Necessity of the ExpFat regularity assumption for circuit estimates

Determine whether the technical regularity assumption ExpFat on an s-embedding is necessary for the existence of FK-Ising circuits in annuli on scales R≥Cδ.

Background

The paper discusses an earlier RSW estimate for the FK-Ising model on s-embeddings established under the assumptions Lip and ExpFat. The Lip assumption expresses a large-scale Lipschitz condition on the associated s-surface and is described as conceptually necessary, whereas ExpFat is treated as a technical regularity condition on the s-embedding.

The authors note that, although ExpFat is used in the known proof, it is unresolved whether the annular circuit estimate itself remains valid without this assumption, at the natural scale range R≥Cδ. Resolving this would clarify whether the technical geometric regularity condition is genuinely needed for the existence of uniformly positive-probability circuits.