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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1511773731

Digital Object Identifier
doi:10.1214/16-AIHP776

Mathematical Reviews number (MathSciNet)
MR3729640

Zentralblatt MATH identifier
06847067

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

Keywords
Critical Bernoulli percolation Slab Russo–Seymour–Welsh theorem

Citation

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. https://projecteuclid.org/euclid.aihp/1511773731


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References

  • [1] M. Aizenman. On the number of incipient spanning clusters. Nuclear Phys. B 485 (1997) 551–582.
  • [2] B. Bollobas and O. Riordan. The critical probability for random Voronoi percolation in the plane is $\frac{1}{2}$. Probab. Theory Related Fields 136 (3) (2006) 417–468.
  • [3] H. Duminil-Copin, C. Hongler and P. Nolin. Connection probabilities and RSW-type bounds for the two-dimensional FK Ising model. Comm. Pure Appl. Math. 64 (9) (2011) 1165–1198.
  • [4] H. Duminil-Copin, V. Sidoravicius and V. Tassion. Continuity of the phase transition for planar random-cluster and Potts models with $1\leq q\leq4$. Available at arXiv:1505.04159.
  • [5] H. Duminil-Copin, V. Sidoravicius and V. Tassion. Absence of infinite cluster for critical Bernoulli percolation on slabs. Comm. Pure Appl. Math. 69 (7) (2016) 1397–1411.
  • [6] G. R. Grimmett. Percolation. Grundlehren der mathematischen Wissenschaften 321. Springer-Verlag, Berlin, 1999.
  • [7] H. Kesten. Percolation Theory for Mathematicians. Birkhäuser, Boston, MA, 1982.
  • [8] C. Newman, V. Tassion and W. Wu. Critical percolation and the minimal spanning tree in slabs. Available at arXiv:1512.09107.
  • [9] L. Russo. A note on percolation. Z. Wahrsch. Verw. Gebiete 43 (1) (1978) 39–48.
  • [10] L. Russo. On the critical percolation probabilities. Z. Wahrsch. Verw. Gebiete 56 (2) (1981) 229–237.
  • [11] P. D. Seymour and D. J. A. Welsh. Percolation probabilities on the square lattice. Ann. Discrete Math. 3 (1978) 227–245.
  • [12] V. Tassion. Crossing probabilities for Voronoi percolation. Ann. Probab. To appear, 2014. Available at arXiv:1410.6773.