# Fires on trees

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.
keyword :
Document type :
Preprints, Working Papers, ...
2010
Domain :
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-00534386
Contributor : Jean Bertoin <>
Submitted on : Monday, November 29, 2010 - 7:42:11 PM
Last modification on : Thursday, April 27, 2017 - 9:46:04 AM
Document(s) archivé(s) le : Saturday, December 3, 2016 - 12:32:01 AM

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firesontrees.pdf
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### Identifiers

• HAL Id : hal-00534386, version 2
• ARXIV : 1011.2308

### Citation

Jean Bertoin. Fires on trees. 2010. <hal-00534386v2>

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