Synchronization and Coarsening (without SOC) in a Forest-Fire Model

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We study the long-time dynamics of a forest-fire model with deterministic tree growth and instantaneous burning of entire forests by stochastic lightning strikes. Asymptotically the system organizes into a coarsening self-similar mosaic of synchronized patches within which trees regrow and burn simultaneously. We show that the average patch length <L> grows linearly with time as t-->oo. The number density of patches of length L, N(L,t), scales as <L>⁻²M(L/<L>), and within a mean-field rate equation description we find that this scaling function decays as e⁻¹/ˣ for x-->0, and as e⁻ˣ for x-->oo. In one dimension, we develop an event-driven cluster algorithm to study the asymptotic behavior of large systems. Our numerical results are consistent with mean-field predictions for patch coarsening.