Abstract
We show that a class of spin models, containing the Ashkin-Teller model, admits a generalized random-cluster (GRC) representation. Moreover, we show that basic properties of the usual representation, such as FKG inequalities and comparison inequalities, still hold for this generalized random-cluster model. Some elementary consequences are given. We also consider the duality transformations in the spin representation and in the GRC model and show that they commute.
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M. Aizenman, J. T. Chayes, L. Chayes, C. M. Newmann, Discontinuity of the magnetization in one-dimensional 1/|x −y|2 Ising and Potts models,J. Stat. Phys. 50:1–40 (1988).
J. Ashkin and E. Teller, Statistics of two-dimensional lattices with four components,Phys. Rev. 64:178–184 (1943).
R. J. Baxter,Exactly solved models in statistical mechanics (Academic Press, New York, 1982).
C. Berge,Graphes (Gauthier-Villars, Paris, 1983).
L. Chayes and J. Machta, Graphical representations and cluster algorithms. Part I: discrete spin systems, to appear inPhysica A.
J. T. Chayes, L. Chayes, and R. H. Schonmann, Exponential decay of connectivities in the two-dimensional Ising model,J. Stat. Phys. 49:433–445 (1987).
E. Domany and E. Riedel, Two-dimensional anisotropicN-vector models,Phys. Rev. B 19:5817–5834 (1979).
F. Dunlop, L. Laanait, A. Messages, S. Miracle-Sole, and J. Ruiz, Multilayer wetting in partially symmetricq-state models,J. Stat. Phys. 59:1383–1396 (1991).
C. Fan, Symmetry properties of the Ashkin-Teller model and the eight-vertex model,Phys. Rev. B 6:902–910 (1972).
C. M. Fortuin, On the random-cluster model II: The percolation model,Physica 58:393–418 (1972).
C. M. Fortuin, On the random-cluster model III: The simple random-cluster model,Physica 59:545–570 (1972).
C. M. Fortuin and P. W. Kasteleyn, On the random-cluster model I: Introduction and relation to other models,Physica 57:536–564 (1972).
C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets,Commun. Math. Phys. 22:89–103 (1971).
G. Grimmett, Potts models and random-cluster processes with many-body interactions,J. Stat. Phys. 75:67–121 (1994).
D. Ioffe, Exact large deviation bounds up toT c for the Ising model in two dimensions,Probab. Theory Relat. Fields 102:313–330 (1995).
L. Laanait, N. Masaif, and J. Ruiz, Phase coexistence in partially symmetricq-state models,J. Stat. Phys. 72:721–736 (1993).
L. Laanait, A. Messager, and J. Ruiz, Discontinuity of the Wilson string tension in the 4-dimensional lattice pure gauge Potts model,J. Stat. Phys. 72:721–736 (1993).
C. E. Pfister, Phase transitions in the Ashkin-Teller model,J. Stat. Phys. 29:113–116 (1982).
A. Pisztora, Surface order large deviations for the Ising, Potts and percolation models,Probab. Theory Relat. Fields 104:427–466 (1996).
J. Salas and A. D. Sokal, Preprint, Dynamic critical behavior of a Swendsen-Wang-type algorithm for the Ashkin-Teller model, Nov. 95.
F. J. Wegner, Duality relation between the Ashkin-Teller and the eight-vertex model,J. Phys. C: Solid State Phys. 5:L131-L132 (1972).
S. Wiseman and E. Domany, Cluster method for the Ashkin-Teller model,Phys. Rev. E 48:4080–4090 (1993).
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Pfister, C.E., Velenik, Y. Random-cluster representation of the ashkin-teller model. J Stat Phys 88, 1295–1331 (1997). https://doi.org/10.1007/BF02732435
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DOI: https://doi.org/10.1007/BF02732435