# Bargmann-Fock percolation is noise sensitive

## TL;DR

This paper proves that planar Bargmann-Fock percolation is highly sensitive to small perturbations, leading to new insights into the near-critical behavior and extending sharp threshold results for Gaussian fields.

## Contribution

It introduces a quantitative noise sensitivity analysis for Bargmann-Fock percolation using randomized algorithms, extending sharp threshold results to more Gaussian fields.

## Key findings

- Planar Bargmann-Fock percolation is noise sensitive under Ornstein-Ulhenbeck process.
- Full restriction of a Gaussian field on one plane provides almost no info about percolation on a nearby plane.
- Near-critical window of level line percolation is polynomially small.

## 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 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.