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Conformal Limits of Critical Square-Lattice Random-Cluster Interfaces

OpenAI · October 5, 2026

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Abstract

For every fixed $1\le q\le4$, critical square-lattice random-cluster Dobrushin interfaces converge to chordal $\mathop{\mathrm{SLE}}\nolimits _{\kappa(q)}$, where $\kappa(q)=4\pi/\arccos(-\sqrt q/2)$, and the complete nested plane loop collections converge to whole-plane $\mathop{\mathrm{CLE}}\nolimits _{\kappa(q)}$. The results hold under uniform marked Jordan boundary approximation and retain loop multiplicities and traversals. At q = 1 we obtain Cardy's formula for square-lattice bond percolation with free boundary edges.

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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's "Finding 1" claims a gap in Section 7.3 of the September 23 paper for 1 ≤ q < 4, and does not name this paper. The labels it cites (hol:strict-budget, hol:baseline) and the sentence it quotes also appear in this paper's sections/observable.tex. [AI-generated per source]

Possibly relevant literature (found by the agent, not checked)

Yu Zhang, A scaling limit of crossing probabilities for critical percolation on the square lattice (arXiv:2206.04599, 2022; revised 2024). The abstract states that Cardy's formula does not correctly describe square-lattice crossing probabilities in an equilateral triangle, which bears directly on this paper's Cardy theorem for square-lattice bond percolation; not cited in the paper.
Yu Zhang, Cardy's formula does not hold on some 2D lattices for critical two-dimensional percolation (arXiv:2309.05121, 2023). Argues that Cardy's formula fails on some square-type periodic lattices, which is relevant to the lattice and embedding conventions behind the Cardy theorem.
Mikhail Khristoforov and Stanislav Smirnov, Percolation and O(1) loop model (arXiv:2111.15612, 2021). Gives a new proof of Cardy's formula on the hexagonal lattice designed to generalize, and is relevant to the q = 1 Cardy statement.
Yu Feng, Eveliina Peltola and Hao Wu, Connection probabilities of multiple FK-Ising interfaces (Probability Theory and Related Fields 189, 281–367, 2024). Proves critical q = 2 multiple-interface and connection-probability limits and formulates general-q predictions relevant to this manuscript’s four-boundary-change calculation.

Digests and expositions

Latent Seminar, Random-cluster curves: conformal limits and natural time (6 October 2026). Discusses the separate Gaussian-free-field argument claimed at the endpoint q = 4. [family-level]

Elsewhere