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2017 Kesten’s incipient infinite cluster and quasi-multiplicativity of crossing probabilities
Deepan Basu, Artem Sapozhnikov
Electron. Commun. Probab. 22: 1-12 (2017). DOI: 10.1214/17-ECP56

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 $d>6$.

Citation

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Deepan Basu. Artem Sapozhnikov. "Kesten’s incipient infinite cluster and quasi-multiplicativity of crossing probabilities." Electron. Commun. Probab. 22 1 - 12, 2017. https://doi.org/10.1214/17-ECP56

Information

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

zbMATH: 1365.60079
MathSciNet: MR3652039
Digital Object Identifier: 10.1214/17-ECP56

Subjects:
Primary: 60K35, 82B27, 82B43

JOURNAL ARTICLE
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