Fires on trees

Abstract

We consider random dynamics on the edges of a uniform Cayley tree with $n$ vertices, in which edges are either inflammable, fireproof, or burt. Every inflammable edge is replaced by a fireproof edge at unit rate, while fires start at smaller rate $n^{-\alpha }$ on each inflammable edge, then propagate through the neighboring inflammable edges and are only stopped at fireproof edges. A vertex is called fireproof when all its adjacent edges are fireproof. We show that as $n\to \infty$, the density of fireproof vertices converges to $1$ when $\alpha >1/2$, to $0$ when $\alpha <1/2$, and to some non-degenerate random variable when $\alpha =1/2$. We further study the connectivity of the fireproof forest, in particular the existence of a giant component.