Exponential excursion tails for topology-preserving Ising disks
For $z \in \mathbb{Z}^2$ let $Q_z = z + [-1/2, 1/2]^2$. A finite collection $C$ of these squares is a clump if its union $|C|$ is contractible and regular: every $x \in |C|$ is an interior point or has $(B_\varepsilon(x) \cap |C|) \setminus \{x\}$ contractible for all sufficiently small $\varepsilon > 0$. Pin $Q_0$. A removal of $Q \ne Q_0$ is allowed if $C \setminus \{Q\}$ is a clump and $|C|$ strongly deformation retracts onto $|C \setminus \{Q\}|$ through a homotopy $H$ with $H(Q \times [0,1]) \subseteq Q$; addition is the inverse move. Start the continuous-time chain $C_t$ at $\{Q_0\}$, with rate $\beta$ for each allowed addition and rate $1$ for each allowed removal. Let $a_n$ be the number of square-lattice self-avoiding polygons of enclosed area $n$, up to translation, and $\kappa = \lim_{n \to \infty} a_n^{1/n}$ their area growth constant. Let $\zeta$ be the first return time to $\{Q_0\}$ after the chain has left that state, and $$\xi = \sup_{0 \le t < \zeta} \#C_t,$$ where $\#C$ is the number of squares of $C$. Baryshnikov and Onaran proved that for $0 < \beta < 1/\kappa$ the chain is ergodic, so that $\mu = \mathbb{E}\zeta \in (0, \infty)$, and that $\eta_T = \mu/\mathbb{P}(\xi \ge T)$ satisfies $\liminf_{T \to \infty} \eta_T^{1/T} \ge 1/(\kappa\beta)$.
Problem. Is $$\lim_{T \to \infty,\ T \in \mathbb{N}} \mathbb{P}(\xi \ge T)^{1/T} = \kappa\beta$$ for every $0 < \beta < 1/\kappa$? Equivalently, is $\lim_{T \to \infty} \eta_T^{1/T} = 1/(\kappa\beta)$?
Source: Baryshnikov and Onaran (2026), Conjecture 7.11.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AT, CO
- Keywords
- AIM Open Applied Problems, rare events, self-avoiding polygons, topology-preserving dynamics
References
@article{baryshnikov2026,
author = {Baryshnikov, Yuliy and Onaran, Efe},
title = {Ising disks: topology preserving {G}lauber dynamics},
journal = {J. Appl. Comput. Topol.},
volume = {10},
number = {3},
year = {2026},
pages = {16},
doi = {10.1007/s41468-026-00248-w},
eprint = {2410.22611},
archivePrefix = {arXiv},
}
@misc{aim-ising-disks-excursion-tail,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Exponential excursion tails for topology-preserving Ising disks}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/504-ising-disks-excursion-tail.md},
note = {Entry 504 at commit 24edaad, 2026-10-15},
}
Resolution claims
We determine the exponential rate at which a topology-preserving planar Ising disk reaches a large area during one excursion from a single square. For addition rate $\beta$, deletion rate $1$, and every $0<\beta<1/\kappa$, the excursion maximum $\xi$ satisfies $\lim_{T\to\infty}\Prob(\xi\ge T)^{1/T}=\kappa\beta$, where $\kappa$ is the area growth constant of square-lattice self-avoiding polygons. This proves Conjecture 7.11 of Baryshnikov and Onaran. The …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/ising_disks_excursion.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00804, "Exponential excursion tails for topology-preserving Ising disks", clopen.solutions.