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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1511406073

Digital Object Identifier
doi:10.1214/17-ECP98

Mathematical Reviews number (MathSciNet)
MR3734104

Zentralblatt MATH identifier
06827047

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

Keywords
frozen percolation near-critical percolation self-organized criticality

Rights
Creative Commons Attribution 4.0 International License.

Citation

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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