14:15
We
consider the forest fire process on Z: on each site, seeds and matches fall at
random, according to some independent Poisson processes. When a seed falls on a
vacant site, a tree immediately grows. When a match falls on an occupied site, a
fire 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.
Next, we consider the more general case where seeds and matches fall according
to some independent stationary renewal processes (not necessarily Poisson).
According to the tail distribution of the law of the delay between two seeds
(on a given site), there are 4 possible scaling limits.
We finally introduce some related coagulation-fragmentation equations, of which
the stationary distribution can be more or less explicitely computed and of
which we study the scaling limit.