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Math-2864 · mathematics · Probability

The exact GFF variance of balanced six-vertex heights

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For square-lattice six-vertex weights $a=b=1$ and fixed $0<c\leq 2$, does the prescribed iterated balanced-torus infinite-volume law exist, and do its rescaled heights modulo constants converge to a planar GFF with variance multiplier 1/arcsin(c/2)? Require joint finite-energy zero-mass measure tests, convergence in the stated negative Besov and Sobolev restriction spaces, and all separated height-increment moments.

Short English statement

For square-lattice six-vertex weights $a=b=1$ and fixed $0<c\leq 2$, does the prescribed iterated balanced-torus infinite-volume law exist, and do its rescaled heights modulo constants converge to a planar GFF with variance multiplier 1/arcsin(c/2)? Require joint finite-energy zero-mass measure tests, convergence in the stated negative Besov and Sobolev restriction spaces, and all separated height-increment moments.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2864-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2864

Balanced-torus six-vertex heights at $a=b=1$ and $0<c\leq2$ are claimed to converge to a Gaussian free field with squared multiplier $1/\arcsin(c/2)$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

The exact GFF variance of balanced six-vertex heights

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Balanced-torus six-vertex heights at $a=b=1$ and $0<c\leq2$ are claimed to converge to a Gaussian free field with squared multiplier $1/\arcsin(c/2)$.

Mathematical statement of the claimed result

Result 1.
Use balanced-torus six-vertex weights $(a,b,c)=(1,1,c)$, $0<c\le2$, conditioning each vertical column to equal left/right arrow counts. Take first horizontal size to infinity, then even vertical size, and height steps of one. The source claims that the resulting plane law exists and, modulo constants, \[ h_\delta\Longrightarrow\sigma(c)\Gamma,\quad \sigma(c)^2=\frac1{\arcsin(c/2)},\quad G(x,y)=-\frac1{2\pi}\log|x-y|. \] For compactly supported zero-mass finite-energy signed measures $\mu_i$, pairings converge to centered Gaussian covariance $\sigma(c)^2\iint G\,d\mu_i\,d\mu_j$. On any bounded open $U$, convergence and tightness also hold in $B^\alpha_{\infty,\infty}(U)$, $B^\alpha_{p,q}(U)$, and $W^{\alpha,p}(U)$, for $-1<\alpha<0$, $1\le p<\infty$, $1\le q\le\infty$, with common pinning. For disjoint endpoint pairs, increment moments converge uniformly on compact configurations to \[ \sigma(c)^k\sum_{\pi\text{ pairing}}\prod_{\{i,j\}\in\pi} [G(u_i',u_j')-G(u_i',u_j)-G(u_i,u_j')+G(u_i,u_j)], \] with odd moments zero. No uniqueness among all plane Gibbs states or slopes is asserted.

Outline of the argument

Finite row scalar products produce thermodynamic transfer kernels and an angular spectrum. The covariance is fixed by these kernels, chiral current identities determine Gaussian higher correlations, and smearing moment bounds give distributional convergence, including the endpoint parameter.

Formal verification

Family 225 has no Lean scope page in this release. Its selected result is not presented as formalized. 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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