The critical threshold for Bargmann-Fock percolation

TL;DR

This paper establishes the critical threshold at zero for percolation in the Bargmann-Fock Gaussian field, demonstrating a phase transition from no unbounded components to a unique unbounded component, with exponential decay of crossing probabilities below this threshold.

Contribution

The paper proves the critical level for Bargmann-Fock percolation is zero and introduces new tools like a KKL-type result for biased Gaussian vectors and a discretization method.

Findings

Percolation occurs at level p=0 with a unique unbounded component.

Below the critical level, crossing probabilities decay exponentially.

Develops new analytical tools applicable to Gaussian fields.

Abstract

In this article, we study the excursions sets $\mathcal{D}\_p=f^{-1}([-p,+\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\mathbb{R}^2$ with covariance $(x,y) \mapsto \exp(-\frac{1}{2}|x-y|^2)$. In [BG16], Beffara and Gayet prove that, if $p \leq 0$, then a.s. $\mathcal{D}\_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\mathcal{D}\_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponential decay of crossing probabilities under the critical level. To show these results, we develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen in [KMS12]) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.