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Math-2870 · mathematics · Probability

Metric–measure universality of subcritical FK and tree-weighted maps

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For each fixed $q\in (0,4)$ in the stated spherical FK map ensemble, and separately for the spanning-tree-weighted ensemble, is there a deterministic $a_n\to 0$ such that the graph metric multiplied by $a_n$ and the specified probability measure converge through all n to the corresponding unit-area Liouville quantum sphere in Gromov–Hausdorff–Prokhorov law? The root and map decoration are forgotten.

Short English statement

For each fixed $q\in (0,4)$ in the stated spherical FK map ensemble, and separately for the spanning-tree-weighted ensemble, is there a deterministic $a_n\to 0$ such that the graph metric multiplied by $a_n$ and the specified probability measure converge through all n to the corresponding unit-area Liouville quantum sphere in Gromov–Hausdorff–Prokhorov law? The root and map decoration are forgotten.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2870-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2870

Subcritical FK-decorated spherical maps and tree-weighted maps are claimed to have their corresponding ordinary Liouville quantum spheres as metric-measure limits through all edge counts. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Metric–measure universality of subcritical FK and tree-weighted maps

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Subcritical FK-decorated spherical maps and tree-weighted maps are claimed to have their corresponding ordinary Liouville quantum spheres as metric-measure limits through all edge counts.

Mathematical statement of the claimed result

Result 1.
For each fixed $q\in(0,4)$, sample rooted $n$-edge planar map/subset pairs $(M,A)$ with weight $q^{\ell(M,A)/2}$, $\ell=2k(A)+|A|-|V(M)|$. Separately, sample rooted map/spanning-tree pairs uniformly. Distances $d_n$ use every primal edge and $\mu_n(v)=\deg(v)/(2n)$. The source claims deterministic $a_n>0$, $a_n\to0$, such that \[ (V(M_n),a_nd_n,\mu_n)\Longrightarrow(S,D_h,\mu_h) \] in Gromov--Hausdorff--Prokhorov law, through all integer sizes. The limiting sphere has the ordinary LQG law normalized to area one: $q=2+2\cos(\pi\gamma^2/2)$, $\sqrt2<\gamma<2$, for FK, and $\gamma=\sqrt2$ for spanning trees. Root and decoration are forgotten.

Outline of the argument

Finite contour records retain every visit and edge incidence needed to read graph distances. Conditional resampling identifies local limiting laws, and metric tightness plus Liouville uniqueness identifies the globally coupled degree measure and intrinsic distance after deterministic normalization.

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.

Sources

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