One-dimensional general forest fire processes

TL;DR

This paper studies a one-dimensional forest fire process with seeds and matches falling according to renewal processes, showing convergence to four possible limit models as matches become rare, generalizing previous Poisson-based results.

Contribution

It introduces a generalized model with renewal processes for seeds and matches, establishing convergence to limit forest fire models and enabling perfect simulation.

Findings

Four possible limit forest fire models identified.

Convergence established under reasonable assumptions.

Models can be perfectly simulated.

Abstract

We consider the one-dimensional generalized forest fire process: at each site of $\zz$, seeds and matches fall according some i.i.d. stationary renewal processes. When a seed falls on an empty site, a tree grows immediately. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding connected component of occupied sites. Under some quite reasonable assumptions on the renewal processes, we show that when matches become less and less frequent, the process converges, with a correct normalization, to a limit forest fire model. According to the nature of the renewal processes governing seeds, there are four possible limit forest fire models. The four limit processes can be perfectly simulated. This study generalizes consequently a previous result of the authors where seeds and matches were assumed to fall according to Poisson processes.