Electronic Communications in Probability

Kesten’s incipient infinite cluster and quasi-multiplicativity of crossing probabilities

Deepan Basu and Artem Sapozhnikov

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Abstract

In this paper we consider Bernoulli percolation on an infinite connected bounded degrees graph $G$. Assuming the uniqueness of the infinite open cluster and a quasi-multiplicativity of crossing probabilities, we prove the existence of Kesten’s incipient infinite cluster. We show that our assumptions are satisfied if $G$ is a slab $\mathbb{Z} ^2\times \{0,\ldots ,k\}^{d-2}$ ($d\geq 2$, $k\geq 0$). We also argue that the quasi-multiplicativity assumption should hold for $G=\mathbb{Z} ^d$ when $d<6$, but not when $d>6$.

Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 26, 12 pp.

Dates
Received: 24 March 2016
Accepted: 2 May 2017
First available in Project Euclid: 6 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1494036081

Digital Object Identifier
doi:10.1214/17-ECP56

Mathematical Reviews number (MathSciNet)
MR3652039

Zentralblatt MATH identifier
1365.60079

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B27: Critical phenomena 82B43: Percolation [See also 60K35]

Keywords
incipient infinite cluster percolation criticality quasi-multiplicativity slab

Rights
Creative Commons Attribution 4.0 International License.

Citation

Basu, Deepan; Sapozhnikov, Artem. Kesten’s incipient infinite cluster and quasi-multiplicativity of crossing probabilities. Electron. Commun. Probab. 22 (2017), paper no. 26, 12 pp. doi:10.1214/17-ECP56. https://projecteuclid.org/euclid.ecp/1494036081


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References

  • [1] Aizenman, M.: On the number of incipient spanning clusters. Nucl. Phys. B 485, (1997), 551–582.
  • [2] Basu, D.: Generalizations and interpretations of Incipient Infinite Cluster measure on planar lattices and slabs. PhD thesis, (2016).
  • [3] Basu, D. and Sapozhnikov, A.: Crossing probabilities for critical Bernoulli percolation on slabs. To appear in Ann. Inst. Henri Poincaré Probab. Stat., arXiv:1512.05178
  • [4] Benjamini, I. and Schramm, O.: Percolation beyond $\mathbb{Z} ^d$, many questions and a few answers. Electron. Comm. Probab. 1, (1996), 71–82.
  • [5] Borgs, C., Chayes, J. T., Kesten, H. and Spencer, J.: Uniform boundedness of critical crossing probabilities implies hyperscaling. Random Structures Algorithms 15, (1999), 368–413.
  • [6] Coniglio, A.: Shapes, surfaces and interfaces in percolation clusters. Proc Les Houches Conf on Physics of Finely Divided Matter, M. Daoud and N. Boccara (Editors), Springer-Verlag, Berlin, (1985), 84–109.
  • [7] Chayes, J. T. and Chayes, L.: On the upper critical dimension of Bernoulli percolation. Comm. Math. Phys. 113(1), (1987), 27–48.
  • [8] Damron, M. and Sapozhnikov, A.: Outlets of $2D$ invasion percolation and multiple-armed incipient infinite clusters. Probab. Th. Rel. Fields 150, (2011), 257–294.
  • [9] Duminil-Copin, H., Sidoravicius, V. and Tassion, V.: Absence of infinite cluster for critical Bernoulli percolation on slabs. Comm. Pure Appl. Math. 69(7), (2016), 1397–1411.
  • [10] Fitzner, R. and van der Hofstad, R.: Mean-field behavior for nearest-neighbor percolation in $d>10$ arXiv:1506.07977
  • [11] Häggström, O. and Peres, Y.: Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously. Probab. Th. Rel. Fields 113, (1999), 273–285.
  • [12] Hara, T.: Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals. Ann. Probab. 36(2), (2008), 530–593.
  • [13] Hara, T. and Slade, G.: Mean-field critical behaviour for percolation in high dimensions. Comm. Math. Phys. 128(2), (1990), 333–391.
  • [14] Heydenreich, M., van der Hofstad, R. and Hulshof, T.: High-dimensional incipient infinite clusters revisited. J. Stat. Phys. 155, (2014), 966–1025.
  • [15] van der Hofstad, R. and Jarai, A.: The incipient infinite cluster for high-dimensional unoriented percolation. J. Stat. Phys. 114(3), (2004), 625–663.
  • [16] Hopf, E.: An inequality for positive linear integral operators. J. Math. Mech. 12, (1963), 683–692.
  • [17] Jarai, A.: Incipient infinite percolation clusters in $2D$. Ann. Probab. 31(1), (2003), 444–485.
  • [18] Kesten, H.: The incipient infinite cluster in two-dimensional percolation. Probab. Th. Rel. Fields 73, (1986), 369–394.
  • [19] Kozma, G. and Nachmias, A.: Arm exponents in high dimensional percolation. J. Amer. Math. Soc. 24(2), (2011), 375–409.
  • [20] Newman, Ch., Tassion, V. and Wu, W.: Critical percolation and the minimal spanning tree in slabs, arXiv:1512.09107
  • [21] Russo, L.: A note on percolation. Z. Wahrsch. Verw. Gebiete 43(1), (1978), 39–48.
  • [22] Schonmann, R.: Stability of infinite clusters in supercritical percolation. Probab. Th. Rel. Fields 113(2), (1999), 287–300.
  • [23] Seymour, P. D. and Welsh, D. J. A.: Percolation probabilities on the square lattice. Annals of Discrete Mathematics 3, (1978), 227–245.