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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.

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Notes

  1. We emphasize that the natural monotonicity of the fluctuations that we could expect with respect to the temperature is not true in general. Indeed, positive temperature may result in reduction of the fluctuations [6].

  2. The boundary of a fixed region \(\Omega \subset \mathbb {R}^d\) must be partitioned in such a way that for each \(n\), on the boundary of \(\Omega _n=\Omega \cap \frac{1}{n}\mathbb {Z}^d\), the number of nearest-neighbor pairs of vertices having different spins is \(o(n^{d-1})\).

  3. More precisely in a region of size \(f(n)\) such that \(\log n\ll f(n)\ll n^{1/(d-1)}\).

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Acknowledgments

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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Coquille, L. Examples of DLR States Which are Not Weak Limits of Finite Volume Gibbs Measures with Deterministic Boundary Conditions. J Stat Phys 159, 958–971 (2015). https://doi.org/10.1007/s10955-015-1211-3

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Keywords

  • Gibbs states
  • Ising model
  • Potts model
  • Boundary conditions

Mathematics Subject Classification

  • 82B05
  • 82B20
  • 82B26