Abstract
We study the homogeneous nearest–neighbor Ising ferromagnet on the right half plane with a Dobrushin type boundary condition—say plus on the top part of the boundary and minus on the bottom. For sufficiently low temperature T, we completely characterize the pure (i.e., extremal) Gibbs states, as follows. There is exactly one for each angle \(\theta \in [-\pi /2,+\pi /2]\); here \(\theta \) specifies the asymptotic angle of the interface separating regions where the spin configuration looks like that of the plus (respectively, minus) full-plane state. Some of these conclusions are extended all the way to \(T=T_{c}\) by developing new Ising exact solution results—in particular, there is at least one pure state for each \(\theta \).
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Acknowledgements
Part of this work has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government program managed by the French National Research Agency (ANR). Part of this work has been carried out by SS at IITP RAS. The support of Russian Foundation for Sciences (Project No. 14-50-00150) is gratefully acknowledged. The research of CMN was supported in part by US-NSF Grant DMS-1507019. The authors thank Alessio Squarcini for help with TeX issues; they also thank an anonymous referee for useful comments and suggestions.
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We join with the other contributors to this special issue of JSP to acknowledge the many contributions to Mathematical and Statistical Physics made over the years by Juerg Froehlich, Tom Spencer and Herbert Spohn. May they each live and be well to one hundred twenty.
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Abraham, D., Newman, C.M. & Shlosman, S. A Continuum of Pure States in the Ising Model on a Halfplane. J Stat Phys 172, 611–626 (2018). https://doi.org/10.1007/s10955-017-1918-4
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DOI: https://doi.org/10.1007/s10955-017-1918-4