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 ${\displaystyle f_{1},f_{2},f_{3},f_{4}}$ are nonnegative functions on a finite distributive lattice such that

${\displaystyle f_{1}(x)f_{2}(y)\leq f_{3}(x\vee y)f_{4}(x\wedge y)}$

for all x, y in the lattice, then

${\displaystyle f_{1}(X)f_{2}(Y)\leq f_{3}(X\vee Y)f_{4}(X\wedge Y)}$

for all subsets X, Y of the lattice, where

${\displaystyle f(X)=\sum _{x\in X}f(x)}$

and

${\displaystyle X\vee Y=\{x\vee y\mid x\in X,y\in Y\}}$

${\displaystyle X\wedge Y=\{x\wedge y\mid x\in X,y\in Y\}.}$

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

The "four functions theorem" was independently generalized to 2k functions in ( Aharoni & Keich 1996) and ( Rinott & Saks 1991).

References

• 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