  We show that planar Bargmann-Fock percolation is noise sensitive under the
Ornstein-Ulhenbeck process. The proof is based on the randomized algorithm
approach introduced by Schramm and Steif and gives quantitative polynomial
bounds on the noise sensitivity of crossing events for Bargmann-Fock. A rather
counter-intuitive consequence is as follows. Let $F$ be a Bargmann-Fock
Gaussian field in $\mathbb{R}^3$ and consider two horizontal planes $P_1,P_2$
at small distance $\varepsilon$ from each other. Even though $F$ is a.s.
analytic, the above noise sensitivity statement implies that the full
restriction of $F$ to $P_1$ (i.e. $F_{| P_1}$) gives almost no information on
the percolation configuration induced by $F_{|P_2}$. As an application of this
noise sensitivity analysis, we provide a Schramm-Steif based proof that the
near-critical window of level line percolation around $\ell_c=0$ is
polynomially small. This new approach extends earlier sharp threshold results
to a larger family of planar Gaussian fields.
