Self-organized critical forest-fire model: Mean-field theory and simulation results in 1 to 6 dimenisons

We argue that the critical forest-fire model (FFM) introduced by Drossel and Schwabl [Phys. Rev. Lett. 69, 1629 (1992)] is a critical branching process in the mean-field approximation where the number s of trees burned in a forest fire is power-law distributed with exponent ${\mathrm{\tau }}_{\mathit{s}}$=5/2. The mean-field model of the FFM in finite dimension is the percolation model and, as in percolation, the upper critical dimension is 6. Simulations show that the FFM becomes increasingly percolationlike with increasing dimension d and is, within error bars, fully consistent with the percolation results when d≥3.