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January 2003 Incipient infinite percolation clusters in 2D
Antal A. Járai
Ann. Probab. 31(1): 444-485 (January 2003). DOI: 10.1214/aop/1046294317

Abstract

We study several kinds of large critical percolation clusters in two dimensions. We show that from the microscopic (lattice scale) perspective these clusters can be described by Kesten's incipient infinite cluster (IIC), as was conjectured by Aizenman. More specifically, we establish this for incipient spanning clusters, large clusters in a finite box and the inhomogeneous model of Chayes, Chayes and Durrett. Our results prove the equivalence of several natural definitions of the IIC.

We also show that for any $k \ge 1$ the difference in size between the $k$th and $(k+1)$st largest critical clusters in a finite box goes to infinity in probability as the size of the box goes to infinity. In addition, the distribution of the Chayes--Chayes--Durrett cluster is shown to be singular with respect to the IIC.

Citation

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Antal A. Járai. "Incipient infinite percolation clusters in 2D." Ann. Probab. 31 (1) 444 - 485, January 2003. https://doi.org/10.1214/aop/1046294317

Information

Published: January 2003
First available in Project Euclid: 26 February 2003

zbMATH: 1061.60106
MathSciNet: MR1959799
Digital Object Identifier: 10.1214/aop/1046294317

Subjects:
Primary: 60K35
Secondary: 82B43

Keywords: Critical phenomena , Incipient infinite cluster , percolation , spanning cluster

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 1 • January 2003
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