OpenAI reports Liouville Brownian limits of walks on critical FK–Ising maps

For critical spherical FK–Ising planar maps with n edges, stationary random walks using every incident half-edge converge jointly with the rescaled map metric and degree measure to Liouville Brownian motion on the unit-area √3-quantum sphere when time is accelerated by exactly n; the convergence holds for any fixed finite number of conditionally independent walks.

Sources for this record

What was already known

Earlier work identified Brownian encodings and quantum-surface limits; walk convergence with time parametrization was known for mated-CRT maps but did not identify the exact clock for finite FK–Ising spheres.

AI involvement AI-led

OpenAI credits itself as manuscript author and states that its internal frontier model produced the results in this release. Individual human input is not detailed enough to establish task-only autonomy.

OpenAI is named as author on the manuscript, and its release attributes the results to an internal frontier model. The individual human input is not detailed enough to establish autonomy.

Importance Notable

Identifies the intrinsic diffusion and exact deterministic clock for walks on finite critical FK–Ising maps.

Limitations

The result uses geometric and electrical convergence from companion manuscripts and any deterministic metric scale satisfying the stated map limit. The linked Lean scope checks a Brownian continuum-random-tree limit for q>4 FK maps, a different phase and claim.

Technical verification Reported

Reported is the default when evidence for a higher level is insufficient. The OpenAI-authored manuscript reports and argues the result, but the reviewed sources do not document a completed check of the full assessed claim.

Human understanding Few

Few is the default when evidence for a higher level is insufficient. No named person is documented as explaining the central argument for this assessed contribution.

Correction and publication history

2026-10-18 · publication

Primary manuscript, attribution, prior-work, and precise Lean-scope or conditional-input review for non-notifying local inclusion.

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