Abstract
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 ([30]) 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.
Highlights
We show that planar Bargmann-Fock percolation is noise sensitive under the OrnsteinUlhenbeck process
1.1 Bargmann-Fock percolation is noise sensitive The planar Bargmann-Fock field is the smooth centered Gaussian field f on R2 defined by the following covariance kernel
We extend our above noise sensitivity result to the dynamical processes t → f (t) defined as follows
Summary
1.1 Bargmann-Fock percolation is noise sensitive The planar Bargmann-Fock field is the smooth centered Gaussian field f on R2 defined by the following covariance kernel. Our main result states that Bargmann-Fock percolation is sensitive to a polynomially small noise. As in [24], we consider a planar white noise W which is the centered Gaussian field ( udW )u∈L2(R2) indexed by L2(R2) with the following covariance kernel. For every t ≥ 0, let f (t) = q Wt. if we assume that q satisfies Condition 1.5, there exists a modification of (t, x) ∈ R+ × R2 → f (t, x) that is continuous. If we assume that q ≥ 0, the near-critical window is polynomially small in the sense that there exists α = α(q) > 0 such that, for every quad Q, 1 − P [Crosss−α (sQ)] and P [Cross−s−α (sQ)] go to 0 as s → +∞.
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