we-solve (beta)

Math-2899 · mathematics · Probability

A GFF limit for triangular odd heights with two boundary arcs

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For uniform odd integer Lipschitz heights on triangular-lattice approximations of a smooth simply connected planar domain with the stated two-arc boundary data, does the height distribution converge to the harmonic boundary extension plus sigma times the domain GFF, for a universal $\sigma >0$, with convergence of every joint test-function moment and mean?

Short English statement

For uniform odd integer Lipschitz heights on triangular-lattice approximations of a smooth simply connected planar domain with the stated two-arc boundary data, does the height distribution converge to the harmonic boundary extension plus sigma times the domain GFF, for a universal $\sigma >0$, with convergence of every joint test-function moment and mean?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2899-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2899

Two-arc odd integer Lipschitz heights on triangular lattices are claimed to converge, after centering and scalar normalization, to the Dirichlet Gaussian free field. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

A GFF limit for triangular odd heights with two boundary arcs

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Two-arc odd integer Lipschitz heights on triangular lattices are claimed to converge, after centering and scalar normalization, to the Dirichlet Gaussian free field.

Mathematical statement of the claimed result

Result 1.
On triangular-lattice polygonal approximations $D_\delta$ of a bounded smooth simply connected domain $D$, take two distinct boundary marks at edge midpoints. Assume uniform convergence of marked boundary parametrizations. Sample odd integer vertex heights uniformly subject to \[ h_\delta=1\text{ on }A_{+,\delta},\quad h_\delta=-1\text{ on }A_{-,\delta},\quad |h_\delta(x)-h_\delta(y)|\in\{0,2\}\ (x\sim y). \] Interpolate affinely over triangles. If $\bar g$ is harmonic with boundary values $+1,-1$ on the respective limiting arcs and $\Phi_D$ is the Dirichlet GFF with covariance $(-\Delta_D)^{-1}$, a lattice-dependent $\sigma>0$ satisfies \[ h_\delta\Longrightarrow\bar g+\sigma\Phi_D\quad\text{in }\mathcal D'(D),\qquad \mathbb Eh_\delta(f)\to\int_D\bar gf\quad(f\in C_c^\infty(D)). \] Consequently $(h_\delta-\mathbb Eh_\delta)/\sigma\Longrightarrow\Phi_D$. All finite joint test-function laws and their mixed moments converge. The normalization is $\sigma=4\pi\sqrt{c_*}$ for the paper's absolutely convergent finite-volume coefficient $c_*$.

Outline of the argument

Exact height cuts are combined with buffered circuit mixing and rare-pin estimates. Reflection positivity fixes the plane covariance and an absolutely convergent normalization; moving boundary pins and conditional Gaussian moment identities identify the field in the prescribed domain.

Formal verification

Family 232 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

Go to contribution →