Sharp phase transition for Gaussian percolation in all dimensions

Severo, Franco

Annales Henri Lebesgue, Tome 5 (2022), pp. 987-1008.

On considère les courbes de niveau de champs gaussiens continus sur ${ℝ}^d$ au-dessus d’un certain niveau $-\ell \in ℝ$, ce qui définit un modèle de percolation lorsque $\ell$ varie. Nous supposons que le noyau de covariance satisfait certaines conditions de régularité, de symétrie et de positivité ainsi qu’une décroissance polynomiale d’exposant supérieur à $d$ (cela inclut notamment le champ de Bargmann–Fock). Sous ces hypothèses, nous prouvons que le modèle subit une transition de phase abrupte autour de son point critique ${\ell }_c$. Plus précisément, nous montrons que les probabilités de connexion décroissent exponentiellement pour $\ell <{\ell }_c$ et que la percolation se produit dans des dalles 2D suffisamment épaisses pour $\ell >{\ell }_c$. Ceci étenddles résultats récemment obtenus en dimension $d=2$ à des dimensions arbitraires par des techniques complètement différentes. Le résultat découle d’une comparaison globale avec une version tronquée (c’est-à-dire avec une plage de dépendance finie) et discrétisée (c’est-à-dire définie sur le réseau $\epsilon {\mathbb{Z}}^d$) du modèle, qui peut présenter un intérêt indépendant. La démonstration de cette comparaison repose sur un schéma d’interpolation qui intègre les corrélations à longue portée et infinitésimales du modèle tout en les compensant par une légère modification du paramètre $\ell$.

We consider the level-sets of continuous Gaussian fields on ${ℝ}^d$ above a certain level $-\ell \in ℝ$, which defines a percolation model as $\ell$ varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than $d$ (in particular, this includes the Bargmann–Fock field). Under these assumptions, we prove that the model undergoes a sharp phase transition around its critical point ${\ell }_c$. More precisely, we show that connection probabilities decay exponentially for $\ell <{\ell }_c$ and percolation occurs in sufficiently thick 2D slabs for $\ell >{\ell }_c$. This extends results recently obtained in dimension $d=2$ to arbitrary dimensions through completely different techniques. The result follows from a global comparison with a truncated (i.e. with finite range of dependence) and discretized (i.e. defined on the lattice $\epsilon {\mathbb{Z}}^d$) version of the model, which may be of independent interest. The proof of this comparison relies on an interpolation scheme that integrates out the long-range and infinitesimal correlations of the model while compensating them with a slight change in the parameter $\ell$.

Classification : 82B43, 60K35, 60G15, 60G60
Mots clés : percolation, sharpness, phase transition, Gaussian fields

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