Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension $d\ge 3$

Abstract

For the Bargmann--Fock field on ${\mathbb{R}}^d$ with $d\ge 3$, we prove that the critical level ${\ell }_c(d)$ of the percolation model formed by the excursion sets $\{f\ge \ell \}$ is strictly positive. This implies that for every $\ell$ sufficiently close to $0$ (in particular for the nodal hypersurfaces corresponding to the case $\ell =0$), $\{f=\ell \}$ contains an unbounded connected component that visits "most" of the ambient space. Our findings actually hold for a more general class of positively correlated smooth Gaussian fields with rapid decay of correlations. The results of this paper show that the behaviour of nodal hypersurfaces of these Gaussian fields in ${\mathbb{R}}^d$ for $d\ge 3$ is very different from the behaviour of nodal lines of their two-dimensional analogues.

61 pages, 4 figures. The previous version contains one error: Prop 1.12 therein is stated with a sprinkling, which we cannot afford in Sect 5, where it is applied at mesoscopic scales. We circumvent this issue by proving a stronger version of Prop 1.12, which contains no sprinkling and is interesting in its own right. We have also added details with references about proofs relying on Morse theory