Higher correlation inequalities


We prove a correlation inequality for n increasing functions on a distributive lattice, which for n = 2 reduces to a special case of the FKG inequality. The key new idea is to reformulate the inequalities for all n into a single positivity statement in the ring of formal power series. We also conjecture that our results hold in greater generality.

This is a preview of subscription content, access via your institution.


  1. [1]

    R. Ahilswede and D. E. Daykin: An inequality for the weights of two families of sets, their unions and intersections; Z. Wahrsch. Verw. Gebiete 43 (1978), 183–185.

    Article  MathSciNet  Google Scholar 

  2. [2]

    N. Alon and J. H. Spencer: The Probabilistic Method, John Wiley & Sons, Inc., New York, 1992.

    Google Scholar 

  3. [3]

    C. Fortuin, P. Kasteleyn and J. Ginibre: Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), 89–103.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    R. L. Graham: Applications of the FKG inequality and its relatives, in: Mathematical Programming: The State of the Art, Springer, Berlin, 1983, pp. 115–131.

    Google Scholar 

  5. [5]

    J. Glimm and A. Jaffe: Quantum Physics: A Functional Integral Point of View, 2nd ed., Springer, New York, 1987.

    Google Scholar 

  6. [6]

    S. Karlin and Y. Rinott: A generalized Cauchy-Binet formula and applications to total positivity and majorization, J. Multivar. Anal. 27 (1988), 284–299.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    J. L. Lebowitz: Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems, Comm. Math. Phys. 28 (1972), 313–321.

    Article  MathSciNet  Google Scholar 

  8. [8]

    D. Richards: Algebraic method toward higher-order probability inequalities II, Ann. of Prob. 32 (2004), 1509–1544.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    Y. Rinott and M. Saks: Correlation inequalities and a conjecture for permanents, Combinatorica 13(3), (1993), 269–277.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    L. A. Shepp: The XYZ conjecture and the FKG inequality, Ann. of Prob. 10 (1982), 824–827.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Siddhartha Sahi.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Sahi, S. Higher correlation inequalities. Combinatorica 28, 209–227 (2008). https://doi.org/10.1007/s00493-008-2249-5

Download citation

Mathematics Subject Classification (2000)

  • 05A20
  • 26D07
  • 60E15
  • 82B20