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Robustness of the Critical Behaviour in the Stochastic Greenberg-Hastings Cellular Automaton Model?
Hugues Berry and Nazim Fates

We study a stochastic version of the Greenberg-Hastings cellular automaton, a simple model of wave propagation in reaction-diffusion media. Despite its apparent simplicity, its global dynamics displays various complex behaviours. Here, we investigate the influence of temporary or definitive failures of the cells of the grid. We show that a continuous decrease of the probability of excitation of cells triggers a drastic change of behaviour, driving the system from an “active” to an “extinct” steady state. Simulations show that this phenomenon is a nonequilibrium phase transition that belongs to directed percolation universality class. Observations show an amazing robustness of the critical behaviour with regard to topological perturbations: not only is the phase transition occurrence preserved, but its universality class remains directed percolation. We also demonstrate that the position of the critical threshold can be easily predicted as it decreases linearly with the inverse of the average number of neighbours per cell.

Keywords: Stochastic cellular automata, reaction-diffusion, non-equilibrium phase transitions, robustness

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