OpenAI · 223-02 · 2026-09-23 · Probability and statistical mechanics

Self-dual random-cluster interfaces below one

preprints/Self-dual-random-cluster-interfaces-below-one-September-23-2026

Family 223. Conformal limits of square-lattice random-cluster interfaces.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Proves chordal $SLE_{κ}$ limits for critical square-lattice random-cluster interfaces for $0 < q \leq 4$ , with $κ = 4 π / arccos (- q /2)$ : bounded Jordan domains are allowed for $q \geq 1$ , and smooth Jordan domains for $q < 1$ , under the stated marked-boundary approximations. For $1 \leq q \leq 4$ , complete nested plane loops converge to $CLE_{κ}$ .

For every fixed $0 < q < 1$ , we prove that the Dobrushin interface of the square-lattice random-cluster model at its self-dual parameter converges to chordal SLEκ, where $κ = 4 π / arccos (- q /2) \in (6, 8)$ . 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.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.