We consider level-sets of the Gaussian free field on Z d , for d ≥ 3, above a given realvalued 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 h(d), respectively describing a wellordered 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) = h(d) for any d ≥ 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.

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We consider the level-sets of continuous Gaussian fields on R d above a certain level −ℓ ∈ R, which defines a percolation model as ℓ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than d (in particular, this includes the Bargmann-Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point ℓ c . More precisely, we show that connection probabilities decay exponentially for ℓ < ℓ c and percolation occurs in sufficiently thick 2D slabs for ℓ > ℓ c . This extends results recently obtained in dimension d = 2 to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice εZ d ) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter ℓ.

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