Equality of critical parameters for percolation of Gaussian free field level sets

Abstract

We consider upper level sets of the Gaussian free field (GFF) on ${\mathbb{Z}}^d$, for $d\ge 3$, above a given real-valued height parameter h. As h varies, this defines a canonical percolation model with strong, algebraically decaying correlations. We prove that three natural critical parameters associated with this model, respectively describing a well-ordered subcritical phase, the emergence of an infinite cluster, and the onset of a local uniqueness regime in the supercritical phase, actually coincide. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the GFF. Due to the strength of correlations, its successful implementation requires that we work in an effectively critical regime. Our analysis relies extensively on certain novel renormalization techniques that bring into play all relevant scales simultaneously. The approach in this article paves the way to a complete understanding of the off-critical phases for strongly correlated disordered systems.