Phase Transition in a Stochastic Forest Fire Model and Effects of the Definition of Neighbourhood

We present results on a stochastic forest fire model, where the influence of the neighbour trees is treated in a more realistic way than usual and the definition of neighbourhood can be tuned by an additional parameter. This model exhibits a surprisingly sharp phase transition which can be shifted by redefinition of neighbourhood. The results can also be interpreted in terms of disease-spreading and are quite unsettling from the epidemologist’s point of view, since variation of one crucial parameter only by a few percent can result in the change from endemic to epidemic behaviour.

💡 Research Summary

The paper introduces a stochastic forest‑fire cellular automaton in which the influence of neighboring trees is not fixed to a conventional 4‑ or 8‑neighbourhood but can be continuously tuned by a single parameter α (0 ≤ α ≤ 1). When α = 0 the model reduces to the classic von Neumann (4‑neighbour) rule, and when α = 1 it becomes the Moore (8‑neighbour) rule; intermediate values represent a weighted mixture of the two neighbourhoods. Each lattice site can be in one of three states – dry, burning, or burnt – and the transition probabilities are defined as follows: a dry site becomes burning with probability p·