Electronic Communications in Probability

Boundary rules and breaking of self-organized criticality in 2D frozen percolation

Jacob van den Berg and Pierre Nolin

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We study frozen percolation on the (planar) triangular lattice, where connected components stop growing (“freeze”) as soon as their “size” becomes at least $N$, for some parameter $N \geq 1$. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites).

Diameter-frozen and volume-frozen percolation have been studied in previous works ([5, 15] and [6, 4], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these “boundary rules” in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 65, 15 pp.

Received: 9 September 2017
Accepted: 2 November 2017
First available in Project Euclid: 23 November 2017

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Zentralblatt MATH identifier

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

frozen percolation near-critical percolation self-organized criticality

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van den Berg, Jacob; Nolin, Pierre. Boundary rules and breaking of self-organized criticality in 2D frozen percolation. Electron. Commun. Probab. 22 (2017), paper no. 65, 15 pp. doi:10.1214/17-ECP98. https://projecteuclid.org/euclid.ecp/1511406073

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