Polynomial equilibration at the four-state planar Potts critical point
On the torus $V_n = (\mathbb{Z}/n\mathbb{Z})^2$, $n \ge 3$, with nearest-neighbour edge set $E_n$, let $\pi_n$ be the four-state Potts measure at its critical inverse temperature $\log 3$, $$\pi_n(\sigma) \propto \exp\Bigl((\log 3)\sum_{xy \in E_n} \mathbf{1}_{\sigma_x = \sigma_y}\Bigr), \qquad \sigma \in \{1, 2, 3, 4\}^{V_n}.$$ In heat-bath Glauber dynamics each vertex, independently at rate one, resamples its colour from its conditional law under $\pi_n$ given the other colours. Let $$t_{\mathrm{mix}}(n) = \inf\Bigl\{t : \max_\sigma \|\mathcal{L}_\sigma(\sigma_t) - \pi_n\|_{\mathrm{TV}} \le 1/4\Bigr\},$$ where $\mathcal{L}_\sigma(\sigma_t)$ is the law at time $t$ of the dynamics started from $\sigma$. Lubetzky and Sly proved a polynomial bound at criticality for the Ising model ($q = 2$). Gheissari and Lubetzky proved for the critical $q$-state Potts model on this torus that the inverse spectral gap of Glauber dynamics is at most polynomial in $n$ for $q = 3$, at most $n^{O(\log n)}$ for $q = 4$, and exponential in $n$ for $q > 4$ in the phase-coexistence regime.
Problem. Are there finite constants $C, a > 0$ such that $t_{\mathrm{mix}}(n) \le C n^a$ for every $n$?
Source: Gheissari and Lubetzky (2018), abstract and Introduction: the inverse spectral gap of critical Glauber dynamics on the torus is expected to be polynomial in n for 1 < q ≤ 4 (critical slowdown). Stated by AIM — Open Applied Problems for the total-variation mixing time at q = 4.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Glauber dynamics, mixing times, Potts model
References
@article{gheissari2018,
author = {Gheissari, Reza and Lubetzky, Eyal},
title = {Mixing times of critical two-dimensional {P}otts models},
journal = {Comm. Pure Appl. Math.},
volume = {71},
number = {5},
year = {2018},
pages = {994--1046},
doi = {10.1002/cpa.21718},
eprint = {1607.02182},
archivePrefix = {arXiv},
}
@article{gheissari2020,
author = {Gheissari, Reza and Lubetzky, Eyal},
title = {Quasi-polynomial mixing of critical two-dimensional random cluster models},
journal = {Random Structures Algorithms},
volume = {56},
number = {2},
year = {2020},
pages = {517--556},
doi = {10.1002/rsa.20868},
eprint = {1611.01147},
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},
}
@misc{aim-critical-four-state-potts-mixing,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Polynomial equilibration at the four-state planar Potts critical point}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/288-critical-four-state-potts-mixing.md},
note = {Entry 288 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-00419, "Polynomial equilibration at the four-state planar Potts critical point", clopen.solutions.