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Fires on trees


Bertoin, J (2012). Fires on trees. Annales de l'Institut Henri Poincaré (B) Probabilities et Statistiques, 48(4):909-921.

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 -> infinity, the terminal density of fireproof vertices converges to I 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.

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 -> infinity, the terminal density of fireproof vertices converges to I 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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12 citations in Web of Science®
13 citations in Scopus®
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Additional indexing

Item Type:Journal Article, not refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:November 2012
Deposited On:22 Jan 2013 08:29
Last Modified:05 Apr 2016 16:18
Publisher:Elsevier
ISSN:0246-0203
Publisher DOI:https://doi.org/10.1214/11-AIHP435

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