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Fires on trees

By J Bertoin

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

Topics: Institute of Mathematics, 510 Mathematics
Publisher: Elsevier
Year: 2012
DOI identifier: 10.1214/11-AIHP435
OAI identifier: oai:www.zora.uzh.ch:70201
Provided by: ZORA
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