A shape theorem for epidemics and forest fires with finite range interactions (Q1317220)

In a model for a forest fire, each point of \(\mathbb{Z}^ 2\) may be in one of the states \{alive, on fire, burnt\}. A burning tree emits sparks at rate \(a\), and such a spark ignites another tree at a site chosen at random from \(\mathbb{Z}^ 2\) according to a mass function \(g\). A burning tree remains on fire for a random time with distribution function \(F\). In the supercritical case (i.e., a single burning tree can ignite the entire forest with positive probability), a shape theorem is proved for the growth of the fire. This generalizes the corresponding theorem in the nearest-neighbour case (when \(g\) is concentrated on the unit vectors) proved by \textit{J. T. Cox} and \textit{R. Durrett} [Stochastic Processes Appl. 30, 171-191 (1988; Zbl 0667.92016)].