The double-dimer loop ensemble converges to \mathrm{CLE}_{4}

Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE₄. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.

Manuscript

The curve scaling limit of half-plane double dimers

September 23, 2026 · source · cite as \cite{OAI:The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026}

We prove that the complete double-dimer loop ensemble for the Temperleyan square lattice in the upper half-plane converges to nested CLE₄. The convergence holds along the full mesh limit and matches every macroscopic loop as an unparametrized curve. This resolves the half-plane Temperleyan form of the double-dimer CLE₄ scaling-limit conjecture.

Result 226 · Probability and statistical mechanics. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.