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

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 -α 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, the terminal density of fireproof vertices converges to 1 when α>1/2, to 0 when α<1/2, and to some non-degenerate random variable when α=1/2. We further study the connectivity of the fireproof forest, in particular the existence of a giant component.

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 -α , 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, la densité terminale des sommets ignifugés converge vers 1 si α>1/2, vers 0 si α<1/2, et vers une variable aléatoire non dégénérée pour α=1/2. On étudie ensuite la connectivité de la forêt ignifugée, et plus particulièrement l’existence de composantes géantes.

DOI: 10.1214/11-AIHP435
Classification: 60J80,  60K35
Keywords: Cayley tree, fire model, percolation, giant component
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Bertoin, Jean. Fires on trees. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 909-921. doi : 10.1214/11-AIHP435. http://www.numdam.org/articles/10.1214/11-AIHP435/

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