Percolation of random nodal lines

Beffara, Vincent; Gayet, Damien

Publications Mathématiques de l'IHÉS, Tome 126 (2017), pp. 131-176.

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 R2 and γ,γ′ 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∈U¯,f(sx)>0} intersecting both γ and γ′, 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∈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.

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