Positive-temperature order in the three-dimensional Edwards–Anderson spin glass
Let $\Lambda_L = \{1, \dots, L\}^3 \subset \mathbb{Z}^3$, and give each nearest-neighbour edge $\{x, y\} \subset \Lambda_L$ an independent standard Gaussian coupling $J_{\{x,y\}}$. The Edwards–Anderson model with free boundary condition and zero field is the Gibbs measure $$\mu_{L,\beta,J}(\sigma) = \frac{1}{Z_{L,\beta,J}} \exp\Bigl(\beta \sum_{\{x,y\} \subset \Lambda_L,\ |x - y|_1 = 1} J_{\{x,y\}} \sigma_x \sigma_y\Bigr), \qquad \sigma \in \{-1, 1\}^{\Lambda_L}.$$ Given $J$, let $\sigma^1, \sigma^2$ be independent samples from $\mu_{L,\beta,J}$ and $R_L = L^{-3} \sum_{x \in \Lambda_L} \sigma^1_x \sigma^2_x$ their overlap.
Itoi, Mukaida and Tasaki proved that the square root of the left-hand side below is a lower bound for the Edwards–Anderson order parameter, so a positive value implies a positive Edwards–Anderson order parameter (2024).
Problem. Does there exist $0 < \beta_0 < \infty$ such that, for every fixed $\beta > \beta_0$, $$\limsup_{L \to \infty} \mathbb{E}_J\Bigl[\mathbb{E}_{\mu_{L,\beta,J}^{\otimes 2}} R_L^2\Bigr] > 0?$$
Source: The expected low-temperature spin-glass order of the Edwards–Anderson model (Edwards and Anderson, 1975) in dimension d ≥ 3, with the order parameter of Itoi, Mukaida and Tasaki (2024), §3.1, Eq. (3.3); see also Newman and Stein (1998), Question 1. Stated by AIM — Open Applied Problems for d = 3, Gaussian couplings and free boundary conditions.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Edwards–Anderson model, replica overlaps, spin glasses
References
@article{itoi2024,
author = {Itoi, Chigak and Mukaida, Hisamitsu and Tasaki, Hal},
title = {Griffiths-type theorems for short-range spin glass models},
journal = {J. Stat. Phys.},
volume = {191},
number = {2},
year = {2024},
pages = {28},
doi = {10.1007/s10955-024-03246-3},
eprint = {2310.04775},
archivePrefix = {arXiv},
}
@article{edwards1975,
author = {Edwards, S. F. and Anderson, P. W.},
title = {Theory of spin glasses},
journal = {J. Phys. F: Met. Phys.},
volume = {5},
number = {5},
year = {1975},
pages = {965--974},
doi = {10.1088/0305-4608/5/5/017},
}
@misc{newman1998,
author = {Newman, C. M. and Stein, D. L.},
title = {Short-range spin glasses},
howpublished = {IAMP open problems in mathematical physics, contribution of 18 September 1998},
year = {1998},
url = {https://web.math.princeton.edu/~aizenman/OpenProblems_MathPhys/9803.SpinGlass.html},
}
@misc{aim-ea-three-dimensional-order,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Positive-temperature order in the three-dimensional Edwards–Anderson spin glass}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/296-ea-three-dimensional-order.md},
note = {Entry 296 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-00427, "Positive-temperature order in the three-dimensional Edwards–Anderson spin glass", clopen.solutions.