Forest fires on ${\mathbb{Z}}_{+}$ with ignition only at 0

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 $v_0$ 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 $G=Z_{+}$, the times between consecutive burnouts at vertex $n$, divided by $\log n$, converge weakly as $n\to \infty$ 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 {\it any} vertex have exponential tails. Finally, we give an elementary proof of an interesting limit: ${\lim }_{n\to \infty }{\sum }_{k=1}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)(-1{)}^k\log k-\log \log n=\gamma$.