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Canonical conformal limits of subcritical FK planar maps
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Abstract
For every fixed $0\lt q\lt 4$, we prove joint convergence of spherical Fortuin–Kasteleyn planar maps in their flag-triangle uniformization to the corresponding unit-area Liouville quantum gravity sphere decorated by an independent conformal loop ensemble. The convergence includes the area measure, deterministically rescaled graph distances between all vertex pairs, and the full nested interface collection, with interfaces converging uniformly up to reparameterization.
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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.
Follow-up works
- Nina Holden, The Tutte embedding of tree-weighted planar maps converges to Liouville quantum gravity (arXiv:2610.10501, 7 Oct 2026). Independent work proving that the Tutte embedding of spanning-tree-decorated Boltzmann maps with a large boundary converges to a √2-LQG disk; it names this paper as one the OpenAI linear-clock paper builds on, and says the OpenAI papers solve several of its open problems, including item 1 on other discrete conformal embeddings and models.
- Nina Holden and Pu Yu, The circle packing and Riemann uniformization embedding of tree-weighted planar maps converge to Liouville quantum gravity (arXiv:2610.10505, 7 Oct 2026). Independent work proving that spanning-tree-weighted maps (γ=√2) converge to LQG disk, sphere and cone under circle packing and Riemann uniformization, a statement of the same type for a different model and embedding; the authors say "none of the OpenAI results imply the results of our paper, or vice versa." [family-level]
Formalization
- Release Lean scope note for family 211 (lean/docs/211.md). The family's Lean formalization covers only the q>4 continuum-random-tree paper, not this one. [family-level]
Digests and expositions
- Orply / Perplexity, Subcritical FK Maps Converge Jointly in a Canonical Quantum-Sphere Coordinate (Orply, 11 October 2026). Explains flag uniformization, area normalization, all-vertex distance convergence and ordered nested interfaces, including the dependence on the intrinsic-metric companion. [AI-generated per source]
Elsewhere
- Emergent Mind: family 211
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026,
author = {{OpenAI}},
title = {{Canonical conformal limits of subcritical FK planar maps}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026/main.pdf}{OAI:Canonical-conformal-limits-of-subcritical-FK-planar-maps-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/211.md. The Lean statements have not been checked by people against the paper's claims.
Family 211: The geometric phase diagram, diffusion, and spectra of random planar maps
Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $0\lt q\le4$ and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.
Other papers in this family:
- Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion · October 5, 2026
- Spectral convergence for critical FK–Ising planar maps · October 5, 2026
- A Linear Clock for Random Walk on Tree-Weighted Planar Maps · October 5, 2026
- The critical Liouville quantum sphere and geometric limits of FK maps at q=4 · September 24, 2026
- Metric-measure limits of subcritical FK and spanning-tree planar maps · September 24, 2026
- Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four · September 24, 2026