Browse › Probability and statistical mechanics › 226 The double-dimer loop ensemble converges to CLE4
The curve scaling limit of half-plane double dimers
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Abstract
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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Possibly relevant literature (found by the agent, not checked)
- O. Schramm and S. Sheffield, Contour lines of the two-dimensional discrete Gaussian free field (arXiv:math/0605337, 2006). Proves discrete GFF level lines converge to SLE(4), the GFF/level-line picture the introduction invokes as motivation and in the Kenyon–Wilson comparison; it is not in the paper's bibliography.
- A. Kemppainen and S. Smirnov, Random curves, scaling limits and Loewner evolutions (arXiv:1212.6215, 2012). A crossing-estimate framework for precompactness of random curves, which can be compared with the paper's Aizenman–Burchard-based compactness step (Section 4).
- M. Lis, L. Rey and K. Ryan, Conformally invariant boundary arcs in double dimers (arXiv:2409.18015, v2 2026-01-09). Conformally invariant limits of arc statistics in double-dimer variants with boundary arcs, related to the Kenyon–Wilson boundary-pairing results the introduction cites as evidence.
- Nahid Ghodratipour and Shahin Rouhani, Connection probabilities in the double-dimer model -- the case of two connectivity patterns (arXiv:1908.07595, 20 August 2019). Computes rectangle boundary connection probabilities with alternating wired/free conditions, whose continuum limit matches the Kenyon–Wilson upper half-plane result; this relates to the introduction's boundary-pairing evidence for CLE₄, under boundary conditions different from the manuscript's half-plane Temperleyan law.
Digests and expositions
- Gradient Damage, Two random tilings make these loops (video, 11 October 2026). Illustrates the manuscript's double-dimer construction using two random tilings of a finite Temperleyan rectangle and explicitly distinguishes the animation from verification of the claimed half-plane scaling limit.
- heyprof, Double-dimer loops and CLE₄ (interactive demonstration, undated). Presents sampling via the Temperley bijection and Wilson's algorithm, and explains the distinction between earlier topological convergence and the curve convergence claimed in family 226.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 226
Independent sites; their content is not reviewed here.
Cite
@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}
}Family 226: The double-dimer loop ensemble converges to CLE4
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.
This is the only paper in this family.