The Ahlswede–Daykin inequality ( Ahlswede & Daykin 1978), also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).
The inequality states that if are nonnegative functions on a finite distributive lattice such that
for all x, y in the lattice, then
for all subsets X, Y of the lattice, where
and
The Ahlswede–Daykin inequality can be used to provide a short proof of both the Holley inequality and the FKG inequality. It also implies the XYZ inequality.
For a proof, see the original article ( Ahlswede & Daykin 1978) or ( Alon & Spencer 2000).
The "four functions theorem" was independently generalized to 2k functions in ( Aharoni & Keich 1996) and ( Rinott & Saks 1991).
The Ahlswede–Daykin inequality ( Ahlswede & Daykin 1978), also known as the four functions theorem (or inequality), is a correlation-type inequality for four functions on a finite distributive lattice. It is a fundamental tool in statistical mechanics and probabilistic combinatorics (especially random graphs and the probabilistic method).
The inequality states that if are nonnegative functions on a finite distributive lattice such that
for all x, y in the lattice, then
for all subsets X, Y of the lattice, where
and
The Ahlswede–Daykin inequality can be used to provide a short proof of both the Holley inequality and the FKG inequality. It also implies the XYZ inequality.
For a proof, see the original article ( Ahlswede & Daykin 1978) or ( Alon & Spencer 2000).
The "four functions theorem" was independently generalized to 2k functions in ( Aharoni & Keich 1996) and ( Rinott & Saks 1991).