Fires on trees

Abstract

We consider random dynamics on the edges of a uniform Cayley tree with n vertices, in which edges are either flammable, fireproof, or burnt. Every flammable edge is replaced by a fireproof edge at unit rate, while fires start at smaller rate n(-alpha) on each flammable edge, then propagate through the neighboring flammable 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 -> infinity, the terminal density of fireproof vertices converges to I 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.