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Forest fires on Z+ with ignition only at 0

Volkov, Stanislav LU (2009) In Latin American Journal of Probability and Mathematical Statistics 6. p.399-414
Abstract
Abstract. We consider a of the forest fire model on graph G, where eachvertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z+, the times between consecutive burnouts at vertex n, divided by log n, converge weakly as n → ∞ to a random variable which distribution is 1−(x) where (x) is the Dickman function. We also show that on transitive graphs with a non-trivial site percolation thresholdand one infinite cluster at most, the distributions of the time till the first burnout of any vertex have exponential tails.

Finally, we give an elementary proof of an interesting... (More)
Abstract. We consider a of the forest fire model on graph G, where eachvertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z+, the times between consecutive burnouts at vertex n, divided by log n, converge weakly as n → ∞ to a random variable which distribution is 1−(x) where (x) is the Dickman function. We also show that on transitive graphs with a non-trivial site percolation thresholdand one infinite cluster at most, the distributions of the time till the first burnout of any vertex have exponential tails.

Finally, we give an elementary proof of an interesting limit:

limn→∞ log log n = γ. ∑ n (Less)
Please use this url to cite or link to this publication:
author
publishing date
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Contribution to journal
publication status
published
subject
in
Latin American Journal of Probability and Mathematical Statistics
volume
6
pages
399 - 414
publisher
Instituto Nacional de Matematica Pura e Aplicada (I M P A)
ISSN
1980-0436
language
English
LU publication?
no
id
b0697f15-78d1-4cc5-af26-88d119c6fa40 (old id 4588064)
alternative location
http://alea.impa.br/articles/v6/06-18.pdf
date added to LUP
2014-08-18 16:12:21
date last changed
2016-06-29 09:11:36
@article{b0697f15-78d1-4cc5-af26-88d119c6fa40,
  abstract     = {Abstract. We consider a of the forest fire model on graph G, where eachvertex of a graph becomes occupied with rate one. A fixed vertex v0 is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing v0 is burnt out. We show that when G = Z+, the times between consecutive burnouts at vertex n, divided by log n, converge weakly as n → ∞ to a random variable which distribution is 1−(x) where (x) is the Dickman function. We also show that on transitive graphs with a non-trivial site percolation thresholdand one infinite cluster at most, the distributions of the time till the first burnout of any vertex have exponential tails.<br/><br>
Finally, we give an elementary proof of an interesting limit:<br/><br>
limn→∞ log log n = γ. ∑ n},
  author       = {Volkov, Stanislav},
  issn         = {1980-0436},
  language     = {eng},
  pages        = {399--414},
  publisher    = {Instituto Nacional de Matematica Pura e Aplicada (I M P A)},
  series       = {Latin American Journal of Probability and Mathematical Statistics},
  title        = {Forest fires on Z+ with ignition only at 0},
  volume       = {6},
  year         = {2009},
}