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Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four

OpenAI · September 24, 2026

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Abstract

We prove the finite-volume continuum-random-tree prediction for critical Fortuin–Kasteleyn planar maps at every fixed q > 4. After rescaling graph distances by a constant times n−1/2, an n-edge map with normalized degree measure converges to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.

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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.

Formalization

  • OpenAI, The geometric phase diagram, diffusion, and spectra of random planar maps: Lean scope (lean/docs/211.md, OpenAI Math Release, 6 October 2026). Describes a Lean formalization of this paper's finite-volume Gromov–Hausdorff–Prokhorov CRT limit for each fixed $q>4$, with normalized degree measure and all positive integer sizes; it does not cover the other six papers of the family.
  • Comparator statement FKCRT.lean (lean/ComparatorChallenges/FKCRT.lean). Comparator statement OAI.FKCRT.finite_fk_maps_converge_to_brownian_crt : MainStatement for the main theorem; in this challenge file the proof is left as sorry.

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Cite

@misc{OAI:Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026,
  author = {{OpenAI}},
  title = {{Brownian continuum random tree limits of finite Fortuin--Kasteleyn maps above four}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026/main.pdf}{OAI:Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-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: