Ising Disks: Topology Preserving Glauber Dynamics

We introduce a dynamic model where the state space is the set of contractible cubical sets in the Euclidian space. The permissible state transitions, that is addition and removal of a cube to/from the set, are closest to Eden model with topological constraints, and, we show, are locally decidable. We prove that in the planar special case the state space is connected. We then define a continuous time Markov chain with a fugacity (tendency to grow) parameter. Using the correspondence between our model on the plane and the self-avoiding polygons, we prove that the Markov chain is irreducible (due to state connectivity), and is also ergodic if the fugacity is smaller than a threshold.

💡 Research Summary

The paper introduces a novel stochastic growth model in which the state space consists of contractible cubical sets—called “clumps”—embedded in the integer lattice ℤⁿ. Each clump is a finite collection of axis‑aligned unit cubes whose geometric realization is both contractible (topologically a ball) and regular, meaning that every point is either an interior point or, after removing the point, the remaining neighborhood is contractible. This regularity condition is crucial because it allows the authors to decide locally whether a proposed addition or removal of a single cube will preserve the global topology.

The authors first formalize the notion of a “community” N_X(X), the union of a cube X together with all its face‑adjacent cubes. They prove that the community is always contractible and that its outer perimeter contains no irregular points (Lemma 3.8). Consequently, checking whether a local move (adding or deleting a cube) keeps the configuration a clump can be done by inspecting only the immediate neighborhood of the affected cube. This establishes the “local decidability” of the dynamics.

A central combinatorial concept is k‑indentability. A clump X is k‑indentable if there exist k distinct cubes X₁,…,X_k such that removing any one of them yields another clump, and moreover there exists a strong deformation retraction from |X| onto |X\X_i| that stays within the removed cube until the final moment. Lemma 4.3 shows that indentability is a property of the community: X is indentable through a cube X_i if and only if its community N_X(X) is indentable through the same cube. This symmetry permits the definition of the inverse operation, “expansion,” where a new cube Y is added provided the enlarged set X∪Y is indentable through Y.

In the planar case (d = 2) the authors prove a powerful structural theorem (Theorem 5.1): every planar clump possesses at least two indentable cubes. This implies that the graph whose vertices are clumps and whose edges correspond to a single admissible addition or removal is connected. Hence any two planar clumps can be transformed into one another through a finite sequence of local moves that never violate the topological constraint.

Building on this connectivity, the paper defines a continuous‑time Markov chain on the space of clumps. Each existing cube independently attempts removal at rate μ, while each admissible neighboring empty site attempts addition at rate λ. The ratio ρ = λ/μ is called the fugacity. Because the state space is connected, the chain is irreducible. The authors then establish ergodicity for fugacity below a critical threshold κ*. The threshold is identified via a bijection between planar clumps and self‑avoiding polygons (SAPs): the boundary of a planar clump is a SAP, and the number of clumps of a given perimeter grows like μ_A = c · A^{−α} · γ^{A}, where γ is the SAP growth constant. When ρ < κ* = γ^{−1}, the addition pressure is insufficient to overcome the entropic cost of creating new perimeter, and the chain admits a unique stationary distribution with finite expected volume. For ρ > κ*, the model enters an “over‑critical” regime where clumps can grow without bound, forming coral‑like structures reminiscent of vesicles.

The authors further analyze rare‑event statistics. They show that the times at which unusually large clumps appear converge, after appropriate scaling, to a Poisson point process. This result parallels classical large‑deviation results for self‑avoiding walks and provides a probabilistic description of the intermittent bursts of growth in the over‑critical regime.

A notable discussion concerns the relationship to Pacch’s animal problem, which asks whether any lattice animal (a connected set homeomorphic to a ball) can be reduced to a single cube via a sequence of admissible deletions while remaining an animal at each step. In two dimensions the paper’s Theorem 5.1 gives an affirmative answer for the stronger class of clumps. In three dimensions the problem remains open; the authors suggest that understanding whether their notion of clumps coincides with Pacch’s animals could shed new light on this long‑standing question.

Finally, the paper situates its contributions within broader research on stochastic growth models (Eden, DLA, ballistic deposition, etc.), emphasizing that unlike those models, which typically ignore global topology, the present framework enforces a global topological invariant through purely local updates. The authors outline several avenues for future work: precise determination of κ* in higher dimensions, quantitative shape analysis of typical clumps (e.g., average width, growth speed), dynamics starting from half‑space initial conditions (interface fluctuations), and extending the theory to three‑dimensional lattices to address Pacch’s animal problem. Overall, the work provides a rigorous foundation for studying topologically constrained stochastic dynamics, bridging combinatorial topology, statistical physics, and Markov chain theory.