Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Crossing probabilities for critical Bernoulli percolation on slabs

Deepan Basu and Artem Sapozhnikov

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We prove that in the critical Bernoulli percolation on graphs $\mathbb{Z}^{2}\times\{0,\ldots,k-1\}^{d-2}$, for each $\rho>0$, the probability of open left-right crossing of rectangle $[0,\rho N]\times[0,N]\times[0,k-1]^{d-2}$ is uniformly positive.


On démontre que dans la percolation de Bernoulli critique sur le graphe $\mathbb{Z}^{2}\times\{0,\ldots,k-1\}^{d-2}$, pour chaque $\rho>0$, la probabilité d’avoir un passage de gauche à droite ouvert dans $[0,\rho N]\times[0,N]\times[0,k-1]^{d-2}$ est uniformément positive.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 53, Number 4 (2017), 1921-1933.

Received: 16 December 2015
Revised: 16 June 2016
Accepted: 28 June 2016
First available in Project Euclid: 27 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Critical Bernoulli percolation Slab Russo–Seymour–Welsh theorem


Basu, Deepan; Sapozhnikov, Artem. Crossing probabilities for critical Bernoulli percolation on slabs. Ann. Inst. H. Poincaré Probab. Statist. 53 (2017), no. 4, 1921--1933. doi:10.1214/16-AIHP776.

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