On the sizes of burnt and fireproof components for fires on a large Cayley tree

Cyril Marzouk

doi:10.1214/14-AIHP640

On the sizes of burnt and fireproof components for fires on a large Cayley tree

Cyril Marzouk

University of Zurich

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

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Résumé

We continue the study initiated by Jean Bertoin in 2012 of a 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 → regarding the asymptotic behavior of p_n.

langue originale	Anglais
Pages (de - à)	355-375
Nombre de pages	21
journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	52
Numéro de publication	1
Les DOIs	https://doi.org/10.1214/14-AIHP640
état	Publié - 1 févr. 2016
Modification externe	Oui

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abstract = "We continue the study initiated by Jean Bertoin in 2012 of a 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 pn or fireproof with probability 1-pn. 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 → regarding the asymptotic behavior of pn.",

keywords = "Cayley tree, Fire model, Fragmentation, Percolation",

author = "Cyril Marzouk",

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AB - We continue the study initiated by Jean Bertoin in 2012 of a 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 pn or fireproof with probability 1-pn. 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 → regarding the asymptotic behavior of pn.

KW - Cayley tree

KW - Fire model

KW - Fragmentation

KW - Percolation

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On the sizes of burnt and fireproof components for fires on a large Cayley tree

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