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Asymptotics of one-dimensional forest fire processes. (English) Zbl 1205.60167
Two families of independent Poisson processes, one with rate \(1\) another with rate \(\lambda \), are combined to model the so-called one-dimensional forest fire process (FFP). It is known that, from the point of view of self-organized criticality, an interesting behavior appears in the asymptotic regime \(\lambda \rightarrow 0\). The asymptotic behavior of such \(\lambda \)-FFP model as \(\lambda \rightarrow 0\) has been studied heuristically and numerically. However, mathematically rigorous results are scarse. In this paper a limit theorem is derived rigorusly, showing that the \(\lambda \)-FFP converges (under suitable rescaling) to some limit forest fire process. Its dynamics is precisely described. The limiting process is unique, can be built by means of a graphical construction and is amenable to reliable simulations. Asymptotic estimates are derived for the cluster-size distribution of the forest fire process.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C27 Dynamic critical phenomena in statistical mechanics
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[1] Bak, P., Tang, C. and Wiesenfeld, K. (1987). Self-organized criticality: An explanation of 1/ f noise. Phys. Rev. Lett. 59 381-384. · Zbl 1230.37103 · doi:10.1103/PhysRevA.38.364
[2] Bak, P., Tang, C. and Wiesenfeld, K. (1988). Self-organized criticality. Phys. Rev. A (3) 38 364-374. · Zbl 1230.37103 · doi:10.1103/PhysRevA.38.364
[3] Bressaud, X. and Fournier, N. (2009). On the invariant distribution of a one-dimensional avalanche process. Ann. Probab. 37 48-77. · Zbl 1171.60022 · doi:10.1214/08-AOP396
[4] Brouwer, R. and Pennanen, J. (2006). The cluster size distribution for a forest-fire process on \Bbb Z. Electron. J. Probab. 11 1133-1143. · Zbl 1128.60080 · eudml:128292
[5] Dhar, D. (2006). Theoretical studies of self-organized criticality. Phys. A 369 29-70. · doi:10.1016/j.physa.2006.04.004
[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. · Zbl 1109.60081 · eudml:127184
[9] Dürre, M. (2006). Uniqueness of multi-dimensional infinite volume self-organized critical forest-fire models. Electron. Comm. Probab. 11 304-315. · Zbl 1130.60091 · eudml:128418
[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. · Zbl 0945.70001
[13] Liggett, T. M. (1985). Interacting Particle Systems. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 276 . Springer, New York. · Zbl 0559.60078
[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. · Zbl 1111.60080 · doi:10.1007/s00220-006-0025-1
[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. · Zbl 1079.82010 · doi:10.1007/s00220-004-1200-x
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