Forest fires on Z+ with ignition only at 0

S Volkov

Research output: Contribution to journalArticle (Academic Journal)peer-review

Abstract

We consider a version of the forest fire model on graph G, where each vertex 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 threshold and 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: lim n→∞ Xn k=1 n k (−1)k log k − log log n = .
Translated title of the contributionForest fires on Z+ with ignition only at 0
Original languageEnglish
Pages (from-to)399 - 414
Number of pages16
JournalALEA: Latin American Journal of Probability and Mathematical Statistics
Volume6
Publication statusPublished - 2009

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