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On the sizes of burnt and fireproof components for fires on a large Cayley tree. (English. French summary) Zbl 1336.60193
Summary: We continue the study initiated by J. Bertoin [Ann. Inst. Henri Poincaré, Probab. Stat. 48, No. 4, 909–921 (2012; Zbl 1263.60083)] of random dynamics on the edges of a uniform Cayley tree with \(n\) vertices in which, successively, each edge is either set on fire with some fixed probability \(p_{n}\) or fireproof with probability \(1-p_{n}\). An edge which is set on fire burns and sets on fire its flammable neighbors, the fire then propagates in the tree, only stopped by fireproof edges. We study the distribution of the proportion of burnt and fireproof vertices and the sizes of the burnt or fireproof connected components as \(n\to\infty\) regarding the asymptotic behavior of \(p_{n}\).

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