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Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion

OpenAI · October 5, 2026

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Abstract

We prove that stationary random walks on critical spherical FK–Ising planar maps converge to Liouville Brownian motion on the ordinary unit-area $\sqrt3$-quantum sphere, using the geometric and electrical results of the spectral companion. The walk chooses uniformly among all incident half-edges, retaining loops and multiple edges. With the corner measure as the stationary law, the deterministic time acceleration is exactly the number of map edges. For any deterministic metric scale giving the metric-measure limit, convergence retains that same surface and any fixed finite number of conditionally independent walks with their time parameters.

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Cite

@misc{OAI:Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026,
  author = {{OpenAI}},
  title = {{Random Walks on Critical FK--Ising Maps and Liouville Brownian Motion}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026/fk-ising-walk-limit.pdf}{OAI:Random-Walks-on-Critical-FK-Ising-Maps-and-Liouville-Brownian-Motion-October-5-2026}},
  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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