Percolation of random nodal lines

TL;DR

This paper establishes a percolation theorem for the nodal lines of random analytic functions, demonstrating that with positive probability, these lines connect specified boundary arcs, extending percolation theory to infinite-dimensional Gaussian fields.

Contribution

It proves a Russo-Seymour-Welsch type percolation theorem for nodal lines of Gaussian analytic functions, linking sign percolation to lattice models in an infinite-dimensional setting.

Findings

Existence of connected nodal components crossing boundary arcs with positive probability

Percolation properties hold for scaled random analytic functions

Sign percolation can be modeled as correlated lattice percolation

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 $\gamma, \gamma'$ 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 $\gamma'$, 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.