Applications of the FKG Inequality and Its Relatives

In 1971, C. M. Fortuin, P. W. Kasteleyn and J. Ginibre [FKG] published a remarkable inequality relating certain real functions defined on a finite distributive lattice. This inequality, now generally known as the FKG inequality, arose in connection with these authors’ investigations into correlation properties of Ising ferromagnet spin systems and generalized earlier results of Griffiths [Gri] and Harris [Har] (who was studying percolation models). The FKG inequality in turn has stimulated further research in a number of directions, including a variety of interesting generalizations and applications, particularly to statistics, computer science and the theory of partially ordered sets. It turns out that special cases of the FKG inequality can be found in the literature of at least a half dozen different fields, and in some sense can be traced all the way back to work of Chebyshev.

[1]  Lawrence A. Shepp,et al.  The FKG Inequality and Some Monotonicity Properties of Partial Orders , 1980, SIAM J. Algebraic Discret. Methods.

[2]  Richard P. Stanley,et al.  Two Combinatorial Applications of the Aleksandrov-Fenchel Inequalities , 1981, J. Comb. Theory, Ser. A.

[3]  Robert B. Griffiths,et al.  Correlations in Ising Ferromagnets. I , 1967 .

[4]  Guy Battle,et al.  The FKG inequality for the Yukawa2 quantum field theory , 1980 .

[5]  Ko-Wei Lih.  Majorization of Finite Partially Ordered Sets , 1982 .

[6]  C. J. Preston,et al.  A generalization of the FKG inequalities , 1974 .

[7]  Richard E. Barlow,et al.  Statistical Theory of Reliability and Life Testing: Probability Models , 1976 .

[8]  D. Kleitman.  Families of Non-disjoint subsets* , 1966 .

[9]  Antoine Brunel.  Une preuve elementaire du lemme de Holley , 1976 .

[10]  P. Erdos,et al.  Old and new problems and results in combinatorial number theory , 1980 .

[11]  Andrew Chi-Chih Yao,et al.  Some Monotonicity Properties of Partial Orders , 1979, SIAM J. Algebraic Discret. Methods.

[12]  P. Winkler.  Correlation Among Partial Orders , 1983 .

[13]  J. H. B. Kemperman,et al.  Review: Albert W. Marshall and Ingram Olkin, Inequalities: Theory of majorization and its applications, and Y. L. Tong, Probability inequalities in multivariate distributions , 1981 .

[14]  John Marica.  Orthogonal Families of Sets , 1971, Canadian Mathematical Bulletin.

[15]  Rudolf Ahlswede,et al.  Integral inequalities for increasing functions , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[16]  S. Sherman,et al.  General Griffiths' Inequalities on Correlations in Ising Ferromagnets , 1968 .

[17]  D. Walkup,et al.  Association of Random Variables, with Applications , 1967 .

[18]  D. Kleitman,et al.  A Monotonicity Property of Partial Orders , 1981 .

[19]  D. E. Daykin,et al.  Inequalities for a pair of mapsS×S→S withS a finite set , 1979 .

[20]  Peter C. Fishburn,et al.  On Unimodality for Linear Extensions of Partial Orders , 1980, SIAM J. Algebraic Discret. Methods.

[21]  Y. Tong.  Probability Inequalities in Multivariate Distributions , 1980 .

[22]  D. E. Daykin,et al.  An inequality for the weights of two families of sets, their unions and intersections , 1978 .

[23]  D. Welsh,et al.  Combinatorial applications of an inequality from statistical mechanics , 1975, Mathematical Proceedings of the Cambridge Philosophical Society.

[24]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[25]  Morris L. Eaton,et al.  A Review of Selected Topics in Multivariate Probability Inequalities , 1982 .

[26]  David E. Daykin,et al.  A Hierarchy of Inequalities , 1980 .

[27]  J. Kemperman.  On the FKG-inequality for measures on a partially ordered space , 1977 .

[28]  Pierre Cartier,et al.  Inégalités de corrélation en mécanique statistique , 1974 .

[29]  Peter Winkler.  Average height in a partially ordered set , 1982, Discret. Math..

[30]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[31]  D. E. Daykin,et al.  The Number of Values of a Boolean Function , 1976 .

[32]  J. Marica,et al.  Differences of Sets and A Problem of Graham , 1969, Canadian Mathematical Bulletin.

[33]  R. Holley.  Remarks on the FKG inequalities , 1974 .

[34]  L. Shepp.  The XYZ Conjecture and the FKG Inequality , 1982 .

[35]  T. E. Harris.  A lower bound for the critical probability in a certain percolation process , 1960, Mathematical Proceedings of the Cambridge Philosophical Society.

[36]  C. Fortuin,et al.  Correlation inequalities on some partially ordered sets , 1971 .

[37]  P. Seymour.  On incomparable collections of sets , 1973 .

[38]  K. Jogdeo,et al.  Association and Probability Inequalities , 1977 .