Forest fires on Z+ with ignition only at 0

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 = .