Electronic Communications in Probability

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

Deepan Basu and Artem Sapozhnikov

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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

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

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

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Zentralblatt MATH identifier

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

incipient infinite cluster percolation criticality quasi-multiplicativity slab

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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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