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2020 Bargmann-Fock percolation is noise sensitive
Christophe Garban, Hugo Vanneuville
Electron. J. Probab. 25: 1-20 (2020). DOI: 10.1214/20-EJP491


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.


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Christophe Garban. Hugo Vanneuville. "Bargmann-Fock percolation is noise sensitive." Electron. J. Probab. 25 1 - 20, 2020.


Received: 4 October 2019; Accepted: 27 June 2020; Published: 2020
First available in Project Euclid: 14 August 2020

zbMATH: 07252730
MathSciNet: MR4136478
Digital Object Identifier: 10.1214/20-EJP491

Primary: 60G15, 60K35


Vol.25 • 2020
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