# Fires on trees

Annales de l'I.H.P. Probabilités et statistiques (2012)

- Volume: 48, Issue: 4, page 909-921
- ISSN: 0246-0203

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topBertoin, Jean. "Fires on trees." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 909-921. <http://eudml.org/doc/271957>.

@article{Bertoin2012,

abstract = {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\rightarrow \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.},

author = {Bertoin, Jean},

journal = {Annales de l'I.H.P. Probabilités et statistiques},

keywords = {Cayley tree; fire model; percolation; giant component},

language = {eng},

number = {4},

pages = {909-921},

publisher = {Gauthier-Villars},

title = {Fires on trees},

url = {http://eudml.org/doc/271957},

volume = {48},

year = {2012},

}

TY - JOUR

AU - Bertoin, Jean

TI - Fires on trees

JO - Annales de l'I.H.P. Probabilités et statistiques

PY - 2012

PB - Gauthier-Villars

VL - 48

IS - 4

SP - 909

EP - 921

AB - 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\rightarrow \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.

LA - eng

KW - Cayley tree; fire model; percolation; giant component

UR - http://eudml.org/doc/271957

ER -

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