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Forest fires on \(\mathbb Z_{+}\) with ignition only at 0. (English) Zbl 1276.60060
Summary: 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 \(v_0\) is hit by lightning with the same rate, and when this occurs, the whole cluster of occupied vertices containing \(v_0\) is burnt out. We show that when \(G = \mathbb Z_+\), the times between consecutive burnouts at vertex \(n\), divided by log \(n\), converge weakly as \(n \rightarrow \infty \) to a random variable which distribution is \(1 - \rho (x)\) where \(\rho (x)\) 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: \[ \lim_{n\to \infty} \sum^n_{k=1}{n\choose k}(-1)^1\log k-\log\log n=\gamma. \]
MSC:
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
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