We consider 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, namely h_{**}(d), h_{*}(d) and \bar h(d), 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, i.e. h_{**}(d)=h_{*}(d)= \bar h(d) for any d \geq 3. At the core of our proof lies a new interpolation scheme aimed at integrating out the long-range dependence of the Gaussian free field. The successful implementation of this strategy relies extensively on certain novel renormalization techniques, in particular to control so-called large-field effects. This approach opens the way to a complete understanding of the off-critical phases of strongly correlated percolation models.
