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 \preceq z, b \preceq z } \} $, $ a \wedge b = \max \{ {z \in \Gamma } : {z \preceq a, z \preceq 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:

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

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].

See also Ahlswede–Daykin inequality; Fishburn–Shepp inequality; Holley inequality.

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