Quantitative quenched Voronoi percolation and applications
[Percolation de Voronoi quenched quantitative et applications]
Vanneuville, Hugo1
1 Institut Camille Jordan 43 boulevard du 11 novembre 1918 69622 Villeurbanne cedex, France
Annales de l'Institut Fourier, Online first, 35 p.

Ahlberg, Griffiths, Morris et Tassion ont montré que, asymptotiquement presque sûrement, les probabilités de traversée quenched pour la percolation de Voronoi critique ne dépendaient pas de l’environnement. Dans cet article, nous montrons un résultat analogue pour les événements à j bras. En particulier, nous prouvons que la variance des probabilités quenched d’événements à j bras est au plus de l’ordre du carré de la probabilité annealed. Les deux principales nouvelles difficultés sont que les événements à j bras sont dégénérés et non monotones. Par ailleurs, nous utilisons ces résultats pour montrer qu’il existe ϵ>0 tel que la fonction de percolation annealed vérifie

∀p>1/2,θ an (p)≥ϵ(p-1/2) 1-ϵ .

Une des principales motivations de cet article est de fournir des outils permettant de faire une étude spectrale de la percolation de Voronoi.

Ahlberg, Griffiths, Morris and Tassion have proved that, asymptotically almost surely, the quenched crossing probabilities for critical planar Voronoi percolation do not depend on the environment. We prove an analogous result for arm events. In particular, we prove that the variance of the quenched probability of an arm event is at most a constant times the square of the annealed probability. The fact that the arm events are degenerate and non-monotonic add two major difficulties. As an application, we prove that there exists ϵ>0 such that the following holds for the annealed percolation function θ an :

∀p>1/2,θ an (p)≥ϵ(p-1/2) 1-ϵ .

One of our motivations is to provide tools for a spectral study of Voronoi percolation.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3710
Classification : 60K35, 60K37
Keywords: Percolation, Random environment, Voronoi tilings, Concentration, Critical Exponents
Mots-clés : Percolation, Environnement aléatoire, Diagrammes de Voronoi, Concentration, Exposants critiques

Vanneuville, Hugo 1

1 Institut Camille Jordan 43 boulevard du 11 novembre 1918 69622 Villeurbanne cedex, France 
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Vanneuville, Hugo. Quantitative quenched Voronoi percolation and applications. Annales de l'Institut Fourier, Online first, 35 p.

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