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

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