Scaling limits of critical Fortuin–Kasteleyn planar maps

Critical Fortuin–Kasteleyn (FK) planar maps are random planar maps weighted by the self-dual critical FK random-cluster model with parameter q > 0. Their predicted large-scale geometry changes at q = 4. For 0 < q < 4 they should converge to Liouville quantum gravity (LQG) spheres decorated by conformal loop ensembles. For q > 4, Sheffield predicted that they converge to the Brownian continuum random tree (CRT). Seven manuscripts in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September and 5 October 2026, make claims across this picture. Their claims cover metric, conformal and tree limits of the maps, and the convergence of random walks on them. This page summarizes the published claims and lists publicly available follow-up work.

Background

As recounted in the manuscripts' introductions:
- FK maps and inventories. On a random planar map the cluster weight also changes the underlying geometry. Sheffield's inventory (hamburger–cheeseburger) model encodes the joint randomness through a word in burgers and orders, with parameter p = √q/(√q + 2). The regime q > 4 is exactly p > 1/2, where the limiting inventory fluctuations are one-dimensional (Sheffield, Quantum gravity and inventory accumulation, Ann. Probab. 44 (2016)).
- The CRT prediction. The finite-volume CRT prediction for q > 4 appears in Sheffield's appendix. Feng restates it as Conjecture 1.2 of Triviality of critical Fortuin–Kasteleyn decorated planar maps for q > 4 (Probab. Theory Related Fields 194 (2026), arXiv:2311.06235), and proves an infinite-volume counterpart in the local Gromov–Hausdorff–Prokhorov topology.
- The LQG regime. Gwynne and Miller's graph-metric conjecture includes FK-weighted maps for γ ∈ (√2, 2) and spanning-tree-weighted maps at γ = √2. Interfaces of FK-decorated maps are predicted to converge to a conformal loop ensemble independent of the LQG surface. A general conformal-embedding universality conjecture is recorded as Conjecture 3.5 of Holden and Sun's survey Liouville quantum gravity: from random planar maps to conformal field theory (arXiv:2510.16431v1).
- Combinatorial ancestry. The finite-map encodings go back to Mullin's enumeration of tree-rooted maps and Bernardi's bijections; the block decomposition used for q > 4 goes back to Tutte.

The original results

q > 4: the CRT limit (24 September 2026). The manuscript Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four claims that for every fixed q > 4, after rescaling graph distances by a constant times n^−1/2, an n-edge critical FK map with normalized degree measure converges to the Brownian CRT in the Gromov–Hausdorff–Prokhorov topology, through all positive integer sizes. The manuscript states that this realizes Sheffield's finite-volume prediction in the compact topology, proved directly under the finite law rather than deduced from Feng's infinite-map limit.
0 < q < 4: metric-measure limits (24 September 2026). Metric-measure limits of subcritical FK and spanning-tree planar maps claims that, after deterministic rescaling of graph distances and with the degree measure, these maps converge in Gromov–Hausdorff–Prokhorov law to the unit-area LQG sphere, for each fixed q ∈ (0, 4) and for uniform spanning-tree-decorated maps. The manuscript states that this resolves the finite spherical cases of Gwynne and Miller's graph-metric conjecture (their Conjecture 1.7), and that it does not address every ensemble or version of that conjecture.
0 < q < 4: conformal limits (24 September 2026). Canonical conformal limits of subcritical FK planar maps claims joint convergence of spherical FK maps in their flag-triangle uniformization to the unit-area LQG sphere decorated by an independent conformal loop ensemble: area measure, rescaled graph distances and the full nested interface collection. The manuscript describes this as establishing the limit for this model and embedding within the broader universality prediction.
q = 4 (24 September 2026). The critical Liouville quantum sphere and geometric limits of FK maps at q=4 constructs the unit-area critical Liouville quantum sphere, with its intrinsic metric, as a limit of subcritical quantum spheres. It then claims joint convergence of FK maps at q = 4, in their equilateral flag uniformizations, to this sphere decorated by an independent nested CLE₄.
Random walks (5 October 2026). Three manuscripts treat random walks:
- Spectral convergence for critical FK–Ising planar maps: for critical spherical FK–Ising maps (q = 2), with time accelerated by the number of edges, joint convergence of the metric-measure space, all ordered eigenvalues and the heat trace to those of Liouville Brownian motion on the √3-quantum sphere. It states that it uses the conformal and metric-measure companion results and "the stated Brownian/Liouville quantum gravity inputs".
- Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion: stationary random walks on these maps converge to Liouville Brownian motion on the √3-quantum sphere, using the spectral companion's geometric and electrical results.
- A Linear Clock for Random Walk on Tree-Weighted Planar Maps: stationary random walk on spanning-tree-weighted planar maps converges to Liouville Brownian motion on the √2-quantum sphere, with time acceleration exactly the number of edges, using the companion contour and metric limits for the same ensemble.
Sources. Commit 1581d4c of the openai/math repository:
Formalization, as described by the repository.
- The scope note lean/docs/211.md links only the q > 4 manuscript. The formalized result is the Brownian CRT limit for critical finite FK planar maps for every fixed q > 4, with distances rescaled by a constant depending on q times n^−1/2 and normalized degree measure, in the Gromov–Hausdorff–Prokhorov topology through all positive integer sizes.
- The Comparator challenge FKCRT.lean states finite_fk_maps_converge_to_brownian_crt: for every q > 4 there is c > 0 such that, for every bounded Gromov–Hausdorff–Prokhorov-continuous test function, the expectations under the finite FK map law converge to those under the Brownian CRT, which is coded through three independent Brownian bridges.
- The LQG, conformal, q = 4 and random-walk results are not among the selected statements.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
2026-10-07	N. Holden and P. Yu, arXiv:2610.10505v1 and arXiv:2610.10480v1; N. Holden, arXiv:2610.10501v1	Related (other embeddings)	Spanning-tree-weighted planar maps converge to √2-LQG under circle packing and Riemann uniformization, with the face path converging to SLE₈ (2610.10505); the mated-CRT map converges to LQG under the same embeddings (2610.10480); under the Tutte embedding, tree-weighted maps with a macroscopic boundary converge to the √2-LQG disk and simple random walk converges to planar Brownian motion modulo time parametrization (2610.10501).	arXiv preprints (v1).
2026-10-08	MathVet (GitHub; review of openai/math, tables at repository commit `c76f1e4`, 14 October 2026)	Review (fidelity of the formal statements)	Rates the family "partial": only the q > 4 Brownian-CRT limit is stated; the LQG limits for 0 < q ≤ 4, the Liouville Brownian motion limits of random walks and the FK–Ising spectral convergence are not. Records a sandboxed Comparator pass for `FKCRT`.	MathVet states that it does not claim any theorem false and reports what was machine-checked as stated.
2026-10-10 to 2026-10-14	con-leche development journal (task #348)	Verification (second kernel)	In its run over all Comparator challenges of the collection at commit `fd4aeeb`, the journal's table of exceptions (time-outs, fuel, memory, errors) does not list `FKCRT`. The challenge was therefore accepted by con-leche and by Lean's official kernel.	Development journal.
2026-10-10	cert-machine, openai/math lane K, run 38051470892	Verification (two kernels)	Records that Lean's default kernel and the nanoda kernel accept the `FKCRT` solution (652 s). The project reports two-kernel certification of 398 of 416 challenges of the collection.	Repository.

Notes

The kernel checks above concern the formal statement finite_fk_maps_converge_to_brownian_crt as written. The other six manuscripts' claims have no formal statement in the collection.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.