  We consider upper level-sets of the Gaussian free field on $\mathbb Z^d$, for
$d\geq 3$, 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 to
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
Gaussian free field. 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.
