Erratum: “Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension d≥3”

Abstract

This note is an erratum to the paper “Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $\mathit{d}\ge 3$” (Ann. Probab. 51 228–276 (2023) https://doi.org/10.1214/22-AOP1594). The published version of this paper contains one error: Proposition 1.12 therein is stated with a sprinkling ${\mathit{R}}^{-2\mathbf{+}{\mathit{\theta }}_0}$ for some (small) ${\mathit{\theta }}_0>0$, which we cannot afford in Section 5, where this result is applied at mesoscopic scales. We circumvent this issue by proving a stronger version of this proposition, interesting in its own right, which contains no sprinkling. We thus obtain that, for a general class of positively correlated smooth Gaussian fields $\mathit{f}:{\mathbb{R}}^3\to \mathbb{R}$ with rapid decay of correlations (including the Bargmann–Fock field), large planar clusters in $\{\mathit{f}\ge 0\}\cap ({\mathbb{R}}^2\times \{0\})$ typically belong to clusters in $\{\mathit{f}\ge 0\}$ which are not confined in thin slabs.