  We consider the level-sets of continuous Gaussian fields on $\mathbb{R}^d$
above a certain level $-\ell\in \mathbb{R}$, which defines a percolation model
as $\ell$ 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 $\ell_c$. More precisely, we show
that connection probabilities decay exponentially for $\ell<\ell_c$ and
percolation occurs in sufficiently thick 2D slabs for $\ell>\ell_c$. 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 $\varepsilon\mathbb{Z}^d$) 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 $\ell$.
