The Annals of Probability

Incipient infinite percolation clusters in 2D

Antal A. Járai

Full-text: Open access

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.

Article information

Source
Ann. Probab., Volume 31, Number 1 (2003), 444-485.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1046294317

Digital Object Identifier
doi:10.1214/aop/1046294317

Mathematical Reviews number (MathSciNet)
MR1959799

Zentralblatt MATH identifier
1061.60106

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

Keywords
Percolation incipient infinite cluster spanning cluster critical phenomena

Citation

Járai, Antal A. Incipient infinite percolation clusters in 2D. Ann. Probab. 31 (2003), no. 1, 444--485. doi:10.1214/aop/1046294317. https://projecteuclid.org/euclid.aop/1046294317


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  • VANCOUVER, BC CANADA V6T 1Z2 E-MAIL: jarai@math.ubc.ca