The critical threshold for Bargmann–Fock percolation

Rivera, Alejandro; Vanneuville, Hugo

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

Dans cet article, nous étudions les ensembles d’excursion ${𝒟}_p=f^{-1}\left(\right[-p,+\infty \left[\right)$ où $f$ est le champ de Bargmann–Fock. Alexander a montré que, si $p\le 0$, il n’y avait presque sûrement aucune composante connexe infinie dans ${𝒟}_p$. Nous montrons qu’au contraire, si $p>0$, il existe une (unique) composante connexe infinie dans ${𝒟}_p$. Par conséquent, le niveau critique de ce modèle de percolation est $0$. Nous démontrons aussi que les probabilités de connexion décroissent exponentiellement vite dans la phase sous-critique. Pour montrer ces résultats, nous utilisons des estimations de traversée de boîtes dues à Beffara et Gayet. Nous développons par ailleurs divers outils, notamment un résultat de type KKL pour des vecteurs gaussiens corrélés (dont la preuve repose sur le résultat analogue dans le cas produit démontré par Keller, Mossel et Sen) et une procédure de sprinkling. Ces résultats intermédiaires sont vrais pour une classe générale de champs gaussiens, pour laquelle nous montrons une version discrète de notre résultat principal.

In this article, we study the excursion sets ${𝒟}_p=f^{-1}\left(\right[-p,+\infty \left[\right)$ 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 $(x,y)\mapsto exp(-\frac{1}{2}|x-y{|}^2)$. 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.

Classification : 60K35, 60G15, 60D05
Mots clés : Percolation, sharp threshold, KKL, critical point, Bargmann–Fock field

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