Abstract
We prove that all the translation invariant Gibbs states of the Ising model are a linear combination of the pure phases for any . This implies that the average magnetization is continuous for . Furthermore, combined with previous results on the slab percolation threshold [B2] this shows the validity of Pisztora's coarse graining [Pi] up to the critical temperature.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Aizenman, M.: Translation invariance and instability of phase coexistence in the two-dimensional Ising system. Comm. Math. Phys. 73 (1), 83–94 (1980)
Aizenman, M., Fernandez, R.: On the critical behavior of the magnetization in high-dimensional Ising models. J. Stat. Phys. 44 (3–4), 393–454 (1986)
Bodineau, T.: The Wulff construction in three and more dimensions. Comm. Math. Phys. 207 (1), 197–229 (1999)
Bodineau, T.: Slab percolation for the Ising model. Prob. Th. Rel. Fields. 132 (1), 83–118 (2005)
Bodineau, T., Ioffe, D., Velenik, Y.: Rigorous probabilistic analysis of equilibrium crystal shapes. J. Math. Phys. 41 (3), 1033–1098 (2000)
Bricmont, J., Lebowitz, J.: On the continuity of the magnetization and energy in Ising ferromagnets. Jour. Stat. Phys. 42 (5/6), 861–869 (1986)
Burton, R., Keane, M.: Density and uniqueness in percolation. Comm. Math. Phys. 121 (3), 501–505 (1989)
Butta, P., Merola, I., Presutti, E.: On the validity of the van der Waals theory in Ising systems with long range interactions. Markov Process. Related Fields 3 (1), 63–88 (1997)
Cerf, R.: Large deviations for three dimensional supercritical percolation. Astérisque 267, (2000)
Cerf, R., Pisztora, A.: On the Wulff crystal in the Ising model. Ann. Probab. 28 (3), 947–1017 (2000)
Dobrushin, R.: The Gibbs state that describes the coexistence of phases for a three-dimensional Ising model. (In Russian) Teor. Verojatnost. i Primenen. 17, 619–639 (1972)
Dobrushin, R., Shlosman, S.: The problem of translation invariance of Gibbs states at low temperatures. Mathematical physics reviews, 5, 53–195, Soviet Sci. Rev. Sect. C Math. Phys. Rev. 5, Harwood Academic Publ., Chur, (1985)
Edwards, R., Sokal, A.: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm. Phys. Rev. D (3) 38 (6), 2009–2012 (1988)
Fortuin, C., Kasteleyn, P.: On the random-cluster model. I. Introduction and relation to other models. Physica 57, 536–564 (1972)
Fröhlich, J., Pfister, C.: Semi-infinite Ising model. II. The wetting and layering transitions. Comm. Math. Phys. 112 (1), 51–74 (1987)
Gallavotti, G., Miracle-Solé, S.: Equilibrium states of the ising model in the two-phase region. Phys. Rev. B 5, 2555–2559 (1972)
Georgii, H.O.: Gibbs measures and phase transitions. de Gruyter Studies in Mathematics, 9. Walter de Gruyter, Berlin, 1988
Georgii, H.O., Higuchi, Y.: Percolation and number of phases in the two-dimensional Ising model. Probabilistic techniques in equilibrium and nonequilibrium statistical physics. J. Math. Phys. 41 (3), 1153–1169 (2000)
Grimmett, G.: The stochastic random-cluster process and the uniqueness of random-cluster measures. Ann. Probab. 23 (4), 1461–1510 (1995)
Higuchi, Y.: On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model. Random fields, Vol. I, II (Esztergom, 1979), 517–534, Colloq. Math. Soc. János Bolyai, 27, North-Holland, Amsterdam-New York, 1981
Lebowitz, J.: Coexistence of phases in ising ferromagnets. Jour. Stat. Phys. 16, 463–476 (1977)
Lebowitz, J.: Thermodynamic limit of the free energy and correlation functions of spin systems. Quantum dynamics models and mathematics, ed. L. Streit, Springer, 201–220, 1976
Lebowitz, J., Pfister, C.: Surface tension and phase coexistence. Phys. Rev. Lett. 46 (15), 1031–1033 (1981)
Martirosian, D.: Translation invariant Gibbs states in the q-state Potts model. Comm. Math. Phys. 105 (2), 281–290 (1986)
Messager, A., Miracle-Solé, S., Pfister, C.: On classical ferromagnets with a complex external field. J. Statist. Phys. 34 (1–2), 279–286 (1984)
Onsager, L.: Crystal statistics I. A two-dimensional model with an order-disorder transition. Phys. Rev. 65, 117–149 (1944)
Pemantle, R., Steif, J.: Robust phase transitions for Heisenberg and other models on general trees. Ann. Probab. 27 (2), 876–912 (1999)
Pisztora, A.: Surface order large deviations of Ising, Potts and percolation models. Prob. Th. Rel. Fields 104, 427–466 (1996)
van Enter, A.: A remark on the notion of robust phase transitions. J. Statist. Phys. 98 (5–6), 1409–1416 (2000)
Zahradnik, M.: An alternate version of Pirogov-Sinai theory. Comm. Math. Phys. 93 (4), 559–581 (1984)
Author information
Authors and Affiliations
Corresponding author
Additional information
We would like to thank C. Pfister and Y. Velenik for very useful comments.
Rights and permissions
About this article
Cite this article
Bodineau, T. Translation invariant Gibbs states for the Ising model. Probab. Theory Relat. Fields 135, 153–168 (2006). https://doi.org/10.1007/s00440-005-0457-0
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-005-0457-0