Gibbs states of graphical representations of the Potts model with external fields

Journal of Mathematical Physics

Volumn 41, Issue 3, 2000, Pages 1170-1210

  Biskup, M ^a,b,c   Borgs, C ^c   Chayes, J T ^c   Kotecky R ^a,c  

^a CHARLES UNIVERSITY   (Czech Republic)

^b RADBOUD UNIVERSITY NIJMEGEN   (Netherlands)

^c MICROSOFT RESEARCH   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034147113     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.533183     Document Type: Article

References (23)

1. C. M. Fortuin and P. W. Kasteleyn, "On the random cluster model I. Introduction and relation to other models," Physica (Amsterdam) 57, 536-564 (1972).
2. 2 Ising and Potts models," J. Stat. Phys. 50, 1-40 (1988).
3. C. Borgs and J. T. Chayes, "The covariance matrix of the Potts model: A random cluster analysis," J. Stat. Phys. 82. 1235-1297 (1996).
4. G. R. Grimmett, "The stochastic random cluster process and the uniqueness of random cluster measures," Ann. Prob. 23, 1461-1510 (1995).
5. C. E. Pfister and K. Vande Velde, "Almost sure quasilocality in the random cluster model," J. Stat. Phys. 79, 765-774 (1995).
6. O. Häggström, "Random cluster representations in the study of phase transitions," Markov Proc. Relat. Fields 4, 275-321 (1998).
7. H.-O. Georgii, O. Häggström, and C. Maes, "The random geometry of equilibrium phases," in Critical Phenomena, edited by C. Domb and J. L Lebowitz (Academic, New York, 1999).
8. M. Biskup, C. Borgs, J. T. Chayes, and R. Kotecký, "Phase diagrams of Potts models in external fields," in preparation.
9. M. Biskup, L. Chayes, and R. Kotecký, "Coexistence of partially disordered/ordered phases in an extended Potts model," preprint.
10. R. G. Edwards and A. D. Sokal, "Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm," Phys. Rev. D 38, 2009-2012 (1988).
11. H.-O. Georgii, Gibbs Measures and Phase Transitions, De Gruyter Studies in Mathematics (De Gruyter, Berlin, 1988), Vol. 9.
12. C. Preston, Random Fields (Springer-Verlag, Berlin, 1976).
13. D. Ruelle, Thermodynamic Formalism (Addison-Wesley, London, 1978).
14. G. Gallavotti, "The phase separation line in the two-dimensional Ising model," Commun. Math. Phys. 27, 103 (1972).
15. M. Aizenman, "Translation invariance and instability of phase coexistence in the two-dimensional Ising model," Commun. Math. Phys. 73, 83-94 (1980).
16. Y. Higuchi, "On the absence of non-translationally invariant Gibbs states for the two-dimensional Ising system," in Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory, Esztergom, 1979, edited by J. Fritz, J. L. Lebowitz, and D. Szász (North-Holland, Amsterdam, 1981), Vol. I, pp. 517-534.
17. A. C. D. van Enter, R. Fernández, and A. Sokal, "Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory," J. Stat. Phys. 72, 879-1167 (1993).
18. R. M. Burton and M. Keane, "Density and uniqueness in percolation," Commun. Math. Phys. 10, 501-505 (1989).
19. J. T. Chayes, A. Puha, and T. Sweet, "Independent and dependent percolation," in Probability Theory and Applications, IAS/Park City Mathematical Series, edited by E. P. Hsu and S. R. S. Varadhan (American Mathematical Society and Institute for Advanced Study, 1999), Vol. 6, pp. 51-118.
20. A. Gandolfi, M. Keane, and C. M. Newman, "Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses," Probab. Theory Relat. Fields 92, 511-527 (1992).
21. A. Gandolfi, M. Keane, and L. Russo, "On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation," Ann. Prob. 16, 1147-1157 (1988).
22. R. Kotecký and S. B. Shlosman, "First-order transitions in large entropy lattice models," Commun. Math. Phys. 83, 493-515 (1982).
23. K. Alexander, "The asymmetric random cluster model and comparison of Ising and Potts models," preprint.