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Natural Occupation Measures for Critical Square-Lattice FK Interfaces
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Abstract
We prove that the rescaled counting measure of a critical square-lattice Fortuin–Kasteleyn Dobrushin interface in the unit square converges to the Minkowski-content measure of its Schramm–Loewner limit, for every fixed cluster weight $1\le q\lt 4$. A single deterministic constant times the predicted power of the mesh gives the normalization. Convergence is joint with the ordered curve and includes the total mass, giving the scaling limit of the interface's total number of steps.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Comments and discussion
- Jim Jardine (Lattice24), openai-math-audit: Independent audit of OpenAI's openai/math release (GitHub repository, audit status dated 2026-10-16). Its "Finding 1a" lists this paper as inheriting the gap claimed in the September 23 paper for 1 ≤ q < 4. The stated reason is that this paper's Inputs 2.1 and 2.4 (the chordal trace and plane-loop results) come from that paper, while its Theorem 1 is stated without conditions. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- Dapeng Zhan, An Exact Hausdorff Measure as Natural Length for SLE (arXiv:2610.15066, October 2026). Using the separate OpenAI exact-gauge theorem, identifies an SLE Hausdorff measure with natural occupation measure, relevant to the continuum measure targeted by this manuscript.
Digests and expositions
- Latent Seminar, Random-cluster curves: conformal limits and natural time (6 October 2026). Explains the claimed counting-measure replacement and distinction between trace convergence and natural time. [family-level]
Elsewhere
- Emergent Mind: family 223
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026,
author = {{OpenAI}},
title = {{Natural Occupation Measures for Critical Square-Lattice FK Interfaces}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026/natural-occupation-measures-critical-square-lattice-fk-interfaces.pdf}{OAI:Natural-Occupation-Measures-for-Critical-Square-Lattice-FK-Interfaces-October-5-2026}},
year = {2026}
}Family 223: Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling
Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for $0\lt q\le4$, with $\kappa=4\pi/\arccos(-\sqrt q/2)$: bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For $1\le q\le4$, complete nested plane loops converge to CLEκ.
Other papers in this family:
- Square-lattice FK interfaces and nested loops for 1 <= q < 4 · September 23, 2026
- Self-dual random-cluster interfaces below one · September 23, 2026
- Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces · October 5, 2026
- Thermal FK–Ising interfaces and massive SLE · October 5, 2026
- Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces · October 5, 2026