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Math-2874 · mathematics · Probability

Liouville Brownian motion as the walk limit on FK–Ising maps

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For the critical spherical FK–Ising n-edge map law, assume that a deterministic $a_n\to 0$ gives metric–measure convergence to the unit-area sqrt(3)-Liouville quantum sphere. Do any fixed finite number of conditionally independent simple random walks, each initially distributed according to the degree probability measure and accelerated by n, converge jointly with that space to independent stationary Liouville Brownian motions, in the curve-decorated Gromov–Hausdorff–Prokhorov topology?

Short English statement

For the critical spherical FK–Ising n-edge map law, assume that a deterministic $a_n\to 0$ gives metric–measure convergence to the unit-area sqrt(3)-Liouville quantum sphere. Do any fixed finite number of conditionally independent simple random walks, each initially distributed according to the degree probability measure and accelerated by n, converge jointly with that space to independent stationary Liouville Brownian motions, in the curve-decorated Gromov–Hausdorff–Prokhorov topology?

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Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2874-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2874

The stationary vertex walk on the same FK--Ising maps is claimed to converge in actual time to Brownian motion of the $\sqrt3$-Liouville sphere, with edge-count acceleration. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Liouville Brownian motion as the walk limit on FK–Ising maps

OpenAI's claim

OpenAI presents a proof in the selected manuscript. The stationary vertex walk on the same FK--Ising maps is claimed to converge in actual time to Brownian motion of the $\sqrt3$-Liouville sphere, with edge-count acceleration.

Mathematical statement of the claimed result

Result 1.
For FK--Ising spherical maps sampled with weight $2^{\ell/2}$, use all-edge distance $d_n$, stationary degree measure $\mu_n(v)=\deg(v)/(2n)$, and any deterministic spatial scale $a_n\to0$ yielding the metric-measure limit $(S,D_h,\mu_h)$ of the ordinary unit-area $\sqrt3$-LQG sphere. Start any fixed number $r$ of conditionally independent simple walks from $\mu_n$, interpolated along unit cable edges. The source claims \[ (\overline M_n,a_nd_n,\mu_n;\overline X^{n,j}_{nt},\ 1\le j\le r) \Longrightarrow(S,D_h,\mu_h;B_t^{h,j},\ 1\le j\le r). \] Conditioned on $h$, the limit walks are independent Liouville Brownian motions started from $\mu_h$, with Dirichlet energy $\tfrac12\int|\nabla f|^2$. Paths converge uniformly on compact time intervals in the common metric embeddings; time acceleration is exactly $n$.

Outline of the argument

The spectral companion supplies unit conductivity, local energy recovery, and the coupled surface. Dirichlet-form and inverse-operator estimates give tight path laws and identify the limiting Markov process, retaining conditional independence for any fixed number of walks on that surface.

Formal verification

For family 211, the selected Lean theorem is the $q>4$ Brownian tree metric-measure limit, with degree measure and the $n^{-1/2}$ distance scale. The subcritical, critical-Liouville, walk, and spectral companions are not included. No Comparator is assigned to the selected manuscript here; the family page selects other papers or supporting statements. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

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