Spreading analysis and finite-size scaling study of the critical behavior of a forest fire model with immune trees

In a forest fire model with immune trees (FFMIT) recently proposed by Drossel and Schwabl (Physica A 199 (1993) 183), the sites of the lattice are either empty, a tree or a burning tree, and at each time step the system is updated in parallel according to the following rules: (i) A burning tree becomes an empty site, (ii) trees grow with probability p from empty sites, and (iii) a green tree becomes a burning tree with probability (1-g) if at least one next neighbor is burning. Fixing an arbitrary grow probability (p) and starting with a small immunity (g), increments of g causes the fire density of the steady state to decrease until the fire becomes irreversibly extinguished at a certain critical point of coordinates {pc, gc}. The set of critical points defines a critical curve gc (p) which, in the L = ∞ limit, divides the {p, g}-plane in two regions: a steady state with fire fronts for g < gc(p) and an unique absorbing state with all sites occupied by green trees for g ≥ gc(p). It is shown, by means of an epidemic spreading analysis that the continuous irreversible phase transition between the stationary regime and the absorbing state belongs to the same universality class as Reggeon field theory. It is found that within the absorbing state time correlations are short-ranged and that the fire density decrease exponentially. Within the stationary regime the irreversible transition is studied in terms of the finite-size scaling theory that was already developed to describe reversible transitions. The results reported in this work show that problems emerging from (apparently) unrelated branches of science such as particle physics, catalysis, directed percolation, and epidemic spreading can be understood by means of a unified description.