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Self-dual random-cluster interfaces below one
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Abstract
For every fixed $0\lt q\lt 1$, we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLEκ, where $\kappa=4\pi/\arccos(-\sqrt q/2)\in(6,8)$. The approximating domains are simple closed nearest-neighbor lattice polygons with distinct marked vertices, whose marked boundary parametrizations converge uniformly to those of a bounded smooth Jordan domain. Convergence holds along the full mesh sequence in the uniform metric on oriented curves modulo increasing reparametrization. The proof combines finite connection comparisons below one, localization in irregular tiled disks, and a boundary observable that determines the limiting Loewner driver.
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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). The audit reports a gap in the companion paper for 1 ≤ q < 4. It states that this paper uses that companion only in the corollary extending to 1 ≤ q < 4, and that the main theorem for 0 < q < 1 is unaffected. [AI-generated per source]
Digests and expositions
- Latent Seminar, Random-cluster curves: conformal limits and natural time (6 October 2026). Explains the claimed below-one comparison machinery and smooth-domain scope. [family-level]
Elsewhere
- Emergent Mind: family 223
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Self-dual-random-cluster-interfaces-below-one-September-23-2026,
author = {{OpenAI}},
title = {{Self-dual random-cluster interfaces below one}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026/paper.pdf}{OAI:Self-dual-random-cluster-interfaces-below-one-September-23-2026}},
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
- Quenched SLE Universality for Weakly Disordered FK–Ising Interfaces · October 5, 2026
- Thermal FK–Ising interfaces and massive SLE · October 5, 2026
- Natural Occupation Measures for Critical Square-Lattice FK Interfaces · October 5, 2026
- Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces · October 5, 2026