Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-26T07:27:28.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Combinatorial applications of an inequality from statistical mechanics

Published online by Cambridge University Press:  24 October 2008

P. D. Seymour  and
D. J. A. Welsh
P. D. Seymour
Affiliation:
University College, Swansea
D. J. A. Welsh
Affiliation:
Merton College, Oxford
Get access Rights & Permissions [Opens in a new window]

Extract

The main part of this paper shows how an inequality of statistical mechanics has several applications in combinatorial theory.

The inequality (known as the FKG inequality) was derived by Fortuin, Kasteleyn and Ginibre(4) in their work on the statistical mechanics of Ising ferromagnets. We first show how it leads to new properties of log supermodular functions, Bernstein polynomials, and log convex sequences.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 3 , May 1975 , pp. 485 - 495
Copyright
Copyright © Cambridge Philosophical Society 1975

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Brylawski, T.A decomposition for combinatorial geometries. Trans. Amer. Math. Soc. 171 (1972), 235–282.CrossRefGoogle Scholar
(2)Crapo, H.The Tutte polynomial. Aequationes Math. 3 (1969), 211–229.CrossRefGoogle Scholar
(3)Ford, L. R. Jr and Fulkerson, D. R.Flows in networks (Princeton University Press, 1962).Google Scholar
(4)Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J.Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 (1971), 89–103.CrossRefGoogle Scholar
(5)Holley, R. Remarks on the FKG inequalities (to be published).Google Scholar
(6)Kleitman, D.Families of non-disjoint subsets. J. Combinatorial Theory (1966), 153–155.Google Scholar
(7)Kurtz, D. C.A note on concavity properties of triangular arrays of numbers. J. Combinatorial Theory 13 (1972), 135–139.Google Scholar
(8)Mason, J. H. Matroids: Unimodal conjectures and Motzkin's theorem. Combinatorics (Institute of Math. & Appl.) (Eds. Welsh, D. J. A. & Woodall, D. R., 1972), 207–221.Google Scholar
(9)Mitrinovic, D. S.Analytic inequalities (Springer Verlag, 1970).Google Scholar
(10)Preston, C. J.Gibbs States on Countable Sets (Cambridge, to appear).Google Scholar
(11)Tutte, W. T.A contribution to the theory of chromatic polynomials. Canad. J. Math. 6 (1954), 80–91.CrossRefGoogle Scholar
(12)Tutte, W. T. A problem on spanning trees. Quart. J. Math. (Oxford), to appear.Google Scholar
(13)Welsh, D. J. A. Combinatorial problems in matroid theory. Combinatorial mathematics and its applications (Academic Press, 1971), 291–307.Google Scholar