Which web is meant by a Gaussian free-field zero set?

Definitions and mathematical statement

Original Question 10 (Scott Sheffield). The note describes a function whose Fourier transform is white noise on the unit circle and asks to understand its web-like appearance, calling it the zero level lines of a Gaussian free field; it says Zelditch and Schramm make the question more precise, but supplies no such precision. These words mix different Gaussian models and do not define a unique random set or requested statistic.

Attached unreviewed model separation. A real monochromatic plane wave has spectral measure on $S^1$, covariance $J_0(|x-y|)$ in a usual unit-frequency normalization, and real-analytic Helmholtz samples satisfying $(\Delta+1)F=0$; it has ordinary nodal curves. Boundary white noise on $S^1$ is a different random distribution. The two-dimensional GFF is itself distribution valued, so canonical level lines are not its pointwise zero set. The aid describes appropriate local-set interfaces as having $\mathrm{SLE}_4/\mathrm{CLE}_4$ laws, with boundary/interface hypotheses not supplied in the summary.

For a precise diagnostic, take zero-boundary GFF $h$ on the unit disk with Green-kernel normalization $G(z,w)=(2\pi)^{-1}\log|(1-z\overline w)/(z-w)|$. Let $h_r(0)$ be its circle average. The aid states that \[B_t=\sqrt{2\pi}\,h_{e^{-t}}(0),\qquad t\ge0,\] is standard Brownian motion. For $N\ge2$ put $F_N=\sum_{n=1}^{N-1}\mathbf1_{\{B_nB_{n+1}<0\}}$. Its reported exact expectation is \[{\mathbb{E}} F_N=\frac1\pi\sum_{n=1}^{N-1}\arcsin\frac1{\sqrt{n+1}}=\frac2\pi\sqrt N+O(1),\] and the aid asserts infinitely many such flips almost surely. The unit-disk base point and normalization matter to this exact Brownian identification.

A separate boundary logarithmic Fourier cutoff is explicitly \[X_N(\theta)=\sum_{n=1}^N\frac{a_n\cos(n\theta)+b_n\sin(n\theta)}{\sqrt n},\qquad0\le\theta<2\pi,\] where $a_n,b_n$ are independent standard real Gaussians. If $Z_N$ counts its zeros over one period and $H_N=\sum_{n=1}^N1/n$, the aid gives \[{\mathbb{E}} Z_N=\sqrt{\frac{2N(N+1)}{H_N}}\sim\frac{\sqrt2\,N}{\sqrt{\log N}}.\] This logarithmic boundary model is not unweighted boundary white noise and is not the monochromatic plane wave.

These finite-cutoff and fixed-point diagnostics show instability of a naive pointwise-zero/sign interpretation in the stated regularizations. They do not prove either convergence or nonconvergence of specially selected unlabelled interfaces, nor replace a precise local-set construction. The intended web and desired limit/statistics remain unresolved as a formulation; all asserted formulas and interface descriptions require independent review.

Short English statement

The source conflates a monochromatic plane wave, boundary white noise and a two-dimensional GFF. The aid separates them and gives exact circle-average sign-flip and boundary Fourier zero-count formulas. Those diagnostics do not determine convergence of selected interfaces or identify a unique interpretation of the original web question.

Sources

• Local UnsolvedMath registry, permanent ID 20000635, source code catalogue entry; aim-analysis-notes.json, notes[166], tag question.
• AIM workshop Random analytic functions, April 2006, Question 10 (Scott Sheffield): https://aimath.org/WWN/randomzeros/arcc1.pdf. Original text also names Zelditch and Schramm without a statement or bibliographic reference. Record marked reconstructed_unverified; no PDF was consulted.
• Attached machine-generated research attempt 1 refers to GFF local sets and $\mathrm{SLE}_4/\mathrm{CLE}_4$ without publication locators. Result partial_result, status partially_solved, rigor label proved, novelty confidence low. It is not peer reviewed and requires independent verification before citation.