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A Linear Clock for Random Walk on Tree-Weighted Planar Maps
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Abstract
We prove that stationary random walk on a planar map sampled with weight equal to its number of spanning trees converges to Liouville Brownian motion on the unit-area $\sqrt2$-Liouville quantum sphere. The convergence retains the conditional path law jointly with the measured metric space. The walk chooses uniformly among all incident half-edges and starts from the stationary degree measure. For total attempt rate one and the continuum Dirichlet form with factor 1/2, the time acceleration is exactly the number of map edges. The result uses the companion contour and metric limits for this same ensemble.
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Notes
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work in progress
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 Tutte-embedding convergence of spanning-tree-decorated Boltzmann maps with a large boundary to a √2-LQG disk, with random walk converging to Brownian motion modulo time parametrization; the author writes that her results "are close to “A linear clock for random walk on tree-weighted planar maps” by Open AI" and that the OpenAI papers solve several of her Future directions items 1, 2 and 4.
- 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 companion work on conformal embeddings of spanning-tree-weighted maps, which says "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]
Possibly relevant literature (found by the agent, not checked)
- Ewain Gwynne and Tom Hutchcroft, Anomalous diffusion of random walk on random planar maps (arXiv:1807.01512, PTRF). Proves subdiffusive displacement n^{1/d_γ+o(1)} for walks on γ-LQG maps including spanning-tree-weighted maps, background for the time-versus-space scaling behind the linear clock of Theorem 1.1; not in the paper's bibliography.
- Zhen-Qing Chen and Yang Yu, Discrete Approximation to Time-changed Brownian Motions (arXiv:2609.07789, 7 September 2026). Its general discrete approximation scheme includes Liouville Brownian motion and may provide a comparison for this paper's time-parametrized walk limit, although the abstract does not identify the finite spanning-tree-weighted map ensemble or its exact clock.
Digests and expositions
- Orply / Perplexity, Tree-Weighted Planar-Map Walks Converge After Exactly n-Fold Acceleration (Orply, 11 October 2026). Discusses the manuscript's exact clock convention, electrical-energy argument, Green-function estimate and retention of the conditional walk law. [AI-generated per source]
Elsewhere
- Emergent Mind: family 211
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026,
author = {{OpenAI}},
title = {{A Linear Clock for Random Walk on Tree-Weighted Planar Maps}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026/linear-clock-random-walk-tree-weighted-planar-maps.pdf}{OAI:A-Linear-Clock-for-Random-Walk-on-Tree-Weighted-Planar-Maps-October-5-2026}},
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
- Canonical conformal limits of subcritical FK planar maps · September 24, 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