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''Fortuin–Kasteleyn–Ginibre inequality''
 
''Fortuin–Kasteleyn–Ginibre inequality''
  
An inequality [[#References|[a3]]] that began a series of [[Correlation inequalities|correlation inequalities]] for finite partially ordered sets. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101201.png" /> be a finite [[Partially ordered set|partially ordered set]] ordered by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101202.png" /> (irreflexive, transitive) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101203.png" /> a [[Distributive lattice|distributive lattice]]: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101204.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101205.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101206.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101207.png" />. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101208.png" /> is log supermodular:
+
An inequality [[#References|[a3]]] that began a series of [[Correlation inequalities|correlation inequalities]] for finite partially ordered sets. Let $  \Gamma $
 +
be a finite [[Partially ordered set|partially ordered set]] ordered by $  \prec $(
 +
irreflexive, transitive) with $  ( \Gamma, \prec ) $
 +
a [[Distributive lattice|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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f1101209.png" /></td> </tr></table>
+
$$
 +
\mu ( a ) \mu ( b ) \leq  \mu ( a \lor b ) \mu ( a \wedge b )  \textrm{ for  all  }  a, b \in \Gamma,
 +
$$
  
and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012011.png" /> are non-decreasing:
+
and that $  f : \Gamma \rightarrow \mathbf R $
 +
and $  g : \Gamma \rightarrow \mathbf R $
 +
are non-decreasing:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012012.png" /></td> </tr></table>
+
$$
 +
a \prec b \Rightarrow \{ f ( a ) \leq  f ( b ) , g ( a ) \leq  g ( b ) \}  \textrm{ for  all  }  a,b \in \Gamma .
 +
$$
  
 
The FKG inequality is:
 
The FKG inequality is:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012013.png" /></td> </tr></table>
+
$$
 +
\left [ \sum _ {a \in \Gamma } \mu ( a ) f ( a ) \right ] \left [ \sum _ {a \in \Gamma } \mu ( a ) g ( a ) \right ] \leq
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012014.png" /></td> </tr></table>
+
$$
 +
\leq 
 +
\left [ \sum _ {a \in \Gamma } \mu ( a ) \right ] \left [ \sum _ {a \in \Gamma } \mu ( a ) f ( a ) g ( a ) \right ] .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012015.png" /> is a [[Boolean algebra|Boolean algebra]] and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012016.png" /> is a [[Probability measure|probability measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012017.png" />, the inequality is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110120/f11012019.png" /> denotes [[Mathematical expectation|mathematical expectation]].
+
If $  \Gamma $
 +
is a [[Boolean algebra|Boolean algebra]] and $  \mu $
 +
is a [[Probability measure|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|mathematical expectation]].
  
 
Related inequalities are discussed in [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]].
 
Related inequalities are discussed in [[#References|[a1]]], [[#References|[a2]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]], [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]].

Revision as of 19:38, 5 June 2020


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].

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
How to Cite This Entry:
FKG inequality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=FKG_inequality&oldid=14368
This article was adapted from an original article by P.C. Fishburn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article