The Ahlswede–Daykin inequality , 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 or .
Generalizations
The "four functions theorem" was independently generalized to 2k functions in and .
History
The story of the discovery of the Ahlswede–Daykin inequality was described in the Introduction to the A. Ahlswede et al. book: "The history of the idea of the AD-inequality is very interesting. As Daykin came to a visit to Bielefeld, Ahlswede was just wallpapering. He stood on the ladder, and Daykin wanted to tell him from a newly proven inequality. The declaration was complicated, and Ahlswede said that probably a more general (and easier) theorem should hold. He made directly—on the ladder—a proposal which already was the AD-inequality."
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