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

TL;DR

This paper proves that for smooth Gaussian fields in dimensions three and higher, the nodal hypersurfaces contain unbounded components near the critical level, contrasting with lower-dimensional cases.

Contribution

It establishes the positivity of the critical level for percolation in high-dimensional Gaussian fields and shows the unboundedness of nodal hypersurfaces at the nodal case.

Findings

Critical level for percolation is strictly positive in dimensions d ≥ 3.

Nodal hypersurfaces at level zero contain unbounded components.

Behavior differs significantly from two-dimensional Gaussian fields.

Abstract

For the Bargmann--Fock field on $\mathbb R^d$ with $d\ge3$, 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 $d\ge3$ is very different from the behaviour of nodal lines of their two-dimensional analogues.