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

NARA GUISONI; ERNESTO S. LOSCAR; EZEQUIEL V. ALBANO

PHYSICAL REVIEW E - STATISTICAL PHYSICS, PLASMAS, FLUIDS AND RELATED INTERDISCIPLINARY TOPICS

APS - American Physical Society

Año: 2011 vol. 83 p. 111251 - 111259

1063-651X

The forest-fire model with immune trees (FFMIT) is a cellular automaton early proposed by Drossel and Schwabl (B. Drossel and F. Schwabl, Physica A extbf{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-absorbingtransition of the model, we locate a transition between theactive percolating and the active nonpercolating phases. Theaverage location and the width of both interfaces, as well asthe absorbing and the percolating cluster densities, obey ascaling behavior that is governed by the exponent $alpha=1/(1+ u)$, where $ u$ is the suitable correlation length exponent($ u_{perp}$ for the directed percolation transition and$ u$ for the standard percolation transition). We also showthat the GM allows us to calculate the critical exponentsassociated with both the order parameter of the absorbingtransition and the number of particles in the multivaluedinterface. Besides, we show that by using the gradient methodthe collapse in a single curve of cluster densities obtainedfor samples of different side is a very sensitive method inorder to obtain the critical points and the percolationthresholds.