Sheffield’s localization conjecture for sloped height functions

Sheffield’s conjecture predicts localization behavior for infinite-volume random height surfaces with prescribed slope. The new result addresses only a substantially restricted regime, not the general statement.

Known results

• Sheffield (2005): for integer-valued heights at irrational slope $\theta$, $\operatorname{Var}(h(x))\to\infty$ as $|x|\to\infty$, with no quantitative bound previously known.
• A 2024 result proves tightness and existence of infinite-volume Gibbs measures for $\varepsilon$-monotone potentials on percolation-transient graphs.
• A September 2024 result obtains Gaussian-free-field scaling and logarithmic variance growth for a perturbed tilted SOS model, not the unperturbed model.
• A May 2025 result proves logarithmic delocalization for $p$-SOS models with $0<p\le 2$ in dimension $d=2$ at high temperature.

September 2026 restricted proof

The paper reports Sheffield’s conjecture in a zero-slope, even-interaction, low-temperature setting, together with uniqueness, extremal-state classification, entropic repulsion, and exponential correlation decay. This is claimed progress rather than a verified resolution of the general sloped conjecture.

Current status (as of September 2026): A restricted zero-slope case is claimed solved, while the general conjecture for nonzero slopes and broader interactions remains open.