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OpenAI · 211-07 · 2026-09-24 · Probability and statistical mechanics

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

preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026

Family 211. Geometry, diffusion, and spectra of random planar maps.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for and to the Brownian continuum random tree for , 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.

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.

Preprint source and Lean scope

The abstract above is the catalogue abstract. The sections below are the full preprint source, 0 sections. This page is not only the abstract. The manuscript file is loading.

This preprint is named in the Lean scope note for 211: The geometric phase diagram, diffusion, and spectra of random planar maps. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.

FKCRT.lean · FKCRT.lean

Statement file

  • Emergence
    A continuum random tree is not spacetime
    Family 211 records finite-volume metric limits of critical Fortuin–Kasteleyn planar maps. For q>4 the stated limit is the Brownian continuum random tree. That metric-measure statement is not a quantum surface, not a Page curve, and not spacetime emergence. Oai211-A marks a conditioned block tree and does not verify the preprint. Hausdorff dimension two is not manifold dimension. The authors are OpenAI. The companion does not verify those proofs. O.1 stays NOT_CLOSED.
    EmergenceOai211-A path-sum certificateOai290-A typed recoveryEn07-C Hawking comparison