A roughening transition in the three-dimensional Ising model
Consider the nearest-neighbour ferromagnetic Ising model on $\mathbb{Z}^3$ at inverse temperature $\beta > 0$ and zero field. Its Gibbs measures are the probability measures on $\{-1, 1\}^{\mathbb{Z}^3}$ whose conditional law in every finite $\Lambda \subset \mathbb{Z}^3$, given the spins outside $\Lambda$, is proportional to $$\exp\Bigl(\beta \sum_{\{x, y\}:\ |x - y|_1 = 1,\ \{x, y\} \cap \Lambda \ne \varnothing} \sigma_x \sigma_y\Bigr).$$ Let $\mu^+_\beta$ be the infinite-volume limit with plus boundary condition and $\beta_c(3) = \inf\{\beta : \mu^+_\beta(\sigma_0) > 0\}$.
Dobrushin proved that for large $\beta$ the model has Gibbs measures that are not translation invariant, obtained with plus boundary condition above a horizontal plane and minus below; van Beijeren extended this to every $\beta > \beta_c(2)$, the critical inverse temperature of the two-dimensional model. Bodineau proved that for $\beta > \beta_c(3)$ every translation-invariant Gibbs measure is a convex combination of $\mu^+_\beta$ and $\mu^-_\beta$. Friedli and Velenik describe the threshold $\beta_R$ of the problem as a roughening transition, at which the interface between the two phases loses its rigidity.
Problem. Does there exist $\beta_R > \beta_c(3)$ such that every Gibbs measure is translation invariant for $\beta_c(3) < \beta < \beta_R$, while a Gibbs measure that is not translation invariant exists for every $\beta > \beta_R$?
Source: The conjectured roughening transition of the three-dimensional Ising model, in the form given by Friedli and Velenik (2017), §3.10.7, after Theorem 3.60; restated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Gibbs measures, interfaces, Ising model
References
@book{friedli2017,
author = {Friedli, Sacha and Velenik, Yvan},
title = {Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2017},
doi = {10.1017/9781316882603},
}
@article{dobrushin1972,
author = {Dobrushin, R. L.},
title = {Gibbs state describing coexistence of phases for a three-dimensional {I}sing model},
journal = {Theory Probab. Appl.},
volume = {17},
number = {4},
year = {1972},
pages = {582--600},
doi = {10.1137/1117073},
}
@article{vanbeijeren1975,
author = {van Beijeren, H.},
title = {Interface sharpness in the {I}sing system},
journal = {Comm. Math. Phys.},
volume = {40},
number = {1},
year = {1975},
pages = {1--6},
doi = {10.1007/BF01614092},
}
@article{bodineau2006,
author = {Bodineau, T.},
title = {Translation invariant {G}ibbs states for the {I}sing model},
journal = {Probab. Theory Related Fields},
volume = {135},
number = {2},
year = {2006},
pages = {153--168},
doi = {10.1007/s00440-005-0457-0},
eprint = {math/0409079},
archivePrefix = {arXiv},
}
@misc{aim-ising-three-dimensional-roughening,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A roughening transition in the three-dimensional Ising model}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/188-ising-three-dimensional-roughening.md},
note = {Entry 188 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00447, "A roughening transition in the three-dimensional Ising model", clopen.solutions.