  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.
