Journal of Statistical Physics

, Volume 159, Issue 4, pp 958–971 | Cite as

Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions

  • Loren CoquilleEmail author


We review what is known about the structure of the set of weak limiting states of the Ising and Potts models at low enough temperature, and in particular we prove that the mixture \(\frac{1}{2}(\mu ^\pm +\mu ^\mp )\) of two reflection-symmetric Dobrushin states of the 3-dimensional Ising model at low enough temperature is a Gibbs state which is not a limit of finite-volume measures with deterministic boundary conditions. Finally we point out what the issues are in order to extend the analysis to the Potts model, and give a few conjectures.


Gibbs states Ising model Potts model Boundary conditions 

Mathematics Subject Classification

82B05 82B20 82B26 



I am grateful to A. van Enter for encouraging me to work on this question, and for valuable comments and suggestions. I also thank Y. Velenik for his advice, a stimulating discussion and a few references, A. Bovier for mentioning reference [3] to me, and V. Beffara for the simulations of Figs. 2 and 3. I am indebted to Y. Higuchi for pointing out reference [33]. This research was supported by the German Research Foundation (DFG) and the Hausdorff Center for Mathematics (HCM).


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© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Institut für Angewandte MathematikRheinische Friedrich-Wilhelms-UniversitätBonnGermany

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