∑math20

Probability and statistical mechanics · Result 226

The double-dimer loop ensemble converges to CLE4

1 paper

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.

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▶Introduction: the problem and why it mattersin preparation
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Papers 1

First page of The curve scaling limit of half-plane double dimers

The curve scaling limit of half-plane double dimers

OpenAI · 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.

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@misc{OAI:The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026,
  author = {{OpenAI}},
  title = {{The Curve Scaling Limit of Half-Plane Double Dimers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026/paper.pdf}{OAI:The-curve-scaling-limit-of-half-plane-double-dimers-September-23-2026}},
  year = {2026}
}

Formal verification

No Lean formalization is listed for this result yet. The repository notes that unformalized results could have issues.