ASYMPTOTICS OF ONE-DIMENSIONAL FOREST FIRE PROCESSES

• 成果类型: Article
• 署名作者: Bressaud, Xavier; Fournier, Nicolas
• 署名单位: Universite de Toulouse; Universite Toulouse III - Paul Sabatier; Universite Paris-Est-Creteil-Val-de-Marne (UPEC); Universite Gustave-Eiffel
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP524
• 发表日期: 2010
• 页码: 1783-1816
• 关键词: self-organized criticality

摘要

We consider the so-called one-dimensional forest fire process. At each site of Z, a tree appears at rate 1. At each site of Z, a fire starts at rate lambda > 0, immediately destroying the whole corresponding connected component of trees. We show that when lambda is made to tend to 0 with an appropriate normalization, the forest tire process tends to a uniquely defined process, the dynamics of which we precisely describe. The normalization consists of accelerating time by a factor log(1/lambda) and of compressing space by a factor lambda log(1/lambda). The limit process is quite simple: it can be built using a graphical construction and can be perfectly simulated. Finally, we derive some asymptotic estimates (when lambda -> 0) for the cluster-size distribution of the forest fire process.