Electronic Journal of Probability

A forest-fire model on the upper half-plane

Robert Graf

Full-text: Open access

Abstract

We consider a discrete forest-fire model on the upper half-plane of the two-dimensional square lattice. Each site can have one of the following two states: "vacant" or "occupied by a tree". At the starting time all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate 1, independently for all sites. If an occupied cluster reaches the boundary of the upper half plane or if it is about to become infinite, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant. Additionally, we demand that the model is invariant under translations along the x-axis. We prove that such a model exists and arises naturally as a subseqential limit of forest-fire processes in finite boxes when the box size tends to infinity. Moreover, the model exhibits a phase transition in the following sense: There exists a critical time $t_c$ (which corresponds with the critical probability $p_c$ in ordinary site percolation by $1 - e^{-t_c} = p_c$) such that before $t_c$, only sites close to the boundary have been affected by destruction, whereas after $t_c$, sites on the entire half-plane have been affected by destruction.<br />

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 8, 27 pp.

Dates
Accepted: 13 January 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065650

Digital Object Identifier
doi:10.1214/EJP.v19-2625

Mathematical Reviews number (MathSciNet)
MR3164761

Zentralblatt MATH identifier
1291.82037

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82C22: Interacting particle systems [See also 60K35] 82B43: Percolation [See also 60K35]

Keywords
forest-fire model upper half-plane self-organized criticality phase transition

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Graf, Robert. A forest-fire model on the upper half-plane. Electron. J. Probab. 19 (2014), paper no. 8, 27 pp. doi:10.1214/EJP.v19-2625. https://projecteuclid.org/euclid.ejp/1465065650


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