Fires on trees

Bertoin, Jean

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 909-921.

On considère la dynamique aléatoire suivante sur un arbre de Cayley uniforme avec $n$ sommets et pour lequel les arêtes peuvent être inflammables, ignifugées, ou brûlées. Au temps initial, toutes les arêtes sont inflammables, et chaque arête inflammable est remplacée à taux $1$ par une arête ignifugée, indépendamment des autres arêtes. Par ailleurs, une arête inflammable peut également prendre feu avec un taux $n^{-\alpha }$, et le feu se propage alors le long des arêtes inflammables voisines et n’est stoppé que par les arêtes ignifugées. Nous montrons que lorsque $n\to \infty$, la densité terminale des sommets ignifugés converge vers $1$ si $\alpha >1/2$, vers $0$ si $\alpha <1/2$, et vers une variable aléatoire non dégénérée pour $\alpha =1/2$. On étudie ensuite la connectivité de la forêt ignifugée, et plus particulièrement l’existence de composantes géantes.

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 $n\to \infty$, the terminal 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.

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