Abstract
This note is an erratum to the paper “Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension ” (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 for some (small) , 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 with rapid decay of correlations (including the Bargmann–Fock field), large planar clusters in typically belong to clusters in which are not confined in thin slabs.
Acknowledgment
We warmly thank David Vernotte for pointing out the error to us and Damien Gayet for indicating reference [4].
Citation
Hugo Duminil-Copin. Alejandro Rivera. Pierre-François Rodriguez. Hugo Vanneuville. "Erratum: “Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension ”." Ann. Probab. 52 (1) 381 - 385, January 2024. https://doi.org/10.1214/23-AOP1661
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