The critical threshold for Bargmann–Fock percolation

Rivera, Alejandro; Vanneuville, Hugo

doi:10.5802/ahl.29

Rivera, Alejandro; Vanneuville, Hugo

The critical threshold for Bargmann–Fock percolation

Annales Henri Lebesgue, Volume 3 (2020), pp. 169-215.

Metadata

DOI10.5802/ahl.29

Keywords Percolation, sharp threshold, KKL, critical point, Bargmann–Fock field

Abstract

In this article, we study the excursion sets $𝒟_{p} = f^{- 1} ([- p, + \infty [)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann–Fock field. More precisely, $f$ is the centered Gaussian field on $ℝ^{2}$ with covariance $(x, y) \mapsto exp (- \frac{1}{2} | x - y |^{2})$ . Alexander has proved that, if $p \leq 0$ , 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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