Phase diagram and critical behavior of a forest-fire model in a gradient of immunity

The forest-fire model with immune trees (FFMIT) is a cellular automaton early proposed by Drossel and Schwabl [Physica A 199, 183 (1993)], in which each site of a lattice can be in three possible states: occupied by a tree, empty, or occupied by a burning tree (fire). The trees grow at empty sites with probability p, healthy trees catch fire from adjacent burning trees with probability (1-g), where g is the immunity, and a burning tree becomes an empty site spontaneously. In this paper we study the FFMIT by means of the recently proposed gradient method (GM), considering the immunity as a uniform gradient along the horizontal axis of the lattice. The GM allows the simultaneous treatment of both the active and the inactive phases of the model in the same simulation. In this way, the study of a single-valued interface gives the critical point of the active-absorbing transition, whereas the study of a multivalued interface brings the percolation threshold into the active phase. Therefore we present a complete phase diagram for the FFMIT, for all range of p, where, besides the usual active-absorbing transition of the model, we locate a transition between the active percolating and the active nonpercolating phases. The average location and the width of both interfaces, as well as the absorbing and percolating cluster densities, obey a scaling behavior that is governed by the exponent α=1/(1+ν), where ν is the suitable correlation length exponent (ν(⊥) for the directed percolation transition and ν for the standard percolation transition). We also show that the GM allows us to calculate the critical exponents associated with both the order parameter of the absorbing transition and the number of particles in the multivalued interface. Besides, we show that by using the gradient method, the collapse in a single curve of cluster densities obtained for samples of different side is a very sensitive method in order to obtain the critical points and the percolation thresholds.