Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-28T13:46:23.304Z Has data issue: false hasContentIssue false
  • English
  • Français

On the duality of interaction models

Published online by Cambridge University Press:  24 October 2008

Norman Biggs
Norman Biggs
Affiliation:
Royal Holloway College, University of London
Get access Rights & Permissions [Opens in a new window]

Abstract

Two kinds of duality arising in studies of interaction models are discussed. The first kind, which has not previously been investigated, is related to algebraic properties of the coefficient ring. The second kind is the well-known geometric duality for planar graphs. The two dualities together lead to a perfectly symmetrical relationship for a general form of partition function.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 80 , Issue 3 , November 1976 , pp. 429 - 436
Copyright
Copyright © Cambridge Philosophical Society 1976

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)Biggs, N. L.Algebraic graph theory (Cambridge University Press, 1974).CrossRefGoogle Scholar
(2)Pauling, L.The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Amer. Chem. Soc. 57 (1935), 2680–2684.CrossRefGoogle Scholar
(3)Potts, R. B.Some generalized order-disorder transformations. Proc. Cambridge Philos. Soc. 48 (1952), 106–109.CrossRefGoogle Scholar
(4)Syozi, I. Transformation of Ising models. Phase transitions and critical phenomena, vol. 1. (London; Academic Press, 1972).Google Scholar
(5)Tutte, W. T.A ring in graph theory. Proc. Cambridge Philos. Soc. 43 (1947), 26–40.CrossRefGoogle Scholar
(6)Tutte, W. T.On the imbedding of linear graphs in surfaces. Proc. London Math. Soc. 51 (1949), 474–483.CrossRefGoogle Scholar
(7)Wannier, G. H.The statistical problem in cooperative phenomena. Rev. Mod. Phys. 17 (1945), 50–60.Google Scholar
(8)Wilson, R. J.Introduction to graph theory (Edinburgh; Oliver and Boyd, 1972).Google Scholar
(9)Wu, F. Y. and Wang, Y. K.Duality transformation in a many-component spin model. J. Math. Phys. 17 (1976), 439–440.CrossRefGoogle Scholar