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Fires on trees

Jean Bertoin (2012)

Annales de l'I.H.P. Probabilités et statistiques

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 - α 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 , the terminal density of fireproof vertices converges...

Fluctuation limit theorems for age-dependent critical binary branching systems

José Alfredo López-Mimbela, Antonio Murillo-Salas (2011)

ESAIM: Proceedings

We consider an age-dependent branching particle system in ℝd, where the particles are subject to α-stable migration (0 < α ≤ 2), critical binary branching, and general (non-arithmetic) lifetimes distribution. The population starts off from a Poisson random field in ℝd with Lebesgue intensity. We prove functional central limit theorems and strong laws of large numbers under two rescalings: high particle density, and a space-time rescaling...

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