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Pré-Publication, Document De Travail Année : 2010

Fires on trees

Résumé

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.
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Dates et versions

hal-00534386 , version 1 (09-11-2010)
hal-00534386 , version 2 (29-11-2010)

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Jean Bertoin. Fires on trees. 2010. ⟨hal-00534386v2⟩
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