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).
Generalizations[edit]
The "four functions theorem" was independently generalized to 2k functions in (Aharoni & Keich 1996) and (Rinott & Saks 1991).
References[edit]
- Ahlswede, Rudolf; Daykin, David E. (1978), "An inequality for the weights of two families of sets, their unions and intersections", Probability Theory and Related Fields, 43 (3): 183–185, CiteSeerX 10.1.1.380.8629, doi:10.1007/BF00536201, ISSN 0178-8051, MR 0491189, S2CID 120659862
- Alon, N.; Spencer, J. H. (2000), The probabilistic method. Second edition. With an appendix on the life and work of Paul Erdős., Wiley-Interscience, New York, ISBN 978-0-471-37046-8, MR 1885388
- Fishburn, P.C. (2001) [1994], "Ahlswede–Daykin inequality", Encyclopedia of Mathematics, EMS Press
- Aharoni, Ron; Keich, Uri (1996), "A Generalization of the Ahlswede Daykin Inequality", Discrete Mathematics, 152 (1–3): 1–12, doi:10.1016/0012-365X(94)00294-S
- Rinott, Yosef; Saks, Michael (1991), "Correlation inequalities and a conjecture for permanents", Combinatorica, 13 (3): 269–277, doi:10.1007/BF01202353, S2CID 206791629