Does stronger ferromagnetic coupling always slow Ising relaxation?
Let $G = (V, E)$ be a finite simple graph with at least one vertex. For $J \in [0, \infty)^E$ let $\pi_J(\sigma) \propto \exp\bigl(\sum_{uv \in E} J_{uv} \sigma_u \sigma_v\bigr)$ on $\{-1, 1\}^V$, the ferromagnetic Ising model with zero external field, and consider the continuous-time heat-bath Glauber dynamics in which each vertex, at rate one, resamples its spin from its conditional law under $\pi_J$ given the other spins. Let $\gamma(J)$ be the smallest positive eigenvalue of the negative generator. Nacu proved that on a cycle the relaxation time $1/\gamma(J)$ is increasing in each coupling. Kargin proved, for the complete graph with all couplings equal, that the relaxation time is nondecreasing in the common coupling.
Problem. For every such $G$ and $J$, every edge $e$ and every $s \ge 0$, is $\gamma(J + s \mathbf{1}_e) \le \gamma(J)$?
Source: Peres (personal communication), as Conjecture 2 in Nacu (2003); also Levin and Peres (2017), Chapter 26, Question 2.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Glauber dynamics, Ising model, spectral gaps
References
@article{nacu2003,
author = {Nacu, {\c{S}}erban},
title = {Glauber dynamics on the cycle is monotone},
journal = {Probab. Theory Related Fields},
volume = {127},
number = {2},
year = {2003},
pages = {177--185},
doi = {10.1007/s00440-003-0279-x},
eprint = {math/0305056},
archivePrefix = {arXiv},
}
@book{levin2017,
author = {Levin, David A. and Peres, Yuval},
title = {Markov Chains and Mixing Times},
edition = {Second},
note = {With contributions by Elizabeth L. Wilmer},
publisher = {American Mathematical Society},
address = {Providence, RI},
year = {2017},
doi = {10.1090/mbk/107},
}
@article{kargin2011,
author = {Kargin, Vladislav},
title = {Relaxation time is monotone in temperature in the mean-field {I}sing model},
journal = {Statist. Probab. Lett.},
volume = {81},
number = {8},
year = {2011},
pages = {1094--1097},
doi = {10.1016/j.spl.2011.03.001},
eprint = {1103.0327},
archivePrefix = {arXiv},
}
@misc{aim-ising-gap-monotonicity,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Does stronger ferromagnetic coupling always slow Ising relaxation?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/266-ising-gap-monotonicity.md},
note = {Entry 266 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-00444, "Does stronger ferromagnetic coupling always slow Ising relaxation?", clopen.solutions.