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Spectral convergence for critical FK–Ising planar maps
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Abstract
Using the conformal and metric-measure companion results and the stated Brownian/Liouville quantum gravity inputs, we prove spectral convergence for critical spherical FK–Ising maps to Liouville Brownian motion on the ordinary unit-area $\sqrt3$-quantum sphere. The discrete walk has total attempt rate one, uses every map edge including loops and multiplicities, and has the corner measure as its stationary law. Accelerating time by the number of map edges gives joint convergence of the metric-measure space, all ordered eigenvalues with multiplicities and padding, and the heat trace locally uniformly at strictly positive times. The conductivity and clock constant are one in these conventions.
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Notes
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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
- 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)
- Nathanaël Berestycki, On the spectral geometry of Liouville quantum gravity (arXiv:2512.02538, Dec 2025). A survey of the spectrum of the Liouville Brownian motion generator (Weyl law, heat trace, open problems), related to the paper's eigenvalue and heat-trace limit H_h in Theorem 1.1; the bibliography cites Berestycki–Wong but not this survey.
Elsewhere
- Emergent Mind: family 211
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026,
author = {{OpenAI}},
title = {{Spectral convergence for critical FK--Ising planar maps}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Spectral-convergence-for-critical-FK-Ising-planar-maps-October-5-2026/spectral-convergence-critical-fk-ising-planar-maps.pdf}{OAI:Spectral-convergence-for-critical-FK-Ising-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
- A Linear Clock for Random Walk on Tree-Weighted 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