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

Metadata

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

Abstract

In this article, we study the excursion sets 𝒟 p =f -1 ([-p,+[) where f is a natural real-analytic planar Gaussian field called the Bargmann–Fock field. More precisely, f is the centered Gaussian field on 2 with covariance (x,y)exp(-1 2|x-y| 2 ). Alexander has proved that, if p0, 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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