Examples of DLR states which are not weak limits of finite volume Gibbs measures with deterministic boundary conditions

Abstract

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.

arXiv comment15 pages, 5 figures. Revised version

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Cited by in corpus (4)

• Gibbs measures of Potts model on Cayley trees: a survey and applications
• Large $N$ phase transitions and the fate of small Schwarzschild-AdS black holes
• Absence of Dobrushin states for $2d$ long-range Ising models
• Nonexistence of spontaneous symmetry breakdown of time-translation symmetry on general quantum systems: Any macroscopic order parameter moves not!