Abstract: Level-set percolation of continuous Gaussian fields on 
has attracted a lot of attention in recent years. While great progress has been made in the planar case 
, higher dimensions remain rather poorly understood. In this talk, we shall prove that fields with positive and sufficiently fast decaying correlations (along with other mild assumptions) undergo a sharp phase transition in arbitrary dimensions. More precisely, we show that connection probabilities decay exponentially for 
and percolation occurs in sufficiently thick 2D slabs for 
, where 
stands for the critical level of the model. In particular, our results apply to the Bargmann-Fock field, for which sharpness was known only in dimension .
The result follows from a global comparison with a truncated and discretized 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 small sprinkling. This approach is inspired by a recent work with Duminil-Copin, Goswami and Rodriguez on sharpness for the (discrete) Gaussian free field.