Source: Ann. Probab. Volume 38, Number 5
(2010), 1783-1816.
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.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
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.
Mathematical Reviews (MathSciNet):
MR949160
[2] Bak, P., Tang, C. and Wiesenfeld, K. (1988). Self-organized criticality. Phys. Rev. A (3) 38 364–374.
Mathematical Reviews (MathSciNet):
MR949160
[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.
Mathematical Reviews (MathSciNet):
MR776231
[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.
