Electronic Communications in Probability

The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions

Artem Sapozhnikov

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This note is motivated by results of Angel, Goodman, den Hollander and Slade (2008) and Da, Sapozhnikov and Vagvolgyi (2009) about global relations between the invasion percolation cluster (IPC) and the incipient infinite cluster (IIC) on regular trees and on two dimensional lattices, respectively. Namely, that the laws of the two objects are mutually singular, and, in the case of regular trees, that the IIC stochastically dominates the IPC. We prove that on two dimensional lattices, the IIC does not stochastically dominate the IPC. This is the first example showing that the relation between the IIC and IPC is significantly different on trees and in two dimensions.

Article information

Electron. Commun. Probab., Volume 16 (2011), paper no. 68, 775-780.

Accepted: 30 November 2011
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Invasion percolation incipient infinite cluster critical percolation near-critical percolation correlation length stochastic domination

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Sapozhnikov, Artem. The incipient infinite cluster does not stochastically dominate the invasion percolation cluster in two dimensions. Electron. Commun. Probab. 16 (2011), paper no. 68, 775--780. doi:10.1214/ECP.v16-1684. https://projecteuclid.org/euclid.ecp/1465262024

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