On the asymptotic density in a one-dimensional self-organized critical forest-fire model.

*(English)*Zbl 1079.82010Summary: Consider the following forest-fire model where the possible locations of trees are the sites of \(\mathbb Z\). Each site has two possible states: ‘vacant’ or ‘occupied’. Vacant sites become occupied at rate \(1\). At each site ignition (by lightning) occurs at ignition rate \(\lambda\), the parameter of the model. When a site is ignited, its occupied cluster becomes vacant instantaneously.

In the literature similar models have been studied for discrete time. The most interesting behaviour occurs when the ignition rate approaches \(0\). It has been stated by B. Drossel, S. Clar and F. Schwabl [Phys. Rev. Lett. 71, No. 23, 3739–3742 (1993)] that then (in our notation) the density of vacant sites (at stationarity) is of order \(1/\log(1/\lambda)\). Their argument uses a ‘scaling ansatz’ and is not rigorous. We give a rigorous and mathematically more natural proof for our version of the model, and point out how it can be modified for the model studied by Drossel et al. Our proof shows that regardless of the initial configuration, already after time of order \(\log(1/\lambda)\) the density is of the above mentioned order \(1/\log(1/\lambda)\). We also obtain bounds on the cluster size distribution, showing that the scaling ansatz of Drossel et al. needs correction.

In the literature similar models have been studied for discrete time. The most interesting behaviour occurs when the ignition rate approaches \(0\). It has been stated by B. Drossel, S. Clar and F. Schwabl [Phys. Rev. Lett. 71, No. 23, 3739–3742 (1993)] that then (in our notation) the density of vacant sites (at stationarity) is of order \(1/\log(1/\lambda)\). Their argument uses a ‘scaling ansatz’ and is not rigorous. We give a rigorous and mathematically more natural proof for our version of the model, and point out how it can be modified for the model studied by Drossel et al. Our proof shows that regardless of the initial configuration, already after time of order \(\log(1/\lambda)\) the density is of the above mentioned order \(1/\log(1/\lambda)\). We also obtain bounds on the cluster size distribution, showing that the scaling ansatz of Drossel et al. needs correction.

##### MSC:

82C22 | Interacting particle systems in time-dependent statistical mechanics |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

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\textit{J. van den Berg} and \textit{A. A. Járai}, Commun. Math. Phys. 253, No. 3, 633--644 (2005; Zbl 1079.82010)

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##### References:

[1] | Drossel, B., Clar, S., Schwabl, F.: Exact Results for the One-Dimensional Self-Organized Critical Forest-Fire Model. Phys. Rev. Lett. 71, 3739-3742 (1993) · doi:10.1103/PhysRevLett.71.3739 |

[2] | Jensen, H.J.: Self-Organized Criticality. Cambridge Lecture Notes in Physics, Cambridge: Cambridge University Press, 1998 · Zbl 0945.70001 |

[3] | Liggett, T.M.: Interacting Particle Systems. Berlin: Springer-Verlag, 1985 · Zbl 0559.60078 |

[4] | Malamud, B.D., Morein, G., Turcotte, D.L.: Forest Fires: An Example of Self-Organized Critical Behaviour. Science 281, 1840-1841 (1998) · doi:10.1126/science.281.5384.1840 |

[5] | Schenk, K., Drossel, B., Schwabl, F.: Self-organized critical forest-fire model on large scales. Phys. Rev. E 65, 026135-1-8 (2002) · Zbl 1046.82031 · doi:10.1103/PhysRevE.65.026135 |

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