Renormalization scheme for forest-fire models

We introduce a renormalization scheme for forest-fire models in order to characterize the nature of the critical state and its scale-invariant dynamics. We study one- and two-dimensional models defining a characterization of the phase space that allows us to describe the evolution of the dynamics under a scale transformation. We show the existence of a relevant critical parameter associated with a repulsive fixed point in the phase space, From the renormalization-group point of view these models are therefore critical in the usual sense, because the fixed-point value of the control parameter is crucial in order to get criticality. This general scheme allows us to calculate analytically the critical exponent nu which describes the approach to the critical point along the repulsive direction and the exponent tau that characterizes the distribution of forest clusters at the critical point. We obtain nu = 1.0, tau = 1.0 and nu = 0.65, tau = 1.16, respectively, for the one- and two-dimensional cases, in very good agreement with exact and numerical results.