The Annals of Probability

Asymptotics of one-dimensional forest fire processes

Xavier Bressaud and Nicolas Fournier

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Abstract

We consider the so-called one-dimensional forest fire process. At each site of ℤ, a tree appears at rate 1. At each site of ℤ, a fire starts at rate λ>0, immediately destroying the whole corresponding connected component of trees. We show that when λ is made to tend to 0 with an appropriate normalization, the forest fire process tends to a uniquely defined process, the dynamics of which we precisely describe. The normalization consists of accelerating time by a factor log(1/λ) and of compressing space by a factor λ log(1/λ). The limit process is quite simple: it can be built using a graphical construction and can be perfectly simulated. Finally, we derive some asymptotic estimates (when λ→0) for the cluster-size distribution of the forest fire process.

Article information

Source
Ann. Probab., Volume 38, Number 5 (2010), 1783-1816.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1282053772

Digital Object Identifier
doi:10.1214/09-AOP524

Mathematical Reviews number (MathSciNet)
MR2722786

Zentralblatt MATH identifier
1205.60167

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

Keywords
Stochastic interacting particle systems self-organized criticality forest fire model

Citation

Bressaud, Xavier; Fournier, Nicolas. Asymptotics of one-dimensional forest fire processes. Ann. Probab. 38 (2010), no. 5, 1783--1816. doi:10.1214/09-AOP524. https://projecteuclid.org/euclid.aop/1282053772


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References

  • [1] Bak, P., Tang, C. and Wiesenfeld, K. (1987). Self-organized criticality: An explanation of 1/f noise. Phys. Rev. Lett. 59 381–384.
  • [2] Bak, P., Tang, C. and Wiesenfeld, K. (1988). Self-organized criticality. Phys. Rev. A (3) 38 364–374.
  • [3] Bressaud, X. and Fournier, N. (2009). On the invariant distribution of a one-dimensional avalanche process. Ann. Probab. 37 48–77.
  • [4] Brouwer, R. and Pennanen, J. (2006). The cluster size distribution for a forest-fire process on ℤ. Electron. J. Probab. 11 1133–1143.
  • [5] Dhar, D. (2006). Theoretical studies of self-organized criticality. Phys. A 369 29–70.
  • [6] Drossel, B. and Schwabl, F. (1992). Self-organized critical forest-fire model. Phys. Rev. Lett. 69 1629–1632.
  • [7] Drossel, B., Clar, S. and Schwabl, F. (1993). Exact results for the one-dimensional self-organized critical forest-fire model. Phys. Rev. Lett. 71 3739–3742.
  • [8] Dürre, M. (2006). Existence of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. J. Probab. 11 513–539.
  • [9] Dürre, M. (2006). Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. Comm. Probab. 11 304–315.
  • [10] Grassberger, P. (2002). Critical behaviour of the Drossel–Schwabl forest fire model. New J. Phys. 4 17.1–17.15.
  • [11] Henley, C. L. (1989). Self-organized percolation: A simpler model. Bull. Amer. Math. Soc. 34 838.
  • [12] Jensen, H. J. (1998). Self-Organized Criticality. Cambridge Lecture Notes in Physics 10. Cambridge Univ. Press, Cambridge.
  • [13] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 276. Springer, New York.
  • [14] Olami, Z., Feder, H. J. S. and Christensen, K. (1992). Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes. Phys. Rev. Lett. 68 1244–1247.
  • [15] van den Berg, J. and Brouwer, R. (2006). Self-organized forest-fires near the critical time. Comm. Math. Phys. 267 265–277.
  • [16] van den Berg, J. and Járai, A. A. (2005). On the asymptotic density in a one-dimensional self-organized critical forest-fire model. Comm. Math. Phys. 253 633–644.