The critical threshold for Bargmann–Fock percolation
Annales Henri Lebesgue, Volume 3 (2020) , pp. 169-215.

KeywordsPercolation, sharp threshold, KKL, critical point, Bargmann–Fock field

### Abstract

In this article, we study the excursion sets ${𝒟}_{p}={f}^{-1}\left(\left[-p,+\infty \left[\right)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann–Fock field. More precisely, $f$ is the centered Gaussian field on ${ℝ}^{2}$ with covariance $\left(x,y\right)↦exp\left(-\frac{1}{2}|x-y{|}^{2}\right)$. Alexander has proved that, if $p\le 0$, then a.s. ${𝒟}_{p}$ has no unbounded component. We show that conversely, if $p>0$, then a.s. ${𝒟}_{p}$ 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 rely on a recent box-crossing estimate by Beffara and Gayet. We also 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) 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.

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