Abstract
In this paper we study a continuum version of the Potts model, where particles are points in ℝd, d≥2, with a spin which may take S≥3 possible values. Particles with different spins repel each other via a Kac pair potential of range γ −1, γ>0. In mean field, for any inverse temperature β there is a value of the chemical potential λ β at which S+1 distinct phases coexist. We introduce a restricted ensemble for each mean field pure phase which is defined so that the empirical particles densities are close to the mean field values. Then, in the spirit of the Dobrushin-Shlosman theory (Dobrushin and Shlosman in J. Stat. Phys. 46(5–6):983–1014, 1987), we prove that while the Dobrushin high-temperatures uniqueness condition does not hold, yet a finite size condition is verified for γ small enough which implies uniqueness and exponential decay of correlations. In a second paper (De Masi et al. in Coexistence of ordered and disordered phases in Potts models in the continuum, 2008), we will use such a result to implement the Pirogov-Sinai scheme proving coexistence of S+1 extremal DLR measures.
Similar content being viewed by others
References
Bodineau, T., Presutti, E.: Phase diagram of Ising systems with additional long range forces. Commun. Math. Phys. 189, 287–298 (1997)
Bovier, A., Zahradnik, M.: The low temperature phase of Kac-Ising models. J. Stat. Phys. 87, 311–332 (1997)
Bovier, A., Merola, I., Presutti, E., Zahradnìk, M.: On the Gibbs phase rule in the Pirogov-Sinai regime. J. Stat. Phys. 114, 1235–1267 (2004)
Buttà, P., Merola, I., Presutti, E.: On the validity of the van der Waals theory in Ising systems with long range interactions. Markov Process. Relat. Fields 3, 63–88 (1977)
Cassandro, M., Presutti, E.: Phase transitions in Ising systems with long but finite range interactions. Markov Process. Relat. Fields 2, 241–262 (1996)
De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Coexistence of ordered and disordered phases in Potts models in the continuum (2008, in preparation)
Dobrushin, R.L., Shlosman, S.B.: Completely analytical interactions: constructive description. J. Stat. Phys. 46(5–6), 983–1014 (1987)
Georgii, H.-O., Häggström, O.: Phase transition in continuum Potts models. Commun. Math. Phys. 181, 507–528 (1996)
Georgii, H.-O., Miracle-Sole, S., Ruiz, J., Zagrebnov, V.: Mean field theory of the Potts Gas. J. Phys. A 39, 9045–9053 (2006)
Gobron, T., Merola, I.: First order phase transitions in Potts models with finite range interactions. J. Stat. Phys. 126, 507–583 (2006)
Lebowitz, J.L., Mazel, A., Presutti, E.: Liquid vapour phase transitions for systems with finite range interactions. J. Stat. Phys. 94, 955–1025 (1999)
Presutti, E.: Scaling limits in statistical mechanics and microstructures in continuum mechanics. In: Theoretical and Mathematical Physics. Springer, Berlin (2008, to appear)
Ruelle, D.: Widom-Rowlinson: Existence of a phase transition in a continuous classical system. Phys. Rev. Lett. 27, 1040–1041 (1971)
van der Berg, J.: A uniqueness condition for Gibbs measures with application to the two dimensional antiferromagnet. Commun. Math. Phys. 152, 161–166 (1993)
van der Berg, J., Maes, C.: Disagreement percolation in the study of Markov fields. Ann. Probab. 22, 749–763 (1994)
van der Berg, J., Steif, J.E.: Percolation and the hard core lattice model. Stoch. Process. Appl. 49, 179–197 (1994)
Zahradnìk, M.: A short course on the Pirogov-Sinai theory. Rend. Mat. Appl. 18, 411–486 (1998)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
De Masi, A., Merola, I., Presutti, E. et al. Potts Models in the Continuum. Uniqueness and Exponential Decay in the Restricted Ensembles. J Stat Phys 133, 281–345 (2008). https://doi.org/10.1007/s10955-008-9603-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-008-9603-2