[Paper Review] Bargmann-Fock percolation is noise sensitive

Christophe Garban, Hugo Vanneuville|arXiv (Cornell University)|Jun 6, 2019

Stochastic processes and statistical mechanics5 references5 citations

TL;DR

This paper establishes that planar Bargmann-Fock percolation is noise sensitive under the Ornstein-Uhlenbeck dynamics, using a randomized algorithm approach to derive polynomial bounds on noise sensitivity for crossing events. The key result is that even infinitesimally small perturbations in the field render the percolation configuration at criticality nearly independent, implying that knowledge of the field on one plane gives almost no predictive power for percolation on a nearby parallel plane—despite the field's analyticity. This noise sensitivity implies a polynomially small near-critical window for level-line percolation.

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.

Motivation & Objective

• To establish noise sensitivity of planar Bargmann-Fock percolation under the Ornstein-Uhlenbeck process.
• To quantify the loss of predictive power for percolation configurations across nearby planes despite analyticity of the field.
• To extend noise sensitivity results to a broad class of planar Gaussian fields satisfying regularity, decay, and correlation conditions.
• To apply noise sensitivity to derive a sharp threshold result for the near-critical window of level-line percolation.
• To demonstrate that restriction of a 3D Bargmann-Fock field to one plane provides almost no information about percolation on a parallel plane at small distance.

Proposed method

• Adopting the randomized algorithm framework of Schramm and Steif ([SS10]), the authors analyze the noise sensitivity of crossing events in the Bargmann-Fock model.
• They use the Wiener chaos expansion of the indicator functional of the crossing event to decompose its variance and sensitivity.
• The proof relies on a variance estimate for the difference f(t,x)−f(0,x) over time t and space x, derived via Taylor expansion and moment bounds.
• A key step involves bounding the supremum of the field difference over small time and space intervals using Dudley’s theorem and the Borell-Tsirelson inequality.
• The authors generalize the result to a class of planar Gaussian fields defined via convolution with a kernel q satisfying regularity, decay, and positivity conditions.
• They establish convergence of discrete white noise approximations to the continuous field, ensuring the validity of the continuous dynamics.

Experimental results

Research questions

• RQ1 Does the planar Bargmann-Fock percolation model exhibit noise sensitivity under small, continuous perturbations via the Ornstein-Uhlenbeck process?
• RQ2 To what extent does the restriction of a 3D analytic Gaussian field to a single plane determine percolation properties on a nearby parallel plane?
• RQ3 How does noise sensitivity in the critical Bargmann-Fock model constrain the size of the near-critical window for level-line percolation?
• RQ4 Can the noise sensitivity framework be extended to a broad class of planar Gaussian fields beyond the Bargmann-Fock case?
• RQ5 What is the quantitative rate of decay of the covariance between crossing events at different times under the dynamical field?

Key findings

• There exists α > 0 such that for any quad Q, the covariance between the indicator of the crossing event at time 0 and at time tn ≥ n−α decays polynomially as O(n−α).
• The restriction of a 3D Bargmann-Fock field to a horizontal plane P(tn) gives almost no information about the percolation configuration on a nearby plane P(0), as quantified by the variance of the conditional probability being O(n−α).
• The noise sensitivity result extends to the event that a nodal line (f=0) crosses a quad, even though this event is not monotone.
• For a broad class of planar Gaussian fields defined by convolution with a kernel q satisfying Conditions 1.4, 1.6, and 1.7 (with β > 2), the same polynomial noise sensitivity holds.
• The near-critical window for level-line percolation is polynomially small, as derived from the noise sensitivity and a BKS-type argument.
• The dynamics of the field f(t,x) can be coupled as f(t) = e−tf(0) + √(1−e−2t) f̃, where f̃ is an independent copy of f(0), enabling explicit pathwise analysis.