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Math-2905 · mathematics · Probability

Interface and field scaling limits of triangular real Lipschitz surfaces

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For uniformly sampled real-valued triangular-lattice Lipschitz surfaces with the specified two-arc boundary values, do the distinguished interface and centered height field converge to chordal $SLE_4$ and sigma times the domain GFF? Require constants independent of the marked domain, $v=2/\operatorname{sqrt}(3)$, $\sigma ^2=1/(vA)$, $\lambda ^2=\pi /(8vA)$, and limiting mean lambda times the difference of the two boundary-arc harmonic measures.

Short English statement

For uniformly sampled real-valued triangular-lattice Lipschitz surfaces with the specified two-arc boundary values, do the distinguished interface and centered height field converge to chordal $SLE_4$ and sigma times the domain GFF? Require constants independent of the marked domain, $v=2/\operatorname{sqrt}(3)$, $\sigma ^2=1/(vA)$, $\lambda ^2=\pi /(8vA)$, and limiting mean lambda times the difference of the two boundary-arc harmonic measures.

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2905-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2905

Uniform real triangular-lattice Lipschitz heights are claimed to give a Gaussian free field, and at a tuned two-arc amplitude their zero contour approaches chordal $\mathrm{SLE}_4$. The question retains these same parameters and conventions. Within these assumptions, this is a claimed affirmative answer to the stated question.

Interface and field scaling limits of triangular real Lipschitz surfaces

OpenAI's claim

OpenAI presents a proof in the selected manuscript. Uniform real triangular-lattice Lipschitz heights are claimed to give a Gaussian free field, and at a tuned two-arc amplitude their zero contour approaches chordal $\mathrm{SLE}_4$.

Mathematical statement of the claimed result

Result 1.
Let $D$ be bounded, smooth and simply connected, with two distinct marked boundary points, and let triangular-lattice polygons $D_\delta$ have uniformly converging marked boundaries. Sample real vertex heights with normalized Lebesgue measure on the polytope \[ h_\delta=\lambda\text{ on }A_{+,\delta},\quad h_\delta=-\lambda\text{ on }A_{-,\delta},\quad |h_\delta(x)-h_\delta(y)|\le1\ (x\sim y). \] Affine interpolation produces a zero-height chord $\gamma_\delta$ between the marks. There exist universal $A,\sigma>0$ and $\lambda\in(0,1/2]$ such that \[ \gamma_\delta\Longrightarrow\mathrm{SLE}_4(D;a,b),\qquad (h_\delta-\mathbb Eh_\delta)/\sigma\Longrightarrow\Phi_D. \] Curves use uniform distance modulo increasing reparametrization, and fields use distributional convergence with $\operatorname{Cov}\Phi_D=(-\Delta_D)^{-1}$. Writing $v=2/\sqrt3$, the constants satisfy \[ \sigma^2=(vA)^{-1},\qquad \lambda^2=\pi/(8vA),\qquad m_D(z)=\lambda[\omega_D(z,A_+)-\omega_D(z,A_-)], \] where $m_D$ is the limiting mean and $\omega_D$ denotes harmonic measure. The coefficient $A$ is determined implicitly by stationary cell flux, rather than by a claimed elementary value.

Outline of the argument

Local slack estimates and reflected stationary comparisons establish a tangent-flux cell problem. Its coefficient determines the bulk variance and tuned boundary amplitude; varying-continuum trace and conditional Dirichlet covariance estimates then identify both field and interface laws.

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.

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