Fires on trees

Jean Bertoin

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

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

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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 $\alpha gt;1/2$, to $0$ when $\alpha lt;1/2$, 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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Bertoin, 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 $\alpha &gt;1/2$, to $0$ when $\alpha &lt;1/2$, 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 $\alpha &gt;1/2$, to $0$ when $\alpha &lt;1/2$, 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 -

References

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