Percolation of random nodal lines

We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let \(U\) be a smooth connected bounded open set in \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) two disjoint arcs of positive length in the boundary of \(U\). We prove that there exists a positive constant \(c\), such that for any positive scale \(s\), with probability at least \(c\) there exists a connected component of the set \(\{x\in \smash{\bar{U}},\ f(sx) > 0\} \) intersecting both \(\gamma \) and \(\gamma '\), where \(f\) is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For \(s\) large enough, the same conclusion holds for the zero set \(\{x\in \smash{\bar{U}},\ f(sx) = 0\} \). As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice.