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Metric-measure limits of subcritical FK and spanning-tree planar maps

OpenAI · September 24, 2026

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Abstract

We resolve the finite spherical cases of Gwynne and Miller's graph-metric conjecture for critical Fortuin–Kasteleyn maps at each fixed $q\in(0,4)$ and for uniform spanning-tree-decorated maps. After deterministic rescaling of graph distances, these maps, equipped with the vertex probability measure proportional to degree, converge in Gromov–Hausdorff–Prokhorov law to their ordinary unit-area Liouville quantum gravity spheres. Distances use every primal edge, and convergence holds through all positive integer edge counts.

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Cite

@misc{OAI:Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026,
  author = {{OpenAI}},
  title = {{Metric-measure limits of subcritical FK and spanning-tree planar maps}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026/main.pdf}{OAI:Metric-measure-limits-of-subcritical-FK-and-spanning-tree-planar-maps-September-24-2026}},
  year = {2026}
}

Lean formalization

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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.

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