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Scaling limits of forest fire processes.

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We consider the forest fire process on the integers: on each integer site, seeds and matches fall at random, according to some stationary renewal processes. When a seed falls on a vacant site, a tree immediately grows. When a match falls on an occupied site, a fire starts and destroys immediately the corresponding occupied connected component. We are interested in the asymptotics of rare fires. We prove that, under space/time re-scaling, the process converges (as matches become rarer and rarer) to a limit forest fire process. According to the tail distribution of the law of the delay between two seeds (on a given site), there are 4 possible limit processes.

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