Existence of an unbounded nodal hypersurface for smooth Gaussian fields in dimension \(d\geq 3\) (Q2105144)

Consider Gaussian field \(f\) on \(\mathbb{R}^d\) which is stationary, isotropic, centered and smooth. Recently, percolation properties of level sets \(\{x\in \mathbb R^d : f(x)=\ell\}\) (where \(\ell \in \mathbb{R}\)) and of excursion sets \(\{ x\in \mathbb R^d : f(x)\geq \ell\}\) have been investigated. Recall that the critical level is defined as supremum of \(\ell\) for which (with probability one) there exists an unbounded component in \(\{f\geq \ell\}\). It is known that, under very mild conditions, the critical level is 0 if \(d=2\), i.e., if \(\ell>0\) then a.s. there is no unbounded component in \(\{f \geq \ell\}\) while such an unbounded component exists a.s. if \(\ell<0\). Moreover, if \(f\) is positively correlated and \(d=2\), it is known that a.s. there is no unbounded component in \(\{f \geq 0\}\). In the paper under review, the authors study these questions in the case \(d \geq 3\). The main result of the paper is that for a class of positively correlated Gaussian fields with fast decay of correlations-including the Bargmann-Fock field the critical level is strictly larger than 0. In particular, contrary to the case \(d=2\), a.s. there exists an unbounded component in \(\{f=0\}\).