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Math-2877 · mathematics · Probability

The continuum tree limit of FK maps above weight four

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For every fixed real $q>4$, is there $c(q)>0$ such that the degree-measured spherical n-edge FK map with graph distance multiplied by $c(q)n^{-1/2}$ converges in Gromov–Hausdorff–Prokhorov law to the Brownian continuum random tree, through all n?

Short English statement

For every fixed real $q>4$, is there $c(q)>0$ such that the degree-measured spherical n-edge FK map with graph distance multiplied by $c(q)n^{-1/2}$ converges in Gromov–Hausdorff–Prokhorov law to the Brownian continuum random tree, through all n?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2877-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2877

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. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The continuum tree limit of FK maps above weight four

OpenAI's claim

OpenAI presents a proof 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.

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