Percolation of random nodal lines

Abstract

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.