  We prove that the connectivity of the level sets of a wide class of smooth
centred planar Gaussian fields exhibits a phase transition at the zero level
that is analogous to the phase transition in Bernoulli percolation. In addition
to symmetry, positivity and regularity conditions, we assume only that
correlations decay polynomially with exponent larger than two -- roughly
equivalent to the integrability of the covariance kernel -- whereas previously
the phase transition was only known in the case of the Bargmann-Fock covariance
kernel which decays super-exponentially. We also prove that the phase
transition is sharp, demonstrating, without any further assumption on the decay
of correlations, that in the sub-critical regime crossing probabilities decay
exponentially.
  Key to our methods is the white-noise representation of a Gaussian field; we
use this on the one hand to prove new quasi-independence results, inspired by
the notion of influence from Boolean functions, and on the other hand to
establish sharp thresholds via the OSSS inequality for i.i.d. random variables,
following the recent approach of Duminil-Copin, Raoufi and Tassion.
