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>^{ -2}M(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.
Categories: cond-mat.stat-mech