Forest fires on $\Z_+$ with ignition only at 0

TL;DR

This paper studies a forest fire model on the positive integers, revealing asymptotic distributions of burnouts and providing new limit results related to the Dickman function and exponential tail behaviors.

Contribution

It introduces a novel analysis of the forest fire process on $ ext{Z}_+$, connecting burnout times to the Dickman function and establishing exponential tail bounds on transitive graphs.

Findings

Burnout times at vertex n, scaled by log n, converge to a distribution involving the Dickman function.

On certain graphs, the first burnout times have exponential tail distributions.

An elementary proof of a limit involving binomial coefficients and the Euler-Mascheroni constant.

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: $\lim_{n\to\infty} \sum_{k=1}^n {n \choose k} (-1)^k \log k -\log\log n=\gamma$.