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

Research output: Contribution to journal › Article › peer-review

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 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.

Original language	English
Pages (from-to)	355-375
Number of pages	21
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	52
Issue number	1
DOIs	https://doi.org/10.1214/14-AIHP640
Publication status	Published - 1 Feb 2016
Externally published	Yes

Keywords

Cayley tree
Fire model
Fragmentation
Percolation

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10.1214/14-AIHP640

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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.",

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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

Abstract

Keywords

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