One-dimensional general forest fire processes

One-dimensional general forest fire processes
[Processus de feux de forêt généraux en dimension 1]

Bressaud, Xavier ; Fournier, Nicolas

Mémoires de la Société Mathématique de France, no. 132 (2013) , 144 p.

Résumé
Abstract

Nous étudions le processus des feux de forêt généralisé en dimension 1 : sur chaque site de $ℤ$ , des graines et des allumettes tombent suivant des processus de renouvellement stationnaires i.i.d. Quand une graine tombe sur un site vide, un arbre pousse immédiatement. Quand une allumette tombe sur un site occupé, un feu démarre et brûle immédiatement la composante connexe occupée autour de ce site. Nous montrons — sous des hypothèses raisonnables sur les processus de renouvellement — que lorsque la fréquence des allumettes tend vers zéro, le processus converge, correctement renormalisé, vers un processus limite. Suivant la nature des processus de renouvellement gouvernant l’apparition des graines, quatre processus limites sont possibles. Les quatre modèles limites peuvent être simulés parfaitement. Cette étude généralise des résultats de [15], où nous supposions que graines et allumettes tombaient suivant des processus de Poisson.

We consider the one-dimensional generalized forest fire process: at each site of $ℤ$ , seeds and matches fall according to 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 previous results of [15] where seeds and matches were assumed to fall according to Poisson processes.

Zbl

DOI : 10.24033/msmf.442

Classification : 60K35, 82C22
Keywords: Stochastic interacting particle systems, Self-organized criticality, Forest fire model
Mot clés : Systèmes de particules stochastiques en interaction, criticalité auto-organisée, modèle des feux de forêt

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Bressaud, Xavier; Fournier, Nicolas. One-dimensional general forest fire processes. Mémoires de la Société Mathématique de France, Série 2, no. 132 (2013), 144 p. doi : 10.24033/msmf.442. http://numdam.org/item/MSMF_2013_2_132__1_0/

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