FKG inequality

In mathematics, the Fortuin–Kasteleyn–Ginibre (FKG) inequality is a correlation inequality, a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method), due to Cees M. Fortuin, Pieter W. Kasteleyn, and Jean Ginibre (1971). Informally, it says that in many random systems, increasing events are positively correlated, while an increasing and a decreasing event are negatively correlated. It was obtained by studying the random cluster model.

An earlier version, for the special case of i.i.d. variables, called Harris inequality, is due to Theodore Edward Harris (1960), see below. One generalization of the FKG inequality is the Holley inequality (1974) below, and an even further generalization is the Ahlswede–Daykin "four functions" theorem (1978). Furthermore, it has the same conclusion as the Griffiths inequalities, but the hypotheses are different.

The inequality

Let

X

{\displaystyle X}

be a finite distributive lattice, and μ a nonnegative function on it, that is assumed to satisfy the (FKG) lattice condition (sometimes a function satisfying this condition is called log supermodular) i.e.,

μ

(

x

∧

y

)

μ

(

x

∨

y

)

≥

μ

(

x

)

μ

(

y

)

{\displaystyle \mu (x\wedge y)\mu (x\vee y)\geq \mu (x)\mu (y)}

for all x, y in the lattice

X

{\displaystyle X}

.

The FKG inequality then says that for any two monotonically increasing functions ƒ and g on

X

{\displaystyle X}

, the following positive correlation inequality holds:

(

∑

x

∈

X

f

(

x

)

g

(

x

)

μ

(

x

)

)

(

∑

x

∈

X

μ

(

x

)

)

≥

(

∑

x

∈

X

f

(

x

)

μ

(

x

)

)

(

∑

x

∈

X

g

(

x

)

μ

(

x

)

)

.

{\displaystyle \left(\sum _{x\in X}f(x)g(x)\mu (x)\right)\left(\sum _{x\in X}\mu (x)\right)\geq \left(\sum _{x\in X}f(x)\mu (x)\right)\left(\sum _{x\in X}g(x)\mu (x)\right).}

The same inequality (positive correlation) is true when both ƒ and g are decreasing.