Uniqueness of the planar spin-glass ground-state pair
For each nearest-neighbour edge $e$ of $\mathbb{Z}^2$ let $J_e$ be an independent standard Gaussian random variable (the Edwards–Anderson model). A configuration $\sigma \in \{-1, 1\}^{\mathbb{Z}^2}$ is a ground state if flipping any finite set of spins cannot decrease its energy, that is, $$\sum_{\{x, y\} \in \partial_E A} J_{\{x, y\}} \sigma_x \sigma_y \ge 0 \quad \text{for every finite } A \subset \mathbb{Z}^2,$$ where $\partial_E A$ is the set of edges with exactly one endpoint in $A$. With $\sigma$, also $-\sigma$ is a ground state.
Newman and Stein (2001) proved that if two ground-state pairs chosen from metastates with, for example, periodic boundary conditions are distinct, then the dual edges where they differ form a single doubly infinite domain wall of positive density. On the half-plane $\mathbb{Z} \times \mathbb{N}$, Arguin, Damron, Newman and Stein proved that the ground-state pair obtained from a particular choice of boundary conditions is almost surely unique, and Arguin and Damron that the set of all ground states has either two or infinitely many elements. In the plane, Newman and Stein (2026) proved that the zero-temperature periodic-boundary-condition metastate is unique and supported on a single ground-state pair.
Problem. Is it almost surely true that the set of all ground states is exactly $\{\sigma, -\sigma\}$ for some $\sigma$? The question concerns all infinite-volume ground states, not only those obtained as limits under particular boundary conditions.
Source: A long-standing belief about the two-dimensional Edwards–Anderson spin glass, posed as a problem by Newman and Stein (2001); for all ground states as in Dario, Harel and Peled (2024), §2.3.2. Stated by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Edwards–Anderson model, ground states, spin glasses
References
@article{newman2001,
author = {Newman, C. M. and Stein, D. L.},
title = {Are there incongruent ground states in {2D} {E}dwards--{A}nderson spin glasses?},
journal = {Comm. Math. Phys.},
volume = {224},
number = {1},
year = {2001},
pages = {205--218},
doi = {10.1007/PL00005586},
eprint = {cond-mat/0103395},
archivePrefix = {arXiv},
}
@article{dario2024,
author = {Dario, Paul and Harel, Matan and Peled, Ron},
title = {Quantitative disorder effects in low-dimensional spin systems},
journal = {Comm. Math. Phys.},
volume = {405},
number = {9},
year = {2024},
pages = {Paper No. 212},
doi = {10.1007/s00220-024-05081-9},
eprint = {2101.01711},
archivePrefix = {arXiv},
}
@article{arguin2010,
author = {Arguin, Louis-Pierre and Damron, Michael and Newman, C. M. and Stein, D. L.},
title = {Uniqueness of ground states for short-range spin glasses in the half-plane},
journal = {Comm. Math. Phys.},
volume = {300},
number = {3},
year = {2010},
pages = {641--657},
doi = {10.1007/s00220-010-1130-8},
eprint = {0911.4201},
archivePrefix = {arXiv},
}
@article{arguin2014,
author = {Arguin, Louis-Pierre and Damron, Michael},
title = {On the number of ground states of the {E}dwards--{A}nderson spin glass model},
journal = {Ann. Inst. Henri Poincar{\'e} Probab. Stat.},
volume = {50},
number = {1},
year = {2014},
pages = {28--62},
doi = {10.1214/12-AIHP499},
eprint = {1110.6913},
archivePrefix = {arXiv},
}
@article{newman2026,
author = {Newman, C. M. and Stein, D. L.},
title = {Ground state excitations and energy fluctuations in short-range spin glasses},
journal = {J. Stat. Phys.},
volume = {193},
number = {6},
year = {2026},
pages = {Paper No. 70},
doi = {10.1007/s10955-026-03631-0},
eprint = {2510.27507},
archivePrefix = {arXiv},
}
@misc{aim-planar-spin-glass-ground-states,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Uniqueness of the planar spin-glass ground-state pair}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/189-planar-spin-glass-ground-states.md},
note = {Entry 189 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-00460, "Uniqueness of the planar spin-glass ground-state pair", clopen.solutions.