Ergodicity of higher-dimensional Ising clumps
For $d \ge 2$ and $z \in \mathbb{Z}^d$ let $Q_z = z + [-1/2, 1/2]^d$. A finite collection $C$ of these cubes is a clump if its union $|C|$ is contractible and regular: each $x \in |C|$ is an interior point or has $(B_\varepsilon(x) \cap |C|) \setminus \{x\}$ contractible for every sufficiently small $\varepsilon > 0$. Let $\mathcal{C}_d$ be the set of all clumps containing the pinned cube $Q_0$, and $\#C$ the number of cubes of $C$. A cube $Q \ne Q_0$ may be removed from $C$ when $C \setminus \{Q\}$ is a clump and there is a strong deformation retraction $H : |C| \times [0,1] \to |C|$ onto $|C \setminus \{Q\}|$ with $H(Q \times [0,1]) \subseteq Q$; addition is the inverse of such a removal. Each allowed addition has rate $p_\uparrow \in (0, 1]$ and each allowed removal rate $p_\downarrow \in (0, 1]$, which defines a continuous-time Markov chain on $\mathcal{C}_d$. For $d = 2$, Baryshnikov and Onaran proved that this chain is irreducible on $\mathcal{C}_2$ and that, when $p_\uparrow/p_\downarrow < 1/\kappa$, with $\kappa$ the area growth constant of square-lattice self-avoiding polygons, it is ergodic with stationary law proportional to $(p_\uparrow/p_\downarrow)^{\#C}$.
Problem. Does the chain satisfy the following extension of the planar theorem: for every $d \ge 2$ it is irreducible on all of $\mathcal{C}_d$, and there is a finite positive constant $\kappa(d)$ such that, whenever $\beta = p_\uparrow/p_\downarrow < 1/\kappa(d)$, it is nonexplosive and ergodic with stationary law $$\Pi_\beta(C) = Z_d(\beta)^{-1} \beta^{\#C}, \qquad Z_d(\beta) = \sum_{C \in \mathcal{C}_d} \beta^{\#C} < \infty?$$ No sharp threshold or mixing-time rate is asked for.
Source: Baryshnikov and Onaran (2026), Conjecture 6.9, that their planar Theorem 6.5 holds for all d ≥ 2 with constants κ(d). Stated by AIM — Open Applied Problems with irreducibility on all clumps containing the pinned cube.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- AT, CO
- Keywords
- AIM Open Applied Problems, cubical sets, Markov chains, 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-higher-dimensional-ising-clumps,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Ergodicity of higher-dimensional Ising clumps}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/500-higher-dimensional-ising-clumps.md},
note = {Entry 500 at commit 24edaad, 2026-10-15},
}
Resolution claims
We consider the topology-preserving cube dynamics in the higher-dimensional extension of the Ising-disk model. The states are all finite regular contractible unions of lattice cubes containing a fixed cube. A removal must admit a strong deformation retraction that keeps the removed cube inside itself, and additions are the inverse moves. In four dimensions, every state is a piecewise-linear manifold with …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: AIM @ 24edaad, paper/cube_dynamics.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00799, "Ergodicity of higher-dimensional Ising clumps", clopen.solutions.