On considère les courbes de niveau de champs gaussiens continus sur au-dessus d’un certain niveau , ce qui définit un modèle de percolation lorsque 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 (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 . Plus précisément, nous montrons que les probabilités de connexion décroissent exponentiellement pour et que la percolation se produit dans des dalles 2D suffisamment épaisses pour . Ceci étenddles résultats récemment obtenus en dimension 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 ) 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 .
We consider the level-sets of continuous Gaussian fields on above a certain level , which defines a percolation model as varies. We assume that the covariance kernel satisfies certain regularity, symmetry and positivity conditions as well as a polynomial decay with exponent greater than (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 . More precisely, we show that connection probabilities decay exponentially for and percolation occurs in sufficiently thick 2D slabs for . This extends results recently obtained in dimension 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 ) 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 .
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Mots clés : percolation, sharpness, phase transition, Gaussian fields
@article{AHL_2022__5__987_0, author = {Severo, Franco}, title = {Sharp phase transition for {Gaussian} percolation in all dimensions}, journal = {Annales Henri Lebesgue}, pages = {987--1008}, publisher = {\'ENS Rennes}, volume = {5}, year = {2022}, doi = {10.5802/ahl.141}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.141/} }
Severo, Franco. Sharp phase transition for Gaussian percolation in all dimensions. Annales Henri Lebesgue, Tome 5 (2022), pp. 987-1008. doi : 10.5802/ahl.141. http://archive.numdam.org/articles/10.5802/ahl.141/
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