Does every strict initial majority win zero-temperature Ising dynamics?
Fix an integer $d \ge 2$ and $p \in (1/2, 1]$. At time $0$ the spins $\sigma_0(x)$, $x \in \mathbb{Z}^d$, are independent, equal to $+1$ with probability $p$ and to $-1$ otherwise. Each site carries an independent rate-one Poisson clock; when the clock at $x$ rings, $\sigma(x)$ is set to the strict majority value among its $2d$ nearest neighbours, and to the value of an independent fair coin in case of a tie. This is zero-temperature Glauber dynamics for the Ising model on $\mathbb{Z}^d$.
Morris calls the assertion that the critical density for fixation at $+1$ is $1/2$ in every dimension $d \ge 2$ a folklore conjecture. By symmetry and ergodicity, fixation at $+1$ has probability $0$ when $p = 1/2$, and Nanda, Newman and Stein proved that for $p = 1/2$ on $\mathbb{Z}^2$ almost surely every site changes state infinitely often. Fontes, Schonmann and Sidoravicius proved that for every $d \ge 2$ there is $\varepsilon(d) > 0$ such that the system fixates at $+1$ almost surely when $p \ge 1 - \varepsilon(d)$. Morris proved that the infimum of the densities $p$ for which it fixates at $+1$ almost surely tends to $1/2$ as $d \to \infty$.
Problem. Is it true, for every such $d$ and $p$, that almost surely every site $x \in \mathbb{Z}^d$ has a finite random time $T_x$ with $\sigma_t(x) = +1$ for all $t \ge T_x$? No fixation time uniform over the lattice is required.
Source: Folklore, as the conjecture that the critical density for fixation is 1/2 in every dimension d ≥ 2; stated as Conjecture 1 by Morris (2011). Stated for every p ∈ (1/2, 1] by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, coarsening, Glauber dynamics, Ising models
References
@article{morris2011,
author = {Morris, Robert},
title = {Zero-temperature {G}lauber dynamics on {$\mathbb{Z}^d$}},
journal = {Probab. Theory Related Fields},
volume = {149},
number = {3-4},
year = {2011},
pages = {417--434},
doi = {10.1007/s00440-009-0259-x},
eprint = {0809.0353},
archivePrefix = {arXiv},
}
@article{fontes2002,
author = {Fontes, L. R. and Schonmann, R. H. and Sidoravicius, V.},
title = {Stretched exponential fixation in stochastic {I}sing models at zero temperature},
journal = {Comm. Math. Phys.},
volume = {228},
number = {3},
year = {2002},
pages = {495--518},
doi = {10.1007/s002200200658},
}
@incollection{nanda2000,
author = {Nanda, S. and Newman, C. M. and Stein, D. L.},
title = {Dynamics of {I}sing spin systems at zero temperature},
booktitle = {On Dobrushin's Way. From Probability Theory to Statistical Physics},
series = {Amer. Math. Soc. Transl. Ser. 2},
volume = {198},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2000},
pages = {183--194},
doi = {10.1090/trans2/198/13},
}
@misc{aim-majority-ising-fixation,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does every strict initial majority win zero-temperature Ising dynamics?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/289-majority-ising-fixation.md},
note = {Entry 289 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-00452, "Does every strict initial majority win zero-temperature Ising dynamics?", clopen.solutions.