Percolation of random nodal lines

Abstract

We prove a Russo-Seymour-Welsch 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 ${\mathbb{R}}^2$ and $\gamma ,{\gamma }^{\prime }$ 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 $\{x\in \bar U, \, f(sx) \textgreater{} 0\} $ intersecting both $\gamma$ and ${\gamma }^{\prime }$, 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 $\{x\in \bar{U},f(sx)=0\}$. 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.