKG inequality

Fortuin–Kasteleyn–Ginibre inequality

An inequality [a3] that began a series of correlation inequalities for finite partially ordered sets. Let $\Gamma$ be a finite partially ordered set ordered by $\prec$( irreflexive, transitive) with $( \Gamma, \prec )$ a distributive lattice: $a \lor b = \min \{ {z \in \Gamma } : {a \cle z, b \cle z } \}$, $a \wedge b = \max \{ {z \in \Gamma } : {z \cle a, z \cle b } \}$, and $a \wedge ( b \lor c ) = ( a \wedge b ) \lor ( a \wedge c )$ for all $a,b, c \in \Gamma$. Suppose $\mu : \Gamma \rightarrow {[ 0, \infty ) }$ is log supermodular:

$$\mu ( a ) \mu ( b ) \leq \mu ( a \lor b ) \mu ( a \wedge b ) \textrm{ for all } a, b \in \Gamma,$$

and that $f : \Gamma \rightarrow \mathbf R$ and $g : \Gamma \rightarrow \mathbf R$ are non-decreasing:

$$a \prec b \Rightarrow \{ f ( a ) \leq f ( b ) , g ( a ) \leq g ( b ) \} \textrm{ for all } a,b \in \Gamma .$$

The FKG inequality is:

$$\left [ \sum _ {a \in \Gamma } \mu ( a ) f ( a ) \right ] \left [ \sum _ {a \in \Gamma } \mu ( a ) g ( a ) \right ] \leq$$

$$\leq \left [ \sum _ {a \in \Gamma } \mu ( a ) \right ] \left [ \sum _ {a \in \Gamma } \mu ( a ) f ( a ) g ( a ) \right ] .$$

If $\Gamma$ is a Boolean algebra and $\mu$ is a probability measure on $\Gamma$, the inequality is ${\mathsf E} _ \mu ( f ) {\mathsf E} _ \mu ( g ) \leq {\mathsf E} _ \mu ( fg )$, where ${\mathsf E} _ \mu$ denotes mathematical expectation.

Related inequalities are discussed in [a1], [a2], [a4], [a5], [a6], [a7], [a8], [a9].

#### References

 [a1] B. Bollobás, "Combinatorics" , Cambridge Univ. Press (1986) [a2] P.C. Fishburn, "Correlation in partially ordered sets" Discrete Appl. Math. , 39 (1992) pp. 173–191 [a3] C.M. Fortuin, P.N. Kasteleyn, J. Ginibre, "Correlation inequalities for some partially ordered sets" Comm. Math. Phys. , 22 (1971) pp. 89–103 [a4] R.L. Graham, "Linear extensions of partial orders and the FKG inequality" I. Rival (ed.) , Ordered sets , Reidel (1982) pp. 213–236 [a5] R.L. Graham, "Applications of the FKG inequality and its relatives" , Proc. 12th Internat. Symp. Math. Programming , Springer (1983) pp. 115–131 [a6] R. Holley, "Remarks on the FKG inequalities" Comm. Math. Phys. , 36 (1974) pp. 227–231 [a7] K. Joag-Dev, L.A. Shepp, R.A. Vitale, "Remarks and open problems in the area of the FKG inequality" , Inequalities Stat. Probab. , IMS Lecture Notes , 5 (1984) pp. 121–126 [a8] L.A. Shepp, "The XYZ conjecture and the FKG inequality" Ann. of Probab. , 10 (1982) pp. 824–827 [a9] P M. Winkler, "Correlation and order" Contemp. Math. , 57 (1986) pp. 151–174
How to Cite This Entry:
FKG inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=FKG_inequality&oldid=46896
This article was adapted from an original article by P.C. Fishburn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article