Electronic Journal of Probability

A forest-fire model on the upper half-plane

Robert Graf

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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

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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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  • P. Bak. How nature works. The science of self-organized criticality. Copernicus, New York, 1996. xiv+212 pp. ISBN: 0-387-94791-4; 0-387-98738-X.
  • P. Bak, C. Tang, and K. Wiesenfeld. Self-organized criticality: an explanation of 1/f noise. Phys. Rev. Lett. 59 (1987), no. 4, 381-344.
  • X. Bressaud and N. Fournier. Asymptotics of one-dimensional forest fire processes. Ann. Probab. 38 (2010), no. 5, 1783-1816.
  • X. Bressaud and N. Fournier. One-dimensional general forest fire processes. arXiv:1101.0480, 2011.
  • J. T. Chayes and L. Chayes. Critical points and intermediate phases on wedges of ${\bf Z}^ d$. J. Phys. A 19 (1986), no. 15, 3033-3048.
  • B. Drossel and F. Schwabl. Self-organized critical forest-fire model. Phys. Rev. Lett. 69 (1992), no. 11, 1629-1632.
  • M. Dürre. Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. J. Probab. 11 (2006), no. 21, 513-539 (electronic).
  • M. Dürre. Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. Comm. Probab. 11 (2006), 304-315 (electronic).
  • M. Dürre. Self-organized critical phenomena. Forest fire and sandpile models. PhD thesis, LMU München, 2009.
  • G. Grimmett. Bond percolation on subsets of the square lattice, and the threshold between one-dimensional and two-dimensional behaviour. J. Phys. A 16 (1983), no. 3, 599-604.
  • G. Grimmett. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6.
  • H. J. Jensen. Self-organized criticality. Emergent complex behavior in physical and biological systems. Cambridge Lecture Notes in Physics, 10. Cambridge University Press, Cambridge, 1998. xiv+153 pp. ISBN: 0-521-48371-9.
  • T. M. Liggett. Interacting particle systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 276. Springer-Verlag, New York, 1985. xv+488 pp. ISBN: 0-387-96069-4.
  • B. Ráth and B. Tóth. Erdős-Rényi random graphs + forest fires = self-organized criticality. Electron. J. Probab. 14 (2009), no. 45, 1290-1327 (electronic).
  • A. N. Shiryayev. Probability. Translated from the Russian by R. P. Boas. Graduate Texts in Mathematics, 95. Springer-Verlag, New York, 1984. xi+577 pp. ISBN: 0-387-90898-6.
  • A. Stahl. Existence of a stationary distribution for multi-dimensional infinite volume forest-fire processes. arXiv:1203.5533, 2012.
  • J. van den Berg and R. Brouwer. Self-destructive percolation. Random Structures Algorithms 24 (2004), no. 4, 480-501.
  • J. van den Berg and R. Brouwer. Self-organized forest-fires near the critical time. Comm. Math. Phys. 267 (2006), no. 1, 265-277.