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Math-2443 · mathematics · Graph theory

Distances in Decorated Random Planar Maps

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Determine the order of typical graph distances in decorated random planar-map models, including $O(n)$ loop models and FK percolation on random planar maps.

Formulation references: [S1], [S2].

Short English statement

Determine the order of typical graph distances in decorated random planar-map models, including $O(n)$ loop models and FK percolation on random planar maps.

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2443-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2443

The main Gromov–Hausdorff–Prokhorov limit uses the explicit n^-1/2 graph-distance scaling for every fixed FK parameter q>4. It determines this FK subclass, but the problem statement also requests q<=4 and O(n)-decorated models. This entry records partial support; complete coverage of the question is not established by this catalogue.

Distances in Decorated Random Planar Maps

OpenAI's claim

OpenAI presents a mathematical result relevant to this question in the selected manuscript. For each fixed FK parameter $q>4$, finite spherical maps are claimed to converge after an $n^{-1/2}$ distance rescaling to the Brownian continuum random tree.

Mathematical statement of the claimed result

Result 1.
Fix $q>4$. Among rooted planar maps with $n$ edges and marked edge subsets $A$, sample with weight proportional to \[ q^{k_M(A)+(|A|-|V(M)|)/2}. \] Loops and parallel edges are allowed. Let $d_n$ use all map edges and $\mu_n(v)=\deg(v)/(2n)$. The claim provides deterministic $c(q)>0$ such that, through every positive integer size, \[ (V(M_n),c(q)n^{-1/2}d_n,\mu_n) \Longrightarrow(\mathcal T_e,d_e,\mu_e) \] in Gromov--Hausdorff--Prokhorov topology. The limit is the Brownian continuum random tree coded by a normalized excursion $e$, with $d_e(s,t)=e(s)+e(t)-2\min_{[s\wedge t,s\vee t]}e$ and $\mu_e$ the pushforward of Lebesgue probability measure.

Outline of the argument

The contour encoding is reorganized into an exact tree of marked blocks. A critical finite-variance offspring law gives the conditioned tree limit, while uniform bounds along ancestral chains show that distances through the actual map blocks have the same scaling.

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. The listed Comparator files are FKCRT.lean. The inventory does not supply a named declaration for these Comparator entries. 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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