Polynomial mixing of planar Ising dynamics with plus boundary
Fix $\beta > \beta_c = \frac{1}{2}\log(1 + \sqrt{2})$, the critical inverse temperature of the Ising model on $\mathbb{Z}^2$. On $\Lambda_n = \{1, \dots, n\}^2$ let $\mu^+_{n,\beta}$ be the Ising measure with plus boundary condition: the probability on $\{-1, 1\}^{\Lambda_n}$ proportional to $\exp(\beta \sum_{x \sim y} \sigma_x \sigma_y)$, the sum over nearest-neighbour edges meeting $\Lambda_n$, with $\sigma_y = +1$ for $y \notin \Lambda_n$. Run continuous-time heat-bath Glauber dynamics: each site of $\Lambda_n$, at rate one, resamples its spin from its conditional law under $\mu^+_{n,\beta}$ given the other spins. With $P_t$ its transition kernel, let $$t_{\mathrm{mix}}(n, \beta) = \inf\Bigl\{t : \max_\eta \bigl\|P_t(\eta, \cdot) - \mu^+_{n,\beta}\bigr\|_{\mathrm{TV}} \le 1/4\Bigr\}.$$
Martinelli proved $t_{\mathrm{mix}}(n,\beta) \le \exp(n^{1/2 + o(1)})$ at low enough temperature (1994), and Martinelli and Toninelli $t_{\mathrm{mix}}(n,\beta) \le \exp(c n^{\varepsilon})$ for every $\varepsilon > 0$ and large enough $\beta$ (2010). Lubetzky, Martinelli, Sly and Toninelli proved $t_{\mathrm{mix}}(n,\beta) \le n^{c \log n}$ for every $\beta > \beta_c$, with $c$ depending on $\beta$ (2013). At $\beta = \beta_c$, Lubetzky and Sly proved polynomial mixing for every boundary condition (2012).
Problem. For every fixed $\beta > \beta_c$, do finite constants $C_\beta, a_\beta$ exist with $t_{\mathrm{mix}}(n, \beta) \le C_\beta n^{a_\beta}$ for all $n \ge 1$?
Source: Levin, Peres and Wilmer, Markov Chains and Mixing Times (2009), open problem on the Ising model with positive boundary conditions, at every temperature; 2nd ed. (2017), §26.1, Question 1. Stated by AIM — Open Applied Problems for continuous-time heat-bath dynamics at β > β_c.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Glauber dynamics, Ising model, mixing times
References
@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{martinelli1994,
author = {Martinelli, F.},
title = {On the two-dimensional dynamical {I}sing model in the phase coexistence region},
journal = {J. Stat. Phys.},
volume = {76},
number = {5-6},
year = {1994},
pages = {1179--1246},
doi = {10.1007/BF02187060},
}
@article{martinelli2010,
author = {Martinelli, Fabio and Toninelli, Fabio Lucio},
title = {On the mixing time of the {2D} stochastic {I}sing model with ``plus'' boundary conditions at low temperature},
journal = {Comm. Math. Phys.},
volume = {296},
number = {1},
year = {2010},
pages = {175--213},
doi = {10.1007/s00220-009-0963-5},
eprint = {0905.3040},
archivePrefix = {arXiv},
}
@article{lubetzky2013,
author = {Lubetzky, Eyal and Martinelli, Fabio and Sly, Allan and Toninelli, Fabio Lucio},
title = {Quasi-polynomial mixing of the {2D} stochastic {I}sing model with ``plus'' boundary up to criticality},
journal = {J. Eur. Math. Soc.},
volume = {15},
number = {2},
year = {2013},
pages = {339--386},
doi = {10.4171/JEMS/363},
eprint = {1012.1271},
archivePrefix = {arXiv},
}
@article{lubetzky2012,
author = {Lubetzky, Eyal and Sly, Allan},
title = {Critical {I}sing on the square lattice mixes in polynomial time},
journal = {Comm. Math. Phys.},
volume = {313},
number = {3},
year = {2012},
pages = {815--836},
doi = {10.1007/s00220-012-1460-9},
eprint = {1001.1613},
archivePrefix = {arXiv},
}
@misc{aim-ising-plus-boundary-polynomial-mixing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Polynomial mixing of planar Ising dynamics with plus boundary}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/192-ising-plus-boundary-polynomial-mixing.md},
note = {Entry 192 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-00445, "Polynomial mixing of planar Ising dynamics with plus boundary", clopen.solutions.