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Fires on trees. (English. French summary) Zbl 1263.60083

Author’s 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 \(n\to\infty\), 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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