Consider the following forest-fire model where the possible locations of trees are the sites of 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 λ, 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 Drossel, Clar and Schwabl (1993) that then (in our notation) the density of vacant sites (at stationarity) is of order 1/ log (1/λ). 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/λ) the density is of the above mentioned order 1/ log (1/λ). We also obtain bounds on the cluster size distribution, showing that the scaling ansatz of Drossel et al. needs correction.
Jarai, A., & van den Berg, J. (2005). On the asymptotic density in a one-dimensional self-organized critical forest-fire model. Communications in Mathematical Physics, 253(3), 633-644. https://doi.org/10.1007/s00220-004-1200-x