Renormalization Group Approach to the Critical Behavior of the Forest-Fire Model

We introduce a renormalization scheme for the one- and two-dimensional forest-fire model in order to characterize the nature of the critical state and its scale invariant dynamics. We show the existence of a relevant scaling field associated with a repulsive fixed point. This model is therefore critical in the usual sense because the control parameter has to be tuned to its critical value in order to get criticality. It turns out that this is not just the condition for a time scale separation. The critical exponents are computed analytically and we obtain $\nu =1.0\mathrm{}$, $\tau =1.0\mathrm{}$ and $\nu =0.65\mathrm{}$, $\tau =1.16\mathrm{}$, respectively, for the one- and two-dimensional cases, in very good agreement with numerical simulations.