Fires on trees

TL;DR

This paper analyzes the spread of fire and fireproofing on large random Cayley trees, revealing phase transitions in fireproof vertex density depending on the rate parameter, and investigates the connectivity of the resulting fireproof forest.

Contribution

It introduces a new stochastic model for fire spread on Cayley trees, characterizing phase transitions and connectivity properties as the tree size grows.

Findings

Fireproof vertex density converges to 1 for er > 1/2

Converges to 0 for er < 1/2

Non-degenerate limit when er = 1/2

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.